LITERATURE CITED i. Yu. P. Gupalo and V. M. Ostrik, "Steady-state operating regimes of chemical reactors with longitudinal and transverse mixing," Prikl. Mat, Mekh., 47, 73 (1983). 2. F. L. Chernous'ko, "Method of local variations for the numerical solution of variational problems," Zh. Vychisl. Mat. Mat. Fiz., ~, 749 (1965).

TRANSONIC GAS FLOW OVER A FLAT PLATE S. K. Aslanov

UDC 533.6.011.35

The solution of the two-sided Tricomi problem in the hodograph plane is constructed with satisfaction of the entire se[ of boundary conditions, which ensures its correct asymptotic behavior with respect to vanishing angle of attack. As a result, it is found that the deviation from the Guderley solution begins with the singular terms. The simplest of nonsymmetric transonic flow regimes is that of a flat plate at a small angle of attack in a sonic flow. While basic, this problem is characterized by the existence of two characteristic scales and belongs to the class of gasdynamic problems with "special" perturbations. Its solution was found by Guderley (1954) but only within a reduced formulation in which the specific boundary conditions for the flow over the forward part of the plate were not taken fully into account and, in particular, the true nature of the branching of the flow at the stagnation point on the surface of the plate was not established in advance. This, however, is needed to select the coefficient in constructing the two-term singularity of the corresponding sonic regime whose influence naturally extends to the flow as a whole. i. Consider a plane unbounded gas flow having the speed of sound a, at infinity and flowing over a flat plate of length s with a small angle of attack ~ (Fig. i). If we use the Tricomi equation, then for the stream function ~(n, 8) in the hodograph plane (Fig. 2) we have the following boundary-value problem:

,~.+~8.=0 ,(%-~)=o, , ( % ~ - a ) = o , 4=o,

~=(~+i)"(l-w/a,)

(i. i) (1.2)

The condition ~ = 0 is satisfied on the characteristics AE and BD corresponding to the deflection of the supersonic flow about the points A and B. Here, 8 is the angle of inclination of the velocity vector to the direction of the free stream at infinity, and ~ is the Frankl' variable [2], which in the near-sonic approximation is expressed in terms of the modulus of the velocity w and the ratio of specific heats 7. The last condition can be carried to the sonic line ~ = 0 in the form of Tricomi integral relations [1 ] 8

#?(0, O) =

3~'r'('/,) .[ ,~(o,t) . 4~ 2 (--~_t-~,iclt

( 1.3 )

valid for the intervals AC (j = 0) and BC (j = i), respectively. The congruence of the regions in Figs. 1 and 2 is indicated by the use of identical notation for the points. The scale of conversion from the hodograph plane (~, 8) to the physical plane (x, y) establishes the multiplicative constant in the solution of the homogeneous problem (i. I)- (i. 3) in conjunction with the given plate length condition xa(0)--x,(0)--l, XA(N)=X(~],--6), XB(~])=X(~],~--6)

(1.4)

Odessa. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 128-137, January-February, 1987. Original article submitted April 8, 1986.

