Aerodynamics

A vortex line is defined to be a curve s constructed at a defmite instant ill a fluid flow and having the property that at each point ·of it the angular velocity vector 0) coincides with the direction of a tangent. A vortex line is constructed Similarly to a streamline (see Fig. 2.1.1). The only difference is that the angular velocities of rotation of the particles 0) (0)1' 0)2' etc.) are taken instead of the linear velocities. In accordance with this delinition, the cross product .00 X dS = 0, Le.

I :~

w~ ~z l=i((o)~dZ-WZdY)-j ((o),;dz-wzdx)

dx dy dz

+k (w%dy-w~dx) =0

Cn. 2. Kinemlltics of II Fluid

l-Ience, taking into account that, for example, COy dz = 0, and so on, we ohtain an equation of a vortex line: dxlw;r;

=

dylw y = dzlw z

91

U1 z

dy =

(2.6.1)

13y consl-tucting vortex lines through the points of an elementary -contour wilh a cross-sectional area of 0, we obtain a vortex tube. The product wo is called the i.ntensity or strength of a vortex tube. Ot· simply the vorticit~·. We shall prove that the vorticity is coustant for all the sections of a vortex tube. For this purpose. we shall use the analogy with the flow of an incompressible fluid for which div V = O. A corollarr is Hle flow rate equation for a stream filament VIS I = V 2 S 2 :-:: • . . . -= VS = const. Let us consider vortex motion (lnd the expression for the divergence or the angular yclocity div

(r)

= QOl;r;IIJx -i- IJUlyfIJy

+ Qooz//)z

Substituting for the componenls of w their values from (2.2.3) and (2.2.fi), we oblain. provided that the second derivatives of l'x, V y , and V z arc continuous, the expression tii,· w ,..." O. using the analogy with the flow rale equation VS = const, we obtain an cClnatioll for the flow of the vector w along a vortex tube in the form 00 1 0"1

=

U1 2(J2

= ... = wo = canst

(2.6.2)

This equation expresses the IIelmholt~ theorem on the constancy of the vorticity along a vortex tube. The properly of a vortex tube consisting in that it cannot break oD suddenly or terminate in a sharp point follows from this theorem. The termination in a sharp point is impossible because at a cross-sectional area of the tube of 0" _ 0, the angular velocity w would tend to infmity according to the IIelmholtz theorem, and this is not real physically.

1.7. Velocity Circulation Coneepl

Velocity circulation is of mAjor Significance in aerodynamics. This concept is lIsecl when studying the flow over craft and, particularly, when determiniug the lift force acting on a wing. L(>t us consider in a fluid flow a fixed closed contour K at each point of which the velocity V is known, and ent\lIate the integral -over this contour: (2.7.1)

92

Pt. I. Theory. Aerodynamics of an Airfoil and


Wing

where V·ds is the dot product of the vectors V and cis. The quantity r determined in this way is known as the circulation of the velocity around a closed contour. Since V=V:J'I-Vllj-j-Vzkandds=dxi+dyj dzk we have

r,.

I V.dx".V,dy··'-V,dz

(2.7.2)

("'

Taking also into account that the dot product V·ds """ -- V cos (V........ds) as '-' Vs ds, where Vs is the projection of the velocity onto a tangent, we obtain

r.~

\ V,ds

(2.7.3)

(K)

II the contour coincides with a streamline in the form of a circle of radius r at each point of which the velocity V. is identical in magnitude and direction. \,,,'C have (2.7.4) r .-= 2nrVs in

The circulation of the vclocity in a vortcx-frec flow can bc cxpressed lerms of the vclocity polential. Assnming that V", dx "I.:~ V: dz =-: dqJ. we obtain for an open contour

-i- VII dy

l'-= ) dq.-='f!A-lJ"n

(2.7.5)

(Ll

wherc
n.......

Let us consider elementary contonr ABeD (see Fig. 2.2.1) and evaluate the circulation around this contour. We shall assume that the velocities are constant along each edge and are equal to the following values:

V.(AB). V,+ ';" dx(8C). V.+ ';'" dy(CD). V,(AD) ConSidering the counterclockwise direction of circumventing the contour (from the x~axis toward the y-axis) to be positive, we obtain

Ch. 2. Kinemllti<:;s of II Fluid

93

for the circulation

d;;, dX)dY-(V

dr:=V:rdx~-(Vv+

dr. =

x

;"

(dVyldx --uV.,)uy)

d~r

",;y'" dy)dx--V"dy dy

According to (2,2.3), the quantity in p
dr,

~

200, dx dy

(2.7.6)

We can prove similarly that

dr", ,.--, 2w", dy dz and dry"'''' 2w g d:c dz In these expressions, the products of the dilterentials arc the areas confmed by the relevallt elementary contours. With Ii view to the rcsults obtained, for the area clement do oriented in space arbitrarily and confmed within an elementary contonr, the circulation is (2,7,i) where Wn is the component of the angular velocity 1I10ng the direction of the normal n to the surface clemen!. du. In accordance with (2.7.7), the "elocity circulation aronnd an elemcHtary closed ('o\tlollr eq\lals the double strength of ,\ vortex in~ide the Helation (2.7.7) can be used for II conlOHr I-tnile conlining a surface S lIt each poilll of whkh thc (If (')" is H(!r<' the veloCity circulation around !itl' \'ontultr i:-:

r-211 '.'.,

d"

(Ii)

Formula (2.7.8) expresses the Sloi;:C'!" titc-Mc-m: the lation arollnd the closed contour 11 equals tlte .]ouhle strt1ngtll or the vOrL ices passing lhrOltg-ll !he slIrface contoul'. The quantity determined hy tlli...; rormu!;) (vol'l('.'( sll'ell~lh) and i:> ,iesignalC'll hy x_

2~ ,

(')n

(2.i .R')

{ia

'(8i

Comhining formulas (2.i.:\) ;lilt! (2.i.X) I'O!' 1'. We' ohlailll\ relalion expn'ssing the integral oyet· the contour f( ill 1f'I'ms of lilt' inli'g'I'I\\ lhe surface S COllflll{'II within this coulollr A': \ V, ds - 2 .\ (i.:)

i

IS)

<0"

du

(2.7.8")

94

PI. I. Theory. Aerodyn.mics 01 an Airfoil and a Wing

Flg.2.7.t

Simply and triply connected regions on a plane: .. -~imply co~n(>cled r('gion (rK _ circulation ov~r th(> conto\lr K); b-triply connected region 0'1, = cJrcula Hon of tbe eJ:tem a J contour, rK, and r x , - clrculatlon~ over the In_ t~ma J conloul':!l X, and X., respectively)

The above expressions relate to 0. simply connected contour (the region being considered is confIDed within a single contour, Fig. 2.7.1a), But the Stokes theorem can also be extended to multiply connected contours (a region confIDed by one external and several internal contours). Equation (2.7 .8~) is used provided that the external contour is connected to the internal ones by auxiliary Jines (cuts) so as to obtain a simply connected region. Now t.he double integral in (2. 7 .8~) is used for the hatched region (Fig. 2.7 .1 b) , while the contour integral is taken for the obtained simply connected region, i.e. around the external contour, along all the cuts, and also around all the internal contours. In accordance with this, by formula (2.7.8"), and also with a view to Fig. 2.7,1b showinJ! 8 triply connected region, we obtain

Jwnda

r K-rK1 -rK.=2 J

(8)

whence the circulation over the contour K of the region being considered is

r K=rKI+ r X.+2

JJ

wnda

(8)

Using this formula when we have n internal contours, we lind I-

rK =

hr ;

Kt -/-

2

Jj

Ctl n

do

(5)

Vortex·lncilieM V.loeHl..

Appearing vortices produce additional velocities in the Ouidfilled space surrounding them. This effect is similar to the electromagnetic influence of a conductor carrying an electric current. In

Ch. 2. Kinemetics ot

110

Fluid

95

d[

accordance with this analogy, the velocities produced hr a vortex arc said to be induced. This electromagnetic analogy is expre5seu ill that Lo determine the vortex-induced "elocity, a Biot-Savart relation i.!' used similar to the one expressing the known law of eleclronwgnelic induction. Let us consider a curvilinear vorlex of all arbitrary shape (Fig. 2.7.2). The velocity vector dw induce(] by the vortex clement dL at point A whose location is rletermiue(\ by the position vector r coincides in direction with the cro."s prodnct r X dL, i.('. the ,"N'.tor dw is perpendicular to the plane containing the \'ectol'S rand dL. The value of dw is determined with the aid of the Biol-Savart rormula, which in the vector form is as follows: dw = (r/4n)

I'

X dL r3

(2.7.n)

where f is the velocity circulation. The derivation of formula (2.7.9)' is given in [101. Since the magnitude of the cross product I r X dL I = r sin 0: dL. where 0: is the angle hetween the direction of a ,'ortex element and the position vector r, the magnitude of the velocity induced at point A is (2.7.10) dw = (f/4n) sin a dLr z Let us use the Biot-Savart relation to calculate the velocily induced by a section of a liue vortex (Fig. 2.7.3). Since r = h/sin 0:. dL = r da:/sin 0: = h do:/sin z 0:, then U)=

4~h

''>

)

.,

sinada=

4~h

(cosa l -coso:Z)

(2.7.11)

For a vortex hoth of whose ends extend to infinity (an inlinite vortex), 0:1 = 0, 0:2 = n, aud. therefore,

w

~

1'/(2nh)

(2.7.12)

96

Pt. I. Theory. Aerodynamics of

r~~'u!:~

a nuid

of a line vortex on p
.n Airfoil .nd •

Wing

~

"

"

fig. 1.1."

Interaction of \'orliccs: o_vorlicrs wilh (In idpntlcat dirPl"

1;011 of mlalion; b · · \·"rUc(·~ with opposite dir,.~tiun~ of motion

For a vortex, one end of which extends to inrmity, and the other has its origin at point A (a semi-infinite vortex), we have a l -::: 0 and a. t •. n/2. Consequently w ~ r / (bh) (2.7.1:1) If a nuid accommodates two or more vortices, they interact with one another, and as a result the vortex system is in motion. The velocity of this motion is determined wilh the aid of the Biot-Savart relation. Let liS tnke liS an example two infmite vortices with the same strength nnd direction of rotation (Fig. 2.7 .4a). These vortices impnrt to each other the velocities V ~ :-:: - l'/(2nh) and VI -:--: = r /(2nh) that
Ch. 2, Kinem.tics of • Fluid

97

in the theory of the functions of a complex variable are known as the Cauch:,r-Riemann eqltations. They express the necessary and sufficient conditions needed for a (',ombinatioll of the two functions cp .L- i'\f.l to be an analytical function of the complex variable cr .= x iy. Le. to be differentiable at all point~ of a certain region. Let us introduce a symbol fOl" this fune.tian:

+

W(cr) -=
+ i'\f.l

(2.8.1)

The function W (0) that is determined if the fUllction of two real variables If ::-; If (x, y) and W=: W(x, y) satisfies differential equations (2.5.9) is called a complex potential. If we recall that the values of the functions If (x, y) or~ (x, y) allow one to determine the velocity field in a moving fluid unambiguously, then, (',onsequently, any two-dimensional flow can be set by a complex potential. Hence, the problem on calculating such a flow can be reduced to finding the function W (0). Let us ('alculate the derivative of the function W (0) with respect to the complex variable cr: dWId
+ iil-f18x

(2.8.2)

Since iJlffiJx = 1':n 8¢/Ox .= - Vy• tlten dWldo = Vo; - iV"

(2.8.3)

This expression is called the complex velocity. Its magnitude gives the value of the ,reiodty itself, I dWfdo I -= VV~ -;- V~ = v. It is evident that the l'eai velOCity vector V = V.., -j- iVy is thp mirror image relative to the x-axis of the vector of the complex velocity. Let us denote by 9 the angle between the vector dWldo and the x-axis and determine the velocities V % = V cos 9 and VII = = V sin O. Using the Euler formula cos 9 -

i sin 9 = e- fi

we obtain dWfda

= Ve- f8

(2.8.4)

2.9. Kinds of Fluid Flows Let us consider the characteristic kinds of flows of an incompressible fluid, their geometric configuration (aerodynamic spectrum), expressions for the complex potentials, and also the relevant potential functions and st.ream functions, 1-01715

98

Pt. I. Theory. Aerodynamics 01 an Airfoil lind II Wing

'.r.II_1 Flow

Assume that the flow of a fluid is given by the complex potential W (0) - V (cos 9 - i sin 9) 0

(2.9.1)

where V and e are constants for the given conditions. According to (2.8.1) (J) + i1p = V (cos e - i sin 9) (x + iy) whence we find the velocity potential and the stream function:
(2.9.2) (2.9.3)

1p= V(ucos9-xsine)

Examination of the expressions for

= V ~ = V cos e,

O!.fJ/8y = VII

=

V sin e, O(J)/8z

= Vz

= 0 (2.9.4)

Here V is the resultant velocity of the flow, and e is the angle between its direction and the x~axis. Assuming the stream function "p in (2.9.3) to be constant and including V in it, we obtain the

equa~

tion y cos e - x sin e = const

(2.9.5)

A glance at this equation shows that the streamlines are parallel straight lines inclined to the x~axis at the angle 9 (Fig. 2.9.1). Since the velocity components V~ and VII are positive, tIte flow will be directed as shown in Fig. 2.9.1. This flow is called a forward plane~ parallel one. In the particular case when the flow is parallel to the x~axis (9 = = 0, VOl: = V, VII ...;; 0), the complex potential is W (0) -

Vo

Two~DlmenslollllJ

_Sink

(2.9.6) Point Source

Let us consider the complex potential (2.9.7) W (0) - (qI2") In 0 where q is a constant. We can write this equation in the form ~ + i'i - (qI2,,) In (,,>0) ~ (qI2,,) (111 r + tel

en. 2. Kinem!ltics of II Fluid

99

v,

Flg.l.9.t

General view of II forward plane-parallel now

Flg.l.9.l

A two-dimeDSional point source

where r is the distance to a point with the coordinates x and y (the polar radins), anti 0 is the polar angle. It follows [rom the obtained equation that ~ ~ (qI20)

In r

~ (q/2n)

'" ~ (qI20)

In V x' -\- y'

e

(~.U.8) (~.9.9)

'Ve find from (2.9.8) that the radial velocity ('omponent (in the direction of the radius r) is iJcp/iJr = V~ = q/(21(r)

(~.9.!O)

and the componellt along a normal to this radius is Vs = O. We thus obtain a flow whose streamlines (pathlines) are a family o[ straight lines passing through the origin of coordinates (this also follows from the equation of a streamline 1j> = const). Such a radial flow issuing from the origin of ('oordinates is called a two-dimensional point source (Fig. 2.9.2). The rate of flow of a fluid through a contour of radius r is 2:nrl'r = = Q. Introducing into this eqllat.ion the value of V~ from (2.9.10),

100

pt. I. Theory. Aerodyn~mi(;$ of ~n Airfoil ~nd ~ Wing

we fmd Q = q. The constant. q is thus determined by the rate of flow of t.he fluid from the source. This quantity q is known as the intensity or strengtb of tbe source. In addition to a source, there is a kind of motion of a Ouid called a two-dimensional point sink. The complex potential of a sink is W (0) = - (qI2~) In 0 (2.9.11) The minus sign indicates that unlike a source, motion occurs toward the centre. A sink, like a source, is characterized by its intensity, or strengt.h, q (the rate of flow). !hree·m.enslon.r Source and Sink

In addition to two-dimensional ones, there also exist three-dimen· sional point sources (sinks). The flux from them is set by the following conditions:

V~= 4~

'7;

Vu= ,~ . ~:;

v¥=

4~' ~

(2.9.12)

whore R = V Xl + y2 + z.2 aud q is the strength of the source (plus sign) or sink (minus sign). The strengl.h of a source (sink) equals the quantity q defined as the flow rate (per second) through the surface of a sphere of radius R. The resultant velocity is V=VV~+V:+V:=±q/(4nR')

(2.9.13)

and coincides with the direction of the position vector R. Conseqnentl~·. the velocity potential depends only on R and. therefore.

