Variational problems of gas dynamics

Variational problems of gas dynamics

U.S.S.R. Compui. Marhs. Math. Phys. Vol. 20. No. 5,pp. 113-127 0041-5553/80/050113-15507.50,‘O Printed in Great Britain C1981. Pergamon Press Ltd...

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U.S.S.R. Compui. Marhs. Math. Phys. Vol. 20. No.

5,pp. 113-127

0041-5553/80/050113-15507.50,‘O

Printed in Great Britain

C1981. Pergamon Press Ltd.

VARIATIONAL

PROBLEMS OF GAS DYNAMICS* Yu. D. SHMYGLEVSKII Moscow

(Received

WORK in the mechanics of continuous

1 I Junuar.r, 1980)

media laboratory

of the Computational

Academy, of Sciences of the USSR. since 1955. on variational

Centre of the

problems of gas dynamics.

is

surveyed. Against a backgr-ound picture of the general development

of optimization

shapes. we survey’ below) work carried out at the Computational

of aerodynamic

Centre of the Academy of

Sciences of the USSR. during its 25 years of existence. Work on optimization of gas dynamics.

of the supersonic

shapes of bodies, on the basis of the exact equations

began in the mid-fifties. The variational

pr-oblems were solved by analytic methods.

and the solutions realized numerically by computer. The method of characteristics was used to solve two-dimensional problems. The later development of optimization methods for threedimensional

nozzles led in turn to new methods of high accuracy.

Even in the fifties there was a big demand for mlprovements flight vehicles, and the previous developments for solving non-classical

time. computers numerical

variational

of gas dynamics.

of solutions with the requisite accuracy. and the equations and quantitative

description

problems of gas dynamics proved to be degenerate. for solving them. Fortunately. on the boundaries, the optimization

for an extremum

and the necessary’ conditions

of one-dimensional

guarantees of gas dy,namics

to use classical methods

[I] had proposed an approach

of the admissible functions

for stationarity

could be satisfied

on pieces of the two-sided

problems. All this prepared the way’ for intense development

of

of aerody,namic shapes.

The problems initially, solved were two-dimensional that can be reduced to one-dimensional

variational

problems. with subsequent

flow and isolation in it of the required body contour.

of a check contour

and methods

of gas flows. The variational

and it was impossible

as early as 1946 Okhotsimskii

whereby, the necessary’ conditions

dimensional

methods.

problems. created a base for advances in this field. At the same

provided a working tool for studies. The method of characteristics

reproduction

provided an excellent qualitative

extremum

in the shapes of nozzles and

numerical

in the field of flow and was named accordingly.

problems of gas dynamics calculation

of the two-

The method involves the use It was only in the early, sixties

that the seemingly simpler approach to the solution of general problems. not admitting of reduction of dimensionahty. was developed. An extensive survq of this topic may be found in Kraiko’s book on variational problems of gas dynamics [?I. In the present paper. in connection with the 25-th birthdab. of the Computing Centre of the Academy. of Sciences. WC summarize the work in this field by associates of the laborator!,

of mechanics of continuous

V. M. Borisov, V. I. Zubov. A. h’. Kraiko (who cooperated *Z/l. rFchisi. Afar.mar. Fi;,

20,5, 1205-1220. 1980.

media. namel).

verl’ closeI> with us till 1964).

Yu D. Shmyglevskii

114

1. E. Mikhailov, I. N. Naumov, A. V. Shiplin, and the present author. The problems solved recently will be discussed in most detail.

1. Check contour method Nikol’skii proposed in 1950 the use of a check contour when minimizing of an axisymmetric

body. based on the linear theory of supersonic

possible in the exact statement. and other conditions

if the equations

of gas dynamics in the form of conservation

of the problem, allow the required quantities

rate through the body contour,

lengths of contour projections

along the lines of the required check contour.

the wave resistance

flows [3]. This approach is also

etc.) to be expressed as integrals

The contour may be e.g., in the case of shockless

flow. the segments (Fig. I, a, b) of characteristics

PC and bc of different

families, passing through

the end points II, b of the required contour of the body, up to their intersection Hantsch

[4] used this approach to reduce the determination

boundar-y value problem for non-linear

laws,

(wave resistance, zero gas flow

ordinary differential

c. Guderley and

of data on the check contour to a equations.