0015-4628/87/2201-0109512.50

9

Plenum Publishing Corporation

109

Y c

ZO=Q4,

9

li C Fig. I

Fig. 2

On the hodograph the solution of the problem ~(n, 8) should have a singularity at the point C, into which is mapped an infinitely remote point of the flow plane: ~ ~ as the singular point C is approached, remaining bounded on the limiting characteristics EC (DC) [2, 3]. The branching of the streamline CO ($ = 0), which abuts against the stagnation point O along the normal to the plate (8 = ~/2 - 6), into two parts (OA from 8 = -6 and OB from @ = ~ -- 6) permits the unique construction of the singularity inherent in nonsymmetric sonic flow over a profile and composed of terms odd and even in @ [3]. The boundary-value problem thus stated, which so far remains unsolved, was first investigated by Guderley [4, 5] in a simplified formulation: the solution was constructed in the form of an infinite series in self-similar particular solutions of Eq. (i.i) with a common center at the point A. As a result of this approach, on the concrete part of the boindary OBD boundary conditions (1.2), (1.3) remained unsatisfied and, similarly, the specified branching of the streamline CO, which necessarily includes precisely this part of the boundary, was not achieved. At the same time, the condition of branching of the flow at the stagnation point O plays an important part in the correct construction of the singularity, which the solution of the problem must possess at the origin C in the hodograph plane (Fig. 2), and hence its influence extends to the entire boundary region. The assertion that within the context of the "passage to the limit 6 + 0 the boundary condition on OBD does not play an important part" [4, 5] can in no way be acknowledged to be correct, since from the standpoint of the theory of the slender profile the problem is, in the terminology of [6], a problem with "special" perturbations: in the case of flow over a plate with a small angle of attack 6 together with "regular" perturbations (commensurable in magnitude with 6) in the neighborhood of the leading edge finite ("special") perturbations are excited; whatever the form of the boundary conditions these cannot automatically disappear from the region of excitation when 6 = 0. Accordingly, if at the limit when 6 = 0 the solution is to satisfy the natural requirement thai all the perturbations vanish (transition to homogeneous flow), it is necessary correctly to satisfy the boundary conditions for the neighborhood of the leading edge and the stagnation point, i.e., on the specifically given line OBD corresponding to the part of the surface of the plate upstream from the stagnation point (Fig. i). In this case the solution for small 6 will be the asymptotic form of the general solution of the problem investigated. The simplified formulation adopted in [5] cannot be acknowledged to be correct, since it does not establish in advance the nature of the branching of the flow at the point O required above. Previously, in [7] an attempt was made to analyze the problem in the complete formulation (1.1)-(1.4). On the basis of the numerical results of satisfying boundary condition (1.3) at individual points it was suggested that the coefficients of the regular part of the solution an ~ 0 together with 6 + 0. In what follows the problem

ii0

of flow over a plate formulated above is completely solved. "2. In the region ~ > 0 the solution is written in the form [7]:

,=271 2

b

[ (n~'+a.)cos n5 + --~,?in n6 ]~ (n'l'n) sin n (0+5)

(2.1)

-=,

b = - ( a , + t ) etg 5,

~(0) = i ,

~ (+=)----0

where X is the Airy function. Boundary conditions (1.2) are completely saisfied; in this case the natural 2~ periodicity of the solution (2.1) with respect to the angle 0 is achieved thanks to the fulfillment of the condition at 0 -- ~ -- 6 ignored in [4, 5]. The solution possesses the required singularity and the correct branching at the point O; the coefficients of the regular part have the following order [2]: an(5) = O(n-4:a). They are determined from the remaining boundary condition (1.3). This requires the continuation of the solution onto the sonic line ~ -- 0, where the series (2.1) diverges. Summation of these divergent series [7] makes it possible to continue them analytically onto the straight line ~ = 0, after having explicitly separated the singular terms :

A ,(0,0) ----~- ~ys[P_,~,(0) + aa

(2~)_V, 2

r,. bSI2> TM 0 t,m z,, ( 0 ) - S2.-,()]+2To(O)

(2.2)

nml

(2n)-v' Z

bSz.m ( 0 ) ]+2T~,(0)

'S t 2~') . - , t o. .) -.z.n

T~(O)----2 m"a' cos m8 sin re(O+5), nt~

s~c~=o~r

P,(O)'=Oq- (-t)~(0+26)q

t

(0),

o~c~)_-- F(m+a/3)~(m+cz/3)m! (2a) ~ ,

~J (0) =

2~ 3,aF~(Vs)

(2.3)

where F and 5 are t h e gamma and z e t a f u n c t i o n s . Then the integrals which appear when conditions (1.3) are satisfied can be reduced to the known types [8] ll u

J'=.-,(~+l~)-~dx,

j" (=+l~)~@_~).-,dx

o

o

ee

u

u

0

(2.4) after having been expressed in terms of complete (F) and confluent (~) hypergeometric functions. Going over in the hypergeometric functions to the argument i/(I -- z), we can finally write the conditions for determining the coefficients am in the following convenient form:

zo(o)=~(o)

~(o)=qD~(~) (-~
o -,, 3bd, 2+--~ V ~-"1~'6-:"+, ' f jJZ ~ ' ~ t

2: ,,: <">

(2.5)

<:,

i!i

2~b6m.. F.+v, --8-

n ,I,a . u~v~.+ .{o~.~v, . . . o. j - w _ . (~} (0, 8) ]cos n6

6z"'v" - 8ai p ~

T'

9~ (0) ----- ~2a

""

(o,

~) + T

o,~,~,

(o,

(2.6)

~)-

~o

{

~215 gb

(,}-r

t

~'(~"-'} (0,6) ---E~-~ a~. r'.+v,(O,6) (2a)-v'X -" n+/3 -- T ~,3.-,(0)-2=bS2. (O)] + o

e+26

o{., =~,

e+==+--T, co)