8
~'ields

'I' = 'F q/(4nR)

(2.9.14)

where the minus sign relates to a source. and the plus sign to a sink.

_lot Let us consider a flow whose complex potential is W (0) = (MI2n) (1/0)

(2.9.15)

where M is a constant. In accordance with this equation, we have

I"

Let us transform the right-hand side of this equation. Taking into account that

r:.8 = r(e08ij~UiD8)

+(cos9-iSin9)

Ch. 2. Kinemlltics of II Fluid

101

FJI.:a.U To the definition of a doublet: B-doubll't streamlines: b-Ionnatlon ut a doublet

we obtain

'P

+ ilP = (l"fl2~r) (cos 9 -

i sin 9)

Hence (2.9.16) (2.9.17)

'P = (MI2:n) cos 9/r lP = - (:rfl2n) sin aIr

+

Assuming that til ~~ const and having in view that r = V x'J. y'l and sin f) "-- ylr ...., Y/V .rt + y'1,. we obtain an equation for a family of streamlines of the flow being ('.onsidered: y (x'

+ y2)

= const

(2.9.18)

The family of streamlines is an infmite set of circles passing through the origin of coordinates and having centres on they-axis (Fig. 2.9.3a). To comprehend the physical essence of this flow, let us consider the Dow that is obtained by !!ummation of the flows produced by a souree and a sink of the same strength located on the x-axis at a small distance e from the origin of (',oordinates (Fig. 2.9.3b). For point M (x. y). Lhe velocity potential due to a sOllrce at the distance of r 1 is 'Ps = (q/2n) In rl and due to a sink at a distance of r2 from this point is
+

:~

+

::~ = al(~~~IfRk)

+

al(~~:'Ip~k) =0

Since the functions fpa and 'PBk satisfy the equations {)'Cf.lfJx"

+ O'Cf81{Jy' = 0

alld O"CfSk/or

+ o'lf98);.I{Jy'l. = 0

PI. I. Theory. Aerodynamics of lin Airfoil lind II Wing

102

then {j2.tf/{jx2. .;- {j2q;/{jy2 identically equals 'zero. Consequently, the resultant function cp satisfies the contihuity equation. The resultant potential from the source and the sink is


= (qI2n})n (r/r2)

Since r l =V(x+e)2+ y2, r2=V(x-l::)2i-Y\ then In*={ln

~;~:~:~:: .0rln~=+ln[1+ (I-~:+Y"J.

The value of e can be chosen so that tile second term in the brackets will he small in comparison with unity. Using the formula for expansion of a logarithm into a series and disregarding the small terms of the second and higher orders, we obtain


(2.9.19)

Assume that the source and sink approac.il each other (e __ 0) and simultaneously their strengths increase so that the product q2e at the limit upon coincidence of the source and sink tends to a finite value J!. The complicated Row formed is called a doublet. the quantity M characterizing this Dow is called the moment of tbe doublet and the x-axis-the axis of tbe doublet. Going o\'er to the limit in (2.9.19) for cp at e __ 0 and 2qe -- M, we obtain the following expression for a doublet M x ({l=2i". x3+yt

M

cos6

=zn'-,-

It coincides with (2.9.16). Henre. the Dow being considered. characterized by the complex potentlal (2.9.15), is a doublet. This can also he shown if we consider the stream function of such a combined flow tllat will coincide with (2.9.17). Differentiation of (2.9.16) yields the doublet velocity components:

~~

= V/"= _

~

•

c~~6

,

+.-;-

= Va = _

~~

.

s~~O

(2.9.20)

Let us consider the three-dimensional ease. For a flow produced by a source and a sink of the identical strength q located along the axis Ox at a small distance s from the origin of coordinates. the potentlal function in accordance with (2.9.14) is cp=

where

4~

(*-

r2 = y2 -:- Z2.

~l)= 4~ [J/(~-:)I+rl

Jf~] (2.9.20')

Ch. 2. Kinemlltlc~ of " Fluid

103

For small values·of e, we have If = (ql'm) 2XE./(x·!

+ r~)3/t

Henct' in a limit pro("·ess with E __ 0, cOlls](iering that the prodnct q2e tends to the fmite limit .'\1, we obtain [or the !low produced by a doublet with tht' moment .11 the v"locity potential

cr : . .:;

(.11/1:1) xl(x2

-;... r'!)~/Z

(2.9.21)

or (2.9.21')

Circulation Flow IVortew..

Let

11S

consider a now

~et

by the complox potential -= -ai In cr

HI (cr)

(2.9.22)

where a is a constant. We can write this equation in the form qJ + i't -ai In (rei8 ) .,.-, a (9 - i In r) Hence ~ =

11'

aB -a l.n r

(2.9.23) (2.9.2/1)

It follows from (2.!1.23) th(lt the rallial ,"elocHy component Fr = = iJ'i,/iJr c"", 0, while the component V, normal to tll" radins equ
(2.9.25)

'Ve obtain lhe equation of streamlines (pathlines) from the condition"¢ ",-. canst, from which in accordance with (2.8.24.) we nnd the equation r . .,. (,OLlSt.. This eqnation represents a family of streamlines in the fOfm of l'.oneenlric. cil'cles. Flow along them is positi\"e if it occurs (,Ollnlerclockwise (from the x-axis to thl' y-axi.s); ill this case the coeflicient a in (2.9.25) has the sign pIllS. A flow in which the particles move (circulate) along concentric circles is said to be a circulation Row (Fig. 2.9.4). The ('ircnlation of the Yeiocity in the flow being considered i~ r = = 2",,. d\f!{)s. Since Bifid.\" air, we have r .... 2:1:a, whence a = = r/(2:t). Hence, the pitysi('.al meaning of the constaut a is that it..."1 value is determinl'd by the circulation of the flow which. as we have alread:\' I'stablishcd, equals the vorticity in turn. The flow produced by a vortex located at the origin of coordinates where V.t = aIr __ co is also called a two~dimensional vortex source or a vortex. '0

FII·1....

A circulation flow (point vortex)

We have treated the simplest cases of flow for which we have exaetly determined the velocity potentials and stream functions. By combining these flows, we can. in definite condition!'!, obtain more complicated potential flows equivalent to tbe ones that appear over bodie!'! or a given configuration. Let tI!'! take as an example a flow formed by the superposition or a plane-parallel now moving in the direction or the x-axis onto the flow due to a doublet. The complex potential or the resultant twodimensional incompressible now is obviously W (a) ~ W, (a)

+ W, (0)

~

Va

+ (M/2n) (1/0)

In accordance with this expression ~ + i1jl ~ V (x + ig) + (MI2n)/(x + iy) whence ~ ~ Vx + (MI2n) x/(:i' + g'), 1jl ~ Vy - (MI2n) g/(x' g')

+

(2.9.26)

(2.9.27)

(2.9.28)

To find the ramily of streamlines, we equate the stream function to a eonstant: (2.9.29) ~ Vg - (MI2n) g/(x' + g')

c

According to this equation, the streamlines are cubic curves. 0 corresponds to one of the streamA value of the constant or C lines. ByEq. (2.9.29), we obtain two eqllationsforslu·.b a zero streamline: (2.9.30) y ~ 0, :i' + g' ~ M/(2"V) Hence. the obtained streamline is either a horizontal axis or a drcle of radius To = fM/(2,nV)J1/2 with its centre at the origin of coordinates. Assnming that the velocity V = 1, and the doublet moment M =- 2n, we obtain a streamline in the form of a circle

Ch. 2. KiMmlllics of II Fluid

1O~.

with a unit radins. If: we assume that this cirde is a ~olid Doundary surface, we can consider an incompr{'ssible now Ilear this surface as one flowing in a lateral direction over a rylinder of infinite length with a unit radius. The velocity potential of sllrh an ineompressible disturbed flow is determined by the first of Eqs. (2.fL2S) having the form lp :-'

X

11 -\-

11(x2

+ y2)]

Introducing the polar (',oordillates 8 and r we obtain ~ ~ ,(1 ~ II,') cos 0

(2.\.1.31) x;ro.~

0 - (x 2 -I-

y2)1/2,

(2.9.3q.

Differentiation yields the components of the ,'cJocity at a point. on an arbitrary streamline along the directionH of the radius rand of a normal .~ to it: V~ ocr/or = (1 - lIr 2) r,os e, V, ~ (II,) a~Ia9 = - (1 + II,') sin 9 (2.9.32)On a cylindrical sHrface in a now (a zero streamline) for which r .; 1. we flJld F r - 0 and y., -2 sin e. The above example illustrates the application of tile princ.iple of now superposition and the concept of a complex potential to the solution of a very simple problem on the now of an incompre.'lsible nuid over a hOlly,

3 Fundamentals

of Fluid Dynamics

3.t. Equations of Motion of a Viscous Fluid The fundamental equations of aerodynamics include equations of motion. forming its theoretical foundation. They relate quantities .determining motion such as the velocity, normal, and sh.ear stresses. The solution of the equations of motion allows oue to determine these unknown quantities. Let us consider the various kinds of equations of motion used in 'Studying gas flows. CIIrtesran Coordinates

We 511311 treat the motion of a fluid partida having the shape of an elemelliary parallelepiped with the dimensions dx, dy. and dz ~onstructed near point A with the coordinates x, y, and z. Let the velocity componenL<; at this point be Vx , Vv ' and Vr A fluid particle of mass flT (here .. "'""" d.r dy dz is the elementary \'olume) moves under the action of the mass (body) and surface forces. We shall denote the projections of the mass force by Xp"t, Yp"t, and Zp"t, .and those of the surface force by PxT, Py"t, and P:"t. The quantities Po;, P y' and P z arc the projections of the surface force vector related to unit volume. The equation of particle motion in the projection onto the x-axis has the form

whence dVxldl '" X

+ (lip) P,

(3.1.1)

where dV,/dt is tile total acceleration in the direction of the x-axis. We obtain similar eqnations in projections onto the y- and z-axes: dV,ldl ~ Y + (lip) P, (3.1.2) dV,ldl ~ Z + (lip) P, (3.1.3)

Ch. 3. Fundamental5 of Fluid Dvnamics

107

FI.3.U

Surface forces acting on a Buill particle

The surface force can he expressed in terms of the stresses acting 011 the faces of an elementary parallelepiped. The difference in the surfar,e forces in romparison witll an ideal (ill\'i~dd) now r.onsists in that not only narmnl. hilt al."o shear strcs.<;{'~ act 011 the fares of a partide. . Every surface forre a(·tiug on a face has three projeclions onto the coordinate axes (Fig. 3.1. t). A unit of snrface area of the leflhand face experiences a surfare fOI'('o \vllO~e projl'('.tirms will be de!lignated hy PU' 'lx:' and Txy. The quantity P.>:", is the normal stress, while T.u and 'l.\'y are the she;tr stresses. It cau be seen thl\t t.he lirst subscripl indicates the axis p4"rpendicuiar to Ihe face being con!:lidered. and the second one indir_ates t.he axis auto which the given stress is projected. The rear face perpendicular to lhe z-axis expel'iencO.<; the ~tress comlJOuenl.. PZf' 'l :_~. and '('z,,: the boUom fare perpelldicular to the Y-Ilxis expl"rien('cs the components Puu' '('y.>:. anci '(',f'

We shall consider normal stresses to be positive if they are lIiteded out of the clement being studied and. consequently, subject it to omnidirecLional tension as shown in Fig. 3.1.1. Pot'Jitivc shear stresses arc present if for the three faces interseding at initial point A these stresses arc oriented along directions opposite to the positivc directions of the coordinate axes, and for the other three facesin the positive direction of these axes. With this in view. let us can· sider the projerliolls of the surface forces onto the x-Ilxis. The lefthand face experienc.es tnc force dne to the normal stress -P:u: dy dz. and the right-hand one-the force Ip.>:", + (iJP",:rJi).E) drl dy tlz. 'fhe ret;ultant of these forces is therefore (iJP;x./iJ:r) d:r dy dz. The components of the forres produced by thl" sheal' stresses actiug on these faces are zero.

108

Pt.

r.

Theory. Aerodynamics of an Airfoil and a Wing

Account must be taken, h9W-e",erj of the shear stresses 'lzx and 'lyx' The rear face experiences the force- -Tzx dx dy, and the front onethe force ['In: + (i1T:z:Jaz) dzJ dx dy. The resultant of these forces is (iTrzx/az) dx dy dz. In a similar way. we find the resultant of the forces acting on the bottom and top faces. It is (a'llIx/ay) dx dy dz. Bence. the projection onto the x-axis of the surface force related to unit volume is Px :- apxx/()x

+ 8TlIx /8y

-I- 8'lzx/az

(3.1.4)

Similarly, the projections onto the other coordinate axes of the sllrfaee force related to unit volume are P y = 8'l:.:y/ax..!. apl/y/ay...;. iJTzy/aZ P z = a'lrz/ax+ rJTyz/ay -I- iJpzz/8z

1

(3.1.4')

A(',eording to the property of reciprocity of shear stresses, the values of those stresses acting along orthogonal faces equal each other. i. e. 'lz:" -= 'lxz. 'lZIl c-::: 'llI l • and 'llI r T rll • llence. of six shear stresses, three are independent. To determine the values of the shear stresses, we shall lise the hypothesis that stresses are proportional to the strains they produce. The application of this hypothesis is illustrated by Newton's formula for the shear stress appearing in the motion of a vise,Otls fluid relative to a solid wall. Dy this formula, 'ly.'I' -= Il (aVyf8x). Le. the stress is proportional to the half-speed E z """ 0.5 (av II/ax) of angle distortion in the direction of the z-axis, whence 'llIr =-: 21lEz (here Il is the dynamic. viscosity). This relation covers the general case of threedimensional motion when tho angular deformation in the direction of the z-axis is determined by the half-speed of distortion E z = '-,- 0.5 (8Vy/ax + aVr/ay). We shall write the other two values of the shear stress in the form T IIZ ----= 21lEx and Tzx -:: 21lEII' Consequently. -,---0

Tyo;;

'llIz

'In

=

21lEz =].1- (8VII /ax

+- avx/ay)}

= 2.u:%:-.=].I- (~VII/8Z: avz/ay) = 21lEII ~-=].I- (ovz/ax, av x/a:)

(3.1.5)

We shall use the ahove-mentioned proportionality hypothesis to establish relations for the normal stresses Pn, PIIII' and PZZ' Under the action of the stress Px.n a fluid particle experiences linear slrain or deformation in the direction of the x-axis. If the relative linear deformation is a;, then Prx -~ E~. where E is a proportionality factor or the modulus of longitudinal elasticity of the fluid. The normal stresses PIIII and Pzz cause the particlo to deform also in the directions of the y- and z-axes, which diminishes the deformation in the direction of the x-axis.

Ch. 3. Fundamentals 01 Fluid Oynemics

109

It is known from the course in the strength of materials that the decrease in the relative magnitude 6.G:T of this deformation for elastic bodies is proportional to the slim of the relative deformations in the directions of .II and Z lInder the action of the indicated stresses. Ac("'ordingly,

6.0x

,= -(11m) (Pu/E

+ PalE)

where m is a constant known as the coefficient of lateral Hnear defor· maUon. The total relative deformation in the direc-tion of tho x-axis is

e;

~ 8~ -1- &8,"" 1',.,1£ - 111m) (p"IE -1- p"IE)

e

(3.1.6)

a.

We can calculate the relative deformations y and along the y. :and z·axes in a similar way. The obtained expressions give us the normal st.resses:

Pn~E~.,(p,,+ p,,)/m PY!I """ E6 y + (Pu -+ pxx)/m PH =

I

(3.1.i)

Ea. + (Pu + P!lu)/m

The relative linear deformations of a particle along the directions Qf the coordinate axes determine its relative volume deformation.