FIG. I

The scope for using the check contour method depends on the avajlable set of conservatron laws. It was only in 1975 that a complete set of conservation laws of gas dynamics [S] and of electromagnetic hydrody,namics (61 could be derived. (The complete system was constructed earlier by Ibragimov [7] for the non-degenerate case of potential gas flows, when Noether’s theorem is applicable.) The results obtained exhaust the application of this approach to the solution of optimization problems. In this context optimization problems were solved for shapes of supersonic

axisymmetric

nozzles of given size or given length with known external pressure,

and also some problems cancer-ned with external flow, and optimization

problems for nozzle

shapes in the case of rotating flows. The three-dimensional problems do not admit of the use of a check surface. To solve these. and also some more general plane or axisymmetric problems, the general method of Lagrange multipliers

is used.

In the Computing Centre work on variational problems of gas dynamics started in the earliest days. and the first publications appeared in 1957. (To limit the list of references, we shall quote joint work of the authors containing the necessary biliography, with an indjcation of the year of first publication.)

In the work of 19.57 (see 181) the statement

of variational

in accordance with the approach used in (41, but relates to both irrotational and the results are in a much simpler form.

problems is

and turbulent

flows,

115

Variarional problems of gas dynamics

In the absence of irreversible processes, the equations

of gas dynamics of axisymmetric

flows may be written as I drQWSin6

drQwCOSti

‘/.?

= 0,

6r

?- rpw2

++Qti%OS26)+

6r

Tds-dh+pdp=O,

sin 6 cos 6 = 0,

d$=rpu:(c,os

h=jc,dT.

Q=Q(T,P),

where x, r are the Cartesian coordinates

tidr-sin

s the entropy,

(9) I

s-s

6dx),

cp=cp CT, P),

in the meridional

plane of flow, w is the velocity modulus,

p is the density, p the pressure, T the tempera

9 is the angle of the velocity to the axis of symmetry, ture, h the enthalpy,

+ h=H(lJl),

f

$ the stream function

and, cp the specific heat at constant p=p/RT, s-c, In (pi'p")

pressure. In the case of an ideal gas, the specific heats are constant, +conat.

x =c,, ‘c,.. c, is the specific heat at constant

contour.

were obtained

to equilibrium

in 1957-

volume. The main results, using the check

1962 for an ideal gas (see 181). In [9] the results were extended

flows of a real gas. These more general results will be discussed here.

The first and third of Eqs. (1) are integrated over the area bounded by the contour abc. The resulting integrals are transformed into line integrals by Green’s formula. From these, the body wave resistance or nozzle thrust, and also the zero gas flow rate through the body contour. are expressed in terms of the unknown

data on the characteristic

terms of the size of the now required characteristic A general Lagrange variational functional

of flow through the body contour is the compatibility determined

in [8,9]

The isoperimetric

part of the check contour.

on characteristic

by the possibility

flow and in

is w(G). A express the absence on thex, r axes

The differential

connection

bc. The domain of admissible functions

is

of the required body contour having a break at the point a

Analysis of the first variation of the Lagrange functional at the point of intersection

function

conditions

and the equality of the lengths of projections

and of the remaining condition

of the incoming

bc.

problem arises. The unattached

of the wave resistance is minimized.

of the body contour

UC

h of the required characteristic

pencil, defined by the break. At supersonic

leads to a transversality

condition

with the extreme characteristic

of the

speeds. by virtue of the properties of the solution of

the hyperbolic system (I), the given condition has to be satisfied on the entire required characteristic. This gives one integral of the system of Euler equations on bh. The first-order equation

for the Lagrange multiplier,

corresponding

homogeneous, while the zero boundary condition entire piece bh. This gives a second integral.

to the compatibility

condition,

shows that the multiplier

is linear and

vanishes on the

116

Yu. D. Shmyglevskii

The required characteristic characteristics

consists of the piece ch, belonging to the zone of the pencil of

cob. and the piece of two-sided extremum rpU” tg a sin’ I?=?.,

I.(pipzL*

i.cos(ti=a)T

-=dl.1 d$

hb, on which are satisfied the equations tg a sin ti cos 6) =/I,

psin(6%~) 7

r’pu sin a

dx -==

cos (0%)

d$

rpusinr.

