Mathematical Models and Methods of Localized Interaction Theory

28

Chapter

1. Introduction

to localized interaction

theory

(LIT)

showed that using the Newton model results in penetrator configurations with good penetrating ability as applied to soil and metal. It should be noted that problems of configuration optimization for penetrating bodies are basically different from those for bodies moving in a medium since the objective function is a certain integral char­ acteristic (generally, penetration depth) which is determined as a result of integration of body motion equation which is set up allowing for changes in the contact surface in the process of body penetration into the medium [263]. In the framework of such approach the problem of configuration optimization for penetrating bodies in LIT was defined in [103]. In particular, it is shown that among conical bodies with geo­ metrical similar cross-sections at predetermined volume, a body featuring minimum drag when moving in the same medium at constant velocity penetrates the maximum depth. This conclusion is valid for a wide range of models. In case of the Newton model as well as for hypersonic flow, star-shaped bodies appear to be the optimal ones. Therefore, it appears that for simulation of physically different phenomena, either identical or similar LIM's are applicable which gives evidence in favor of development of common approaches in the framework of LIT.

Chapter 2 M e t h o d s of calculation of integral characteristics of environment effect on body moving in it 1

Differential equations method

The local nature of model of interaction between a medium and a body surface enables one to develop calculation methods which often appear more convenient as compared with direct integration of local characteristics with respect to the exposed area. The efficiency of such methods is especially evident when the need arises to obtain func­ tional relationships for components of forces acting on the body as against angles which define the orientation with respect to the direction of its motion in the sur­ rounding medium. Such situation is characteristic, in particular, for aeromechanics where the body aerodynamic properties are evaluated by the results of aerodynamic characteristics analysis at different positions of the vehicle in a flow.

1.1

Basic equations. General solutions

The initial expression for coefficient CV of force exerted on a body by a flow may be written in the following form (1, 1.4), (1, 1.8): C F = - ^ ff[nn(t)-n°

+ nv{t)-v}dS,

t = arccos(v • n°),

(1.1)

(S)

whereas unlike Section 1, Chapter 1, for unit vector of free-stream flow velocity vj^, subscripts are omitted, S" is a characteristic area and (5) is an exposed area. The value of Cjr will be considered as a function of angles a and tp which are defined as shown in Fig. 1.1; x°, y° and z° are unit vectors of the body axis coordinate system. In general case, both parameter t and boundary of exposed area (S) depend on a and ip. 29

30

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 1.1: Definition of angles a and
IJJmds (S)(S) (S)

dn(t)

dS, -IS-in?* 86 ' d5

(S) (S)(S)

where 5 is one of the angles a or ip, 0 is some differentiable function. Two cases when this equation is valid, are mentioned below. T h e first one occurs when ( 5 ) does not depend on 8 in some vicinity of the considered point. This is true for some range of angles a and
0(0) = 0 . Later on, attention will be paid to conditions which are required in t h e second case but not in the first case. Further, t h e need will arise for expressions for force and m o m e n t components in air-path coordinate system. T h e unit vectors of this coordinate system are expressed in terms of t h e unit vectors of body-axis coordinate system as follows: x a = v = cos a x° + sin a cosy? y ° — sin a simp z°, y a = v a / | v a | = — sin a x° + cos a cos ip y° — cos a sin y? z°, z° = —vv>/\vv\

= sin ip +cos ipz°

(1.2)

31

Section 1. Differential equations method while

dv | v a | = l, 6 = a,
|v,,| = s i n a ,

0<
v{ = j - : ,

-T/2
(1.3)

Considering (1.1)—(1.3), t h e formulae for calculation of force coefficients may be w r i t t e n as

Cxa(a,v) = CF-x° = j-tJJilD(t)dS, (S)

Cya(a,?) = CF • y° = j-JJnn(t)tadS, (S)

C„(a,
(10,(0 U dS,

(1.4)

JJ

where U = - o 7 ( v - n ° ) = ws - n ° ,

86

6 = a,ip

a n d function fie has t h e form (A, 1.2) As follows from (1.4): —=— = — sin a—dip da

cos a C2a

provided functions Qn satisfy t h e condition fi„(0) = 0 or t h e exposed area boundary r e m a i n s unchanged when varying a and
np(0 = £ M i=0

Rr

nT(0 = v T ^ ££,**',

(i.5)

i=0

where Ai, 5 , are constant coefficients. As shown in Section 2, C h a p t e r 1, models (1.5) cover a large n u m b e r of specific situations. T h e expression (1.5) may be considered as an a p p r o x i m a t i o n relationship as well. It follows from (1.5) t h a t R„

n„(0 = £>*'', i=0

RD

M')-EM;, j=0

(i.6)

32

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

where

a, = Ai — Bi-i, Oi^Ai-Bt-i,

bj bj^Aj-i = Ai-x + +BjBi-Bj-, Bj-t

(1.7) (1.7)

whereas coefficients A, with subscripts i < 0 o r i > Sp and coefficients Bj with subscripts j < 0 or j > R? are assumed t o equal zero; p a r a m e t e r s Rn and RD are chosen on t h e condition t h a t begins with i = R„ + 1 and j = RD + 1 all a, and bj equal zero. Let us introduce functions which will be called "support":

Gk(a,V) = —JJMt)ds, (S) 0 M) O = M = t**k = = (v • nii0 ))*, *,

v v == v(a v(a))V >). V >).

(1.8) (1.8)

It then follows from (1.4) t h a t Cxa(ot,
^Cxa( (a, a ,V)^ ) == 'Eb ^ 6iGi{a,
(1.9) (1.9)

v=o

while Cya and Cza, through of derivatives

Cya{*,v) = = jpjj \T,a,ti • ■ ttj ds C».(",f) J ; / / (l><* aj ds l = S'f-ii+lJJ y ai If ddat^ds US *-° (S) *-°

(S)

Rn+l

= fa? £(TV i) / !

„ ,

,

1=1

1 ^

3ada

(1.10)

'

fa,.1\dGi(a,f)

ana r ^V z /

c^

(1.11)

whereas the change in the sequence of integration and differentiation is justified as far as t h e integrands on the boundaries of area ( 5 ) vanish at t = 0. Let us consider the operator:

1\

d2

d2

c o t a

d

a +Q ++ cot a a— ++ m(m+ 1) =- sin .. 22a '' "5 "o9- ^1 o<^> 1" da da m(m + 1) sin a dip da1 da displaying the property [114]: ■^m. L-m

=

(m - i/)(m + + i/v ++ 1l)j/v v{v --' 11)0,-2, £mVv = Lmlpv = (m-v)(m ) ^ ++ 1/(1/ )^-2,

2 << i v/ <<mm, ,

(1.12)

Section

1. Differential

equations

33

method

where functions i\>v are of the form (1.8). The validity of (1.12) may be verified directly by means of transformations employing (1.2). After integration on both sides of (1.12) with respect to the exposed area of the body, division by 5* and change of sequence for integration and differentiation operations (which may be done since i/>„(0) = ^(O) = 0, v > 2), the following expression [114] is obtained: v){m + v+ \)GV + v{v - 1)G„_».

Lm Gu = (m-

(1.13)

In the particular case when m = v = k, (1.13) is transformed into the system of recurrent relations for determining Gk'. (1.14)

LkGk = k(k-l)Gk-2

which was obtained in [114] independently of [254] and later reiterated in [222]. On the basis of (1.13) equations [80] for C°a may be derived, where RD

C£. = X >

G

"

C„ = C^a + b0G0 + blG1.

(1.15)

It is sufficient to write (1.13) for n = RD, " — 2,3, — , RD, to multiply i/-th relation­ ship by 6„ and to add the equations obtained. The result is: = *,

LRDC°S

(1.16)

where * = bRv ■ RD{RD - \)GRD

+ £

b„[(RD - v)(Ro + v + 1}GV + v{v - l)Gv-i\.

(1.17)

i/=2

For many models, R^ + 1 = RD and expressions (1.10) and (1.11) may be presented in the following form [80] allowing for (1.15):

M Cya =-5—, da.

i

a

*

G2a = — : •-=—, sin a Of

U-ioJ

where

SoPJL + £ h*<<«. iGu ** = SoPJL ■=i 1

tt0 = y°'

RD ORD

,

ai = ao,

a, = ^ - - 6 i a o , I

i = 2 , 3 , . . . ,RD - 1.

(1.19)

34

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

From t h e form of functions (1.17) and (1.19) it follows t h a t to calculate Cxa, Cya and Cza on t h e basis of equations (1.16) and (1.18) it is sufficient t o know functions G o , G\,.

■. , G / j D _ i .

It is easy to see t h a t t h e above relationships allow several m e t h o d s of application which practically do not differ between themselves as far as efficiency is concerned. Equations (1.14) written for k = 2 , 3 , . . . , max(.R n + 1, RD) m a y be represented as two independent recurrent systems, for even and odd k. To determine t h e "even" Gk it is enough to know Go while to determine the "odd" ones, it is sufficient to know Gi while

G0(a,ip) = j-JJ dS (S)

is the per-unit area of t h e exposed surface region and

Gl(a,
is t h e per-unit area of projection of the exposed region on a plane perpendicular to vector v . Therefore, t h e problem is reduced to sequential solution of equations (1-14) and calculation of Cxa, Cya and C2a using formulae (1.9)—(1.11). Another solution may be employed which implies calculation of G o , G i , . . . , G H D _ I , determination of Cxa by solving equation (1.16) and of Cxa from expression (1.15). Provided Rn < Rp—2, Cya and C 2 a may be immediately calculated by using formulae (1.10) and (1.11). If Rn = RD - 1 formulae (1.18) and (1.19) may be applied to determine Cya and Cza. If Rn > RD then GRD may be calculated from (1.9) and calculation of Gk of higher orders may be proceeded with. For the Newton model (1, 2.1), RD

= 3,

b0 = 6j = b2 = 0,

63 + 0,

Rn

= 2,

do — 0,1 = 0,

C2 = 631

Cxa = Cxa = 63 G3 and both modifications of the m e t h o d which are considered above appear identical. From expressions (1.16)—(1.19) straight away follow equations of the same form, for which t h e theory was developed [173-174, 250-253]: L3 Cxa = 663G1, r

_1 3

dCx* da

r

1 3 sin a

dCxa dip

Methodically, it is more appropriate to further discuss t h e version of t h e m e t h o d when t h e support functions Gk are determined following which Cxa, Cya, Cza calcu­ lated using relationships (1.9)—(1.11). Therefore, t h e problem is reduced to solution of equations (1.14).

35

Section 1. Differential equations method The bounded general solution of equation (1.14) has the following form [168]: G*(a, V) = G° + ^.Tk(cos

a) + £ T^cos a) [pQ c o s ( ^ ) + $

sin(^)]

where Tk are Legendre polynomials

T

¥¥/^2-^

^ =

Tk are associated Legendre functions

TZ(u) = (l-u*f£;Tk(u), ft\J and /ijj.,, are arbitrary constants, G° is a particular solution of equation (1.14). Obviously, for bodies of revolution, dependence of Gk on ip is lacking and equation for Gk is of the form (1.14) where J2

J

Lk = —

+ cota—

+ k(k + l).

(1.20)

The general bounded solution of this equation is as follows: Gk(a)

= fiik0iTk(cosa)

+ G°k(a).

One useful feature of equations (1.14) is noteworthy. Should Gk-2 (k = 2,3) satisfy the corresponding homogeneous equation that is Lk-2Gk-2

= 0,

k = 2,3,

(1.21)

Gk 2

(1.22)

then functions Gk

~ 2(2k - i)

'

are particular solutions for inhomogeneous equation (1.14) which follows from pre­ sentation of linear operator in the form: LkGk = Lk-2Gk

+

2(2k-l)Gk.

To find particular solutions, more general formulae [252, 254] may be used. So, the problem of calculation of force component coefficients as a function of angles a and ip is reduced to determination of particular solutions for equations (1.14) and calculation of constants which appear in the general solution of corresponding homogeneous equations.

36

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

T h e constants may be naturally determined by calculating t h e values of Gk for a necessary n u m b e r of points by directly using formulae (1.8). T h e r e is another m e t h o d involving determination of constants by existing d a t a of experiments or calculations conducted by other m e t h o d s which may be assumed at least as precise as t h e calcu­ lations using LIM, however, this m e t h o d cannot be recommended as universal due to three m a i n reasons. Firstly, t h e "outside' 1 d a t a may be used only by means of integral characteristics t h a t is t h e constants will be calculated only after determination of all Gk- So far as t h e equations for Gk are of recurrent nature, the number of constants on the right side of t h e equations grows with increasing k and essentially impedes the search for particular solutions. However, this objection is not valid when "free" constants are a d m i t t e d only in equation (1.16) employed to determine Cxa. Secondly, determination of constants is reduced to t h e problem of calculation of parameters of t h e function by its known values for a certain n u m b e r of points, while statistical characteristics of errors inherent in such values are usually not known. T h e problem of obtaining reliable estimates of desired parameters in such problems is not trivial and its analysis requires application of special m e t h o d s . Thirdly (and closely related to the above), t h e desired constants are determined by a body configuration in t h e framework of t h e initial LIM used. By interpreting this constants as free parameters and selecting t h e m by "outside" d a t a for a body of pre-determined configuration, we introduce unwanted controversy to t h e m e t h o d . T h u s , it is preferred to determine constants on t h e basis of formulae (1.8). As far as the search for particular solutions for equations (1.14) is concerned, it is accomplished by rather traditional methods. In [86-87] some features concerning their application for t h e case are discussed. Below, t h e particular case of small angles a and

1.2

T h e case of small angles a and
Should a body have a flat m a x i m u m mid-section (base section) perpendicular to axis x of t h e body-axis system and the range of variation of angles a and ip is such that the boundary of exposed area is the boundary of t h e m a x i m u m mid-section, then formulae for Go and G'i are of the following form: G0(a,if)

= Ssur/S+,

Gi (a, ip) = cos a,

where SSUT is a body side surface area, 5+ = S* is a base section area which is assumed as a characteristic area. Therefore, in this case conditions (1.21) are valid

Section 1. Differential equations method

37

and particular solutions G% and G° are calculated by formulae (1.22). Thus, the general solution is of the form: G2(a,ip) = G2(a) + 3 sin a cos a(fi2\' sin

+ /4„» cos2y>), o

£3(0-, y) = 63(0) + - sin a(5 cos2 a — 2)(/J 3 V sin

) + 15 sin2 a cos a(/4» sin 2y? + /4«, c o s 2v) + 15 sin 3 a cos a(/i 3 , sin3y + / 4 . . cos3yp),

(1-23)

where c

(0)

G 2 (a) = 3 5 - + — ( 3 c o s

!

)'

(0) 3 G 3 (a) = - c o s a + ^ c o s a ( 5 c o s 2 - 3 ) (1.24) 5 2 whereas G2(oi) and G3(a) are general solutions of equation (1.20) that is support functions for bodies of revolution. As follows from formulae (1.7), it is sufficient to know functions G2 and G3 to calculate integral characteristics in case of models (1.6) ai Rj, < 2, RT < 1, whereas

a0 = A0,

a.i= A\ — Bo,

60 = Bo,

b1 = A0 + B1,

a2 = A2 — ft, b2 = A1-B<>,

h = A2-Bl.

(1.25)

Such a subclass covers many specific models described in Section 2 of Chapter 1. Analytical solutions written below are obtained in the framework of the subclass. Expressions for C xo (a,y>), Cya{a,'<'>(cosa)\ i=0 3

Cya(a) = s i n a ^ A ' y^ c o s a ) ' , i=o

where

^' = §r(*-f) + T^-*>.

38

Chapter 2. Methods of calculation of 1C of environment effect on body moving in it 3 KP = Hft1 ++ \AU \{A B^H 2 + Kp = AA0 0--lB -(A 2 -2-B^tl 2+

55

K^

5 5

=

Z Z

\^.{M-BQ),

*Af'^.U^-ft), w=!,&<*-*), A

KfT) ) = = T j ( ^ - f t ) ( ±(A 5 ^ 2°-Bi)(5 . ) . - ft£\-2)-Ao, 2)-A0,

x ™ = ! *§*&(*-ft), &(*-ft,), K™ ArW , - f 2t-)Bl.). K™==- ^ W ( A-\^M In t h e case discussed above when a body side surface is completely exposed, for­ mulae for calculation of Gk in explicit form may be obtained directly for t h e general class of models (1.6) as applied to bodies of revolution. We shall m a k e use of this to b e t t e r d e m o n s t r a t e t h e features of the differential equations m e t h o d . Should a body configuration in t h e body-axis system be determined by equation x = $ ( p ) , where x is a coordinate along t h e body axis and p is t h e one in t h e perpendicular plane. T h e n , t h e expression for calculation of Gk follows from t h e formulae of Section 3 of Appendix and has t h e form: 1 2TT

Gk(a)

= - j J p{¥+ 1 ) ^ ( 0 0 3 0 - * sin a cos 0)* dp dO

(1.26)

0 0

whereas transition is m a d e to dimensionless variables using t h e common characteristic size equal t o t h e radius of t h e base section of t h e body. T h e area of t h e base section is adopted as a characteristic one and t h e dot denotes differentiation with respect to PAs a result of transformations which are omitted, t h e formula is reduced to t h e form:

m ...

= Y, liJ\smafj(cosa)k-2\

Gk(a)

(1.27)

j=0

where ,0) k

s fc! i 2{k-2j)\{j\f

jU) k '

I

/<J» = f p ^ i ^ + l)^

dp

Section 1. Differential equations method

39

whereas 6j = 4 if j = 0, and <5; = 1 at j / 0. It is evident that between integrals Ig* at k > 2 there are relationships of the form:

lij) + I{kj+1) = / & ,

i = 0,l,...,[fc/2]-l.

Therefore, for any m > 0, all integrals J ^ may be expressed in terms of inte­ grals 4 , 4 J • • • i ?%m while integrals 4 „ + 1 — i n terms of integrals l\°\ j | , . . . , /Im+i whereas there are no linear relationships inside these groups of integrals. Thus, in order to calculate the next Gk, only the new integral Ik should be calculated, while

40) = ^ ( o ) . If the values of independent integrals in terms of which the body configuration effects the support functions Gk are interpreted as indeterminate coefficients, it ap­ pears that, in expression for Gk the number of such coefficients (including SSUT and S + ) equals k. The same situation also occurs in the case when Gk are determined by solving equations (1.14). It should also be noted that the form of particular solutions of equation (1.14) is defined by (1.27). Therefore, in the considered case both approaches appeared almost equivalent although in this situation significant advantages of the support functions method are pronounced. Firstly, it provides a reasonably universal algorithm of obtaining functions Cxa(a), Cya(a) in the form of series. Secondly, the above analysis which led to formula (1.27) was essentially based on the possible use of formula (1.10) for calculation (i.e. on one of the attributes of the support functions method). Otherwise, the difficulties associated with validation of formulae for Cxa(a) and Cya(a) would grow substantially. In a general case, when the boundary of exposed surface is natural, that is defined by the condition (v • n°) = 0, the formulae similar to (1.6) are not convenient for obtaining the analytical relationships and preference should be given to the differential equations method. Let us consider again the case of three-dimensional bodies and assume that A0 = A1 = B0 = 0. Then the calculations become simpler and the formulae acquire a more evident form. In this case, it is convenient to write the formulae for calculation of force coefficients in the form: C*„(a,vj) = 5 , cos a + (A2 Cya(a^)

_ ,

= -(A2-B1)^ ,

A2-B, 3 sin a

B^G^, = - - ^ ^ +

dG3 da

1 3 sin a

BlSina, dCia dip

40

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

After insertion of G 3 from (1.23) and transformations, the following formulae are obtained: Cxa(a, + p3J,cos

2f)

2

+ 10sin a(fi3J sin 3y + /4«, cos 2ip)\ , Cya{a, tp) = C y a (a) + -{A2 - Bx) [3cos a (5cos 2 a - 4)(/4V sintp + fi3,\ cosy?) + 10 sin a(3cos 2 a — l ) ( / 4 . sin2<^> + ^ J , cos2
^-r—- [(5cos 2 a-2)(^ 1 » ) cosv?-/i^ ) .sin¥!) + 20 sin a cos a(/x3„ sin 2y? — /x3„ cos 2<^>) + 20 sin2 a ( ^ sin 3<^ - p | « cos 3y>)] ,

(1.28)

where (?*a(o) = - c o s Q [ ( 5 c o s 2 a - 3 ) C i a ( 0 ) + (3X2 + 2fi 1 )sin 2 a] , Cya{a) = - s i n Q [ ( l - 5 c o s 2 a ) C I 1 1 ( 0 ) + (3>l2 + 2 B , ) c o s 2 Q - y l 2 ]

(1.29)

whereas Cxa{a) and Cya(a) are the corresponding characteristics of the body of rev­ olution having a drag coefficient which equals C xa (0) for symmetrical flow around. Cx„(0) is a drag coefficient for a three-dimensional body when the free flow velocity is directed along axis x. Should the body be symmetrical, formulae (1.28) are further simplified. If the horizontal plane xz is the plane of symmetry, then Cxa(a,ip) — Cxa(a,ir —
= -cosa{(5cos2a-3)Cxo(0), + [oA2 + 2B, + 30(A2 - Bx )$}, sin2 a ] } ,

41

Section 1. Differential equations method Cya(a,0) = - s i n a { ( l - 5 c o s 2 a ) C I a ( 0 ) + [3A2 + 1BX + 30(A2 - Bi)/4!l cos2 a] -

[A 2 + 1 0 ( A 2 - B 1 ) 4 2 . ) . ] } -

(1-30)

By using formulae (A, 1.10) relating Cxa and Cya to d and Cy, simple expressions may be obtained for coefficients of force components in a body-axis system: Cx(a, 0) = fio -f ^i cos 2a, CJ,(Q,0) = /i 2 sin2a,

(1.31)

where

/*o = i[CUO) + A]» ft = \^ [3Cr(0) - A], 11 _ _, ft = fift-ft + 2#i> -Bu 0- m + ft = (2)

£ = A2 + 1 0 ( A 2 - A M ; ; -

(1-32)

As a matter of fact, in case of three-dimensional body, C x (0) and A may be considered as independent constants defined by a body configuration, while for a body of revolution, £ = A2.

1.3

Analysis of integral characteristics behavior at small angles a and
To analyze the IC behavior on the basis of formulae (1.30) and (1.31), additional information is needed on the constants depending on a body configuration. The necessary expressions are easily obtained if the equation for a body surface is specified in form (A, 2.1) and formulae (A, 1.14), (A, 1.15) and (A, 2.2) are used. Taking into account the body symmetry relative to the horizontal plane, we get for the model (1,2.15) lx0

= ^{A2-B1){h + l2)+l-Bu

jil = !(Aa-B 1 )(Jx-/ 2 ) + i B l , H2 = (A2-Bi)I2 + -Bi, L = hlh,

f=l,2

(1.33)

42

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

as well as t h e expression for coefficient of pitching m o m e n t : "* z = Hz sin 2 a , li3

= (A2-B1)I3

+

/„ = IJ(IQL),

-Bj4,

U = 3,4,

(1.34)

where L »/2

IQ=]-S+= j j Wdxd8>0, 0 T/2

L

r r W3 I I —dxd.j

h=

/ l3 =

-TT/2

> 0,

raw J -ljj-dxd6>0, 0 -,r/2

*

jjH*trW(xt7 + *W osO)W + $Wcos 6)W = C ax atdxde

'--IS

2

0 -*/2 £

I4=ff

T/2

$(t7Cos8

/ / 0

+

x$x)dxd8,

-„/2

55$ $ #. = _ ,

W = **x,

w = ^x,

d$ a$ ** = -

». = —, *, = -

(1.35) (1.35)

whereas functions <2 and £7 are determined by formula (A, 2.2) and L is t h e body length. For t h e considered class of models ( 1 , 2.15) the following holds true: M > 0,

Bi>Q,

A2>BX.

So we get from (1.31), (1.33) and (1.35) Ho > 0,

fi2 > 0.

So far as t h e following estimate

w/ 0

-,11

WdxdO

. + (i/*,) 2 + (*«/w)2 -

/ / wdxde = i0 0 -»/2

Section 1. Differentia/ equations method

43

Figure 1.2: Characteristic variation of coefficients C*, C„, m, against angle of attack a at a < a, for bodies with horizontal symmetry plane. is valid for t h e integral h,

then

C „ ( 0 ) = Cx(0)

=110 + ^

= (A2 - 5 , ) / ! +

B1
while it m a y be easily shown t h a t C x (0) = A2 for a plate perpendicular to flow direction. T h e curves shown in Fig. 1.2 illustrate t h e p a t t e r n of t h e IC behavior as a function of t h e angle of a t t a c k at a < a, where a, is a limiting angle at which no shadow is formed on a b o d y side surface. T h e s e regularities are confirmed by numerous experimental d a t a and "precise" calculations. Some d a t a are given in Fig. 1.3 a ) , b), c). By excluding t h e angle of attack from equations (1.31), t h e equation of polar curve of t h e second kind is obtained:

(C*-/'o) 2 , Cj n K

.

,,.

|«| < a .

which defines t h e elliptic curve (Fig. 1.4 a), b ) ) . Point A on t h e ellipse corresponds t o a = 0. M o v e m e n t from point A to point C along t h e ellipse corresponds to increase of a in t h e region of positive values while movement from point A to point

44

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 1.3: a) "Precise" calculation data [239] for cones with spherical bluntness (6 is the radius of bluntness divided by maximum cross-section radius).

C" corresponds to decrease of a in the region of negative values. T h e position of point A depending on t h e sign of \ix is shown in Fig. 1.4 a), b). Further analysis is m a d e for bodies of revolution. In such case, more demonstrative results are obtained concerning the features of t h e IC behavior. T h e corresponding formulae are derived from (1.30) at n\;„ - 0 and from (1.31) and (1.32) at u = A2

Section 1. Differential equations method

45

Figure 1.3 (continued): b) Experimental data [164] for blunted cones with half-angle of opening of 20° in hypersonic rarefied gas flow. The notation is the same as in Fig. 1.3 a). For t h e sake of clarity, t h e formulae appear as follows: Cx{a)

= j [C,(0) + A,] + j [3C,(0) - A2] cos 2 a ,

Cv(a) = - [A% + Bt - Cx{0)] sin 2a,

(1.36)

Cxa{a) = - c o s a [ ( 5 c o s 2 a - 3 ) C I ( 0 ) + ( 2 5 1 + 3 A 2 ) s i n 2 Q ] , Cya(a) = ^ s i n a [ ( l - 5 c o s 2 a ) C r ( 0 ) - r ( 2 B 1 + 3 A 2 ) c o s 2 a - y l 2 ] .

(1.37)

From equations (1.36) and (1.37) one useful property of IC follows: should drag

46

Chapter

2. Methods

of calculation oflC

of environment

effect on body moving in it

Figure 1.3 (continued): c) Experimental d a t a [155] for three-dimensional star-shaped bodies with four "cycles" (f is the radius of the circle inscribed into the maximum cross-section divided by the radius of the base of the cone having the same length and maximum mid-section area).

coefficients of two bodies of revolution be equal at zero angle of attack, then values of Cx, Cy, Cxa and Cya of these bodies are equal in the entire range of angles of attack \a\ < Q , . Therefore, t h e p a t t e r n of variation of IC for a body of revolution is defined only by t h e value of Cx(0) and it is not possible to change the p a t t e r n of plots of Cx(a), Cy(a), Cxa(a), Cya{a) by changing a body configuration only but retaining t h e value of C x (0). It is evident from t h e curves in Fig. 1.2 t h a t for three-dimensional bodies both increase and decrease of Cx{a) can occur. T h e first equation (1.36) enables to specify the criterion for t h e behavior p a t t e r n of function Cx(a) for bodies of revolution. T h e

Section 1. Differential equations method

47

Figure 1.4: Polar of the second kind for bodies with horizontal symmetry plane at |a| < a,; a) /ii > 0; b) in < 0. increase takes place when /i > 3 while t h e decrease occurs at \i < 3 where H =

A2

CM

>i,

a t fi = 3, Cx{o) does not depend on the angle of attack. T h e plots of Cy(a) depending on t h e p a r a m e t e r values fi is illustrated in Fig. 1.5 a), b ) . Values of p a r a m e t e r \i vary in t h e range 1-2 for highly blunt-nosed bodies and

48

Chapter

2. Methods

of calculation of IC of environment

effect on body moving in it

Figure 1.5: Characteristic plots of Cx(ct) and Cv(a) at different values of /i for a < a , . A — "precise" calculations [214] for spherical segments (Moo = oo, f = 1.4); O — "precise" calculations [239] for cones with spherical bluntness (Moo = 8, <S = 1.015; the meaning of parameter S is explained in the caption for Fig. 1.3 a).

49

Section 1. Differential equations method

Figure 1.6: Relationships between parameter // and half-angle of opening of spherical segment u> on the basis of "precise" calculations [214], increase for sharp-nosed configurations. T h e relationship between /J. and u (which is a half-angle of t h e spherical segment opening) plotted on t h e basis of t h e calculation results [214], is presented as an e x a m p l e in Fig. 1.6. T h e value ft = 1 corresponds to flow a r o u n d a p l a t e perpendicular to the free-stream flow. For sharp cones, [i ~ (sin P)~l where j3 is half of t h e angle of cone opening. Let us analyze t h e variation of force component coefficients in air-path axis system ( a > 0). A t B\ = 0 it is completely determined by t h e value of p a r a m e t e r /x. Drag coefficient Cxa (Fig. 1.7 a) ) decreases with increasing angle of attack if 1 < (i < 2. A t fi > 2, t h e behavior of function Cxa(a) is more complex. At small a, an increase takes place which occurs up to t h e angle of attack a = min(a 0 ,
/.-I 3/i

Should a, be reasonably large (a, > aQ), then at a > a0 drag coefficient begins to decrease and at fairly large angles of attack may become less t h a n Cx(0).

50

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 1.7: Characteristic plots of CIa{c<) and Cya(o) at different values of ft; A, O — see the caption for Fig. 1.5; • — "precise" calculations [48] for sharp cone of 35° P a r a m e t e r fi effects the behavior pattern of lift coefficient Cya (Fig. 1.7 b) ) to still greater extent determining, in particular, its sign at small angles of attack. At 1 < fi < 2, function Cya(a) is decreasing, however, the character of t h e corresponding curves may be different. If 1 < fi < 11/7 w 1.57 and a , > o i , where

a i = arcsm

2Q*-2p\ 3(3// - 5 ) )

then at a , = Q] transition from decrease to increase of Cya(a) takes place. When 11/7 < fi < 2, Cya(o) decreases in the entire range of angles of attack a < a,. At fi > 2, C v a ( a ) increases to angle of attack a = m i n ( a i , a « ) . If Qi < a,, then further decrease on (a > ct\) occurs which may reverse the sign of lift coefficient at fairly large angles of attack.

51

Section 1. Differentia/ equations method

25 Figure 1.7 (continued). The revealed regularities which are valid for any bodies of revolution (of course, to the extent of validity of the used model) enable one, in particular, to predict the IC behavior pattern at different angles of attack only by information on body drag coefficient at symmetrical flow around. The conducted analysis enables one to estimate the aerodynamic properties for bodies of different configurations having minimum initial information with the aim of selecting only the most promising ones for further more detailed examination.

52

2

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Invariant relationships method

In t h e present section, method for obtaining relationships between IC's for bodies of different configurations is developed. T h e validity of t h e relationships transpires from t h e basic LIM representation and does not depend on selection of specific model. Such relationships take place for "identical" conditions, when t h e "global" p a r a m e t e r s which characterize t h e process of interaction between bodies and a m e d i u m conserve t h e values. As the "similarity laws" often imply the regularities in which t h e change in conditions of such interaction plays an important role, to avoid terminological uncertainty in t h e developed approach, the t e r m "correspondence relationships" is used. T h e purposes of subsections 2.1-2.4 are as follows: In Subsection 2.1, t h e m e t h o d is developed which can be used to d e t e r m i n e con­ figuration of pairs of bodies with IC's which are related between themselves by cor­ respondence relationships valid irrespective of the selection of t h a t or another LIM. Such relationships will be called "universal" ("invariant"). Various classes of invariant relationships for typical classes of configurations are constructed. In Subsection 2.2, the method is further developed by generalization of several bodies which enables one, in particular, to calculate IC's for a wide class of configu­ rations on t h e basis of information about IC's of two or three bodies without using d a t a on t h e specific model of interaction between a flow and a. body. A number of such relationships are constructed for different bodies. In Subsection 2.3, t h e invariant relationships m e t h o d is developed in modification concerning its direct use for calculation of IC's for bodies of predetermined configu­ ration based on results of "precise" calculations or experiments conducted for bodies of different configuration. In t h e framework of this technique, calculation formulae are obtained and specific examples are given as illustration. Generalization for a general class of three-dimensional bodies is presented in Sub­ section 2.4.

2.1

Invariant relationships for a pair of bodies

Let us consider a pair of bodies—"the original" one (its characteristics designated in superscript by zero in parentheses, sometimes not) and "the corresponding" one (figure one in parentheses). Assume the expression for some IC both for the original and the corresponding bodies may be presented in the form:

P,M =

/ 7f)[a;W1*M,$M]-H(i>M)dxl"\ J o

$<»> = f * l dx^"'

( 2 .1)

where function $' t ''(:r'''') characterizes the configuration of a body of length L'"' in longitudinal direction, function H includes information on t h e LIM type; functions

Section 2. Invariant relationships method

53

If" a r e defined by t h e class of body configurations under consideration; subscript indicates t h e IC n u m b e r out of w-"> considered; t h e rest of t h e functions and parameters characterizing t h e body configuration and its orientation are not specified as arguments. As follows from A p p e n d i x , formula (2.1) may incorporate components of force of m e d i u m effect for a wide range of bodies including wings with curvilinear edges, bodies of revolution and others. Consider t h e problem dealing with t h e search for t h e transformation which can be used to o b t a i n t h e corresponding body (also in some predetermined class of configurations) for a r b i t r a r y original one of t h e given class in such a way t h a t IC's of bodies of form (2.1) satisfy t h e relationship: 1

n(">

E E « . M ^ = ». i/=o ; = i

2 2 ((2.2) -)

i/=o ; = i

where a) are certain arbitrary chosen coefficients. T h e requirement of invariance (2.2) with respect t o choice of model is a principal one, i.e. expression (2.2) should b e valid irrespective of t h e selection of t h a t or another function H. Let us prove t h e existence of invariant relationships (2.2) and validate the m e t h o d for d e t e r m i n a t i o n of functions which characterize the configuration of t h e corresponding body. Transition to p a r a m e t r i c definition of function $ W ( x M ) in t h e form:

$0) $(1) = = * $ (( x* )) ,,

x ( 1 ') = x(x), x(x), x'

x(0)= x(0) = 00 ,,

x(L) x(L) = L Lw(,1 \

x<°> x < ° >== x

enables one to transform expression (2.1) to t h e form: p . ( i ) = / j i W Lx,$,— $ *1 J L x i x' 0 o

H\%]-x'(x)dx, H 7 • x'(x) dx, ix.I .J

ii = =

M ,, . , It l,2,-.\,2,...,n

where derivatives with respect to x of functions characterizing t h e corresponding body configuration are m a r k e d by prime. Requiring t h e observance of condition {x)

*' -= 6(x) ¥$ *(*) 9{X x'(x) x'[x)

(2.3) (2-3)

>

t h e following combination is obtained: l

1

»W Til"'

^E EEL«f!t M i/=0 i = l

L L

") p(")

r

„(D

) = / #/ /((*§)) x' x'J^a'i £ i)4TyTP(X, {x,^,i)

0

i=l

11

„«»

0) #, 4) ++ Y£iai°a!)T0)t{0)r/(x,^,i) (x, $, $) i=l

rfx dx

54

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

whence it transpires that condition G = 0 is valid irrespective of the form of function H, should the following expression hold true: x> £ af^d,

*,*)

= -£

aflj^x,«, *).