Designating the- magnitude of this deformation by

0,

we obtain

(3.1.8) Summating Eqs. (:t1.7) and taking into account the expression for 0: we have PX.T -:- PUJI

Determining tho sum PUy ing it into (3.1.6), we find

+ PH

= mEfi/(m -

2)

(3.1.9)

+ PH from this expression and introduc(3.t.tO)

For our following transformations, we shall use the known relation between the shear modulus G, the modulus of longitudinal elasticity E, and the coefficient of lateral deformation m that is valid for elastic. media including a compres..<;ible fluid:

G

~

mE/12(m

+ I)]

(3.1.11)

Substituting this rolation into (3.1..1.0)J we obtain

Pxx= 2G8:c+

m~2 8

(3.1.12)

110

Pt. I. Theory. Aerodynamics of an Airfoil and a Wing

Let us introduce the symbol (3.1.13) "7r = (Pxx + PIIY + Pu)/3 Adding and subtracting on the right-hand side of (3.1.1Z), we have p:x:x=a+ZG6 x + m.2~2 e-(px';+PlIy·-!-pu)/3

a

Transformations involving Eqs. (3.1.9) and (3.1.11) yield p" - cr + G (28, - 28/3) (3.1.14) Similarly, p" _ + G (28,- 28/3); Pu - +G (28, - 2il/3) (3.1.14') It has heen established for an inviscid fluid that the pressure p at any point of the flow is identical for all the areas including this point, i.e. with a view to the adopted notation, in the case being considered we have p""" = PIIII = Pn = -p; consequently, P = - - (p" + P" + Pu)/3. Hence, when studying the motion of a viscous fluid, the pressure can be determined as the arithmetical mean, taken with the minus sign, of the three normal stresses r.orresponding to three mutually perpendicular areas. Accordingly, := -p. The theory of elasticity establishes t.he following relations for the shear stresses acting in a solid body: 'til'; = Gy yX ' 't IlZ = GYIIZ> 't zx = G'V:,; (3.1.15) where 'VIIX' yy:. and 'Vu are the angles of shear in the directions of the axes z, x, and y, respectively. A comparison of formulas (3.1.15) with the corresponding relations (3.1.5) for a viscous fluid reveals that these formulas can be obtained from one another if the shear modulus G is replaced by the dynamic viscosity Il, and the angles of shear y by the relevant values of the speeds of distortion e of the angles. In accordance with this analogy, when substituting Il for G in formulas (3.1.14) and have to (3.1.14'), the strains (relative elongations) e,;.6 11 , and be replaced with the relevant values of the rates of strain 9" = by = lJV,;llJx, 9 11 = lJVyllJy. 9: = aV:llJz, and the volume strain the rate of the volume strain 9=fJV:/lJx+lJV ll lay + lJVz/Jz =div V. The relevant substitutions in (3.1.14) and (3.1.14') yield:

a

a

a

e:.

e

P,;:r:=-p+v.(Z 0:Xx --}divV) pyy=-p+.v.(2 0:VY --}divV) PZI= -P+1l

(z

O~' --}div V)

(3.1.16)

Ch. 3, Fundlimenfliis 01 Fluid Dynllmics

111

The second terms on the right-hand sides of expressions ,(3.1.t6} determine the additional stresses due to viscositv. By using relations (3.1.4), (3.1.4'), (3.1.5), and (3.1.16), we can evaluate the quantities P x , PilI find Pl' For example, for P x from (3.1.4) we obtain

Px=-*+*[~(2 8Jxx -{-divV)J+*[~l( &::11 + a~x)J +*[~ (a~z + a:z%)]= -*+~liV. +t-~ div V + :~

(2 aJ.x --}diV v)

where

li

+*

(a:ux + 8:: ) +~ (a:sx +8;;)

is the Laplacian operator: li iP/ax'J. + a'J./ay1.

+ 82/az"

Particularly, liV.~

-eo

a2VxiaX~

+ iPVx /{Jy2 + a2v.",'oz'J.

In a similar way, we ran lind the expressions for P II and Pz' The relations for the stresses due to the surface forces in a fluid have been obtained here by generalizing the laws relating stressesand strains in solids for a fhlid having the property of elasticity and viscosity. We shall obtain the same relations proceeding from a number of hypothetic notions on the molecular forces acting in • fluid (see [9, 1\1). By compiling projections or the total accelef expressions found for P"" P y , and P r , we can obtain the equations of motion (3.1.1)(3.1.3) in the following form:

8~x+Vx 8:;~ +Vy a;/ .. it·: a;;",

-~.vdVx+f.+XdiVV-:+[~~

(2

==-x-+.-;;-

a;~x

+% ( a:;~ + Q;~, )+ ;~ (8:;T

-+diVV)

)J a;;" -r-Vr a~" =Y-f.* +vliVy-l"Y.*divv+f[*{2 8:; --}diVV) T

a;:~

a;~y ~,v,t a:~y +Vy

+*( a;:u + ~:")+ ~~I (iJ~y -[- iJ:Vx)J

I

(3.1.17)

~.

I. Theory. Aerodynamics of

q~Z +V"

aa:. +Vy a:u.

til"

Airfoil lind II Wing

+Vr a::~

=z-+.

~~

+\I~V:+i.~divV++r ~~ (2 a:~. -fdivV)

+ :~ ( :: + a:;" ) + :~ ( ~~. + a~u )] t

where " ~-:: J.l/p is the kinematic viscosity. The dillerelltial equations (3.1.17) form the theoroticalfoundation -of the gas d ),oamics of a viscous compressible fluid and are known as the Navier-Stokes f'qu8tions. I t is assumed in the equations that the dynamic viscosity 11 is a function of the coordinates x, y. and %, i.e. 11 = t (x, y, z). Presuming that /..L = const, the NavierStokes equations acquire the following form:

d:r x =x=+.*+'V~V",-!-f·*diVV

d:!J '=Y-*'*7v~Vu+T.*divV

a:r =Z--f,.. ~~ z

+\'~V~+~.~diV

(3.1.18)

V

When studying gas nows, the mass forces may be left out of ac~ount, and, therefore, we as..'iume that X = Y = Z -= O. In this -case, we ha\·e

~=-...!..~+v~V -C..~.~divVl dt p ih >: ' 3 QX dVy

=-.!.. . .it.+vl'.VII ..L~ • ....£....divV p ay . 3 iJy

d:'r

=

dt

-t'*+V~Vr+-i-''*

divV

I

(3.1.19)

J

For two-dimensional plane motion characterized by a varying -dynamic viscosity I' =F const, equations are obtained from (3.1.17) .as a result of simple transformations:

.!!i. at ~ _L.!'.E. p f J x+L...'C.. p f J x [~(2 ~-~diVV)] fJx3

1

+f·-;;(J.lez) fjf

1

fJ

1

fJ

8V

2

~~-..L+_._[~(2-2__3 divV)] fJt p iJll P 811 811

+f·-:X (I'ez)

1(3.1.20)

Ch. l. Fundamentals of Fluid Dynamics

113

When div V = 0, and 1.1. = canst, these equations are reduced to equations of motion of a viscous incompressible fluid. Let us consider the equation of motion of an ideal (in viscid) compressible Ouid. Assuming the coefficients ~ and v in (3.1.17) to be zero, we obtain:

8~x +Vx 8~x +V"iJ~:J:

-I' V"

8::

+V"

8:'11

+V:x;

~fI

8J,z ..i- V:x; 8'fzz

+V"

a: =x-f·* 1 z'"

8;'Y

+ Vy 8Ju' + V, 8~"

=Y-f·* =Z-f·*

t (3.1.17') J

These equations were first obtained by Leonard Euler, which is why they are called Euler equations. They arB the theoretical cornerstone of the science dealing with the motion of an ideal gas whose hypothetic properties are determined by the absence or negligibly small influence of viscosity. Vector Form

01 the Equations of Motion

By multiplying Eqs. (3.1.18) by the ullit vectors i, i, and k. respectively. and then summating them. we obtain an equation of motion in the vector form: liVid.

=G-

(lip) grad p

+ v f1 V + (v/3) grad div V

(3.1.21)

where the vector of the mass forces in Cartesian coordinates is

G = XI + Yj +Zk the pressure gradien t is grad p = (liplliz) I -I- (liplliy) j + (liplilz) k the vectors

av = aV;oi + aVyj + aVzk grad div V (Ii div VlOz) I + (Ii div V/liy)j + (Ii div V/Ii,)k II a fluid is incompressible, div V = 0, and. consequently, dVld, = G - (lip) grad p + v f1V (3.1.21') In the absence of mass forces, G = O. therefore dVld, = - (lip) grad p + vf1V + (v/3) grad dlv V

(3.1.21")

The vector of the total acceleration can also be expressed as dV/d. _ liViIi. + grad (V'12\ + curl V X V (3.1.22) 8-011U

114

Pt. I. Theory. Aerodynllmics of 111'1 Airfoil lind II Wing

With account taken of (3.1.21") and (3.1.22), we can write the equation of motion as

~: +grad ~-+ curl V xV= -fgrad p+ v6V +igrnd divV (3.1.22')

For an inviscid compressible nnid, the equation of motion is

~~ +grad'::;;" + curl V X V = --/;-grad P

(3.1.22")

Curvilinear C_rdlnates

Let us transform Eq. (3.1.22') using the concept ot generalized curvilinear coordinates qn' This allows us to go over r,omparatively simply to eqnations of motion containing a specific form of curvilinear orthogonal coordinates, as was done with respect to the continuityequation. Let us consider the transformation of separate terms in (3.1.22'). :For the second term on the left-hand side, and also for the first and third terms on the right-hand sides, with a view to (2.·1..18), W(J have the following expressions: ,

grad

~=

h

(grad';:

n-I

Li

3

n=

~ ~. a~~~2)

in

(3.1.23)

n=1

3

3

g"dp~."2; (gradp),i,~2; -hI '-.'P i, ,,~I

,,=1 3

3

,,=1

n-.I

graddivV=~ (graddivV)"i,,=h

n

-&-.

(3.1.24)

q"

a~~:v

in

(3.1.25)

For transformation of the vector product curl V )( V, it is necessary to Hnd the form of the exprc;.;sion of the vector eurl V in generalized coordinates. For this purpose, let us calculate the curl of both sides of Eq. (2.1.19): 3

curl V = L~ (V n curl i" + grad V nXin)

(3.1.26)

n=1

where (3.1.27) grad V" )( ill ,.. (grad Vn)Ji m - (grad V,,)mij Here the projections of the vectol' grad Vn onto the relevant eoordinate directions are determined from expres'ion (2.1.18) with V" substituted for <\'>. Introducing (3.1.27), and also the expressions for curi in from (2.1.22) into (3.1.26), we obtain the following ex pres-

Ch. 3. Fundllmental$ of Fluid Dynllmics

sion for the curl of the velocity: curlV:.-= hzlh3

>:

[1l(:;~3)

[a(;;:l) _ a(!~:'3) ]i2-! h:"2

-

11~

"I:;:.) }I-T-~ .- ()(:;~:\) :1 i3

[J(~~:")

(3.1.28}

Consequently. the erosr- pl'Oduri 3

cud V)(V=

L

n=t

(cud VXV)" '"

(31.29)

where the projection of tllis veetOI' 011 to a tangent to the corresponding coordinate curve is (curl VXV)n"'"

h~hi" [(IU~~")

_

iJ~;~J)

J (3.1.30)

Rerall that the reialioH as follows'

U('I\\'C{,H

j • ••

(he subscript!! m, j, and n is

2

3

3 1

1 2

The left-hand side of Eq. (:i.1.:!2) is the vcctor or the total alion W . dVldt thnt has t.he form

acceler~

,

\V="~l W"i"

(3.1.31)

where W" is the projcction of the fll'celE'l'atio!l H·ctor onto the direction of a tangent to the coordinntc line q". Each quantity W" ean be eonsidered as the slim of the relevant projections of tile vectol's aVld/, amI also of the vcctor~ (3.1.23) and (3.1.29) onto the indicated directions. Aerordillgly,

Wn~~ iJ~"

+ h:~n

.

iJ

+ ~: . ~~';, + h~;m

(~~~n)

-

h:;m •

~::

-

.

a(;;::n)

.

h~t :;~

(3.1.32}

Let u>; apply the Lap/adall operator to the vector V, and use the expression

,

.6,V= ~ l!Vnin=graddivV-cllrlcllriV ,,=1

(3.1.33)

where dF n arc the projeetions of Ihe vector along the coordinatelines qn'

Pt. I. Theory. Aerodynam!c$ of lin Airfoil and a Wing

116

The first vector on the right-hand side of (3.1.33) has been determined in the form of (3.1.25), while for calculating the second one Eq. (3.1.28) should be used. Taking the curl of both sidos of this equation. we obtain

, ,

curl curl V = ~ (curl curl V)"i n

._1

=~ "~hJ{<'I"J~:IVlJl

a[hm(~:;lV)ml}i"

(3.1.34)

.~I

where the corresponding projections of the vector curl V are found from (3.1.28). Having these data at hand, we can consider the transformation of the equations of motion as applied to specific forms of curvilinear orthogonal coordinates. C,IIJMlrle.' CooJCIIMfes

In accordance with (2.4.12), (2.4.16), and (2.4.25), we have ql = Z, q'l. = r, qs = y, hi = t, h'l. = t, ha = r, VI V~, Vz = Vrt Vs = V,. Consequently,

W = 8:t '+,Vs 1

8~s

+v,"8:rr +~.

a:"\~;

(grad

P)l =

ClplClx

Next we find (grad div Vh = CI diY V/ax Since div V is determined by formula (2.4.26), we have

8d:;V =i; ("~s + a;; ++. a:; +.!f) From (3.1.28), we have (curl Vh =

8Jzr - "::

(curl V)'I. =

+(0:V:c - r

~)

Introducing these expressions into (3,1.34), we obtain

(curl

,.urIV)I-+{'H~-~)] '[W;:-.~)]} ••

=

iJy

::ir - o;~: ++ Uat- ':: )_+.- (O;~: -r ::~;)

Ch. 3. Fund.mentals 01 Fluid Dynamics

117

Since (grad div V)l ~ iJ div Vlox in accordance with (3.1.33) we have (dVh = (grad div Vh - (curl curl Vh j

-7- aa2~:c +~.a;;z~++_

_ a;;.'C

a;;

\Vc tIm:! compile an equation or motiou in a projeetion onto th' x~axis

of a cylindrical coordinate system:

8~; '~-V:'~'¥'Vr~':~' D;y' ~=

-to :: (3.1.35)

We obtain the oLher two C(luat.iolls in projections onto the coordinate lines ramI y in a similar way:

~+ V:c D;: +Vr =

a;; -7-~' a:,: -

_.!. !.1.....1...\-. (.1.V p

Dr

'

1

~~

_.!..~_.:r..)-l_":::'. ddivV r

iiy

1'2

1'2

'3

Ii

or

.!2.+V !!!..1...V av? ¥L.!:L.~+ r,vy 8t ~a:·r8r·rfh' I'

(3.1.35')

= -f,:.*+v (flVy¥i_~. ai)~ -;r) + ir· Dt~~VV J In these equations, we have introduced a symbol for the Laplacian operator in cylindrical coordinates: .1.

=;;-t :,22+..!,. ~: ++-fr

We determine the divergence of t.he velocity by formula (2.4.26). For an axisymmctric Row. the equations of motion are simplified: aV:':.i-V iJt

.

:.:

DV~_:...V 8V:I::=_.-!._~-t--'\-'.1.V (Jz

.

r

(Jr

p

iJx.·

+....!... (JdivV .

3.%

r -i' V:. ~"";'Vr~:":'~ -..!....-~-:...vflVr ov iJt 0:': Dr p Dr' I

+-i-' adi: v

:.:

I II

1

(3.1.36)

J

where div V :.= iJV"liJx + iJVr/{Jr + Vrlr; fl = iJ 21iJ:x;'! + lPliJr" + (1Ir) alar. For a steady Row. one must assume in the equations that aV"Iat - aV,lat - av,tat _ 0

+

118

PI. I. Theory. Aerodynamics of an Airfoil and a Wing

Spherice' Coordinates

The spherical coordinates, Lame coefficients, and the projections of the velocity vector onLo the directions of the coordinate lines are related by formulas (2.4.13), (2.4.17), and (2.4.33): ql =-: r, (/2, ~ e, q3 .-" tp, hi -= 1. h z = r, h g ""=' r sin e. VI

. Vn V z

~

Vo, V 3

0...