(2) dr -=T d$



sin (era)

rpwsina

where Q is the Mach angle. h = const, and ccis the Lagrange multiplier.



The upper signs are taken

when solving the exterior flow problem? and the lower, when solving the nozzle optimization problem. Ii point h is known. then we know the initial values ofx, I at this point, while the initial value of p is given by the second equation at point h. The constant coordinates

of point II are found from the conditions

checked after integrating

r, p, p, W, a, 6

of (2) with known quantities

h is found from the first equation Eqs. (2). When optimizing

of (2) from the data at point h. The F($)

S(I/‘~) =xb,

=rb,

which can be

the shape of a nozzle of given length with

known pressure p+ in the external flow. instead of the condition

r($~~) = rb we take Buzemann’s

condition

pbwb2 sin In the case of iso-energetic equations

26b-2

(3)

flows. the solution is even simpler. Instead of (2) we have the

tg a sin’ O=i,,

rpw’

ctg f&=0.

(pb-p_)

u: cos (6*a)/cos

a=D,

which define Q(T). 19(r). if h. E are found from the data at point h. The coordinates

(4) of point h

ate found from the condrtions ‘b

J

rpu+

lb

sin a

Sin (0=x)

dr=$o-t$k.

J

ctg (4Ta)

dr=x:--xh.

the first of which expresses the impermeability of contour ab. while the second requires that a given length of projection of ab on the x axis is obtained. If, instead of rb. we are given p+. then Eq. (3) is added to Eqs. (4) and (5). The solution of (4) and (5 1 for an ideal gas was obtained in 1957 [8]. For a real gas it was found in [lo] : the incorrect statement of the problem in [ 1 I] fortunately yielded a correct solution. The problem conditions may include a restriction on the curvature of the required contour. In this case (Fig. 1, c, d). instead of a break at point u, there is a section ad with maximum curvature. To solve the problem. instead of the characteristic nh, we take the characteristic hd.

(5)

Variational problems of gas dynamics

The simplicity

of the solution obtained

velocity hodograph construct

117

is such that it can be written in the plane of the

[8]. It is easily shown that all the stream lines in abh are extremals. To

the required body contour or flow stream line, we have to find the solution of Eqs. (1)

with known data on

ah

[ 121 . Solutions

and bh. It is found by the method of characteristics

are

also obtained for the plane case in [8, 91. In Fig. 2, in the plane of the velocity hodograph, the extremals are shown for different values oft for an ideal gas with K = 1.4, in the case of extremal irrotational

flows. They fill a ring of supersonic velocities.

To “reject”

the solutions obtained.

of the Legendre condition

it is useful to have necessary conditions

type. The use of the classical Legendre condition

since it is of no interest in the case of a degenerate variations of the free function expression

problem. On the characteristic

on a small element of length are considered.

light of all the connections

here,

bh, the

In this case the

for the second variation of the resistance is reducible to an interesting

resistance or maximum

for a minimum

is impossible

form, in the

of the problem, and of relations (4) and (5); in the case of minimum

thrust. it gives

(6)

sin 2a

A = arctg

wcr, tg a ’

cos 2a-

Here. all the upper signs refer as before to the external flow. n is an integer or zero. and the Mach angle a and its derivative are given by equations

a=arcsin”

LL’:

^ a-=-.

