Mathematical Methods in Aerodynamics

LAZAR DRAGON

MATHEMATICAL

METHODS IN

AERODYNAMICS

ll KLUWER ACADEMIC PUBLISHERS

EDITURA ACADEMIEI ROMANE

MATHEMATICAL METHODS IN AERODYNAMICS

Mathematical Methods in Aerodynamics by

LAZAR DRAGO$, Roqsonime Academy.

H charm Romania

L7 KLUWER ACADEMIC PUBLISHERS

EDITURA ACADEMIEI ROMANE

DORDRECHT/BOSTON FLONDON

BUCURE$11

A C.I.P. Catalogue record for this book is available from the Library al' Congress.

ISBN 1-4020-1663-8 ISBN 973-27-0986-3

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Table of Contents xiii

Preface

I The Equations of Ideal Fluids

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The Equations of Motion 1.1.1 Elements of Kinematics 1.1.2 The Equations of Motion ... 1.2 The 1.2.1 Helmholtz's equation. Bernoulli's integral 1.1

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Potential Flow .............................

1.2.2 1.2.3

The Linear Theory .........................

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3 5

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6 8

... .......... .. ..... The Shock Waves Theory ............ ............. 1.3.1

The Jump Equations ........................

11 11

ilugoniot's Equation

13 15 16 17 19

1.2.4

1.3

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The Equation of the Potential ...................

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1.3.2 1.3.3 1.3.4 1.3.5 1.3.6

The Acceleration Potential

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The Solution of the .lump Equations ..............

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Prandtl's Formula ....................... .. The Shock Polar .......................... The Compression Shock past a Concave Bend

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2 The Equations of Linear Aerodynamics and its Fundamental Solutions 21 2.1

2.2

21 21

2.1.2 2.1.3 2.1.4 2.1.5 2.1.6

22 24

26 28 29 30

The Steady Solutions ....................... 30

.... ... ................. 34 ............ ... The Fundamental Solutions for the Fluid at Rest ........ On the Interpretation of the Fundamental Solution ....... The Fundamental Solutions of the Steady System ............ 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7

2.4

. ......... The Equation of the Potential ..... ... ........... .. ............ . The Linear System ...... The Uniform Motion in the Fluid at Rest ..... ... ... . The Equations of Motion ........ ... The Equations of Linear Aerodynamics .............

The Fundamental Solutions of the Equation of the Potential ...... 2.2.1

2.3

................. . .........

The Equations of Linear Aerodynamics 2.1.1 The Fundamental Problem of Aerodynamics

Oscillatory Solutions ...... ...........

Oscillatory Solutions for Al = 1 The Unsteady Solutions . . . . . The Unsteady Solutions for At = I .

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42 43

2.3.1

The Significance of the Fundamental Solution

2.3.2 2.3.3 2.3.4 2.3.5 2.3.6

The General Form of the Fundeunental Solution .........

2.4.2 2.4.3

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45 46 47 48 48 50

The Determination of the Velocity Field .............

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Other Fortes of the Components V and W

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The Subsonic Plane Solution .... .... The Three-Dimensional Subsonic Solution The TwoDimensional Supersonic Solution

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The Three-Dimensional Supersonic Solution ...........

The Fundamental Solutions of the Oscillatory System 2.4.1

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The Determination of Pressure ... ... ............ 50

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2.4.4 2.4.5

2.5

The Fundamental Solutions in the Case Af = I ......... Fundamental Solutions of the Unsteady System I .......... ..

55 57 57 58

.. Cauchy's Problem ........ ............ . .. .. The Perturbation Produced by a Mobile Source ... . ..... Fundamental Solutions of the Unsteady System If .. ... ..... 2.5.1

2.5.2 2.3.3 2.3.4 2.6

The Incompressible Fluid ..................... 55 Fundamental Solutions . Fundamental Matrices .

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62 64 64 66

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2.6.1

2.6.2

The Fundamental Matrices .. ............. ..... The Method of the Minimal Polynomial .... .... ... ..

3 The Infinite Span Airfoil In Subsonic Flow 3.1

3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.1.9

3.2

3.3

69

. ...... . .... ... ... .. ... .... .... .. A Classical Method ........................ The Fundamental Solutions Method . ....... ... .. ..

The Airfoil in the Unlimited Fluid 3.1.1 The Statement of the Problem

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The Function f (z). The Complex Velocity in the Fluid The Calculation of the Aerodynamic Action . . . . . .

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72 75 76

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77 Examples ........ . The General Case ......................... 80 Numerical Integrations ... ...... .............. 81 .

The Integration of the Thin Airfoil Equation with the Aid of Gauss-type Quadrature Formulas . ...

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81

The Airfoil in Ground Effects ....................... 82 3.2.1 The Integral Equation .. ..... ....... .... ..... 82 3.2.2 A Numerical Method .................. . ... .. 85 3.2.3 The Flat Plate ........................... 85 3.2.4 The Symmetric Airfoil .. ..................... 86 The Airfoil in Tunnel Effects .. ... . .... ... . ......... 88 . . .. . 88 3.3.1 The Integral Equation . .. 90 3.3.2 The Integration of the Equation (3.3.9) .

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The Numerical Integration .............. ......

92 92 95

The Integral Equation ................. ......

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Numerical Results . . . . . . . . . . 3.4 Airfoils Parallel to the Undisturbed Stream 3.4.1 The Integral Equations . . 3.3.3

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3.4.2 3.5 Grids of Profiles 3.5.1 3.5.2 The Numerical Integration 3.6 Airfoils in Tandem . . . . . . .

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3.6.1 3.6.2 3.6.3

3.6.4

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The Integral Equations The Determination of the Functions f, and ff

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The Lift and Moment Coefficients Numerical Values

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4 The Application of the Boundary Element Method to the Theory of the Infinite Span Airfoil In Subsonic Flow 4.1

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... . .... ... .............. 109 .. ... ... . .......... ....... 109 .. . ............ .. 110

The Equations of Motion 4.1.1 Introduction 4.1.2 The Statement of the Problem .

4.1.3

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The Fundamental Solutions .. ..

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vii 4.2

Indirect Methods for the t idimited Fluid Case ............. 113 4.2.1

The integral equation for the Distribution of Sources ...... 113

4.2.2

The Integral Equation for the Distribution of Vortices

4.2.3 4.2.4 4.2.5 4.2.6 4.3

4.4

The Determination of the Unknowns .............. . The Circular Obstacle

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115 115 117 120

The Elliptical Obstacle ...................... 121

4.3.1 4.3.2 4.3.3 4.3.4 4.3.5

The representation of the solution ............... .

122

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125

4.3.6 4.3.7 4.3.8

Appendix ...... ................... ... .. 129

The Integral Equation ............. ... .. .... 123 .

The Circulation ..

The Discretization of the Equations ............... 126

The Lifting Profile ......................... 126 The Local Pressure Coefficient .................. 128

Numerical Determinations ................. ..

. .................... .. The Representation of the Solution . ............. .. 131 131 131

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The Airfoil in Ground Effects 4.4.2 4.4.3 4.4.4 1.4.5

The Integral Equation .................. .. .. 134 The Computer Implementation ... .......... ..... 135 . 136 .. The Treatment of the Method . .. .

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The Circular Obstacle in a Compressible Fluid ......... 137

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.1.4.6

Appendix

4.5.1

The Representation of the Solution ............. ..

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The Airfoil in Tunnel Effects .... ................ ... 140 4.5.2 4.5.3 4.5.4 4.5.5

4.6

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The Direct Method for the Unlimited Fluid Case ............ 122

4.4.1

4.5

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The Boundary Elements Method ................ .

The Integral Equation ................ ....... 144

The Verification of the Method ...... ... ......... 146

Appendix ...... ................... ..... 149

Other Methods. The Intrinsic Integral Equation .......... . 4.6.1

140

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Green Functions ...... ......... ........... 142 150

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The Method of Regularization .................. 150

5 The Theory of Finite Span Airfoil in Subsonic Flow. The Lifting Surface Theory 5.1

The Lifting Surface Equation 5.1.1

5.1.2 5.1.3 5.1.4

155

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... ... ............. .. 155 .

The Statement of the Problem .................. 155 Bibliographical Comments ..................... 158

The General Solution ................ ... .. .. 159 The Boundary Values of the Pressure ... ........... 161 .. 163 .The U.S Boundary Values of the Component w . . .

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x.1.6

The Integral Equation

=..1.7 .5.1.8

The Plane Problem ....... ..... .. ... ... ...

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164

Other Forms of the Integral Equation .............. 166

5.1.9 The Aerodynamic Action in the First Approximation

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168 169

5.1.10 A More Accurate Calculation ................... 171 5.2

5.1.11 Another Deduction of the Representation of the General Solution 173 Methods for the Numerical Integration of the Lifting Surface Equation 175 5.2.1 5.2.2

The General Theory ........................ 175 Multhopp's Method ............. ........... 178

Viii

5.3

5.4

.. 179 180 The Third Method ......................... 181

5.2.3

The Quadratum Formulas Method

5.2.4 5.2.5

The Aerodynamic Action ... .. .

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.............. 184 . ...................... 184

Ground Effects in the Lifting Surface Theory 5.3.1

The General Solution

5.3.2

The Integral Equation

5.3.3

The Two-Dimensional Problem .... .... ....... ... 188

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The Wing of Low Aspect Ratio

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The Integral Equation ......... ........ ... .. 189 The Case h = h(x) .......... . .. ... .... ... 192

5.4.3

The General Case

5.4.1

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6 The Lifting Line Theory 6.1

186

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Prandtl's Theory .............................. 197 6.1.1 The Lifting Line Hypotheses. The Velocity Field . ...... 197 6.1.2 Prandtl's Equation . .. ..... .............. 200 6.1.3 The Aerodynamic Action ...... ........ . ...... 202 .

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...................... 203

6.1.4 The Elliptical Flat Plate 6.2 The Theory of Integration of Prandtl's Equation. The Reduction to

Fredholm-Type Integral Equations ....... ..... ........ 205

6.2.2

The Equation of Trefftz and Schmidt ............... 205 Existence and Uniqueness Theorems ............... 209

6.2.3

Foundation of Glauert's Method

6.2.4 6.2.5

The Minimal Drag Airfoil

6.2.1

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210

Glauert's Approximation ..................... 212

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6.3 The Symmetrical Wing. Vekuas Equation. A Larger Class of Exact Solutions

... ....... .... . ... .. . ............. 214 The Integral Equation ....... .. . ... .......... 215

6.3.1

Symmetry Properties ........................ 214

6.3.2 6.3.3

Vekua's Equation

......................... .

217

The Elliptical Wing ........................ 220 6.3.5 The Rectangular Wing . . . 221 6.3.6 Extensions .................. . ...... ... 222 Numerical Methods . .. . 223 6.4.1 Multhopp's Method ........................ 223 6.4.2 The Quadrature Formulas Method .. ... .. .. ...... 228 6.3.4

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.. ... ................. 231 Various Extensions of the Lifting Line Theory .............. 234 6.5.1 The Equation of Weissinger and Reissner .. .......... 234 6.5.2 Weissinger's Equation. The Rectangular Wing ......... 236 The Lifting Line Theory in Ground Effects .. ... .. 2.18 .

6.5

6.6

6.4.3

The Collocation Method

6.6.1

The Integral Equation .......... ...... ....... 238 The Elliptical Flat Plate ...................... 240

6.6.2 6.6.3 6.7

Numerical Solutions in the General Case

............. 241

The Curved Lifting Line ................ 6.7.1

6.7.2 6.7.3

The Pressure and Velocity Fields The Integral Equation . . . .

.... ...... 242

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7 The Application of the Boundary Integral Equations Method to the 251 Theory of the Three-Dimensional Airfoil In Subsonic Flow 7.1 The First Indirect Method (Sources Distributions) ........... 251 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5

7.1.6 7.1.7 7.1.8 7.2

The Integral Equation The Integral Equation

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The Discretixation of the Integral Equation ........... 255 The Singular Integrals

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The Velocity Field. The Validation of the Method ....... 258

The Incompressible Fluid. An Exact Solution .......... 259

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ible Fluid .................................. 265 7.2.1 7.2.2 7.2.3 7.2.4

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The General Equations ... . .................. 251

The Integral Equation ..... ... ............. .. 265 The Flow past the Sphere. The Exact Solution ......... 267

The Velocity Field ......................... 268 The Velocity Field on the Body. N. Marcov's Formula ..... 268

The Direct Method. The Incompressible Fluid ............. 271 7.3.1 The Integral Representation Formula ........ .. ..... 271 7.3.2 7.3.3 7.3.4 7.3.5

..... . ................ .. ..................... ... 275

The Integral Equation Kutta's Condition

274

The Lifting Flow .......................... 276

The Discretization of the Integral Equation ........... 279

8 The Supersonic Steady Flow 8.1

8.1.1

8.1.2 8.1.3 8.1.4 8.1.5 8.1.6 8.2

283

The Thin Airfoil of Infinite Span .. ................ ... 283

The Analytical Solution ...................... 283

The Fundamental Solutions Method ....... .... .... 286 The Aerodynamic. Action

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The Graphical Method ....................... 289 The Theory of Polygonal Profiles ................. 290 .. 294 Validity Conditions . Ground and Tunnel Effects .. .. .................. .. 295 8.2.1 The General Solution ............... .... .... 295 298 8.2.2 The Aerodynamic Coefficients ................. . The Three-Dimensional Wing .............. ... ... .. 300 .

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8.3.1

Subsonic and Supersonic Edges ..... ............ .

8.3.2 8.3.3 8.3.4 8.3.5 8.3.6 8.3.7 8.3.8 8.3.9

The Representation of the General Solution ........... 302 The Influence Zones. The Domain Di ....... .. ..... 304 The Boundary Values of the Pressure ........... . .. 305 The First Form of the Integral Equation .

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The Equation D in Coordinates on Characteristics .... .. 308

The Plane Problem ......................... 310 The Equation of Heaslet and Lomax (the 11L Equation)

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The Deduction of HL Equation from D Equation ...... 313 8.3.10 The Equation of Homentcovschi (II Equation) . .... .... 318 8.4

... .... .... 320 Abel's Equation .......................... 320

The Theory of Integration of the H Equation .. 8.4.1

8.4.2

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The Solution of the H Equation in the Domain of Influence

of the Supersonic Trailing Edge ...... ........ ... .

321

X

8.4.3

The Solution in the Domains of Influence of the Subsonic Lead-

ing Edge .......................... .. ... 323

The Wing with Dependent Subsonic Leading Edges and Independent Subsonic Trailing Edges . . . . . . . . . 324 8.4.5 The Wing with Dependent Subsonic Trailing Edges . .. ... 326 8.4.6 The Solution in the Zone of Influence of the Subsonic Edges under the Hypothesis that the Subsonic Leading Edges are letdependent . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.4.7 The Wing with Dependent Subsonic Trailing Edges . . . . . . 337 8.4.4

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The Theory of Conical Nloticaas ...... ...... .......... Introduction ............... .. ... .. ... . 8.5.1

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8.5.2 8.5.3

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8.5.4 8.6

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The Wing with Supersonic Leading Edges ..... ....... The Wing With a Supersonic Leading FAlge and with Another

Subsonic Leading or Trailing Edge .. .... . ...... ... The Wing with Subsonic Leading Edges .... .........

Flat Wings ... .

... .. .. .... .... ... ...... .. ...

The Trapezoidal Wing with Subsonic Lateral Edges ...... 352 The Trapezoidal Wing with Lateral Supersonic Edges ..... 355

The Triangular Wing. The Calculation of the Aerodynamic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 .

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The Equations of the Transonic Flow .......... .... ... ..

359 :3.59

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9.1.1

The Presence of the Transonic Flow ....

9.1.2

The Equation of the Potential .. ... .... .... . ..... 361 The System of 'transonic Flow . ... . ....... . ... .. 364 The Shock Equations

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The Plane Flow ..... ............... . .......... 369 .

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The Fundamental Solution The General Solution . .

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The Lift Coefficient The Symmetric Wing The Solution in Real .

9.2.6

The Symmetric Wing

9.:3.1

The Fundamental Solution .......... . .. .... ... .M1 The Study of the Singular Integrals ........ ....... 386

9.2.1

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9.4

347

8.6.3 8.6.4

9.1.3 9.1.4

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The Angular Wing with Supersonic Leading Edges .... ... 347

9 The Steady Transonic Flow

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8.6.1 8.6.2

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... ... ... . ....... ..... 387 .......... :389 The Lifting Line Theory .. ... ........ .. .......... 392 The General Solution

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Flows with Shock Waves .. ...... .. .. .

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9.4.2

The Velocity Field ......................... :392 The Integral Equations ......... . .. .. 394 . .

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10 The Unsteady Flow

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... ... . .... . .. 397 10.1.1 The Statement of the Problem ..... . .. .... .... .. 397 10.1.2 The Fundamental Solution ..... . ............ .. 397

10.1 The Oscillatory Profile in a Subsonic Stream

10.1.3 The Integral Equation

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10.1.4 Considerations on the Kernel ..... ....... .. ..... 402

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10.2 The Oscillatory Surface in a Subsonic Strewn .....

10.2.1 The General Solution ............ ......... . 404 10.2.2 The Integral Equation ....................... 405 .

10.2.3 Other Expressions of the Kernel Function ......... .

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10.2.6 The Plane Flow ........ . .................. 414

10.3 The Theory of the Oscillatory Profile in a Supersonic Stream ..... 415 10.3.1 The General Solution 10.3.2

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10.3,3 Formulas for the Lift and Moment Coefficients

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423 424

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10.3.5 The Oscillatory Profile in the Sonic Flow

10.4.1 The General Solution ............ .... ... . ... 426

10.4.2 The Boundary Values of the Pressure .............. 428 10.4.3 The Boundary Values of the Velocity, The Integral Equation

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10.4.4 Other Expressions of the Kernel ................. 433

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10.4.5 A New Form 10.4.6 The Plane Problem . . . . . . 10.5 The Oscillatory Profile in a Sonic Stream

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10.5.1 The General Solution. The Integral Equation .. ........ 438 10.5.2 Some Formulas for the Lift and Moment Coefficients

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441

10.6 The Three-Dimensional Sonic Flow .................... 442

.. .. .. 442 .. .................. 443 10.6.3 The Plane Problem ......................... 446 10.6.4 Other Forms of the Kernel . .. . ... . 447 10.6.1 The General Solution

. .

.

. .

.

.

.

. .

.

.

.

.

.

11 The Theory of Slender Bodies 11.1 The Linear Equations and Their Fundamental Solutions ..... ..

.

.

.

.

.

.

.

.

.

.

10.6.2 The Integral Equation ...

.

. . .

. .

11.1.1 The Boundary Condition. The Linear Equations ... 11.1.2 Fundamental Solutions .

.

449 449

. .... 449

... .... ........ . .... .

11.2 The Slender Body in a Subsonic Stream ... ............. .

452 454

. ... . ...... ........ 454

11.2.1 The Solution of the Problem

11.2.2 The Calculus of Lift and Moment Coefficients .....

..... 456

11.3 The Thin Body in a Supersonic Stream ................. 458

11.3.1 The General Solution ..... .................. 458 11.3.2 The Pressure on the Body. The Lift and Moment

. ............... 461 . ... .. ........ . ........... .. 463

Coefficients .............

11.3.3 The wing at zero angle of attack .... ............. 463 11.3.4 Applications

A Fourier Transform and Notions of the Theory of Distributions

465

.. ............... .. 465

A,1 The Fourier Transform of Functions

A.2 The Spaces V and S .. ................... .... .. 466 A.3 Distributions

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

467

A.4 The Convolution. Fundamental Solutions ................ 470

A.5 The Fourier Transform of the Functions from S ............ 472 A.6 The Fourier Transform of the Temperate Distributions ......... 473

xii

............ 475 .............. 478

A.7 The Calculus of Some Inverse Fourier Transforms A.8 The Fourier Transform in Bounded Domains

B Cauchy-type Integrals. Dirichlet's Problem for the Half-Plane. The Calculus of Some Integrals 481

Cauchy-type Integrals ........................... 481 .................. 482 B.3 Plemelj's Formulas ............................. 483 B.l B.2 The Principal Value in Cauchy's Sense

................ 483 ........ 485

B.4 The Dirichlet's Problem for the Half-Plane B.5 The Calculus of Certain Integrals in the Complex Plane B.6 Glauert's Integral. Its Generalization and Some

Applications ................................. 489

B.7 Other Integrals

............................... 491

C Singular Integral Equations C. i

493

The Thin Profile Equation ......................... 493

C.2 The Generalized Equation of Thin Profiles

................ 496

C.3 The Third Equation ............................ 498 C.4 The Forth Equation ............................ 502 C.5 The Fifth Equation ............................. 504 D The Finite Part

509

D.1 Introductory Notions ............................ 509

............................. 509 ................ 510

D.2 The First Integral D.3 Integrals with Singularities in an interval

D.4 Hadamard-Type Integrals ......................... 513

D.5 Generalization ............................... 515 E Singular Multiple Integrals

517

F Gauss-Type Quadrature Formulas

521 ............................. 521 F.2 Formulas of Interest in Aerodynamics .................. 525 F.3 The Modified Monegato's Formula .................... 529 FA A Useful Formula .............................. 530 F.1 General Theorems

Bibliography

533

Index

571

Preface

The researchers in Aerodynamics know that there is not a unitary method

of investigation in this field. The first mathematical model of the airplane wing, the model meaning the integral equation governing the phenomenon, was proposed by L. Prandtl in 1918. The integral equation deduced by Prandtl, on the basis of some assumptions which will be specified in the sequel, furnishes the circulation C(y) (see Chapter 6). Using the circulation, one calculates the lift and moment coefficients, which are very important in Aerodynamics. The first hypothesis made by Prandtl consists in replacing the wing by a distribution of vortices on the plan-form D of the wing (i.e. the projection of the wing on the plane determined by the direction of the uniform stream at infinity and the direction of the span of the wing). Since such a distribution leads to a potential flow in the exterior of D and the experiences show that downstream the flow has not this character, Prandtl introduces as a supplementary hypothesis another vortices distribution on the trace of the domain D in the uniform stream. The first kind of vortices are called tied vortices and the second kind of vortices are called free vortices. On the basis of this model one developed later the main theories of Aerodynamics namely the lifting surface theory (after 1936, more precisely in 1950, when Multhopp gave the equation of this theory), the lifting surface theory for the supersonic flow (after 1946) and the lifting theory for oscillatory wings and surfaces for the subsonic, sonic and supersonic flow (after 1950). In the framework of the last theory the wing is replaced by doublets distributions. From a physical point of view, there is no reason for replacing the wing with vortices or doublets distributions. It is true that the vortices are detaching from the wing, but these are effects , not causes of the presence of the wing. The fact that these replacements lead to correct results shows how subtle was Prandtl's in-

tuition. We specify that the distributions on D and its trace do not result from the equations of motion (they have been introduced outside the mathematical model). Taking into account this inconvenient, we have shown in (5.7) how it can be removed. We have to consider that the wing and the fluid constitute an interacting material system. If we want to study the fluid flow, then according to Cauchy's stress principle

xiv

(the principle of the internal forces; see for example [1.11), p.35), we have to assume that there exists it forces distribution on the boundary, which has against the fluid the same action like the wing itself. We shall replace therefore the wing with it forces distribution instead of a vortices, sources or doublets distribution and we shall find out the density of this distribution such that it should have the same action against the fluid like the wing itself. We shall proceed by imposing to the fluid flow determined by the forces distribution to satisfy the slipping condition on the wing, condition which is also satisfied by the flow determined by the wing. In this way it follows an integral equation for determining the forces density. This equation constitutes the mathematical model for the wing we have in view. This method is an unitary one and it is based only on the classical principles of mechanics (in fact. Cauchy's stress principle). It may be applied to all configurations: see [5.7) for the wing in a subsonic stream, [8.4] for the wing in a supersonic stream, [10.15], [10.16], [10.17) for the oscillatory wings in subsonic, sonic or supersonic stream etc. All these results are given in this book (see chapters 5, 8, 10, 11). We called this method (in[5.7)): the fundamental solutions method. It may be utilized to all cases in which one can calculate the fundamental solutions of the equations of motion. We have to notice that in the framework of this method, the existence of the vortices downstream the wing follows from the model (i.e. from the equations of motion) and it must not be introduced artificially. In the sequel we shall present some of the models of aerodynamics. For two-dimensional configurations, in a subsonic stream, the models are one-dimensional singular integral equations considered in the sense of Cauchy's principal value. One may integrate analytically only the equation of thin profiles in a free stream. For other geometries one determines numerical solutions with the aid of Gauss-type quadrature formulas (see Chapter 3). For three-dimensional wings in a subsonic stream, the models are two-dimensional integral equations with strong singularities, which are defined in the sense of Finite Part (see Chapter 5). For other geometry (for example the wing in ground effects) the models are generalized equations. All these models are solved only numerically. For the wing in it free stream, Multhopp's method is available. In this book we introdace a more general method - the quadrature formulas method. In the last part of Chapter 5 one presents the theory of low aspect wings which was extended by the author to the general case of asymmetrical wings. The lifting line theory may be deduced from the lifting surface theory with the aid of Prandtl's assumptions (6). This theory is developed

xv

by presenting analytical and numerical methods for solving Prandtl's equation; one considers also extensions of this theory, all the methods representing one-dimensional integral-differential equations. The author shows how these equations may be reduced to integral equations with strong singularities and for this type of singularities he gives a Gausstype quadrature formula, which allows the equation to be reduced to a linear algebraic system which is solved numerically. This method, which is very general. allows to obtain numerical solutions both in the case of the lifting line (Chapter 6) and the case of the lifting surface (Chapter 5). In the case of supersonic flow, the integral equations are solved analytically. For the three-dimensional wing (the lifting surface) we present in Chapter 8 a nice solution given by D. Homentcovschi in 18.16]. The integral equations describing the flow past oscillatory wings and profiles (chapter 10) have the same nature like the equations utilized in the case of steady flow but the kernels are more complicated. However for the sonic and supersonic flows these equations may be solved exactly by means of the Laplace transform, as it is shown in [10.17]. Chapter 9, devoted to the transonic motions. begins with a new asymptotic deduction of the equations of motion. The two and three-dimensional integral equations are obtained following the papers of the author and D. Homentcovschi. The theory of subsonic and supersonic flow past slender bodies (in Chapter 11) relies also on the fundamental solutions theory. In Chapter 2 one deduces the equations of the linear aerodynamics, on the basis of an asymptotic analysis. assuming that the small parameter depends on the thickness of the profile. In the classical aerodynamics this deduction is performed under the assumption that the unknowns and their derivatives have the same order of magnitude, but this fact cannot he a priori assumed. Then one calculates the fundamental solutions for the equation of the potential (paper [2.11]) and the fundamental solutions for the systems of equations of aerodynamics : the steady system[2.8], the oscillatory system [10.17], the unsteady system [2.6], (2.7]. On these solutions will rely the theories from the forthcoming chapters. The models we have already presented are the so called classical or linear

models. They are suitable for the thin wings and thin profiles because they rely on the following assumptions: 1) one uses a linear boundary condition, 2) the boundary condition is imposed on the support of the wing (the segment (-1.1] for the profile, the plan-form D for the three-dimensional wing), :3) the equations of motion are linearized. The development of the scientific computing allows us to develop more exact methods. Indeed we can give up to the first two assumptions using

xvi

the boundary integral equations method (BIEM). also called the boundary element method (BEN), which was employed for the first time by Hess and Smith [7.9], [7.10]. The integral equations on the boundary are obtained imposing the exact boundary condition on the boundary of the wing. The integral equation is discretize d using, for example, the collocation method. One obtains an algebraic system which is solved numerically. The linearization of the equations of motion is necessary only in the case of compressible fluids. The theory that we have developed is thus valid for every body in an incompressible fluid and for at thin body in a compressible fluid. Two chapters from this book, Chapter 4 for the 2d airfoil and Chapter 7 for the 3d airfoil are baLsed on our papers (L. Drago.1 and A. Dinu). The comparison between the known analytical results and the numerical results shows a very good agreement. In the Appendices we give some results concerning The Distributions Theory, The Singular Integral Equations Theory, The Principal value and The Finite Part, Gauss-type Quadrature Formulas, etc. In every work one finds, in a certain measure, both the achievements

of the predecessors and of the researchers contemporaneous with the author. Among the people which have directly collaborated with tae, I have to mention at first my professors Victor Va lcovici and Caius lacob, who introduced me in the field of aerodynamics. I also mention my younger colleagues Nicolae Marcov, Liviu Dint, Dorel Homentcovschi, Adrian Carabineanu, Victor Tigoiu, Vladimir Cardoi, Gabriela Marinoschi, Stelian Ion and Adrian Dinu. They were my students at the University of Bucharest, but I learned a lot from their papers. Some of them were my fellow - workers in the aerodynamics research, many of them stimulated me with their youth and their way of thinking in our seminars from the Faculty of Mathematics of the University of Bucharest. I am very grateful to all of them. My special gratitude goes to Adrian Carabineanu for his work in performing the English translation of the book, to Adrian Carabineanu and Stelian Ion for typesetting the monograph in Latex and to Victor 'igoiu for his activity in finalizing the 195D Grant with the World Bank. I acknowledge that the hook was sponsored by MEC-CNCSIS Contract 49113/2000, Grant 195D with World Bank. LAZAR DRACOS

Chapter 1 The Equations of Ideal Fluids

1.1 1.1.1

The Equations of Motion Elements of Kinematics

In this Chapter we present the equations governing the flow of ideal fluids. On the basis of these equations we shall develop the theory in the forthcoming chapters of the book. It is well known (see, for exannple, [1.11J), that the fluid flow is defined by the di eomorphism

x = X(t, X),

(1.1.1)

where X is the vector of position of a particle P in the reference configuration (for fluids this is the initial configuration), and x is the vector of position of the same particle at the moment t. For a fixed X and a variable t , the equation (1.1.1) furnishes the motion law for the particle having the vector of position X. Hence the velocity and the acceleration of the particle will be given by the formulas

v (t, X =

It X (t, X ),

a (t, X =

cit.

v (t, X).

(1.1.2)

The fluid is a continuum medium. It means that the support of the initial configuration is it domain Do. The image of this domain by the diffeomorphism (1.1.1) will be denoted by D and one demonstrates [1.11J that it is a domain. The functions X;(t, Xi, X2i X3) appearing in (1.1.1) belong to the class C2(D0) and the Jacobian is 0(X1, X2, Xa)

V(XI, X2, X3)

# (}

(1.1.3)

The velocity field defined in (1.1.2) may be discontinuous in isolated points, on abstract. curves or across abstract surfaces. This kind of surfaces will be named shock waves.

THE EQUATIONS OF IDEAL FLUIDS

2

The fluid flow may be described by functions defined on Do, i.e. functions having the form 46(t, X) , or by functions defined on D, i.e. functions having the form F(t, x). The first presentation is called the material description, because it utilizes quantities attached to the material particles, the second is called the spatial description, because it furnishes information about the particles which are located at the

moment t in the points of a domain D. The material derivative of the quantity 0 attached to the fixed and it is given by the formula particle X, is denoted by 46(t, X).

(1.1.4)

In order to obtain the derivative of F when X is fixed, we have to take into account that F depends on the coordinates X, through the functions Y{. Using the derivation rule for the composite functions, we obtain for the material derivative OF OF d X; OF OF OF +(v V)F. (1.1.5) F(t' x) dt + as d t +V'r7x; t For studying the motion of fluids we employ the spatial description. The main quantities are: the density (or specific mass) p(t, x ), the pres-

,

T.

sure p(t. x ), the velocity field v (t, x ), the temperature T(t, x ), the entropy s(t, x) etc. The acceleration which is the material derivative of the velocity is obtained by means of the formula:

a = at + (v 0)v .

(1.1.6)

Utilizing the derivation rule for the determinants, from (1.1.3) one obtains Eider's theorem

J = J div v

(1.1.7)

and then (see for example [1.11]) Reynolds's formulas for continuous inotions (i. e. motions characterized by fields belonging to the class) C1((to,tII X DWff

dtJDF(t,x)dv= r

-

1

ID

(F'+Fdivv)dv= r

(

1.1.8)

BtF dv+ / DFv nda

fD [OF + div(Fv)J dv =

JD and Reynolds's formula for motions with shock waves

dtI

F(t,x)dv= J_a_dv.f-J

E++E_

F v neda-J fFlld da, s

( 1.1.9)

3

THE EQUATIONS OF MOTION

- F_, and d is the displacement velocity of the surface of discontinuity S (see formula (2.5.1) and figure 2.5.1 from where OFJ = F.,. (1.110.

The Equations of Motion

1.1.2

The principle of conservation of mass is d

dt,Dpdv=O (V)DCD.

(1.1.10)

For continuous motions one utilizes the formula (1.1.8) and one obtains the equation of continuity

p+pdivv =0.

(1.1.11)

The general expression for the principle of variation of the momenturn is t L"= dPwdv

-jDpnda+ JDpf dv (V)DCD,

(1.1.12)

f representing the force per unity of mass. For continuous motions, from (1.1.11) and (1.1.6), one obtains Euler's equation p[

+ (v V) v] =pf -grade-

(1.1.13)

The balance equation of the energy (the first principle of the thermodynamics) is

j

( aD JOD

+ /Dpf - vda- /D q nda (V)DCD,

(1.1.14)

e representing the specific internal energy, and q, the flux of heat vector. For continuous motions, taking into account (1.1.11) and (1.1.13), we deduce

Pe = -pdiv v - div q . (1.1.15) In ideal fluids the processes are reversible. Eliminating div v by the aid of equation (2.1.11) and employing the second principle of thermodynamics (1.11], p.54, one obtains the fundamental equation of thermodynamics

de=Tds - pdv,

(1.1.16)

THE EQUATIONS OF IDEAL FLUIDS

4

where s is the specific entropy and v is a notation for 1/p. Eliminating div v from (1.1.15) and taking into account (1.1.16), we deduce the following remarkable form of the equation of energy

pTs= - divq.

(1.1.17)

This shows that if it is possible to neglect the change of heat (it is the case of aerodynamics where the velocities are great), then

s = 0.

(1.1.18)

The equation (1.1.18) indicates that s is constant on trajectories, the constant varying from one trajectory to the other. One calls such a motion isentropic motion.. If there exists a configuration where the entropy constant is the same everywhere, then in every configuration arising from the first one, the constant will be the same everywhere. Such a motion is called homentropic or isentinpic everywhere. The perfect gas is characterized by the following equations of state

p=pRT,

(1.1.19)

The first one (which is obtained from the laws of Boyle-Mariotte and Gay-Lussac) is the thcrmic equation (or Clapeyron's equation) and the second is the caloric equation. It is easy to prove (see for example [1.11) p. 57-58) that for this gas we have

p }+Co, e= P7 //J

1

-+C1.

-Y-1p

(1.1.20)

c, being the so called specific heat at constant volume and y = cp jc where cP is the specific heat at constant pressure. For the air 7 = 1.405. From the expression of a it follows for the homentropic motion:

p = kp'',

(1.1.21)

k representing a constant. The quantity c, defined by the formula

=

(dp/

'

(1.1.22)

has the dimension of a velocity. One shows (see for example [1.10)) that it gives just the speed of propagation of the surfaces of discontinuity of the pressure (sound waves). For the ideal gas in homentropic motion it follows c2

= 7plp.

(1.1.23)

5

THE POTENTIAL FLOW

The Potential Flow

1.2 1.2.1

Helmholtz's equation. Bernoulli's Integral

If the tnassic forces possess a potential f = grad II (in aerodynamics these ford (representing the weight of the air) are neglected) and if the fluid is characterized by a thermodynamic law having the form p = p(p), where p is a derivable function, defined for p > 0 such that ;7(p) > 0, then we deduce

f- Igrad p= grad

II -1 pp)

Utilizing the identity (v V) v = curl v x v + grad (v 2/2), Euler's equation (1.1.13) becomes

/ f +curlvxv=grad(n-1 dp-2v2). jt P \\

(1.2.1)

This is Hetnahoftss equation.

The flow of a fluid is irrotational in a domain D if everywhere in D. we have curl v = 0. This equation constitutes the necessary and sufficient condition for the existence of a differentiable function jp(x, x), such that

v=grad 44.

(1.2.2)

Such a flow is called potential. Applying the operator curl in (1.2.1), we eliminate the term in the right hand side of the equation. The resulting equation is integrated [1.11] as follows 0

P

(.VX) =

X(t,X),

(1.2.3)

WO and m) representing the vortex (2w =- curly) and respectively the density in a reference configuration. The formula (1.2.3) shows that if the motion is potential in a configuration, it remains potential in every theorem). configuration arising from the first one

We can therefore put curl v = 0 in (1.2.1). It results Bernoulli's integral

Ot+2vz+ J LP -n=C(t) 4t=8W Ot.

(1.2.4)

One may give to the spatial constant C(t) the value zero ([1.11], p.90). Neglecting the messic forces, Bernoulli's integral is

tt+2vz+

J

Pp=0.

(1.2.5)

6

THE EQUATIONS OF IDEAL FLUIDS

The motion of a fluid is stationary or steady if the velocity field does not depend explicitly on t. Nowr (1.2.4) becomes

2v2+ J ap-n=C.

(1.2.6)

P

In this case the constant cannot be zero.

For the perfect gas in homentropic motion we have the formula (1.1.21). With a convenient notation of the constant, from (1.2.6) we deduce v- + 'Y P = 7 PO = _ , (1.2.7) 2

T::_1 -P

7-lpo 7-1

pa, pn and co representing the pressure, the density and the square of the sound velocity for the fluid at rest. From (1.2.7) one deduces that in the compressible fluid there is a superior limit v of the velocity which is obtained for p = 0 and a critical value v, which is obtained for v = c. These values are VCr = Co

Vmax = Co

V7 +2

_

(1.2.8)

For v < v ., the flow is subsonic and for vrr < v < vmm, the flow is supersonic.

1.2.2

The Equation of the Potential

The equation of the potential is obtained from the equation of continuity (1.1.11), Bernoulli's integral (1.2.5) and the equation (1.1.21), assuming that v has the form (1.2.2). Calculating the material derivative of (1.2.5), we obtain: dt(5t+2tr2)+P=0.

(1.2.9)

Taking into account that p depends on t through the agency of p and utilizing the notation (1.1.22), we deduce

_± d dpP P

(Oe

+ 2v2)

(1.2.10)

Replacing in the equation of continuity, we obtain the equation of the potential

c2A -(v .V)(v -V)o -2(v V).6e-#u=0,

(1.2.11)

*r11F: POTENTIAL FLOW

where c' depends on 40. We find this dependence in the case of the perfect gas in homentropic flow. Taking into account (1.1.21), Bernoulli's integral (1.2.5) becomes v2

4+

+

)-i

{n7-t -1710-t}

= o,

(1.2.12)

representing the density in a reference state. From (1.1.21) and (1.1.22) it follows cc = kyp0-1, and from (1.2.12) we deduce:

po

1

(+v2)]7_1

=p

11 P=poll-7 1 (fit+2v2)Jy-1

(1.2.13)

,

(1.2.14)

CS

+2v2),

(1.2.15)

representing the pressure and the sound velocity in the CO reference state of density po. For the steady flow one obtains the equation

Po and

c2A4'-(v .V)(v cI)4'=0

(1.2.16)

and the formulas //

1

P=po1--y21

}

Jy-1

y P=poll-y 21V1\-V-1

,

(1.2.17)

c2 = c9

-

y-1 v2 2

(1.2.18)

where po, po and co are quantities corresponding to the fluid at rest (v = 0). One demonstrates [1.10], p. 207, that the equation (1.2.11) is hyperbolic and the equation (1.2.16) is elliptic for the subsonic flow (v2 < c'-) and hyperbolic for the supersonic flow (c2 < v2). The equality v2 = c2 occurs only on curves in the two-dimensional flow or surfaces in the case

of three-dimensional flow. These varieties are separating the domains where the flow is subsonic from the domains where the flow is supersonic. This kind of motion is called transonic. We denoted tit = I v 12.

8

THE EQUATIONS OF IDEAL FLUIDS

1.2.3

The Linear Theory

When the equations (1.2.11) and (1.2.16) are utilized in order to determine the perturbation produced by a thin body in a fluid having a known flow, they may be linearized Let us consider for example, that the uniform flow of a fluid having the velocity U,,, i, the pressure p,, and the density p,,,, is slightly perturbed by the presence of a body. We shall denote by

v = UU i + v'

(1.2.19)

the velocity field for the perturbed flow ( v' is the perturbation of the velocity field). We assume that all the coordinates of v' have the same order of magnitude a (e representing a small parameter which characterizes the body). Hence we assume that. (1.2.20)

where I- VI is bounded and e << 1. According to Lagrant o-Cauchy's theorem, the perturbed flow is potential, since it arises from a potential

flow. Hence, (3)¢, such that v = grad O and from (1.2.19), it follows

v' = grad p and -0 =U,,,r+V.

(1.2.21)

The condition (1.2.20) is equivalent to Sit) =

(Yr .

;Py, s5z) .

(1.2.22)

Replacing in the formulas above and neglecting the terms O(E2), we deduce v = (U,o + E ;5;r, E v, 63z), 2J2 = U200 + 21J. pa + 0022)

(1.2.23)

and then (1.2.24) U p7= - 2U.v*1 - pc1 = 0, (1.2.25) (1 - Af`)Vls +Wv + Vz_ = 0, where Af(= U,,,/c,,,) is Mach's number in the perturbed flow. One easily find that the equation (1.2.24) is hyperbolic, and the equation (1.2.25) is elliptic if Al < 1 and hyperbolic if Al > 1. The motions characterized by Al < I are named subsonic, and the motions with

Af > 1 arc named supersonic. The pressure is deduced from Bernoulli's integral, (1.2.5), which may be written as follows 1

2

actual

v2) 0o

+

p

f

clp

-;p

= 0.

(1.2.26)

THE POTENTIAL FLOW

9

if we neglect the massic forces. It obviously results

de+2 (v2-U2)+f dr=0,

(1.2.27)

Let us assume that the relation

p=Poo+0(e).

(1.2.28)

is also valid. Then we deduce p(p) = poo +

J

r dp

rx 1

=

I

loo (p

`poo)

dr + 1 --

J

pW JJ j 0

I

P

0(e2) = p + P c2px

pay

+ O( r2)

0(6

c?C p

2) .

poll

(1.2.29)

Replacing (1.2.21), (1.2.23) and (1.2.29) in (1.2.27), we find in the linear approximation the formula for the determination of the pressure

P - P,x = -f).G%e + U,otp,.),

(1.2.30)

Since the previous formulas will be deduced in a different manner in Chapter 2, in dimensionless variables, we are going to introduce here

these variable (x', y'. z', t'), by means of the formulas (x, y, z) = Lo(r`, y', z'),

t = (Lo/UU)t',

y: = Uaip'p - P = PoUp',

(1.2.31)

Lo representing a reference length which is not specified yet. So, the equation (1.2.22) becomes j112Vt.e. = 0.

(1.2.32)

and the formula (1.2.30). l), = - (vi. + co=.) .

1.2.4

(1.2.33)

The Acceleration Potential

Another function, often utilized in aerodynamics, is the acceleration potential ;', introduced by Prandtl. Its existence is ensured for the unlimited fluid characterized by the lawn p = p(p), by the equation

a = -brad / .11

dam'

P

(1.2.34)

TILE EQUATIONS OF IDEAI. FLUIDS

10

resulting from (1.1.6). (1.1.13) and (1.2.1). Denoting

a= grad !, ,

(1.2.35)

we get

v" = -

J

clp'

+ C(t) .

(1.2.36)

where C(t) is an arbitrary function depending on t, which can be determined Lw imposing the value of 0 for a certain state. For example, if the fluid is incompressible and one consider, that t/' vanishes at

infinity (where p = p,), it follows C = p,/p, whence (1.2.37)

V, = (P. - P)//).

Since t' represents the perturbation of the pressure, it is usually called the prrssum function. The relation between 4) and 0 may be deduced from the equations (1.2.5) and (1.2.26). We have

r

= grad

grad (wt + (1/2)v2) _ -grad J

(1.2.38)

If the flow is uniform at infinity, with the velocity U., it results tl =Vet + (1/2){tom - U )

(1.2.39)

In the framework of the linearized theory this relation becomes

V'= P +U.oP.-

(1.2.40)

Taking into account the permutahility of the operator a/Ot + U,,(9/0x with the operators appearing in (1.2.24), it follows that '0 satisfies the sane equations (1.2.24) or (1.2.25). From (1.2.29) and (1.2.39), it results that ,, has the same expression (1.2.37) in the case of the linear approximation for compressible fluids. In order to write the boundary conditions, we have to indicate the relation between 0 and the coordinates of the velocity. For steady motions, we obtain from (1.2.40), (taking into account that rv must vanish at infinity) : Uc

= jW4)dx,

Ucw

as-dX.

(1.2.41)

For unsteady flow, deriving (1.2.40), we get .2.42) (1.2.42)

THE SHOCK WAVES THEORY

11

Performing the change of variables t, x --+ r, £ :

r=t,

t;=x-U".t

(1.2.43)

and denoting

w(t, x, y, -) = w (T, 0(t, x, y) z) _ V"'(r,

U,

,

y, z) ,

(1.2.44)

we deduce ay//Or = N//&r, whence, integrating

Ta

(1.2.45)

00

Returning to the variables t and x, we get w(t, x, y, z)

=

f

+U.r',y,z)dr' _ (1.2.46)

fe

1.3 The Shock Waves Theory 1.3.1

The Jump Equations

The principles of motion for the continuous media are expressed in terms of balance equations. The general form of a balance equation is dt,npQdv -

Jen

R nda+JnpSdv,

(1.3.1)

where D is a material domain, Q and S are tensors having the same order and R is a tensor having the order greater with a unity. For the ideal fluid, the principle of conservation of mass (1.1.10) has the form (1.3.1), with

Q=I, R=0,S=0.

(1.3.2)

The principle of variation of the momentum (1.1.12) is obtained for

Q=v, R=-pI, S= f,

(1.3.3)

I representing the unity tensor, and the balance equation for the energy (1.1.14) for

Q=a+(1/2)v2,R=-pv-q,S= f v.

(1.3.4)

THE EQUATIONS OF IDEAL. FLUIDS

12

Let us see now the balance equation (1.3.1) in case that D is crossed by a surface of discontinuity with the displacement velocity d ([1.11), p.28). Applying the derivation formula (1.1.9), the equation (1.3.1) gives

.I [

UL 51

pQ)-AS1 l dv + i;:++E.. pQ(v

n)-Rnda-

llpQQdda=O.

We pass to the limit superposing F,+ and F_ on S (fig. 1.3.1). In this case, n+ becomes n, and n_ becomes -n.

Fig. 1.3.1.

If we assume that the integrand from the first integral is bounded, the first term tends to zero, because vol D± ---. 0. Hence one obtains:

jDPQ(v.n_d)_Rnoda=07

(1.3.5)

where we denoted (1.3.6)

The limit values encountered here are continuous on S (they are not continuous across S), and S is arbitrary, because the balance equations are formulated for every D. By virtue of the fundamental lemma ([1.11) p.31), from (1.3.5), we deduce

OpQ(v n - d) - Rn = 0.

(1.3.7)

13

THE SHOCK WAVES THEORY

This is the equation which leads to the balance equation. It establishes the connection between the limit values from the two parts of the shock waves. If across a surface the fields have discontinuities, this surface is a shock wave. Utilizing (1.3.7) for (1.3.2), one obtains;

=0,

(1.3.8)

for (1.3.3)

Opv(v n - d) +pn, = 0,

(1.3.9)

and for (1.3.4), (1.3.10)

The conduction laws utilized in practice (particularly Fourier's law) make impossible the discontinuity for q. Hence we shall utilize the equation (1.3.10) as follows 1 p(e+2v2

=0.

(1.3.11)

To the previous equations we shall add the inequation: Ops(v . n

- d)Q > 0,

(1.3.12)

coming from the Second Principle of Thermodynamics ([1.111, p.50). For the ideal fluid we have an equality.

1.3.2

Hugoniot's Equation

The jump Equation (1.3.11) is quite complicated. It can be replaced by a simple equation (Hugoniot's equation), which establishes a connection only between the thermodynamic quantities from the two sides of the shock wave. In order to deduce these equations, for the sake of

simplicity we introduce the propagation speed P = d - v n. In this way, the jump equations become

r

OPPI =0,

(1.3.13)

OpPv-png =0,

(1.3.14)

=0,

(1.3.15)

THE F.QUATIOSS OF IDEAL FLUIDS

14

[pPsO <0.

(1.3.16)

From these equations we have to find the unknowns behind the shod; wave and its propagation speed if we know the state of the fluid in front of the shock wave. For the salve of simplicity too, we mark by the index 1 the limit values (on S) in front of the shock wave and by the index 2

the limit values behind the shock wave. The normal to S is positively orientated from the region 2 toward region 1. We exclude the situation Pi = P-2 = 0, because in this case the surface of discontinuity does not cross the fluid (it is a material surface moving together with the fluid). For deducing Hogoniot's equation, we notice that from (1.3.13) it follows that the quantity p P is continuous across the shock wave. N e denote by

rn=pjP1=P2P2

(1.3.17)

this quantity. Since in the jump equations we employ only the velocity on the shock wave in a current point of the wave, we write

v=v,,n+vet.

(1.3.18)

v'2 =

vn representing the normal component and vi the tangential component

(situated in the plane determined by n and v). We denoted by t the versos of the tangent.

Projecting (1.3.14) on the tangent and on the normal to the shock wave and taking into account that by hypothesis, in T 0, we deduce

UVt =0,

rnOv,, 11

- H =0.

(1.3.19)

The first equation shows that the tangential component of the velocity

remains constant across the shock wave. From the definition of the propagation speed P and from the conservation relation (1.3.17), we deduce

Vn =d-P=d-nzT, r- 1/p,

(1.3.20)

the second equation from (1.3.19) becoming rn2OTO +

0.

(1.3.21)

From (1.3.15) taking into account the equations (1.3.19), (1.3.20) and (1.3.21), we notice that:

pPvt 9 =III

vie) = 0,

OpP'el1 = -21n2dOrl + rn3[r20 = 2dOpE

- Innllpll (rl + r'2)

THE SHOCK WAVES THEORY

15

From these equations and from (1.3.15), we get: 2Ohl = OpO(T1 +r2),

(1.3.22)

where h = e + pr is the enthalpy ([1.11], p.55). This is the first form of Hugoniot's equation. Performing the calculations in (1.3.22), we get also the form (1.3.23) 2Oej + rlJ(PI + P2) = 0, and. for the perfect gas, when e is given by (1.1.20), one finds the third expression: P2 P1

=

b+ 1)p2 - (7 - 1)PI

(1.3.24)

(7 + 1)pt - Of - 1)P2

which is useful in applications.

The inequality (1.3.2) shows that if m > 0 (the shock wave surpasses the fluid), then 81 < 82; if in < 0 (the fluid surpasses the shock wave), then 82 < s1. In all cases behind the shock wave the entropy is not decoeasing. It increases or remains constant. 1.3.3

The Solution of the Jump Equations

In the sequel we are going to demonstrate, for the perfect gas, that giving the state of the medium in front of the shock wave and the propagation speed Pt, one may determine completely the state behind the shock wave. To this aim, we introduce the numbers:

M1 = Pt/cl ,

M2 = P2/c2,

(1.3.25)

where c1 and c2 represent the sound velocity in front of the shock wave respectively behind the shock wave (CI = 7Pi/Pl, 4 = 7P2/p2). By hypothesis, M1 is known and M2 is unknown. From (1.3.21), we deduce:

P2 - P1 = in2(rl _,r2) = p1P1 (1 - r2/rl) = 7P1Mi (1 - 72/r1) .

The function (pa/pl) -1 may also be obtained from (1.3.24). Comparing the expressions we get:\

T2-1= 71

2 (M1-11,E-1= 27 7+1 / Pt 7+

(M2 -1).

(1.3.26)

From the relation P1r2 = P2r1i it follows:

f

-1)I(1-Jt't2)=11th-1

(1.3.27)

THE EQUATIONS OF IDEAL FLUIDS

16

From the perfect gas law p = pRT and from (1.1.20), we obtain: - 81

82

C11

fi-C )(p1)'

= in

J

(i) r

111

r2/rl and p2/pt being replaced from (1.3.26).

Prandtl's Formula

1.3.4

For the stationary wave (d = 0), we have v = -rnr, such that the equation (1.3.11), for the perfect gas, becomes Q1v2+

ry

PO

y-1P

2

=0.

(1.3.29)

The expression written here intervenes in Bernoulli's integral (1.2.7).

Taking into account the meaning given there to

v,,,,,x

, we have the

relation

P1=1 2 2+ iv max=l 2vt.7-1pt 1

P1=12v2msx 2

11

(1.3.30)

which shows that vmax is invariant across the shock wave. Taking into account that 0, from (1.3.30) we deduce: 1

y

P1

PI

2 t[ 2 0 _ 'r' - l pj

P11}2

1l

(1.3.31)

/

We eliminate p2/pt by the aid of equation (1.3.24) and pre/pl, by the aid of relation PIV1.. = P2v2n which results from (1.3.17). Employing once again (1.3.30), we deduce: ('y + 1)v1nv2n - (/ - 1)v? = 2yP1/Pl

1)(v 1

- vl ).

Writing vu = v2i = vi, we obtain Prandtl's fonnula v1nt'2n=

-llvmax-Vi y+1

ry+1

)=t2.r(1.3.32)

the significance of vt. being that of (1.2.8). In the expressions vma and va. utilized here, cp is the sound velocity for the fluid in front of the shock wave (at rest). The relation (1.3.32) gives c2,, when the state of the fluid in front of the shock wave is known. For the normal shock According to this relation, if (:;( = 0), the relation becomes vttt2 = vt is supersonic, then v2 is subsonic.

THE SHOCK WAVES THEORY

17

The Shock Polar

1.3.5

Buselnann (Vortrage arcs dem Gebiete des Aerod amik, Aachen, 1929), gave a graphic method for constructing v2 if the angle with v1 is known. To this aim, we consider in the current point Al from the shock wave (fig. 1.3.2) the AIX axis, pointing in the direction and sense of v1 and the AIY axis in the plane determined by v1 and the normal n to the shock wave.

Fig. 1.3.2.

We use the normal pointing towards the state (2); the change of the sense of the normal does not affect the jump relations (excepting the relation (1.3.16), whose signification is known), because in these relations the normal intervenes linearly. Because of the continuity of the tangential component of the velocity, v, lies also in the plane

determined by v1 and n and it will have on the tangent versor t the same projection AIQ like vl. We denoted by PI and P2 the extremities of the vectors v 1 and v2 and by 0, the angle between v2 and v1. Our aim is to find the geometric locus of the point P2 when 0 is varying. We shall use, obviously, Prandtl's formula (1.3.32). Denoting by (X, Y) the coordinates of the vector v2, from the figure, it results X = v2 cos 0, Y = 1v2 sin 0, vl = v1 sin o, vt = v1 cos B , v2., = v-± sin(v - 0) = X sin Or - Y cos a .

(1.3.33)

So, Prandtl's formula becomes:

Xsin2a-YsinorCosa= vi

- 7+1 VICOS2a

(1.3.34)

THE EQUATIONS OF IDEAL FLUIDS

18

From the triangle P1P2R, where the angle P1P2R = a, it follows tana = (vi - X)JY, such that (1.3.34) becomes (1.3.35) Y2(b- X) = (vl X)2(X - a),

-

%%'here we denoted 2

a=

Vt

,

b=v`,+ vl

2

y+1

vi.

(1.3.36)

From the assumption that vt is supersonic, it follows that a < vi, we deduce vj < b < v.. and from vi < The curve representing eq. (1.3.35) is symmetric with respect to MX, it intersects the MX axis in the points A(a) and Pt (v1) (which are inverse with respect to the sonic circle x2 + y'' = v,2,,), the last point

being double and has a vertical asymptote in B(b). The curve is real for a < X < b and it is represented in figure 1.3.3. It is called foliurn of Descartes. It is the shock polar.

Fig. 1.3.3_

Let us show now how we construct v2 when we know the angle

0 made with vl. First of all one constructs the shock polar. This is determined only by the state (1). The same state determines the angle ©p made by the tangent MT with the MX axis. If 0 < Bo, there exist three intersection points 1,1_,13, of the radius vector with

the polar, but only one is P2. The points on AT are eliminated because they correspond to instable shocks. One eliminates the points

THE SItOCI' WAVES THEORY

19

1:1 because. generally, the shocks are compressive. (v2 < vt ). P2 will coincide therefore with 12. If C is the intersection of the polar with the sonic circle and 9° is the corresponding polar angle, then v2 will be supersonic if 0 < B, and subsonic, if 0 > O. Once 11 determined, it results the direction of the versor t (the orthogwua from M on P1P2) and then the angle a. The density, the

pressure. the temperature and the entropy behind the shock wave is obtaincNi front (1.3.26) and (1.3.28), setting M1 = -vln/et = -MI sin or, where .111 = v1/c1 is Mach's number in front of the shock wave.

1.3.6

The Compression Shock past a Concave Bend

We consider a supersonic flow having the velocity v1 i, the density p) and the pressure P1 in the presence of a concave bend having the opening b (fig. 1.3.4a)). The wall ME produces a compressive shock, the discontinuity line MM' being characterized by the unknown angle a. Behind the shock, the velocity which has to be tangent to ME, will make the angle 6 with MX . If 6 < Bo, the position of P2 will be given by 11 from the polar corresponding to this motion. It follows like above, a and the flow behind the shock. If 6 > B(), one cannot satisfy for v2 the condition to be parallel to AI E. As the experience confirms, the assumption of a rectilinear shock wave Al Al' cannot be taken into consideration. In this case one admits the existence of a detached curvilinear shock wave which is formed in front. of Al (fig. 1.3.4b)). The experience confirms this assumption. Flence the detached shock waves are formed in front of the dihedron (S > A() (fig. 1.3.5a)), or in front of the bodies with rounded leading edge (fig. 1.3.5b)). As an information we give the following values : (10 = 10° for M1 = 1.42 and Bo = 22.55°, for M1 = 2. Depending on

Fig. 1.3.4.

the shape of the body, i.e. on the values of 0, behind the shock wave,

THE EQUATIONS OF IDEAL FLUIDS

20 C

0<00

Pig. t.3.5.

we meet regions where V2 is subsonic (11M2 < 1), if 9 > 9, and regions

where u2 is supersonic, if 9 < O. When 9 is big, the compression is big. When 9 is passing to small values, it appears a detente and the velocity becomes again supersonic. On the direction MV the shock is normal (vi v2 = v,2 and v2 is subsonic. The subsonic regions are separated from the supersonic ones through sonic lines CD and C'D'. Behind the shock wave, the flow is transonic. The shock waves theory will be present in the transonic flow and the hypersonic flow.

Chapter 2 The Equations of Linear Aerodynamics and its Fundamental Solutions

2.1 2.1.1

The Equations of Linear Aerodynamics The Fundamental Problem of Aerodynamics

The fundamental problem of aeroclynamica consists in determining the perturbation produced in a given state of a fluid by a certain motion of a body. The given state of the fluid is called basic state or unperturbed

state. The unperturbed state of the fluid may be the rest state, the uniform flow state, or more generally, the state given by the flow with an imposed non-uniform velocity field. In this book the unperturbed state will be either the rest or the uniform flow. In his turn, the body may he fixed, moving uniformly or may have a general imposed motion. Obviously, a fixed body in at rest state of a fluid does not produce any perturbation. The most common are the case when the unperturbed fluid moves uniformly and the body is fixed and the case when the fluid is at rest and the body has a given uniform motion. As we shall see in 2.1.6, these cases are equivalent, from the mathematical point of view. In both cases, the resulting perturbation will he stationary. If the perturbing ixxly has a non-uniform motion, the perturbation will be non-stationary, whatever should be the state of the fluid. In the sequel we shall consider only the cases when the unperturbed fluid is at rest or has an uniform flow. The problem of determining the perturbation may be practically solved only in the case of small perturbations, when we can neglect their products (we keel) only the principal parts of the equations). In these cases, the linear systems of equations obtained for the perturbations may be investigated either with the methods of the classical analysis or with the methods of the theory of distributions. The systems will be linear

22

LASEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

with constant coefficients in case that the basic state is uniform and they will be linear with variable coefficients in case that the basic flow is not uniform. Since, as we have already mentioned, the last case will not be treated in this book we give here some references, [2.4], [2.14], [2.15], [2.20], (2.22), for the reader interested in this subject.

The order of magnitude of the perturbation is determined by the basic flow and by the shape of the body. For some basic flows, a slender body with it small incidence changes slightly the flow, i.e. produces small

perturbations (governed by linear systems of equations). We cannot establish in advance the conditions of validity of the linear theory. This will be done after determining the solution of the linearized equations, imposing not to obtain results which cannot be accepted from a physical point of view. In this way, from Chapters 3 and 8, it will follow that in the case of steady flow, the linear theory is not valid when the basic flow

has approximately the sonic velocity (M = 1), or has an hypersonic velocity (M > 3) even if the body is slender with a small incidence. In these cases one has to employ the non-linear equations for determining the perturbation. A special feature has the unsteady flow because the system of equations of motion is hyperbolic and it is well known (see, for example, [1.6]), that for this sort of problems, Cauchy's problem is correct, i.e. the solution depends continuously on the initial data. If these data are small, the perturbations will remain small at every instant.

2.1.2 The Equations of Motion First of all we assume that the basic motion (the unperturbed motion) (the Ox axis is of a fluid is an uniform motion with the velocity taken to be on the direction and in the sense of the unperturbed stream

and we denote by i the versor of this axis), the pressure po,, and the density p,,. We denote by xi, yi, zi the generic spatial coordinates, by t i the time and we introduce the dimensionless coordinates x, y, z, t by mean., of the relations (2.1.1) (xi, yi, zi) = Lo(x, y, z), Uwtl = Lot. where LO is a characteristic length which has to be specified in every problem we have in view. We assume that the uniform motion defined above is perturbed by the presence of a body which has a prescribed motion. Let

F(ti, xi, yi, zt) = 0.

(2.1.2)

23

TILE EQUATIONS OF LINEAR AERODYNAMICS

be the equation of the surface of the moving body. Denoting by V 1, p1 and pt the velocity, the pressure and the density for the perturbed flow, we shall write:

V 1 = UU(i + v), Pt = P.- + P"Uoop, pi = Poa(1 + P),

(2.1.3)

the first terms defining the basic motion, and the last ones, the perturbation. Obviously we have: lien (v, p, p) = 0. _-.-OD

(2.1.4)

We neglect the heat changes (in aerodynamics this assumption is plausible, because the variation of the phenomena are very rapid and there is not enough time for the heat change) and we assume that the fluid obeys to the perfect gas law. In these conditions, the perturbed flow will be determined by the following equations:

01+p1divlV1 =0, sl =0, P1V1+grad, pi =0,

(2.1.5)

s1 representing the specific entropy and "the point", a notation for the material derivative: (2.1.6)

s 1 = c,.1n(Pl /Pi) + C, f = of /0t1 + (V 1 V 1)f .

The index 1 attached to the differential operators indicates that the derivatives are calculated with respect to x1, yl, zl Taking into account the expression of the entropy, from the equation (2.1.6), we get: P1p1 = 'YPt11,

(2.1.7)

so that we can eliminate pl from the first equation (2.1.5). Hence, we have to take into consideration the system (2.1.8)

01 + ypldiv3 V1 = 0, p1Vl +grad, p1 = 0 which, taking into consideration (2.1.1) and (2.1.3) becomes: A!2[pt+(1+u)p=+VVlpy+wp;]+(1+-tM2p)(u?+vy+w=)

+p)[ut+(1+u)U=,+ ruy+wu,]+ps=0, +p)[Vt++u)vs+vuy+wvZ]+Py=0, (1+p)[wt+(1+u)w2+trwy+wWz]+p: = 0,

= 0, (2.1.9) (2.1.10) (2.1.11)

(2.1.12)

24

LINEAR AERODYNAMICS. F'UNDA'MENTAL SOLUTIONS

where (u, v, w) = v, pt = dp/at, Ps = Op/tax,... and M = U00/c., coo =

'YPoo/Poo ,

(2.1.13)

Al representing Mach's number in the basic motion and 'y, the ratio of the specific heats The condition for the perturbation surface (2.1.2) to be a material surface is F = 0 (the Euler-Lagrange criterion)and it may be written as follows

Ft +(l+u)F,+vFy+wF =0.

(2.1.14)

This condition must be satisfied for F(t, x, y, z) = 0. The system of equations (2.1.9) - (2.1.12), together with the boundary condition (2.1.14), will determine the perturbation (p,v).

A. The Linearization around the Uniform Motion 2.1.3

The Equations of Linear Aerodynamics

We assume now that the equation of the perturbing surface is z = h(t, x, y) = ei (t, x, y),

(2.1.15)

where e is a small parameter and h(t, x, y) is a known function with continuous first order derivatives (fig. 2.1.1a)). If the perturbation surface is cylindrical with generators parallel to Oz (fig. 2.1.1b)), then

the equation of the profile determined in the xOy plane is assumed to have the form y = h(t, x) = J (t, x) . (2.1.16) In this case, the perturbation will be plane. For the surface (2.1.15), we write F = eh(t, x, y) - z in (2.1.14). One obtains Eht + E(1 + u)h= + evhy = w (2.1.17) which has to be satisfied for z = Eh(t, x, y). The principal part from the left. hand side of the equality (2.1.17) has the order of e. We deduce that the right hand part must have the same order of magnitude. Hence it follows

w(t,z) y,eh) = e'w(t)x,y,eh). We consider that this relation is valid all over the fluid, whence w(t, x, y, z) = e1 F(t, x, y, Z).

(2.1.19)

25

TIIE EQUATIONS OF LINEAR AERODYNAMICS

--s 0

a)

Fig. 2.1.1.

Taking into account (2.1.18). the principal part of the boundary condition (2.1.17) is h, (t, x, y) + h..(t, x, y) = w(t, x, y, 0) .

(2.1.20)

Taking into account (2.1.19), we deduce that the principal part of the product from (2.1.12) has the order e. It follows therefore that p has the same order. Hence, p(t, x, y, -) = Ep(t. X, r, =)

,

(2.1.21)

the residual equation from (2.1.12) being

Dw+p: =0,

(2.1.22)

where D is the material derivative operator for the unperturbed motion:

D = a/at + 9/ax .

(2.1.23)

Taking into account (2.1.21), from (2.1.10) and (2.1.11), we deduce u(t, x, y, z) = F-u(t, x, y, z) , v(t, x, y, :) = e'u(t, x, y, z)

(2.1.24)

and the residual equations

Du+p,,=0, Dv+py=0.

(2.1.25)

Now, the equation (2.1.9) becomes

?J2Dp+ dive = 0.

(2.1.26)

This equation together with the equations (2.1.22) and (2.1.25) which have the vectorial form Dv + grad p = 0,

(2.1.27)

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

26

constitute the fundamental system of the linear aerodynamics. Obviously, this system has to be integrated with the boundary condition (2.1.20) and the condition at infinity upstream

lim (p, v) = 0.

(2.1.28)

If the perturbation is plane, the condition (2.1.20) will be replaced by

ht(t, x) + h=(t, x) = v(t, x, 0)

.

(2.1.29)

If the perturbation surface is fixed (i.e. h does not depend explicitly on t), the perturbation will be stationary, determined by the system 1v12p= + div v = 0 ,

vx + grad p = 0

limo(p, v) = 0

2 30) . (.1

and by the boundary conditions w(x, y, 0) = hr (x, y) ,

(2.1.31)

v(x, 0) = hr (x) .

(2.1.32)

Obviously, the equations (2.1.26), (2.1.27) and (2.1.29) could be easily obtained from (2.1.9)-(2.1.12) and (2.1.14), supposing that all the perturbations have the same order of magnitude. But this thing has to be demonstrated. In the case of transonic flow, for example, the perturbations have different orders of magnitude (see Chapter 9).

2.1.4

The Equation of the Potential

Applying the operator curl in (2.1.27), we obtain:

Dcurlv = 0.

(2.1.33)

Taking into account the significance of the operator D, we deduce that curl v is constant on the parallels to the Ox axis, the constant generally varying from one parallel to the other. Since at infinity upstream (x =

= -oo) the constant vanishes on every parallel to Ox, it follows that curl v = 0 on every trajectory coming from --cc. This property is not true for trajectories detaching downstream from the body. This fact will be better put into evidence by the fundamental solutions. In the irrotational zone we have v = grad cp(t, x) .

(2.1.34)

THE EQUATIONS OF LINEAR AERODYNAMICS

27

Taking into account this representation of the field v, from (2.1.27) we deduce

p = -D

(2.1.35)

The function f (t) which should be added in the right hand side of the equation may be considered equal to zero, since W is determined with the approximation of an arbitrary additive function of t. From (2.1.26) and (2.1.35) it follows Dip = M2D2co or, explicitly (1 - M2)fpxs + Wy, +'pz:. - 2M2WfZ - M2Vu = 0.

(2.1.36)

This is the equation of the potential. In case of the stationary flow, from (2.1.33) it results curly = f(y,z),

f representing an arbitrary field which must be considered zero because

for x -- -oo we have curl v = 0. Again it is true the observation that curl v is not zero on the parallels at the Ox axis detaching downstream from the body. In the irrotational zone we have therefore v = grad W(x) .

(2.1.37)

From the first equation (2.1.25), taking into account (2.1.28), it follows

p(x) = -u(x) .

(2.1.38)

Replacing (2.1.37) and (2.1.38) in the first equation (2.1.30), it follows

(1 - M2)9u + ,pyy + ,P:: = 0.

(2.1.39)

This equation may be obviously deduced from (2.1.36). For the incompressible fluid (M = 0) these equations become Al P = 0.

(2.1.40)

Applying the operator grad in (2.1.39), we get: (1 - M2)v22 + vyy + USX = 0.

(2.1.41)

Hence, the coordinates of the velocity (and the pressure) satisfy the same equation (2.1.39). The equations (2.1.36) and (2.1.39) have been obtained in another

way in Chapter 1. There we utilized the Lagrange-Cauchy theorem in order to prove that the perturbed flow is potential. Here, without

28

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

utilizing this theorem, we demonstrated that in the first approximation the perturbation is potential. In addition, we see here that downstream the perturbing body, the perturbation does not possess this property any longer.

B. The Linearization around the Rest State 2.1.5

The Linear System

Let us assume now that the basic state of the fluid is the rest state, which is perturbed, as above, by a prescribed arbitrary motion of a body. We denote by po and po the pressure and the density of the fluid at rest. We denote also by ca = -ypo/po the square of the sound velocity for the same state. Since, in this case, there is no characteristic velocity, it is not recommended to use the variables (2.1.1) and (2.1.3). It is natural to use dimensional coordinates and to put V1 = V, P1 = PO + POP, Pt = Po(1 + p),

(2.1.42)

litn (p, v, p) = 0. (x}-00

(2.1.43)

The system of equations of motion has the shape

p+(1+p)divv=0, (1+p)P=(c2 +7p)p, (1+p)v+grad p=0.

(2.1.44)

If the equations of the perturbing surface have the form (2.1.15), we shall have the boundary condition Eli, + EU) X + Evii, = w

(2.1.45)

which has to be satisfied for z = Eh(t, x, y). We deduce w(t, x, y, d) = eii (t, x, y, ch)

(2.1.46)

and the boundary condition w(t, x, y, 0) = ht (t, x, y) .

(2.1.47)

Assuming like in (2.1.3), that (2.1.46) is valid all over the fluid, w(t, x, y, z) = ew(t, x, y, z),

29

THE EQUATIONS OF LINEAR AERODYNAMICS

from the projection of the second equation from (2.1.44) onto the Oz axis, we deduce

P(t,x,v,z) = C PO, z, v, z)

and the residual equation

wt+pz=0. Acting in the sequel like in subsection (2.1.3), we obtain at last

p t +divv = 0 , p t = c2pt , vt + grad p = 0

(2.1.48)

We deduce the system

pt+c2divv=0, vt+gradp=0 lim (p, v) = 0 III--=

(2.1.49)

and the solution p = c2p. The system (2.1.49) has to be integrated with the boundary condition (2.1.47). From (2.1.49) one obtains the fundamental equation of acoustics

pa - cRAp = 0. 2.1.6

(2.1.50)

The Uniform Motion in the Fluid at Rest

Let us consider the particular case of the uniform motion of the per-

turbing body in the fluid at rest. We assume that the motion is performed with the velocity U0 in the negative sense of the Ox axis. Putting

V, = Uov, pi = poll + A, Pt = po + poop

(2.1.51)

and utilizing the variables (2.1.1), we get the system:

Mop+(1+yAf,p)divv=0, (1+p)Mop=(1+yM p)P, (1+p)v+gradp=0,

(2.1.52)

where MQ is Mach's number for the fluid at rest. As above, one deduces the boundary condition (2.1.47) and the system (2.1.52) reduces to the residual form

Af2pt+divv=0, vt+gradp=0 and M p = p.

(2.1.53)

30

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

In a frame of reference R' = Cx'y'z' (fig. 2.1.2) solidary with the body, having the axes parallel and having the same sense with the axes

of the frame R = Oxyz (the frame R' is inertial), the system (2.1.53) has the form of the steady system (2.1.30). Indeed, we pass from the frame R to the frame R' by means of the Galilean transformation

t'=t,

=x+t, Y ,=Y, z'=z.

(2.1.54)

With this transformation, the system (2.1.53) gains the form of the system (2.1.26). (2.1.27) and the condition (2.1.47), the form (2.1.20).

But, since we deal with a translation of the body, h in (2.1.47) has the form h(x + t, y) which is transformed in h(x', y'). The boundary condition in the variables x', y', z' will have therefore the form (2.1.31), so that it will determine a steady motion. The system of equations of motion in z', y', z' will have the form (2.1.30). Hence we have demonstrated that the problem of determining the perturbation produced by a body moving uniformly with the velocity -Uoi, in a fluid at rest, is equivalent to the problem of determining the

perturbation produced by the same fixed body in an uniform stream with the velocity Uoi.

2.2

The Fundamental Solutions of the Equation of the Potential

2.2.1

The Steady Solutions

This subsection is written on the basis of the paper [2.111. The fundamental solutions of the equation (2.1.39) are the solutions of the equation

(1 -

e ,+

(2.2.1)

a representing Dirac's distribution. These solutions are, obviously, dis-; tributions. Utilizing the Fourier transform method, we shall obtain ternperate solutions. Taking into account (A.5.1), valid also for temperate solutions, we obtain using the notation a2 = IaI2, (a2

- M202).? = -1,

(2.2.2)

whence

E_

1

a2-M2ni

--V-'

1

[(1

-A12)a +a2+a

(2-2.3)

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 31

In the subsonic case M < 1, we shall denote

fl= 1-M2,

(2.2.4)

and in the supersonic case (M > 1),

k= M2-1.

(2.2.5)

In the subsonic case one utilizes the formulas (A.7.10) and (A.7.11). One obtains the following fundamental solutions:

x+

4w

E

1

2Aft

In

1(y

+z)

,

n = 3,

x f2 +y2, n = 2,

(2.2.6)

(2.2.7)

For the two-dimensional case we have not written the additive constamt C - in j3 appearing in (A.7.11), because the fundamental solution is determined with the approximation of a solution of the homogeneous equation. In the supersonic case, one utilizes the formulas (A.7.14) and (A.7.15). One obtains the following fundamental solutions:

1 H(x - ky2+z2)} 27r

.6 = -

x-k(y+z)

n=3,

H(x - klyl), n = 2.

(2.2.8)

(2.2.9)

H repn-senting, as we have considered in Appendix A, Heaviside's func.

tion. From the definition of this function (A.3.13), it follows that the three-dimensional solution is different from zero only for

x > kk/,/2 +z2.

(2.2.10)

This inequality implies x > 0, xs > k2(y2 + zs).

The set of points from the space verifying these inequalities constitutes

the interior of the cone with the vertex in the origin of the system of coordinates (the perturbation point) and the symmetry axis along the Ox axis (fig. 2.2.1a)). This cone is called Mach's cone. In fact, it is the characteristic cone associated to the partial differential equation (2.1.39)

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

32

Y

b)

a)

Fig. 2.2.1.

in the hyperbolic case. The radius of the cone is x/k, and the angle p (the semi-opening) is determined by the formula tan y =

xxk

=k

= Ml -11

(sin µ = M J

.

(2.2.11)

In the two-dimensional case, the solution is different from zero only

inside the dihedrou made by the plane; x = ±ky where x > 0 (fig. 2.2.1b)). This is Mach's dihedrun. 2.2.2

Oscillatory Solutions

The fundamental oscillatory solutions are defined by the equation

(1 - M2)En + Cm + e - 2M26 - M2Ea = 6(x)exp(iwt) , (2.2.12) associated to the equation (2.1.38). They will have, obviously, the shape

E = E(x)exp(iwt),

(2.2.13)

where

(1- M2)E;= + E,,,, + E.: - 2iwAf2E= + w2M2E = 6(x).

(2.2.14)

Performing the change of functions E - e: E = exp(Ax)e

(2.2.15)

and nullifying the coefficient of the derivative e=, one obtains the equar tion: (1 - M2)e", + eyy + e1t + ae = exp(-Ax)6(x),

(2.2.16)

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 33

where A

ikM,

w2M2

a= 1-M2

(2.2.17)

In the subsonic case (M < 1) one performs the change of variable (x, y, 0 (X, Y, Z):

X =x, Y = fly, Z=Az

(2.2.18)

and one takes into account (A.3.11). One obtains the equations:

cxx + cyy + e z + k2e = exp(-AX)8(X ), n = 3, e x x + eyy + k2e = ftuexp(-)LX)b(X), n = 2,

(2.2.19)

where k = WA1/02 = wM. In (2.2.19) we have Helmholtz's non-homogeneous equations. It is well known (see, for example, [A.12], §9), that Helmholtz's equation has two fundamental solutions in the three-dimensional case and the same number of solutions in the two-dimensional case, the choice of the solution depending on the type of oscillation defined by the equation. Taking into account that we obtained Helmholtz's equation looking for oscillations having the form (2.2.13), it follows (see, for example, [1.41) Chapter 7, §2) that the following solutions have a physical meaning: ea

=

exp(-ikIXI) 4i(XI

1 fr(2)

e2 = 4

(kIXI) ,

63 representing the solution for the three-dimensional case and 'e2 the solution for the two-dimensional case. Hoe) is Hankel's function. Performing the convolutions of these solutions with the right hand member from (2.2.19) (A.4.6 formula), we get the following fundamental solutions e2 = 4'-Ho2)(kjXI),

e3 =

so that, taking into account the changes already made, we find: E3

ik(Mx - R) , (2.2.20)

E2 = -'blIt HQ(kR)exp (kMx) ,

with the notations

R= a +f2(y2+z2), 17 = Vx2 ++Q2y2.

34

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

In the supersonic case ; (Al > 1) one performs the change of variable

X =x, Y=ky, Z=kz.

(2.2.21)

The equation (2.2.16) becomes

exx - eyy - ezz + v2e = -exp(-AX) 6(X),

(2.2.22)

where v = wM/k2. In (2.2.22) we have the non-homogeneous KleinCordon-Fock equation. The fundamental solutions of this equation are also known (JA.8J [A.111). Performing the convolutions of these equations with the right hand member of (2.2.22), we get: Y Y

X27tH (

e3

C2

y- -2+-Z Z2) cos X

X.

Z.

V

Z2

--y2) 2kH(X - IYI)Jo (vvX2

such that finally it follows

-

E E3

1

H `x - k Vy2 +zs)

cosv

x -k y +z)

x_k (

kf.M2

.i.i)exp,

1-M=x) (2.2.23)

(v 2L,

H(x - IyI)Jo

x2 - k2y2) exp (12X)

Due to the presence of Heaviside's function in (2.2.23), it is obvious

that these solutions will be different from zero only in the interior of Mach's cone with the vertex in the origin for x > 0, respectively in the interior of Mach's dihedron (with the edge on Oz) for x > 0.

2.2.3

Oscillatory Solutions for M = 1

If M = 1, the equation (2.2.14) becomes

Ev + E..s - 2iwE= + w2E = 6(--).

(2.2.24)

Introducing the Fourier transform 2 with respect to the variables y and z one obtains A

2iwE=-(w2-az-a3)E_-6(x).

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 35

The solution of this equation (see (A.21)) has the shape E = H(x)E, where

2iwE. - (w2 - az It follows

0,3 ) E(o) E = O , = -1/2iw.

E - - H(x)

w2

2iwP

- a3 - a3.

[

(2.2.25)

2iw

For obtaining E we take into account that

L

0

exp (-au2)d u =

(2.2.26)

we notice that +00

exp (-iAa - Ba2)da =

FOG

exp -B (a + 2B

l)a

-

da

///

42

I.

One obtains:

E

4irx) p

(- i'd r2) ,

(2.2.27)

with the notation r2 = z2 + y2 + z2. In the two-dimensional case we have:

[__(x x 2+y2)], 11

(2.2.28)

where 2o- 2Riw = 1.

The solutions (2.2.27) and (2.2.28) may also be obtained as limits of the subsonic solutions (2.2.20) for M - 1. For obtaining (2.2.27), we notice that M2

2

jj(Mx - R) _ - lim M+ R = -U , limy

(2.2.29)

x being positive, as it follows from the supersonic solution (2.2.23). For obtaining (2.2.28) one takes into account the asymptotic behaviour of Ho2) for great values of the argument ((1.42)). HO(2(k1)

t/2

(-h)

exp (-ik R +

i)

(2.2.30)

and one performs a calculcalculus similar to (2.2.29). In the two-dimensional

case, the limit of the supersonic solution for M --+ 1 is obtained in (10.171.

36

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

The Unsteady Solutions

2.2.4

In the sequel we are going to determine the fundamental solutions of the equation (2.2.14), i.e. the solutions of the equation

(I - M2)e11 + £yy + E, - 2Af2£u - Af 2£tt = 6(t, x)

.

(2.2.31)

This equation determines the perturbation produced in the uniform stream, defined in 2.1, by a source of potential acting at the moment t = 0 in the origin. The problem is plane (two-dimensional) if the source is uniformly distributed on the Oz axis. Applying the Fourier transform, we deduce M2d2£/d t2

- 2M2ial d E/d t + (a2 - A12c )£

(2.2.32)

We know from Appendix A that the solution of this equation has the form £ = H(t)E, where H(t) is Heairiside's function and k, is a solution of the problem

MV P/dt2 - 2M2ia1 dE/d t + (a2 - Al2a2 )E = 0, k(0) = 0 (dE/d t)(0) = -AI--2.

(2.2.33)

We get:

A'IE =

_sin J a jf- ti t

e t°'t

.

(2 . 2 .34)

Utilizing the formulas (A.7.14) and (A.7.15), we deduce

£(t , x) = -- Irt 6(t-MR) , 1

2r where

R=

H(t - MR)

n=3 , n = 2,

(2.2.35)

(2 . 2 . 36)

t2 - M2R`

(x-t)2+1,2+z2, R= (x-t)2+12.

We shall not write any longer the factor H(t) from the right hand member, because obviously this member is different from zero only for

t>0. Further we shall perform a detailed investigation of the solutions (2.2.35), (2.2.36). We denote

h(t)=t-AIR.

;'.2.37)

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 37

We are interested to find the zeros t; of this function, as we are going to utilize the formula: (2.2.38)

from [A.10], page 20. We have also to know the sign of the function h, because the two-dimensional solution differs from zero only for h > 0. One notices in (2.2.37) that the zeros t; are positive. Obviously.

h(0) < 0 and h(oo) = (1 - M2)oo. For the graphic representation of the function h : (0, oo) - R, we have to separate the cases M < I and M > 1. The zeros of the function h(t) are tt-

M2x+M x +(1(2.2.39) 1

112

In the subsonic case (Al < 1), h(t) has a single positive root namely h(t)

t,

t

Fig. 2.2.2.

t+. The graphic of the function h is represented in figure 2.2.2. Utilizing the formula (2.2.38) and taking into account that

t+=Ai (x-t+)2+y2+z2, we deduce

--

1 t+

4r t

x

b(t - t+) + (y` + x )

(2.2.40)

,

in the three-dimensional case and

H(t - t°.)

1

27r

t -A

(2 ,

.

2 41) .

38

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

in the two-dimensional case. Here we denoted to = t f(z = 0). The solution (2.2.40) was given for the first tine' in [2.11], and the solution (2.2.41) is given in [1.2]. A given point P (fig. 2.2.3) perceives in different manners the perturbation produced at the moment t = 0 in the origin of the system of coordinates in the three-dimensional case and the perturbation produced at the moment t = 0 uniformly on the O: axis, in the two-dimensional case. In the three-dimensional case the perturbation is perceived at the moment t+ and only at this moment (see the solution (2.2.40)). In the two-dimensional case, the perturbation is perceived by P continuously, beginning from the moment t° (see the solution (2.2.41)). The explanation of this difference is that in the case of the plane problem, one admits that, at the moment t = 0, the entire Oz axis emits perturbations, to representing the moment when the perturbation emitted by the origin reaches the point B(x, y, 0) and t > to , the period when the perturbations emitted at the moment t = 0

by the other points of the Oz axis reach P. In figure 2.2.3, Q is the position of the source (moving with the stream) at the moment to, the distance QP being given by the formula J(x - t.)2 + y2 = M'lt+

Fig. 2.2.3.

In the supersonic case, we have to determine the zeros of the function

h'(t). They are given by the equation

(x-te)2+y2+z2 = M(to-x).

(2.2.42)

Noticing that x < to, it follows that there exists a single zero, namely

to = x +

y2 -+z2/k.

(2.2.43)

A simple calculation gives h(to) = x - k

y2 -+z 2 .

(2.2.44)

FUNDAMENTAL SOLUTIONS OF TILE EQUATION OF THE POTENTIAL 39

We shall distinguish three cases (fig. 2.2.4):

h(to) = 0,

h(to) > 0,

h(to) < 0.

(2.2.45)

Ni)

t,

2 x>k(y+x)

b)

a)

c) Fig. 2.2.4.

In the first case the fundamental solution is £3

1 t+b(t - t+) + t_8(t - t_)

t x- ky+

41r

0, E2

_

_

1

2;r

1

t -M (-x

t)

t
(2.2.46)

,

(2.2.47)

Ea was given for the first time in (2.11]. One interprets this solution as follows: given a point P(x, y, z) in the interior of Mach's cone, there exist two and only two moments t+ and t_ when the perturbation produced in the origin at the moment t = 0 affects this point (fig. 2.2.5). The moments t+ and t_ are calculated as functions of the coordinates of P by means of the formula (2.2.39). In the two-dimensional case, P(x, y, 0) is affected during the interval It, to_) the explanation being similar to that given in the subsonic case. In this interval, the expression under the square root is positive.

40

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

In the three-dimensional case, figure 2.2.5 presents the intersection of the z = 0 plane with Mach's cone and with the spheres having the centers Q1 and Q2. In the two-dimensional case the figure indicates the intersection of the same plane with Mach's dihedron and with the cylinders having the radii M- t9+ and Mtoi . Since from the relation sing = Q,P;/OQ; we deduce sinp = M-1 it follows that p is Mach's angle, i.e. the two spheres (cylinders) are inscribed into Mach's cone (dihedron).

Fig. 2.2.5.

If P belongs to Mach's cone, there exists only one moment t+ = = t_ = to = xM2/k-2, when the point P is affected by the perturbation. At that moment, the source is located in the intersection of the Ox axis with the normal at the cone in the point R.

If P is in the exterior of the cone, it is not affected by the perturbation. Hence, for the unsteady flow, the perturbed zone is also in the interior

of Mach's cone (respectively Mach's dihedron), but unlike the case of steady or oscillatory flow (when an arbitrary point P from the zone was affected all the time) in the unsteady motion it is affected only at two moments t+ and t_ (in the interval between these moments in the two-dimensional flow) or in a single moment if P is on the cone (dihedron). This happens because in the first two cases the fundamental solutions correspond to sources acting all the time in the origin (on the Oz axis), while in the last case the solutions correspond to sources acting only in the moment t = 0.

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 41

2.2.5

The Unsteady Solutions for M = 1

If M = 1, the solutions of the equation (2.2.31) will be (2.2.35) and

(2.2.36) with M = 1. The function h(t) = t - R has an unique zero (it' > 0) namely the moment tj = r2/2x. It follows that the fundamental solutions E3

-_

5 ( t8;rt x2

) 2x

)

4rrx a (t

- 2x (2.2.48)

1 H(2xt-x2-y2) 2n

2zt. _ x2 _ y2

vanish only for x > 0. Every point P from the half-space (half'-plane) x > 0 is affected by the perturbation. In the three-dimensional case, the perturbation emitted by the origin at the moment t = 0 is received in the point P(x > 0, y, z) at the moment ti, and only at this moment. If the perturbation is emitted by the Oz axis (the two-dimensional flow), an arbitrary point P(x > 0. y, 0) receives this perturbation continuously, beginning with the moment tj = (x2 + y2)/2x.

Fig. 2.2.6.

The moment of the first reception tj is given by the equation (x - ti), + y2 + z2 = ti.

(2.2.49)

The graphic of this equation is a sphere (a cylinder in the two-dimensional

case) with the center on the Ox axis in the point tj and the radius

42

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

t1 (figure 2.2.6). The point t1 is obtained graphically, taking the intersection of the Ox axis with the mediator plane of the segment OP (the mediator line of OP in the two-dimensional case).

2.2.6

The Fundamental Solutions for the Fluid at Rest

As we have seen in subsection (2.1.5) , the propagation of a small perturbation in the fluid at rest, is described by the equation (2.1.50). The fundamental solutions of this equation will be defined by

Eu-c2(E +Eyy+t )=6(t,z).

(2.2.50)

They will give the pressure produced by an unitary perturbation in the origin of the system of coordinates at the moment t = 0. Applying the Fourier transform in (2.2.50), we get: d2 ?/d t2 + c2a2E = d(t) .

(2.2.51)

Utilizing (A.3.17), we deduce that the solution of the equation (2.2.51)

is E = H(t)E, where d2E/d t2 + c2a2E = 0, E(0) = 0, k(0) = 1.

(2.2.52)

Solving this problem, we find: sinclcxIt

cla By virtue of formulas (A.7.14) and (A.7.15), we obtain:

6(ct - r) C3

=

£2 =

- 6(ct - r)

4zrc2t 1

2rc

H(ct - r)

41rrr (2.2.53)

c t- -r

E1 representing the solution in the three-dimensional care and t2 the solution in the two-dimensional case. The solution E shows that the perturbation produced in the origin at the moment t = 0 is concentrated, for t > 0, on a spherical surface having the radius ct and the center in origin. The perturbation propagates as a spherical wave having the velocity c. On the sphere of radius r the perturbation is only at the moment t = r/c. We notice the validity of Huygens' principle. The

FUNDAMENTAL SOLUTIONS OF THE EQUATION OF THE POTENTIAL 43

solution E2 shows that the perturbation emitted on the Oz axis at the moment t = 0 lies at a moment t > 0 in the interior of the cylinder of radius ct. There is a foregoing front of the wave propagating with the velocity c, but there is no posterior front. Unlike the three-dimensional case, behind the foregoing front, the perturbation differs from zero at every moment t. In this case Huygens' principle is not valid any longer. We know from 2.2.4 the explanation of this fact.

2.2.7 On the Interpretation of the Fundamental Solution The equation

p+pdivv = pgb(x)

(2.2.54)

may be interpreted as the equation of continuity when there is a source with the intensity pq in the origin. Indeed, integrating on every domain D containing the origin and taking into account (A.7.3), we get:

(p+pdivv)dv=pq.

(2.2.55)

ID The integral gives the variation of the mass from D in the unity of time. This is given by the intensity of the source. In every domain D which does not contain the origin, the mass is preserved. The presence of the term b(x) in (2.2.54) represents the cause of the motion. Since the term has a spherical symmetry, it follows that the flow will have this property too. Hence v = F(r)x/r and v = grad V with YP = f F(r)dr. An uniform flow, with the velocity U,,i is also potential, hence the flow resulting by overlapping the uniform flow over the flow due to a source is also potential. According to the calculus from subsection 1.2.2, it follows c20O -

(V - V)(V V)¢ = 95(x) Setting q = eq, the equation may be linearized and for q = 1 one obtains (2.2.1). The solution of the equation (2.2.1) could therefore represent the perturbation produced into the uniform stream defined by M, by a mass source of intensity p, placed in the origin of the system of coordinates. In the same way, the equation (2.2.31) could be obtained from the equation A+ pdiv v = pgb(t, x), (2.2.56) which would represent the equation of continuity in case that a source

of intensity pq is acting in the origin at the moment t = 0.

44

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

2.3 The Fundamental Solutions of the Steady System 2.3.1

The Significance of the Fundamental Solution

In order to put into evidence the physical significance of the fundamental solutions of the steady linear system, we consider the equations: Af 2px + div v = 0,

vx + grad p = F, (2.3.1)

lim (p, v) = 0.

As it is already known, these equations determine, in the first approximation. the perturbation produced into the uniform stream defined in 2.1 by a force density F. For being able to utilize these equations in the case of a force of intensity f = (fl, f2, f3), applied in a point C, we have to define a density F whose action against the fluid must have the same torsor (resultant and resultant moment) like the force f. We state that this density is

F = f b(x - t),

(2.3.2)

where 6 is Dirac's distribution. Indeed, taking into account (A.7.3), we deduce that the torsor of this density is

f fb(x - lr)dx = f, f x x fb(x - F)dx = E x f ,

(2.3.3)

i.e. just the torsor of the force f. Hence the system

M2p,, + div v = 0, v= + grad p = f b(x) lim (p, v) = 0,

(2.3.4)

x--.-oo

determines the perturbation produced in the uniform stream, defined in 2.1, by the force of intensity f = E 7f, applied in the origin of the system of coordinates. By definition, this system determines the fundamental solutions of the steady linearized system of aerodynamics. One obtains the plane solutions if one considers that the force having

the constant intensity f is uniformly distributed along the Oz axis, being parallel to the xOy plane, i.e. f = (fl, f2i 0). In this case we have the same conditions for every z; and the perturbation is plane.

THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM

45

The General Form of the Fundamental Solution

2.3.2

We are interested in those solutions of the system (2.3.4) which can be obtained by means of Fourier transform. They are, obviously, distributions. Utilizing the formulas (A.6.4), from (2.3.4) we deduce:

A12alp+et v=0, ialy+iap=-f.

(2.3.5)

From (2.3.5) we deduce: P" =

is f

(2.3.6)

a2 - M2a'I

Then, from the second equation (2.3.5) it follows

f ink

(-ia)(-ia f)

(2.3.7)

iai(02-M2a1)

Utilizing (A.6.9) and (2.2.3) from (2.3.6) we deduce:

P(x, y, z) = -(f . V)F_i [a2 - A12ai = (f V)E. I

(2.3.8)

Taking into account (A.7.7), and (2.3.7) we obtain:

v = fH(x)5(y,z) - V(f 0)S-' [ini(cr2 1 Af2c 2)J

(2.3.9)

But

X

[iaia

a2 - A1201] =

_ AI2a2)J

E,

(2.3.10)

whence, integrating with respect to x, by virtue of Lebesgue's theorem [A.9}, it follows J.--`1

ial(a"-

' .1(2 "

1fI

=

f Ed x.