(2.4)

The corresponding function can be determined from equations (2.3) and (2.4). The following conditions may be assumed as the initial ones: x(0) = 0,

1(0) = r « ,

(2.5)

where r' 1 ' is some parameter which, as a rule, characterizes dimension of correspond­ ing body nose section. Therefore, the problem of search for the corresponding body is reduced to solving the system of differential equations (2.3), (2.4) with initial conditions (2.5). For many classes it may be easily solved. We outline the modification of the problem dealing with determination of body configuration having n integral characteristics Pi related between themselves by in­ variant relationships of the form:

£>.Pj=o. 1=1

In this case, function $(x) which defines body configuration may be found from

£>7i[*,»(*),*(*)] = 0 i=i

with the desired initial conditions. Function H may be of an even more general form than that shown above. This modification of the method is further used when examining the problem of designing the bodies with invariable static stability factor. To explain the essence of the method validated above using formal transforma­ tions, some considerations are given below regarding the simplest situation when velocity vector of a free-stream medium flow with respect to a body is parallel to the axis of a body of revolution effected on by the medium. In this case, the angle of tangent inclination at each point of frontal part of the body surface conclusively defines the angle between vectors vj^ and n° Condition (2.3) provides mapping the points of initial curve described by function $(x) onto the points of the corresponding curve described by function $i l '(a^*') so that the angles of tangent inclination are equal in the corresponding points and consequently, for the considered classes of bod­ ies (the original and the corresponding ones) quantitative characteristics of medium effect are identical. As far as relationship (2.4) is concerned, it represents additional requirements for the corresponding points (they may be tentatively called geometric ones) which are intended to validate (2.2).

Section 2. Invariant

relationships

method

55

Figure 2.1: Coordinate system and notation for bodies with affine cross-sections.

Now let us turn to using the method for solving specific classes of problems. It appears that its applicability stretches far beyond the framework of the simplest situation just considered. The terminology used below is typical for aerodynamics and deals with the effect of a flow on a body. This does not limit the general nature of results but makes them more demonstrative. As an example, a situation is considered when both the original and the corre­ sponding bodies have affine cross-sections (Fig. 2.1) while the configuration of trans­ verse contour of the bodies is known and identical. Bodies of revolution, in particular, belong to this class. In cylindrical coordinate system x'1'' p'"' 9^ the surface for such a body may be given in the form: pM = n{0M) ■ $ ( ">(x ( "'), where function r) characterizes the transverse contour configuration and function $'")—the longitudinal contour configuration. Being divided by dynamic head, projection of aerodynamic force F'"' on some direction defined by vector 1° lying in plane i'">, j / ' " ' and forming angle x with axis a;'"', is considered as characteristic P ' a body-axis coordinate system, free-stream flow velocity vector v ^ lies in plane x'"', 2/(l/) at angle a to axis x(") (Fig. 2.1). The effect on the body side surface is considered.

56

Chapter 2. Methods of calculation of IC of environment effect on body moving in it The expression for P'"' using formulae (A, 2.6) may be written in the form: PM(a,x)

= (F^/qoo) C\"H] = (F ( ">/ 9oo ) ■ 1° = S*<"> ■ C, L<") < " '

= = //

M ( i MM ) )-w • WMMdx^de^\ dxW dOM, J/ $ ¥"\x

(2.6) (2.6)

o jW

where W(,/> is of the form (A, 2.6) while functions U are calculated by formulae (A, 2.5) at / = $<"). In particular cases, formula (2.6) yields expressions for aerodynamic force compo­ nent such as drag (x = a) and lift (x = 7r/2 + a) as well as axial force (x = 0) and normal force (x = jr/2) in the body-axis system. In a general case, a shaded region is formed on the body side surface. Its bound­ aries are determined from condition t7 = 0 which may be written in the following form by using formulae (A, 2.5): M 2 cosa-^-ri(0 -sin cos a ■ 6 (l/) •) 17(9'"') — sina a ^ ^ - s i n e^+

T){6M)-cos dM

dd^i

= 0.

(2.7)

Should the body configuration and the range of considered angles of attack be such that any straight line which is parallel to v ^ crosses the boundary surface in no more than two points and longitudinal and transverse contours are smooth enough, then equation (2.7) yields solutions defining side boundary contours: W M == 0W[# (a: W")], }), 9^ )[*(" >(x<

M 6 e

4p

- « >[$<">(x< ">)1-

The fact that functions 0_ and 9y realize relationship on i'"' through relation­ ship to derivative $'"' is absolutely important. Similar situation takes place also for function W^v' which follows from formulae (A, 2.5). Therefore, the expression for may be written in form (2.1) should the following hold: T ;; H = $<">, ) B[¥" #($<">]]= =

J/

^ 1, n "M' = l, u

f^ = 00,1, ,1, M

r (, , w^[¥ \e^]de W <")[* ' ,^ ) ]^.( '' ) -

(2.8) (2.8)

It is apparent that the given formula is valid also when there is no shadow region on the body surface exposed to a flow; in such a case, 9_' = 0, 0+ = 27r. It should again be noted that to obtain correspondence relationships, the specific form of function H is not important. In this case it has a fairly complex structure and depends on both

57

Section 2. Invariant relationships method

specific LIM and configuration of transverse contour of bodies as well as angles a and It must be stressed that relationship (2.2) connects any components of integral aerodynamic force of the same name (that is in fact the vector of the force) at any angle of attack identical for both bodies. From formulae (A, 2.8) and (A, 2.9) it follows that at W<"> = l|"5 and allowing for (2.8), P<"> in (2.1) yields the expression for calculating the body exposed surface -.(") whereas in the particular case, when 6 (») 0 and e{+] 2w, the expression area S}"J, for the body side surface area S ^ . Therefore, along with relationships between the force components P(1)(a,*) =

fcP(a,x)

(2.9)

there exist relationships between areas -d) S}!j(a) =

kSlig{a),

1

S^ sur ' - —* where k = —a\ /a[ Assume that WH

_ $ M . vy(»

0,

9+=2w,

i/ = 0,l,

then £,(">

pi-)

=

J $MibMdx-

fr,(6^)2de^ 2-ir

= [*M(L) 2 -*'">(0) 2 _ e(") _ §{*>)

">) d0M

2 L

0

whence the following relationship transpires: S{1] - S(y = k(S+ - S_), where S i , S_ are areas of base section and nose section respectively. In particular, from the relationship between areas it follows that drag and lift coefficients for exposed surfaces of all corresponding bodies are identical and equal to those for the original body, provided either a side surface area or an exposed side surface area or a difference between the areas of base section and nose section is taken as a characteristic area. Let us assume that the original body does not have a flat nose section, while for the corresponding body it is characterized by parameter r*1', that is conditions (2.5) are

58

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

valid. T h e n invariant relationships (2.9) take place should t h e functions characterizing t h e longitudinal contour of the corresponding body be determined from t h e system of equations: $ ' = x' 4>(x),

x' $ =

fc$(x).

Taking into account initial conditions (2.5), the solution may be written in t h e form: X

x(x) = k / $ ( < ) • o

dt/${t),

$(s) = [k$(x)2 + rW2]1/2

(2.10)

In order to demonstratively illustrate the variation in body shape as a result of the transformation, let us consider the following specific situation: k = 1 (forces acting on t h e original and corresponding bodies should be equal) and t h e equation for longitudinal contour of the original body has the form: $(x)=x1/2

(2.11)

whereas here and further on all linear dimensions are presented (retaining desig­ nations) in dimensionless form using common characteristic size L. O m i t t i n g the calculations, t h e following equation of the corresponding function is obtained which results from (2.10), (2.11): x

(

ij>(t) = Wt2

*

\

$

f

1 11/2

- 1 - \n{t + Vt2 - 1).

(2.12)

In Fig. 2.2 it is shown how t h e body longitudinal contour configuration is changed as a result of transformation (2.12) for different values of r' 1 '.

2.2

Generalization for several bodies

In order to avoid blocking up the discussion with excessive calculations, one integral characteristic per body is considered. T h e body number is indicated by a subscript. Assume function $ „ ( ! „ ) defines t h e configuration of t h e i^-th body out of n + 1 of the considered ones. Similar to Section 2, Chapter 1, the expressions for IC's are presented by functionals of the form:

P , = / T „ {X„, * „ , * , ) ■•tf(&„)«!a:M, J

o

$ „ = -f^, dXv

(2.13)

Section

2. Invariant

relationships

59

method

Figure 2.2: Variation in longitudinal contour configuration as a result of the transformation (for bodies with affine cross-sections); the original body, the corresponding bodies.

where T„ are given functions, H is a function including description of interaction between a flow and a body surface, $„(£„) = r„,

0 , = 0,

0 < x, < Lv.

(2.14)

Functions $„(£„) and parameters L„ which characterize the configuration of the first, the second,..., the n-th body, are given. It is desired that for function $ 0 (zo) and parameter L0 (subscript 0 will be omitted, as a rule), P0 = P be linearly expressed in terms of characteristics P„ [y — 1,2,... ,n) irrespective of the choice of function

60

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

H, that is

(2.15)

P = J2P»P„.

It is assumed t h a t functions $ „ ( x „ ) («/ = 0 , 1 , . . . , n ) have the following properties: # * ( £ „ ) = u'1

$„(0) = i C \

(2.16)

and retain the character of convexity at 0 < xu < L„. If, for example, function $„ characterizes t h e generator configuration for a body of revolution, then t h e specified conditions mean t h a t bodies under consideration are those with bulging surface and angles of generator inclination to axis xu are equal in the initial and final points respectively. If conditions (2.26) are fulfilled, then functions §u{xu) may be presented in para­ metric form choosing the variable reciprocal of t h e current value of derivative as a. p a r a m e t e r , t h a t is $*, = * „ ( u ) ,

x„ = x„(u),

&u = u-x-x'u,

i/_
(2.17)

where derivative with respect to u is marked by prime. Upon substitution of derivatives, expressions (2.13) for Pu may be presented in the form: P„=

I Tu [*„(«), $ „ ( « ) , u'1] ■ Hiu'1)

i'^u)

du.

(2.18)

Taking into account equations (2.17) and (2.18) the following combination is formed:

u 1 1 I £/?xMw)A{tO,«_1,j-*'>) G=Y,P»p»-p= --IJH(H(U~ ) ){E/Wi„(u),ft„(tO,u].£'» - T[x{u),§„{u),u-l\-i\v.)\du.

(2.19)

It is evident from relationship (2.19) t h a t condition (2.15) is valid irrespective to choice of function H if the expression in brackets identically equals zero, that is functions <&(u) and x(u) satisfy t h e system of differential equations:

Tix,®,!!-1)-*' x ■ u"

= 0(u), 1

= <&',

(2.20)

61

Section 2. Invariant relationships method where

V>(«) = £ A,V,(«), ^(«)«T,[i„(u)t4„(ii)Ju-,I.4'1,(ti)1

i/ = 0 , l , . . . , n .

(2.21)

The initial conditions follow from (2.14): £(u_) = 0,

$(u_) = 0.

(2.22)

Parameters Lv and r„ may be determined from formulae: r

Lu = £„(«+)>

" = *•-(«+)■

For a number of practically important cases, the solution for system of equations (2.20) may be obtained in closed form. 1) Assume T is a function of the first argument only T = T{x). Then the solution for problem (2.20), (2.22) may be formulated as follows: x(u) =
$(u) = /

x'(w)dw/w,

z

z

T(z) = I i>(w) dw,

ip(z)= JT(w)dw.

u-

(2.23)

0

Should i(u)/u < oo, then the expression (2.23) may be presented through integrating by parts in the more convenient form: 6{u) =

£H _ *(«_) _ u

v.-

lHv^dw J

w2

u-

2) Now assume T = T($). Then the solution may be written as follows: $(u) = tp~l[Ti(u)],

r,w = / ip(iv) dw.

x(u)=

w$'(w)dw,

62

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

If u $ ( u ) < oo, we get u

x(u) = u $ ( u ) — u_ $ ( « _ ) —

$(w)dw.

(2-24)

u—

Let us directly proceed with deriving the invariant relationships between IC's of different bodies. At first, we shall consider the class of bodies with similar cross-sections described in previous subsection. Longitudinal contour configuration is defined by function $ » ( i « ) {? = 0 , 1 , . . . , n ) and there is no flat nose. As before, divided by dynamic head projection of aerodynamic force on some direction defined by vector 1° is considered as a characteristic. Presentation of IC's in the form similar to (2.13) was discussed before so the expression for function T„ may be presented straight away: lUcc,,, $ „ , # „ ) = # „ ,

v =

Q,l,...,n.

Equations (2.5) are written in the form:

*(«).£'(u) = J2 / W « ) ■*'(«). #'(u) = «_1i'(«), whence with regard to (2.17) and (2.22), existence of t h e first integral transpires:

«(«)a = E/?,*,(u)2.

<2-25)

In order to make t h e meaning of equality (2.25) more evident, relationship between $ „ ( u ) and the area of the corresponding body cross-section is used: 2TT

2n

ssec„(u)=l-$i,(u)2

J^efde.

0 0

After t h a t , relationship (2.25) is transformed to the form: n

Snc(u)

= Y.P»SKc

•>(*)•

(2-26)

T h u s , the original and the corresponding bodies have the following property— in sections perpendicular to axis xv which are characterized by the same angles of contour inclination to t h e axis, section areas are connected by expression (2.26).

Section 2. Invariant, relationships method

63

As follows from (2.24) and (2.25), the corresponding body configuration is determined by functions:

a

1/2

*(«) = [i;/wu) l . *(B)

=

L

J

i/=i

u

x[u) x(u) = iu**(«) *(u) -—«_ u_**(u_) [v,-) — j/ *(w) $(w)dw. dw. U—

Similar to how it was done in previous subsection, it may be shown that the following relationship is valid: n

2 27 ((2.27) - )



where o-„ is maximum mid-section area of either an exposed (side) surface or a full body surface. Therefore, between aerodynamic force components of the same name (in particular, drag and lift) for the original and the corresponding three-dimensional bodies of the considered class flowed around at arbitrary equal angles of attack irrespective of the choice of LIM (identical for all bodies), the following relationship holds: n

(2.28)

P{a,*) = £ A A ( a , « ) . i/=i

So far as the body base section areas are connected by a similar relationship (2.27), then formula (2.28) remains valid also for the case when the effect of a flow on a back part of a body is taken into account. It is also valid when in the framework of LIT, the effect a flow has on a shaded part of the body surface is allowed. In Fig. 2.3 a), b), calculation results are given for the case when configuration of longitudinal contours is defined by exponential law and n = 2. All bodies have the same length, m„ is an exponent in generator equation,r„ = rul L, r„ = *„(L„). Let us consider now the case when the original bodies are cylindrical wings of span bu which are symmetrical with respect to the plane xuiu (Fig. 2.4 a) ) while the corresponding one is a body of revolution. Angle of attack a = 0 and there is no flat nose. Expressions for drag of wing and body of revolution divided by dynamic head follow from formulae (A, 2.15) and (A, 5.4). Respectively, they are presented in the form:

X,

^ = 2= 62b , vl/ / /H(* ( * „u)dx„, ) dx„, J

J


A'

?L

0

*„ .**, P, == £ . ^--—, Zby °v

l

? *00-Hi^dxo, • # ( * o ) dx0, = ^L *° ==2-K 2*[$

=

^oo loo

Po ~ Po== ^ 7^~

64

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.3: Longitudinal contour configuration for the original and the corresponding bodies with affine cross-sections. The case of two basic original powerlaw bodies; — the original bodies; the corresponding bodies; a) mi = 1/2, m2 = 1/3, Tj = 1.0, Ti = 3/2.

Therefore, characteristics are presented in form (2.13), whereas T lu

_ I Lm, if v - l,2,...,n, ]*„, ifi/ = Q.

T h e drag for a body of revolution is expressed in terms of values of wing drag as

Section 2. Invariant relationships method

65

Figure 2.3 (continued): b) mi = 1/2, m 2 = 1/3, n = 1.0, r 2 = 3/4.

follows:

x = % Y^, -jf-flvXu, while t h e equation of its generator is such: 1/2

$(u) =

x(u)

•zJ20»i"M<

= u$(u) — /

$(w)dw.

Therefore, it becomes possible to calculate t h e drag for a body of revolution by t h e results of e x p e r i m e n t s conducted for wings (airfoils). T h e case of powerlaw contours of t h e original body is illustrated in Fig. 2.4 b). Similarly, relationships between t h e characteristics of bodies belonging to other classes m a y be derived.

66

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.4: a) Cylindrical wing.

2.3

The inverse problem

T h e m e t h o d developed above allows to define a wide class of bodies of a different configuration with characteristics which may be simply recalculated on t h e basis of predetermined several (in general case) bodies with known IC's. Many such bodies are defined by variating the coefficients /?„ in t h e correspondence relationships. Let us call t h e problem where the corresponding body surface is not predetermined, a direct one. Among the bodies having IC's which may be calculated by the above method, specific configurations may be chosen with certain design features. A number of requirements to the corresponding bodies may be taken into account beforehand and realized through limitations on coefficients j3„. For example, t h e existence of relationships similar to (2.28) makes it possible to require t h a t the corresponding body characteristics assume predetermined values at some values of a and x. Let us consider now an inverse problem dealing with calculation of IC's for bodies of predetermined configuration on the basis of known IC's for several basic (original) bodies.

Section 2. Invariant relationships method

67

Figure 2.4 (continued): b) Longitudinal contour configuration for the original (cylindrical powerlaw wings) and the corresponding (bodies of revolution) bodies; the original bodies (mi = 2/3, r\ = 1.0, m 2 = 4/5, T2 = 5/6); the corresponding bodies.

So, let IC's for several basic bodies be known. It is necessary to d e t e r m i n e charac­ teristics of s o m e b o d y of p r e d e t e r m i n e d configuration. In order to retain t h e accepted system of n o t a t i o n , p a r a m e t e r s of basic bodies are denoted by subscripts 1 , 2 , . . . , n . For a b o d y with desired IC's, subscript 0 is used or o m i t t e d altogether. T h e r e is no a c c u r a t e solution for this problem in general case since p a r a m e t e r s B\ 8% ...,0n which provide validity of conditions (2.20) for functions x(u), $ ( u )

68

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

characterizing t h e body configuration with desired IC's may not be found. However, selection of a sufficient number of basic bodies allows to compensate for calculation error to a great extent while the absence of requirement of defining t h e specific model is an essential positive factor. So, functions 1) and t h e body with desired IC's (1/ = 0) satisfying conditions (2.16) are given. Also, IC's of basic bodies P„ are known. T h e corresponding integral characteristic P has to be found. T h e problem is reduced to determination of coefficients /?„ in expansion of function i/>o(«) into basic functions ip„(u). Here we will not go deep into t h e problem of selection of o p t i m u m approximation procedure since from m a t h e m a t i c a l standpoint, this problem is sufficiently formalized and there is a wide arsenal of m e t h o d s to study it. It should only be noted t h a t physical features of the input problem suggest that it is desirable to estimate t h e quality of approximation not only by function values but by derivative values. In t h e simplest case, integral method of least squares may be used to determine coefficients f3„. According to t h e method, the desired parameters should correspond to t h e m i n i m u m of the function

?r s i2 *(A,ft,...,A.) = / IEM»-*»(«) du and /3„ are determined by solution of the system of linear equations: n

] T OiuPv = a.o,

i = 1 , 2 , . . . , n,

i/=i

(tw = / « M « ) • Mu)du-

(2-29)

If it is desirable to avoid integration procedure, then /?„ may be determined either from t h e condition of m i n i m u m deviation of values of functions V'o(u) and 51" 0uipu(u) for m > n of points, or from the condition of agreement of their values for n points, or by using combinations of the methods. In principle, the above is enough to use the m e t h o d directly. However, to facili­ t a t e its understanding, the method is described as applied to blunt-nosed bodies of revolution without reference to the general results of previous section. Bearing in mind t h a t its validity will be further illustrated as applied to aeromechanics while not limiting its general n a t u r e , the adequate terminology will be used. Assume function <JV(x„) characterizes a configuration of a generator of a body of revolution, $^(0) = 0. By using, as before, a cotangent of angle of generator inclination to t h e body axis as p a r a m e t e r u changing for all bodies in the range from

69

Section 2. Invariant relationships method U- = 0 up to u + , the generator equation may be presented in the form: $* = r„ •

$„(u+,u/u+),

xu = Lu ■

x„(u+,u/u+),

_

«„(*„) = t » \

$„(«+, l) = x „ ( u + , l ) = 1,

(2.30)

where r„ is a base section radius, L„ is a body length, the dot denotes differentiation with respect to x„, 5*'"' = BT'"' . The expression for coefficient of aerodynamic force projection on some direction defined by a unit vector 1° may be presented as follows: F 1° C
= — f uW{u'\a,x)—^l,(u+,u/u+)2du, 2TT J du o

(2.31)

where function W without considering a constant factor has the same meaning as in Section 2 of Appendix as applied to bodies of revolution. Therefore, the desired integral characteristic C\ is expressed in terms of the known ones in the form

Ci = £AA,

(2-32)

*(«+, «/«+)2 = £ &*„(«+, u/u+)2

(2.33)

should the following identity hold

which transpires from (2.31) by taking into account $„(0) = 0. Using the integral methods of least squares, the expression for coefficients of sys­ tem of linear equations (2.29) may be written in the form: ai„ = J $„(«+, ID)2 $t(u+,w)2 i = 0,l,...,n;

u = l,2,...,n.

dw, (2.34)

It should be noted that the coefficients at unknown parameters /3„ are defined only by basic body configuration.

70

Chapter 2. Methods

of calculation

of IC of environment

effect on body moving in it

From (2.33), by taking into account (2.30) at u = u+ transpires t h e following a p p r o x i m a t e equality: n

£/?, = ! which may be used to estimate t h e approximation quality as well as to be introduced as an additional condition when determining /?„. T h e fact t h a t coefficients /?„ in (2.32) do not depend on t h e model (in t h e frame­ work of LIT), the angle of attack, and t h e kind of aerodynamic force component considered as integral characteristic, is of basic importance. Therefore, calculation of Ct becomes possible on the basis of known C\u without using t h e information on the specific model of flow around. T h e results obtained are valid also for bodies with affine cross-sections. Function $„(a:„) characterizes the body configuration in longitudinal direction while the body configuration is t h e same for all bodies in transverse direction. If a flow acts only along t h e normal to the body surface, then invariant relation­ ships may be derived which relate pitch m o m e n t coefficient mz for some body of revolution to characteristics mzu or Cyl/ for basic bodies. Therefore, it is enough to know the values of coefficients of aerodynamic force components for basic bodies to calculate force and moment coefficients for bodies of a different configuration. Let us apply the method for calculation of IC's for powerlaw bodies of revolution and show t h a t the use of three basic bodies provides sufficient accuracy at least for the case of supersonic dense gas flow. T h e choice of the class of bodies is explained by their wide use as vehicle elements and the existence of systematic "precise" calculations [11, 262] which enable to assess the method efficiency. For basic powerlaw bodies (rnu is an exponent in the generator equation), l

xv(u+,u/u+)

=

(u/u+)l-m",

$„(u+,u/u+) = (u/u+)1_m", and formulae (2.34) yield the following expressions for coefficients a,-„ (u > 0) [33]: (1 - m , ) ( l

-m„)

1 + m,- + m„ — 3m/m„

(2.35)

It appears t h a t powerlaw bodies have a useful property which consists in the fact t h a t when taken as basic ones, they lead to t h e system of equations for determination of /?„, the left-hand side of which does not depend both on configuration of the body with desired IC's and on « + , that is values of a;„ [y > 0) are universal in the framework of the given set of basic bodies. Should t h e b o d y with desired IC's also be powerlaw, then t h e right side of e q u a t i o n s (2.34) are calculated by using formulae (2.35) at v = 0 a n d coefficients /3„

Section 2. Invariant relationships method

Figure 2.5: Comparison of calculations by the correspondence relationships with "precise" calcula­ tions [11] (o = 0, 7 = 1.4) for powerlaw bodies of revolution; — "precise'' calculations for basic bodies; — "precise" calculations for the corresponding bodies; • — calculations by the correspondence relationships. do n o t depend on u + . T h e expressions for calculation of coefficients of drag Cxa, Cya, axial force Cx, normal force Cy may be written in t h e following form: Cx(a,m) Cy(a,m) Cxa(a,m)

Cya{a,m)

53#,(m;

m1,m2,...,rn„

Cx(a,mu) Cy(a,mu) Cxa(a,ml/ .C,«(tt,m„),

lift

(2.36)

whereas IC's are considered at t h e same values of u+ = \u/mu (i/ 0 , 1 , . . . , n), \„ = Lujru. Let us consider bodies with exponents m j = 0.5, m2 = 0.6, ra3 = 0.7 as basic ones

72

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.5 (continued).

and calculate IC's for bodies with m = 0.65 and m = 0.55. Solving the system of equations (2.29) having coefficients which are calculated by formulae (2.35), we get TO = 0.55, ft = 0.374, ft = 0.758, ft = -0.134, m = 0.65, ft = -0.121, ft = 0.734, ft = 0.389.

(2.37)

Let us compare calculation results obtained by using the suggested method with published data as applied to the case of supersonic dense gas flow. In Fig. 2.5 a), b), c), solid lines represent the results of "precise" calculations [11] of drag coefficient for bodies of revolution in the case of symmetrical flow around. The values obtained by using formulae (2.36) and (2.37) are denoted by circles. The calculated values are practically located on the "precise" curves, while good accuracy is evident at low values of Mach number M ^ > 1 where the local approach is not

Section 2. Invariant relationships method

73

Figure 2.5 (continued). sufficiently substantiated. At the same time, direct calculation on the basis of the theories of Newton and Newton-Buseman results in noticeable errors. To illustrate this, the curves which appear in [11] are reproduced in Fig. 2.6. The values of Cx/Cx0 (where Cxo is a drag coefficient for a cone of the same aspect ratio) are laid off along

74

Chapter 2. Methods

of calculation of IC of environment

effect on body moving in it

Figure 2.6: Errors for the Newton model and Newton-Busemann model for powerlaw bodies [11] (see text).

t h e ordinate axis while t h e per-unit body volume V/Vo (where V is a volume of a powerlaw body and VQ is a cone volume) is laid off along the abscissa axis. T h e solid line corresponds to "precise" values for hypersonic velocities, the contour lines—to those calculated on t h e basis of the Newton model and the dash-dot line—to those calculated on t h e basis of t h e Newton-Buseman model. T h e vertical contour lines indicate the value of the exponent in t h e generator equation for t h e body. Although the Newton-Buseman model does not belong to t h e LIM class as far as t h e definition given in Section 1, Chapter 1, is concerned, it is not by chance t h a t the solid line which relates to the model is shown in Fig. 2.6. It appears t h a t relationship (2.31) is valid also in t h e framework of this model. In any case, it is t r u e for drag in o = 0. It may be shown by writing t h e expression for calculation of pressure p„ on a body surface in t h e following form (see, for example, [278]): pu(xu) __ p„(s„) ?oo ?co

2$l 2$^

+ •* J; " i1 +

2<j>„ 2$„

|

+

"t $ „Xf§ # iy,$uAx dxu

$,(i + i>ifi* I

v /TTij'

T h e n t h e formula for calculation of a body drag coefficient is obtained by integration: F Cxu =

-x° 1 ", - = — - —

L

f / (2TT $ „ $ „ ) p„ dx„

and assumes t h e form of (2.31) should the following hold W = 4TT $ „ { 2 $ „ - [1 + • „ ( £ „ ) * ] - , ( l + * ' ) _ 1 } In the case considered, function W explicitly depends on the angle of generator inclination in t h e final point of the contour which does not contradict t h e system

75

Section 2. Invariant relationships method

of prerequisites when deriving the correspondence relationships since this angle is identical for all bodies and its cotangent equals u+. The fact that calculation by the correspondence relationships provides higher ac­ curacy than calculation by specific models should not be a surprising one. In fact, the particularities of local interaction between a medium and original body surfaces at predetermined conditions of flow around are "automatically" accounted for in calcu­ lation of the corresponding bodies. In general case, when using experimental results for the original bodies, a number of additional factors are taken into consideration such as the existence of tangent force component on a surface which depends on a surface properties. It should be noted to this end that attempts of model adaptation accounting for additional information is a fairly widespread technique. The simplest example is the transition to the modified Newton formula for calculation of pressure by introducing the normalizing factor which provides the "precise" value of pressure in the frontal point of a body surface. The method of invariant relationships is based on striving to use experimental or "precise" values of the component of the force or flow effect in all corresponding points. Let us proceed with comparison between results of calculation by formulae (2.36) and (2.37) and results of "precise" calculations for the case of flow around at the angle of attack a = 10° Calculations [33] are based on exact equations, the method is described in [30]. Considered are bodies of revolution having generator in the form of circular arc which at u = «o ~ 1.5 gradually becomes a powerlaw curve. As the configuration of bodies somewhat differs from powerlaw one, let us con­ sider the problem of the effect of spherical bluntness on the procedure of calculating coefficients /?„ in somewhat more general case when the generator configuration at u > u0 is characterized by function $* = r 1 / $*(« + ,u/u + ). Then the following may be written for function $„(u,u/u+)

u+

M '^)

—

u/u+

$*(u + ,u/u+)

\/ u o ■

+ 1

A*, i \ t n^ ^ ■ $l(u+,u/u+) for 0 < u < u0, for u0 < u < u+,

whence it immediately follows that it is sufficient to determine coefficients of expan­ sions of function 4>*2 in terms of 4>*2 in segment u 0 < u < u + to validate identity (2.32) for 0 < u < u + . If /?„ is determined from the condition of the lest approxi­ mation in segment [0,u + ] D ["o,^+] then it is natural to assume that the accuracy of approximation, as a rule, should not decrease considerably which simplifies the calculation procedure somewhat and enables the use of coefficients (2.37). In Fig. 2.7 curves are drawn by solid lines which represent the relationship between coefficients of axial (Cx) and normal (Cy) force and u+ = X/m at different values of m. The curves are based on "precise'' calculations for perfect gas. In Fig. 2.8, similar relationships are presented for the case of flow around by equilibrium dissociating air. As before, circles for Cx and triangles for Cy in Figs. 2.7 and 2.8 denote results

76

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.7: Powerlaw bodies with spherical bluntness; #, A — calculations by the correspondence relationships.

,

"precise" calculations [33];

Figure 2.8: Same as in Fig. 2.7 but for flow around by equilibrium dissociating air at altitude 60 km, "oo = 7.4 km/s.

Sect/on 2. Invariant relationships method

77

of calculation by formulae (2.36) and (2.37). The agreement between calculated and "precise" values is good including the case of flow around by equilibrium dissociating air although the specific local models are not known for such conditions. Therefore, in case of supersonic flow around bodies of revolution of exponential configuration in the range of Mach numbers from 2 up to oo, calculation based on the correspondence relationships can be used to obtain estimates of aerodynamic force coefficients close to the results of "precise" calculations. The calculations given confirm the invariance of the correspondence relationships, that is their independence of the model of flow around, the angle of attack and the kind of force component under consideration. Due to the lack of published data on results of systematic experimental or nu­ merical study of IC's for powerlaw bodies in rarefied gas, it is not possible to expand accordingly the realm of considered conditions of flow around when analyzing rela­ tionships (2.36) and (2.37). However, it should be noted that the kind and parameters of the model of interaction between a flow of rarefied gas and a body surface are es­ sentially effected by the properties of a surface material and free-stream flow, a local angle of attack and other factors. As the method developed is based only on informa­ tion on locality of interaction and the information on the specific kind of model is not used, increase in accuracy of calculation conducted on its basis may be expected due to automatically allowing for the indicated and other factors difficult to formalize.

2.4

The case of three-dimensional bodies

The salient feature of definitions of the problems dealing with obtaining the invariant relationships considered above was the requirement of variation of only one function of an independent variable when defining the corresponding body configuration. The method allows generalization also for the case when body configuration is defined by functions of two independent variables. As applied to three-dimensional bodies, the direct and the inverse problems may be defined, whereas for the sake of clarity, we shall consider the case of two bodies (the original and the corresponding ones) and one characteristic for each of them. In the direct problem, the configuration of the original body is known and that of the corresponding body having the same value of IC's is to be found. In the inverse problem, the original body and the surface on which a region has to be assigned featuring the same value of IC's are known. The direct problem is examined below. Let us consider two bodies: T' 0 ' and T' 1 ' (in the direct problem, superscript " 1 " in parentheses indicates the original body while "0" or the lack of it indicates the corresponding one). The surface which is effected upon by a surrounding medium may be described for the i/-th body in parametric form by functions x (l/) = x(">(f<"',77(">), j / ( " ' = ^'"'(f'"'.*?'"') an< J **"' = •^"'(^"'i 7 /'"') whereas parameters fM a n d 77'"' vary in a

78

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

bounded closed domain (
P<"> = JJ H(U #({/„)0) • [/<"> U& #<"> dqW, dn(v\ > (i) >

M

°* = as • w

=

6 = (M,nM,

xM,yM,zM;

r/„ — _ cfo^ i7„f-» x (") „(") ,/") z. M z,(") \ Co a;,(„), y^), y^»)« y^*), ^{(")' zvw), Co — ,(-)i %»)i 0(.a: £(^), T (M, ^»)/t f («>>

(2.38) i^.oo;

where [/'"' is a function of t h e same arguments as Uo, function H characterizes t h e model of interaction between a medium and a body surface, functions U0 and (/'"' are known. T h e objective of t h e study is to develop t h e a p p a r a t u s which can be used to determine t h e configuration of t h e corresponding body having t h e same value of characteristics irrespective of t h e choice of function H. Assume t h a t there arc continuously differentiate functions ip and ip which provide a one-to-one mapping of region (*»£,) on region (o-£n):

m tf» £ = ¥»(£>'?), ¥>(£,»»), v ij = (&*) = *HU) (1)

(2.39) (2.39)

w

and t h e Jacobian of t h e transformation does not equal t o zero. T h e n expression (2.38) may be reduced to t h e form: p ( i ) - ff H\lUrm P « = JJ H [u0{x%,

r(1) f/(1)

.,£«)]• | J | . £ / « ^ ,

x%, 4 » , . . . , «§>,)] • | / | ■ V™ di dr,,

where

*P } =- [ds J ^w V(f U>*> ^ }\ ( c m w Ws

while w and 8 have t h e meaning as in (2.38). T h e characteristics will be identical irrespective of t h e kind of function H, should t h e following system of equations be valid

U Uo{xi-,x 0(xi,xrin,,y i,. n) yi,...,z f/(0)(i{,a:„,j/?,.

1

(1)

(I)

(1)

- Ua(£#}),£$ nv ofr' - , Z ,T , J1 — = l ^ a ': ^ r,m,V§}),..-J%), , ^ , ,-, , , ^ , , , . . . ,?^ , , ; ^ ^ ^ 1 - l/l-f/' 1 '^' 1 '

z ( I ) u (1)

(2.40)

i(1M

Therefore, if there is some three-dimensional body T ' 1 ' then by defining different transformations (2.39) and solving t h e system of differential equations (2.40) of the first order functions i'(£,J?), 2/(£,'/) a n d 2 ( £ , ' / ) which define the surface configuration of t h e corresponding body T ' 0 ' m a y be determined.

Section 2. Invariant relationships method

79

Functions ip and tp define, in particular, the transformation of boundaries of the region ( c ^ ' ) which is a basis for defining the initial conditions for systems of equations (2.40). A note should be made of modification of the method when the problem is to determine a configuration of three-dimensional body having two characteristics P ' 1 ' and P ' 0 ' which are connected, P ' 1 ' = fcP'0', with respect to LIM. It is apparent that the relationship is valid if functions x(£, n), y(£,r)), z((,rj) satisfy the differential equation « (1) (*«t ^»2/«.ft»< Z{> z>,) = H / ( 0 ) ( x f , . . . , 2 „ ) whereas the body surface may be particularized by defining the adequate initial con­ ditions. The property of unchanged location of center of pressure for some surface elements which was substantiated otherwise in Chapter 3, may be verified by P'"' as a normal force and pitch moment by way of a characteristic. To make the procedure of determining the corresponding bodies more specific and, in particular, to substantiate the initial and the other conditions for system of differential equations (2.40), it is more convenient to use the particular coordinate system. For clarity, the value of body drag in a flow divided by dynamic head will be considered as a characteristic. In Cartesian coordinate system a;'"' j / ' " ' t'"> (Fig. 2.9) the surface equation for body TM is defined by the function i W = #M(y<«0,*W). The free-stream flow velocity vector may be considered as directed along axis a:'"' without loss of generality. The body surface is bounded by contour I"'"' described by equation xM

= x^(S),

y(u) = y£\6),

, = 0,1.

zM = zi»\6),

Projection 7'"' of contour r'"' on plane pM*M bounds region {cryuJ). The formula for calculation of body drag is of the form (2.38) whereas as follows from (A, 4.3), (M = yW,

j,M = 2<"),

U0(w1,w2,-..)