fj(p*s) ip

zL, =

da (WY 3) dw

The dependence a(~, $) follows from the finite equations of system (1). This result u’as obtained for a perfect gas in 1962 (see [8] ). It is interesting that. at points of the curve 9 = T A. the derivative &3/d+ becomes infinite on the exrremals [ 131. A second scheme of solutions of the problem was found in 1961 (see [8] ) for cases when the data at point h of the pencil of characteristics cab do not satisfy inequalities (6). This scheme is shown in Fig. 3, a, b. The body contour on the piece ok stipulates focussing of the characteristics at point h. The discontinuity arising at point h leads to the formation of a shock wave hn, of a contact discontinuity discontinuity

hnz, and of a pencil of characteristics

at point h, the variation of the Lagrange functional

lhb. In the light of the

of the problem leads to two

auxiliary conditions, which mean that. given the quantities at point h as we approach it from the side of point c (superscript -). we can evaluate the corresponding quantities at point h as we approach it from the side of point b (superscript

(p--p&) (u.- coSfl-_u.’

sin 6-tp-

t):

(u.-)‘tg

‘%- Sin ii- sin (V--6’)

cos ti1+) = (p--/,-,)

sin(ti-Ta-) p-u.- sin a-

sin (a--O+ta-)

+p+(u.+)‘tga’sin6’ [

p-u;-

=O.

Sill ‘1-

i-

(7) 1 p+u’+

=o. I

118

Yu. D. Shmyglevskii

It should be noted that the shock wave hn and the contact discontinuity triangle, abh, so that the solution may be termed a discontinuous

FIG. 2

FIG. 3

FIG. 4

hm are outside the

shock-free solution.

1 ariationa!

problems

01 gas dwamics

119

As we have said. Eqs. (7) enable us to find the required quantities

at the point h on the

length b/z, from the data at the point h on the length ch. As a result. the wanted characteristic

bh

may be found in the same way as in the case of a continuous

of

solution.

To obtain the solution

Eqs. (1) with known data on ah and bh, we need to compute the flow configuration This configuration

at the point h.

includes the centred waves ahk and Zhb, the shock wave hn, and the contact

discontinuity

hnz. Use of the well-known

discontinuity.

and use of the quantities

Prandtl-Meyer

relations in the flow and on the lines of

found at point h, gives the directions

lines kh and Zh at the point h, and all the requisite quantities

of the tangents to

at the point h as it is approached

along these lines. After this. system ( 1) can be solved easily in the domains ahk and lhb. From the characteristics

kh and Ih, the solution in the domain khl may be found.

In [ 141 the case is considered when the nozzle or body of rotation (Fig. 3. c. d). and the pressure on it is known. This situation

has an end-piece

can occur e.g. when flying in empt)’

sapce. if the jet from the nozzle cannot be developed up to the end-face (third scheme). It can be shown that. in this case. the first and second schemes of solutions condition

are realized, while at point b’

(3) must be satisfied. in which I)+ is the given pressure at the end-face.

In the same walk. for an ideal gas with adiabatic exponent K = 1.4, the contours of optimal nozzles of all possible sizes were computed. It happens that the three schemes described enable the optimal solution to be found for :ny dimensions. In Fig. 4 we show the domains of the different

solutions.

The scale of the figure depends on the radius of critical section Oa. The

domains AaB of the end-points

b refer to a nozzle with uniform

flow at the output.

In this case

the greatest thrust is realized for a given rb. while nozzles with point b on line af3 are the shortest. When point b lies in the domain BaED, the solution is given bl, the first scheme. which does not include discontinuities. Domains CED refer to discontinuous shockless flows. If. given the pressure p+ on the end-face. the piben point b is above the nozzle end-pomt calculated

(3) in it I then it has an end-face. and the piece of contour ab refers to

with condition

the first or second scheme. depending The Introduction

u (the entrap)

for the resistance to be stationary

(see [8] ). The quantity condition

on p+ and xt,.

of shock waves into the flow naturalI}, influences

the bodl A new free function condition

b’. when the nozzle is

CJis introduced

for an extremum

increment)

the wave resistance of

then makes its appearance

A necessar!

with respect to the entropy was derived in 1962

by the relation s =S($)

+ u. The correspondmg

necessary

serves 10 define u. and has the form

6=nnT_M3 p; =

_lI=arctg

ap (u, 3: 0) 60

>

Po ctg a ctg’ a ’

pou2-pO

po =

(8)