(2.3.11)

The integration limits have been appropriately imposed in order to satisfy the last condition from (2.3.4). It results therefore v(x, y, z) = f H(x)6(y, z) + VV, where

r

f o0

Ed x.

(2.3.12)

(2.3.13)

46

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

The formula (2.3.12), which is valid both in the subsonic and supersonic

cases, shows that the perturbation is potential, excepting the Ox axis for x > 0 (in the two-dimensional case, one excepts the xOz plane for x > 0), where the first term does not vanish. Hence the perturbation is not potential downstream the point (or the fine) where the perturbing force is acting. From (2.3.8) and (2.3.12) it also results:

u(x, y, z) = ft H(x)ft, z) - p(x, y, z) .

(2.3.14)

With the exception mentioned above one obtains (2.1.38). In the sequel we shall utilize also some expressions of the components of the velocity which do not result from (2.3.12). In the two-dimensional

case, performing the change az = a2 - M2a1 - (1 - M2)ai in the component v resulting from (2.3.7), we deduce:

V=-

a

ia2f

+ (1 - M2)

iaif2 a2-1b12a

(2.3.15)

whence (2.3.16) v(x, y) _ - ftEy + (1 - M2)f2El . Analogously, in the three-dimensional case, replacing a3 = Maa1 - (1 - M2)a1 a2, in the component w resulting from (2.3.7),

-

we obtain: w

_

ioaf1

-a2-M2

1

ia1f3 +(1-M2 )a2-M2a1 (2.3.17)

1a2f3 ia2a3f2 crl(a2 M2a1) a1(a2 - M2at) '

-

whence ,,2

w(x, y, z) = -f1Es + (1- M2)f3e2 - f2 ys

Ed x + f382yy f Ed x ,

'f-C00

00

(2.3.18)

with the notations

= 82/8y8z, .. .

2.3.3 The Subsonic Plane Solution If the perturbation is plane and the free flow is subsonic, E has the expression (2.2.7). From (2.3.8) one obtains

P(x, y) -

1 xf1 + 132yf2 27x/3 x2 + /32y2

(2.3.19)

47

THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM

and from (2.3.16), 0x

(2.3.20)

- yf +g2y 2 If the force f is distributed on a straight line parallel to the Oz q), then the axis intersecting the xOy plane in the point 2nx2

v(x, y) =

.

perturbation will be determined by the system

12px+divv=0, v=+gradp= f6(x-t,y-q), (2.3.21)

litn (p, v) = 0.

After performing the change of variables (x, y) -, (xo, yo):

xo=x-e,yo=y-17 ,

(2.3.22)

the system (2.3.21) is transformed into the system (2.3.4). Hence, for (2.3.21) one obtains I

P(x, y) = 2T'8

xofl + X02

02yof2

+ p2ya

,

v(x, J) =

Q x0f2 - yofl 27r XT o

2y02

-

(2.3.23)

These solutions have been obtained in [2.10].

2.3.4

The Three-Dimensional Subsonic Solution

In this case, E is determined by (2.2.6). From (2.3.8) one obtains

4ap(x, y, z) = --(fia: + flay + f30z)(1/R) ,

(2.3.24)

where

R=

x2+f2(y2+z2).

(2.3.25)

Taking into account that

_

dx

J (x2 + a2)3/2

x a2(x2 + a2)1/2

(2.3.26)

we deduce

j'a()dX=_22 (2.3.27) 1

R dx

-y+z2

48

LINEAR AERODYNAMICS. EUNDAMEN"TAL SOLUTIONS

Employing these results, we deduce from (2.3.13) the expression of the potential cP(x'y,z)

1

fi _

= 4zr R

(1+j)12-22 (1+)f3]

y

y2+z

(2.3.28)

which gives the possibility to calculate the components of the velocity. From (2.3.18) we also obtain:

/

4-1r8,(I?)-4 fsa=[

w(x,y,z)=

(2.3.29)

- -,(f2a: - f3ay)y2 + z2

Cl

+R

.

Considering A = 1, we find the solutions for the incompressible fluid. These results have been obtained in [2.8).

2.3.5

The Two-Dimensional Supersonic Solution

In this case, E is given by (2.2.9). Taking into account (A.7.17), from (2.3.8) we deduce

2kp(x, y) _ (-f, + k f2sign y)a(x - klyl)

(2.3.30)

,

and from (2.3.16),

2v(x, y) = (-f1sign y + kf2)6(x - klyl).

(2.3.31)

Obviously, the perturbation differs from zero only in Mach's dihedron. The solution was given and utilized in [2.10].

2.3.6

The Three-Dimensional Supersonic Solution

In this case, E is given by (2.2.8) and (2.3.32)

P(x, y, z) = (flame + fear + fsa.)E .

Denoting s = k y2 + z2 and taking into account (A.3.9) and (A.3.14), we deduce

ay T Ed x = a H(x - s) o

H(x - 8)(9y

J

rl

dx

Js I X7 9

2 z2E - = 11+

x 7X=F== R

(2.3.33) f

49

THE FUNDAMENTAL SOLUTIONS OF THE STEADY SYSTEM

such that,

By means of formula (2.3.12) we determine the velocity field. The similarity between this potential and the subsonic potential is striking (2.3.28).

Let us show now that both p given by (2.3.32) and the velocity field resulting from the potential (2.3.33), are different from zero only in the interior of Mach's cone. The assertion follows immediately if we use the formula d [H(x) dx xa )j

= x(x .A) = -aH(+i

,

A # 0,1,2,...

(2.3.35)

demonstrated in the theory of distributions (see, for example, [A.5], §2.2). Indeed, we can write:

1 H(x - s)

1

(2.3.36)

27.- (x - s)1/2 (x + s)1/2

such that the derivatives of £ will have the factor H(x - s). From (2.3.18), it follows for w : (2.3.37)

w(x, y, z) _ -(fl(9t + k2f3a=)£ + (f2as - f3ay)./2 +/z'£

In the sequel, we shall utilize also another expression of the component w. This follows writing W

ia3f1

a2 - M2c,

_

ia2a3f2 al (a2 - J%f 2a1)

k2iai - ia2 oil (a2 - Af 2ai)

f3

instead of (2.3.17). Utilizing (2.3.11), one obtains: Fx

w(x, y, z) =

f2t.1

£dx - f3(k28L - 01y) J

z

£d x. (2.3.38)

50

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

The Fundamental Solutions of the Oscillatory System

2.4 2.4.1

The Determination of Pressure

The fundamental oscillatory solutions are defused by the system: M2(pt + p=) + div v = 0

vt + yr + grad p = f d(x)exp (i wt)

(2.4.1)

lim (p, v) = 0.

z-+-oo

They will be complex. Since the system is linear, the real part of the solutions will correspond to the case when exp (i wt) will be replaced by coswt, and the imaginary part will correspond to the case when exp (i wt) will be replaced by sin wt. The solutions of the system (2.4.1) determine the perturbations produced in the uniform stream defined in

2.1, by a force having the periodic intensity f exp (i wt), applied into origin in the three-dimensional case and uniformly on the Oz axis in the two-dimensional case. Obviously, the solution of the system (2.4.1) has the form

p = P(x)exp (iwt), v = V(x)exp(iwt) ,

(2.4.2)

P and V satisfying the system AI2{iwP-}-Ps)+divV =0 iwV + V,, + grad P = f 8(x)

(2.4.3)

Urn (P, V) = 0.

s-+-oo

Applying the Fourier transform in (2.4.3), we get

A12(w-as)P=a'V, (w--al)V-aP=-if, whence

P

_

ia- f M2(as - w)2 - a2'

V

_

if

a

ia- f

as - w + as - w M2(as - w)2 -- a2 (2.4.4)

Applying the Fourier transform, from equation (2.2.14) we deduce:

E = -F-s [a2 - M2(as - w)2j

,

P = (f V)E.

(2.4.5)

THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM

51

The pressure P will be expressed by means of the solutions (2.2.9) and (2.2.23). In the two-dimensional subsonic case, we shall have:

P= -0(f V)[Ho2)(k1)exp(ikdlx)] ,

(2.4.6)

and in the throe-dimensional subsonic case, P

Q)

{expJux - R)]

(2.4.7)

4r(f k being given in (2.2.19) and R, R in (2.2.20). In the two-diinensional supersonic case, we have:

P = -(f' V) [H(x - klyl)Jo (v x2 - k2y2)exp (-ivMx)] ,

(2.4.8)

and in the three-dimensional supersonic case, 1

P

(f 0)[H(x-Z: 21r

y2 +z2)

1 cosy x--k2( +z7) exp(-ivMx)]

\/X- 2 - k (y +:)

(2.4.9)

v being given in (2.2.22). Taking into account the formula (2.3), we deduce that the solution (2.4.9) differs from zero only in the interior of Mach's cone.

The Determination of the Velocity Field

2.4.2

Since the velocity field V vanishes at -oo, we deduce like in (2.3.12):

f

_ i

_

iai(a2-M'-a? +2alw+w2) - f 1

"

Gdx,

where

G = -f- 1a2

1

- M2a1 + 2alw + w2 Using the change ai - ca -+ aI, we get:

(

]

r Gdx. -i a 1 w NIz a lW)2 - a2] = -ie'""d 00

(2.4.10)

(2.4.11)

If we replace al by al - w in (A.7.9) we find

ai

- w ] = e-"`H(x)b(y, z) i

(2.4.12)

52

LINEAR AERODYNAMICS. FUNDAMENTAL. SOLUTIONS

Utilizing (A.6.9), on the basis of formulas (2.4.11) and (2.4.12), from (2.4.4) we deduce:

V = fe-k"H(x)b(y,z) + Vcp,

(2.4.13)

where

_ -(f

0)e-k'1X

/

G dx.

(2.4.14)

f oc

We notice that the perturbation is potential, excepting the Ox axis for x > 0, i.e. excepting the trace of the source in the uniform stream. Let us determine now the distribution G. We notice, applying the Fourier transform, that it is the solution of the equation

(1 - M2)Gu +

G - 2iwGx - w2G = 6(x).

(2.4.15)

The solution of this equation may be obtained like in 2.2. So, in the two-dimensional subsonic case we find

G = _ ZI-0 Hoe) (k1) exp (iox) ,

(2.4.16)

and in the three-dimensional case G

41rR

exp

[ik(_R)]

(2.4.17)

.

In the two-dimensional supersonic case one obtains-

G = -- 2k H(x - klyl)Jo (ui/2 - k2y2) exp (lJx) = H(x - kEyl)g,

not (2.4.18)

and in the three-dimensional case

G = - 2 H (x - k

y2 + --2)

x= - k (y' + z-)

cos v X-

r

- k-(y- + -)

exp (izux)

-

not

H(x-k. Vy2 -+z2) g, (2.4.19)

w being defined in (2.2.19). Taking into account the behaviour of Ho2) for great values of the argument (2.2.30), we deduce that in all cases we have:

lim (G, G=) = 0.

s----oo

(2.4.20)

THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM

53

2.4.3 Other Forms of the Components V and 14 In the two-dimensional case from the component V given by (2.4.4) one eliminates a2 by means of the identity

a2=a2-M112(al -w)2+M2(a1-w)1-ai. Thus one obtains

if, ai -W

ft, a2 h12(a1-W)2-a2

+ if2 (M2 -1)a, - 2M2wa1- M2w2 a, -w M2(a1 - w)2 - a2

whence, utilizing the inversion formulas (A.6.9) and (2.4.11), V (X, y)

{f,

,

+ f2 [(M2 -1)O + 2M2iwt7, - M2 W2] }

(e-iWX

X

J

Gdx = -f,e';v,x Gy - iw 00

J

m

-fee-i4" llI2iwG - (I - M2)Gx + w2 J X G d xl

.

J

oo

(2.4.21)

We also obtain this form using the expression of V given in (2.4.13) and (2.4.14),

V=

`H(x)6(y, z) - f14 f/

X

x

a-wx

J

\

G d x)

-

00

Gyy, dx,

;aid eliminating Gy, with the aid of the equation (2.4.15). Indeed, in the two-dimensional case, utilizing (2.4.20), we deduce

G., d x = H(x)d(y) - (1- it f2)Gx + 2iwG + w2 J

Gdx .

From the last taro formulas we find again (2.4.21).

For the supersonic solution, taking into account (2.4.33), with the definition of g given in (2.4.18) we have s r

-

8y

Gdx=H(x-kIyI)

r

gdx,

f Ivl

r r (x Gdx=H(x-kjyj)OOJ x gdx,

J oo

lyi

(2.4.22)

54

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

such that V given by (2.4.21) is zero outside Mach's dihedron. In a similar manner, in the three-dimensional case we deduce for the component W

W (x, y, z) = - f ie " (G c - iw rx

G$ d x)

J oo

- f2e'11x foo G,, d xx

-he-i- 2"-(1-M2)Gx+(w2-822y)

x

Gdx, 00

(2.4.23)

We notice that using the notation:

s=k y 2 ++z2

(2.4.24)

we may write in the supersonic case W (x, y, z) = H(x - s)w(x, y, z) ,

(jX)

(2.4.25)

where taking into account the definition of g from (2.4.19),

w = -fie

ir'"

k.ix

(.

- f3e-u.,x [2iwg+k2g+(,2_ay)jXgdxj whence we deduce that W is different from zero only in the interior of Mach's cone with the vertex in the origin. Indeed, utilizing the formulas (A.3.9), (2.3.35) and noticing that s8/8y = k2y8/8s, we may write :

82yy /x Gdx=822

[H(x-s)

JT gdxJ =

00

= [H(z

- s)8(k2y/s)8,j gd= H(x - s)8J g d x

and from

GdxJ =H(x-S) {g=-iw f g2dx)

8s\\`G-iwJ 00

it follows the formula (2.4.25).

.J.

55

THE FUNDAMENTAL SOLUTIONS OF THE OSCILLATORY SYSTEM

2.4.4

The Incompressible Fluid

The (oscillatory) solutions for the incompressible fluid may be obtained considering Al = 0 in (2.4.4), or directly in the subsonic solution. So, in the two-dimensional case we deduce from (2.4.5): P(x, Y) =

I xf! + 02

(2 . 4 . 26 )

2

The general form of the equation (2.4.15) for M = 0 is G = g exp (iwx),

(2.4.27)

where Og = 0. One obtains therefore: G(x, y) =

exr (i x)

ln(x2 + y2) .

(2.4.28)

For the three-dimensional problem we have:

(f V) T ,

P(x, y, z)

with the notation r = 2.4.5

G(x, y, z)

ex (iwx)

(2.4.29)

x2 + y2 +-2.

The Fundamental Solutions In the Case M = 1

If Al = 1, E is the solution of the equation (2.4.23), and G, the solution of the equation

Gyy + C_. - 2iwG,. - w2G = 8(z) .

(2.4.30)

C is determined like E. They have the form:

E = H(x)e, G = H(a)g.

(2.4.31)

In the two-dimensional case. e and g have the expressions:

e=-

a

.exp

(x+ T

\ g=-

2

exp

)J , l2 (x- (2.4.32)

with the notation utilized in (2.4.26) for a. The solution given by (2.4.5) and (2.4.21) is P(x, Y) = H(x)(fies + f2eg) (2.4.33)

V (x, y) = H(x)v(x, y),

56

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

where

v(x, y) = - f le-iwx [gy - iw J xgy (r, y) d r]

-

Q

(2.4.34)

[2ig + w2

f2C-

g(r, y) d TJ

Ix

J0

In the three-dimensional case, utilizing (2.2.6) we deduce:

p-(

1

C = - ----- ex 4irx

iw

2x

t2

_

1

J

,

-1

4'rr

2 2 x- y+Z

iw

exp

x

2

(2.4.35)

Taking into account (2.4.5) and (2.4.23), it follows

P(x, y, z) = H(x)(fie. + f2e,, + f3e=) (2.4.36)

W(x, y, z) = H(x)w(x, y, z), where -fie-Lox

w(x, y, z) =

rgz - iw

jx1

''j

x { - fee

gy d x-

//

- f3e-twx [2iwg + (w2 - ayy)

x

g dx

.

J

Q

The integrals from (2.4.37) have a strong singularity in the origin, but they are convergent, as it results from the following calculus indicated by V.Iftiinic:

!

/Oz

gd r

fJf

47r

bin ,

4T

hm e\O

r -exp

!02

r

I i (br `

fT r exp { i (6T F

lL

r= exp (ibr) - 1 exp

2l) l d r -2\1Jff

T!

f

J

dT 1

(-i- I d r + EimQ fJ T exp f\-- i

idt.

The first limit exists because exp (-ice /r) is a bounded function. In the second integral we perform the change of variable r = 1/t and we integrate by parts. We obtain:

inn J T

ce/z

(_!)dr=j

exp (-iC'-t)

jt =

57

FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1

x

iC2

= 2 exp (x )

f

0c

2

dt

exp (-ic t) t2 .

~ ir2 The last integral is obviously convergent. The solution (2.4.33) was given in {10.20) and the solution (2.4.36), in [10.17]. They can be also obtained as limits of the subsonic solutions.

Indeed, for P given by (2.4.6) and (2.4.7) one utilizes (2.4.27) and (2.4.28). A similar calculus may be performed for G given by (2.4.16) and (2.4.17). Passing to the limit for M \ 1, we get g given by (2.4.32) respectively g given by (2.4.35). This section was written entirely on the basis of the papers [10.17]-[10.20].

Fundamental Solutions of the Unsteady System I

2.5 2.5.1

Fundamental Solutions

In this section we determine the fundamental solution of the system (2.1.51), i.e. the solution of the system

pt+c2divv=0, vt+gradp= f6(t.x) (2.5.1)

liim(p,v)=0. 00

We already know that this system determines the perturbation produced in the fluid at rest, having the density po and the pressure po, by a force having the intensity f, applied instantaneously (at the moment t = 0) in the origin of the system of coordinates (on the Oz axis in the case of the two-dimensional problem). Utilizing Duhamel's principle, we deduce that the solution of the system (2.5.1) has the form:

(p, v) = H(t)(P, V) ,

(2.5.2)

where H(t) is Heaviside's function and (P, V) is a solution of the system

Pt+c2divV =0, P(0, x) = 0,

lim(P, V) 00

Vt+gradP=0 V (0, a) = f 6(x)

(2.5.3)

0.

Applying the Fourier transform, we deduce the system (2.5.4)

58

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

which has to be integrated with the conditions

P(0) = 0, V (O) = f .

(2.5.5)

The solution of this system is f)(1-coscla(tl

P=cia. f5inlcall It, V = f+ia(ia

J

2

,

such that, by inversion, we have:

P(t,x) _ -c(f . V).F-i

Csinclaltl

.

)

1C O

(2.5.6)

(1- Cos cla It 1

V (t' x) = fo(x) + V (f V ).F`'

C,,2

.

Utilizing the formulas (A.7.14) and (A.7.22), with the notation r = lxl, we deduce the following fundamental solution for the three-dimensional problem: P(t,x)=-(f.V)d(ct-r)

41rt (2.5.7)

V(t,x) = f6(x) + V(f V)

H(ct - r) 47rr

and for the two-dimensional problem,

P(t, x) = -

c

H(ct - r)

(f V) ct2-r' 21r (2.5.8)

r) Inct+ VCr t -r

V(t,x)=f6(x)+V(f

These are the solutions of the system (2.5.3).

2.5.2

Fundamental Matrices

The solution of the system (2.5.1) has been determined for the first time by means of matrices in (2.6]. As we shall see, in some applications it is preferable to use fundamental matrices. We introduce therefore the matrices:

0c200

00c20

000c2

0000 1000 ] 0000 Q= 0000 ,R- 1000 ,S= 0000 1000 0000 0000

,

(2.5.9)

FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1

VT

= fir,

U, V, W1, FT

59

= (0. f1, f2, f3J,

u, v, w representing the components of the vector v and fl, f2, f3, the components of the vector f. The system (2.5.1) is written as follows:

Vt+QV=+RV,+SVL =F6(t,x) (2.5.10)

V(t, oo) = 0.

Introducing the matrix L. such that

V .- LF,

(2.5.11)

we deduce:

Le + QL,, + RLy + SL,. = E6(t, x) (2.5.12)

L(t, oo) = 0,

E representing the unit matrix with 4 x 4 elements. Utilizing Duhamel's principle, we deduce that the solution of the equation (2.5.12) has the form

L = H(t) K(t, x), where

(2.5.13)

Kt+QKx+RKy+SKs=0 (2.5.14)

K(0, x) = EJ(x)

,

K(t, oo) = 0 .

Applying the Fourier transform to the problem (2.5.14), we deduce:

k,=.U, k(0,a) = E,

(2.5.15)

where 0

A=icr1Q+i02R+ia3S=i

C2a1 C202 C2a3 01 0 0 0 a2 0 0 0 0 a3 0 0

The solution of the problem (2.5.15) has the form:

K = Eexp(At).

(2.5.16)

It is well known that for determining the function exp (At) one may employ two classical methods: the method of matrix functions and the

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

60

method of the minimal polynomial. We shall utilize herein the method of the minimal polynomial described in 2.6.2. The eigenvalues of the matrix A are Al = 0, \2,3 = ±i laic, the first one being multiple of order two. One deduces that the minimal polynomial has the form

a(A - a2)(-\ - a3) = a(a2 + a2c'2) . Comparing this form with (2.6.13), we obtain

(?p=a2=0,

ai =-a2c2

Hence, the equations (2.6.18), (2.6.1.9) and (2.6.17) become

-

9oft

2c2921

go = 0,

92(0) = 9'2(0) = 0, !f2(0)

go (0) = 1,

9i = -c12c292 + 90,

91(0) = 0 .

K has the form Ego + Agl + A-2g2. We get:

k = E+ Asmclalt + A21 - COSCktIt C202 Clal

Using the inverse Fourier transform we obtain:

K(t,x)=E6(x)+AY-1 I

sinclalt+A2.._t

1 -c:asclaltclal

c2a2

J

where, for the three-dimensional case, 0 A

C2ar Clay c2as

ox ay

0 0

0 0

0 0

a_

0

0

0

Utilizing the formulas (A.7.14) and (A.7.22), we deduce that the matrix K has the form: H(ct - r) (2.5.17)

K(t,x) = Eb(x) + 4tAb(ct - r) + 4ac2A2

71

In the two-dimensional case we have:

K(t, x) = Eb(x) + +

1

27rc2

1

21rc

A

H(ct - r) +

c t - r' VC-It-

A2 H(ct-r)Inct+ r

-r

(2.5.18)

FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1

A=-

61

0 CZC7z c28y 0 0 ax a, 0 0

The formulas (2.5.13), (2.5.17) and (2.5.18) give the fundamental matrices.

2.5.3

Cauchy's Problem

We shall prove in the sequel that the solution of the problem Vt + QVz + RVt, + SV. = F(t, x)

V(0, x) = 0,

(2.5.19)

V(t, oo) = 0

(2.5.20)

is c

V(t,x)=1 K*Fdt,

(2.5.21)

0

where

K*F= f 3K(t-r,x-4)F(r,4)d4=f K(r,4)F(t-r,x-4)d4 is a convolution. Indeed, taking into account (2.5.14), we have:

Vt=K*FLt +

t

f K,*Fdr, 0

t

QVz+RVv+SVs=J (QKx+RKv+SK=)*Fdt, 0

V +QVs+RVy+SVz

F(t,x). fR3

The conditions (2.5.20) are obviously verified, because K(t, oo) = 0. Taking into account (2.5.17) and (2.5.18), with the notation p = _ fit', it follows for the solution of the problem (2.5.19), in the three-

62

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

dimensional case,

V(t,x)

.fu F(-r,x)d-r +

4c A

r

c

Jr3 F(t-

-r, x - t)6(cr - p) d t +

, A2 I d r fa F (t-

irC"

-r,x - ls)

3

H(cr-P)d P

(2.5.22)

and in the two-dimensional case, r1

V (t, x)

= foF(r,x)dr+2xc AI

u

-T,x~4)

H(crr _

I dTI2F(te

p) d4+2 VR-T P2

A2

p2dt.

-r,x-E)H(cr-P)III Cr+ P

(2.5.23)

2.5.4

The Perturbation Produced by a Mobile Source

We shall apply the above formulas for determining the perturbation produced in the fluid at rest (having the density po and the pressure po) by a force of intensity f . whose application point moves uniformly, with the velocity v, in the direction of the Ox axis and in the negative sense. This problem, also considered in [2.6), is important in aerodynamics because it models the leading edge of an uniformly moving airplane in the air at rest. The solution is obtained from (2.5.22) replacing

F(t, x) = Fo6(x + vt)6(y, x),

(2.5.24)

63

FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM 1

where Fo' = (0, fl, f2, f3). One finds V(t,x)=FDH(x,+vt)o(y,a)+4I

R,)

AFD.I D

t

l A2Fo f +zirc-

H(

dr+

b(cr

_ RT) d

(2.5.25)

where

R, _

x + vt - vr)2 + y2 + z2 .

(2.5.26)

For calculating these integrals we have to study the behaviour of the function h(r) = c-r - R,r , (2.5.27) on the interval (0, oo). We shall proceed like in 2.2.4. Because h(O) < 0

and h(oo) = (c2 - v2)oo. we have to separate the subsonic flow case (v < c) from the supersonic flow case (v > c). The zeros of the function h(r) are given by the formula (c2

where

- v2)r+ = -v(x + vt) ± R,

R = [(x + vt)2c2 + (c2 - v2)(y2 + z2), 1/2

(2.5.28)

.

(2.5.29)

In the subsonic case, only r+ is positive. If the velocity of the source

is supersonic (v > c), the function h(r) has two zeros (r+ < 'r-) if /to > 0 and no zero if !to < 0. We denoted ho = h(ro), ro representing the zero of the derivative of the function h,

vro=x+vt+

cr0

Vv-

(2.5.30)

It results,

vho = c(x + vt) -

(v2 - c2)(y2 + z2).

(2.5.31)

Utilizing the relation crfh'(r ) = ±R which may be verified directly, and the formula (2.2.38), we deduce

f

' 6(cr - R') T

0

dr =

cR-1 H(t -,r+),

if v < c,

cR-1JH(t-r+)+H(t-r_)J, if v>candho>0, 10,

if v > c and ho < 0.

(2.5.32)

64

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

In the same time, taking into account the definition of Heaviside's, function we have:

JtH(CRr) dRr v

- v In c(x +

vt)

c(x + vt) + R R,

if(vc,ho>0,t c, ho > 0, t >T-,

-;-

if v>c,ho<0.

0,

(2.5.33)

Now the solution (2.5.25) is explicit. The solution of this problem may be also obtained directly (without using the formula (2.5.22)) [1.10].

The interpretation of the solution is similar to the interpretation from (2.2.4).

Fundamental Solutions of the Unsteady System II

2.6 2.6.1

The Fundamental Matrices

In this section we determine, using the papers [2.71 and [2.9], the fundamental solutions of the system (2.1.26), (2.5.27), i.e. the solutions of the system

M2(pt+p=)+divv=0 vi + v;r + grad p = f a(t, x)

(2.6.1)

limp, v) = 0. 00

Introducing the matrices

Q=

1

1

0 0

0 0

00 10 ' 01

000M-2

00x1'20

1 M'200 R=

00

0

10

0

00 0

0 0 0

_ ' S

000 0 000 0 100 0

and the matrices V and F defined in (2.5.9), the system (2.6.1) may be written like in (2.5.10). We shall proceed like in the previous section. Considering

V = H(t)K(t, x)F,

(2.6.2)

65

FUNDAMENTAL SOLUTIONS OF THE UNSTEADY SYSTEM II

one obtains the following equation for k : Kt = AK, where

K(0, a) = E,

(2.6.3)

al a1M-2 02M -2 a3M-2 al

A=i

a2 CM

1

0

0

0 0

1

0

0

1

The roots of the characteristic polynomial are:

Al = ial,

A2,3 = i (al ± lalM-1),

(2.6.4)

the first root being double. One verifies that the minimal polynomial has the form

(A-A1)(A-A2)(A-,13) = (A-ial)3+a21tf-2(a-ia1). Comparing with (2.6.13) it results: a2 = 3i a1, al = 3a2 - 02M-2, ao = i a1(a2M-2 -

a2).

(2.6.5)

k has the form K = Ego(t) + Ag1(t) + A2g2(t) ,

(2.6.6)

the functions 90, 91, 92 being determined by the equations gg2 - a292 - a19o2 - aog2 = 0, 92(0) = g2(0) = 0, 92(0) = 1, (2.6.7)

goo = ao92, 90(0) = 1,

901 = a1.92 + go, 91(0) = 0.

Determining these functions, .%v find:

go = I 1 -

ioM

lal t la) sin M

M2 - a?a2

/ I

1 - o0c1

lalt`1

M JJ

/ m sin lal t - 2ia?M2 a2 1 -cos lal t J

91

=

9'2

(al 1 exp(ialt). = a/ - aft

_

M

M

exp(i alt),

exP(i a1 t),

1112

1

(2.6.8)

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

66

In the three-dimensional case one utilizes the formulas (A.7.15) and (A.7.23). From (2.6.6) it follows the fundamental matrix:

K(t, x) = Eb(2- t)b(y, z) + 4 +

(E

t

R)+

(EOY +

+2A8,.+A')11(tA1R

-R} (2.6.9)

where R = V(2--- t)2 + YT-A- z2 and

A = -

AI'28y M 2(9:

8r A1'2O 0. a2 0

8=

0

0

0

0

Sr

0

ay

In the two-dimensional case one utilizes the formulas (A.7.15) and (A.7.23). From (2.6.6) one obtains the following fundamental matrix: A12

K(t, x) = Eb(x - t, y) + -(EO + A) 2"

+

where R=

(

A122

2-r

H(t -- AM) jt72

+

A12R2

(O_ + 2A87 + A2}H(t - AIR) In t i

+_

t2 - A12

nip

(2.6.10)

- t)- y2 and 8? A

AI'28,. M-28y

8=

0.1

0

8y

0

8t

lit [1.10] and [2.7] one gives some applications which are important for aerodynamics. We mention the mobile source on the direction of the unperturbed stream and the mobile source on the direction perpendicular to the same stream.

2.6.2

The Method of the Minimal Polynomial

In the sequel we present the method utilized for determining the function exp (At). Let A be a matrix with n x n elements and let At, ... , Ak(k < n) be its eigenvalues, i.e. the roots of the characteristic polynomial

A(A) = det (AE - A)

.

FUNDAMENTAL SOLUTIONS OF TIIE UNSTEADY SYSTEM II

67

Obviously,

0(a) = (A - A1)m1

... ()1 - A )mk ,

(2.6.11)

where ml + ... + ink = n. The minimal polynomial associated to the matrix A is by definition the polynomial with the smallest degree which has the property

P(A) = 0.

(2.6.12)

Let

-... - a1,\ - ao P(A) = am be this polynomial. Obviously, m < n, because .(A) = 0. am-la,n-I

(2.6.13)

From (2.6.12) and (2.6.13) it results m-1

Am=

EajA''

j=0

and step by step m-1

Am+p

= E aln) A3 ,

p = 0, 1, 2, .. .

jO Taking into account that the exponential is a power series, we deduce

that there exist the functions go(t), ... , g.-1(t) .

such that

exp(At) = goE+g1A+...+gm_1A"-1

(2.6.14)

We have:

A exp (At) = goA +... + +9,n-1(ao E + al A + ... + a,,,-1 Am-1 ) , gm_2A'n-1+

( 2. 6 .15)

On the other hand,

A exp(At) = (d/(It) exp(At) = g(E+g'A+...+g',,,-lAm-1. (2.6.16) Since P(A) is unique, from (2.6.15) and (2.6.16) we obtain:

90=ao9m-1791 =a19+n-1+go,...,9in_1 =a+n-19m-1+9m-2 (2.6.17)

68

LINEAR AERODYNAMICS. FUNDAMENTAL SOLUTIONS

We deduce that 9m_1 is the solution of the differential equation 1rn?1

= am-1g(n1 1' -#- ... + olgVn- 1 + (109,n-1 ,

(2.6.18)

with the following initial conditions: g,,

1(0) = girl -1(0) = ... = girl1(0) _ 0, g

(in-1) 1' (0) = 1 .

(2.6.19)

The functions go, ... , gin-2 are determined from (2.6.17) with the cu conditions 90(0) = 1,91(0) = 0,---,9,,.-2(0) = 0-

(2.6.20)

imposed by (2.6.14). In fact, for determining the minimal polynomial, one takes into ac-

count that it is a divisor of the characteristic polynomial, hence it has the form:

P(A) = (A - A0," (A - A2)u'2 ... (A - Ak)' = = A"-on,-1Arn-1 -...-ao,

(2.6.21)

nj < mj and n1 + n2 + ... + nk = in. To conclude, we try with divisors having the form (2.6.21), beginning with the simplest where

(ill = n2 = ... = nk = 1) in order to satisfy the equation P(A) = 0. from the University of Bucharest We mention that dr. *t. has communicated us this method.

Chapter 3 The Infinite Span Airfoil in Subsonic Flow

The Airfoil in the Unlimited Fluid

3.1 3.1.1

The Statement of the Problem

In this subsection we determine the perturbation produced by a thin infinite cylindrical body in a subsonic uniform stream having the velocity UQ,i, the pressure pQ, and the density p,,. We assume that

the generators of the cylinder are perpendicular on the velocity of the unperturbed stream which is parallel to the xOy reference plane. The cylindrical bodies are bodies with constant cross-section and the infinite cylinders are cylinders which are long enough, such that the effects of the end conditions over the flow in the rOy plane may be neglected. Under these assumptions ( the conditions which determine the perturbation are not varying in time and they are the same in every plane parallel to xOy), the perturbed flows of the fluid will be stationary and plane. Ua

Y

Fig. 3.1.1.

% Te utilize the variables r , y, z introduced in (2.1.1) and the per-

turbations p and v defined by (2.1.3). In figure 3.1.1 one presents the profile determined by the intersection of the cylindric body with the xOy plane. We denote by

y = la(x) ± hl(x)

(3.1.1)

the equations of the curves C+ (the upper surface) and C_ (the

70

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

lower surface). The curve having the equation y = h(z) gives the so called skeleton of the profile. For imposing the boundary conditions we use the projection of the profile onto the Ox axis (the direction of the unperturbed stream). Defining Lo from (2.1.1) as the half length of this projection and taking the origin of the axes of coordinates in the middle

of the projection, it follows that the functions h(x) and h1(x) have to be defined on the segment [-1,+1]. We assume that these functions have continuous first order derivatives and that the derivative h(x) satisfies Holder's condition (see Appendix B). The cylinder is thin if h(x) = Fh(x),h1(x) = Eh1(x)

e representing a small parameter. We notice that whatever would be the equations of the curves C+ and C_, they may be written like in (3.1.1). Indeed, if the equations would be y = h±(x) = ehf(x), they might take the form (3.1.1) putting

h+(x) + h_ (x) = 2h(x), h+(x) - h- (x) = 2h1(x). Taking into account the equations (3.1.1) the following conditions have to be satisfied v(x, ±0) = h(x) ± 14 (x), Ix I < 1

(3.1.2)

as it follows from (2.1.32).

3.1.2 A Classical Method The classical methods for solving this problem rely on the assumption that the perturbation is potential. We have therefore v = grad Sp(x, y).

(3.1.3)

The potential p which is defined in the exterior of the segment [-1, +11 from the xOy plane, is the solution of the equation (3.1.4)

P"P:: +'p1,s, = 0

with the boundary conditions jpi,(x, f0) = h'(x) ± h4 (x) ,

JxJ< 1

(3.1.5)

and the condition at infinity z

lim00(P_, spy) = 0

(3.1.6)

71

THE AIRFOIL IN THE UNLIMITED FLUID

which follows from (2.1.28).

Using Clauert's transformation x, y -, X, Y:

X=x,

Y=fiy,

(3.1.7)

one obtains from (3.1.3) and (3.1.4)

u=six -- U,

v =A(PY=QV

sOxx +'PYY = 0

(3.1.8)

(3.1.9)

in the exterior of the segment [-1,+11 from the XOY plane. The boundary conditions (3.1.5) and the conditions at infinity (3.1.6) become

V (X, fO) = h'(X) f hi (X),

1X 1 < 1

lira (U,V)=0. X -»- cc

(3.1.10)

The problem was solved by many authors (see for example Birnbaum [3.51, [3.61, Sohngen [A.34), Sedov [1.38], p.47, Iacob [1.21), p.661), by means of one of the following two methods: reduction of the problem to an ntegral equation which is then solved or the reduction of the problem

to a boundary value problem in the complex plane. We also mention the methods relying on the expansion of the solution in a Fburier series. (Glauert [3.211 and Atei. inger (3.491).

In the sequel we give one of the most natural methods for reducing the problem to an integral equation. We represent the holomorphic function W = U - iV in the Z = X + iY plane with the cut [-1, +1] on the real xis, by means of the Cauchy integral

W(Z)=-L

r -f(t)+I t(t)dt,

(3.1.11)

where f(t) and fl(t) are unknown real Holder functions. It is easy to understand the significance of this representation. W(Z) Is the sum of the complex velocity determined by a continuous superposition of squrm having the intensity fm on the segment [-1, +1) and the complex velocity determined by a continuous superposition of vortices having the circulation f, on the same segment. In fact, the integral containing f may also be regarded as a continuous superposition of doublets on the segment [-1, +1] from the (Z) plane. Obviously; W(Z) is a holomorphic function in the exterior of the

segment [-1, +11 and it vanishes for Z -+ oo. Hence, U and V

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

72

satisfy the equation (3.1.9) and the condition (3.1.10). In order to prove this we pass to the limit considering Z - X ;k i 0, where X E (-1, + 1). Using Pleme(j's formulas (B.3.1) and separating the real part from the imaginary one we obtain:

2f(X)+2- /

U(X,±0)= V(X,±0) =

2ft(X) -

t

2zr,

it()dt

(3.1.12)

tfit) dt.

(3.1.13)

The mark"' " indicates the principal value in Cauchy's sense (Appendix B). From (2.1.38), (3.1.8) and (3.1.12) it follows

p(x, +0) - p(x, -0) = U(X, -0) - U(X, +0) = f (x).

(3.1.14)

This relation puts into evidence the significance of the function f (x).

It will be fundamental for the determination of the lift and moment coefficients.

Imposing the conditions (3.1.10) in (3.1.13), adding and subtracting the relations just obtained, we find Q

T

/ tdt _

= -2h' ( X)

JX i < 1 ,

Oft(X) = -2h4(X) JXi < 1.

(3 . 1 . 15)

(3.1.16)

The relation (3.1.16) determines the function Of, and (3.1.15) represents an integral singular equation for the determination of the function Of. It is the well known equation of thin profiles which was put into evidence by Birnbaum in 1923 and solved by Sohngen in 1939. Its solution, given in Appendix C, will be written in (3.1.25).

We notice that the thickness of the profile is ensured by the distribution of sources (ht = 0 ft = 0) and the discontinuity of the pressure (hence the lift and moment coefficients) is ensured by the distribution of vortices.

3.1.3

The Fundamental Solutions Method

There is no physical justification for replacing the wing by a continuous distribution of sources and vortices. It is more natural to replace the wing by a continuous distribution of forces and to determine the

73

THE AIRFOIL IN THE UNLIMITED FLUID

intensity of these forces, in order to obtain the same action against the fluid like in the case of the wing. Indeed, the fluid and the wing are two interacting systems. If we want to determine the motion of the first subsystem, we have to replace the action of the second subsystem by a distribution of forces. This is the basic idea of the fundamental solutions method. We shall replace the wing by a continuous distribution of forces f = (fl, f) on the segment [+1, -1], having the intensity f a priori unknown (the intensity will depend on the point). In fact, for obtaining a plane perturbation, the forces have to be uniformly applied on parallels to the Oz axis, intersecting the xOy plane in the points of the segment [-1,+11. Taking into account the formulas (2.3.23), it will result the following perturbation in the fluid:

p(x,y) = (x, v) _

1 27r#I f

+1 xof1() X02

+ #2y2 (3.1.17)

/+t xof(t) -yfi(E)d 21r ./

1

xo + 82y2

In view of imposing the boundary conditions (3.1.2) we have to calculate

ylimo v(x, y) for jxj < 1. Denoting by v(x, ±0) these limits, we have

u(x, f0) =

Q

lim 2ir y--±o

f+1 xof W 1

(3.1.18)

xo + 02y2

The first integral represents the tangential derivative of a simple layer potential and the second the normal derivative of the same potential. Although the formulas which give the limits of these derivatives are known, we prefer to calculate them directly. If we pass to the limit in the first integral making y = 0, we cannot

simplify with x0 because for

= x it is zero. We shall divide this

integral in three parts. Since for a e small enough, we may approximate

f

on the segment (x - e, x + E) with f (x), we deduce performing the substitution - x = u,

:+C xoM) _

#2y2

+`

ud u

I-C U2 +Qty 2

74

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

the integrand being an odd function. It remains +1

lim

xof W

d+

r:- E f

d- -lim E-.0 l J (

V-±0 J1 xo2+ 1

zo

1

r1 f(_)

Ji x-4

f

d J

=

d

excepting

zero,

The second limit from (3.1.18) is

f1

Jx.c xo

the

(3.1.19) interval

(x - s, x + e) where the integrand is infinite for y = 0 and f = x. Approximating again f j (C) by f, (x), or applying the mean formula, since f i is integrable and multiplied by an integrable factor having a constant sign, we get: Yli

m

J

lm Y

114 +a

fi(x)vuy f,

+ uZ

v(x,

0)

°J

y

-Z0'

f

±f1(x) -

d

(_)

Imposing the boundary conditions (3.1.2), adding and subtracting, we get: Q

f

(4)

d _ -2h'(x),

(3.1.21)

(x).

(3.1.22)

fi(x) _

We also have

P(x, fU) =

fi(e

f2f(x) + 2WQ

(3.1.23)

such that f (x) has the same significance like in (3.1.14). The sntegral equations (3.1.15) and (3.1.21) coincide. In fact, the representation (3.1.11) may also be obtained from the fundamental solution. This representation is valid for the points which do not belong to the semi-axis Ox (x > 1). Indeed in these points we

have u = -p, such that

u- v=

1

1

2

i -f (S) + i P

J_i

-Z

.11(x)

df

'

(3.1.24)

75

THE AIRFOIL IN THE UNLIMITED FLUID

where Z = x + i fly. We see from (3.1.16) and (3.1.22) that instead of Of, from (3.1.11) we put here fl. The equivalence just established shows that the formula (3.1.11) is not valid on Ox (x > 1). On this half-line the flow is not irrotational. We mention finally that the proof of Plemelj's formulas (which are necessary for obtaining the integral equation (3.1.15)) is anyway more difficult than the proof of the formulas (3.1.19) and (3.1.20). 3.1.4 The Function f (x). The Complex Velocity in the Fluid The solution (bounded in the trailing edge) of the integral equations (3.1.15) and (3.1.21) are obtained from the formula (C.1.9). We have therefore: (x)=

Of

2 V -x' 44 /iTih'(t)d t. tt x l+xf

(3.1.25)

1

Obviously, for the symmetric profile (h = 0) it results f = 0. One obtains the complex velocity substituting (3.1.16) and (3.1.25) in (3.1.11). Since, taking into account the formulas (B.5.5) and (B.5.7), we deduce

_

dt 11-t 1+t (t-Z)(s-t)

+1

1

+1

1

t

1

a

1 Z+t(tZ+st)dt=

Z

1

s-Z Z+1'

it follows

W(Z)

h'

I

-- A J-11 t'(Zdt+ i

+zr(3

Z

1

Z+1

f,

+1

(3.1.26)

1+ t h` (t)

1-tt-Z d t

For the incompressible fluid (f3 = 1) one obtains the formula (t) u-iv=-1AJf 1 hit-z dt+ i

+

-

z_ i

+1 J

+l /rTih (t) 1

(3.1.27)

1-tt-zdt'

where z = x+iy. The formula (3.1.27) was given for the first time by Sedov [1.38], p.51 and deduced in a different manner by lacob [1.21), p.664,

76

THE INFINL'I'E SPAIN AIRFOIL IN SUBSONIC FLOW

which solved the problem (3.1.9), (3.1.10) reducing it to a boundary value problem in the complex plane. It is not simpler to solve the bound-

ary value problem than solving the singular integral equation (C.1.1). Moreover, the complex velocity field is not of much interest in aerodynamics. It is utilized only for determining the jump of the pressure on the profile (which is calculated directly in the framework of the method of the integral equations).

3.1.5

The Calculation of the Aerodynamic Action

In some papers the aerodynamic action is calculated by means of a curvilinear integral on the contour of the profile. This calculation is wrong for contours with angular points. A correct calculation is performed using a control contour (surface in the three-dimensional case) surrounding the profile (wing). We shall perform this calculation where it will be absolutely necessary. Here we shall give a simple calculation, observing that in the first approximation, the action of the fluid comes

from the jump of the pressure pl(xl, -0) - pi(xi, +0) = Dpi 1, which gives the lifting force, parallel to the Oy axis. Taking into account the formulas (2.1.1) and (2.1.3) on the unity of length of the cylinder, it follows the lifting force

L= I

(IPi I dxI = P.UULo

J

I i [!pO d x

(3.1.28)

and the following momentcalculated with respect to the point x°:

MI

=fJ to (xi - x°) x pt ?d xi ,

where j represents the versor of the Oy axis. We obtain MI = Mk, where

rfi

M = poU020Lo J

(r - x°)OpOdx.

(3.1.29)

We denoted

Bp0 = P(x, -0) - p(x, +0)

(3.1.30)

Instead of the dimensional quantities L and M, it is preferable to use the dimensionless quantities CL and chl, named the lift coefficient respectively the moment coefficient. These aerodynamic coefficients are defined by the formulas: L M CL,

= (l/2)PooU2 (2Lo) '

cA'

= (1/2)p.U,

(3.1.31)

77

ME AIRFOIL IN THE UNLIMITED FLUID

It is preferable to use the dimensionless aerodynamic coefficients because

the numerical calculations are performed for dimensionless quantities. From (3.1.14), (3.1.28)-(3.1.30), and (3.1.31) it results CL

=-

f

+1

f (x)d x, cAt = i

-12

J

'

(x - x°)f (x)d x .

(3.1.32)

i

Finally, utilizing the solution (3.1.25), (B.5.4) and (B.6.9) we get

cL-

h(3.1.33)

r+i JY

CAf=-i13 J-1 f V

±tth'(t)dt- 1

2x°cL.

1

(3.1.34)

In the case of the profiles which are symmetric with respect to the Ox

axis, (h = 0), cL and cM vanish. Obviously we cannot use the method for calculating the drag because

it has the order e2

3.1.6

Examples

The flat plate. For the flat plate having the angle of attack (fig. 3.1.2a)), the equation (3.1.1) is

E

y = -xtgE = -Ex. It follows

h(x) = -.ex whence CL=

27re

ire

CM= -(x° + 2)A 1

(3.1.35)

These formulas were given for the first time by Glauert (1928) and Prandtl (1930). For the incompressible fluid (M = 0) one obtains CL = 27rE,

1l

cAf = - z° + 2) Ire

(3.1.36)

We notice that CL is increasing (c,%f is decreasing) for M / 1 (fig. 3.1.3) (the lift is increasing because of the compressibility). In the vicin-

ity of M = 1 (starting approximately with Al = 0.8) the lift has very great values, in contradiction with the reality. We deduce that in the vicinity of M = 1 the linear theory is not valid any longer. Therefore, for the transonic flow (Chapter-9) we shall utilize other equations. From (3.1.25) one obtains the jump of the pressure

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

78

U.

b) r

c) Fig. 3.1.2.

CI

2ttE

M 0.8

1

Fig. 3.1.3.

/if = -2e

1

V +

x

(3.1.37)

The parabolic profile (fig. 3.1.2b)). This profile is obtained for h(x) = e (I - x2), h1 = 0. Utilizing (B.6.9) we find CL =

Ze 73

°f=-

7r

Cm

1 - xa

(3.1.38)

with the same interpretation for cG.

The S-profile (fig. 3.1.2c)). For h(x) = e(x - x3) and hl (x) = 0, we obtain CL

- x°)

A , CH = 4,6 (

j

,

f = V-x + Tx 1+x

--X 2 (3.1.39)

1

The profile with thickness, having the shape of an elliptic sector For a profile having the shape of a sector bounded by two arcs

THE AIRFOIL IN THE UNLIMITED FLUID

79

of ellipse having the small semi-axes e2 < ei (fig. 3.1.4), we have

h+(x) _ -62 VI -X

h_(x)

,

x7, (3.1.40)

2h(x) = -(i, + s2) 1 - x

2h1(x) = (El -- 62W1 -X*

,

.

uw

Fig. 3.1.4.

For f, and f ,

taking into account the representation (3.1.16) and

(3.1.25) we deduce

Aft = (F1 - E2)

lx x v-

3f =

,

-

dt

t

1-t 1-t t -x

X

e2

1

?r

+xI

f' 1-x t-x

where

1+t

£l

x

dt +

1

1-x

(3.1.41)

+1 dt 1

1-t

Per forming the change of variable 1 -- t = u and taking into account (D.2.3), we get 7* +l 1ddu 1

,/ 1- t = `r2

u = in 2.

(3.1.42)

Utilizing (B.5.8) we get 1(3f =

El i Ex

/ I

\

x . In 2 + x In l + x)

(3.1.43)

Ale also have £1 +F2

2-

dt

_ 61+12(2--1n2), (3.1.44)

CMS _ -

£12x°(2 - In 2). 2/3

The action of the fluid is equivalent to a lifting force passing through the origin.

80

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

3.1.7 The General Case It is well known that the Chebyshev polynomialsof first kind

T,,(t) = cos(n arccost)

(3.1.45)

constitute a basis on the interval (-1, +1). We shall consider the series

expansion of h(t) in this basis: h'(t) = E an cos(n arccos t).

(3.1.46)

t=ooso

(3.1.47)

Putting it follows

co

h'(cos a) _ E an cos no n-U

whence j" h'(cos8)d6,

ao

an = 2

0

7r

7T

jh1(cos0)cosn0d0, (3.1.48)

n= 1,2,...

Substituting (3.1.46) in (3.1.33) and (3.1.34) and performing the change of variable (3.1.47), one obtains 2n cf,

ao +

al 2

a2\ 7r V cdt = - a ao + al + 2 J.

(3.1.49)

Theses formulas were given by Homentcovschi in (A.201.

It is important to calculate the distribution of the jump of the pressure on the profile. Using the change of variable (3.1.47) and Glauert's formula (B.6.6) we deduce +l 1 + t h'(t) °Q (1 + cos a) cos no-

J

1-tt-xdt=EanJ n-o

coso-cos9

U

=

(3.1.50)

= 7r cot

8 °O

an sin n9,

x = coo 8.

-o

Replacing in (3.1.25) it follows:

pf(x)=2Eansinn8 00 n=0

It is Glauert who proposed this type of solution.

(3.1.51)

THE AIRFOIL IN THE UNLIMITED FLUID

3.1.8

81

Numerical Integrations

In the case of an arbitrary profile, for calculating the integrals (3.1.33)

and (3.1.34) one may employ quadrature formulas (F.2.24). Using the notations

_

tQ-cos

2a-1 a=1,...,n 2n + 1'

(3.1.52)

,

one obtains n

> (1 +

41r

13(2n + 1)

cL

(3.1.53)

CM

-

n

2n -T3

1

a

(1 + ta)tali (tQ) - 2 CL.

(2n -+I) a-

1

For calculating the integral from (3.1.25) one utilizes the formula (F.3.1). In the collocation points 2i7r

xj =

t = 1,...,n,

Cos Fn + 1 ,

(3.1.54)

it results

f(T,)

4

p(2n

+'

1)

X. E 1

+

t

Q=1 Q

- x1 h (tQ)

(3.1.55)

The results obtained with these formulas are much more accurate than the results obtained with other methods.

3.1.9

The Integration of the Thin Airfoil Equation with the Aid of Gauss-type Quadrature Formulas

We may use the quadrature formulas from Appendix F to create and extremely efficient method for determining the solution of the integral equation (3.1.15). There were made many such attempts in various papers but nowhere one can find the good solution because there is not

prescribed the behaviour of the solution in the points ±1. We know from Appendix C that the solution of the above mentioned equation depends on the behaviour imposed in ±1. The solution satisfying the Kutta-Joukovsky condition in the trailing edge is

/3f (t) _ -E

+ t F(t) 1

(3.1.56)

82

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

Using (F.2.19) and posing h = -Eh, we reduce the equation (3.1.15) to the algebraic system n

Ai,F, =hl, j = 1,...,n,

(3.1.57)

a=1

where Fa = F(ta),

h1

= h (xj), A,a =

t-

2n + I '

2n + 1 Qa-

z1 =

x,' (3.1.58)

+ 17r' In (3.1.57) we have a linear algebraic system with n equations for n unknowns Fl, ... , Fn. We can create a computer code for Cos

solving the system. The coefficients AJa, the weight points to and the collocation points xj are the same for every profile. For a given profiles we have to change only the column with the elements h1, ... , K. After determining the unknowns Fa, the lift and moment coefficients result from the formulas (3.1.32) and (F.2.18). CL = - +1 f(t)d t = F t F(t)d t =

j_1I

l+t

Q

(3.1.59) 1

cl=-Z J

+1 1

WE

tf(t)dt=-2ENA1,

where n

ta)F0,

NL a=1

(3.1.60)

a=I

For verifying the method we utilized the analytic solution for the flat plate (3.1.37). From (3.1.56), it follows that for hJ = 1 we must have F = 2. The results obtained with the numerical method described above, with n = 20 gave for F. values situated between 0.999 and 2.001 and for NL and NM the value 0.999. Hence the method is extremely efficient [3.12].

3.2 3.2.1

The Airfoil in Ground Effects The Integral Equation

When an airplane is landing or taking off we have to take into account the ground effects. Some of the first papers devoted to these effects belong to Tomotika and his fellow-workers (3.44], [3.45]. We have also to

83

THE AIRFOIL. IN GROUND EFFECTS

mention the papers of Pancenkov [1.33], [1.34], [3.36], and especially the papers of Plotkin and his fellow-workers [3.37]-(3.39}, where one gives the integral equation of thin profiles in ground effects and one proposes

approximate solutions. The is considered a small parameter. Widnall and Barows (5.37J and Tuck [3.46] used asymptotic methods for investigating the problem. In fact one encountered two small parameters: the arrow of the airfoil and the distance from the airfoil to the ground. The fluid was considered incompressible. In the sequel, following 13.131, we shall utilize the method of funda-

mental solutions, in order to obtain the integral equation for the compressible fluid and for the airfoil with thickness. The small parameter is the arrow of the airfoil. In case that the distance from the airfoil to the ground is also small, we have to elaborate a new theory. We use the notations from the previous subsection. We denote by a/2 the distance from the airfoil to the ground (fig. 3.2.1). The perturbation has to satisfy the following boundary conditions: v(x, f0) = h(x) ± hI (x),

v(x, -a/2) = 0

U.

Ix1 < I.

(3.2.1)

- oo < x < oo .

(3.2.2)

-I

+1

x

T a/2

an

-I

---

- -

+1

Fig. 3.2.L.

According to the method of fundamental solutions, we have to replace the airfoil by a continuous distribution of forces (ft, on the

segment [-1, +1] from the y = 0 axis. For satisfying the boundary condition (3.2.2) we shall consider a symmetric distribution of forces

(fl, -f

on the symmetric segment (-1, +11 of the y = -a axis

and we shall determine the intensity of the distributions from the condition (3.2.1). Taking into account that the perturbation produced within the uniform stream having the velocity U,i by the force (fl, f2) is (2.3.23), it follows that the two distributions will determine in the fluid

84

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

the perturbation

1 J-tf

+t

P(-T' y) =

+

v(x, y) =

1

xoft W + J32yf (t) x02 +,82y2

f

+t

xofi(Q - 132(y + a)f (C) + M(y + ;)-2

A f+t z0f(0 - bf1(0 d

xa +U

1

-

(3.2.3)

1 r+t xofW + (y + a)ft (_) 21r

x0#2(N+a)2

1

f and f j

having to be determined. One easily verifies that this representation satisfies the condition (3.2.2). We denoted xo = x - . Only the first integrals become singular. when passing to limit for y

±0, Using the formulas (3.1.19) and (3.1.20), we deduce

p(x, +0) - p(r., -0) = f (x) ,

f(o d t-

t

v(x, f0)

_

2ft(x)

+ 27r

(3.2.4)

,

(3.2.5)

+1

io+m

11-r

where in = fla. adding, we get:

Imposing the conditions (3.2.1), subtracting and

ft(x) _ -2hi(x) ar

FI fro)

2l

jxi < 1

t1 oo+ in

dt = H(x),

(3.2.6) (3.2.7)

where

H(x) = h'(x) -

A ,l

+t,,,2d 1

xo

(3.2.8)

The equation (3.2.7) is the integral equation In Appendix C it was called the generalized equation of the thin airfoil. For the incompressible fluid (/3 = 1) and for the airfoil without thickness (ht = 0) it coincides with the equation given by Pancenkov and Plotkin.

85

THE AIRFOIL IN GROUND EFFECTS

3.2.2

A Numerical Method

The equation (3.2.7) is obviously a singular integral equation. As we have shown in Appendix C, it may be reduced to a Fredholm equation, but the problem remains still unsolved because there are not available general methods for solving this type of equations (excepting the method of successive approximations). As we have already shown in 13.131, the equation (3.2.7) may be solved numerically utilizing the quadrature formulas from Appendix F. Looking for the solutions of (3.2.7) having the form (3.1.56), putting H = -e and using (F.2.18) and (F.2.19), one obtains the linear algebraic system n 7 1j

.

(3.2.9)

a=1

where FQ = F(t0),

T-1 j = H(xj),

B ra

1

(t° - 1)(xj - t0)

2n + I (xj - tu)2 + m2

(3.2.10)

Aj,,, t,, and xj being given in (3.1.58). From the system (3.2.9) we determine the unknowns F1,. .. , F,,. Now the lift and moment coefficients (3.1.32) (for x° = 0) will be obtained from the formulas CL

J_ !+1

CAI

2{3

I.

(3.2.11)

_

+

t tF(t)d t

2 Nit ,

where we have utilized the notations (3.1.60). One may write computer programs for solving the system (3.2.9). The coefficients Aja and Bj0 do not depend on the shape of the

airfoil, such that the program may be utilized for every airfoil. One changes only the matrix with one column Wj . 3.2.3

The Flat Plate

In the case of the flat plate with the angle of attack we have 77j = 1. We solve the system (3.2.9) and then we determine NL and

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

86

NM with the formulas (3.1.60). We write the aerodynamic coefficients (3.2.11) as follows CL = cL NL,

(3.2.12)

CAf =

where c' and cM are the coefficients for the free stream (a = oo). 5.

NL1 4

3

2.

1.0.6 Mp0

}

1.

0.0

4.0

2.0

6.0

8.0

0.0

4.0

2.0

6.0

8.0 a

a

a)

b)

Fig. 3.2.2.

The variation of NL and NAf versus a is shown in figures 3.2.2a) and 3.2.2b). The compressibility effects on the lift and moment are greater than it is shown in the above mentioned figures, because A also intervenes in c -L4 cy. Obviously, the ground effect is great when a is small and it quickly decreases when a increases. 3.2.4

The Symmetric Airfoil

As we have seen in (3.1.5), for the symmetric airfoil in a free stream with zero angle of attack, the lift and the moment vanish. The situation is different for the wing in ground effects. As we may observe from (3.2.7) and (3.2.8), i i the case of symmetric airfoil (h = 0, hl 0 0), H does not vanish, hence the solution of the equation. (3.2.7) is different from zero. For the Joukovsky symmetric profile considered in 11.28), [3.37], [3.13) y = +E(1 - X) VI - x2

(3.2.13)

THE AIRFOIL IN GROUND EFFECTS

87

we obtain using the notation hl = -(1 - x) 1- x' and the formulas (F.2.12) and (F.2.18):

H =-m

;r J-1 (xj - e)2 + m2 in

P

d_

1 - (-2,

'1

2m

to (1 - trt )

+1F-(,)2+fn2 - 2n+I

E(xj-tn)2)+m2'

(3.2.14)

In this case one presents NL and N11 versus a in figures 3.2.3a) and 3.2.3b) for n = p = 20. The lift coefficient is where

cos y7r

p+

negative i.e. the resultant is a force pointing towards the ground. Hence when the airplane is landing or taking off it becomes heavier. The pilot has to take into account this fact.

- al.bo

- - - Msean6

I

: _

F-

a'l 0.001 l

000

1.60

/

-=-

l

4.00

5.20

a30

410

OAO.

I

000

400

I

1

a 1.60

'20

4.90

640

900

b

a) Fig. 3.2.3.

The fact that both in the case of the flat plate and the case of the symmetric profile, the lift and the moment become very great when a is very small, is not true in reality. Hence, for small values of the parameter

a we have to elaborate a new theory based on two small parameters (see [5.37]).

88

3.3 3.3.1

THE INFINI'T'E SPAN AIRFOIL IN SUBSONIC FLOW

The Airfoil in Tunnel Effects The Integral Equation

The experiments for determining the aerodynamic parameters are performed in wind tunnels. We have therefore to take into account the influence of the walls of the tunnel an the aerodynamic characteristics, of the airfoil. Since many papers dedicated to this subject are secret, we cannot give the history of the research in this field. We have cited in the bibliography some authors, without consulting their papers. We shall present therefore, only the model that we gave in [3.14]. This model can be easily obtained with the method of fundamental solutions and it is in the spirit of the theory previously presented in this book. We formulate the problem as follows: an uniform stream, having the

velocity U ,i, the pressure pao and the density p,,, flowing between two infinite flat plates parallel to the O.T. axis, encounters a thin airfoil of infinite span with the generatrices parallel to the Oz axis. The fluid is compressible, and the velocity of the uniform stream is subsonic. One requires to determine the perturbation and the influence of the stream on the airfoil. We utilize the variables (2.1.1) and (2.1.3). Let (3.1.1) represent the equations and a the distance between the plates (walls) (fig. 3.2.1). For determining the perturbation, we have to impose the following boundary condition: v(x, ±0) = h'(x) ± hl(x),JxJ < 1

(3.3.1)

v(x, ±a/2) = 0, -oo < x < oo

(3.3.2)

Y

+11 w2

xx

I /2 Fig. 3.3.1.

In order to utilize the method of fundamental solutions, we shall f)(t) replace the airfoil by a continuous distribution of forces defined on the segment [-1, +1). For satisfying the boundary conditions (3.3.2), we have also to take into account symmetric distributions on the

images of the strip [-1,+1) in the planes y = ±a/2 and symmetric distributions on the images of the images in the planes y = ±3a/2 etc.

89

THE AIRFOIL IN TUNNEL EFFECTS

(the method of images). In this way one obtains the following general representation of the perturbation: +oo

P(x, Y) - 27rp

na)f(e) d xo + 02(y - na)2

+1

1

J-

,

(3.3.3) (

v x, y) =

p }OO f 27r

1

(-1)"Xof(t) - (y ro + (y - na)2

with xo = x - . We can easily verify that v given above satisfies (3.3.2). For imposing the conditions (3.3.1) we have to pass to the limit considering y -+ 10, -1 < x < 1. The only singular integrals

correspond to n = 0 and they are calculated using the formulas (3.1.19) and (3.1.20). Taking into account the equalities (1.161: 00

_

1

k2 + n2 n=1

zr cosh kn 2k sinh kir

1

2k2 '

A

2k sink ka

"-1 k2 + n2

2k2 '

we deduce: 1

p(x, f0) = f 2 f (x) +

v(x,±0) _

f xo

)d

+

(3.3.4)

+1

1

7rp

-L

J_1

2f1(x) + 2r ,

rLd+ (3.3.5)

+p

where m = ap Kl (xo) _

I fi

K(xo)f(t)dt`,

-o

K(xo) = m sink-' (m xo' -

,

coth \\(m xo//

,

\

XO

(3.3.6)

From (3.3.4) we deduce again the significance of the function f :

f (x) = p(x, +0) - p(x, -0),

(3.3.7)

90

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

and from (3.3.5) and (3.3.1),

fl(x) = -2hi(x),

(3.3.8)

+I

x

2, K(xo)f()d= h'(x)

.

(3.3.9)

1

The formula (3.3.8) determines fl and the equation (3.3.9), the function f. The integral equation of the problem is in fact the generalized equation of the thin profiles. The kernel K has no singularity for t = x. We notice that for a - oo (m -+ oo) the equations (3.3.4) and (3.3.9) are reduced to the equations corresponding to the airfoil in a free (unlimited) airfoil.

3.3.2

The Integration of the Equation (3.3.9)

We utilize the quadrature formulas from Appendix F for integrating

the equation (3.3.9). Putting h = -eh, we shall look for solutions f having the form (3.1.56). Utilizing (F.2.18) and (F.2.19), we reduce the equation (3.3.9) to the system n

j=1,...,n,

>2CfaFo =hj.

(3.3.10)

0=1

where F. = F(ta), hi = h(xj), C'° = m 2n

1

sinh-1 I -(xj

- to,)}

,

(3.3.11)

§i xj being given in (3.1.58). After determining the unknowns F1,.. . , F, from (3.3.10), the lift and moment coefficients may be obtained by means of the formulas: tQ

(3.3.12)

cA, _

CL =

NL and NA, having the expressions given in (3.1.60). The distribution of the pressure on the airfoil may be obtained from (3.3.4), taking into account (3.3.8). We get p(to,T0)=ff&

2J3

l+ta

1 rr/3

1 hl(x)dx1

tp - x

THE AIRFOIL rN TUNNEL EFFECT'S

91

The values of the pressure in the points to, may be determined in the wind tunnel by means of pressure plugs. We have thus the possibility to verify the theory presented herein. We may use a computer for solving the system (3.3.10). Since the coefficients C1,, do not depend on the shape of the airfoil, we have to change in the program only the column containing the elements 1 .

Numerical Results

3.3.3

= 1. In

For the flat plate having the angle of attack e we have

this case, the formulas (3.3.12) become cL = cL NL, cm = cMNM, cOLO

and cc representing the coefficients corresponding to the unlimited fluid. - M. b 0

»

1

MWI"06

za

-I

I

1.4

1.3

a

00

02

0.4

0.6

01

1.0

'10.1

00

01

0.4

06

02

1.0

to"

h)

a) Fig. 3.3.2.

The figures 3.3.2 present the variations of NL and Niv versus the width of the tunnel for M = 0 (incompressible fluid) and M = 0.6. We notice that the lift in the wind tunnel is greater then the lift in the unlimited fluid and it decreases when the width of the tunnel increases. For the same width, the lift is an increasing function of M. The theory presented herein allows us to determine these variations.

92

3.4 3.4.1

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

Airfoils Parallel to the Undisturbed Stream The Integral Equations

In the literature, the problem of the uniform flow past a configuration of airfoils has been solved for the incompressible fluid and for airfoils whose exterior may be mapped conforma.lly on the exterior of a circle. There were given especially solutions for the biplane [1.36], [3.38]. We shall consider in the three forthcoming subsections, following the paper [3.15], the general problem of the compressible fluid in subsonic flow and

airfoils with thickness of arbitrary shape. YA

UM

a/2

x

Fig. 3.4.1.

We shall consider in this subsection, two airfoils parallel to the undisturbed stream (fig. 3.4.1), not necessarily identical. The free stream has

the velocity U,,,i, the pressure p,, and the density pp,;,. One utilizes the variables (2.1.1) and (2.1.3) and one considers that the equations of the airfoils are: y = h(x) . h1(x), Y = 1(x) ± 11(x),

krl < 1

(3.4.1)

1 XI < 1

(3.4.2)

the first corresponding to the upper airfoil and the second to the lower airfoil. The dimensionless distance between the projections of the airfoils is denoted by a. Utilizing the method of fundamental solutions, we

shall replace the upper airfoil by a distribution of forces of intensity and the lower airfoil by a distribution (gl, g)(t), both of them (fl, defined on the segment [-1, +1]. Starting from the above mentioned perturbation produced by a force in the free stream (the fundamental solution (2.3.23)), we deduce that the perturbation produced by the two

AIRFOILS PARALLEL TO THE UNDISTURBED STREAM

93

distributions is: +t

1

Ax, U) - 2yrQ

xof 1(e) + QZ(y - a12)f (t)

1- 1

xa +,Bs(y - a/2) s

d t+

+t

X091 (t) + O2(y + a/2)9(t)

1

+ 27rp J-1

d

(y + a/2)2

xo +

(3.4.3)

x, y) = (

p T,,-- 1

+1 xof (e) - (y - a/2)ft(e) d t+ xp +,02(y - a/2)2 1

+ Q r+t xo9(t) - (y + a/2)gt () d t,

2tr f 1

xo + /32(y + a/2)2

where x0 = x - .. The distributions of forces will be determined from the boundary conditions

When y

v(x, (a/2) f 0) = h'(x) ± hi (x),

fix( < 1

(3.4.4)

v(x, -(a/2) f 0) = !'(x) ± li (x),

jxj< 1

(3.4.5)

(a/2) ± 0), the first integrals from the representation

(3.4.3) become singular and they have to be calculated with the formulas (3.1.19) and (3.1.20). One obtains

P(x, (a/2) f 0)

2 f (x) +

+1

1

+ 2aQ

' fl

,

270 T-1

-;To

(3.4.6)

xo.91 + 7n,89

xo + jn2

_1

1

v(x, (a/2) 10) _

1 ft (x) + 2a

T-1

d+

d

f (')d t+ xo

(3.4.7) 1

1 Qxog -

f

+ 2rr ,l

1

mgt

xo + m2 d

,

where rn = ap. From (3.4.6) it follows f(x) = P(x,(a/2) + 0) - p(x,(a/2) - 0),

(3.4.8)

and from (3.4.4) and (3.4.7)

fi(x) = -2h'1(x)

(3.4.9)

94

THE INFINITE SPAN AIRFOIL ICY SUBSONIC FLOW

J

t f x()) d

t

+ 2 , i t xo +(r roe d = H(x),

where

(3.4.10)

+I

H(x) = h'(x) +

2

d

.

(3.4.11)

In the same way, passing to the limit in (3.4.3) when y -i -(a/2)±O (in this case the second integrals become singular), from the boundary condition (3.4.5), we obtain

g(x) = p(x, -(a/2) + 0) - p(x, -(a/2) - 0),

'(mot 9(e)

1

gi(x) = -21c(x), +t f( )2dC +

I

0

I

X2 +,fn

(3.4.12) (3.4.13)

= L(x),

(3.4.14)

0

where

L(x) = l'(x)

-

,1

t

I x('2d.

(3.4.15)

The distributions of forces on the two chords are determined by solving the system of integral singular equations (3.4.10) and (3.4.14). From

(3.4.9) and (3.4.13) the functions H(z) and L(x) are known. The field of pressure on the profiles are given by the formula (3.4.6) and the corresponding formula for p(z, -(a/2) ± 0). Symmetric Airfoils. If the two airfoils are symmetric with respect

to the Ox axis, then

l(z) = -h(x), 1I (x) = hI (x) .

(3.4.16)

It follows L(x) = -H(x) whence

xz(+tn

r,

I

If

gdt;+ o

J

(3.4.17)

I

Since the solution of the generalized equation of thin profiles is unique, it results f = -g, and the 'system of equations (3.4.10) and (3.4.14) reduces to a single equation

2

Ifro) d 1

2

f

+I I

+(n) 'TO f s

d4= H(x)

(3.4.18)

o

which is just the equation (3.4.7) of the airfoil in ground effects. The result is natural. The lift and moment coefficients for the upper airfoil

95

AIRFOILS PARALLEL TO THE UNDISTURBED STREAM

are given by the formulas (3.2.11). For the lower airfoil they have the opposite sign. For the entire configuration we have cL = cAt = 0. Identical Airfoils. If the airfoils are identical, then 1(x) = h(x) - 2a,

it(x) = hj(x).

(3.4.19)

Moreover, if the airfoils have no thickness, then L = H. Subtracting the equations (3.4.10) and (3.4.14), it results g = f whence A

f

I_ z2 +(m)2d = h'(x). 1

1

27r

(3.4.20)

This equation has the form of the generalized equation of thin profiles and it can be integrated numerically like in (3.2.2) and (3.3.2).

3.4.2

The Numerical Integration

In order to solve numerically the system (3.4.10) and (3.4.14), we shall use the quadrature formulas from Appendix F. For thin airfoils,

the functions H(x) and L(x) have the form H(x) = -eH(x), L(x) _ = -iL(x), hence we shall look for the following type of solutions

Qf (t) = -E V 1

+t

F(t),

3g(t) = -E

1 + tG(t)

(3.4.21)

which satisfy the Kutta-Joukowsky condition on the trailing edge. Utilizing (F.2.18) and (F.2.19), the system (3.4.10) and (3.4.14) is reduced to n

E(Aj.F.-Bj.Go)=Hl, ,7=1,...,n, a=1

(3.4.22) n

>(A1oGo-BjoFa)=1i, a=1

A,,,, tQ and x, are given in (3.1.58) and Bj, in (3.2.10). Like always,

F. = F(t0), G. = G(t0), H, = 77(x,), LJ = L(x,). We have to find out the unknowns Fl,. .., Fn, Gl, ... , Gn from the system (3.4.22). The

coefficients Aj0 and B 0 are the same for all the airfoils; only H and L, are varying. The lift and moment coefficients for the entire configuration are given by the formulas

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

96

CL

=-1

1+1

+1

(f+9)dt=

/(F+G)dt--Ni 1+t (3.4.23)

+1 CAI

--2J-,

+1

t(f+9)dt=2Q11

where

t

l+t(F+ c)dt =

-ASE2Q1V2,

n

N, =

1

2n + 1

:(1 - ta)(F. +G(), Q=1 (3.4.24) n

N2

2n 1 1

1)(F.+G.). 0=1

We notice that N, and N2 contain only the unknowns Fa + Ga which may be determined from the system

Aja(Fa + Ga) - Bja(Fa + Ga) = Nj + Lj

(3.4.25)

a=l

of n equations (j = 1, ... , n) with n unknowns. The unknowns Fa and Ca are separating only when we calculate the pressure on the two airfoils. For example, the field of the pressure on the upper airfoil is obtained from (3.4.6) with the formula

p(t,., (a/2) f 0) = f

29E

f

1 -to F.

0 - 4)G(ti) f3(2n + 1)

(ta - t{)2 + m2

- x)li(x) h', (x) dctQx(ta-x)2+m2d x, iro f!" 1

t

cos

2iIr

2n+1'

(3.4.26)

to being given in (3.1.58). In the same way we may determine the field of the pressure on the lower airfoil. The numerical determinations have been performed for the biplane

having the angle of attack E(h' = 1' = -E) (fig. 3.4.2a)) and for the symmetric biplane (h' = -1' = -e) (fig. 3.4.2b)) taking n = 10. In

GRIDS OF PROFILES

97

a)

b) Fig. 3.4.2.

the first case HJ = L, = 1, and in the second 77J = -Tj = 1. The coefficients A, , Bin depend only on the parameter m = a#. In the second case we obtain N1 = N2 = 0 for all the values given to m (it is natural). In the first case, the values of N1Q'1 and N2Q-1

depend on a and M. They are given in tables 1 and 2. In the first line we may find the values of 13-1N1 and 0-1N2 for the monoplane. We notice that the lift coefficient for the biplane is much greater then the lift coefficient for the monoplane and it increases with a. The lift increases also with M. The same conclusions are true for the moment coefficient. Table 2

Table I

The values of N,$-1

a=0.5 a=1 a=5

3.5 3.5.1

M=0

M=0.6

1.00 1.461886

1.25 1.741425

1.70842 1.981013

2.034937 2.46087

The values of N2

a=0.5 a=1 a=5

1

M=O

M=0.6

1.00 1.624288

1.25 1.941250 2.209987 2.477736

1.827670 1.991025

Grids of Profiles The Integral Equation

The classical problem may be also solved in a simple manner by means of the method of fundamental solutions. One obtains again the generalized equation of thin profiles. Let us consider a grid of identical airfoils having the equations y = na + h(x) ± hl (x) ,

n=0,±1,±2,...,

(3.5.1)

98

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

which perturb the uniform subsonic stream defined in section 2.1.1. Since every airfoil produces the same perturbation, we shall replace every profile by the same distribution of ford (fl, f) (c). Since the perturbation produced by the force (fl, f2) in the uniform stream is (2.3.23), we deduce that the perturbation produced by the grid is

1 C`

P(x, U) =

+t xoft(s) + Q2(y 1-1 (3.5.2)

Q 21r

u(x, A =

+°° -00

f

+' xof W - (U - na)fl W d x0 + (32(y - na)2

L

where xo=x-t;. The profile corresponding to n = 0 will be called the reference Passing to limit, when y -, ±0 the integrals corresponding to n = 0 become singular and they are calculated with the formulas profile.

(3.1.19) and (3.1.20). Employing the notation k = xo/m. we get +00

n*o

-00

x° A0 + m2n2

- 2x°

-00

nx

n

-

1

2x°

1r

k2 - n2 - m2 \2k

m2

= +00

00

n

coth -xo m 7T

coth k7r -

1

2

1

K(xo),

r

TO

2n=0' (3.5.3)

whence: '6 P(x, ±0) = f 2 f (x) + j,-

f x°) d t+

f1

(3.5.4)

+z

1

+2- j h (xo)fi (4)d t, P(x, +0) - P(x, -0) = f W,

(3.5.5)

d+

1

u(x, f0) _ 2 Q

ft (x) + 2 J

t

o (3 .5.6)

+i

+2 fl

99

GRIDS OF PROFILES

Imposing the boundary condition v(x, f0) = h'(x) ± M, (x)

(3.5.7)

fi(x) = -2hi(x)

(3.5.8)

we deduce +1 2 j K(xo)f(t)dt

If (o) d t +

= h'(x)

(3.5.9)

1

This is the integral equation of the problem. It has the same form like the equations (3.2.7), (3.3.9) and (3.4.10), (3.4.14), excepting the non - singular kernel. Obviously, from (3.5.3) we deduce that the kernel K(xo) is not singular. After determining f by means of (3.5.9), the field of pressure on the reference profile will be obtained from (3.5.4), and (3.5.5) will be utilized for the calculation of lift and moment coefficients.

In order to obtain the pressure on the profile corresponding to n = I and to impose the boundary condition on this profile, we must pass to the limit in (3.5.2) y - a :E 0. In this case, the integrals corresponding to n = 1 become singular. With the change of variable y - a = Y we have for example

rx, +t

Ax, a t 0) =

lim

to J- xxo + #2Yf d +

27r,3 Y

+

+00

1

n#1

27 r3

2

-00

+1 xof1 +,82(1 - n)af d ,l-1 x0 +1112(1 -n)2

Putting I - n = nl in the last integral we deduce +00

n'At

-00

_ +ao

1

To' + m2(1 -

-a0

00

1-n

+00

"#1

n)2

n,00

X.2

+ m2(1 - n)2

1

m2n'

-0

such that, taking into account (3.5.3), we obtain for p(x, a t 0) the same expression like for p(x, ±0). In the same way, for v(a, a f 0) one obtains the same formula like for v(x, ±0). In this way, imposing the boundary condition

v(x,a±0) = h'(x) ±14(x),

(3.5.10)

100

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

one obtains the equations (3.5.8) and (3.5.9). We draw the conclusion

that if f1 and f are determined by (3.5.8) and (3.5.9), the boundary conditions arc satisfied on each profile and the field of pressure coincides with the field on the reference airfoil. The lift and moment coefficients are calculated on every profile by means of the formulas

cj =-J

1

1

2

1

3.5.2

r

f(t)dt, cal=---

+

tf(t)dt.

(3.5.11)

t

The Numerical Integration

If f haa, the form (3.1.56) and It has the form -sh, using the quadrature formulas (F.2.18) and (F.2.19), we obtain from the integral equation (3.5.9) the system n

Di,,Fa = Itj, j = 1, ....:a ,

(3.5.12)

a=1

where

T 1 - t Goth

in 2n + 1

71

in

(xj - ta) ,

(3.5.13)

to and xj being given in (3.1.58). Considering for CL and cAf the form (3.1.59), we obtain for NL and N,%j the expressions (3.1.60). For example, for a grid of flat plates having the angle of attack E(hj = 1), the variations of the coefficients Nf and N,tt versus a, are given in figures 3.5.1.

We notice that the coefficients cL and cAt are increasing functions

of a and Al. They tend asymptotically to I when a - cc, hence the lift and moment coefficients tend asymptotically to the taken in the case of a single profile. The pressure in the control points (nod(s) is obtained from the formula

P(tu>f0) =

213

1

7T,9

x-

tQda.-

(3.5.1 l)

/' ,ro

-1

AIRFOILS IN TANDEM

101

.o

N 12

1.02

a%

IN 09$

014

091

066

011

US

0n

00

a

000

020

010

0.60

030

040

1.00 X10-,

a)

000 am 040

060

0.*0

1.00 X10'

b) Fig. 3.5.1.

3.6 3.6.1

Airfoils in Tandem The Integral Equations

In the sequel we shall determine the perturbation produced in the uniform stream defined in 2.1 (under the assumption that it is subsonic (M < 1)), by a configuration of airfoils in tandem (fig. 3.6.1), having the equations

y=11(x)±h1(z), al <x
(3.6.1)

a2 < z < b2 .

(3.6.2)

y =12(x) f h2(z),

For the incompressible fluid this problem was studied by Chaplygin and Sedov (1.38) who determined the complex velocity. If we reduce the problem to integral equations and we solve them like in 13.151, we can determine directly the quantities of interest in aerodynamics (the field and the jump of the pressure on the airfoils, the lift and moment coefficients). Moreover, the fluid may be compressible and we don't need to use elliptic integrals. In the sequel, we shall present the results from the last paper cited above. Replacing the airfoils by distributions of forces having the intensities (gl . fl) respectively (g2, f2) and taking into account the fundamental

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

102 Y

K

a,

b,

a3

b,

Fig. 3.6.1.

solution (2.3.23), we get the following representation of the perturbation:

jbX0g1(t)+#2yf1(C)

1

P( x, y) = 2aQ 1

x0 + /y2 r"' xo92(

+2Q 1

S

) + 02Yf2(t) d

xo + Q2 y2

,

(3.6.3)

v(x,y)=

xofiW --y91( )d xo + 132y2

2,r J a

Q rb'

+ 2a az

x0 +

dtt

y2

We kept the same notations like above. Passing to limit for al < x < b1 and taking into account (3.1.19) and (3.1.20), one obtains:

P(x,f0) = 2fi(x) + 27r#

xo

J a2

(3.6.4)

p(x, +0) - p(x, -0) = f 1(x)

(3.6.5)

f,a,

0

Q, f bi fi(t)

v(x, t0) =

bz

o1( )dC+

91(x) + 27r J

n,

xo

d+

fb aZ

xo d '

(3.6.6}

Imposing the boundary condition

v(x, ±0) =1 (x) ± hi (x) it follows 91(x) = -2hi(x)

Q

fl(O)

27r1 axo 1

d

2a Jxo

2xo L d=11(x) (3.6.8) Q

f

+

+

(3.6.7)

AIRFOILS IN TANDEM

103

In the same way, for a2 <x < b2 we deduce:

P(x,±0)=f2f2(x)+2a V(x, f0) = -T 192W + 2

27r

-

C

g 0{dt+2 )

., f )dt+ 2 Mo

22(i)dt(3.6.9) Y J., x0

f oQd(, z

whence, utilizing the boundary condition

v(x, f0) = !'(x) f h2(x) we obtain

p(x, +0) - p(x, -0) = MX),

(3.6.10)

92(x) _ -214(x),

(3.6.11)

2f

a,

i

t

'f

: _ (x).

f

( 3. 6 . 12)

Hence the functions fl(x) and f2(x) will be defined by the system (3.6.8) and (3.6.12). This is an interesting mathematical problem, since the first equation is defined for al < x < bl and the second is defined for a2 < z < .b2.

3.6.2

The Determination of the Functions f, and f2

For f, the equation (3.6.8) has the structure (C.1.5) which has the solution (C.1.9). Utilizing (B.6.11) and the last formula from (B.5.4) we deduce:

Of, (x) = Lt (x) + x

xt-

t

2(f)d

f

V

- bt

(3.6.13)

where

L1(z) = a

b-a t

n,

(3.6.14)

i

Substituting fl given by (3.6.13) in (3.6.12) and taking (B.6.11) and the last formula from (B.5.4) into account, one obtains the following integral equation for f2:

2 r.2 V/F - a'

f2(t)dt=L2(x), a2<x
(3.6.15)

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

104

La(x)

r

x-

,(x)

=

a1 b1

-!

- a,

h'

7r Ja,

Y

-- x

4b1 --

clt

.

(3.6.16)

The type of the equation (3.6.15) is (C.1.1) and its solution has the form (C.1.9). One obtains:

2 ((x - b1)(b2 - x)11/'r,ll (x - a t)( - aa)

/3f2(x) =

/t- a

l.

L2(t)dt

t-x

t

l

or, taking (3.6.16) and the first formula from (B.6.11) into account, 13f2(2) =

2

(x - bi)(b2 _ x)

7r

x - '1)(x - n2)

+

(

( -a l}(aa (b1 - .)(b2 -

t-n 1)(t-a- I.' )

(t-b1)(b2-t) t - .r

a

)

- x`l +

a2 < x < b2 .

dt

(3.6.17)

Once 13f2 determined, from (3.6.13) it results f3f1: (b1 - x)(ax -

fllx) ! 2

a) r

(x - al)(x -

J a2

2

V(()

- ) li(S} xd +

--a1}(a2

(t - a1)(t - a2) 12(E)

(t - b1)(ba - t) t - x

a1 < c < b1.

dt

(:3.6.18)

In this way we determined the solution of the system (3.6.8) and (3.6.12).

3.6.3 The Lift and Moment Coefficients From the formulas (3.6.5) and (3.6.10), we deduce for the lift and moment. coefficients: b;

1 f;(x)dx, cAr

cL = -

1

b,

xfi(a)dx

(3.6.19)

Utilizing the solutions (3.6.17) and (3.6.18), changing the order of integration and taking (13.6.5) into account, we find

c1

bi

-2j +

a, b=

102

E)1t(ti)dti

(b1 --

/(t - al)(t - a2)12(t)cit (t - b1)(b2

- t)

.

(3.6.20)

AIRFOILS IN TANDEM

105

Utilizing the same solutions, changing the order of integration and taking (B.G.6) into account, we get for the moment coefficient:

C.%/

( - al)(a2 - )

L

-13

(bi

b2 FLI- al)(t-a2)

bi)(t - t)

+,._ n

alt (od +

,

tt2(t)d t

+

al+as-b1 -b2 4

CL C. (3.6.21)

Obviously, the formulas (3.6.20) and (3.6.21) are generalizations of the formulas (3.1.33) and (3.1.34). Utilizing (B.5.14) we may prove by induction that in the case of n profiles in tandem we have:

r

cL

b

kt

/i_ak1(t)d k=1

1

(3.6.22)

"

n

cet=-1R fj11 k=1

3.6.4

k

t

t

Q

I n

bktlJ(t.)dt+(ak-bk) k=1

Numerical Values

In the case of two flat plates having the angle of attack fs

(fig.

3.6.2a) or 3.6.2b)), the integrals (3.6.20) may be expressed with the aid of elliptic functions. The first one has the form of the integral 252.21 from (1.4), page 105, and the second has the form of the integral 256.21 from the same book, page 122. For more complex configurations, these kinds of expressions are not available. However, in all cases, the integrals intervening in the expressions of c z, and ctit (for every n), may be calculated numerically by means of the quadrature formulas (F.2.24). For example, if n = 2, performing the change of variable C -+ t:

=

b1-al al+bl 2 + 2

-1
(3.6.23)

THE INFINITE SPAN AIRFOIL IN SUBSONIC FLOW

106

a)

b) Fig. 3.6.2.

and utilizing (F.2.24) we deduce:

_

(b - ) (b'i - ) d - 2n+ 1 or=l (1

Jo 1

I1

Jl =

n

bl - at

al)(a2 -

1

1

(a2f)) {d f t)(62

al + bl

2

(3.6.24)

-

-

it

a to b - to

+ ta)

(bi - al)2 n 1 + 2(2n E to(1 + to) + 1)

:to

0=1

a = 2a2 - (al + bl)

b ,

=

bl - al

252 - (al + bi) bl - al

where to are given by (3.1.52). Similarly we get: '2

1

2

1

J2

,

(t - al)(t - a2)

(t - bl)(b2 - t)

=a2 2 C=

-' 1 + t 2n+1 am1

(t - bl)(b2 - t)

7r

J2

(t-al)(t-a,) = b2-a2 n

to

(3.6.25)

c-tom

2(2n+1)

2al - (a2 + b2)

b2-a2

td t

c

d=

Vd-to'

0=1

251 - (a2 + b2)

b2-a2

Hence for two flat plates in tandem having the angle of attack -- we

107

AIRFOILS IN TANDEM

have 0ir2

CL =

£(11 +12),

cM = re`(Jt +J2)

(3.6.26)

13

where 11, 12, J1, J2 may be calculated numerically using a computer. For example, for at = 1, b1 = 2, a2 = 3 and b2 = 4 one obtains: 4 11 = 0.370671 and 12 = 0.629328. For a1 = 1, b1 = 2, and b2 = 5 one obtains: I1 = 0.415458 and 12 = 0.584541. For a1 = 1, b1 = 2, a2 = 6 and b2 = 7 one obtains: I1 = 0.449746 and 12 = 0.550253. In all the cases we have 11 + 12 = 0.999999, hence the lift has practically the value of the lift for a single plate and this is true even if the distance between plates increases. Since --11 + 12 takes the values 0.258657, 0.169083, 0.100507 it results that if the first plate

has the angle of attack -c and the second e, the lift decreases when the distance between plates increases. If the first plate has the angle of attack -s and the first has the angle of attack e, the lift becomes negative.

The previous formulas enable us to obtain numerical values for cL and cAl in the case of n airfoils with different shapes.

Chapter 4 The Application of the Boundary Element Method to the Theory of the Infinite Span Airfoil in Subsonic Flow

4.1 4.1.1

The Equations of Motion Introduction

The theory exposed in the previous chapter relies on the following three assumptions: 10 The linearization of the boundary condition. 2° The linearized condition is imposed on the chord of the profile, not

on the boundary (as it is natural to do). 3° The linearization of the equations of motion.

The assumptions are plausible for thin airfoils, otherwise they may be the cause of great errors. The application of the boundary integral equations method (BIEM), which is also called the boundary elements method (BEM), enable its to give up the first two assumptions. Hence we shall utilize the non-linear boundary condition which will be imposed on the contour (boundary) of the airfoil. For the incompressible fluid, we shall employ the exact equations (4.1.4) i.e. the equation of continuity and the equation of irrotationality, such that, in this case, the mathematical model (the equations of motion and the boundary conditions) will be valid for every airfoil (not only for the thin ones). In the case of the compressible fluid, we shall utilize the linearized equations, hence the results will be valid only for thin profiles, but even in this case the results are better then the results obtained by adopting the first two assumptions.

110

THE BOUNDARY PLEXENTSNIFTHOD

The first step in solving a problem with the boundary integral equations method (or the boundary elements method) consists in reducing the boundary value problem to a boundary integral equation (whence it follows the first name of the method); the second step consists in solving approximately the integral equation, replacing the boundary with a polygonal line (whence it follows the second name). In addition we have to mention that there exists a direct method and an indirect method. In, the direct method we utilize the equations of motion for deducing a representation of the solution in the domain occupied by the fluid with the aid of the solution on the boundary. Then we pass to the limit on the boundary. In the indirect method, we replace the wing with a distribution of fundamental solutions (sources, vortices, doublets) on the boundary and and we determine their intensity imposing the boundary conditions to be verified; in this way one obtains an integral equation for the intensity of the fundamental solutions; the method is therefore indirect. As we shall see in 4.2, it is easier to use the indirect method, but the results are less accurate. It is more difficult to utilize the direct method (4.3) but the results are more accurate. Therefore, in the forthcoming subsections we shall use mainly this method.

4.1.2

The Statement of the Problem

The problem was already formulated in the previous chapter. A subsonic stream having the velocity U,,.i (the Ox, axis has the direction of the unperturbed stream), the pressure p. and the density p is perturbed by the presence of a cylindric body with the generatrices perpendicular on the x1Oyu plane, having a given cross section (see fig. 4.1.1). One requires to determine the perturbation and the action of the fluid against the body. Denoting by X, Y the dimensionless coordinates introduced by means of the relation

(xi,yi) = L0(X,Y),

(4.1.1)

Lo being an unspecified reference length, and by U,,,,V and pc,.U"'2'P, the perturbation of the velocity and the perturbation of the pressure, we have:

V1 =UU(i+V), P1 =p +pWU,P.

(4.1.2)

The system which determines the perturbation is (2.1.30), written with capital letters. Projecting the second equation on OX and taking the damping condition at infinity into account, we deduce

P = -U.

(4.1.3)

111

THE F . UNrIONS OF MOTION

y

U., x Fig. 4.1.1.

In this way we deduce the equations which determine the perturbation:

/28U/ax + aV/aY = 0, 8V/ax - au/ay = 0,

(4.1.4)

U and V representing the coordinates of the vector V. and ,0 having the usual significance ([f = l - Af ). On the boundary C, of the airfoil we shall impose the condition

(l +U)N,y +VNy = 0,

(4.1.5)

N,V and Ny representing the coordinates of the inward normal to C1. Performing the change of variables

x=X. y=0Y (4.1.6)

u= f3U, v=V, the system (4.1.4) becomes au/ax + aV/ay = 0,

av/ax - au/8y.

(4.1.7)

In order to transform the condition (4.1.5), Ave notice that if the

boundary C1 has the parametric equations X = X (S), Y = Y(S) with S increasing when the curve C1 is traversed like in figure 4.1.1, then

N = (dY/dS, -dX/dS).

(4.1.8)

Performing the change of variable (4.1.6), we have x = x(X (S)), y = y(Y(S)) and

THE BOUNDARY ELEMENTS METHOD

112

Cl and the transformed curve C being traversed in the same sense (a increases with S). It follows NX

- # dS'

Ny

=n

a-S

(4.1.9)

,

such that (4.1.5) becomes

(,3 + u)nx + l2vny = 0 on C.

(4.1.10)

One also imposes the damping condition at infinity lim(u, v) = 0.

(4.1.11)

00

4.1.3 The Fundamental Solutions The first step in applying the BIEM consists in determining the fundamental solution of the system of equations (in our case, the system (4.1.7) ). We call source-type solution the solution of the system

au'/ax + 8v /ay

a(x - t, y - 11), (4.1.12)

&v'/Ox -au.la!l = 0. The fact that the perturbation represented by 5 intervenes in the equation of continuity justifies the name of the solution. We apply the Fourier transform and like in Chapter 2 we obtain: U0

1

x-t

27r(x-e)2+(y- n)2'

v*

_

I

y

q

21r(s-t)2+(y-17)2

(4 1.13)

We call vortex-type solution the solution of the system

au /Ox + W /ay = 0, (4.1.14)

Ov'/Ox Obviously one obtains:

W = -v*, v = u' .