»M = $ M ,

= w\ + w\

and function H is of the form (A, 3.9). Conditions (2.40) takes the following form:

8 *; + •: ==[•ft + •ft 1 ] y<"=WM *;+** l^.v+^,j l"=v(l/,*) z - y = ± 1 .

'

(2.41) (2 41) " (2.42: (2.42)

The transformation featuring property (2.42) is called an equiareal one [256].

80

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.9: Coordinate system and notation for the case of three-dimensional bodies.

T h e condition of identity of characteristics P ' " ' which takes place irrespective of t h e kind of function H yields (when H = 1 and H = i / [ / 0 + 1) respective conditions of identity of areas of regions ( c ' y ) and the side surface of t h e original and the corresponding bodies. Therefore, if one of these areas is chosen as a characteristic one, 5 * ' " ' , then t h e body drag coefficients C^J = p ( " ) / 5 * ' " ' appear equal. T h e relationships between t h e characteristic areas of the original and the corre­ sponding bodies validated here and above are useful, in particular, when comparing the transition region of rarefied gas flow with wettable surface area which is used to calculate the main similarity criterion which is Reynolds number. T h e procedure of determination of t h e corresponding body configuration is re­ duced to t h e following. Let t h e surface equation for body 7 1 ' 1 ' be given. T h e surface is bounded by contour D 1 ' and its projection on plane z = 0 is a closed curve 7 ' 1 ' which bounds region (er£''). If transformation t/' 1 ' = ip(y, z), z' 1 ' = ip(y,z) which transfers {(T^J) to (cr^J) (and 7 ' 0 ' to 7' 1 ') is found, then the corresponding body sur­ face is determined by solving equation (2.41). It should pass through contour T' 0 ' (the initial value problem for partial differential equation of the first order). T h e example may be found in t h e work [133]. Let us consider now the problem which was called an inverse one. To m a k e it more evident, we examine the case when drag value divided by dynamic head is considered

Section 2. Invariant relationships method

81

as a characteristic and body configuration is defined in the coordinate system shown in Fig. 2.9. Assume that the body T(0> is bounded by surface S<°> on free-stream flow side and some surface described by the equation x' 1 ' = $W(yW,aW) are given. The problem is to determine region S' 1 ' on the latter surface having the drag which is identical to the drag of body T<°> for any LIM. So far as functions $(">(y("), z (l/) ) are known and functions tp and i/> are to be determined, condition (2.41) becomes an algebraic equation of the form: E^My,z),rP(y,z)] El»[
= E(y,z), E(y,z),

(2.43)

where £P

W » ) == •%• + •$? 2

2

E(y,z)-£<*,*) == *$; ++»$; .. If function

=

R[E(y,z)^(y,z)]

then upon substituting into (2.42), linear partial differential equation of the first order is obtained with respect to function tf> 0E(y,z) dE(y,z) dy

di> dj>_ dE{y,z) dE(y,z) dz dz d~z dz

djP dV _ dy dy ~

1 gj&gil dt dt

I t=E(y,z) \t=E(y,z)

The general solution of the equation may be obtained by the standard method [192]. For many classes of surfaces, solution of equations has a rather simple structure. For example, if fid) = Em(ym), Em = fiP»(»W),

E = E(z), E(z),

(2.44)

R2(z) = E^E^1[E(z)], -'[Eiz)},

(2.45)

then the solution is
= T-Fyr-; + R3(z), H2(z)

(2.46)

where R3 is an arbitrary function. For illustration, a specific example is considered below. A wing section located in the region x > 0, y > 0, 2 > 0 is considered as the original body T' 0 ' whereas here and further on all linear dimensions are divided by the length of the original body having the base section perpendicular to axis x.

82

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

Figure 2.10: Configuration of: a) the original body; b) the corresponding body in sections x'"' = const. Let the original body surface equation have the form: „2

$ = b-y + -(z + cY-

2

(2.47)

while cylindrical surface with axis parallel to axis 2' 1 ) is considered as t h e correspond­ ing body surface described by t h e equation

$M = £i (2/ w + C i r _ ° lC "'

(2.48' (2.48)

whereas in formulae (2.47) and (2.4S), a, b, c, a i , cY are positive constants, n, > 1. Then £(')

=

a2(?/(i) + Cl)^-D,

E = P + a2(z + cf

Section 3. On some properties of integral characteristics resulting from body symmetry that is, case (2.44) takes place and solutions (2.45) and (2.46) assume the form: _i_

__j

y>(2) = a 1 1 - n ' [62 + a 2 ( 2 + C ) 2 p(n,-i)

-cu

l Cn, — l l r t " 1 - 1

=

l

2ni-3

i \ • V ■ [b2 + a2(z + c ) 2 ] ^ . - » (2.49) a'(z + c) whereas in (2.46) a positive sign is chosen and R3 = 0. To make it more graphic, let us choose specific values of parameters a = b = c = 1, ni = 3/2, a.i = 4, C\ = 1/8. The original body configuration sections by planes x = const are shown in Fig. 2.10 a). Projection of the base section on plane x = 0 is bounded by coordinate lines y = 0, z = 0 and by curve 7 described by the equation y = 1 — 1/2 z2 — z. Equations (2.49) defining the transformation of regions which are projections of 5'"' on plane x'"' = 0 assume the form M.*)

2,

j,(i> = * » + ! , , , w = _?_„ y 16 8 ' z+ \y and therefore, boundary intercept [0,1] on axis y is transferred to boundary intercept [0,8] of axis z' 1 '. Boundary intercept [0,V3 —1] on axis z is transferred to boundary intercept [0,1/8] on axis yl1' while contour 7 is transferred to contour 7' 1 ' described by equation of the following form: zW

=

8(1-8yO)

Configuration of region S' 1 ' is illustrated in Fig. 2.10 b) where horizontal lines indicate sections of the surface by planes x' 1 ' = const. The inverse problem does not always exist. Such situation, in particular, occurs when function UQ cannot be realized on the corresponding surface some of the values it has on the original body surface.

3

On some properties of integral characteristics resulting from body symmetry

We should admit that this chapter was included in the book after long hesitation. Therefore, we would like to share the basic "pro" and "con" arguments and to explain why the positive decision was eventually taken. The doubts basically emerged for the following reasons. A vast number of pub­ lications are devoted to studying of the IC behavior for bodies at slight deviations from conditions of symmetrical effect of a medium. Naturally, we have taken no­ tice (in any case for some specific situations) of some of the regularities which are

83

84

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

substantiated in t h e present section. However, we are not able to handle t h e corre­ sponding body of information to provide t h e necessary references as required by the traditional approach. The second "con'' argument is a simple study technique which becomes fairly easy-to-grasp after t h e objectives a n d prerequisites of t h e analysis will have been formulated. T h e basic "pro" a r g u m e n t is t h e fact t h a t u p to now p u b ­ lications systematically appear where the regularities of general n a t u r e presented in this section are established for some particular cases and interpreted as character­ istic ones exactly for such specific situations. It suggests an idea t h a t such general regularities are not obvious for many researchers m a y b e simply because they did not ponder over the possibility of their existence. It occurs to t h e authors t h a t this reason alone justifies the inclusion of the present section in the book be it for methodological considerations only. T h e presentation essentially follows our work [135] and we are apologetic in ad­ vance for possible lack of references to the publications devoted to different aspects of the problem and probably containing some of t h e presented results. So, we even do not assume that interaction between a m e d i u m and a body sur­ face is necessarily of local n a t u r e although, of course, t h e results are also valid in the framework of LIT. T h e requirements for t h e n a t u r e of t h e interaction are the most general ones. Essentially, they are reduced to natural assumption of the closed character of system "medium-body", i.e. t h a t there are no outside effects of any kind which prevent s y m m e t r y considerations from being applied. Let us consider a body axis Cartesian coordinate system xyz (Fig. 3.1 a) ). Angle of attack a is an angle between v^, and plane xz, slip angle /? is an angle between axis x and projection of v ^ on plane xz. T h e body is located in region x > 0. T h e body surface is divided into four sections S* 1 ', 5 ( 2 \ S ( 3 ' , S<"> (Fig. 3.1 b)) so t h a t S ' 1 ' is located in region y > 0, z > 0; 5 ' 2 ' — i n region y > 0, z < 0; S' 3 '—in region y < 0, z < 0; SW—i n region y < 0, z > 0. T h e superscript in parentheses will denote t h e n u m b e r of surfaces referred to by t h e relevant characteristic. Let us denote axis projections of coefficients of force acting from the side of a medium in a point of the i-th section as c^\a, /3, x, y, z), c^(a, /?, z, y, z), 4 ' ' ( a , /?, at, y,z). Meanwhile, it is assumed t h a t coordinates x,y,z are such t h a t the point is located on the relevant section of the surface. Integral force coefficients are calculated from t h e formula: Cw(a,l3)

1 4 = — £

rr JJ c${a,l3,x,y,z)dS,

W =

x,y,z,

' = 1 s(') where S" is t h e characteristic area. Now suppose the body is symmetric about plane xz. T h e n it is apparent that for W = x,y,z, functions cv|;( — Q , / 9 , X, y, z) may be expressed in terms of functions c\y(a,/3,x, —y,z) (i,i/ = 1,2,3,4) in the following form: Cw(-a,P,x,y,z)

= 6W -c{w~'\a,P,x,-y,z),

W = x,y,z,

(3.1)

Section 3. On some properties of integral characteristics resulting from body symmetry

85

Figure 3.1: Illustration to definition of angles a and /? and division of body side surface into sections S^\ S< 2 \ S<3> S<4>.

where 8w = I \i W = x,z and 8w = —I if W — y. For some function i^{x,y,z) the following identity holds: jj $(*, V, z) dS = JJ 4>(x, -y, z) dS.

(3.2)

5

s< -)

S(')

Taking this into account for integral characteristics, properties of being evenness (oddness) may be established for the functions: Cw(-a,

P) = j-.Y,JJ

$(-<*,

0, x, y, z) dS

'=1S(')

= &w ■ -gill

Jj

■=i S (.)

c(w"'){a,^x,y,z)dS

86

Chapter 2. Methods of calculation of IC of environment effect on body moving in it 1 = $w "Fill

4

ft JJ

''(«c%" i » , z)dS r )}M((X,fi,x,p,z)dS

•"-= 11 5(5-) S<5-)

= 6w6Cw{oi, 0). wCw(a,0). Let us t u r n now to m o m e n t s of forces. Expressions for local m o m e n t characteristics in the body-axis coordinate system are of the form: X. >P,/?, > *X.I >»> Viz*) ) == !/4''(ay <^\ot, ,/?, 0,x,y,z)-z t^(a,0,x,y, z), /?, X. y, *), ™*L( m Q m Q x z z a X Z X 0 x X , < / , *), l»L( ' L ( >- J, ft X,i 3/y,> ») ) = ct\ i .0, Pi X,I VIv,)~* ) - i c4 | ' (' («a.. ,/8, Pi iVi z), = «
m z m ii'L( ' L ( QQ> ,0, Pix,y, ) *) == ^ xc^y'\ct,0,x,y,z)-y X. >v, ' ' ( a ,/?, X ,y, * ) -

4 ° ( a , ,/3, 0, X. x, ,v, y, *). z), ■ s/4''(«

while projections of integral m o m e n t are obtained by integration

1 4 rr m mw(a ,/M mw(a,P)./») == f o i l£//«$«*(* l JJ wioc(a,2/, iPix2)d5, iyiz)dSi i=I '-' s(') where /* is the characteristic dimension. Let angle of attack changes to the opposite one. T h e n with account of (3.1) and (3.2) we get !

4

1

-*4°(-

m mxt ((-a,0) - -a,/3) == T^ciJl JJ [yc{')(-a,0,x,y,z)-zc 7 ■EjJ[y4Ha, yA,)(-a,0,x,y,z)]dS «,», 2 )]
I ^ i=1

,=1

= ^

1

S(') s(')

''(a, ■E//hr E jj[y i "'V,P, \a,0,x,-*, -y, + 4 - ( ,/?,x, -2/,) o+*<£*" c 5

z

z 5 ]

a

1=1

1=1

i = fsifl

Aar, -»,*)] - „ , « ) ] «tS ds

S(')

s(')

E JJ/ / [-yc b^ "W.< t'\ ,/3,x, ,z)

"'(a *)+*4*" +

r,J/,

{

a

o>

y

,/3,x ,j/,z)] c?5 zc^(a,0,x,y,z)]dS

x(a,0), == -m x ( -m>,£),

m,(-o ,/?) == m v (amy(a,0), ™y(-(Xil3) ,0), m -a. ,0) == -m.(~rna,/?). ™2 z(-{-a,0) z(a,0). T h u s , on the assumption of possibility of expansion of C x , Cy, C z , m „ my, considered as a function of a , into Maclaurin series, the following is obtained: 2 4 CW(Q) Cw{a) = = CV(0) Cw{0) + + a[v»a a$a2 + O(o O ( o 4 ), ), ,,

33

33

mz

ID z, w = = r ,x,z,

55

C „ ( a ) = a( »a C„(a) »a + a< >a >a + 0 ( QQ ) , 3) 3 3 5 m-w(a) = m-w(a) = b$a b$a + a[, ay3)aQ + 0{a 0{a5),),

m myy(a) (a)

2 = = m m„(0) + b^ b^aa2 y(Q) +

4 4 + 0(a0(a ), ),

w-x,z, w-x,z, (3.3) (3.3)

Section 3. On some properties of integral characteristics resulting from body symmetry 87 where an), bw' (W = x,y,z; v = 1,2,3) are some functions depending on /?, body configuration and conditions of interaction between a medium and a body surface. Let us cancel demand of symmetry of a body about plane xz and consider sym­ metry about plane yz. Then, c

3 x Cw~ *>w • & i+1■c lv +j)3)((aa>,-P, -Zxiy,z) , i y, *) == $W \a,w~1+1 /?,\a,l3,x,y,-z), x, y, — z\ 8 = 1,3; j = 0 , 1 ; W = x,y,z,

where Sw = 1 HW = x,y and Sw = — 1 if W = z. Allowing for identity

JJ ^

-IS

JJ ip(x,y,z)dS
ij>(x,ip{x,y,-z)dS ,U-> -z)dS

s(—i+n

S('+J>

similar to (3.2) it is easy to determine the properties of being evenness (oddness) for functions Cw and raw with respect to /3: Cw(a,-/3) 6wCww(a,P), (a,l3), Cw(a, -0) = 6w-C (a, -j3) ww==x,y,zx,y,z mww(a, -f3) = = -S -6Ww •rnw(<x,fi), ■ mw{a,P), whence the following expansions into a series transpire: ] 22

O(/344),), + O(fi

Cw(/3) ==--Cw(O)+dw Cw(0)+d$(30 CwW)

C>((3)== d^(3 rf«j8 ++d.W/3 d^/3 cm

33

)

)

mw(/3) ==--eWp ew l3 + efff ew f mw(fi)

mm --

2

w = x,y,

++ 0(13% 0(f),

+ OiP 0(l35), 2

w = x,y,

4

m2(/?) == m 2 (0) + e< >/? + 0(/? ), where the meaning of the coefficients is the same as in expressions (3.3). Should a body have two planes of symmetry, xy and xz (which is equivalent to symmetry about axis i ) , the following expressions hold:

C,(a,0) = C CxI(O,O)+ (0,0)+ ££*<**/?« A£**>a*0*, C,(a,0) i,j>0 i,j>0

Cvy(a,0) = C (a,0) =

£ , l (2.'+l,2 J +l) Q 2, + l / j2,+l i :,i>0 :,i>0

2, 2j+i) 1 2 2j+i iJ+ C,(«,f3)= CM,P) x ^ihi*•Wa*/**+\ ''ft^+QV'/3 C,(a,P) = E Y. i8 \,

i,j>0 i,j>0

{a,P) = i,j>0 nxx(a,P) !C ^ 2 i + 1 ' 2 j + 1 ) a 2 i + V 2 ' 7 i .j i>> 00

ny{a,0) =

£<42' ,2j+l V '•/?««,

i,j>0 i,J>0 •J>0

C /3A fl2j 2j n (aj) = V5Z^ „(2i+l,2j) sl 2ffi ' +1 ' 2:,)a„a 22t+l ' +1 /?/■> -= 2^

_

mzz(a,f3)

i'J>0

87

88

Chapter 2. Methods of calculation of IC of environment effect on body moving in it

It may be shown t h a t t h e properties of being evenness (oddness) which are sub­ stantiated for force and moment components in coordinate system xyz are valid for t h e relevant components also in air-path axis coordinate system. T h e analysis may also be continued as applied to derivatives of integral character­ istics used in stability analysis as well as other characteristics which are combinations of those considered above (left-to-drag ratio, longitudinal static stability factor, etc.) however, this does not present major problems so we will not write t h e relevant for­ mulae. T h e fact t h a t a similar approach may also be used for obtaining useful estimates concerning local characteristics should be noted. An example of this is described below. Assume a body is symmetrical about the plane xz and p(ot,P,x,y,z) is a pres­ sure exerted by outer medium in a point of a body surface with the corresponding coordinates. T h e n it is apparent t h a t p(-6,P,x,y,z)

= p(0,P,x,-y,z),

6 =

+a,-a

and function 9 = p(a, P, x, y, z) + p(a, P, x, -y,

z)

is even whence (observing the smoothness condition) expansion of ip into Maclaurin series in terms of a follows having the form: p(a,P,x,y,z)

+ p(a,P,x,-y,z)

= 2p(0, fi, x, y, z) +

0(a2).

Therefore, half-sum of pressures in symmetrical points for low angle of attack differs from pressure for the same points effected upon symmetrically by a medium, by t h e value of order a2 T h e above regularities are confirmed by numerous experiments for different con­ ditions of the effect of a medium on a body surface. It is also appropriate to take note of t h e fact t h a t available analytical expressions for t h e specific models appear to yield t h e expansions of the above form.

4

Generalization of area rules

Assume configuration of a body of length L is defined by function p — $(x,8) in cylindrical coordinate system shown in Fig. 4.1. In general, t h e body may have a flat nose. Angle of attack and t h e body configuration are such t h a t t h e frontal part of its surface is fully effected upon by a medium (no shaded region). Formulae (A, 1.11), (A, 1.22) yield the following expressions for projection of force of t h e effect of a medium on direction of its velocity (in aerodynamics, drag) Xa L 2w

P[*(.T,0),O]

= XJqM = 5* • Cxa =

5-0Z)(COSQ)

+JJ 0

nD(t)t2dx dd, 0

89

Section 4. Generalization of area rules

Figure 4.1: To prove the area rule. The coordinate system. where function ftrj is defined by formula (A, 1.12); t, t2 are defined by formu­ lae (A, 2.2); S- is a body nose area; S* is a characteristic area. A s s u m e there is some body of revolution having a generator defined by function p = r(x) and similar by configuration to t h e considered three-dimensional body in t h e following sense: $(x,0)

= r(x) + e£{x,0),

£
{(M)~r(*)~l and angle of attack is of order e

1+l/

Q=

(4-1)

, t h a t is

fce1+",

fc~l,

v > 0.

T h e objective of t h e study is to s u b s t a n t i a t e t h e conditions for which t h e difference in drag values for such bodies is of order e2, P[9(x,0),a]

- P[r(x),a]

= 0(e2)

(4.2)

whereas it is evident from the results featured in Section 3 of t h e present book t h a t condition (4.2) is equivalent to t h e following: P{S>(x,6),a}-P[r(x),0}

= O(e2).

Let us proceed directly with derivation of area rules. Considering P [ * ( x , 0), a] = P[r{x) + e ((x, 6), fc£1+" ] = G{e)

(4.3)

90

Chapter 2. Methods

of calculation

of IC of environment

effect on body moving in it

as function of £ and assuming that expansion of G(e) into convergent Maclaurin series is possible, it may be written that /»[•(«,0),a]= P[r(x),0] + G"(0) ■ £ + 0(£ 2 ) whence it transpires that estimates (4.2) and (4.3) are valid, should the following hold: G'(0) = ^-P{r(x)

+ eax,e),ke1+"]

~ e.

<7£

E=0

Upon the necessary calculations, expression for derivative G'(0) may be presented in the form: G"(0) =

r(O)-SlD[l)J{(O,0)M o L

2TT

+ I j [Ri(x) ■ £{x,e) + R2{x) ■ Ux,6)]dxdO,

(4.4)

0 0

where

R-i(x) = n D (u) ■ uj, T T

Ri{x) (u), Ri{x) = = —= —= Ofl(u) Ofl(u) + + ruO ™0 DD(u), / , , i u = Vr'22 + 1 ?(22 = v r ' 2 + l ,

„ = -f , u = —, «2

i ( dfi D Q ilDD{u)U ) == ——,

(4.5) OX

Using the relationship j R2(x) ■ Ux,0)dx o

-

= R2{x) ■ i{x,6) =0

[R!i{x)-((x,S)dx Jf

expression (4.4) may be written in the form G"(0) = [r(0) ■ n 0 ( l ) - R2(0)} ff(0) + H2(X) ■ <x(L) + / , *» L

I = J JR3(x)t{x,0)dxdO, 0

R3(x) =

R1(x)-R'2(x),

0

2TT

a(x) = J t(x,0) dO.

(4.6)

91

Section 4. Generalization of area rules

Integral / in expression (4.6) may be presented as follows: L

I = f R3{x)a{x)dx o

(4.7)

and from the expression (4.6) it transpires that G'(0) = 0 and therefore, charac­ teristics of a three-dimensional body of revolution differ by the value of order e2 if o-(x) = 0, i.e. in each section perpendicular to axis x the average value of radius vector for a surface point of the three-dimensional body coincides with the radius vector of the body of revolution. To complete derivation of area rules in traditional formulation, the condition of equality of body section areas is written below: 1

-j[r{x)

+ e((X,6)]2d6

=

nr(x?

o from which the following estimate transpires with account of (4.1):

e

'<*>-{*> 7 ^ " } ~

e

and therefore, also estimates (4.2) and (4.3) as a result of taking into account the presentation for G'(0) in the form (4.6) and (4.7). Thus, for the general case of local interaction, the following area rule is substan­ tiated: should any three-dimensional body be similar to a body of revolution by configuration and areas of sections by planes perpendicular to the symmetry axis of the body of revolution be equal while the angle of attack a ~ e 1+ " (v > 0), then the difference in the values of the effect of a medium along the symmetry axis (drag) for such bodies is of order e 2 , where e is an order of difference in body configuration. The above analysis can be used to extend the regularities which are known for some forms of conditions of dense gas flow to the general case of LIM. Regrettably, we do not succeed in uncovering data which would enable the examination of the area rules for other conditions of interaction between a medium and a body, in particular, for rarefied gas flow, hence, the authors are forced to confine themselves to illustrative example. The value A of the per-unit difference in drag values for pyramidal bodies (with regular n-gon as a base) and right circular cones which are calculated from model (1, 2.14) for the case of hypersonic rarefied gas flow around bodies, is shown in fig. 4.2. Lengths of bodies and areas of body bases are equal, a = 0. The parabolic nature of the plots may be noted in the graphs. Already at e < 0.5 (n > 6) the per-unit difference in drag values becomes less than 5% for the entire range of regimes of flow around, from dense gas flow to free-molecular flow.

92

Chapter 2. Methods of calculation ofIC of environment effect on body moving in it

Figure 4.2: Per-unit difference A in values of drag for o cone and n-gon inscribed in it as an illustration of the area rule. T h e modified area rule is derived below for the case when a three-dimensional body is similar to a cone by configuration, i.e. r(x) = r(0) + ax, where a is constant. From (4.5) and (4.6) it follows that function Rs(x) becomes constant (its value is denoted by 6) and expression (4.6) may be written in the form: G'(0) = r(0) • ( n D ( l ) - e]
I = bf 6(O)d0, 0

+ I,

I

5(6}=

f[Z{x,0)+£(xtT

+

0

where e = aifiD(ai) + - i f i r j ( a i ) ;

a;

x/^TT

6)]dx,

Section 4. Generalization of area rules

93

If the areas of sections of a cone and a three-dimensional body by planes passing through the cone axis are equal then 6(6) = 0 and to satisfy condition G'(0) ~ e, it is sufficient to add the conditions for the nose (if the body of revolution is not a cone) and the base section. Such conditions are identically the same as that formulated for the case of the "classic" area rule for arbitrary section perpendicular to the body axis. It must be kept in mind that the three-dimensional body surface includes concave portions of curves. In particular, if nose and base section areas of bodies are identical and a = 0, then the modified area rule may be formulated as follows: should a configuration of some three-dimensional body be similar to a cone and areas of sections of these bodies by planes passing through the cone axis be equal, then the difference in drag values for the bodies is of order e 2 , where e is an order of difference in configuration. Most evidently this rule is displayed for bodies of revolution since to satisfy the required conditions it is sufficient for the areas in one of the sections to be equal. In the framework of LIT, other modifications of area rules may also be substan­ tiated, in particular, for three-dimensional bodies having surface which is similar to cylindrical one with axis perpendicular to velocity of outer medium [135].

Chapter 3 Design methods for bodies with invariable longitudinal static stability factor 1

Problem statement

Let some body be symmetric about plane xy (Fig. 2, 4.1) in which vector w^ of free-stream medium velocity lies. Further on, aeromechanics terminology is used and a body in a gas flow is considered. Formula (A, 1.2) for moment coefficient may be written in the form: m(r 0 ) = m(0) - (r 0 /Z) x CF, where m ( r 0 ) , m(0) are moment coefficients with respect to the point of radius vector r 0 and the origin of coordinates, respectively; CF is the integral force coefficient; L is the body length. As follows from symmetry of flow, m(0) = m 2 z°, CF = Cxx° + Cyy°,

(1.1)

where mz is the pitching moment coefficient (projection of moment coefficient on axis z relative to the origin of coordinates). It transpires from (1.1) that CF • m(r 0 ) = 0, i.e. the main vector and the main (resultant) moment of the system of forces are per­ pendicular to each other. Therefore, the line of action (straight line) of the resultant of forces is described by the end of vector r 0 which satisfies the condition: m(r 0 ) = 0. 95

(1.2)

96

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

It is evident t h a t in our case this straight line lies in plane xy and abscissa xcp of t h e point of its intersection with t h e axis may be determined from equation m 2 z ° = (xcp X°/L)

x (Cx x ° + Cy y°)

which is obtained by substituting (1.1) into (1.2) and allowing for ro = Z C P • x ° . T h e solution of the equation has the form: C C P = I C P / L = mzjCy.

(1.3)

P a r a m e t e r xcp is called a center of pressure (CP) and Ccp, a coefficient of center of pressure ( C C P ) . Therefore, C P is a point of intersection of line of action of the resultant of forces (in this case, forces of effect of a m e d i u m on a body surface) with a characteristic line. Longitudinal body axis (axis x) is chosen as t h e one. Index A = XQM/L

—

CCP,

where XCM is t h e abscissa of mass center of a body, is called a longitudinal static stability factor. At A < 0, positive stability takes place; at A > 0, negative stability; at A = 0, neutral stability. In t h e general case, C C P a n d consequently A are dependent on angle of attack and conditions of m e d i u m effect. As far as ensuring t h e flight stability is concerned, body configurations for which there is no such dependence are of special interest. T h e present Chapter is devoted to study of this problem on the basis of LIM. It should be noted t h a t term "center of pressure" notwithstanding, formula (1.3) allows for a flow effect along the tangent to a body surface. Prior to proceeding with formulation of general theory, it has to be a d m i t t e d that for three-dimensional bodies having two planes of s y m m e t r y in case of a valid model (1, 2.15), it follows from formulae (2, 1.31) and (2, 1.34) t h a t CCP = /i2//*3,

(1.4)

i.e. C C P does not depend on angle of attack a at | a | < a, t h a t is when there is no shaded region on a body side surface. Figure 1.1 shows results taken from [155] and illustrating the connection between C C P and angle of attack. Insignificant effect of a on position of C P for star-shaped bodies was noted in [154, 155]. This problem was also studied on the basis of the Newton theory in [242] where a p p r o x i m a t e estimates for C C P were obtained. As applied to star-shaped bodies, the phenomenon of slight effect of angle of attack on the C P position may result not only from viability of t h e Newton model but also from features of body configuration. Turning again to the general case, we shall assume t h a t a m e d i u m acts along the normal to a body surface.

97

Section 1. Problem statement

a)

b) Figure 1.1: Experiment [155], e — see caption to Fig. 2, 1.3 c); * , H four cycles; • — cone; A — powerlaw body with exponent 0.65.

star-shaped bodies with

Let us consider t h e flow effect on some surface element AS = A S ' L I A S " , where AS' and AS" are surface elements symmetrical about plane xy (Fig. 2, 4.1). It follows from t h e s y m m e t r y of flow t h a t line of action of integral force lies in plane xy. T h e point of intersection of this line with axis x is considered as C P of element AS and its coordinate xCp is d e t e r m i n e d from t h e following expression similar to (1.3): AY

■ xCp =

AMZ,

(1.5)

where AY, AM2 are n o r m a l force and pitching m o m e n t acting on surface element AS (or AS' or AS"). Moreover, considering a body surface or its element lying in region z > 0, we shall always assume t h e existence of similar surface (element) symmetrical about plane xy. In particular, C P of surface element AS is considered as C P of surface element AS'. Some properties of C P necessary for further study are outlined below. 1. Let us consider some elements AS' on a surface of revolution. T h e axis of t h e surface lies in plane xz and parallel to z. From geometric considerations it follows t h a t t h e line of force acting in each point of t h e surface passes through t h e axis of

98

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

revolution and therefore, C P of t h e element is located on t h e axis and its position does not depend on configuration of t h e element, angle of attack and model of interaction between a m e d i u m and the body surface which m a y be absolutely arbitrary (not necessarily LIM). In particular, t h e C P position for bodies of revolution in case of transverse flow around remains t h e same irrespective of arbitrary change in t h e n a t u r e of flow, emergence of arbitrary shaded regions (symmetrical about plane xy), etc. Among t h e surfaces of t h e considered class, elliptical (spherical) and cylindrical ones, which are often used in design of vehicle, are of major interest. It should be noted t h a t as applied to spherical segments, effect of Mach n u m b e r for free-stream flow and angle of attack on t h e C P position in case of supersonic flow was observed in experiment [179]. T h e faint dependence of the C P coordinate on angle of attack is indicated also by results of experimental investigation of supersonic flow around segmental-conical bodies [47] in case when the flow essentially acts on the segmental section. Of course, at low angles of attack, the phenomenon of t h e C P position stability m a y be explained on the basis of formula (1.4) 2. Assume body surface S may be presented in t h e form

S--= S--

U5« 1

whereas i c p = xCp is t h e C P coordinate for surface S ' 1 ' while for the rest of surfaces one of the following conditions holds: either XQP = xcp or y ' 1 ' = My' = 0. T h e n , xcp is t h e C P coordinate for surface S. T h e surface for which normal force and pitching m o m e n t equal zero may be exemplified by a cylindrical surface element with t h e surface axis parallel to axis y t h a t is symmetrical about plane xz. T h e particular case is a flat element parallel to plane yz. T h u s , if surfaces with the invariable C P position (with possible inclusion of cylindrical surfaces symmetrical about planes xz and xy and having axis parallel to axis y) are taken as body surface elements and combined so t h a t their C P ' s coincide, then the C P position for such a, body remains invariable. This property is a basis for "design" of bodies of complex configuration with t h e invariable C P position. T h e simplest example is a spherical zone of arbitrary configuration having surface which is a combination of spherical and flat elements (Fig. 1.2). Point C marks the C P position. 3. Assume t h e fact t h a t independence of C C P on angle of attack is established for a body at 0 < | Q — a0\ < e whereas mz(a0) = Cy(a0) = 0 but Cy(a) ^ 0 if a / a0. In particular, such situation takes place at Qo = 0 for symmetrical bodies. T h e n t h e following is adopted as C C P : Ccp(ao) == lim l i m

n

'i

== \ ~~

lim lim CCcp == CCcp. Cp = C p.

a—*ao

T h e conducted study is carried out in the following sequence. In Section 2, t h e search for surface elements with the invariable C P position is continued. Subsequent sections

99

Section I. Problem statement

a)

0

b)

d)

e)

0

Figure 1.2: Simplest bodies with unchanged position of the center of pressure.

100

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

deal with substantiation and application of t h e m e t h o d for obtaining t h e bodies of complex configuration featuring this property.

2 2.1

Surface elements with the invariable center of pressure position General solution

Let equation of some surface be defined by function p = $(x,8) in cylindrical co­ ordinate system px9 (Fig. 2.1). T h e n , by using formulae appearing in Section 2 of

Figure 2.1: Regarding the definition ot surface element. Appendix, expressions may be derived for normal force AY and pitching m o m e n t AMZ for t h e surface element cut out by planes 8 = 0_, 6 = d+, x = 0, x = L (gener­ alization is possible for the case when t h e curve bounding t h e "back section" of the element is not a flat one) S+ L

A>7 9oo = - I jcv-t7dxdO, e_ o

(2.1)

Section 2. Surface elements with the invariable CP position

101

»+ L

AM2/qoo

= ~ f J <=p(a;<7 + $ 2 $x cos 9) dx d9, «_ o

(2.2)

where cp is the local pressure coefficient which may be calculated by the following formula in terms of LIM:

cP = n p (t,/t a ) whereas expressions for t%, t2, t7 are derived from formulae (A, 2.2). Conical bodies with the following side surface equation are considered: $(x,9) = (ax + b)r){9),

iy(0) = 1

(2.3)

on assumption that the value of Cp on a body surface does not depend on coordinate x. The assumption which is valid in the framework of LIT takes place for conical flows. It should be noted that lines of intersection of a body surface with planes 9 = 6L, 6 = 9+ may be both the virtual boundaries of its surface and the boundaries of the exposed section on the side surface. In the latter case, 9- and 9+ depend on angle of attack. For the considered class of bodies and regimes of flow around, inneral integrals in (2.1) and (2.2) may be calculated. Upon transformations, we get *+

AY/q^

= -kt

f Cp(0)(r? sin 9 + 7/ cos 9) d9, «_

(2.4)

A M 2 / 9 o o = - / cv(9)[k2(Ti sin 9 + TJ cos 9) + k3 if cos 6] d6, L

r; = — ,

L

ki = I [ax + 6) dx, o

k2 =

(2.5) L

x(ax + b) dx, o

k3 = / (ax + b)2 dx. o

By substituting expressions (2.4) and (2.5) into (1.5), a condition is obtained for the CP position independence on the model of flow around (function cp) and angle of attack (k2 — fciicp)(^sin0 + 7/cos0) + fc3773cos0 = 0.

(2.6)

By using the second condition (2.3), solution of equation (2.6) may be written in the form: 7,(61) = (1 -fi sin 9)~1/2,

ii = const

whereas the following condition should be valid: L

j [(x - xCp)(<2x + b) + a(ax + b)2] dx = 0

(2.7)

102

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 2.2: Triangular plate

which enables one to determine t h e C P coordinate. Therefore, t h e equation for a body surface with t h e invariable C P position is of the form: *(s,fl)=

,aX + \ ■ v/1 — ^ s i n 6

(2.8)

It should be noted t h a t xcp does not depend on angles 0_ and 9+, i.e. on configuration of t h e element cut out by planes 8 = 8_, 6 = 9+. Some special classes of surfaces defined by equation (2.8) are studied below.