V¢

According to these data, from (3.1.32) we find the projection of the acceleration onto the direction of the coordinate line r;

WI =

a~r + Vr

a;r

.l-..!! .~+r~;~a'~- ve~v; (3.1.37)

The projection of the pressure gradient is (3.1.38) (grad ph = 8pl8r Wilh a view to the expression for div V (2.4.34), we lilld the relation (grad djv V). ':=: v

ad;;

+, [~.a(V;:2) +rs:nu.~+rs:ntl. a~wJ (3.1.39) We shall lise this relation for determining the projections (11 Vh of the vector 11 V. To do this, we shall calculate the valne or (curl .curl Vh in (3.1.33). Fl·om (3.1.28), we have (curl

Vh =

+ (!:r) - a: J' [a

11

(curl

Vh =

rs:n

0 [~_

a(V¢~:in 0) ]

lntroducing these relations into (3.1.34), we obtain

'"

(3.1.40)

With a view to expres5ions (3.1.39) and (3.1.40), we find

(I1V)j = (grad div Vh -(curl curl V)j =I1V r _ 2~': (3.1.41) where

I1Vr=~·-:r (r2 a;r) + rls~lle'~ (sinS a:a')

+ Si~2 O. ~2:: ri

(3.1.42)

Ch. 3. Fund.!lment.!lls of Fluid Dynemic$

119

Taking into al'Coulit (3.1.37)-(3.1.3H) and (:U.!J1), we find the eqnation of motion ill a projeclion onto lhe toorriinal(' line r:

iJ;/~"J',u~;'i-~' _ r~~ rJ -- r!

°:0+

f.* + v

=

s~n U • J~~~.

_

2:: _

r:;~o' JJI~~

--h-. J~~o

(AVr

2Vllr~OlO) ~_

T' a d~; \.

(3.1.43)

We obtain the olher two equations in a similar' way:

~

iI,:·ro ..; ~. d~;~

;- V r

J1rl'o-:'~COlfl -

r2

__

+,.~

~~ n . o;~~

(AYo+-#r. o;j

~';~2 t! - r~ ~~~~AU' ~JI~~') + -1-.

O!';r ':-Yra~>

I'~. J;'~'

l'",(!·r-·· '"0 ("otO)

·:·v

r

-;-V

·1·

-+-.

1 '""' -pr sin

(AV~--r~~;~tu~ r2$~n9' ~I;

{J

(~~ \'

r!;~e'~

t

(3. t.43')

iJp

f)'

aif

r;~~:}~ iJnl~~)

,. ildiv \' ...,- 3r sin 0 '----aij:""

where tl](> Lapiaciull opel'alor in spherical coo['dinalc-s is 11

-~ ~ . .;.;. (r~

-:r) + r~

$:11 t:I •

-to (sin 0 -k) . r! ,'1111" o· ~ (a.1.4,'f)

For lwo-dimellsiollal l-lpatial gas nO\\'5 charal"terizNI uy a change in the parameters (\'elodty, pressUI'e, density. e\c.) in the riirection of only two {~oordilJale lines, let liS write the eqllation of motioll - a simpler form:

aV r

.

dt~~-

+v

--.;..~ _

~-

au .

21'; _ 21'ge~)tO) r • r

-I-...!..... &div V

~-i ~.~+ VrVO ... .~ ~-.-V at r ar r ali r -

+v

1

' \'(1 in', \"~ l"p ra, r-;:-'&o--,=-p'o,

V ar r

(CiV r _

_..!....21I

3

br

rp

(CiVo -:--'}. a~j -r2~~20) +*.bd~;\"

bO

J

(3_t.45)

HIO

Pt. I, Theory. Aerodynamics of an Airfoil and a Wing

where

Il=-J,.f, (r2

-!r) +

r<si1nll

.-:0 (sin e ~)

diVV"""~. U{~;r2) i ~.a(\'9a~inO)

(3.1.46) (3.1.47)

Equations of Two·Dlmenslonal Flow

of a Gas Hellr II Curved Surface

For the Case I eing ton..,idereu, the curvilinear coordinates. Lame coefficients, 311d Yelo('ily ("omponents are determined by formulas (2.4.40), (2.4.12), and (2.4.43), respeC'th·ely. We shall adopt the further condition that motion occurs near II. wall. and. consequently. y« R. Accordingly, q1 = X, 12 = y, q3 = y. hI = 1, h, = 1. h$ = r VI = dx/dt = V=, V, = V r, V, = 0 With a view to these data, the acceleration component is

W,

OV,lal

+ V, OV';O$ + V, aV,;ay

and the projection of the preSfmre gradient is (grad ph = fJplfJx

(3.1.48) (3.1.49)

In addition, (grad div Vh

=

(3.1.50)

iJ div V/iJx

where the divergence of the velocity has been determined by relation (2.4.44). Consequently, (graddiVV)I=-1;{+[O(:x"r),.

O(;r

l ]}

(3.1.51)

We use (3.1.28) to calculate the projections 01 the velocity curl vector: (3.1.52) (curl V), = fJV/fJx - iJV,,/fJy, (curl Vh = 0 Introducing these expressions into (3.1.34), we obtain

(curl curl V),=+.fu-[r (~_

U;y=)]

(3.1.53)

With a view to relations (3.1.51) and (3.1.53lt we have (.6.V), = (grad divV)I- (curl curl V),

=fz{+ [u(::r) (3.1.54)

Ch. 3. Fundamentals of Fluid Dynamics

Similarly, when considering the coordinate line

Q2,

+ v x aVylax + l'y av /ay = aplay

W z = aV,,,at

(3.1.55) (3.1.56) (:·U.57)

(grad ph (~Vh

(grad div Vh --: a div v/ay = (grad eli\' Vh - (nl1'l curl

12t

we have

vh

(::r) ]}-+.:z [r (a:: _":;T) J(3.1.58)

= -i;-{+ [8<:t) + a

Using the obtained relations, we ha\'e:

a~:t + Va:

"!: + V" a:V:t ~:~ -f'"*--: v(L\V).

I

V:t~+Vy~=- -+.*-:'v(~Vh

I J

7T'~

a~~

-t

1

a div \'

'Y

.

,.

8,1 vV

-:'T'---ay

} (3.1.59)

where (~Vh and (~Vh are determined by formulas (3.1.54) and (3.1.58), respectively, We have thus obtained various forms of the equations of Iliotion for a viscous liquid. Experience shows that this is needed because in some cases, when studying thc laws of interaction of gas flows with bodies in them, it is com'cuicnt to use one form of the equation, and in others, a different form. l.l. Equations of Energy and Diffusion of a Gas Diffusion Equation

The motion of a dissociating viscous fluid can be ill\'estigated with account taken of tIle influence of gas diffusion on this mot.ion. This is expressed. particularly, ill that diffusion i~ taken into consideration when deriving the equation of energy-one of the fundamental equations of gas dynamics. By dlft'usion is meant the levelling outoftheconcentration because of the molecular transfer of a substance. This is a thermodynamically irreversible process, and it is one of the reasons why a gas in motion loses mechanical energy. The diffusion oquation is an equation of the transfer of the i-th component in a gas mixture (it is a continuity equation for t.he same component).

122

Pt. I. Theory. Aerodynollmics of an Airfoil oIInd

011

Wing

To simplify onr investigation, we can assume that the intem,ity ·of thermal and pressure diffusion is negligibly small and determine the diffusion flux of the i-th component ill a direction n by the ·equation (3.2.1) where Ci is the concentration of the i-tll component, and Di is the diffusivity \.hat determines the diffusion flux when a concentration gradient is present. For a mixturo of gas components, we mnst take into eOllsideratioll the binary diffusivities corresponding to each pair of components, for example, to oxygen atoms and molecules, or to nitrogen atoms and molocules in the air. In approximate calculations, we ·can proceed from a certain value of the binary difiusivity TJ that is the same for all the pairs of component.s. Taking this into account, we have Qi.rI.n...." -pD oc;lon. Considering the directiOlls x, y, and z along which diffusion occurs, we have Qi.d.x = -pD oc;J(}.r, Qi.d.U .; -pD (}c;l(}y, Qi,d .• -: -pD (}c;loz (3.2.2) ·or in the vector form (3.2.3) Qi.d -, -pD grad Ci The diffusion of a substance is directed into a region with a reduced concentration, thorefore (}CiJiJn has a negative sign. Since the -right-hand side of (3.2.3) contains a minns sign, the quantitr Qi.d is positive. Let liS eOllsider the derivation of the diffusion equation in a erlindrical c,oordinate system assuming the motion to be steady. ,"Ve shall assume the flow to be three-dimensional and symmetric ahout the x-axis, i.e. such in which the velocity component Vl' .....:: O. Let liS separate an elementary volume of tho gas in the form of a ring with dimensions of dr and dx (Fig. 3.2.1) constructed near point P whose coordinates aro x and r and whose velocit}' components are V", and V r . \Ve shall assume that the substance diHusos only in a radial direction. ConsequenLly, the nux of the i-th component through the internal surface of tlle element is mr -= 2nrflV.ci dx ~ + QI.d2nr dx, where Ci and Qi.d are the c.oncentration and the diffusion flux of the i-th component, respectively, ovaluated per unit area. The flux of the i-tll component through the outer surface is m r ';' a;;r dr = mr -:- a (p~;rcil2n dr dx -\- a (~I;rlr) 2n dr dx

Consequently, the flow of the component into the volume being ·considered is 10 (pV,rc,)lad 2n dr dx +110 (Q',dr)lod 2" dr dx

v,

Fig. l.l.1 An elementary gas particle in an a:dsymmetric thr<'c-dim('nsional flow

DisregRrding the clirtll~ion nux of the suhslam'('
mx

--co

2:trV~c;rdr

and through the right-hand one m~ -:- (dm.,./(Ix) d.r m~

+ Ii) (p V~rc;)ldJ·l 2:1 dr dx

lIenco, the flow of til(' component inl_o t.he vol1lme is

2:t la (p V.,(cf)iO.rl dr d:r Since the amount of ga~ in the voilime does 1I0t change, tllC total flow of the component into the volume equals its outflow because of chemical reactions. If the rRte of formation of the i-th eomponent in a unit volume caused hy chemiC-al reaetions is (W ~h)" the consumption of the component in the elementary volume is 2:t (W dl);r >: X r dr dJ.:. Therefor£', the balance of mass of tllC i-th component in the volume being considered is

8 (pl'."rcJ/aJ.: -:.

a (pV rrc;).'8r

-= -8 (Q;.dr)i8r:.... (W~hLr (3.2.4)

This equation is known as the diffusion equatiun ill a cylindricnl coordinAte !"ystem. In A similar Wily, we can obt.ain a diffusion equation in the Cartesian coordinates x alld y for u pi nne flow:

+

rJ (pV"c;)/8x + rJ (pV vc;)J8y "'"' -8Q,.dJ8y (Welt); (3.2.5) where Qi.d is found from (3.2.3). If we consider a binary mixture of atoms and molecules. then L C; ,...., CA +- CM ..--' 1 and, consequently, QA.d .~ -QM.d . We determine the value of (Weh ); for a given reaction in a dissocialing gas by the formulas of chemical kinetics.

124

Pt, I, Theory, AerodynllmiC5 of lin Airfoil lind II Wing Energy EqueHon

Together with the equations of state, motion. and continuity, the energy equation belongs to the system of fundamental differential equations as a result of whose solution we completely determine the motion of a gas. Let us consider a system of rectangular Cartesian c.oordinates and compile an energy equation for a fluid particle in the form of an elementary parallelepiped. This equation expresses the law of energy c.onservation, according to which the change in the total energy of a particle, consisting of its kinetic. Rnd internal energy. during the time dt equals the work of the external forces applied to the particle plus the influx of heat from the surroundings. The kinetic energy of a particle of volume 't - dx dy ch is (p VI/2) 't, and its internal onergy is up't (u is the internal energy of a unit mass of the gas). Consequently. the change in the total energy during the time dt is

1t [( p;1 'r- pu) '(] dt

=

p't

*

(.x;.;. u ) dt

The work of the external mass (volume) forces in the displacement of the particle during the time dt can be represented in the form of the dot product G· V multiplied by the mass of the particle p't" and the time dt. The mass force vector is G Xi + Yj -:. Zk, consequen tly, -0":

(G·V) P' dt

~ (XV, -

YV, -;. ZV,) P' dt

Let us calculate the work of the surfac(' forces. First we shall consider the work done during the time dt by the forces induc.ed by the stresses acting on the right-hand and left-hand faces. The work done by the forces acting on the lelt-hand face equals the dot product 0,,' V multiplied by the area dy dz and the time dt. In the dot product, the vector of the surface forces is 0" =-= p,%,%i -,- 't.q,j + 't:uk The work done by the surface forces acting on the right-hand [ace is

[O.zV + 8

<:;V) dz] dy dz dt

Having in view that the forces fOT the left-hand and right-hand faces are directed oppositely, we must assume that the work done by these forces is opposite in sign, for example, positive for the right-hand and negative for the left-hand face. Accordingly, the work of all tbe surface forces applied to tbe left-hand and right-hand faces is

8<:t)-Cdt=[-!;(Puv,,+t:ruVu-i-'t"IV:] tdt

(3.2.6)

Ch. 3. Fundamentals 01 Fluid Dynamics

125

We obtain expressions for the work done by Lhe ~mrface forces acting on the lower and upper, and also on t.he front and rear fates, in a similar way: [a (au V)/fJyJ • dt and [a (a,V)/fJzl

'C

dt

Considering that the vectors of the surface rorces acting on the lower and rear faces are, respectively,

obtain tlte following expressions for LllC work: a

(::V)

Q(~;")

1:

dt =

['*

.dt= [-;;:

('C~ .•Y lI:'r PUyV ~ + 'ty~Vz)] .. dt

(3.2.6')

J -rdt

(3.2.6~)

('CuV.o:+'t~yV!l+PuVz)

In expressions (3.2.6), (3.2.6'), and (3.2.6"), the stre~ses arc determined by relations (3.1.5) and (3.1.16), respectively. The inl1ux of heat to the particle occurs owing to hea\. conduction, diffusion, and radiation. Let q", dy dz (where q", is the specific hent flow) be the amount of heat due to heat conduction or diffusion transferrer! to the particle through the left-hand face in nnit time. During the time dt, the heat flux qo: dy dz dt is supplied to the particle. The heat nux through the right-hand face is - Iqx ...l. + (dq."ld.r) dxl dy dz dt. The amount of heat transferred to the partic]" tnrough both. faces is - (fJq,,/fJx) 'C dl. We oMain similar expressions for tile faces perpendir.1I1ar to the y- and z-axes. Hence, the total heat flux transferred t.o the particle is - (aq:Jdx + fJq~/fJy '';-

+ agz/az) • dt.

If we consider the supply of heat caused by conduction, the specific heat nows, equal to the heat fluxes along the relevant coordinate directions through a unit area, are expressed by the Fourier law qT ..>: = -A. aT/ax, qT.u = -A. aTlfJy, qT"

= -A. fJTlfJz

(3.2.7)

With this taken into account, we have -(fJqT.x/aX

+ fJqT.ifJy + aqTjfJZ) 'C dt

= div (). grad T)

'f

dt (3.2.8)

where gcad T ~ (aT/ax)i + (aT/ay); + (aT/Oz)k. The energy supplied to a gAS particle at the expense of diRusion is qd.,.,=

f Qt,

rI. xiI>

qd. U=

f Qt.

d,

~i" qd, z = ~ Qf. rI. zit

where ii is the generalized enthalpy component of the gas mixture.