(4.1.15)

113

INDIRECT METHODS FOR THE UNLIMITED FLUID CASE

Indirect Methods for the Unlimited Fluid Case

4.2

The integral equation for the Distribution of Sources

4.2.1

Taking (4.2.13) into account, we deduce that if we replace the airfoil

having the contour C (fig 4.1.1) with a distribution of sources of intensity f (x) apriori unknown, the perturbation of the velocity in the point Al(t) from the fluid will be given by the formulas:

v( ) _

2-4

-1

(4.2.1)

,Cf(x)Ix-4I2ds.

In order to determine the intensity f (x) we impose the boundary condition (4.1.10). To this aim, we have to calculate the boundary values in (4.2.1) when C -+ zo a current point Qo of the boundary C. We are going to prove that if f (x) satisfies the Holder condition on

C, i.e. there exist two positive constants A and p (p < 1) such that, for every two points Q(=j) and Q(x2) belonging to C, we have: If (z1) - f (w2)1 < Alaet - x21',

then

- 2 . PC f(x)

v(xo ) = lim

_ - 2 f (xo)n °

-2

- 2a Pf (m)

(4.2.2)

d8= (4 .2 .3)

X Isa

X0 d =012

it,

in every regular point Geo. We denoted PC

= li , Pi-C

(4.2.4)

where c is the arc cut out from C by the circle of radius E and center Qo (the arc QI Q2 from fig. 4.2.1). We denoted by no the inward normal at C in Qo. Indeed, taking the definition (4.2.4) into account, we notice that it suffices to evaluate the integral on c. We have limo

Eli

( _ 2 - f f (x)

\

=

Ii

n

{

L= liX0 PC

12-4

TV JC I-T-C

ds= f

[f (x) - f (xo)) I

d aJ - f Z

o)

44.2.5)

THE BOUNDARY ELEMENTS METHOD

114

Fig. 4.2.1.

Taking into account that f satisfies the condition (4.2.2), we deduce that the first integral from the right hand side of the equality (4.2.5) vanishes when E -- 0. For calculating L we shall replace the arc Q1Q2 with the segment on the tangent Q'Q'2 (fig. 4.2.1) and we shall put on this segment: x = xO + STO,

s E [-E, +EJ .

(4.2.6)

We assume, for the sake of simplicity, that M(C) tends to Qo(xo) on the direction of the normal n°. We have therefore

4 =xo--9n° cu q>0.

(4.2.7)

It follows

L= Um

17-0

r

+C E

o+ r1no s2 .

,2

q.e.d. Imposing the condition (4.1.10), we obtain from (4.2.1) and (4.2.3) the following integral equation: (flo2

+ N2n02) f (xo)+

+

(x - xo)n° +f2(y - bo)n° Uds=2pn2. cf(x) Ix-xo12

(4.2.8)

INDIRECT' METHODS FOR THE UNLIMITED FLUID CASE

115

From this integral equation we are going to determine the intensity of the sources. For the incompressible fluid (M = 0) it may be solved exactly for the circular obstacle (4.12).

4.2.2

The Integral Equation for the Distribution of Vortices

If we replace the airfoil with a distribution of vortices having the intensity (circulation) g(x) , defined on C, then, according to the formulas (4.1.15), we obtain the following expressions for the components

of the perturbation u and v in a point M(t) from the fluid: da,

u(0 =

2

where l; = (1:, rl), x = (x, Y) Taking the formula (4.2.6) into account, we obtain: Y-YO d s W012

U(X0) = 2g(xo)ny +

9(x) Ix

-

, (4.2.10)

f

2 g(xo)n:

v(xo)

- -1 P9(m) 1 -Z

X0 12

d

and from the boundary condition (4.1.10):

-M2g(xe)niny+ +1 ir

'(x)p2(x-xo)ny-(V -bb)nrd8=2f3n=, c

(4.2.11)

Ix - xol

where M is Mach's number in the unperturbed flow. For the incompressible fluid (A! = 0) we have a first kind integral equation.

4.2.3

The Boundary Elements Method

One utilizes the following collocation method in order to solve the equations (4.2.8) and (4.2.11): one approximates the boundary C by it polygonal line {!,,}(j = 1 T), with the end points on C (fig.

'TIME BOUNDARY ELEMENTS METHOD

116

Fig. 4.2.2.

4.2.2) and one approximates on every segment Li, the unknown f (respectively g) by the value fj (resp. gi) from the midpoint of the segment. Denoting by :xf the vector of position of the midpoint, we have fi = f (x°), (j = L N). We may also consider a linear variation of f on L. In this case we have to determine the two constants from the values of f in the end points of the segment L,. i.e. from the values

of f on C. Now the equation (4.2.8) may be written as follows:

( 02

+/32802) f(xo) + 1

+_7

r

(x - X0 710 + )`12(y - ?ko)tty

'Y

fi

L,

ds = 28nr .

Ix - xo1Imposing

the equation to be satisfied in every midpoint, i.e. putting successively xo = x9 . i = 1N, we get the system: N

ai fi +

Ai i f) = Ai,

i

(4.2.12)

where

tai = nr(xP) + '3ly1y(x)) (4.2.13)

AiJ

n'r(x:))Uii + /3lnv(x°)V i

+

Ai = fln (xn) ,

117

INDIRECT METHODS FOR THE UNLIMITED FLUID CASE

with the notations

Ucj = / u'(x,x°)da L (4.2.14)

Y1 = jv(zz?)ds1 f

u' and v' being given in (4.1.13). The linear algebraic system (4.2.12) consisting in N equations will determine the N unknowns f{. Now the perturbation of the velocity in the points Po(x°) follows from (4.2.3) N

1

vi

=-2f.n(x°)-EV1jfj J-1

(4.2.15)

V>, = (U+f,V ) In the same way, from the equation (4.2.11) we get the system N

hi9, + 1: Bij9j = A, , i = 1,(4.2.16) j:1

where

2b; _ -1tnz(x°)n,,(x°) , (4.2.17)

Bij = #'nr(x°)U,j - n(x°)Vij. From (4.2.10) we obtain the following velocity field N

1

w = 29jnr(x0)+Evj9j j..1

(4.2.18)

1

Vf

4.2.4

N t'4s(x'i')-Uijgj. -

The Determination of the Unknowns

The integrals (4.2.14) may be computed numerically using quadrature formulas. They may be also calculated exactly for every shape of

THE BOUNDARY ELEMENTS METHOD

118

the contour C. Since the integrals UUJ, V, are singular we shall calculate their Finite Parts. For avoiding the singular integrals, we may utilize the regularization method 4.8. For obtaining a parametrization of the segment L,, we shall denote by (x1, yl) and (x2,112) the coordinates of the end points PI, respectively P2, considered in the sense of traversing C (fig. 4.2.2). Then, the coordinates of the generic point on L., will be

y°+-2ylt, -1<-t<1,

x°-i-x22xlt,

(4.2.19)

where, obviously,

x°

X2 2 x1

yj=112

2 111

(4 . 2 .20)

.

Utilizing the formula

ds = 1(&-)2 -+ (dy)

,

we deduce

gds=l,dt, lJ =

(x2-x1)2+(112-111)2.

(4.2.21)

Denoting

4a = I,b= (x°-x°)(x2-xl)+(i4 -Y?)(112-Yi), c = (xjI - x? )2 + (yp - y1)2,

-

+1

k 1

tkdt

ate+bt+c'

k = 0,

(4.2.22)

1,

we get (x - x°)2 + (y - y°)2 = ate + bt + c

(4.2.23)

whence, U,,

=

Io 47f [ (xj_ x°)

+x2 2

x1J l]

, (4.2.24)

ii4i[(y°- y° ) Io+2 yi

V

119

INDIRECT METHODS FOR THE UNLIMITED FLUID CASE

Taking the running sense on C into account, we shall have in (4.2.13), (4.2.15), (4.2.17) and (4.2.18): n,,(x°) _

_-V1

r+y(x 0) = x1 !J X2 1

ii V2

(4.2.25)

1 11z`)

n,,(xo) _

n (x?) = x I

j.

where (xl'), yi')) and (X( 2" Y2()) are the coordinates of the end points of the segment Li and 1; is the length of this segment, i.e. xz')

- xi'1)2 + (yz;l - y1'))2]1/2

(4.2.26)

.

We have to notice that in (4.2.24) the integrals Ik may be calculated exactly. Indeed, since

<0,

A =- b=-4ac = we have

to =

V'--5

Il - I 2a

In

arctan.

(4.2.27)

dc-a'

a+b+c -

(4.2.28) b

a-b+c a/

arc t an vr---s

c-a

Taking (4.2.23) into account, we deduce a + b + c = (x2 - x°) + (y2 (4.2.29)

a-b+c= (xl -x°)2 +(yi -y°)2. For i = j we have b = c = 0, such that the integrals Ik become

singular. Utilizing the formulas (D.2.3) we obtain

Io = -2/a,

Il = 0.

(4.2.30)

So, the coefficients ai, At,, A{, b1 and B1j depend on the coordinates

of the end points of the segments L, on C. Hence, we may solve the systems (4.2.12) and (4.2.16) using a computer.

120

TllE BOUNDARY ELEMENTS METFIOD

4.2.5

The Circular Obstacle

For testing the two methods presented above and the direct method from the following subsection, we shall use the exact solution for the incompressible flow past a circular obstacle. This solution is already

known. If 0 is the origin (center) of the obstacle and R its radius, then the complex velocity in dimensionless variables is (see (5.2.4) from 11.111)

R UI -iVI =U01- Z2) \\

.

(4.2.31)

t

Taking R as reference length and putting z = exp(iO) it results u = - cos 2O ,

y _ -sin 2O

(4.2.32)

In figures 4.2.3 and 4.2.4 we compare the numerical values obtained for u by mens of (4.2.15) and (4.2.18) with the values obtained by

means of (4.2.32). We observe that the solution obtained by means of the distribution of vortices is better. It is also useful to compare the numerical values obtained on the three ways for a quantity of aerodynamic interest. This is the local pressure coefficient defined by the formula

A P.

Cy _ = (1/2)P..oUx ,

(4.2.33)

with the notations from (4.1.2). Taking into account that for the incompressible fluid, the system (4.1.7), for which we have determined the approximate solutions, is the exact system of the equations of motion, we shall calculate also Cn exactly. This may be obtained from Bernoulli's integral 1 V2 Pi 1 r r2 . Poo _ (4.2.34)

Poa

2

Poo

It follows

Cp=1-V2=1-(1+ u)2-t'2.

(4.2.35)

Using this formula, from (4.2.15) we get C,, in the case of the distribution of sources, from (4.2.18) we get. C,, in the case of the distribution of vortices and from (4.2.32) we get C,, exactly. We also calculate Cn with the direct method from the following subsection. The graphic rep-

resentations show that the direct method gives better results than the indirect methods and among these ones, the method based on the distributions of vortices is better.

INDIRECT METHODS FOR THE UNLIMITED FLUID CASE

121

t 1.

I

es

m

As

.03

.1s

.1s

a

.1

a

n 6e m IJI If Im )m >r )10 1L ))a us

a

W b b Ip Ip 110 7q 30 M RD 30 No

Fig. 4.2.3.

4.2.6

Fig. 4.2.4.

The Elliptical Obstacle

The profiles with smooth boundaries (for example the circle or the ellipse) are non-lifting profiles. The profiles with angular points determine a circulatory motion of the fluid and therefore it appears the lift. The circulation (and the lift) are determining from the Kutta-Joukovsky condition (see for example, (1.101 p. 179), hence these profiles are lifting profiles. In the first case we encounter d'Alembert's paradox. As we shall see in the following subsection, the indirect methods are utilized mainly for non-lifting bodies, while the direct formulation is suited for

writing the k - J condition. It is well known the exact solution for the elliptic obstacle in the incompressible fluid. The complex potential from the (Z1) plane is given by the formula (see for example, (1.101 p.178).

F(zi) =

2 {( zi + za'°+

zl

+ (zi '

at + bt e1°l

(4.2.36)

t

at - bj

where at and bl are the semiaxes, 4 = at - b1, and a is the angle of attack. Deriving (4.2.36) we deduce the complex velocity which has the same form in dimensionless variables z = Z, a, b, c . Hence it follows

1+u-iv=

]e`1°+

2[(1+

(4.2.37)

+(l `

z

a+bei°l a - b

122

THE BOUNDARY ELEMENTS METHOD

We determine the components u and v separating the real part from the imaginary one. The graphic representations for the exact solution and for the solution obtained with the first indirect method, are given in [4.7J. We notice that the results are almost identical.

4.3

The Direct Method for the Unlimited Fluid Case

4.3.1

The representation of the solution

As we have specified in 4.1, in the framework of the direct method we represent the velocity field in the fluid with the aid of its values on

the boundary of the body and then we pass to the limit. In order to obtain this representation, we shall consider a domain D exterior to the boundary C and limited by a circle CR having the radius big enough, such that C should be situated in the interior of CR. For f and g continuously differentiable function C', we have the identity

[f (aulax + avlay) + 9(8v/ax - 8u/8y)Jda = 0

(4.3.1)

which follows from (4.1.7). Noticing that

fau/ax = a(fu)/ax - ua f /ax, .. . and applying Gauss's formula, we obtain the identity J[u(Of/ax - 89/8y) + v(8f/ey + 09/8x)Jda = (4.3.2)

=

j[u(fn

- 9ny) + v(9nx + f ny)Jds .

the integral on 47R vanishing when R - oo, because of the behaviour

at infinity of f and g which will be identified with u' and v' and because of the condition (4.1.11).

For f = u' and g = -v' (4.1.13), taking (4.1.12) into account, we deduce:

u(f) =JC u(x) [u'(x, )n (x) (4.3.3)

+v(x) [u'(x, Ony(x) - v'(x, )n=(x)] }ds

,

THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE

123

and for f = v' and g = u' 2L(x) [v`(x, S)ns(x) C

(4.3.4)

+v(x)[u'(z,4)n=(x) + v'(x,C)ny(x)]}d8.

This is the integral representation of the solution, valid for every point M(4) from the exterior of C.

4.3.2

The Integral Equation

For obtaining the boundary integral equation, we have to pass to the limit in (4.3.3) and (4.3.4), letting the point M(r;) to tend to the generic

point Qo(zo) belonging to C. To the limit, the integrals containing the fundamental solution (u', v') become singular. Hence we have to adopt the definition (4.2.4). We calculate the integrals on c, as follows: lim

Ju(z) [u*(x,F)n (x) + v*(x,C)nyr(x)]ds lim

J [u(x) - u(xo)] [u'(x,4)n=(x)+ (4.3.5)

+v (x, 4)nr(x)]ds + u(xo)Li L 1 = lim

t-M. 1c

[u' (x,

,

where

v`(x, 4)ny(x)]ds .

If c =

is replaced by the segment Q'1Q'2 (fig. 4.2.1), then we Z have the parametrizations (4.2.6) and (4.2.7) and n = no. Hence,

L=

27r re

s2

+

ds = 2

.

Passing to the limit, the first integral from the right hand side of the equality (4.3.5) vanishes when e -+ 0.

124

THE BOUNDARY ELEMENTS METHOD

Similarly we have: Eli

m

J

v(x)[u*(x,Lr)nv(x)

- v`(x,t)nt(x)]ds

_ slim J [v(x) - v(xo)] [u`(x, )ny(x)-v*(x, t)nz(x)]ds + v(xo)L2 , where

L2 = lim

=

(4.3.6)

J[u*(x)n(x) +v(x,)n(x)]ds = ny - -r,0, no.)

r

s2 it

+

ds = O. 172

Therefore, from (4.3.3) we deduce:

2u(xo) _

{u(x) [u*(x, xo)ns(x) + v*(x, xo)ny(x)]+ (4.3.7)

+v(x) [u* (x, xo)nv (x) - v* (x, xo)nz(x)] }ds

.

In the same way, from (4.3.4) we obtain: 2V(xo)

{u(x)[u*(z,xo)nz(x) - u*(x,xo)nv(x)]+ (4.3.8)

+v(x) [u* (x, xo)n=(x) + v*(x, xo)ny(x)] Ids. The formulas (4.3.7) and (4.3.8) constitute a system of two singular

integral equations on C for the unknowns u(xo) and v(xo). Now we introduce the function G = (13 + u)ni, - vnx .

(4.3.9)

We may utilize on C the condition (4.1.10). It follows

Q+u= n2n2G, v= -nz v

2

2C.

(4.3.10)

nv

Replacing these relations in (4.3.7) and (4.3.8) and taking into account the relations PC. (u* n,,

+ v*nv)ds = -1/2;

P(u*nv - v*nr)ds = 0,

(4.3.11)

THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE

125

which will be demonstrated in (4.3.7), we get: u ( xo )

= ,3 + 2

P C

(v '

n,n V u ')Gd s - M2 n2 + #2n2 x

y

4.3.12) u (xo

)=2TS

and then G(xo) - 2

\

- u '-A12 n2+

2n2v ' Gds

x

y

)

I

PC

+ hf2

[u'nox + v'rty+ n22

nzny + g2nyh (v n o

'

- u ' ny° )JGd s = 2An y ,

(4.3.13)

with the notation no = n(xo). The equation (4.3.13) is the singular integral equation of the problem. Putting M = 0, one obtains the equation for the incompressible fluid.

4.3.3

The Circulation

In the case of the profiles with angular trailing edge, we have to utilize the circulation. Taking the sense defined on C1 into account and denoting by stl) and n(') the (dimensional) versors of the tangent and inward normal, we have 41),

sod = -11y1), 41) =

(4.3.14)

such that

rl =

V1 si'idwi'

(U1nyll

- Vinzll)ds(') =

= -U0LoP[(1 + U)Ny - VN,;Jds. With he changes of variables (4.1.6) and (4.1.9), we deduce

[' 1 = -

U

0 ° j[(0 + u)ny - vn=Jds.

(4.3.16)

Denoting r1 = U,,,Lor, r representing the dimensionless circulation and utilizing (4.3.9), it results the simple formula:

r = - PCds.

(4.3.17)

For the incompressible fluid one obt ins:

r=-

G ds.

(4.3.18)

These formulas also emphasize the significance of the function G.

THE BOUNDARY ELEMENTS METHOD

126

4.3.4

The Discretization of the Equations

Acting like in 4.2.3, we reduce the equation (4.3.13) to the algebraic system N

i = 1, 2, ... , N ,

(1/2)G{ + F A;jG,i = J3;,

(4.3.19)

j=1

where

Ci = C(x°), A = Onv(x°),

A,j = -nh(x°)Utj - n,(x°)V J--

(4.3.20)

nz(x°)n1r(x°)

n2(x') nr(x°)V, -- nv(x°)U,j]

_M2 n2(xj),

Uy and

being defined (4.2.14).

After determining the unknowns G1 from (4.3.19), we may determine the circulation from the formula: N

-/3I' = E 1jGj ,

(4.3.21)

j=1

representing the length of the segment Lj. We deduce the components of the velocity 1j

+ u; =

A2ny(x°)G=

n2(x°) + A2n2(x°)

ns (x°)Gi

n2(xe) +

n2(x°) '

(4.3.22)

from (4.3.10). Obviously, for n(x°) and n(x°) we shall utilize the expressions (4.2.25) and for U;j and [;j, (4.2.24).

4.3.5

The Lifting Profile

If the profile has an angular point (fig. 4.3.1), or a cusp, like the Joukovsky-type profiles, then we have to determine the circulation using the Kutta-Joukovsky rule. The circulation determines the lift. For determining the circulation,

we impose the equality of the pressure in the points P, (on the upper side of the angle) and P; (on the lower side of the angle) when

THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE

127

Fig. 4.3.1.

these points (Pp and Pi) tend to the angular trailing edge P1. From Bernoulli's integral it follows that the equality of the pressures implies the equality of the tangential components of the velocity, (according to the slipping condition the normal components are null). Taking into account the orientation of the versor of the tangent s(1) on the upper and lower sides of the angle (fig. 4.3.1), we deduce the condition: Vi(P,).stt}+V1(Pi),alt}

=0.

(4.3.23)

With the notation (4.3.9), we have:

V1 slit =

U)sal + VsY)J = UM

+u J"'J 13

+vn=1ds

pJdS

d5

such that (4.3.23) implies G(Pi) = 0.

(4.3.24)

Consequently, for such a profile, we have to add the condition (4.3.24) to the system (4.3.19). In order to solve this problem we introduce [4.18J

the regularization variable A and we obtain the system:

A+(1/2)Gi+AijGj =;3i, i = 1,...,N, (4.3.25)

G(P,) + G(Pi) = 0, for the unknowns C1,. .. , Gtr , A. in this way the number of equations equals the number of unknowns. For big values of N we obtain a small A i.e. (A - 0).

THE BOUNDARY ELEMENTS METHOD

128

4.3.6 The Local Pressure Coefficient As we have seen in 4.2.5, the local pressure coefficient for the incompressible fluid (4.2.33) is (4.2.35). Let us deduce the expression of the local pressure coefficient for the compressible fluid. For the potential flow we have the formulas 11.101 page 206 which become, using the notations from this chapter:

Vi ;fir

Pi = Pu

2 Pl-potl-ry-1

l C?

: c2o

V12\i

(4.3.26)

J

2

_co2t1-72

For the uniform flow that we have in view, these formulas give

7-1UW 7-l 2

cam)

7-1 U2

Pco - Po

2

T

(4.3.27)

co

=cfltl-7-1 U"

cclo

2

czo

From the last relation we deduce k

,y-1

1

Uz

7-tlli2+

2

(4.3.28)

such that, denoting

r=72142,

(4.3.29)

it follows 1

k(7 - 1)

=1+1. z

(4.3.30)

Introducing the parameter k in the formulas (4.3.26) and (4.3.27), we

THE DIRECT METHOD FOR THE UNLIMITED FLUID CASE

129

get:

P1=PO I-

V"k 00

\

r

(4.3.31)

Poo=P0(1-721k}T

,

7-1

Poo = Po 1-2

k)

T

These formulas will be replaced in (4.2.27). Taking out the factor (7 - 1)k/2 and utilizing (4.3.30), it results _ 2 2 ( )]'-T-11. (4.3.32) Cp = j -- 1+ ry 2 1 M2 ( U2 .yM2

For the incompressible fluid (M = 0) one obtains the formula (4.2.29). For the compressible fluid, V12/U.2 from (4.3.32) will be replaced by (1 + u/#)2 + v2. We shall employ the local pressure coefficient in order to define the lift coefficient which gives a global measure of the action of the fluid against the profile CL =

'f Li

Cl,nylidat ,

(4.3.33)

where Ll is the length of the contour C1. It is also important to get the expression of the critical velocity v, because the subsonic character of the flow is maintained only if Vl < vi,, . This velocity is obtained from (4.3.26) putting vl = cl and utilizing the definition of k. We obtain: U02

4.3.7

11+721 (M2 -1)]

.

(4.3.34)

Appendix

In this subsection we shall prove the formulas (4.3.11). To this aim, we denote by C, the arc (semicircle) of the circle of radius e with the center in Q0, interior to the profile (fig. 4.3.2) and we notice that we have the parametrization

x-xo=Ecos9, y-yo=EsinO (4.3.35)

00<0<00+ir

THE BOUNDARY ELEMENTS METHOD

130

Fig. d.3.2.

which, for u', z}' (4.1.13) implies: U* (x, xa)

1 cos 0 2;r

v' (x, x)

E

1 sin y 27,,

(4.3.36)

E

We also consider the circle CR with the center in Qo, and the radius R big enough, such that the profile C is in the interior of the circle. On this circle we have the parametrization

x-x0=Rcos4

,

,-

such that

9l'

lcoscp

= in R

y-yp=RsincF, V*

(4.3.37)

1Sill

27r R In the domain D. exterior to the contour C - c + Cs and interior '

to the circle CR, we may integrate the equation: au'/ax + av'/(9y = a(.r - x0, y - yo)

(4.3.38)

Applying Green's formula we get:

(u'n., + v'ny)ds + Prr (0n, + v'ny)ds+ (4.3.39)

+

(u'n2 + Ti'ny)ds = 1.

THE AIRFOIL IN GROUND EFFECTS

131

or, PC

lu*(x, xo)ns(x) + v'(x, xo)ny(x)jds + 1

+ 2n

f"' o

2a

1

dO

+-

j dip = 1. ,

0

One obtains the first of the relations (4.3.11). Integrating on D the equation

8u'/8y - 8v-/8x = 0, one obtains the second of the relations (4.3.11).

4.3.8 Numerical Determinations In figure 4.3.3 we compare u calculated by means of the direct method with u calculated exactly. We observe that the values are almost the same. In figure 4.3.4 we give 4 graphic representations: u

exact, u direct, u sources, u vortices. We notice that u obtained by means of distributions of sources (4.2.15) does not give very good results and u obtained by means of distributions of vortices (4.2.18) gives better results. The values of u obtained with the direct method are constantly in the vicinity of exact values of u . Is

tS

as

a3

.1

Ti rr-FT lTF

43

TG

as a

a o IN laI10us 110

210 210 010 110

a

to

.a

10 Ib ISO to ]b M 7s Im in IM

Fig. 4.3.3.

4.4 4.4.1

Fig. 4.3.4.

The Airfoil in Ground Effects The Representation of the Solution

When the airplane is landing or taking off we have to take into account the influence of the ground. We shall consider this problem herein

'1'm; BOUNDARY ELEMENTS METHOD

132

Y;

x> Fig. 4.4.1.

replacing the ground with the x10:1 plane (fig. 4.4.1). We have the same problem like in the previous sections. Art uniform subsonic stream,

having the velocity U,,,, the pressure p and the density p", flowing in the yt > 0 half-space, is perturbed by the presence of an infinite cylindrical body, having the generatrices parallel to :r, IOzt, the Oxt axis having the direction of the stream. The x10y1 plane determines a cross section whose boundary C1 is assumed to be a thin profile. Using the smite notations like in 4.1 we have to integrate the system (4.1.7) in

the ha lf-plane y > 0, with the boundary condition (4.1.10) on C and the supplementary condition zr(x, 0) = 0,

(4.4.1)

(V)X.

In this case, instead of the fundamental solution (u', v') from 4.1.3 we shall utilize Green's distribution (U', V') determined by the system au; 119X l aVV/ay = 6(x -

y -11)

(4.4.2)

(9DU/ay = 0,

I) representing a point front the y > 0 half-plane, and the boundary condition

U'(a,0) = 0.

V+(r.0) = 0.

(V)x.

(4.4.3)

This distribution is obtained by means on the fundamental solution (4.1.13) using the method of images. Denoting 2r,

(x - )2 + (y - rj)2 (4.4.4)

vt (T, y, , tl) _

1

27r (x

y-rI

- )2 + (y - 1)2

133

THE AIRFOIL IN GROUND EFFECTS

we have

Uf = u*(x,11,Cq) f u*(x,y,C-q) (4.4.5)

Vf = v'(x,y, ,q) ±v'(x,y,C -q) Indeed, u' (x, y, t, q) and v'(x, y, t, q) verify the system (4.1.12) for ev-

ery t in the xOy plane. Obviously, u'(x, y, t, -q) and v' (x, y, t, -q) verify the system

8u'/8x+ 8v'/811= b(x - C y +q) (4.4.6)

8v'/8x-8u'/Oy=0. When t is in the superior half-plane (y > 0) this system becomes

8u'/8x + 8v'/ay = 0 (4.4.7)

8v'/8x + 8t,'/8y = 0. With these specifications one easily finds that (4.4.5) verifies (4.4.2). It is also very easy to verify the conditions (4.4.3). Let us return now to the representation of the solution. Denoting by D the domain from the superior half-plane y > 0 which is exterior to C and delimited by a half-circle CR with the center in the origin, the

diameter on Ox and the radius big enough in order to contain C in the interior (fig. 4.4.1), we deduce like in 4.3, for R - oo, the identity [u(8f 1,9x

- 8g/8y) + v(8f/8y + 8g/8x)] ds =

(4.4.8)

IL) =:

j [(u f + vg)n= + (v f - ug)ny] ds +

J

u(x, 0)9(x, O)dx. 0000

We took into account the fact that on the diameter of the half-circle we

have n = (0, -1) and that the integral on CR vanishes for R - oo when f and g are distributions having the form U., V. Putting now f = U+ and g = -V;, from (4.4.2) and (4.4.3), we deduce

u( ) = J (u(x) [U+(x, 4)n1(x) + V+(x, )ny(x)] +

c

+v(x) [U+(x, C)n,(x) - V+(x, 4)ns(x)] Ids.

(4.4.9)

THE BOUNDARY ELEMENTS METHOD

134

In the same way, for f = V. and g = U , from (4.4.9) we obtain

vW =

PC {u(x) [V -*(x, F)nr(x)

- V*

(4.4.10)

+v(x)[U-w (x, )ns(x) + V`(x,Ony(x)]}ds. Formally, the solution is identical to (4.3.3) and (4.3.4). It differs only by Ut, V}.

4.4.2

The Integral Equation

Acting like in 4.3, we deduce f

(1/2)u(xo) = PC {u(x)[U+(x,xo)n:(x) + V+ (m, xo)ny (x)] +

+v(x)[U..(z,xo)ny(x)

(1/2)v(xo)

_'

- V+(x,xo)ny(x)]}ds, (4.4.11)

{u(x) [V=` (x, xo)n: (x) - U* (x, xo)ny(x)] +

+v(x) [U' (x, xo)n=(x) + V! (m, xo)nv(x)} }ds.

Utilizing the identities PC (U}nt + V.ny)ds =

-2, P(U_ny - V'n=)ds = 0

(4.4.12)

which will be demonstrated at the end of this section, we obtain for G defined by (4.3.9), the following integral equation:

(1/2)G(xo) - PC {U (x, xo)n.(xo) + VV (x, xo)ny(xo)+

+1bllni(z)+

(ny,x) [V_(x,xo)n=(x)-

(4.4.13)

-U, .(x, xo)n.,(x)] }Gds = Qnv(xo), where the mark "prime" has the same significance like in (4.2.4).

The formulas (4.3.10) which determine u and v in function of C, remain valid.

135

THE AIRFOIL IN GROUND EFFECTS

4.4.3

The Computer Implementation

The equation (4.4.13) is discretized like in 4.3.4. One obtains the system N

(1/2)G; +

i1

A;}Gj= B;,

I = 1, ... , N ,

(4.4.14)

where

AU = -n,,(x°)U,j o

o

-M2ny(x')+Q2n2(xq)( (x)V

(4.4.15)

+

with the notations

U, =

J Lf

U; (Z' x0)ds , V j =

j

V(x, x9)ds,

(4.4.16)

s

B; remaining unchanged. The system (4.4.14) will determine the unknowns C1,.. . , GN. From (4.3.22) it follows u;, v;, (i = 1, ... , N). The circulation is obtained with the formula (4.3.21). For lifting airfoils, we also consider the Kutta-Joukovsky condition (4.3.24). We solve the problem introducing the regularization variable A and acting like in 4.3.5.

In order to obtain the expressions of the unknowns U, j and Vtj*, we use the notations (4.2.22) and the relations Y= (xs -

x°)(x2 - x1) + (y° + y°)(y2 - y1),

c1= (xi -x°)2+(y;+y°)2

Ik =!-i1

(4.4.17)

ate+bit+

'

In this way one obtains

(z-x°)2+(y+y?)2=ate+b't+c', 4[(xjo -x0)(Io

U,3

Vi'*

=

-Lj

41r

kla)+x22x1(Ilfli)f

12 yl (Il f li)J [Y-v?Iodv?+Y?I6+ (o

(4.4.18)

THE BOUNDARY ELEMENTS METHOD

136

n(x0) and n(xti) being defined in (4.2.25). For It) and It one obtains the expressions (4.2.28). Since A = b2 - 4ac'

(?1j + Y9)(x2

- XI) - (xj - xe))(?M -

x!1)]2

< 0,

it follows

4, =

a

2Qr arctan

,

C' -a

(4.4.19)

1t=2a III +b+c'-afwhere

,arctar.

-©

c' - a

,

a+b'+c' = (>'2-x°)2+(J2+J9)2 (4.4.20)

a -b<+cl = For i = j the integrals Io and It become singular and they have the values (4.2.30).

4.4.4

The Treatment of the Method

The problem presented in this subsection was studied in (4.10] and we present with small modifications the solution given in that paper. For testing the method we use the exact solution which is available for the circular obstacle in ground effects (fig. 4.4.2), (4.12]. If the center of the circle is in the point (0, b) and the radius is a (a < b), then, employing the usual notations, we have the exact solution:

u - iv= -2 E Z2 C" +

n>1

where the sequences

+4

E n>t

cn b» Z2

+ b2

(4.4.21)

and {cn} are defined by the following

{bn}

formulas: 2

2

b"+1=b-b+a

c,++1=(b+bC11

b,,,

n)2n--0, > (4.4.22)

b1=b,

c1=a2.

Decomposing (4.4.21) in simple fractions, we get

at - iv = - E Gn [(z - bn)2 + n>1 i1

(1

(4.4.23)

+ ibn )2}

THE AIRFOIL IN GROUND EFFECTS

137

YA

U-

Fig. 4.4.2.

whence, separating the real part from the imaginary one, we deduce u

= - L'` n>1

x22-(y-b,,)2 + x2-(y+bn)2 f x2 + (y

- bn )212

[x2

+ (y + b.)212 (4.4.24)

v= -2xEG& rx y - b + (y - bn)212 + n>1

y +bn Jx2

+ (1/ + bn)?12

Putting

r=acos8, y=b+asinO, we find uIC and vJ,. Using the formula (4.2.35), we represent in figure 4.4.3 the exact solution by a continuous curve and the approximate solution (in the case

of the incompressible fluid, rt = 0) by small squares. We considered r = 0, a = 1 and b = 1.1. Since the two solutions practically coincide, we conclude that the method presented herein is very good. It may be applied for every shape of the airfoil.

4.4.5

The Circular Obstacle in a Compressible Fluid

In this case there is not available any exact solution, such that we have only the information provided by the BIEh1. We have to take care to use

THE BOUNDARY ELEMENTS METHOD

138 4

C,

0 -4

.8

1

-12J

ace METHOD EXACT

-16.J -20 J -24. r

0

90

180

I

270

360

Fig. 4.4.3.

for the radius of the circle values that are small with respect to the other characteristic lengths (for example, the distance to the ground) because only in these circumstances we can employ the linearized equations. In figure 4.4.4, we represent the coefficient CL defined by (4.3.33) for the circular obstacle with the radius a = 1, considering 60 nodes on the boundary, . Obviously, the compressibility determines the increase of the lift (as we have already seen in the case of the free fluid). We considered 7 = 1.405. Giving the distance to the ground, we may calculate (in the absence of the circulation) the values of M for which the flow remains subsonic (one utilizes vQ given by the formula (4.3.34).

4.4.6

Appendix

In order to prove the formulas (4.4.12) we take into account that in the entire plane, the solutions (U7, VV) satisfy the systems

aUU/ax+av}/ay=a(x-t,y-17)± 5(x-C y+ ii) (4.4.25)

aVV/ax - aUl/ay = 0, where 7> 0. Like in 4.3.7, we denote by Cf the arc of circle interior to the profile

C, having the center in Qo(zo) and the radius a and by CR the

THE AIRFOIL IN GROUND EFFECTS

139

000C,

-0.02 -0.01

-0.06

-0.08 -0.10

-012 -0.14

-016.

-02000

10

0.5

1.5

2.0

30

2.5

3.5

H

Fig. 4.4.4.

circle having the center in Qo and the radius R big enough, such that the profile C belong to the interior of the circle. The domain exterior to the curve C - c + CF and interior to the circle CR will be denoted by D. We shall integrate on this domain the first equation from (4.4.25) and and we shall apply Green's formula. One obtains: 2 = P[U+(x, xo)nr(x) + V. (x, xo)nv(x)Ids+ + find P (U+r1r + V. n.,,)ds + -00

f)(U*nr +V. nv)ds . 1z

(4.4.26)

With the parametrizatiou (4.3.37) we deduce on CR: :c - :r.0

COs

(r-xo)2+(y- ))2 - R x-x'0

_

(x - xo)2 + (y + yo)2

Cos

/

'

_

T. - xU R2 + 411o(y - yo) + 4Jo

I 1 +4sin R

z

R +42)

1

=

Cos

+O(R `)

whence,

u+ =

RV

-

+ O(Rr2), V+

si

+ O(R--)

140

THE BOUNDARY ELEMENTS METHOD

V. = O(R-2) . Utilizing the parametrization (4.3.35) on C1 we have for example

U! = O(R-2),

y-yo+2yo y+yo (x-xo)2(y+yo)2 (x-xo)2+(y-vo)2+4yo(y-1m)+4y 2 (,+,,sin 2yo j (1 + Esyoe +

2yo

2

bI

+ 0(c).

Hence, Uf

_

1

=tar

C0619+0(6),

s

_

V1

sine 1 e + 4Ayo

+0(6).

Taking into account that on C. we have d s = Ed 9 and on CR, d s = Rdw, we get: 6o+x

lim

(U+n= + V nv)d s = PC.

limo PC

tar ,

(U..n + V+ny)d s=

1

d9

2t

I f0 de = 2.

Replacing all these in (4.4.26) it follows the first relation (4.4.12).

4.5 4.5.1

The Airfoil in Tunnel Effects The Representation of the Solution

Assuming that the airfoil is situated between two planes parallel to the direction of the unperturbed stream (fig. 4.5.1), we have to integrate the system (4.1.7) in the domain exterior to the contour C and bounded by the two planes, with the boundary conditions (4.1.10) and the supplementary condition (4.5.1) v(x, 0) = 0, v(x, a) = 0 (V)x, Denoting by D the domain of motion delimited by the sides x = fd (fig. 4.5.2), we deduce for d - oo the identity ID [u(Of I8x - 89/8y) + v(8f/8y + 89/&)I da =

= JR/u + gv)n, + (f v - 9u)nv)ds+

+J

+00

00

q(x, 0)u(x, 0)dx 00

g(x, a)u(x, a)d x . +00

(4.5.2)

141

THE AIRFOIL IN TUNNEL EFFECTS

Fig. 4.5.2.

We took into account (4.5.1) and the fact that for x = td the integrals vanish when d --+ oo if f and g behave like the solution of (4.1.13) and it and v have the property (4.1.11). For f =1.t and g = -V+, where (17+, V+) verifies in the domain occupied by the fluid, the system (4.1.12) and the relations V+(x, 0) = 0,

V+(x, a) = 0,

(4.5.3)

we get from (4.5.2):

u(4) =

lc 1u(x)[U+(x,01=(x)

(4.5.4)

+v(x) [U+(x, 4)ny(x) -V+(=. F)n7(x)] )d s.

Analogously, for f = V_ and g = U_, where (U_,V_) verify the system (4.1.12) and have the supplementary property U_(x,0) = 0,

U_(x,a) = 0,

(4.5.5)

we get from (4.5.2):

v() _ (4.5.6)

+v(x) [U-(x, t)rns(x) - V-(x, Ony(x)] }d s.

142

THE BOUNDARY ELEMENTS METHOD

Hence we obtain for u(x) and v(4) the same representation like in (4.4.1), with the difference that (UU, V V ) become (U_+, V ).

4.5.2

Green Functions

The system (4.1.12) determines the fundamental solutions of the system (4.1.7). These are distributions. The fundamental solutions which are defined only in a portion of the plane and satisfy the boundary conditions are named Green functions, or, more properly, Green distributions.

The solutions (U+,V+) and (U_,V_) are therefore Green functions. For obtaining these solutions we employ the method of images. If M(t)

is a point of the strip -oo < z < +oo 0 < y < a (fig. 4.5.1), then the solution (without restrictions) of the system (4.1.12) is (4.1.13). As we already know, it represents the perturbation produced in the entire plane xOy by a source located in M. For satisfying the first condition of (4.5.3), we have to place a source having the same intensity (taken to

be equal to the unity) in the symmetric point M(t, -ri). For satisfying the second condition of (4.5.3), we have to place sources having the same intensity in the points which are the symmetric points of M and

M with respect to the y = a axis, i.e. in the points Mi (t, 2a - rl), Mi (t, 2a + q). These sources disturb the condition on y = 0, such that

we have to place sources in the symmetrical points of Mi and Mi with respect to the y = 0 axis, i.e. in the points M+1 (t, -2a + i) and -2a - ,) etc. The perturbations produced by these sources are

x-t

1

+

V(")+

27r (x-t)2+[(y-(2na±q)12 1

y-(2nat9)

(4.5.7)

27r (x - t)2 + [y - (2na ±, )12

and they satisfy the system OUP }/Ox + OV n)/Oy = a(x - t, y _ (2na ± ii)) (4.5.8)

OVA" i /Ox + OD'") /ay = 0.

In the strip -oo < x < +oo, 0 < y < a, all the perturbations satisfy the homogeneous system, excepting the perturbation (Uo , V+)

THE AIRFOIL IN TUNNEL EFFECTS

143

which satisfies the corresponding non-honwgeneous system.

lI+(x, ) = 2;r

(x'

+ [y - (2nn + 11)]'

+

(4.5.9)

r:

I 'V-4'

{ (r

=-00

y - (2nn + rl)

}

(x - )2 + [y - (2nn + n)]2

2,r n=-00

y - (2na - r1)

[y- (2na-17)]2

+{:r.

satisfies the system (4.1.12) and the boundary conditions (4.5.3). It is just the solution utilized in (4.5.4). In the same way we deduce that the solution

U-(x, ) = 2;r

{ (x - C)2 + [y - (2na + 11)12

-

x-t (4.5.10)

)-2,

L

I

n=-1

(2na

+ 11)

(a-ti)2+ [y-(2na+1))]2+

y-(2na-11) + (x - )2 + [y - (2na -11)]1

satisfies the system (4.1.12) and the boundary condition (4.5.5). This is just the solution utilized in (4.5.6). The series (4.5.9) and (4.5.10) are the Green functions of the problem. The sums intervening in (4.5.9) and (4.5.10) may be calculated using the formula = ircot az,

(4.5.11)

where z = x + iy. Separating the real part, from the imaginary one, we

144

THE BOUNDARY ELEMENTS METHOD

get from (4.5.11): +oo

y

y2 + (x - n)2

= 7r

sink 27ry cosh 2iry - cos 21rx (4.5.12)

+00

0

x-n

sin 2irx

n=-ao y2 + (x - n)2 = A cosh 2iry - cos 27rx

Using these relations we obtain the expressions of the Green functions:

sinhQ(x-

(x - ) - cos Q (y - q)

cosh Q

71

f=4n cosh 1

sinh -(x

- .)

a (x - ) - cos a (y + +)) (4.5.13) 7r

_

V f (x, ) =

sin -(y - q) n

1

sin

7;(y

cosh Q(x - e) -

4.5.3

+

A

`a

+,q) 7r(y + n)

The Integral Equation

In the sequel we shall proceed like in 4.3.2, i.e. we pass to the limit

-i xo E C in (4.5.4) and (4.5.6), we introduce the function G by means of the formula (4.3.9) and we utilize the relations

+(x, xo)n.(x) + V+ (x, xo)ny(x)]d s = - 2 (4.5.14)

lu-(x,xo)ny (x) - V-(x,xo)n.(x)1ds = 0,

145

THE AIRFOIL IN TUNNEL EFFECTS

which will be demonstrated in Appendix 4.5.5. One obtains the following integral equation:

(1/2)G(xo) - c {U_(x,xo)n(x)

V+(x, xo)nt,(x)+

+M2ni(X) + f(X) [V-(x, xo)nx(xo)-

(4.5.15)

-U+(x, xo)n,(xo)]}Gds = Onv(xo) , and discretizing, we deduce the system N

(1/2)Gi + E AijGj = Bi, j = 1, ... , N,

(4.3.16)

i.t

where Aij is A!. from (4.5.15) with the notatipns

U= f Dx, x°)ds, l= J V f(x, x°)d s

.

(4.5.17)

;

..

In order to determine these coefficients we consider on Lj the parametrization (4.2.19). Further we have two possibilities: the exact calculus like in 4.2.4, or the approximate calculus, with Gauss-type quadrature formulas (A.55], for the resulting integrals +l Ut(t)d

1-'

Uii =

2

t, Vj = 1_j t1 f V±(t)d t . 2

t

(4.5.18)

t

In case that i = j, these integrals become singular for t = 0, because of the terms sinh a [x(t) - x?d t

+1 t

cosh

a

sin

t ui

[x(t) - x°] - cos'-` [y(t) - A a

a

(y(t) - y°) d t

[x(t) - x°] - cos [y(t) - y°j

cosh

a

a

146

THE BOUNDARY ELEMENTS METHOD

but, taking into account that the integrands are odd functions, we deduce that they vanish. We have therefore 1.

+1

8a

i

Uf=± ` }-

`" -

sink - [x(t) - x9 'd t

a

cosh

[x(t) -;J - cos a [y(t) - y°] (4.5.19)

a

sin

+e

Co.),

a

[y(t) - y,)jd t

n 7r [x(t) - x,)[ - cos [y(t) - yp[ a a

These integrals may be also calculated numerically, using the Gauss-type quadrature formulas.

4.5.4 The Verification of the Method In order to perform this verification the shall use the solution given by the complex potential

f (z) = z + Ctanh

(4.5.20)

z

where C and a are real constants. Let us study the flow determined by this potential. Employing Euler's formulas we deduce the identities sinh iy = i sin y,

cosh iy = cos y

cosh 2 Ax cos2 Ay + sinh2 Ax sins Ay = 2 (cosh 2Ax + oos 2Ay),

cosh2 Ax cost Ay

- sinh2Ax sin Ay = 2 (1 + cosh 2Ax cos 2Ay), (4.5.21)

such that, separating the imaginary parts, we get from (4.5.20): C sin 2Ay

- y + cosh 2Ax + cos 2Ay '

(4.5.22)

We denoted A = 7r/a. Obviously the straight lines y = 0 and y = a are streamlines ('P = 0 and 'I' = a). Another streamline is the line y = a/2(41 = a/2) but not entirely because for x = 0 the fraction from (4.5.22) is not determined. Further we shall study the velocity field. From (4.5.20) it results

w(z) = u

iv = I +

AC cosh2 Az

,

(4.5.23)

THE AIRFOIL IN TUNNEL EFFECTS

147

whence, utilizing (4.5.21), u.

= 1 + 2AC

1 + ch2 Ax cos 2Ay (cosh 2Ax + cos 2Ay)2 (4.5.24)

V = 2AC

sinh2 Ax sin 2Ay (cosh 2Ax + cos 2Ay)2

Now we return to the streamline y = a/2. This line may be continued ('I' = a/2) on the set of points (x, y) where we have: a z

= -y+

C sin 2Ay cosh 2Ax + cos 2Ay

(4.5.25)

This curve is symmetric with respect to the Oy axis, because the equation is even in the x variable. It is also symmetric with respect to the y = a/2 axis, because the points having the ordinates y = a/2 + yo and y = a/2 - yo, where

_ yo

C sin 2Ayo

cosh 2Ax - cost Ayo

(4.5.26)

simultaneously belong to the curve. Hence the curve having the equations (4.5.25) is an oval which orthogonally intersects the axes y = a/2

and x = 0. In the point P(-a, a/2) we have therefore u = 0. From (4.5.24) it follows the equation

sinh2An = AC,

(4.5.27)

for the half-diameter a as a function of C. The semi-diameter 6 on the Oy axis may be determined from (4.5.26) as follows

_

C sin 2A,0

1-cos2Af3' whence it results a simpler equation

$tan AQ = C.

(4.5.28)

We draw the conclusion that the complex potential (4.5.20) characterizes the uniform flow with the velocity (1, 0), in the channel having the sides

y = 0 and y = a, in the presence of the oval having the diameters 2a and 2/3 with the center on the Oy axis (fig. 4.5.3). It is not difficult to see how this potential was obtained. One knows that the uniform flow at infinity, in the presence of a doublet, determines

THE BOUNDARY LLPMENTSMEITHOD

148

y=0

L. H=8110

x=0

Fig

the flow past a circular obstacle. Let us study now the circular flow in the presence of a doublet in a channel. For the sake of simplicity, we

consider the doublet situated in the origin and y = a/2, y = -a/2, the equations of the lines which are the walls of the channel. These right lines become streamlines for the flow determined by the doublet, if we add, according to the method of images. symmetric doublets. The potential f0(z) describing the flow produced by these doublets is

A(z)_...+

I z+ia

+1+

+...

1

=

_ro

_ let -cot Trz Ia IT

1

Ir

irz

u

a

-coth

Ave take

fo(z) = CcothA>. A = 7r/a,

C representing it real o constant. If the Ox axis coincides with the lower wall. as we have previously considered. then the potential becomes

fo(z) = Ceoth A

(:_ i a2)

= Ct.unh Az .

Adding the potential of the uniform flow having the velocity (1,0) we obtain (4.5.20). In the paper [4.9] that we utilized for writing this subsection, we made tests for the local pressure coefficient C,,, defined by (4.2.35). In figure 4.5.4 we present the values of Ci, calculated exactly (by means of (4.5.24) and the values calculated with the numerical method presented

above with H = a/10. The maximum relative error is 0.73% in the case of a discretiration with 80 nodes. For the circular obstacle, symmetrically placed in a tunnel, we present the compressibility effect in figure 4.5.5 where one indicates the variation of the maximum velocity

THE AIRFOIL IN TUNNEL EFFECTS

149

i-

I -o-

-

na i

\ 0 J Fig. 4.5.4.

_

Fig. 4.5.3.

against the distance to the walls for 'r = 1.405 in the cases M = 0 and M = 0.3 and in the absence of the circulation (F = 0). We considered 60 equidistant nodes on the circumference. In [4.9) one presents the graphic representations for C, in the case of the NACA-4412 profile at 0 angle of attack and Mach=0.5 in a free stream and in a tunnel. The example is instructive, because, in this case the trailing edge is angular, and we have to impose the equality of the pressures on the two sides of the edge in order to determine the circulation.

4.5.5 Appendix In order to prove the first formula (4.5.14) we shall notice that (U+, V+) (x, y, xo, yo) satisfy in the domain D, exterior to the contour C-c+C1,

and bounded by the walls y = 0 and y = a and by the segments x = ±d, the equation:

aU+/49x+aV+/Oy=6(x-xo,y-yo). Integrating this equation over D and applying Gauss's formula one obtains:

J (U+nx + V+n1)d s + ii;t a 1c,

+ d im

ra

J0

(U+n" +

s+

U+(d, y, xo, yo)d p + slim J t7+(-d, y, xo, yo)d p = 1, -oo J.a (4.5.29)

THE BOUNDARY ELEMENTS METHOD

150

the integrals on y = 0 and y = a vanishing by virtue of the relations (4.5.1). The integrals on C6 are calculated utilizing the parauiietrizntions (4.3.35) which imply

sin0+C,

cos0+0(e),

Uf=

I

I

where C_

1

2ryo

yn

1

1 + a2

n=0

(yo/a)2 - n2

To the limit, when a -i 0, this is 1/2. One calculate the last two integrals taking into account that d t Id

_

2

(a. - b)tan (t/2)

a -b

a+bcost

arctan

a -b'

To the limit, when d -- 0, every integral is 1/2. One obtains the first formula (4.5.14). Analogously, we obtain the second formula if we integrate the equation aTJ_/ay

- 8TH-/ax = 0

on the same domain D.

4.6 4.6.1

Other Methods. The Intrinsic Integral Equation The Method of Regularization

The methods that we have already utilized have been given by L. Drago§ [4.6] and by L. Drago§ and A. Dinu [4.7], [4.8], [4.9], [4.10]. Specific to these methods is the fact that one utilizes physical variables the velocity and pressure fields), such that from the solution one obtains directly the elements of interest in aerodynamics (the velocity and the pressure on the profile). Other methods (Biudolino a.o. [4.2], Morino and Luo [4.20]) utilize the real potential and others (Griello a.o. [4.15],

Carabineanu [4.4]), the stream function. All these methods may be utilized only for incompressible fluid. The theory that we are going to present in the sequel will be also applied to the incompressible fluid, but it has the advantage to utilize the physical fields. Moreover, the singular integrals are avoided and one utilizes the intrinsic elements of the flow (the tangential component of the velocity). Therefore it is suitable to call this method the method of regularization [4.11].

OTHER METHODS. THE INTRINSIC INTEGRAL EQUATION

151

Denoting

V = Un(i+v),

(4.6.1)

we shall determine the perturbation from the equations

dive=0, rotv=0

(4.6.2)

with the boundary condition (4.6.3)

and the condition at infinity limn v(x) = 0.

(4.6.4)

We may write the equations (4.6.2) as follows

div (v - c) = 0, rot (v - c) = 0,

(4.6.5)

c representing a constant vector. We put into evidence the normal and tangential components of the velocity:

v = (v n)n + (v s)s =

v,8.

(4.6.6)

By virtue of the condition (4.6.3) we may write

v = -n?n + v,s.

(4.6.7)

Rewriting the scalar formulas from 4.3 as a vectorial formula we deduce that for every two continuously differentiable functions or distri-

butions f and g , by virtue of the equations (4.6.5) (k - the versor of the Oz axis), we have:

ID if div (v - c) + (gk) rot(v - c)]da = 0,

(4.6.8)

D representing, like, in 4.3.1, the domain exterior to C and interior to the circle Cn having the radius R big enough. Utilizing the formulas

div f (v - c) = (v - c) grad f + f div (v - c) (4.6.9)

div [gk x (v - c)] = (v - c) rot (gk) - gk rot (v - c) and applying Gauss's theorem, we obtain the identity:

J(v -c) gradf-rot(gk))da=

f(v - c) (fn-nxgk)ds, }- CR

(4.6.10)

152

THE BOUNDARY ELEMENTS METHOD

n being the normal (pointing outwards the domain D (i.e. the inward pointing normal with respect to C and the outward pointing normal with respect to CR). We shall write the distribution (4.1.13) as follows

v,-

1

x - xn

2r,

Ix - xo12

(4.6.11)

and the system (4.1.12) that it satisfie :

divv'=b(x-x()), rote'=0,

(4.6.12)

the equations being valid for every point xn from D + C. For (f, g) (u', -v'), one obtains from (4.6.10) the projection of the identity

(v - c)divv'da = J

{[n (v - c)]v + [n x (v -- c)] x v}ds, +CR

JJJD

(4.6.13)

on the Or axis. For (f,g)

(v'.u') we obtain the projection of the same identity on the Oy axis. Hence the relation (4.6.13) is valid. By virtue of (4.6.12) we obtain from (4.6.13)

v(xa) - c = f

{[n (v - c)]v' 4. [ri x (t+ - c)] x v}tl s. (4.6.14) TCu

llere we shall put c = v(xo) - vo and we shall evaluate the integral on CJt. Utiliziug the parametrization (4.3.37), we shall deduce for the projection of this integral on the Ox axis

xli

{n - (v - vo)u' + (u - ua)(u'n7 + v'n,)2m

-riI[(ei - 110W + (v - vo)v']}d3 =

1

2=r j (u - t!o)dV = -110,

because for R , oo, u vanishes according to the condition (4.6.4). One deduces a similar formula for the projection on the Oy axis. Utilizing these results one obtains from (4.6.14) the following representation

V(x0) = PC {[n (v - vo)]v' + In x (v - vo)] x v' }d s.

(4.6.15)

The integral is not singular because the factor v - vo tends to zero when x xo E C. This is a regularized integral. The formula (4.6.15) is valid for both xo in the fluid and on the boundary C.

OTHER METHODS. THE INTRINSIC INTEGRAL EQUATION

153

We notice utilizing (4.6.7) that on C we have

nxv=v,k (4.6.16)

In - (v-vo)]v' = (-n=+ such that, after elementary calculations, from (4.6.15) one obtains

V(xo) _

P(v,k x v' + v;((v' . s°)n-(n v')s° - (n s°)v'] - nxv'-

(4.6.17)

-79x((n° v`)n - (n . v')n° - (n n°)v']}d s. For obtaining the unknown v, outside the integral, we shall perform the operation no x (4.6.17) and we shall take into account that we have no_ soy, n o_ y - -soy,

(4 6 18)

no x v' = (so v')k

(4.6.19)

:-

.

.

We deduce

no x s°

whence 11°= 4.6.20)

((s° . v')n' - (so v')ns]d s. This is the integral equation of the problem. It is a regularized (nonsingular) integral equation because when x - x°, the denominators and the numerators (n° v')v°) of the integrand simultaneously vanish (see (4.6.11)). The equation (4.6.20) was obtained in a different manner by V.Cardng (4.5]. It is, obviously an intrinsic equation. This equation may be also solved by means of the boundary elements method. One obtains, like above, a linear algebraic system of N equations with

N unknowns. In the case of lifting airfoils the following boundary condition is added (see also (4.3.23)).

-+Pf.

(4.6.21)

154

THE BOUNDARY ELEMENTS METHOD

Taking (4.6.1) and (4.6.18) into account, we apply this condition as follows:

v. (P.) + v.(A) = ny(P.) + ny(Pi) ,

(4.6.22)

P, and Pi being sufficiently close of Pf. From the equation (4.6.20) and the condition (4.6.22) we shall determine v, on C. 'Since we have more equations than unknowns, we shall treat the problem like in 4.3.5, introducing the auxiliary variable A. The reader may find examples in [4.11].

Chapter 5 The Theory of Finite Span Airfoil in Subsonic Flow. The Lifting Surface Theory

5.1 5.1.1

The Lifting Surface Equation The Statement of the Problem

We assume that an uniform flow, having the velocity (the Ox axis has the direction and the sense of the uniform flow), the pressure p,, , the density p,, and Af(= U,./c,.) < 1, is perturbed by the presence of a finite span wing, perpendicular on the flow direction. Any body which has a characteristic dimension much larger then the two other dimensions is considered to be a wing. We call the span of the wing, the length of the wing taken along the direction of the large characteristic dimension (figure 5.1.1).

Fig. 5.1.1.

One requires to determine the action of the fluid against the wing. In fact. as we have already seen when we studied the two-dimensional case, for determining the action, one has to calculate the perturbation

THE LIFTING SURFACE THEORY

156

fields p(x) and v(x). We employ the Cartesian variables x,y, z introduced in (2.1.1) and

the fields p and v introduced by (2.1.3). The Oy-axis has the span direction and the origin 0 is situated in the middle of the wing. The Oz-axis is taken perpendicular to the Ox and Oy-axes in order to detennine a Cartesian, positive oriented frame.

We assume that the projection of the wing onto the xOy-plane is a simple connected domain D, with a piecewise smooth boundary &D such that every straight line parallel to the Ox-axis (which has the direction of the unperturbed flow) should intersect the boundary in almost two points Q. and Q f. As we may see in figures 5.1.2 and 5.1.3, the lateral edges of the wing may be exceptions from this assumption if

they are parallel to the Ox -axis . The intersection of 8D with a lateral edge parallel to the Ox -axis consists of a point, two confounded points or a straight segment. Assuming that the two lateral edges consist

of a point (denoted respectively by B and B') we notice that they divide the boundary into two arcs

The front arc BB' which is attacked by the stream is called the leading edge and the rear arc is called the trailing edge. The equation

of the leading edge (consisting of the points Q,) is x = x_(y), and the equation of the trailing edge is x = x+(y). Hence Qa has the Cartesian coordinates (x- (y), y) and Qf (x+ (y), y). For wings with lateral edges consisting of simple or confounded points we obviously have:

x+(±b) = x-(±b),

(5.1.1)

where 2b is the span (in dimensionless variables). We shall call these ones I wings, and the wings whose lateral edges consist of straight segments will be called II wings. For the II wings the condition (5.1.1) will be replaced by the condition of the continuity of the pressure along the edge (the Kutta-Joukowski condition (see (5.1.30)). In figures 5.1.2 we indicated the domain D for some I wings having lateral edges consisting of simple points (the delta wing, figure a), the gothic wing, figure b), the trapezoidal wing, figure c), the rhombic wing, figure d), the swallow tail wing, figure e) or lateral edges consisting of double confounded points (the elliptical wing, figure f)). In figure 5.1.3 we present the domain D for the arrow shaped wing (with lateral edges consisting of straight segments). The domain D is named the plane form of the wing or the planar form of the wing.

157

THE LIF"TINC SURFACE EQUATION

b)

a)

d)

f)

e) Fig. 5.1.2.

We notice that the wing has two surfaces: the upper surface, denoted by S+ and the lower surface denoted by S_ . The equations of these

THE LIFTi G SURFACE THEORY

158

surfaces may be written (for the sake of simplicity) in the following form

z = h(x, y) f hj (x, y) ,

(x, y) E D .

(5.1.2)

Indeed, considering z = f (x, y) the equation for S+ and z = g(x, y) the equation for S_, then, setting 2h(x, y) = f (x, y) + g(x, y) (5.1.3) 2h1(x,y) = f(x,y) - g(x,y),

we obtain (5.1.2).

B.

Fig. 5.1.3.

The wing is thin if and only if h and h1 have the form h(x, y) = A-(x, y),

h1(x, y) = Fhl (x, y)

,

(5.1.4)

E(<<1) being a real parameter (max h, h1 in D), and !t $i hl bounded functions. We also assume that h(x,y) and hl (x, y) are defined on D and have the first order derivatives with respect to x, denoted by h,,, hi=.

The two surfaces S+ and S_ do not intersect each other if and only if h(x, y) + hl (x, y) > h(x, y) - hi (x, y).

5.1.2

(5.1.5)

Bibliographical Comments

The bibliography concerning this subject is extremely rids and it is not possible tp mention it entirely. In fact, this is not necessary

'ME LIFTING SURFACE EQUATION

159

because the fundamental solutions method, utilized herein is different from the previously utilized methods. As we have already mentioned in the introduction, the methods utilized so far rely on the equation of the potential (we assume that the perturbation is irrotational, the wing being assimilated to a distributions of bound vortices or doublets. Downstream, behind the wing, where the experience shows that the flow is not irrotational, one introduces in an artificial manner a free vortices

distribution. The perturbation of the velocity results from the velocity field induced by the two distributions. This is Prandtl's model [6.21) for the lifting line theory, which assumes that the wing may be replaced by the segment BB'. The lifting surface theory, which does not replace

the domain D with the segment BY, was developed after 1936 by Prandtl [5.26], (5.27), Weissinger [5.35], Reissner [5.29], Multhopp [5.24], Flax, Lawrence [5.12), Truckenbrodt (5.32), Mangler and Spencer (5.22),

etc. We also mention the unpublished paper [5.6] of P.Cociirlan. Some synthesis of the research in this domain may be found for example in [1.1), [1.2], (1.38) and [1.41]. D. Homentcovschi [5.16] presents, for the compressible fluid, a theory relying on the equations of motion written in distributions, without assuming that the flow is potential. Another theory is given in (5.7]. Here one introduces the fundamental solutions method. It is not necessary to assume that the flow is potential and a vortices sheet is formed behind the the wing. These properties result from equations.

5.1.3

The General Solution

As we have already mentioned in 3.1.3, there is no physical reason to replace the wing by a distribution of vortices. It is reasonable to replace the wing by a distribution of forces which should have against the fluid

the same action as the wing. Indeed, the fluid and the wing must be considered as an interacting material system. According to Cauchy's tension principle (see [1.11), p.35), there exists a ford distribution f on the wing (or on D) which has the same action against the fluid like the wing itself. The perturbation of the fluid flow will be determined by this distribution and by the slipping condition on the surface of the wing. We try to employ a forces distribution on D, having the form

f = (f1,0,f)(,t1) If we manage to satisfy the boundary conditions, on the basis of the uniqueness theorem, we deduce that we found the solution of the prob-

160

THE LIFTING SURFACE THEORY

Obviously, there exist various distributions which enable us to reach the solution. In chapter 7 we shall employ distributions on the boundary of the body. From (2.3.24) we deduce that the perturbation of the pressure due 1em.

to the force f applied in the point (t, q) is

ft(c rt)8 + f(c n)8 u-.) ,

(5.1.6)

xo=x-t,yo=y-v1, Ri= xo+(yo+z2),

(5.1.7)

P(x,IV,x)

J

where

A continuous superposition of such forces on D, will give

[fi(ii)- + f

P(x, y, z) = -1

n) ez

] JJD since the problem is linear. For the potential jp, from (2.3.28) we deduce ( X y, z) = T 7r-

JD

[f

(R1

) -- uo+

z z2

(_) d t d n ,

+ RI) f (t,

n)1 d

(5.1.8)

dn (5.1.9)

and for the velocity field from (2.3.12)

v(x, y, z) = 6(z) Jf f(, t7)H(xo)b(yo)d d r< + Oip

(5.1.10)

which can be written explicitly:

u(x, y, z) = 6(z) Jj h (, r1)H(xo)a(yo)d d r- p(x, y, z)

v(x,y,z) =

AD I fl

,

(f,n)' (i;)

-f(t,rl)Z [y

(i+9.)] }ddt 2 (5.1.11)

w(x, y, z) = 6(z)

IfD f (t, tt)H(xo)5(yo)df d n+

+4- fD f fi(t,n)g ()dd_ r1

-4Rffnf(F,n)az

[1l0

z2

THE LIFTING SURFACE EQUATION

161

For w, from (2.3.29) we get:

w(x,y,z)

4rr

lID

[fi(i); (RI}

-Q2f(E,rl)

CRI)

JdC dq+

+4 AD f(C17)Zy

(5.1.12)

(l+ Xg

Jo2+YO

Ri)J

z2

dtdn

From formula (5.1.10) we deduce that the perturbation is potential,

excepting the strip from the xOy - plane, -b < y < b, x > x_ (y), where the first term does not vanish. We notice that the existence of the vortices sheet behind the wing, which was a hypothesis in the classical theories (see Prandtl [6.21]), represents herein a consequence following from the equations of motion. As the experience confirms the existence of such a sheet, it means that the experience confirms the mathematical model based on the fundamental solutions method.

5.1.4

The Boundary Values of the Pressure

In view of calculating the boundary values of the pressure when z -' ±0 and (x, y) E D, we notice that the first term from (5.1.8) represents the tangential derivative of a simple layer potential and the second, the normal derivative of the same potential. From the potential theory one knows (see for example, [1.6), [1.39]) how to obtain the boundary values of these derivatives but we prefer to calculate them directly. After deriving we have to calculate

It

limo

f

i

(s, y) E D

(5.1.13) =-

lim

z-'.±O JD

(x, y) ED

dd

f(t,n)

.

1

We notice that setting z = ±0 and (x, y) E D, Rl vanishes when the generic integration point Q(t, n) coincides with M(x, y) E D. The integral Il becomes singular. Denoting by DE the disk with the radius

THE LIFTING SURFACE THEORY

162

E and the center h1, we shall write and we shall consider by definition ,

hm

(5.1.14)

U IJD 110 = JJ-D, +AD, if the limit exists. For calculating the integral on Dt, one makes the

change of variable

O7D,

q -+ r, 0: - x = Or c os 0, tj - y = r sin 0 (5.1.15)

0
0<0<21r

and one notices that if f is a continuous function, for a small enough, we may approximate f on the disk with f (x, y). We deduce:

AD, 1 and then, the formula

dtj=

fJ

r2 cos

f(xJ 1!) 12,,

0c (

f1

r2

+z 2)3/2

d9dr=0

=o1 f(e,r1)0dedt, R

(5.1.16)

D

where

R= Vxg+

(5.1.17)

As it is shown in (E.0.12), the last integral exists. As regarding 12, it vanishes outside a vicinity DD of M. We have therefore I2

Zo :

(x. y) E D

.f(C,t]) z dCdti

lID

74

2s

=f

li

n)

fo

z

Jo

(5.1.18)

rd 0d r = f2tr f (x, y) Jo (r2 + z2)3/2

(;, v) E D

Taking into account these results, from (5.1.8) we deduce:

p(x, y, f0) =

-

0

ID f i (4, r]) ? d d of 2f (x, y), (x, y) E D. (5.1.19)

This is the formula for the pressure on the upper and lower surfaces of the wing. We deduce the formula (5.1.20) p(x, y, +0) - p(2, y, -0) = f (x, y) , which gives us the significance of the function f (x, y). Like in the twodimensional case, f is the jump of the pressure on the wing. This is a fundamental result for the calculus of the aerodynamic action.

THE LIFTING SURFACE EQUATION

5.1.5

163

The Boundary Values of the Component w

If one utilizes (5.1.11), then taking the limit, when z -+ ±0, the first term disappears because 5(±0) = 0. It is more difficult to calculate the limit of the last term because it contains the derivative with respect to z. The calculation is performed in (5.221 and in 11.111, p.208, 209. We utilize here the expression (5.1.12) for w. In this expression, the limits of the first two terms are given by the formulas (5.1.17) and (5.1.18) and the derivative with respect to y from the last term commutes with the limit. We have therefore, lira

13

a8y yo + z

I1

a

z

(1 + La Rl )] dC drl =

(s, y) E D

(5.1.21)

8

_

j22K(xty,z7ii)dt,

y) ED +Cr,

with the notation x+(7)

K(x,y,z,r1)=1_

(5.1.22) 00

After setting z = 0 in the last expression of 13, we notice that we may not simplify with yo, because it vanishes for 77 = y. We shall divide the integral into three parts:

J-bb

_ 1b £+Jy+E+

JyyEa

The last integral vanishes because for F small enough we may consider K(x, y, z, v7) = K(x, y, z, y) whence

+E

TY-C

yoz

K(x,y,z,rl)dn=-K(x,v,z,y) f +E +z =0. E u 2du2

For the integrals on (-b, y - s) and (y + e, +b) we may pass to the limit, setting z ± 0 and we may then simplify by yo, because it does

THE LIFTING suitFACE THEORY

164

not vanish. It follows 13 =

a

Hen I I

`J b

3y

= line

+ r rb\ ! (r., rl, 0, 0 d rJ

Jy+il

1C(x,'y, 0, y - E)

l

f

+b\l

,,...s

+

t3

=

Yo

1i

(x, y.

L'

+ \J-b + Jv+c/ ay \ yo /

0,

y + `)

(5.1.23)

d tJ

Expanding into a Taylor series the first two terms and taking into account the definition of the "Principal Value" (D.3.1) and the definition of the "Finite Part" (D.3.6), we deduce 3'/`' b

1

+1 tb

h'(x,

II

_JJDf -7( o- `1+

Rddy+2

ayh(x,y,0,J1)di, _

JJ,

f(e,iJ)x (5.1.24)

Taking into account the expressions of 11,11 and 13 from (5.1.12) we obtain: u,(x, y, ±0)

5.1.6

= f fl (?', y) - 41, 1fn f (y, i) (l +

)

d d r1 . (5.1.25)

The Integral Equation

F om the boundary conditions (2.1.33) and the equatiolLS (5.1.2) we deduce: ua(x, y, f0) = Jr=(3:. y) t h' (x, y)

,

(:r, y) E D .

(5.1.26)

Replacing the last relation in (5.1.25), subtracting and adding the two relations, the first corresponding to the upper sign (the condition on the upper surface) and the second corresponding to the lower sign, we get (5.1.27)

4

_ ly JJ n

C1

yo

\

+

0) d diJ = hr(x,y), R

(x, y) E D.

(5.1.28)

THE LIFTING SURFACE EQUATION

165

The relation (5.1.27) determines directly f, (w, y) and the relation (5.1.28) constitutes an integral equation for determining the function f . This is the well known equation of the lifting surface. It is obviously a

singular integral equation with a strong singularity. The mark * is for the "Finite Part". For the incompressible fluid, The equation has been written for the first time by Multhopp in 1950 [5.24]. A rigorous demonstration is given by Mangler in the Appendix of the above mentioned article. The demonstration is quite complicated because the author utilizes the representation

w(x,y,z)=-47r

(5.1.29)

which follows from (5.1.11). Short deductions are presented in [1.1], [1.38].

For wings with lateral edges consisting from straight segments (figure 5.1.3) the equation (5.1.28) is integrated with the conditions

f (x, ±b) = 0

(5.1.30)

which, taking into account (5.1.20), means the continuity of the pressure (the Kutta-Joukowski condition). On the trailing edge

x = x+(y), -b < y < +b

(5.1.31)

one imposes the boundedness condition for the function f (x, y). An equation equivalent to (5.1.28) was given by Lawrence and Flax in 1951 [6.11]. Formally, it can be obtained from (5.1.28) using the identity

)= R

(I+0

yo

xo+R _ 8 rxo+Ro xpyo) 8y 1 \ xoyo 8

(5.1.32)

It follows: 4v

1

A

f(

AC 17)

(1+)ddtih'(xY)1

(x, y) E D.

(5.1.33)

Sometimes it is preferable to use this equation instead of (5.1.28), because it has weaker singularities (Cauchy-type singularities). A rigorous deduction of equation (5.1.33) from (5.1.28) must start from the definitions of the Finite Part and Principal Value . Using the function

C(y) = -

/

+(n)

s . (n)

f (t, t1)d e ,

(5.1.34)

TILE LIMNG SURFACE THEORY

166

whose significance will be given in 6.1, one demonstrates the identities

r f(art) d z; d q - -

= Iv

yJD dy

ff

X

f(C rl) d

(5.1.35)

d d(5.1.36)

ff

aUo

d

e

which lead to the equivalence in view (see also 6.1).

5.1.7 Other Forms of the Integral Equation 1. We are going to give another form for the expression (5.1.12) of the component w . In this expression the integrals are not singular, so we may employ Green-type formulas. We have

(1+.)]dd1l=

fin

_aly 1 70, +_ - i

yo

z2 {1

+

a)}

drl

d

(1+)]ddY! 1111 Lf YD_2 R1 + y02

+

JJ D

a yoZa YO (1 + RI ) d d l = r 1 Jo J

= PODf(4,r!)yo

a

.a

YO

1/o+z2

(5.1.37)

Using the identity:

(

ax `Ri)

1

l/ + 3 R z

8 (_ i xo

Of

oR

(5.1.38)

and Green's formula, we deduce

I

fDf(c,rl)8x

(_)ddT!= -POD M,17)

+132z2ffD x(oR1)dedn-JJ1

l

Lf

(5.1.39) YO T9Ridr dry.

167

THE LIFTING SURFACE EQUATION

Replacing (5.1.37) and (5.1.39) in (5.1.12) and passing to the limit, we obtain 1

1

w(x, y, ±O) = t 2 f 1(x, y) +47rJaL -

f (c n) ( yo

\l +

R moo) d e+

(5.1.40)

+4n JJD 5; yo \1 +

o} dt dn,

whence it follows the integral equation:

R

f (C n)

1

47rJoD

yo

(5.1.41)

of +4R JJD

yo

+

J de do ° h=(x,y)

This is the equation given by Homentcovschi equation herein, utilizing

Green's formulas before passing to the limit is preferable to that of Homentcovschi which utilizes Green's formulas for singular integrals. Formally, the equation (5.1.41) is deduced from (5.1.28) by means of the identity (5.1.32) as follows: ACT))

AD

(i+ xo

_

1r

8

(1+

R

1

R

(1+

l o)d /

`

dn(5.1.42)

2. In other papers (see for example, 11.21) we find the equation:

47r

AD 1 yo

I+ o) d

do - h=(x, y) .

(5.1.43)

It can be obtained from Homentcovschi's equation, imposing the condition f (x, y) = 0 on 8D. (5.1.44)

Usually this condition may be imposed either on the trailing edge or on the leading edge and on the lateral edges. Indeed, imposing the

168

THE LIFTING SURFACE THEORY

condition (5.1.44) both on the leading and trailing edges means that for the plane profile obtained through it section of the wing with a plane parallel to the xOz-plane, one has to determine the jump of the pressure which vanishes at the both edges. But it is well known that the jump of the pressure for the plane profile satisfies the equation (C.1.1) which

has a bounded solution at both edges only if the free term Satisfies the condition (C.1.13). Hence the condition (5.1.44) is valid only after imposing some restrictions for hx(x, y). However, for wings whose trailing (leading) edge is a straight line perpendicular on the Ox-axis, like, for example, the delta wing with the base representing the trailing (leading) edge perpendicular on Ox, or the trapezoidal wing with the big base representing the trailing (leading) edge perpendicular to Ox, the entire integral equation (5.1.41) reduces to the integral equation (5.1.43), because on the perpendicular trailing (leading) edge we have d = 0, and on the lateral edges we may impose the condition (5.1.41). For the rectangular wing the equation (5.1.43) is exact.

5.1.8

The Plane Problem

In order to obtain the solution of the plane problem from the threediniensional solution we assume that the intersection of the wing with each plan parallel to the xO.-; -plane is a profile having the same shape (figure 5.1.1). This means that the equations (5.1.2) have the form z = h(x) ± hl (x) ,

(V)y,

(5.1.45)

and the domain D is the rectangle (- I < x < 1, -b < y < b) with no (half of the chord of the wing wsas taken as the reference length La). From these hypotheses it follows the general representations of the solutions (5.1.8) and (5.1.12), b

fi = fi(b),

f = fW

Taking into account that

J

dy

y

(a2 + ff2y2)3/2

(12 f a'- + / y2

we deduce

lim r b-..oo_b

_ (!)(17)_$(229) R 2:co

169

THE LIFTING SURFACE EQUATION

lin)

b- -oo ./ b

Oy

1

y02

V+

(1 +

Z2 2

=- lim r+b0 (

1,

R

y°

)J cl vi = (5.1.47)

(i+)Jdi=o.

Jl[yo

So, the above mentioned representation becomes

I

+I

nQ

xofi(o)+42 #2Zf

dE

Q r+1 xofg) - zpIWd, w(C z) = 2a JI xO+/32z2

(5.1.48)

and v(x, z) = 0. Taking into account that here Oz has the position of the Oy -axis from the problem considered in 3.1,we find again the formula- (3.1.17).

The integral equation is obtained from (5.1.28). For calculating the integral

)r1 b

b yo

(1 +

Lo dtl, R

we use the definition of the Finite Part and the identity (5.1.32) valid on each of the intervals (-b, y - s), (y + E, b). Thus we obtain I+b 1 b

yo

(1 +°J d

lint f x0 + ,

e-'0 l xoyn

R

+x yRl+b c -

2

(1+

Uf

Ix011

/-bby ( l+

R)drl=-o.

whence

urn

(5.1.49)

The integral equation r educes to (3.1.21).

5.1.9

The Aerodynamic Action in the First Approximation

We indicated in [1.11], p. 79 the way to calculate the action of the fluids against the bodies by applying the transport equations of the momentum and the moment of momentum to the fluid filling the domain

THE LIFTING SURFACE THEORY

170

bounded by the body surface E and the control surface E0 surrounding E. This calculus is absolutely necessary when the body has corners, like in the case of thin bodies. However, in a first approximation the calculations may be done taking into account that the action of the fluid is given by the jump of the total pressure OPi 0 = Pt (x1, yl, -0) - PI (x, yi, +0)

which determines the lift L. On the unit of area the force is pljk (fig. 5.1.4), k representing the versor of the Ozl-axis, and on the entire area of the domain D1, L = Lk, where

L=

f 1 pl I

1dy1 =

f fD POdxdy .

(5.1.50)

Fig. 5.1.4.

For the resulting moment one obtains the formula

M

=

zl x Opt I kdxtdyl = PooUZLo fJ z x QpO kdxdy, (5.1.51)

where according to (5.1.20),

64 = P(x, y, -0) - p(x, y, +0) _ -f (x, b) .

(5.1.52)

171

THE LIFTING SURFACE EQUATION

Denoting by At the area of the domain DI (which is the domain D in dimensional coordinates) we have

Al =

dx1dy1 = LoJ1 dxrdy = LOA.

(5.1.53)

U

IL,

For the lifting coefficient et, one obtains the formula cl

def

L

(1/2)paoU.A1

= - A AD f(x,y)dxdy,

(5.1.54)

and for the moment coefficients (5.1.55)

2 Aao

dot

der

_11f

=

Z

Aao

l,layf(x, J)dxdy,

xf (x, y)dxdy,,

(5.1.56)

where al = a0Lo is the medium chord of the domain D1 on the direction of the unperturbed stream. As we shall see in the sequel, the drag (the projection of the resultant on the direction of the unperturbed stream ) and the gyration moment (the projection of the moment Ozlaxis) have the order of magnitude e2, for this reason they are not present in this calculation. M? is the rolling moment, and Afy, the pitching moment.

5.1.10 A More Accurate Calculation We obtain a more accurate calculation if we have in view that the resultant and the resultant moment are determined by the jump of the tension vector i.e. by the formulas

R = - Jf pt nOdxtdyl , M = - fJ x1i x OplnO&rtdyl ,

(5.1.57)

I

w here n is the outward normal on E. The presence of the normal enables us to take into account the shape of the wing. If the equation of the wing is h(x, y) - z = 0(h = FIT), the normal n is

+h k n=kri1+hr+hy =hri+hyj-k+

ta'' + h2 2

172

THE UFTINC SURFACE THEORY

Neglecting the terms of order O(E3) and taking into account that h is h(x, y) f h1(x, y), it follows

- p'nD = 2pco [h1=i + hlyj + (hxh1 + hyhlu)k] -POOU,2, IpI (h=i + by j - k) + pooU.2, < p > (hi1i + hlyj)

,

where we denoted

< p >= p(x, y, +0) + p(x, y, -0) .

(5.1.58)

Hence the lift is:

R. = -PpoUULo ff .f (y)dxdy + 2p.Lo ff

(h,,hl +

hyhiy)dxdy,

(5.1.59)

and the drag Rr = 2PooL2

IfD h1:dxdy + p UULo Jf f (x, y)h,,dxdy+ D

(5.1.60)

f < p > hudxdy.

+p,oUO2OL0 ifD

For the rolling moment one obtains the formula:

h!= = -P.UULo LID yf (x, y)ddy +2pLy(hh+ (5. 1.61)

for the pitching moment one obtains

= pULx f (x, y)dxdy - 2pLx(hhl,, + hhly)dxdy,

jj

(5.1.62)

and for the gyration moment, AI. = 2pooL03

Jj ( xhl- yhl)dxdy+

+PoU;,Lo JJ f(xhy - yhx)dxdy+ .l D

(5.1.63)

+PooUo,LO f f < p > (xhty - yh1m)dxdy, D

where

=--

xo

h1xR3dEdrl, D

(5.1.64)

THE LIFTING, SURFACE EQUATION

173

as it results from (5.1.19) and (5.1.27). If we keep only the terms of order O(E), it follows: R.- _ -Poo U.2 Lo2

Rr = 2pxL

JJ

JJ D f

(x, y)dxdy,

1i1=(x, y)dxdy, (5.1.65)

(A fr, Afy) = p

A-Y'X)f (x, y)dxdy, D

M, = 2pmLo Jf (xh- yhtr)dxdy. For the lifting surface (hl = 0) the formulas (5.1.55)-(5.1.59) become R.

fD f (x, y)dxdy,

R: = -PU Lo f f f(x,y)h.(x,y)dxdy, D

(5.1.66) (M1. A.fy)

=p

LG

ff(-y, x)f (x, y)dxdy,

A9, = 2pxLo ff (xhy - yhr)dxdy.. D

Rz, Air §i My being O(s), and Rr and Al;, O(E2). 5.1.11

Another Deduction of the Representation of the Gen-. eral Solution

In the sequel we shall deduce again the representation of the solution

(5.1.8), (5.1.12). We start from D. Homentcovschi's idea (exposed in (A.81) to utilize the Fourier transform for bounded domains (A.6]. The method synthesizes the problem of determination of the fundamental solution and the problem of replacing the wing with a forces distribution.

In addition it justifies the assimilation of the wing with a distribution of forces having the form (fl, 0. f ). Indeed, employing the formulas

THE LIFTING SURFACE THEORY

174

(A.8.2) to the system of linearized aerodynamics (2.1.30) and taking into account that D is a surface of discontinuity, we get

0=-ia1A (5.1.67)

0=-iali) - iap- Pk, with the notations [jpj)e,(QI x+a2Y)d x d

y,

(W' P) = JJ D ( where, taking into account (5.1.20), (5.1.25) and (5.1.27), awj = w(x, y, +0) - w(x, y, -0) = 2hi=(x, y) (5.1.68)

W = p(x, y, +0) - p(x, y, -0) = f (X., Y) From (5.1.67) one determines first p and then w. One obtains

ia3P - ia1W

a2-M201

(5.1.69)

ia3W W =ate

+,Q2 a

ia1P

ia2P

+ a ( 2 - M22)

Utilizing the expressions of P and W, we find: 1Q3

p-

J fp(t, r?) a2-A?2aj e(-t4+0tin)9 dv7is l

dd d11,

-2 [1D hit (e, t?) a2 - M2a2

whence, taking into account (A.6.9), P(x, y, z)

fJD'02

,-I

i(o,+ozq)

a2 -

2ai d dy+ (5.1.70)

rrrr +2JJD

hlt

,rl)

az

a2 - M2a2

Employing (A.7.2) we obtain

P(x,y,z) =-4a

+ 27r

f fD rr

JJD

(_)d+ (5.1.71)

hlt (, q) 8x (R1 d dr7

NUMERICAL I`TECRATION OF THE LIFI1NC SURFACE EQUATION

175

where R1 is that from (5.1.7). We obtained in this way the representation (5.1.8). Similarly, utilizing (2.3.11) and (2.3.27), we have:

F`1 fly

_

ial(a2 - M1ai)

(5.1.72) r

0 aJ-110

1)dr

47r

ya+z2 yo

(i+) jai

such that R, having the same expression, it follows:

w(x,y,z)

=-T7r

-

1,! n

Yj)

dFdij-

/32

+ 4T 11

(5.1.73)

(R1) di;d;+

0 [ ?100+Z2 yo

(1+)].d rl,

which is just the representation (5.1.12). We have to notice that from (5.1.70) one deduces the inversion formulas 1

[Y2

at(G2 - h12a21) 1

4z y2

y

1+

X

r.' +

(a" +z-) (5.1.74)

which could be utilized in 2.3.

5.2 5.2.1

Methods for the Numerical Integration of the Lifting Surface Equation The General Theory

There are not yet known exact solutions of the equation (5.1.28). The first numerical solution was given by Multhopp in 1950 15.24]. Previously the same author had given in 1937 the approximate solution (to which one assigned his name), for the lifting line equation (Prandtl's equation (6.1.16)). Multhopp's method relies on the Gauss-type quadrature formulas for non-singular integrals. At that time there were not available quadrature formulas for singular equations. For the singularity appearing in the lifting surface equation, Multhopp utilizes a series expansion

176

THE I..IFTINC SURFACE THEORY

with Chebyshev polynomials of first kind, which is truncated in order to obtain an algebraic system. The method is analogous to Glauert's method for Prandtl's equations, except that the sin functions are replaced by Chebyshev polynomials. In 1958, at a Meeting in Fort Worth, Hsu gave a quadrature formula for integrals with a strong singularity and employed this formula for the singularity from the equation (5.1.28). However, in Hsu's formula the unknowns are present even in the collocation points and this is a drawback. Starting from a formula given by Monegato [A.52), Drago§ gives [6.5] the formula (F.3.5) where there are present supplementary unknowns in the collocation points. One utilizes successfully this formula for solving Prandtl's equation in 6.5 and for solving the lifting surface equation in [5.10] and [5.11]. These solution will be presented in 5.2 and 5.3. In 5.2 we shall sketch the solution of the equation S via the collocation method. We have to solve the equation: I 4rr

N(xo, lyo)dt dr1= -hr(x, y),

i f( Q

(5.2.1)

o

where

N(xo,yo)=1+

To

V2'or+

Y4

(5.2.2)

with the following conditions:

f (t, ±b) = 0,

x-(fb) < < x+(±b)

(5.2.3)

-b < rl < +b. (5.2.4) The first two conditions mean the continuity of the pressure on the f (x+(rl), q) = 0,

lateral edges in case that they are straight segments (if the lateral edges are represented by points the conditions disappear by virtue of (5.1.1)). The condition (5.2.4) ensures the boundedness of the pressure along the trailing edge. One performs the following reasoning: each intersection

of the wing with a plane parallel to xOz determines a thin profile; as it is known from 3.1 for such a profile one imposes the boundedness condition on the trailing edge.

In order to utilize the quadrature formulas we shall perform the change of variables

(x, y) - (u, v)

TI) --4 (a.

defined by the equations x = a(y)u + c(y) Zr = a(ri)a + c(q) (5.2.5) y

by

rl = b13

177

NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION

with

a(y) =

x+(y)

- z-(y) 2

,

c(y) =

x+ (y) + x-(y) 2

(5.2.6)

Writing the equation (5.2.1) as follows:

r

1

T

UFO

+(n)

f

q)N(xo, yo)d

do = -4xhz(x, y)

and taking into account that e(t,g)

= a(F)b,

.9(a, /3)

we deduce

f

a(A)

1

(v - A)2

f (a, /3)N(u, v, a, Q)da] dfl = 9(u, v),

(5.2.7)

where denoting by a(v), c(v), f (a, /3) etc., the functions a(y), c(y), f((, q) in the new variables, we have

a(v)u + c(v) - a(/J)a - c(/3)

N(u, v, a, /3) = 1 +

1/2

{[a(v)u + c(v) - a(Q)a - c(f)J' + k2b2(v - /3)21

(5.2.8) (5.2.9)

g(u, v) = -4rrbhi(u, v),

with the notation k

The Kutta-Joukowski conditions

(5.2.3) become

-1 < a < +1) , f(1,/3)=0, -1
(5.2.10) (5.2.11)

The solution of the equation (5.2.7) depends on the behaviour that we impose at the extremities of the intervals (±1). For a fixed 0 , one obtains within the wing a thin profile. For such a profile we have imposed

in 3.1 the behavior given by

++ a . Along the span one imposes in

the behaviour 1 --02. Such a behaviour ensures that the conditions (5.2.10) are satisfied. We shall seek for solutions f (a/3) having the shape

f(a,A) = f -l

++aF(a,A),

(5.2.12)

178

THE LIMING SURFACE THEORY

with F bounded. For F one obtain the equation

JT (p1- v a(13) I- t '1 + a

F(Q, 3)N (ti: vj rx $)da1 d8 _

-1
=y(u.v),

(5.2.13)

Utilizing the quadrature formula (F.2. [8), this equation becomes ni

1 ^F

t

2. (1 - a)

2

r)

,

a(;T):"(u.. v,(xi,O)F(ai, 1)d 8 = (5.2.14)

= (tin + 1)g(u.v),

where

`=cos2m= (5.2.15) i= lm. a +1' From now on Nttlithopp's method and the quadrature: formulas method are separating.

Multhopp's Method

5.2.2

Using the formula (F.4.2) for (f3 m

v)-2

,

equation (5.2.14) becomes

p

--4irEE(1+k)(1i=1 k-i

J1

-or)Uk(u) JJJ

- 3 `u(f3)N(a. v,

!

= (2m + 1)g(u,v).

(5.2.16)

One calculates the integral from this equation utilizing (F.2.12). One Obtains

r -.1-r

it

E(l+I)(I

r=1 k-l,t=1 (5.2.17)

=(2m+ 1)(n.+ 1)9(u, V),

-1 < u,v <+1,

NUMERICAL INTEGRATION O THE LIFTING SURFACE EQUATION

with

cosn+l'

7

=r

.

179

(5.2.18)

Introducing the notation

H(ai,f3j) _ (1 - ai)(1 -QJ)a(#j)F(ai,f3j),

(5.2.19)

the system (5.2.17) becomes m rn

p

-4irrG.rL.r

(1

k)Uk(v)Uk(Yj) (u, v,ai,Qj)H(ai,QJ) _

i=1 j=1 k-i

(5.2.20)

=(2m+1)(n+1)g(u,v),

-1
There are m x n unknowns, H(ai, /3j ). In order to obtain the same number of equations in (5.2.20) we shall assign m values to u and n values to v, obviously, all of them in the interval (-1, +1). As we shall see later, the aerodynamic coefficients are functions of H(ai, /3j ). It is sufficient to find out these unknowns. We may write computer programs for solving the system (5.2.20).

5.2.3

The Quadrature Formulas Method

Utilizing in equation (5.2.14) the formula (F.3.5), we obtain the equation

m n

tar E F` [1i=1 j=1

(1 - ai)(1 ()3j - Ak)2 m

-7r(2m + 1)(n + 1)2 E(1 - ai)a(.3k)N(u, Qk, ai, Qk)F(ai, Qk) _ iml

= (2m + 1) (n + 1)g(u,13k) , (5.2.21)

where

Aj=ten+1, for k = T.

(5.2.22)

180

THE LIFTING SURFACE THEORY

In (5.2.21) we have a system of n linear algebraic equations with m x n unknowns F(a1, /3j ). Imposing the system to be verified in m points ul

-

oOS

Vir 2m+1

=al,

t=T,

(5.2.23)

the number of equations equals the number of unknowns. The system will be written as follows

m n

Bik, a=Tm k=I-n(5.2.24)

Aekij Ha 13 i=1

where we denoted Alki;

(2 [1- (-1)j+k]

=t

()3j -)3k)2

-(2m + 1)(n + 1)21

j

-A

}N(atflkoi/3i)

(5.2.25)

Btk = (2m + 1) (n + 1)9(a1,Pk)

5.2.4

The Aerodynamic Action

The lift and moment coefficients are calculated by the aid of formulas (5.1.54)-(5.1.56). Using (5.2.12), (F.2.18) and (F.2.12) we obtain: CL

A

ff

D

Af

+1

=Af

f

+L +1

f(t,

1- fl2a(Q) 1

41r2b

m

[f

1

l f(a,

d/3 =

+1 1

1+

aF(a, l)da] dQ =

(5.2.26)

"

A(2m+ 1)(n+ 1) FF H(a"Pj)' w1 j=1

where H(a;, (3j) are (5.2.19). Formula (5.2.26) is valid for both Multhopp's method and the quadrature formulas method.

NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION

181

In a similar way we obtain: 2

=

IL of 4R

Aao

+I

J_ I

ja(,3)f (a,13)da df3 =

n

in

2b2

f

+I

aoA(2m + 1)(n + 1) E E Qj H(ai,13j) i=1 j=1

,

(5.2.27) cy

Aao 2b - Aao

11

f

r')dCdr' =

+1

+I

J-1 4-Ir

2

ta(,B)a + c(R)I a(A)f (a, Q)da d/.3 =

b

aoA(2m+1)(n+1) ao

m

r( ( it

(5.2.28)

q

ti_1 j=1

representing the length of the medium chord and

di =

CD = - A !ID 41r2 b

A(2n+1)(n+1)

n

(5.2.29)

n

i=1 j=1

For the flat plate of incidence E, we have h(x, y) = -Ex then g(u, v) _ 47rb. For the rectangular flat plate we have x_ = -1, x+ = 1, a = 1,

c=0, u-a

N(uvapl=1+ , , , 5.2.5

(5.2.30)

The Third Method

Some numerical experiments show that it is not always indicated to

impose the behaviour along the direction a under the form We shall use in (5.2.7) the following quadrature formula +1

m f (a, Q)N(u, v, a, /3)da = > f(ai, i9)Ki (u, v, f3) ,

1

i=1

(5.2.31)

182

THE LIFTING SURFACE THEORY

2i

where a; = - m , i = 0, 1'.. . , m are equidistant nodes on the interm val (-1,+1), and K1 (u, v, /3) =

joi-I

N(u, v, a, fl)da =

2

m

1

a(Q) /[a(v)u + c(v) - a(A)a; -- c(/3)1+ k2b2(v

+

- p)2+

(a(v)u + c(v) - a(/3)a{-1 - c(,6)J2 + k2b2(v - /3)2.