2.2

Flat wings

At fi = 1, equation (2.8) defines a plane parallel to axis z and located an angle arctan a with respect to plane xz. Depending on combination of values of p a r a m e t e r s a and 6, different classes of flat wings with the invariable C P position are obtained

Section 2. Surface elements with the invariable CP position

103

such as rectangular (a = 0, b > 0), trapezoidal (a > 0, b > 0), triangular (a > 0, 6 = 0). Of course, for flat wings considered as separate bodies, CP is defined as a point of intersection of the resultant of forces with a wing plane. However, the fiat element is of interest mainly as a part of a surface of more complex configuration for which a point located on axis x is considered as CP. Let us discuss in more detail the case of a triangular plate (Fig. 2.2) which is of interest as far as its use as an element in "design" of complex bodies with the invariable CP position is concerned. By letting b = 0, a = p0 cos 0O/L in (2.7) (the notation is described in Fig. 2.2) and integrating, the following is obtained: CCP = XCP/L = { [1 + (Po/L)2 cos2 BQ] = |

[1 + (h/L)2].

(2.9)

Should a approach zero, it transpires that CP for a delta wing is located on the symmetry axis at distance of (2/3) L away from the vertex, that is CCP = |

(2.10)

and its position does not depend on the angle of attack nor on the model nor on the wing configuration in wing plan form. By using the Cartesian coordinate system, the same result may be substantiated in a simpler fashion. The invariable CP position for flat wings of the configurations discussed above as applied to different classes of flows was mentioned in publications. For example, in [203] this property is substantiated as a result of calculation of conical supersonic flow around plates. Also, [208] should be noted, where it is shown that there is a slight shift in the CP position for a triangular wing due to non-uniformity of chemical reactions in the air in a wide range of flight conditions. It has to be admitted that the result obtained by the authors transpires from the assumption of a conical character of flow over a body rather than the requirement of invariability of the pressure over the entire body surface which follows from LIM, i.e. is a more general nature.

2.3

Cylindrical and conical surfaces

Let us consider other surface elements of practical interest featuring the configuration described by equation (2.8). If a — 0, b > 0, \i < 1 then side surface equation is obtained for an elliptical cylinder having the axis which coincides with axis x. By using (2.7), it may be shown that the following formula is valid for CCP: r

Ccp

ZCP =

1

~L ~ 1'

where L is the cylinder length. In case o > 0, b > 0, ft < 1, (2.8) is a side surface element equation for a truncated elliptical cone. The expression for CCP using (2.7)

104

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 2.3: Comparison of formulae (2.11) with "precise" calculations [214]. Details appear in the text. may be written in the form:

CCP = lT=v>(r,/?)= W > + B tan j3, B

= i[l

+

r

—+ir

I'

(2-11)

where r, R are vertical (in t h e direction of axis y) semi-axis of ellipses of nose and base sections, respectively, /3 is angle of inclination of a conical body generator to axis x in plane xy. T h e C P coordinate does not depend on body dimensions in transverse (along axis z) direction. Therefore, an arbitrary side surface element of a t r u n c a t e d elliptical cone which is cut out by a planes 6 = Q_ and 9 = 6+ has the property of the invariability of C P po­ sition. As far as model of interaction between a flow and body surface is concerned, it is assumed t h a t pressure is invariable along cone generators. It is interesting to com­ pare t h e obtained results of general nature with t h e results of "precise'' calculations for specific conditions of flow around. To t h a t end, d a t a [214] are used which apply

105

Section 2. Surface elements with the invariable CP position

Figure 2.4: Comparison of formulae (3.12) with data [271] for sharp elliptical cones. to aerodynamic characteristics (AC) for side surfaces of truncated circular cones. In Fig. 2.3, solid line represent the relationship between CCP (denoted as CCP) and the per-unit dimension of the body nose which is plotted using formula (2.11) at /? = 10° The circles correspond to calculation results [214] for perfect gas, adiabatic exponent 7 = 1.4 and Mach number M^ = 6 at angles of attack a = 10° (light-colored circles) and a = 5° (dark-colored circles). The triangles exhibit calculation results allowing for physiochemical transformations at altitude 30 km at velocity w^, = 5000 km/s. It is evident that the CP position is virtually invariable and its coordinate may be determined by formula (2.11) with reasonable accuracy. By assuming that pressure value does not depend on coordinate of a point (that is, LIM is valid) in a nose and base parts of elliptical cone surface or by neglecting the contribution of such forces to generation of pitching moment, a conclusion may be drawn that elliptical cones with flat bluntness feature the property of the invariability of CP position. In case of sharp elliptical cone (r = f = 0), formulae (2.11) may be transferred to the form: CcP = ^XCP = ^ s e c 2 / ? = ^ - [ l +

( / W

(2.12)

106

Chapter

3. Design methods

for bodies with invariable longitudinal

static stability

factor

Figure 2.5: Parameter variation regions providing the position of center of pressure inside or outside of the truncated cone. Notation is given in Fig. 2.3.

In Fig. 2.4 data from [271] are presented for sharp elliptical cones flowed around by supersonic "dense" gas flow. Dashed lines correspond to calculations by formula (2-12). The analysis of formulae (2.11) indicates that CP of a cone may be located de­ pending on its geometry both inside it and beyond the base section. In the former case, the following condition should be valid: Rp{ftP)

< L = (R-r)

-cot/?

which may be reduced to the following form after simple transformations: 0 < fl < Po,

Po = arctan

( l - f ) ( 2 f + 1) 2(r 2 + f + 1 )

(2.13)

In Fig. 2.5, a boundary curve (2.13) is plotted which highlights two regions of variation of parameters f and 0. Should the point lie below or on the curve, CP is located inside the cone, otherwise, outside of it.

Section 2. Surface elements with the invariable CP position

107

Figure 2.6: Flat triangular element.

2.4

Flat elements

Let us consider a flat element assuming that pressure at each point of its surface is the same. Attention is essentially paid to triangular plates which are of major interest as elements of vehicle although similar results may be obtained for plates of another configuration. Assume the vertex O of a plate is located at the origin of the coordinates (Fig. 2.6) and side AB lies in plane i = L. The plate configuration is unambiguously defined by parameters L, 6, 0+, p_, p+ and its equation in Cartesian coordinate system xyz is of the form: x = ®(y,z) = ay + bz, p+ sin0 + — p- sin#_ p+p- sin(# + — 6-) '

p+ cos 0+ — p_ cos 6_ P+P- sin(# + — 0-)

The expression for the CP coordinate is easily obtained by using (A, 4.2)

//

c p ( $ $ v + y) dy dz

//

cp$ydydz

JJ [(a2 + l)y + abz] dy dz a / / dy dz

108

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 2.7: The case of two adjacent sides. where integration is considered for the region confined inside AOA'B' which is a projection of region O A B on t h e plane yz. O m i t t i n g awkward transformations, the expression for C C P of a plate may be w r ' t t e n straight away in t e r m s of p a r a m e t e r s P-, />+, 0 - . 0+> L C C p - xCp/L _

2

3

= /(/»_,/>+,0_,0 + ) P-P+ 3

2p-P+ +

sm2(6+

{pi — P+){p+ cos 9+ — /?_ cos ( sin(0 + — #-)(/>_ sin 0_ — p+ sin 6+)

6.

p i - p i - 2p+p+ cos{6+ P-IL,

p+ =

0_y (2.14)

p+/L.

Should a triangular plate lie in a plane passing through axis z, formula (2.14) is transformed to form (2.9) and (2.10). Along with (2.14), a different formula for C C P will be used further on:

h.

2 Ccp =

= r-/L,

I

+

3 ( f " "" f+) '

f+ = r+/L,

h =

h/L

(2.15)

whereas t h e notation is described in Fig. 2.6. It follows from formulae (2.14) and (2.15) t h a t by adding a surface section sym­ metrical to t h e original one about plane xz, the C P position remains invariable. Let us derive t h e conditions which provide coincidence of C P ' s for two adjacent sides symmetrical about some plane 8 = x (Fig. 2.7), i.e. condition of validity of t h e

Section 3. Pyramidal bodies

109

following equality: f(?-l,P+U*

- &,*) ~ f{p+l,P-l,*,x

+ 6).

(2.16)

By using expression (2.14) for function / , condition (2.16) may be transformed to the form: (pit -P~li){pli

-2p-ip~+i cos 6+ plx) mn2x = Q.

Therefore, the CP position for the considered configuration is invariable provided points A[, B[ (B'2), A'2 are located on a circle with a center at the origin of coordinates (p-i = p+i = p+2) or the plane of symmetry 9 = x is one of the planes 0 = 0,0 = w/2, 0 = IT (sin2x = 0).

3

Pyramidal bodies

Let us turn now to formation of combined bodies of complex configuration with the invariable CP position which comprise the elements discussed above as having the same property. According to Property 2, Section 1, the surface of such bodies may be made up of the elements for which AY = AMZ = 0. At the beginning, the analysis is conducted for bodies of pyramidal configuration, i.e. for conical bodies having a polygon lying in plane x = L as a base section. If CP's of the pyramid triangular sides coincide, the body has the desired property.

3.1

Typical geometric configurations

At first, the case is considered when the pyramid base is a polygon symmetric about plane xy and inscribed in a circle of radius p0 = poL. By letting p_ = p+ = p0, the following formula is obtained from (2.14) to determine the CP position for a side of a pyramid: „

Ccp =

ZCP

2

T ~1

2

1

1 + p0 cos

+

—

Thus, the CP position for a side of a pyramid does not depend on its orientation. Upon considering the possible alternatives for location of a regular n-gon in the pyramid base, the conclusion is made that CP's of the sides coincide and CCP of the pyramid is determined by the expression: £CP

CCP = ~Y

2 / , , _2 2 w\ 2 = y ( ^ l + p 0 2 cos 2 -J = - sec 2 /?,

where /3 is the angle between a pyramid side and a perpendicular to its base.

(3.1)

110

Chapter

3. Design methods

for bodies with invariable longitudinal

static stability

factor

Figure 3.1: Pyramids with rhombic mid-section. Comparison with "precise" calculations [195].

Therefore, the CP position for a pyramid is invariable. CCP depends only on an angle of a side with a perpendicular to the pyramid base. As n —> oo, a pyramid turns into a sharp circular cone and formula (3.1) is transformed into formula (2.12). Let us consider a wing of pyramidal configuration having a rhombic base (Fig. 3.1 a) ). CCP of side OAB (where O is a vertex of the pyramid) is deter­ mined by formula (2.14) at 0_ = 0, 9+ = 7r/2 which yields the expression: C CP = | ( l + | t a n 2 7 ) .

(3.2)

So far as the wing is symmetrical about plane xz, CCP of side OBC is also determined by formula (3.2). Consequently, the CP position for a diamond-shaped wing does not depend on the model of flow around, the angle of attack and the wing span, and is defined solely by the angle of inclination of its surface to axis x in plane xy and the property of the invariability of CP position for diamond-shaped wings substantiated in [242] on the basis of the Newton model is valid in general. In Figs. 3.1 and 3.2 b), c), results of processing the data which appears in [195] are presented respectively for diamond-shaped bodies (Fig. 3.1) and pyramidal bodies with an isosceles triangle as a base (Fig. 3.2 a) ). The solid line represents the

111

Section 3. Pyramidal bodies

a)

Figure 3.2: Pyramids with isosceles triangle as a mid-section; a) body configuration; b) comparison with "precise" calculations.

112

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 3.2 (continued}: c) comparison with "precise" calculations.

relationship [195] between pitching m o m e n t coefficient and angle of attack. T h e circles correspond to the calculated values of this quantity obtained by t h e formula CCP

■ Cy

(3.3)

where Ccp was determined by formula (3.2) and t h e values of Cy were calculated on t h e basis of t h e values of Cxa, Cya from [195]. Although it follows from comparison of formulae (2.10) and (3.2) t h a t the bodies displayed in Fig. 3.2 a) do not exhibit the property of t h e invariability of C P position, in case of slender wings (7
113

Section 3. Pyramidal bodies

3.2

General algorithm

Let us consider now the general procedure for formation of pyramidal body transverse contour. The CP position for a body is predetermined and should remain invariable for various models of flow and angles of attack. To that end, the sides should be taken so that their CP's coincide. It is convenient to reduce the problem to the search of contour which is a projection of the pyramid base on plane yz while using coordinates y, z for its analysis §=-7-,

5

= T->

A0 = L^/A,

A = 3CCP-2.

(3.4)

A-o AQ All parameters normalized in this way will be marked by the "caps'". Assume r o , r i , . . . , r „ are ordinates of points of inflection of the base section con­ tour, hi is an ordinate of the point of intersection of extended i-th segment of the broken line with the body symmetry plane (Fig. 3.3). Conditions (2.15) may be written in the form: *

hi(fi - fi-i) = ± 1 ,

*■

hi

hi = — ,

T'

?i = — ,

AQ

(3.5)

AQ

where "plus" corresponds to the case when CCP > 2/3, "minus"—to the case when Ccp < 2/3. Below, the problem is considered for the former case and the latter may be analyzed in a similar way. As follows from the condition of body symmetry about a plane xy, fo = hi,

fn = hn.

(3.6)

Relationships (3.5) and (3.6) show that for coordinates y, z analysis of possible con­ tours may be conducted irrespective of defining the specific value of Ccp- Prior to describing the general procedure, some particular cases should be discussed. Let n = 2; then the conditions (3.5) and (3.6) may be presented as follows: ro(r 0 + f i ) = l ,

f2{fi + h) = 1

whence h = ?0~1-r0,

f2 = -r 0 _ 1 .

The basic kinds of contours for n = 2 are shown in Fig. 3.4. Cutpoints of separate segments are denoted by solid circle. So long as a symmetrical mapping a contour about axis z does not result in the CP position changes, should its property of the invariability of CP position be proven for the original body, then this property for configuration c) follows from the one for configuration a). By varying parameter f0 and the contour "width", a family of pyramidal configurations may be obtained which

114

Chapter

3. Design methods

for bodies with invariable longitudinal

static stability

Figure 3.3: Illustration of a general algorithm for pyramidal bodies.

factor

115

Section 3. Pyramidal bodies

Figure 3.4: Typical configuration, n = 2.

116

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

significantly differ in shape. If the pyramid base is a hexagon (n = 3), then conditions (3.5) and (3.6) are transformed to the form: (r0 + fi)fo

= i,

( n + r 2 ) f 2 = 1,

(r 2 + f 3 ) r 3 = 1,

whence .-i n = r 0 - rQ,

. .-i r 2 = r3 - r 3 ,

; r0r3 ft2 = —— ; . . r-rr(r0 + r 3 ) ( l - r 0 r 3 )

Some contours are shown in Fig. 3.5. In case n = 3, three p a r a m e t e r s fp, r 3 and the contour "width" are available. By varying these p a r a m e t e r s , configurations of different shape may be obtained. Let us consider now the general case when the pyramid base is a 2n-gon. T h e following relationships should be valid:

(n + n+ijn = l, i = o,n, (f3-_j + fj)hj

- 1,

j =

l,2,...,n-l.

Assume t h e values of the ordinate of points of inflection of t h e contour generator r 0 , f2, . . ., r „ _ 2 , r„ are predetermined. Below, the technique for contour formation is described which ensures t h e property of the invariability of C P position for a pyramid. Numbers from 0 to n are assigned to cutpoints of adjacent segments while number i is allocated to the segment located between points t — 1 and i (Fig. 3.3). At first, ?i, r n _ i , hj are determined by formulae h = ro1 ~ r0,

f„_i = f~' i =

- rn,

hj = (r^_i +

fj)'1,

l,2,...,n-l.

By drawing an arbitrary (but intersecting straight line y — rx in region z > 0) straight line from node 0 at angle 70 to axis y, node (1) and segment (1) are obtained. Further on, another straight line is drawn through node (1) and t h e point having coordinate z = 0, § = h%. T h e intersection of the line with straight line y = f2 is node (2). T h u s segment (2) is formed. Proceeding with the technique, a part of t h e contour is obtained which is formed by segments 1 , 2 , . . . ,n — 1. T h e last, n-th segment is formed by connecting nodes n — \ and n. By varying n parameters 70, r0, f2, . . . , r n _ 2 , fn, different configurations may be obtained. Some contours for n = 4 are shown in Fig. 3.6.

4

Bodies with surface comprising conical and flat elements

Let us proceed with discussion of conical bodies by expanding the class of configura­ tions analyzed. Assumption is m a d e t h a t the base section bounded by t h e contour

Section 4. Bodies with surface comprising conical and flat dements

a)

b)

Figure 3.5: Typical configuration, n = 3.

117

118

Chapter 3. Design methods

for bodies with invariable longitudinal

static stability

factor

Figure 3.6: Typical configuration, n = 4.

may contain "sections" of elliptical curves apart from straight lines. Formula (2.12) enables one to obtain the equation for ellipse frontal section boundary with CCP equal Ccp and with vertical semi-axis R in the form: y2 (AJ/2)

z>__ 2

+

Rl

~

'

where L is the ellipse length, Ro is the arbitrary length of horizontal semi-axis, pa­ rameter Ao has the same meaning as in (3.4). In coordinates y, z defined by (3.4), the ellipse equation assumes the form: 2 i2 = 1, -y — i — R? + Rl '

Section 4. Bodies with surface comprising conical and flat elements

119

where -- __ vv // 22

-- 7T *i 0c

Therefore, the conical body contour with the invariable CP position may contain "sections" of ellipse. The vertical semi-axis of the ellipse equals V2/2 in coordinates y, z, while the horizontal semi-axis may be of arbitrary length. The general technique for obtaining the body base section contour including straight and elliptical sections is similar to that described in previous section. The difference is that the i-th segment (|r,"-il < \/2/2, |r,| < -v/2/2) may be an ellip­ tical curve. The general view of one of the possible configurations is presented in Fig. 4.1 a). Conical bodies which are combinations of an elliptical cone and a triangular wing located at an angle to plane xz (Fig. 4.1 b) ) also belong to the bodies with the invariable CP position. Winged elliptical cone does not belong to the body class with invariable CP position. It follows from comparison of formulae (2.10) and (2.12). However, at small angles /?, this property is observed with considerable accuracy which is illustrated by data on Fig. 4.2 b) taken from [177] (body aspect ratio is 1). It should be noted that the invariable CP position for the component bodies and consequently with certain accuracy for the body itself (Fig. 2.4 a) ) follows from the assumption of conical nature of flow around. As applied to construction complex bodies with invariable CP position it should be remembered that geometrical sizes and region of angles of attack must rule out the possibility of casting shadow of different elements of body against each other. Let us consider now the winged vehicle which are shown in a general fashion in Fig. 4.3 a) and in a plan view in Fig. 4.3 b). For definiteness, assume that the wings are of triangular configuration. CP for such body is invariable if given CP's of the frame (elliptical cone) and wings coincide, that is the following condition holds true: 2 2 — (1 + tan 2 (i)L = x0 + — (xi - x0) which may be otherwise presented in the form: (2 + a)xi cos2 0 = 2,

(4.1)

where xo = xo/L,

xi = Xi/L,

xo = ox\

whereas to ensure the viability of the configuration, the following relationships must be valid: 0<<7<1,

0 <«!
(4.2)

120

Chapter 3. Design methods for bodies with invariable longitudinal

static stability

a)

b)

Figure 4.1: Configuration of some bodies comprising conical and flat elements.

factor

121

Section 4. Bodies with surface comprising conical and flat elements

b)

Figure 4.2: Winged elliptical cone; a) body configuration; b) data [177]. Joint consideration of conditions (4.1) and (4.2) yields the following inequality: P
(4.3)

Therefore, if ft < /3o, a wide class of bodies, namely winged elliptical cones with the invariable CP position may be formed while CCp = y ( l + tan 2 /9) and CP is not shifted when cone and wing dimensions along axis z are changed. Condition (4.3) may be omitted should the wings be allowed to protrude beyond the cone base section.

122

Chapter

3. Design methods

for bodies with invariable longitudinal

static stability

factor

Figure 4.3: Example of winged vehicle; a) general appearance; b) plan view of configuration.

5

Bodies of complex configuration with elliptical cross-section

Let us further consider different class of bodies with the invariable CP position. Analyzed are configurations made up of elements of ellipsoids, conical and cylindrical surfaces. Although bodies with ellipse as cross-section are discussed, for the sake of clarity, bodies of revolution are considered.

Section 5. Bodies of complex configuration with elliptical cross-section

123

Figure 5.1: Cylinder with flat or spherical bluntness.

5.1

Cylinder and cone with spherical bluntness

It was shown above that the CP position for an element of spherical and cylindrical surfaces does not depend on the model of flow around and the angle of attack, whereas for the bodies of the former class, CP is located in the center of the sphere while for the latter, in the middle of the cylinder. Therefore, CP for a combined body which is a cylinder with spherical bluntness is invariable if the sphere center coincides with the middle of the body cylindrical section whereas that section may be both flat and spherical (Fig. 5.1). From obvious geometric consideration follows the relationship between cylindrical section length L, cylinder base radius r and bluntness radius p, r2 + L 2 /4 = p2 Should spherical bluntness (Fig. 5.2 a) ) be added, the CP position for cone C remains invariable while the sphere center should coincide with CP for the conical sur­ face. From geometric considerations, the expression is obtained for radius of bluntness PoPo/L = y/r* + [V(r,/3)]*, where function ip is of the form (2.11). The relationship between bluntness radius p0 and half-angle ft of the body conical section opening is shown in Fig. 5.2 b). Functions p0((3) are concave, p0(ft) is decreasing at ft < ft.{r) and is increasing at ft > ft,(f); the point of minimum = arctan ( w 1 — E(r

(^

E(f) =

f/B{f).

So long as E(f) > 0 and E'(f) > 0, increase in bluntness radius begins at the values of ft which are less than 45° whereas the point of minimum is shifted to the left when the increase takes place.

124

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 5.2: Cone with spherical bluntness: a) body configuration; b) relationship of dimensions.

By using expression (2.11) it may be shown t h a t xcp/r > t a n / ? , i.e. spherical bluntness does not cast a shadow on the body conical section.

5.2

Segmental-conical and some other bodies

Let us require the fulfillment of condition (2.13), t h a t is C P of a cone should be located inside it. T h e n bluntness radius po of the rear part of segmental-conical body

Section 5. Bodies of complex configuration with elliptical cross-section

125

Figure 5.3: Segmental-conical bodies; a), b), c) configuration versions; d) relationships between geometric dimensions ensuring the unchanged position of center of pressure.

126

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

Figure 5.4: Combined bodies.

(Fig. 5.3 a)-c) ) should be expressed as follows:

El R

-WR

\

ZCPJ

+ R2

1 + B(r) t a n ;

B(r) - 1 tan/3

which provides t h e coincidence of C P for a sphere and C P for a conical surface. In Fig. 5.3 d) curves are presented which represent t h e relationship between spherical bluntness radius for t h e body rear part and semi-angle j3 of t h e conical surface. Segmental-conical bodies with the invariable C P position exist in the range of angles 0 < /S < A), where /30 is determined by formula (2.13). T h e configuration shown in Fig. 5.3 b) corresponds to the case when f = 0. It is obvious t h a t the C P position for bodies with flat nose and spherical bluntness of the rear part (Fig. 5.3 c) ) is also invariable when the model of flow around and the angle of inclination are varied.

Section 5. Bodies of complex configuration with elliptical cross-section

127

Figure 5-5: Double cones.

If /?o < P < T / 2 , t h e n C P for t h e conical surface denoted as C lies beyond t h e rear section, therefore, t h e C P position remains invariable once a cylinder of length 2a = 2 ( i c p — L) is added as shown in Fig. 5.4 a ) . Flat nose and rear part m a y be s u b s t i t u t e d for spherical ones. Sphere centers should be located at point C (Fig. 5.4 b) ). T h e configurations which are double cones with possible flat or spherical bluntness on t h e side of nose and rear p a r t s (Fig. 5.5) also belong to bodies with the invariable C P position (with a d e q u a t e selection of geometric dimensions). T h e list of examples of configurations with t h e invariable C P position created

128

Chapter 3. Design methods for bodies with invariable longitudinal static stability factor

on the basis of cylindrical, conical, spherical and fiat elements may be continued, however, the apparatus for determination of configuration of such bodies is developed to the extent which is sufficient for its application without basic difficulties in each specific case allowing for particular requirements specified for a body configuration.

Chapter 4 Variational problems of optimum body configuration determination 1

Some general solutions for local models

In the present section two problems of configuration determination for bodies char­ acterized by the minimum value of the component of effect of a medium along the direction of its motion (drag value) are considered. Such problems are typical and, of course, were repeatedly studied as applied to the specific models describing inter­ action between media and bodies moving in them. However, here these problems are considered in the framework of the general LIM class. It can be used to establish existence of body classes among which the optimum may be chosen for the specific model by selecting one of the parameters. Moreover, in a number of cases, optimum body properties are retained when LIM is varied which makes it interesting as far as applications are concerned.

1.1

The class of the optimum three-dimensional bodies

Assume in the coordinate system xyz (Fig. 1.1), a three-dimensional body surface is defined in the form x = $(j/, z) and its back section is bounded by curve T. The equations of the curve are x = x0{6), y = yo(6), z = z0(<5). The projection of contour T on plane yz bounds region {a). Based on formulae of Section 4 of Appendix, expressions for body drag value may be given as —

= a Uxa

=

JJllD[
¥>($„ #,) = [*; + * l + 1]" 1/2 ,


129

130

Chapter 4. Variational problems of optimum

body configuration

determination

three-dimensional bodybody. Figure 1.1: Regarding the configuration optimization problem for a three-dimensional

where function fir; is of the form (A, 1.12) while the area of region (
Ssvr= m

z

(1.1) (1.1)

Naturally, in this case, the area of region (a) is also known. The problem is reduced to minimization of the functional:

I[*(y,z)] = JJw[
cu(i) = cu(() = nDD((II) + + yy

(1.2)

by allowing for condition (1.1). The study of the variational problem will be carried out on the basis of the necessary conditions in the form described in [278]. The Euler-Ostrogradsky equation for functional (1.2) may be presented in the form: — [di(tfi>] (fi^j + +g -—\u(ifi) \u(ifi)■ •v* v* ] = 0, [di(tp) ■ ■¥>*,] =z ]2 0,

Hv) -

dip

Section 1. Some general solutions for local models

131

whence it follows that function $(j/, z) satisfying the differential equation of the first order <£>(#s,, # e ) = Ho,

Ho = const

(1.3)

is the solution if fi = Ho is the extremum of function w(/i), i.e. CID(HO) - T

= 0.

(1.4)

Ho The Legendre condition for a weak minimum has the form: <S(0o) > 0

(1.5)

that is H — Ho should be the point of minimum for function UI(H). The Weierstrass condition for a strong minimum brings about the requirement that H — Ho should be the point of global minimum. Therefore, as transpires from (1.1) by allowing for (1.3),
Ho = -z— < 1 iJsur

while parameter \ s is determined from (1.4) as follows: As =

HI&D{HO)

and condition (1.5) is transformed to the form: &D{fk>) + — &D{HO)>0-

(1-6)

Ho The drag coefficient C™ n for the optimum body C™"

= 7 / / WHO)

dy dz =

QD(HO)

does not depend on the base section configuration should a be taken as a characteristic area. The body surface equation is obtained from the solution of the differential equation

* ; + ^ = «2, with the initial conditions x = x0(8),

-a-r

y = j/0(<5),

z = Zo(6),

132

Chapter 4. Variational problems of optimum body configuration determination

it has t h e form [192] z(t,S)

= p0{6) ■ t + z0(<5),

y(t,6)

= q0(6)-t

$(t,<5) = a2-t Po(o) =

z0xQ±y0P

+ +

,

y0(6),

x0{6),

qo(o) =

p = (a27 - xlf\

7

2/oioTfo/

= i* + yl

whereas t h e sign is selected from t h e condition t h a t the outer side of t h e surface faces t h e outer m e d i u m . If t h e base section is a circle lying in plane yz, t h e o p t i m u m b o d y is a cone with half-angle of opening equal arcsin/j 0 T h e analysis of different alternatives of defining t h e isoperimetrical conditions and t h e conditions on a body boundary is conducted similarly to t h a t performed in [176, 278] as applied to t h e Newton model. In [278], t h e o p t i m u m b o d y configuration is graphically described for the case when x0(S) = 0 and we shall not dwell on these problems. It is i m p o r t a n t t h a t in t h e framework of LIM there is a one-parameter family of o p t i m u m bodies and the concrete definition of t h e model affects only t h e se­ lection of p a r a m e t e r /xo- Moreover, t h e o p t i m u m body configuration does not depend on t h e model of interaction between a body surface and a m e d i u m and t h e model is necessary only to check t h a t condition (1.6) is satisfied. Of course, these conclusions transpire from t h e study of t h e problem on t h e basis of t h e necessary conditions of optimum. Let us illustrate t h e technique for obtaining t h e solution for hypersonic flow around a body by gas of different rarefaction as an example. Model ( 1 , 2.14) is accepted as a basis and function Qo(t) in this case has t h e form: VlD{t) = {A2-Bl)t2 A2>

Bi>

+ Alt + 0,

Bu

Ax > 0.

Condition (1.6) taking t h e form 2v4i 6(A 2 - Bi) + — - > 0 Ho is valid throughout and t h e o p t i m u m body drag coefficient is

c™ n = p t 2 - B i ) / 4 + A i f i o + B 1 . In Fig. 1.2 curves are presented illustrating variation in t h e o p t i m u m b o d y drag coefficient with respect to cr/Ssur. T h e "experimental" points correspond to a right circular cone which is t h e o p t i m u m body in t h e case of a circle back section.

Section 1. Some general solutions for local models

133

Figure 1.2: Drag coefficient for the optimum bodies. O, • , 4, A — experimental data for cones [164].

1.2

The optimum wing class

Let us consider the problem of obtaining the least drag wing in the conditions of local interaction between a medium and a wing surface. This problem was studied previously [247] on the basis of the Newton model allowing for friction while the principal goal of the authors is to reveal the properties of the solution which are common for LIM's as well as its specific features defined by selection of the specific model. The coordinate system to be used is shown in Fig. 1.3. The body configuration is

134

Chapter 4. Variational problems of optimum body configuration determination

Figure 1.3: Regarding the problem of wing configuration optimization. determined by function y = $ ( x , z ) , the trailing edge equation is y = t{z), x = n(y), t h e equation for t h e given leaning edge is x — m(z), y = 0. A 1/4-th p a r t of t h e body is considered and it is assumed t h a t the velocity vector is directed along axis x. In agreement with the formulae of Section 5 of Appendix, t h e expressions for body drag and side surface area may be presented in the form:

JJnD{h/t2

9oo

£2 dx dz,

?xz)

SSUr = / / h dx dz,

where functions t\, t2 are of t h e form (A, 5.3) at x = 0, crxz is t h e area of t h e 1/4-th of t h e wing in a plan view, i.e. the area of region {crxz) defined by t h e inequalities: 0
m(z) <x


Assuming t h e smallness of $ z as compared to t h e unit, t h e expressions for drag, wing area oxz in a, plan view, side surface area SsnT m a y be given by

—- =

l[u>0($x)dxdz,

(Cxz)

<JXZ =

//

("«)

dxdz,

Section 1. Some general solutions for local models

135

Ssur ==JJ^l Ssur / / yf«l ++ ldtedz, ldxdz,

uiJi\

— t O,-, I

(1.7) ( L7 )

1

n <**(*) =
The considered problem of minimization of wing drag in the general case allowing for isoperimetric conditions (1.7), is reduced to the study of the following functional

7[*(x I[9(x, z)] = JJ w($ „ ff, ,A,)cix(iz, A,) dx dz, f(*x, x , \A a; u= = ^o(#r) ub(«s) + + X„^l X . ^ J + 11 + + A,,. V The study is carried out on the basis of the necessary conditions. The Euler-Ostrogradsky equation has the form: d9 \ dx )x

$

[

y/*l + l. Ufa =

= 0, *■*]

du

—

whence the following relationship transpires ^ 00(($ w $ rr) + A Xs S - =** L = = A(z),

(1.8) (1.8)

where A(z) is an arbitrary function. The conditions of transversality are presented in the form:

(1.9)

£ r <5x + £„ Sy + £* 6z = 0,

Ex 6x + Ev Sy + Ez Sz = 0,

(1.9)

where / , )0„(#*) C * 1 -- $* (6 ■ui„(<S> ) i+ i+ ^o($x) 7 *' Exx = zi w -KK , ^ • 2 + 1 v*s +* 7

Ey = i

r

[

'

v / ^ + lJ v*j+ij

< * ( * ,+ ) +- _ £E2z = = -- jwo(*») | «„($,) + +W A s ,/$« *, a uo($x) ?^ = =i i=K+ + «i$z * S ++11 + K whereas i is the inclination of the trailing edge contour in plane xz.

1

(1.10) J> (iio)

136

Chapter 4. Variational problems of optimum

body configuration

determination

Let us consider the possible alternatives of denning the configuration of the trailing edge. If n(z) is predetermined and t(z) is free, conditions (1.9) and (1.10) are reduced to the requirement of validity of the following relationship for the trailing edge, «*(*«) + A .

,$*

=0.

(1.H)

x/*!TT If t(z) is predetermined and n(z) is free, then **(»,) - ** • wb(»«) + - T = £ = + K = 0.

(1.12)

And finally, if the conditions on the trailing edge are natural ones, the following relationships should hold true "o(*x) + K^l w 0 ($ r ) +

+ 1 + K = 0, '

= 0.

So far as function u does not depend on $ 2 , the Legendre condition becomes simpler and is reduced to the inequality:

o. Specific classes of solutions for different alternatives of defining the conditions (1.7) and the trailing edge configuration are discussed below. 1) Assume the trailing edge configuration, that is functions t(z), n(z), is predeter­ mined and Ssur is not limited. Then allowing for the predetermined conditions, the solution is presented in the form: ${x,z)

= R(z)[x -

m(z)],

, !{Z) , ; n[z) — m(z) while the following formula holds for the optimum body drag R(z) =

■faW'

-^r-it{z)-RizT

whereas the Legendre condition should be valid, w[R(z)] > 0.

(1-13)

137

Section 1. Some genera.! solutions for local models

2) Assume now that t(z) is predetermined and S,UT and o-xz are free. Then it transpires from condition (1.12) that the solution of equation (1.8) may be presented in the following form by accounting for the conditions on the leading edge, $(x,z)

= C[x-m(z)]

(1.14)

where C = V>-1(0),

0(i)

and function n(z) is defined by the equation n z

( ) = -jjKz) +

m z

{ )-

The following formula holds for the optimum body drag coefficient rm ,„

_ *r' B _ MC)

where cryz is the area of body projection on plane yz. The Legendre condition has the form Wo(C) > 0.

3) Once t(z) is predetermined, whereas t(b) = 0 and the wing area in a plan view is o~xz, then the wing surface equation is of the form (1.14) where parameter C is determined from condition (1.7) for cxz and equals

C=

^. °~xz

The Legendre condition is w 0 (C) > 0. The expression for the optimum wing drag coefficient and the equation for its surface may be written as follows: nmin

_ XT" qcaVyz

$(x,z)

_ <MC) <-/

= C[x-m(z)].

(1.15)

4) Assume now that 5 s u r and function n(z) are predetermined. Then in agree­ ment with equalities (1.8) and (1.11) the following holds for the trailing edge (the

138

Chapter 4. Variations! problems of optimum body configuration determination

corresponding parameters are marked by "caps") ^o(*x)

V^ + i Wo($B) +

*'**

= 0

whence it transpires that A(z) = 0 and from (1.8) it follows that $ x = C, i.e. $(x,z) = C[x — m(z)], A, = W f ^ .

(1 .16)

Parameter C is obtained from isoperimetric condition (1.7)

c = \l(^)2-l>

*** = j H*) - ™(2)1dz-

The Legendre condition is of the form:

^+<0kio while the optimum body drag is given by Vmin «B(C). Q00&XZ

It is evident from formulae (1.13)—(1.16) that the optimum wing configuration (should the Legendre condition be valid) does not depend on the specific LIM that is the wing retains the optimum characteristics in a wide range of flight conditions.

2 2.1

Bodies of revolution of minimum drag in gas of different rarefaction Problem definition

The class of bodies of revolution featuring the generator configuration described by function p = $(x) is considered. The free-stream flow is directed along axis x and the body may have a flat bluntness. Except for function $(x), the body geometry is characterized by length L and radii of the nose, p_, and of the base section, p+.