126

Pt. I. Theory. Aerodynamics of an Airfoil and a Wing

Hence. _ ( uqa; .\'

+ a~d; y + aq: / ) 't dt

=" _

(~

it

lJQ~zd. x

i

+h

aQ~!ld.1J

if

+ hit iJQ1at;) "dt= (- ~ i,divQ"d) "dt

i

i

i

Introducing into this equation the value of Qi,d from (3.2.3), we obtain - (Oqd. :iox i·iJqo. gloV+ Oqd. z/oz) 'rdt = ~ it div (pD grad e,)

't

dt

(3.2.9)

In addition to the energy transferred to a particle by conduction and diftusioll, il also receives heat owing to radiation, equal te> n dt (e is the heat flux due to radiation absorbed by unit volume). By equating the change in the energy of a particle during the time dt to the sum of the work done by the mass and surface forces and the influx of heat ber3use of c:ondllction, diffusion, and radiation, we oblain the energy equation:

PTt (~+ u) + "zgl' v-I

'----"p (XV;,: ,!. YVy +ZV1 )

+-tr (PnV;,:

'rXZ V2 ) -1'-;'; ("!I~V;,:+ PnVv + "VIV,)

+.;;. ('r;:tVx..L "zVVy+ pzzVz);- div (A grad T) + ~ i,dh'(pDgraric,l+e

(3.2.10)

i

Upon the motion of a chemically reacting mixture of gases, the energy equation expresses the condition of the heat balance including the heat that may appear as a result of chemical rcacLions. If we take into account, however, that upon the proceeding of reactions the generalized enthalpy of the gas mixture LCJii does not change, then upon int.roducing the generalized enthnlpy inte> Eq. (3.2.10), we cnn no longer take into account separately the release or absorption of heat because of chemical reactions. Introducing into (3.2.10) the expressions for the st.resses (3.1.5) and (3.1.16) and excluding the term taking into account the work done by the mass forces, after transformations we obt.ain an energy equation for a gas: p

1t (~-!. u)

= -

div (pV)_.. -} div (jlV div V)

+-f; (~~) +fu- (~

u:;) +-i; (~ u:!)

*

Ch. 3. Fund"mentals of Fluid Dyn"mics

2

-tx

f,!{VyCt7 Vzcy)I-i- 2

127

IfL (V."€l ;. V.f x)]

:- 2 "';;'I~ (V.~ey -; Vllc:rl/·;...div (Agraa T) +~ ijdiv(pDgradc/)+€

(3.2.11)'

Equation (3.2.11) sl10ws what causes the kinetic energy of a nuid to change. III addilion to conduction, {Jifrllsion, alld nuliatiotl. this energy changes at the expense of the work of comprC'ssion di,· (pV) and tho work of tho friction forces (the tcrJII~ of t.ho eqllation ("ontaining the dynamie viscosity pl. The di!'sipatioll of ell('I'gy i.'" 1ISSOciated with the losses of mechani(:l1I en('rgy ror on'('{'ollling the friction forces. j':nergy dissipation {'onsists in that the mcehanica\ energy in part tram;forms il'l'eYersihJy into heat. Accordingly. the friction forces arc ('alled dissipath·e. The terms on till' right-hand side of (3.2.11) ('ontaining 11 form the dissipative fUllction. For two-dimellsiollal plnllC' molioll of a \'i!'colls nuid. thC'· energy equation is p

~ (~-;. u) =

-div (pV)-i- div

(~lV div V)

+fx(~l~)~*(~W) 2 -if; (~lVllez) ._- 2

i; (fLV ",ell.1- div (Agrad T)

+ ~ i l div (pDgl'ad ci ) -: where grad T

V "'" V.ti (oT/B.d i

€

(0.2.12)1

+ V"/L di\' V '"'" iJV)iJx + (}r"ldy, + (oT/uy)j, grad C; -" (8c;:(h)i ~- (IJc;lay)j

Let us transform the energy eqllation (:~.2.12). To do this, we' multiply the first equation (3.1.20) hy V", the second olle hy V", and sum up the results. \Ve obtain

p

_ Vx~[~(2

-t- (~).~ - (V"~'.Vy*)

u:~~-+diVV)]'i-Vyi;[~(2 U~',_~ di\"V)]. -: 2

[l"o~ ~ (j.l€z) -I· VII

-;;. (Il€.) ]

We can show by simple tran.!'fol'mations thnt l'" up/ax VII iJp/ay = di" (pV) - P (Jiv V

+

(3.2.13)-

128

Pt. l. Theory. Aerodynemics 01 en Airfoil end eWing

where we find p div V by using the continuity equation p div V ~ -(pip) dpld" or p d'v V ~ P (dldt) (pip) -

dpldt

We transform the sum of the other two terms on the right-hand side of the equation to the form

V%.:x[~ (2 ;t--}diV v)] +V.. ';; [~

(2 ~ --} div V) J

~ -h (~~) +.:,- (~W)-2~[( '::)' +( '::

n

-{-diV (~V div V) ++~(divV)t For the last term in (3.2.13), we have the expression 2

[v" ;;'(fJo!,;z) l· V y ';;

(!-LBz)]

=2* (I-'V"e

l )

+~ (",V:t:ez) - 4",e~

Let us make the relevant substitution in (3.2.13) and subtract the obtained equation from (3.2.12). Having in view that i = = It + pIp, we obtain

p~=*+21-'{[(~r

+ea;y" r]-i- (divV)2+4e~}

-I- div ().grad T) + h i, div (pDgrad cI ) + e

(3.2.14)

In the absence of heat transfer by diffusion and radiation. we can write the energy equation in the form

p~

=%+21-'{[( 0;; r + ( 0;,;1 rJ --} (div V)Z+!ie~} +div{).gradT)

(3.2.15)

At low gas velocities. when the work of the friction forces is not great. we may disregard the dissipative terms. In addition. the work done by the pressure forces is also insignificant (dp/dt 1::J ~ 0). In the given case, instead of (3.2.15), we have dTld'

(Alpc,) d'v (grad T)

(3.2.16)

The quantity A/(pC p ) = a, called the thermal dillusivity,characterizes the intensity of mol ocular heat transfer.

Ch. 3. Fundllmenlotls or Fluid Dynllmics

129

3.3. System 01 Equations of Gas Dynamics. Initial and Boundary Conditions

The investigation of the motion of a gas, Le. the determination of the para.meters characterizing this motion for each point of space, con~ists in soiYing the relevant eqnations that relate these parameters to one another. All these equations are independent and form a system of equations of gas dynamies. We determine the number of independent equations by the number of unknown parameters of the gas being sought. Let us consider the motion of an ideal compressible gas. If the velocities of the Row are not high. we may ignore the change in the specific.' heats with the temperature and take 110 account of radiation. In this case. the gas How is a thermodynamically isolated system and is adiabatic. The unknown quantities for the now being considered arc the three "elocity component$ l'x, VII' and V z• and also the pressure p, dl>Jlsitr p, nnd temperilture T. Consequently. the system of equations of gas dynamics must include six independent- onc.'i. Among them arc the eqllation~ of motion, continllity. slate, IIlld ell(>rgr. which art' ('USlomllrily called the fundamental equations of gas dynamics, B(>fore compiling this s-y:-;(ern of equations, let us cOIl~idel' separately the energy eq1lation. III ac('onlnncc willi ollr m':-;umption on the nrliabatif: nature of (he Ilo\\", we \I'Hn:-;fol'lll til(' ('1I('rgy equation (:1.:!,14) tI!! follows: di

= dp/p

(3.3.1)

If we take into cOII$lideralioli the cquatiorl di -, c/• dT, aud also the ('xpressions el' - C p ..." Rand p -'" RpT. from which we can nnd

then (3.3.1) is reduced to the form dp/p = k drIp. Hence (3.3.2)

where A is a constant characteristic of the given conditions 01 gas flow. Equation (3.3.2), is known as the equation of an ndiabat (isentrope). Hence. in the case being considered, the energy equation coin('ides with that of an adiabat. Having the energy equatioll in t.his

130

Pt, I. Theory. Aerodynamics 01 an Airfoil and" Wing

form, we shall write all the equations of the system: 1

dV:\, _

8p

-p'Tz'

---cit-

dV lI ~=

1

ay

-p'Tu

~~:=-+,~, -*+pdivV=O p=RpT.

(3.3.3)

Aph-I=RT

Let 11S ('.onsider the system of equations for the more general case of the motion of an inviscid gas at high speeds when the specific heats change with the temperature, and dissoC'-iation and ionization may orcur in the gas. For generality, we shall retain the possibility of the heated gas radiating energy. ~ow the thermodynamic process in the gas flow will not be adiabatic. Accordingly, the quantity £ determining the radiation heat flllx remains on the right-hand side of energy equation (3.2.14). We shan note that the equation of state must be adopted in the form of (1.5.8) taking into ft(',count the change in the mean molar mass Ilm with the temperature and pressure. In accordance with the above. and also taking into ar,r.ount that the equations of motion and continuit.y do not change in form. we shall give the fundamental equations of the system: dV."\"

_

I

8p

8V"

1

8p,

(ii"'=--p'&%' d't-:'-p'Ty

d!'t%

:;;~

-+'*: ~+pdiVV=O

p=!;-pT,

f (3.3.4)

p-¥:-=*+e

We can see that in the given system in addition to the six unknown quantities indicated above (V", V y • V z • p, p, and T) three more have appeared: the enthalpy i, mean molar mass of the gas 11ru. and the heat flux e: produced by radiation. Besides these quantities. when studying the flow of a gas, we must also determine the entropy S and the speed of sound a. Hence. the total number of unknown parameters characterizing a gas flow and being additional1y sought is five. Therefore. we must add this number of independent relations for the additional unknowns to the system of hlndamental equations. These expressions can be written in the form of general relations determining the unknown quantities as functions of t.he pressure and temperature: i ~ /, (p, T) (3.:1.5) (3.3.6) S = t. (p, T) (3.3.7) ~m = t. (p, T) (3.3.8) • =t.(P, T) (3.3.9) • = t. (p, T)

Ch. 3. Fund~ment~l$ of Fluid Dynllmics

131

The linding of these functions is tlte subject of special branches of physics and thermodynamics. The solution of Eqs. (:1.3.1)-(3.3.0) determines the parameters of flow of an inviscid dissociating and ionizing gas with account taken of the radiation effect. Snch flow is studied by the uerodynamics of a radiating gas. Let us consider a more general clIse of flow chara(·tcrized hy the action of friction forces and heat transfer. \Ve shall assume that chemical reactions occur ill the gas. Therefore. Llw fllndamental equations of th(' sYiitcm (for simplirlcation we shall ("onsider twodimensional plane flow) will include two differential equations (3.1.20) of motiou of a yisrous compressible fluid \vith a varying dynamir viscosity (Il =t= cOllsI), and also l'ontinuity e(Iuation (2.4.1). These equations mUl'it be l'iupplcmented \vail equation of slate (1.5.8) relating to the general cuse of a dis~ociated and ionized gas, and with expression (3.2.11) that iii the ('nergy equation for a two-dimensional compressible gas rlow in which heat transfer by diffusion anc1 radiation occur. These equations describe the gelleral case of unsteady motioll and characterize the unsteady thermal processes orcufring ill a ga~ now. !fence, we have

~== _+.~++'~['I (2

iI;;. --}diV v)J

++~(Ile:)

~

=

-+.* r+'+v [~(2 a{~;1 ._+ div v)] +4·*(~eJ

(3.3.10)

47-, pdivV=O; fJ%=~

-'-2M

{[(~r .L(~r]-+{clh'VP· 4e~}

+div ("-grad T)+ ~ i, div (pD~rad cJ) :.

t·

This system mllst be slIpplelllelltf'd with relations n.3.5)-(:·I.~Ul), and: also with tbe gener,]} relations for the thermal conductivit.y

).

~!. (p, T)

"

~

(3.3.11)

the dynamic viscosity

!,

(3.3.12)

(p, T)

.md the specific heats

c,.

= /,

(p, T),

C1'

'-'

/9

(p, T)

(3.3.13)

182

PI. I. Theory. Aerodynamics of an Airfoil and a Wing

The last t\\'o quantities are not contained explicitly in Eqs. (3.a.10). but they fire RCvertheless used in solving them because when studying the flow of a gas its th(>rmociynamic characteristics are determined. Since the energy equation also t.akes into account heat transfer by diffusion. equation (3.2.5) has to be included additionally. It must be taken into accollnt simultaneollsly that the COIlcentration Cj in the energy and diffusion equations is a function of the pressure and temperature, and it can be written in the form of the general relation

c,

~

/" (p, T)

(3.3.14)

The abo\'e system of equations including the fundamental equations of gas dynamics and the corresponding number (according to the number of unknowns being sought) of additional relations is considered in the aerodynamics of a viscous gas and allows one to fmd the dist-ribution of the normal and shear streiises, and also the aerouynamic heat nnXeii from the heated gas to the wall over which it Hows. In spedflc ("aRCS, for which a definite schematization of the now process is possible, the above system is simplifled, and this facilitates the solution of the differential equations. When soh'jng the equations, it becomes necessary to in\'olve additional relations used for determining the characteristic parameters of motion. Among them are. for example, relations for determining the ~pE"cinc heats and the degree of dissociation depending on the pressure and tcmpcrature, 0.11(1 formulas for calculating the shear stress depcnding on lhc \'clocity. The solution of a system of gaswdynamic equations describing the flow onr a given snrfarc mustsatis[y defInite initial ami boundary conditions of this now. The initial conditions al'e determined by Lhe values of the gas parametcl'!'! for a certain instant and ha\'e sense, evidently, for unsteady motion. The boundary conditions arc superposed on Lhe solution of the problem both for steady and [or unsteady motion and must be obsen'ed at eury ins/ant of this motion. According to one of them, the solution must be SUeil that the parameters determined by it equal the values of the parameters for the undisturbed Dow at the boundary separating the disturbed and undisturbed now regions. The second boundary c.onditioll is determined by the nature of gas flow oyer the rele\'sllt :"urfnce. If the gas is in viscid and does not penetrate through such II surface. the flow is said to be without separation (a fre{' streamliop. now). In accordance with this condition, the normal velocity component at eac·h point of the surface is zero, while the vect.or of the total velocity coincides with the direction of a tangent to the surface.

Ch. 3. Fundamel'ltals of Fluid DYl'Wllmics

133

It is general knowledge that-the "ector grad F [nerc F (qt. q'J' q3) = = 0 i,!l the equation of the sllrfaC'c in the 00\..... and qh q'J' q3 are the generalizcII C'urvilinear C'ool'(linat~l coincides in dircction with a normal to the snrlaC'e. lIen(,f-, ror conditions of Row without sep· aralion. the dot prodnel of 111i~ vcC'tor and the "clority vector V is zero. Conseqllcmtly. the condition of Row withont separation can be written in a mathematical form ,IS follows: VgradF ·0 Taking into account that gradF

""-&. :~ i

l • -/;;.

;~

i'J

~-*. :~

13

(3.3.15)

the ('ondition of Row without separation can be wl'iUcn as

-&' :~ VI+i· :~ V2~-*' :~ V =O 3

(3.3.16)

For Cartcsian coorninate."', we haye grad F , . (iJF.'iJ,r) i l + (;)F.'iJy) i'J -:- (iJpiiJz) i.1 COII!lt'qllcntly. r.r (3.3.16) i'.,. ilFliJx ~- 1'" OFhJy --:- 1": flF/i)z ,... ()

(:1.3.17)

For two-dimensional plane flow 1""

V;"". -

aF/iJ.r; aFlog

(3.3.17')

If the equation of lhc surface is gh'en in q;lindrical ('oordiIl8tes. we haye

gradF-:-~il+*j'J-:-

+.*i

3

thererm'f', the ('omlilion of flow wilholll scpal"ulioll has Illc rorm (3.3.18) In a particulllr case. whNI a :'lUl'face of rc\'olnt.ioli is ill the flow, we obtain the equation Fx OF/O:r -;... {)F/i)r - 0 (:3.:3.18')

'·r

froll1 whieh we find the condition for the vclo('itr ratio:

f.; - ~~~:;

(3.3. "19)

Other bOlilldal'Y cOlldilioll~ ('an also he rormulated. They are determined for each spccinc problem, the boundary conditions ror

1.34

Pt. I. Theory. Aerodynamics of an Airfoil and a Wing

a viscous gas differing from the cOlHlitions for an ideal fluid. ParliClllarIy, witli'll studying the flo,v of a viscolls gas in a boundary layer, the solutions of the pertinent equations must satisfy the conditions on the sllrface of the body and at the edge of the boundary layer. According Lo experimental data, the gas partides adhere, as it were, to the sllrface, and therefore the velocity on it is zero. At the boundary layer edge, the velocity be{'~omcs the sallie as in free (inviscid) flow, and the shear stress equals zero.