(5.2.32)

The behaviour in the span direction remains the same (like in the previous methods), hence:

1--#2F(&,,6),

f (a, A) =

(5.2.33)

where F(a, 3) is finite for rQ = ±1. We shall use the same formula (F.3.5) with respect to p. So, the equation (5.2.7) furnishes the following discretized form of the equation (5.2.7) ,n

n

HtkiiF(a (3:) = bh=(ailk) ,

(5.2.34)

i=I j=1

where

Htk;j _ =

1

4(n + 1)

{i_(_i+k]

2 Ak)2a(13i)Hj(ac,,6k, Qj)

(Qj -

,

k (5.2.35)

and

n +l 1 H10919 =

a(/3k)Ha(at, Qk, /3k) .

(5.2.36)

The algebraic system (5.2.34) has m x n linear equations with m x n where #j =coo 17r unknowns

n+l*

NUMERICAL INTEGRATION OF THE LIFTING SURFACE EQUATION

183

The lift and moment coefficients are calculated by means of formulas CL

_ -

2b

+1

A

11

+1

1-,2a(fl)F(a,A)da dQ =

J_1

1

2bir

(n + 1)A

j

(1-1,)a(13j) J

F(a, $$)da

1

(5.2.37)

1

n

4bnr

(1 in

n+ 1 )rr1A Jul i=1 C,

22 2

1

1+1

#a(13)

J1

=

1- J32F(a, /3)da dfl _ (5.2.38)

nm

4b2ir

E I6ja(0j)(1 - AJ)F(ai, Qi) = (n + 1)mAao, Jul i=1 Aao

f

1111

a(#)[a(A)a + c(Q)]

1- 02fla,,O)da d/3 =

n cm

-4b,r In + 1)mAan Jul iml

EEa(AJ)(1-

_Q1)(2i-'m-1a(pi)+c(fii),F(ai,QP)(5.2.39)

In the paper 15.101, from which we presented this method, there are presented computer programs for the elliptical flat plate h(x, y)

= -cx, x2 + e < 1, b = 2

and for the wing whose projection on the Oxy - plane is an rectangle

-b
h(x,y) = e(1 - x2),

e <
(5.2.40)

THE LIFTING SURFACE THEORY

184

The fluid in considered incompressible. For the rectangular wing one obtains an analytic solutions in the framework of the theory dealing with the wings of low aspect ratio. In this way we have a test solution for the third method. One deduces that this method furnishes very good results.

5.3 5.3.1

Ground Effects in the Lifting Surface Theory The General Solution

The ground effects in the lifting surface theory were taken into consideration in (5.311 and [5.371 where one utilizes asymptotic methods. An approach of this subject in the framework of curvilinear lifting line theory may be found in [1.32], [1.331. The present subsection is elaborated following (5.8], where one gives the general theory. One utilizes the fundamental solutions method. The geometry of the problem is presented in figure 5.3.1. The origin of the reference frame is located in the middle of the span, the Ox-axis

has the direction of the unperturbed stream and the Oy-axis has the span direction. The ground is considered to be the plane 1I having the equation z = -d/2. The unperturbed flow is characterized by the velocity Ui , the pressure p(,,, and the density po and it is considered to be subsonic like in 5.1. The field of velocity v1, the pressure p1 and

Fig. 5.3.1.

the density p1 for the perturbed flow have the form (2.1.3). One utilizes

185

GROUND EFFECTS IN THE LIFTING SURFACE THEORY

dimensionless variables. The equation of the upper and lower surfaces are given by (5.1.2). It results the following boundary conditions w(x, y, ±0) = hr(x, y, ±hiz(x, y)

(5.3.1)

(x, y) E D

(5.3.2)

w(x, y, -d/2) = 0, (V )x, y.

For satisfying the last condition we shall utilize the images method. This means to replace the wing with a forces distribution f + = (fl, 0, f )

defined on D and with a symmetric distribution f - = (fl, 0, -f) defined on the domain D', which is the symmetric of D with respect to the plane II. In order to write the general form of the perturbation fields p and w we have at first to write the form determined by the

concentrated forces f+ in P+(4+) and f- in P_(4-). We have therefore to determine the solution of the equations

M28p/8x + divv = 0 (5.3.3)

8v/8x + grad p = f+b(x - 4+) + f -b(x - 4-) corresponding to the system (2.3.4). The system (5.3.3) is linear and its solution is the sum of the corresponding solutions from (2.3.4). Taking into account (2.3.24) and (2.3.29), and the form of the fields f + and f -, we deduce: p(x, y, z)

=-4

w(x* y, z)

=4

0T (f,

+fa ) `R+l

8z ( R+)

4 f8 x

47r

(fie -fe)

(i)'

R+) +

+47r 5; [(y_+)2+(z_z+)2 (i+xR+ )J+ 2

+4 Oz

f

R_)+4 f8x (R_)y - 17

4ir 8y (y - 17- )2 where

R+=

(x - t:1)2+

(z - C-)2

+

-- (1 +

x R-

2n*)2 +(z - (±)2).

l

)J

' (5.3.4)

(5.3.5)

186

THE LIFTING SURFACE THEORY

The points P+ and P- are symmetric if £+ _ {- = f, tl+ = n- = n, t+ _ C, C- _ C - d. When P+ are in D((= 0), the symmetric points

P-((- _ -d) are in V. Assimilating the wing with a continuous superposition of forces defined on on D, it results the following general solution:

p(x, y, z) _

- 47r JJv [fi,-+f,q)-}

()d_

d-9(5.3.6)

rr

f (e, n) 8 l ((Rid) dkc di J

w(xy, z)

4s

IL

+47r'ID

[f(t,

n)ex

JJDf(t,rl)

4r AD

A2f(t, n)

] (R,'

) de cI +

(L)d+ (i+)}d_ [yo-0

If,V,17)z +

1

+47r

-

02f (f, n)

8J

!0-

Lyo+

dC

(1+ -10-d/J dt

dn,

(5.3.7)

where xo=x-t, yo=y-Hand Rt =

xo+(32(yo+z2), Rld= 90 +p2[yo+(z+d)2J.

(5.3.8)

This is the general representation of the perturbation, the functions f, (t, r)) and f (t, t) being determined from the boundary conditions (5.3.1). The condition (5.3.2) is obviously satisfied.

5.3.2 .

The Integral Equation

Acting like in 5.1, we shall pass to limit considering z -- ±0, (x, y) C-

E D. To the limit, only the integrals related to R are singular. Using the formulas (5.1.16), (5.1.18) and (5.1.24), from (5.3.6) and (5.3.7) we

CROU\l) EFFECTS IN THE LIFTING SURFACE THEORY

187

get

P(x, y, ±0) = 4rID fi(f,t,)Rgdt dry ± 2f (x, y) +

+ 4a JJ f1(, q)

ddi 4j r

R3 dti,

,

(5.3.9)

P(x, y, +0) - P(-T, V, -0) = f (x, y),

and

w(x, y, f0) =

2 f1 (x, y)

47

if

-

47r ,IJD

R (i+)d_

f

ff f (t, t)N(xo,

(Z' Y)

-

d'l

41r

3Nn)dt dr1,

(5.3.10)

where

N

yo) _

(d2-#

) + (d + ;'F d

2

2

(1 + xd° )

I

(5.3.11)

with the notations

R=

xo + $2y$ , Rd =

xo + 02 (y02 + d2) .

(5.3.12)

Imposing the boundary conditions (5.3.1) we deduce

fi (x, y) = -2hi:(x, y)

(5.3.13)

and then

4Rv

f

(YOT

L1+'0 R)

dk drl + 4,r

AD

f (f, rl)N(xo, yo)dC dy =

(5.3.14)

The equation (5.3.14) may be called the generalized lifting surface equation. It was given in [5.8]. The sign "*" is for the Finite Part. Using

188

THE LIFTING SURFACE THEORY

the identity (5.1.32) we may write the equation as follows:

4n 8y

jf f (,'1)

1+

o)

= h' (x, y)

drl

-

4a

If f (t, r!)N(xo Uo)df do =

.JJD hlfR3

1) dt

dtl . (5.3.15)

The Two-Dimensional Problem

5.3.3

Like in 5.1.7, in order to obtain the representation of the solution and

the integral equation for the wing of infinite span we assume that the normal sections in the wing determine profiles having the same shape. It means that the equations (5.1.2) have the form z = h(x) ± hl(x)

(V)y

and the domain D is a rectangle (-1 < x < 1, -b < y < b) with oo. We notice that in the representation of the general solution

b

(5.3.6) and (5.3.7) ff and f have the form f,(£) and f(t). Relying on formulas like (5.1.46), from (5.3.6) we deduce

P(x, z) -

1

21rp 1

)324f (t) dt+ f,2 xoft Wxo ++ $2z2 r+1 xofl (t) +,32(z + d)1 (t)

0 +$2(x2+d)2 which are just (3.2.3)1. Using formulas like (5.1.47), from (5.3.7) it results: w(x,z) __

r11 x01(0+

z2(C)dc _2 J_li xOf +fl

(

+d)2( )41

2a

i.e. just (3.2.3)2.

For obtaining the integral equation we shall use (5.1.45). We also have oo do

J

ydq

2 ,

l

00

The integral

1=

+00

L -0o (y[+

d2 _ yo

xo dn

)2 R1

= 0.

(5.3.18)

189

THE WING OF LOW ASPECT RATIO

reduces to integrals having the shape

Jo

(u2 +

r

du d2) VU-T-+31

Je

du (u2 + d&)Z u +

which may be calculated using the change of variable u -+ v where

u

V

(5.3.17)

VU-27 71

One obtains

I=

2,0

d

1-

xo

xo

arctan

xo dQ

(5.3.18)

Using the same change of variable (5.3.17) one obtains also +O0

xo d

1 (plc

01?

11

9

1

+ ;01

o

xo

x

arctan d

(5.3.19)

such that we finally obtain oo

-l

(5.3.20)

xo + d2 02

Taking into account the previous results, the integral equation (5.4.12) becomes

,r+i 1 Jl

1

xo

_

dt

-

r+i xof ()

27r JJ

1

xo + d2#2

L" /+1

clk

= (5.3.21)

h'(1)

which is just (3.2.7).

In (5.111 one extends the method from 5.2.3 also to the equation (5.3.14).

5.4 The Wing of Low Aspect Ratio 5.4.1

The Integral Equation

In the sequel we shall pay attention to the ratio A = (2b)2/A,

(5.4.1)

190

THE LIFTING SURFACE THEORY

called the aspect ratio. We denoted by 2b the span and by A the area of the domain D. For wings characterized by a small A (wings of low aspect ratio) one develops herein a theory which leads to the integration of the lifting surface equation (5.1.33). For wings characterized by a big A we shall develop in the following chapter the lifting line theory. These

are the two asymptotic theories of the lifting surface equation. If the square of the span is small with respect to the area A, we deduce that on the greatest part of the wing we have yo << xo (see fig. 5.4.1) and we may write R

Ixoi

xo

xo l

L1 + O (

)J ,

(5.4.2)

o

where R is (5.1.17). On the contrary, when the square of the span is big with respect to A (see figure 6.1.1), on the greatest part of the wing we have xo << yo, so that we can make the hypothesis (see 6.1).

Fig. 5.4.1.

R = kayo)

(5.4.3)

Returning to the wings of low aspect ratio, we notice that under the hypothesis (5.4.2) the equation (5.1.33) may be approodmated by 4rr 8y J JD

f (n) (j+LX-01 df drl =1. (z, y)

(5.4.4)

Noticing that the integrand of this equation is different from zero only on t < x, we shall consider only wings whose trailing edge is a straight line perpendicular on Ox like in 5.4.1. Introducing the unknown function F(x, q)

JS-(n)

f(s, q)d,

(5.4.5)

THE WING OF LOW ASPECT RATIO

191

utilized for the first time by Jones 15.171, 15.18), the equation (5.4.4) becomes

8

1

v+(=)

{

dry = h=(x, y) ,

2w 8y Jy_ (x)

(5.4.6)

yo

where y = y+(x) represents the equation of the leading edge OB, and y = y- (r), the equation of the leading edge OA. From (5.4.5) and from the figure one notices that F(x, y- (x)) = F(x, y+ (x)) = 0.

(5.4.7)

In (5.4.6), utilizing the definition of the principal value, we derive taking into account (5.4.7) and then we integrate by parts. It follows a

F(x, n)

ay

_f dq -

e F'

1

yo! d

_ 'v+ SF do

- Jy-

an

'

(5.4.8)

where, from now on,

y- = y-(x), y+ = y+(x) In this way equation (5.4.6) becomes + OF dr)

1

Jv-

8q

- h. (x, y)

(5.4.9)

This equation can be also obtained starting from Homenteovsch's equation (5.1.41) which, in case that one imposes the condition (5.1.44) on the leading edge, becomes equation (5.1.43). Then we can see that from (5.4.5) it follows

8F_

8f

(5.4.10)

The equation (5.4.8) is a thin profiles-type equation (C.1.1) and it has one of the solutions (C.1.9), (C.1.10), (C.1.11) or (C.1.14). The solution of the equation (5.4.8) must satisfy the condition

J

+OFdq=0 y_ a!

(5.4.11)

arising from (5.4.7). The only solution satisfying this solution is (C.1.11)

with C = 0. Hence, 8F(x,TI)

-2

1

(t-y-)(y+-t)h=(x,t)dt. t-77 (5.4.12)

192

THE LIFTING SURFACE THEORY

Integrating this solution on the interval (y_, y) and taking into account (5.4.7) and (B.6.11), we get F(x, y)

_

'

dq

2ly

(n-y-)(y+-q)

-

+

(t - V -)(y+ -' t)

t

y-

(x, t)dt .

-q

(5.4.13)

As from (5.4.5) it results f (x, y) = F, (x, y), we deduce f (x, y) =

=2

8

8x

ydq

ru+

(t -

(qy-)(y+-q) y-

ly-

t)

t-q

(x, t)dt . (5.4.14)

Employing (B.6.11) ones easily check that this solution satisfies the vanishing condition on the leading edge

f(x,y-)=f(x,y+)=0. 5.4.2

(5.4.15)

The Case h = h(x)

In case that

h does not depend on the variable

y, taking into

account (B.5.6) and performing the change of variable q = y_ cos20 + y+ sin20, we get

: Jv-

0 < 0 < 7r/2,

(5.4.16)

' V+

dt7

At - Y-) (y+ - t)dt = t-q n - y-)(y+ - q) Iv_7r

Y+-Y- r 2

-a J

y+

v-

dq

On - y-)(y+ - r!) ydq=sV(y-y-)(y+- y)

Hence,

f(x, y) = 2

[h'/i_

y)]

.

(5.4.17)

193

THE WINO OF LOW ASPECT RATdO

Utilizing (5.1.54)-(5.1.56) and Green's formula we deduce: CL

- A If

19

[W(x)Vv - v-}(v+ - Y1

f (y-a (b-y)dy=-2Ah'(1)(b-a)2,

A

-S

= - A ,,D ybx

= - Ah'(1)1

cv

A

v=

Y

[h!(x)-,/(y -

v-)(v+ - v)] dxdy =

(v - a)(b - y)dy = -

bWrh'(l)(b

- a)z,

I/I; [zWv'?j- v-)(v+ - v)1 dx dy_ A

If h'(x) vir(v -y-)(Yu)dxdy = -

_ -CL -

1

0

[v+(z) - y_(x)12 h'(x)dx, (5.4.18)

the reference chord ao bung 1. For example, fnr the rectangular flat plate of incidence -e with the span 2b one obtains CL = wk,

c, = -

c= = 0,

!

,

(5.4.19)

and for the triangular flat plate of incidence -£, we have,

CL=xe(b - a), because y+ = bx,

c, = (b2 - a')

r

,

cy = - 3 xc(b - a),

(5.4.20)

y- sax.

5.4.3 The General Case We shall write (5.4.14) as follows

f(x, y) =

2b x8s

r_

(t - -y-)(y+ - t)h.(x, t)I(y, t)dt,

(5.4.21)

THE LIFTING SURFACE THEORY

194

where

Y

d-q

I(y,t) =

(5.4.22)

(r1-y-)(y+-rl) t - n with y- < y < y+, y- < t < y+. This integral may be calculated l

fv-

explicitly.

For y < t the integral is not singular.

For

t < y the

integral is singular but noticing that we have 1(y+, t) = 0 (it results from (B.5.8)) we get IY+ 1

I(y,t)

d'l

y+-rl) t-17

(5.4.23)

which also is not a singular integral. Since

A

(n-a)(b-n) t-11 (5.4.24)

(t-a)(b-rt)+ (b-t)(q -a)

1

in

(t - a(b-it)- -v(((r1-a)

(t - a)(b - t)

we deduce the fundamental formula

f(x,y) = =

2 8 f+h

(t-y-)(y+-y)+ (y+-t)(y-y-) dt.

(x,t)

a 8x v-

(5.4.25)

For determining the lift and moment coefficients we have to calculate the integral

(t

V+

G(a, t) =

- y-)(y+ - y) +

(y+ - t)(y - y-) dy.

J(t-y-)(y+-y)- V(-y+---4(y -y-)

(5.4.26)

Performing the changes of variables

t=y++y-+y+-y-moo, y=y++y-+y+-y-cos9, 2

2

2

2

(5.4.27)

we deduce

Gxt

- y+ 2 y- fo X in

sm2 sm

B+a 2

sin 8 d9,

(5.4.28)

195

THE WINC OF LOW ASPECT RATIO

the integral having an integrable singularity for 8 = a. The integrand from (5.4.28) is the kernel from the equation (6.2.15). Using for 0 91 a the expansion (6.2.17), we get: G(x, t) = w y+

y- sin o =

2

(y+

- t)(t - y-) .

(5.4.29)

It follows

(t - a(b-y)+ (b-t)(y-a)

fb

G(I, t)

=J

In

Ja

IWa)(b-y)-'(b-t)(y-a)

(5.4.30)

=x (b - t(t-a). Similarly one deduces b

(t - a) (b - y) +

JQ y

(b - t) (y - a)

dy=

IV'(t-a)(b-y)-i(b-t)(y-a)

(5.4.31)

=att+b2a) (b-t)(t-a). Utilizing Green's formula and the expression (5.4.25) for f (z, y) taking into account (5.4.30), (5.4.31), we get:

A

CL

-

-4

,

lb 41

h`(1, t)G(1, t)d t = rb

a

_

b

cs = - AJa t (b - t)(t - a)h=(1, t)d t + !-+-b ct, Cy = -CL -

r1

4

rA

Ju

4

r2

- -CL - A

dx

y+

y_

h.(x, t)G(x, t)d t =

ry+

dT o

f

(5.4.32)

hs(z, t)

(Y+ - Wt - y-)d t .

When h(x, t) does not depend on t we find again the formulas (5.4.18).

196

THE LIFTING SURFACE THEORY

This problem, for wings symmetric with respect to the Ox-axis, was studied in [5.171. It was also included in [1.1], [1.2]. The general problem for the arbitrary wing as it was presented herein, was solved in 15.91.

Chapter 6 The Lifting Line Theory

6.1 6.1.1

Prandtl's Theory The Lifting Line Hypotheses. The Velocity Field

Prandtl's theory is the first mathematical model for the three - dimensional wing (the finite span airfoil). It was elaborated in 1918 16.211 and it remained until the years '40 the only theory for this wing. The german

scientist, gifted with an extraordinary engineering intuition, guessed very well the simplifications which may be performed. Prandtl's method consists in replacing the wing with a distribution of vortices on its plane

- form (the domain D from 5.1). Since the experience indicates that downstream the wing the flow is not potential, Prandtl introduced a vortical distribution defined on S, the trace of the plane-form D in the uniform stream (figure 6.1.1), the velocity field in the fluid being determined by the two distributions. The vortices on D are called tied vortices and the vortices on S are called fee vortices. This idea continued to dominate the aerodynamics, the models concerning the subsonic, supersonic and transonic, steady or oscillatory flows, elaborated in the years '50, '60, being conceived on the basis of this method. In 1975 D. Homentcovschi, utilizing the theory of distributions proved that the hypothesis of the existence of free vortices is not necessary because it follows from equations. L. Dragon [5.71 obtained

the same result utilizing the method of the fundamental solutions. In this subsection, we deduce the lifting line theory from the lifting surface theory, as we proceeded in [5.7], utilizing the following three hypotheses (the hypotheses of Prandtl's theory): 10 One neglects the thickness of the wing, therefore in (5.1.2) one considers hl = 0. From (5.1.27) it results fl = 0, such that the representations (5.1.8), (5.1.11) and (5.1.12) give

P(x,y,z) = -4 7r-

IL f(CTT)1 (-) dcdi 19Z

(6.1.1)

THE LIFTING LINE THEORY

198

u(x,y,z) _ -p(x,y,z) v(x,y,z)

_-

+

jo

z2(1+Rl)]dfdi7

(6.1.3)

)Io

qJ

w(x, y, z)

(6.1.2)

fD f (C, 17)

8x 1)dt d t7+

JJD0Y

(6.1.4) -+ z2

(1 + 11 )]d t d n;

2° One considers that the unknown is the circulation C. along the contour c, resulting from the intersection of the wing with an arbitrary

plane n parallel to the xOz plane at the distance y(-b < y < b) (fig. 6.1.1). This will be, obviously, a function of y

Fig. 6.1.1.

Pudx,

C(y)=

because on this contour d y = 0 and wdz = O(E2) (it results from (2.1.17) and (2.1.21) ). Taking into account (6.1.2) and (5.1.20), we obtain

r C(y) =

Jx

u(x, y, -0)d x + / + u(x, y, +0)d x x

We have also

f X

f(x,

y)d x .

(u)

(6. 1 .5)

x+ (y)

C(n) = -

x_ (y)

f (f, n)d t.

(6.1.6)

PRANDTL'S THEORY

199

From (5.1.1) or (5.1.30) it results

C(±b) = 0;

(6.1.7)

30 The domain D, i.e. the projection of the wing on the xOy plane, is replaced by the segment [-b, +bJ (fig. 6.7.1); for this reason we call this theory the lifting line theory. when For studying the behaviour of the integrals (6.1.1)-(6.1.4) x_(n) -+ 0 .- x+(il), we notice that in the vicinity of this segment, for a given n , the function f (e, n) keeps a constant sign, such that we may apply the mean formula. For a function h(z, y, z, n), continuous in f, when x- (q) -+ 04-- x+(n), we have therefore f ( n)k(x, y, z,

limJ1

r+b

hm

ti)d d r! _

r+(n)

J-b [ Js_ty)

-lim

,

f (t, n)k(x, y, z, t, r1)d

d r! +b

-b

k(x, y, z, (', n)C(n)d n = - fb k(-T, y, z, 0, n)C(n)d n , (6.1.8)

where £ E (x_(rl), x+(n)). Applying this formula in (6.1.1)-(6.1.4) and taking into account (6.1.7), we obtain

2

P (x,

y, z) = -

4 rb C(n)

d n = -u(x, y, z) ,

f+b

v(x, y, z) = 4

.! b

_- a 4n

w(x,y,z) =-Q2 41r

ly2+z2

J b C('l)

f

1

)]dii

C(17)ay [ o + z2 (1 +

= rT

o

(6.1.9)

C'(n)yo+z2 ( 1 + C(n)

rb

-4 J b

Ro

}n, d

drl-

b

0,07) y02

YO

Z2

(1 + X

d

200

THE LIFTING LINE THEORY

where we denoted RD =

x2 +#(y2 + z2)

(6.1.10);

Fbr Q = 1 we obtain Prandtl's representation ((1.21], p. 708).

6.1.2 Prandtl's Equation In the sequel we shall deduce the equation for C(y). This will obviously result from the lifting surface equation by virtue of the above hypotheses. We shall employ the equation (5.1.33). Taking (6.1.6) and (D.3.6) into account it results

8

f( , n) d

8y JD y-n 1

fd n= dy -d- f' +b

C(q)

-b n-y

-

d

rJ

C(q) d n.

O

-b (n - y)2

Utilizing (D.3.7) and (6.1.7), we also deduce:

.

&

c(_)

= l1+b C'(n)

J

b

'n-yd

(6.1.12)

In the second term from (5.1.33) we take into account that when

x_(n) - 0 4- z+(n), on the greatest part of the domain D, we have so << 14, before E which intervenes in the definition of the principal value of the integral with respect to n becomes zero, i.e. before yo becomes zero. In the principal value we shall perform therefore the approximation R = fl I. An exact evaluation of this approximation is, made by D. Homentoovschi in [6.9) as follows

jjf(r)

ryxTo +

dd

XDYO

Taking into account that sign yo = 2H(yo) -1,

H'(yo) = 6(y - q),

H representing Heaviside's function and 6 Dirac's distribution we

201

PRANDTL'S THEORY

have: Z_

JJ

20

f (f, n) xovo +b

I J-b

JJ

d {d ~ Q f (txon) t

LJ

d C,

f (on) s yod fd n =

' f=+ (v) f (t'17) _ d

5(y - n)d n= 2Q

xo

(y)

(6.1.13)

Hence the equation (5.1.33) becomes: 1

+(v)

,rte C'(n) d n +

sr f b n - y

f (c Y) d

'r s_(v)

xo

= 2h' (a, y)

(6.1.14)

x]-1/2

(6.1.15)

Multiplying this equation by [x - x-(y)]11.2[x+(y)

-

and integrating it with respect to x on the interval obtain from the first formula (B.5.4): AC(y) =

(x-

a(2) 'f+b C'(n)dn+1(y),

J bn-y

(y), x+(y)), we

(6.1.16)

where we denoted

a(y) =

J(y) _ -2

W

x+ (y) - x- (y)

(6.1.17)

2

IT

x(6.1.18)

a(y) representing a half of the chord of the profile at the distance y. The equation (6.1.16) is the well known Prandtt's equation (the lifting line equation). This equation, together with the conditions (6.1.7), has to determine the unknown C(y). It is an integro-diferential singular equation, with a Cauchy-type singularity. Utilizing (6.1.12), the equation

for C(y) may be written QC(y) =

a 2y)

t +b

r

b

(n

+ (ny)2 d n

9(y) .

(6.1.19)

In this form, the equation is not integro-differential any longer, it is only integral, but with a stronger singularity. The mark "s" is for the Finite Part. As we have already shown in [6.5] [6.6), this form is more adequate for the numerical integrations (see also 6.4.2, 6.4.3).

202

THE LIFTING LINE THEORY

z

A

Fig. 6.1.2.

The significance of the function j (6.1.18) is given in [1.2J. In the case of the flat plates having the angle of attack c (fig. 6.1.2) we have

hx = -e whence (6.1.20)

j(y) = 2irea(y).

6.1.3

The Aerodynamic Action

Taking into account (6.1.6), from (5.1.54) and (5.1.56) we deduce 2 CL

2

+b

C(y)d y, c

+b

Aao 1-b

yC(b)d y, cv = 0.

(6.1.21)

The expression of c, is natural because for the wing reduced to the segment [-b, +b] the fluid cannot create a moment which should rotate this axis. By virtue of the first hypothesis (h1 = 0), in the framework of the lifting surface theory, the quantities Rs and M, which have the

order of magnitude 0(e2) are reduced to those given in (5.1.56) and (5.1.59). Utilizing (6.1.6) and (6.1.7), we obtain:

R. = -p.U.Lo

C(y)h=(0, y)d y, b

(6.1.22)

+b

M: = P,,. U,2,. L. f

yC(y)hi(0, y)d y b

But, from the boundary condition

w(x,y,0) = h,(x,y)

6.1.23)

and from (6.1.9) it results h=(0, y) = w(0, y, 0) _ -

1

lim (

41r c-.o \

f

V-s b

+

b y+c

)

C'M d Yo

rl ,

203

PRANDTL'S THEORY

because, with the change of variable n - y = u, we have y+E

Eiti y-E

C(rl)' 11o

+-Z 2 d r! _ -C'(y) lim E-0

f

+6

z = 0.

1 y+bCl(I1)d b

tl

y

rJ ,

( 6.1.24)

we deduce 2 CU

f

A J-b

C(y)w(y)d y, (6.1.25)

2

MS cZ

A

(1/2)Poo

`k

2 Alai

Aa_

Jb yC(y)w(y)d y,

representing the area of the domain D and ao, the length of

the dimensionless medium chord along the direction of the unperturbed stream. In fact, for w we may also utilize the expression

w(y) = QC(y) - ?(y)

(6.1.26)

2wa(y)

which results from (6.1.18) and (6.1.26).

6.1.4

The Elliptical Flat Plate

Until the appearance of the papers of Magnaradze [6.16] and Vekua [6.28] in the years 1942, 1945 it was available only one exact solution of Prandtl's equation corresponding to the elliptical flat plate wing. Usu-

ally this solution is obtained as an answer to the following minimum problem: "To determine among the wings having the same lift, the wing corresponding to the minimum drag" (see, for example, (1.201). Sometimes one utilizes Glauert's approximate method (see 6.2.4). In the sequel we shall present a simple method which does not need any special considerations on Prandtl's equation. Let us consider a flat plate with the angle of attack E, whose projection on the x10y1 plane is an ellipse having the semi-axes Lo and bLo (figure 6.1.3). In dimensionless coordinates, the equation of the ellipse is x2+ +y2/b2 = 1. For the edges and the chord it results y2

xf = f

1 -

2 52

,

a(y) =

1 -

2'

(6.1.27)

204

THE LIFTING LINE THEORY

YI

bLo

L=am

Fig. 6.1.3.

For a flat plate j(y) has the form (6.1.20) hence it is proportional to a(y). From Prandtl's equation it results that C(y) is also proportional to a(y). We shall look therefore for solutions having the form:

C(y) = k 1-

a Eb2

,

(6.1.28)

k being a constant which will be determined by imposing (6.1.23) to verify (6.1.27). Using the change of variables q = boos 0, y = b cos o, and Glauert's formulas (B.6.6), it results: di7

11-y such that

k

(2TE

A + 7r/2b)

_ir,

(6.1.29)

(6.1.30

Since the area of the ellipse is w b, from the formulas (6.1.23) and (6.1.27) we obtain s cL = k, CD = , cj = cs = 0. (6.1.31) 46

Obviously, cD = 0(0). For b - oo one obtains the infinite span flat plate. From (6.1.23) and (6.1.27) it results: CL =

21r8

(6.1.32)

205

THE INTEGRATION OF PRANDTL'S EQUATION

6.2 The Theory of Integration of Prandtl's Equation. The

Reduction to Fredholm-Type Integral Equations 6.2.1

The Equation of Trefftz and Schmidt

The general method of solving the integro-differential equations con-

sists in reducing them to Fredholm-type integral equations. As it is well known, for the last ones a general theory is available (existence and

uniqueness theorems, exact and approximate methods for solving the equation). The first investigation of Prandtl's equation was performed by Trefftz in 1921. He reduced the problem of solving Prandtl's equation to the problem of determination of a harmonic function in the superior half-plane with mixed conditions on the boundary. We shall prove in the sequel that such a problem may be reduced to a Fredholm-type equation. To this aim we shall consider the harmonic function U(y, z) in the superior half-plane z > 0 from the yOz plane, with mixed boundary conditions in the Z = y + iz complex plane (fig. 6.2.1). We shall also consider the complex function

F(Z) = U(y, z) + iV(y, z) =

1

2m

: ZdR. Jb+1 COO

(6.2.1)

z (Z)

-b

+b

y

Fig. 6.2.1.

Obviously this is an holomorphic function in the (Z) plane with the cut (-b, +b) and it vanishes to infinity. U(y, z) is therefore a harmonic function in the half-plane z > 0. It results obviously

U(y, +0) = 0, y E (-oo, -b) U (b, oo).

(6.2.2)

The limit values on the segment (-b, +b) are obtained by means of Plemelj's formulas. We deduce

2U(y, +0) = C(y), y E (-b, +b).

(6.2.3).

206

THE LIFTINC LINE THEORY

Taking the conditions (6.1.7) into account, we get d

-dn= f

C(t1)

ZJ

bbtj

b

=-f C(rl)'3;

rl

b

C(rl)aZ

Z)dr)_ Cn

1 Z) d+l -

f

b 71- Zdrl

Deriving we deduce from (6.2.1):

F'(Z) =

I-i

8 = 2m J-b W(Zd q

(6.2.4)

and with Plemelj's formulas ' +b

az (y, +0) = d (y, +0) = 2a f

d,

(6.2.5)

n representing the inward pointing normal of the superior half-plane on

the segment (-b, +b) of the real axis. Taking (6.2.3) and (6.2.5) into account, Prandtl's equation is transformed into the following condition on the segment (-b, +b): dU

21U(y, +o) A(y)

- J(y),

(6.2.6)

A(y) = ra(y), A(y)J(y) =1(y) ,

(6.2.7)

in

(y, +0) =

where, from now on, we denote

So, Prandtl's equation was replaced by the following mixed boundary value problem: "to determine the harmonic function U(y, z) in the z > 0 half-plane, vanishing at infinity and satisfying the boundary conditions (6.2.2) and (6.2.6)". Then, the function C(y) will result from (6.2.3). The mixed problem is reduced to a Fredholm-type integral equation. Proceeding like in [1.211, p. 713, we notice that because of the condition (6.2.2), the function F(Z) may be extended by symmetry in the lower half-plane z < 0. In this way, on the lower margin of the cut (-b, +b1 we shall have the condition dU

d n (y, -0) =

2QU(y, -0) + Ay), A(y)

(6.2.8)

So, we reduced the above mixed problem to the problem of determination of the harmonic function U(y, z), in the yOz plane, with the cut

THE INTEGRATION OF PRANDTL'S EQUATION

207

[-b, +b) and the conditions (6.2.6) and (6.2.8) on the two margins of

the cut. It is known (see for example, 11.111 [1.201 [1.31J) that the Joukovsky-

type conformal mapping Z - W: Z = I

(W + W

(6.2.9)

maps the exterior of the cut [-b, +b] from the (Z) plane onto the exterior of the circle of radius b and the center in origin from the (W) plane (fig. 6.2.2), the superior margin of the cut being mapped on the

superior half-circle from the superior half-plane. The points fb are double and singular. One obtains the correspondence of the boundaries

putting W = be''. We have

y=boost, a=0.

(6.2.10)

Fig. 6.2.2.

For 0 < a < r, (6.2.10) gives the correspondence between the halfcircle r+ and the superior margin of the cut, and for -W < a < 0, the correspondence between r- and the inferior margin (fig. 6.2.2). Since from the extension by symmetry it results U(y, +0) = -U(y, -0), we deduce that U(a) is an odd function on F. With the same application (6.2.10), the functions A(y) and j(y) become even functions. We denote them by A(a) , respectively j(o). Now we shall see how the boundary conditions (6.2.6) and (6.2.8) are transformed. To this aim we remind to the reader that, after performing a conformal mapping,

208

THE LIFTING LINE THEORY

the ratio of the lengths is given by the modulus of the derivative. More

precisely, let W = f (Z) be a conformal mapping and let M(Z) and N(Z+AZ) be two neighboring points and M1(W) and N1(W+AW) their images. Obviously we have AMN' lim I

1

= lim I1Z 1 = lim

I

AZ = If'(Z)1

Hence, returning to our problem and denoting by N the outward normal to 1', we shall have

dU_dU dN_dUIdW do

dN (a) do

TN- d

Taking (6.2.10) into account, the boundary conditions (6.2.6) and

(6.2.8) on the two half - circles r+ and r_ give dU (a) = 2/3I sin a)

dN

A(a)

U(a) - J(a) sin a

(6.2.12)

Denoting W = it + i v, the harmonic function U(y, z) in the yOz plane becomes the harmonic function U(u, v) in the exterior of the circle r. This one vanishes at infinity and has the normal derivative (6.2.12) known on 1'. It is a Neumann problem. Its solution is given by Dini's formula (see for example (1.20] p. 31). We obtain

U(u,v)=n J_" U(a)Inlbei°-Wlda+ko dN

(6.2.13)

being an unknown constant. Considering that W tends to a point be' ° from C, one obtains U(s) =

b

r+ir J JJJ

aN (o )1n I2 sin s

2 a I da + kO .

(6.2.14)

:

Since U(a) is an odd function it results dN (a)

dN

(-a) ,

and integrating only on the interval (0, 7r), we deduce U(s)

b f0" ddN (a)S(s, a)d a,

7r

(6.2.15)

THE IN'I'ECRATION OF PRANDTL'S EQUATION

209

where we denoted S(s, o) = In

(6.2.16) am

2

We have (see, for example, 11.16]): 00

S(s, o) _ -2 F sin ks sin ko km1

(6.2.17)

k

the series being absolutely and uniformly convergent (s 0 o). In (6.2.15) we did not encounter ke, because we have U(s) _

_ -U(-s) whence U(0) = 0. Taking into account the relation (6.2.12) in which 2U(o) is replaced by C(a) according to the condition (6.2.3), one obtains the following Fredholin-type integral equation C(S) = A J

C(°)

si) S(s, a)d or + Jo(a)

,

(6.2.18)

where A=

2bQ

z

Jo(s)

,

2b

x

J(o)S(s, o') sin o d o .

(6.2.19)

To

Hence we reduced Prandtl's integro-differential equation to the Fredhoimtype equation (6.2.18). The kernel of this equation has an integrable singularity. Equivalent integral equations have been given by Betz and Gebelein in 1936 and Trefftz in 1938.

6.2.2

Existence and Uniqueness Theorems

For proving the existence and uniqueness of the solution of Prandtl's equation , we shall use the first theorem of Fredholm. This may be enunciated as follows: the equation b i

P(s) = A

K(s,)p(o)ds + f(s),

(6.2.20)

fa has an unique solution for a given value of A and for every fee term f if and only if the corresponding homogeneous equation admits only the trivial solution V(s) = 0. Hence, we must show that the equation

210

THE LIFTING LINE THEORY

(6.2.18) which is homogeneous (Jo = 0) has only the trivial solution. But the homogeneous equation corresponds to the boundary problem (6.2.12) which is homogeneous (J = 0). Applying Green's formula

/ JD

U)2du dv = -JOD Ud S,

(6.2.21)

where D is an annulus, exterior to the circle r, bounded by an concentric circle of radius R > b and observing that for R -- oo the last term vanishes (see, for example, §6.1 from

1 fx A

we deduce

UdNda < 0.

Utilizing the homogeneous condition (6.2.12), we obtain: W IAsin

0.

(6.2.22)

This inequality implies U = 0, because, as it results from the definition (6.1.17), we have A > 0. Hence, Prandtl's equation has an unique solution. This result is very important, because, as we have already seen in 6.1.4 and as we shall see in the sequel, we manage, on various ways, to determine a solution of this equations. The above result ensures that if we find a solution, this is the unique solution of the equation.

6.2.3

Foundation of Glauert's Method

The integral equation (6.2.20) is a Fredholm-type equation of the second kind. This equation has a symmetric kernel if

K(s, a) = K(a, a).

(6.2.23)

The equation (6.2.18) has not a symmetric kernel, but it can be symmetrized. Indeed, multiplying the equation with

(on the inte-

A) gration interval the quantity under the radical is positive) and, taking the function as an unknown c(8) =

sins C(s) A(s)

(6.2.24)

211

THE INTEGRATION OF PRANDTL'S EQUATION

one obtains the following equation:

c(s) _ -2a

Ja

c(a)

VsTn,; i "in a A

00

sink sin ka

a ()k_i

d a + F;i(S) A(s) (6.2.25)

00

K=-2E k i1

sin s smo sin ressin rca k A(s) A(a)

is obviously symmetric. The kernel is even degenerate, but not of finite rank. As it is known (see, for example, [1.22], vol.3, p.193), the integral equations with degenerate kernel may be reduced to infinite algebraic systems and the equations with degenerate kernel of finite rank may be reduced to linear algebraic systems with a finite number of equations. We are not in this situation, and according to (6.2.25)we shall take only the property of symmetry of the kernel into account. According to the theory of Hilbert and Schmidt (see, for example, (1.22] v. 3, p. 243) the solution of the integral equation may be expanded, with respect to the eigenfunctions of the kernel, into absolutely and uniformly convergent series. Hence the solution of the equation (6.2.25) has the form: 00

C(a) =

A(a E At, sin ka. )k-1

Taking (6.2.24) into account, it results: 00

C(a) _ E Ak sin ka.

(6.2.26)

k=1

The (constant) coefficients Ak will be determined replacing (6.2.26) in the equation (6.2.18), or easier, performing this replacement in Prandtl' equation (6.1.18), which, with the change of variables y = b cos s,

(6.2.27)

1 ) = b cos a

and with the notation C(s) for the function C(y) composed with (6.2.27)1 etc., becomes 2b/3C(s + a(s)

C'(a)d a=

casa-cuss

2itbaOff( s s

)

(

6.2.28 )

Before discussing about how to determine the coefficients A, from (6.2.26) and (6.2.27), we must notice that the form (6.2.26) of the so-

lution C(a) may result directly from (6.2.18), without utilizing the

212

THE LIFTING LINE THEORY

theory of Hilbert and Schmidt. Indeed, taking (6.2.17) into account, the equation (6.2.18) becomes: _'inks

C(s) = -2a

fo"

k=1

A(( )) in k sin a d a+

V +4rb

sin ks

Air)

sin ko sin od a .

I0 The integrals are constants and the solution will have the form (6.2.26).

6.2.4

Glauert'e Approximation

We shall return now to the problem of determination of the coefficients Ai from (6.2.26). Replacing C from (6.2.26) in (6.2.28) and using Glauert's formula (B.6.6) (herein is the origin of this formula), we deduce: E Ak[2bfl sins + k7ra(s)) sin ks = 21rba(s) j (s) sins .

(6.2.29)

k=1

Glauert's approximation consists in keeping the first n terms from the expansion (6.2.26) and then imposing (6.2.29) to be satisfied for n distinct values of the variable s. The coefficients Ak are the solution of an linear algebraic system, but we cannot evaluate the error of the approximation. Many other approximations have been given in the literature (see Lotz in [1.24), Carafoli in [1.5)).

6.2.5

The Minimal Drag Airfoil

The foundation of Glauert's method,which consists in establishing the formula (6.2.26) gives the possibility to give an answer to the following problem of practical interest: to determine among the wings with the same lift, that one which has the minimum drag. In view of this determination we shall calculate, utilizing the formulas (6.1.21), (6.1.25) and (6.2.26), the lift and drag coefficients cL and CD. For determining the lift and the drag, we multiply these coefficients by the same factor 1 2 p00UUA1. Since

2! sin ka sin lad v = irdkl ,

k,1=1,2,...

213

THE INTEGRATION OF PRANDTL'S EQUATION

we deduce +b

f Jb

ct, = A

C(y)d y = A f C(a) sin ado, = JO

'At v.

(6.2.30)

Utilizing Glauert's formula (B.6.6), we obtain:

()

C'(s)s a

1

1

4ab To Cos a - oos or

such that: CD

/

2

C(V)w(y)d, y = -

b

A °°

1

2A

1 (ksmka) ksI

f

4b

sin ka kAi sin a

w(a)C(a) sin ad a =

o0

Alsinio do =

(6.2.31)

!:1

oc

kAj. k=t

The formulas (6.2.30) and (6.2.31) indicate that among all the wings with the same lift (with the same A1), the minimum drag corresponds

to the wings for which A2 = A3 = ... = 0. The solution of Prandtl's equation for these wings is

C(o) =A1sin or =C(y)=At V, where Al = C(O) may be determined obviously from the equation (6.1.26). We have _C(_)

A(N)

5

= 2w(y) + J(y) _ - 1 At + J(y).

(6.2.33)

In the case of the flat plate, j has the form (6.1.20). It results J = 2e whence we deduce that the member from the left hand side of (6.2.33) is constant. Hence,

a(y) = ao 1 -

,

(6.2.34)

the constant ao being determined by the relation

0

+

2b) Al = 2E,

(6.2.35)

214

THE LIFTING LINE THEORY

if one gives Al. The same relation determines Al if one gives ao.

For example, when Al = k like in (6.1.28), it results ao = 1, like in (6.1.27) and vice versa. The expression (6.2.33) shows that the wings which have the above property are the elliptical flat plates.

6.3

The Symmetrical Wing. Vekua's Equation. A Larger Class of Exact Solutions

6.3.1

Symmetry Properties

Very often in aerodynamics we encounter the case when the wing is symmetric with respect to the xOz plane. In this situation we have

x*(y) = xt (-y) ,

-b < y S +b

h(x, y) = h(x, -y) ,

(6.3.1)

From (6.1.19) and (6.1.20) it results

a(y) = a(-y),

j(y) = j(-y).

(6.3.2)

Let us prove that we also have

C(y) = C(-y) .

(6.3.3)

Indeed, changing in Prandtl's equation (6.1.18) y by -y and taking (6.3.2) into account, it results

AC(-y) =

a2 y)

1c (q) d q +.7(y) . Cc(q)

n

(6.3.4)

y

Putting in the integral n = -u and observing that

C(q)d17 = dC = C'(u)du, we deduce r -16 C'(rl)d

f b 1+31

_

'/'_b

C(u)du =,+b Cl(,i)d+l

J.fb -u+y

b

7I-y

(6.3.5)

Introducing this relation in (6.3.4) and comparing with (6.1.18), we get (6.3.3).

THE SYMMETIUC'AL WING. VEKt'A'S EQUATION

215

The Integral Equation

6.3.2

We shall present in the sequel the simplest method for obtaining the equation (6.2.18). The demonstration is inspired from [A.27), where, on his turn, it was taken from Magnaradze (6.16] and Vekua (6.28). With the notations (6.2.7) Prandtl's equation is I 'r+b

C"(n) d

27r ,!-b rl - y

n = 0C(y) - J(y) .

(6.3.6)

A(y)

For the existence of the principal value we have to assume that C'(y) satisfies Holder's condition on the segment (-b, +b). We shall invert this equation assuming that the right hand member is known. As it is known from (C.1.1) the solution C'(y) depends on the behaviour imposed in the points ±b. We know that we cannot obtain a bounded solution in the two points without imposing a restriction to the right hand member. In the same time, because of the symmetry of the wing,

we cannot consider C bounded only in an extremity. Hence C' is unbounded in the two extremities, i.e. the solution has the form (C'.1.11). Further, for inverting the equation (6.3.6), A(y) and J(y) have to satisfy Holder's condition on [-b, +b]. If A(y) and hs(x, y) (with respect to the y variable) have this property we deduce the same thing for J(y) . Moreover, a(y) must not vanish on (-b,+b). If all these conditions are satisfied, then, using the formula (C.1.11), we obtain

C'([/) _

-2

'r+b

1

b2y2

b

bz -

I' [#'(") - J(n)] d n+ n-y A(q) (6.3.7)

B representing a constant which has to be determined. It is zero because from (6.3.3) we have C'(y) = -C'(-y), whence C'(0) = 0. Imposing this in (6.3.7) and observing that the integrand is an odd function, it results the assertion. Utilizing now the identity

P) --r12 d dy

In

-y2- 62-n2 i(y-n)+ i(y-n)+ b2-y2+ P-

(6.3.8)

216

THE LIFTING LINE THEORY

and integrating (6.3.7) on the interval (-b, y), from (6.3.3) one obtains:

C(y) _

+b

2 W

1

110A(+l)

b

(6.3.9)

i(y-tl)+ &2y2i(y-q)+ b -y + /bbl-

-J(17)] In

because the modulus is equal to the unity for y = -b. Performing the change of variable y = b cos s ,

tI = b cos a

(6.3.10)

and taking into account that in

-y2-

(y-tl)+

(y-17)+ b2-y2+ b2-q2

is - e is = In l e-i s _ eio I

e

- S($ a)

we obtain obviously the equation (6.2.18).

Using the notation A(tr-a) we have A(bcoe(7r-a)) = A(-boos a) _ = A(-y). Hence, taking (6.3.2) and (6.3.3) into account, it results A(tr - a) = A(a),

J(tr - a) _ i(a),

C(tr - a) = C(a)

(6.3.11)

whence:

sin ad a =

I"/2 [#A(a) - J(a)J In

s+a --/z

pc(o') - J(a)J In

o

cos cos

2 s - a sin ad a . 2

But, .

sin In

sin

$-a 2 s+a 2

cos

s+a

2 s-a cos 2

= In

sins - sin or sin s+sin or

THE SYMMETRICAL WING. VEKUA'S EQUATION

217

In this way, the equation (6.2.18) for the symmetric wing becomes 26

C(s) _

J"/2 [(;) (6.3.12)

- sing -J(a) In sins sinada, sins + sin a J

with s in the interval (0, n/2).

Vekua's Equation

6.3.3

In 1945, I.N.Vekua [6.28] gave for the symmetric profile whose chord

has the form bz

a(y) =

2

-y

with p(y) = p(-y) > 0,

,

P(y)

(6.3.13)

(where p(y) is an analytic function on [-b, +b]), a Fredholm-type integral equation which has the great advantage that it may be integrated exactly for a large class of profiles. Vekua's method was extended immediately by Magnaradze [6.16] to wings for which the function p(y) is not necessarily analytic on the interval f-b,+b]. Since we had not the occasion to read this papers, we present herein a a synthesis due to Muschelisvili [A.27]. To this aim, we write the equation (6.3.7), where we considered 13 = 0, as follows 20

A(y) C'(y) +

+b

C(11 f-b ' '

1

(6.3.14)

= A(y)J1(y) -

tb

a(y) b2 - y2

R(y,n)C(rl)d>j b

where 2(3

R(y, rl) =

Ji (y) =

1

(i-r a(n)

7r 17 - y 2

-

_

a(R-7-ill

y)

(6.3.15)

Ifb +b

62 - y2

J(q)d i1.

(6.3.16)

Ji (y) _ -J1(-y) .

(6.3.17)

rl - y

Obviously we have:

R(y, ) _ -R(-y, -y)

THE LIFTING LINE THEORY

218

Further we shall assume the continuity of the first order derivative of the function

P(y) =

(6.3.18)

a(y)

In this case, R(y, i) will be a continuous function. Since according to (6.1.7) we have: C(q

ay i-bb

r1

11

=

dq =

o d y \Jb ` + Jy+a, v

11

tab

do _

J-b

17

-

dn,

from (6.3.14) it results

dy [A(y)C'(y)] +

2Q ' +b

J-b rl (y d q

B(y),

(6.3.19)

where

B(y) =

dy [A()Ji() -

lr+b

a(y)

J

R(y, n)C(r1)d n I

.

(6.3.20)

Obviously,

B(y) = B(-y)

(6.3.21)

Eliminating the integral from (6.3.19) by means of Prandtl's equation, we obtain the following differential equation: A(y)

b [A(y)C'(y)] + 4/32C(y) = A(y) [B(y) + 4i3J(y)I .

(6.3.22)

Assuming that the right hand member is known, we have in (6.3.22) a differential linear equation for C(y). The homogeneous equation has the linear independent solutions cos s(y), sin s(y), where 20

8(y) =

(6.3.23)

Jo a(rl)

Utilizing Lagrange's method of variation of constants, we deduce that the equation (6.3.22) has the following solution: C(y) = Co cos s(y) + Cl sin s(y)+

y sin [s(y) + 2Q f [B('l) + 4QJ(n)]

- s(n)]d n,

(6.3.24)

THE SYMMETRICAL WING. VEKUA'S EQUATION

219

Co and C, being constants. Obviously, Co = C(0). Calculating C(-y), taking into account that s(y) is an odd function (its derivative is an even function) and B(q) and J(n) are odd functions, and imposing (6.3.3), it results C1 = 0. Introducing B given by (6.3.20) in (6.3.24), performing an integration by parts and observing that the integrated term is zero because

JI(0) = 0,

1-b

it results the following integral equation: +1

C(y)

K(y, ii) =

a

f

+b

(6.3.25)

K(y, q)C(t7)d n= g(y) ,

R(qj, q)

cos [s(y) - s(rh)]d nl ,

(6.3.26)

To

+2 10yJ

g(y)=Cocoss

sins

s

d+ (6.3.27)

+ [iivi)ccs[a(v) - s(n)1 d q.

The equation (6.3.25) for J9 = 1 is the equation given by Vekua and Magnaradze. Unlike the equation of Trefftz (6.2.18), this is regular (the kernel has no singularity). Moreover, in case that the function p(y) given by (6.3.18) is a rational function, more precisely in case that a(y) has the form:

a(y)=a

0

-y21+ply2+...+pny2n

9

1+qly +...+gny2"

(6.3.28)

as we shall see in an example, the equation of Vekua and Magnaradze reduces to an algebraic finite system. This form for a(y) is suitable for approximating every wing of practical interest. We have to mention that, for the wings having the form (6.3.28), the case when qt = 92 =

... = q,, = 0, has been solved by H.Schmidt in 1937, 16.241, and the case when pi = pl = ... = pn = 0 belongs to a larger class, considered by the author of the present book in 1958, (6.41. For this class one obtains the exact solution.

Before passing to applications we notice that if constant, then, taking (B.5.6) into account, we deduce g(y) = Co cos s(y) + 2k1,

J(tl) = k is a (6.3.29)

220

THE LIFTING LINE THEORY

where

I (y) _

{sin[s() - s(rl)) -

cos[s(y) - s(n)J

q

dq. (6.3.30)

6.3.4 The Elliptical Wing

Denoting by a and b the semi-axes of the ellipse from the xOy plane, we deduce s

1-b V

a(y)=aa b2-y2, as=a/b.

Obviously, R = 0 whence K(y, r)) = 0. The equation (6.3.25) gives directly the solution C(y) = g(y), where g is calculated with the formula (6.3.29). In I one performs an integration by parts. Since from (6.3.23) it results 2fi s'(tl) = ira(y)

we deduce

+wao

I=

b2-y2-bcoos(y).

Since from (6.1.20) and (6.2.7) it results J = 2e, using the notation 4wea o

aao+2/9' we deduce

C(y) = Co cos s(y) + k

- kb cos s(y) .

(6.3.31)

For determining the constant Co we shall employ the condition C(b) = 0. Since from (6.3.23) it results

s(y) =

20 wao

arcsin b ,

we deduce Co = kb whence

C(y)=k b2 -y2. For ao = 1/b one obtains exactly the solution (6.1.30).

(6.3.32)

THE SYMMETRICAL WING. VEKUA'S EQUATION

221

The Rectangular Wing

6.3.5

We shall consider now that a(y) has the form a( E/)=ao

b2 -

y

21+

(6 . 3 . 33)

,

the real numbers p and q being chosen in order to ensure only positive values of the fraction one [-b, +b]. In the sequel we shall see that one

imposes pb2 > -1 whence qb2 > -1. From (6.3.15) and (6.3.33) we deduce

R(nt, n) =

c(q + Th)

(1+pip)(1+pni)'

20(g -

C

p)

(6.3.34)

irao

and from (6.3.26) K(y, r1) =

' wo(y) +'P1(y)

(6.3.35)

1+pr7

where acs{s(by} -_aq(m)) 1 +

MY) = c

1

1

d'ri

(6.3.36)

Taking into account that for pb2 > -1 we have d ,q

q 1 + pb2

1

I (1+p) b2-

1+

arct&n

from (6.3.23), for p # 0, we deduce s(y)

2Q f q arcein EI + _ 7rao Lp b

1 +'2 J

p- q p

arctan

y

1+ pb2 l

(6.3.37)

and for p = 0, (I +

s(y) _ L

c) aresin b - Zy

vfb2 --y2 l

.

(6.3.38)

Replacing K(y, q) given by (6.3.35) in the integral equation (6.3.25) and observing that the first term vanishes because the integrand is an odd function, we deduce:

C(y) + tPiny)

r-b

1

C +(p

dt = Cocoss(y) + 2k1(y).

(6.3.39)

THE LIFTING LINE THEORY

222

The integral is a constant Cl which may be determined by multiplying (6.3.39) with (1 + py2)-1 and integrating with respect to y on the

interval (-b,+b). We obtain

J

III[

b

1 + P?1

1 (6.3.40)

bbl+py2dy.

-Col bb +a( dy=2A;

Imposing (6.3.39) and the condition C(b) = 0, we deduce the relation Co cos s(b) - 91(b) CI = -2k1(b).

(6.3.41)

Determining the constants Co and Cl from the system (6.3.40) and (6.3.41), we find the exact solution of Prandtl's equation

C(y) = Cocoss(y) - !C1SOj(y) +2kI(y).

(6.3.42)

Using the inverse method, i.e. considering various values for the con-

stants p and q and calculating the form of the chord, we may find important wings for which the exact solution (6.3.42) is valid. So, in (A.271, considering q = 0 and pb2 = 0, 9, one obtains an almost rectangular wing (the variation of the chord versus the span is very small). Indeed, we have y/b

a/boo

0. 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00

1.02

1.03

1.05

1.06

1.06

1,03

0.95

0.75

Importantresults ofthis meth method are givenin [1.331. 6.3.6

Extensions

Modifying Vekua's method, we managed in (6.4) to give the exact solution of Prandtl's equation for wings whose chord satisfies the relation ys

a(y) =

P(y)

,

(6.3.43)

where p(Z) is a holomorphic function in the Z = y + i z complex plane, excepting the vicinity of the point at infinity where one admits

223

NUMERICAL METHODS

the following series expansion: k>O

p(Z)

=n=-oo E PAZ"

(6.3.44)

.

The determination of the solution of Prandtl's equation is reduced to solving a Hilbert-type problem, whose exact solution is given. The polynomials belong to the class (6.3.44), whence the great importance of this solution. According to Weierstrass's theorem (see for example, [6.13], p.61), every continuous function on the interval [-b, +b] (i.e., every possible form of the wing) may be approximated by polynomials. Practically, for this one may employ an interpolation method (for example, Newton or Lagrange's method). Even if p(y) is a polynomial, in this method it is not necessary to be symmetric, like in the theory of Vekua and Niagnaradze.

Numerical Methods

6.4 6.4.1

Multhopp's Method

The idea of biulthopp's method consists in approximating the func-

tion C(a) by the trigonometric polynomial P"(c) obtained by the Lagrange interpolation in the basis (sinkc)kl,...,n. For determining P", we notice that after introducing the matrices

ST = (sin c,...,sin nc),

aT

= (al,...,an),

it may be written as follows

=

P.

n

(c)

ak sin ka = sTa.

(6.4.1)

kal

The points where one imposes for PP(o) to coincide with C(o) (the nodes) are given by the uniform grid aI

n+1 -o,

a2

2a

n+1

-2a,...,c"=

na

n

1 =nor,

(6.4.2)

whidi is usual in the theory of interpolation [6.13] p.20. These are equidistant on the half-circle with the diameter on the span (fig. 6.4.1). The points xJ = cos a1

(6.4.3)

THE LIFTING LINE THEORY

224

+z

x

-1

1

b

y

Fig. 6.4.1.

are the zeros of the Chebyshev polynomial of order two (F.2.8) on the interval (0,7r).

Denoting c; = C(a1) we have to determine the matrix a from the system Sa = c,

(6.4.4)

where

sin ai sin 2a1 ... sin nal S = sin a2 sin 202... sin nag

c=

(6.4.5)

sin a, sin 20,E ... sin na

One obtains [6.131, p.21

S-1 =

IS. n2

(6.4.6)

NUMERICAL METHODS

We shall present at the end of the section this calculus. Hence,

n+l

sin v1... sin na1

Cl

... sin na

c

(sin a...sin no) I sin on

1Ecinko l 2

n+

(6.4.7)

k=1

1(sin a... sin na)

ctsinka 2

+

sin koj s i n j=1

k=1

For C(a) we have the Wowing expansion:

C(a)

= n +2 1

n

n

ck

sin ka j sin jo.

(6.4.8)

We must notice that this expression could be also obtained from (6.2.26) approximating the Fourier coefficients according to the definition of the integral for an equidistant division a. Indeed, f om (6.2.26) we have:

C(Q) sin yoda

Aj TO

+1

k,i

C(ck) sin jai, .

Since jak = jka = kof, eraplaying Glauert's approximation Al sin jo,

C(Q) _

j-1 one obtains (6.4.8).

Utilizing Multhopp'e expansion (6.4.8), from Prandtl's equation (6.2.28)

226

THE LIFTING LINE THEORY

and from Glauert's formula, we deduce: sin s

4b,3

n+1

n

n

ck > sin jka sin js+ k=1

j=1 (6.4.9)

27r

s sins.

sin jka sin js = 2irba s j=1

k=1

Giving to s the successive values La(£ = 1, ... , n) and taking into account the formulas (6.4.17) and (6.4.19) from below, we deduce the

system (£ = I,-, n), BtkCk = Bt,

Oct +

(6.4.10)

k=1

where

2Bt = (n + 1)ira(£k)j(£a).

(6.4.11)

Fbr k $ t, Btk

-

lra(£a) 2bsin£a

I - (-1)n-t 8

_

I sine

(k + t)a 2

1

sine (k - £)o`

' (6.4.12)

2

and for k = e, Att =

7ra(£a)

n(n + 1)

2b sin la

4

(6.4.13)

Determining the unknowns c1, ... , c from the system (6.4.10), one obtains the solution of Prandtl's equation from (6.4.8). Since Atk vanishes when k - £ is an even number, the system (6.4.10) may be separated; more precisely, the unknowns with odd indices may be expressed by means of the unknowns with odd indices and vice versa. This fact was proved in [1.2]. As it is already mentioned in (1.2] and in (6.17] an iterative procedure for determining the unknowns is

also established. The author had not at his disposition this paper. The procedure simplifies in the case of the symmetric wing (C(y) = C(-y)), i.e. when ck = Cn+l _k (k = 1, 2, ... , [(n+ 1)/2]) or in the case of the antisymmetric wing (ck = -c,+1_k). In the first case the system reduces to [(n + 1)/2] equations, and in the second to ((n - 1)/2]. The square brackets indicate the integer part of the number from the interior.

227

NUMERICAL METHODS

In the sequel we shall calculate the sums that intervened in the above formulas. Let it be for the beginning n

r

j=1

i=1

11.Ecosrja, 12 - Esinrja.

(6.4.14)

Denoting z = e1r`r, for r 96 0, we obtain

_

z - zn+i

-

=

l- z

1

whence, separating the real part from the imaginary one,

I1 = -1 +

1 - (-I}r ,

2

12 =

1 - (-I)r 2

cot

ra . 2

(6.4.15)

One obtains the sum n

r

Ejcosrja-

1

j=1

4

I

era

sin 2

(6.4.16)

12 with respect to a. We deduce therefore, noticing that k - f and k + f are odd or even simultaneously,

deriving

n

n

2 E sin kja sin eja = E cos(k - e) jaj=1

j=1

(6.4.17) n

-Ecos(k+f)ja= (n+l)6kt. j=1

With these formulas, the relation (6.4.6) written as follows S-1S

n2

S2=1

may be immediately proved, since we have: n

Sz

n

_ E sine ja F sins ja sin 2ja ... sine ja sin nja j=1 ial j=1

(6.4.18)

THE LIFTING LINE THEORY

228

Utilizing now (6.4.16), for k q& a we deduce n

2 E j sin kja sin eja = j=1 (6.4.19)

4

sin2

and for k = e:

(k+2 e)a

sin2

e

(k 2

n

4 E j sine kja = n(n -}- 1).

(6.4.20)

j=1

Analogously it results: n

2 E sin jka sin ja = > cos j(ka - a) - > cos j(ka + a) _ j-1 j,1 j

-Re z-Zn+1

Z-Zn+1

_Re 1- f

I-Z

s,,01(ka+.)

(-1)k+1 sin(n + 1)a sin ak COs a - COS ak

(6.4.21)

If we utilize this identity, for the formula (6.4.8) which gives the solution of Prandtl's equation , we obtain the final form

C(a) __

1

n+l

n k=1

k+1

`

sin(n + 1)a sinker . coo a - cos ka

(6.4.22)

In [A.23], p. 98-111 one gives a mathematical justification of this method. More precisely, one demonstrates that under certain circumstances, the iterative procedure that one utilizes for solving the system (6.4.10) is convergent and the solution (6.4.22) converges uniformly to the solution of Prandtl's equation.

6.4.2

The Quadrature Formulas Method

In [6.5) we gave a numerical method for solving Prandtl's equation by means of Gauss-type quadrature formulas. It is well known that these

229

NUM ERICAG METHODS

formulas give the best approximation. The key of this method consists in writing Prandtl's equation in the form (6.1.21). With the change of variables

y = bs , q7 = bz

(6.4.23)

and, keeping the notations C(s), a(s) and j(s) for C(bs), a(bs) and respectively j(bs), the equation (6.1.21) becomes .

/3C(s)

2b)

1

J-1 (z

(s)s d x + j (s) ,

(6.4.24)

and the conditions (6.1.7),

C(±1) = 0.

(6.4.25)

The solution of the equation (6.4.24) has the form

C(s) =

1

-_3 2 c(s) .

(6.4.26)

Employing the quadrature formulas (F.3.5), the equation (6.4.24) reduces to the algebraic system n

EAkici =7k, k = 1nn,

(6.4.27)

i=1

where we denoted ak = a(zk), Ck = C(xk), era k

A

2b(n + 1)

(-1):+kl

1-

Akk=fV1 -xk+6

jk = j(zk)

(x1x2

-1

i#k,

zk)2

(6.4.28)

n41; zi=Cosn+1,t I'n'

the unknowns being c1,... , c,t. The system has to be studied theoretically and solved numerically using a computer. After determining the unknowns, the lift, drag and moment coefficients, defined in (6.1.21) and (6.1.25), CL

C=

2b

win the form

f

+1

1 - s2 c(s)d s, cD = - A

1

= mA J

1

`s

1 - s2 c(s)d s, c;

r1

ZA-

j

1 - s2 w(s)c(s) ds,

11

s

1 - s2 w(s)c(s) ds, (6.4.29)

230

THE LIFTING LINE THEORY

give with the formula (F.2.12) CL =

(n + 1)A E(1 i=1

- x?)c" CD = - (n + 1)A isl

" _ {n + 1)mA E(1- x; 2)xc:, 27rb2

i_I

x?)w,c{

>,

2nb2

2 c= (n + 1)mA E(1 - xd )x:wig i_1

(6.4.30)

where

w; =

13 VI - x, .

2aai

(6.4.31)

In (6.4.30) we used, like in (6.1.21), the notations A for the area of the domain D and m for dimensionless length of the mean chord (in the direction of the unperturbed stream). For verifying the method, we applied it in [6.5) to the elliptical Sat wing, for which the exact solution is known (6.1.28), (6.1.30). With the notation (6.4.26) it results c(s) = k. Putting in (6.4.27) Aki =

1- x4Ak;,

c; = k4,

it results that the system Ak.;ci

= Q+ it/(2b)

(6.4.32)

must have the solution dl = d2 = ... = c;, = I. For b = 10 one obtains numerically dl = 4 = ... 1000. We deduce therefore that the proposed method is very good. In the sequel we shall give the numeric solution for the rectangular wing (in this case there exists no exact solution). Taking the reference

length Lo in the definition of the dimensionless variables (2.1.1), to coincide with the half of the chord, we deduce xt = :L1. It results

a(y) = 1 and j(y) = 27rc. Putting c; = 27red', the system (6.4.27) becomes Akic!

= 1, k = I- n,

(6.4.33)

where Ak; are given by (6.4.28) where we put ak = 1. The quantities

NUMERICAL 1METHODS

231

of interest in (6.4.30) are

1

kl =_ 71+1

(1-xi)c',

k2=-.1 (1-x?)3,2(c,')2, (6.4.34)

V(1-x?)x,c;, k4=

n+1 i_i

A

n+1

E(1-x?)3/2x,(cI')2. i=1

One obtains CL = 7r2ek1 ,

CD = 7r2e2(kl - k2), (6.4.35)

2c= = 7r2bek3 , 2c; = 7r2bC2(k4 - k3)

.

For b = 10 we obtain the following values:

fi

1

0.8

kl

k2

k3

k4

0.2347 0.2544

0.0907 0.0856

0

0

0

0

The result c= = c. = 0 is natural because of the symmetry of the wing. The lift and the drag increase because of the compressibility. This result is also natural. The value 2.307c obtained here for cL in the case of the incompressible fluid is smaller than the values 7.29 e, 5.28 £, ... ,

obtained with Glauert's method [1.12], but the values obtained with Glauert's method come closer to the values given here if A(= 2b/m) increases, i.e. the span is great with respect to the chord. Just in this situation the lifting line theory is valid. We may think therefore that the method we have just exposed is at the same time very simple and very efficient.

6.4.3

The Collocation Method

The simplest numerical integration method is certainly the collocation method [6.6]. In case that Prandtl's equation has the form (6.1.19) and satisfies the conditions (6.1.7), the solution has the form

C(y) =

b2

- y2 c(y) .

(6.4.36)

According to the collocation method, the segment [-b, +b] is divided into iV elements L; and the function c(y) is approximated on each

232

THE LIFTING LINE THEORY

element with its value c; from the mid - point y° of the segment. So, the equation (6.1.19) gives

,

N

[

4

c(y) = a(y)

2,8

fd

ix1

(b - y)2 d rI + 2j(y),

(6.4.37)

where, as we have already stated, Cj = c(y?). Imposing this equality to

be satisfied in every point yk, k =I,-, N, one obtains N

2

Vr2b2

- yk2ck = ak E Aloe: + 2jk k = I ,

,

(6.4.38)

i=1

where ek = c(yk), ak = a(yko), ?k = ?(yk),

Ak, -

w+1 f J(y

(6.4.39)

b-q

-

y y;+1 representing the extremities of the segment L;(y1 = -b, yN+1 = = b). So, (6.4.35) represents an algebraic linear system consisting of N

equations with N unknowns cj. For calculating Ak{, we notice that for e V (a, ?'J we have

I

a

( 7 7-e )2

d

,

y-e

a -- e

+

nee In [ (ry-e)2 + ( b -e +

+aresin

a

b2_.72s)2 (a-e)

- ry b --a- 7 (6.4.40)

For e E (a, 7) we shall write

II = o

-

(71e)2 d q b2-772

s

=J

66

(n - e)2 v° 'I

d1 (6.4.41)

T

(77 -

e)2 d

b -V2 7

(rl - e)2

d ij.

We use the formula (6.4.37) for calculating the last two integrals. Since (D.3.9)

r-b (q - e)l

d n= -7r ,

(6.4.42)

233

NUMERICAL METHODS

it results that I1 is also determined. Utilizing these formulas, for k 96 i we obtain: Aki

V - y?

b2

y

- 77i+I

yk0

yi+1 - yk

yl

(yi+1

02is

b2

72 + (Vb2_YA.

b2-t/i+l

In

y`

(Vi-Yk)2+ yi

+arcsin

(\

b2-yk2

- yko

yi+1 - yk

y,2

+

+

R--i/i+l - -Eli F1

b2

y,2

P

(6.4.43)

and for k = is y2

b2

Aii =- -7r -

Ii yt I

-

Vbi

b2 - y<+1

y2

+

+ (y +1 I Vb2 - y2

+arcsin b2

1,2+

- yI

`2

Vb2-y°2+ b2-y, )

/

2]n

t? +

(VFb2

2b (6.4.44)

Vb2

y°2

+

tJ+i

where we denoted 2ti = yi+1 - yiFor testing the method one utilizes also the elliptic flat wing. For this wing the exact solution is (6.1.28) with k determined in(6.1.30). Comparing (6.4.33) with (6.1.28), it results c = k/b. For E = 0.1,,0 = 1

(the incompressible fluid) b = 10 and N = 50, the numerical values obtained for ci are situated between 0.050 and 0.053, and the value of k/b is 0.052. For E = 0.0872, A = 1, b = 20 and N = 50 the values obtained for ci are situated between 0.0243 and 0.0252 and the value of k/b is 0.0252. The results given by this method are more accurate when the angle of attack is small. The accuracy also increases when N increases.

THE LIFTING LINE THEORY

234

Since the method is simple and can be easily applied one can accept the error of order 10-3 that it contains.

6.5 Various Extensions of the Lifting Line Theory

The Equation of Weissinger and Reinner

6.5.1

In the sequel we shall study new integral equations in order to improve

Prandtl's model. In the past the researchers did not pay too much attention to these equations because the integrands contain both the unknown C and the derivative C'. Now we may use successfully these equations because some fundamental results concerning the Finite Part are known and, as we have mentioned in 16.5], after replacing the

Principal Value of an integral which contains C' and has a Cauchy type singular kernel by the Finite Part of an integral which contains C and has an hypersingular kernel, we may employ the numerical methods. First we shall study the equation of Weissinger [6.30] and Reissner [5.29]. For obtaining it, we return to the lifting surface equation (6.1.28). Taking (6.1.6), (6.1.11) and (6.1.12) into account, we deduce +b

rlydn. l Cdn-7dt J-e 9 ff bod JJ dq+2

(6.5.1)

Utilizing (5.1.35), from (5.1.28) it results the first form of the integral equation

r+b fi(n) d n +

1

ri - y

1'rj

(1 4w JJD lJo \

_ z o d td r1= rt, (x, y). R

(6.5.2)

Multiplying this equation by (6.1.15) and integrating it on the interval [x_(y), x+(y)] one obtains

j+b

a(y)C(rl)d71+

brl - Y +

1

+b

2a J-b

+(v) x+(n)

Tx-)

x-x-(y) Mm) (i_i) dx d d E rl f(v), x+(v)-x (6.5.3)

j(y) being (6.1.18).

235

VARIOUS EXTENSIONS OF THE LIFTING LINE THEORY

Further we shall perform an approximation which consists in replac-

ing f(f,9) by C(n) A(n)

f (f, rl)

x+(n) - f

(6.5.4)

This is just the solution of the two-dimensional problem when h' is constant. Indeed, according to the formula (C.1.9), for a given y , the equation of the two-dimensional problem

1+(v) f(f,y)

T

( 6 .5.5 )

= hx (x,y),

n ,-(,) f - x

becomes, in case that h' does not depend on x: f (x , y)

x+(y) - x

= -h(y) x - x_(y)

(6 5 6) .

.

Multiplying (6.5.5) by (6.1.15), integrating the result over (x_ (y), X+ (y)] and utilizing the formulas (B.5.4) one obtains: -+(v)

C(y) __

- J:-(v)

(6.5.7)

f (f, y)d f = ira(y)h'(y)

Eliminating h' from (6.5.6) and (6.5.7) one obtains (6.5.4). With this approximation, the equation (6.5.3) becomes:

a(y) J bbC n)di? +27r YO

T-b

(6.5.8) TO

where

N (y, n)

:+(v)

_+(»)

xo rIl R)

x+(ti) - F x- x-(y) - x-(+))

x+(y) -- x

dxd

f

.

(6.5.9)

Using the notation r)) No ( y, n) = N(y,

A(y)A(n) = 1

x(y)

f A(y)A(q) J=_(vl 1

(6.5.10)

_+(,) J=-(»)

x+(r1) - f

x -x_ (y)

f- x-(rI) V x+(y) - x

d xd

f,

THE LIFTING LINE THEORY

236

we may write the equation (6.5.8) as follows: CYO

- f eb

)dn+

2A

fC

o

N°(b,n)dn = J(y).

(6.5.11)

and J(y) being defined in (6.2.7). This equation has been deduced by Reissner [5.29). Other simplifications are performed in the papers [5.24] [6.30] in order to obtain an approximate solution. The A(y)

difficulty consists in the presence of both C(n) and C'(n) in the integrand (the equation is integro-differential). Employing, as we have already shown, the Finite Part, i.e. taking (6.1.12) into account, we may write the equation (6.5.11) as follows C(q)

2 _1

y! `Ni(y, n)d, = J(y) ,

(6.5.12)

where we denoted NI = No - 2. This is the equation we are going to use in applications. This is an integral equation.

6.5.2 Welssinger's Equation. The Rectangular Wing In the case of the wings for which the lifting surface equation cannot have the form (5.1.41) (this class is established in 5.1.7, and it contains the rectangular wing) one obtains an equation which may be easily utilized in applications. This equation was given by Weissinger [6.31), and received his name. Starting from the definition (6.1.6), we have:

-i',}(n)f (x+(n) n) + x'-(n)f (x-(n), vi)-

f - f=-(")

x+(+r) a f

8dt

rx+(n) 8 f

_ J=-0)

(6.5.13)

dt.

We are going to show why the sum

-x+(n)f (x+(n), n) + x' (rl)f (x- (n), n) can be neglected. The first term vanishes by virtue of Kutta-Joukowski condition. The last term vanishes for a straight leading edge, perpendicular on the Ox axis (for example this is the case of the rectangular wing). It also can be neglected when the leading edge is slightly curved (x'_ (q) = 0). In general we can neglect this term considering that we have prolonged the theoretic leading edge in front of the real leading

237

VARIOUS EXTENSIONS OF THE LIFTING LINE THEORY

edge and in this zone f = 0. With this approximation, the equation (5.1.41) may be written as follows:

'+b

1

C'(17)

Ud,1

-a

2w

AD 09 xo

-d

fd

=-

"(x,

N)

(6.5.14)

We make a second hypothesis, having in view to be satisfied in the case of the rectangular wing of width, let's say, 2a, namely we substitute

a' instead of zo from R. This approximation is justified by the fact that (x - jj varies from 0 to 2a, hence on the greatest part of the domain D we have IxoJ = a. So, the equation (6.5.14) becomes:

'+b

1

27r

1-6

dn -

we

-

f

1

27r

f+A

"+

_. aq

(6.5.15)

x0yo

Multiplying by (a +- x a - x) and integrating with respect to z on the interval (-a, +a) one obtains

a

-d n+ L jr a C 'm

2Ja

2

30(y) _ -a

-a

C (rl)d rl

rah(y) ,

a±xh"(x,y)dx.

(6.5.16)

(6.5.17)

Introducing the non-singular kernel

K(pa) =

f, fd

a+

_ a

yo

go

(6.5.18)

the equation (6.5.4) may be written as follows C' r7)

Ir

YO

n+

Zax

f C'(rj)K(ya)d tj_ jo(y) .

(6.5.19)

This is the definitive form of Weiseinger's equation. It was created mainly for the rectangular wing. Before presenting the method of integration of this equation, we have to notice that it is not necessary to determine C(q). It suffices to find C'(t) because from (6.1.9), integrating by parts, it results CL

b yC'(y)d y, es = -aQ = -- A J b

y2C`(y)d y.

(6.5.20)

238

THE LIFTING LINE THEORY

The solution of the equation (6.5.19) has the form Cr(y)

tiP(J)

(6.5.21)

with sp satisfying the equation '''+6

1

d i7 b2

b

2 Yo

1

+ lira

f

(t ) h (Ju)d q = lo(U) . (6.5.22) b2 - rf2

6

Performing the change of variables 'i = brl', y = by' one obtains 1

rrb

'r+t

J 777=7 it

y

sp(rl)

1

d>>

;p(aT)

27ra J

i

i - >>

h0(bJo)d n = .?o(bJ) (6.5.23)

This is Weissinger's equation. It looks like the generalized equation of thin profiles (C.2.1). The numeric solution is determined by means of the formulas (F.2.5) and (F.2.6).

6.6 6.6.1

The Lifting Line Theory in Ground Effects The Integral Equation

The lifting line theory in ground effects is obtained from 5.3, as well as Prandtl's theory is obtained from the lifting surface theory. We are not interested in finding the velocity and pressure fields. The integral equation is obtained from (5.3.13) utilizing the formulas (6.1.11)-(6.1.13)

and the relation (6.1.8). One finds

1 T7r

+b E + f.?+ (y) f y, y) ! d d U f $ b 7-y y n (y)

+ (6.6.1)

1

+b

+2- J C(t)i(x, yo)dTl = 2/is(x, y) , where d2fl2x

d2

- yo

yo) _ (d2 +y()2 )R3 + (dl + y02)2 ( \1 + R2 =

x + e2 ,

e2

= 02(y2 + d2).

x

RI) '

(6.6.2)

239

THE LIFTING LINE THEORY IN GROUND EFFECTS

Multiplying (6.6.1) by (6.1.15) and integrating the relation just obtained

with respect to x on the interval (x_(, ), x+(rj)), we obtain, taking (B.5.4) into account: C'(11) d

2#C(y) + a(y) Lb

Y-17

q

r

+

,l

C(n)No(y,

yo)d n = 2j(y) , (6.6.3)

b

where we denoted s+(v)

x - x_ (y) N(x,

J:^(v)

x+(y)

1

No(y, yo)

(6.6.4)

yo)d x,

j being (6.1.18). The equation (6.6.3) is the lifting line equation in ground effects. It was given in (5.8J. It is obviously an integro-differential equation which generalizes Prandtl's equation (6.1.16) (Um N = 0). Further we shall make some considerations concerning the kernel No(y, yo)-Taking (5.3.11) into account and introducing the functions

Iv(y, yo) _ -

+(v)

r

J

z+(y) (yx (xz + e2)"/2

(v)

d x, v = 1

,

(6.6.5)

we obtain:

No(y, yo) = (a +

y)2 [I, (y, yo) - a(y)] +

#2 13(y, yo).

(6.6.6)

Performing the usual substitution x = s(y) + a(y)t,

(6.6.7)

where

s(y) = x+(y)

x-(y) , a(y) = x+(y)

X-(Y)

(6.6.8)

2

2 the integrals I become +i

1

IN = -na(y)s(y)

J

l+ t 1

t

dt P(t)v

aZ(y) 7r

r+i

.!-1

-1+-t

td t

1-t

P(t). , (6.6.9)

where

P(t) = a2t2 + last + e2 + s2 . (6.6.10) It is well known that the best approximation of these integrals is given by the Gauss-type quadrature formulas (F.2.24). With this form (6.6.9) one may study the asymptotic behaviour of the kernel No(y, yo) given by (6.6.6) depending on the parameter A2 = d2/b2.

240

THE LIFTING LINE THEORY

6.6.2

The Elliptical Flat Plate

Without considering the ground effects, the solution of this problem is (6.1.28), (6.1.30). In the sequel we shall determine the influence of the ground. We assume that the wing has the equation x2/e2 + y2/b2 = 1. We deduce

xf(y)=fb b2-y2,

a(y) =b Vb2

- y2s(y) = 0.

(6.6.11)

Assuming that the span is much larger than the chord (e << b), we shall neglect the terms of order (e/b)2. From (6.6.9) we deduce I = 0, whence

`

Integrating by parts, we deduce:

f

+b

+b

C(n)No(y,yo)dn=-a(y)f C'(n)&?yodn, b

b

and the integral equation (6.6.3) becomes:

2fC(y) + a(y) J b C,n) d q - 2a(y)

J

bb

C'(1) d2 + y02 d n =

4_-Tra(y).

(6.6.12)

For d - oo one obtains Prandtl's equation. (6.1.16). Taking into account the shape of a(y) from (6.6.11), we shall look for solutions of (6.6.12) having the form

C(y) = k b2 - y2.

(6.6.13)

The first integral was calculated in (6.1.29). The second may be calcu-

lated with the substitution y = by, n = biz. If b < < d we neglect the terms O(b/d)4 and we obtain

f

b 2ir \a)2

b

(n)d2+yody=k

2

We assumed that the span is much smaller than the distance to the ground. Replacing (6.6.13) in (6.6.12), we obtain the relation which determines k: (b `2 Tr = 1Ezr6 . (6.6.14) k 2p+ -n b

-

-I

241

THE LIFTING LINE THEORY IN GROUND EFFECTS

Denoting by k the value of the constant when there are no ground effects, i.e. (6.1.30), we deduce

k=kak, where

(6.6.15)

2f3 + (t/b)ir k0 = 20 + (t/b)ir - (e/b) (b/d)27r

(6 .6 . 16)

Obviously, k > 1. The lift, drag and moment coefficients are given by the formulas (6.1.21) and (6.1.25), where for w(y) = w(0, y, 0) we have 15.81: W(y)

C ,(n)

J 47r

bb

d

rl - 4n

C'(Tl)

b

& yo d n.

(6.6.17)

These formulas (6.1.21), (6.1.25) and (6.6.17) are valid for every shape of the wing. For the elliptical flat plate wing we obtain:

w(y) = - 4

1

2

+

C

!

(6 .6 . 18)

such that

cL=kocL, cD=klcp, c==c.=c, with the notation

(6.6.19)

2

k1=ko li(1+2 )>1,

(6.6.20)

lift

Cr and cD representing the

respectively drag coefficients in the absence of ground. Obviously, both the lift and the drag are increasing in the presence of the ground. The coefficients ko and k1 depend on M, t/b, b/d. The numerical calculations from 15.81 show that for the lift the increase is not significant

but for the drag it is considerable. The ground effect is a decreasing function of d. For the same values of the ratios t/b and b/d the influence coefficient ko is an increasing function of Mach's number M.

6.6.3

Numerical Solutions in the General Case

Utilizing the formula (6.1.12), one may write the equation (6.6.3) as follows:

2#C(y) - a(y)

f

+b b

c(n) d n+ (n - y)2

f

+b

C(n)N(y, yo)d n = 2?(y) b

(6.6.21)

242

THE LIFTING LINE THEORY

As it is known, this is an integral equation, not an integro-differential one, but the singularity is stronger than in (6.6.3). In (6.6.3) we have a Cauchy - type singularity and in (6.6.21) we have to consider the Finite Part of a hypersingularity. But for this kind of equations there are available quadrature formulas. In order to apply this method, we have to perform the change of variables y = by, q = bq' for calculating the integrals on the interval (-1, +1). We obtain 2bIC(y) - a(y)

f

+t C(n) Y)2 d n + b21 C(i7)N(y, yo)d =

+1 1

(n -

j(y)

1

(6.6.22)

Since the solution of this equation has the form C(y) = V171- y2 c(y),

(6.6.23)

we obtain

1 - yj cj - aj

2b/3

1 --17C(17) d n+

Ti

(n - yj)

1

1 - i2c(yl)N(yj, yj - n)d n = 2bjj , yj = cos n

+b21+

+ 1,

t

j=T

.

(6.6.24)

Using (F.3.5) and (F.2.12) one obtains the system: n

Ajcj + E Ajkck = 2bjj , j = 3n-,

(6.6.25)

k=1

where

A.;= MO 1-y;+aj7r Aak

-a L

n+1 2 b2k.

7r

n + 1

- yj)2 + 1(yk - (-1)'+k

n + 1

N(yj, yj - yk) (1 - yk2) J

(6.6.26)

In the first term from Ajk one excepts k = j. The system (6.6.25) is solved numerically.

6.7 6.7.1

The Curved Lifting Line The Pressure and Velocity Fields

In this subsection, we shall pay a special attention to the aspect ratio A = (2b)2/A introduced in (5.4.1). Usually, if A is small, one applies

243

TILE CURVED LIFTING LINE

the theory from 5.4 concerning the wings of low aspect ratio. If A is large one applies the lifting line theory. These are the two asymptotic theories of the lifting surface theory. As it is known, one of Prandtl's hypotheses consists in replacing the

domain D by the segment [-b, +b] taken along the span (the Oy axis). This hypothesis is plausible for the wings having the shape of an ellipse, triangle, trapezium or rhombus (see fig. 6.7.1) but it can be the source of great errors in the cage of the wings having the shape of a

swallow tail or the shape of an arrow. In the first case it is natural to replace the wing by the curvilinear median (see fig. 6.7.2), and in the second case one approximates the wing by the median broken line (fig. 6.7.3). For birds, the nature preferred the curvilinear median. These are enough reasons for studying in this subsection the curved lifting line. In YA

b

j'I i

Y

Y.

b

i

b

rt 1

1

0 x

-4 i

t

aX

0 1 1

b)

1

Fig. 6.7.1.

Fig. 6.7.2.

Fig. 6.7.3.

244

THE LIFTING LINE THEORY

this case, one starts too from the general representation (5.1.8)-(5.1.12) and Prandtl's hypotheses. We assume therefore that the wing is without

fl = 0) and that the unknown is C(y) defined thickness (hl = 0 by (6.1.5), with the conditions (6.1.7). When the domain D reduces to the curved line r (fig. 6.7.2) having the equation x = x.(y), the formula (6.1.8) is replaced by

r`

/ JD f

lim /

il)k(x, y, z, t, q)d d y = -

f

+b

k(x, y, z, x. (q), n)C(i )d +1,

b

(6.7.1)

for x-(n) -. x. (n) - x+ (n) So, with the notation R. =

ix - x.(n)]2 + A2(y02 + z2),

(6.7.2)

the formulas (6.1.9) become z

P(x,y,z) _ -4

C(ri) z dn.

(6.7.3)

rt

v(X' Y, z) = -

I

:Fir =fl

lim.

o-)]

J 1D f (C, n) gEl l yob z

.1 JD f (C W 1+ RI

(1 + R-

)

z

y2 + z2

d

d

)dtdn+

+XOZ

C(n)11+x(n)lal\

l Jb

47r

L

o+z2)dn-

JJ

p2yo

+6

[x - x.(n)]z - T7r 14 C(rl) yof-' R3 d n 1

(6.7.4)

Utilizing now the identity

r

x - x` R. R

_ 8(

8

) &1 z

z

z(x-x') /2yo y

x-x.`1 R* jl +

z2 R3 _ (6.7.5)

32z ,

THE CURVED LIFTING LINE

245

we obtain, after performing an integration by parts,

j

f

v(x, y, z) =

+b CI(rl)

f1+

x R'(q)] d,,R.

C

(6.7.6)

Q2 47T

/ +b

J

b

zZl=

C(i) R. d q .

Analogously,

w(x,y,z) =wi +w2,

(6.7.7)

where

=4a

fffri)_ j+bx R,

x}m.(_)ddtl=

(n) d n

4nx*-x.JJDf(C v?) 2[y02

W-2

47r xlim- . If,,, f (C rT) l 1 + Rt 1

lr ,ffDf( , l) o+Z2 YO X02

lice

=

art

l 6r+.2) d d'r

(6.7.8)

(_)dd= R 1

r+b

4 J b C(") ['x_x,(,i)l 09l ( T2 + Z2 l d YO

Y7+

J

+Q rb

c(n) yo +'Z2)

dn.

Introducing the identity

$2(x _ R.

yg

+Z2)

110+z2

1

T3

(6.7.9)

Jyoyo

ly +z2

[1+x-x.(n)1I+Q2yO2X1 Rl J Rs

)

246

THE LIFTING LINE THEORY

and, integrating by parts, we obtain u2. In fact, it results: x w(x' y, z)

('l) d

4a fb C(r1)

R3

+b

-4

f

b+6

q+

4n

bb C(17)

-

--d n-

r

(+I)] d>1.

y0+2 I1 + x

r

b

l`

(6.7.10)

6.7.2

The Integral Equation

We start from the lifting surface equation having the form (5.1.28). Utilizing (6.1.6) and (6.1.12), we deduce:

Y=dtdn

Eb

(q)

0

(6.7.11)

dr=

C

1 =;F

+b C'(rl)dq.

1

4'rJ-b 11-y

0

b

Taking (6.2.1) into account, we obtain:

12 = -

lim

1

47r

11D

f(

x0 d l;d

R

To

= (6.7.12)

C(n) x-x.(rl)drl,

1

R°

where

R; =

[x-x.(n)12+(.32y .

(6.7.13)

Utilizing the formula (D.3.7) and taking into account that C(±b) = 0, we deduce Iz

1

'r+6

41r ,l -b n

8r

1

u ON {C(rl)

x- x. (n)1 Ro.

Jd_ (6.7.14)

1

'' +6 C11) -x-

4ir.!_b n-y

x. Ro

d'l+la

247

THE CURVED LIFTING LINE

whore 1

13 = -

Co 8

-

n v 8n

_-

d *!

R°

(6.7.15) C(n)xx

x(n

4(n)dn

4sr

From (5.1.28), (6.7.11), (6.7.12) and (6.7.14) we obtain the equation: t bb

K(x, y, n)d n + 4A ,!_b n (ny

air

C('i)L(x,

y,,7)d n = hs(x, v) (6.7.16)

where

K(z,y,n)=1+ (6.7.17)

x - x. (n) - ray'. (n)

L(x, y,

(R.)

are non-singular kernels. The equation (6.7.10) was obtained in another way by Prosadorf and Tordella (6.22]. It is a singular Integro-differential equation.

For the straight line (x.(q) : 0) one deduces Ci(r!)

Ib n +b

47r

j' C'(n) x

- yd n + Oar J

iJ n - y Rd n - 4

x C(q)

n = h: (6.7.18)

x+ . This equation is a first approximation of the lifting line equation. Fbr deducing the equation (6.1.16) we had A < < yp on the greatest part of the domain D, such that we might where R

consider Re = i4jyo(. Here we cannot perform this approximation.

6.7.3 The Numerical Method Using (D.3.7), one denaozutrates the identity

J-bb (n -

y2K(x, y,

n)dn = Jn vK(x, wn)dn+ (6.7.19)

C(tl) 8 + 14 n-y b K(x,y,y)dn,

248

THE LIFTING LINE THEORY

such that (6.7.16) becomes: C(rl) K(x, y, ri)d rl - 1 7 T-b (11-y)'"

} C(q) 8 K(x, b, y)d q+

J b n-y q

(6.7.20)

This is an integral equation (not an integro-differential one) but with a strong singularity, for which the Finite Part is considered. We denote

AI (x, y, y) =

8

5; K(x, y, y)

(6.7.21)

and we perform the substitution y = by', v = br1'. The equation (6.2.14) becomes:

'(

A

l

C(eta

2 K (x, y, r1) d 1l -

I

b

iJ

1 C(+1)

+

n-y

M(x, y, y) d r1+ (6.7.22)

+ b2 r+1 C(i) 7r

_

L(2:, y, +1)d 17 = 4bh'. (x, y) , I

where

K(x , y , rl) = L(x, y, n)

xx.(rl)

1+

1(x - x,)2 + b2R2yo11/2

x-x,('7)-byox:

_ -F 2 1(x - x.)2 + 62Q2yo13/2

(6 . 7. 23)

M(x,y,y) = 6 W (T, y, Y) Utilizing the quadrature formulas method, we shall take into account that the solution of the equation (6.7.22) has the form:

C(n) =

--q2 c(r1)

(6.7.24)

and we shall utilize the formulas (F.2.12), (F.3.4) and (F.3.5). Denoting IM

=cosnk+l, k=in-,

(6.7.25)

one obtains from (6.2.16) the algebraic system

'AJ,ec* = 4bh2(x,g3), j = ln,

A,cj + L= I

(6.7.26)

249

THE CURVED LIFTING LINE

where

AJ =

+1 2

K(x, n;, nj),

Ask= (1 - rlk)

1[1-(-1) +k

:i+1

(nK(k x, nj n,,)nk)

-

I

2+

(6.7.27)

+b [1- (-i)+k1 M(x,17;, nk) + b2ajkL(x, n;, nk) J

rlk - ni

For writing explicitly this system we have to know the shape of the wing. For example, for the flat plate having the shape of an arrow with the angle of attack e we adopt the broken line model (fig. 6.7.3). We

have f = -E and utilizing the substitution y = by,

0
Y,

K(x,y,n) =

L(x,y,n) =

K1=1+xRn, 0
=1+

x+r7,

R

-1 < n < 0,

(6.7.28)

L,=-#2x-"-yo 0
where

1/2 R= [(z_y)2+b22(y_q)2]

The system (6.7.26) may be solved numerically using a computer.