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

139

As follows from formulae (A, 2.15), the expression for body drag coefficient in the conditions of LIM upon transition to dimensionless parameters may be presented as follows:

cxa = - JxaL - = l\±.£n,[i) ^ I O

+ /#(§) .ids],

*p\
(2.1) (2.i)

0

where Xa is the body drag coefficient and function H is determined by formula (A, 2.16), Ppi _

xX

x = —, V

L '

$

pp+ +

# = —, V

-

L'


ax

The length and radius of the body base section are given, i.e. Xf *f = h1,

Pf PS = = TT.-

(2.2)

The discussed problem lies in determination of configuration of the body revolution of minimum drag. When the problem is studied as applied to transition region of rarefied gas flow, model (1, 2.14) is used. In this case, the expression for drag coefficient upon simple transformation is given by

Cxa -= ~I[^] + B1, T T

where

m{x)\ imi^-pf-Po+jF&idx, = ^p?p0 + JF{k>)
(At2-B^ {A -B^

pm m

~

(HI

+

At? AJ v^+I'

P . 1-B Po0 = = n flpP(l)-B (l)-Bl l = =A A,2 + + A1A -B 1. 1

(2.3)

The problem of configuration determination for the body of minimum drag is reduced to minimization of functional / for conditions (2.2). In particular, the nose bluntness radius is also to be determined. The requirements of condition of increasing of $(x) and convexity of the optimum body generator are not included into mathematical problem definition in explicit form. At Re0 = 0 and ReQ = oo, the problem is reduced to the one considered in [278] for free-molecular regime and regime of "dense" gas flow, respectively.

140

2.2

Chapter 4. Variational problems of optimum body configuration determination

Analytical study of variational problem

The study of problem (2.2)-(2.3) is conducted on the basis of the necessary conditions of a minimum. The subsequent numerical calculations show that the solution realizes a strong minimum of drag. The overscribed "bars'' of the dimensionless parameters are omitted below. The function which determines the generator configuration for the optimum body should satisfy the Euler equation having the first integral

$ •• iE($) iJ(4) == C, C, R(t) R{t) ==tF'(t) tF'(t) -- F(t), F(t), C = const, where

R(t) =

2(A2-B1)i3 2

(< + l)

AS

2

^

2

3 2

(< + l ) '

The bluntness radius is not predetermined, so the transversality conditions are of the following form: (2.4)

* [Po - F'(Pi = 0, Pi Pi)] = where

+ -B^f ^+A<^l>0. (< (t (< + l) (i + l )

F\t) = {A K 2

;

22

2

22

3/2

So far as i?($) < oo and C / 0, equation (2.4) yields the following condition for bluntness radius determination, T{(H) T( Pi) = 0,

T(t) = T{t) = F'(t) F'(t) - A22 - A, + Bi. BL

(2.5)

Function $(x) should satisfy the Legendre condition, F"($) > 0,

™-**-*>{?$+ \fi+if ' %££«• \t' + lf>F"(t) = 2{A2 - Bl z?)i(3-<2)

i Ai

2

-<2

M(2.6)

< >

Let us show that equation (2.5) with account of (2.6) always enables one to unambiguously determine Pi. Indeed, T(0) = Bx - A2 - Ax < 0, T(l) = J 4 . I ( 1 . 5 / \ / 2 - 1 ) > 0 and T(t) is continuous and increasing function whence the existence and uniqueness of solution transpires whereas 0 ,-< 1. Condition p\ = 1 is valid at Ax = 0, in particular, for Newton model. For the analysis, it was taken into account that M > 0, Ai > 0, 5 ] > 0, A2 > BL

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

141

Thus, the optimum body of revolution has a flat bluntness and the angle of gen­ erator inclination in the initial point does not exceed 45°. The value of the angle is unchanged for various values of r . Should $ = t be chosen as a parameter, the Euler equation solution may be given by * = CA(t),

x = CB(t),

where 1 {t)

1

iF>(t)-F(t) > °'

~W)~

B(t) = f F"(u)A(u)2 du,

t;
Since c

$


F"(t)A{t)

the body generator is convex. When a point moves over the contour from x = 0, parameter £ varies from p\ to tj. The unknown parameters C and it are determined from boundary conditions (2.6) which give the equations CB'tj)

= 1,

A(tf) = rB(tj),

(2.7)

Let us study the second equation (2.7). To that end, curve $ = A(t), x = B(t) at 0 < t < to = p, should be considered. When t decreases, the values of A(t) and B(t) infinitely increase while the curve is convex and the inclination of its tangent line tends to zero as t —* 0. Apparently, this curve has a single point of intersection with straight line $ = TX thus proving the existence and uniqueness of a solution of system of equations (2.7) which may be presented as follows: tj = D-\r),

C= - L B(tf)'

v

'

D(u)-Hl!] B(uY

The bluntness radius may be determined from the formula: MPJ)

The expression for the optimum body drag coefficient is given by

C™n = [A(tf)}-2

\A{pi)

Po + 2

J [tF>(t)-F}*j

+ B

'

142

Chapter 4. Variatiomil problems of optimum

body configuration

determination

Some limiting cases are considered below. At fairly high Re0 in t h e t r u e range of variation of p a r a m e t e r s tw and 7, it may be assumed t h a t A\ = 0 as shown in substantiation of formula ( 1 , 2.15). T h e n t h e minimized functional is presented in the form:

I[<S>] =

r <J> $ 3 dx

p? (A2-Bl) 0

$2+1

whence it transpires t h a t for t h e accepted model of flow around t h e configuration of t h e body of m i n i m u m drag does not depend on parameters Reu, tw, 7 at sufficiently large values of Re0. On the other hand, at small values of Re0, it may be assumed t h a t A2 = Bi = 2. T h e n

m=AM+]*pL i.e. in this case a similar situation takes place which was mentioned in [278] when dealing with free-molecular flow. Therefore, in t h e region of large Re0, stabilization of configuration of t h e bodies of m i n i m u m drag occurs at smaller values of Re0 t h a n stabilization of drag. At both fairly low and fairly high values of Re0, the o p t i m u m body configuration does not depend on p a r a m e t e r s 7, tw.

2.3

The numerical study method

T h e results of t h e study conducted in previous section provide a guide t o t h e con­ figuration features of the body of minimum drag. However, as far as studying this problem in case of arbitrary LIM is concerned, it should be noted t h a t analytical m e t h o d s along with their advantages (the possibility to qualitatively study the prob­ lem and t o obtain t h e "standard" solutions, efficient solution of problems in simplified definitions aimed at formulating the directions for a more detailed study etc.) have a n u m b e r of disadvantages. Among those, the following are outlined: • for reasonably intricate LIM, analytic study becomes difficult so simplifications are employed (slender body, etc.) depreciating t h e significance of t h e results obtained and requires additional study into t h e bounds of their validity; • for t h e practical used models, the study on t h e basis of analytical m e t h o d s of calculus of variations results, as a rule, in the need to use a computer to obtain specific numerical values. In such a case, working hours are often comparable to those spent on solution of the problem by numerical methods;

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction • as a rule, the analytical study of variational problems is conducted on the basis of the necessary optimum conditions while the correct feasibility check of the global optimum entails significant difficulties; • the analytical methods specify the fairly rigid requirements for smoothness of functions included into the objective functional; in particular, tabulated defini­ tion of LIM is not allowed. The abovementioned considerations indicate that to solve the problems of body configuration optimization on the basis of LIM, it is expedient to use the numerical methods along with the analytical ones. To solve the problems of the discussed class, the technique [220] is rather conve­ nient. The procedure originates from the known method of dynamic programming [71] which is generally described, for example, in [233]. Thus, the search for the global optimum is ensured. Let us set forth the method as applied to the problem of opti­ mization of bodies of revolution for a general case of LIM. Below, digitization of the problem is carried out. The generator configuration for the optimum body of revolution is to be obtained in the class of piecewise linear functions. To that end, the region of variation of the independent variable and the function made dimensionless are divided respectively into n and m equal intervals of length Ax and A $ Ax = —, n

A $ = —. m

Thus a grid is obtained having nodes with coordinates vAx, /iA$ (y = 0 , 1 , . . . , n ; H = 0,1,...,m). If for each i' 1 ' = J A i ,

z' = 0, l , . . . , n

the following is known, §U«) = j , A $ then some broken line is thus defined which passes through the corresponding grid nodes (Fig. 2.1). Therefore, the broken line which represents the generator config­ uration for some body of revolution is identically defined by defining the numbers Expression (2.1) for body drag coefficient may be presented as $0'o)2

c^-^iyi) T

*

r(0

>=1 .j (. -_i )n

$0.) _ $0—i)

Ax

U) _

*


' + - — 7 T — ( * - z''"1': dx.

143

Chapter 4. Variational problems of optimum body configuration determination

144

Figure 2.1: Presentation of contour of a body of revolution as a broken line.

Upon t h e transformation, we get

cia = -^n p (i) + H w ( i . - ji-i) ■ (i. - ji-i) i—i

where

w(fc)

rmn

\ m J

(2.8)

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

145

Therefore, the original problem is reduced to finding integers j„ which give the min­ imum of functions (2.8). In such a case, a number of limitations must be taken into account. The condition for the non-decreasing of the function and the boundary conditions may be given by 0 < Jo < k < - - • < 3~e < ■ ■ ■ < jn - m. Moreover, it should be required for the surface of the body of revolution to be located outside of the cone having the same length and base section diameter ji>i~,

m n

i = 1,2, ...,n

- 1.

The necessary condition of convexity also has to be stated. It is realized as a require­ ment for the straight line which is an extension of any segment of the broken line to intersect the straight line x = 1 not below the point $ = r Ji-i <

ji{n - i +

:

l)-m

,

i = 1,2, ...,n

- 1.

n —i

To solve the problem on the basis of the dynamic programming methods, the multistep optimization process is used. At the i-th step (i = 1,2,..., n) for fi = ip(i),..., m the following values are determined:

G<;> = miniGir1) + g , G<0) = ~ n„(i), Ln = {v + n) -w(// - i/),

0(x)

.m n

i —

(2.9)

whereas exhaustion by v is conducted by allowing for the limitations Mi - 1) < v < mm(m,^n V

- -+ ^ n—i

m

), I

i = 1,2,... ,n.

(2.10)

As a result, the minimum value is determined for body drag coefficient C ^ m = Gj*' as well as ordinates $ ' J ' ' of the break points, r, ^(5(n,m), ^S(n — l,i5(n,m)), . . . , connected with the abscissas 1, (n —l)/n, (n — 2)/n, — The value of v providing the minimum Gt'' in problem (2.9)-(2.10) is denoted as S(i,fi). The parameters which appear in formulae (2.9)—(2.10) may be defined as follows: Gfi {i = 1,2,... ,n) is the minimum value of drag coefficient for a body of length i/n and base section radius fir/m, lulx is the drag coefficient for truncated cone side surface (nose and base radii are Tu/m and Tfi/m), Ge is the drag coefficient for a body nose of radius 9/m. Meanwhile it is assumed that all linear dimensions are divided by L and value itp^. is selected as a characteristic area for calculation of drag coefficients.

146

Chapter 4. Variational problems of optimum body configuration determination

It should be noted t h a t conditions (2.10) do not assure t h e search in t h e class of convex functions. Strong allowance for t h e condition of convexity results in t h e increase of dimensionality of t h e problem. It is worth to introduce it only when t h e solutions obtained without taking t h e condition into account, yield non-convex generators of t h e desired bodies. Should t h e convexity be unquestionable, it is expedient t o add t h e following con­ dition to economise t h e calculation time, >

S(i-

l,i/) 2

+jt

(2.11)

which must be taken into account when selecting t h e values of v. P a r a m e t e r s n and m are selected from the condition of achieving t h e predeter­ mined accuracy of calculations. Two m e t h o d s are generally used: either study of deviation of t h e solution for various n and m or comparison with the precise "stan­ dard" solutions for any particular cases. It is evident from relationships (2.9)—(2.11) t h a t t h e basic optimization procedure employs only information on the values of function F defining t h e specific LIM in (m + 2) points while there is no need in calculating t h e values in each step which is rather i m p o r t a n t as far as calculation time is concerned. T h e m e t h o d does not impose any restrictions to the model type; for different models t h e calculations are carried out using t h e same algorithm with varying u>(k) which are calculated in some outside procedure. This property features enabled to develop t h e universal program for optimization of bodies of revolution in t h e conditions of LIM.

2.4

Calculation results

Below are the results of the studying the effect of Reynolds n u m b e r , t e m p e r a t u r e factor and given form factor T of a body of revolution on configuration and drag of the o p t i m u m body. It is assumed t h a t 7 = 1.4. T h e configuration for the o p t i m u m body of m i n i m u m drag is shown in Fig. 2.2 a),b) for various Reo and tw. Decrease in value of Re0 (increase in gas rarefaction) at the fixed value of tw or increase in value of tw at the fixed Re0 results in the fact t h a t t h e o p t i m u m body generator passes higher. Bluntness radius pi is i m p o r t a n t and convenient for analysing characteristic of the o p t i m u m body configuration. T h e behavior of t h e radius is shown in Fig. 2.3 a), b) as against Re0 and tw. It is evident from Fig. 2.3 a) t h a t t h e per-unit value of bluntness (its radius divided by the base section radius) at the fixed value of Reo increases with rising t e m p e r a t u r e factor for any values r. At high r , this variation is negligible but when shortening the given body length while retaining its m a x i m u m d i a m e t e r , t h e variation of relative dimensions of t h e bluntness becomes noticeable and accounts for 11.6% at tw varied from 0.01 to 1.0 (Re0 = 1). W i t h increasing Re0 at t h e fixed tw,

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

147

Figure 2.2: Optimum body contour.

t h e per-unit bluntness radius decreases whereas t h e rate of t h e variation increases with decreasing r (Fig. 2.3 b) ). So far as flat bluntness is an essential element of the o p t i m u m body configuration, it is e x p e d i e n t to use t h e experimental d a t a [164] on t r u n c a t e d right circular cones for additional check of t h e model used. T h e expression for drag coefficient C°a for a t r u n c a t e d cone is given by

C°xa = (A2 + A1)a2

+

J

[(A2 - BJsin20

+ AlSm6

+

BJUnO,

148

Chapter 4. Variational problems of optimum body configuration determination

Figure 2.2 (continued). where 9 is the angle between the cone generator and the rotation axis, Pi

tan 8 1 -a'

For the cases when ReL = 0.92, 9 = 10°, a = 0.5, 0.35, 0.26 the calculations conducted by using this formula yield the values of C°a which equal 0.93, 0.85, 0.82, respectively, coinciding with the experimental data presented in [164] to an accuracy of the curves. Rer, is Reynolds number calculated over the body length, Re0 = 2rRei. The agreement between the calculated and the experimental values confirms the validity of the model for the bodies similar to the optimum ones.

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

Figure 2.3: Radius of the optimum body flat bluntness as a function of flow parameters.

149

150

Chapter 4. Variational problems of optimum body configuration determination

Figure 2.4: Optimum body drag coefficient as a function of flow parameters and the given relative body thickness.

Let us now consider t h e effect of parameters Reg, tw and T on t h e drag value for the o p t i m u m bodies. In Fig. 2.4 a), curves are given which represent t h e relationship between the drag coefficient for t h e o p t i m u m body of revolution, C ™ n , and Re0 at different values of tw. At increasing Re0, the m i n i m u m achievable value of drag C™ m descends monotonously while approaching, in t h e framework of t h e considered model, the limiting value which corresponds to the solution of optimization problem for the modified Newton model. Characteristic curves illustrating t h e variation of the o p t i m u m body drag coefficient C ™ " with respect to tw, are presented in Fig. 2.4 c). For all values of Re0, function C™n(tw) is increasing while t h e rate of t h e increase is rather high; t h e calculations show t h a t at Re0 = 0.1 and tw increasing from 0.01 to 1.0, t h e m i n i m u m drag coefficient for a body of revolution for r = 1 increases from 2.03 to 2.53, t h a t is by 25%. For t h e values of Re0 which equal 1, 10, 100, t h e drag increases by 27%, 12%, and 1 1 % , respectively. Therefore, t h e decrease of tm is t h e reserve for down in body drag in a flow. T h e variation of drag coefficient C™a'n with respect to T at different values of tw and Re0 is shown in Fig. 2.4 b)

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

b)

Figure 2.4 (continued).

151

152

Chapter 4. Variational problems of optimum

body configuration

0 Figure 2.4

(continued).

Figure 2.5: Ratios of drag coefficients for the optimum body and a sphere.

determination

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

153

whereas the variation of the body length at the fixed mid-section radius corresponds to variation of r. Function C™n(T) is the increasing one and as r —» oo tends to the drag coefficient value for a disk which is perpendicular to flow direction. It should be noted that the calculated value for the disk is Cxa = 1.8 at Reo = 73.4 and agrees with the experimental one from [164]. The variation of ratio £ of the optimum body drag coefficient (T = 1) to the corresponding characteristic C°a for a sphere with respect to Reo is given in Fig. 2.5. The hatched region corresponds to variation of £ as against variation of tw between 0.01 and 1.0. It is apparent that tm does not affect e significantly. At high rarefaction of the flow, the optimum body and a sphere are virtually equivalent as far as drag is concerned. In case of increasing Re0, the difference becomes more significant even considering the inevitable errors of the accepted model.

2.5

Study of powerlaw bodies

Powerlaw bodies of revolution, i.e. the bodies described by generator equation which is exponential term s

P+

$ = rxm,

r = ^ ,

■

$

* = y,

x

x =

-

take an important place in aerodynamics. It is explained by the fact that such bodies have optimum or close to optimum properties as far as drag and heat flux [11] are concerned, in flight at high supersonic speed in dense atmosphere. It triggered the study of aerodynamic properties of powerlaw bodies in the entire range of flight altitudes. It is convenient to present expression (2.1) for drag coefficient for powerlaw bodies as follows, C . = (A2 - B1)I1 + Axh + B,

(2.12)

where i

dz J (1+^jr")1'*'

I

a =

1 , r ■n

m =

v = 1,2,

1 -n , n

n < 1.

, 2.13)

It transpires from the form of the integrals that 0 < I„ < 1, which yields the estimate, #1 < Cxa < M + A2. The calculations were conducted to better than 0.1% accuracy. The accuracy was monitored both by diminishing the step of break-down of the interval of integration

154

Chapter 4. Variations! problems of optimum body configuration determination

Figure 2.6: Drag coefficient for powerlaw body as a function of flow parameters and the given relative body thickness; • — "precise" calculations [11], O — experimental data [164].

and by comparison with the results of calculations by analytical expressions for the cases when the integrals in (2.13) enable one to perform the calculation in explicit form. For example, for n = 3/4 we get

C

Bx

+

3(/l

2

-B

)(|-a

2

+

4 a

lna'

> ) + 2/l 1 a(0.66 5 - 2a 2 6 3 + 3a 4 6 - 1.66 5 ), a = -r, 4

As Re0 —> co, the model up to within Let us turn now Cxa(Re0), Cxa{tw),

1

a

6 = ( a 2 + l)1/2

calculation results coincide with those in [262] for t h e Newton a normalizing factor. directly to the analysis of calculation results. Curves of functions Cxa(r) for body $ = T Z 0 7 5 at different values of t h e rest of the

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

155

Figure 2.6 (continued).

parameters characterizing the body configuration and flight conditions are given in Fig. 2.6 a), b), c). As evident from Fig. 2.6 a), the values of Cxa monotonously decrease at diminish­ ing gas rarefaction. From Fig. 2.6 b) it appears that the effect of the tw on Cxa barely manifests itself for Re0 > 10. With diminishing gas rarefaction, Cxa(tw) increases noticeably while with tw growing from 0.01 to 1.0, Cxa increase from 2.07 to 2.63, i.e. by 26% (r = 1) for Re0 < 1. The shown regularity takes place at any values of T. The curves in Fig. 2.6 c) represent the relationship between Cxa and r at tw = 0.01. For the fixed mid-section diameter and decreasing body length (xj = Pf/r), drag coefficient increases tending to the value of A-[ + A2 as r —► oo which may be obtained from (2.12) and (2.13) by the limiting transition as fj. —> 0. The effect of r on Cxa is significant at high values of Re0 and essentially is dimin­ ished at higher gas rarefaction. As far as the body configuration optimization is concerned, the study of the effect

156

Chapter 4. Variational problems of optimum body configuration determination

C)

Figure 2.6 (continued).

of the exponent n on the body drag coefficient is a m a t t e r of interest. Plots of functions Cxa{n) are given in Fig. 2.7. It is evident t h a t functions Cla(n) have a minimum, however, in t h e vicinity of the m i n i m u m , the curve is rather gently sloping whereas t h e more clearly the m i n i m u m manifests itself, t h e less is t h e value of Reo. Shown in Fig. 2.8 is the region of values of n bounded by curves n = " - ( T ) and n = n + ( r ) and including inside the o p t i m u m values of exponent n. W h e n n varies from n_ to n+, t h e calculated body drag coefficient deviates from t h e o p t i m u m value by no

Section 2. Bodies of revolution of minimum drag in gas of different rarefaction

Figure 2.7: Comparison of powerlaw bodies with the optimum ones; — optimum bodies; # — "precise" calculations [11],

157

— ppowerlaw bodies;

m o r e t h a n 1% in a wide range of variation of p a r a m e t e r s Re0 and tw (Reo = 0 — oo, tw = 0.01 - 1). To analyze t h e "loss'' in drag when changing-over to body configuration optimiza­ tion in t h e class of powerlaw bodies, a comparison is m a d e between t h e o p t i m u m powerlaw bodies and t h e "absolutely" o p t i m u m ones by Cxa- In Fig. 2.8, t h e values of CXa f ° r t h e "absolutely" o p t i m u m bodies are shown by t h e dashed lines at t h e s a m e values of Re0, tw, T for which t h e curves were obtained representing t h e varia­ tion of drag coefficient for powerlaw bodies with respect to n. It is evident t h a t t h e difference between t h e values of Cxa for t h e o p t i m u m powerlaw bodies and the "ab­ solutely" o p t i m u m ones does not exceed 3 % at Re0 < 100 (tw = 0.01, r = 1) whereas this difference tends to grow with decreasing gas rarefaction and increasing r . More graphically it is illustrated by t h e curves given in Fig. 2.9, where C ™ n and C ™ " are drag coefficients for t h e o p t i m u m powerlaw and "absolutely" o p t i m u m bodies, respectively. For comparison, this figure displays t h e curve (dashed line) representing

158

Chapter

4. Variational problems

of optimum

body configuration

determination

1.0

0.5

Figure 2.8: Field of values of exponent in the generator equation which are close to the optimum ones (see text).

Figure 2.9: Relative loss in drag value when optimizing in the class of powerlaw bodies (see text).

Section 3. On the optimum configuration of penetrating bodies

159

Figure 2.10: Relative loss in drag for some powerlaw bodies. the per-unit difference by Cxa for a sphere and the optimum body of revolution. The curves presented in Fig. 2.10 enable one to estimate the relative deviation of drag coefficients Cxa for bodies of exponential configuration from their "absolutely" optimum values for some values of n at different Reo-

3 3.1

On the optimum configuration of penetrating bodies Specific character of penetration problems

In this section we consider the problems of simulation of a process of penetration (i.e. gradual immersion) of a hard body from the air into some medium (metal, soil)

160

Chapter 4. Variational problems of optimum body configuration determination

which is assumed to be a semi-space with non-existent boundary effects. P e n e t r a t i o n directs along normal to t h e plane dividing the media. Modelling p e n e t r a t i o n of small bodies (impactors) entering hard media at high velocities is discussed below. There­ fore, t h e gravity effect may be o m i t t e d which anyway is not of principal importance when solving t h e considered problem. It is assumed t h a t any outside forces also do not affect t h e penetration process so t h e process of body introduction into a media is defined only by its drag. T h e problems of body penetration modelling and especially t h e problems of body configuration optimization for such conditions essentially differ from t h e problems considered above and dealing with body motion in media (for brevity sake, t h e latter will be called t h e problems of flow around). Some basic differences are outlined below. A problem of flow around is essentially a part of a more general problem of body motion modelling when t h e motion may be considered as a sequence of stationary translational motions ( S T M ) . Such S T M ' s are discussed in traditional liquid and gas dynamics when interaction between a, body and a m e d i u m at t h e given law of its motion (in the present case, a uniform translational motion) constitutes t h e object of study. T h e body configuration optimization problems for its certain S T M are formu­ lated accordingly. More sophisticated motion may also be considered (as done, for example, in C h a p t e r 5), however, this does not affect the gist of t h e problem as long as the effect of a medium on variations in body motion p a t t e r n is not studies. Such an approach is valid for relatively slow process when the real non-stationary process may be presented as a sequence of stationary ones while the "stationary'' optimization problems are apparently considered only when optimization of the a d e q u a t e "station­ ary" stage or combined optimization of several stages have a pithy meaning. Such an approach is traditionally used as applied to t h e problems of search for t h e o p t i m u m aerodynamic configuration [278]. A different situation takes place when modelling t h e body penetration [263]. In this case, fast processes with unknown body motion occur while it is clear beforehand t h a t t h e motion is more sophisticated than the uniformly decelerating one since in t h e initial stage of penetration the surface of body contact with a m e d i u m varies, t h a t is t h e motion does not take place under the action of a constant force. As far as t h e objective functional is concerned, the integral parameters (at first, penetration depth) characterizing t h e process as a whole rather than its separate stage are of interest in this case. T h u s , when dealing with body penetration at high velocities, t h e study of t h e problems of search for their o p t i m u m configuration involves t h e need to consider t h e m together with motion modelling.

3.2

Modelling of body penetration into media on the basis of LIM

Assume in cylindrical coordinate system shown in Fig. 3.1 t h e configuration of a convex three-dimensional body is defined by p = $(x,6). To retain t h e form of

Section 3. On the optimum configuration of penetrating bodies

161

Figure 3.1: Regarding the problem of modelling of penetration of bodies into media. the formulae usually used to calculate body drag, we assume that isotropic mobile medium passes over a still body. The body features two mutually perpendicular planes of symmetry passing through axis x (that is, axis x is a symmetry axis) and its flat base section is perpendicular to axis x, L is a body length. The body motion in the penetration process occurs along axis x beginning from moment T = 0 when the body nose touches the medium. Upon entering the medium to depth h, the part of body side surface which is located in region x < h, = min(L, h) is affected by the medium. A flat region on the boundary of the medium which is limited by the contour of contact between the medium and the body is denoted as (a). Let us introduce the following localized model of the effect of a medium in some

162

Chapter 4. Va.riationa.1 problems of optimum

body configuration

determination

point of the body surface, „2

dE

?°°"

P (0)

, „p(l)

dS

where dF is the force acting on the surface area dS at some moment characterized by medium velocity v ; a > 0 is a constant factor; T° is determined by equality (1, 1.3) at v ^ = v°; expressions for CF are of the form (1, 1.1) but functions fi p , ftr appear with superscripts. Then, for some moment characterized by penetration depth h, the expression for projection on axis x of the force of the medium effect on the part of the body which is in contact with it may be given by Fx=P-?fcV

+ aC?\

(3.1)

where

c^\h) = J jn{3(t)-t2 dx de > o, whereas functions ft^,,
(3.2)

ft'0) = 1, ft<°' = 0

(3.3)

as well as

then the following takes place

Cf\h) = a(h) =

I

- J^2{h,0)d6 at h
a(L)

(3.4)

at h > L,

-J&(h,e)d6 at plane h L, the model is reduced to the one described in [287, 288] (a is dynamic hardness of a medium) and substantiated by the results of a wide series of experiments with rigid conical penetrator conducted by varying their configuration in a wide range and at penetration velocities of the order of 100-1000 m/s. The Newton model also formed the basis of theoretical and experimental studies [78] devoted to simulation of penetration of star-shaped bodies into metals and soils.

Section 3. On the optimum configuration of penetrating bodies

163

Let us turn back to the general case. The equation for motion of surface boundary (ff) in the body-axis coordinate system has the form: (3.5)

mh + Fx = 0, where m is the body mass. Upon substitution from (3.1), we get h + ipi(h) ■ h? +
-Cl°\h)>0,

0.

(3.6)

The initial conditions are given by h(Q) = 0, ft(0) = w0, where |t>o| is the body velocity immediately before the start of penetration. By substituting h2 = /JI(/(), the equation (3.5) is reduced to the linear one with respect to hi(h). Allowing for the initial conditions, the solution may be presented as follows,

[h(h)f =

1 G(h)

n

»o- JMP)G{P)df}

(3.7)

where G{P)=exp(2j
(3.8)

From (3.7) and (3.8) the equation is obtained by determining H, xl>(H) = 0, where 2

2 4>(W) i>(W)==MW) MW) - -v u00, ,

w

1>l(W) = Jtp0{p)G(p)dp.

(3.9)

Chapter 4. Variations! problems of optimum body configuration determination

164

Let us show that equation (3.9) always has a single solution; this fact among other things, may serve as one of corroborations of correctness of the model used. Below, the behavior of function ij>{W) is studied. At fairly high W > L

i>(W) = f(L) + i>{W), w

t(W) = j
0

= tp0(L) ■ G{L) ■ Jexp 2ipi{L)J do\d/3 0

) G L)

= ^] [

L

{exp [2^{L){H -L)]-l)

(3.10)

whence it follows that 0(+oo) = +oo. So far as

0(0) = -vl < 0, j'(W)

= 0,

the formulated statement is proven. From (3.9) it follows that VQ(H) and H(VQ) are increasing functions. Therefore, formulae for calculation of H may be presented as follows by allowing for (3.10), i/.fVo) H = L+

2*

at v0 < y/ML),

hA™*™LKi}

**> ^

(3-H)

(3 12)

-

whereas the upper integration limit, when calculating the function 0 a in (3.11), is always less than L. Basically, formulae (3.11) and (3.12) provide the basis for defining and solving the variational problems of configuration determination for bodies ensuring the maximum penetration depth. However, the complex character of relationship between H and a body configuration significantly complicates their study. Considered below is the im­ portant particular case of conical bodies where interesting and demonstrative results are obtained. The case is also of special interest because the study and validation of the original model [287] were carried out for conical bodies.

Section 3. On the optimum configuration of penetrating bodies

3.3

165

Configuration optimization for conical three-dimension­ al bodies

Let us consider the class of bodies for which function 0(x, 6) has the form: • ( * , * ) = kxn(9),

7,(0) = 1,(2*) = 1,

where k is the constant parameter. If (3.3) is valid, then h2Ii[n] at h < L,

9iW

(3.13)

L2Ii[rj\ at h > L,

where functionals Io, A are given by

/oW = gioW, AM = ^ A M , 2ir

2ir

Jj00== Jr,*de, frj2 dd, A^/og^f A ^ / n g y ^ f dd n

(3.14)

n

whereas in compliance with the formulae appearing in Section 2 of Appendix

<<°> = 7,*, 4°) = [(JbV + i y + -/' 2 ] 1 / a ,

t(o)

_

*JL. 71 _» _= *L lL

22

Te-

The expressions for functions G, $i are essentially simplified and assume the form:

Go(0) G(/3) =

at /? < L,

G0(i)exp[2Z,2/1(/?-I)]

a.tp>L,

Go(^) = e x p ( | / 1 / 9 3 ) , Go(^) = e x p ( | / 1 / 9 3 ) ,

Thus the following formula is obtained for the maximum body penetration depth.

2 for the / 3 maximum body penetration depth, Thus the following formula is obtained

M

# =

f W'

AH A

2,

vl

(1hvl\

a t Do < DQI

(3.15) (3.16)

166

Chapter 4. Variational problems of optimum body configuration determination

where X(W)

= W-1 ln(W + 1),

•H^Mf^-H

,1/2

}

(3-17)

whereas the values of H are calculated by using relationships (3.15) and (3.16) at vo = VQ coincide. Let us show that function x(W) decreases at W > 0. To that end, the following relationships are used, X{ Xl(W)

w2(w + iy

'

= W - (W + 1) ln(W + 1),

xi = - l n ( ^ + 1).

So far as xi(W) < 0 at W > 0, Xi(W0 decreases and its values Xi{w) < Xi(0) = 0. Therefore, x'(W) < 0 at W > 0, i.e. x(W) decreases. In the problems of body configuration optimization, it is more natural to predetermine density po of the material the body is made of instead of its mass. Thus, the mass is calculated through p0 and volume V as follows, m = p0V = p0ja{h)dh

= ^^J0[v).

(3-18)

0

By substituting into (3.14) we get

*-,£•

7

(3 19)

-

'=2PLVW'

where

7M=^j.

(3.20)

Upon substitution of (3.19) and (3.20) into (3.15)—(3.17), the following is obtained: ( 2

bo

\J + J X(W) at J < bu H L ~ 1/3 {[hx(hl)} l[*»X(V)] 1/B a t / > 6 1j , where h = -rx 63

(V)>

h

?= — . a

6

foPoo

3 = ak2 '

Section 3. On the optimum configuration of penetrating bodies

167

The fact that 62 and 63 serve as similarity parameters should be noted, that is at the given body cross-section configuration, they determine the relative penetration depth. Considering the penetration depth H as a function of J, the conclusion is drawn that the maximum depth occurs at the minimum value of integral / . The solution of the relevant variational problem yields the desired value of the maximum relative penetration depth Hmax/L. Should the requirement be specified concerning the body having the predetermined length and volume (or the base section area), it appears that the body featuring the minimum drag when moving in the same medium at constant velocity ensures the maximum penetration depth. This assertion is proven below. By allowing for (3.1) and (3.4), the expression for body drag X may be given by X = gooCll\L)

+ ao-(L),

qoo

=

*!!&,

where v^ is the constant velocity of the free-stream medium. The formula may be transformed by accounting for (3.6), (3.13) and (3.14) to the following form: X = y [
(3.21)

Should the body volume Vb be given and allowing for (3.18), this condition may be presented as follows, ^j-Mv] = ^0

(3.22)

and functional (3.21) is equivalent to the following one, X = ^(qcaI[r,]

+ ak2).

(3.23)

Therefore, the same function yields the minimum value of body drag X and the maximum value of penetration depth H. Should the base section area be given, P L 2

i r i

then functional (3.21) is transformed to the form: x

= % fo-'M +ak2)

which also confirms the validity of the abovementioned property in this case. It should be stressed that up to now the form of functions ffi1', fiW has not been particularized and the analogy obtained for the general case essentially sim­ plifies the search for the optimum configuration of penetrating bodies which should already be conducted for the specific model. In particular, in the case of the Newton model (3.2)-(3.3) suggested in [287, 288], the solution of the problem is known from aeromechanics [152]. Bodies of star-shaped configuration are the optimum ones.

Chapter 5 Generalization of LIT 1

Generalized model

In the case of translational motion of a body, velocities of all its surface points are equal and such velocity is naturally included into the list of global parameters while the quantitative characteristics of the effect of a medium in the given point of a body surface are considered in the framework of LIT as depending only on the angle between the velocity vector and the normal to the body surface. Retaining the requirement for independent effect of a medium on different surface elements for nontranslational motion, the local velocity of a surface point (or that of a medium if the situation is considered where a medium comes to a still body) should also be included into the local parameter list. The abovementioned independence of the effect of a medium in surface points is interpreted so that the local velocities may be calculated from kinematic considerations assuming that different surface elements interact with the medium apparently on their own. When generalizing LIT for the case of nontranslational motion of a body, we mean, of course, the approximate allowance for nonstationary condition. The nonstationary motion is interpreted as a sequential change in a body position while at any moment the situation is defined by the instantaneous values of the angles defining the body orientation, by angular velocity values, etc. In other words, "locality in time'' is postulated. Such an approach is widely used in mechanics [70, 74]. The following arguments will be more illustrative should a body moving in a still medium be considered instead of a still body and incoming medium. As applied to this case and with further analysis in mind, the modified formulae (1, 1.1), (1, 1.3) are given by dF = <7oo[-np(a, t)-n° + ttT(a, t) • r ° ] dS, o v2

v° - t ■ n° <=

v

co"

n

.

r

A — ^

'

9<x>

~

—

2~'

( L1 )

where dF is the force of the effect a medium has on a unit element of body surface 169

Chapter 5. Generalization of LIT

170

having area dS; v,^ is t h e velocity of translational body motion; n ° is t h e external normal to t h e body surface; superscript "zero'', as before, denotes unit vector; t h e sign in t h e expression for T ° and in front of fl p is inversed as a result of motion reversal. Let us consider t h e problem of generalization of model (1.1) with m i n i m u m com­ plications for the case of different velocities for body surface points. Essentially, such generalization is reduced to transferring t h e number of p a r a m e t e r s from t h e "global" to t h e "local" ones. D y n a m i c head q^, used for transition from physically natural p a r a m e t e r "the force" to t h e more convenient dimensionless p a r a m e t e r "force coefficient", is a fullvalue element of t h e interaction model. Therefore, v^ should be s u b s t i t u t e d for v in t h e formula for calculation of t h a t parameter, PooV2

9oo —► q =

2 In t h e expression for t h e local angle of "incidence" between t h e m e d i u m and t h e body surface, local velocity v should appear instead of ««,, t h a t is t = v°-n°

(1.2)

and naturally,

,° = -

V

°-^°.