6 p (U’. 3.0) ao



where the functionsp and p depend. in accordance wtth (I), on two variables. for which we choose HJ and $: and they also depend on the enttop)- increment. In accordance with (6) and (8). the hncs 13= A and 9 = M arc the lines USC’ and L%‘l of Fig. 2. If pieces of the extremal bh are below curve USV. the result obtained is unsuitable. It can be seen from Fig. 2 that there are domains in which condition (6) is satisfied. bet -3 - .V<6 <. - hl In these domains we can expect to obtain solutions in which the resistance minimum can be reduced as a result of increasing the entropy. since In this case. for 6 u > 0. the first variation of the resistance is negative. Yet experimental

attcnipts

to find such solutions (see [8] ) yield no

Yu. D. Shmyglevskii

120

results: they have always led to unrealistic variation. The domain of discontinuous solutions.

was constructed

situations

on the discontinuities,

shockless solutions,

for

K =

I

defining the entropy

.4, leading to realistic

in [ 151. For the analysis of flows in nozzles, we have to consider the

mirror image of Fig. 3 with respect to the horizontal

axis.

Some other ty’pes of problem can be solved by using a check contour. In works of 1958 and 1960 (see IS]) particular which the surrounding

classes of optimal plane profiles and bodies of rotation were found, for

flop, has an associated shock wave. A similar solution was constructed

[ 161 for the case of non-uniform check contour.

incoming gas flow. It is to these classes that application

consisting of a shock wave and adjacent characteristic,

in

of the

is restricted. The general

problem of this t!pe was solved later on the basis of a different approach. In [17], the class of solutions obtained

was used to optimize bodies with specific blunting

the optimal shapes of the stern part of a body of revolution were computed.

In [ 18. 191: estimates were obtained

of the head part. Also in [17]

in the equilibrium

of the maximum

flow of a real gas

resistance of the head part

of a body of revolution

and the least resistance of the stern part, without describing in detail the

resulting flow structure.

In (201. for an ideal gas with constant

circulation,

the optimization

problem for a nozzle was solved, similar to the problem discussed here for untwisted

flows. It is

shown that the nozzle thrust force can be increased as a result of twisting the flow. In short: some fundamental

solutions, based on reduction

form. had their origins in the Computing

of problems to one-dimensional

Centre.

The check contour method has provided the solution of important

optimization

problems

for supersonic shapes. The approach to the solution of problems of a general type was developed in 1962-- 1964.

2. The general method of Lagrange multipliers By no means all the constraints terms of quantities

imposed on the required optimal contour are expressible in

on the shock wave and the characteristics

entering into the check contour. An

example is the volume bounded by the generating body. The check contour cannot be employed usefully in problems with an associated shock wave. A check surface is not suitable for stating three-dimensional

variational

problems of gas dynamics.

The development of a general method is due to CuderIe),. Armitage, and Kraiko. Later, associates of the Computing Centre used the method to solve some important and complicated problems, and for developing a general approach to three-dimensional problems of nozzle shape optimization. In [2 1, 221 the minimized functional is written as an integral over the generating body. while all the constraints are associated with it in the form of connections. jointly with the equations of gas dynamics in the approach flow domain, In the plane and axisymmetric cases, the variational problem remains two-dimensional. The Lagrange functional is formed, and variation of it is performed. The necessary conditions for an extremum include a boundary value problem for the Lagrange multipliers. while by, solving the resultant problem as a whole we can find the required generator. In ]2 I] the shape of supersonic nozzle with given lateral surface area is optimized. The problem for the Lagrange multipliers proves to be solvable in the class of continuous functions.

lhriarional

After

nothing:

[? 1. 221, attempts

ofgas

for an extremum

the situation

each of the partial differential

multiplier,

dependent

seemed only to lead to unsolvable

equations

on two independent

and certain boundary

two intersecting

characteristics

arise at the boundaries

conditions

conditions

of the

may be In the

are defined on the entire closed contour

If we state the gas-d) namic problem with conditions

for the Lagrange multipliers

linear; their coefficients

for

on

and the required body contour. while finding the solution dow-flow, the

The problem for the multipliers

hyperbolic

These conditions

equation

ac and bc (Fig. 1) or a shock wave and a characteristic. the boundary

embracing the domain of determinacy. the initial characteristic contour.