Chapter 7 The Application of the Boundary Integral Equations Method to the Theory of the Three-Dimensional Airfoil in Subsonic Flow

7.1 7.1.1

The First Indirect Method (Sources Distributions) The General Equations

The superiority of this method in comparison with the classical meth-

ods has been exposed in Chapter 4. Using this method we succeed to impose the non-linear boundary condition just on the boundary of the wing. Moreover, it allows to solve numerically the integral equation of the problem, approximating the boundary by a polygonal line in the two-dimensional case, or by a polyhedral surface (consisting of panels) in the three-dimensional case. We deal with the problem considered everywhere in this book. A subsonic stream, having the velocity U,,.i, the pressure poo and the density per. is perturbed by the presence of a fixed body, having a known surface E. One requires to determine the perturbed flow and the action of the fluid against the body. Introducing the dimensionless variables X, Y, Z related by the dimensional variables xl, yl, zl as follows

(XI, y1, zt) = Lo(X, Y, Z)

and putting

V1=Uoc(i+V), P1=p,, +p,oU,2,.P

(7.1.1)

we obtain the system for the perturbed fields: R12OP18X + Div V = 0, OV/OX + Grad P = 0.

(7.1.2)

Projecting the last equation on the OX axis, we deduce:

P=-U

(7.1.3)

252

HIEM. THREE-DIMENSIONAL AIRFOIL

Taking into account this result, the first equation from (7.1.2) and the projections of the second on OY and OZ, give fl29U/aX + OV/aY + 8W/0Z = 0 (7.1.4)

av/ax - are/ay = o, aw/ax - av/az = 0, U, V, W representing the coordinates of the vector V. Denoting by F(X, Y, Z) = 0 the equation of the boundary E, we have to impose the condition

(1+U)Nx+VNy+WNZ=O, F=0, where

N_

(7.1.5)

Grad F (Grad F1

(7.1.6)

With the change of variables

x=X, y=AY,z=QZ u= j3U,v=V, w=W.

(7.1.7)

the system (7.1.4) becomes

au/ax +8v/ay + aw/az = 0

(7.1.8)

au/ax + 8u/ay = 0, raw/ax - su/az = 0.

(7.1.9)

Performing also a change of variables in F we have

aF/8X = OF/0x, (9F/8Y = (30F/8y 8F/8Z =130F/az , such that the boundary equation (7.1.5) becomes

un=+ f32(vny+wns) = -13b?,F = 0,

(7.1.10)

where grad F (7.1.11)

Igrad F1

We agree to utilize the inward pointing normal to the body. We also impose the damping conditions at infinity lim(u, v, w) = 0. 00

(7.1.12)

The first equation from (7.1.9) represents a necessary and sufficient condition for the existence of a function io(x, y, z) such that

u=

v = acp/ay .

253

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

From the second equation and from the damping condition it results

w=8o/8z. With this representation, the equation (7.1.8) gives App= 0. It is well known that the fundamental solution of this equation,

tp(x)_-47r 1, r r=Ix - tI,

(7.1.13)

represents the potential of the flow determined by a source of intensity f, having the position vector . The velocity field is

v = grad p = 7.1.2

4 xr

(7.1.14)

The Integral Equation

Replacing the body with a continuous distribution of sources on E, having the unknown intensity f (x), the velocity field in fluid will be

U( ) _

-4; Iff(x) Ix -

3da,

(7.1.15)

t representing the position vector of the generic point M in the fluid.

7.1.3

The Integral Equation

In order to impose the boundary condition (7.1.10) we have to pass to

the limit in (7.1.15) considering that M(4) tends to the generic point Qo(xo) E E. To the limit, the integral from (7.1.15) becomes singular. Following the procedure from the two - dimensional case (see 4.2.1), we

shall prove that if f (x) satisfies Holder's condition on E, then v(xc)

4o - \ 4r

jf f

(x) I x - 41 3 d a1

=-2f(xo)r+o-,- 11 (X)Ix-xolda,

(7.1.16)

254

BIEM. THREE-DIMENSIONAL AIRFOIL

where (7.1.17)

6-+O1 JE-O

J JE

a representing the surface cut from E by a sphere E, having the center in Qo and the radius E. Indeed, writing

IL =

J

(7.1.18)

f,0+0

we have to calculate

L= lim

JJf(z)x x3 da,

i.e. the last term from (7.1.18). Writing this term as follows

L= lim

t-.z0 J

where Lo

d a + f (zo)Lo

J (f (z) - f (xo)l x

li n f f f

IX

,

(z)1013da,

(7.1.19)

we notice that the first integral from the expression of L tends to zero

when a - 0 because f satisfies Holder's condition. For calculating Lo we shall replace o by A, the projection of the surface a on the plane 11 which is tangent to E in Qo (fig. 7.1.1).

Fig. 7.1.1.

255

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

On this projection we shall use the parametrization

x = xo + r(cos 9io + sin 9 jo), 0 < r < e, 0 < 0 < 27r,

(7.1.20)

io and jo being versors orthogonal to the plane n. Also, taking into account that the limit is the same on every path on which 4 --+ xo, we consider the limit on the direction of the inward

normal no = n(xo) to E in Qo. We have therefore

f =xo-qno, q>0.

(7.1.21)

Hence r 12" r(cos 9io + sin 9jo) + rlno rd rd 8 = 2zrn (r2 + 712)3/2 'i-.0 0

L0 = lim

.

(7.1.22)

Now the formula (7.1.16) is demonstrated. Imposing the condition (7.1.10), we deduce the following integral equation {n2(xo) + li2En2(xo) + n2(xo)]}f (xo)+

+T7r

A Zf ff

(x) (

x-xo)no+fit[(y-yo)noo+(z-zo)noj da= 2 Ix - xo13

.

(7.1.23}

For the incompressible fluid it becomes Lf = f (-To) +

27r

A

f (x) (

Ix

no

xxo13 d a = 2n0, .

(7.1.24)

The integrals (7.1.23) and (7.1.24) are singular.

7.1.4

The Discretization of the Integral Equation

We shall approximate the surface E of the body by a set of triangular panels TJ (j = 1, ..., N) and we shall approximate on every panel Ti , the function f by the value f; of the function in the center of mass G2 (x°) of the triangle. The equation (7.1.23) reduces to {(no)2 + f 2[('nV)2 1

//r

+ (n°)2]}f(xo)+

(x - xo)n? +,02[(y - yo)ny + (z - zo)n?}

Ix-xo13

0 da-21nx.

256

BIEM. THREE-DIMENSIONAL AIRFOIL

Imposing this equation to be satisfied in the centers of mass GG, i.e. putting xa = x°(i = 1, ... , N), we obtain the linear algebraic system N

a,f,+EA,jfj=b;, i=1,...,N,

(7.1.25)

i-1

where we have no summation with respect to i. We denoted

a = n2 (mg) + p2n?(x°) + #2n2 tx°), bi = 2 n (x°) A=i

= X,in.(x9) + O2Yjnv(x°) + Q2Z,jn:(x1

(7.1.26)

(7.1.27)

0

X`, = 2.-r, 1 M

W1

13da.

(7.1.28)

For determining the quantities n(x°) and X,j we shall denote by mil, xt2 and x,3 the vectors of position of the vertices of the triangle T{ ; we choose the sense on the sides of the triangle such that the normal n(x9) is positively oriented towards the interior of the body. Taking into account the definition of the vector product, we obviously have

n(xo) =

(x12 - Wit) X (X,3 - Oil) 2S;

'

(7.1.29)

Si being the area of the triangle Ti expressed by means of the coordinates of the vectors x,l, xi2, x13. The integrals X;j are singular when i = j. We consider at first the

case i 0 j. Denoting by xjl, xj2i xj3 the vectors of position of the vertices of the triangle T1 j, we shall consider the parametrization of the triangle

x=xjl+(xj2-xj1)Ai+(xj3-xjl)A2.

(7.1.30)

F\xrther we shall proceed like in 5.3. Introducing the polar coordinates by means of the formulas

Al = rcos0,

A2 = rsin8 0 < 0 < r/2, 0 < r < p,

(7.1.31)

where p is defined (fig. 7.1.2) by the relation (cos 0 + sin 0)p = 1

(7.1.32)

and denoting e(O) = (xj2 - xj1) cos 0 + (xj3 -

0,

(7.1.33)

257

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

Fig. 7.1.2.

we deduce

x-x9=xj1-x°+re(O) (7.1.34)

I0 -x,012=ar2+br+c, where

a = lel2, b = 2(xjl - x°) e, c = jxj1

- x912 (7.1.35)

6 = b1- 4ac = -4(xj; - x9)e

- (vjl

<0.

For the element of area of the triangle Tj one obtains (7.1.36)

d a = 2Sjd Al d A2 = 2rSjd rd 8,

S; being the area of the triangle. Hence,

X;; = S. f"21(xii - x)I1(e) + e(e)I2(8)]de where

_

rdr

P

Jo (ar2 + br + c)3/2

=

2bp + 4c 6(ap2 + by + c)1 2

_

(7.1.37)

4f 6

(7.1.38) P

alt (O) - f (art + br + C)3/2 with the notations

J - IP

dr

are+br+c

-

2

%r-7

J1 - b11 - C/C ,

( arctan Zap + b vr-7

17)

258

13IEM. THREE-DIMENSIONAL AIRFOIL

K

_

dr

)r°

J0 (art + (n + c)3/2

4ap + 2b 2b 6(ap2 + by + c)1/2 + W c

Ones employ the formulas (7.1.29) and (7.1.37) for calculating the coefficients A1.

7.1.5

The Singular Integrals

The integrals (7.1.28) are singular when i = j. Following the model from 5.3 we shall write: T,j = T(12)

+ T(23) + 7 31)

(7.1.39)

where T!kl) is the triangle GjPkP, and we shall consider the parametriza-

tion of the triangle T,(12) putting

x - x° = (x,1 - x° )J11 + (xj2 - x°)a2 .

(7.1.40)

Passing to polar coordinates and denoting

E12=(xj1-x°)cos6+(xj2-x°)sin8,

(7.1.41)

we deduce

x - x,, = rE12,

Ix - x°I = rIE121

(7.1.42)

Utilizing also (D.2.3) it results X(12)

»

=

0

1

2n

T.02) I x

X0 13

da= (7.1.43)

=

1

Ir ' J JO 3S

X?j =

s/2 E12 (0)1np(8)

UU

(E121'

(7.1.44) + These expressions are utilized for determining the coefficients A.,,.. X123)

7.1.6 The Velocity Field. The Validation of the Method The numerical values of the velocity field are obtained from (7.1.16) with the formula N

2v(x°) _ - f

X y f3 j=1

.

(7.1.45)

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

259

For testing the method we shall use the exact solution in the case of the sphere placed in an uniform incompressible stream. We know (see, for exaunple, [I.11 J, p.163) that if the sphere has the radius a and the center in the origin of the coordinate axes, and the uniform stream has the velocity Uk , then the potential of the perturbed flow is

a3)

(7.1.46)

and\\\V1

Calculating V1 = grado = U,,(i+ v) which results from (7.1.1) under the hypothesis that the fluid is incompressible (A = 1) and using for tv the values (7.1.45), it results the comparison from figure 7.1.3 for IVI I /U,,. We notice that the approximate method and 1.

V 1.0

Exact

...._. I talho d +

a

ao Fig. 7.1.3.

the exact one give very closed results. In the paper [7.2], that we have utilized for writing this subsection, we may find the approximate results for the ellipsoid for various angles of attack and for various Mach numbers. We may also find approximate results for the wings whose cross section is a NACA - 64 - A - 008 profile.

7.1.7

The Incompressible Fluid. An Exact Solution

We have seen that, for the incompressible fluid the integral equation is (7.1.24). In this subsection we shall determine the exact solution of

DIEM. THREE-DIMENSIONAL AIRFOIL

260

the equation in case that the perturbing body is a sphere (7.4).

INg. 7.1.4.

Considering that the points on the sphere (x and Qo(xo) have the coordinates (fig. 7.1.4): X = R(sin q, cos 92, sin ql sin 42, cos ql )

0
(7.1.47)

xo = R(sin qi cos 92, sin qi sin gZ, cos qi)

we deduce for no (the inward pointing normal to the sphere in Qo) no = -(sin g° cos q2, sin q, sin qz, coe g°) .

(7.1.48)

We have therefore:

f (x) - f (ql, q2),

f (xo) - f (q°, q2) (7.1.49)

da = R2 sin gldgidq2,

no -- no = - cos qo

and Ix - xo12 = 1X12 + Ixo12 -- 2x0 xo = 21122(1 - cos9),

(7.1.50)

cos 6 = sin q1 sin q, - cos(Q2 - q2) + cos ql cos go.

(7.1.51)

where

Hence it results

(x - xo) (no) - (x-xo) (x - xo) no Ix-x013

da=

= R(1-cosB),

(1 - cos 8) sin gldgl dq2

-sing1dgldq2

2(1-cos0) 2(1-cos9)

2 1-(oO) (7.1.52)

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

261

With these formulas the equation (7.1.24) becomes 2w

I

f(gi,gz)+47 rJo Jo

f(gi,g2)sin2(igld)q2

=-2cpsq0j.

(7.1.53)

the sign "' indicating that one eliminates the vicinity of the point Qo, i.e. 0 = 0 according to the definition (7.1.17). Denoting 1

K(qi, qq, q1, 42) _

4ir

2(1 - cos 9) '

(7.1.54)

D = (0, 7r) x (0, 2r),

we deduce that the integral operator of the equation which determines the density f (q,, 92), is a bounded operator with respect to the uniform convergence norm for real functions, continuous on the closed rectangle b. This boundedness follows from the property

KsingldQ1dg2 = 1.

(7.1.55)

We sludl also prove that the integral operator has the invariance property

J

j(cosqi)Ksintdqidq2 = cosq°.

(7.1.56)

3 the change of variables For proving these properties we shall perform (qj, q2) -- (0, A) defined by (7.1.51) and by the following relation, which is a direct consequence of the sine rule from the spheric trigonometry

sin A sin 8 = sin q1 sin(g2 - q2),

(e, A) E D .

(7.1.57)

Hence we choose the spherical coordinates relative to the point x for which, instead of the Oz axis we take the direction -no, and instead

of the xOz plane we take the plane of the versors k and nO (fig. 7.1.5). In this way, for the element of area in the generic point Q one obtains d a = R2 sin Q1dQ1dQ2 = R2 sin 6dOdA.

(7.1.58)

Using the cosine theorem from the spherical trigonometry, we obtain cos q1 = cos q° cos 0 + sin q0 sin 9 cos A. Hence,

r

1 ID

KsingtdQldg2 = lID Ksin8d9dA

(7.1.59)

BIEti1. THREE-DLMMENSIONAL AIRFOIL

262

Fig. 7.1.5.

(2" d A /"` Jo o

=I

sin 8d 9

2(1 - cos9)

J f(cosqi)Ksinqidqidq2 = 1 (cos

Jo

n

cos) d A) / (sin 8)K sin dd 9 = 0

/R 2-(

J(cosqi)KsinodldA =

J ID cos 6K sin Bd9d.1+

r2x

+(sin q°) (

-_ 1 '

sin8cbs9

de_

1

qo

qo) 0

3 (1 - CM e) Utilizing the identity (7.1.56), we may solve the equation (7.1.53) by

means of the successive approximations method. Indeed, putting

f1 _-2cosg0,, we deduce

f2 = -

f

fnf1(41,92)K(4i,42,91,92)sing1dg1d42

=

_ -3(-2cosgi) fs =

-If

U

(,)2 (2(41,42)h'sin41d41d92 =

2cosgi)

263

THE FIRST INDIRECT METHOD (SOURCES DISTRIBUTIONS)

I

fk+I = `J

f

Jofk(q ,gs)Ksingtdqidqs =

(3

\k J

(-2cos401)

whence 00

f(gl,g2) =Ffk+1(gi,92) _ k=o 00

= (-2oosgt)

(7.1.60)

(1)k

=

-i cogt.

k=O

This is the exact solution for the spherical obstacle.

7.1.8

The Expression of the Potential

For testing the integral equation, we must prove that the potential calculated with the density (7.1.60) coincides with the exact potential given by (7.1.46). The potential in the generic point M(F) determined by a source having the intensity f placed in Q(x) is according to (7.1.13),

WW _ 4r Ix

i

EI '

whence it results that the potential determined by a continuous distribution, of intensity f (c), on the surface E, will have the expression

w(f) = --I Jf IE IM-41 da. (x)

(7.1.61)

In the case of the sphere E = S(O, R), with the density (7.1.60), the potential is

Arr) =

3 87r

cos

fJ Ix - qlEI d a .

(7.1.62)

Considering the point M(L) exterior to the sphere, having the coordinates

f = r(sin gicosgs,sinq°singscoo q),r > R, we get

Ix - Cl' = R2 - 2rReoe6 + r2

(7.1.63)

BIEM. THREE-DIMENSIONAL AIRFOIL

264

where cos O is defined by (7.1.51). Taking (7.1.58) into account, it results 3

8n 3

R2

cosglsing1dqldQ2 - '?r R cos 8+ r ID

f

c

2

8?

R-

l Jj,

qi sin Od Od A

R - 2rR cos 0 + r-

On the basis of the formula (7.1.59) and the periodicity of cos A, we obtain 3 V 3

2m

R2

_ $R

cosrqo1

ff

d A fo

8n(coc; gi)

cosOsinOd0

J

r

=

[(R2 + r2)(R2 - 2rRcvsO + 1.2)1/2_

o

-(R2 - 2rRcosO + r2)1/2 I sinOd0 = 3 16

lr- RI)- -(r+R)3- Ir-R131

and finally, since r > R,

2 T6 This is also the perturbation potential given by (7.1.46).

(7.1.64)

THE SECOND INDIRECT' METHOD (DOUBLET DISTRIBUTIONS)

7.2

265

The Second Indirect Method (Doublet Distributions). The Incompressible Fluid

7.2.1

The Integral Equation

The theory in this subsection follow the paper [1.11]. We denote by po(x) the potential of a known flow in the entire space. We assume

that this flow is perturbed by the presence of a body whose support is the simple connected and bounded domain D. We assume that the boundary of this domain, E is smooth, such that we may apply Poisson's

formula. We denote by n the outward pointing normal to E. The potential po is a harmonic function, such that we may write (1.11) (D.3.2)

J JE [fi(x) n !x 1 F{

1doi

2(pe(4),

IX-41 tin

4ED

to

(7.2.1)

EE

EE,

E representing the exterior of D. When we write F E D we understand

that the point Al whose vector of position is t belongs to D. Now we assume that the perturbation is produced by the body having the support D and let So(x) be the potential of the perturbation in E. According to the equation of continuity (7.1.8), cp will be a harmonic function which has to satisfy to infinity the damping condition line

S'(F} = 0.

141-Under these conditions, V will have the representation (D.3.10) from [1.11)

2T

J JE

- (x

x7-

x'

tEE

1

1

t)

0 T+p(x)1 d a=

So(t), 0

the normal being the same like in (7.2.1).

F= xo E

LED,

(7.2.2)

266

BIEM. THREE - DIMENSIONAL AIRFOIL

Adding (7.2.1)3 to (7.2.2)1 we obtain W(E) =

f

4a J

{ [,Po(x) + v(x)]

1 41

Ix

(7.2.3)

(s o+9)(x)}da, t E E.

1 I

Determining V in order to have A

A

(7.2.4)

W E E,

co(x) =

the representation (7.2.3) becomes

00 =

f

[PO(x) + W(x)]

Ix

1f

du.

(7.2.5)

Subtracting (7.2.1)2 from (7.2.2)2 one obtains ,P(X0) - Po(xo) = .1

g (Ix 1

f [,Po(x) + P(x)J n

Introducing the function

j u:

x01

)

E E.

d a, xo (7.2.6)

E -+ R by means of the formula

p(x) = fi(x) +

(V)x E E,

(7.2.7)

the equation (7.2.6) reduces to .U(X0) _

2zr J

Jl U(x)

,

\ Ix 1 xol d a =

JPo(xo), (V)--o E E, (7.2.8)

and (7.2.5) to

da. (7.2.9) WW = 4r ,/ / The equation (7.2.8) is the integral equation which determines the function µ, and the equation (7.2.9) shows that this function is just

h (); (Ix 1

the doublets density E which replaces the body. We may write the equation (7.2.8) as follows

lt(xo) +

2a J Js

µ(x)n(x) IxX

x013 d

a = 2cp(xo), (d)xo E E, (7.2.10)

n representing the normal pointing towards the fluid (the outward normal to E). This equation which determines the doublets density, is similar to the equation (7.2.21) which determines the sources density. Only the free terms and the normals from the kernels are different. In (7.1.24) it appears n(xo) while in (7.2.10) this is replaced by n(x).

THE SECOND INDIRECT METHOD (DOUBLET DISTRIBUTIONS)

267

The Flow past the Sphere. The Exact Solution

7.2.2

Considering the problem from (7.1.7), i.e. the integral equation in the case of the uniform flow with the velocity U,,,k past the sphere S(O, R) and utilizing the formulas (7.1.47) we have:

oo= -- =Rcosq°

(7.2.11)

such that the equation (7.2.10) becomes ir

i

II ( g , q0 )

±

47r

2w

j1 0

II (gl, 92, ) s1

-

-la- gc

V 2(l

d. B )

= 2R cos q a .

(7 . 2 . 12)

This equation differs from (7.1.53) only by the right hand side term. The equation (7.2.12) has therefore the exact solution

it = 3 R cos q1

(7.2.13)

.

For testing the integral equation (7.2.10), we shall prove that the potential ip, obtained from (7.2.9) with the density (7.2.13), coincides with the exact perturbation potential (7.1.64). Indeed, (7.2.9) becomes

- 4-r1 I Jz

u(x) '

4 d a. ix 413 Using the notations (7.1.47) and (7.1.63) we have cp(l;) =

(7.2.14)

(x - la) n = R - r cos 9

x F = Rr cos 6,

da = R2 sin glclgldq2,

such that, utilizing (7.2.13), we deduce

1 3k 1P 4ir 2

0 2n'

(R-rcos0)cosgjsingtd

(R2 - 2Rr cos 9 + r2)3/2

J0

91 d q2.

Passing to the variables 0 and A and taking the formulas (7.1.58), (7.1.59) and the periodicity of the function cos A into account, it results 3

'P = - 4 R 3cos ql0

fW sin O cos 0(R - r cos 9) (R2

J

- 2Rr cos 0 + r2)3/2

Performing the change of variable 0 -+ u : it = (R2 - 2Rr cos 0 + r2)1/2

we deduce

f

x sin0cos0(R -

Jo (R2 - 2Rr cos 0 + r2)3/2 d and finally, (7.1.64).

6

2

- - 3r2

d0,

268

I3IEM. THREE - DIMENSIONAL AIRFOIL

7.2.3

The Velocity Field

Since the potential has the expression (7.2.14) we deduce:

v(t) = grade = - 4 Jj;(x)gracI£ 47r

n(x)] d a =

L lx - 41

l I µ(x) (n grad) 1 Ix -

X12

Ida

and finally

vW =

4zr !

j1`(x) [n(x) - 3(x - ,)(IX -

I2nl

Ix - 413

(7.2.15)

This formula gives the velocity field in the fluid. We deduce that far away

the kernel has the order I4I-3, as it is natural to be (see for example (6.1.16) from 11.11]). On the boundary of the body, the integrals from (7.2.15) have strong singularities. We have therefore to transform these integrals.

7.2.4

The Velocity Field on the Body. N. Marcov's Formula

In the beginning we have to notice that the double layer potential of constant density equal to 1 defines a piecewise constant function whose gradient is obviously zero in the continuity points. Indeed, from (7.2.1) it results: 1

1,

fr

x 1 t)

TT(1)da=

1/2, 0

4ED E

(7.2.16)

EE,

and for the gradient we have 1

4n

ff n(x) - 3(x -)

da

Ix - tp2 ] Ix - CI3

(7.2.17)

in E (and D). Multiplying this identity by the constant jt(xo) and subtracting it from (7.2.15), it results

J

P( X)

!t(xo)

Ix-EI

- n n-3(:c-F) (xIx-412

da

Ix-EI2

(7.2.18)

TILE SECOND INDIRECT METHOD (DOUBLET DISTRIBUTIONS)

269

for AI(D) in E.

For calculating the velocity when AI(F)

Qo(xo), a point on the boundary E, we shall denote again by or the portion cut from E by the sphere with the center in Qo and the radius s and we shall use the definition of Cauchy's principal value lim

JJa-

JJE

(7.2.19)

if it exists. With no the outward normal in Qo, we put t = xo +qno (q > 0). For the limit value of the velocity in this point, we have: =bin

tje(xo) = 1-0 v(xo + qno) = It + 12,

(7.2.20)

for every F > 0. We denoted

11 = lira

1 if

ti--o 47r

-3(x I2

no4r

12(x) -11(xa) _o

x - 41

da

j

Ix - 412

Ix _ X12 ' (7.2.21)

11 !l( Ix -

4Ixa) [n-

-3(x-4) (x - t) - n

da

Ix-XI2,

Ix-£12

where = xo + q no. The first limit and the integral may be interchanged because the integrand has no singularities on E - a. Hence

one obtains 1t

=

/e(x) _ 14(x0)

1

47r

E-o

Ix - xol

[n (7.2.22)

(x - xo) n Ix-xo12

da ix-x012

For every e > 0 this integral exists if µ(x) satisfies Holder's condition. So, we deduce v(xo) = Jim (1) + 12) = w(xo) + link 12 ,

(7.2.23)

270

RIEM. THREE - DIMENSIONAL AIRFOIL

where

w(xo) =

1

Jf_

/L(x) - JL(xo)

Ix-xol

-3(x - xo)

L

(7.2.24) J

Ix - xo12

da IX - X012

For calculating the last limit from (7.2.23), when s is small enough,

we may replace a with its projection A on the tangent plane in Qo and we shall utilize (7.2.17). It results that we have

x - t = r(cos Oio + sin Ojo) - i'no,

Ix

- t12 = r2

+.q2

,

(7.2.25)

µ(x) - /&(xo) = (V /)(x0) ' (x - xo) + ... _ = rvp(xo) (CDs NO +Sill 8jo, where V1z(xo) = (V/c)(xo), according to the usual notation in Analysis. Considering the scalar product Vy - (cos Bin + sin Ojo) ,

in the basis io,)o we obtain the identity [V -(cos Oio + sin 8jo)](cos Oio + sin 8jo) =

= [(Viz io)cos8+ (V/z jo)sinO](cos8io + sinOjo) = (7.2.26)

= (Vp - io)(cos2 Oio + cos8sin 6jo)+

+(Vp . jo)(sin Ocos 8io + sine` 0jo) .

Noticing that some terns vanish after integrating with respect to 0, from 12 it remains 3 12

t

2-x

r- 0 47 r Jo JO

r3)?

(r2 + i,2 )5/2 X

(Qµ io) x (cost 8io + cos O sin 8jo)+

drdO,

x

+(QJJt jo) x (sill 8 cos 8io + sine 8jo)

I2 = lim 4

J'

(r2 r3'1

[(VJt - io)io + (ViA - ?o)7o)Id r .

't'ilt: l)IRECT ME IIOI). THE INCOMPRESSIBLE FLUID

271

One obtains (Vi)(xo) in the square bracket and we take it off from the integral. So, t I., =

r3

17

(r2+ 2)b/2dr=

Op(io)

Hence. taking (7.2.23) into account, we obtain the velocity on E by ineans of the formula

v(xo) = 2(V1)(xo) + w(xo),

(7.2.27)

w(xp) being given in (7.2.24). This formula was obtained by N.Markov in a unpublished paper. Since P(x) and Qo(zo) from (7.2.25) belong to the plane which is

tangent to r in Qo, we deduce that Vp(zo) is in the tangent plane. Hence, the boundary condition

(i+v).n=O pe E determined the following integral equation: 1

-1s(xo) f Fi(x) Ix - xol

4A E

da =-n=, VxoEE -3(x-xc) no Ix-moll Ix-xo12 which is an alternative to (7.2.8).

7.3 7.3.1

The Direct Method. The Incompressible Fluid The Integral Representation Formula

For writing this subsection we used the paper (7.3]. Since we study the same problem like in the previous subsections, we shall utilize the repre-

sentation (4.6.1). the equations (4.6.2), the boundary condition (4.6.3) and the condition to infinity (4.6.4). The difference is now that the prob-

lem is three - dimensional. In (4.6.3), C will be replaced by E, the surface of the perturbing body B. For avoiding the singular integrals, we shall replace the equations (4.6.2) by the equations

div(v-c)=O, rot(v-c)=O,

(7.3.1)

272

BIEM. THREE - DIMENSIONAL AIRFOIL

c being a vectorial constant. The system (7.3.2) div v' = 6(x - xo) , rot v' = 0 for every point xo E D + E, D representing the domain occupied by

the fluid (the exterior of the body B), defines the fundamental solution V* =

x - xO 41rIx-x013 1

(7.3.3)

From the equations (7.3.1) we deduce the identity

4,

[ f div (v - c) + g rot (v - c))d v = 0

(7.3.4)

for every two functions, or regular distributions, f and g. We denoted by Do the exterior of B, bounded by a sphere S(O, R), R being great enough, such that the body B is included into the interior of the sphere. Utilizing the identity (4.6.9)1 and the identity

rot [g x (v-c)]

(7.3.5)

and applying Gauss's formula, from (7.3.4) we deduce

( v - c) (grad f - rot g)d v = (7.3.6)

(v - c) (fn - (n x g)id a, E

n being the outward pointing normal on Do. Substituting successively

(f, g) -' (7 - v*, -9 x VI)

(7.3.7)

we find the projections on the axes of coordinates of the following identity:

J (v - c)div v'd v = LER {n (v - c)v' + [n x (v - c)] x v' }d a , Do

xoEDo+E (7.3.8)

273

THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID

proving in this way that it is correct. Taking (7.3.2) into account, it results

V(x0) - c = f

{n (v - c)v* + [n x (v - c)j x v' }d a .

(7.3.9)

+ER

The integral on ER may be written as follows

v)v+ (n x v) x vjd a - hR

c)v' + (n x c) x v*] d a .

IS,,

T he first term vanishes when R

oo, because of the condition

(4.6.4). For calculating the second term we use the spherical coordinates: R, 9, cp:

x = R sin O cos V

y = R sin O sin p

0<9<7r.

z=Rcos9, We obtain

lim. JR [(n c)v' + (n x c) x v')d a = C 47C

f

0

2A

f

A

sin 9d 9d W = c.

0

Now, the representation (7.3.9) becomes

v(xo) =

j{n

(v - c)v+ [n x (v - c)] x v}d a ,

xoED+E. Setting c = v(xo) = vo we obtain the representation formula V(x0) =

j{n. (v - vo)v* + [n x (v - vo)) x v' }d a

.

(7.3.10)

This is a regularized representation, valid both for xo in D and for xo on E. Obviously when xo E D, the integrand has no singularity. This property is also valid when xo E E because of the factor v - vo which vanishes for x -- xo. This representation formula is fundamental [7.1j. Utilizing the boundary condition (4.6.3), the formula (7.3.10) becomes

vo=1 E

(v-vo)) x v*}da.

(7.3.11)

274

7.3.2

RIEM. THREE - DIMENSIONAL AIRFOIL

The Integral Equation

The vector

F- nxv

(7.3.12)

which intervenes in the surface circulation

C= J nxvda.

(7.3.13)

E

introduced by Pascal and utilized again by R.von Mises 11.271, [1.20], has a great importance herein. We shall deduce the equation for P. To this aim we write the formula (7.3.11) as follows: (7.3.14)

vo =

After the vectorial multiplication by no one obtains

Fo = J[_(nz + n - vo)(n° x v') + (n° v')F(7.3.15)

-(n° F)v - (n v')Fo + (vo . v')(n(' x n)ld a. But, from the boundary condition n v = -it, and from (7.3.12) we

deduce v = -nxn - n x F such that n - vo = -non ' no - (n, n°, Fo)

(v' vo)(n° x n) = -n°(v` n°)(n° x n) - (n°, Fo, v*)(n° x n). (7.3.16)

One obtains therefore

Fo = J(._(ni - nn n°)(n° x v°) + (n, n°, Fo)(n° x

+(n° v')F - (n° - F)v' - (n v')Fo-n°(v' n°)(n° x n) - (n°, Fo, v)(n° x n)Jd a . Utilizing the double vectorial product formula

(UxV)xW

(7.3.17)

275

THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID

it results

(n, n°,

x v°) - (n°, Fo, v*)(n° x n) _

= no x ((n°, Fo, n)v' - (n°, Fo, v')nJ =

(7.3.18)

= no x [(n x v') x (n° x Fo)J = -(no ,n,v')(no x Fo). Hence, the equation (7.3.10) becomes

F°+

J,

[(n° F)v' - (n° v')F + (n v')Fo] d a+

+ no x Fo) f (n° x n) v'd a = j {n(v" n°)(n x n°)+ + (7.3.19)

this representing the integral equation of the problem. This is a regularized equation. The integrals are not singular. Although the denominators of the distribution v' vanish for x -+ x°i we have to take into account that in these points the numerators vanish too. Indeed,

no -.0,

7.3.3

Kutta's Condition

In the sequel we shall study the flow around lifting bodies (with smooth surfaces). In order to solve this problem, the classical theory, beginning with the lifting line theory (Prandtl 1918), introduces a vortical surface behind the body. In this way the flow in no longer irrotational in the exterior of the body, hence the above solution cannot be utilized. For defining the fluid How with the aid of the solution given herein, we shall consider a discrete distribution of vortices located in certain points from the interior of the body [7.3]. In this way, the external flow remains everywhere irrotational and it will be characterized by the solution given herein. Nloreover, the circulation C defined by (7.3.13) does not vanish on the trailing edge. It remains to discuss the location of the vortices. The Kutta-Joukowskicondition concerning the continuity of the pres-

sure on the trailing edge will be imposed, like in the two-dimensional

BIEM. THREE - DIMENSIONAL AIRFOIL

276

case, writing that the velocity in plane cross-sections, perpendicular to

the trailing edge, is continuous when P, and P, - Pf (4.6.21). Considering in figure 7.3.1 the point P, and the local frame n, a and

Fig. 7.3.1.

(3 = n x s and knowing that V = V,s, we deduce

n x V = V,/3.

(7.3.20)

Hence, the vectorial form of (4.6.21) is:

(n x V)(P,)+(nx V)(P,)-o 0 (7.3.21)

P P; - PI.

Taking into account that V has the form (4.6.1) it results the condition

+ j- k=0,

(7.3.22)

<¢>=q(P,)+ b(P.).

(7.3.23)

with the notation

7.3.4

The Lifting Flow

One knows from the theory of the two - dimensional potential in compressible flow that the lift is generated by a vortex of intensity rk placed in a point from the interior of the body. For example, in the case of the flow past the circular obstacle with the center in ;.o (see [1.11) p. 117), the potential Ua, f(z - zo)e-io +

l

R2 _

Z-20

1l

et° I

characterizes the non-lifting flow and the potential

r

27ri

In (z - zo)

277

THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID

defines the lifting flow, the lift being -pU,:1'. According to this model,

we shall try to generate the lift by means of some vortices from the interior of the body, the number of vortices being determined by the number of pairs of panels adjacent to the trailing edge for which we have to write Kutta's condition (7.3.22). To this aim we need at first the velocity field w' generated by a vortex line of constant intensity r. This will be determined by the system div w' = 0,

rot w' = r6(x) ,

(7.3.24)

w' being a distribution (for this reason it was marked by *).

For

determining the solution we apply the Fourier transform, like in the case of the system (7.3.2). Using the formulas (A.6.5) we deduce

axiv'=ir, whence, utilizing (A.6.9),

w'=rx(-

w' =rxVF-'

2

and then (A.7.10) and (7.3.3)

w` = v' x r.

(7.3.25)

If the vortex is located in the interior of the body, in a point having the vector of position xi, then

w(x) = v'(x - xi) x r,

(7.3.26)

and if there are L lines, since the equations (7.3.24) are linear, it results L

w'(x) = E v'(x - xk) x rk .

(7.3.27)

k=i

Returning to our problem and considering, for the sake of simplicity, a single line, we shall write the formula (4.6.1) as follows:

V=U_ (i+w'+v).

(7.3.28)

From div v = 0 and rot r = 0, taking (7.3.24) into account, it results

div v = 0, rot v = -ri(x - xh),

(7.3.29)

278

BIEMM. 'THREE . DIMENSIONAL AIRFOIL

and the equations (4.6.5) will be replaced by

div(v-c) =0 rot(v - c) = -ra(x-xl).

(7.3.30)

The identity (7.3.4) will be written as follows: Joo

[fdiv(v-c)+g.rot(v-c)]ctv=

,xl E D° ,x1 D°.

0

(7.3.31)

Since the point having the vector of position xl is in the interior of the body, we shall obtain a second equality identical to (7.3.4), such that all the consequences follow like above. We have therefore (7.3.10). From the boundary condition v - n = 0 we deduce (7.3.32)

such that the formula (7.3.11) becomes

vo=

R

xv'}da (7.3 .33)

xoED+E. The relation between v and F is now

v=-(n,, +wn)n-nxF,

(7.3.34)

and the integral equation (7.3.19) becomes

Fo+

JN

[(n° - F)v* - (n° - v')F + (n v')Fo]da+

+ (n° x Fo)

x n) v'd a + wn(xo)

I [(n° v')(n° x n)-

-(n°.n)(n° x v`)]da+J wn(x)(n° x v')da = E

= f, { n2(n° v*) (n x n°) + [n0. (n - n°) - n.,] (n° x t

v')}da, (7.3.35)

where, taking (7.3.27) into account, we have:

wn(xo) _ I no x v'(xo - xi)] r (7.3.36)

w,,(x) = [n° x v(xo - xi) ] . r. L

THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID

279

In (7.3.35) the dependence of F and r is linear. In the case of L vortices, the equation (7.3.35) remains unchanged and the expressions (7.3.36) become L

Wn(x0) _ F [no X v'(x0 - xk)] r 11

k=1

(7.3.37) L

W4(z) _ E In X v'(xo k=1

- xk)1

r.

In equation (7.3.34), F, r1i rL are unknown. For specifying the conditions (7.3.21) we shall write the product n x V as a function of the principal unknown F. From (7.3.28) and (7.3.34)

it results

nxV=Uoo(F+nzj-nyk).

(7.3.38)

The condition (7.3.21) will be written as follows 0

F + (nx -ny

0

(P,) +

[F+ (n,

(P;) = 0,

(7.3.39)

-ny

for every pair of panels from the trailing edge. It will result therefore L equations. These ones, together with the equation (7.3.35) discretized below, will determine the solution F, F1,... , r,. The numerical results are more accurate when the points P, and P; are closer from the corresponding point Pf from the trailing edge (see [7.31, p. 364).

7.3.5

The Discretization of the Integral Equation

Like above, one approximates the surface of the body by N panels (triangles and quadrilaterals) and we approximate on every panel III

the function F by the constant value F,,, that it has in the center G, (x9) (the collocation method). If we impose to the equation just

obtained from (7.3.35) to be verified in all the centers G,(x°) and we denote

X;;=J v'(x-x°)da

(7.3.40)

280

BIEM. THREE - DIMENSIONAL AIRFOIL

we deduce, for i = 1 N 1

Fi+ L.. [(ni Fj)Xji - (iii X ji)Fj + (nj Xji)Fi] + j=1

N

L

+ (ni x Fi) J(ni x nj) Xji + j

[(ni x x;k) . rk]' k=1

1

[(ni x nj)(ni Xjj) - (ni x X ji)(ni ' nj)] + E j=1

[(nj x xjk) rk] (ni x xji) =

+ j x1 k=1

{

(7.3.41)

- ni(ni.

,j=1

X ji)(ni x nj) + {n(rsi.nj)_n}(ni x Xji) }, In this system, the singular terms Xii disappear, because the integrals from (7.3.35) are regularized. Taking into account the relations

(nj Fj)X3i - (ni ' Xji)Fj = ni x (Xji x Fj)

(ni nj)X ji - (ni Xji)nj = nj x (Xji x N), we deduce that the system (7.3.41) may be written as follows N

L

LA,,F,f+EBk{lrk=Ci, i= 1,...,N j=1

(7.3.42)

k=1

where

AiiFi = Fi+FnEnj Xji+(ni x Fi)E(ni x nj) Xjie j#i

j#i

AijFj = ni x E(Xji x Fj), j#i Bki?rk = [rk(na x Xik)j

E(ni Xji)(ni x nj)+ ;oi

+ > {r. j#i

[ - (ni x Xik)(ni nj) + nj X Xjk] }(ni x Xji), 1

THE DIRECT METHOD. THE INCOMPRESSIBLE FLUID

281

Ci= nixE[n (njxXji)xn1-n Xji! i0i the unknowns being F1,... , FN,

r1, .... r1. To this system of N

equations, we add the equations (7.3.39) written for every pair of panels

adjacent to the trailing edge. The number L of vortices equals the number of pairs; in this way, the number of (N+L) equations obtained from (7.3.42) and (7.3.39) equals the number of unknowns F1,... , FN,

rl,... , ri.

For calculating the coefficients Xji (7.3.40) one utilizes either the quadrature formulas (for example, the Gauss-type formulas), or the analytical formulas given by Hess and Smith in [7.10]. Denoting by x, =21, x23, (24) the vectors of position of the vertices of the panel lIl which is a triangle (or a quadrilateral), with the definitions

Vill

- y)2 + ( k+1 - 4)2

(xik)2+(yiyk)2+ (Z, -k)2,

Tk= mk,k+1 =

'.)I + (yll +

(k+1

dk,k+1 =

+1 -

, ek = (xi-2'k)2+(zi-k)2 , hk = (xi-

)2(Yi-yk)2 ,

k

the formulas of Hess and Smith are

Xji = (Xji,Yji,Zji), +1 - Ylk

Xji =

dk,k+l

k=1

Yji = -

3(4)

-4+1

4s k=1 4 = -

-

` k In rk + rk+i - dk,k+1

dk'k+l

rk + rk+1 + dk,k+1

3(4)

1

41r

rk + rk+1 + dk,k+1

arctan

k=2

'mk,k+lek - hk (zi

-arctan mk,k+1 + ek+1 '- hk+1 (zi - k)rk+1 k)rk

In fact, Hess and Smith deduced the formulas assuming that the panels IIj are situated in planes parallel to xOy (4 = 0). An example is presented in [7.3].

Chapter 8 The Supersonic Steady Flow

8.1 8.1.1

The Thin Airfoil of Infinite Span The Analytical Solution

In this subsection we study the same problem like in 3.1. The only difference is that the unperturbed flow is assumed to be supersonic (M > 1). Hence the uniform flow, having the velocity Ui, the pressure p,. and the density p,, is perturbed by the presence of an infinite cylindrical body. The xOy reference plane represents a cross section and the leading edge is the origin of the axes of coordinates. The length of the projection of the airfoil on the direction of the unperturbed stream (which does not always coincide to the length of the chord) will be taken as reference length (fig 8.1.1). Utilizing the coordinates defined

Fig. 8.1.1.

by (2.1.1), we shall denote the equations of the profile determined by the xOy plane

y = hf(x),

(8.1.1)

284

THE SUPERSONIC STEADY FLOW

the functions h..(x) being defined on the interval

(0, 1J

and and

possessing first order derivatives. Taking into account the formulas (2.1.3), we deduce that the perturbation (which is obviously steady), will be defined by the system (2.1.32) and by the boundary conditions (2.1.33). From this system, taking into

account that the perturbation is plane, it results:

u = -p, v= + py = 0, M2pz + ut + vy = 0 ,

(8.1.2)

whence one obtains

k2vtt - vyy = 0,

(8.1.3)

with the usual notation k = M - 1. The boundary condition (2.1.32) becomes

v(x,±0)=ht(x), 0<x<1,

(8.1.4)

and the damping conditions at infinity lim (p, u, v) = 0.

(8.1.5)

The equation (8.1.3) is hyperbolic and the families of characteristics C+

and C_ are defined by the equations

x-k-y=c+, x+ky=c_.

(8.1.6)

Obviously, the first family consists of parallel half-lines which make the

angle µ with the Ox axis, the second consists of parallel half-lines which make the angle -µ with the Ox axis, where tan p

=1 (sins =

,

oos p =

(8.1.7)

For determining the characteristics passing by a certain point, we write that the coordinates of the point verify the equations (8.1.15) and and then we determine the constants c+ and c_. For example, the characteristics detaching from the leading edge have the equations x - Icy = 0 and x+ky = 0. We assume that the profile is in the interior of the angle having the opening 2µ which is determined by these characteristics. It is well known that the equation (8.1.3) has the general solution

v(x, y) = F(x - ky) + G(x + ky),

(8.1.8)

such that from (8.1.2) and (8.1.5) it results kp(x,y) = F - G

(8.1.9)

ku(x, y) _ -F + G.

'I'III: TIIIN AIRFOIL OF INFINITE SPAN

285

'I'll(- function F is defined on the characteristics from the superior half-plane, and the function G on the characteristics from the inferior half-plane. We have in view the characteristics detaching from the segment [0, 1) which replaces the profile that represents the source of perturbations. The solution from the strip I will be determined by F(x - ky) which will be denoted by F+(x - ky) and the solution from the strip II will be determined by C(x + ky) which will be denoted by F_. (x + ky). Hence the solution has the shape

vf(x,y) = FF(xT ky), kpf(x, y) = ±F±(x T ky),

(8.1.10)

ku±(x,y) = T Ff(xT ky), the upper sign corresponding to the solution from the y > 0 half-plane and the lower sign to the solution from the y < 0 half-plane. In fact one may demonstrate that if upstream of AOA' the flow is not perturbed, then the solution of the equation (8.1.3) has necessarily the forum (8.1.10). Indeed, if the assumption is true, it means that OA and OA' are lines of discontinuity where we have to impose the jump relations (1.3.8), (1.3.9) and (1.3.10) Op1V1nO = 0,

Opl'IVln +P1nO = 0,

(8.1.11)

with the notations (2.1.1) and (2.1.3). Taking into account the order of magnit.ucle of the perturbations, by linearization it results

=0, Ovn,, +pnO =0.

(8.1.12)

On the line OA we have n = (sin p, - cos µ). Taking into account that in front of OA and OA' the perturbation vanishes, from (8.1.12) we obtain:

0=using-vcosa+psina (8.1.13)

U=vsina - pcosa, the relationships being imposed on the line x = ky. Replacing here u, v and p given by (8.1.8) and (8.1.9) and taking (8.1.7) into account, from the second condition we find C = 0. Since in the linear steady theory we have p = M2p, it follows that the first condition is identically verified.

In the sauce way one demonstrates that in the zone II v = G(x + ky). Hence the solution of the equation (8.1.3) has the form (8.1.10).

286

THE SUPERSONIC STEADY FLOW

Imposing the boundary conditions (8.1.4), it results

F±(x) = 14(x).

0 < x < 1.

(8.1.14)

These relations determine the functions F±(x) on the segment 10, 11. In the exterior of this segment we take F±(x) = 0 because v and p are continuous functions on the Ox axis in the exterior of the segment [0,11 (it results from (8.1.12), setting n = (0, 1)), such that F+ = F_, F+ = -F_. Hence the perturbation zones are I and II. For c E [0, 1], on the half-line having the equation x - ky = c we deduce: v(x, y) = h+(c), and on the half-line x + ky = c

kp(r, y) = h+(c),

(8.1.15)

(8.1.16) u(x, y) = h' (c), kp(x, y) = h' (c) . Along a characteristic, the velocity and the pressure take the values that they have in the point where the characteristic intersects the segment [0,11.

In the sequel we shall prove that all these results may be obtained directly with the fundamental solutions method, as we could expect, taking into account that this is a global method.

8.1.2

The Fundamental Solutions Method

We know that. the perturbation produced in the uniform stream hav-

ing the velocity Ui, the pressure p,., and the density po, by a force density (fl, f2) uniformly distributed along an axis which is parallel to Oz and intersects the xOy plane in the point of coordinates (F, 0), is given by the formulas (2.3.30) and (2.3.31). Replacing the profile from figure 8.1.1 with such a density applied on the segment [0, 1) from the Ox axis (i.e. on the strip whose cross sect ion is the segment [0, 1] from the Ox axis), the perturbation will be given by the formulas i

2kp(x, y) = in [ - fl(E) + kf ()sign y] b(xo - klyl)d (8.1.17)

2v(x, y) =

J0

[

- fl ()sign y + k f

6(xo - klyl)d

Taking into account the property of the distribution

f

f (t)6(x

- )d

=

f (x), 0,

if:

ifx E (a, b) ifx E C(a, b),

(8 . 1 . 18)

287

THE THIN AIRFOIL OF INFINITE SPAN

we deduce that if x - klyl = c E (0,1), then we have: 2kp(x, y) = -ft (c) + k f (c)sign y (8.1.19)

2v(x, y) = - f, (c)sign y + k f (c) .

Conversely, if c E C(0,1), then p = 0, v = 0. The set of points for which x - kly) = c determines the characteristic half-lines

x f ky = c

(8.1.20)

which detach from the point x = c belonging to the segment (0,1). On these half-lines, the pressure and the velocity have constant values, equal to their values in c.

8.1.3

The Aerodynamic Action

From the representation (8.1.10) we deduce that the jump of the perturbation pressure on the profile

p(x, -0) - p(x, +0) is given by the formula

kOP1 = -[F+(x) + F-(x)] = -[h+(x) + h'-(x)]

(8.1.21)

which results from (8.1.14). In this way, the lift and moment coefficients CL

=

L

CM =

(1/2)po,U2Lo'

M (l/2)poU2Lo'

(8.1.22)

are given by the formulas CL = - 2 1'[W+ (x) + h'_ (x)Jd x CA! = -

2

k

(8.1.23)

rx[h+(x) t

+ h'(x)]d x

the moment M being calculated with respect to the origin of the axes of coordinates.

We notice from the formulas (8.1.10) and (8.1.23) that the linear theory cannot be applied in the vicinity of the value M = 1(k = 0).

288

THE SUPERSONIC STEADY FLOW

As a first example, we shall consider again the case of the flat plate having the angle of attack e with E < y (fig. 8.1.2). Since the equation of this profile is y = -Ex, we deduce h.4. = h' = -E whence

v = -E, ku = ±e,

kp = T- e,

(8.1.24)

the total velocity and pressure being A

Pig. 8.1.2.

vl

U [CI± ) i - Ej]

P1 = poc

,

_T P00U2 k

.

(8.1.25)

For CL and cM we obtain cL =

4e ,

T

2s CM=T.

(8.1.26)

These formulas have been obtained for the first time by Ackeret in 1925. The torsor of the aerodynamic forces reduces to a lifting force L, applied in the middle of the chord of the profile, force which tends to rotate the airfoil in the trigonometric sense. The graphic representation of the function cL(M) (fig. 8.1.3) shows

that the linear theory is not valid in the vicinity of the value M = 1, because the lift cannot be infinitely great. One estimates that the validity of this theory begins from approximately M = 1, 2 and finishes to approximately M = 2, 5, because the lift cannot be extremely small.

For the zone M = 1(0,8 < M < 1,2) one elaborates the theory of transonic flow (see Chapter IX), and for M > 2, 5, the theory of hypersonic flow.

289

THE THIN AIRFOIL OF INFINITE SPAN

CO

O Fig. 8.1.3.

8.1.4

The Graphical Method

This method relies on the following fundamental theorem: in every point the perturbation velocity is perpendicular to Mach's line (the characteristic line) which is passing through that point. The demonstration results at once if we take into account that the characteristic lines have the versors i of coordinates (cos p, ± sin µ) and we utilize the representation (8.1.10) and the formulas (8.1.7). Indeed, we have

v-if=uoosµfvsinµ=

kF*

±Ftj=0.

(8.1.27)

Let us show for example how we can utilize this theorem for obtaining the graphical representation of the perturbation in the case of the flat plate. We consider at first the region I from the 8.1.2. In an

arbitrary point M from OA we draw the vector MP = Ui and a vector V1 parallel to the plate and having an arbitrary magnitude; the perpendicular on OA which passes through the vertex P of the vector

Ui determines the magnitude of the total velocity V 1 = MQ; the perturbation velocity Uv is given by the vector PQ. For determining the perturbation velocity from the region II, in a certain point M' from OA' ones draw the vectors M'P' = Ui and V'1, the last being parallel to the plate and having the magnitude unprecised yet; drawing from P (the vertex of the vector Ui) the perpendicular line on OA' we determine the vector V' = M'Q' and then the perturbation

Ua,v=P'Q'. Beyond FB and FB' the flow becomes again parallel to Ox. One knows the non-linear solution of the supersonic flow past a convex dihedron(the Prandtl-Meyer fan). See for example [1.21], [1.34). In the framework of the non-linear theory, the velocity which is parallel

290

THE SUPERSONIC STEADY FLOW

to Ox before OA , becomes at last parallel to the plate, crossing the Prandtl-Meyer fan. In the framework of the linear theory this fan is reduced to the half-line OA.

8.1.5

The Theory of Polygonal Profiles

In the case of polygonal profiles, the solution may be easily determined starting from the solution for the flat plate. The solution, which obviously is piecewise constant, may be analytical, graphical or mixed.

Ui

Fig. 8.1.4.

We shall consider at first the profile from figure 8.1.4 (E < p) with and e1CO8E1 + e2COSE2 = 1. According to the linear theory, the last relation will be replaced by el + t2 = 1. From the formulas (8.1.14) it results

OP = el, PQ = e2

F+=-El, 0
f1COSE1^'e1 <x<1,

(8.1.28)

FI=E, 0<x<1. We deduce, k Up0 =

E+E1, C + E2,

(8.1.29)

291

THE THIN AIRFOIL OF INFINITE SPAN

whence CL =

I(e+E1e1 +E2e2), CM =

a

kE2(1- e,).

(8.1.30)

Further we shall determine the pressure in different zones starting from the solution for the flat plate. Taking (2.1.3) and (8.1.24) into account, we deduce in zone I that P1 = Poo - POQU2

k

.

(8.1.31)

In zone II we take into account that the deviation with respect to the direction of the uniform stream from zone I is 62 - E1. Using again the formula (8.1.24) it results P2=PI-PI

V

(8.1.32)

Using the formulas (2.1.3), we deduce P1V2 = p ,,,,U2 +... Taking into account that the factor (E2 - E1) has the order of magnitude of the perturbations, in the framework of the linear theory we have to retain E E E P2=P1-Po,U2 2k 1 =Poo_PooUT

In zone III we have: p3 = poo +

po0U2

.

(8.1.33)

(8.1.34)

For the jump of the pressure we get the formulas (8.1.29). In fact the formulas (8.1.33) and (8.1.34) coincide with the formulas given by the analytic method. We shall calculate now, in a different way, the resultant of the pressure on the wing. The perpendicular to the plate OP oriented towards the body has the director cosines

nl =

coos (El +

2) , - sin (El + 2 )) _ (-E1i -1).

(8.1.35)

In the linear approximation, the perpendicular to the plate PQ has the director cosines (-E2, -1), and the perpendicular to OQ pointing towards the body (e, 1). Hence the resultant of the pressure is R = (pit, n 1 + p2e2n2 + p3n3)Lo

(8.1.36)

We took into account that the real lengths of the plates are I, LO, 12L0, L0.

For the lift we obtain the formula 2

L = Ry - paoUk'0(e+f1e1 +E2e2).

(8.1.37)

292

THE SUPERSONIC STEADY FLOW

In the framework of the linear theory the drag (8.1.38)

R. = (E - 1191 - e2E2)LoPoo

vanishes because, from the projection of the contour OPQ on the Oy axis we deduce E = 11E1 + 12E2

In fact this is a known result. The drag has the order of magnitude two CD =

2 (92 + E1t + 6212) k

(8.1.39)

In the sequel we shall consider the polygonal profile from figure 8.1.5,

symmetric with respect to the OR axis. We assume that all the E-s are small. Obviously, the profile is considered to be in the interior of Mach's angle with the vertex in the origin. We deduce like above that in zone I the pressure is given by the formula P1 = Poo +

p00U2

(8.1.40)

E,

in zone II by the formula

86

Fig. 8.1.5.

k E2

p - p00U2

,

(8.1.41)

P3=Pa-PzV2E3 kE2 =Poo-PooU2k ,

(8.1.42)

P2 = P1 - P1VzE1

k

in zone III by

293

THE THIN AIRFOIL OF INFINITE SPAN

in zone I', P1

=Poo+p00U2!1

,

Ei = 2E+e1,

( 8.1.43)

in zone IF, -P'1V1' k

Ps=Pi

=Poo+p,,U2k2,E'2=2E-E2

(8.1.44)

2kes"'p.-.U2k3,E's=ES-2E

(8.1.45)

and in zone III', P'P2P',Vi2C1

We denote by n 1, n2, n3, n1, n; and n3 the perpendiculars to OP, respectively PQ, QR, OP, P'Q' and Q'R pointing towards the body. In the framework of the linear theory they will have the coordinates

(E1,-I),(-E2r-1),(-E3,-1),(es, 1),(E'2,1) and (-E'3i1). The resultant of the pressure is

R = Lo(£1pini + t2P2n2 + 13p3n3 + tlp/ini + t21/2ns + tsp3n3)

.

(8.1.46)

The lift is

p _ Poo 2 Lo

[6 (E' - E1) + 12(C'2 + E2) + t3(E3

- 6'3))

.

(8.1.47)

Taking into account El, c'2, r'3, and the relation t1 + t2 + t3 = 1 , we deduce CL =

(8.1.48)

In the framework of the linear theory, the drag Rt = 2p ,Lo(E + t1E1 - t2E2 - t3e3)

(8.1.49)

vanishes, because from the projection of the polygonal contour ORQ'P'

on the Oy axis we deduce E = titJl + t2bJ2 - tstJ3

(8.1.50)

whence, taking into account the expressions E'1, E12, E13 we get

E + ties - t2E2 - 1363 = 0.

The formula (8.1.39) is very important. It shows that cL has the same expression like in the case of the wing consisting only of its skeleton OR. The result is natural, because of the symmetry of the wing with respect to OR.

294

THE SUPERSONIC STEADY FLOW

8.1.6

Validity Conditions

We shall investigate, for the flat plate, the limits of validity of the linear theory. Since the flow is defined by the formulas (8.1.25) in I and II (fig.

8.1.1) and by Vt = Ui, pt = p in the regions upstream of AOA' and downstream of BQB', we find that pi < pil and pi < p,,. Hence, in order to have a positive pressure all over the fluid (this is a physical condition), we must have (8.1.51) Pi > 0, this representing a first condition of validity. Taking into account the

expression of pf given in (8.1.25), we deduce, with the notation kX = = M2e,

0
(8.1.52)

A second condition must ensure us that the flow is everywhere supersonic. Since from (8.1.25) or from Bernoulli's integral it results IVtlf'2 < (Vt1 12, we deduce that we must have ( 1)2 < (V ')2, whence

'7Pf'/pfl = (cl'2 < U2 [(i - k)2 + x,2]

.

(8.1.53)

Taking into account that in the framework of the steady linear theory we have p = M2p (it results from (2.1.7)), we deduce the inequality

11+X equivalent to

k)2

<M2 (1

+s2,

1 +yX < (1+X)(M2 - 2X + X2).

(8.1.54)

(8.1.55)

In the linear approximation with respect to e, this inequality becomes

1-M2<(M2-2--y)X.

(8.1.56)

Further we shall deal with the inequalities (8.1.52) and (8.1.56) when

the fluid is the air ('y = 1, 405). For M2 < 3,405, the two inequalities are satisfied in the shaded region from the (M2, X) plane (fig. 8.1.6), where M2 - 1 2 f (Af 2) = 3,405 - M

(8.1.57)

295

GROUND AND TUNNEL EFFECTS x+

0.711

............. ......... ........... ............

0 Fig. 8.1.6.

This result shows that the linear theory is valid only in the zone 1 < M2 < 2. For a certain value given to M2 in this zone, E is subjected to the restriction (M2

e<

- 1)3/2

M2(3,405 - M2)

(8.1.58)

The last ratio increases when M2 increases, hence the superior limit of the angle of attack is increasing with M2. Other results concerning this subject are given in [8.17] [8.32].

8.2 8.2.1

Ground and Tunnel Effects The General Solution

We assume now that the wing lies between two parallel planes having the equations a

y=-2 and y=Z

(8.2.1)

In this case, the general solution (8.1.8) and (8.1.9)

ku(x, y) _ -F(x - ky) + G(x + ky) v(x, y) = F(x - ky) + G(x + ky)

(8.2.2)

P(x, y) = -u(x, Y), with the damping condition

lim F(x - ky) = lim s--00 x--ooG(x + ky) = 0,

(8.2.3)

296

THE SUPERSONIC STEADY FLOW

has to satisfy besides the slipping conditions on the profile, v(x, ±0) = h4 (0),

x E [0,1] ,

(8.2.4)

the slipping conditions on the planes

,-a) = 0,

v

v

(x, 2 1 = 0,

x E R.

(8.2.5)

We assumed that the equations of the profile are

y = ht(x)

(8.2.6)

In the sequel we have to calculate the velocity field in the domain

D-={(x,y):x+kyE[0,1),0>y>-2}.

(8.2.7)

We consider the natural number n such that nka < 1 < (n+ 1)ka and the segments (0, ka), (ka, 2ka), ..., ((n - 1)ka, nka), (nka, 1) on the chord of the profile which is the segment [0,1] from the Ox axis. We consider the characteristic having the equation - kq = C E (0, ka).

2).

For n=-2 wehave

The characteristic having the equation x + ky = E

(- ka ka

2,

2

ka

2, t E

does not intersect the chord of the profile, i.e. the

segment y = 0, x E (0, 1]. We have, by virtue of condition (8.2.3) G

ka) =

G(x + ky) = 0, t E

lim

ka, ka (8.2.8)

From (8.2.5) it results

0=va)

2

and from (8.2.8) and (8.2.9)

F(f+

Z)=0,

2

)

t E(-2, 2j.

(8.2.9)

(8.2.10)

Utilizing the notations

G(x+ky) = G- (x + ky), F(x - ky) = F"(x-ky),(x,y) E D" (8.2.11)

GROUND AND TUNNEL EFFECTS

297

we have on the upper surface of the profile

F- (x) = 0,

x E (0, ka) .

(8.2.12)

From the previous equation and from the slipping condition on the profile (8.2.4)

h'_ (z) = v(x, -0) = F- (x) + G` (x), x E (0,11

(8.2.13)

it follows that

G-(x) = h'_(x) x E (0, ka) .

(8.2.14)

Further one demonstrates by induction that for x E (jka, (j+1)ka) , j E N, j < n, we have:

G-(x) = h'_(x) +h'_(x-ka)+h'_(x-2ka)+... ... + h'_ (x - jka), F- (x) _ -h'-(x - ka) - h' (x - 2ka) - ... - h'_ (x - jka) .

(8.2.15) (8.2.16)

Indeed, for j = 0 the relations are true. Assuming that the relations are true for j, we shall demonstrate that they are also true for j + 1. Let us consider x E ((j + 1)ka, (j + 2)ka). From (8.2.5) we obtain:

0=v(x-k2,

-G'(x-ka)+F-(x).

Since x - ka E (lka, (l + 1)ka), by the induction hypothesis we have

F-(z) =-G- (x-ka)=-h! (x-ka)-h! (x-2ka)-... (8.2.17)

...-h'(x-(j+1)ka). From the slipping condition on the profile

h'_ (x) = v(x, -0) = p-(x) + C-(z) and from (8.2.14) it follows:

G-(x) = h' (x) +h'_(x-ka)+h'_(x-2ka)+...+h'_(x- (j+1)ka), (8.2.18)

whence we deduce by induction the validity of the relations (8.2.15) and (8.2.16).

Taking into account that

P(x, y) = -u(x, y) = k [F(x - ky) - G(x + ky))

,

(8.2.19)

298

THE SUPERSONIC STEADY FLOW

we may calculate the velocity field on the lower surface of the profile [h'_ (x) + 2h' (x - ka) + ... + 211- (x - jka)]

p(x, -0)

,

(8.2.20)

x E (jka.(j+1)ka)n[0,1]. Considering the domain

D= (x, y) : x - ky E [0,1), 05Y:5 Z

(8.2.21)

and utilizing the notations

G(x + ky) = G+(x + ky), F(x - ky) = F+(x - ky), (x, y) E D+, (8.2.22)

we can prove, like in the previous case, that for x e (jkb, (j + 1))kb) fl [0, 1) we have

G+(x) = -h+(x - kb) - h+(x - 2kb) -

... - h+(x - jkb),

(8.2.23)

F+(x) = h+(x)+h+(x-kb)+h+(x-2kb)+...+h+(x- jkb), (8.2.24) whence it results

P(x, +0) = k [h+(x) + 24' (x - kb) + ... + 2h+(x - jkb)] (8.2.25)

x E (jkb,(j+1)kb)n[0,1).

8.2.2

The Aerodynamic Coefficients

As we know from section 8.1, the aerodynamic coefficients are given by the formulas

cL =

j rt

[p(x, -0) - p(x, +0)) d x (8.2.26)

cA( = J x [p(x, -0) - p(x, +0)1 d x. ..110

In the sequel we shall consider some examples. For the profile in a free stream (a = oo, b = oo), it results the already known formulas (8.1.23).

299

GROUND AND TUNNEL EFFECTS

For the thin profile in ground effects (b = oo), taking for example 1 > ka > 1/2, we deduce 1

CL =

j[14(x) + h'_ (x)] d x- k

jh..dx (8.2.27)

1

2

1

cM= --

x[h+(x)+h'_(x)]dxo

k

f

1-ka

(x+ka)h'dx.

o

For the thin profile in a wind tunnel, taking for example 12: ka >

1>_kb>1/2,we get CL = - I

J

1 [h+ (z) + h' (x)] d x - k

-2 JI

h+(x)d x,

k 0 1

CM = -

k1

f 1-ka h'_ (x)d x-

1

2

j1-ka

x (h'+ (x) + h'_ (x)] d x - 2

(x + ka)h'_ (x)d x-

-2 11-kb (x + kb)h+(x)d x. k

(8.2.28)

For the flat plate with the angle of attack e, we have h+(x) _ = h_ (x) = -Ex, x E (0,1], whence 2E

E

L= k, CM=k7

(8.2.29)

for the profile in a free stream,

(2 - ka), cm = k (2 - k2a2) ,

CL =

(8.2.30)

k

for the profile in ground effects (12: k > 1/2) and cL =

(3 - ka - kb),

cu= k (3 - k2a2 - k2b2) ,

for the profile in tunnel effects (1 > ka > 1/2, 12: kg > 1/2). This subsection was written following the paper [8.8].

(8.2.31)

300

8.3 8.3.1

THE SUPERSONIC STEADY FLAW

The Three-Dimensional Wing Subsonic and Supersonic Edges

In this subsection we present the general theory of the thin wing in a supersonic stream. We shall utilize the coordinates (2.1.1) and the fields (2.1.3). The free flow is by hypothesis supersonic. Like in the subsonic case, we shall denote by z = h(x, y) ± hi (x, y) (8.3.1)

= er(x, y) ± hi (x, y)} the equations of the upper and lower surfaces of the wing. The projection of the wing on the xOy plane will be the domain D, assumed to be simple connected. On the boundary r of this domain we have:

hi (x, y) = 0.

We assume that r is smooth. Then there exist a point F where the tangent to r makes with the direction of the stream at infinity the angle of Mach p defined by the formulas (8.1.7) and a point A, where the tangent to r makes with the direction of the stream at infinity the angle -µ (fig. 8.3.1). The point of intersection of these tangents will be considered the origin of the frame of reference. There also exist two

Fig. 8.3.1.

Fig. 8.3.2.

points B and E where the tangents are parallel to the direction of the stream at infinity. As we know from the subsonic case, the points

B and E separate the boundary r in two portions: the leading edge

301

THE THREE-DIMENSIONAL WING

EFAB and the trailing edge BCDE (C and D are the points where the tangents make the angles p respectively -p with the direction of the free stream). Obviously, we assume here again that every parallel to

the direction of the stream at infinity intersects the edge r in at most two points at a finite distance. Definition. We name supersonic (subsonic) part of the leading or trailing edge, the part for which the absolute value of the component normal to the edge of the velocity of the free stream is greater (smaller) then the sound velocity. We shall prove that this definition is equivalent to the following one: If in a certain point of the leading or trailing edge the angle of the tangent to the edge with the direction of the unperturbed stream is greater (respectively smaller) than Mach's angle, then in that point the edge is supersonic (respectively subsonic). Indeed, from figure 8.3.2 it results that the component normal to the

edge of the velocity of the free stream in the generic point P has the magnitude U sin p1. If this is greater than the velocity of the sound in the unperturbed flow we have U sin p > c, whence sin pl >

1

-

= sin p

Utilizing the second definition, it results that the edge FA1 A from

figure 8.3.1 is a supersonic leading edge, the edges AB and FE are subsonic leading edges, the edges BC and DE are subsonic trailing edges, and the edge CB'E'D is a supersonic trailing edge. It is known from the theory of hyperbolic partial differential equations (see also the plane problem from 8.1 and 8.2) that the zones of influence are the zones delimited by the characteristic lines. For example, in figure 8.3.1, the zone of influence of the subsonic leading edge

FE is FF'E'E, FP and EE' being parallel to OA. Definition. We name wing with independent subsonic leading or trailing edges, a wing for which the zones of influence of these edges are disjoint.

It results therefore that a wing has independent subsonic leading edges if the Mach lines AN and FP do not intersect in the domain D and independent subsonic trailing edges if BY and EE' do not intersect in D. For example the wing from figure 8.3.1 has dependent subsonic leading edges and independent subsonic trailing edges and the wing from figure 8.3.3 has only independent subsonic edges.

THE SUPERSONIC STEADY FLOW

302

Fig. 8.3.3.

8.3.2

The Representation of the General Solution

Like in the subsonic case, we shall replace the wing with a continuous

distribution of forces having the form f = (fl, 0, f) defined on D. We shall see that we may determine such a structure of f, in order to satisfy the boundary conditions. The perturbation of the pressure determined in the uniform stream by the distribution f will be, according to the formula (2.3.32),

P(x,y,z) = -T"

fi)a-+f(Tl)8

'ID

J

G(ro,yo,z)ddq, (8.3 2)

where G(zo, yo, z) =

H(zo - s) ' z0-9

s = k ya + za.,

zo=z-t,

yo = Y-17. (8.3.3)

For the velocity field, from (2.3.12) and (2.3.34), it results

v(z, y, z) = 6(z) lID f (rl)II (zo)6(yo)ddi+ V ,

(8.3.4)

where ( X, y, z) =

2Tr

ff

fi

n) - f (t+ o

G(zo, yo, z)dc dTl

z2

J

(8.3.5)

Obviously, the perturbation is potential excepting the trace of the domain D in the uniform stream, where the first term from the expression

of v does not vanish.

303

THE THREE-DIMENSIONAL WING

For the component w from (2.3.12) and (2.3.34) it results:

fJ f (, n)H(xo)6(yo)de dt7+

w(x, y, z) = 6(z)

+ 2a 8z JJD [h (t, n) - f

rl) 0 +

y

G(xo, I, z)dtdo z2 J

(8.3.6)

For w we also have the representation: w(x, y, z) = -'ID'n) 2

8 G(xo, y, z)d (8.3.7)

+2x 11 f(t,n)N(xo,yo,z)dt dtl, D

where N(xo, yo, z) = k2

G(xo, yo, z) +

8

xo+G

(xo, Y0,

[-yol-+-Z2

z)(8.3.8) J

which results from (2.3.37) and the representation

w(x, y, z) = ' AD ft (t, n) 8z G(xo, yo, z)dF dn+

1(2_02 _ (k

2n

8x2

2

2) AD f (C,

r(

)

H -7.1 [1-0*077

d11 d drl , (8.3.9)

which results from (2.3.38). Each of these representations determine an integral equation for the function f (x, y). All the known representations (Evvard (8.9], Ward (8.34), Krasilscicova (8.20], Heaslet and Lomax (8.15], Homentcovschi (8.16] and Dragoq (8.7)) are found in the formulas

(8.3.6)-(8.3.9). Prolonging the functions fl and f with 0 in R2\\, the above representations may be written as convolutions. For example, p(x, y, z) and rp(x, y, z) have the following form:

p=-2A 8 fl*G+BZf*G

if h #G-f * y2+z2G) 7r

where the sign x, y.

*

(8.3.10) ,

indicates the convolution relative to the variables

304

8.3.3

THE SUPERSONIC STEADY FLOW

The Influence Zones. The Domain Di

First of all we must notice that the perturbation my be represented by integrals whose integrand contains the factor H(xo - 8). Indeed, for (8.3.5) this is obvious. Taking into account the formulas (2.3.35) and (2.3.36), it results that the assertion is also valid for (8.3.2) and (8.3.7). In (8.3.9) we have

f 7

f zp H(r - s) d-r = H(xo - s) J 2 T -s .ll:

rdT-s

(8.3.11)

Since the above integrands contain the factor H(xo - s) we deduce that for a point M(x, y, z) from the domain occupied by the fluid, the

integrals on D are in fact calculated only on the domain Dl where we have:

xo > s

(8.3.12)

This inequality implies 4 < x (x - 4)2 > k2[(y

-

11)2

+ z2].

(8.3.13)

The points from D verifying these inequalities are situated between the leading edge and the hyperbola C which has the equation (x - )2 = k2[(y

-

q)2

+ z2)

(8.3.14)

and the branches to -oc because < x (fig. 8.3.4). The hyperbola C (the variables are and t) has the axis parallel to Ox. In fact, C represents the intersection of the cone having the equation (x - C)2 = k2 [(y - t1)2 + (Z -

()2)

with the plane C = 0. This is Mach's cone. It has the vertex in Al and the axis parallel to Or. From the mechanical point of view, this result represents a consequence of a fact known from the hyperbolic partial differential equations theory [1.6] namely the fact that in Al one can receive only the perturbations produced in the points belonging to the interior of Mach's cone with the vertex in Al. When Al will be on the wing (z = 0), the hyperbola C will be reduced to the half-lines

x-==±k(y-,). These are the characteristics issuing from M (fig. 8.3.5).

(8.3.15)

THE THREE-DIMENSIONAI. WING

FIg. 8.3.4.

305

Fig. U.S.

We may easily explain why the points from D - D1 do not affect the perturbation in M if we have in view (the significance of the fundamental solution) that the perturbation produced in a point Q E D propagates only in the interior of Mach's cone with the vertex in Q. The point 141 is in the interior of all the cones with the vertices in Dl and in the exterior of all the cones with the vertices in D - D1. It also results that in the fluid exterior to the envelope of the posterior

cones with the vertices on D, the perturbation is zero. Hence we can give up the factor H(xo-s) in the integrals expressing the perturbation if we replace the domain D by D1. Prolonging the

functions f, and f in the exterior of D with the value zero, for a given i;

,

ij will vary between Y_ and Y+ defined by (8.3.14) through

kYf = ky ±

xo - kszs .

(8.3.16)

The vertex of the hyperbola C has the coordinates tI = y, = x - kjzj (obtained for Y+ = Y_). 8.3.4

The Boundary Values of the Pressure

For the integrals having the form: I (x, y, z)

=

JJ

rl) 8zG(xo, yo, z)d>; dry

(8.3.17)

306

THE SUPERSONIC STEADY FLOW

we have: Jz-kJ=I

1 _ 8z

d

f Y+

fy_

r1)

ro-s

(8.3

dt) .

Performing the change of variables rq -+ 8: kn = ky - Jxo - k2 z2 cos 0

(8.3.19)

we obtain jx_&.izi

I (x, y, z) =

k.

z

d/f

rR

!

f

y - rx.2 - k2z2 cos 9) d8 = C

z -L-1-TI

_ -sign z !O f (x - kjzj, y)d©+ JJ

dC oo

Of

Jo 8rr

noose

xo - k z

dO.

Hence,

I (x, y, f0) = F it f (x, y) .

(8.3.20)

Using this formula, from (8.3.2) we obtain

Ax, y, ±) = - 2I j

jD f1 (e, n)

G(xo, No, 0)d do f 2 f (z, y)

whence, (8.3.21) f (x, y) = F(x, y, +0) - P(x, y, -0) . This result puts into evidence the significance of the function f.

8.3.5

The First Form of the Integral Equation

The simplest way to obtain the lifting surface equation relies on the representation (8.3.7). Taking into account the derivation formula (2.3.35), we may write the kernel (8.3.8) as follows: N(xo, yo, z) =

-k2

H(xo - s) _ xo(yo - p2) H(xo - s) xo - s yo + z2 (xo - s2)312 (yo x022

(8.3.22)

We notice that for z = ±0 it appears the singular line n = y. Detaching

from D the domain DE defined by y - e < n < y + E in Dl - D, (fig. 8.3.5), it is possible to simplify by yo after putting z = ±0. Performing this.operation we deduce N(xo, yo, f0)

xO

H(xo

o

xo

so) so

H(xo, yo) .

(8.3.23)

307

THE THREE-DIMENSIONAL WING

Adopting the definition

f-'0 11.

f- D2

(8.3.24)

,

we shall prove that

j = lim lID. f ( r1)N(xo, Uo, z)ddq = 0 .

(8.3.25)

Fore small enough, we may perform the replacement n = y in the integrand. Hence, rx

= C-0

0o

lirn

(

x-kIzl ao

[f-" f (F, n)N(xo, yo, z)di7 I d _ c

v-c d= 0 [f()N(zotz)jdfll 1

-

.

Using the form ula (8.3.20) from (8.3.7) we deduce:

w(x, y, +0) = T 2 f1(x, y) +

27r

JJ f(f, n)N(xo, yo)dtdo . (8.3.26) D

Adding and subtracting the boundary conditions w(x, y, f0) = h..(x, y) ± h1x(x, y) (x, y) E D ,

(8.3.27)

one obtains:

h71 f(f,n)N(xo,yo)dt do = h.(x,y), ft (x, y) = -2h1x (x, y) ,

(x, y) E D.

(8.3.28)

(8.3.29)

The equation (8.3.28) is the lifting surface equation in the supersonic stream. It can be also written as follows:

f(")

_ ,

yo

o::ll

xox k

(8.3.30)

o

D1 representing the shaded domain from figure 8.3.5. The analogy of this equation with equation (5.1.28) is obvious. The equation (8.3.30) was given in [8.7). For the sake of simplicity we shall name it the equation D.

308

THE SUPERSONIC STEADY FLOW

The Equation D in Coordinates on Characteristics

8.3.6

We know from (8.3.15) that if the current point M is on the wing (x, y) E D, z = 0, the hyperbola C from (8.3.4) degenerates in the characteristics MMl and MM2 (fig. 8.3.5) having the equations

t -- krt = x - ky respectively t; + kr' = x + ky ( and q are the variables, x and y are the coordinates of the point M). Performing the change of variables C, ,q - a, Q, x, y -+ a, b defined by the formulas

- kn = a

x-ky=a

t;+kn =Q

x+ky=b,

(8.3.31)

we deduce

2kdt; drt = da d#.

The characteristic MAi1 has the equation a = a and the characteristic MM2, the equation Q = b. The domain Dl given in the old variables by the inequalities

t-k, <x-ky, +krl<x+ky

(8.3.32)

will be characterized in the new variables by a < a;

f3 < b .

(8.3.33)

The axes OA and OF will be the new axes of coordinates because

on OA we have /3 = 0 and a is variable, and on OF, a = 0 and 0 is variable. The coordinates of M with respect to the new axes are (a, b) (fig. 8.3.6). Since the new axes are characteristic lines, the coordinates a and /3 will be named coordinates on characteristics. In these coordinates the equation (8.3.30) becomes

f(a,

k 2rr

Dl

a - a + b -,8 (a - a - (b - Q)(2

dadQ = H(a b), (a a)(b - /3) (8.3.34)

where we denoted by f (a,,6) the unknown f in the new variables, H(a, b) the function given by h=(x, y) in the new variables and Dl, the domain D in the new variables, i.e. the shaded domain from figure

8.3.6 (a
THE THREE-DIMENSIONAL WING

309

Fig. 8.3.6.

and denoting by D6 the parallelogram indicated on figure 8.3.6. We adopt in (8.3.34) the definition

Jf

,

C0

(8.3.35)

if, -n. *

In this way, the non - integrable denominator from (8.3.34) does not vanish (even if a = a). In Dl - D6 we may utilize the identities

a - a+b _

1

-a-(b- /3)]2 n-a b-

2

0

a

s

a- a -(b-Q)-

P-i

a-a+_/_-_

82

-2 8a8b In

_

I

a --a - VS --- I I ' (8.3.36)

such that the equation (8.3.34) may be also written as follows

k0

('Pv, f(a, Q)

dadf

T-,6 a-a- (b-Q)

it 8a k 82 ;r 8aOb

a --a

JID,Di

H(a, b)

,

a --a + V- da d,B = H(a, b)

(a, Q) In I

a --a - ../b-/31

(8.3.37)

(8.3.38)

310

THE SUPERSONIC STEADY FLOW

in order to have weaker singularities. In [8.7) one gives another form for equation (8.3.37).

The Plane Problem

8.3.7

We act like in 5.1.7. In order to have a plane-parallel flow, we must consider the case when the conditions that determine the flow are iden-

tical in every plane parallel to xOz. We assume therefore that the equations (8.3.1) have the form z = h(x) ± h1(x),

(8.3.39)

and the domain D is rectangular, such that the span 2b tends to infinity (b - oo). Hence we assume that D is characterized by the inequalities 0 < x < 1, -b < y < b. These conditions imply ff = fi(t) and f = f (t;). Noticing that 2 100 H(xo

+0o

Ed

xp

k

1.00

_

- u + k z) du

-k'z2 u

°-k I 2T

2

kH(x0 -

lID f()8!!

(y02

(8.3.40)

xo - k222 - u

o

= kH(xo - kizi)

du

ldv v

10

= -H(xo - kjzi),

+z2E/dj?=

f [.1:(j2 E)cli}df 1

we deduce the representation (8.1.17).

(8.3.41)

=0,

THE THREE-DIMENSIONAL WING

311

8.3.8 The Equation of Heaslet and Lomax (the HL Equation) This equation may be deduced from the representation (8.3.6). To this aim we denote L(x, y, z)

8z

(/

f f fx° Di

6z

.

!l

2

do =

82.+8

+

(8.3.42) d

+z2

Y_ 747-77

the functions Y+, Y_ being defined in (8.3.16). Here the change of variable (8.3.19) is not indicated, because the term that one obtains by deriving the superior limit of the first integral becomes oo for z --+ 0. We act therefore as follows:

r-k L = limm

Y+

as J-oo

f (4, q)

x0z

a yp + z2

fY_

(8.3.43)

so Ll (x, y, z) + Twki,1 xot(x0, Y, 44,

where, after performing the change of variable 9 -, it : q - y = u, we have:

V+cf(x-k[+ £ q)

L1=-k2Jim c-+o lyt -

e-

do + z2

+s

_ -kz2f (x - kizl, y) hm

t(xo, y z )

F,

f

ft

-ki f (x - kjzj, y), e - u (u2 + z2) = (8.3.44)

Y+

z

f (F, n)

Y_

x0 - a Y + z

d ry =

a(s)

8 as

a(:)

F(u, z)zdu,

+z

(8.3.45)

with the notations a(z) _

x8 - k2z 2

,

F(u z) ,

M,u+y)

(8 . 3 . 46)

x - k u2 + z)

We deduce therefore

L(x, y, ±0) _ -ka f (x, y) + f xot(xo, y, ±0)d.. 00

(8.3.47)

312

THE SUPERSONIC STEADY FLOW

For determining the function L(xo, y, t0) we notice that a'(+-0) =0. It results that the derivative from (8.3.45) and the integral interchanges. We have therefore: l(xo, y, ±0) =

_ limo sL m

8z

+* OIF

f

8uOz

8 zUM

+°

8.. J C r

a

{ [F(a, z) + F(-a, z)] arctan

z-»f0

J

+a OF

f4

=

}-

U2

+ z-.f0 lim

a z2 + a2

z

+a OF

arctan z -d u + yli

lim (F(a, z) + F(-a, z)(

u

F(u, z)d arctan

u

+0

f

+ Z2

8F

du=

z

87Z u2 + ..2 d u+ c

u2+z2du.

8u

But lim z* .O lim z

0

r+n OF

f

a

J

du=

z

J

z-.f0 8z

8Z u2 + z2

+° OF

8F (0,z): /+` du

lim

u

du

a 8u u2 + z2

_ - el 0 z

0

f7rBF

f u2 + z2 =

8z (

+c OF

r

(-a

j+d019F

)

+ Ja

u

8; u2 + z2

0, 0)

du=

du

(u, U)

u

where ao = a(0). Finally, taking the formula (D.3.7) into account, we obtain:

t( x0, y, f0 ) _ + W OF ( 0 , 0) =

F(oo, 0)

- F(-ao, 0) +

ao

ao

r

'/ +°O OF (u, 0) d

1

8u

u±

u (8.3.48)

F(u2 0)d u

-,-0

f

BF(0 , 0) .

Having in view the expression of F(u, z) (8.3.46), we deduce: L ( x, y, f0 )

= -k rf (x, y) +

f oo

x0

T y

f(t, n)

do x =k-2-y- d on d t (8.3.49)

313

THE THREE-DIMENSIONAL WING

Hence,

w(x,y,±0) _ T 2ft(x,y)+ Zf(x,y)(8.3.50)

[*

T7r

v- s

f xk'y yo

L771 d

such that, imposing the boundary conditions (8.3.27), we find: f

kf (x, y) - ! J

xo [

T

ok2yo

24(x, y), (8.3.51)

d

ao J

I1 (x, y) = (h' -

(8.3.51) is the HL equation. Integrating in (8.3.42), at first with respect to respect to y, we obtain

w(x,y,±0) = T 2ft(x,y) -

oo

«

2zr

(8.3.52)

(x, y).

.

°

r-kh#o)

d2y b0

and then with

[.! ro-k o

dl .

8.3.9 The Deduction of HL Equation from D Equation This deduction was performed by V. Iftimie. To this aim, we shall extend the integral from (8.3.30) from the shaded domain Dl, in figure 8.3.5 to the infinite domain D&, situated between the characteristics

MMl and MM2 extended to infinity, putting f = 0 in the exterior of the arc M1 M2. We denote by I the integral obtained in this way and we perform the change of variables t, n -+ u, v:

u=x-Z;, v=y-y.

(8.3.53)

In the new variables, the domain Df, will be transformed into the domain Du,, (fig. 8.3.7) determined by the angle Mi MMz. Denoting g(u, v) = f (x - u, y - v), we deduce f(x,y) = g(0,0).

(8.3.54)

THE SUPERSONIC STEADY FLOW

314

Fig. 8.3.7.

The HL equations result from the formulas

g(u'

v)

u

dudv

v2

u-k

_ d

=-k,rg(0,0)+or duTAOU,v 00

=

it

v)

J

u

u

(8.3.55)

k

akv

which have to be demonstrated (the demonstration was given by V.1ftimie). For proving the first formula, we isolate the singularity by

two parallels to the Mu axis (fig. 8.3.7) at the distance e from this DD axis. We denote by D' and D" the domains composing and by I' and I" the integrals I corresponding to these domains. Using the definition (8.3.24) we get I = limo(I, + 1") .

Putting v = tu, we deduce:

I'-

11 JfJJJJ

g(u,v)u

v2

dudk u

_ r0O 1

=J. u[u /k

oc

v

-lip uI,J

(8.3,58)

r

JJ

t2

g(u,tu)

1-k t

dt du.

u

1

dv'du

315

THE THREE-DIMENSIONAL WING

We expand g in a Taylor series:

g(u,tu) = g(u,0) + tug"(u,0) + t2u2m(u,tu). and we introduce this series in the above integral. Acting analogously for 1", we deduce:

- °° u F f-`

I"

9(u, v)

d vl d

v2 u -k

J

1-v-1g(u,tu) kE u l Jl

t2 1- k t

u=

dt1du

whence,

f

00

1' + 1" =

g(u, 0)

[ JiY

U

Jibe

r

t2 1- k t

+Jits g°(u,0)LI=" t lack t

+

i m(u, tu)

uf

LJ4

+r

dt

1-k t

d

The factor which multiplies

L

J_

+J-I

t+!J_

1t

dt du+ t2 1- k "t

t I-

m(u, tu) 1

t

t,du+

d t d u.

is zero if we replace in the second

integral t by -t. An elementary calculus gives

rk1-k t

dt 411

_r

dt

_ u -k E

t2 1- k t - J_ t2 1- k t

whence,

P+ I"

E

e

k2E2d u+

u

(8.3.57)

+fJ-u(j1+j-'U) m(u,tu) dtdu When a -+ 0, the first integral is a Finite Part. Hence 2 /'O g(u, 0) C JkE

U

1 9(u1 0) - 9(01 0)

u2 - k2E2d u E

fkC

U

u2

k2E2+

316

THE SUPERSONIC STEADY FLOW

u u

+E g(0,0}

ke du+E foo 9( L+0)

u2 - k2F2 d u .

JEc

The first and the last integrals from the right hand member are finite. When we consider the Finite Part, we neglect these integrals because they h a v e the factor E t . One calculates the integral from the middle and one obtains

FP 2

0o

E Jke

u2 - k2e2d u = -kirg(0, 0) .

g(u' 0) U

Hence, from (8.3.57) we deduce: r00

I= lim(I`+I") =-k7rg(0, 0)+J

o

oJ

I

m1(uk tut)

,

d t]d u. (8.3.58)

We set now

J= f 00 [1" 9(u,v)v f

dUk

Jdu.

u

(8.3.59)

Using the same Taylor expansion we deduce:

J' _

J6

u v)

g('

[Ilk

u

d v=

- Js

v2v

Y

d tk

9(u. 0)

t2 VI

U

t +9u(u1 0)J°

f I m(u, tu)

+u f

1 - 00

0

J =Jr 9(u, v)

dv

u - kv

+gv (u,

Y

0)

J

dt g(u, tu) u t2 1 - k t

t

latkt

-

+

d t,

u dt

t 1-k t

f_1 -{-

t2 1-k t+

.. .

whence,

J' + J" =

2 g(u, 0)

u - k e + u{

m(u, tu)

U 43

1-kt

dt.

317

THE THREE-DIMENSIONAL WINC

When we consider the Finite Part (FP) we neglect the first term and we get li

m

(J' + J") = u

m(u, tu)

f

J= /

1-k t

u

Jo

[

JI

m(u, tu) d t d u.

1-k t

(8.3.60)

From (8.3:58) and (8.3.60) it results

I = -k7rg(0, 0) + J,

(8.3.61)

which demonstrates the first equality from (8.3.55). For proving the second equality, we denote: 00

K

I

T0,

J

u

g(u,v)

kv

dudv= lim(K`+K"),

where (we may change the order of integration)

K' - r-E r r J_O0

sr

fI

g(u, v) d v2 7u!=-=kFJug(u, v)

kv v2 u - u v

0o

v du = I', (8.3.62)

K" = 100 [/

r - Jkr

d ul d v =

ky v2 u- JC v

h

00 u

Jk

uy(u, v)

du] d v =

c

u

g(u,v)

i v2Vu- 72 -k v

dv du = l"

Hence K = I and the demonstration is finished. The HL equations resulting from (8.3.27) are

kf (x, y) - a

-1

f

a'° 00

l vklvo!

co

p

o°

f(")

d? l d t = 2h.(x, U),

k Uo V5

o-n)k o

F--7

d

t

(8.3.63)

= 2h= (x, y)

These are the equations (10-21) and (10-22) from (8.151.

(8.3.64)

318

THE SUPERSONIC STEADY FLOW

8.3.10 The Equation of Homentcovschi (H Equation) This equation results from the representation (8.3.9). Taking into account (8.3.20), we deduce:

w(x, y, t0) = T 1h (x, y)+

+2

f

l rr

(k2 52 +J

2/

_

=o

Id f H(T ^s)drdn,

L

.0

o

(8.3.65)

such that imposing the boundary conditions (8.3.27), we obtain the equation (k 2

ll

axe - r7y2)

ff f

n)

f

r:o d T d dn = 2 (x,b) J H(T - so) J

(8.3.66) which was given for the first time (in a different way) by D.Homentcovschi (8.16].

We have to notice that after evaluating the interior integral with the formula (8.3.11) one obtains the equation

!

..

(k2..02

_

vx-!::

02

aye) Jf,

f (E, n) In xa + 40 ,

sod d r1= 2h (x, b) (8.3.67)

In the variables on characteristics this equation becomes (8.3.38). However, Homentcovschi does not follow this way. Inspired probably by the papers of Evvard (8.9], Ward (8.33] and Krasilscicova (8.20], he intro.

duces the unknown N, related to f by the equality N(x, y) = 1-00 f (t, y)d t,

(8.3.68)

f (x, y) = N= (x, y) .

(8.3.69)

whence it results

We have seen in (5.4.5) that o function similar to N also intervenes in the theory of low aspect wings in a subsonic stream. Taking (8.3.69) into account, the integrand from (8.3.66) becomes:

N

:° H(T - so)dT J

-

= 0

[NJ° H(T - so)dT

9[NH(x(, -4o) f°u

x2 -4

o

dT

a_

so) 0J

V0-

o

_

319

THE THREE-DIMENSIONAL WING

[N(f , n) In 'To + sp o - so] +

= H(xo - so) {

F(I,,,) 1. 0

0

In this way, after applying Green's formula, the equation (8.3.66) becomes

111I N(C, g) d i d n+

a_ 82

(kz

xo-8o

81l

+ POD N(t, Y)) In

sod

xo +

so

1

(8.3.70)

n} = 2h.(x,1!)

In variables on characteristics this equation becomes k

a dQ

02

8

AD

l

- 0) +

V'r(a - C

v1_a --a

+ JeD; v(a' Q) In

a-a

+ VT--71

d ( - a) } = H(a, b). (8.3.71)

From (8.3.68) it follows that N(£, n) vanishes on the leading edge, hence

on EFAB.