2

(--t)"n+~'a,,[

P

(0,8)

(0, a) ]cos n8

t,

(0,6)=(0+8-=~)~'O.

s _,_,.)

t

F.(z)=3--F(t,i-2n,~;-t-z) 2 F,., ,~, (0, 6)= (~-6)

3

]

;•

'

3

'

2n-V,l~Lt-2n'~ -2n;'-z)

)

-~ ;o_ - ( - I ) ~ ( = + 6 ) - ~ F (~& 3

q

~ . O' §

'3

'3

)

F. ~'>(0, 8)= (~t-8)~F (I, l-2n, 4 - 2 n ; 0 - ) - (-I)~ (u+8)'F (i, !-2n,--~--2n; 0+ )

(2.7)

Of course, the reason for the disappearance from these conditions of the terms singular at ~ = 0 is connected with the nature of the singularity, which cannot extend along the characteristics CE(CD). After odd continuation of the piecewise-analytic function on the interval [--~ -- 5, ~ -- 6] by the Fourier series

9(0) = Z

(A~,§

~(e), we represent it

m(0+8)

0

s--d

--6

0

(2.8) Then

from (2.5) we obtain the infinite system of equations

a,/=A,~,+A,.2 ( r e = l , 2 . . . . ), a,/=a=cos m6

(2.9)

3. In order to solve this system we make use of the fact that the angle of attack 6 is small. As a result, correct to 0(6) in (2.6) there remain as dominant terms those represented by the curly bracket. For the arguments 8+_ in (2.7) we carry out expansions in powers of 6/~ and isolate the dominant terms in the hypergeometric functions. Then, after using for the latter the Gauss recursion formula [8], we finally have

r m3 (o) = TV~ =- v,[2 L s a r ,, { Is) Fm ~ss (o, o) + ~8 ~z~=

l-

112

~2n--1

o ~:', ~~ 7 0(2._,) (,, . 27,-- r/~ ; 4 ) co, o) - 2-~/, ~ -if-/

."7,

S

els) •

~(5) A2n-1

-2~-~ ~

j

~'

+

As a result, standard form:

(3.i)

(- I)n n'l'an''[~n) (8, 0)- ~)(_x n)(0,0)] -~O (6)

r(v )

,

correct to 0(6) (2.9) gives an infinite system of the following

c,,,.a,/+ b.,

(re=t, 2. . . . )

(3.2)

]}

(3.3)

i

3 /L o

r

(+)[

(x+o-w, + T

(z+l)-'/,

sin mOz dz

1

[ +

(=z,O)- r

(~z,O)]sinm~zdz

(n~>2)

0

[ 15 ~ I ~

' ,8, i~,

+ ~

.~-,

,=

r

~h-, (i - ~)'/, 2 k - ~i~

.(8)

~2;r-i ]

3F~ r 1

(3.4) 0

We will show that the substitution A m = mZ-=a~ makes the system (3.2) quasiregu!ar [9]. The constant terms and the coefficients of the new unknowns in the transformed system, respectively, take the form Cmn = Cmn(m/n) I-~, B m = bm m1-~. For the purpose of an asymptotic estimation of (3.4) as n ~ ~ we arrive at !

o

Using the asymptotic forms of the confluent functions ~ appearing after integration of the last expression [8], we finally conclude that as n ~ = Cmn decreases not more slowly than O(n~-4;~). To estimate (3.3)-(3.4) for large values of m we use the asymptotic expressions Fourier integrals [i0], as a result of which we have Bm, Cmn ~ m -~ (n = i, 2 . . . . ).

for

Thus, when 0 < ~ < i/3 the estimates obtained above with respect to n and m make it possible, for any fixed 6, to satisfy the condition of regularity of the system in all the rows, starting with N + I, since from ICmn[ the series in n converges for all m, and when m > N its sum can be made less than u n i t y b y an appropriate choice of N. Satisfaction of the quasiregularity requirements makes ~ possible to find the solution of the infinite system in question by the reduction method [9]. Using the smallness of the parameter 6, we can calculate the constant terms bm exactly for a finite number m of Eqs. (3.2), if in the integral (3.3) we replace the sine by its argument and use the known formulas [8] for expressing integrals of the type i