3.4. Integrals of Motion for an Ideal Fluid The differential equations deriver! for the general case of motion of a gas are non-integrable ,in the finite form. Integrals of these equations can be obtained only for Lhe particular case of an ideal (inviscid) gas flow. The equation of motion of an ideal gas ill the vector form is 8V/fJt + grad (V2/2) + cud V x V = -(tip) grad p (3.4.1) This equation can be obtained from vector relation (3.1.22') in which the terms on the right-hand side taking into acconnt the influence of the viscosity should he taken equal to zero. In its form (3.4.1), the equation of motion was first obtained by the Rmisian scientist prof. I. Gromeka. With a view to the mass forces, Gromeka's equation becomes 8V/8t -;- grad (V2/2) eurI V X V ~ G - (lIp) grad p (3.4.2)

+

Let us assume that the unsteady now will he potential, hence curl V = 0 and V = grad 'P. In addition. let us assume that the mass forces have the potential U, therefore the vector G = -grad U

+

where grad U = (8U/ax)1 (aU/uyH --1_ (uUliIz)k. If a fluid has the property of barotropy characterited by an unambiguous relation between the pressure and density (this occurs, for example, in an adiabatic flow, for which p -:-:: Ar"), then the ratio dp/p equals the differential of a certain function P and, therefore, (lIp) grad p = grad P \Vith a view to this equation, expression (3.4..2) becomes 8 (grad rp)18t grad (1'1/2) = -grad L: - grad P

+

Substituting the quantity grad (8(r/8t) for the deri"ali\"e ,ve obtain grad (8rpI8t) grad (V212) = -grad U - grad P (3.4.~)

u (grad
+

Ch. 3. Fundementels 01 Fluid Dynemics

13~

Going o\'er from the relation for the gradients to one hetween the corre~pollding scalar fnne.tions. we rlild df{'ldt

+ V~/2 + P

where

-i- C -= C (t)

(:{.4.4)

Jdpip

p.=

(3.4.5)

Expression (3.4.4) is known as the Lagrange equation or integral. The right-hand side of (3.4.4) is i\ function that depends on the time, but docs not depend on the coordillates, i.e. is identical tor any point of t1 potential flow. The term:: on the left-hand sic/£' of (3.1.4) have a ~imple physical meaning: 1'2/2 is the kinelic energy, P "-:: ) dplp is tlle potential energy d\le to the pressure for a unit mass, and U is the potential energy due to tlte position of the fluid particles and related to their mass. To re\'eal the physical meaning of t.he Iirst term, let liS lise the expression for the potenlial function rJ(rlas -:-: T"s. where V" is the projection of the velocity "eclor onto a certain direction s. We can dctermine the function (f from the cOlUlition (f So and s are the coordinates of a

lix~d

-1

V..ds (where

and an arbitrarr point. respect-

i\·ely). The deriyathe lh(dl .~ .~ (dFs'r)l) d.~.

,.

Thl' local acccleration aV,,/at cau be considered as the projection of tllc inl'rtia force due to the presence of local acceleration and related to unit ma~s, and the product (iH's/dl) ds as the work done by this force on the se('tioll d.'1. Accordingly, the derivative a
+

+

If, parti('ularly, the y-axis is directed \'('rt.ically upward, then L' = gy, alld (3.4.6') Qlf'dt + ~rll/2 pIp gy =-= C (t)

+

+

136

PI. I. Theory. Aerodyn!!mics 01 lin Airfoil lind

II

Wing

Of major practical significance is the particular case of a steady potential [low for which aqJliJt = 0 and the function C (t) = tonst, Le. is independent of the time. In this case, Eq. (3.4.4) is reduced to t.he form

V 2 /2+

Jdplp--!·U=const

(3.4.7)

This partial form of the Lagrange equation is called the Euler equatiOD. It expresses the law according to which the total energy of a unit mass is a constant quantity for all pOints of the steady potential flow. Hence. the constant in the Euler equation is the same not only for t.he entire region of a flow, but also, unlike the function C (t) of the Lagrange integral, is independent of the time. The Euler equation for an incompressible fluid (p = canst) has the same form as (3.4.7), the only difference being that instead of dp/p it includes the ratio pip. Let us considC'T the more general case of a non-potential steady flow of a gas. The equation of this motion has the form curl V X V = -grad P - grad U (3.4.8) grad (V2/2)

J

+

or grad (V2/2

+ ) dplp +

U) "'... -curl V X V

(3.4.8')

The right-hand side of (3.4.8') equals zero if the vectors curl V and V are parallel, Le. provided that a vortex line and streamline coincide. In this case, we have

V'/2

+

Jdp/p

+U~

C,

(3.4.9)

This equation was ftrst obtained by I. Gromeka. The constant C1 is the same for the entire region where the condition of the coincidence Of the vortex lines and streamlines is observed. Such regions, to study which the Gromeka equation is used, appear, for example, in a flow past a finite-span wing. This flow is characterized by the formation of vortices virtually coinciding in direction near the wing with the streamlines. A flow does not always contain such regions, however. A flow is customarily characterized by the presence of vortex lines and streamlines not coinciding with one another. The family of vortex lines is given by Eq. (2.6.1), and that of the streamlines (pathlines), by Eq. (2.1.6). The flow being considered is described by Eq. (3k8'). Let liS take the vector of the arc in the form as -:--= dxi + dyj -++ dzk. belonging to a streamline or vortex line, and determine the dot product ds·grad (V2/2 + dplp + U) = -cis-(eurl V X V)

i

Ch. 3. Fundamentals 01 Fluid Dynamics

137

The left-hand side ill this equalion is tlJ(' 101,1\ differenlial of the trinomial ill pl'lrcntheses, consequently.

d CV2/2

+ ) dp/p

-:- L') = -ds· (curl V X V)

(3.4.10)

The product (Ourl V X V is A. veclor perpendicular to the \'f'('tors curl V and V. The dot product of this y('ctOI' aTld the \,prlor d'3 is zero in two cases: when the ('ector d~ COiIlCidl'S wi/It. 11'1' directioll oj a streamline (pathlinc) 01' {",hen (hi,) reef 01' coilleides Il"ilh the direction of a vortex. III these two ca,<:e~. the following '<:o\l1tion of tile equation of motion is valid:

P/~ -.:. \ dpip

+

C ....: C 2

(3.-1.11)

where the vallle of tlte cOI1..-\.alll C z clepc/HII< 011 what palhJine or vortex line is beillg considered, Relation (3.4.11) is known as the Bernoulli equation. It is obvious that for yariolls \'ortex lines passing through points on a given streamline, the conslaut is lhe :;iUlie as fOl' tile streamline. III eXfletly the same way, Ihe ('onsl.ants fire id~ntical for II rllllliiy of ~tr(>amlincs (pathlines) find the vortex through whose points a slrenmline pfl~~(lS. One must deflrly understand Ihe distinction between the Grollleka and Bernoulli equations con~idcred abo\'(·. TIley nrc both derived for a vortex (non-potential) nO\\', howen~r the iiI'S!. of them reJ1p(,\.f'. the fact that the totfll energy of a IIllit llli1~S 01' the IS in the entire region where the \'orlex liHl'S and while the second equation ('."'tabl [ghes Ihe law this energy is constant along fI giH'n streamline lIr L·or/c.! ingly. in the Gromeka equation. tlw COll;.:talll, is thp ."amp entire flow region being consi(ll:'red. wllere
138

Pt. 1. Theory. Aerodynllmics of lin Airfoil and a Wing

we shall write this equation ill the form

V 2 /2

+ pip + yg =

C"

(3.4.12)

When studying the motion of a gas. we may disregard the [nflu· ence of the mass forces. Consequently, we mllst assume that L' ,,= 0 in Eq. (3.4.11) and the other int.egrals. Particularly, instead of (3.4.12). we haye (3.4.13) Lel \lS consider the motion of an ideal compressible gas. In such a gas. heat transfer processes due to viscosity (heat conduction, diffusion) are absent. Assuming also that the gas does not radiate energy. we shall consider its adiabatic (isentropic) molion. Exam· ination of energy eq\lation (a,2.14) reveaL-, that Eq. (3.3.1) holds for s\lch fill inviscid gas in the ahsence of radia\io[] (€ - 0). Consequently. the Bernoulli integral is (3.4.14) In this form, the Bernoulli integral is an ene!'gy eqnation for an isentropic now. According La this rqlLalion. the slim of the kinetic energy and enthalpy of a gas partide is COlLstanL Assuming that i 0' cl,T cl,p/(pR), cp - CD c- R, and k - c1>'"c(., we lilld i ~ lk/(k - 1)1 pip (3.1,.15) Con~equently,

V'/2 V 2/2

+ [k/(k -

1)[ pip ~ C

+ kRT/(k -

1) --" C

(H.16) (3.4.16')

The Bernoulli equation for an ideal compressihle gas is the theoretical foundation for investigating the laws of isentropic flows of a gas. 3.5. Aerodynamic Similarity Concept 01 SlmUlrHy

The aerodynamic characteristics of craft or their individual elements can be determined twlh theoretically and experimentally. The theoretical approaches are hased on t.he llse of a system of equations of gas dyuamics that is solved as applied to a hody in a flow, the hody ha\Oing a given configuration and arhitrary absolute dimensious. \Vhen rUllning experiments intended for obtaining aerodynamic parameters that can he used directly for further ballistic calculations or for verifyiug the resnlts of theoretical investigations. it is not always possible to use a fnll-scale body because of its large size, and a smaller-size mod "I of the body has to be used. In tlds con-

Ch, 3, Fundamentals 01 Fluid Dynamics

139

Jl('ction, Ihe question appem's on Ihe poss.ihility of trans.fel'fing the l'xpl'rimental 1'('''-lIitS obtained 10 full-scllle ])ollts and indicates dU'lracit'ri!UTnll that measurl.'Olent!< in a wind tUJlIl(>1 :riel(led II drag fo\"t'e which ill a('('ol'(lance with (1.:1.5) is Xm"~ c,,',modQmodSmnd' l\OW let us s('e wilen we ('all use the l'e,'-'11l1 ohlaillcd to determine the dl'flg rorre of a full-!>cal(' bod~' in a('('o\"llallre witll the formula Xf,; ..., (,..t~qtsSts in whid\ the Ilrag- ('oel1i('il'Jlt c.\',fs ror this hody is an unknown qlHlIItitr, while tile \"elocitr hl'afl qt~ and the rererence or dHiraeterislic Mea SfS an' gi\'(,Jl. FI'OIll the Iwo rormulHs for Xt~ lind X,"od' we oblain X,s '-' X mod (cx.fs:C.,·.'HOd) (lJ,sSf~'lJn'''dS'''''d) (3.5.1) A glance at this (!xpl'ession r('\'('als thai tlte experimental \"lIlue X mCld ran be used to e\"aluate the fllll-,"('il.le force X", only upou the ('quality of the aerodynamic {"oeffidput~ (,·.lIIool Hnd (\'.Is because the qUlIlIlities q"'(ldSmod and qtsSr~ are detCl'lIlined ullamhiguously by the given yaillt's of the wlo('ity h('uris and tILL' referellce areas. lIere both flows-the model and fl\ll-~cale on('s-haw the property of dynamic similarity. It consisls. in thi~ ('lISC ill Ihflt the preset fOt'ee characted~lic or one flow (the drag X ,urld) i!< lIsed to rInd the rhllr(lr leristi(' of till' olher flo\\" (thc force .Yr.) by simple cOHYersion, similar to tit(' IransitiOH from olle systf'm or units of mea~lll'emeHt to anoLher. The re(ll1irements whose satisraclion in the gin'll ('ase (,IlSlll'es the equality C.\"."".d - c.\"'~' ,11\(1 in Ihe gPlleral ra~e of other OilllC'llsiOllless aerodynamic coet"iicil'nls 100, are eslabli..-lted in the dimellsiOHal analysis allfl s.imilarity method pro(·('pding eilllCr from the physicAl natlll'c of the phenomenon bping stlldi('d or from the corresponding differential equation!> of aerodynamil's. Con..-iderLng the expression rOt' the ll(']'()(lyuamic coefiieiellt

(fx,,)1*

C'\·8= ) [iJeos(Il.T D )·;·(',."cos (3 ..'5.2) <s, Qhtnined from (1.3.2), we can !':'CE.' Ihat this coefficienl depends on dim('n~ionless gcomet!'ir parllnlell'I's. and also on (limensionless qlllllltities sudL as th(' pl'eSSlll'(' rQPllidont 11IIt! Jocal fricLion facioI'. Hellc(' it follows Ihat thc aerO(IYllllmic co('ffirienl..- fol' a fuJl-sclIle ohjed [lIHI an eXpl'fillll'lltal onl' with diffNent ahsolute dimell!: with 11 gi\'en configuration

p

140

PI. I. Theory. Aerodynllmics of an Airfoil and a Wing

of the body in the flow known \"alues of the angle of attack lind the sideslip ,mgle. as of the rudder and elevator angles, will be functiolls of the stream velocity l' the pressure poc, the density p "', lIH: dynamic yj~co!>ity .~oe. the specilic heats c" "" and C,,(IO of the gas, as \I'cll as of a certain characteristic (reference) linrar dimension of thf' hody L ConsequenLly. the drag coefficient will also depend all tltese parameters, and we can compile a functional relation for it in the form ex:...: t (V"',, p"", p"" fl=, c)lx" C,._. L). Since this coefficient is a dimensionless quantity, it mnsl also be a function of dimensionless parameters. From the general considerations or the dimensional method. it follows thnt the seven different argnments of the funclion C x call be reduced to three. The latter are dimensionless combhHltiolls compiled from V.", p"", poo, fix-, cl'''''' c,_ ,." and L because there arc fOllr independent units of measnrement, namely, mass, lcngth, time, ancl temperaturE'. These dimensionless combinations h,,\'E' the following form: II "" Vk~p<:.Jp"" '- V ",,/a ~ = = Moo-the :vrnch numiler for an Hlldisturbcd flow; lloop""LifLoo = = Reoo-thc Reynolds number bi\~ed on t.he parametl'rs of an undisturhed flow and the cilflracleristic linear dimension L; cp ...,Ie,."." = koc-the adiabatir exponent. In the expression for Jl,,,,, it is assumed that Vk""p""."p", = am is the spet'd of sOllnd in Lhe undisturbed flow. Indeed, in accordance with the gt'l)pral t'.\pre~sion for the speed of sOlmd a? ""'" dp.dp. and also with a dew to the adiflhatic nature of propagation of sonic disturbances in a gas, according to which p ==- Apk, we have 0 2 = = d (Ar~)/dp '----' kp'p. Ac('ordingly, the square of the spepd of sound in an l1udislurbed flow is a;' ".., k ...poo.p"". Hence, the ratio V .../Vk~p"",p= '----' All other dimcnsionlcss combinlltions except for M"", Re"", k"", formed from the seven paramet.ers indicated aho\·e or in general hom any quantities that can he determilled by them are fllnctiou!'; of the combination!'; Moe, Re"", and k,.,. Consequcntly, the drag coefficient is ex --= t (Moe, Re".., /c",,) (3.:i.;{} Similar expressions can he obtained for the olher IIcrodYll1l1nic coefIidents. It follows from these expressiolls that when tile nllm~ her." M "'" Re.., and the parAmeter k"", for a model and a flll1~scalc­ flows are equal, the aerodynamic coefficients for geometrically similar bodies arc the same. Hence, an important concillsion can he made in the dimensional analysis and ~imilarity method in accordance with which the necessary and sufficient condition for aeroclynamic similarity is I.hr constancy of the numerical \'alues of the clim(>nsionie:o;s comLinntion~ forming what is called a bns(>, i.e. a system of dimellsionless ql1antilies determining all the other parametPfs of a flow The~e (lirnensionless combinations are ("died Similarity eritNia. (10,

Ch, 3, FUfldamentab 01 Fluid Dynamics

141

The ~imilaritr critt'ria gi\'t~n above have a delillite phrsical meaning, In accordance with the e:o.:pression a:! = dp/dp, the speed of sound can be considered as a criteriou tlcpeuding 011 the property of compressibility, i.e. on the ability of a gas to change its densit}' with a change ill the pressure. Consequently, the Mach number is the similarity criterion that is used to characterize the influence oj compressibility 011 the flow of a gas, The Heynolds number is a parameter used to appraise the influence of ~'I$C'osily Oil a gas ill motion, while Lhe parameter k _ = cp ""fc r "" determines the features of a flow due to the thermodynamic properties of Q gas. Similarity (rU.ria Taking Account of

the VIKGsU, and Heat Conduction

In the more genernl conditions of flow characterized by the influence of a number of other physical and thermodynamic parametel's on the aerodYllamic properties of craft, the drnamic similm'ily criteriA are more complicated and diverse. To establish these criteria, we can usc a different approach of the dimensional analysis and similarity method based on lh(> use of the eqUAtion of motion of a "bcons Iwnl-conducting ga~, Lel us wrile these (!qtUltioll~ in the dimclI:-:iolli(>ss form, i.e, in a form such that the paramett'fs (vciociLr, presslIre, temperature. etc,) in the equations arc rclal('d to I'd('rt'llco panUl1l'lt'I'S. The illtter arc constants for f\ gh'en flo\\' alHl dl'lt'rmillt' iLs scal(>. \\'e shall take as references the paramnlers of lhl' (1'(>(' stream: it,;; Yeioci~y r ... , pressure p_, dCllsitr p_. lempf.'l'al1ll'C T...,. f[YIl
;; ~ .IL. Y ~ yiL. Z ~ ,no t

~

tit.