N = 0,

(8.3.72)

Indeed, for every point

n) E EFAB, the integral (8.3.68) is calculated on the half -line having the applicate q upstream of the wing, where f = 0. We also notice that the curvilinear integral from (8.3.71) vanishes on the characteristics a = a and (3 = b (fig. 8.3.6), because the logarithm vanishes. Hence, for a given M(a, b) , the curvilinear integral from (8.3.71) will differ from zero only in those points of the trailing edge belonging to 0DI. For example, for the wing from 8.3.1, the curvilinear integral will differ from zero only for M situated in the zones of influence of the subsonic trailing edges BC and DE, respectively in the domains BCB' and EE'D. Utilizing the definition (D.3.6) for (8.3.71) we also have the equivalent forms: k 4ir

N(a) p)d a d.8

f JD1

(a - or)-111(b - 0)312

+ (8.3.73)

+_7r 8a JD a - a - (b, Q)

-d(A-a) =H(a,b)

320

THE SUPERSONIC STEADY FLOW

k "f 4;r J

N(a,A)da dA 7D) (a - c')3/2(b - 10)3/2

8

f

N(a,

A) d

(8.3.74)

+k 8b oD1 b-,C-(a-a) a-a (A - ) = H(a, b), the curvilinear integral being different from zero only in those points of

the trailing edge where a < a and A < b (see the definition of D1 in coordinates on characteristics (fig. 8.3.6). The equation (8.3.71) is the equation of Homentcovschi (H equation). In the following subsection we shall present the solutions of this equations as they were determined by the author himself.

The Theory of Integration of the H Equation

8.4 8.4.1

Abel's Equation

Abel's equation has the form x f"Jdx=h(Y)1 -x

a< y.

(8.4.1)

This was the first equation encountered in applications. Multiplying (8.4.1) by (A - y)-1/2 and integrating with respect to y on the interval (a, A) we obtain:

dy

J

rV

Ja

f(x) dx

y-x

rA h(y) dy

-JQ Vx --y

Changing the order of integration in the first member (fig. 8.4.1), we have: A

A

f (x)

J=

dy

(A - y) (y - x)

and then

f f(x)dx = I

fA

dx=

Jay

h

dy.

A

h(y)

dy

Deriving this relation and utilizing the definition of the Finite Part (D.4.3), we deduce: d

f (A) = a d A

f'\

hid

y

I

21r

,

{A

h(y)3/2 d y.

(8.4.2)

THE THEORY OF IN BGRATION OF THE X EQUATION

321

y

a

x

1)

Pt

8.4.1.

It results that the solution of the equation (8.4.1) is (8.4.2) and conversely, the solution of the integral equation (8.4.2), where h is the unknown, is given by the formula (8.4.1).

8.4.2

The Solution of the H Equation in the Domain of Influence of the Supersonic tailing Edge

If M is in the zone of influence of the supersonic trailing edge (fig. 8.4.2), as we noticed at the end of the section 8.3., the curvilinear integral from the equations (8.3.73) and (8.3.74) disappears. We see in

fact on the figure 8.4.2 that on the arc BM2 we have a > a and on the arc Ml E, /3 > b. It remains to integrate the equation: k

4a

N(a, /3)d a d,6 JDI

(a - a)3/2(b -

= H(a, b) .

(8. 4 . 3)

/3)3/2

Denoting:

a = A(,B), the equation of the arc DEFA and (8.4.4)

/3 = B(a), the equation of the contour FABC, t h e equation (8 .4 . 3 ) may be wr itten as follows

k'b

Ni (a, fl)

4ir JBi,l (b -8)3/2 d

= H( at b) ,

(8. 4 . 5)

THE SUPERSONIC STEADY FLOW

322

Fig. 8.4.2.

where

r

Nt (a, Q) _

N(a' a)

A(p) (a - a)3/2

(8.4.6)

d c t.

Utilizing (8.4.2) and (8.4.1), we deduce from (8.4.5) 2

_

(a, Q)

J

°)

H(Nl 3d

(8.4.7)

br

In this way, the equation (8.4.6) becomes _

1

27r

N(a,fl) da =

1A(8) (a - a)3/2

T11

H(a,b')db'.

(8.4.8)

$(°)

The equation (8.4.8) is similar to (8.4.2), so that 1V(a, A)

ra

d a'

JA(9)

a--a,

1 r H(a', b') dg.

Irk ,lg(a') 13

b'

For a = a, 6 = b we obtain N(a,b) _

1

"

Irk IA(b)

da' H(a',b')db, b - b' a --a' J Br(.') b (8.4.9)

1 If r(ab)

H(a',b')da' db' a - a' (b - d)

THE THEORY OF INTEGRATION OF TIIE H EQUATION

323

This is the solution in the zone of influence of the supersonic leading edge.

8.4.3 The Solution in the Domains of Influence of the Subsonic Leading Edge We assume that M(a, b) is situated in the zone of influence of the subsonic leading edge AB (fig. 8.4.3). In this case, the curvilinear

integral also vanishes, because on BM2 we have a > a, and on M1E, ,8 > b. The equation which has to be integrated is also (8.4.3). Obviously, A(b) < a1 < a, B(a) < b < b1. The equation (8.4.3) may be written as follows k 47r

J

N(a, /3)

da

(b) (a - a)3/2 B(a) (b - 0)112

d p = H(a , b) ,

(8.4.10)

such that we denote

N2 ( a, b)

(

2 JB(a) (b

we obtain the integral equation

Fig. 8.4.3.

0Q/2 d

1

(8 .4 11) .

324

THE SUPERSONIC STEADY FLOW

k

N2(a, b) JA(b)

(a - a)3/2

da

= H (a, b )

( 8 . 4 . 12)

whose solution is

N2(a, b) =

-1

k

H(a'' b)

J

A(b)

a- a' d a

(8.4.13)

for a > A(b). From (8.4.11) we obtain for N(a,/3)

the following

integral equation

2 r r l B(o) (b

(IB 2 d 1 3 =

7rk IA'

(b)

Haas'

d a'

(8.4.14)

Utilizing again the solution of the equation (8.4.2), we deduce:

N(a, R) = irk 1 J B(a)

for a > A(b) and

N(a,6) -

d &'

H(O' b') d a , f(- b JA(bl) a - a'

> B(a). Putting a = a, Q = b, we obtain: 1

db' JB(4) b - b'

(8.4.15)

H(a', b')d a' d b'

D]

Irk

fN)a='

H(a,b')dIrk

(0)

V'r(a,

- a) (b - b')

Dl being the shaded domain from figure 8.4.3 We notice that in this case D1 is not the entire domain determined by the leading edge and the characteristics issuing from M. From this domain one eliminates the strip where b' < B(a). This result was obtained for the first time in 1949, independently, by Evvard and Krasilscicova and it is called in some books the theorem of Evvarrl and Krasilscicoua

The solution is obtained analogously when M is in the zone of influence of the edge FE, with the difference that in this case one eliminates from Dl a strip parallel to the 0/3 axis. 8.4.4 The Wing with Dependent Subsonic Leading Edges

and Independent Subsonic Trailing Edges For a wing with dependent subsonic leading edges and independent subsonic trailing edges (fig. 8.4.4) the solution in the domain bounded

TIE THEORY OF INTEGRATION OF THE H EQUATION

325

by the curve AHFA is given by the formula (8.4.9), the solution in the domain bounded by ABH'HA - by the formula (8.4.15), and

the solution in the domain bounded by FHH"EF - by a formula analogous to (8.4.15). The case when M is in the common zone of influence HH'B'E'H"H of the subsonic leading edges is presented in the sequel.

Fig. 8.4.4.

We notice at first that in this case the curvilinear integral from

(8.3.73) also vanishes, because on BM2 we have a > a, and on All E'3 > b. Hence, we have to integrate the equation (8.4.3). Denoting by R1, R2 the domains bounded by the curves FM"RF, respectively FMIt'M"F and by Q1, Q2 the domains bounded by the curves

AQA1', respectively AM'M'A, we notice that on Rl and Q, we have N = 0 (it results from (8.3.68)). Hence, the domain of integration from (8.4.3) may be prolonged to the domain bounded by MRFAQM, i.e. to the domain

D2+R1+R2+QI +Q2, where D2 is the shaded region from figure 8.4.4, i.e. the region bounded by the contour Iii"'AMA! A.!"Al't'. The leading edge of this domain is entirely supersonic, such that the solution of the equation (8.4.3) is given by the formula (8.4.9)

N(a,b)

k

[Jf.+JJRi+fJR,+ff, +ff. (8.4.16)

H(a', b') d a' d b' . (a - a)(b - b')

THE SUPERSONIC STEADY FLOW

326

Let us consider now that M belongs to the zone of influence of the subsonic leading edge AB. From formula (8.4.15) we deduce: 1V(a'b)

irk IJJ

+

f+[ ,

R, ,

(a-a)(b-b')

da'd (8.4.17)

Similarly, taking into account that M belongs to the zone of influence of the leading edge FE, we deduce: N(a, b)

Tk-

f

ffDz +If +f fQz, Q1

H(a', b')

(a - a) (b - b')

dadbl. (8.4.18)

From the formulas (8.4.16) - (8.4.18) it results

N(a,b)

irk

ff

bi)da'db',

(8.4.19)

(aH(a',b')

this representing the solution in the common zone of influence of the two subsonic leading edges. We notice that the solutions from the previously considered domains have the same form, differing only the domains of integration. But we

may establish a common rule for determining the domains of integration. They are bounded by the parallels to the characteristics issuing from the points where the first parallels intersect the leading edge and by the remaining portion of the leading edge. In the first case, when M is in the zone of influence of the supersonic edge, the parallels to the characteristics issuing from the points where the parallels from M towards infinity upstream intersect the leading edge, do not intersect any longer this edge.

8.4.5

The Wing with Dependent Subsonic Trailing Edges

We consider now a wing for which the parallels M'M", M"Mf V

intersect in a point P belonging to the interior of the domain D (fig. 8.4.5). In this case, denoting by R2 the domain bounded by FM'PM"F and by Q2 the domain bounded by AM'PMIVA, (the domains R1 and Ql keeping the same definition) and performing the same reasoning like in the previous subsection, we obtain successively

N(a,b)=1(D3+R1+R2+D4+Q2+ Q1) N(a, b) = I(Ds + Ri + R2) ,

N(a, b) = I(D3 + Q2 + Q1) ,

327

THE THEORY OF INTEGRATION OF THE H EQUATION

R'%G A

Fig. &4.6.

I representing the symbol for the integral appearing in (8.4.19). It results the solution N(a, b)

AD,

(a H(a,' '){

b

d a' d b'(8.4.20)

kiDD.

H(a',b')

(a-a' (b-b')

dadb'

The result may be generalized for every wing having finite dimensions.

For example, for the wing from figure 8.4.6 the solution in the point M(a, b) is

N(a,b)=I(D3-D4+Ds-Ds).

(8.4.21)

We stop here the presentation of the solutions in the zones of influence of the leading edge.

8.4.6

The Solution in the Zone of Influence of the Subsonic Edges under the Hypothesis that the Subsonic Leading Edges are Independent

For the wing from figure 8.4.7 the solution is determined in the domain bounded by the curve ABB'E'EFA. It remains to determine the

THE SUPERSONIC STEADY FLOW

328

solution in the zones BCB'B and E'DEE'. To this aim we shall use the method of Homentcovschi [8.16].

Fig. 8.4.7.

Let M(a, b) be situated in the last zone. In this case, the curvilinear integral from (8.3.73) does not vanish. More precisely, it is zero on BM2,

where a > a, it is zero on M1M", where 6 > b, but it is not zero on WE. We assume at first that N is known on the trailing edge. Since M is in the zone of influence of the subsonic leading edge FE, the double integral may be inverted with a formula similar to (8.4.15). We have therefore

N(a ,b) =

fl EM"

1

Irk AD, (a,b)

/a'-a

H(a',br)da'db' (a ---a') -(b- b'

b'-p a'-a-(b'-p)

+

1

W2

d(!g-a)1

AD

'

8a'.

da'db'

(a-a')(b-b') (8.4.22)

where D1 (a, b) is in the shaded domain from figure 8.4.7, i.e. the domain A(b) < a' < a, b' < b. In front of the last integral we have the

329

THE THEORY OF INTEGRATION OF THE H EQUATION

sign + because we have changed the sense on WE to EM". On WE we shall denote N(a,10) = N(A(Q), Q) = N(3) ,

(8.4.23)

because the equation of the edge DEFA is a = A(13). The curvilinear integral imposes to eliminate from D1 (a, b) the points where b' < p. D1 from the second integral (8.4.22)is therefore the shaded domain from figure 8.4.8. Denoting this integral by T and interchanging the curvilinear integral and the integral on the domain we get:

T=

--In, a 8a' a[,! a da' -a')(b _ )(b-b') rb

[ a' - A(/3) /

N(Q) V

1

J

[I-A'((3),d0 1. a'-A(l3) b'-(3 a -A(Q)-(b'-0)J +

bN(f3)[1-A'(A)]dfJ

2

db' (b' -)3)(b - b') u - b',

da' a A(b) 7a =' r?a'

1

,

(8.4.24)

where u=a'+A-A(/3).

Fig. 8.4.8.

330

THE SUPERSONIC STEADY FLOW

For calculating the interior integral we shall prove first that b < u, i.e. that

b
(8.4.25)

Denoting by P, R, S the points having the ordinates 3 namely, P on the trailing edge, R on the la intersection with the direction of the unperturbed stream and S on the boundary of the domain Dl (fig. 8.4.8), we have R(A(fl),,3), S(A(b), (3), 3 RM"S =3 SRM" = it, p representing Mach's angle (the angle made by the characteristic lines with the direction of the unperturbed stream). It results

PS = A(b) - A(#).

RS = SM" = b - f3,

Since M"P is an are on the trailing edge we have RS < PS whence

6 < A(b) + A - A(#) < a' +,0 - A(/3) ,

(8.4.26)

because a' > A(b). Employing the substitution b'-b+A+b-Qcose,

2

u=b+A+b-Qs 2 2

2

we obtain b

db'

1

d0

2

b-# 0 cosO-s'

(b'_/3)(b--b') u-b'

the inequality b < u implying 1 < s. This integral has the form (B.6.1) and the solution is given by (B.6.4). One obtains:

Il =

(u -b (u--/B) 7r

7r

a'-A(A)][a'+Q-A(Q)-

(8 . 4 . 27 )

The following integral has the form

°

8(

1

1

a' -v J

8a'

da

'=-1

Q

1

2 JAM (a'-v)3/a

da'

a-a,

where v = b + A(fl) - Q < A(b). This inequality results from the first inequality (8.4.26). Noticing that 1

ZA(b)

(a'-v)3/2

da'

_

2

fa - A(b)

a-a' a-v A(b)-v

331

THE THEORY OF INTEGRATION OF THE H EQUATION we deduce

T=-

1- A'(#)

N(/3)

a - A(b) ) 6

f3-A(f3)+A(b)-b 13-A(a)+a-b

62

(8.4.28)

whence

N(a,b) _

H(a',b)da' dN +T

1

Irk

(8.4.29)

ADi(a,b) -vf(-a - a' (b - 6')

in T, N(fl) being unknown. The form (8.4.29) is given in [8.16].

We consider-that M tends to the position M" on the boundary. It means that we make in (8.4.29) the substitution, a integral is 1

1=

-

ll

H (a', b')d a' d b' r( ab)

-

(a - a')(b - b')

where

L(a')

= jb

A(b). The first

L(a')d a' I irk JA(b) a - a'

H(al'b')db'

(8.4.30)

(8.4.31)

(a)

b = B(a) representing the equation of the edge FABC. Obviously, this integral vanishes when a - A(b). We shall perform in the expression of T the substitution 0 - t 13 - A(/3) = t and we shall denote N(Q) = N1(t). We also denote

a-A(b)=E, b-A(b)=c, b2-A(b2)=c2.

(8.4.32)

From the first inequality (8.4.26) it results 13 - A((3) > b - A(b) whence C2>C. Hence,

T,=

ff Ni(t) 1r

c

dt t - C+E

We shall integrate by parts setting

u=N1(t), dv= 7tIm td(t-c) - c+E It results V=

arctan rLe--f = The integrated term vanishes because t = c2 implies A = b2 and .

N1(c2) = N(b2) = 0.

332

THE SUPERSONIC STEADY FLOW

r

We obtain therefore:

T=--J and 2

f_m T = -- 2 f

Ni (t) arctan

Vt

e

cd t

(8.4.33)

C2

Ni(t)d t = Ni (c) = N(b).

In this way, passing to the limit in (8.4.29) we obtain

N(A(b),b) = N(b), i.e. an identity. This means that for an arbitrary given N(b)

, the function N(a, b) given by (8.4.25) is a solution of the integral equation (8.3.73). One obtains an indetermination like in the subsonic case). This indetermination exists in the zones of influence of the subsonic

edges EE'DE and BCB'B. In these zones the integral equation of the problem is not sufficient for determining the solution. Like in the subsonic case we remove this indetermination imposing

the Kutta-Joukovsky condition. Imposing a finite velocity on EE', it results that the jump of the velocity on EE' is finite. Since the jump of the component u is given by the jump of the pressure with the changed sign, and the jump of the pressure by f (x, y), it results that it is sufficient to impose for f to have finite values on the subsonic trailing edge. We have

f(x, y)=8xN(x,y)_

\8a+

bN(a'b)=(8a+gb)(I+T).

(8.4.34)

Performing in (8.4.30) the change of variable a' -- s : a - a' = _ (a - A(b)]s and keeping the notation a - A(b) = s we deduce I

I=

rkoJ

L(a - es) f =

2

L(a) - O(£3/2) _

a V - A(b)L(a) - O(e312) ,

whence

a

+

)

I

k .1-AI(b)

=

a fk

Ate)) L(a) +

jb (a)

H(a, b')d b' + 0(E) b - b'

(8.4.35) .

333

THE THEORY OF INTEGRATION OF THE H EQUATION

From (8.4.32) we deduce

T=- 2 J 7t

(b2-A(bz)

t- b+ A(b)

NfI (t) arctan

&-A(b)

a - A(b)

dt

whence,

s

a

as + ab

b:-A%)

1- Al(b)

T=

t- b+ A(b)

N' (t)

Jb

?r/e-

t-b+a

b-A(b)

V

d t + O(,/E .

But

t-+A -b

(b) t-b+a

t --b + A(b) =

1

t-b+A(b)+e

t-b+A(b)

[1 + O(e))

,

such that finally we have: a +

:b) T (aa

N'(t)dt

1 - Al(b) f*2-A(b2)

7r f

It - b + A(b)

-A(b)

+O(f). (8.4.36)

We obtain therefore

f(x,3!) =

A'(b) 1

1

[k

b B(a)

H(a, ) db,+ %lb

(8.4.37) -A(b z)

LAb)

N(t)d t

1

t --b + A(b)

+ O(f) .

The function f (x, y) has finite values on the trailing edge (a -+ A(b), e -+ 0) when the square bracket vanishes i.e. when Ni (t)d t

pb:-A(b,)

1 fb

t - b + A(b)

Jb-A(b)

k JB(A(b))

H(b(b), ) d b'

-G(b) . (8.4.38)

This condition is an integral equation (of Abel type) for determining the unknown N' (t). Using the notations (8.4.27) we may write this equation as follows

j2 Ni (t)d t

(8.4.39)

J

v where Gl (c) = C1 (b - A(b)) = G(b). We deduce f'2

z

dc C

(C3 N'(t)dt

-xJc

t-C

-

t z

C, (C)

xdC.

334

THE SUPERSONIC STEADY FLOW

Changing the order of integration in the left hand member we get: C2

NI(t)dt J

do (c - x)(t - c)

- _ f C2 Gi(c) dc. c-x

s . 1. Since Nl(c2) = N1(b2 - A(b2)) = N(b2) = 0 (on the leading edge N vanishes), we obtain

j Glt f t --X d t.

1

N, (x)

7r

For x = b - A(b) we deduce l Jbz-A(bz) _A(b)

Gl dt (t) t - b + A(b)

or, using the change of variable t

i3 : 3 - A(,3) = t,

N(b)

G(i3)[1

1

N(b) _ R

A'(/3)l d /3

VP - A(/3) - b + A(b)

rb

(8.4.40)

K(fl)[1 - A'(i3)ld/3 (3 - b + A(b) - A(I3)

I f' bz

where

8 H(A(#), b') db k D(A(#)) VF-7 1

(8.4.41)

.

This formula determines N in every point of the subsonic trailing edge ED. Replacing this expression in (8.4.28), we may find out T. In the sequel we shall give an explicit expression of T. Changing the order of integration (fig. 8.4.9), we deduce:

T= 12 --A

b

dJ3 ,3 - A(/3) + a

b

Jbzb2

A (fl) + A(b) - b a

_b. [1

K(E3')(1-A'(f)ldA'

-b+A (b) -A(f3)J

_-

(8.4.42)

b

_

2

a --A (b)

b

K(A')I (1Y)(1 - A'(Q'))d // ,

where, using the substitution p -+ t:,3 - A(0) = t,/3' - A(i3') = t',

IV) =

f

1

(t - c)(t'-

dt (8.4.43) (b - a)

THE THEORY OF INTEGRATION OF THE H EQUATION

335

0

b

be

I

0

be

Fig. 8.4.9.

From (8.4.26) It results b - A(b) < i - A(8) for every j6, hence for

_ P. This implies c < t'. The integral from (8.4.43) has the form (B.6.11). It results Ir

1=-

[a-A(b)]

-A(.')-b+a

whence T=_1

K ) [1-A'(B)]d3

(8.4.44)

rb, /3-A(P)-b+a Using this form of T, we express the solution in the domain bounded by the curve EE'DE by means of the formula (8.4.29). 2. If M(a, b) is in the domain bounded by the curve BCB'B (fig. 8.4.10), then we shall utilize the integral equation (8.3.74). Obviously, the curvilinear integral does not vanish on the arc BM'. Utilizing the solution (8.4.15), we deduce: N(a, b) where Tl

= plc 1 JJD1(.,b)

H(a', b')

a - a' b -

d a' d b' + Ti

(8.4.45)

a' d b' =- 2 JJJD, ad-a(b-Y) (8.4.46)

ef° 8b' f

FL- B(a) N(°`)

1]d a

-a b'-a'-B(a)+a

THE SUPERSONIC STEADY FLOW

336

Taking into account that the equation of the edge BC is Q = B(a),

we denoted N(a, B(a)) = N(a). Dl (a, b) is the shaded from figure (8.4.10), i.e. the domain bounded by MM', the parallels 0 = B(a) and A = b and the leading edge included between these parallels. D] is the

domain bounded by the contour MPQM' imposed by the condition

a < a'. Interchanging in (8.4.45) the curvilinear integral and the double in-

337

THE THEORY OF INTEGRATION OF THE H EQUATION

tegral we obtain

JaN(a)[1

T, =

T1

a 5-b'

- B'(a)] az

f

(

a)

b - b'

do

({/_B(a)j

' w-all

db' d1

I

(8.4.47)

where

w=b'+a-B(a). The similarity of the expressions (8.4.47) and (8.4.24) is obvious. TI

may be obtained from T replacing b, /3 and A by a, a and B and conversely. It results therefore

N(a) 1-B(a) da T,=- Ir b-B(a)f \/a - B(a) + B(a) - a a - B(a) + b - a az (8.4.48)

and then 1

T1 = - -

r° Kl (a) (1 - B'(a))d a aZ

where R 1 (a )

I

a-B(a)-a+b

k A(B(.))

H(a B( )) d a ' Q

(8.4.49)

(8 . 4 . 50)

The solution in the domain bounded by BCB'B is (8.4.45) where Tl is given by (8.4.49).

8.4.7

The Wing with Dependent Subsonic Trailing Edges

For a wing with subsonic dependent trailing edges (fig. 8.4.11) the solution in the zone AHF is determined by the formula (8.4.9), the solution in the zone ABPH by the formula (8.4.15), the solution in the zone FHA'E by a formula analogous to (8.4.15), the solution in the zone HF'IA' by the formula (8.4.19), the solution in the zone BCE'IB by the formula (8.4.45), where T1 is (8.4.48), and the solution

in the zone E113'DE by the formula (8.4.29), where T is (8.4.44). It remains to determine the solution in the zone IE'B'I, i.e. in the common zone of influence of the subsonic trailing edges.

Noticing that the curvilinear integral does not vanish on BM' and WE and utilizing in the first case the expression from (8.3.74), and in

338

THE SUPERSONIC STEADY FLOW

the second case the expression from (8.3.73), we deduce that the integral equation has the following form:

N(a , Q)

k

TJ , (a -

k Of +7r 8b

0)3/2(b - /3)3/2 a d p+

N( a)

al,a)

+k f ir 8a AruE 0

- B(a)

(B'(a) - IId a

a-a b-a-B(a)+a+

a - A(#)

(8.4.51)

[1 - A'(Q)]d 0

b-(3 a-b-A(O)+#

the integral on BM' representing in fact the integral with respect to

a on the interval (a2, a) and the integral on WE representing the integral with respect to 13 on the interval (b, b2). 03 b

b3

M'

a.b)

O-

Fig. 8.4.11.

In this case, the point M(a, b) is in the common zone of influence of the subsonic leading edges AB and FE. Hence the double integral may be inverted according to the formula (8.4.19), D2 representing the shaded domain from 8.4.11, i.e. the domain bounded by the curve

THE THEORY OF CONICAL MOTIONS

339

Af AI"AI""A1I,` AI'A1.

N(a,b)=

1 %TA'

ff , n

V

H (a' ) da'db'+Tt+T, (a - a')(b - b')

(8.4.52)

where Tt has the expression (8.4.43), and T, (8.4.27). Setting Al AI"(a -+ A(b)), the integral on BM' vanishes (as we can see on the figure), such that from (8.4.52) one obtains N(A(b), b) = = N(b). Imposing the Kutta-Joukovsky condition, we deduce that T has the form (8.4.43). Similarly we deduce that Tt has the expression (8.4.49).

Now the problem is completely solved. In the end, it is at pleasant duty for me to mention that for elaborating this section I utilized especially Homentcovschi's paper [8.16] and the license thesis of my former student Luminita Berechet [8.2].

8.5 8.5.1

The Theory of Conical Motions Introduction

The theory of conical motions was initiated by Busemann in 1943, 18.31. It refers to wings bounded by conical surfaces with the vertex in the origin of the system of coordinates, the body being placed downstream.

The surface of such a body is a smooth surface consisting of half-lines issuing from the origin and leaning on a closed curve situated in the plane

x = 1(xi = L I). According to the boundary conditions the velocity is constant along every half-line passing through the origin and belonging to the boundary of the body. The hypothesis of conical flow leads to the assumption that the velocity has everywhere in the fluid this property. We have therefore

v(mx, my, mz) = v(z, y, z)

(8.5.1)

for every in . real and positive. It means that the velocity is a homogeneous function having the zero degree. Under this assumption the equation of the potential becomes simpler, the unknowns depending not on three but on two variables. After Busemann, many authors (Langerstrom [8.22]. Germain [8.11], Poritsky [8.28], Ward [8.34], Heaslet and Loniax [8.15], Iacob [8.18], Carafoli [1.5] $.a.) have contributed decisively to the development of this theory. In all this theory, which will be called

the classical theory, we make the hypothesis that the motion is conical.

340

THE SUPERSONIC STEADY FLOW

Starting from the lifting surface equation in a supersonic stream, one may prove that if the wing is conical, then the solution of the integral equation is conical. For the equation (8.3.30) this thing is done in (8.5], and for equation (8.3.71), in [8.161. In the present subsection, utilizing the solution from the previous subsection, we shall give the solution of the conical motions by particularization. We shall also give the basic elements of the classical method, because they may be obtained directly, without knowing the solution of the lifting surface equation.

8.5.2

The Wing with Supersonic Leading Edges

We assume that the surface of the wing is a conical surface. From

z = h(x, y) it results that Az = h(ax, ay) and, with A = (l/x), h(x ,y ) = xh (1 , x ) = xg (x )

(8 . 5 . 2)

We deduce therefore

hr =.9 (x)

- r9 \x)

and then

H(a, b) = F (a )

Fig. 8.5.1.

.

8.5.3)

341

THE THEORY OF CONICAL MOTIONS

We shall consider now a wing with supersonic leading edges (fig.

8.5.1). Denoting by b = mla the equation of the edge OA and by b = m2a the equation of the edge OF, it is obvious that m1 < 0, m2 < 0, because on OA we have a > 0 , b < 0, and on OF, a < 0 , b > 0. Since the entire domain D is only in the zone of influence of the supersonic leading edge, for every M(a, b) the solution is given by the formula (8.4.9) where D1(a,b) is the domain limited by the curve

OAMFO, and H(a',b') will be replaced by F(µ), where b' = µa'. We have therefore to put b = ma and to replace the variables b and b`

by m and p. For M in the zone OCD we shall denote N = N2 N2 (a, ma) = N21 + N22 + N23,

(8.5.4)

where

k 1., Y-f*Jo

N21(a,m) --

1

-

ad 'a'

\J°

k

+

N23 (a, m)

m F()

1

N22(a, m) =

aa')(al-c)}dµ, c=

(a

Pa,.

'dµ+

(a - a) (c - a)

r°O F(µ)

r °'d a'

1dµ ,

Fv

14lJco

oo

aa''-c))df

(a

(8.5.5)

N21 representing the integral on OAA', N22 on OA'MF' and N23 the integral on OF'F. Performing the calculations we find: a irk

N21 = -

µ m+11

C,

arctan

it m

-

VM--\ Jt

/I F(µ)d µ

N23 = - a /"'.W 2 /m+p

ak J7-71 arccot rr m -

N22

a irk

f o

µ

,

)F(P)d,

(8.5.6)

\m+µ '+' -'1F(µ)dµa.

2µf If - /I

it

It is obvious that

f(x,Y) = (Oa + 8b)N2

(8.5.7)

342

THE SUPERSONIC STEADY FLOW

is constant on every half-line issuing from the origin. The flow is conical.

If M(a, b) is in the zone OBC, then the solution is

kJmt F(-) Jco

N2(a, m)

(a

aa')(a' - c) )

dy (8.5.8)

a

f

m

F(p) m+1dp

M17-p

A

and if M is in the zone ODE, then Ns(a, na) -

1

wk

-a 8.5.3

(p) z FlN

f.

V---14

a d a'

C

(Z

(a - a')(a' - c) (8.5.9)

"'2 P((µ) m+{A dµ, Im 11 fl

V-

The Wing with a Supersonic Leading Edge and with Another Subsonic Leading or Trailing Edge

Further we shall consider a wing having a supersonic leading edge (the

edge OB from figures 8.5.2) and another subsonic leading edge (fig. 8.5.2a)), or a subsonic trailing edge (the edge OE from fig. 8.5.2b)). In this case, the solution is obtained with the formula (8.4.15) where Dl is the domain limited by MFF'AM. For M belonging to the interior of Mach's cone i.e. M in the zone OCEO, noticing that the equation of the line FF' is a= d (it is obtained from the intersection rn of b' = b with b' = m2a') where d = a, we deduce N2 (a, m) w

i

m2

F'(µ) d

ak

mi

+.1, J0 1

+ ;k J,,,

µ

ad a' (a - a')(a' - c)

d

7

F)dµJ ' F(p)d µ rjA

(a

sa a'

+ +

(8.5.10)

a'd a' Jd

(a- a) (c - a') '

c being defined in the previous section. The interior integrals are elementary.

THE THEORY OF CONICAL MOTIONS

343

Oa

b)

a) Fig. 8.5.2.

For M in the zone OBC noticing that the intersection of the line b' =pa' with b' = b has the abscissa c, we obtain

Ni(a,m)

- ak a

ffl

dJ

,

(8.5.11) rm

F(p) m + µd

/ml 7 8.5.4

(a-&)(a'-c)

L

The Wing with Subsonic Leading Edges

When the two leading edges are subsonic, it is difficult to utilize the solution from the previous subsection. We shall use therefore Homentcovschi's idea concerning the direct integration of the equation (8.3.73).

Assuming that N(a, µa) has the form

N(a, µa) = aN(µ),

(8.5.12)

344

THE SUPERSONIC STEADY FLOW

the equation we have in view reduces to k a2N(µ)d a d µ _ 47r JJD (a - a)3/2(ma - Na)3/2 - F(m)

(8.5.13)

D being the shaded domain from figure 8.5.3. With the same notation for c (= ma/p), the equation (8.5.13) may be written as follows

/° , µ3/2 fmjV('2

4a

0

a2da d is (a - a)3/2(c - a)3/2 J (8.5.14)

N(p) J.r`

+k 47

r

a2d a {a - a)3/2 (c - a)3/2

µ'/2 Obviously, in the first integral a < c, and in the second, c < a. The interior integrals are considered in Hadamard's Finite Part sense. Taking into account the formula da

ml

f

aa

d,,=-'2

a a

0 (a

f

a)3/2

d a,

(8.5.15)

given in (D.4.3), the equation (8.5.14) may be written as follows k

a2do I -N(p)(02 )dp+ 3/2 f a a (a a)(c - a) ,ni o

k,

14

(8.5 .16)

ml N(

492

cf

143/2

0

2

a2da ld it (a - a)(c - a) / µ

We notice that in the first case (c > a), we have

f

- 4ac In fc + f - 34 (a + c) fac+ 3(a + c)2 8 f - fa

a2d a

(a - a)(c - a)

Jo

and in the second (c < a),

f V(a -a)(c-a) a2d a

`

_

3

4(a+c) ac+

3(a + c)2 - 4ac f- + f In 8

.

Vc-

(8.5.17)

The results are the same if we put under logarithm (/ - . Performing the calculations, it results that the integral equation (8.5.16) may be written as follows k

T' N) In

-2 mµ(M

+14)2}dµ=F(m) 1

(8.5.18)

THE THEORY OF CONICAL MOTIONS

345 0,

(N Fig. 8.5.3.

for ml < m < m2. We put N(p) = µNI(µ). Denoting

H(m) =

k

.,

In .IA

/./ + ///7;, t%' V''

d µ,

on the basis of the equation (8.5.18) we obtain the following differential equation

m2H"(m) + mH'(m) - 4H(m) = -F(m).

(8.5.20)

The homogeneous equation has the linearly independent solutions v/ and l//. Hence, the general solution of the equation (8.5.20) is: H(m) = 2cl frra --

202

+ Fo(m),

(8.5.21)

cl and CG2 representing constant which have to be determined, and Fo representing a particular solution of the non-homogeneous equation. From (8.5.19) and (8.5.21) we deduce the following integral equation

for Ni k fm

Nl(N) In

/ + `dµ = 2c1 VG

+ Fo(m).

(8.5.22)

Deriving with respect to m we obtain k

Ni (l+) d

7r,'µ-m

p = cl +

- + /o (m) , m

(8.5.23)

THE SUPERSONIC STEADY FLOW

346

which is the classical equation of the thin profiles. As we have already observed, from the definition of N it results that Nl (N) vanishes on the leading edge. The solution of the equation (8.5.23) which vanishes

for m = ml and m = m2 has the form (C.1.14) with a condition having the form (C.1.13). Taking also into account (B.5.8) it results ' mZ

1

fFa(µ)

Nl(m) = -7r (m - ml)(m2 -

+c2

m1){m2 -

dp N-m+

(m - ml)(m2 - m) m mlm2 (8.5.24)

The condition (C.1.13) will give

x cl +

rm

C2

mlm2

Voµ

+f

m, ( nl)(m'l -P)

d y= 0

(8.5.25)

and will be useful for the determination of the constant cl, after determining the constant c2. In fact, the constant cl is of no interest. The constant c2 which intervenes effectively in the solution (8.5.24) will be determined imposing for the solution to verify the integral equation (8.5.18). This condition is necessary because the solution was determined after some derivations. Writing the equation (8.5.18) as follows 2

T",

Nl (µ) d µ+

N1(l1) d

1

µ-m

2m

(8.5.26)

- 4m jrn, ' Nl (i)K(m, it)d /1 = - k

F( m) M3/2

where K(m, µ) is the symmetric kernel K(m, u) =

+ 1

Inl%FM-VjAI'

(8.5.27)

we notice that the equations (8.5.24) and (8.5.26) will determine the un-

knowns Nl and c2. The replacement of Nl from (8.5.24) in equation (8.5.26) leads to difficult calculations. It is necessary, for example, to know the formulas for interchanging the FP (Finite Part) with PV (Principal Value) and PV with FP (the formula of Poincar&Bertrand [A.27]).

FLAT WINOS

347

The equation (8.5.26) may be also solved numerically using the Gauss-type quadrature formulas (because N has the form N1(µ) =

(8.5.28)

(µ - ml)(m2 - A)n(p),

n(p) representing the new unknown).

Flat Wings

8.6

The Angular Wing with Supersonic Leading Edges

8.6.1

For the flat wings having the angle of attack e we have F = -e. The theory of the angular airfoil with supersonic leading edges may be

obtained from 8.5.2, putting F = -2. Since m 6 m=-, U= b,W, c=-a, a it

(8.6.1)

it results

°

ad a'

(a-a')(a'-c)

Jo

N21 (a, m)

a r 1 + m) arctan F-!A+aFT, ( 8.6.2)

-

k-v [ m -mli arctan

-

(8.6.3)

Similarly one obtains N23

m)

2ac (M -m 2

arctan

(8.6.4)

We have also a

L

a'd a'

(a-a')(c-a')

-a

Iv-

m + a rI + ml 1i) V u 2`

VrM +

and then, from (8.5.5) or directly from (8.5.6),

N22(a,m)ael,I J' ffm-1 kzr

2 .

m in Vm- +f ddµ

1

+

µ/

i

i- V M (8.6.5)

THE SUPERSONIC STEADY FLOW

348

In I we make the change of variable In this way one obtains

l1 = x and we denote vM- = q.

I=I1+g212-2g13,

(8.6.6)

where r4

11= J In x+qdx+ i:° In x+qdx=111+112, 12-

Ix-g1

ix - qj

v

2In

x

13= 11 d. +

f

x

Ix - q)

=131+132.

The integral 11, is elementary (it has an integrable singularity). One obtains (8.6.8)

111 = 2q In 2 .

The integral 121 has a strong singularity in x = 0. It must be considered in the Finite Part sense. On the basis of the formula (D.2.2) we have

12i =

f9ln(x+q)-1n(q-x)-2x/q + 2Inq x2

q

Integrating by parts, one obtains 121 =

2(ln2q+ q

1).

(8.6.9)

Using (D.2.3) we deduce (8.6.10)

131 = In q.

For calculating 112,122 and 132 we make the substitution x = 1/y and we utilize the results (8.6.8) - (8.6.10). One obtains 13 = 0, 11

2

q( -+1 q

=2g1u2+2

In

(8.6.11) 2

12=2g11n2+1) +g1n2, such that

N22(a, m) = -, /m-(2 2 + 1) .

(8.6.12)

349

FLAT WINGS

In this way, taking into account that m = b/a, the formula (8.5.4), together with (8.6.3), (8.6.4) and (8.6.12) give a

b

N2(a, b) _ -

- Zee (

-m VM1

am

b

_n2-aarctan v72

a -b -

arctan

2

-

)! -

ab) v a (2 in 2 + 1) . (8.6.13)

This is the solution when M is in the zone limited by the characteristics

OC and OD . If M(a, b) is in the domain limited by OD and OE, then we use (8.5.9). We deduce that N3 (a, b)

k

£ 7n2 (m2a

- b),

(8.6.14)

and if M is in the zone OB, OC (fig. 8.5.1) N1 (a, b) _ - k

8.6.2

b

mla

(8.6.15)

The Triangular Wing. The Calculation of the Aerodynamic Action

In order to obtain a finite action, it is necessary to consider a wing having a finite area. We assume that in the physical plane it has the triangular form from figure 8.6.1. In order to obtain a well determined wing we must give the coordinates of the points A and F. Let mi be the inclination of the line OA and ai the ordinate of the point A in the frame of reference Oa,J and m2 the inclination of the line OF and a2 the ordinate of the point F in the same frame of reference.

Then the equations of the edges OA and OF will be b = mia respectively b = n2a, and the coordinates of the points A and F respectively (ai, bi = ml al) and (a2i b2 = m2a2). The equation of the line is b = (m1 - m3)a1 + mia, where m3

_

7Tb2a2 - miai

a2 - a1

(8.6.16)

(8.6.17)

350

'rilE SUPERSONIC STEADY FLOW F(a,.b,)

Ap .e

.0

D(O.b4)

A(a,.b,) Fig. 8.6.1.

Denoting by (a3, b3) the coordinates of the point C and by (a,,, b4) the coordinates of the point D, we deduce a3=1-m1,

b3=0; a4=0, b4=(m1-9n3)a1.

(8.6.18)

As we already know, the lift is given by the formula

= L

- ffD(x.y) Bpi I d x, d yi = -PmU,2,,,Lo J1D(=.y) epO d x d y,

and the lift coefficient cL, by the formula CL = 1 L 2P0U00A

,

(8.6.19)

where A is the area of the wing and Lo, the reference wing. Taking (8.3.69) into account, passing to coordinates on characteris-

FLAT WINGS

351

tics and applying Green's formula, it results CL = - A

rf

2 o JJD(= v)

2L A

f

f (x, y)d x d y = 8 N(x, y)d x d y ax (8.6.20)

A

IJD(a, b)

(8a + ab

)N(ab)da d b =

- - Ac /eD(a,bjM (b - a)

IA 1,

where

1=11 + 12 + 13, It = JOA+AC+cd"1 d(b - a), (8.6.21) 12

13 = JoD+DF+F 3 d (b - a).

- JO C+CD+DO-2 d (b - a),

Taking (8.6.15) into account, it results

Nid(b-a)loA=0 N1d(b-a)IAC_N1d(b-a)

b=(ml -m3)a1+msn

-E(m3 - m1)(m3 -1) (a k

m1

N1d(b-a)I co=Nid(b-a)lb_o-

- ai)da

kmlada,

such that

I1

= _E(m3 km1)(m3 - 1)

at)da - £'nl

3

ada =

To

S(m3 - m1)(m3 - 1) (a3 - at)2

k -m1

2

E -m1 a3 k

2

(8.6.22)

In the same way one calculates 13. Taking into account (8.6.14), one

352

THE SUPERSONIC STEADY FLOW

obtains 13

- k -m2 2 E

443

E(n3 - 1)(m2 - 1n3) k --r n2

a!+ 2

(8.6.23)

+Eata2(m3 - 1)(m1 - m3)

k -rn2 and the problem of calculating c1, is solved.

8.6.3

The Trapezoidal Wing with Subsonic Lateral Edges

We assume that the projection of the wing (which is flat and has the angle of attack s) on the plane xOy is the isosceles trapezoid ABEF from figure 8.6.2, having the bases 21, 2L and the height h (dimensionless quantities). The direction of the unperturbed stream is perpendicular to the bases.

Fig. 8.6.2.

We consider the case a <,u. The leading edges AB and FE are subsonic and the edge AF is supersonic. We also assume that h is such that the subsonic edges are independent (for the sake of simplicity).

Obviously, in this case the fluid motion is not conical. In the domain D2 the solution may be obtained from (8.4.9), and in D1 and D3 from (8.4.15). For developing these solutions, we must characterize the domains in coordinates on characteristics. We must specify at first the

FIAT WINGS

353

physical coordinates. We introduce the parameter m by the formula

0<

h

tangy =

< tan p

.

(8.6.24)

It results therefore 0 < m < 1. The distances dl and d2 are defined bu the formulas

d1=lk,d2=d1+h.

(8.6.25)

The equations of the straight lines AB and FE are respectively

-y=l+ m(x - dj),y = t + m(x - d,).

(8.6.26)

The equation of the wing (see figure 8.6.3) is

d12d2 )

x

(8.6.27)

the function being defined on the domain D + D1 + D2 + D3 from the XOy plane:

d1 <x
-yi(x) < y < y1(x),

where (x - d1).

y1(x) = e1

(8.6.29)

k

Fig. 8.6.3.

It obviously results

H(x, y) = -c.

(8.6.30)

Passing to coordinates on characteristics we put

b-a

a+b 2

'y

2k

(8.6.31)

It results

H(a, b) = -e

(8.6.32)

354

THE SUPERSONIC STEADY FLOW

and the following equations for the sides of the domain D

AF:a+b=2d1,

BE:a+b=2d2 (8.6.33)

AB:a=2d1+mob,FE:b=2d1+moa, where we denoted mo

=

i+ m

(8 . 6 . 34)

For the vertices we deduce the following coordinates:

A = (2d1, 0), B = (d2 + kL, d2 - U), (8.6.35)

F= (0,2d1),E= (d2 -kL,d2+kL). Taking (8.4.9) into account, we deduce for the solution in D2 a kir

N2(a,b)

da'

1-b

db' f a - a' J2d1 -a, %Fb - b'

(8.6.36)

=-1(a+b-2d1). The solution in D1 is given by (8.4.15). Noticing that A(b) = 2d1 - b,

B(a) = 2d1 - a,

(8.6.37)

it results Nl

(a'

er

b)

krr

d b' , -4

b

a

da' b' Jet -V a - a'

(8.6.38)

Obviously, this expression may be obtained from (8.6.36), changing a with b. Hence

N1(a,b) = -1(a + b - 2d1)

(8.6.39)

and an identical expression for N3(a, b) (because of the symmetry). According to (8.6.20), for cL we have the expression 2

CL = - kA

(8.6.40)

where

I= POD N(a + b - 2dj)d (b - a)

-

(a + b - 2d1)d (b - a) = (1- m02)(d2 - kL)2. AB+BE+EF

FLAT WINGS

355

The Trapezoidal Wing with Lateral Supersonic Edges

8.6.4

We consider an isoeoelea trapezium with the bases having the length

2L respectively 2t perpendicular on the direction of the stream at infinity (fig. 8.6.4). We Introduce here again the parameter m defined by the relation:

k-tana=Lht>tanw=k

(8.6.41)

F. 8.6.4.

Obviously, m > 1. Denoting by dl the distance from the small basis to 0 and by d2 the distance corresponding to the great basis, we obviously have d2 = dl + h, and for the c arte ian coordinates of the vertices of the trapezium

A = (d,, t), B = (da, -L), F = (dl, t), E = (d2, L)

(8.6.42)

The coordinates on characteristics are obtained from the formulas,

a=x-ky, b=x+ky. No ti c i ng

th a t

(8.6.43)

d2 = kL, we ded uce A = (di + kt, dl - kt),

B = (2kL, 0)

F = (dl - kt, dl + kt),

E _ (0, 2kL)

(8.6.44)

356

THE SUPERSONIC STEADY FLOW

Using the notation

d1-k£ d1+2h-k8

k

2

k-1

m+1

(8.6.45)

and observing that from figure 8.6.4 we have the compatibility condition ao < p, which implies

dl = tan ao < tan p =

1

dl > kt,

(8.6.46)

we deduce 0 < k < 1. Now, the equations of the sides of the trapezium may be written as follows

AB :a+kb=2d2iFE: k-la +b=2d2, (8.6.47)

BE:a+b=2d2,AF:a+b=2d1. The entire leading edge is supersonic. The solution may be expressed by means of the formula (8.4.9). To this aim it is necessary to specify

the functions a = A(#) and A = B(a). We have: - on the edge BA, b .- BI(a) = -- on the edge AF,

2d2

a,

a = Bi 1(b) = 2d2 - kb, (8.6.48)

b = B2(a) = 2d1- a,

a=

t

1

}=Li_b.

We must also observe that for determining the lift coefficient we do not need N(a, b) on the entire wing, but only on the trailing edge BE. Indeed, this may be expressed with the formula (8.6.20), and N on the leading edge BAFE vanishes as we have already mentioned in formula (8.3.68). The domains of influence are (fig. 8.6.3) : D1 = ABA'A,

D2 = AA'A"A,

Do = AA"F"A,

D3 = FF"F'F, D4 = FF'EF Hence, we shall put

I=11-+-12+I3+ 14,

(8.6.49)

FLAT WINGS

357

where

r

it = J

BA'

I. = r

JA"F"

14 =

JF"F.

12 = f

Ni (a, b)d (b - a),

JA'A"

No(a, b)d (b -- a). 13 = J

Ni (a, b)d (b - a)

"F'

N2 (a, b)d (b - a),

N3(a, b)d (b - a),

(8.6.50)

.

Using the formula (8.4.9) and the equations (8.6.47), we deduce

__ 8

N1 (a'b)IBA' =

N2 (a, b)IA'A#l

a

k

,l Bt I(b)

E

Tr -

Bi (b)

No= -h, N4=

-,-(I -

da' Ilrb dillu a -a' B,(u') vb - v 1,9A'

da' a - a'

b

JB2(n')

_eb(l-k)

db' b - b' L'A" - ... ,

)a

After elementary calculations we deduce: E(I - k)

11 = 14

and finally,

k2

k.

f

d2-"-h

bdb

e(1 - k) (dl - kf)

= k2 Vk-

2

4Eh = - E(1- k) (d1 -2kt)2 ' to =-k2 (k£ - h),

2'

kv-.

Chapter 9 The Steady Transonic Flow

The Equations of the Transonic Flow

9.1 9.1.1

The Presence of the Transonic Flow

We call transonic flow the flow which is subsonic in a domain of the space and supersonic in the adjacent domain. One demonstrates (for the potential flow - see [1.21] pp 517, 518) that the equality v = c comes true in E2 only on curves separating the domains where the flow is subsonic from the domains where the flow is supersonic, and in E3 on the surfaces which separate such domains. The name of transonic flow was introduced by Th. von in 1947.

In the present paper the transonic flow has been encountered in several situations. At first, we have to mention the one-dimensional flow [1.11] §4.5.

The formula (4.5.8) which gives the variation of the velocity against tile variation of the cross section indicates that, in the subsonic flow (Al < 1). the velocity increases when the area decreases and decreases when area increases (like ca in the incompressible fluid), while in the supersonic flow (M > 1) the variations are produced in the same sense. This circumstance leads to the conclusion that in a tube having the shape from figure 9.1.1 the flow may become transonic. To this aim it is sufficient for the upstream subsonic velocity to have the critical value in the section of minimum area. Further since the area of the section increases, the velocity also increases, remaining supersonic. In the linearized theory we deduced for the aerodynamic action the

formulas (3.1.33) and (3.1.34) in the subsonic case and (8.1.9) in the supersonic case. It is obvious that these formulas are not valid in the vicinity of A! = 1. For the flat plate these formulas become (3.1.35) and (8.1.22). The figures (3.1.3) and (8.1.3) are very suggestive. In the cause of the subsonic flow with great velocity past thick bodies like in figure 9.1.2. the flow may become transonic. Indeed, considering

THE STEADY 'TRANSONIC FLOW

360

Fig. 9.1.1.

the flow between the streamline which includes the boundary and a neighbor streamline L, we shall find that the flow is like in a tube. Since the domain between these lines narrows because of the body, it

Fig. 9.1.2.

follows that in the vicinity of the body the flow nay become supersonic. The transition from the supersonic flow to the subsonic flow is performed

by a shock wave S according to the scheme described in 1.3.6. Until S the flow is transonic. We shall deduce in the sequel the equations which describe this flow. The flow with great subsonic velocity past thick bodies is described by the scheme from 9.1.3.

-> V
y« r- Loctaau Fig. 9.1.3.

Finally, in the supersonic flow, for great velocities, practically in the

hypersonic regime, it appears, as we noticed in 1.3.6, detachedor at-

361

THE EQUATIONS OF THE, TRANSONIC FLOW

tached shock waves (figure 1.3.5). Behind these waves the flow is transonic (it passes from the subsonic regime (A,12 < 1) to the supersonic one (A12 > 1)). As we could see, in modern aerodynamics the transonic regime is frequent. So one explains the great number of papers devoted to this subject in the last years. We mention especially the papers of Bauer,

Garabedian and Korn [9.1] devoted to the theory of minimum drag wings. There are three dominant methods for studying the transonic flow, namely:

1° the hodagmph medwd, suitable only for the plane steady jet flow (see for example Ferrari and Tricomi [9.11], Manwell [9.30], [9.31] etc.);

2° direct analytical methods, based on the semi-linearized equation of the potential. They lead to integral equations which may be solved numerically;

3° numerical methods applied directly to the system of equations which describes the fluid flow (we mention especially the finite elements method). In this chapter we present some direct analytic methods.

9.1.2

The Equation of the Potential

The reasoning based on the assumption that the independent variables x, y, z have the same role in the structure of gyp, (utilized for deducing the equation (2.1.39)), is not valid for the flow in the vicinity of M = 1. Indeed, in this vicinity M2 - 1 becomes itself a small

parameter. If, for example AI2 - 1 = O(E), then for V_: = 0(cp) it results +p.y and W.'. = 0(E2). One imposes an analysis of the order of magnitude of the perturbations depending on the geometry of the body and the conditions which determine the flow (Mach's number Al, the thickness and length parameters, the angle of attack, etc.). In fact, the idea that the variables y and z do not behave like the variable x.

results from the special property of the Ox axis (which is parallel to the direction on the unperturbed stream). We shall introduce therefore the variables y = u(E)y.

4 = V(C)Z,

(9.1.1)

expecting for vv(e), like for q(r) from the expansion Or, Y. Z' `) = U (x + T (E),(x, J, <_) +

... ]

to be determined by comparing the orders of magnitude.

(9.1.2)

362

THE STEADY TRANSONIC FLOW

Coining back to the linearized theory 2.1, we notice that for Al = = 1 a catastrophe is produced (it disappears terms from the equations). The lift and moment coefficients become therefore infinite. But this catastrophe has only a mathematical nature, not a physical one. It is

determined by the fact that in the vicinity of the value Af = 1, the order of magnitude of all the first order derivatives is not the same (E). One imposes therefore (9.1.1). It results

0r=U(1+1150:+...),OS,=U1)v

+...,O:=Urpi (9.1.3)

Orr = UtWrr,

Ory = UTJVP,,y,

0, = Ugv2 ;OW .

From (1.2.17) we deduce c2 = c2 -

(7 - l )U2>);pr + 0(112) ,

(9.1.4)

and from (1.2.16) written explicitly as follows

(r.2¢r).o +(C?-0y)Oyy-20=.ysOv+...=0,

(9.1.5)

we deduce

[1 - M2 -(y + 1)(M2 -

(y +

+ [1 _ (y - 1)(Af

0(7)2)l(p=r+

(y - I)W=)v2 y-

(9.1.6)

- 2M2vrl2"o + 0(1j2v2) + ... = 0. For a fixed Al , we see that the equation is consistent if ii --+ 0 when

q-'0,so

v2, t), 1- M2_ 71.

(9.1.7)

We introduce now the boundary condition. If z = eh.(x, y)

(9.1.8)

is the equation of the perturbing surface, imposing the condition to be a material surface i.e. Eh=dOr + Eh,,Oy = &

which implies, taking into account (9.1.2) Eh. =

(9.1.9)

THE EQUATIONS OF THE TRANSONIC FLOW

363

whence

E = 9V.

(9.1.10)

Taking (9.1.7) into account, we deduce

t)=E2'3

V=Et13

(9.1.11)

When M - I

we have to compare. in (9.1.6) the terms of order immediately superior to those which gave (9.1.7). It results I - M2 = = Kv2 whence

K-

_ hl2

1 .

(9.1.12)

.

K is called the parameter of the transonic similitude. In this way, the first approximation from (9.1.6) (the dominant equation) is

[K - (7 +

rpyp + 4p: = 0.

(9.1.13)

This is the equation of the transonic flow (the equation of the potential). It is elliptic if Cpl < K/(7 + 1) and hyperbolic if gyp= > K/(y + 1). The relation V_ = If/(-y + 1) is verified on the surface where

V2 = c2. Indeed, using the notations (2.1.3), and taking (1.3.32) into account, the condition V2 = c2 becomes

V2=c2=co-7 v;2=cam- 1 o (Vt2-U2).

(9.1.14)

Here, the dominant relation is u2 1+2 u, } = c2 - (7 - 1)U2vu,

(9.1.15)

T

whence ii = K/(7 + 1). Now it is clear that the non - linearity is necessary for making this transition possible.

The first study of the transonic flow has been performed by von Karmsui [9.26). By various methods the problem was investigated by Ovsiannikov [9.531, Guderley [9.15), Cole & Messiter [9.6[ etc. Cole's study from 1975 relying on the method of perturbations was continued by the same author in 1978. In the last study one proves that if we denote by s the thickness parameter and we set for the cross sections

y-Et13y, =-E113' , then the potential 0 has the following structure [9.54]:

4)(i',y,4;Al ,Q,b,b) = U[z+E2/39(x,V,z;K,A,B)+ (9.1.16)

+E413 t'2(x, y, Z; K, A, B) + ...,

,

364

TilE STEADY TRANSONIC FLOW

where K is the transonic parameter (9.1.12), A, the parameter of the angle of attack = aft, and 13, the span parameter = 6e113. For p one obtains the equation (9.1.13).

9.1.3

The System of Transonic Flow

It is rigorous to perform the asymptotic analysis on the system of equations and not on the equation of the potential which has been obtained from the system by derivation with respect to the x, y, z coordinates. We present here such an analysis which was performed together with professor A. Halanay in the years '80. We utilize the coordinates y and r in the form (9.1.1) and we denote E)

=v

\x' V(E) , +'7 (E)

)'

h(x, y,

E) = h (x,

(9.1.17)

It results r"(X,j/,E)

= it , x,

h,

V(£)

(2-,

y(E)/ V(E)

and the boundary condition Elts(x, y) 11 + u(x, y, Eh(x, y))] + Eh' ,(x, y)v(x, y, eh(x, y))

_

= w(x, y, Eh (x, y))

becomes

X.

(r, y,

c) [1 + u (1', lI, - h(x, P, E), E)] +

+sv(E)hV(x, N, 0V (x, /,

TU

_ S,fj,

E

_

v(E)-(x'v,

ll

v(s)h(x,p,E),EJ

The dominant term in the first member would be Ehx(x4, E) if 9 would not disturb. But for a small p we have hr (x, y) E) =1t= (T, vVE) )

=

hx (x, 0) + Y( h=y(x, 0) +

r)

...

From the physical conditions of the problem it results that le, (x, 0) 74 0.

365

THE EQUATIONS OF THE TRANSONIC FLOW

The condition (9.1.18) suggests that the right hand and member has the order of c. Hence, w (x,

v(E)

h(x, y, e) J = Eii (X,

h(x, b,

v{s)

E)>

E

(9.1.19)

We assume that this is valid in the entire domain occupied by the fluid, i.e.:

i (x,y,',E) = ew(x,y,;F,E).

(9.1.20)

Taking (9.1.19) into account, from (9.1.18) we retain in the first approximation, under the hypothesis that v(E) --+0

111 +u(x,y,0,e)] = ii (x,y,0,E)

(9.1.21)

Using the notations (2.1.3) the system which determines the perturbation produced by a fixed body in the uniform flow of a compressible fluid characterized by M is determined (see (2.1.10) - (2.1.13)) by the system (9.1.22) (1 + p)M2p = (1 +7M2p)p M2p + (1 + yM2)div v = 0

(9.1.23)

(1+p)v+gradp=0

(9.1.24)

where

[(1+u

a

8

)8x+vp,....

(9.1.25)

We notice now that from the structure

p x,

(9.1.26)

v(E)

it results the formulas Op Ox

_ Op ap _ Ox ' 8g

Op Op v(e) 8y ' 8~ I

1

Op

v(e) Oz '

(9.1.27)

which will be replaced in the projections of the equation (9.1.24) on the axes of coordinates. In this way, the projection on Oz gives r

here

Comparing

thu8x

8w Op +evVOw +e2vw +v !=0.

(9.1.28)

the dominant terms we deduce that P(x, y, =, E) =

v(E)

'lx,

, z, £)

1

(9.1.29)

THE STEADY TRANSONIC FLOW

366

and from (9.1.28) one retains

(1+p)(1+u)8 +-=0.

(9.1.30)

Analogously, from the projection of the equation (9.1.24) on the Oy axis, it results

+v(E)v-+EV(E)w-J +E

v8/

=0,

(9.1.31)

From this equation it follows v(x, y, z, E) = 6(x, T, z, E)

and then (1 +;5)(1 + u)

(9.1.32)

e + 5i = 0.

(9.1.33)

At last, the projection of the equation (9.1.24) on the Ox axis gives (1 + P)

}- EYi

1(1 + u)

GIV

zi + EvUI

v(E)

= 0,

whence we obtain 11(X,

Y(E)

u(x,

(9.1.34)

and then

89 = 0 . (9.1.35) + P) Fx + 8x The behaviour (9.1.34) determines for (9.1.33) and (9.1.30) the forms: (1

8; + 8y =0,

(1+P)LW

(1+P)WV +-=0,

8x

(9.1.36)

and the boundary condition (9.1.21) determines the equality

K(x,y,E) = 10(x,9,0,E) which implies hx(x, y) = w(x, y, 0) .

(9.1.37)

Knowing that M2 = 1 constitutes a singularity, we shall consider in (9.1.22) and (9.1.23) M2 = I + µ and we shall keep the dominant

367

THE EQUATIONS OF THE TRANSONIC FLOW

terms for a small p . Utilizing the previous results, the equation (9.1.22) becomes

l+YU1a+£vu+evrI [1+-1(1+µ)EPI 11

p

+EVWF

K

whence we deduce

4

7

v e Pox,

,

(9 . 1 . 38)

and then

89 = ap . (9.1.39) 8x 8x Having in view the damping condition at infinity for the perturbation, from the last equation we deduce

p =P-

(9.1.40)

Taking the relation (9.1.38) into account, it results that the dominant parts in the equations (9.1.35) and (9.1.36) are

8

+8 =0, 8 + =0, 8 F=0,

(9.1.41)

whence it results

u=-p,

ex

-=0, -=0.

(9.1.42)

Z

Finally, from the equation (9.1.23) written as follows

(1+µ){I1+ _u) 2ff +EVv V

ax

+£VW

+

+[l+ry(1+µ)ipj[+2+v] =0 ax ay 6F we obtain, if we have in view

(µ+vu+µv +'Y(1+µ)vpe

(9.1.41)1,

+(1+µ)EV(v +10 Lv) + ax 8Y ft 4rV

&0

0.

368

THE STEADY TRANSONIC FLOW

The dominant part is obtained from the linear terms. We may write therefore P

\µ+;u}

49V

+7vpax+uj( O-V + E

E

v

v

az-

J

0,

(9.1.43)

and the residual equation

(-K+u) 2E +YpBx+ a +a =0. Bxp

(9.1.44)

At last, from (9.1.43), (9.1.44) and (9.1.42) one obtains

K=

v(£) _ £1/3

try

r?x

1 - M2 £2/3

tr =0.

(9.1.45)

(9.1.4G

)

This equation, together with the equations (9.1.42) constitutes the gen-

eral system of equations of the steady transonic flow. In the x, V, z space the equations (9.1.42) give the irrotational conditionof the velocity of coordinates (u, u, is). Introducing the potential jp(x, ji, -_,F) by means of the formulas u = c0*,

V = cpy,

w = Pz'

(9.1.47)

one obtains (9.1.13) from (9.1.46).

9.1.4 The Shock Equations In the case of the flow with shock waves, from the integral form of the equations of motion (9.1.42) and (9.1.46), written in the conservative form (by means of the div operator),

vi+(-u)y=0, vas+(-u): =0 r f Kii I

(9.1.48)

- L+-'iP 2

+i%+urf=0, 1 :

369

THE PLANE FLOW

integrating on every domain which contains the shock surface and passing to the limit as usually, it results [i1n+: [IKu

-

- Qulny = 0, 9wf ns - QiiOnr = 0, 7+ 2

(9.1.49)

421% + Ovlny + OwOny = 0,

where n=, ny, n1 are the coordinates of the normal to the shock surface, i.e.

n;r = (d7jdz)

ny = (d-zdx)

ns = (dxd-y), .

(9.1.50)

If, for example, the parametric equations of the shock surface are x = x(A1, A2),

11=

A2),

z = {ai, 2) ,

then from n = da1z x da3X, it results n=

9.2

"2

8a1 812

"Z

8a1

dJ11da2i

... .

(9.1.51)

The Plane Flow

9.2.1

The Fundamental Solution

\Ve consider. like in Chapter 3, that an uniform stream, having the Mach number M is perturbed by the presence of an infinite cylindrical body, with the generatrices perpendicular on the direction of the stream

which coincides with the Ox axis. The Oy axis is in the section perpendicular to the generatrix. Let

y=h*(x),

1xI <1

(9.2.1)

be the equations of the profile determined by the cross section. Our aim is to determine the perturbation and the action of the fluid against the profile.

The flow is obviously plane. The velocity will lie in the xOy plane and will not depend on the variable z. Taking into account the orientation of the axes, we deduce from (9.1.37) the boundary condition v(x, ±0) = h' (x),

jxI < 1.

(9.2.2)

370

THE STEADY TRANSONIC FLOW

According to (9.1.42) and (9.1.46), the perturbation will be determined by the equations

il = -n i y - vs = 0,

(9.2.3)

Kit= + ij = ( + 1) uit= .

(9.2.4)

Using the change of variables

y=fl?i, u'=VY u,

(9.2.5)

we reduce the system (9.2.4) to

NO 8 ay-ax=0' ax

a

86

NO

2 gay=kaxu,

9.2.6

)

where we denote ry + I = 2kK3/2. In the sequel we shall integrate the system (9.2.6) without writing the marks *;' any longer. The fundamental solution of this system is determined by the equations au ay

av

8u

a

8u

- ax = ld(x, Y), ax + ay = k ax u2 + m (x, y) .

(9.2.7)

For the Fourier transforms u and v one obtains the formulas

-ial£+ia2n+kala2F

tl =

a2

a2

,

(9.2.8)

where F = F[u2 ], a2 = a + a2. We take into account the formulas F-'

a2,

-

i _ 2a In ro - e, ro = !LA

x2 + y2 (9.2.9)

F-1

[F]

4",

u2 * e

11

t d ri,

tt

where u2 * c is the convolution product and r =

+ yo,

xo = x - C YO=Y-71-

(9.2.10)

From (9.2.8) it results

1 mx + ly

u(x, y) = 21r x2 + 2 +

k

8J

ax

xo rJ)

r2

d (9.2.11)

1 My-ex

v(x, y) = 2ir x2 + y2

u2 s

+ 2 ax

fJasr

*1)

d d il .

371

THE PLANE FLOW

Obviously, the last integrals are singular. They have the shape (E.3). Isolating the singular point (x, y) with a circle having the radius a and setting f = xo/r and respectively yo/r we deduce that the condition (E.5) is satisfied and the integrals are singular. Also, from the formula (E.10) we getr

I = Ox / f z =

+7) r0 d d rl =

2U2

JJ.

J=a JIz

(n)df dn,

II2

(9.2.12)

or, performing the calculations,

=Jju2(,q)dxd,7_,ru2(x,y) 17y0r4x0J

I

(9.2.13) U2(t,11)x2

=-ff 9.2.2

2dOdx

dn.

The General Solution

Replacing the profile by a perturbing distribution defined on the segment (--1, +1) (the chord of the profile), it results the following general representation of the perturbation:

u(x, y) =

1

1

27r

40Y + m(Oxo

P-1

O2

-}- y 2

k d t + 2n I , (9.2.14)

1

v(x, y) = 2a 1

+1 m(E)v - f(F)xn 1

0

-

k -

T2r

Taking into account that xo + y2 = (xo - i y)(xo + i y),

xo + yo = (xo - i yo)(xo t i yo) ,

for complex velocity

w(z) = u(x, y) - i v(x, Y),

372

TUE STEADY TRANSONIC FLOW

it results the following formula: W(y)

=

1

+i

2k L(-),

M(C) +i

2

(9.2.15)

where

ata JJ

=

u2( F''1)L(z)

Z-(

1,

and

_=x+ir1, In order to impose the conditions (9.2.2) we shall pass to the limit on the segment [-1, +11. Using Plemelj's formulas we obtain from (9.2.15)

u(x, f0) - i v(x, f0) = T 2 [rrt(x) + i e(x)] + (9.2.16)

1

+ 'IrJr+l m(O + i e(f) xo

k 0 If r u2(t, 71) d

d +27r OxJf T x-

do

whence

k 0 f/ 2 +2;r 8a R su

xa

4

,dl; r1`

(9.2.17) ' +1

v(x, ±0) = ± m(x)

- 2n 11 zn) d £-

49

-2 a_JJ

2d dn.

It results

e(x) = u(x, +0) - u(x, -0) = K (p(x, -0) - p(x, +0)1.

(9.2.18)

The function l(x) will determine therefore the lift. Imposing the conditions (9.2.2) we shall obtain

m(x) = h+(x) - h' (x) twt h`(x),

lxj < 1

(9.2.19)

373

THE PLANE FLOW

' +t

w 8x fir ul{t,*))

J-t

dd dn+

o

(9.2.20)

C(Z), IxI < 1,

+h+(x)

where we have utilized the notation h+(x) = h+(x) + h'-(x). The relation (9.2.19) determines the unknown m(z). Considering that G(x) is known, the equation (9.2.20) determines f with the aid of the formula (C.1.9)

t(x) -_ 1

Vi- a

l+z

7r

1-z

1

/;:E t h+(t) d t-

l+t 8

+1

l+xJ_

n

1

-t t - x k

d

1-t 8t

)2+tp}

dt

t-x' (9.2.21)

Noticing that according to the formulas (B.5.3) and (B.5.4) we have

f+l 1-t dl: t-.z(1+ t-z

dt

1

(9.2.22)

-J

1-4 df ) a 1+4f-t t-z

+1

1 VEZ+ 1

and, after replacing (9.2.21) in (9.2.15), it results W(Z)

+t

1

C (9.2.23)

z- 1 1 tai z+

+l 1

1+ t h+(t)

1-t

dtk

k

t-z+2L(z)-wM(z),

where we denoted

M(z)

-M

z

+1J 1 1

1+trd fr 1

n dCdr1 ( - t)2 +V2

z

t d t Jf u ('*1) (9.2.24)

dt

--:

the formula coinciding with (4.11) from [9.21].

TIIE STEADY TRANSONIC FLOW

374

In the sequel we shall deal with M(z). We may write

M(')

7ri

z + I f12 u2(f, q)

ltd

f+1 {

,

q

1-t8t (t-t)2+q2,

dt

t-z1dq

1

7n

x

1

z+1

+1

ILl (F, q)' 2

The last integral is calculated by Homentcovschi by means of the residue

theorem. It may be calculated elementary noticing that we have q

1

1

1

t-z - 2i((-z) t - (-

(t-e)2+

rt

1

1

1

2i((-z) t-(+((-z)2+q2 t-z and taking (B.5.1) into account. It results

_ M(z)

1

z+lIir z

1

u2((,ri)8

1

1

I(1_ +

We write the first integral as follows

If u'(t,n) I

1

dt drt_

1

z

C1z s±i u2(E,17)-u2(t,-17) C-z JJR2

r

1

_

and the second

8z 11.2 u2(t,q)( (1-z

(1 z}d(dq

1

}dtdn,

THE PLANE FLOW

375

=

Jf/f

u2(

((- z)2

JJR2

We have therefore +00

M(z) = 2 If [u2(.7,) - u2(C, -t1)] K(z,C)dC d tj-00

(9.2.25) 1

+00u2(t, n)

-2rJ -00

u2(C, -n)

((-z)2

d d tj,

where

K(z'

}

_

z

1

z+1

(+1

1

_

C - 1 L(S - z)2

1

(9.2.26)

(C - z)(C2

Taking also (9.2.13) and (9.2.14) into account, we have that y02 -x02 +2ix°y°

1

(C - z)2 ,

H

L(z) = I - i J = -7ru2(x, y) - IL

(9.2.27)

' 'ddrj.

(9.2.28)

(C - Z)2

We replace the expressions (9.2.25) and (9.2.27) by (9.2.23) and we obtain the complex velocity. Separating the real part of the complex velocity, we obtain the integral equation of the problem. We have:

'j,

+1

u(x, y) =

--fl 1

h

1

l

4tr

l+th+(t) t

x2 + y2

d

z+l 1- z+l z

1t-z

VE

1

i t-; jdt+ (9.2.29)

+00

+2I(x,Y)-$ JJ

q)

[K(z,c)

-00

+K(,)]d drj+ !T[u2(1

yor4x°dC

dn

This equation, was given in a slightly different form by Homentcovschi in [9.211.

376

THE STEADY TRANSONIC FLOW

9.2.3

The Lift Coefficient

From (9.2.18) we deduce the following formula for the lift coefficient +1

1

c

+I V=x

1 fKl-te(x)dy

t+( )dt

' +t 1+ t

r

l+z .!_t

dx- ;2/ k J

+t v=1

x

l+x

i-t (9.2.30)

We have

J

R3u2(frn)5

[(t-t)2+

n21dfdtl

u2(f,8tn)t-( '( 1-t--( 1)dtdn

2i1 JJ .2 1

-2i J2

(9.2.31)

u2(f.n)-u2(f,-n)df do

(t-()2

From (B.5.1), derivating, it results:

l+t

t

1

dt

1

1-t (t-f)2

it

1

(9.2.32)

f-1 (2_1

Using these results we obtain cp

7K 1

2i

9.2.4

j k

I-t

hf (t)d t(9.2.33)

rJ u2(f+n) - u2(f, -n) d f do S-1

JJets

f-1

.

The Symmetric Wing

If the wing is symmetric, then the equations (9.2.1) have the form

y = ±h(x)

(9.2.34)

377

THE PLANE FLOW

From the boundary conditions (9.2.2) we have the relation

v(x, +0) = -v(x, -0) which suggests that the solution of the system (9.2.6) has the property

v(x, y) = -v(x, -y) . (9.2.35) We easily check that if u(x, y), v(x, y) is a solution of the system (9.2.6), then u(x, -y), -v(x, -y) also have this property. By virtue of u(x, y) = u(x, -y),

the uniqueness theorem it results (9.2.35) whence u2(x, rj) = u2(t, The integral equation (9.2.29) receives the form +1

ff(t)

-To

d4+ (9.2.36)

+2

Z

if

dn-u2(x,y),

JJ00

given for the first time by Oswatitsch (9.491. Obviously, the lift coefficient vanishes. The result is natural because the angle of attack of the wing is zero.

9.2.5 The Solution in Real We shall present a solution which is different from Hamentcovschi's solution which utilizes the complex velocity. Using the formulas (3.1.19) and (3.1.20), one obtains from the general solution (9.2.14): u(x, ±0) =

±-t(x) + Zx I

(ssd s +

J

v(x,f0) = f2m(x) - 2R ,r+t

k I (X, ±0), T7r

ds+ 2 J(x, ±0),

(9.2.37)

-1<s<1, where

I(x, t0)

.2

J(x, ±0) = 2JJu

2- Z

0° + n2)2 d t d it

- iru2(x, f0)

,

dri, -00 < ,n < 00. (9.2.38)

378

THE STEADY TRANSONIC FLOW

We obtain the formula (9.2.18), and from the boundary conditions (9.2.2)

m(x) = h- (x) 1

e(s)

JJ

s-x

7r

(9.2.39)

d s = h+(x) - kJ(z, 10), w

1xI < 1.

(9.2.40)

By means of the formula (C.1.9) we deduce

r+

£(z) - -V1:11 x

l+x

n

+ t h+(t)d

1-tt-x

t+ (9.2.41)

1+ t J(t, f0)

+as 1+z

1-i t-x

1

dt

.

For obtaining the integral equation of the problem, we shall replace m and t in (9.2.14)1. We shall denote by s the superposition variable from the first integrals (9.2.14) and xt = x - s. Changing the order of integration, we deduce

1+t 1&+(t)N(3, t)d t-

+1

u(x, y) = 27ra

J-1i

y

-27r y T2

,

f

1

± t J(t, ±0)N(x, y, t)d t+

(9.2.42)

k +2R I+1x2+ 2ds+2-I(x,y), x1 r -- s, 1

I

1

where we denoted

N(x, y, t) =

f+i V1 -s 1+s J 1

Since 1

x

I

ds

y2 s-t

(9.2.43)

1 fl is 1J + s-t z-z t- s-z t-z s-x 1

1

1

1

1

1

(x-t)2+y2 s-t' taking the formulas (B.5.3) and (B.5.4) into account, we deduce

N

n

x-z

(

1

t-I

'z-1+1

t

1 `+111. z -x

(9.2.44)

379

THE PLANE FLOW

Let us calculate nw the term

T =1

i + J(t, f0)N(x, y, t)d t . t

i

(9.2.45)

To this aim we shall specify J(t, f0). Taking into account (9.2.13) and the identity 2h7

1

t-C - t

(t-w +

1

we deduce

J(x, ±0) =

- JJ

rlLe2(F,

18 iT 3i 1

1+I

2i

1C-t C)dCdq=

f

_

and then I

ldfdq=

v))!

IL

ifm

(9.2.46)

u2(Cn)-UU(E,-11)ded+l=

t-(

u2(E,11)

- U'(C, -WdC dq

( t-C)2

dil} (t- u2(e,-17)dt - C)2iy

l +t VI u2(F,17)

1-t

1

,

x

.

(9.2.47)

s

1

l

1

1

t -z z+l t--l+1

cit.

Changing the order of integration and denoting

IPTE

To(C, Z) = El N 1

1

t, t (t i()2 t zd

(9.2.48)

we deduce

T

If 4Y

JJ*3

[ j2((, +1)

- U2(t, -n)1 [To((z)J4_ (9.2.49)

-To(CIT) FT 711

3+1 jddti

380

THE STEADY TRANSONIC FLOW

for specifying the function To we notice that I

t

1

(t-C)2 j---Z 1

+[Z

1

_

1

(t

((Z

1

- ()2 1

1

1

z-( (z-S)2 t-C+(z-()2

Taking (B.5.1) into account , one obtains

1+1 -

A To(C,z) _ (z - ()2

,r

-

1

-

z+1

n (z - C)2

z

1

- 1

-

1+S

d

z-C dC C-1' (9.2.50)

The term T which intervenes in the integral equation (9.2.42) has therefore the form (9.2.49) where To It is given by the formula (9.2.50). Using the expression (9.2.41) for the lift coefficient cL =

JJt(x)dx

(9.2.51)

one obtains the formula 1

CL

k +i

ai

-VrJ1

FI+- ft+(t)dt+7Vn f

i+tJ(t,±0)dt.

1

(9.2.52)

The function J(t, f0) is given in (9.2.46). For calculating the last term we change the order of integration and we take the formula (9.2.32) into account. One obtains the formula (9.2.33).

9.2.6

The Symmetric Wing

We gave in (9.2.36) the equation of Oswatitsch for the symmetric wing. We obtained this equation from the general theory presented in (9.2.2). We present here the direct deduction based on the equation of the potential cpn + lpyy = lops

which may be obtained from (9.1.13) or (9.2.6).

(9.2.53)

381

THE PLANE FLOW

The fundamental solution is defined by the equation

Eu+EI,o, =k

49 E.3

+m6(x,y).

(9.2.54)

Applying the Fourier transform it results

(a2+a4)E=iaikF-m, F- 9e ], whence

ia1F

al

m

aFI+ a

and then

-F rl -k 2 ax

F

f f

ll

- mF-1 [-

]

.

(9.2.55)

1

al +a2 -71+072 Applying the convolution theorem (A.6.13) and taking (A.7.11) into account, we obtain 8

F-l

r. (_ '

[-'r

a14J

s

47r

(+

+ao

--4, f f E=(t,q)ln(xo+14)dCdn, -00

such that +*0 E

= + 7r JJ

-6. 2(4, n)

ax

f d f d n + 4n ln(x2 + y2).

(9.2.56)

-00

Replacing the wing with a continuous distribution of perturbation sources defined on the chord of the profile (only in the symmetric case

the sources distribution is sufficient), from (9.2.56) it results, in the domain occupied by the fluid, the following general representation of the solution of the equation (9.2.53):

'P(z,y) = I

fl

m(f)ht{so+y2}dC +

j;

ffR?u2(t,n)

X0

PO +

y dt dn, (9.2.57)

where in(k) is a function which has to be determined by the shape of the profile, i.e. by the boundary condition

v(x, f0) = ±h'(x), x E [0, 1],

(9.2.58)

382

THE STEADY TRANSONIC FLOW

which is imposed by (9.2.34). From (9.2.57) we deduce

1+1 0 + y2 (9.2.59)

f+1

v=SPY= 2rr

xo

y2

where

I = ff

po d 4' d ii

u2 (4, n) x2

(9.2.60)

.

The singular integral I has the shape (E.9). Applying the formula (E.10) we deduce

1= fju2(f,rl)8x (x o+ f2fR2 x2

-

u2(4, 11)

Jdt;dn-aru2(x,y) o

2

/

(xd {

d *1- 7ru2(x, y)

+ y02)2

(9.2.61)

,

I=-2 ffzu2(t,,1)(xo Hence we have the following representation of the solution

u(x, y) =

1

+1

/-1

xo

z0 + y2 d

2 f fR2 u2 v, r+t

v(x, y) = T

!-1

- +y 2

rl)

(x o

z

d Cd +1-

k

d t- fJ uz

m(C) xo

2uz(x, y)

,

x2y2 n) (xo + y02)2 d

dn

(9.2.62)

which is also given in (9.2.14). The boundary values of the first integrals are given in (3.1.19) and (3.1.20). We obtain therefore

v(xi f 0) = f2rn(x) +

11 ir

u2(f,rl)

2

(xz + 2)2dt d>l. o

(9.2.63)

383

THE THREE-DIMENSIONAL FLOW

Imposing the boundary conditions (9.2.58) we find +oo

II

u2V,77)

22'°'1d d drl> (ro+r1)

(9.2.64)

u2(x, y) = u2(x, -y)

(9.2.65)

whence we deduce 17) = u2(s, -+l)

Taking into account (9.2.63) and the previous relation, it results

m(x) = 2h'(x).

(9.2.66)

In this way we determine the distribution m. Coming back to (9.2.62), we obtain the equation: u(x, y) + k u2(x, y) +

2 11.2

2_ 2 (xUO+ y02)2

+i

d

d j?

(9.2.67)

rr -1 which coincides with (9.2.36).

9.3 9.3.1

The Three-Dimensional Flow The Fundamental Solution

In the last 40 years, a great number of papers was devoted to the steady transonic flow past thin bodies. Usually one assumes that the flow is irrotational, the potential satisfying a non-linear equation having the form (9.1.13). For deducing the integral equations of the problem, we apply Green's formula to the equation of Poisson and we assume that a vortices layer is present downstream the wing. Derivating, we obtain the non-linear integral system for the components of the velocity (see for example [9.36]). In the case of the symmetric profiles the system reduces to a single

equation for the component u(x, y, z). In this sense, after the initial paper of Oswatitsch [9.50] where one defines a principal value for the singular integral which intervenes in the representation, it followed the paper of Heaslet and Spreiter [9.17] where one gives a general representation which in the symmetric case reduces to an equation. The

384

THE STEADY TRANSONIC FLOW

representation is valid both for the flow with shock waves and the flow without shock waves.

For the lifting wings the forms of Norstrud [9.41] and Nixon [9.39] are available. In this case, the problem reduces to a system of two nonlinear integral equations. At last, we mention the paper of Ogana [9.471 where one shows how the integral equations depend on the definition given to the principal value of the singular integrals. A new point of view, belonging to D. Homentcovschi [9.20], [9.21] and L. Dragog [9.8] ]9.9] does not assume that the flow is potential. Utilizing the system of equations of motion it is necessary to assume the existence of the vortices layer downstream. In the sequel we shall utilize the method of fundamental solutions [9.8]. The system which determines the perturbation is (9.1.42) and (9.1.46). Performing the change of variables

u`=

,

(9.3.1)

and omitting the marks * and A, the system becomes

uy-v==0 U,-w2=O (9.3.2)

us + vy + w. = k(u2)s,

where k has the same significance like in (9.2.6). We shall see further that employing a fundamental solution similar to the fundamental solution of the system

uy-uz=eo(x,y,z), us-ws=0 (9.3.3)

u. +vy+w2 = k(u2)=+mb(x,y,z) we may satisfy all the conditions of the problem. This solution will be determined in the manner described in 2.3. Applying the Fourier transform, solving the algebraic system just obtained and considering the inverse Fourier transform, on the basis of the formulas from appendix

385

THE THREE-DIMENSIONAL FLOW

A, we obtain: u(x,y,z)

)r

41r (m&x + t

v(x,y,z) - 4w 8z k

4w 8y

r 82

_ M 49

-4w 8

r,

8zsuz

r

8

as

8z

alai

1

4w 8xOy w(z, y, z)

82

4w

8 I_ m 8 1+

t

_

__

us

r, k

2

02 2

1

(r} ^18z [ate, - 4 8x8zu * r, 1

(9.3.4)

where r =

+ y + z and,

-.

I 8, C) dv , (9.3.5) u r Iffits I z - 41 with the notation dv = d£ d, d(. This integral is called the aonvolntion of the functions u2 and 1/r. TaIdng M = 0, from the formulas (2.3.11), (2.2.6) and (2.3.27) it results us

al l = -.f`1 8 1 J i alas

[ala2J

= (9.3.6)

4Ir 8y

r

j.

4w y2 + z2 11

+r

and a similar formula. In fact one obtains the following form of the fundamental solution:

u(2,y,z)_-4w (mf +e ) r 1

v(x, y, z) = 4w

K

jr02 w(z,y,z) =

8 r 8z

[y2+z2 (i. +

r

Io,

r-

-m

1

4

t

8 + 4w 8a

r), r 4w 8xjo' y

+z

(1

x) +w

k 8K° 47r 8x, (9.3.7)

386

THE STEADY TRANSONIC FLOW

where we denoted

lo= au2*1, r Jo= ay TX

au2#1, Ko= aZ r

r

(9.3.8)

P. and m being constants. 9.3.2

The Study of the Singular Integrals

The integral (9.3.5) has a weak (integrable) singularity. The integral exists, (it is convergent) (u2 is zero far away) and it may be derived (the convolution, if it exists may be derived (A.3.7)), such that we have

I° = u2 * -

-u2 * Ix13 = J u2(4) 14 - I3dv

-r=

and similar expressions for J0 and Iio. Since the integral has the form (E.3), it is convergent. With the notation

E -- xl, x

f __ IC

has the form (E.9) and may be derived according to the formula (E.10). For calculating the last term one utilizes the spherical coordinates with the center in the point having the vector of position x: 1°

-x=sin9c s rl - y=sinOsinV

( -z=cos9. One obtains

Jfces(n,x)dn =

Ir

whence it follows the formula

8xlo = J ul()FZ (::,1) dv - 43 u2(x) _ (9.3.9) U2(t)2=02

0

"°dv - 4xu2(x) 3

.

THE THREE-DIMENSIONAL FLOW

387

Analogously one demonstrates that

J a to = 1, 49X

u2(,) Ix?o lsdv, (9.3.10)

K = ±Ko =JR3 u2(F)Ixxo{Isdv, where xo = x - t, yo = y- rt, zo =z-(. The integrals we have obtained are convergent if u2 satisfies Holder's condition and if its behaviour at infinity is u2(C) = O(ItI-`) with I > 1.

9.3.3

The General Solution

Denoting by D the projection of the wing on the xOz plane and by

y = h(x, z) ± hl(x, z),

(x, z) E D

(9.3.11)

the equations of the wing (which is assumed to be thin), we shall be able to satisfy the conditions of the problem with a continuous superposition of solutions having the form (9.3.7), defined on D. It results the following general representation:

(1R)

u(x,y,z) =-4a JJ [M(C 0ax

dt dC

(9.3.12)

-4 J(x,y,z), v(x,y,z) = 47r

11D

[e(C) ax

-

(R)

d d(-

{2:

47rf

(9.3.13)

k

ir- Ax' Y, z), u !(x, Y, Z)

(R) +

T" AD

f

(9.3.14) 1

D

K

I y +:.p Y

dt d(,

THE STEADY TRANSONIC FLOW

zo=z-(, R- xo+y2+zo.

(9.3.15)

Taking the formulas (5.1.16), (5.1.18) and (5.1.24) into account, it results

u(x,10,z) =±t(x,z)+4Ao,, m(E,() D

v(x,±0, z) = f1m(z,

z) + 4-r

no dEdC- 4x1(x't0,Z)

(9.3.16)

1.

(z((,C)2

(i + x0)d{ d(-

- 4 J(x, ±0, z) (9.3.17)

where

Ro= zo+zo

(9.3.18)

and the mark * indicates the Finite Part like in (5.1.24). From (9.1.42) and (9.3.16) we deduce the significance of the function

t(x, z) : t(x, z) = p(x, -0, z) - p(x, +0, Z).

(9.3.19)

Hence, t(x, z) gives the jump of the pressure. This function will be utilized for calculating the aerodynamic action. From the expression of v(x, ±0, z) and from the boundary condition

v(x, f0, z) = h'(x,z) ± h' (x, z) (x, z) E D,

(9.3.20)

where the mark "prime" indicates the derivative with respect to the x variable, it results after subtracting and adding m(x, z) = 2h' (x, z),

4ir,1Dt(

(9.3.21)

(1+)dd(+

+ 2k s u2(4) (xo + q2

+

Z02

T,,-, d v = h'(x, z), (x, z) E D. (9.3.22)

389

THE THREE-DIMENSIONAL FLOW

The formula (9.3.21) determines directly the unknown m(x, z). In the equation for t(x, z) it intervenes the values of u2 in R3. They are obtained from (9.3.12) after replacing m by (9.3.21). We deduce

u(x)- 3u2(x) +

k

4I

U2(4)2xix MI5 -o d v(9.3.23)

-41T f t((,()R3d(dq

2,1

D

f Dh'(k,()R3dCd(

Hence, for determining the unknown f(x, z) on D we have to solve the system consisting of the equations (9.3.22) and (9.3.23) where u(x) is defined on R3. For u(x, f0, z) we shall utilize the values (9.3.16). The mathematical problem is extremely difficult and there are not known any attempts for solving it. For the symmetric wing (h = 0), the solution is obtained for t = 0

and u(x, y, z) = u(x, -y, z) if k

u(x2)- u2(x) + 4 /

2x x05 d v =

U 3

(9.3.24)

3

= 21r JDhi((,()Rd(d

9.3.4

Flows with Shock Waves

In the case of the flow with shock waves, the general solution has also the form (9.3.12)-(9.3.14). We can see it in the simplest way if we utilise

the notion of Fburier transform for bounded domains, introduced by D.Homentcovschi 19.191. Indeed, in the fluid domain D the equations

vx-uy=0, wz-uz=0, (9.3.25)

uz + vy + w: = k(u2). , with the notations from (9.3.2) have to be satisfied. On the shock waves E one imposes the relations OvOnz

- OuOny = 0,

llwOn: -

0, (9.3.26)

uOnz + [lvjny + Uu'Qn: = kjJu21nz,

THE STEADY TRANSONIC FLOW

390

deduced from (9.1.49), and on the borders S+ (upper surface) and S_ (lower surface), the conditions (9.3.20). Applying the Fourier transform for bounded domains, we shall utilize the formulas of the type (A.8.1). From (9.3.25) we deduce

-iaiv+ia2u=S1+T1i -iaiw+ia3u=S2+T2,

(9.3.27)

-ialu - ia2v - ia3w+kialu2 = S3 +T3i where, taking into account that on S+ we have n = (0, 1, 0), and on S_, n = (0, -1, 0) (vnz - uny)e' a'xd a =

S1 = s++s_

I Juleiaxda = -JDt(x,z)e'("'+"'s)da,

_-

D

S2 = J S3=

; +s_

s++s_

(wns - un=)e' a'Zd a = 0,

(9.3.28)

f(urn+vny+wn;-ku2nr)e'c.'da=

=f OvOe1a xda _

fom(x,z)e'(",+"':)da.

t(x, z) and m(x, z) having the signification from (9.3.19) and (9.3.21). The integrals

Ti =

J (OvOn. - Julnw) T3 =

JE

T2 = f£ (OwOn.=

(Duin. - Ovlny + Owonz

- Juln=)

et°r'ada

- klu2Onz)

vanish because of the relations (9.3.26). Hence, the system (9.3.27) reduces to

-ia1v+ia2u=S1

-ialw+ia3u=0 -i&- ia2i;-ia3w= S3-kialu2,

(9.3.29)

391

THE THREE-DIMENSIONAL FLOW

which has the solution

u= `ia2sas al Ss + i a2 S3

V=

op, +

10 S1+ka t

W=

where a2 = ari + d

u2,

3s

u,

(9.3.30)

a S1 + ka y-

i

1

+ &23.

Considering the inverse Fourier transform and utilizdng the formulas (A.6.9) we obtain 1

[83]

- k-t-r-I [a,] (9.3.31)

Cf2

f is a2]

+8 -k.Ozox jr8a

1

us

8°

-

u2

1

i-k_

8

air

By direct calculations, we deduce

.F-1

f

(2703 OY J22

=

t la {atE+wt)d dC L_ L f

e '(a1x+a*v+o*z)d a

da]dt ds fD[(211)3mnhJP.3 47s

f

a

_

O R ()dd(. (9.3.32)

where, with the notation

R= V;i.

,

(9.3.33)

THE STEADY TRANSONIC FLOW

392

we utilized the formula (A.7.10). From (2.3.11) and (9.3.27), it also results a 'r- j (

I_ 1 8 1-

1

iata2

t?z

1 47r

4ir Oz J

oo V2.2

ao

+ y2 + z-2

_

zd x

1

f

dx

(x3 + y2 + z2)3/2

1

z

4-x y= + z2 (1 +

x) r (9.3.34)

'On the basis of this formula we deduce

_

1

8 jI

(27r)3 Vz st3

I

e't(alx-+a,y+ass)da=

P

i0la2

OZ .3

(2n)3

1L

4;r

i

ala2

dct]d d( =

'y2+z2 (1+)ded(. (9.3.35)

At last, taking into account the definition of the convolution product, it results u2 i u2(t) d (9.3.36) a2

4:r a3 Ix - f 1

With these formulas and with the similar ones it is not difficult to see that in (9.3.31) we have just the solution (9.3.12)-(9.3.14).

9.4 9.4.1

The Lifting Line Theory The Velocity Field

The lifting line theory in the transonic flow is studied in (9.55) and [9.$). In the last reference, it is obtained, as it is natural to do, from the Lifting surface theory. This method is also utilized herein.

We shall deduce the equations of the lifting line theory from the lifting surface equations using the assumptions 10,V,3" (Prandtl's

393

THE LIFTING LINE THEORY

hypotheses) from 6.1. Hence we shall take hl = 0 and we shall consider that the unknown is the circulation

C(c) = +

(9.4.1)

e(4, ()d

C(±c)=0

(9 .4.2)

and we shall utilize the formula tim

=

f

()k(x, y, z, 4, ()d e d ( =

fJ

s-(()-O-s+(()

(9.4.3) }r.

C(C)k(x, y, z, 0, <)d

2c representing the span of the wing on the direction of the Oz axis. On the basis of the first hypothesis (hi = 0), from (9.3.21) it results m = 0 and the representation u(x, y, z) = T"

fJ

C) R3

d d(

T-

I (x,

y, z)

v(x,y,z) =- 1 ID f t 4sr

(1+

j)]ded(-

- J(x,y,z),

y (i+ az y2 + s02

R) ,did( -

I(x,y,z)

a ly +

AD f

0

ev,oa[

1 ffD

w(x, y, z) = 4z

(9.4.4)

or, using also the formulas (9.4.1) and (9.4.3):

u(x, y, z) =

1

4zr

+`

J

k

C(() R3 d _4 I (x, y, z) , 1

v(x, y, z) _ -

c

C(S)

d

I - 4?r

-C

C`(C) yz

zo

(1

+

+C

w(x,y,z)-:i!. f

C(S) 2+x2 y o

Ri) d(

4 J(x, y, z)

,

k

(1+R )dC- K(x,y,z), 1

(9.4.5)

THE STEADY TRANSONIC FLOW

394

where we denoted

x2 +y2+; 2.