(1.3)

T h e problem is more complicated for t h e "global" p a r a m e t e r s . Some of t h e m (for example, similarity parameters) may be left among t h e "global" ones. Anyway, this problem cannot be discussed in the framework of t h e LIT generalization since it is a characteristic for specific models. However, we must draw a t t e n t i o n to t h e fact t h a t velocity v^ frequently appears among t h e "global" p a r a m e t e r s only as a result of translational motion and is "local" by the essence. T h e characteristic example of such situation is t h e model of free-molecular flow around ( 1 , 2.5). T h u s , in general, it is unacceptable to limit oneself to allowance for t h e effect of t h e value of local velocity v only through dynamic head and v should be introduced as an independent variable into functions Q p , fi T . A similar situation also occurs as applied to t h e r m a l characteristic. Therefore, t h e generalized "local" model may be presented in t h e form

d¥ =

^cFdS,

cF = - n „ ( a , ^ , t ) n ° + n T ( a , ^ ,

<)T°,

dQ = — - — cQ dS,

CQ = n«,fa, p i V

(1.4)

171

Section 1. Generalized model

where v* is some characteristic velocity; Q p , ClT, £IQ are the functions defining the model; a is the vector of the "global" parameters. In case of translational body motion, it may be assumed that v = v^,. Then variable v/v* = v^/v* is entered to the "global" parameter list and model (1.2)—(1.4) is transformed into the "classic" LIM. It should be noted that the above analysis is of illustrative nature. It could simply postulate the model of some form for nonstationary body motion by specifying the requirement that in particular case it would become "stationary". As far as the above deliberations are concerned, the importance of correct derivation of local velocity from a number of "global" parameters should be stressed although it is frequently difficult to accomplish for empirical models and usually generalization has to be carried out by substituting v^ for v in the expressions for t in (1.2) and q^. Of course, for empirical models as well as for any other, theoretical constructions cannot replace the traditional methods of verifying their adequacy, in the first place, the experimental ones. In principle, model (1.2)-(1.4) is a generalization of LIT for the case of nonstation­ ary body motion and may apply as a basis for studies of the problems of mechanics. However, the analysis shows that a number of behaviors for body integral charac­ teristics determined, for example, for the case of free-molecular flow around, cannot be reproduced in this model. On the other hand, the experience of studies in the framework of the classic LIT shows that many regularities are the consequence of fairly general representations for the functions which define it rather than for specific analytical expressions. It appears that such situation also occurs in the considered case. The analysis shows that many known local models belong to the following subclass,

""(£'*) = A"(u)' "=£)•' M^<W(^) M ^ W G O22--u2Mu) " 2 ^'

(L5) (1.5)

where A p , AT are the functions which define the model. It is easy to see that v o "» u = — • n = —, v' V

^ (I.) '...^i^.i,

(L6)

K \v'J v' v* ' where v„, vT are the normal and tangent components of local velocity v, respectively. It should be noted that models (1, 2.1) and (1, 2.5) describing respectively the effect of supersonic flow of "dense" and rarefied gas in free-molecular regime and fea­ turing the fairly high level of theoretical validation are reduced to form (1.5) assuming

172

Chapter 5. Generalization

of LIT

Figure 2.1: Coordinate system and notation.

t h a t in t h e first case v* = v^ while in t h e second case, v* is t h e most probable velocity of chaotic molecule motion in a flow.

2 2.1

Calculation of integral characteristics for bod­ ies in combined motion Problem definition

Let us now calculate integral characteristics for bodies of revolution in nontranslational motion t h a t is, when at any m o m e n t , rotation at low angular velocity u> takes place along with translational motion at velocity v^,. Assume t h a t radius vector for body surface point in body-axis coordinate system xj x2x3 shown in Fig. 2.1, is given by r = $ ( / ? ) x i + pcos0x°

+ />sin0x°,

where x? (t = 1,2,3) are t h e unit vectors of the corresponding axes, function $(/>) defines t h e general configuration, while $ ( 0 ) = $'(0) = 0,

*'(J?) < oo,

* ' ( » > 0,

0 < p < R,

#" > 0 ,

Section 2. Calculation of integral characteristics for bodies in combined motion

173

where R is t h e radius of t h e m a x i m u m mid-section perpendicular to axis m%. T h e system of axes is selected so t h a t t h e vector of translational motion velocity Voo lies in plane xiXi forming angle TT — a with axis X\, Voo = — Uoo cos a x? + Voo sin a x°

(2.1)

whereas, for definiteness, we assume t h a t 0 < a < TT/2. T h e velocity of a surface point is a combination of two c o m p o n e n t s , 3

v = v00 + u > x r ,

u>=^WiX?,

(2.2)

where w; are t h e components of angular velocity in body-axis coordinate system Xl

X2X3.

Converting from dimension p a r a m e t e r s t o dimensionless ones, dS

— it

—>dS,

v

v^

— —► v , V

— V

—► Voo,

— V

uR

>w

and relating linear dimensions t o R, the relationships for local force and m o m e n t coefficients m a y be presented in t h e following form by using formulae (1.4)—(1.6), cF = A„(«) • n ° + A„(u) • v ° , m, DC = r x cF,

(2.3)

vhere n ° = ( - x ? + $ ' cos 0 x ° + $ ' sin 0 x f ) / / * i , u = T/fH, T(u

,p,0)

/i, = v V 2 + 1,

= —H2 sin 0 OJ2 + (/^3 + ^2^3) cos 0 + Uoo cos a,

H2 = $ $ ' + />,

|<3 = Woo sin a ■ $ ' ,

A B (u) = - [ A p ( « ) - u - A r ( « ) ] ,

A„(u) = - A T ( u ) .

(2.4)

Upon transformations on t h e basis of (2.1), (2.2), (2.4), t h e expressions for integral characteristics (2.3) are given by,

C,,(w ) = x? / / CF dS = 1 / A,-(w , p, 0) dpd0, (S)

W)

mx, (w ) = x° / / m , o c dS = JJ Bi(u,p,0)dp

d0,

Chapter 5. Generalization of LIT

174 where Ai/p

= - A „ ( « ) + ^iA„(u)(—I*,*, cos a + p sin 0 u>2 +

A2/p

= $ ' cos 6 A„(u) + A„(u) ( $ ^ 3 - p sin 0 wi + u<x> sin a ) ,

A 3 / p = $ ' s i n 0 A n (u) + pl\v(u)(pcos9ui

pcosOu>3),

- $u; 2 ),

(2.5)

Bi = p(A3 cos 0 — Ai sin 0), £ 2 = psinfl A\ - <J>v43, Z?3 = $ A 2 — pcosO Aj.

(2.6)

T h e exposed part of t h e surface (S) is characterized by the condition T>0,

(2.7)

where (a) is t h e projection of ( 5 ) on plane x 2 £3. It should be noted t h a t function T does not depend on u»i, i.e. t h e value of the component of angular velocity along the body s y m m e t r y axis does not affect the position of t h e boundaries of the body exposed area. Moreover, function Ai in (2.5) also does not depend on u>i and consequently, CX1 does not depend on u)\. Our main objective is to calculate the rotary derivatives of t h e first and t h e second order, t h a t is derivatives of force and m o m e n t coefficients with respect to components of angular velocity.

2.2

The case of body rotating about its axis

At first, let us consider the case when rotation occurs around axis x\, t h a t is u>2 = u>3 = 0. Then formulae (2.4)-(2.6) are simplified and t h e expression for calculation of IC assume t h e form: CX1 = - 2 / / p[An(u)

+ Voo cos a Hi Av(u)] dp d8,

(
CX2 = 2 Up [<E>'cos0 A„(u) + Voajii sin a A„(u)] dpd9, (a/2)

Cxz = 2uii II p2 p\ cos 6 A„(ii) dpd6, (S/2)

mxi

= 2u>] / /

wm

p3p1Av(u)dpd8,

Section 2. Calculation of integral characteristics tor bodies in combined motion mX2 = —2uii / / p2 fix $cos6

175

Av(u)dpd0,

("72)

mX3 = 2 / / p[fj.2CosdAn(u) + iiiVoo($sma

+ pcosacos0)Av(u)]dpd0,

(2.8)

("72)

where (ff/2) is half of area (CT) which is located in region xi > 0; it is defined by the inequalities, 0 < 6 < arccos [max(—1, — cot «/#*)], 0 < p < 1,

(2.9)

which transpire from condition (2.7). Superscript "zero'' indicates that u>2 = ^3 = 0. The formula for calculation of u in the considered case is simplified and assumes the form: u = (p3cos6 + UooCosa)/^].

(2-10)

The properties of evenness and oddness for some components of the integrands and the symmetry of the domain of integration about plane X\ x?, are used hereafter. For some functions ip and angle /? the normalization of those properties is expressed as

J\p(0)d6 = 26- I\p{6)d0, -p

0

where 8 = 0, if function rp(0) is odd, and 6 = 1 if the function is even. As transpires from formulae (2.8)-(2.10), in the case of body rotation about its axis of symmetry, not only CX1 does not depend on angular velocity but also CX2 and mX3. For the case of free-molecular gas flow, this result was established earlier in [171]. Rotation about a body symmetry axis does not affect the position of the center of pressure. The apparatus for obtaining the invariant relationships (Section 2, Chap­ ter 2) may be generalized as applied to rotating bodies. It also follows from formulae (2.8)-(2.10) that when there is no tangent force component with respect to the body surface (A„ = AT = 0, A„ = Ap, CX3 = mXl = mX2 = 0), the effect of rotation about the symmetry axis is defined only by function AT.

2.3

General case. Basic relationships

Whereas at ui2 = ^3 = 0, the exposed area boundary appears not to depend on u^, in general this is not true and the main difficulties result exactly from this factor. Since the general LIM class is considered, it would be rather difficult to substantiate the

176

Chapter 5. Generalization of LIT

possibility to adopt any assumptions simplifying the problem. We shall try to study the problem in "precise" definition. Let us consider some part of body surface between sections by planes x\ = a.\ and x2 = a2 (a,\ < a2). That part is called "fully exposed" if there is no aerodynamic shadow on it, so that every where v ■ n° > 0. If on every circle which is the intersection of plane x1 = a, (ai < a, < a 2 ) with the body surface there are both exposed (v • n° > 0) and unexposed (v • n° < 0) points, then such area is called "partly exposed". At sufficiently low values of w, three basic types of exposed area structure may be highlighted. At cot a > $'(1), the side surface is fully exposed. At $'(0) < cot a < $'(1), fully exposed area changing over to the partly exposed one takes place at the vicinity of nose. At cot a < $'(0) only partly exposed area of side surface is present. In the last case (high angles of attack), which is considered in detail, the exposed area (c) is given by 0M(u,p)

<6<6M(u>,p), 0
(2.11)

whereas functions $"• defining the area boundary satisfy the equation u = 0, i.e.

r[«W(w,p)i = o, „ = i,2. It should be noted that in general, area (a) is not symmetric about plane X\ x2 and its boundaries depend on u>. This general case differs in principle from the case when &2 = <^3 =

0.

Let us write the formulae for differentiation of double integral of some function allowing for definition of integration domain (er) in form (2.11),

d n,, „. „ r f * i . i M -JJ^,P,o) dpde = jj-dp( 2 2

}

• ^(-lr/^w.p.^w.p)] "=1

x

d2

0

^-^dp

-Jjnu,P,o)dpdo =JJrr +_ =1

2.12

crtf Jj^

dbJlcdulj

dpde

■dq(u>,p,QW) ggM dujj

duk

111

Section 2. Calculation of integral characteristics for bodies in combined motion 2d9{u,p,^))

80 dV(w,p,QM) ggW

dwj

dix>k + .*(

w

dwj /)2fl(") i , p , f l ( " ) ) . ^ - dp

dwk

(2.13)

t h e first c o m p o n e n t s on t h e right of formulae (2.12) and (2.13) deal with variation of t h e integrand function while t h e ''additional" ones, with variations of boundaries of t h e integration region. Similarly, rotary derivatives may be presented, pi — "(~'z< _ pi

_

Cjk

■ i

Cm; = "3— ULOj aw,-

\pi

= Cfk + ACf,

a Qr,

dujduik m

i

M/ + AM/,

OUJjOUlk du/jdujk OU>jOU>k

= M1^ + AMf,

where A denotes correction due to variation of boundaries of ( = 0 which is indicated by zero superscript at t h e relevant p a r a m e t e r s . F u r t h e r study provides t h e sequential use of formulae (2.12) and (2.13) for $ = Ai and $ = S , (i = 1,2,3), whereas t h e expressions for Ai and 5 ; are yielded by formulae (2.5) and (2.6). It involves fairly complicated transformations and t h e relevant calculations are not given below. T h e discussion is restricted, as a rule, to t h e major issues.

2.4

General case. "Main" components of rotary derivatives

R o t a r y derivatives are calculated at u = 0 . Therefore, area o used to calculate t h e double integrals in (2.12) and (2.13) is symmetrical about plane X\,x2 and it is convenient to change over to integration with respect to area (
c\ = jj Aj dPde = 2 jj A{,, dp de, (#)

k

(S/i)

ci = JJ A? dPde = 2 JJ A*{ dp de,

178

Chapter 5. Generalization of LIT

' ~

\ Ft, . / \ OUlj / IjJ—o

—

J /.iu>=o —n \au>jOijjk/ \ niiiifliiit.

*• "*" ■ " • • « >

= MH + A^i

(2.14)

whereas hereafter one and two asterisks denote odd and even components, respec­ tively. Let us use the formulae, (%£=) V ou>j 1 u> =o f Q'AW \ dwjduj ) u =o

=McosO-Sism9)-k'w,

l4(4k H =

sin2

Pi'

8 + 4h cos2 0 - Si% sin 0 cos 6) • A'w, u> = n,u,

where A' = .„

=

dA4«(w,p,0)] rfu rf2Am[M(w,/>,0)] du2

Then, the expressions for Alit A,,;, A3ti, A,,,- may be obtained from (2.5) by direct differentiation. The resulting formulae are as follows, A

'.\ = SJ2psm0{fi4An + p,2vx cos a A'v + ntpAv),

^ . 2 = S2psin8( —p4$' cos8 A„ — ^v^sina

A'v — fiipAv),

A{3 = 6J3p4$'p sin 8 cos 8 A'n, A J „! = -<5^/9cos 0(^4 A'n + fi2vx cos a A'u + fiipAv), K,2 = 633p{p4$' cos2 8 A'n + //2i>oc.sinacos6> A'v +/*i$A„), 4 U = -*2/9(^4*'sin 2 0A^ + /i 1 *A„) + ^ 1 / ) 2 c o s 0 A „ , AJ,i = <5^3/9sin0cos0(//2A" + /i 2 ^ 4 « 0O cos a A" + 2p,2pAv), Mk2 = 6ik3psm 9(-fil®' cos2 0 k'^ - fii^Voo sin a cos 8 A" — 63l3p2p2 sin 0 cos 0 A'„,

fi2$A'v)

179

Section 2. Calculation of integral characteristics for bodies in combined motion 2 2 2 e{u\V + 2/z Aikh3 = 632kppsin 9kl 0 A^ + 2p22$A'J $k'v) S\nd{p 4Vsmsin

+ 6ikpn\& sin 9 cos2 9 A" - <5j[W 2 sin 0 cos 0 A;, Mil = ~p{nlK

+ P2/*4»oo cos a A; + 2/i 2 /»Ai)(*j i sin2 9 + 4" cos2 0),

^4«2 = *2*/> sin2 0(/i2 cos 9 A" + PiP^v^, sin a A") + 63 />cos0(/i 2 ^ 4 u oo sinacos0 A" + 2p2$h'v + p4$'cos9 2

A")

2

+ ^>2p sin 0A'l,, A(t 3 = ^ p c o s 0( M 2 $' sin2 0 A* -

M2$A'V)

+ 6\k3p2p2 cos2 6 k'v,

where 6l = 1 if j = k = v and £j* = 0 if otherwise; £j* = 1 if j : = f and fc = p or j = p and A: = J/; Sjj*, = 0 in the rest of the cases; <5£ is the Kronecker symbol. A simple analysis yields the following list of nonzero values: CJ, Clk: C\, C\, C\, A2 A22 /^33 fi\2 _ Al\ All Azi A\3 _ A?,\ /Oi23 _ /Oi32 ° 3 ' °1 ! °1 > °2 — U 2 o ' l 2 i l , 2 i ° 3 — '-/3 i ° 3 — 0 3 •

The expressions for M/ and Af/ may be presented in the form similar to (2.14),

Mj = 2 11 Bl„dpd9, (Si*) k

Ml = 2Jj Bitidpdd, where Biti

= 8}p{A{,3 cos 9 - Ai2 sin 0) +

Sf(pAilSm9-Al3)

+ 6f(*Ai,2-pcos9

Al„)

and formulae for B3,fi are obtained by substituting superscript j for pair jA;. The following non-zero values of Mk and Mf are obtained by an analysis: M], 2 M , M\, M\, Ml, M]2 = M21, M 2 3 = Mf 2 , M\2 = M\\ M 2 3 = M 3 2 , M312 = M 2 1 ,

if22, M33 2.5

Corrections resulting from the exposed area modifica­ tion

Let us now calculate the corrections related to the dependence of exposed area bound­ ary on U3 . The calculations are carried out by formulae (2.12)—(2.13) and involve the

180

Chapter 5. Generalization of LIT

following expressions for derivatives of function 6^ T = 0: 0(2> = 0o(p),

0<»> = -e0(p),

=0

d28M{u,p)

=

which may be obtained from

e0 = a r c c o s ( - cot Q / $ ' ) ,

eijj>)+(-!)"«♦(/»:

= 4w+(-irCw.

duijdu>k

(2.15)

where V0++ —

^

°2>

^oto^; 3i

A*3/*5 ,2

C = - ^ S ^ M^* - ^(2$'2 - cot2 o)^*], "^1

*2*3

^7 cot 2 a ( 5 ^ ) ,

= \ / $ ' 2 — cot 2 a .

/'5

(2.16)

It is also convenient to represent functions A, and 5 , as a sum of even and odd components,

A, = i . ! + i..,, 5, = 5.,+ 5..,, where A,i=0, A,2 = 0,

/ 4 „ i = - p ( A „ + /iit; 0 0 cosQ: A„), X „ 2 = p(<&'cos#A„ + / / 1 u 0 O s i n a A„),

i . 3 = p$'sin6> A„, B,i

= pcosQ At3 — psin9 5.2 = psin0A,„i - * i ,

5,3=0,

A..3 = 0, A+,2, 3

,

5»»i = 0, 5 „ 2 = 0,

5 . . 3 = $A»,2 - /3COS0 A . , ] .

T h e n , considering A*;, /4»»;, 5 . , , 5«,{ at ut = 0 as functions 9 (p is not marked as a r g u m e n t ) , the expressions for corrections AC;, AM- with allowance for (2.15)—(2.16)

Section 2. Calculation of integral characteristics for bodies in combined motion

181

and properties of evenness and oddness may be written as

AC? = J {[A.i(0o) + A„i{e0)] ■ [ < 4 + flj+] 0

- lA.i(-e0) + A„i(-80)] ■ [«£++ - eU}dP i

= 2 J[A.i(e0) ■ 930++ + A.,;(0o) •flj+]dp, 0 1

AM! = 2 J[B.i($0) ■ «j++ + £„.-(«„) •fljjdP0

More illustrative formulae may be written for nonzero corrections, AC? = 2ri cot a h A„(0) + 2cot 2 a 72 A„(0), ACf = 2T? cot2 a h A„(0) - 2 cot a J2 A„(0), A C 2 = - 2 T ? ( / 3 - cot2 a /i)A„(0), A M 2 = 2(J, - cot 2 a J 2 )A„(0), A M | = 27?(J3 - cot2 a J 4 )A„(0) + cot a ( J : - cot 2 a J 2 ), A M | = 2 cot 2 ar]J4 A„(0) + cot a(cot 2 a J 2 - J 4 ), where

i2 = JtW2 + Up,

h=jidp, 0

0 1

r) = (v00sina)-\

t W

^5

I

dp,

$V*'2-cot2a'

J^f^dp, J u 0

I3 = J£*'2 o

J2 J=* ' u J^dP, 2

J

,2

J 3 = / — - dp, J us

o

1>

J4=

* * ' * »5

„„2

-r~- dp J 9' 'us

(2.17)

182

Chapter 5. Generalization of LIT

J $* >%5 Therefore, t h e expressions for rotary derivatives of t h e first order forces a n d m o m e n t s have t h e following structure:

C1, C2, m3 : ( : ) ■

Ci, fh{, fn\ : (!)■ where symbols "plus" and "asterisk" refer t o nonzero (in general) values of rotary derivatives, whereas t h e "asterisk" indicates t h e nonzero corrections related to al­ lowance for changes of exposed area boundaries. Corrections AC/'*, AAf/* are yielded by t h e second integrals in expression (2.13) calculated respectively for \t = A,- and * = B,at ui = 0. By allowing for symmetry properties, for A C / , t h e following formula m a y be obtained, I

Ac/* = 2 / [ek0++Ai, + flj++yi;,. + eh0+At., +

ei.Al,

0

+ 2(«* 2(«*++flj flj++ «£++C+)4.,C+)^.,- + 2(«o* 2(«o\++ «J++ %AMli ++ + «5 +*o ++ + OoAML + <4i,,- + < i . , « ] ^

(2.18)

while t h e formula for A M / is obtained from t h e previous one by substituting symbol A for symbol B as follows,

A* = f ^ ) '

\d6Ju=o

'

\d9Juma

= M,+Al„ = B!i + B!.i,

Ki = P7P(K +/»i«oo cos a A^),

A»,!=0,

^ 2 = -/»(*'AR + /
if, 2 = 0,

if 3 = o, i ! , 3 = p*'( cos M„-/j7sin 2 6>A„), B.«. = 0 , i. 8 . 1 =/»[(^.3 + ^ . 2 ) « » » + ( ^ 2 - ^ 3 ) « ' » » ] .

Section 2. Calculation of integral characteristics for bodies in combined motion 0,

Bt

Bl2

Bi3 = $K ^7 =

2

183

= p(A,*i cos^ + AU sine) - cMt.3,

- p sin 0 A., i - p cos
flf.,

= 0 (2.19)

Pi

When substituting expressions (2.19) into (2.18) and similar expression for Mj , it should be assumed that 8 — 90. Formula (2.18) and the similar one for corrections to moment characteristics may be transformed and presented in the form similar to (2.17), however, we shall not write the relevant awkward formulae but only the matrices illustrating the result of the analysis,

6?: /^tjk

o jk

o jk

0 0 0 0 * 0 0 0 *

5

0 0 * 0 0 * * * 0

C?:

0

+

+

* 0

0 0 •> *

0

+

+

* 0

0

mJ3

:

0

0 0

+

Allowance for changes of exposed area boundaries does not result in additional nonzero rotary derivatives of the first and second order. The comparison with the results of [70, 171, 172, 295] dealing with free-molecular flow around gives evidence to coinciding lists of nonzero derivatives. One of the essential issues which deal with consideration of the general class of LIM must be stressed. When varying the conditions of interaction between a medium and a body surface in a wide range, a number of rotary derivatives retain zero value.

Appendix A Basic formulae for calculation of integral characteristics on t h e basis of LIM A.l

General relationships

Should representation of local coefficients of force in form (1, 1.4) be adopted as the basis, expressions (1, 1.8), (1, 1.9) for coefficients of integral force and moment may be respectively given by -1

= j-JJWn(t)n0

+ nv(t)v°JdS,

(A, 1.1)

(S„s)

m=

n xv s W / ({"»WK -WKr - r°) x n°l°I + n-WK n-WKr r "- -rr °)0 )xvl]}dS, °°°\}dS' L 21.2) )

where, as in Section 1, Chapter 1, S* and /* are the characteristic area and length, respectively; n°, vj^ are the unit vectors of interior normal to a body surface and velocity of a free-stream flow around a still body; r is the radius vector of a surface point; To is the radius vector of a point with respect to which moment is calculated; fi„, fi„ are the functions related to Qp, UT by relation (1, 1.5); (Si;9) is the exposed body area in which

t= = vl vl •• n°n°>>0.0.

1.3) (A, 1.3)

Unless stated otherwise, further, by moment we mean the moment relative to the origin of coordinates, i.e. rp = 0. The bodies with a flat base section perpendicular to axis x are considered. Therefore, in the general case, the exposed area boundary is made up of curvilinear segments of natural boundary and a segment .of the. base, section boundary. 185

186

Appendix A. Basic formulae for calculation ofIC on the basis ofLIM

Figure A, 1.1: Body axis system xyz and air-path axis system

xayaza.

So far as t h e primary objective is to arrange in an ordered fashion references to formulae in t h e m a i n text of t h e book, only those formulae are given below, on the whole, which can serve t h a t objective. Two basic coordinate systems are considered (Fig. A, 1.1): t h e body axis system xyz and t h e air-path axis system xayaza. Axes z and za are aligned. Vector v ^ lies in plane xy (xaya) and forms angle of attack a with axis x, t h a t is vj^ = x° = cos a x° + sin a y ° , y° = - s i n a x 0 + c o s a y 0

(A, 1.4)

Coefficient of projection of force on some axis I lying in plane xy (Fig. A, 1.1) and described as 1° = cos x x ° + sin

J<

y°

(A, 1.5),

187

Section 1. General relationships

and coefficient of projection of moment on that axis are expressed by the integrals

c,M = cF-i° = j^JJ [nn(t)(n° ■ i°) + n.(0(v£, • i0)] ds, m z = m • z° = ^

jj

{ n „ ( 0 [ r n V ] - ilv(t)[r ■ y°a}} dS.

(A, 1.6) (A, 1.7)

In aerodynamics, different terms for coordinate axes, components of forces and mo­ ments and their coefficients are used. However, the terminology is not fully unified. We used the following terms: in the body axis system, Cx is axial force coefficient, Cy is normal force coefficient; in the air-path axis system, Cxa is drag coefficient, Cya is lift coefficient; moment coefficient mz = mza is called "pitching moment coefficient". It is obvious that formulae for Cx, Cxa, Cy, Cya may be obtained from general formulae (A, 1.6), Cx = C,(0), C„

= Q{a),

Cy = C|(ir/2), Cya = Ci(a

+ TT/2).

(A, 1.8) (A,

1.9)

In particular, the following take place Cx = Cxa cos a — Cya sin a, Cy = Cxa sin a + Cya cos a.

(A, 1.10)

Formulae (A, 1.6) and (A, 1.9) give the expressions for Cxa and Cya,

cxa = j;JjnD(t)ds, (S„s)

Cya = j-JJnn(t)(y0a-n0)dS,

(A, 1.11)

(Si„)

vhere

nD(t) = tSK(t) + n,(t) = mp(t) + \/i-* 2 n T (<).

(A, 1.12)

Assume r = r(£, £) whereas £ and £ have their values in domain ( o ^ ) , where (<7f() is the set including only those values of parameters £ and £ which provide for location of the end of vector r on the exposed part of side body surface. The following formulae hold true [192] n° = <5y^,

dS=\N\d£d(,

N = rfxr(,

(A, 1.13)

188

Appendix A. Basic formulae for calculation ofIC on the basis of LIM

where 8 = ± 1 while the sign is selected so that n° appears as an interior normal to the body surface. Upon transformations with allowance for (A, 1.13), expressions (A, 1.6) and (1, 1.7) may be written in the form: Ct(*) = y.jj

[fl„(l)<3 + n„(<)<4 ] di d(,

£pJf[(ln{t)i, + av{t)t*]dtdC,

m2

(A, 1.14)

(A, 1.15)

C'ec)

where t =

h/U,

fj = S • ( v ^ • N) = 8 ■ (cos a ■ Nx + sin a ■ Ny), *2 = |N|, t 3 = 8 • (N • 1°) = 8 ■ (cos * • TV* + sin x ■ Nv),

(A, 1.16)

0

U = (v^-l )|N| = cos(*-a)-i2, h = <5-[rNz°], U = - ( r ■ y a ) | N | = ij [sina(r ■ x°) - cos a(r • y 0 )], N% = N • x,

JV„ = N • y°

Moreover, the following formulae are valid,

Sm= JJt2dtd(,

(A, 1.17)

Si,9 = jjudidQ,

(A, 1.18)

where S s u r , Sug are the areas of the side and exposed surface, respectively; (o~°,-) is the region (&(() in the case when the entire side surface is exposed. Condition (A, 1.3) takes the form: ti>0.

(A, 1.19)

As a rule, the situation when (cr^) is bulging is considered and described as

U0<£<£+(0, (-<(<

C+

(A, 1.20)

189

Section 2. Cylindrical coordinate system—I so that the following substitution may be made in the integrals, C+ MO

II- I I

{"(() i- £-(0 The effect on a body having flat bluntness ("nose") is allowed for by additives; for coefficient of projection of the force on axis 1°, the following holds,

g_

AC|(x) = — [fi p (cosa)cos x + flT(cosa)sin x]

(A, 1.21)

■->+

and, in particular, at x = a, AC„, = | p n D ( c o s a ) ,

(A, 1.22)

where £_ is the body flat nose area. Let us now consider specific coordinate systems.

A.2

Cylindrical coordinate system—I

Assume the body side surface is described in cylindrical coordinate system xp6 (Fig. A, 2.1) in the form P = *(M),

* = *,

C = 0.

(A, 2.1)

Then, r(x,9) = x ■ x° + $ cos 9 y° + $ sin a z°, rx = { l ^ c o s ^ f ^ s i n t f } , r« = {0,i 8 ,< 7 }, N = {*«„-«,,«,},

<5 = 1,

i = *i/
tj = $ $ r cos a — t-j sin a,

*a = >/* 3 (*S + 1 ) + *2.

#

*=^>

t3 = $ $ ! cos x — ty sin x, <4 = cos(x — a)t2, tf,= —x-t7 — $ 2 # T cos 6, t6 = ( x s i n a — $ cos 6 cos a) t2, if = $ 9 sin 0 + $ cos 0, i g = $ s cos 0 - $ sin 0

*e = ^ '

(A>2-2)

190

Appendix A. Basic formulae for calculation ofIC on the basis ofLIM

Figure A, 2.1: Cylindrical coordinates—I. and formulae (A, 1.14), (A, 1.15), (A, 1.17), (A, 1.18) remain valid with allowance for (A, 2.2). T h e formula for the a r e a of b o d y section by a plane p e r p e n d i c u l a r to axis x is as follows: 2T

SSec(x)

= ~ J

&(x,9) d9.

(A, 2.3)

Should t h e body be such t h a t all its sections by planes perpendicular to axis i are geometrically similar, then *(x,0)

= f(x)-ri(0),

(A, 2.4)

1,(0) = 17(2*) = 1

and t h e following is obtained from formulae (A, 1.16) t = fj/lj, *i = f'rj1 cos a — t7 sin a , ,

df

^ywv+i)??, r =dx/ t3 = f'r/2 cos x — t-r sin x, <4 =

C O s ( x — a)

• f2)

<5 = —x ■ t-j — f f ' r j 2 cos 9, t§ = (x sin a + fr] cos 6 cos a) t2,

, _ dt]

dff'

Section 2. Cylindrical coordinate system—I

191

t7 = r\ sin 8 + 7/ cos 8,

iu = U/f,

e=l,2,...,7

(A, 2.5)

whence

c, = — y y /(i) w dfl
(A, 2.6)

0 fl_(x)

L M»)

\x)[nn{t)i5 + nv{t)i6]d9dx,

(A, 2.7)

0 9 Is) L %it

= y y /(x) f(x)t f{x2)ifa2d8dx, d8 5SUT <*
(A, (A,2.8) 2.8)

J

0

0

L M»)

5 H , = y y f{x)i2d8dx,

(A, 2.9)

0 »_(*) ^7T

5ec

[/(s)]! fv2d0, " 2

(A, 2.10)

0

where 8-, 8+ are the functions which appear in the description of the exposed part of body surface in the form (A, 1.20). For bodies of revolution, 7,(0) = 1

(A, 2.11)

and formulae (A, 2.6), (A, 2.7) are further simplified since t\ = / ' cos

Q

— cos 6 sin a,

i2 = y/r* + l, t3 = / ' cos x — cos 8 sin x,

(A, 2.12)

<4 = COs(>f — a ) • t2,

tS = (-X+ff)

COS 6

while formulae (A, 2.8) and (A, 2.10) may be written as L

Smr=2wjfjf>* o Ssec = ir[f(x)}2

+ ldx,

(A, 2.13) (A, 2.14)

192

Appendix

A. Basic formulae for calculation

ofIC

on the basis of LIM

Figure A, 3.1: Cylindrical coordinates—II.

If for body of revolution a = 0, then mz = Cy = Cya = 0 and only force coefficient Cx = Cxa = C;(0) is of interest. To calculate that characteristic, the following formula may be obtained from (A, 2.12) at a = x = 0,

CI = Cxa =

2ir

r

—jf(x)H(f)dx,

(A, 2.15)

where

H(W) = n„( w„) w + v/w2 + i n„( Wo) = wnp(w0) + nT(w0), w0 =

A.3

w

v/W 2 + 1'

(A, 2.16)

Cylindrical coordinate system—II

Assume the body side surface is described in cylindrical coordinate system xpd (Fig. A, 3.1) as

* = *(/»,«),

f = p, C = «-

Then, r(p,0) = $ ■ x° + pcosd y° + psinO z°,

(A, 3.1)

193

Section 3. Cylindrical coordinate system—II r„ = {$„,cos0,sin0}, r« = {$0, —

psm6,pcos8},

N={p,t7,-ts},

,5=1,

11 = cos a • p + sin a t7,

h=jp m+!)+•?, h = jP%2(n+1) + n, t3 = pcos x + i 7 sinx,

(A, 3.2)

£4 = cos(a — x) <2, f5 = $ • f7 — p2 cosd, t6 = t2 ( $ s i n a — p cos 0 cos a), t7 — $ e sin 5 — /)$ p cos 6, ig = $e cos 0 — p
9$

_ 5$ d$

and expressions (A, 1.14), (A, 1.15), (A, 1.17), (A, 1.18) with allowance for (A, 3.1), (A, 3.2) yield the desired formulae. In case of body of revolution, x = $(/»)

(A, 3.3)

and it transpires from (A, 3.2) that

t = iilh, ti = cos a — $ ' cos 6 sin a,

t2 = v/$' 2 + l, t3 = cos x — # ' cos # sin x, £4 = COs(x — a ) • ^2;

U = - ( $ $ ' + *>) cos 0, t6 = t2 ($ sin a — p cos 5 cos a), <7 =

— p COS # ,

#'=-j-:

1 = -,

f = l,2,...,7

(A, 3.4)

and formulae (A, 1.14), (A, 1.15), (A, 1.17) take the form

c, = j ; JJ p[nn(t)i3 + nv(t)h}dpd0, (
(A, 3.5)

194

Appendix A. Basic formulae for calculation ofIC on the basis ofLIM

m2 = —

JJp[Sln{t)k

+ nv{t)k]dpdB,

(A, 3.6)

L

Ssur = 2wjp

V * ' 2 + l dp.