(3.5.4)

while those for the velocity, pressure, density, viscosity, and mass forces have the following form:

V.~VJV••

V,~I'~v•.

V.~V.IV •. p~plp•• }

p ~ pip.

;="'/""'"

v==v/""'"

X=XIH. Y=Ylg.

(~.5.5)

Z=Z/g

1+2

PI. I. Theory. Aerodynamics of an Airfoil and a Wing

Let us introduce dimensionless variables into the equation of motion (3.1.17) and the continuity equation (2.4.2). We shall use only the first equation of system (3.1.17) for trallsformation becau.sc the other two equations are compiled in a similar way. Let us write the indicated equations using dimensionless variables:

~.'iV.T+ t~

dl

V!(v /,

~+V cti\. "'c1l: U

iJfx),,~gX au'.;..y'iJi

_~ . ..!...~+ \'~v~ poeL

it c)x

+-'-[.i(2 P iJx

L~

{".1V +...!.3 ...!.. ax div V ;r.

iJT!"_..!..diVV)+~( av,:. +~)

+*

og

fJ.t;j

((l~;T

-+-

b~,

iJ,I

lJr

)]}

~.~+p""roe [(l{P~'Tl +iJ(P~ul +a(p~l)]=O h"

L

"l

()x

ag

bz

where div V '- afix/ax -I- aVuMi'· aV,/aJ. From the reference quantities in these equations, we can form dimensionless numbers charaeteriling the similarity of gas flows. These numbers, named after the scientists who were the first to obtain them, are given in the following form:

Sh .,--, V oot~/ L - the Strouhal number; Fr - V~/(gL) - the Fronde number;

M = V""la~-the Mach number; He = V ""p""L/~"" = V""U""" - the Reynolds number (the subscript "00" on M and Re has been omitted).

(3.5.6)

Here a .... = 'Vk ....RT"" is the speed of sound in an undisturbed now (k .... and T .... are the adiabatic exponent and the temperature of gas in the undisturbed flow, respectively). IntrodUCing these numbers into the equation of motion and the continuity equation. we obtain:

8

Sh-I u~~ _V%U~T+VJl u.~:r

in

ux

II

+V:

uV.:r =_p'

uz"

X

-k",,1M'+'~ +~ {VAV:r+4-.fz-diV V

Ch. 3. Fundilmilnillis of Fluid Dynllmics

143

...:..~[~ (2 8~~ -~div V) +.i (8~x -+- J~!I '1 P

ax

3

ax

ay

8y

a,x

I

+*( 8~~", ~ a:r~ )J}

(3.;J./)

Sh~+a(p~x) _1_a(p~!I) +ii(p~.) ,-,,0 ,1/ ux (ly ,h

(:{.;l.8)

Let us transform the energy equation (:1.2.14) in which we shall exclude terms taking into account radiation and diffusioll. \tVe introduce the dimcllsionless variablcs

'.i' '-"

TiT <>0,

c

p ': "'"-' C"ICI' ""

"'l -

A A",

(3.5.9)

where cp~ and A~ are the specilic heat and the heat conductivity of a gas in the 1Indisturbed flow, respectively. Having in view that di -_ c p dT and expanding tllt! total derivative dTJdt, we obtain after the corresponding substitutions:

P<>oC:~"'T~PCp*,

P.... C1JOO;.'<>oToo

pCp (Vx-f

l-VY~-i-Vz ~~ ).~ ~': .*+fl~,~~~ x {[(

U~[)'..;- (":n'J-fi'ldivV)" 4i',:) ...!.-

Let

liS

~"" div (i'grad T)

(3.5.10)-

introduce the dimensionless Prundtl nllmber Pr ,-= J.I. oocp ""iA<>o

(3.5.11)

by means of which we I>hall compare the relatiu eoect of the riscosity and heat conduction, or, in other words, appraise tlle relation he tween the heat flux due to skin friction and the molecular transfer of heal. Hence, taking into account that pooJp<>o= ilT.., = c,.,., (1 -1.k..,» X T <>0, we have:

:r -*)

PCp(Sh-'*~_V:<*-7-V" -r Vz koo - ' • .<0. _, ,\1'(1'00-" {[( a~,)" (al:, )'] dt· Ife rJ,x iJy

~Sh-'

1100

- {- ~ (eliv V)2 + 4~e~} + Ife'lpr div (Kgrad

T)

(3.5.10')

144

Pf. I. Theory. Aerodynamics of .!In Airfoil and a Wing

Let us converl several additional reilltions of the system of equa· :tions of gas u}'lIamics (see Sec. ::l.:~) to the dimensionless form:

p = pT ii"" = t. (p~p, T "'1) (1;"m~) i: = t. (p~p, T.T) (1J).~) ~=

'p

=

(3.5.12) (3.5.13) (3.5.14)

t, (p~p, T .1') (1t"~)

(3.5.15)

I. (p~i,

(3.5.16)

T ~1'i (l/cp~)

Assume that we are investigating two flows over geometrically similar surfaces. For such surfaces, the dimensionless coordinates of analogous points are identical, wbich is a necessary condition for the aerodynamic similarity of flows. To observe the sufficient condition for such similarity, we must ensure equality of the dimensionless values of the gas-dynamic parameters (velocity, pressure, density, etc.) at analogous points. Since the dimensionless variables are simnltaneotlsly sol1ltions of the system or equations (3.5.7), (3.5.8), (:1.5.10'), and (3.5.12)-(3.5.16), the indicated equality is evidentJ~' ohserved provided that the systems of dimensionless equations, allli also the dimeu.e:ioniess boundary and initial conditions for each flow are the same. ''''hell considering syst.ems of dimel1~ionless eqllations for two flows, we see that both these systems are identical if: (1) the similarity criteria are eljllfll:

1<'r l = Fr 2 , ReI = Re z , M 1 = M z. Sill

=

Sh 2

(3.5.17)

(2) the equality of the Prandtl numbers Pr l = Pr 2 is observed, i.e. (cp~"~/)'~), =

(cp~~"j),~),

(3.5.18)

and also the equality of the specific heat ratios for the t,,·o gas £Io\,'s (3.5.19)

(3) each of the equations (3.5.13)-(3.5.16) determines the depend.ences for the dimensionless variables -;:im, I, ~, or p on the relative quantities p and f. and also on the variables (3.5.17)-(3.5.19). Such relations do not exist for a dissociated gas because similarity criteria of the form of (3.5.17)-(3.5.19) or of some other form cannot be found. This is why the corresponding dimensionless equations (3.5.13)-(3.5.16) are not the same for a full-scale and a model flows, and complete dynamic similarity cannot be ensured. Two particular cases can be indicated when this similarity is ensured. The first is the flow of an undissociated gas for which the mean molar mass remains constant (f!ml = P-ma). while the specific

c

en.

3. FundallUlntals of Fluid Dvnamics

148

heals. heat conductivity, and viscosiLy var)' depending on the temperature according to a power law of the kind y = aTr.. In this case, Eqs. (3.5.14).(3.5.16) for the quantities 1. "ii. and p are replaced by the corresponding dependences only on the dhnension)rss temperature T = TIT .... The second case is the Do\\' of a gas at. low speeds when the parameters A., "', and c p do not depend on the temperat.ure. The corresponding values of these parameters are identical for the full-scale and model flows. For this case, the system of equations includes the dimensionless equations of Kavier-Stokes. continuity, energy, and also an equation of state. The boulldary conditions imposed on the solutions of the dimensionless equations gi\'e rise to additional similarity criteria. This does not relate to the condition of flow without separation. which does not introduce new similarity criteria. Indeed, this condition in the dimensionless (orm is

c

V. 8Flar + Y, aFiay + i', aF,a: = 0 and is the same both for a full-scale and for a model flow because of the geometrical similarit.y of the surfaces. Bul the temperature boundary condition according 10 which the solution for the temperature mu~t satisry the cquality T .... Til' (hcl'e T" is the temperature of the waUl introduce~ IlJl ndtlitional ~imilnl'ity criterion. In reality, it follows from the hOlindillT ('ourlilions for a fllllwscale and model sUl'faccs haYing the forlll T1 • (1',,"h Mltl 1"~ "" (1'w)~. rt'spccth-ely, lhnt the dimcn~ionle~s tt'lllpNatllr(lS arc TJ "'"' (T1,h and 1'2 = = (TW)2' From tht' cOIHlilion or ~illlilal"ilr. 1") '" 12, Iht'refore the equality (Twh ..... (Twh musi IJ~! nhsencl!. ifl'lIcl'. tiH:' boundary condition for the wuJlIC'lllpel'1lLlII'c 1:~'Hls to nu 1I11diliOilHi ~imilarity criterion: (3.5.20) Tho dimensionless gas-d:mamic yariablcs on the surface of a body in a Dow, as can he seen Crolll the sy~tem of dimen"ionl(>ss equations [provided that Eqs. (3.5.13)-(;~.3.Hi) determine il'm, [, ~, and Cp as a function of 11 depend on the dimensionless cool'dinates anu the time. and also on the similarity criteria (3.5.17)-(3.5.10). Particu· Jarly, the dimensionless pressure can he represent~d a~ the (ulldion pip. = CJll (F,., Re, M, Sk, p,., k.,

Tw ,

XI

y, Z, t)

(:~.5.21)

We can \l.SC the known pressure distrihuLion to dctC'l'lIIine the dimensionless drag force coerrident for a gh'ell inslanl: c:C" = X./(q_S) =
146

Pt. I. Theory. A.rodynMnies of .n Airfoil end a Wing

This expression determines the dependence of the aerodynamic drag coefficient on the dimensionless similarity criterion more completely than (3.5.3). But relation (3.5.22) does not reOeet all the features of fiow of a dissociated gas because it was obtained from the simplified equations (3.5.13)-(3.5.16). Consequently. formula (3.5.22) is less accurate for such a Row than for that of an uudissociated gas. and determines only partial similarity. The similarit.y criteria on which a dimensionless aerodynamic variable depends havo a definite physical meaning and characterize the real factors aRecting Lhe aerodynamic force. The Froude number Fr is a similarity criLerion taking into account. the infiuence of the mass force (force of gravity) on the drag. It can be seen from the equation of motion given in the dimensionless form t.hat the number Fr equals the ratio of the quant.ity V!oIL due to the infiuence of t.he inertia forces t.o the scaling of t.he mass forces g. The equality of the Frollde numbers Fr for a full-scale body and its geometrically similar model signifies that they have identical drag coefficients due t.o the influence of the force of gravity of the fluid. This similnrit.y criterion is of no siguirlcance when studying gas flows because the influence of the force of gravity of a gas on motion is negligibly small. But the importance of this criterion may be appreciable in hydrodynamics, particularly in the experimental investigation of the wave resistance of various navigable ves.c;els. When a body moves in n real fluid. the aerodynamic forces dt'pcnd on thE! viscosity. The viscous force is characterized by the Reynolds number Be that can be obtained as the ratio of the quantity V~/L describing the influence of inertia forces to the parameter v"" V "",IV' determining the influence of the viscosity. If the equality of the Reynolds numbers for two geometrically similar flows is observed. the condition of partial aerodynamic Similarity with account taken of the influence of viscosity is satisfied. In this condition. the friction factors for a full-scale and a model bodies are equal. The similariLy with respect to the Mach number is obtained from the ratio of the quantity V:"IL to the parameter p,,"/(p,,"L) that takes into account the influence of the pressure forces depending on the compressibility of tho gas. The partial similarity of two flows of a compressible gas flowing over geometrically similar hodi(>$ is observed when the Mach numbers are equal. When investigating unsteady flow, similarity with respect to tho 5trollhal nllm her is signilicant. It is obtained by comparing the iU('rtia forces and the forces due to the influence of tbe unstt'adiness, I.e. from the ratio of the quantities v:.,n and Veo.t"". Two. unsteady flows around n full-scale body and a model have partial aerodynamic Similarity with identical values or the Strollhal number. The similarity criteria with respect to the Prandtl number P,.