~ z=(l-z)~F(=,~,~;zz)dz 0

in terms of hypergeometric series with argument equal to --i and generalized hypergeometric series 3F2 with the argument i. The value of the former can be expressed in F functions by means of two known relations [ii]. In order to use them it is first necessary to transform these hypergeometric series by repeated application of the identity

113

F(i, ~, ~; z) =

~-i

i

- - [F(i, ~-i, ~-i; ~) - i]

whose validity is fairly obvious. Similarly, it is possible to establish a formula for expressing the generalized series sF~(=, ~, i; 7, 2; z) in terms of the hypergeometric function and its differentiation rule. The application of this rule and subsequent passage to the limit z = 1 make it possible to employ the known relation for a hypergeometric function at the point 1 [8] and express the values of the generalized series appearing in bm in terms of F functions. As a result, for the constant terms of the system we have b~ = -2a - m 6 '~ - 5= -F

2[

r

=o.mo~'~z

(3.5)

The termwise integration of the power series and the hypergeometric series F and in (3.4) reduces to the evaluation of integrals of the same type as the last integral in (2.4) for u = i, so that the dominant terms O(i) of the coefficients Cmn are expressed in the form of series in confluent hypergeometric functions with argument • some of them elementary. The last integral in Cml is evaluated in the same way as (3.3) and is a small quantity O(6115). The fairly rapid convergence of the series forming Cmn permits the numerical calculations to be made without difficulty. Carrying out the calculations for m = i, 2, 3 led to the following: al=O,020~'~; a 2 = O . l ~ ' ~ ;

(3.6)

a3=0.114~'~

4. It remains to find the scale factor A from the satisfaction of condition (1.4). Integrating for approximation (i.i) the formula [5] for conversion from the hodograph plane to the flow plane along the plate intervals OA and OB (Fig. i), we determine

:.,(n)

Of+i)"p.<,.J' n,olo--~an.

(~+t)'h

~. (n) --- - -

co

il*0i0-,,-,d'l'l

(4.1)

ao

respectively. Here, 0, is the density of the sonic free stream, XA(0) and XB(0) = --s express the distance of the stagnation point O from the trailing and leading (s edges of the plate. On substitution io (4.1) solution (2.1) leads to integrals which can be evaluated, if for the Airy function we use the dependence [ii]

2 ]FT x (0 - ~/~r ('is) ~'s' "~{-~t'1, ) 4A (~ + , y '

,

fQ~> ( 0 ) - : ' ~ ' Q~') (o)+ r (%) ~

% }

(4.2)

c Z nlIK'/,(n ~)(oo,} 2,,,l' Q{s}(~l)=~'q sin (n, 6); ~=,-~-I p

(4.3)

where Kv is the Macdonald function. In the serids expressing x B only the factor cos n~ is added. In this case the attainment of the upper limit of integration in (4.1) for the singular part of the solution represented by the quantities QU should be understood in the limiting sense (q ~ 0), since these series diverge at q = 0. The above-mentioned limits were found [7, 12] in the form of expressions for the summation of such series, i.e., their continuation to the point q = 0. The use of the latter finally gives (y+1) V'

114

2

4

2

~-Ir,2=~_,l,~(4)2n-~62~-

an

2

4

zB(o)= ~ A (V+ ~)V'r ( 2 ){ I T

O,a,

~[ ,,=~

a,+~

-6-*~-'/, + T

R-'/, "- ~

an

(- i)" ~

2~q-~

2

zn\ ="-- 6 2 n + l

2

•

"'2n+l]J--

Rq = (= + 8)q+ (- i)q(= - 6)q

Confining ourselves to the dominant terms in the small quantity 6, in XA(0) we retain the terms with 6 -413 a n d XB

p.a.

(o)

~

. . . . . ,. La'/'r r l ta=~ {%} + p %,~ -

a,+

{- ~}" ~

(4.4

n=l

so that in conformity with the accuracy with respect to 6 assumed in (3.2) the equation for A from (1.4) takes the form XA(0) = s The relative distance s163 of plate can be estimated by means of surmned, if we substitute in it the tion theory [Ii]. Convoluting the

the stagnation point from the leading edge of the (4.4). The first series of this expression can be integral representation of ~(s)F(s) from zeta funcseries in the integrand, we arrive at

~ x v' exp (--x) eth

x dx

@

which, in accordance zeta function ~(4/3, have

with 1/2)

a known f o r m u l a [ 8 ] , c a n b e e x p r e s s e d in terms that reduces to an ordinary Riemann function.