(A, 3.7)

0

At a = 0 we get

Cx = Cxa = yt

j

PH(*')dp,

(A, 3.8)

where R+ is the radius of the body base section, H(W)

= n»(w 0 ) +

Wo =

A.4

=

^ 4 T ^ ,

"PC^O)

+ H/nr(VK0),

*

(A, 3.9)

Cartesian coordinate system—I

Assume the body side surface is described in coordinate system xyz (Fig. A, 4.1) as x = $(y,z),

i/ = £,

z = (

(A, 4.1)

whereas the requirement is not specified t h a t a body has flat nose section. T h e n , r{y,z)

= $-x°

+ yy°

+

z-z0,

VV = { # „ , 1, 0 } , r 2 = {**, 0, 1}, N = = {!,-*„-4,}, {1,-*B,-*X}, N (i = cos a — 4 y sin a,

t2 = y*J + * 1 + T , t3 = cos x — $5, sin xr, t4 = cos(a — x) <2,

h = -(y + •*„), <6 = ^2 ( 4 sin a — y cos a ) ,

- 9$ _

(A, 4.2)

Section 5. Cartesian coordinate system—II

195

Figure A, 4.1: Cartesian coordinates—I

F o r m u l a e (A, 1.14), (A, 1.15), (A, 1.17), (A, 1.18) are valid when allowance is m a d e for with allowance for (A, 4.2). If a = 0 t h e following formulae are obtained for Cx = Cxa (x = 0), 2 + $l)dydz, Cx = Cxa = j-JjH($ - JJ H($l *J) dy dz, y «)

Ssur ==SSlitltg==j j-.JJ ; JJ ft* dz,dz, Ssur \/n ++*»n++1 *dydy

(A, 4.3) 4.3) (A, (A, (A, 4.4) 4.4)

where function H is of t h e form (A, 3.9). It should be noted t h a t in t h e general case, Cy ^ 0, Cya ^ 0, mz ^ 0 if x, z is not a plane of s y m m e t r y .

A.5

Cartesian coordinate system—II

A s s u m e t h e body side surface located in the region y > 0 is described in coordinate system xyz (Fig. A, 5.1) in t h e form

y = 9(x,z), $(x,z),

tC == x,x

, (C == zz

(A, (A, 5.1)

196

Appendix A. Basic formulae for calculation ofIC on the basis ofLIM

Figure A, 5.1: Cartesian coordinates—II. whereas t h e flat form of the base section is not indispensable condition. T h e n , r ( i , z) = x ■ x° + $ • y° + z • z°, r , = {1, $ x , 0 } , r 2 = {0, « „ 1}, N =={ #{#«,-!, « , - 1 , # #,}, ,}, N U

V f i = $>x cos a — sin a ,

(A, 5.2)

U = y/9l y/H ++ *l#2 ++ 1, t2 = h *3 = <&x cos x — sin x, (4 = cos(a — x) £2, t6 = - ( 1 + $ $ * ) , i 6 = t2 (sin a — <J> cos a), r9$ *r

^

3$ = ozIT-

Formulae (A, 1.14), (A, 1.15), (A, 1.17), (A, 1.18) are valid with account of

197

Section 5. Cartesian coordinate system—II (A, 5.2). If a = 0 then

Cx = Cxa = j ; Jj [n„ ( ^ ) ■*, + «„ ( ^ ) • ta] dx
2

(A, 5.3)

2

In case when the body is a cylindrical wing element of span 6, i.e. $ = $(s) and (ff°2) is the region where 0
0

<x
formula (A, 5.3) is transformed to the form:

cx = cxa = —jH(&)dx, whereas function H is defined by equality (A, 2.16).

*' = ^r

(A, 5.4)

This page is intentionally left blank

Bibliography [1] Abramovich, Yu. V. and Shirokopoyas, E. P., Computer-aided engineering method of calculation of aerodynamic characteristics of space vehicles in hy­ personic flow, Trudy TsAGI 1580 (1974) 3-29. [In Russian] [2] Abramovskaya, M. G. and Bass, V. P., Investigation of aerodynamic charac­ teristics of circular cones in the transitional regime of flow, Uchonye Zapiski TsAGI 11 (1980), no. 1, 122-126. [In Russian] [3] Abramovskaya, M. G. and Bass, V. P., Aerodynamic characteristics of rotating bodies in a multicomponent stream of rarefied gas, Fluid Dynamics 16 (1981), no. 6, 943-946. [4] Abramovskaya, M. G. and Bass, V. P., Calculation of bodies aerodynamical characteristics in the transition flow regime, in Kosmicheskie issledovaniya na Ukraine [Cosmic Investigations in the Ukraine], issue 16, Kiev (1982), pp. 29-34. [In Russian] [5] Abramovskaya, M. G., Bass, V. P., Limanskii, A. V., and Timoshenko, V. I., Calculation of bodies aerodynamical characteristics in the transition flow regime, in Prikladnaya aerodinamika kosmicheskikh apparatov [Applied Aerodynamics of Space Vehicles], Naukova Dumka, Kiev (1977), pp. 69-75. [In Russian] [6] Abramovskaya, M. G., Bass, V. P., Perminov, V. D., and Shvedov, A. V., Some improved algorithms for calculation of aerodynamic characteristics of astrovehicles in free-molecular flow, Trudy TsAGI 2436 (1990) 68-74. [In Russian] [7] Ackeret, J., Luftkrafte auf Flugel die mit Grosserer als Schallgeschwindigkeit Bewegt Werden, Z. Flugtech. & Motorluftschiffahrt 16.3 (1925) 72-74. [8] Adams, J. C , Jr., Calculated hypersonic viscous aerodynamics of a sharp 7deg. cones at incidence, AIAA-Paper 76-361 (1976) 1-8. [9] Adams, J. C , Jr. and Griffith, B. J., Hypersonic viscous static stability of the sharp 5-deg. cone at incidence, AIAA J. 14 (1976), no. 8, 1062-1068. 199

Bibliography

200

[10] Aerodinamika razrezhennikh gazov [Rarefied Gas Dynamics], Collected Articles, issues 1-11, Leningr. Univ. Publ., Leningrad (1963-1983). [In Russian] [11] Aerodinamika sverkhzvukovogo obtekaniya tel vrascheniya stepennoy formi [Aerodynamics of Supersonic Flow past Bodies of Revolution of Powerlaw Form], Mashinostroenie, Moscow (1975). [In Russian] [12] Aerogidromekhanika (1993). [In Russian]

[Aerohydromechanics],

Mashinostroenie,

Moscow

[13] Aks'onova, 0 . A., Dependence of regime coefficients on regime parameters in local interaction theory, Vestnik Leningrad. Univ., ser. 1: Math., Mech., Astron. 1 (1989), no. 1, 100-102 [In Russian]. [14] Aks'onova, O. A. and Khalidov, I. A., Accuracy of aerodynamic calculations on the basis of local theory, [127], p. 152. [In Russian] [15] Aks'onova, O. A., Miroshin, R. N., and Khalidov, I. A., Inverse problems of local interaction theory for rarefied gas, [126], p. 76. [In Russian] [16] Alekseeva, E. V., Calculation of regime coefficients in the local aerodynamical method for finite values of Mach numbers, dep. VINITI, issue 2941-B (1987), pp. 1-15. [In Russian] [17] Alekseeva, E. V. and Alekseeva, S. N., Taking into account the influence of finite quantity of Mach number on the coefficients of regime in the local interaction theory, [10], issue 11 (1983), pp. 243-247. [In Russian] [18] Alekseeva, E. V., Anolik, M. V., and Eshov, A. T., Computer program for calcu­ lation of aerodynamical characteristics of convex bodies of revolution according to local method, [276], 1, p. 95. [In Russian] [19] Alekseeva, E. V., Anolik, M. V., and Naydanova, L. A., Errors in regime co­ efficients and their influence on the accuracy of aerodynamical calculation by local method, [126], p. 72. [In Russian] [20] Alekseeva, E. V. and Barantsev, R. G., Lokal'nyi metod aerodinamicheskogo raschota v r a z r e z h e n n o m gaze [The Local Method of Aerodynamic Calculation in Rarefied Gas Dynamics], Leningr. St. Univ., Leningrad (1976). [In Russian] [21] Alekseeva, E. V. and Khokhlova, L. D., Taking into account the influence of finite quantity of Mach number in the local interaction theory for unisotropic flow, [10], issue 10 (1980), pp. 196-202. [In Russian]

Bibliography

201

[22] Alekseeva, E. V. and Miroshin, R. N., Two-component statistical model of working experimental results on rarefied gas, [10], issue 6 (1972), pp. 5-8. [In Russian] [23] Alekseeva, E. V., Vasilyev, L. A., and Gashtol'd, L. P., Calculation of forces and moments of light pressure on a body, [10], issue 9 (1978), pp. 163-179. [In Russian] [24] Alekseeva, S. N., Barantsev, R. G., and Ender, I. A., Local interaction the­ ory with consideration of finite quantity of Mach number, [10], issue 8 (1976), pp. 162-171. [In Russian] [25] Alekseeva, S. N. and Miroshin, R. N., On dependence of local interaction pa­ rameters on Knudsen number, [10], issue 7 (1974), pp. 180-190. [In Russian] [26] Andrews, J. S., Steady state airload distribution on a hammerhead shaped payload of a multistage vehicle at transonic speeds, Boeing Co., Rept. D222947-1 (1964). [27] Anisimov, S. I., Imas, Ya. A., Romanov, G. S., and Khodyko, Yu. V., Deistvie izlucheniya bol'shoi moschnosti na metalli [Influence of Great Powerful Radiation on Metals], Nauka, Moscow (1970). [In Russian] [28] Anolik, N. V., Balueva, N. F., Khalidov, I. A., et al., First interactions of rarefied gas atoms with rough surface, [282], Part 4, pp. 55-59. [In Russian] [29] Anolik, M. V., Tabolkina, E. N., Khalidov, I. A., and Eshov, A. T., Aerodynam­ ical calculation using a local method taking into account the surface roughness of body, [126], p. 46. [In Russian] [30] Antonets, A. V., Three-dimensional supersonic flow past blunt-bodies with con­ tour discontinuities, with account for equilibrium and frozen state of the gas in the shock layer, Fluid Dynamics 5 (1970), no. 2, 324-328. [31] Antonets, A. V., Easy approximations of functions with additional conditions, U. S. S. R. Comp. Math. & Math. Phys. 25 (1985), no. 11. [32] Antonets, A. V., Approximation of complex dependences of quantities by se­ lection of weights in averaging equations, Fluid Dynamics 21 (1986), no. 1, 138-143. [33] Antonets, A. V. and Dubinskii, A. V., A method of calculating the aerodynamic characteristics of bodies on the basis of invariant relations of the theory of local interaction, Appl. Math. & Mech. 47 (1983), no. 5, 700-702.

202

Bibliography

[34] Antonov, S. G., Ivanov, M. S., Kashkovsky, A. V., and Chistolinov, V. G., Influence of atmospheric rarefaction on aerodynamic characteristics of flying vehicles, in R a r e f i e d G a s D y n a m i c s , Proc. of 17th Int. Symp. on Rarefied Gas Dynamics, Veincheim (1991), p p . 522-530. [35] Apshtein, E. Z. and Kolotilova, N. G., Area rule for integral quantities, Fluid Dynamics 16 (1981), no. 4, 621-623. [36] Apshtein, E. Z. and Pilyugin, N. N., Area rule for heat transfer coefficient of three-dimensional ablating bodies in heat flow t h a t depend locally on t h e angle of attack, Fluid Dynamics 14 (1979), no. 2, 223-227. [37] Apshtein, E. Z. and Pilyugin, N. N., Stable shape of bodies ablating under the influence of heat fluxes which depend on t h e local slope of t h e surface, Fluid Dynamics 16 (1981), no. 6, 915-920. [38] Apshtein, E. Z. and Pilyugin, N. N., About stability of solutions of t h e equations of taking away t h e bodies mass under influence of t h e r m a l fluxes depending on local incidence angle, in G i p e r z v u k o v i e t e c h e n i y a pri o b t e k a n i i t e l i v s l e d a k h [Hypersonic Flows Over Bodies and in Tracks], Mosc. St. Univ., Moscow (1983). [In Russian] [39] Apshtein, E. Z., Pilyugin, N. N., Sevast'yanenko, V. G., and Tirsky G. A, Radiative heating during t h e entry of the probes into t h e atmosphere of t h e earth and planets with superorbital velocities, I t o g i n a u k i i t e k h n i k i , s e r . " M e k h a n i k a z h i d k o s t e i i g a z o v ' ' [Advances in Science and Technology, ser. Mechanics of Fluids and Gases], Vol. 23, V I N I T I (1989), p p . 116-236. [In Russian] [40] Apshtein, E. Z., Sakharov, V. I., and Shevoroshkin, A. V., Investigation of su­ personic radiative flow past bodies at low altitudes, Fluid Dynamics 26 (1991), no. 3, 470-473. [41] Apshtein, E. Z., Vartanyan, N. V., and Sakharov, V. I., Distribution of radiant heat fluxes over surface of three-dimensional bodies in a supersonic ideal gas flow, Fluid Dynamics 2 1 (1986), no. 4. [42] Apshtein, E. Z., Vartanyan, N. V., and Sakharov, V. I., Distribution of radiant heat fluxes over the surface of three-dimensional and axisymmetric bodies in a supersonic ideal-gas flow, Fluid Dynamics 2 1 (1986), no. 1, 78-83. [43] Apshtein, E. Z., Vartanyan, N. V., Sakharov, V. I., and Tirsky, G. A., Radiant heat fluxes in supersonic inviscid gas flow past three-dimensional bodies, Dokladi Akademii Nauk SSSR 2 8 6 (1986), no. 3, 579-582. [In Russian]

Bibliography

203

[44] Arguchintseva, M. A. and Pilyugin, N. N., Extremal problems of heat transfer to three-dimensional bodies at hypersonic speeds, Appl. Math. & Mech. 56 (1992), no. 4, 545-558. [45] Arguchintseva, M. A. and Pilyugin, N. N., Body shapes with minimum surface heating during supersonic travel through an atmosphere, Cosmic Research 30 (1992), no. 5, 499-510. [46] Arguchintseva, M. A. and Pilyugin, N. N., The optimal slender bodies forms with limit on local heat flux, Cosmic Research 31 (1993), no. 3. [47] Artonkin, V. G. and Petrov, K. P., Investigation of aerodynamic characteristics of segment-conical bodies, Trudy TsAGI 1361 (1971) 1-25. [In Russian] [48] Babenko, K. I., Voskresenskii, U. P., Lyubimov, A. I., and Rusanov, V. V., Three-dimensional flow of ideal gas past smooth bodies, TT F-380, NASA (1966). [49] Barantsev, R. G., On the choice of momentum flux dependence on the inci­ dence angle in the local interaction theory, [10], issue 7 (1974), pp. 168-179. [In Russian] [50] Barantsev, R. G., Some problems of gas solid surface interaction, in Progress in Aerospace Science 13, Pergamon Press, New York, pp. 1-80. [51] Barantsev, R. G., Vzaimodeistvie razrezhennikh gazov s obtekaemimi poverkhnostyami [Interaction of Rarefied Gas Flows with Surfaces], Nauka, Moscow (1975). [In Russian] [52] Barantsev, R. G., Hypersonic gas motion. Steady unviscous gas flow over bodies, Itogi Nauki i Tekhniki, ser. "Gidromekhanika" [Advances in Science and Technology, ser. Mechanics of Fluids], Vol. 9, VINITI (1976), pp. 5-53. [In Russian] [53] Barantsev, R. G., Present state of the gas-surface interaction, [281], pp. 221248. [In Russian] [54] Barantsev, R. G., Local method in rarefied gas aerodynamics, Rozprawy Inzynierskic [Engineering Transactions], 26 (1978), no. 1, 3-9. [55] Barantsev, R. G., Analytical methods in the rarefied gas dynamics, Itogi nauki i Tekhniki, ser. "Mekhanika zhidkostei i gazov" [Advances in Science and Technology, ser. Mechanics of Fluids and Gases], Vol. 14, VINITI (1981), pp. 3 65. [In Russian]

204

Bibliography

[56] Barantsev, R. G., Local theory of impulse and energy transmission on t h e body surface in a rarefied gas, in M a t e m a t i c h e s k o e m o d e l i r o v a n i e , a n a l i t i c h e s k i e i c h i s l e n n i e m e t o d i v t e o r i i p e r e n o s a [Mathematical Modelling, An­ alytical and Computering Methods in t h e Theory of Transfer], Minsk (1982), p p . 90-98. [In Russian] [57] Barantsev, R. G., Analytical studies of gas-surface interaction, [259], p p . 6 4 5 652. [58] Barantsev, R. G., A variant of local method for slender bodies in rarefied gas, Vestnik Leningrad. Univ., ser. 1: Math., Mech., Astron. (1982), no. 13, 99-101. [In Russian] [59] Barantsev, R. G. and F'odorova, V. M., Taking account of surface curvature in t h e local interaction theory, [10], issue 9 (1978), p p . 195-202. [In Russian] [60] Barantsev, R. G. and F'odorova V. M., Local interaction theory with regard to surface curvature, [128], P a r t 2, p p . 43-48. [In Russian] [61] Barantsev, R. G. and F'odorova, V. M., Anisotropic effect of curvature in local interaction theory, [10], issue 10 (1980), pp. 192-197. [In Russian] [62] Barantsev, R. G. and F'odorova, V. M., Specific character of local theory for slender bodies of revolution, Vestnik Leningrad. Univ., ser. 1: Math., Mech., Astron. (1987), no. 4. [In Russian] [63] Barantsev, R. G. and Naritsa, V. S., Development of local m e t h o d for slender bodies, [282], P a r t 2, p p . 94-98. [In Russian] [64] Barantsev, R. G., Vasil'yev, L. A., Ivanov, E. V., Kozachek, V. V., Minaichev, A. D., and Murzov, N. V., Aerodynamic calculation in a rarefied gas flow on t h e basis of locality hypothesis, [10], issue 1 (1963), p p . 170-184. [In Russian] [65] Bass, V. P., Calculation of flow of highly a t t e n u a t e d gas around a body, taking into account interaction with the surface, Fluid Dynamics 1 3 (1978), no. 5, 729-733. [66] Bass, V. P., Brazinskii, V. I., Kariagin, V. P., et al., Aerodynamic character­ istics of the probe for investigation of Halley comet, in P r i k l a d n i e v o p r o s i a e r o d i n a m i k i l e t a t e l ' n i k h a p p a r a t o v [Applied Problems of Aerodynamics of Vehicles], Naukova dumka, Kiev (1984). [In Russian] [67] Bass, V. P., Kovtunenko, V. M., and Chepurnoi, V. N., Determination of the aerodynamic characteristics of a complexly shaped body in a free molecular flow with consideration of shadowing effect, Cosmic Research 12 (1974), no. 1, 34-38.

Bibliography

205

[68] Bass, V. P. and Timoshenko, V. I., Application of local interaction method to the calculation of aerodynamic characteristics of complex form bodies in hypersonic rarefied gas flow, Trudy TsAGI 1833 (1977) 28-37. [In Russian] [69] Beletskii, V. V., Dvizhenie iskusstvennogo sputnika otnositel'no tsentra mass [The Motion of Satellite around the Centre of Mass], Nauka, Moscow (1965). [In Russian] [70] Beletskii, V. V. and Yanshin, A. M., Vliyanie aerodinamicheskikh sil na vraschatel'noe dvizhenie iskusstvennikh sputnikov [Effect of Aerody­ namic Forces on the Rotational Motion of Satellites], Naukova Dumka, Kiev (1984). [In Russian] [71] Bellman, R., Dynamic Programming, Princeton Univ. Press, Princeton, New Jersey (1957). [72] Bellomo, N., Riganti, R., and Vacca, M. T., Stochastic Dynamics of a Flat Plate in Molecular Flow, in Rarefied Gas Dynamics, Part 2, Technical Papers Selected from the Twelth Int. Symp. on Rarefied Gas Dynamics. Charlottesville, Va., 1980, published by AIAA, New York (1981), pp. 1063-1080. [73] Belotserkovskii, S. M., Presentation of unsteady aerodynamic moments and forces by means of coefficients of rotary derivatives, Izvestiya Akademii Nauk SSSR, otdel. tekhnicheskikh nauk (1956), no. 7, 53-70. [In Russian] [74] Belotserkovskii, S. M., Kochetkov, Yu. A., Krasovskii, A. A., Novitskii, V. V., Vvedenie v aerouprugost [Introduction to Aeroelasticity], Nauka, Moscow (1980). [In Russian] [75] Berd, G. A., Molecular Gas Dynamics, Clarendon Press, Oxford (1976). [76] Berdichevskii, V. L., On the shape of a body of minimum drag in a hypersonic gas flow, Moscow Univ. Mech. Bull. 28 (1974), no. 3-4. [77] Blick, E. F., Aerodynamic coefficients in the slip transition regimes. AIAA J. 1 (1963), no. 11. [78] Bondarchuk, V. S., Vedernikov, Yu .A., Dulov, V. G., and Minin, V. F., Op­ timization of star-shaped penetrators, Izvestiya Sibir. otdel. Akademii Nauk SSSR, seriya tekhnicheskikh nauk 13 (1982), no. 3, 60-64. [In Russian] [79] Boylan, D. E. and Potter, J. L., Aerodynamics of typical lifting bodies under condition simulating very high altitudes, AIAA J. 5 (1967), no. 2, 226-232. [80] Bunimovich, A. I., Relations between the forces acting on a body moving in a rarefied gas, in a light flux and in a hypersonic newtonian stream, Fluid Dynamics 8 (1973), no. 4, 584-589.

206

Bibliography

[81] Bunimovich, A. I., Aerodynamic characteristics of axisymmetric bodies in a flow under "localization-law" conditions. Moscow Univ. Mech. Bull. 29, no. 3-4 (1974), 63-67. [82] Bunimovich, A. I., The law of locality in aerodynamics and rarefied gas dynam­ ics, [281], pp. 399-404. [In Russian] [83] Bunimovich, A. I., Local methods in aerodynamics, in Problems of Compu­ tational and Applied Mathematics, issue 60, Tashkent (1980), pp. 23-31. [In Russian] [84] Bunimovich, A. I., Local interaction theory in rarefied gas dynamics, in Aerotermogazodinamika v razrezhennikh potokakh [Aerothermodynamics in Rarefied Streams], MAI, Moscow (1988), pp. 25-39. [In Russian] [85] Bunimovich, A. I. and Chistolinov, V. G., Aerodynamical forces acting on con­ vex three-dimensional body in a rarefied gas stream, [281], pp. 411-417. [In Russian] [86] Bunimovich, A. I. and Chistolinov, V. G., Analytical computation of aerody­ namic characteristics of bodies of revolution under locality law conditions. Fluid Dynamics 10 (1975), no. 5, 784-789. [87] Bunimovich, A. L and Chistolinov, V. G., Analytic method of calculating aero­ dynamic forces in a three-dimensional problem in conditions of the "localisability" law, Appl. Math. & Mech. 39 (1975), no. 3, 442-448. [88] Bunimovich, A. I. and Chistolinov, V. G., Analytic method of calculating aerodynamic characteristics of bodies in hypersonic flow of various rarefaction, Trudy TsAGI 1833 (1977) 11-27. [In Russian] [89] Bunimovich, A. I. and Dubinskii, A. V., Generalised similarity laws for flows around bodies in conditions of "localizability" law, Appl. Math. & Mech. 37 (1973), no. 5, 812-818. [90] Bunimovich, A. I. and Dubinskii, A. V., Generalised similarity laws for bodies in a rarefied gas stream, [281], pp. 405-410. [In Russian] [91] Bunimovich, A. I. and Dubinskii, A. V., Generalized similarity laws in flows past solid bodies, Appl. Math. & Mech. 39 (1975), no. 4, 709-713. [92] Bunimovich, A. I. and Dubinskii, A. V., Use of generalised similarity laws in computing the aerodynamic characteristics of three-dimensional bodies, Moscow Univ. Mech. Bull. 31 (1976), no. 1-2, 48-53.

Bibliography

207

[93] Bunimovich, A. I. and Dubinskii, A. V., Optimal blunt-nosed figures of revo­ lution in a gas flow of different rarefactions, Fluid Dynamics 15 (1980), no. 3, 457-460. [94] Bunimovich, A. I. and Dubinskii, A. V., The localised stream-surface interac­ tion methods in gas dynamics, in 5 Vsesoyuznii s'ezd po teoreticheskoi i prikladnoi mekhanike [Ail-Union Congress on Theoretical and Applied Me­ chanics], Alma Ata (1981), p. 79. [In Russian] [95] Bunimovich, A. I. and Dubinskii, A. V., On the center of pressure of a body, Fluid Dynamics 17 (1982), no. 5, 760-764. [96] Bunimovich, A. I. and Dubinskii, A. V., Local methods in rarefied gas dynamics, [259], pp. 431-438. [97] Bunimovich, A. I. and Dubinskii, A. V., Center of pressure of pyramidal bodies, Moscow Univ: Mech. Bull. 38 (1983), no. 5, 33-36. [98] Bunimovich, A. I. and Dubinskii, A. V., Aerodynamic design of bodies rotating in a stream, based on local flow interaction models, Cosmic Research 23 (1985), no. 4, 463-467. [99] Bunimovich, A. I. and Dubinskii, A. V., Aerodynamic characteristics of arbi­ trarily rotating bodies in a gas of variable rarefaction, Cosmic Research 27 (1989), no. 2, 147-152. [100] Bunimovich, A. I. and Dubinskii, A. V., Bodies of minimum drag in rarefied gas, in Volnovie zadachi mekhaniki deformiruemikh sred [Wave Problems in Mechanics of Deformable Media], Part 1, Mosc. St. Univ., Moscow (1990), pp. 118-125. [In Russian] [101] Bunimovich, A. I. and Dubinskii, A. V., The localized-interaction theory in the case of arbitrary bodies moving, [127], p. 158. [In Russian] [102] Bunimovich, A. I. and Dubinskii, A. V., Calculation of rotational derivatives for "local" interaction of a flow with the surface of a body, Appl. Math. & Mech. 56 (1992), no. 1, 45-50. [103] Bunimovich, A. I. and Dubinskii, A. V., Modelling problems of a penetration of bodies into various media and optimization of penetrating bodies shape in localized interaction theory (in print). [104] Bunimovich, A. I., Dubinskii, A. V., and Kuzmenko, V. I., Aerodynamics and thermal characteristics and optimum shape of bodies in rarefied gas, [128], Part 2, pp. 49-54. [In Russian]

208

Bibliography

[105] Bunimovich, A. I., Dubinskii, A. V., and Pavlov, A. V., Aerodynamics of arbi­ trary revolving bodies in a gas flow of raring rarefied, [277], p. 86. [In Russian] [106] Bunimovich, A. I., Dubinskii, A. V., and Yakunina, G. E., Three-dimensional bodies of minimum drag in a rarefied gas stream, [276], Part 1, p. 82. [In Russian] [107] Bunimovich, A. I. and Khots, 0 . A., Calculation of aerodynamic characteristics of convex bodies of revolution on the basis of local theory, in Gazovaya i volnovaya dinamika [Gas and Wave Dynamics], issue 3 (1979), Mosc. St. Univ., Moscow, pp. 84-90. [In Russian] [108] Bunimovich, A. I. and Khots, O. A., Analytical method of calculation of aero­ dynamical forces acting on bodies of revolution in a stream under conditions of local interaction law, in Gazovaya i volnovaya dinamika [Gas and Wave Dynamics], issue 2 (1979), Mosc. St. Univ., Moscow, pp. 146-154. [In Russian] [109] Bunimovich, A. I. and Kuzmenko, V. I., Aerodynamic and thermal characteris­ tics of three-dimensional star-shaped bodies in a rarefied gas, Fluid Dynamics 18 (1983), no. 4, 652-654. [110] Bunimovich, A. I. and Kuzmenko, V. I., Aerodynamic and thermal character­ istics of star-shaped bodies at the angle of attack in a hypersonic rarefied gas flow, Moscow Univ. Mech. Bull. 39 (1984), no. 2, 15-18. [Ill] Bunimovich, A. I. and Sadykova, L. G., Unsteady speedy motion of a class of three-dimensional bodies, in Gazovaya i volnovaya dinamika [Gas and Wave Dynamics], issue 2, Mosc. St. Univ., Moscow (1979), pp. 140-145. [In Russian] [112] Bunimovich, A. I. and Sazonova, N. I., Force and moment characteristics of three-dimensional bodies in gas stream under local interaction conditions, in Gazovaya i volnovaya dinamika [Gas and Wave Dynamics], issue 1, Mosc. St. Univ., Moscow (1975), pp. 32-39. [In Russian] [113] Bunimovich, A. I. and Sazonova, N. I., An analytical method for determin­ ing aerodynamic forces and moments in the unsteady motion of bodies in gas of various rarefaction, in Gazovaya i volnovaya dinamika, [Gas and Wave Dynamics], issue 2, Mosc. St. Univ., Moscow (1979), pp. 32-43. [In Russian] [114] Bunimovich, A. I., Vorotyntsev, M. A., and Sazonova, N. I., Determining aero­ dynamic characteristics of bodies translating under "localization law", Moscow Univ. Mech. Bull. 30 (1975), no. 5-6, 49-53. [115] Bunimovich, A. I. and Yakunina, G. E., Investigation of the shape of the trans­ verse contour of a conical three-dimensional body of minimum drag moving in rarefied gas, Fluid Dynamics 21 (1986), no. 5, 771-776.

Bibliography

209

[116] Bunimovich, A. I. and Yakunina, G. E., The shapes of three-dimensional minimal-resistance bodies moving in plastically-compressible and elasticallyplastic media. Moscow Univ. Mech. Bull. 42 (1987), no. 3, 59-62. [117] Bunimovich, A. I. and Yakunina, G. E., On the shape of minimum-resistance solids of revolution moving in plastically compressible and elastic-plastic media, Appl. Math. & Mech. 51 (1987), no. 3, 386-392. [118] Bunimovich, A. I. and Zinatulin, I. S., Aerodynamic characteristics of body in nonstationary motion, in Konferentsiya po rasprostraneniyu uprugikh i uprugoplasticheskikh voln. Tezisi dokl. [Conf. on Propagation of Elastically and Elastical-Plastic Waves. Book of Abstr.], Part 1, Frunze (1983), pp. 121-123. [In Russian] [119] Busemann A. Aerodynamischer Auftrieb bei Uberschallgeschwindigkeit, Liiftfahrtforschung (1935), no. 12, 210-222. [120] Cercignani, C , Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburg k. London (1975). [121] Chapman, G. T., A Simple relationship between the drag near zero lift and initial normal-force-curve slope obtained from Newtonian theory, AIAA J. 3 (1965), no. 6, 1194-1195. [122] Chernyi, G. G., Introduction to Hypersonic Flow. Academic Press, New York, London (1969). [123] Chin, J. H., Shape change and conduction for nosetips at angle of attack, AIAA J. 13 (1975), no. 5. [124] Chistolinov, V. G., Calculation of aerodynamic characteristics for bodies of revolution in conditions of law of locality, in Issledovaniya po mekhanike zhidkostei i tverdikh tel [Investigations on Mechanics of Fluids and Rigid Bodies], Mosc. St. Univ., Moscow (1977), pp. 44-49. [In Russian] [125] Deev A.A., Levin V.A., and Pilyugin, N. N., Body shape with minimum total radiant energy flux toward its surface, Fluid Dynamics 11 (1976), no. 4, 560564. [126] Dinamika razrezhennikh gazov. Tezisi dokladov 10 Vsesoyuznoy konferentsii [Rarefied Gas Dynamics. 10th Ail-Union Conf. on RGD (Moscow, 1989). Book of Abstracts], Moscow (1989). [In Russian] [127] Dinamika razrezhennikh gazov Tezisi dokladov 11 Vsesoyuznoi konferentsii [Rarefied Gas Dynamics. 11th All-Union Conf. on RGD (Leningrad, 1991). Book of Abstracts], Leningr. Mekhan. Inst., Leningrad (1991). [In Rus­ sian]

210

Bibliography

[128] D i n a m i k a r a z r e z h e n n o g o g a z a . T r u d i V I V s e s o y u z n o i k o n f e r e n t s i i [Rarefied Gas Dynamics. Proc. of 6th All-Union Conf. on R G D (Novosibirsk, 1979)], P a r t s 1-2, Novosibirsk (1980). [In Russian] [129] Donov, A. E., Two-dimensional airfoil with sharply-pointed edges in a hyper­ sonic s t r e a m , Izvestiya Akademii Nauk SSSR, ser. Matem. 3 (1939), no. 5-6, 603-626. [In Russian] [130] Draper, N. R. and Smith, H., A p p l i e d R e g r e s s i o n A n a l y s i s , J. Wiley & Sons, New York, London, Sydney (1966). [131] Dubinskii, A. V., General similarity laws for semi-bodies of revolution in stream, in I s s l e d o v a n i y a p o m e k h a n i k e z h i d k o s t e i i t v e r d i k h t e l [Investigations on Mechanics of Fluids and Rigid Bodies], Mosc. St. Univ., Moscow (1977), pp. 26-28. [In Russian] [132] Dubinskii, A. V., T h e universal relationships of aerodynamic characteristics of bodies in s t r e a m in t h e conditionals of t h e "localised stream-surface interaction model", in G a z o v a y a i v o l n o v a y a d i n a m i k a [Gas and Wave Dynamics], issue 3, Mosc. St. Univ., Moscow (1979), p p . 160-162. [In Russian] [133] Dubinskii, A. V., Relations between forces acting on bodies of different shapes moving in a gas, Appl. Math. & Mech. 4 4 (1980), no. 1, 125-127. [134] Dubinskii, A. V., Research and optimisation of aerodynamic characteristics of bodies in a rarefied gas on the basis of analytic and c o m p u t a t i o n a l methods, in D i n a m i k a r a z r e z h e n n o g o g a z a i p o g r a n i c h n o g o s l o y a [Dynamics of Rarefied Gas and Boundary Layer], Mosc. St. Univ., dep. V I N I T I , 4218-80 (1980), p p . 50-68. [In Russian] [135] Dubinskii, A. V., Some regularities of behavior of aerodynamic characteristics in a flow determined on t h e basis of the "localised stream-surface interaction models", in D i n a m i k a r a z r e z h e n n o g o g a z a i p o g r a n i c h n o g o s l o y a [Dy­ namics of Rarefied Gas and Boundary Layer], Mosc. St. Univ., dep. VINITI, 4218-80 (1980), p p . 13-32. [In Russian] [136] Dubinskii, A. V., Some classes of o p t i m u m 3-D bodies in a stream determined on t h e basis of the "Localised stream-surface interaction models'', in D i n a m i k a r a z r e z h e n n o g o g a z a i p o g r a n i c h n o g o s l o y a [Dynamics of Rarefied Gas and Boundary Layer], Mosc. St. Univ., dep. V I N I T I , 4218-80 (1980), p p . 81-101. [In Russian] [137] Dvoretskii, V. M., Ivanov, M. Ya., Koniaev, B. A., and Kraiko, A. N., On the "equivalence" rule for flows of perfect gas, Appl. Math. & Mech. 3 8 (1974), no. 6, 951-962.