Ch. 3. Fund!lmenlals of Fluid Dynamics

147

and the ratio of the specific heals are due to dclinill' I'cquirements to the physical properties of the gases in the full-scale and model nows. Tho gases may differ, but their physical properties mllst k00 2 • The Praucltl observe the equalities Pr l = Pr2 and k""l number depeuds on the dynamic viscosity and tit(' heat conductivity, The d}'namic viscosity reflects the properties of II gas which the molecular lran~fer of the momenlum depends Oil, while the heat conducth'i!y chnracterizt·s the intensity of the molecular transfer of heaL. Consequently, the Prandtl Jlumber Pr "-, I1c poo Arin nrc contradictory, Lt,t II~ consider, for t'xampl<>, the R<'YllOlc[s, Froudt', all(1 Mach 1l11llllwrs, 1"01' th(' ohserv!lllce of simililrily with rN,pecL to the skin frirtioll f()i'(~es, ii, is pss('nti,,1 thut F 1 L" \"I::..o V/L~ \'2' If we a:::sume thal for 1\ fllll-~calt· 1111(1 mo(lel nows the coefficients \'1 . , \'2' then Ill(' spN'd of tll(' III {)(I I' I now 1"2 1'1 (LliL2)' Lt'. il is greater than the !lpI'('d of the full,scnl(' flo\\' the same llumlwr of times that the mod('l of the hocl~' ill th(' flo\\' is l('ss than the full-scale 011(', To enSllr<> similarity with r('specl to the forc('s of gra\'i[.y, it is necessary to ellsme I'quality of [,he Frolldc nllmber!l, i.e, V~ (LICI) '-" ,.- I'i (/J3g~), wlu:"'nce it follows that if experimt'lIls were rlln at idE'ntical values of g. tll('n tlw speed of the 1I101lei flow is 1-'2 = = F, V /,/1. 1 , We can see Lhat in the given case the speed V 2 for t.he smAlI-sizc model must be smaller lhan \/1 instead of grcater as in the previous eXllmple, lipoll equillity of the ,Mach numhers, we havc 'V,."a l :..~ 1"2.02' Assuming for simplilicatioll that a 2 ,.... a l , we oht.ain the condition for ('qnality of the speeds of the l1loc1('1 all(1 fulJ-selllc flows, II is n8tural that all these conditions for t1l(' speed CallIlot· be ohser\'ed simultaneously, th<>refore we cao (ollsi
148

PI. I. Theory. Aerodyntlmics of tin Airfoil tlnd a Wing

If at the same time the speeds are not great, then the influence

of the pressure forces due to compressibility of the gas is negligibly small, and, consequently, no account may he taken of the similarity criterion with respect to the Moch number. assuming Olat the aerodynamic coefIicient depends on the Reynolds number. The aerodynamic force. moment, or hoat flux from a gas to a surface is the result of the actiou of a moving gas on a body. Various processes occur simultaneously in the gas: skin friction. compression (or expansion), heating, a change in the physical properties, etc. Therefore, one must try to satisfy the maximum number of similarity criteria. Itor example. it is expedient that the equality of thfl Reynolds and Mach numbers for a full-scale and model flows be retained simultaneously, i.e. Rei = Re'l and Ml = M'l' This is especially important when studying aerodynamic forces, which for bodies with a large surface may consist of equivalent components depending on the friction and pressure due to compressibility. This condition can be ensured wheu running experiments ill variable density wind tunnels. If tests are boing performed in a gas flow in ..... llich the speed of sound is the same as in the full-scale flow (a 2 = a l ). it follows from the equality of the Mach numbers that V t = 1'1' Having this in view and using the equality Rei = Re 2 or, which is the same, VtP2L2/[.L'l = VIPILI/IlI' we obtain L2P2i!l-2 = L1Pl'f11' Assuming that 112 = !l-l' we lind that the density of the gas in the wiud tunnel Dow is P2 = Pl(L 1IL 2). Assuming that t.he temperature of the fullscale and model flows is the same (T2 = TI ) and using an equation of state. we obtain the condilion P2 = PI (LIIL,). Hence. to simultaneously ensure similarity with rcspecl to t.he {Ol'ces of skin friction and of pressure with account taken of compressibility, i.e. to observe the equalities Re l = Re 2 and Ml = M 2 • it is essential that the static pressure in the flow of a gas produced by a wind tunnel be greater than the pressure in the full-scale flow by the same number or times by which the model is smaller than the fullscale body. The design o{ a wind tunnel makes it possible within known limits Lo control tile static pressure in the model flow of a gas depending on tile sizc of the model. With a known approximation, the influence of heat transfer may noL be t.aken into account when determining the force interaction. lIere the aerodynamic coefficients will depend on the numbers Re • •V. and Sh. l{ in addition the tests are conducted in a gas for which k""2 -" 1.:- 1' we ha,-e c. ~

f

(Re, M, SI.)

(3.5.23)

~

f (//e,

(3.5.24)

For a steadr flow

c.

M)

Ch. 3. Fundamentals 01 Fluid Dynamics

149

3.6. Isentropic Gas Flows Conflgur~on

01 Gas Jet

Let us consider the steady motion of an ideal (in viscid) gas in a stream or jet with a small expansion and a slight curvlIture. Motion in such a jet can 1lC studied as one-dimensional, cha.racterized by the change in tbe parameters depending on one linear coordinate measured along the axis of the jet. For a steady flo\\', the parameters determining this now :lrc iclentical in each cross section at any instant. If the width of the jet is small in comparison with tJu.- radius of curvature of the ('('ntrl' line, Lhe lateral prc~lIre gradient may he disregarded. and one mar consider that the prC!ssure nt ench poiut of the jet cro~., section is til(' same. Thl' consideration of slIch olle-dimensional sl,elldy now of 1I ('Olllpr('ssiblp gflS IN\(ls to the simplest approximate solution of the equatiolls of gas dynamics. Condition (2.4.51) of n constant mass flow is retained along the jet, i.e. PIVISI = P2'V2S~ "-' PaVaS., ~ ... , or pFS = canst, where th(' subscripts 1, 2, 3 signify the COl're~ sp~nding paramcters of the gns at the control surfaces. Cross .!'lections of lhe channel with an area of SI> S2' ant} S3 have been chOSl'n as these slIrfflceg. Taking logarithms, wc obtain In p In V :- In S = = const. Differentiation of this l'xprl'ssion yields

,I

+

dpip

-;~

dvrV -;- dSIS

= 0

whence

(J.G.I) Le~

us use relation (3..1.11), by dilTcrentiation of which we have

IT. dV = -di

(3.6.:.!)

Substituting dp r for di. we fmd VdV ,-' -dp/p

(3.6.3)

Taking the speed of sount! fl :-..= V dpldp into accollut 1IIul llsing r~lalioll (3.0.3). we transform (;{.6.1) to the form dS;dV \\'hel'(' M

'-~

~

(Sir) (M' -

1)

(3.0..1)

V,'a is the }Iach Ilumber in the given cross section of

the jet. LeL lIS IlSS11lnC! (hot l.hl' vclocity along the jet grows (dV> 0), 1mt, remains subsonic and. CO[]~l·q1Lent.ly, M < 1. A glance at (:':1.0.4) rcyeals that for Illis caSt' the derivative dS/dV < O. This indicntes that the jet couverges downstream. COIl\'('rsciy, for a subsonic

1~O

Pt. I. Theory. Aarodynolllmics of an Airfoil and

0lil

Wing

V.p.~

p.r.d

v,;

''Fig. U.t

Parameters of

d

p~

to

Po '.

gas flowing over a body

Dow with a diminishing velocit.y (M < 1. dY < 0). the cross section increases. which is indicated br the inequality dS > 0 following from (3.6.4). Let us consider a supersonic Row (M> 1). If the velocity decreases, then. as can be seen from (3.6.4). the differential dS < 0, and. consequently. the jet converges. Conversely, when the velocity grows. the value of dS > O. Le. the jet diverges. Let us take a nozzle that first has the shape of a converging. and then of a diverging channel. In defmite conditions in the converging part of the nozzle. a subsonic now is accelerated, reaching the speed of sound in the narrowest cross section [here dS """ 0 and, as follows from (3.6.4), M = 1J, and then becomes supersonic. This is how nozzles are designed in rocket engines, gas turbines, and wind tunnels intended for obtaining supersonic flows. Flow V.locHy

Let us consider a gas jet Rowing oyer a surface (Fig. 3.6.1). We shall denote the free-stream parameters by Y .... P 00. P 00, T ,." L.,. and a .... and the parameters for the part of the jet in the disturbed region by the same symbols without subscripts. To lind the \'elocity in an arbitrary cross section of the jet. we shall lise Eq. (3.4.14) in which we shall determine the constant C according to the preset parameters of the frce stream: C = Y!.12 1~ (3.6.5)

+

Wit.h this in view. we have yIl2 i = V!./2 .:. too whence

+

V-VV:.-,.2(1~

I)

(3.6.G)

At the stagnation point. V:....: 0, consequently the enthalpy is t = io = V!..:·2 -:- i... (3.0.7)

Ch. 1. fynd~m.,nt.ls of Fluid Dyn~mics

151

Hence. t.he constant C as regards its physical meaning lcan be considE!red as the stagnation enthalpy. With a "iew to this valne of C. the "elocit r in t he jet is V~V2(i,-i)

(3.6.8)

The stagnation pressure Po and density Po correspond to the enthalpy i o• They are determined from the condition i o"'""

k:1'~'=- ~~~~. i ..

=.x;.+ k~l

.

~:

(3.6.9)

We can rewrite Eq. (3.6.9) as follows:

';:+ k~1 ·f:·

k~l .~

Sillce the flow is isentropic, we have pip' .~ p,lp\

(3.6.10) (3.6. \I)

consequently (3.6.12) Seeing that for conditions of stagnation the speed of sound is ao = VkPo/po. we fmd (3.6.13) Examination of (3.6.8) reveals that the velocitylalong the jet grows with diminishing of the enthalpy, and therefore more and morc or the heat is convcrted into kinetic energy. The maximum velocity is reached pro\'ided that the entbalpy i = 0, i.e. all the heat is spent to accelerate the gas. The \'alue of this velocitr is Vmu:= V2i;,

(3.6.14J

or with a view to (3.6.9):

VIPU=Vk~1·~=ao l/k~l

(3.6.15)

Accordingly, the velocity in an arbitrary section is V~Vm .. VI-iii,

(3.6.16)

V=Vmu. 'Vi_(pIPo)(II.-t)ill.

(3.6.17)

or In the narro'.... est. critical, cross section of the jet, the ,·elocity equals the local speed of sound. The latter is caUed critical nnd is

162

Pt. I. Theory. Aerodynillmics of ilIn Airfoil lind II Wing

designated by a*. The critical pressure p* and density p* correspond to the critical speed. It follows from the Bernoulli equation that for the critical section we have a· 2 , k

p.

-y -.- k=T' p* =

Po

k

-r=T 'P;-

Having in view that kp*/p* = a*z, the critical speed is a*=

V k~1 '*=00l / k!l

(3.6.18)

or, taking into account (3.6.15),

a* ~ Vron Y"(k:---C1")!"'(k"~c"1)

(3.6.19)

To determine the local ~ound speed a, we shall use Eq. (3.6.10). Performing thc substitutions a2 = kp/p and ~ = kpo/Po in it, we obtain (3.6.20) or a'1=

k-;i

a*'1_

k"2l

V2

(3.6.21) (3.6.22)

a2= k-;l (vfullJ.- V2)

Let us introduce the velocity ratio (relatiw ,'elocity) '}. = Via"'. Dividing Eq. (3.6.21) by VZ and having ill view that the ratio Via = = M, we can establish the relation between'}. and M: 1.' ~ I(k

+ 1)/2IM'/{1 + I(k -

1)/2IM')

(3.6.23)

Hence it follows that in the cross section of the jet where Vmu is reached, the number M = 00. We find the corresponding value of '}. = Amal< from (3.6.23) pro·vided that i.V -+ 00: 1.ruu~ Y(k '1)/(k

1)

(3.6.24)

Evidently, in the critical section where M = 1, we
1<;;; 1.<;;; Y;;;(k"+--'1"')/"'(k:--"1)

(3.6.2:,)

Pr.....,.., DensIty. lind , emp.rature It follows from (3.6.17) that the pressure ill an arbitral'Y cross section of the jet is (:1.6.26)

Ch. 3. Fundamentals of Fluid Dynamics

1~3

By (3.6.22), the difierence

1- l'I:x = (1 -- "-;-1 M~ f'

(3.6.27)

consequent! y, p=Po(1.-

k;1

;lf~)-II/(II-I)=Po .• (.b/)

(3.6.28)

where the function it of the arguDlPllt M is d~termined by the prt'SSure ratio plPo. For the conditions of the rT(,Il-~tream no\\", fOl'mula (3.G.2:8) yields: Po=poo

(1..:...-YM';,)II.lk-ll,./;.,..

(3.G.29}

Consequently I

p=poo {

~~',i::.~)).;',·~~ f.-III-I) =p,~ ;~..~~)

{3.6.30}

From the equation of an acliabat p ph ,... PO'P:' in which P isreplaced in accordancll wilh formulas (3.fS.2:G) And (3.6.28), we find a relation for the density p = Po ( '1 _. r~:J 1,lk-1) .-, Po ( 1 _;' "'-;- I 'l/~) - I;I/!_ I) ~ POE (;it') (3,G.3'1}

where the function e of the argument ,If is determined hy the fntiq. of the densities pIpo. Using the equation of slate pipo = (p.po) T, T L) (;-\.1).:-\2) and also Eqs. (3.6.26) and (3.6.28) fol' the pressure and l3.G.31)· for the density. we have

T=To

(1-

r~:x) ~~-]'o

(1+ k;-I

.1lrl.~·roT(.,,) (3Ji.~J3)

where the function 't of the argullll'llt .11 is rll"'t('rmineci by the mtio oC the temperatures TIT o. Tables of the gas-dynamic fnnctions :t (,It). e (.1I), ilud t (.U) for values of Lhe exponent k from 1.1 to 1.67 fll'C given ill 11:!!. We shall determine the corresponding fl'tations for llU' dlmsity P&and temperature To by means of expressions (:{.6.:~1) and (a.fi.:i3). For the conditions of a free-sLrNun now, lh('.o;c expl'('~~ion.s yield:

Po=p""

(1 + k;1

,"':or/{k-I)'--f:~~~"")

O.tl.34.) (3.G.o;,)

184

PI. I. Theory. Aerodynamics of an Airfoil and II Wing

III the critical cross section of the stream, M = 1. Consequently. f~om (a.(}.28), (3.6.31), and (3.6.33), we obtain the following formulas fo~ the c~itical yslues of the pressure p*, density p*. and tempe>rature 1'*: P* = Po ( k~t )"/{h_l) P*=Po

(k!t

= Port (1)

(3.636)

)t/(Il-t)=P08(l)

(3.6.37)

T·~T.( k~1

)=T.T(I)

(3.6.38)

where n (1). e (1), and 1" (1) are the yalues of the gas-dynamic functions at M = 1. The above formulas, suitable for ally speeds, can give us approximate relations for cases when the numbers M arc very large. A glance at (3.6.30) reveals that when M>- 1 a.nd M <10 >- 1. we have (3.6.39) Similar relations for the density and temperature have the form plp_ = (M .IM),,,h-.) (3.6.40) TiT _ - (M .IM)' (3.6.41) Using relation (3.6.27) for the velocity of the free-stream flo ...... ,

(3.6.27') for the conditions Moo >- t and M >- 1, we obtain the following .approximate formula for the local velocity:

(3.6.42) Flow of • O.s from • Reservoir

Formula (3.6.12) allows us to determine the velocity of a gas .flowing out through a noule from a reservoir (Fig. 3.6.2) in which the parameters arc determined by the conditions of st.agnation corresponding to a velocity of the gas in the reservoir of V ~ O. Such conditions are ensured in practice if the critical cross section of the nozzle is suffiCiently small in comparison with the cross:iectional dimension of the reservoir. In the critical section of the nozzle, which the value of the derivative dS!d~' = 0 corresponds

Ch. 3. Fund.ment.1s of Fluid DyMmic'

Fig. 3.6.1

Paramett'fs of

It

1I!1f~

gas nowiog from a res('n·oir

to, the Ma ch H1Imber, as can be seen from (a.!i.4), equals unity. i.e. Lit£' \'elocity in this section equals the local speed of sound:

V=a*=

V

k::J·

~:

"-'no

l/·I"-~ 1 ··'l /" 2:~~o

Such n speed is ensllred providc(1 t hat the pressure in t he reservoi r accordiug to (~.(l.3G) is not lower than thc vaillc

Po .• p*(¥)I,(I'-I)

(3.6.103)

If the nozzle communicatt>s with the utmospheric air whose prelSSure is p,. ,-: 1O~ Pa (this pressu.·e is call1'd the backpressure), now through the uonlc is possible whcll p* ;;;;:, 1O~ 1',1. Assuming that p* =- 10:' Pa and k = 1.1, we obtain the pre5sIIre needed to ensure the sp('('d of sound at the ant let of a eO[l\-erging nozzle:

1'0 = 10"( 1.4;-1 )1 ."(t.'-t)~".fI :-: J(lS Pa When the critical partuncters are re-uelle-d in the throat of a nozzle, a further reduction of thl' bnckpressure (pa < p*) no longer nfleets the \·;IIIIt'S of a* und 1'*, whi el. Ilepclld on the stAte of the gas in the reservoir. But Ill're conciitiolls (\fl' crl:'ated fot' the nppearance of a s uprrsonic specil of th e gAS in the diYergillg port of the TlO1.zil'. In Fig. ;Uj.2 . tllt'se c()J\d iHo[1~ of iscutropic motion are charaet('rhed br l' IIt'VPS 1 d(·termilling thr cilaugc ill tlt~ ralio " lpo and in the lIumber M