of a generalized As a r e s u l t , we

ea

--'---"

z

2 ,--V7~ l--~z,)"'

+~

Z (--~)nan

o=,

6%

or in accordance with (3.6) s = 0.0086 51a which, in particular, for 6 = 13 ~ is 0.07%. From the numerical solution of the problem carried out for this case in [13] by a finitedifference method on the basis of the exact Chaplygin equation, we have the value 0.16%. The discrepancy between these results is attributable to the inadequacy of the Tricomi approximation [i] near the leading edge of the plate, where the angles @ are important. 5. We note the degeneracy of the solution constructed above when 6 = 0, in conformity with the fact that in this limiting case it is not possible to use the hodograph method for solving the problem, since the one-to-one relation between the planes (n, 8) and (x, y) should disappear together with the disappearance of all the perturbations. In fact, let us consider solution (2.1) on the sonic line AB (Fig. 2). Then when 6 = 0 the interval AC degenerates, and along the interval CB the solution is represented by (2.2), if we substitute the factor A found in Sec. 4. Expanding the functions (8 + 26) ~ in powers of 6/8 < 1 and taking into account a~=a$)6 ''~ from (3.6), as the dominant terms we have the following:

,(o, o) = [a(~+~)]',, r(5/a)

(__t)" y',o,o, o=~-,1+8',,s '~ r 2ax a,,-tv j a,~

(sl

Thus, when 6 = 0 the latter vanishes on the entire interval CB, i.e., identically with respect to 8, thereby ensuring the degeneracy of the solution along the entire sonic line AB and hence throughout the boundary region, since any of the streamlines = const intersects the sonic line ~ = 0. The behavior of the solution obtained near the singular point C is described by the following terms dominant in 8/6 << i:

,(o,o)

p.a.Z~-'~ ~ (i

3 o _vo,)(T)

-"'

-

a, +v)(})-'+='+

}

(5.2

115

which correct to 6 x1~ completely coincide with the results of the approximate aproaches

[7,

14]. At the same time, the solution described in [4, 5] correspondingly gives

p,a,15-V"

3

F('13)f( O ~-'i' ' 9 ( ~ ~,'

,

which for the fundamental reason mentioned above differs even in the singular part from that constructed asymptotically with respect to 6 and with satisfaction of all the boundary conditions of the problem. The difference between the terms written down in (5.2) and (5.3) is 6, 12, and 42%. The sonic lines BC(AC) (Fig. I) are found by integrating the formulas for conversion from the hodograph plane [5] along n = 0 9x e = - -

[ (~+ i)'h/p,a, ],..

Near t h e l e a d i n g e d g e t h e y c o o r d i n a t e o f by ( 5 . 1 ) . The x c o o r d i n a t e can be r e p r e s e n t e d m e n t i o n e d summation o f t h e s e r i e s from ( 4 . 4 ) . corresponding to values of the angles 0 = 0(1) a r e c h a r a c t e r i z e d by t h e s c a l e s163 = O(~SJ~).

y= (tip,a,) ~

( 5.4 )

t h e s o n i c l i n e BC i s e x p r e s s e d d i r e c t l y i n a s i m i l a r f o r m by u s i n g t h e a b o v e Thus, the interval of the sonic line o c c u p i e s a f l o w r e g i o n whose d i m e n s i o n s

The v e l o c i t y d i s t r i b u t i o n on t h e s u r f a c e XA(q) o f t h e p l a t e i s f o u n d from ( 4 . 1 ) in t h e same form as ( 4 . 2 ) . In this case the principal part of the distribution, expressed c o r r e c t t o 6 x/s by t h e s e r i e s c o m b i n a t i o n Q ~ ' ~ ( ~ ) - ( t / 8 ) Q ~ ' I ( ~ ) , can be summed i n c l o s e d form. For this purpose we use Mellin integral transforms in exactly the same way as in [7, 12]: o+ioo am

Z l(n)

' I

:

(5.5)

2ai

ll~l

O--iv:

S-,,<.<,,>,.-,<,,: o

0

(~+~)'+"

r(s+%)

F

(

s+v,~

+ ~'- , s + - - ' 2

'

(5.6)

Res > IRe~l, Re (~ + ~)> 0 Each of the series obtained after replacement of the trigonometric functions in (4.3) by exponentials can be represented by a contour integral of type (5.5) with the right side of (5.6) as the function F(s). In our case ~ : +_i6 and ~, in accordance with the notation (4.3), guarantees the legitimacy of using (5.6) for ~ > 0. The contour integrals introduced can be evaluated in accordance with the residue theorem if the contour is closed on the left by a semicircle of infinitely large radius, in order to ensure the convergence of the integral. All the poles of the integrand function (generated by the F function) lie on the left of the imaginary axis, except for the pole of the zeta function (at the point s = i). For ~ ~ 6 precisely this pole gives the residue dominant with respect to the small quantity 6, since all the others contain its positive powers and should be dropped together with O(6Zz3). As a result the dominant term of the velocity distribution takes the form:

x..