R.1 =

(9.4.6)

The Integral Equations

9.4.2

Using the identity

I(1+xo\^8 xo+Ro Ro J

8z

(9.4.7)

rozo

where Ro is (9.3.18), we deduce for (9.3.22)

JJn

1+'0

r(io

RD / d

d

td (=

8z

AD e(c o

X0 0ze

td C =

FC(C)Zd

-d

Using the calculations (6.1.11) - (6.1.13) we deduce in the sequel

T

-j

C'() d C az ff4;)s)flzOd t d ( = e

+(1) t(co )d( 10

Je

' +` C(() d

Lc

zp

L(()

e

a(z-S)d(

(9.4.8)

2 , _+(_) t(F, z) d

J=-(z)

xo

Substituting this in (9.3.22), multiplying the obtained equation by

x - x_ (z)

Vx+(z) - x and, integrating with respect to x on the interval (x- (z), x+ (z)), one obtains the following equation:

C(z) =

a(2)

'

'((z d(+

is u2(C)S(=, F, n, ()d v+ H(z) , (9.4.9)

THE LIF"TINC LINE THEORY

395

where

x - x_(z)

j_+(`)

3xogdx

S(z, , , C) = flV x+(z) - x (xo +, 2 + 4)512 , (9.4.10)

H(z) = 2

s+(_

- x-(Zh (x, z)dx . x

77

The equation (9.3.23) for the unknown C(z) is

u(z) - 3 u2(x) +

4J [

y +`

-^ 4

2X.2 U

-

x

2

y

2

zo is

dv (9.4.11)

C(C)d C

, (x2 + y2 + 40)3/2

We have therefore the equations (9.4.9) and (9.4.11) for the unknowns C(z) and u(x, y, z), C being defined on [-c, +c], and it, on R3. The continuation of this reasoning may be found in (9.8). Obviously, we have to consider only numerical solutions.

Chapter 10 The Unsteady Flow

10.1 10.1.1

The Oscillatory Profile in a Subsonic Stream The Statement of the Problem

As we have already mentioned (see Chapter 2), the general problem of aerodynamics, i.e. the problem concerning the determination of the perturbation produced by an arbitrary moving body, in a fluid whose state is known, is very difficult. A presentation of this subject can be found in the papers of Kiissner [10.37], [10.38), [10.39]. We consider in this chapter, the particular case of the wing which is oscillating harmonically in an uniform stream. The problem is important in the flutter theory. For the incompressible fluid the plane problem was investigated by Theodorsen (10.75). For the compressible, subsonic fluid in a plane flow, the problem was studied at first by Possio [10.59) and then by Dietze [10.14) and Haskind 110.30). These authors used the potential of accelerations i1', replacing the body by a doublets distribution defined on the chord of the profile. The integral equation which was obtained made the object of many researches [10.22], [10.23]. In [1.1], [1.3], [1.8], [1.18] one may find syntheses and references. Considering that the replacement of the wing by a doublets distribution has no physical justification, in [10.15] we deduced the integral equation starting from the idea that the wing must be replaced according to Cauchy's principle by a distribution of forces. In the same paper, we gave a solution for the integral equation for small values of the frequency (w K 1). The present' section is written on the basis of this paper.

10.1.2

The Fundamental Solution

We utilize the dimensionless variables x, y, z, t introduced by (2.1.1), and for the velocity field V 1 and pressure P1 resulting from the super-

398

THE UNSTEADY FLOW

position of the perturbation over the basic flow, we set

V1 =UO +V), Pi = P.+p,0U;pP.

(10.1.1)

Like in the previous sections Umi is the velocity of the unperturbed stream, and p,,. and pa, the pressure and the density in that stream. Imposing the condition that these fields verify the equations of motion, it results (see the equations (2.1.26) and (2.1.27)) the system

oV /at + 9V/Ox + grad P = F,

A2(aP/at + OP/ax) + div V = 0,

(10.1.2)

lim (V, P) = 0,

ixl

where F is the force density assumed to be small (the system is the result of a linearization), and M is Mach's number for the unperturbed stream. If the uniform flow of the fluid is perturbed by harmonic forces having the form f Cos (iwt), f Bill (iwt),

applied in the origin of the axes of coordinates, then we shall put in (10.1.2)

F=

(10.1.3)

f5(x)ek'Ji

.

Solving the system, we shall obtain that the real part of the solution is determined by the force f eos(iwt) , and the imaginary part by f Bin(iwt)

.

The perturbation produced by (10.1.3) will obviously have the shape

V = v(x)ei"' P = P(x)e"

(10.1.4)

Replacing in 10.1.2 one obtains the system iwv + av/ax + grad p = f 5(x) M2(iwp + ap/ax) + div v = 0,

,

(10.1.5)

lim (v, p) = 0.

Ixi-. Q

The solutions of this system are given in 2.4 (they are the solutions of the system (2.4.3)).

399

THE OSCILLATORY PROFILE IN A SUBSONIC STREAM

In the case of the subsonic, two-dimensional flow, assuming that f has the form (0, f) , it results from (2.4.6) (10.1.6)

P(x,y) _ -fly Go(x,y), and from (2.4.16) and (2.4.21)

v(x, y) = f e-"' [2iwG - f32G= + w21 x

G(7-, y) d 71

.

(10.1.7)

00

We denoted

Go (x, y)

- 4I

G (x, y) =

4if3

)

H

°

( k/ x2 + My2) e»

H(2) (k

(10.1.8)

x2 + /32y 2) e1

where

k=wM, 0 =w/$2, a=kM

(10.1.9)

Hoe) being Hankel's function.

10.1.3 The Integral Equation Assuming that the perturbation of the fluid is determined by an oscillatory wing having the equation y = h(x)e'`'t , 1x1:51,

(10.1.10)

from (2.1.29) it results the following boundary condition

v(x, 0) = h(x) + iwh(x) = H(x), jx) < 1,

(10.1.11)

Let us replace the action of this wing against the fluid by the action of a continuous forces distribution having the form (0, f) defined on the interval (-1,+i] . From (10.1.6) and (10.1.7) it results the following general representation of the perturbation P(x, y) = -

J

+1

v(x,y) = f f(t)e

11

f (t)

Go (xo, y) d

r.,xo(2LG(xo,y)

,

(10.1.12)

- f32GX(xo,y)+ (10.1.13)

+w2J

G(r,y)dr]d,

THE UNSTEADY FLOW

400

where xo = x - f

.

The function f has to be determined from the

condition (10.1.11).

For obtaining the limit values p(x, 0) and v(x, 0) , we shall take into account that Hankel's functions H0(2)(u) and H12)(u) satisfy the relation (10.1.14) Hoe) (u) = (u) , du and have the following asymptotic behaviour for small values of the

argument [1.16], [1.40]:

H421(u) . I - 1n u , r = 1 -

(10.1.15)

?l 2(C- 1n 2) ,

Hi2l() _i u ,

(10.1.16)

where C(= In y) is Euler's constant (= 0.577215) . We deduce y

P(x, y) =

i::' f

+ p2y2)

(k

exp (iaxo)

o + 2y

d£

.

When we calculate p(x, 0) we observe that, because of the presence of the factor y, the product y f+1 vanishes excepting the vicinity (x - e, x + t) where the integrand becomes infinite (for t = x ). If a is (iaxo) may be apsmall enough, in this vicinity the function f proximated by its value in the middle of the interval (we assume that f (C) is continuous), i.e. with f (x). It results rte , f (x) P(x, ±0) = y 1 Hill(k xo + [32y2) vlli

4i

x+y

_e

-+ t :

and, performing the substitution into account,

dt

+C

AX, f0) _

f (x)Y--+O limy 2w

J

-x = t , and taking (10.1.16)

t2 + /32y2

1 f (x). 2

(10.1.17)

Hence we obtain the significance of the function f (x) . We have P(-T' + 0) - P(x, - 0) = f (x) .

(10.1.18)

The component v(x, y) is the sum of three terms vi , V2, V3, which are represented as follows +i v1 (x, y) =

J

f

j f We-

G(xo,

y)d

""r° Hoe) (kx2 +/32y2)ei0xo d

401

THE OSCILLATORY PROFILE IN A SUBSONIC STREAM r+1

1

VI (X'0) =

H(21

4iQ 1-1 f (

(kjxol)e0'0 dE,

(10.1.19)

the singularity from j 2) being integrable. We have also, v2 (s, Y)-

J

+1

f(e)e-I° 8 G(xo, II) d t =

k 4iQ

+i

f

1

(k

xq + Qom)

x

-07

0

°d t+

f+1

+ 4Q3

fl f

(k

and taking (10.1.16) into account 1

k V2 (x, 0)

Q J-1

f

(klxol) f!l e i2-0 d E

OHo (k1xo1)e

f+' f(t)-

Odd,

[JZ0G(r,y)dr] 0

00

1

V3 (r,0)

(10.1.20)

I

From

one obtains

,

,

(f)e-6-0 H(2)

+403 f-+1

v3(x,U)=

xo + Q2y2)e'D*° d

dt,

1 f(t)e- '-O [JXOJ;42)(kfrI)ev1d1] dt,

_!_ f

where one performs the change of variable r -+ u : for = u and one takes into account the formula J

J

Hoe) ( M [u1) eh'd u

=

2 n 1 Mf I

,

(10 . 1 . 21)

given in [1.2[. One obtains V3(X,0)

26;

If(4)e!

0 [! 1M6

+ Zr o

(M1u[)ej"duJd

J

(10.1.22)

After all, from (10.1.13) it results I

v(x,0) = T f(C)Ni(xo)dC+J

11

f(F2(xo)dt,

(10.1.23)

402

THE UNSTEADY FLOW

where we denoted

'

Nk(xo) = nk(xo) e'

nl ( xo ) =

4 pM

k''X0

,

k=1,2

(10.1.24)

wr0 Hi2)(k (xoI) } o_ e

n 2 ( Xo ) = A`-'A HO(2) (kl xol)e °''z0

fQC)

4

- 2W-

In

1+,3

(10.1.25)

M

H2)(MIuI)eudu.

Imposing the boundary condition 10.1.11 it results the following singular integral equation r

I

i

f

d

H(x) , IxI < 1

(10.1.26)

where

N(xo) = N,(xo) + N2(xo)

(10.1.27)

In fact the kernel N depends on the variable xo and on the parameters M and w. We deduce therefore that the kernel has the following explicit expression (a = kM)

N(xo,

_iw

M'w)

e_

-- --Hi2)(klxol) Ixol etQTp +

°r

QHQ2)(klxol)e' O-

e'i'' 0 In (10.1.28)

The equation (10.1.26) is Possio's equation [10.591. There are a lot of papers devoted to the kernel (10.1.28) in the literature 110.5], [10.22), [10.301.

10.1.4

Considerations on the Kernel

Taking (10.1.16) and (10.1.16) into account, we deduce

N(xo, M, 0) = urn N(xo, M, w) =

2

n

403

THE OSCILLATORY PROFILE IN A SUBSONIC STREAM

which is the kernel for the steady flow. With this kernel the integral equation (10.1.26) reduces to (3.1.15). The kernel for the incompressible fluid is obtained from (10.1.28) calculating (10.1.29) N(xo, 0, w) Mhm.(xo, M, w) .

Integrating by parts we transform the formula (10.1.28) into

et '° {Mi42)(kIxoI)

N(xo. Al, w) =

- iHi2)(k(xol) (X(, +

4A + t!ft e-6'0 lim Hoe) (Maul) 4

- W4M e-6'0-

Utilizing now the relations (10.1.15) and (10.1.16) the notations (1.16) 00

J;

Ci(z) =lnyz+

u-Idu =lnryz+-1)"(

JJ

Si(z) =

1sill u u

n)t

,

n-1 °O du = > (-1)r (

n=O

2n+1

+ 1)(2n + 1)! ' (10.1.30)

called, the first, integral cosine, and the second integral sine, we obtain N(xo' 0, W)

2

2 + Si(wxo), }

.

(10.1.31)

This is the kernel for the incompressible fluid. One demonstrates in (10.15] that for small values of the frequency (w a 1), the integral equation (10.1.26) has the form

a

xd w

+

mQJ_11f(t)(In (lx-t(+r))dt=2H(x),

(10.1.32)

where r is a constant. This kind of equations are solved in (A.16). We leave to the reader the task of writing explicitly the solution. In (10.22) one shows that the general kernel (10.1.28) has the form

N(xo,Af,w) = Ao(xo)+Ai(xo,M,w)In(lxol)+A2(xo,M,w), (10.1.33)

404

THE UNSTEADY FLOW

where

Ao =

_0 2,

A,=- 2 CBI(xc,M,w)e ,mxo (10.1.34)

A2

-iwxo

2. B2 (xo, M, w)e

,

A, and A2 being analytic functions with respect to x0.

10.2 10.2.1

The Oscillatory Surface in a Subsonic Stream The General Solution

The problem presented in this subsection was studied in many papers (10.86], (10.87), (10.45], (10.35], (10.83] where the integral equation was obtained by means of the potential of accelerations, replacing the wing by a distribution of doublets. A slightly different investigation was given in (10.12]. We studied this problem in (10.6] utilizing the fundamental solutions method which will be presented in the sequel. The problem is the following-, an uniform stream having the velocity

the pressure p and density po , is perturbed by a surface, oscillating according to one of the laws

z = ho(x, y) coo (wt), z = ho sin (wt), (x, y) E D.

(10.2.1)

One requires to determine the perturbation. One utilizes the dimensionless variables introduced in (2.1.1) and the notations (10.1.1). The problem is simplified if we replace the laws (10.2.1) by

z = ho(x, y)e"" ,

(x, y) E D .

(10.2.2)

In this case the real part of the solution will give the perturbation produced by (10.2.1) and the imaginary part the perturbation produced by (10.2.1b). The boundary condition (2.1.20) and the linearized system (10.1.2) lead to solutions having the form (10.1.4) where the functions v and p are determined by the system (10.1.5) and by the boundary conditions w(x, y, 0) =

8

ho(x, y) + iwho(x, y) = H(x, y), (x, Y) E D. (10.2.3)

The solution of the system (10.1.5) under the assumptions that f = _ (0, 0, f) and the unperturbed stream is subsonic (M < 1) is obtained from (2.4.7) and (2.4.17) as follows a (10.2.4) P( x,Y,z _ - f 8z 0o(x,Y,z),

405

TILE OSCILLATORY SURFACE IN A SUBSONIC STREAM

w(x, y, z) = f e

i"x

l

-2 . ) G + (w2

[(2i

) f G(T, Yz) d r 00

J

where

1 exp [io(x - M RI )l

(10.2.5)

R,

x +

RI(x,y,z) =

(y T .

As we already know, the formulas (10.2.4) define the perturbation produced in the uniform stream by the force (0, 0, f) exp(iwt) applied in the origin of the axes of coordinates. Replacing the wing with a continuous distribution of such forces, defined on the domain D, we obtain the following general representation of the perturbation

P(x,y,z) _ w(x, y, z) =

J J°

f(f,rr)

a8 G

(10.2.6)

G(xo, yo, z)+ f (t, -T) e-'"' [(2iw - 021) ox 0-2

(10.2.7)

+(w2-a-y2)

where, as usually, xo = x-t, yo = y-7) .The function f is the unknown.

10.2.2

The Integral Equation

In order to determine the unknown f , we shall impose the conditions (10.2.3). At first we shall prove that if f (x, y) is a continuous function, then

Zlim0JJ f(E,rr)a Go(xo,yo,z)dfdi' = r2f(x,y),

(10.2.8)

D

Indeed, we have

j f fl) iwM + Ro j )q aik(Mx°-RO) d e d >) rJ° a

P(x, y, f0) =

4Z

J

z = 0, the integrand will be zero excepting the point Q(x, y) E D. Denoting by DE the disk where Ro = Rt (xo, yo, z) . We notice that if we

/set

having the center Q and the radius E and assuming that t is small

406

THE UNSTEADY FWW

enough in order to approximate f t)) with f (x, y) (this is possible if f is continuous) and the exponential with the unity, it results P(x, y, ±0)

4lymo

Jf 4 (ic?vf +

)ddr

.

Performing the change of variables t, q -* r, 0:

-x = Qrcos0, r/- y = rsin0,

0
P(x, y, ±0) = 2p f (x, y) zluuo z

i

(iwM +

r .8 z } r2+

r

z2 =

/!

=fif(x,y), (10.2.9) and then

P(x, y, +0) - p(x, Y. -0) = f (x, y) , (10.2.10) The formula (10.2.9) proves (10.2.8) and (10.2.10) gives the significance of the function f (x, y) . In (10.2.7) we may interchange the limit and the derivations (with

respect to x and y). It results therefore

w(x,y,0) =

nj

' fj f

26 -,62

x

(,)enw [n+n2]ddtl,

(10.2.11)

eR ,

(10.2.12) oo e(1iEl=d,,

F00

E

=j

R,

d

dT

,02X = (xo R,. =

=J

T2 +

MR), R = xo + /yo ya

(10.2.13)

407

THE OSCILLATORY SURFACE IN A SUBSONIC STREAM

In E1 we perform the substitution r

A:

r + AIRT = /32A ,

(10.2.14)

Using the notation Iyot = r, we deduce

dr

dA

7,\T+:+

vrr2 -+#2r2

(10.2.15)

7,

and then

r

El =

00

a-WA

Mr/ A +r

a wA +r

00

da = fo

Mr

lp

dA

A

a-iWA

A +r

o

r

M

lo

= Ko(wr) - 2 [Io(wr) - Lo(wr)] - f

dA

dµ,

el + µ2

(10.2.16)

Ko and to being Bessel functions, and Lo the Struve function [1.16]. The expression of El may be derived taking into account that 02/0y2 = c72/c')r2 . Using the relations between Bessel's functions and their derivar tives, we deduce

-w2

rtir/$

1 + µze''wrs`dµ .

0

In E2 we make the notation iIyoj = u and then the change of variables

r--# B . r=usinh9.

(10.2.17)

Observing that 82/8y2 = (3282/&u2 , we deduce

8

e&oRX

=7 W2 1 /32 u2

+ j92 T3 e x + iwM R2 e-x - yo R2 e wx K2-

/'s0 (r - MR,)2efr-Mx.) dr J0 Rr

where

/ 2X = ao - MX, R =

xo + u2 ,

Rr =

r2 -+U2.

408

THE UNSTEADY FLOW

Performing inside the integral the change of variable T --- A::

r - MR. = Q2A ,

(10.2.18)

we notice that the formula (10.2.15) remains valid because it depends only on A,12. One obtains the identity 1

I

F2+ (j4

s0 {r -

dT

R

yo Jo

= (10.2.19)

1

z/Q'

A2+yoe'"'xdA, Y6

-Mlyoi/$

Taking these results into account we obtain

xoe ,x

n(xo,yo)=ni+n2=-Q2 R +iw R-xoM R K1(wr)

2

w,x #2y02e

-

III (wr) - L1(wr)) +W2

-w2 jJ11/1)

1 + µ2ek`'r0dp

f ar s

A, A2 + r2e d (10.2.20)

where, for the sake of simplicity we maintain the notation r = 1yol, and X and R are given in (10.2.13). Obviously the line yo = 0 is singular.

Employing this form of the expression nl + n2 in (10.2.11) and imposing the boundary condition (10.2.3) we get the following integral equation

4a jf f

n)N(xo, ?ro)d d r) = H(x, y) , (x, y) E D,

(10.2.21)

where

N(xo, yo) = e-"n(xo, yo)

(10.2.22)

is the kernel. Since w 4z 1 we have the asymptotic expansions

Ki(wr) = 1-, +O(w), Il(wr) = O(w), Ll(wr) = W

7r

+O(w) (10.2.23)

we deduce

limn=-4(1+...), t/0 R

(10.2.24)

409

THE OSCILLATORY SURFACE IN A SUBSONIC STREAM

i.e. just the kernel from the steady case. In the case of the incompressible fluid one obtains the kernel setting M = 0 in (10.2.20). From (10.2.13) it results

X=r0, R= xo+yo,

(10.2.25)

whence,

n(xo,yo) =-XoeR +iw y-'e4'0 - iKI(wr)2

-2w

(I1(wr)

- LI (wr')) + Uf ;07 J

0 so

,\2

+ r2eWX d A, (10.2.26)

This is the kernel for the ineompressible fluid

10.2.3

Other Expressions of the Kernel Function

Because of the Bessel functions, the expression (10.2.20) is considered to be complicated. However it does not contain divergent integrals. The

idea concerning the introduction of these functions may be found in [10.86] and it was also taken into consideration in (10.17J. We obtain another expression of the kernel function, starting from the fundamental solution (2.4.13). Indeed, with this one, the component w(x, y, z) is written as follows

w(x,?!,z) = a

fj f (.,?1)n(xo,yo,z)d dn,

where 2

,2

n(xo,yo,z) = a

f

=o 000

R1T

dr

(10.2.27)

( 10.2.28 )

with the notation

Rir = Vr2 + /32(y02 + z2) .

(10.2.29)

The passage to the limit and the derivation with respect to z do not interchange (only the passage to the limit and the derivations with respect to y interchange; for this reason we gave the expression (2.4.23)).

Therefore we shall derivate at first with respect to z. The expression

410

THE UNSTEADY FLOW

(10.2.28) may be found at Watldns [10.87], Williams [10.89], Dowell and others. Performing the change of variable r --t A : ,Q2A = T

- MRir,

(10.2.30)

we deduce as we have already seen, dT

dA

'

A' +s'

s2yp+z2.

(10.2.31)

Hence,

E(xoyo, a)

110

eMR,) R11.

0o

/X1

dr=J

dA

el"

(10.2.32)

A+

oo

where 32X1 = xo - M xo + p2s2.

(10.2.33)

Derivating we obtain

8E _ _ 8z

ek'X,

Me R /X+

(10.2.34)

iwX1

:-.0 8z2

dA

I+s

xo +

E

fim

Ma

ewa

JA1 0

dA

W (A2 + y02)3/2

where

R

V xo+/32Uo, A2X1

=xo--MR.

(10.2.35)

Imposing the boundary condition (10.2.3), we get the integral equation (10.2.21), where

M ew'X1 R Xl +

n(xo, yo)

x, J-oo (A2 + g2)3/2

d A,

(10.2.36)

Obviously the line yo = 0 is singular. This expression of n(xa, yo) is simpler than the expression (10.2.20), but here the integral is no longer convergent. We must therefore consider the Finite Part. The expression (10.2.36) is utilized by Ueda and Dowell (10.80), Ando [10.3], etc. Utilizing the identity 1

X?+yo

Mxo + R

M

xo+yo -x0 -X1,

(10.2.37)

411

THE OSCILLATORY SURFACE IN A SUBSONIC STREAM

one obtains two other expressions for n(xo, yo) . For the incompressible fluid (M = 0), we deduce eirra

1XO

n(xo.ieo)=- J

(,\2 + y02)3/2

To

d a.

(10.2-38)

If the unperturbed flow has the sound velocity, we obtain the kernel considering M - 0. Since Al

lim X = lim x° -

At--1

Xo +

yo

02

AM-1

= xo - yo _X1

(10.2.39)

2r

it results

xl,

xo 2+ 2 e

a nw

N(xo, yo) = -e

a

'

' + J oo (A2 + y 2)3/2

dA

1 .

J

(10.2.40)

This is the kernel for the sonic flow. One may verify the similarity of the representations (10.2.20) and (10.2.40), if we employ the relation

eiA

X

e"

J1 =

Ix j,\2 + yo)3/2 d J1 + Kl (wr) + r

(10.2.41)

+i 2 r ill (wr) - Li (wr)J which can be easily proved if we take into account the formulas

1 °°

( t 2 CD s 2r ) 3 / 2

d t = r KI wr) , (10.2.42)

sill Wt

Joy

ir W

(t2 + ,.2)s/2 d t = - 2 r

[11

- Ll (fir)]

given in the tables dedicated to Fourier transforms [1.16]. From the integral representation of the function Ko we deduce the identity rX

ei'.3(r-MRjT)

1-,0 (t2 + r2)3/2

dT

2Ko(w

yo + z2)

jf

oo

R1,.

d T' (10.2.43)

with the notation (10.2.25). Putting this in 10.2.24) and deriving according to the formula

8z2 82

X32 8

s 8s '

(10.2.44)

THE UNSTEADY FLOW

412

we obtain to the limit 2w

-iWxo

K t (wr) -

/x

(iWN1 +

eo(r-rlRa.)

Q2

dT

Ix (10.2.45)

where r2 + 012

Ror =

(10.2.46)

This is another expression of the kernel.

The Structure of the Kernel

10.2.4

The identity (10.2.41) makes possible to deduce from (10.2.32) the structure of the kernel in the vicinity of the singular line yo = 0. Indeed for a small r we have (1.40] 1

K1(wr) =

1+

wr 2

In r + ...

,

Ii (wr) = wr + ... , Lt(wr) =

2

a

(10.2.47)

+ ... ,

such that

X

eiWX

sadA+iI - Jo (ACos Jo + yo) /2

J'00 (A + yo)

3/2dA=

+72 -

tr

(A2

tn AdA+ 3/0)3/2

+

2

+ 2 IAr + ... , (10.2.48)

the points representing series of integer powers of r. If X > 0 the last two integrals have no singularities. Indeed, using the expansion formulas for

LA

d a .. . (A2 + yo2)v+1/2

given by Ueda (10.81], we deduce X

°T

(A 10

)3/2 d

x sinwA J0

00

(A2 + r2)3/2

A =E (

(2n)1 2n

n=0

00 (_1)nw"+i d A - nL-.o

(2n + 1)! 12n+1,

(10.2.49)

413

THE OSCILLATORY SURFACE IN A SUBSONIC STREAM

where

=1rX (A2 + r2)3/2 d A = Atm

Im

Xm-I

1))(,-3 X2 + r2 - (m - 1)(m - 3)J,.-4.

(M

X +*

(10.2.50)

Using the notations

Jm=

o 0

+r2dA=X--i(X2+r2)312-(m-1)r2Jm-2,

Am/A2

(10.2.51)

we may calculate the coefficients I,,, step by step, noticing that

IX

1 r2=rs +

0=

11 = VTr+--r-l -r. The formulas (10.2.48) and (10.2.49) give the structure of the kernel.

10.2.5

The Sonic Flow

If the velocity of the stream equals the sound velocity (U. = ca ), then one obtains the solution of the problem from (10.2.6), (10.2.7) and (10.2.12) setting M --# 1. We have shown in (10.2.27), that in this case there are perturbations only in the x > 0 zone. When we shall pass to limit we shall consider therefore x0 > 0. We have limp Go (xo, Spin, z) =

exp 4

[iW

Jim M2x0 _MR,1

z0exp-iw

2

+2xo

=

i - Go (10.2.52)

Z2

X02

lime G(x0, y0, z) =

4

lim X = lim

M-.1

M-.1

exp I

-i.

w J

z0-MR =

hml E2 = 110 exp

T-

u.

2(

(xo 2

= Gl ,

- 14),

)T

(10.2.53)

414

THE UNSTEADY FLOW

From (10.2.16) it obviously results lim E1 = 0 . Hence, the representation obtained from (10.2.6), (10.2.7) and (10.2.52), gives the perturbation in the sonic flow and (10.2.12) gives the kernel of the integral equation. This is s

r

nl+n3 = oexp 1 2 (x0 -

zo

92

J

L

10.2.6

iW 1

2 (T -

y02

)1

dT .

(10.2.54)

The Plane Flow

Acting like in 5.1 we obtain the formulas for the plane flow from the formulas which characterize the three - dimensional flow. We assume that in the equation of the perturbing surface (10.2.2), ho(x, y) has the

form ho(x), i.e. it has the same form for every y. With a suitable choice of the reference length Lo, the domain D will be rectangular with -1 < x < 1, -b < y < b. In (10.2.6) and (10.2.7) we shall have f = f (r;) . Considering b - oo we obtain +00

go (X0,

z) =

J

Go(xo, yo, z)d r1= 00

eik(Mxo-Ro)

1

=97r ED

dr1=

Ro

e

QHoe)(k Vxo+ 62z2)=

00

G(xo, yo, z)d r1=

g(xo, z) = f

xo +

Q

e(xo, z) = r-00 (E1 + E2)d r1= roo g(r, z)d r , +00 02 81/2G(xo, yo, z)d q = co

+oo 02

J

G(xo, yo, z)d r1= 0 . ao

X7

(10.2.55)

22

(10.2.56)

Using these results we obtain from (10.2.6) and (10.2.7) the representations (10.1.12) and (10.1.13). Taking into account the formulas of Fresnel

1 sin x2d x = fo

0

00

oos x2d x

YY

OSCILLATORY PROFILE IN A SUPERSONIC STREAM

we deduce

415

r+ e-In"'du= Vjj(1-i),

/ .

(10.2.57)

o0

and then, with (10.2.54) +00

=

it

-oo

(ni + rn)d q _

+ i)

Trw

OX-0

iw exp(xo) - wi2

JL Zo

iw

exp(2 r)

d7

r

(10.2.58)

This is the kernel of the two-dimensional sonic flow.

10.3 10.3.1

The Theory of the Oscillatory Profile in a Supersonic Stream The General Solution

One considers that Carrick and Rubinow 110.241 have investigated for the first time this problem. They have utilized the method of the pulsating sources potential. The presentation which follows relies on the fundamental solutions method [10.17]. The integral equation is solved explicitly. One obtains finite formulas for the lift and moment coefficients. One utilizes the dimensionless variables (x, y) defined like in (2.1.1) and the representations (2.1.3) for the velocity and the pressure, in order

to deduce that in (2.4.8), (2.4.18) and (2.4.21) the perturbation pressure and velocity determined by the action of an oscillatory force having the

form felt, applied in the origin of the axes of coordinates, have the form (10.1.4), where

P(x,y) = _(f V)Go(x,y),

v(x, y)

=f

(10.3.1)

1e-k.% [G(r, y)d r] + (10.3.2)

r] +f2e-k''x [2k4G + k2Cr + w2 f C(r, y)d r]

Utilizing the notations W = w1k2 , v = wAf and H for the function of

THE UNSTEADY FLOW

416

Heaviside, in (10.3.1) and (10.3.2) we have

Go(x, y) = H(x - klyi)so(x, y), (10.3.3)

G(x, y) = H(x - klyl)g(x, y), where

a = vM 2kgo(x, y) = Jo

(''- k2y2) e`ia"

2k9(x, y) = Jo (vVx2 -

k

)

(10.3.4)

e-k:,x.

As it is shown in (2.4.22) the perturbation (10.3.1) and (10.3.2) is zero outside Mach's dihedron with the edge on the Oz axis and the opening 21,t . This is the fundamental solution of the problem.

Let us consider now that the uniform flow having the Mach number M, is perturbed by the presence of a profile whose equation is y = h(x)exp(k it) .

(10.3.5)

Taking the origin of the reference frame on the leading edge and the length of the chord as reference length L0 , the function h(x) will be defined on the interval [0,1] . Replacing the profile by a forces distribution (0, f)(t)e''t defined on [0,1] , one obtains the following general representation of the perturbation P(x,y)

_-/

f o

+1

i

f (f )e

iWxo

L2iwG(xo, y)+

(10.3.6)

-G(xo, Y) + W2 f G(r, y)d rJ d 00

.

J

Taking into account the definition of the function H(xo - kIWI) and the formulas (A.3.15), with the notation X = x - kIyI,

(10.3.7)

417

OSCILLA'T'ORY PROFILE IN A SUPERSONIC STREAM

we deduce

jf()Go(xoiY)d t _ = H(X)

y f t f (E)9o(xo, y)d = H(X) Jo, f W

`""°G(xo, y)d t = H(X) f f

9o(xo, y)d

,

°g(xo, y)d t ,

o

0

f

(xo, y)d =

=f1

f

f

J=

X

f' f A

[H(x) I f (t)9o(xo y)d

(t)e-'.."`og(xo,

ff(t)-iW8 ax (H(xo-kI yI)9(xa, y)ld

y)d(xo - kjyj)d4 + H(X) f f (f )e-k"19. (xo, y)dk, o

:o

= H(X) f to9(r,y)dr.

G(r,y)dr,

kM

00

(10.3.8)

Hence the solution (10.3.6) is

P=O, u=0, ifX<0,

(10.3.9)

X

P(x,y) =

-Jo

u(x,y) = fo f

if X > 0, (e)e-iwxonl(xo,y)dt+ Zf(x)exp(ialy)),

ifX > 0, (10.3.10)

where xo

n i (xo, y) = 2iwg(xo, y) + k29=(xo, y) + a'Z

f 9(r, y)d r .

The formulas (10.3.9) show that the perturbation produced by the profile propagates only in the interior of Mach's angle with the vertex in 0, and (10.3.10) that in a point M(x, y) from the interior of this angle one receives only the perturbation produced by the segment OMo (fig. 10.3.1).

THE UNSTEADY FLOW

418

Fig. 10.3. 1.

The Integral Equation and Its Solution

10.3.2

For 0 < x < 1 we deduce

v(x, 0)

p(x, +0) - p(x, -0) = f (z)

(10.3.11)

= if (x) + 2 I f (t)N(xo)d t,

(10.3.12)

where

N(xo) = e-" =

ym

n1(xo, y) _ s

k

(Jo(vzu) + iMJ1(vxo)J e

+ke

J

Jo(v-r)e rd z ,

0

(10.3.13)

Imposing the boundary condition (10.1.11) one obtains the following integral equation

k f (x) + J f (t)N(xo)d t = 2H(z) ,

0 < x < 1.

(10.3.14)

0

This is a Volternz type integral equation of first order. We solve it using the Laplace transform. One knows (see for example (1.32)) that the Laplace transform of a certain function g(x) is the function g(p), defined by the operator

£(g) = jg(x)e_Pxdz

(10.3.15)

419

OSCILLATORY PROFILE IN A SUPERSONIC STREAM

where p can be a complex number(p = pi + ip2) whose real part is positive.

[JX]

Applying the operator G in (10.3.14) we obtain

kf + fo

aPxdx= 2A(p) .

Here we shall change the order of integration. In figure 10.3.2 we observe that the domain of integration is D (for a given x , C goes from 0 to

x). But D can be also covered integrating at first with respect to x to oo and then with respect to . We have therefore from

k7+

f (t;) [f°°e_P0tN(xo)dx]d C= 2R(p)

.

(10.3.16)

XAK

0 F'ig. 10.3.2.

Using the change of variable z -+ u : x - = u, we deduce from (10.3.16)

(k+R)7=2f1.

(10.3.17)

In order to determine the transformation R we shall utilize the formula 1

(10.3.18)

G(Ja(vx)1 =

which may be found in the tables with Laplace transforms (1.161, [1.32], 11.331. So, using the notations Nl (x)

=

Jo(vx)e-i.x

, N2(x) = Ji (vx)e ix" , (10.3.19)

N3(x) = e-16'" I = J0(vr)e W-dr, 0

THE UNSTEADY FLOW

420

with a = vM , we obtain 00

C(Nl} = f Jo( vx )a- (p+")xd x = (p + ia} + v2 p + is + 00

£(N3) =

(p+ ia)2 + v'j' v

1

G(N2) =

1

e-pie-"[f

dxJ

(p + ia) + v2v2 '

Jo(pT)e-a d r ]

0

=

(10.3.20)

°O

fc*

Jo(vr)e`+E [je

ir-"'i"dxI dr

r00 e-(P+",)udu =

Jo(t r)e_u/M dT

0

0 1

1

(p+ivM) +v P+iw With the notations

iwM '

Pt =

M+1'

UJAf

Af-1'

(10.3.21)

it results

1

(p + ivM) + v =

(P + P1)(P + p2)

k(k+N) [(P+iw) (P+Pt)(P+P2), =k(P+Pi)(P+p2),

A=

+k[p+ivM-Vll»PI)(P+P2),+

P11

+

)(P + P2) w2

1

+ kP+iw} (10.3.22)

and then

OSCILLATORY PROFILE IN A SUPERSONIC STREAM

not

g

_

k

_ p+P2-P2+iw

p+iw

k + N

(p + p1)(p + p2)

M1)1/2(p+

421

(p + P1)(p + p2)

M+M1)-1/2_

(p+

M

1w+1) M -1/2

iw

M - I (p

iwM

1/2

(p+M+1)

M

whence

kf = H+ g.

(10.3.23)

From the tables with Laplace transforms [10.581 it results g = L-1(g) and then with the convolution theorem x

kf (x) = 2 f H (xo)g()d = 2H(x) - 21-w f H(xo) [Jo(ve) + iMJ1(v4)1 a `t d t. 0

(10.3.24)

This is the solution of the integral equation (10.3.14). It was given in [10.171.

10.3.3

Formulas for the Lift and Moment Coefficients

The lift and moment coefficients have the form

CL = cLexp(iwt), CM = cmexp(iwt),

(10.3.25)

where, because of the formula (10.3.11), CL = -2

f

1

1

f (x)d x, cM = -2 fo x f (x)d x .

(10.3.26)

0

We considered that the length of the chord Lo is the reference length and we defined CL =

P ' CAt = (1/2)poU*20Lo (1/2)poU.2Lo '

P being the lift and M the moment on the direction Oz z.

(10.3.27)

422

THE UNSTEADY FLOW

Utilizing ((10.3.24) we find

4iv f1 kAl 0

CL =

-4

CM =

-411 xH(x)dx +

J0

H(x)dx +

1

4iv

e 'a`d

(Jo(vl) - iMJI

,

f 1 G(t) (Jo(vt) - iMJ1(ve)J a '&(d 0

(10.3.28)

where

J

1

it x H(x - )dx

H(x - t)d x,

1

E

(10.3.29)

.

The coefficients (10.3.28) may be calculated numerically on a computer. Another method consists in approximating the function h(x) by polynomials whence one deduces that CL and cAf may be expressed by means of the terms having the form

f" (M, a)

=10

"Jo(vl;)e'°(d

1

(10.3.30)

9n(M,a)=

frJi(z)e'd7

Taking into account that Jj(z) = -Jo(-) and integrating by parts one obtains that

vg, _-Jo(v)eis+nfi_1-isf", n=1,2,..., (10.3.31)

vgo = -Jo(v)e is + 1- lab. These formulas show that g" may be expressed in terms of f,,,. Integrating f by parts, we deduce

laf" = -Jo(v)e-' + of"-1 - v

ivM j '

f

1

0

F"Jlv )e-'f d

,

(10.3.32)

_ -Ji(v)e+ of+ (10.3.33)

+(n - 1)11 0

Substituting (10.3.33) in (10.3.32) we find for f" an expression which contains the last term from (10.3.33). This may be eliminated with the

423

OSCILLATORY PROFILE IN A SUPERSONIC STREAM

aid of the relation (10.3.32) where n was replaced by n - 1. After all one obtains na

+ (n

a

Jo(v)e-i. - 1 Ji(P)e ia+

1)

(10.3.34)

1)2A-2 + 1(1 - 2n)fn-1

This formula shows that all the terms fn may be expressed by means of fo. This result was given for the first time by Schwartz in [10.70]. In the same paper one gives the following expansion for fo 00

[,

fo=e

n=O

IMM1:Jn(a)+iJn.f.1(a)w".

(10.3.35)

2"ni(2n + 11

In [10.701 one gives tables with the numerical values of fo, with eight

exact decimals, for 1 < M < 10 and 0 < a < 5. In [10.33] one gives the numerical values of the functions fn for n = 0,... , 11. 10.3.4

The Flat Plate

For the flat plate having the angle of attack -E (h = -ex) we deduce H = -2E (1 + iu;x) such that it results s

CL = -

E

[ 2 f2 + iw(2 + iw) fi - (1 + 2iw -

s ,

2 2e

CM = -

)f of

(10.3.36)

z

[f3+f2_2+2)f1 - 2(iw - 3 )fo] w2

These formulas are sufficient if we utilize the tables for , f1fo, f2' fs For w -+ 0 one obtains the well known formulas of Ackeret 4e

2E

CL = k , cM= k . Obviously, cL and cm may be expressed only by means of fo if one utilizes (10.3.34).

Noticing that

f = fn + ifn ,

(10.3.37)

where f1

f, = J

1

fn = -f eJo(4)sin(a)de,

424

THE UNSTEADY FLOW

we deduce from (10.3.36) CL = c'L + icL , cAf = (.! + icM

,

(10.3.38)

If the equation of the plate has the form

y = -ex coswt = Re [-ex exp(iwt)J ,

(10.3.39)

then

CL = crL cos wt - CIL sin wt,

(10.3.40) CAS =

cos wt - ciAf sin wt,

these formulas give the variation of the lift and moment coefficients versus the time. For example, for w = 7r and M = 2, we obtain CL = e(-9.2060 cos art + 11.8941 sin zrt)

,

(10.3.41)

CM = e(-6.8779 cos in + 17.5209 sin irt)

10.3.5

,

The Oscillatory Profile in the Sonic Flow

We are interested in the behaviour of the formulas of Nl and N2 when M 1 (k -- 0). It results v -+ oo such that we shall utilize the well known asymptotic expressions Jo(z) =

F2

Coo

7r

(z - )+0(.-,), 4

Jl

(z) =

(10.3.42)

r2z ooe (z - 34) + O(z-1)

for great values of z. In this way, we deduce [Jo(vxo) + iMJI (vxo)J exp(-iaxo)

NJ = _

k 1

=w xo

[(1- iM) cos axo cos vxo + (M - i) sin axo sin vxo-

-i(1 - iM) sin axo cos vxo + i(M - i) oos axo sin vxoJ , (10.3.43)

425

OSCILLATORY PROFILE IN A SUPERSONIC STREAM

and analogously

N2(x) = k

_

10"

Jo(vre)e'du

1-i

coe0(M-1)u+isinW(M- 1)ud u +I,

2fJo

Mu

(10.3.44)

where, with the change of variable u -+ t : u = (M -1)t, we have

_ cosw(M+1)u-isin'(M+1)u duI _ 2l+i two Mu

(10.3.45)

xo

l+i M-1 IMexp(-iwt)dt 2 xrw ft M Jo Taking into account that we also have xo

f

M -1 gyp( Ld)dt slim Ja

- Jo

)d

t = w (1 +

it results that

,

No(xo) = J m1 N(xo)

=(i+1)

we

r 1xoexp(2iwT O ) wi L

YYY

iw

2 Jexp(2 u)

I

du

,

(10.3.46)

he. exactly (10.2.54).

The integral equation (10.2.14) reduces to

fo f (4)No(x - 4)d4 - 2H(x)

(10.3.47)

This is also a Volterra-type equation of first kind. 'Lbt integrating it we shall use again the Laplace transform. Applying this transformation we deduce

(1 + i)% 7 = 2$b0, where § o'=

(P+ 1w

2)

1f2

iwf

+ 2 (P+

iw 2

}`

(10.3.48) 1/2

(10.3.49)

426

THE UNSTEADY FLOW

From tables (see for example [10.581) we have that G

p+iw/2,=exp(-)G[vii=-2exp

(-) (10.3.50)

such that we obtain go

2 Rx(i``' -

x)exp (

i2

x)

(10.3.51)

and using the convolution theorem, from (10.3.48) we deduce

After determining 1(x), the lift and moment coefficients result from (10.3.25) and (10.3.26). We shall give calculation formulas in 10.5.2 when we shall consider again this problem.

10.4 10.4.1

The Theory of the Oscillatory Wing in a Supersonic Stream The General Solution

The theory of the oscillatory wing in a supersonic stream, was conceived according to the model of the theory in the subsonic stream. The papers of Kussner [10.37J, [10.38] represent the starting point of this theory. We mention then, the study of Garrick and Rubinow [10.25] where the potential of the pulsating source is determined, the paper of Miles where one considers the symmetric arrow - like wing, having the leading edges outside Mach's cone [10.53, the paper of Nelson for the triangular wing [10.57], etc. But the fundamental work in this domain is the paper of Watkins and Berman [10.85). Here one may find for the first time the integral equation of the problem and various forms of the kernel. The method is similar to the method from the subsonic case. From the potential of accelerations of a pulsating source, one obtains, deriving with respect to z the potential of accelerations of a pulsating doublet. The potential of the flow is obtained superposing the doublet potentials. The boundary condition gives the integral equation of the

OSCILLATORY WING IN A SUPERSONIC STREAM

427

problem. In the following papers, due to Ashley, Windall and Landahl [10:4], Landahl [10.44), Stark [10.72], Harder and Rodden [10.29], Ueda and Dowell [10.81] the theory was developed and numerical methods for the integrations of the equation of Watkins and Berman were given. We shall indicate in this subsection how one may also solve this problem by means of the fundamental solutions method. Assuming that the equation of the wing is (10.2.2), we shall use distributions having the shape fe".,c

= (0, 0, f)e' .

(10.4.1)

Utilizing (2.4.9)we deduce that the perturbation of the pressure determined by such a force applied in the generic point (t, n, 0) is given by the formula (10.4.2) p(x, y, z) = f azGo(xo, yo, z). For the component w, it results from (2.4.13) w(x, y, z) = - f e "" `0H(xo)b(1M)6(z)+ 82

+f

:o 8z2a-""`O, co

(10.4.3)

G(r,yo,z)dr,

and from (2.4.23) fe",,xa[(2iw+k28x) G(xo,yo,z)+

w(x,y,z) =

8z

02)

+

(10.4.4)

IZO G(7-, yo, z) d r ] , 00

where we denoted Go(xo, yo, z)

G(xo, 3fo, z)

= 2A =

H(xS

s)

H(xSo

s)

cos (LS)e-iaxo , (10.4.5)

cos (vS)e-o"

1

k= M -1, 1=w/k2, v=OM, a=vM, s=k yo+x , S= xo-s , ST= 'r -3 ,

(10.4.6)

H being the function of Heaviside. One may prove, taking into account the formulas (2.3.35) and (2.3.36), that the perturbation given by (10.4.2)-(10.4.5) vanishes in the exterior

428

THE UNSTEADY FLOW

of Mach's cone with the vertex in the point ((, i, 0) and with the axis on the direction of the unperturbed stream (the Ox axis). Using a forces

distribution having the form (10.4.1), applied on the domain D - the projection of the wing on the rOy plane, the perturbation will be given by the formulas p(x, y, z) =1

JD

w(x, y, z) = -6(z)

+21-

f (C OF Go(xo, yo, z) d d rl,

J JD

f (t,

i7)e-'

°H(xo)b(yo)d

(10.4.7)

d q+ (10.4.8)

f f f(e,q)e D

w(x, y, z) = 2 . wh ere nl (xo, yo, z) _

IL f (t, q)e-'"" °n2(xo, yo, z)d t d i

82 f.0 z2

H(T - s)

0o

co$ (vSr) e S

r

(10.4.9)

,

dr= (10.4.10)

82

_ z2 H(xo - s) J n2(xo, yo, z) = (2iw + k2 40)

+[w2-

\

10.4.2

02 2

Cos (VSr)

H(xS- s)

r,

cos (vS)e"'Ww'+

)H(xn-s) r a

°sr)e-OrdT,

Cos

Sr (10.4.11)

The Boundary Values of the Pressure

They may be obtained writing p(x, y, z) =

o - s)

Cos (vS) a ;

, d t d q,

27r C7z S (10.4.12)

and noticing that because of the presence of the factor H (xo - a), the integrand differs from zero only in the domain DI defined by the inequality xo > s for a given M(x, y, z). This inequality is equivalent to

(t -x)2-k2(q-y)2> k2`z2,

429

OSCILLATORY WING IN A SUPERSONIC STREAM

which are solved in 8.3.3. Denoting by M'(x, y, z) the projection of the

point M on the xOy plane and X = t - x, Y = rI - y we deduce that Dl is the foregoing branch of the hyperbola X2 - k2Y2 = k2z2 (fig. 8.3.4). When M' is in D, the hyperbola degenerates into the half-lines X = ±kY (fig. 8.3.5) Since the function f is defined only on D, we shall prolong it in the outer region taking it equal to zero. It follows that in the perturbed region from the fluid we have

I

f (C, n)Go(.To, yo, z) d d q=

Qi

l

J

ly- f()

cos (vk

(Y+-,)(,-(Y+

- 77)(9- Y_) (10.4.13)

where

Yf = y

xok-2

With the change of variable q - 0:

2(Y++Y_)- 2(Y++Y_)cos0 = y-

- z2 .

(10.4.14)

P-

ok-2 - z2cos0, (10.4.15)

we deduce

21rk JO

cos

a-iwco 110 f (C y

-

xpk-2

z2cos0) .

(- z) 2sin 0d BJ d

whence, if f (x, y) is a continuous function,

P(x, y, ±0) _ _]

az I = Tf(x, Y)

(10.4.16)

P(x, y, +0) - P(x, y, -0) = f (x, y) .

(10.4.17)

and then

Hence, like in the previous sections, f represents the jump of the pressure on the wing.

430

THE UNSTEADY FLOW

10.4.3 The Boundary Values of the Velocity. The Integral

Equation For z 0 0 the first term from (10.4.8) vanishes (5(z) = 0). It has to be considered in the same way in the limit values for z - ±0. The remaining term is the kernel given by Watkins and Berman [10.85). Elementary calculations give 0 az

_ k2 z e

82

8s

22

8 k4z2 82 - k2s (1 _ k2z2 -s2-)T. + 2 8s2 .

(10.4.18)

In the cited papers one considers that the terms which contain the factor z2, vanish when z - 0. But this is not always true (see for exampie (3.1.20)). This is true when the factors which multiply z2 remain bounded when passing to the limit. In the following we shall see that for (10.4.10) the form obtained under this assumption is correct. Hence we shall consider the kernel

Cos(1S')e 'dTJ

82

nI(xo, yo, z) °-`

8 {H(xo_s)

(10.4.19)

s

The derivation is performed according to the formula (A.3.15), but we

have to take care that for s = xo the integrand is unbounded. We eliminate this inconvenient writing

(20Cos('1r2 -s')e '''rd7= T -8 is 0 e-0 - e+

/s

T

-s

V7-r- S

Js

d T + CO°

TO

1

e' d-r+

dr

T -8

12

(10.4.20)

After all k2

ni (xo, yo, z) =

s

H(xo - s)1,

(10.4.21)

s ) e "°TdT ,

(10 .4 .22 )

where

0 8s,

=o Cos

(

r

r

the derivation being possible if we utilize the equality (10.4.20), but we have no interest to do it. The integral may be calculated with the

OSCILLATORY WING IN A SUPERSONIC STREAM

431

substitution r - A : r = scoshA. Deriving one obtains 1

- -rpcos(vxo-3 )e S

O

V xp zp

-.%1

-r

e'"''r

s

d

dr [sin (v

r2 - s2)] d r-

esin(v r2 - s2)d r

(10.4.23)

.

We integrate by parts in the second term from the right hand side. Passing to the limit in (10.4.21) we notice that, like in the steady case,

it appears the singular line yo = 0. After eliminating from D the domain D, defined by the inequalities y - E < tj < y+E we shall put in the remaining domain z = 0. One obtains the following singular kernel It 1(ro,yo) = lim nt(xo,I/o,z) _

Yo

+

ff

e-V"o sin (vS) + M

Cos(,S)e_k-xo+

u)

-H(x

J

1X0

e iur sin (vSS)d r =

= H (xo - u)n(xo, yo) , (10.4.24)

where, u = kIyol and

S=

x0-u2,

Sr =

Jr2-u2.

(10.4.25)

For w = 0 one obtains (8.3.23). This will be the kernel of the integral equation. In the sequel we shall give a demonstration where the terms which contain tht factor :2 are not neglected. As we have already noticed, acting in the classical manner, we have to calculate the limits for z --+ f0 of some kernels which contain derivatives with respect to this variable (see Nlangler [10.52] for the subsonic steady flow, Heaslet and Loomax for the supersonic steady flow, Watkins, Runyan and Woolston [10.86} for the oscillatory subsonic flow, Watkins and Berman for the supersonic flow, etc.). Since generally, for performing this calculation we have to evaluate at first the derivatives, the passage to the limit becomes difficult. In

order to avoid this, we gave other expressions to the component w

432

THE UNSTEADY FLOW

((2.3.29), (2.3.37), (2.4.23)). In the general solutions built on the basis of these expressions it appears only the derivatives with respect to y. The passage to the limit interchanges with these derivatives. In the actual case from (10.4.11) we obtain n2(xo, yo) = line n2(xo, yo, z) _ 2-.o _ (2iw + k2)

H(x u) cos (vS)e-'"'%O+

(10.4.26)

cos vS,. ecWrd T,

82

+(w2 - 8y2 )H(xo - u)

S,)

where u = k1yoI . Since we have 02/0y2 = k282/c9u2 , with the notation

J_

cos(vST)e-,,tdT,

49

au

(10.4.27)

r

u

we deduce 1,2

-y2 H(xo - u)J =

k2 82 H(xo - u)J = k2 0 H(xo - u)J

(10.4.28)

where J, calculated like 1 ,is

J = - xo caos (ys)e-'ixo S

u

-!

e

MU

sin (vS)(10.4.29)

2°

W

Mu j

a w''r sin (vST)d r.

For determining H(xo - u)J we take (2.3.35) into account. In this way, from (10.4.25) one obtains rigorously (10.4.24). If the equation of the oscillatory surface is

z = h(x, y)e'

,

(x, y) E D

then one imposes the boundary condition

w(x, y, 0) = 8 h(x, y) + iwh(x, y) _- G(x, y), (x, y) E D

(10.4.30)

One obtains the following integral equation

J

1n,

f

n)e`"n(xo, yo) d 4 d n = 21rG(x, y),

(10.4.31)

DI being the domain marked in figure 8.3.5 (the domain where xo > u).

OSCILLATORY WING IN A SUPERSONIC STREAM

433

Other Expressions of the Ker e1

10.4.4

We have

rro

L =1 e-'

sin (vST )d r = 2i (L_ - L+) .

(10.4.32)

u

where we denoted

Tn

=

(10.4.33) U

In L+ we perform the substitutions r --- A:: T

MS, = kiA.

(10.4.34)

Taking into account that T is positive in both cases we deduce

kT=-uA+uM

1+A2,

(10.4.35)

such that

LT- = tyo(

f/

1

1+A'

e-'+Iyolad A

(10.4.36)

e-i:wlyola d A.

(10.4.37)

whence

L

= -- j (yo'

2i

(=o+MS)/ku

AtA

z o-AjS)/ku

1+A

-1

)

Since, on the other side, (x +MS)/ku

eWroad A =

'

4yo

jxo_MS)/ku

sin (PS)

.

(10.4.38)

from (10.4.24) we deduce Cos (VS) a-iUX0- iw

n(Xo' yo)

yo

S

2Iyo)

=o+MS)/ku

L0MS)/kU

A

e-i&+lYOI-%

dA.

V1--+-A2

(10.4.39)

This is the kernel given by Watkins [10.851. Obviously for w = 0 one obtains the steady kernel.

434

THE UNSTEADY FLOW

Performing the change of variable A -> v : Jyp)a = v and and integrating by parts, we deduce

' _IYoEa (x04-MS) f ku

a

2iyol

=-

1+A

.Y+

1

y2+X+

2y0

-

d.1=

iirJ

JX*

ye-n.rv

+L dt

-

2-y6'

X-

-

e--iwx_

+

TY0127X2

c- iwv

1#

1

e-'wx+

e

-d v'

2

x_ {Up + v2)312

(10.4.40)

where

k2X,=xo± MS.

(10.4.41)

Observing now that

2e-0'0 coss(vS)=e-+e-"'x+

(10.4.42)

utilizing (10.4.40) and the identities xo

yQ + .X+

- SX+ = xo yo + X? + SX_ = Myc'

(10.4.43)

,

we obtain for n(x0, yo) the following form given by Harder and Rodden [10.20)

2YL(xo, IM)

a-iwx..

e--W,x}

M

e-Iwv

.

V7=` 0+

1j,x

+2

(YO-1

+172)3/2

(10.4.44)

Another form of the kernel is obtained if one utilizes the identities

_ yO+X2

Mxo:F S xp+yo

_

Al

xe+X

(10.4.45)

One obtains the relation L12

2n.(ro,rlo)

S

a-ivx«

a swx

o+X_ + xo+X+)

X..

a-iwv

{3lo+tr2)3/2dv, (10.4.46)

utilized by Ueda and Dowell [10.811 for obtaining the numerical solution of the integral equation.

OSCILLATORY WINC IN A SUPERSONIC STREAM

435

One obtains the sonic limit at once from (10.4.44), or (10.4.46) notic-

ing that lim

A!-l

X_=-12 (xOlxo/ -X,

limN--1X=00.

(10.4.47)

One obtains the following kernel

H(ro)

2

2

12g + yo

(xu, Ilo)

+e

iWx

la + J/

oo

e'""

(Jp + t')3/2d vJ (10.4.48)

which coincides with (10.4.26).

A New Form

10.4.5

We utilize the formuhas (see for example 11.30], pp. 406, 422) with real parameter O° cos,

cos

"i

(pr)dT = ko(u v2 -12), (10.4.49)

°° cos (vSr )

i-Sr

sin (pr)d r = 0.

For p = 0 we obtain the identity

r°

ST)e-wTd r. _ Ko(wlyol) T

u

00 Cos(VST)e

- f.0

rrrdT,

(10.4.50)

T

as follows 2

fi Cw2 - k2

0' f J

rn Cos (VS_)

u

a lord r =

r

1 Iw--k2ou2/

[Ko(-ku)- f

T7r

.J=f

T

J

In the last part we derive without any difficulty. Deriving, the kernel (10.4.25) becomes

n2(xo, yo) = H(xo - u)n(xo, yo)

n(a'o,yo) _

r

- wM kkt(Au) u T J=°

. Cos(yST)e_Ord

-k.2 To

ST

sin(vST)e-` S2

r,

TdT-

(10.4.51)

'TIE UNSTEADY FLOW

436

where u = kJy I . This is the new form of the kernel. Having in view the behaviour of lit , for small values of the argument, this is 1

-

22

(I,, Iyul + r1) ,

r, = in 2 + -y

2

(10.4.52)

,

yo

-y being Euler's constant. An additive constant r2 also appears from the two integrals (10.4.51).

The kernel of the integral equation in the case Al = 1 is obtained from (10.4.51). We have

illl A2 I\1u) =

1yu1Rt(wIyoI)

Denoting rao

It = 2iJ

Sr -

7)]d T

ao

I2-2i f exp(-i (MSr+r)1S

1117-

we have.

°O sin (yS1) xp

r 2

_rd T = It + I2.

But

It

_

1

exp

2l

-

1

21

nhinl It =

f.0

[1'A12S2 - r2

Al ST + r

r2

exp iw AI

(IT

S; = dr

Aft 2

r2 - k2y + T

T2

,

- k2y0

li

10"exp Pww (r - TA) ] T L.

One obtains after all (10.6.13).

10.4.6

The Plane Problem

such that in the repIn this case, the density f q) becomes f resentation (10.4.7)-(10.4.9) we can calculate the integral with respect

437

OSCILLATORY WING IN A SUPERSONIC STREAM

to r l. We have 90 =

f

00

Co(xo,Uo,z)d9=

+oo

1

J

= 27

H(xo - k r2 + z2)

cos[v xo-k (r +x - k (r + z) zo

)J

dr.

(10.4.53)

Because of the presence of the function H, the integrand differs from zero only for xo > k r + z . This inequality implies zo > 0 and k2r2 < x2 k2z2, whence xo > kizl and -c < kr < c, where c = xo - k z2 . After all

-

oos{v

2-e'H(xo-klzl)f

90

-

C

-k r dr. r

(10.4.54)

Utilizing the formula 11.16] ` Cos (p

c- x-x)

(10.4.55)

gxdx= 2Ja(cjp2+g2),

it results

go =

H(xo - klzl)Jo(v xo - k2z2

(10.4.56)

and analogously 9

Gd q = gH(xo - klzl)Jo(v

roo +00

e

=

J

Edrt = ao

1 jr-a° H(r - kjzj)Jo(v

_ H(xo - kjzj) 2k

k2z2)e'

xo

Jo(v

0,

(10.4.57)

r2 - k2z2)e''O'd r =

r2 - k2z2)e rdr.

fk1Z1

(10.4.58)

For obtaining the results from 10.3 we have to consider the chord of

the profile on the Ox axis (0 < x < Lo) and to take Lo as reference length. Observing that

j

+oc &2 00

OY

2E(xo,yo,z)dq=-J

+00

02

8 Edt =-AEI 00=0,

438

THE UNSTEADY FLOW

we deduce 1

p(x,z) = - I f(e)e sodt, w(x, z) =

f

(10.4.59) 1

f we-k""°

[(2iw+k2)g+,2c]d

which is exactly the solution (10.3.6). We obtain too +00

n(xo) =

J

n2(xo, yo)d tl = +m G(xo,

= 2iw

yo, 0) d >1 + k2

J00

+w2J

a TX

+x G(xo,

y , 0)d n+

100

G(r,yo,0)dil = 2a(xo)Jo(z'xo)eui""`0+ 00

+H(xo)no(xo),

(10.4.60)

where no(xo) =

T-

T

[Jo(vxo) + 121 Jt(aod

e1 .lo(vr)ed T , o

(10.4.61)

i.e. (10.3.13).

10.5 10.5.1

The Oscillatory Profile in a Sonic Stream The General Solution. The Integral Equation

We proved in (10.2.54) that there exists the limit of the subsonic solution for M / 1, and in (10.2.45) that there exists the limit of the supersonic solution for M \ 1, and in addition, the two limits coincide. We shall prove now that there exists also the solution for M = 1, and this one coincides with the two limits. It will result therefore that the flow is continuous to the passage past the sonic barrier, unlike the case of the steady flow. We shall consider therefore the oscillatory profile (fig. 10.5.1) of the equation

y = h(x)exp(iwt),

(10.5.1)

439

OSCILLATORY PROFILE IN A SONIC STREAM Yt

0x

I Fig. 10.5.1.

which perturbs the uniform flow which has the velocity Ua, = cm (M = 1). With the notations (10.1.4), the boundary condition (2.1.27) gives

v(z, t0) = h'(x) + iwh(z), 0 < x < 1.

(10.5.2)

The fundamental solution in the two-dimensional sonic flow is given by the formulas (2.4.32)-(2.4.34). If the profile is reduced to the skeleton like in figure 10.5.1 it is sufficient to replace it by a forces distribution having the form (0, f)exp(iwt). (10.5.3)

It results therefore

P(x, y) = --f

a Go, (10.5.4)

v(x, y) = f ei`''°` [(21&G + w2 T G(r, y)d rj

o

where r

Go(x, U) = H(x)

exp I _ !(X

+s

)J E H(z)So(x, p) , (10.5.5)

1

G(a, Y) = H(x) -OM

[(x -

)}

H(x)9(z, v) ,

with the notation 2f/ = 1. A continuous superposition of forces having the shape (10.5.3) on the segment [0, 11, will give the perturbation

i

P(X, U)

I

v(x, p) =

J0

1

f (() gpGo(xo, y)df ,

f (C)e^'"'° [(2iWG(Xo, y) + w2

Jzo

0

G(r, y)d rJ dl;

440

THE UNSTEADY FLOW

Taking the significance of the function H(x) into account. it results that for x < 0 we have

P=O, v = O,

(10.5.7)

and for x > 0

f 2fV)ryyo(ro.y)d

P(x,y)

To

fr

v(x, t!) =

(10.5.8)

(2iw9(xo y) + w2 f 9(T, y)d r d ti

fWe-ten

I

.

0

(10.5.9)

This is the general solution of the problem. It was given in (10.18. Performing the change of variable F -« u : u = y'/xo one obtains °G

iw1Z Jy2

P(x. ,) =

f (x -

y-'

)exp

;r.

[_!f + u)]

ll

Ou-

and then

P(x, ±O) _ T-iwfl f (x) f exp

-'2 It

`

= +,l f (x )

(10.5.10)

It results therefore

p(x, -0) - p(x, +0) = f (x)

(10.5.11)

.

For 0 < x < 1 we deduce

v(x,±0) =

jj

f(Z;)e-'"`°n(xo)dy.

(10.5.12)

ro

27 [

exp(2 xo) - 2

( } J exp 'T)

1

.

(10.5.13)

Imposing the boundary condition (10.5.2) it results the following integral equation

jf()n(xo)d

A(x)

(10.5.14)

where A(x) = h'(r) + iwh(x). The kernel (10.5.13) coincides with (10.2.54) and (10.3.46).

441

OSCILLATORY PROFILE IN A SONIC STREAM

10.5.2

Some Formulas for the Lift and Moment Coefficients

Taking into iweount that

1-i

=

I

D- 2

(10.5.15)

4 fa-w

the solution (10.3.52) may be written as follows

3(x) = 2iwft

fA

)exp(- Z l;)d 10.5.16)

-2fl J e 12

ex p(-

2 t)u

,

the sign * indicating the Finite Part (Appendix D). Denoting

f

F = Ir A(xo)eXp(- +t)dt

A(,)

2

xo

o

P(_!2xo)df ,

(10.5.17)

with the definition formula (D.4.2), we deduce

1z o

5.7r

[A(')

l so gyp(

_ iw 2

]

xo)d=

dx '

(10.5.18)

and then

-J

exp(

- 2 )d t - iw F+2df .

(10. 5 . 19)

After all the Jforinula (10.5.16) becomes

1(x) = ti

(ii' + dx

(10.5.20)

The lift and moment coefficients are given by the formulas (10.3.29) with (10.3.30). Utilizing (10.5.20) we find cL - SS2 J

1 A(1 - 0 + WB(t)

0

P(- 2 (10.5.21)

c,tt = 80

ri

A{1-

) + iwD(>;)

2 )d

where we denoted I

1

B(4) _ I A(xo)dx, D(t) = J xA(xo)dx.

(10.5.22)

442

THE UNSTEADY FLOW

Approximating the function h(x) by polynomials, we deduce that the integrals from (10.5.21) have the form 1

f(

2exp(-2 )d

,

it = 1, 2, ...

(10.5.23)

Integrating by parts we obtain

-.2 exp(-12)+

2n

1W

n=1,2,...

iW

(10.5.24)

It results that all the integrals from (10.5.21) may be expressed as functions of

Io =

J

exp(-

)

Vq

[c(/) - iS(

=2

11

7r

)]

,

(10.5.25)

where C(x) and S(x) are the integrals of Fresnel [1.30):

C(x) =

fS(x) =

1

2.

j

(10.5.26) -

the notation z = irx2/2. In the case of the flat plate having the angle of attack c (h = -E.r. ) one obtains CG = -4f k(1 + iw) CM

[2exP(_) +(I+ iW)I01

=-aQEK-iw+3tw+e P(12)+(2i

4)i].

+1

(10.5.27)

10.6 The Three-Dimensional Sonic Flow 10.6.1

The General Solution

In the three-dimensional sonic flow the fundamental solution is (2.4.36)

and (2.4.37). A force having the shape (0, 0, f)exp(iwt) , applied in the origin, will produce the perturbation

P = H(x)p, W = H(x)w,

(10.6.1)

where

P(x,y) _ -f8 go, w(x, y) = f e'w,x [2iw9 + (w2

-

z) / JJJ

(10.6.2)

g(r, y, z)d Tl J

TIIE THREE-DIMENSIONAL SONIC FLOW

443

with the notations 1

go (x, y, z) =

, zy)_

9(x,

4;rrexp

1

.rr

(-

2r.' 2

e`priw IL 2

(10.6.3)

y2+z

z-

x

The perturbation produced by a distribution of forces having the form (0, 0, f (s, r1))cxp(iwt) ,

defined on the domain D (the projection of the wing on the xOy plane), will be characterized by the formulas

P(x,y,z) w(x, y, z) =

8

J JD

f (E,

r?}e-"'"`" 1(2iwg(xo,

+(wl - ay ,)J

10.6.2

yo, z)+

(10.6.4)

9(r,yo,z)drJdedr.

The Integral Equation

Assuming that D is such that every parallel to the span (the Oy axis) intersects the boundary OD in at most two points and denoting by y_ and y+ the ordinates of these points (fig. 10.6.1), we obtain

110 With the change of variables

x-

s1=y+l:I v;

f

V}}"'d'1 dC

(10.6.5)

r1) -p u, v :

zJdudv

(10.6.6)

THE UNSTEADY FLOW

444

Fig. 10.6.1.

we deduce

P(x,y,±O) = 47r

iw

lim x

lim

fp

tE) f(c 17)

+ yo + ~ zo

2

/

d

dn

22

Z

47r :-.to 2

(X0 I

xo

7- (F)

exp

f (x - -, y + I= Iv)

I

U

(f)-SI/I=I

exp

iw

z2

2

u

+u(v2+1)]}dudv f /

=

exp [ - Z uv2)dv] du.

Utilizing the integrals of Fresnel 00

I

cosx2dx =2 f

0

20

Cost

dt=

°sinr2dx = 1

f s'n dt = 2

2

(10.6.7)

we deduce

P(x,y,f0)_T2f(x,y)

(10.6.8)

Hence P(-T, y, -0) - P(x, y, +0) = f (x, y) .

(10.6.9)

445

THE THREE-DIMENSIONAL SONIC FLOW

Assuming that the equation of the oscillatory surface is

z = h(x, y)exp(iwt),

(10.6.10)

we obtain the boundary condition w(x, y, ±0) = h'(x, y) + iwh(x, y) _- A(x, y) , (x, y) E D.

(10.6.11)

It results the following integral equation 2n

lID f( t, ri)N(xo, j) d t d 0 = A(x, y),

(x, y) E D ,

(10.6.12)

where

N(xo, yo) = amp

[iw(

2x o

y1 + (10.6.13)

\W2

1

02

'O=p iw

yl

dr

2

We have also obtained this kernel from the kernel corresponding to the

subsonic flow (10.2.58). On this formula we cannot observe yet the singular part of N We shall calculate therefore the last term. Using the formulas

I

0

zo

J

exp

2

Ocein coo

(r -

(ax- z )dx _ {2Ko(2v)

\l LT

(10.6.14)

2K0(11p0D)-1°O exp

f2

(r

}] dr

The last integral may be derived with respect to y interchanging the derivation with the integration. One obtains

b

Texp ( 2r0)8'r' lrexp (

"4 exp

2

In this way we deduce the final form of the kernel

(.r

N(xo,yo)=-1 ! K1(wlllol)+ 2

_)1]J 7d 1, 2

(10.6.15)

446

THE UNSTEADY FLOW

The principal part is in the first term. Taking into account that for small values of the argument we have

+zlnz+...,

K,(z)=

(10.6.16)

it results

N(xo,yo)=----w2(lnlyol+r)+....

(10.6.17)

A

The singularity has therefore the same order like in all the other spatial problems.

10.6.3

The Plane Problem

We remind that one obtains the solution of the plane problem if we assume that D is a rectangle having the dimensions Lo and bL0 and we consider b --+ oo. Moreover we assume that every section with a plane parallel to xOz determines the same profile, hence in (10.6.10) h depends only on x. Taking Lo as a reference length, the domain D will be defined by 0 < x < 1, -b < y < b. Considering b oo we have to obtain +00

n(xo) _ j

N(xo,yo)dri,

(10.6.18)

o0

where n(ro) must be (10.5.13), and N(xn, yo) (10.5.15). Indeed, utilizing the representation 11.30] !OD

J

(t2

C} ``'dt,

(10.6.19)

-2)3/2

we deduce rc+Oht(_u)du=2

w)+'Kt(wlyol)dyl_2w

00

o

lyol

u

0

=2'(r0°coswt-1+Idt-7rw Jo

t2

zrw+2J

o

t

J" coswtdt t.

+2J, t-

2 16 f00dt

+ 10 7rw

2

=

447

THE THREE-DIMENSIONAL. SONIC FLOW

Utilizing Fresnel's fortnulas (10.6.7) and integrating by parts, we obtain exp

T2 011171d"

T

=21 cxp(i"T) [f (1-i)

2 7.ro=

Pxp(

`

2

IIIJII

-10

4:

u2JduJ

tw expTol+

T)dr

+ iw J exp (t2 =27ri+

27-

vfT

- tw

I

exp (t2 T)

vfT-1

(1-i)[ 2 exp(t-xo)-iwJrexp 2TdTr 7x=o

`\'2

o

We deduce therefore

-'.w [

ri(x0)=(1+i

1

yrr-o- x

p

o

iw 2x o1 2

ex

p iw ) d T 1

of

2T sfJ

'

(10.6.20)

i.e. just (10.5.13).

10.6.4

Other Forms of the Kernel

From (2.4.13) and (2.4.35) it results the following representation of the component w of the velocity w(X ,

, z) = 5(`) JJ f(. rt)H(xo)6(yo) d d i+

t f

.{.i-

(10.6.21)

n

Jf

where 2

11(x0. yo.

a'2 J

exp

2 (Ir _ yQ+z2/J T

LT (10.6.22)

One may demonstrate that this integral is convergent. Denoting r = iyol and performing the change of variable r --+ A:

7.

T - - = 2A

dT

-r

dA (10.6.23)

77 7

448

THE UNSTEADY FLOW

one obtains

n(xo,yo,z) =

8 J-.

d.1,

+r

(10.6.24)

where

X=2(TO-xo).

(10.6.25)

Observing that

02 _1 1_z29+za -2 r

Z2

(

r2) th

r2 5r2

we deduce that the line r = 0 is singular. Eliminating a vicinity of this line we obtain lim

82

_

10

r Or

O 0z2

whence

n(xo, yo, z) =:-to lim n(xo, ?b, z) _ 2

fA

(1u.o.lo)

eWA 00pt2+r2)3/2da.

This kernel was obtained in (10.2.36) as a limit of the subsonic kernel and in (10.4.48) as a limit of the supersonic kernel (Ueda and Dowell [10.80]). This fact proves that the oscillatory perturbation is continuous to the passage of the sonic barrier. f0, the first term from (10.6.17) Passing to the limit, when z vanishes because 6(f0) = 0.

Chapter 11 The Theory of Slender Bodies

11.1 11.1.1

The Linear Equations and Their Fundamental Solutions The Boundary Condition. The Linear Equations

In this chapter we study the aerodynamics in the presence of slender bodies (fig. 11.1.1). The axis of the body is considered the Ox axis

and the Oz axis lies in the plane determined by the velocity of the unperturbed stream V. and by the axis of the body. We denote by a the angle of attack of the stream and we assume that a = e, where e characterizes the thickness of the body. We employ the cylindrical coordinates x, r and 0 which are related to the cartesian coordinates x, y, z by the formulas

x=x,y=rcosO,z=rsinO (11.1.1)

xER,rE (0,oo),0E (0,21r).

Fig. 11.1.1.

The equation of the body has the form r = h(x,0) = eh(x,0).

(11.1.2)

450

THE THI3OKY OF SIZNDER BODIES

Denoting by i, j, k the versors of the Ox, Oy and Oz axes and i,., ie (fig. 11.1.2), we shall have the following formulas

i,. = jcos0+ksin8

j = ircos9--ipsin0 (11.1.3)

ie=--jsind+kcos8

k=i,.sin8+iscosO.

Fig. 11.1.2.

The velocity of the unperturbed stream is

V,, - U,,,(i cosa + ksina) = UU(i + ak) + 0(a2).

(11.1.4)

Denoting by

V1 = Uvv,

P1 = Poc + P-U«,P,

P1 = Poo(1 + P)

(11.1.5)

the fields which characterize the perturbed flow and using the cylindrical coordinates

V=Ui+Vi,.+Wi$

(11.1.6)

v=Ui+I ,. +Wie, we deduce

U=1+u, V =asin8+v, W =acoa8+w.

(11.1.7)

THE LINEAR EQUATIONS AND THEIR FUNDAMENTAL SOLUTIONS

451

Obviously, the perturbed flow will be steady because the conditions which determines it do not vary time. On the boundary we shall impose the condition

V grad F = 0,

(11.1.8)

where F = Eh(x, 0) - r. Taking into account that we have

gradF=

5s+

Vii.+Tiei

(11.1.9)

from (11.1.8) we deduce the condition

(1 + u)F as + (a cos 0 + w) r L = a sin 0 + v,

(11.1.10)

which must be satisfied when r = h. Comparing the orders of magnitude we deduce:

v(x, h, 0) = ev(x, h, 0)

.

(11.1.11)

We assume that this structure is valid everywhere in the fluid. We have therefore

v(x,r,9)=e'v(x,r,0).

(11.1.12)

From (11.1.10) and (11.1.11) we deduce the condition

v(x,r,0) +asin0 = hz(x,0),

(11.1.13)

which will be imposed for r = h. In fact, this condition must be imposed

for r = 0, but here v is not defined. In cylindrical coordinates the equations of motion are (1.11]: AP [Up.. + Vp. + (W/r)pg)+ +(1 + ryM2p)(U,+Vr + (1/r)V + (1/r)We] = o

(11.1.14}

(1+p)(UU=+VU.+(W/r)Ue]+p,, = 0 (I + p)[UV + vv. + (W/r)Ve - (W2/r)] +p. = 0

(11.1.15)

(1 + p)[U11=+VW.+(W/r)We+VW/r]+(1/r)pe =0,

(11.1.17)

(11.1.16)

where U, V, W will be replaced by (11.1.7), and P. =00/8x'... With the reasonings from 2.1, the equation (11.1.16) gives

p(x,r,9)=E (x,r,0), v=+pr=0,

(11.1.18)

the equation (11.1.17) w(x, r, 9) = sw(x, r, 9),

rwz + pe = 0 ,

(11.1.19)

452

THE THEORY OF SLENDER BODIES

and the equation (11.1.15)

7t(r,t,B)=SYf{e,r,B),

71r+p. =0.

(11.1.20)

Keeping the terms having the order of E, from (11.1.14) we deduce

AM2pr+u,.+vr+(1/r)v+(1/r)wo

0.

(11.1.21)

One observes that in the linearized system o does not intervene. The system coincides with the system for cr = 0. It. is the system (2.1.32) in cylindrical coordinate:.

One may also obtain the equation of the potential. Indeed. from (11.1.18) -- (11.1.20) it results:

yr - ur = 0, rw;r - uy = 0.

P = --u.

(11.1.22)

The last two equations prove the existence of the function V(:r, r, 0), a.I. 7r. _ (j.,

V = 4'r i

to = (1/r);pe ,

(11.1.23)

and the equation (11.1.21) gives (1 - Al `)

11.1.2

a"-

10

t1:r.2 +

r Or

r

a(pl

C Or

1

+

j92 = 0.

(11.1.24)

Fundamental Solutions

We shall utilize for the solution of the system (2.3.4) the intrinsic form (2.3.8), (2.3.12) which will be written in cylindrical coordinates. From the equality

ft: + f29 + f3k = f1 + frtr + fOZO,

(11.1.25)

taking (11.1.3) into account , we deduce

f2 = frcus0 - fosin0,

fr = f2cas0+ f3 sill 0 (11.1.26)

fi = fr S1118 - fB COS B,

f0 = -12 sin 04- f 1 cos 0 .

Writing the inner product in cylindrical coordinates, in the subsonic ease and Taking (11.1.26) into ac( -ount, from (2.3.4) it results

r) _

-

I

+fri0) ` ) 1

(11.1.27) 1

THE LINEAR EQUATIONS AND THEIR FUNDAMENTAL SOLUTIONS

453

where

R, =

x2 +#2r,2,

(11.1.213)

From (2.3.13) we deduce

_

1

47r

_

/

dx

fr J x

(x2 + #2x2)3/2

2r

f=

R, -

(11.1.29) 1

4:s

fr

Rl -

fr 1 + x

r(R,)

Taking into account the expression of the distribution 6(x) in cylindrical coordinates [A.7J, [A.10). we obtain yr

=

f'.

L. a

H(x)d(r) 21rr

+ 4r, 8r

fr a

1

x

1

(11.1.30)

R, } 4;r 8r r

In the supersonic flow we have

P(x,r) _ -i-. (fr_8x +fr where

E(x. r) _

i

l

J E(x ,r),

(11.1.31)

H( x-kr) x- -k- r

(11.1.32)

Since

a f" H(r-'-r) d r =H (x- kr) 0 8r oo T- k r

dT

=

,r r- - k r

2:E (x,r),

r

from (2.3.13) it results

p(x, r) = 1 (f= - x f,.) E(x, r) (f..

19

+ r fr - r fr5T) E.

(11.1.33) (11.1.34)

Taking into account the formula (2.3.35), p and Vr will be: x Vr

_

- fr

H(x - A-r) r

H(x)b(r) 27rr

x

81

+

27r

h

8

+

1kr, x r2

fr-

(11.1.35)

1

fr 5T x -k r These formulas show that the perturbation propagates only in the interior of the cone x = kr. -r

454

THE THEORY 0FSLENDER BODES

11.2

The Slender Body in a Subsonic Stream

11.2.1

The Solution of the Problem

In the case of slender bodies of revolution, the equation (11.1.2) has the form

r=h(r). 0 < x < 1.

(11.2.1)

Considering that the unperturbed flow has the angle of attack z in the xOz plane, we deduce that. this plane will be the plane of symmetry of the flow. We shall replace the body with a distribution of forces defined on 10.11

with f2 = 0. From (11.1 26) we deduce fr = f sin0 (we

denoted f3 = f). Taking (11.1.27) (11.1.30) into account, we deduce that perturbation produced by this distribution may be represented by means of the formulas

p(.r, r, 0) .

" { f,

Rd

Jx + f (l;) Sin A

J

47r-

(11.2.2)

W(arrtd)=

xj 1

'r(X, r, O) =d(l)

R

sin 0

d

4irr

sin 0

,-TI

'

x0

(1+ R)f(E)d

Jhit(0

0

)

(11.2.3)

sill 0 y rl (1 + x0) d 4 it 8r Jo r R where

R= 4+Ir'r2.

.T.ox-

Imposing the boundary condition (11.1.13), we notice that d(h) = 0 because h does not vanish for 0 < x < 1. Separating the variables we obtain the following integral equations:

f

f

0(I)d

41rh'(x)

(11.2.4)

(11.2.5)

for O < x < 1 and r = h(x). In order to solve the first equation we shall utilize the identity

v fir

1

R)

1

8 ((xol _ 1 5-x

iI_

((xo R} ra(RI

(11.2.6)

455

THE SLENDER BODY IN A SUBSONIC STREAM

Integrating by parts, we obtain: x - 1 47rh(x)h'(x) = ft(l)

x -1) + Q h (11.2.7)

-fi (0)

r2

+

= - [fl(1) + fr(o)] [1 + 0(h2)]

h

where

_ t I=f'

xo

h

xo+

fi(:;)dt.

(11.2.8)

For calculating the principal part of this integral, we notice that we have xo

+

lio' [i + 0(h2)]

h

(11.2.9)

excepting the vicinity of the point £ = x where xp = 0. In [1.1] one utilizes this approximation on the entire interval (0, 1). Correctly, the integral I must be written as follows

(/

I=

+ / 7+1

J

(11.2.10)

In the first and last integrals we may utilize the approximation (11.2.9).

For ,i small enough, in the second integral one may replace fl (t) by f ' (x). We obtain therefore

I = lro

[ r:o

fi (t)d t -

J:+n fi (t)d C]

[1 + 0(h2)] +

1

+ f (x) lim

+''

,j- .0

q

dt

p

+h

We calculate the last integral with the substitution x - = u and we observe that it vanishes. Hence,

I = [fl (x) - fi(0) - ft(1) + fl(x)J [1 + 0(h2)] .

(11.2.11)

Neglecting the terms of order h2 with respect to 1, from (11.2.8) and (11.2.11) we obtain

fi(x) = -27rh(x)h'(x) = -S'(x),

(11.2.12)

with S(x) = 7rh2(x) the area of the crass section a of the body in the point having the abscissa x.

456

THE THEORY OF SLENDER BODIES

Deriving in (11.2.5), we find

j' (1 + R)

f(E)d F + f32h2

"' f J. 1T4

-4iah2 .

(11.2.13)

Calculating the integrals by means of the formula (11.2.10), we have

J1(1+ O)f(:)dF=21 f(t)dF[1+0(h2)] Hence, neglecting 0(h2) with respect to 1, we obtain

f(e)d t = -2Tah2 = -2nS(x) 10

or, deriving,

f (x) = -2aS'(x) .

(11.2.14)

For the profile with zero angle of attack (a = 0) we deduce f = 0 whence

N(x,r) _ -4 0,

xo+

r

dF.

(11.2.15)

This representation of the potential is known in the literature [1.11, (1.38]. We have also, from (11.2.2) (for r # 0), (11.2.16)

P(x, I') = -4r, Vr = SFr

11.2.2

The Calculus of Lift and Moment Coefficients

We shall calculate at first the pressure for r = h. It can be obtained from (11.2.2), (11.2.12) and (11.214). Utilizing the identity (11.2.6) and the calculations (11.2.8) - (11.2.11), we deduce

jf(0

.

d

(R)

(11.2.17)

[1 + O(e2)]

.

Hence, sin 0

P(x, h(x), e) = pi (x, h(x)) - a Wh(x) S`(x),

(11.2.18)

157

THE SLENDER BODY IN A SUBSONIC STREAM

where

r fl (E)' (1?)

P1 (X. h(.r)) _ - 1

J

1r,,iid E

(11.2.19)

1f1 4r

:rOS'(E)d E (%2+132,12)3/'2

Taking into account that fr = O(V2), we calculate the principal part of pl as follows: I

4;rP1(x, h(x))

W) r_I,d

fl (S)3

_

11(1)

(x11-)

f1(0) x'-+f32h2

-j3 h=

9 r,

+J fl(E) I-E

S'(1) + S'(0)

+f

d

z*F7

+ fi(x) lim

r.

1-x

T

J0 x -

-

01 S"(r)

Er

>

r0+th`

+O(Fd)-

- S"(E)d

x-E

x

t

J

S"(x.)

E--x

1

+ S(x) In

1-c r.

+ O(c4) ,

the principal part which was written being O(_2). The lift coefficient may be calculated with the formula

cL=

(11.2.21)

- As

where S is the surface of the body, and n, the outer normal and with the notation F = r - h(x), given by the formula grad F - -h'i + i, (11.2.22)

n =

Egrad FI

V1-1+ h'2(a)

Taking into account the element of area on the surface S, and the relation (11.2.13), we obtain r1

A

CL =- / / ,r p(x, h(x), O)h(x) sin 0d rd 0 = o

=(I Es(1)-s(0)j.

(11.2.23)

458

THE THEORY OF SLENDER BODIES

For the drag coefficient one obtains

CD =- Jjpn id a =

jj(x, 1

h(x), O)h(x)W (x)d xd 0 = (11.2.24)

pl(x, h(x))S'(x)d x = O(E4) .

The drag coefficient does not depend on a. At last, the moment coefficients are the scalar components of the product

-2 /f xxpnda, S

where x = xi + h(x)ir. Taking into account (11.2.2) and (11.2.22), we obtain

cr =0, cs=0, (x + hh')h(x)p(x, h(x), 0) sin Od x d 0 =

f1I

-2a / o

1

Lx +'

(11.2.25)

S'(x)d x = O(E3) . J

Obviously, for a = 0 one obtains cy = 0. Neglecting the term (S')2, for the moment coefficient c, we obtain the approximate value

ri

c, = -2a J xS'(x)d x = 2aV, 0

b ecause S(1) = 0 and the term

of the body.

11.3 11.3.1

j

S(x)dx represents the volume V

The Thin Body in a Supersonic Stream The General Solution

In this subsection we consider the same problem like in the previous subsection, but now the unperturbed (free) flow is supersonic (M > 1).

459

THE THIN BODY IN A SUPERSONIC STREAM

titre consider again that in this case the xOz plane is a symmetry plane, such that we have f; = 0 whence fr = f sin 0. The fundamental solution is (11.1.31), (11.1.34). and the corresponding potential is (11.1.33). For a continuous superposition of forces on the segment [1).11. the perturbation will be given by 1

+f w sin ©J E(.,ro, r)d

P(x. r, 0) = - T f p(x, r, 0) =

[f=w

2-r J0

- rl f (5) sin 0] E(r(j, r)d i

X

t'r(X, r, 0) = (5(1') in0

2'r

JU

.E(xa)d ,r

+

sin0

ff U

1

err f0 roI

0Y

.

l - J)F(xor)d r

9r,

(11.3.1)

The derivatives may interchange with the integrals and. taking (A.3.15) into account, we have for example

f 1 f W E(xo, r)d t= ax f

_

a

x-kr

H(.r-kr) o

f x

1

f

E(xo, r)(1 _

FL-r H(x-k'')r3

4.

`I

f (4) ==

zo

k-12-7-11

Hence, the solution (11.3.1) maybe written as follows

_

2r TX o =

r-kr

1

r

_ yr

x-kr

8

1

P

x

sin 0_ 8 d42z Or x2 - k r(4)

dE-

in

xo - k 2 rI

sin 0 2rrr

f()cI+ 2 r

b(r)

+siI 0 27tr

1

Of

sin 0

8

f V) dC xo - k2r2

o

r-kr xQ

2,rr Jo 1

r

r-kr

fx (50)

ro

r-

d

jr-kr x xo

xo-k'r

k2r2

d (11.3.2)

460

THE THEORY OF SLENDER BODIES

valid for x > kr and

p=W=vr=0 forx
(11.3.3)

Imposing the boundary condition (11.1.13) we shall notice again that

6(h) = 0, because h # 0. Separating the variables we deduce the following integral equations

8r 0 1 Cr

r

x0 _

Vd t)

rah

= 27rh'(x)

x-kr a x0 ar} Jo xo - k ref (t)dtIrah = -27rah(x) .

(11.3.4)

(11.3.5)

In order to put into evidence the principal part of the integrals for r = h = O(e), we must notice that we cannot derive directly the integrals, because the integrands become infinite for { = r - kr. For avoiding this situation, we shall perform the change of variable 4 -- u:

x - a=krcoshu.

(11.3.6)

One obtains a of=-kr

a'

8

fr (S)

xo - k r

d = 8r

x r

x -k r

x

f:(0)

fx(0)

x- k r

,

-kJ

f. (x

UCh fi(x - kr cosh u)cosh u d u

o

_

_r

1

r f'(0) [1 + O(ff)]

- krcashu)du =

rx-kr 1 0

-r

0

xo

' xo - k r f (t)d -kr

f=(t) [1 + o(e2)] d. (11.3.7)

Neglecting O(e) with respect to 1, we deduce that the principal part of the integral from (11.3.4) is - f=(x)/r. Hence, from (11.3.4) we deduce

ff(x) = -2whh' = -S(x),

(11.3.8)

S(x) giving the area of the cross section of the body in the point x.

TILE THIN BODY IN A SUPERSONIC STREAM

461

Similarly we obtain

0

r-Ar

xO

Or Jo

t-tr

1

r

X0

r \/x - 1 1

r2

z-kr

0

1f(01d --1

x -

0

+

f(0)

:c2

f(c do :c3 - k2r2

r

ra

u

f'(Od (11.3.9)

Neglecting O(r-2) with respect to 1 we find at first 1

(7

z-kr

xp

.r

f,-Ir

f

f (E)d

-r-kr rx`t

-rI 1

x-kr

f ()d + r

f (0)

fz_kr

f'(e)d

r1-jZkr

Ja

f(4)d4.

The equation (11.3.5) becomes S kr

kr f (x - kr) + 0

f

-2-xnrh.

(11.3.10)

where we put r = h. In the left hand part of the equality, we expand into series for obtaining the principal part. We have r kr f (:x) + 0(r2) + kr f (x) + 0(r2) = -2 rorh . f 0

Hence,

s

J

f

--2riS(x)

whence

f (x) = -2aS'(x),

(11.3.11)

the analogy with the subsonic case being obvious.

11.3.2

The Pressure on the Body. The Lift and Moment Coefficients

In order to put into evidence the principal part and the expression of the pressure, we notice that utilizing the change of variable (11.3.6), we

TILE THEORY OF SLENDER BODIES,

462

deduce x-kr

8

o -krx

f=(t)

Ox

x -k r r-kr

+f=(x)

xo-k r

o

f=(a)

o-k r

0

x +f:(x)arccosh kr

__

rs

-

x

x - f., t

to

2s

= fs(x)ln kr

xo -kr

Jo

f'(C) - fr(x)d4+

df

f

ff(t)d t

r=-kr

-k r

+

f=(0)

+

f1(0)

d t=

-J

+

Jo

r f.'W - f.'W d t +

x-

d t+fx(x) In

x + x- k r e kr

fi(x) - fY(0dt.

x-t

Taking into account the calculation performed at (11.3.7), we deduce

p(x, h, 0) = pi (x, h) - a sin 0 S'(x),

(11.3.12)

where

pi (x, h) = -

S"(x) 21r

kh _

2x

1

S"(x) - S"(4)

27r fo

X-

d

(11.3.13)

The lift, drag and moment coefficients may be calculated with the formulas (11.2.23) - (11.2.25). We have therefore, like in the subsonic case,

CL =a [S(1) - S(0)1 = 0(e3)

CD =Ji p,(x,h)S'(x)dx=0(e4) 4

(11.3.14)

Gr=az=0, cy = - Cr f ' 0

Ix +

2(x) J

S'(x)d x = O(s3) .

THE, THIN BODY IN A SUPERSONIC S1'IREAAI

11.3.3

463

The wing at zero angle of attack

For a = 0, we deduce f = 0 whence P('r., r) =

9(r,

r)

t'r(x, 7.)

a

1

S'()

/'x-rrr

2;, Tx Jo

zo ` J. r d

=-kr S

-o

xo - kr'

1 of r-kr

2

j

S (e)d o

d

S'(S)dS

.4-k r'

x-kr

1

2rr J0

Sit ( )d k2r2 (11.3.15)

This is the solution between the wave from the louring edge and the wave from the trailing edge i.e. in the region denoted by II, where a > kr (fig. 11.3.1). In I (:c < kr) p = yr = 0, and in IN the solution is

p(x, r) _

1

oz-kr

2r,

2irr

(11.3.16)

z-kr

1

Vr(x,I-) -- -

S"()d t xo-k-r'2

0

dc. x _ 13-1

1A w 1

Fig. 11.:3.1.

This explicit manner of representing the solution is also valid in the general case.

11.3.4

Applications

At first we shall consider the thin body having the shape of a cone (fig.

11.3.2) of equation r = ex, 0 < t < 1. We deduce S = 1re2x2

THE THEORY OF SLENDER BODIES

464

whence 2 CL = JrE at cy = -

21u E2 3

ti

,

CD = -7rE' ln(k/2).

(11.3.17)

Fig. 11.3.3.

Fig. 11.3.2.

For cy and CD we have retained only the principal part. If the angle of attack is zero, then in II we have

d _d C = 62 arccosh j _

z kr

p(x, r) = E2

E2 In

yr (x, r) = -

xo - k"jr

Jo

x+ x2-k r x-kr

EZ

r

(11.3.18)

kr

o

x0

xkr-d t _ -8

2

:r - k r r

We notice that on the radius vectors r = cx, c < 1/k, the pressure and the velocity are constants. The flow is conical.

The second example

the double cone from figure

is

11.3.3

Since Ex,

h(x) =

0 < x < 1/2

(11.3.19)

r(1-x), 1/2<x<1,

it results that S' has a discontinuity in the point x = 1/2. We suggest to the reader to establish the formulas of derivation for fW dta rx-kr

xo-k r and to write the solution.