Bibliography

211

[138] Etkin, B., Dynamics of Flight. Stability and Control, New York, London (1959). [139] Evans, W. J., Aerodynamic and radiation disturbance torques on satellites hav­ ing complex geometry, J. Astron. Sci. (1962), no. 4. [140] Farrar, D. E. and Glauber, R. R., Multicollinearity in regression analysis. The problem resisted, The Review of Economics and Statistics, 49 (1967), no. 1. [141] Ferrari, C., Aerodynamic problems of re-entry, Meccanica 6 (1971), no. 1, 2342. [142] Filippov, B. V., Bodies in the stream of high rarefied plasma, [10], issue 4 (1969), pp. 133-141. [In Russian] [143] Follet, M. I., Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flow, Fluid Dynamics 24 (1989), no. 6, 938-943. [144] Follet, M. I., Supersonic flow past a star-shaped body without symmetry planes at angle of attack, Fluid Dynamics 28 (1993), no. 4, 560-567. [145] Fundamentals of Gas Dynamics, ed. H. W. Emmons, Princeton Univ. Press, Princeton, New Jersey (1958). [146] Galkin, V. S., Determining moments and forces acting on rotating bodies in a free-molecular flow and in a light flow, Inzhenernii zhurnal (1965), no. 5, 954-958. [In Russian] [147] Galkin, V. S., Erofeev, A. I., and Tolstikh, A. I., Approximate method of calcu­ lating the aerodynamic characteristics of bodies in a hypersonic flow of rarefied gas, Trudy TsAGI 1833 (1977) 6-10. [In Russian] 148] Galkin, V, S., Erofeev, A. I., and Tolstikh, A. I., Approximate method of aero­ dynamic calculation for rarefied gas flow, Trudy TsAGI 2111 (1981) 27-35. [In Russian] 149] Galkin, V. S. and Zvorikin, L. L., Rotating derivatives of bodies in a hypersonic rarefied gas, Trudy TsAGI 2220 (1984) 3-15. [In Russian] 150] Garsia, F. Jr., Comment on the location of the center of pressure of the right circular cone using Newtonian impact theory. AIAA J. 4 (1966), no. 6, 1150— 1151. 151] Gil'man, 0 . A. and Pilyugin, N. N., Pareto-optimal forms of axisymmetrical bodies moving at high supersonic speeds, Appl. Math. & Mech. 55 (1991), no. 2, 232-237.

212

Bibliography

[152] Gonor, A. L., Conical bodies of m i n i m u m drag in hypersonic gas Appl. Math. & Mech. 2 8 (1964), no. 2, 471-475.

flow,

[153] Gonor, A. L. and Cherniy, G. G., Minimum-drag bodies at hypersonic speeds, Izvestiya Akademii Nauk SSSR, otdel. tekhnicheskikh nauk (1957), no. 7, 89-93. [In Russian] [154] Gonor, A. L., Kazakov, M. N., and Shvets, V. I., Aerodynamic characteristics of star-shaped bodies at supersonic velocities, Fluid Dynamics 6 (1971), no. 1, 86-89. [155] Gonor, A. A., Zubin, M. A., and Ostapenko, N. A., Experimental in­ vestigation of aerodynamic characteristics of star-shaped bodies in super­ sonic stream, in N e r a v n o v e s n i e t e c h e n i y a g a z a i o p t i m a l ' n i e f o r m i t e l v s v e r k h z v u k o v i k h p o t o k a k h [Nonequilibrium Gas Flows and O p t i m u m Shapes in Supersonic Streams], Mosc. St. Univ., Moscow (1978), p p . 28-39. [In Russian] [156] Gorenbukh, P. I., Correlative relation for drag coefficients of convex bodies in hypersonic rarefied gas flow, [282], Part 2, p p . 51-55. [In Russian] [157] Gorenbukh, P. I., Correlative relation for drag coefficients of bodies in hyper­ sonic rarefied gas flow, Uchonye Zapiski TsAGI 1 7 (1976), no. 2, 99-105. [In Russian] [158] Gorenbukh, P. I., Approximate calculation of aerodynamic characteristics of bodies of a simple form in a hypersonic rarefied gas flow, Trudy TsAGI 2 4 3 6 (1990) 28-43. [In Russian] [159] Grishina, R. B. and Petrov, V. P., Application of local m e t h o d in t h e case of nonuniforming stream, dep. VINITI, 8564-B (1986), p p . 1-11. [In Russian] [160] Guschina, N. A., Omelik, A. I., Pomarzhanskii, V. V., and Shvedov, A. V., Experimental determining aerodynamic characteristics of some grids in a. hy­ personic free-molecular stream, Uchonye Zapiski TsAGI 1 7 (1986), no. 4, 16-23 [In Russian]. [161] Gusev, V. N., Hypersonic modelling depend on variation of Mach number and Reynolds number, Uchonye Zapiski TsAGI 10 (1979), no. 6, 19-29. [In Russian] [162] Gusev, V. N., High-altitude aerodynamics, Fluid Dynamics

2 8 (1993), no. 2.

[163] Gusev, V. N., Erofeev, A. I., Klimova, T. V., Perepukhov, V. A., Ryabov, V. V., and Tolstikh, A. I., Theoretical and experimental investigations of rarefied gas flow past bodies of a simple shape, Trudy TsAGI 1 8 5 5 (1977) 3-43. [In Russian]

Bibliography

213

[164] Gusev, V. N., Klimova, T. V., and Lipin, A. V., Aerodynamic characteristics of bodies in transitional region of hypersonic gas flow, Trudy TsAGI 1411 (1972) 1-53. [In Russian] [165] Gusev, V. N., Kogan, M. N., and Perepukhov, V. A., A similarity and change of aerodynamic characteristics in hypersonic flow transition regime, Uchonye Zapiski TsAGI 1 (1970), no. 1, 24-32. [In Russian] [166] Hayes, W. D. and Probstein, R. F., Hypersonic Flow Theory, Acad. Press, New York, London (1959). [167] Hindelang, F. J. and Meisinger, R., Anwendbarkeit der Newtonischen Theorie zur Berechnung Aerodynamischer Beiwerte und Stabilitatsderiativa, Raumfahrtforschung 2 (1974), no. 18, 79-84. [168] Hobson, E. V., The Theory of Spherical and Ellipsoidal Harmonics, The Univ. Press, Cambridge (1931). [169] Hui, W. H., Unified area rule for hypersonic and supersonic wing-bodies, AIAA J. 19 (1972), no. 7, 961-962. [170] Ivanov, S. G., and Yanshin, A. M., Forces and moments acting on bodies rotat­ ing about a symmetry axis in a free molecular flow, Fluid Dynamics 15 (1980), no. 4, 449-453. [171] Ivanov, S. G. and Yanshin, A. M., Forces and moments acting on axisymmetric bodies arbitrary rotating in a free-molecular stream, in Kosmicheskie Issledovaniya na Ukraine [Cosmic Research in the Ukraine], issue 16, Kiev (1982), pp. 25-29. [In Russian] [172] Ivanov, S. G. and Yanshin, A. M., Nonlinear effects associated with unsteady rotation of body in rarefied gas, in Gidrogazodinamika i teploobmen letatel'nikh apparatov [Hydrodynamics and Heat Exchange of Vehicles], Kiev (1988), pp. 114-117. [In Russian] [173] Jaslow, H., Aerodynamic relationships inherent in Newtonian impact theory, AIAA J. 6 (1968), no. 4, 608-612. [174] Jaslow, H., Lift and drag of flat-top configurations of arbitrary cross section, AIAA J. 6 (1968), no. 10, 2039-2040. [175] Jaslow, H., Nonaffine similarity laws inherent in Newtonian impact theory, AIAA J. 8 (1970), no. 11, 2062-2064. [176] Jaslow, H., Nonaffine similarity laws for powerlaw bodies, AIAA J. 9 (1971), no. 12, 2466-2468.

214

Bibliography

177] Jorgensen, L. H., Estimation of aerodynamics for slender bodies alone and with lifting surface at a ' s from 0 to 90, AIAA J. 11 (1973), no. 3, 409-412. 178] Kameko, V. F., Nikiforov, A. P., and Omelik, A. I., Experimental investigation of t h e m o m e n t u m entering t h e surfaces of various matherials in supersonic freemolecular s t r e a m , Uchonye Zapiski TsAGI 10 (1979), no. 5, 43-52. [In Russian] 179] Koriagin, V. P., Loshakov, A. V., and Shvets, A. I., Flow over blunted cones and segmental bodies, Zhurnal Prikladnoi Mekhaniki i Tekhnicheshoi Fiziki (1978), no. 5, 98-102. [In Russian] 180] Keel, A. G., Jr., Viscous effects on center-of-pressure location for sharp and blunted cones, J. Spacecraft and Rockets 1 3 (1976), no. 5, 316-317. 181] Khabalov, V. D., Influence of the roughness of surface on the regime coefficients in t h e local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1989), no. 1, 80-83. [In Russian] 182] Khalidov, I. A., Dependence of parameters of local interaction on rougness of surface, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1981), no. 19, 118— 120. [In Russian] 183] Khalidov, V. D., Plane variant of local interaction theory for rarefied gas flow, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1988), no. 1, 72-74. [In Russian] 184] Khalidov, I. A., Linear model in plane variant of local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1988), no. 4, 71-74. [In Russian] 185] Khmel'nitskii, A. A., Engineering method of calculation of aerodynamic char­ acteristics of bodies in a rarefied gas stream, [126], p. 63. [In Russian] 186] Kholyavin, 1.1., Determining Regime Coefficients of the Theory of Local Interac­ tion on t h e Basis of Plate Data, Vestnik Leningr. Univ., ser. Math., Mech., As­ tron. (1986), no. 2, 125-128. [In Russian] 187] Kogan, M. N., Drag of bodies t h a t like to bodies of revolution, zhurnal 1 (1961), no. 3, 156-158. [In Russian]

Inzhenerniy

188] Kogan, M. N., R a r e f i e d G a s D y n a m i c s , Plenum Press, New York (1969). 189] Kogan, M. N. and Perminov, V. D., Recovery of the normal and tangential ac­ commodation coefficients on the basis of results of force m e a s u r e m e n t , Uchonye Zapiski TsAGI 12 (1981), no. 3, 140-142. [In Russian]

Bibliography

215

[190] Korchagina, M. A. and Pilyugin, N. N., Choice of shape of bodies with minimum aerodynamic heating during motion in the atmospheres of planets in the solar system, Cosmic Research 29 (1991), no. 2, 257-266. [191] Korchagina, M. A. and Pilyugin, N. N., Optimal shapes of slender bodies for minimum aerodynamic heating in flight in solar system planetary atmospheres, Cosmic Research 29 (1993), no. 3, 340-345. [192] Korn, G. A. and Korn, N. V., Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Corp., New York, San Francisco, Toronto, London, Sydney (1968). [193] Koshmarov, Yu. A. and Rizhov, Yu. A., Prikladnaya dinamika razrezhennogo gaza [Applied Rarefied Gas Dynamics], Mashinostroenie, Moscow (1977). [In Russian] [194] Kostrov, A. V., Dvizhenie asimmetrichnikh ballisticheskikh apparatov [The Motion of Asymmetrical Ballistic Vehicles], Mashinostroenie, Moscow (1984). [In Russian] [195] Kosykh, A. P. and Minailos, A. N., Aerodynamic characteristics of simplest form wings at supersonic speeds, Trudy TsAGI 1891 (1977) 1-35. [In Russian] [196] Kotov, V. M., Lychkin, E. N., Reshetin, A. G., and Schelkonogov, A. N., An approximate method of aerodynamic calculation of complex shape bodies in the transition region, [259], pp. 487-494. [197] Kotov, V. M., Lychkin, E. N., Reshetin, A. G., and Schelkonogov, A. N., Calcu­ lation of aerodynamic characteristics of complex shape bodies in transitional re­ gion, in Chislennoe modelirovanie v aerogidrodinamike [Computational Modelling in Aerohydrodynamics], Moscow (1986), pp. 115-124. [In Russian] [198] Kovtunenko, V. M., Kameko, V. F., and Yaskevich, E. P., Aerodinamika orbital'nikh kosmicheskikh apparatov [Aerodynamics of Space Vehicles], Naukova Dumka, Kiev (1977). [In Russian] [199] Kraiko, A. N., Variatsionnie zadachi gazovoi dinamiki [Variational Prob­ lems of Gas Dynamics], Nauka, Moscow (1979). [In Russian] [200] Kraiko, A. N., Minimum wave drag body head of given volume in an approxima­ tion of Newton's drag law, J. Appl. Math. & Mech. 55 (1991), no. 3, 310-315. [201] Krasnov, N. F., Aerodinamika tel vrascheniya [Aerodynamics of bodies of Revolution], Mashinostroenie, Moscow (1964). [In Russian]

216

Bibliography

[202] Krasnov, N. F., Borovskii, E. E., and Khlupnov, A. I., Osnovi prikladnoi aerogazodinamiki [Foundations of Applied Aerodynamics], Part 1, Visshaya Shkola, Moscow (1990). [In Russian] [203] Krasnov, N. F. and Koshevoi, V. N., Upravlenie i stabilizatsiya v aerodinamike [The Control and the Stabilization in Aerodynamics], Visshaya Shkola, Moscow (1978). [In Russian] [204] Krasovskii, V. M., Experimental investigation of flow over blunt-nosed bodies in hypersonic helium stream, Inzhenernii Zhurnal (1961), no. 2, 240-253. [In Russian] [205] Kurishev, A. P. and Filippov, B. V., Aerodynamical coefficients of bodies of revolution in a high rarefied plasma, [10], issue 4 (1969). [In Russian] [206] Kurishev, A. P. and Filippov, B. V., Influence of thermal velocities of ions on the aerodynamic coefficients of bodies in high rarefied plasma, [10], issue 4 (1969). [In Russian] [207] Kuzmenko, V. I., Approximate method of calculation of convective heat ex­ change in hypersonic rarefied gas flow, in Dinamika razrezhennogo gaza i pogranichnogo sloya [Dynamics of Rarefied Gas and Boundary Layer], Mosc. St. Univ., dep. VINITI, 4218-80 (1980), pp. 33-41. [In Russian] [208] Kuznetsov, M. M. and Polyanskii, 0 . Yu., The position of centre of pressure on a triangular wing in a hypersonic nonequilibrium air flow, Uchonye Zapiski TsAGI 14 (1983), no. 5, 97-99. [In Russian] [209] Ladizhenskii, M. D., Generalization of hypersonic area rule, Izvestiya Akademii Nauk SSSR, otdel. tekhnicheskikh nauk, Mekhanika i mashinostroenie (1961), no. 3, 188-189. [In Russian] [210] Ladizhenskii, M. D., Hypersonic area rule, AIAA J. 1 (1963), no. 11, 2696-2698. [211] Levin, V. A. and Pilyugin, N. N., Optimal'nie aerodinamicheskie formi po radiatsionnomu teploobmenu [Bodies with Optimal Thermal Radiation], in Aktual'nie problemi mekhaniki [Current Problems of Mechanics], Mosc. St. Univ., Moscow (1984), pp. 59-61. [In Russian] [212] Levina, 0 . A., Aerodynamic forces acting on body of revolution and determi­ nation of form of optimal bodies in hypersonic Newtonian flow and in flow of rarefied gas, Moscow Univ. Mech. Bull. 30 (1975), no. 3-4, 32-37. [213] Lun'ov, V. V., Giperzvukovaya aerodinamika [Hypersonic Aerodynamics], Mashinostroenie, Moscow (1975). [In Russian]

Bibliography

217

[214] Lun'ov, V. V., Magomedov, K. D., and Pavlov, V. G., Giperzvukovoe obtekanie prituplennikh konusov s uchetom ravnovesnikh fiziko-khimicheskikh prevraschenii [Hypersonic Flow past Blunt-Nosed Cones with Account of Equilibrium Physical and Chemical Reactions], publ. by Comp. Cen­ ter of Soviet Acad. of Sci. (1968). [In Russian] [215] Malmuth, N. D., Hypersonic flow over a delta wing of moderate aspect ratio, AIAA J. 4 (1966), no. 3, 555-556. [216] Malmuth, N. D., A New area rule for hypersonic wing-bodies, AIAA J. 9 (1971), no. 12, 2460-2462. [217] Martino Heat transfer in slip flow, UTIA Rep. 35 (1945). [218] Matting, F. W., Approximate bridging relations in the transitional regime be­ tween continuum and free- molecule flows, J. Spacecraft and Rockets 1 (1971), no. 1, 35-40. [219] Maxwell, J. C , On the stresses in rarefied gases, the Scientific Papers, 2A, Paris (1890). [220] Mikhalevich, V. S., Consecutive algorithms of optimization and their applica­ tion, Parts 1-2, Kibernetika (1965), no. 1, 45-56; no. 2, 85-89. [In Russian] [221] Miroshin, R. N., Linear regression analysis of experiments in rarefied gas flow, [10], issue 5 (1970). [In Russian] [222] Miroshin, R. N., Remarks on the local interaction theory in rarefied gas, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1983), no. 13, 51-57. [In Russian] [223] Miroshin, R. N., Optimum aerodynamic shapes in rarefied gas flow, Vestnik SPeterburg. Univ., ser. Math., Mech., Astron. (1993), no. 1, 77-82. [In Russian] [224] Miroshin, R. N., Solving some internal problems of rarefied gas aerody­ namics using methods of nonlinear dynamics. Vestnik St.-Peterburg. Univ, ser. Math., Mech., Astron., (1993), no. 2, 94-98. [In Russian] [225] Miroshin, R. N., Mathematical problems of local interaction theory, Dokladi Akademii Nauk SSSR 285 (1985), no. 5, 1078-1081. [In Russian] [226] Miroshin, R. N., Local interaction theory in a rarefied gas dynamics—a variant of "tangent cones method", Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1985), no. 15, 103-105. [In Russian] [227] Miroshin, R. N., Linear independence of the form functions of sharp cones in the local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1986), no. 2, 69-76. [In Russian]

218

Bibliography

[228] Miroshin, R. N., T h e simplest model of local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1987), no. 1, 63-67. [In Russian] [229] Miroshin, R. N., Some m a t h e m a t i c a l problems of the local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1987), no. 4, 41-46. [In Russian] [230] Miroshin, R. N., Series for aerodynamical coefficients in t h e local interaction theory, Vestnik Leningr. Univ., ser. Math., Mech., Astron. 4 (1988), no. 22, 65-68. [In Russian] [231] Miroshin, R. N., Dependence of convex bodies drag coefficients in rarefied gas on t h e Reynolds number, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1989), no. 2, 48-51. [In Russian] [232] Miroshin, R. N. and Khalidov, I. A., T e o r i y a l o k a l ' n o g o v z a i m o d e i s t v i y a [Local Interaction Theory], Leningr. St. Univ., Leningrad (1991). [In Russian] [233] Moiseev, N. N., E l e m e n t i t e o r i i o p t i m a l ' n i k h s i s t e m [Elements of Optimal Systems Theory], Nauka, Moscow (1975). [In Russian] [234] Monaco, R. and Orsi, A. P., Molecular gas flow over convex bodies: a physicom a t h e m a t i c a l model for heat transfer calculation, Appl. Math. Model. 4 (1980), no. 5, 225-230. [235] Musanov, S. V., Nikiforov, A. R , Omelik, A. I., and Freedlender, 0 . G-, Ex­ perimental determination of m o m e n t u m transfer coefficients in hypersonic free molecular flow and distribution function recovery of reflected molecules, [259], p p . 669-676 [236] Naritsa, V. S., Determining regime coefficients for slender bodies accord­ ing to the d a t a of computational and experimental investigation, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1983), no. 7, 53-56. [In Russian] [237] Naritsa, V. S., Aerodynamic computation of slender bodies in rarefied gas, Vestnik Leningr. Univ., ser. Math., Mech., Astron. (1987), no. 4, 46-50. [In Russian] [238] Naritsa, V. S., Analysis of approximations of aerodynamic characteristics of slender bodies in rarefied gas. Vestnik Leningr. Univ., ser. Math., Mech., As­ tron. (1987), no. 2, 114-116. [In Russian] [239] Nersesov, G. G. and Shustov, V. I., Tables of aerodynamic characteristics of spherically blunted cones (/? from 2.5 to 35 deg., M = 2 - 20 ), Trudy TsAGI 1 6 2 9 (1975) 1-21. [In Russian]

Bibliography

219

[240] Nikolaev, V. S., Approximations for local aerodynamic characteristics of bodies like wings over a wide range of similarity parameters of viscous hypersonic gas flow, Uchonye Zapiski TsAGI 12 (1981), no. 4, 143-150. [In Russian] [241] Omelik, A. I., Measurement of momentum transport to surfaces of different form in a hypersonic free-molecular stream. Fluid Dynamics 12 (1977), no. 4, 633-635. [242] Ostapenko, N. A., On the center of pressure of conical bodies, Fluid Dynamics 15 (1980), no. 1. [243] Ostapenko, N. A., Bodies of minimal wave drag in a swirling hypersonic flow, Fluid Dynamics 18 (1983), no. 1, 86-95. [244] Ostapenko, N. A., Conical bodies with star-shaped section which possesses a, reserve of static stability, Fluid Dynamics 19 (1984), no. 6, 930-937. [245] Ostapenko, N. A. and Yakunina, G. E., Least-drag bodies moving in media subject to the locality hypothesis, Fluid Dynamics 27 (1992), no. 1, 71-80. [246] Oswatitsch, K., The area rule, Appl. Math. Rev. 10 (1957), no. 12, 543-545. [247] Perminov, V. D., Wings with optimal characteristics in hypersonic flow, Fluid Dynamics 4 (1969), no. 6, 42-46. [248] Perminov, V. D., Application program "Altitude-2", [127], p. 164. [In Russian] [249] Perminov, V. D., Gorelov, S. L., and Freedlender, O. G., Khmelnitsky A. A., Approximate aerodynamic analysis for complicated bodies in rarefied gas flow, in Rarefied Gas Dynamics, Proc. 17th Int. Symp. on Rarefied Gas Dynamics, Veincheim (1991), pp. 554-561. [250] Pike, J., The lift and drag of axisymmetric bodies in Newtonian flow, AIAA J., 7 (1969), no. 1, 185-186. [251] Pike, J., Newtonian lift and drag of blunt-cone cylinder bodies, AIAA J. 10 (1972), no. 2, 176-180. [252] Pike, J., Newtonian aerodynamic forces from Poisson's equation, AIAA J. 11 (1973), no. 4, 499-504. [253] Pike, J., Moments and forces on general convex bodies in hypersonic flow, AIAA J. 12 (1974), no. 9, 1241-1247. [254] Pike, J., Forces on convex bodies in free molecular flow, AIAA J. 13 (1975), no. 11, 1454-1459.

220

Bibliography

[255] Pilyugin, N. N., Tikhomirov, S. G., and Chernyavskii, S. Yu., Area rule for the wake behind a three-dimensional body, Fluid Dynamics 15 (1980), no. 4, 447-448. [256] Pogorelov, A. V., Differentsial'naya geometriya [Differential Geometry], Nauka, Moscow (1974). [In Russian] [257] Ponomarev, V. Ya. and Seregin, V. S., Calculation of the aerodynamic charac­ teristics of standard bodies on the local interaction conjecture for steady and unsteady motion in a rarefied gas, [281], pp. 392-398. [In Russian] [258] Rakhmatulin, Kh. A., Sagomonyan, A. Ya., Bunimovich, A. I., and Zverev, I. N., Gazovaya dinamika [Gas Dynamics], Visshaya Shkola, Moscow (1965). [In Russian] [259] Rarefied Gas Dynamics, Revised Versions of Selected Papers from the 13th International Symposium (Novosibirsk, 1982), Plenum Press, New York (1985), Vol. 1-2, 1424 pp. , [260] Reshetin, A. G., Lichkin, E. N., Kotov, V. M., and Schelkonogov, A. N., Viscous gas Flow over Complex shape bodies, Chislennie metodi mekhaniki sploshnoi sredy (Novosibirsk) 11 (1980), no. 6, 110-122. [In Russian] [261] Rizo, A. E., Empirical formulas for cones in subsonic aerodynamics, Vestnik Leningr. Univ., ser. Math., Mech., Astron. 2 (1988), no. 8, 116-117. [In Russian] [262] Rusanov, V. V. and Nazhestkina, E. I., Volnovoe soprotivlenie tela vrascheniya stepennoi formi (osesimmetrichnoe obtekanie). [Wave Drag of Bodies of Revo­ lution of Power-Law Form (Axisymmetrical Flow)], preprint, Inst. Prikladnoy Mekhaniki, Akademiya Nauk SSSR (1972), no. 33, pp. 1-28. [In Russian] [263] Sagomonyan, A. Ya., Pronikanie [Penetration], Mosc. St. Univ., Moscow (1974). [In Russian] [264] Sazonova, N. I., Not fully established flow around bodies under the conditions of the law of localness, Fluid Dynamics 12 (1977), no. 3, 485-489. [265] Schrello, D. M., Approximate free-molecule aerodynamics, ARS J. 30 (1960), no. 8, 765-767. [266] Sedov, L. I., Similarity and Dimensional Methods in Mechanics, Mir, Moscow (1982). [267] Shumskii, A. V., Engineering method of calculation of the aerodynamic heating on bodies in a transitional regime, [127], p. 173. [In Russian]

Bibliography

221

[268] Shvedov, A. V., Calculation of aerodynamic characteristics of bodies with grid surfaces in a hypersonic rarefied stream, Trudy TsAGI 2436 (1990) 44-60. [In Russian] [269] Shvets, A. I., Aerodinamilca sverkhzvukovikh form [Aerodynamics of Su­ personic Shapes], Mosc. St. Univ., Moscow (1987). [In Russian] [270] Shvets, A. I., Sverkhzvukovie letatel'nie apparati [Hypersonic Vehicles], Mosc. St. Univ., Moscow (1989). [In Russian] [271] Shvets, A. I. and Shvets, I. T., Aerodinamilca nesuschikh form [Aerody­ namics of Lifting Bodies], Vischa Shkola, Kiev (1985) [In Russian]. [272] Stenson, K. F. and Lewis, A. B., Aerodynamic compression of the conical and biconical re-entry vehicle, in A I A A — A t m o s . Flight. Mach. Conf., Holly­ wood, Fla. (1977), pp. 333-338. [273] Stenson, K. F. and Lewis, A. B., Static stability compression of a conical and biconical configuration, J. Spacecraft and Rockets 15 (1978), no. 5, 257-258. [274] Stulov, V. P. and Shapiro, E. G., Shock layer radiation in hypersonic airflow past blunted bodies, Fluid Dynamics (1970), no. 1. [275] Tactical Missile Aerodynamics, eds. M. J. Hemsh, J. N. Nielsen, American Institute of Aeronautics and Astronautics, New York (1986). [276] Tezisi dokladov VIII Vsesoyuznoi konferentsii po dinamike razrezhennikh gazov [8th Ail-Union Conf. on Rarefied Gas Dynamics and Molecular Gas Dynamics (Moscow, 1985). Book of Abstracts], Parts 1-2, Moscow (1985). [In Russian] [277] Tezisi dokladov IX Vsesoyuznoi konferentsii po dinamike razrezhennikh gazov [9th Ail-Union Conf. on Rarefied Gas Dynamics and molecular Gas Dynam­ ics (Sverdlovsk, 1987). Book of Abstracts], Parts 1-2. Sverdlovsk (1987). [In Russian] [278] Theory of Optimum Aerodynamic Shapes, Acad. Press, New York, Lon­ don (1965). [279] Timoshenko, V. I., Using the differential correlations of local interaction the­ ory for forces and moments for more accurate determination of aerodynamic characteristics of bodies in rarefied gas, [128], no. 2, 104-109. [In Russian] [280] Tirsky, G. A., To the theory of the hypersonic viscous chemically active multicomponent gas flow past plane and axisymmetric bodies with gas injection, Nauchnie trudi inst. mekhaniki MGU (Mosc. St. Univ.) (1975), no. 39, 5-28. [In Russian]

222

Bibliography

[281] Trudy IV Vsesoyuznoi Konferentsii po dinamike razrezhennogo gaza i molekulyarnoi gazovoi dinamike [Proc. of the 4th Ail-Union Conf.on Rarefied Gas Dy­ namics and Molecular Gas Dynamics (Moscow, 1975)], TsAGI, Moscow (1977). [In Russian] [282] Trudy VIII Vsesoyuznoi Konferentsii po dinamike razrezhennikh gazov [Proc. of the 8th Ail-Union Conf. on Rarefied Gas Dynamics (Moscow, 1985)], Parts 1-4, Moscow (1986). [In Russian] [283] Vapne, V. D., Accommodation coefficients as a random variables, [10], issue 6 (1972), pp. 8-10. [In Russian] [284] Vasil'chenko, V. I., Decrease in the wave resistance of the combination of a body of revolution with a wing in a supersonic gas stream, Fluid Dynamics 16 (1981), no. 3, 464-467. [285] Vasil'ev, L. A., Opredelenie davleniya sveta na kosmicheskie apparaty [Calculation of Light Pressure on Space Vehicles], Mashinostroenie, Moscow (1985). [In Russian] [286] Vedernikov, Yu. A., Gonor, A. L., Zubin, M. A., and Ostapenko, N. A., Aero­ dynamic characteristics of star-shaped bodies at Mach numbers M = 3 — 5, Fluid Dynamics 16 (1981), no. 4, 559-563. [287] Vitman, F. F. and Stepanov, V. A., Effect of the strain rate on the resistance of metals to deformation at impact velocities of 100-1000 m/sec, in Nekotorie problemi prochnosti tverdogo tela [Some Problems of the Strength of Solids], publ. by the USSR Acad. of Sci., Moscow, Leningrad (1959). [In Russian] [288] Vitman, F. F. and Zlatin, W. A., Collision of deformable bodies and its mod­ elling. Status and theory, Soviet Phys. Tech. Phys. 8 (1964), no. 8, 730-735. [289] Voronkin, V. G., Lunov, V. V., and Nikulin, A. N., Stationary shape of bod­ ies during their rupture because of aerodynamic heating, Fluid Dynamics 13 (1978), no. 2, 274-281. [290] Vorotyntsev, M. A. and Sazonova, N. I., Determination of the moments of aerodynamic forces acting on three-dimensional bodies that move under the "law of locality". Moscow Univ. Mech. Bull. 31 (1976), no. 1-2, 30-34. [291] Wang, Ch. T., Free molecular flow over a rotating sphere, AIAA J. 10 (1972), no. 5, 713-714. [292] Widhoph, G. F., Heat-transfer correlations for blunt cones at angle of attack, J. of Spacecraft and Rockets 8 (1971), no. 9.

Bibliography

223

[293] Wittlift, C. E., Correction of drag coefficients for sharp cones, AIAA J. 6 (1968), no. 7. [294] Yanshin, A. M., Average aerodynamic characteristics of satellite, Cosmic Re­ search 21 (1983), no. 5, 535-539. [295] Yanshin, A. M. and Ivanov, S. G., Calculation of unsteady aerodynamic forces and moments in free-molecular flow, Cosmic Research 27 (1989), no. 3, 323329. [296] Yanshin, A. M. and Zabluda, S. M., Optimal program for control orientation of space vehicle with planar panels of solar batteries, Cosmic Research 27 (1989), no. 6. [297] Yaroshevskii, V. A., Dvizhenie neupravlyaemogo tela v atmosfere [Mo­ tion of Uncontroled body in the Atmosphere], , Moscow (1978). [In Russian] [298] Yaskevich, E. P. and Filatov, E. I., Additional aerodynamical forces and mo­ ments in the free-molecular flow acting on a body of revolution rotating around the symmetry axis, in Gidrodinamika i teoriya uprugosti. Respublikanskii nauchno-tekhnnicheskii sbornik (Kharkov), issue 6 (1967), pp. 17-23. [In Russian] [299] Zabolotnev, 0 . A., Approximate determination of the rotational derivative co­ efficients of bodies of revolution at great Mach numbers, in Aerodinamika neustanovivshegosya dvizheniya [Aerodynamics of Unsteady Motion], Col­ lected Articles, issue 2; Trudy VVIA im. N.E.Zhukovskogo 828 (1961) 114-132. [In Russian] [300] Zmievskaya, G. I., Aerodynamics of satellites and determining the surfacestream interaction parameters on the basis of observing the braking of the "Kosmos-166" and "Kosmos-230" in the higher atmosphere, in Prikladnie zadachi kosmicheskoi ballistiki [Applied Problems of Cosmic Ballistics], Nauka, Moscow (1973). [In Russian] [301] Znamenskii, V. V. and Zubarev, A. V., Calculation of heat conversion for threedimensional flow past body in the case of given pressure distribution, Fluid Dynamics 22 (1987), no. 3. [302] Zubin, M. A, Lapygin, V. I., and Ostapenko, N. A., Theoretical and experimen­ tal investigation into the structure of supersonic flow past bodies of star-shaped form and their aerodynamic characteristics, Fluid Dynamics 17 (1982), no. 3, 355-360.

224

Bibliography

[303] Zubin, M. A. and Ostapenko, N. A., Aerodynamic and static stability of starshaped conical bodies in supersonic flow, Fluid Dynamics (1992/1993), no. 6, 867-872. [304] Zueva, E. Yu., Komarov, M. M., and Sazonov, V. V., Representation of the aerodynamic moment in problems of mathematical modeling of the rotational motion of satellites, Cosmic Research 30 (1992), no. 6, 622-628.

Subject Index airfoil, 5, 6, 65 area rules, vii, 26, 89, 91, 93

global parameters, v, 1,8, 52, 170, 171 grid, 27, 143

center of pressure (CP), 25, 79, 96 characteristic area, 29, 36, 57, 80, 84, 89, 131, 145, 185 characteristic dimension, 4, 14, 15, 86 coefficient of a center of pressure (CCP), 96 cone, 7, 14, 15, 18, 25, 49, 73, 91-93, 104-106, 110, 119, 121, 123,124, 127, 132, 145, 147, 148 continuum mechanics, 10, 11 convective heat transfer, 19

heat flux (transfer), 2, 4, 19, 20, 23 heat flux (transfer) coefficient, 26 integral characteristic (IC), v, vii, viii, 4, 6, 20, 21, 25, 26, 41, 43-46, 51-53, 58, 62, 66-68, 70-72, 77, 174 invariant relationships, 24, 52-54, 58, 62, 70, 75, 77, 175

elliptical cylinder, 103 Euler equation, 140 Euler-Ostrogradsky equation, 130, 135 exposed area (region, surface), 2, 3, 22, 29-31, 33, 36, 174-176, 179, 182, 183, 185

Legendre condition, 131, 136-138, 140 light flux, v, 9, 10, 21, 27 local force coefficient, 1 local heat flux coefficient, 2 local interaction hypothesis (LIH), 23 local method, 10, 12, 27 locality, 5, 9-11, 19, 20, 23, 24, 26, 77, 169 localized interaction model (LIM), vviii, 3-7, 9-13, 15, 18-24, 2628, 36, 52, 57, 63, 74, 79, 81, 91, 96, 98, 101, 103, 105, 129, 132, 133, 138, 139, 142, 143, 146, 171, 175, 183 localized interaction theory (LIT), vviii, 1,3-5, 7,10-12, 14,20,21, 23-28, 63, 70, 93, 101, 169-171

free-molecular flow (regime), v, 5, 10, 14, 15, 18, 20, 22, 27, 91, 139, 142, 170, 171

metal, v, viii, 27, 28, 159, 162 moment coefficient, 4, 70, 95, 112, 173, 174

dense gas flow, 5, 10, 13-15, 19, 20, 22, 23, 70, 72, 91, 106, 139, 171 density, 1, 166 drag coefficient, 15, 40, 49, 51, 72-74, 80, 131-133, 137, 139, 141, 143, 145, 147, 150, 153, 155-157, 159, 187 dynamic head, 1, 55, 62, 63, 79, 80, 170

225

226 natural boundary, 2, 3, 30, 39, 185 Navier-Stokes equation, 10 negative stability, 96 neutral stability, 96 Newton model, 5-7, 10, 21-28, 34, 74, 96,110, 132, 133, 140, 150, 154, 162, 167 paraboloid, 20 penetration, v, 27, 28, 159-165, 167 plasma, v, 27 positive stability, 96 powerlaw body, 153 pressure coefficient, 2 pyramid, 25, 91, 109, 110, 113, 116 radiative heat transfer, 19, 23 regime coefficients, 12, 15, 18 rotary derivatives, viii, 21, 22, 26, 174, 177, 182, 183 segmental-conical body, 25, 98, 124 shaded area (region), 2, 3, 6, 56, 63, 88, 96 shape functions, 12 soil, v, viii, 1, 27, 28, 159, 162 sphere, 14, 15, 19, 123, 126, 127, 153, 159 star-shaped body, 25, 28, 46, 96, 162, 167 static stability, vii, 25, 54, 88, 96 supersonic gas flow, v, 5, 6, 70, 72, 77, 98, 103, 106, 171 tangent force coefficient, 2 tangent-cone method, 7, 10 temperature factor, 14, 15, 146 transition region (regime), vi, 5, 10-15, 18, 20, 22, 80, 139 Weierstrass condition, 131 wing, 25, 53, 63-65, 81, 102, 103, 110, 112, 119,121, 133-135, 137, 138 winged elliptical cone, 119

Subject Index