~ l= ~ 3

--~-! t ~

x

3

)

'

6 ' ' 2 ";q~ +

1+

F

----'qo

))

' 6 ' 2 '

i-i~/8 q0---- i+i~/8 If by means of relations for the hypergeometric functions we go over to the argument 1 -- r~ = 1 -- 4q0/(l + q0) 2 = --(~/6) 2 and then use a Kummer transformation, we can reduce all the hypergeometric series to algebraic functions and represent the expression for the velocity distribution on the plate in the final form: 3

116

where the subsequent terms are expressed in terms of the coefficients an . The first term coincides with the distribution found in [14] found from the approximation by selfsimilar solutions of (I.i) without satisfaction of the boundary conditions of the problem on the parts of the boundary OBD and AE which, as seen above, serve precisely for determining the coefficients an . LITERATURE CITED I. F. G. Tricomi, Lectures on Partial Differential Equations [Russian translation], Izd. Inostr. Lit., Moscow (1957). 2. R. G. Varantsev, Lectures on Transonic Gas Dynamics [in Russian], LGU, Leningrad

(1965). 3. F. I. Frankl', "Two gas dynamic applications of the Lavrent'ev--Bitsadze boundaryvalue problem," Vestn. Mosk. Univ. Ser. Fiz.-Mat. Estest. Nauk, No. ii, 3 (1951). 4. G. Guderley, "The flow over a flat plate with a small angle of attack at Mach number I," J. Aeronaut. Sci., 21, 261 (1854). 5. K. G. Guderley, The Theory of Transonic Flow, Oxford (1962). 6. M. D. Van Dyke, Perturbation Methods in Fluid Mechanics, New York (1964). 7. S. K. Aslanov, "Profile with a plane underface moving at the speed of sound," Tr. Kuibyshev. Aviats. Inst., No. 12, 259 (1961). 8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (1965). 9. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York (1959). i0. A. Erd41yi, Asymptotic Expansions, New York (1956). Ii. Handbook of Special Functions [in Russian], Nauka, Moscow (1979). 12. S. K. Aslanov, "Drag of a tapered profile in a sonic flow," Prikl. Mat. Mekh., 20, 756 (1956). 13. W. G. Vincent, C. B. Wagoner, and N. H. Fisher (Jr), "The flow over a flat plate at an angle of attack at free-stream Mach number i," Actes 9, Congr~s Intern. Mec. Appl., Vol. 2, Brusselles (1957), p. 5. 14. S. K. Aslanov, "Approximate solution of the problem of sonic flow over a flat profile," Izv. Vyssih. Uchebn. Zaved. Aviats. Tekh., No. 2, 8 (1982).

CONTRIBUTION TO THE SOLUTION OF THE VARIATIONAL PROBLEM OF THE OPTIMUM SHAPE OF SUPERSONIC NOZZLES A. A. Sergienko and A. A. Sobachkin

UDC 533.6.011.5

The requirement that one of the components of the first variation of the Lagrange functional outside the integral vanish is removed; the region of existence of a solution of the boundary-value problem is considered; and calculation results are presented for one example with the identification of both an extremum of known type and boundary extrema. In the general. case in seeking the boundary extrema it is necessary to assume that all the terms of the first variation of the Lagrange functional are nonzero. However, in this formulation there are no algorithms for solving extremal problems for which the boundaries of the region of existence of the solution are not known in advance. At the same time, the approach proposed makes it possible to identify new, previously unknown boundary extrema. In [i] the problem of the optimum shape of an axisy~etric supersonic nozzle is formulated using a control characteristic contour and the necessary conditions for an extremum are obtained in the form of a boundary-value problem for a system of firstorder ordinary differential equations. In [2] this boundary-value problem was solved Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 138-142, January-February, 1987. Original article submitted January 2, 1986.

0015-4628/87/2201-0117512.50

O 1987 Plenum Publishing Corporation

117