' arJ10

xo -k'r

d

(11.3.20)

Appendix A

Fourier Transform and Notions of the Theory of Distributions

A.1

The Fourier Transform of Functions

The following definitions will be given in R3. Their expressions in R1, R2 or RI will be easily deduced. The Fourier transform of a function f : R3 --+ R is denoted .F[f], or f and it is defined by the formula (A.1.1)

F[f](a) = f(a) = j3 where

x = (x, y, z),

a = (al, a2, 03) (A.1.2)

a.x=alx+a2y+a3z,

dx =dxdydz. The operation F is called the Fourier transform. We notice that i f (x) is absolutely integrable on R3, he. if

3 (f )d x < oo,

(A.1.3)

then the Fourier transform exists. Moreover, if f satisfies certain conditions of regularity, for example, if f E C0(R3), or if f is piecewise smooth with respect to every variable [A.10], then f may be obtained from f using the following inversion formula

(2;r)3 f (x) _ (2-,,)3-F-1 [j(x) f

a= (A.1.4)

_ F[fl(-x) _ [f (-a)J(x) , where ci a = d a1 d 02 d a3. The last two expressions show that the inverse of a Fourier transform may be obtained by a direct transformation.

466

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

One proves (Lebesque's theorem [A.9], (A.51) that j tends to zero when ICkI -+ .x}.

The condition (A.1.3) is restrictive enough. For example, the function f = 1 does not verify this condition. The theory of distributions allows to enlarge the class of the functions which admit a Fourier transform.

Notions of the Theory of Distributions A.2 The Spaces D and S The theory of distributions relies on the notion of space of the test functions. The space V = D(R3) consists of the set of functions W(x) infinitely derivable, with compact support in R3. We say that a sequence of functions {V,} from D, converges to a function S' E D, if for every multi-index k = (k1, k2, k3), k1, k2, k3 being non-negative integers, we have: DkW11 rya Dkcp,

(A.2.1)

where, with the notation IkI = k1 + k2 + k3, we have put Dk

-0jkj

= --,-

k,

(A.2.2)

The set D endowed with this convergence law is a linear space-

Ale may we that the Fourier transform of a function from D is not a function from D. Indeed, the Fourier transform is an analytical function [A.12), but the support of an analytical function which cannot be compact. There exists another space which is invariant to the Fourier transform and plays a basic role in the definition of the Fourier transform

of a distribution. This is the space S = S(R3), i.e. the space of the rapidly decreasing test functions. We call rapidly decreasing function an infinitely derivable function V(z) in R3 which, together with its derivatives decreases when Ixl - oo faster then every power of Exl`t. This means that (A.2.3) IxkD'v(x)i < CL-j,

for every nmlti-indices k and 1. We denoted xk = xklyk'2zka. The convergence in S is defined in the following manner: the sequence of

467

DISTRIBUTIONS

functions {V } from S converges to the function cp E S if for every multi-indices k and 1 , we have: X,

xkD'pn(X)

xkDrv(x).

(A.2.4)

The set of the rapidly decreasing functions, endowed with this convergence law. is a linear space. The convergence in D implies the conver-

gence in S. We have D C S. but S does not coincide with D; For example. the function exp. (-x2) E S, but

A.3

D.

Distributions

The distributions are linear and continuous functionals on D or on

S. This means that a distribution f detennines the correspondence between a test function V and a number denoted by (f, q) and we have:

(A.3.1) (f,Anpi +A 2) = an (f,Sit) +A2(f,p2) , for every two real or complex numbers an, A2 and that if P>s I V in D (or S), then

(f, °Pn) - (f, (P)

.

(A.3.2)

One denotes by D' the set of the distributions defined on D and by S' the set of the distributions defined on S. S' is the space of the temperate distributions. Obviously, S' C D'.

We say that the distribution f is equal to zero in the domain f2 (it is denoted by f = 0) if (f, gyp) = 0 for every p from D (or S) with the support in 11. The distributions fn and f2 are equal in n if fl - f2 = 0 in 11. i.e. if (f1, 0 = (f2. (P)

,

(A.3.3)

for every v E D.

We call the support of the distribution f and we denote it by supp f the set of the points which have a vicinity where j is not equal to zero. Every locally integrable function f (x) defines a distribution by means of the functional

U. P) = j f (x)4%(x)d x .

(A.3.4)

A distribution f is it regular function-type distribution if it may have the form (A.3.4). Every other distribution is singular. The best known

468

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

example of singular distribution is the distribution of Dirac. The distribution of Dirac with the support in 4 is denoted by bo(x) or b(x and is defined by the functional (b(x -

V, (x)) = rW .

(A.3.5)

Formally, it may be written as follows

J

v(x)6(x - 01 x = pW .

(A.3.6)

If rn(x) is an infuiitely derivable function, then mv is infinitely the distribution f and the function m., denoted by tnf, by means of the formula

derivable and we may define the product

rnv)

(inf, 4,)

(A.3.7)

It results

m(x)b(x - ) =

(A-3.8)

whence,

ni(x)b(x) = 0,

if rn(0) = 0.

(A.3.9)

For the existence (A.3.8) the continuity of the function is sufficient in in t;.

We shall consider the three-factor c and the homothety cx = c3z). If h(x) is a locally integrable function, then h(cz) is also locally integrable, such that the distribution corresponding to h(ex) is defined by the functional

_ (cix.

(h((x), 4"(x))

=f

h.(cx)p(x)d x

Setting cm = and observing that the integration limits interchange when c; is negative, we deduce, with notation lei = (clc-2cy) IcC (h(cz), 5'(x)) - j

h(E),p(k/c)d

_ (h(x). 4p(x/c))

For o distribution f (x), one defines the distribution f (cx) by means of the formula ICI (f (cx) s'(x)) = (f (x), cp(xl (:))

.

(A.3.10)

The definition may be extended to a non-singular linear transformation. As a particular case, it results: Ic16(cI x. C 2Y, car) = b(x ah z) .

(A.3.11)

469

DISTRIBUTIONS

f

The derivative of the distribution determined by the formula

is denoted by

f'

and is (A.3.12)

(f', V) _ - (f, ) . Defining the function of Heaviside H(x) by the formula

x<0 H{r.) = 10, 1,

(A.3.13)

x > 0,

it results 00

(x)dx=5P(0)

i

0

Taking into account (A.3.5) we deduce

H'(x) = 6(x).

(A.3.14)

Let us establish now the formula

(m(x)H(x)' = m(0)6(x) + in'(x)H(x) for a function E C'(R). We have ((mH)'. y,) =

= m(0M0) + f

- (mH. vp')

x

(A.3.15)

r

x=

m'(x)cp(x)d x = (m(0)6(x) + m'(r)H(x), v')

.

0

On the basis of (A.3.8) and (A.3.15) we deduce that the solution of the differential equation

Lu = 6(x),

(A.3.16)

where Lu = u("')(x) + O, (X)u('"-I)(x) + ... + am(x)u(x). is

u(x) = H(x)v(x),

(A.3.17)

where v verifies the equation Lv = 0 and the conditions v(O) = v'(0) ...

= v,(m-2)(0) =

1

0,

We call the direct product of the distributions f(x) E D'(R") and g(y) E D'(Rm) the distribution f (x) g(y) E D'(R"+') defined by the functional

(f (x) g(y), r'(x, y)) = (f (x), (g(y), W(x, y)))

,

(A.3.18)

470

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

where V E D(R"+'"). The definition is valid also and when D' is replaced by S. The direct product is commutative, i.e. f (x) g(y) _ = g(y) f (x). As an application, we shall prove that in R2, we have (A.3.19)

b(x) = a(x) - a(y) Indeed, applying the definition, we have: (OX), AX, y)) = 4'(0, 0)

.

But we have also, (a(x) a(y), Ox, y)) = (a(x), (a(y), (x, y))) _

= (6(x),;?(x,0)) = V(0,0)

A.4 The Convolution. Fundamental Solutions For two functions mi(x) and m2(x) absolutely integrable in R3, we define the convolution m(x) = (ini * in2)(z) by the formula (ml * MAX) =

=1

3

J pa

"11)mx - t)d : =

m( m2(x - t)dt = (in2 * ni)(x)

The function m(x) is absolutely integrable, such that it generates a function-type distribution (A.3.4). Writing explicitly that functional, we are determined (A.51 to define the convolution 0= fi * f2, of two

distributions fi and f2 by means of the formula

(fi * f2, ') = (fi (x) f2(O, V(x + f)) ,

(A.4.1)

fi(x) f2(4) representing the direct product of the distributions fi and f2. The convolution (A.4.1) exists if, for example, one of the distributions have a compact support (A.5j. The equality f j *f2 = f2 *fi results from the commutativity of the direct product. As an application, we shall calculate b * f. We have

(6 * f, P) = (6(x) . f W,'?(x + 0) = (f (0, (6(x),'P(x + i)))

_(fW"PW)

THE CONVOLUTION. FUNDAMENTAL SOLUTIONS

Hence,

a*f=f*6=f.

471

(A.4.2)

We shall establish now the formula

D(fi*f2)=Df,*f2=fi*Df2,

(A.4.3)

the operator D being defined in (A.2.2). We have (D(f1 * f2), Sp) = (-1)Ikl (fh * ff, DW) _

= (_ 1)lkl (fi (x) - ME), Dpp(z E)) _ (-1)Ikl (f2(E), (fi(x), Dip(x + i))) _ _ (f2(E), (Dfh(x), D& + F))) = (Dfi * f2, sp) , and the result proves the first equality from (A.4.3). The second equality results from the commutativity of the convolution. We shall consider now the linear differential operator of order m, M

L = E a*Dk , Ikl-O

where ak are constant. We call fundamental solution of this operator, the distribution t E D' which verify the equation

LE = 6(c).

(A.4.4)

Obviously, the fundamental solution is defined with the approximation

of an arbitrary solution of the homogeneous equation. We have the following theor_mr. If f e D' and there exists in ?Y the convolution f s E, then the solution of the equation

Lu = f

(A.4.5)

exists in D', is given by the formula u = f *.C

(A.4.6)

and it is unique in the set of distributions from D', for which there exists the convolution (A.4.6). Indeed, utilizing (A.4.2) and (A.4.3), we deduce

Lu=L(f*E)

a1Dk(f*E)=f*LE= f$6= f. lkI-O

472

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

In order to demonstrate the uniqueness, we denote by u another so. lution. Obviously, no = u - u satisfies the homogeneous equation Luo = 0, such that we have:

uo=uo*6=uo*Le=L(uo*e)=(Luo)*e=0.

A.5 The Fourier Transform of the Functions from S Since the functions from S are absolutely integrable, they have Fourier transforms F[ pj defined by (A.1.1) and these are bounded continuous functions on R3. Integrating by parts and taking into account (A.2.3), we deduce:

-F(VACt) = (-ial)FIVJ(a).

(A.5.1)

On the basis of the same condition, we may derive inside the integral, such that it results

(O/Oai).('PJ(a) = .F(ixtipj(a).

(A.5.2)

By recurrence, it results:

.F(Dk(pJ(a) = (-(a)k.F(V](a),

f((ix)'4 j(a),

(A.5.3) (A.5.4)

for every multi - indices k and L. The last relation show that F(VI is infinitely derivable.

We shall prove in the following F(cpJ E S. To this aim we replace cp with (ix)rcp tin (A.5.3) and taking into account (A.5.4), we deduce

F(Dk((ix)`p)J(a) = (-ia)k.F[(zx)l ,](a) = = (-i)IkjakDI,F[Wj(a) whence

]akD'F('j(a)I <_ fx$ IDk(xt4o)IdZ < 00, i.e. Y[w] satisfies (A.2.3).

Taking into account that the Fourier transform F(spj is integrable and continuously differentiable, we deduce that cp may be obtained from cp sing a formula like (A.1.4).

THE FOURICR TRANSFORM OF THE TEMPERATE DISTRIBUTIONS

473

A.6 The Fourier Transform of the Temperate Distributions The Fourier transform of the distribution and it is defined by formula

f = .F[ f l

f E S' is denoted by

(f, 0) = (f. y i) , for every V E S. Since

(A.6.1)

E S. it results [A.121 that the Fourier

transform of a temperate distribution is also it temperate distribution.

For f E S' we define the inverse Fourier transform F--I, by the formula(27r)'-F-' [f (x)1 = -F[f (-x)] .

(A.6.2)

We shall prove that

f

j:-- I [.[fl]

(A.6.3)

.

Indeed, taking (A.3.10) into account, we deduce for c = -1 (27r)3 (,F-1[.[f]h

(.F[.F[f](--c )l,

F-'

(-a)) = (2r)3 (27r)3

(f,.Fp,-' [.,III)

_ (f )

Similarly we prove the second equality. We shall establish now the operations (A.5.3) and (A.5.4) for temperate distributions. We have:

(F[f=], } = (f=, ) _ - (f, a,.F[pl) =

[iaip]) _ (F[f].-iniy,) _ and we deduce, whence

(_iaiJ,)

.F[f] =-iaif,

(A.6.4)

T[grad f) = -i a f , F[div f ] = -i a f (A.6.5)

.F[rotfl=-iax f. From (A.6.4) we obtain by recurrence a similar formula to (A.5.3). Analogously, we have:

(

0 _771fl, V(a)

Ylfb

f..F

an.,

474

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

(f, i x.F[d) = (i xf, Y[w)) = (f [i xf ], io) hence, (A.6.6) .F[i:rf) and, by recurrence, a similar formula to (A.5.4). From the definition (A.6.2), it results that relations similar to (A.5.3) and (A.5.4) are available for the inverse transformation too. Hence,

f-1

(i x)" f (x)

ia)!f](x) = D'f(x).

(A.6.7)

(A.6.8)

We shall frequently utilize relations of the form

.F-t[--t«tf1 = f:, F-t[cto2f) _ -f=y.

(A.6.9)

We shall prove in the following the formula (A.6.10) 17-t(f(ca))(x) = ICIf 1 cl c representing a three-factor. Performing the change of variable a =

= cf3, we have for a function

lip\alJ (x)=,l

(

cxd A = lclw(cx).

lei

For a distribution f we shall utilize the formula (A.3.10). We have lei (F-t [f(Ca)1 P) (f(a),F-tso(a/c))

lei (f(ca), -F- 1,P)

_

(f(x), 0(cx)) = (f (x/c), w) whence (A.6.10).

For the direct product f (x - g(y)) we have: (A.6.11) F[f(x) - 9(y)] = Y[f)(a) - F[9)(0) , and for the convolution, if one of the distributions has a compact sup-

port, we deduce .Fff * 9) _ F[f).F[9] .

From (A.6.11) it results f-I

[f (a) 4(/3)) = f (x) . g(y) .

(A.6.12)

(A.6.13)

THE CALCULUS OF SOME INVERSE FOURIER TRANSFORMS

475

A.7 The Calculus of Some Inverse Fburier TSransforms We shall calculate at first the Fourier transform of the distribution of Dirac. For cp E S we have

(716(x - 4)], Sp) _ (a(z - ), r) ='P(A) (A.7.1)

(el a-4, SP)

,

Hence

.F[a(x - a; )] = exp (i a ), b(a: - F) _ '-t [exp {i a }j , whence, for

(A.7.2)

= 0,

7A =1,

_ 1 [1] _

(A.7.3)

It results therefore

F[1](a) = (2a)3d(a) .

(A.7.4)

Hence the temperate distribution I has a Fourier transform. Denoting by F2 the Fourier transform in R2 and by f'1 the transformation in R1, we have on the basis of the formula (A.6.13): .Fz 1 [1/i a] _ .F_ [l/i a1](x) '

t [1]{3l)

(A.7.5)

Applying the Fourier transform to the equation

du/dx = b(x),

(A.7.6)

we deduce -i o1i = 1 whence:

U = -X1[1/ialj(x).

(A.7.7)

The equation (A.7.6) has the form (A.3.16) and its solution has the form (A.3.17). It results s=X(x). (A.7.8) From (A.7.2) - (A.7.8) we deduce

T1 [1/i all = -H(x)6(y).

(A.7.9)

For the determination of the fundamental solution of the equation of Laplace we need the following results (demonstrated, for example, in [A.12], §97, [A. 131, §6.6, or [A.71, §5.3). I

= 4x [ ['

n = 3,

(A.7.10)

476

FOURIER TRANSFORM. THEORY OF DISTRIBUTIONS

YV [i'i'_}

n=2,

2 (ln lxl + C),

(A.7.11)

C being a constant determined in [A.13], but without any importance here. Taking into account (A.6.10) with c = (,3,1,1.) we deduce, on the basis of the formulas (A.7.10) and (A.7.11): 1

Jrz 1

_

1

#2aI+a2+a3 - -]

I FP fl202 t

1

+

4n

(A.7.12)

(J +z )

= - 2n (ln x'- + 02y= + C - In (i) ,

2] 2J

i

1

(A.7.13)

,6

FP representing the symbol for the Finite Part (Appendix E), and p, a positive constant. Also, for determining the fundamental solution of the wave equation we need the formulas

f-1 sinalalt

6(at - Iii), 4,rat

lal

T-I sin aloe lt1 H(at - l xI ) I

Ial

1

21r

a t- -- Ixl

n = 2,

(A.7.15)

demonstrated, for example (A.7.14) in [A.5] Chapter 2, §3.4, and (A.7.15) in (A.12) §9.7. We shall prove in the sequel the following formulas 1

_ 1 H(x - k f2 + z2)

1

[-k2a"i + a2 + a3 J F2 1

-k2

2a

x-k

(y

+ a2, = 2k H(x

+ s)

kl>!i).

,

(A.7.16)

(A.7.17)

In order to prove (A.7.16), one considers the following partial differential equation: k2u." - uyy - u=, = 6(x) . (A.7.18)

Applying the Fourier transform, we deduce (-k2a? + az + a3)i = 1, whence, u

[_ k2al2 +a2+a321 1

.

(A.7.19)

It results therefore that the first member from (A.7.16) is just the solution of the equation (A.7.18). We shall determine in a different manner this solution, namely applying the Fourier transform only with

THE CALCULUS OF SOME INVERSE FOURIER TRANSFORMS

477

From (A.7.18) one obtains the

respect to the variables y and equation

(k2d2/dx2 + w2)ta = d(x) ,

(A.7.20)

where w = + , and u represents the two-dimensional transformation. The equation (A.7.20) is like (A.3.16), and its solution has the form (A.3.17). We deduce

u.-

H(x.) sin(wx/k) k

w

Utilizing (A.7.15) one obtains (A.7.16). Acting similarly, in the two-dimensional case, we find:

u= ii

H(x) sin(a2x/k) k

(Y2

t

being the notation for the Fourier transform with respect to the

variable y. Utilizing the formula F1 sinXax11 7r

(A.7.21)

I(Inl>aa ,

0 given for example in (A.1], p.202, we obtain (A.7.17).

In the unsteady aerodynamics we shall meet the following type of formulas

co alc tl

Mat

ct2

47r Ix

1 - cosalaltl U2

- Iml),

(A.7.22)

a t - Ix+

Ix()

2r

= 3,

(at +

(A.7.23)

IxI

n=2. We shall prove these formulas (following an idea suggested by V.Iftiinie) using the results concerning the Cauchy problem for the non-homogeneous wave equation. To this aiin we shall denote:

v(t, a) =

l - cosalalt Ck2

We deduce u.(0, a) = 0,

u((0, a) = 0 ,

irtc(t, a) = a' arsa.lalt = -(i2a2ii(t. a) + (L2.

478

FOURIER TRANSFORM. T'IIEORY OF DISTRIBUTIONS

Applying the operator F-', we deduce: 11(0, x) = 0,

ut(0. x) = 0

utt = a2Aat + u26(x)

.

For determining u we have therefore to solve a Cauchy problem for the non-homogeneous wave equation. The solution of this problem in the three- and hi-dimensional cases is given by Poisson's formula [A.12]. Utilizing this formula, we find (A.7.22) and (A.7.23). At last, replacing x by x. - t in the formulas (A.7.9), (A.7.14), (A.7.15), (A.7.22) and (A.7.23), we obtain in the three-dimensional Case,

with the notation R = F-I

(c - t) + y' + = 5(x - t) - a(t,.z)

Ceinit. I

sin aIctIt einat

f-I

_

I a5(at. - R), )' = TI-rat

(a1

n247rR

1 - c)sajait

iaIt - II(at

(A.7.24)

R)

and in the two-dimensional case, with the notation I#

F-a ei,«t] f_1 Il

I

b(x - t) - (5(71), I

ial

JI

27

(1 - cosa'aHt dolt I

rte

_

H(at - R)

a' '' --V H(at -- 7) 2-r

In

at +

n."2t2 - f2

R

(A.7.25)

In [1.101 we may find direct. demonstrations of the formulas (A.7.25).

A.8

The Fourier Transform in Bounded Domains

In this last part, we return to the Fourier transform of the functions and we give, following Homentcovschi's idea [A.61, the transformation formulas in case that the function f (x) is defined on a bounded domain D. We assume that D is bounded by a surface S which closes a domain D' and by a surface of discontinuity E . We prolong f in DY, giving

479

THE FOURIER TRANSFORM IN BOUNDED DOMAINS

it the value zero. We make the same thing in the domain D° which closes E. Applying the flux-divergence formula we obtain:

fxID=fx6:' =fTIR3

rrfn - JS

fie'

f,,n

fnleia'xda- J'; Of llnleiaxda.

(A.8.1)

F(grad f[D = -ic"f - 1 fne'o

F[div fJD = -ice f F[rot f]D = -ia x

where (A =f+ -f-.

J

xda- r

f nei°t'xda - J QfIne'a-xda

f f x ne`a'xda- J s

DfOne`a'xda,

E

(A.8.2)

Appendix B

Cauchy-type Integrals. Dirichlet's Problem for the Half-Plane. The Calculus of Some Integrals

B.1

Cauchy-type Integrals

We consider in the z = x + i y complex plane a smooth curve r, i.e. a curve which has the parametric equations

x = x(8), y = y(s), sl

a

82,

(8.1.1)

where x(s) and y(a) are continuously differentiable functions, whose derivatives do not vanish simultaneously in the same point. The curve

r may be closed or open; if it is dosed, then z(81) = z(82); if it is open, then we assume z'(sl) = z'(sl + 0) and z'(s2) = z'(32 - 0). By definition the positive sense on r is the sense corresponding to the increase of the parameter s. The smooth curve is obviously rectifiable, such that we may consider as parameter 8 the length of the arc measured from sl (= 0) to 82(= 1). In this case, we obviously have zi2 + y'2 = 1. Let f(t) be a complex function depending on the complex variable 1, defined on r and Riemann integrable. The integral

F(z) = 2Ai

J

t (t) d t

(B.1.2)

is called Cauchy-type integral. As we lmow from the books of complex analysis, the function F(z) is holomorphic in the interior of the contour

r. if r is at a finite distance, then F(z) behaves at infinity like 1z4'1. We shall investigate, in the following, what happens with the integral (B.1.2) if z = to E r. In this case, the integrand has obviously a nonintegrable singularity in to and generally the integral has no sense. There exists however a large class of functions (we are not interested here in the largest class) for which we may give a definition to the integral,

482

CAUCIIY-TYPE INTECRALS

namely the class of the functions which satisfy the so called Holder's condition.

We say that the function f (t) satisfies Holder's condition on r if there exist two positive constants (different from zero) A and µ(µ < 1), such that, for every two points tl and t2 E r sa we have (B.1.3)

If (ti) - f(t )I < Ajtl - t2VY.

Obviously, the functions f which satisfy Holder's condition are continuous on r. If u = 1, the functions satisfy Lipschitz's condition.

B.2 The Principal Value in Cauchy's Sense We shall give now the definition that we have mentioned before. We assume at first that to does not coincide with any extremity of the arc r (if it is open). We consider the are of circle with the center in to

and the radius s which cuts the curve F in two points tl and t2 and (t) d t we denote by - the are t1t2. If for a -+ 0 the integral Jr-y tf to has a finite limit, then this limit will be called the principal value in Cauchy's sense. We denote

f(t) Iifot--f(t) dt - frto t - d t. Y

to

(B.2.

t.

The principal value is it distribution (A.12], [A.14]. We shall prove in the sequel the following theorem: "If f (t) satisfies Holder's condition in the vicinity of the point to, then the limit (B.2.1) exists and it is unique. For the proof we shall write:

f

f(t) dt=

tto

fr_1,

f(t)-f(to)(I t+f(tu) f t - to

r1F t dt - to

(B.2.2)

Having (B.1.3) in view, the limit of the first integral from the right hand member exists and it equals the usual improper integral on r. The last integral is calculated as follows:

f

tt

-7 t

o

= 111(t - t)la` + ln(l. - to)It = In u

- to+

+ ln(tl - to) - 111(t2 - to) .

Diu tl - to = Iti - tolein, t2 - to = It2 - tole'' and Itl - toI = It2 - tot.

483

PLEMELJ'S FORMULAS

It results In(tt - to) - ln(t.2 - to) = i (a - 0). Passing to limit, when

E-0, ct - W=ir,weget dt

Inn

_

dt

e- 0

= In

b - to

a-t0

+ i a = In

b- to to - a

(B.2.3)

Since the last integral from (B.2.2) has a well determined limit, the theorem is demonstrated. The case when to coincides with one of the extremities of the are F. depends on the behaviour of the function f in that point (,See for extunple [A.27], §29--32). If t9 coincides with an extremity and f (to) = 0, we are in the previously considered case., because we may

extend arbitrarily the contour t beyond to, setting f = 0 on the extension.

B.3

Plemelj's Formulas

We shall investigate the behaviour of the Cauchy-type integral in the vicinity of the curve F. To this aim we shall give at first the following definition [A.271: we say that F(z) is continuously prolongable on r in to (different from the extremities) at left (right), if F(z) tends to

a well determined limit F+(to)(F_(to)) when z - to on every path situated at left (right). With this definition we may give the following fundamental theorem : verifies or, t Holder's condition. then F(z) is continuously

If f (t)

prulongable on rat left and at right, excepting the extremities where f (to) 76 0 and Fi. (to) =

t2f(to) + 2ri I tf (t ndt.

(13.3. 1)

formulas. They have been The formulas (B.3.1) are called given in 1908 [A.29]. Their demonstration may be found for example in [A.18], [A.271.

B.4

The Dirichlet's Problem for the Half-Plane

tine shall solve in the. sequel the following problem: We seek for the function (B.4.1) F(z) = u(x y) + i v(x. y).

484

CAI;CIIY-TYPE, INTEGRALS

holornorphic in the half-plane y > 0 and continuously prolongablct on the Ox, axis. which reduces at infinity, to an imaginary constant i C (C may have the value zero) and whasc ir_al part is imposed on the above mentioned axis, i.e.

u(x,O) = f(x),

(13.4.2)

where f is a function with a compact support compact which satisfies Holders condition. At first we have to mention that there exists a single function with the above mentioned properties. Indeed, assuming that there exists two

functions F1 and r2 with these properties. their difference F = = F1 - F2 is holotnorphic in the superior half -plane and it vanishes at infinity. The real part of the function F is therefore harmonic in the half-plane y > 0. zero on the boundary y = O and zero at infinity. According to the maximum principle for the harmonic functions, the real part of the function is identical zero. F reduces therefore to an imaginary constant which is zero because F is zero at infinity. We shall prove that the function F(z)

7i

f }a

t

(t) d t + i C

(13.4.3)

CO

satisfies the conditions of the problem and it is therefore the solution we are looking for. Indeed. the function F(z) defined by (13.4.3) is holomorphic in the superior half-plane because it is a Cauchy-type integral and it is continuously prolongable on the real axis. At infinity it reduces

to the constant i C because, if we denote by (a, b) the support of

f(x),wehave I-M

t

f (I

f () dt

(t) dt

/-

tl

dt.

The integral is therefore zero at infinity.

Passing to the limit with

z

x a point from the real axis, and

using Plemelj's formulas we obtain: 11(.x..0)+11'(x,0)= f(.1;)+ 11 1

J+x -x ., tf(t)dt+1C.

Taking the real part of this relation we obtain (B.4.2). Hence the problem is solved. The real part of the solution (13.4.3), i.e. UI(X. Y)

y ,or

fa+OC

f(t) (t - T)2 + ry2

dt

(B.4.4)

THE CALCULUS OF CERTAIN INTEGRALS IN THE COMPLEX PLANE. 485

determines the harmonic function in the half-pl.uie y > 0, vanishing at infinity, continuously prolongable on the Or axis and satisfying the condition (13.4.2). This is the solution of Dirichlet's problem for the half-plane concerning the harmonic function u.

B.5

The Calculus of Certain Integrals in the Complex Plane

At first we shall prove that G

1

a tit

t.

a

(B.5.1)

b - tt

where the determination of the radical is the positive one for z = r > b, and (a, b) is an interval on the real axis. Indeed, with the mentioned determination, we have

x
a<:x
rb6

After all.

r
().

Re(i

r-a

b.....

z

a<x
Since the real part of the holomorphic function i 1/: - n is known on G

the real axis, the function will be determined by the formula (13.4.3). We have therefore

Urb-t t.. i

a

-zzz-

b

-t--a dt

-ya'

+C.

Considering z oc it result., C = 1 and then the formula (B.5.1). Analogously we deduce that ' f it 1

b

t

a

at

dt

=

-l + JfT b

(8.5.3)

486

CAUCHY-TYPE INTEGRALS

Passing to the limit and taking into account Plemelj's formulas; (B.5.1) and (B.5.3) we obtain that:

'b fl -a dt

i t- x-

b

:r

1j

V

a t-

t

a

-

b=-t dt

(B-5-4)

-1

We must notice now that the above considered function

i

z-b

Y YYY

reduces to an imaginary constant at infinity (this is i). If we want to apply the same procedure for the function (z - a)(z - b), we must consider the expression

(z-a)(z-b)-z. To this aim we have

x>b, x
n,

Re [i ( (z - a)(z - b) - z)} =

-(x-a)(b-x), a<x
whence

(z - a)(z - b) - z =

+- r

b

(.t

a)(b - t)

d t + C.

The integral becomes zero to the infinite. This imply a+b 2

We have therefore 1

(t - a)(b

7r,"b

t-z

- t) d t = V"(-z-- a) (z - b) - z +

a

+b

(B.5.5)

and, passing to the limit and applying Plemelj's formulas,

rrt x 1

L

(t - a)(b - t)

d t = -x +

a

+

b

z

(B.5.6)

The following example is studied starting from the expression i (z - a)(z - b) '

which vanishes at infinity. Hence, the constant C will be zero. One obtains the integrals rb

V

a

(b - )(t - a)

t tz-

a)(; - b)

(B.5.7)

THE CALCULUS OF CERTAIN INTEGRALS t\ '1'flE COMvLI;Y PLANE 487 6

dt =

1

ir Ja f h - t) (t - a) t -

(B.5.8)

Xt) .

We consider now two intervals (a, b), (c, d) on the real axis. The

method can be extended to an arbitrary number of intervals, or even disjoint arcs from the complex phuie, in the last case the integral being solved in another way in [A.18], p.88. The result is useful in the study of the grids of profiles. Denoting Q(z) = (z - a)(-- - b)(z - c)(z - d),

(B.5.9)

for P(z) and R(z) arbitrary polynomials, we have

P(..)

n<x
- i li'(z)

Q(z)

C<x
The function from square brackets is holomorphic in the half-plane y > 0. Utilizing the formula (B.4.3), we deduce

- R( _ V Q(=)

t

-r J

d tz

PQ(t)I

t

a t4

+C.

-Q(t)I

(B.5.10)

Taking into account. the behaviour of the integrals for z - oc, we deduce that at; infinity we must. have the expansion

P(z) Q(Z)

R(z) C+ a2+....

(B.5.11)

This formula allows to determine R(Z) when P(z) is known. For example, for

P(z) = (z - b)(z - d) it results R = 0 and C = 1. and for P(z) = z(z - b)(z - d)

it results

n+c-b-d 2

CAUCIIY-TYPE INTEGRALS

488

After all, we have the results (b - t))("_t) (1_ (d - t)

°

1

dt

(t - b)(d - ') t) d t

rt

1

(t.

t -z (B.5.12)

-

(z - b)(z - d}

-

(z-a)(z-c) r:

(b - t)(d - t)

1

rr,

_

td t

t-

1

+%r

l

d

1

(t - b)(d - t)

td t

(t-a)(t-c) t -, (B.5.13)

a+c-b-d

(z-b)(z-d)

Passing to the limit for z x E ((L, b), we apply Plemelj's formulas to the first integrals. One obtains:

jd /(t-b)(d-t),t) dt (b-t)(d-t) d t (r- -C.) t-x (t-a)(C-t} t-x}7f

6

1

q

b

tt

JG

(b - t)(d - t) td t

(t - b) (d - t) td t

(tc) t-x

1

(t-a)(c-t) t -

-1

x+

-x- a+c -2 b - d (B.5.14)

and for z - x E (c,d), employing Plemelj's formulas for the integrals on this interval, we obtain from (1.5.12) and (13.5.13):

(t-b)(d-t)

dt _ -1

1' J (t-b)(d-t) (t-a)(t- c) s,f J(t_a)(c-t)t_-x +r,

t _ t-xdt-

1

ti

1

(t-a)(c-t) t-x + r

7r

1

(b-t)(d-t) dt

b

'

`t

(t-a)(t-c) t-x

(b-t)(d-t) tdt

a+c - b - d 2

489

t:LAUERT'S INTEGRAL

B.6

Glaucrt's Integral. Its Generalization and Some Applications

We shall calculate the integral 1

_n

f -do e ne

f

_

27r f r

cos e - 8

os ne

cos 9 - s

do,

(B.6 .1)

where s is a real number. For -1 < s < +1 the integral was calculated by Glauert in [3.19]. The general case is considered in [5.38]. The method consists in passing to the complex variable. Denoting exp (i 0) and noticing that we have

1

+" sin no d8=0, , cos B - s

because the integrand is an even function, it results 1

27r

++

f

x

-J=1_tz2-2sz+1

e ne

oos9-sd

1

9

/'

z"d z

iri

(B.6.2)

z"d z in

where

lsl=1

(z - 0Z - Y)

1

a=s+ s2-1, Q=s -

82-1.

The last integral is calculated with the residue theorem. Since aj3 = 1, It results that the poles of the integrand are situated either one in the interior and the other in the exterior of the circle z = 1, or both of them on the circle.

In the case s < -1, the pole z = a is interior and according to the residue theorem, we have:

I = 2tr1 "

Q'"

1

= (3 +

S - 1)"

MQ

(B.6.3)

and for s > 1, the pole z = j3 is interior and it results

I.

1)"

(B.6.4)

At last, in the case -1 < 3 < 1, using the substitution s = cosa, we have a = exp (i o), 0 = exp (-i a) and from the semi-residue theorem [A.221. p.320 (the poles are situated on the integration path) f (z) d z = ai f (zo),

z - :w

(B.6.5)

490

CAUCIIY-TYPE INTEGRALS

we deduce

In =

an

/i"

sin na

a-,Q

jj-rt

sin a,

Hence, Glauert's formula is

1 r* rr ffff0

Cos no

Sill I

cos 0 - cos a d 0

sin o

n=0,1,2

(B.6.6)

Glauert's formula has many applications in aerodynamics. For example, using the substitutions

t=c+ecosO, x=c+ecoso,

(B.6.7)

2c=a+b, 2r.=b-a,

(B.6.8)

where

one obtains the formulas (8.5.4), (8.5.6), (8.5.8) and other similar ones, like for example

Ft- t Jt/---dt=ir(x+e).

(B.6.9)

b

Using the substitutions

t=c+ecosO, x=r.+es

(8.6.10)

and the formulas (B.6.3) and (13.6.4) we also obtain

l b t-a dt 71'

o

b-t t-x

4b-Xx 1,

l x-a x - b

xb

b-x - 1

1

7rI'.

b

t dt

t-a t-x

I va - x

1

1

n fL a

dt (b--t)(t-a) t - x 1

(a - x)(b - r,) 0.

1

Vf(x etc.

x
491

OTHER INTEGRALS

B.7

Other Integrals

In the boundary elements methods one may encounter the following integrals

cosne+i sin no

1k -1

(B.7.1)

for k = 1, 2. They are also calculated with the residue theorem putting z = ei0. It results

sin8=(z-z) ,oos8(z+z), 21

Ii 12 -

_

z"d z

2

i b + Jc I:(z - zi)(z - z2) ' 4i

(i

z"f 1d z rJi:i_1 (z - zt)2(z - z2)2

where, with the notation r = a - b zi

_

-a+r

b- ic'

z2

_

,

(B.7.2)

, we have.

-a-r b- is

(B.7.3)

Obviously, J--1z21 = 1, such that either a root is in the interior of the

unit circle and the other in the exterior or both of them are on the circle.

We are interested in the case when a2 > b2 + c2 and a > 0. In these conditions, the root z1 is in the interior of the unit circle, such that, utilizing the residue theorem, one obtains:

71= F(r), 12 = G(r),

(B.7.4)

where

F(r) =

27r(-1)" (b+ic\" G(r) = 2w(-1)" a+nr (b+ic)" r3 a+r r r , M IC)

(B.7.5)

Separating the real part from the imaginary one, we obtain a series of

492

CAUCIIY-TYPE INTEGRALS

integrals, like, for example 2a

d0 r A a+bcos0+csin0

I

'"

a+bcos0+csiuO

o

f-

f I

cos 0

r d0

sin 0

cos 20

a+bcos0+csin9

'

=- 2:rr

b

c

d0=-'';r

r a+r'

rr+bcos0+csin0

'T

Jo

2.

d 0 = 2;.

b-'

r

(B.7.6) d0

27ra.

Jo (a+bcos0+csin9)2 - r=s .3T

Jci

(a+bcos0+csin0)2

o

(a+bcos0+csin0)2

'

d0=-rbi

sin 0

2:r

J0

- c2

cos 20

(a+bcos9+csin9)2

c

-; a} 2r r 2+r1 r2 \d0

b2

- c2

(a+r)2.

The case a2 > b2 + c2 and a < 0 is studied in the same manner (in this situation, z2 i3 in the interior of the circle). One obtains

Il = F(-r). 12 = G(-r).

(8.7.7)

If n2 < b' + c2, the two roots are on the unit circle, such that the integrals are calculated by means of the semi-residue theorem. One obtains

21, = F(r) + F(-r), 212 = G(r) + G(-r).

(B.7.8)

At last, if a2 = b2 + c2, the roots coincide and they are situated on the unit circle. The integral will be calculated using the Finite Part (see Appendix D).

If a = -s, b = 1. e = 0 and k = 1, one obtains the re cults from the previous section.

_

Appendix C

Singular Integral Equations

C.1

The Thin Profile Equation

The thin profile equation in a free stream has the form 1

'b

f (t) d t= h(x), 7rt-x

a < x< b,

(C.1.1)

where 1(t) is the unknown, and h(x) is a given function. It is sufficient to assume that f satisfies Holder's condition on the interval (a, b) for deducing the existence of the principal value from (C.1.1). The solution of the equation also depends on the behaviour imposed to the unknown at the extremities of the interval [a, b]. From the expression of

the solution it follows that h also has to satisfy Holder's condition on the interval (a, b). The equation was pointed out by Birnbaum in 1923 [3.5) and solved for the first time by Sohngen in 1939 [A.34]. An ample study devoted to this equation is due to Schroder [A.33]. The solution was found again, with different. methods by many other researchers (Weissinger [3.49], Cheng [A.30], Homentcovschi [A.19], Carabineanu [A.151 etc.). In the sequel we start from the method of C. lacob [A.20]. For solving the equation (C.1.1) we have to determine the function

F(z) = u(x, y) + i v(x, y), z = x+iy,

(C.1.2)

holomorphic in the superior half-plane y > 0, vanishing at infinity and continuously prolongable on the real axis, with the conditions u(x, 0) = 0,

for x E (-oo, a) U (b, oo) (C.1.3)

v(x,0) = -h(x),

for x E (a,b).

In order to ensure the uniqueness of F(z) we have to impose its behaviour

in a and b. The function F may be bounded or not in these points.

494

S1N ULAR IN1 CRAIG l:QU VI`IONS

We w.,sume. for example that F is bounded in b, and it behaves in the

vicinity of a like

F(_)=0
(C.1.4)

where F` is bounded We must notice that if we have determined such a function and if we denote by f (x) the boundary values of a on (a, b), then It results that the function f is just the solution of the equation (C.1.1). Indeed, the function F(:), holomorphic in half-plane y > 0 and vanishing to infinity, whose real part on Ox is

u =0,

for :t E (-nc,a)U(b,x) (C.1.5)

it = f (.r.),

for :r E (a, b),

is given by the formula (8.4.3). i.e.

F(z) =

i

r

I

(t) t -

dt

(C.1.6)

.

Utilizing Pleanelj's formulas %,.-c obtain for ar< :r. < b

it (X, +0) + i v(r. +0) = AX) +

J t, t !

cl

(C.1.7)

t..

Separating the imaginary part and taking (C. 1.3) into account.. Ave obtain (C.1.1) for v We shall solve in the sequel the problem (C.1.2) - (C.1.4). Having (C.1.4) in view, we shall consider the function (8.5.2). With the determination of the radical precised there and with the conditions (C.1.3), we deduc c:

Re

_ tt 1

rE

0..

[r(z) Y-bJ --

T---a

b-x

a) U (b, ;,c)

x E (I a, b)

Hence, we may utilize the formula (13.4.3). It results a

a1tV)

Cit.

tt rri F(`) z-Q`-1 the constant C vanishing because I' must v:uaish for z -- x. We deduce

rz --b

t

(C. 1.8)

495

TIIE THIN PROFILE EQUATION

Applying Plemelj's formulas and separating the real part we obtain

Ax)

1

n

bx-

a a s

_ta b

t

IL (1)

t, - x

d t.

(C.1.9)

This is the solution of the integral equation (C.1.1) which satisfies the boundedness condition in x = b. The solution which is bounded in a and unbounded in b may be

-b

obtained starting from the function We obtain

f(x)=-

'

W

r-a

b

6-x f

7r

.

4 - a

c

v

(C.1.1O)

(txat.

If we wish to obtain an unbounded solution the both end-points, - a)(z - b), we have

using the function

Re [F(z)

e

(z - a)(z - b)j = x E (-oo, a) U (b, oo)

ol

(x-a)(b-x.), xE(a,b)

h(x)

whence

!

f (x)

'b r (x-a)(b-x) Ja

(

1

i (t) d t+ t - a)( b - t ) t-x a

(C.1.11)

C

jr - a)(b - ar) At last, for the solution bounded in the both end-points, we deduce b

(z - a)(z - b) 1.

(t -a)(b -

t)

(C.1.12)

t t,

When z is great enough, this becomes 1

F(")

_

i

(z - a) (z - b)

ri

(1-a)"2 (i_)"jb

/b

Jn

h(t) (t - a)(b - t)

dt

1- t/

_

-

h(t)(1+...)(lt

496

SINGULAR INTEGRAL EQUATIONS

such that imposing the zero value for F at infinity we have

f

h(t) d t = 0. (t - a) (b - t)

(C.1.13)

If this condition is satisfied, the solution of the equation (C.1.1) (bounded at both end-points), is obtained passing to the limit in (C.1.12). Taking into account the determination of the radical, we find

f(x)=-1R (a-a)(b-x)Ja

b

h(t)

(t - a)(b - t)

t dt -r

(0.1.14)

Because of the restriction (C.1.13), this solution cannot be utilized in aerodynamics.

C.2 The Generalized Equation of Thin Profiles This has the form: b

- Ja

(tad t +

f (t)K(t - x)d t = h(x) a < x < b,

(C.2.1)

n

where K is a non-singular kernel. We assume that f (x), K(x) and h(x) satisfy Holder's condition on (a,b). This type of equations are encountered in the theory of thin profiles in ground effects, in the theory of grids of thin profiles, in the theory of grids of thin wings etc. From the mathematical point of view, the equations (C.1.5) are extensively studied in (A.27). In order to ensure the uniqueness of the solution we have to impose the behaviour of the unknown f at the extremities of the intagration interval. The method of investigation consists In their regularization, i.e. they are reduced to Fredholm-type equations for which existence and uniqueness theorems are available. For the equation (C.2.1) this may be performed utilizing the solution (C.1.1). If we are interested by the solution of the equation (C.2.1) which vanishes in the trailing edge b, we shall utilize (C.1.9). Assuming that the last two terms in (C.2.1) are known and knowing that a singular integral interchanges with a non-singular integral (A.27), we obtain the following Frcdholm-type integral equation

f(x) +

! f f (C)M(t, r)dt = H(x), a < x < b,

(C.2.2)

497

THE GENERALIZED EQUATION OF THIN PROFILES

where

b =x f

z) _ -1

t wa k(t - L)dt'

aQ

x

b

-1 bz [

t-a

_x- s,

W

t- x

t

(C.2.3)

h(t)

-C--t t - x

dt.

Sometimes we may specify the kernel M . For example, in the case of the thin profile in ground effects, we have:

k{t -) =

(t

-)l

+

(C.2.4)

mz .

The integrals

t

1_

a

t-t

dt

) (C.2.5)

b-t (t-t)2+m2 t-z

a

may be calculated with the residue theorem [A.20], or considering the integral

_ =

f

b-a m dt b-t (t-4)2+m t-x

8

and noticing that if we denote S = t + i m, we have: 6

t

dt

a

Fa

dt

V=t

t (t-()

l

-

(dt

f n VLb_at

Utilizing (B.5.1) and (B.5.4) it results

1+iJ=z*C

C-b

)b (C.2.6)

t-z

(C.2.7)

whence

1- a J

2

1

S-a+

1

z-C-b x-S x1 C

C-

-a -b c -b

(C.2.9)

498

SINGULAR INTEGRAL EQUATIONS

and con."juently,

b= x

1

1ll(f r) =

- a+

1

1

C/z:).

C.3 The Third Equation We shall consider the equation [A.1G]:

-

b

1

J

f (t)(In it - xj + ro)d t = g(x) ,

(C.3.1)

defined on the interval (a, b), where f is the unknown and r o is a constant. We intend to determine the general solution and to investi. gate the conditions which are necessary for the solution to satisfv the restrictions

f(a) = f(b) = 0.

(C.3.2)

With the change of variables t, x -. 0, a defined by the relations t = c + e cos 9, x = c + e cosa , where

c=

a+b

-2'

e= b-a 2

,

(C.3.3)

(C.3.4)

the equation (C.3.1) becomes 1

-

f

F(9)(lnelcos0-cosaI +1'o)csinOd0=G(o)7T

With the notation

(t - a)(b - t) f (t),

(C.3.5)

'fF'(0)(lncjcm0-cosOl+l'o)d0=G(a),

(C.3.6)

F'(0) = eF(O) sin 9 = this may be written as follows 1

where we have

1 coe m9 cos crier .

In 2l cos O - cos al = -2 m>1

(C.3.7)

499

THE TN1R1) EQUATION

The function F''(6) may be prolonged on the interval (-ir,O) such that the result is an even function on (-pr, +7r) and it may be therefore expanded into a trigonometric series, with the aid of the even functions F' (9) = ao +

an cos no,

(C.3.8)

F` (9) cos nod 0,

(C.3.9)

n>1

where an =

ao

r

ao =

I f,P(O)dOi f F(9)e sin 9d 9= 1Jaf f (t)d t. 7r

o

Analogously, the function G(a) may be expanded into the series

G(a) = bo + E bn aog no,

(C.3.10)

n>1

where

b,, = -

rx G(o) coo nod a = - 21 " G,(0,) I sin nad Q , 0

9(x) dx.. (C.3.11) 4 1o (x a) (b - x) a Substituting these series into the integral equation (C.3.6), it results bo

G(a)da = 1 A

r*

ao+E an coo no x

0

n>I

x

n>1

(_2!cosmOcosmo+F)d9=

(C.3.12)

m>_ 1 m

-aor + n>1

i

a. cos no,

where we denoted

r = ro + In e = ro + I n (b 4 a) .

( C . 3 . 13)

Identifying the coefficients we find:

I

bo = -aor, bn = + an .

(C.3.14)

500

SIN(:ULAIt INTEGRAL EQUATIONS

In the sequel. we have a) (b - t) g,(t)d

T

t..

(t 1 u)(b - 1 1 ( l (

fb

x

77

sin0

1

tt

r Jot COS a -, Cclti 0

_

1

sin a

r

0 COS a - C

ci

sin0 (C(4))

cos a - cos 0

(

(g)d0=

sin n0 d0=

7 cos(n + 1)0 - cos(n - 1)0 d

O

cos or - Cos 0

n

or, utilizing Glauert's integral (B.6.6),

i

(t.-a)(b-t),(t)dt

%b

ra

t -- x

sngit - 1)a

Stn(n + 1)a n>1

n>1

-E a,, cosna+-F'(a)+tatt = V12!1

(x - a)(b - x)f(x) - bo/r,

= --

(C.3.15)

where bo is (C.3.11). From (C.3.15) it results

f (s) =

-

1

rb y/ (t - (z)(G -

1

(x, - (1) (b

t-r

X) JJ4

t)9

(t)d t,(C.3.16)

_

1

r1'

1

(x - a)(b - x)

6

g{t) dt. (t -- -a) (b - t)

This is the first form of the general solution of the equation (C.3.I).

-

T11F,ru1RD EQUATION

501

One obtains another form if one utilizes the identity (t - a) (b - t)

(x - a)(b - x)

(x -- a)(b - x)

(t - a)(b - t)

(C . 3 . 17)

(x-t)(x+t-a-b)

(r. - a)(b - x)(t - a)(b - t) Substituting the first ratio in (0.3.16), it results the final solution

f(x)

(z

g'(t) dt + -a)(b-x.) (t-a (b-t) t-x a 1

1 1

-a)(b-a) J

g(t)1

I'

J

[(.

+ t - a - b)g'(t)-

(C.3.18)

dt

(t-«)(b--t)

We notice that from the relations (C.3.9), (C.3.14) and (0.3.11) it results

1 Jn

bf(t)dt=TJ b

-a)(b-r)clx,

(

(C.3.19)

which is an useful relation in applications.

We shall determine in the sequel the conditions which have to be satisfied by g, such that the relations (C.3.2) are satisfied by the solution (C.3.1). At first we notice that when the parameter r vanishes, a necessary condition for the existence of the solution (C.3.18) is b

L

g(t) d t = 0. (t - a)(b - t.)

(0.3.20)

The last integral from (C.3.18) defines for x real a polynomial of the first degree. f vanishes in a, is this polynomial has the root a

(the first term from (0.3.18) vanishes for x = a). In this case, the last integral has the order of (x - a), while the denominator of the fraction has the order of (x - a) 1/2. Hence, we must have -",)(b

f' [(t - b)g'(t)

grt)J

(t

- t) = 0.

(C.3.21)

502

SINGULAR INTEGRAL EQUATIONS

Analogously, f (b) = 0 implies fb

[(t - u)9'(t) - 9(t) I

(t - u)(b - t)

0.

(C.3.22)

Imposing the both conditions, subtracting and adding, we obtain 91 (t)

L b

J

[(t

a

9rt - - tad

- c)9 (t)

(C.3.23)

(t-a)(b-t)dt __ n,

(

t/b )1

t)

0.

(C.3.24)

This is the answer for the proposed problem.

C.4 The Forth Equation At least in nragnetoaerodynarnics [3.9] [1.9) p.208, in the theory of oscillatory wings [10.15) and in the theory of the wing in fluids with chemical reactions [3.10) [3.24], it intervenes the following singular integral equation [A.16) fb

1

f(t)dt+- j

f(t)(lnt-x)+ro)dt=h(x), a<x
where f is the unknown, ro an arbitrary constant and It, a given function. The solution is unique if one imposes the behaviour of f in one of the end-points of the interval. We shall impose the boundedness

of f sa in b. Denoting

1b

g(x) = -,- f f(t)(ha it - xI + ro)d t,

(C.4.2)

a

we shall notice that

g'(x) =

r

ad t .

t So, the equation (C.4.1) is reduced to the differential equation

(C.4.3)

a

(C.4.4)

whose general solution is

9(x) =

Aee('-`) +go(x),

(C.4.5)

503

THE FOIri'H EQUATION

where A is an undetermined constant, c is (C.3.4), and (C.4.6)

The unknown f may- be found from the equation (C.4.2) which is exactly the equation (C.3.1). Its solution is therefore (C.3.18) where g will be replaced by (C.4.5). For specifying it, one observes that

performing the substitutions (C.3.3), denoting by lo(w), Il (w), .. . Bessel's functions of imaginary argument and taking into account that II(w) = Il (w), one obtains

(

CW(t

T1

)

a)(b -- t)

dt

rr" e

a

,

x

b

T2

'" °d 6 = 7rIo(M)

(t - a)(b - t)

dt

(c+ecosO)e 06Od0

(C.4.7)

o

= irc1o (w) + rrel i (w) .

From the expansion

exp(0cos9) = lo(Z3) +

(C.4.8) n>l

and from Glauert's formula (B.G.6) we deduce e`'(t-c)

b

dt

1

'

-x

-

2-,r

e

Sill it (T

n>1

smo

ejjr

ewcoo0

cos0-cosad8 (C.4.9)

-

2-,r

V-r(b

E 1,, (0) sin laor

- x)(x - a) ,>I

where w = ew. For T; and x we have the parametric representation (C.4.9) and (C.3.3). We deduce therefore

f(.x)=-'A (x-a)(G-x)7'3+ x

[w(x_2c)_]Ti+_ 17

7,-

(x-a)(G-x)

x

--a) (x - (b - x) (C.4. 10)

504

SINGULAR INTEGRAL EQUATIONS

fo being obtained from f by replacing g with go. In aerodynamics f represents the jump of the pressure and we need therefore the integrals

j

rb

b

f x f (x)d x

f (x)d x,

(C.4.11)

4

which may be easily calculated without utilizing (C.4.9). Directly, the first integral is obtained from (C.3.18). In the sequel we shall determine the constant A. Imposing the condition f (b) = 0, we have from (C.3.22) and (C.4.5) A [ (aw

+ ,) Tt - wT2]

(C.4.12)

= GO,

where b

Go

[(t - a)9o(t) - Tgo(t)

(t

u)(b

t)

(C.4.13)

Now the problem is solved.

C.5

The Fifth Equation

In the theory of oscillatory wings [10.20) it intervenes the following integral equation (A.16[:

b(tf(t))2dt+

2

Jf(t)(lnIt-x[+I'o)dt=h(x), a < x < b,

b

(C.5.1)

which has to be solved together with the following conditions

f(a) = f(b) = 0.

(C.5.2)

The sign "' from (C.5.1) shows that. one considers the Finite Part. Introducing the notation (C.4.2) and taking into account the formula (D.3.6), from (C.4.3) we deduce . b o

(t - x)2 d t ,

(C.5.3)

a

such that the integral equation (C.5.1) reduces to the following differential equation g" - w2g = h. (C.5.4)

505

TILE FIFTH EQUATION

The homogeneous equation has the solution g = Aocosh w(x - c) + Bosinh w(x - c) .

Applying Lagrange's method of variation of constants, one obtains for the solution of the differential equation (C.5.4), g = Acosh w(s: - c) + Bsinh w(x - c) + go(x) ,

(C.5.5)

where

h(E)siuhw(x - E)dt;

r(l(x)

(C.5.6)

The solution of the integral equation (C.5.1) may be obtained from (C.3.18) where one replaces (C.5.5). For specifying it we notice that setting

t = c+eu,

u E f-I,+1J,

we deduce (t - a)(b - t) = e2(1 - u2) and then sinhw(t - c) d t = r+1 sinh (Foil)

So n

J

(t - a)(b - t)

1

d u = 0,

(1 -- uu

the integrand being an odd function,

r6 Coshw(t - c)

(t - a)(b - t)

n

b tcosh w(t - c)

C1

(t - a) (b - t)

Ja

ar.

=

Cosh Pu}

_rdu=glo(w)

fJ

J1 - u2

1

dt

(C.5.7) Yt

J Sl

/b

(c+eu)

cosh (Cu)

1-u

tsinhw(t - c)

(t-a)(b-t)

du=crrlo(w) 1 usinh (Ou)

dt=e f-i

1-u

d u = eirl1( ,.r ) .

Also, using the change (C.3.3) and taking into account (C.4.8), Clauert's formula (B.6.6) and the relations

(-1)' I,,(w),

(C.5.8)

SINGULAR INTEGRAL EQUATIONS

506

we deduce ?,s

b

sinh w(t - c)

dt

- a)(b - t)

r' sinh (iv cos 9)

e

o

cos0-cosy

d8

sina

(b - x)(x - a)

f[

1

sinner

=E [1. (0) e n>1

TC

t-x

-

Eli n>1

cashw(t - c)

ner

(C.5.9)

dt

(t-a)(b-t) t-x n

sin nay.

(b - x) (x - a) n>1[1 -

These formulas, together with (C.3.3), give the parametric representation of the integrals T., TT and the variable x. With these results, the solution of the equation (C.5.1) is

f (x) _

+

(x - a)(b - x)(ATT + BTT)+

(x - a)(b -

x)I A(wS1 \\

11,Co)

+wB (x-2c)Co+C1I I, L

(C.5.10) where fo is obtained from (C.3.18), replacing g(t) by go(t) given

by (C.5.6). As we have already mentioned in the case of the equation (C.4.1), in aerodynamics we need the coefficients (C.4.11). They may be obtained easier utilizing the form (C.3.18). For determining the constants A and B we impose the conditions (C.3.23) and (C.3.24) where g' has the expression (C.5.5). One obtains the system wCOA=Gm, (C.5.11)

w(C1- cCo)A+ (wS1 -- r-1Co)B = G2,

507

THE FIFTH EQUATION

where '' C1

-L

9-'o(t)

dt (t-a)(b-t), (C.5.12)

rb

G2 = -

JJ

[(t - 09' W) -

l TYO(t)J

(t.- a)(b - t)

Taking into account (C.5.7), we deduce G1 A- w7rlj1(D) B,

r-110(w)[

(C.5.13)

In many applications we encounter the situation when h = -c (the case of the flat plates with the angle of attack e). In this situation, from (C.5.6) it results w290(r) = e[1 - cosh w(X - c)),

such that G1 = 0, W2G2 = Frr[wl1(i)

- r-`lo(ay) + r-.1[

whence,

A=0;

13= 4

[1+zr11(0)-IQ(W)1

.

(C.5.14)

Appendix D

The Finite Part

D.1

Introductory Notions

The notion of "Finite Part" of a improper integral has been introduced by Hadamard in 1923 [A.39], in order to give a significance to the divergent integrals which appear in applications and to utilize them. Hadamard studied integrals having the form b

f Ja

fW

d

(b-X)*,+1/lr,

(D.1.1)

where n = 1,2,3,.... There exists however many integrands with non-integrable singularities which appear in applications especially in aerodynamics. It exists therefore different manners for treating this problem. In the subsonic aerodynamics one utilizes especially the definition of Mangler [A.45], but we had not the possibility to read this paper. A less cited contribution, but very adequate to aerodynamics belongs to Ch. Fox [A.37]. Here, the notion of "Finite Part" appears like a natural extension of the concept of "Principal Value" in Cauchy's sense. We shall present in the sequel some results of this author. For the integrals having the shape (D.1.1) we shall utilize the paper of Heaslet and Lomax [A.44]. These ones appear in the supersonic aerodynamics. Important results concerning the notion may be found in the papers of Kutt [A.42) and Kaya and Erdogan [A.41]. At lasst, the theory of distributions give an unitary method for the study of this notion [A.5].

D.2 The First Integral We shall consider at first the integral

Il -

fa

dx,

n=0,1.....

(D.2.1)

510

THE FINITE PART

If f admits derivatives up to the order n + 1 in the origin, then we may write =Jara

[f(x) -

(dx

I

yt

fl')(0) + ft'?(o)} d+1+

[.rx)-E

o

ft,,(0)]

-, xi-^ f(i)(0) a i - it

=

i=0

r=0

=

dx

it

14)

xla

+ n!

In 0

The integrated part for x = 0 becomes infinite. Neglecting these infinite constants, one obtains the so called "Finite Part" of the integral 1. Hence, indicating by an "asterisk" the Finite Part, we have:

f(X) xn{i(Ix =

in [1(x)

-

"- t

JO

dx

.n+1 + im0

"

f(i)(O)

(D.2.2) a'

f In) (0)

In a.

i-0

For f = 1, it results

Td'x =lna,I-!+i J

»

1 it a

(n.> 1).

(D.2.3)

D.3 Integrals with Singularities in an Interval We shall consider the integrals having the form 12

rb

I( X) +idx,

11 =0,1,...,

(D.3.1)

where a < u < b. For n;-- 0, we consider the "Principal Value" of the integral in Cauchy's sense.

,(J G-E + c-» u \ , x-u d x = lien

lb

1L f (x)

r1

f (?) d x xu

.

(D.3.2)

511

I? TECRALS WITH SI`CULARITIES IN AX INTERVAL

«Vc know from (B.1.2) that this limit exists if f satisfies Holder's condition in the interval (a, 6). Let us derive now (D.3.2) with respect to the variable u. In the right hand part, the derivation is performed according to the derivation formula for integrals containing the variable in the limits. We have therefore d(,(

b

d r =1im U C-0 d u Jn X - u f 0x)

t. s+

b

f (X)

Ju+E) (:r - fl):.

-

d x-

-f(u-0 _ f(u+=) or, expanding into a Taylor series the functions f (u - E), f (it + E),

,

+

x (xatd x

Tit J

en o

(.1:

L+J (z )2J

(D .3.3)

If the limit from the right hand part exists. we denote it by

Ja T.

f 0-) (T-u)2

(D.3.4)

d X.

and we have

f

d x def

(x - u)2

lim s-'0 l

f u-e + fb l C a

d rh f(x) dA, J du a X - It a

f (X) d (:C - u)2

2--1

x-

(Xf(T) -.11)2 dx.

(D.3.5) E

(D.3.6)

The limit (D.3.5) defines the Finite Part of the integral from the left hand part. The Finite Part is a distribution {A.14]. Ex. 11. One proves [A.37{ that if there exists f'(x) on (a, 6) and this function satisfies holder's condition, then the limit from (D.3.5) exists. We notice that this theorem constitutes the extension of the theorem of existence of the limit (D.3.2). We indicate now how one may reduce the calculation of the integral (D.3.1) to the calculation of an integral with a weaker singularity. We consider the case n = 1. Hence, we demonstrate that in the same conditions like above (f' is defined and satisfies Holder's condition on (a, b) ), we have

f(T) ,

(x-n)2

f(u)

f(6)

a-u 6-it

f'

, :r - u

THE FINITE PART

512

for every of from (a, b). Indeed, employing for the left hand side member the definition (D.3.5) and integrating by parts, we obtain

r f () a

+

ii

(x-u)2 fb

t+

U -C

d x = lim

e-'0

CL

I-

+

Zb +t )

a

f 2f (u)} XU d a - c

f (a)

f f (x), d x+

a; x - rt

-

f (b) +

a-u b-u

f.b

t(x)

f da X-u

.

The extension of the definition (D.3.5) and theorem (D.3.7) to an arbitrary value of n , is performed in (A.37]. In the same paper one gives the respective definitions in the complex plane and also Plemelj's formulas for integrals having the form

F(z)

7ri I 21

(t

1(t +I d t

(D.3.8)

.

Utilizing (D.3.6) we may calculate (by derivation) the Finite Part when we know the Principal Value. So, from (D.4.1) it results the integral often used in Appendix C, 1

7r

(t-a)(b-t)dt=-i, u<s
(D.3.9)

From (B.5.3) we deduce

1`/b ia

dt

1

0

(b - t) (t -a) (t - x)2

(D.3.10) ,

and from (B.6.9),

t

1

it Ja

t

a

b - t (t

x)2

dt = 1,

(D.3.11)

and the sequence of examples may be enlarged (see also (A.41]). We also observe that we have day

(x - u)2

_

1

a - it

- b - it 1

which may he obtained considering f = 1 in (D.3.7).

(D.3.12)

513

HADAMARD-TYPE: INTEGRALS

D.4

Hadamard-Type Integrals

We shall consider the integrals having the form

I3=Ja

a

(s x)'+tf2dx,

n=1,2,...

(D.4.1)

which intervene in the supersonic flow. As we have already seen in (D.2.1), the basic idea in the definition of the Finite Part consists in leaving apart the infinite values from the structure of the integrals. In order to see the significance of the integral denoted by (D.4.1) in the case n = 1, we shall observe that d ds

f'

78

{ (s

s(x)x

d. = sic dds

- e)

f +J

a -C 8 a

f (x)

s-xJ dxJ

a8

.

If f is continuous in a, the first term from the right hand part becomes infinite such that we must leave it apart. We shall consider by definition

B(x)xl d x = ds.

Ja a

I

s(x) d x

(D.4.2)

or

like in (D.3.6). For f = 1 we deduce

dx a

(s-x)3/2

-

2

s-a

(D.4.4)

Let. us prove now that if f is continuous in s and admits bounded first order derivatives on [a, s), then we may give a formula for calculating the member from the right hand part of (D.4.3). Indeed, we have d ds

f f(x))s f(a)dlmf(x)-f(a)+ [1(x)-f(s)dx. ifs s - x Ja as ,/ -x 3-x J (D.4.5)

But, with the above hypotheses, on the basis of Lagrange's formula

f(x) = f(s)+(x-s)f`[s+9(x-s)),

THE FINITE PART

514

we deduce that the limit from (D.4.5) is zero. Hence, d

f(x) dx= ds / s-x

is

f(x)-f(s)dx+

,/Rx

dx R-x J =

!( s ),.

dx

--1JNf(x)-f(s)dx- f(s) f 2 a

(s - x)3/'

(s - x)3/2 (D.4.6)

2

If we also utilize (D.4.4), then (D.4.3) becomes:

' AX)

(s - x)3/2

d x=

fa f(x) - f(s) dx-

2f(s) s --a

(s - x)3/2

(D.4.7) '

this representing the calculation formula for the Finite Part. It is similar with the formula (D.3.5). From (D.4.5) it also results a derivation formula, in fact, the formula analogous to (D.3.7), which reduces the calculation of the integral with a strong singularity to the calculation of an integral whose singularity is weaker with an unity. In the case we had in view (n = 1), the weaker singularity will be integrable. Indeed, taking (D.4.4) into account, we have: ds

f" f (r) - f (s)

fa

s-x

d x=

d

ds

f (x) d x--

s-x d

(D.4.8)

x - 77=7= f (_)

lax V-9 - i

Noticing now that 8/8R = -8/8x, integrating by parts and taking into account (D.4.6), we obtain:

1f(z) - f(s) 8s s-x

dx S^x,

d a

-1NIf(x)-f(s)]dx (7s l a f (g) Z

N

dx 797-77

x

)dx=

+f(a)-f(s)+

s-a

+f8 f'(x) dx. s -x. (D.4.9)

515

GENERALIZATION

Equating the first members from (D.4.8) and (D.4.9) according to the formula (D.4.5), we obtain: d

dx_ f(a) +1fs fI (x) dx,. 8-x 8-at a s - x

f(.T)

as

ds

(D.4.10)

This is the formula of reduction to a weaker singularity, analogous with (D.3.7). It also proves that the term from the right hand part of the

equality (D.4.2) is finite. Analogously one obtains d ds

x(x)8dx =

I'

-

f (b)s

fx(x)sdx, .

+b

(D.4.11)

The generalization of the definition (D.4.2) is performed as follows

I's 88" an [ f(x) s-x ]d, =

dn

(D.4.12)

d8" Q s-x if f is continuous in a, n times derivable and with the derivative of order n bounded in (a, s) (A.40J.

Generalization

D.5

In the theory of oscillatory wings one may encounter integrals having

he shape (D.4.1) where f depends also of 8. We shall establish for these ones the derivation formula (i

rb AX'S) dx = - f (b, 8) + J/'b fs (x, 8) + fi (x, 8) {D.5.1) x x-e b-s s Vime from which one obtains (D.4.11) in case that f does not depend on s. d4

Indeed, we have d

dsJ, xx-s

fdx 1 ds[rf(x's

L

+f(8,s)

x fss's)dr.+

X

JJ,I

-

(D.5.2)

sJ

But,

d bf(--, cly

y

s,s)dx=\sf(z,8)x

fs8,d)+

z (D.5.3)

+

/'b 8 [f(x1s)_f(ss)1dx8x-8

J

THE FINITE PART

516

If f (x, s) is continuous in x = s and admits the derivative

f=

bounded in the interval Is, b], then, from

f(x,s) = f(s,s) + (x - s)f [s,s+d(x - s)] we deduce that the limit from (D.5.3) is zero. After all, changing 8/8s

by -a/Ox and integrating by parts, we obtain

f

f (x, s

r

d s J,b

+ Jb ff(x,s)

f"[f(x.s) - f (s,

s) d

- f(s,s))sdx =

-f(x,s) -

x-s +

_

g s'

v

ad x+

x

fb f=(x's)dx+

777-

x-s

J3

Xd-f(s,4) j xs b fi(x, s) + fr(x, 9)

f (b, s) - f (s, s)

+le

b-s

1

s)1849

s

b

e x dx-8sr(s'8) x-s

Replacing in (D.5.2) and taking into account that d

s

TS

dx

x-s

b - s'

we deduce (D.5.1). The established formula (D.5.1) shows that the first member is finite. Hence, we may set by definition

r O f(x ss '

d

ds= ds,f f

(D.5.4)

d x,

for every function f (x, s) continuous in the point (s, s), derivable, with bounded partial derivatives in x and a, where s < x < b. In the general case we shall define I

n f(x,8)dx..

dsb

f

(x,s)dx.

(D.5.5)

Appendix E

Singular Multiple Integrals

In the euclidean space with n dimensions E, one considers a domain D bounded or not ( D may coincide with and a function If F(,c) defined on D. We denote t = (fi, ... , r;,,), a = (XI, ... , 1) is unbounded we shall assuine that F tends to zero when Itl -+ 00 in a certain iuanner which will be precised in the sequel. We admit that there exists a point Q(r) in D, such that in every vicinity D. (having

the diameter E) F is unbounded. while, in D - DE, F is bounded mica integrable in the usual seas. Then we set

E-U D-D, IIDFdC=Urn

If this limits exists, it is finite and does not depend on the shape of D,, then the integral will be convergent. Otherwise the integral will be diverrgeiit.

We consider now a divergent integral. If there exists a certain shape of D, (for example, sphere or cube) for which the limit exists and is finite (hence it is unique for every sequence of spheres or cubes contracting towards Q). then the integral will be called semi-convergent. The limit (which depends on the shape of DE) will be called principal value of the integral.

Utilizing spherical coordinates one demonstrates that the integrals having the form

JD'' f

(E.2)

where r = IC - xi, and is bounded in D, are convergent for a < n and divergent for a > n. The case a = n will be investigated

separately. We shall consider the integral having the form V(X) = JD

(X, M)

where m =

r T,

(E.3)

SINGULAR MIXTIPI.E IN1'E<;RALS

518

where f will be called the characteristic, u will be called the density

and x the pole of the integral. The ratio K(x1) = f /r" will be called kernel. All the integrals that we utilize in this book have the form (E.3). The first who studied this type of integrals was Tricomi [A.351 who gives some results in the case a = 2. An ample presentation, which will guide us in the sequel, of the theory of integrals having the forth (E.3), may be found in the books of Mihlin (A.251 and [A.26]. In this presentation. D, will be spheri cal the convergence of the integral will be investigated with respect to this form. We assume that:

satisfies Holder condition in D; if D is unbounded, we 10 assume that u(t;) = 0(It;l-t), k > 0. Holder's condition means that there exists two positive constants A and a, 0 < a < 1, such that for every two points 1;1 and 42, front D sa we have ju(f1)

-

Ajf 1

- 21' ;

(E.4)

2° The characteristic f (x, in) is hounded and for a fixed x it is continuous in in. Under these assumptions, we have the following theorem: The necessary and sufficient condition for the existence of the inteyrnl

(E.3) is to have '

is f(x,rn)dS=0,

(E.5)

where S is the surface of the unit sphere centered in x. In order to make the demonstration, we isolate the pole with a sphere

included in D, having the radius 6 and the center in x. Obviously,

f

(r, n)d,

1)

1

Litre


r

JII--Ui

u(h)f

r"

(u(E) - u(x)J f (T.t-) d o + u(x) l t o J

t
f (rnm)

The first two integrals from the right hand part of the equality are absolutely convergent. In the third one utilizes spherical coordinates

relative to the pole x. Since d = r" 1 d rd S, we obtain J

Cr"

f (x, err)

iss

f (x, rn) J

= In

r

f f (x. rn)d S.

F fs

It results the necessary and sufficient condition (E.5).

519

SINGULAR MULTIPLE INTEGRALS

if this condition is satisfied, the integral (E.3) may be represented as follows

L v.(o

r" (E.6)

r L - nj u(h)f (x,m)(it + inh Mu(d) - u(x)]f (x,m)dt. r

As an application, we shall consider the integral:

f

0

J p 1(c,n)

(E.7)

1 Ox Rcwhere

I? is (5.1.11). With the change of variables q - y = sin 0, the condition (E.5) gives

-x=

008 0,

21r

X (19

I2-.T

0

=

Ro

J0

cos 8d 9 = 0.

(E.8)

Hence, the integral (E.7) exists. The second important theorem that we utilize (for the transonic flow) is the following: If, in addition to the hypotheses 10 and 20, we assume that grad.K(:r. ) = O(r-"° 1), then the singular integral (E.3) (as a function of x) satisfies Holder's condition, with the same exponent like u, in every domain which is bounded, closed and included in D. The theorem was proved for the first time by Giraud in 1934, and for n = 1 by Privalov in 1916. As we can see in (A.251-and (A.261, the demonstration is not simple. The last theorem refers to the derivation of the integrals with weak singularities having the shape

v(x) =

JD

u(h)f

(E.9)

which lead to integrals of the type (E.3). Like in the previous theory, D may be a domain bounded or not of the space E,,, or it may coincide with the entire space. W e assume that the function f (x, m) is continuous and bounded together with its first derivatives (the first order derivatives with resped to the cartesian codrdiriates of the points x and na). We also assume that u(i:) satisfies Holder's condition and

at infinity (if D is unbounded)

o('t;-'J), t > 1. Under these

520

SINGULAR MULTIPLE INTEGRALS

assumptions, there exist the first derivatives of the integral (E.9), and they are given by the formula axk = ID

xk

[1t]

d

, - u(x)

j

s

f(x, m)

cos(n,

xk)d S, (E.10)

where, like above, S is the surface of the unit sphere centered in x, and n, the outer normal to the sphere. Olviously, the first integral from the right hand part of the equality is singular.

Appendix F Gauss-Type Quadrature Formulas

F.1

General Theorems

This appendix relies on the paper [A.49] and it is completed with some results due to Monegato [A.52). Gauss-type quadrature formulas give exact evaluations for the integrals of polynomial functions, multiplied by a weight function w. In aerodynamics it also meet integrals with singularities. Approximating an arbitrary function (according to Weierstras's theorem) by a polynomial function, we may utilize these evaluations. Practically, the approximation is performed by a Lagrangetype interpolation formula. We shall consider in the sequel that to : 1-1, +11 - R is a positive integrable function. THEOREM 1. We have the exact evaluation

+i 1

n

Af (x.)

f (x)w(:c)d x

(F.1.1)

0=1 if.

10 f is a polynomial of degree < 2n -- 1; 2e the points x = x,,, a = 1, n are the n zeros of the polynomial P,,(x) of degree n from the orthogonal system of the weight w(x) on [-1, +1j

w(x)P,(x)Pj(x)dx = 0,

i 0j;

(F.1.2)

30 Using the notation

Wt) = fP(x)!.Q3dx,

(F.1.3)

the coefficients A. are given by the formulas Qn(2a)

4a = Pn(xa) .

(F.1.4)

CAUSS-TYPE QUADRATURE FORMULAS

522

Proof. Taking into account that f is a polynomial of degree 2n-1, and P is a polynomial of degree n, we may write

f

n

a,

E 27 -xa + Fn_1,

T-

(F.i.5)

where Fi_1 is a polynomial of degree < n - I. We determine the coefficients a° multiplying (F.1.5) with x - x° and putting x = xa. It results (F.1.6)

as = f(z°)/PP(xa)

whence

n __

f

P.

f(xa)

=1 Pn(X-)(Z - xa)

+ A,

Since

dx =

J- 11 n( ) )(x

)

( )`vn(xa) a)

=

and because F,,-, may be written as a linear combination of P0,.. . , Pi_1, such that +1

Pn(x)FF-1(x)w(x)dx = 0, we deduce (F.1.1). THEOREM 2. We have the following exact evaluation:

rt

J- t

x; d x = E A° f (x° n

f (x)

a=1

.

(F.1 (F.1.7)

x

where j = 1, 2, ..., if 10 f is a polynomial of degree < 2n; 2° the points x = x°t a = T-, n are those defined in Theorem 1;

3° the points t = t,, j = 1;2,..., are the zeros of the function Qn defined by (F.1.3);

40 the coefficients A are defined by (F.1.4).

523

GENERAL THEOREMS

Prof. Reasoning like before, we have:

f P_E ac. +Fn, X X0, A-1

where the degree of F, < n. It results that as has the form (F.1.6). Setting F = (x - tj)Fn_1(x) + A, where the degree of Fs_1 < n - I, we deduce

f

P'

L-i

whence

+1

P"(x)f(x0)w(x)

LI f (xa

Pn(x.)(x-xQ)(x-tj)dx__

'(+1 P,(x)w(x)

P., (X-)

A

P (xQ)(x - x.) + (x -

1

A. (:r.) _ QXQ - tj

xQ - tj

[_jdx= 1

1

f (zz)Qn(t5)

1n(xa)(xa-tj)

_ A. f (x,)

xQtj

because Q,, (t,) = 0. THEOREM 3. We have the exact evaluation

I

(f(x)

(F.1.9)

t

where the "asterisk" is for the Finite Part (I).3.5), if.-

10 f is a polynomial of degree < 2n + 1;

20 the points x = x.., a = 1, n are those defined in Theorem 1;

30 The points t = tj, j = 1, 2.... are those defined in Theorem 1; 4e the coefficients An are given by (F.1.4), and A = Qgn(tj) Pn(tj)

(F.i.10)

Proof. We shall notice at first that, on the basis of the definition (D.3.G), we have for t E (-1, +1),

t0adx,

(F.1.11)

524

GAUSS-TYPE QUAUR .ATURE FORMULAS

whence

t

Q;,(tJ) =

jPn(x)(')2dz.

(F.1.12)

We shall write m above is

PA

f - P" LF-

(F.1.13)

)x-)

being a polynomial of degree :5 n - I and B, C, constants. For x = tj it results

I(ti)

C

F&

f(3.-*)

(F.1.14)

0-1 Fnxa)(tj - xa)

P"(tj)

Since

_

1

(x - xn)(x - tj)2 2

1

1

tj)2 x

(rct

1

x,.

1

1

(z _tj)2 x-tj + tj -xo (x-tj)2 Taking into account that

0 and that Ave have (F.1.12),G

we deduce

1+`

P"(x)f(x.)w(x)

f-1 Pn(x.)(x - X.)(x - t j)2

= f(x.) (x.-tj)2( x-x P (x.) (' +1 P"(x)w(x) 1 A 1

dx=

- r. 2- t3/ d x+

' +1

+J-1

(F.I.15)

dx

ty-xo

(x-ti)2

f(xa) Q.(xrx) + Q;z(tj) P.(x0) (x,, - tj)2 tj - Xw Having in view the definitions of the Principal Value and Finite Part,

it results t

T'I

x - tj fl+' P.(r)u'(x)dx

x - tj

4n (tj) = 0

Utilizing (F.1.14) and (F.1.10), we deduce

1Pn(xQ)(tj +!

C

+

_1

P"(x)(xu,(x)

- tj)

2dx = AI(tj) -

y 1{XQ}'^Ln(tj)

- xa)

525

FORMULAS OF INTEREST IN AERODYNAMICS

whence it results (F.1.9). The integrals having the form

wx ( )n :

+1

1 f

(x) (x

n>2

'

(F.1.16)

1

are studied in [A.49] and [A.52].

F.2

Formulas of Interest in Aerodynamics

It is well known that on the interval (-1, +1] Jacobi's polynomials (x) constitute an orthogonal basis with the weight function w(x) _ (1 - x)°(1 + x)' ( see, for example, (A.561). Obviously, the zeros xo and t) from the theorem--, from F.1 do not depend on the factor of normalization of the polynomial P,,. One can simplify this factor in A,,

from (F.1.4) and A from (F.1.10). We shall utilize therefore

Jacobi's polynomials without the constant factor. 1°. For the weight w(x) = (1-x2)-1/2, Jacobi's polynomials reduce to Chebyshev's polynomials x = cos9.

TT(x) = cosn9,

(F.2.1)

F o r 0 < 0 < jr, the polynomials T. (n = 1, 2, ...) vanish when

9° - 2a-1 w , n

2a-lir ' a = Tn.

xQ_

- oos

2

n

Utilizing the notation t = coax

2

and Glauert's integral (B.6.6), we

deduce:

sin or

(F.2.3)

Qn(t) = x sin or

with the zero

a,=? , tj=cos

r(x) _ -

Since

1

-1-,n n

(F.2.4)

sinn9 sin9

dTn

sing dO from (F.1.4) it results that A. = 7r/n. Since n(t))

V

(F.2.2)

d sin na

- -slam do sino'

= nir cos ja cf

sill2

526

GAUSS-TYPE QUADRATURE FORMULAS

from (F.1.10) it results that A = -n7r/(1 - t). We deduce therefore the following formulas: tt

411

f(x) dx = T Ef(xn), it 1-x 0

r J-

i

+I

+1

f (x)

dx

7r c- f (x, )

y:"]X-tj

dz f(x) (x _ tj)2 1-Z

J-1

(F.2.5)

1

n i-.

(F.2.6)

n.Lt f (v)

tt7r

(xa - t;)2

1 --

t=f(tj)

(F.2.7)

for j = -, n- 1, where .r are given by (F.2.2) and tj, by (F.2.4). 211. For the weight function w(x) = (1- x2)1/2, Jacobi's polynomials reduce to Chebyshev's polynomials of second order sin( e1)9 sin

Cyr

+

e"

I'

1

x=

xa - coy re[+ 1 '

(F.2.8)

a = 1, nrt .

(F.2.9)

Using the notation t = cos a and Glauert's integral (B.6.6), we deduce:

-ircos(n j- I)a

(P.2.10)

whence

2, j=1,ri+1.

tj=cvs't

(F.2.11)

1

Utilizing the formulas given in the theorems F.1 for A,, and A, on the basis of Glauert's integral, we obtain:

- n4-1 (1 -" xa),

A = -r (n + 1)

.

Hence, the following quadrature formulas are established: n

+'

1-1`.f(x)dx=n+1E(I-

+1

xt

f(rn),

(P.2.12)

es=1 f(r.)

x -tj

dx=

7r

I - xp f( :ra ),

-t;

(F.2.13)

527

FORMULAS OF INTEREST IN AERODYNAMICS

+1

1- xa

ir

f (x)

1-z (x-tj)2dx=n+1

J-1

(xQ-tj)2f(xo)

awl

(F.2.14)

-ir(n + 1)f (tj), where j = 1, n + I, xa are given by (F.2.9) and tj by (F.2.11). We notice that the numbers tj given by (F.2.4) are the zeros of the and the numbers ti given in (F.2.11) are the zeros of the polynomials Tn i(t). x)1/2(1 + x)''/2, Jacobi's 3°. For the weight function w(x) polynomials

polynomials are [A.56].

Pn(x)

=

with the zeros

sinj(2n + 1)8/2]

_

2aar

x = cos 0,

,

a=1-,n.

(F.2.15)

(F.2.16)

With the aid of Glauert's integral, we deduce

/2j'

-,'[(2n

Qn(t) =

cos(

CO6C

/Z)

whence

9=l .

(F.2.17)

Utilizing (F.1.4) and (F.1.10), it results:

A. =

2n 2n+1(1-xa),

A

_ 2n+1x

1+tj 2'

Hence, we established the following formulas: +1

J-

1

r+i . +1

_0

2w

fin

1f(x)dx 2n+1` a-i

J

- x f (x)

_

(1-x.)f(x.),

27r

+x x-tjdx 2n+1 a-i -a

1-z

f(x)

-I V 1+x (x - tj)2 n

(F.2.18)

(F.2.19) 13

dx= _

(F.2.20)

j 2f(tj)

2n+1r(Q-tj)2f(x.)- +

528

GAUSS-TYPE QUADRATURE FORMULAS

j = 1 n,

for

being given by the formula (F.2.16) and tj by

x,,

(F.2.17).

40. For the weight function w(.r.) = (1 - x)'"112(1 + r)h/2, Jacobi's polynomials are

'n(s) with the zeros

cos((2n + 1)9/2J cos(0/2)

2a -1

r

x = cos9,

o= ln.

2n+1T,

(F.2.21)

(F.2.22)

Utilizing Glauert's integral, we deduce:

Qn(t) _

sin[(2n+ 1)Q/2] s in(a/2)

,

t=CADS a,

polynomials which have the zeros

t'-

(F.2.23)

2n+1'

On the basis of the formulas (F.1.4) and (F.I.10) we obtain 2'7r A"2n+1(1+xn),

2n + I An=-1-t,

ar

such that one establishes the following formula, +1

J

1+x 1

'

n

L(1 +x,,)f(F.2.24) f (x)dx = 2n + 1 27

J1 + x f (x)

21r 1-xx-tjdx=2n+l

1 + x0

1+x

ar 2n+1 it 1-t; ?f(tJ)

s"-tjf(n),

(F.2.25)

1

f

1--x {x-tJ)2 (F.2.26) tar

n

2n+I

owl

(ca-tf)-f(x°)'

for j = 1, n, the zeros xa being given by the formula (F.2.22) and tj by the formula (F.2.23). We have to notice the relations between the formulas from 30 and

40. The points x,, from 3° coincide with the points t1 from 40, and x,,, from 40 with t,, from 3°.

529

TILE MODIFIED MONECATO'S FORMULA

F.3

The Modified Monegato's Formula

It is preferable sometimes to replace the formula (F.1.9) which contains the numbers f (t;) by the formula given by Monegato (A.52] p. 279. t1,(r)

f(r)

dx=>

(F.3.1)

(x - t)2

J-1 where

Q1. (Z.) - Q. (t) - Q/n(t)(x° - t)

Pn(x°)(x° W

(F.3.2)

,

the formula containing only the numbers f (x°). The formula (F.3.1) is exact, i.e. Rn (f) = 0. if f (x) is a polynomial of degree n - 1.

In fact, in applications one utilizes not the formula (F.3.1) [5.10), [6.5], but another one which may be obtained as follows.

In (F.3.1) we isolate the term corresponding to a = j and we pass to the limit, +1

t -.

We find

n,

ut(x) (x f(x) dx = E wa' (xj)f(x°)+ - xj)2 °=1

+wj(xj)f(xj) + Rn(f),

(F.3.3) .

where the mark at F, means. that one excepts the term corresponding to a = j. The factor w'°(xj) is obtained from (F.3.2). Using the rule of I'Hospital u J(xi) we find that q., (xj)

(F.3.4)

In applications one utilizes (F.3.3) for w(x) = (1-x2)1/2. The numbers xo are given therefore by (F.2.9). From an elementary calculation it results Qn(s'a) = -7r(-1)° , Q'n(xj) = 0,

+ 1)(-1)° X'2

,

Qn(xj) = 70 + 1)2 01 s 1-x12

530

GAUSS-TYNE QUADRATURE FORMULAS

whence f(x)

+'

1

_-2

E [1 (-1)j]

(xa -

0=1

2f(:rn) -

n2

1f(xr)+ R,,. (F.3.5)

This is the modified formula. From

f d_

+t

w(x)

1.-t

uwa(t)f (xo) + Rn(f)

a

(F.3.6)

a=1

where (A.50J page 275, ww(t)

Rn (xo)

Qn(T )

- Q.t (t)

(F.3.7)

,

isolating in E the term for which a = j and passing to limit t --+ xj, one obtains the formula ry1

-t

Tl

u

w(x) f (x) d x =

x - xj

',)f (.r.)+

owl

xa)

(F.3.8)

f(x '

For the weight function w(x) = (1 - x2)1/2 we find: +t 1

n

1 - x2 1(x) cl x =

amt

n+1

2 f( X( -xj x° ) (F.3.9)

F.4 A Useful Formula We shall establish the following series expansion:

= -2E(j + 1)Uj(y)Uj(ri)

1

(n

X1)2

1

(F.4.1)

531

A USEFUL FORMULA

where U (x) are Chebyshev's polynomials (F.2.8). The series is divergent, but it has a first Cesaro finite sum: n

(tl

-2

1 y)2

I

U + 1)Ui (y)Ui(q) .

(F.4.2)

Indeed. setting q = ooeO and y = oosa on the basis of Glauert's integral, we deduce

r+1Ua(q)dq=-aoos(n+1)o, (F.4.3)

and deriving and taking (D.3.6) into account, we obtain the formula

J

(q - y)z

d q = -w(n + 1)U*(y).

(F.4.4)

1

Expanding now the function (q - y)'2 on the interval [-1, +11 in a Chebvshev-tvpc polynomial series, we have: 1

aJ(y)Uitq).

(F.4.5)

(q - y)" =

Taking into account (P.4.4), and the orthogonality condition

2!

UJ(1)Uk(q)dq =

10 JiAk J7r

,1 = k,

(F.4.6)

we obtain:

a,(y) = -2(j + 1)UJ(y). Replacing it in (F.4.5), we obtain the formula (F.4.1).

(F.4.7)

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Index

Abel's equation, 320 Acceleration

as material derivative , 2 of the particle, I potential, 9, 397, 404 Ackeret's formulas, 288, 423 Acoustics, equation of, 29 Aerodynamic action drag, 171, 172 lift, 76, 170, 172 moment, 76, 170

gyration, 171, 172 pitching, 171, 172 rolling, 171, 172 Arrow of airfoil, 83 shaped wing, 156 Attached shock waves, 361

Complex velocity, 71, 75, 120, 371, 375 Compressibility effects, 86, 148, 231 Conduction law, 13 Cone body's shape, 463 double. 464 Conical flow, 339-347, 464 Conservation laws, 11 Convolution, 33, 370, 470

Critical velocity, 6, 359 Delta wing, 156 Descartes, folium, 18 Detached shock waves, 19, 360 Distributions, theory of, 465.479 Doublet

Bernoulli's integral, 5-7 Bessel's functions, 407, 503 Biplane, 92

density, 266 flow induced by, 148 potentials, 426 Drag coefficient co, 181, 203, 204, 213, 229, 231, 241, 292, 458, 462,

Body, theory of slender, 449-464 R":rndarv conditions

Duhamel principle, 57

airfoils in tandem, 1:,Y airfoils parallel to the undisturbed stream, 93

grids of profile, 99 ground effects, 83 material surface, 24 rest state, 28 tunnel efects, 88 uniform motion, 25, 70 Boundary conditions (nonlinear case),

464

Energy, 3, 4 Enthalpy, 15 Entropy, 4 Euler -Lagrange criterion, 24 constant, 400 equation, 3 formulas, 146 theorem, 2

111, 112

Flutter, 397 Caloric equation, 4 Cauchy integral, 71 principal value, 482, 510 problem, 61 Characteristics coordinates on, 308 variables on, 318 Chebyshev polynomials, 80, 525, 526 Circulation, 125, 126, 274, 393 Clapeyron's equation, 4 Collocation method, 115, 231, 279 Complex

potential, 121, 146

Flux-divergence formula, 479

Forces, continuous distribution of, 73 Fourier, transform, 465-479 Fredholm integral equation, 209 Fundamental matrices, 61 Fundamental solutions equation of potential fluid at rest, 42

M = I oscillatory, 34 M = 1 unsteady, 41 subsonic oscillatory, 33 subsonic steady, 31 subsonic unsteady, 37 supersonic oscillatory, 34

INDEX

572

Fundamental solutions equation of potential supersonic steady, 31 supersonic unsteady, 38 oscillatory system

pressure formulae, 50 velocity formulae, 52 steady system general form, 45 plane subsonic, 46 plane supersonic, 48 3-D subsonic, 47 3-D supersonic, 48 unsteady system, 57

Causs-type quadrature formulas, 521530

Clauert integral, 489 method, 210 Gothic wing, 156 Green function, 142, 143 Ground effect, 82, 86, 136, 184, 238, 299

Hadamard finite part, 509 type integrals, 513 harmonic forces, 398 Heat, specific c,,, c.,, 4 Helmholtz's equation, 5, 33 Henkel's function, 33 Homentropic motion, 4 Huygens' principle, 42 Ideal fluid, 1 Instable shock, 18 Integral equations steady subsonic flow

lifting line, 201, 209, 219 lifting surface, 165, 167, 187 airfoils in tandem, 103 grids of profiles, 99 ground effect, 84 parallel airfoils, 94 tunnel efect, 90 thin profile, 72 steady supersonic flow lifting surface, 307, 313, 320

Integral equations

steady transonic flow lifting line, 395 lifting surface, 389 unsteady flow sonic profile, 440 sonic wing, 445 subsonic profile, 402

subsonic wing, 408 supersonic profile, 418 supersonic wing. 432

Invariant, 16 Irrotational flow condition, 368 definition, 5 equation of, 109 Isentropic motion, 4 Joukovaky profile, 86

Lagrange

-Cauchy's theorem, 5 interpolation, 223 variation of constants, 218 Leading edge, 62, 156, 301 subsonic, 301 supersonic. 301

Lift coefficient, 129 Lift coefficient CL, 76- 78, 81, 82, 85, 86,90, 96, 100, 104, 107, 171, 180, 183, 193, 195, 202, 204, 213, 229, 231, 237, 241, 287, 288, 291, 293, 298, 299, 350, 354, 357, 380, 421, 423, 441, 457, 462, 464

Mach angle, 40 cone, 31

dihedron, 32 number, 8, 19 Moment coefficient cm, 76-78, 81, 82, 85, 86, 90, 96, 100, 104, 107, 287, 288, 291, 298, 299, 421, 423, 441

Moment coefficient cs, 203, 204, 229, 231, 241, 458, 462

Moment coefficients c1, cv, 171, 181, 183, 19.3, 195, 2(r2, 229, 231, 237, 241, 458, 462. 464

INDEX

573

Plemelj's formulae, 483 Prandtl's theory, 197-205 Prandtl-Mayer fan, 289

Source mass, 43

Pressure coefficient Cp, 120, 129, 148

Swallow wing, 156

Rhombic wing, 156

Thermodynamics, equation of, 3 Trailing edge, 156. 301

Shock waves

Hugoniot's equation, 15 jump equations, 13 Prandtl's formula, 16 shock polar, 18 Sonic barrier, 438, 448 Sonic circle, 18, 19

potential, 36

Kutta-Joukovski, 81, 165, 275,332 subsonic, 301

supersonic, 301 Trapezoidal wing, 156

Vortex, 5, 71

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