problems. On

we obtain a partial differential

which we referred to above as the check contour. The boundaries

system of resulting equations,

boundary

with a Lagrange

variables in the plane and axisymmetric

conditions.

problems.

the Lagrange

of system (1) is introduced

equating to zero the first variation of the functional, the multipliers,

121

in greater detail. When formulating

functional,

domain of influence.

d),rlamics

were made to solve other problems. but initially, they came to

the necessary conditions

It seems worth describing

problems

arise on the closing characteristic

is solved up-flow. The equations

depend on the gas-dynamic

and the body

for the multipliers

variables w, 9, p, p, and the equations

are are of

type in the case of supersonic flows. In the physical plane x, T, the characteristics

Eqs. (1) and of the equations

for the multipliers.

When solving problems in the general case, the necessary conditions to be unsolvable,

if we demand continuity

of the Lagrange multipliers.

general method into a working tool by introducing dynamics

came up against similar discontinuous

with discontinuous

boundary, conditions,

Similar discontinuous

on the discontinuities.

importance

is the derivation

discontinuous

multipliers.

were derived. Also of

at break points of the required contour.

problems solvable. Later. successful application

initiated

problems that were awaiting solution.

Within the limitations

Specialists in gas

by Kraiko for the Lagrange multipliers.

made the optimization for numerous

prove

made the

flow past a sharpened body.

travelling along the characteristics.

c!f conditions

for an extremum

Kraiko [23,2]

solutions in linear problems of supersonic flow

e.g. when computing

solutions were introduced

Conditions

of

are the same.

These results

of the method was

of a survey’ paper. we cannot give the procedure

for solving problems

by the general method. The approach is laborious, due to the complexity of system (1). The solution of each problem demands intelligent variation of the Lagrange functional on the boundaries

of the domain of influence

and on the singular lines in the field of flow. The process

of solution and the results depend very much on the concrete problem. The system of necessary conditions for an extremum always looks unexpectedly complicated and only a careful study of it allows the solvability of the problem to be established, and the solution algorithm to be constructed. In view of this, we are obliged to confine ourselves to just basic examples of the problems solved at the Computing

Centre. Numerous

applications

of the general method are

described in detail in the book [?I, and the reader should refer to it for any necessary information. As a preliminary. let us note another general result. which has likewise been obtained beyond the walls of the Computing Centre. For twodimensional variational problems of supersonic gas dynamics,

the necessary condition

for a resistance minimum

has been derived [24]

To check this

condition at a point of the body contour obtained. we have to evaluate definite integrals along certain lines in the flow field. In problems to which the check contuur method can be applied, the condition of [24]. derived on the basis of variation of the body contour element, transforms in a natural way to (6).

Yu.D. Shm)$evskii

122

When solvmg problems which cannot be converted to one-dimensional possible to isolate. in the domain of influence. end-point

form, it is generally

a domain of the form ahb (Fig. I), including

b of the contour. With fned characteristic

ah, the end piece ab of the contour must

have minimal resistance in the context of the problem constraints,

since small variations of ab do

not effect the up-stream flow from ah. If the particular problem of optimizing fixed data on ah admits passage to a check contour, accordance with Section multrpliers.

dependent

partial differential

the

the piece ab with

then the data on bh may easily be found in

1. To solve the complete problem. we have to find the Lagrange

on two variables. on the line ah; but these are subject to the appropriate

equations

and boundary

conditions

on ab and b/r. An analytic solution of this

problem was found in [Z5] . and thereby the Lagrange multipliers which simplifies the solution as a whole. The integrals obtained simplified ior other more complicated

on the line ah were found, enable the solutions to be also

problems.

One of the main problems in body, shape optimization

is to construct

body contours

passing

through two given points. when an associated shock wave is present. In [26, 271. uniform incoming flow of an ideal gas is considered. In accordance with [2 1. 231, the system of necessary conditions for an extremum is derived. It turns out that these conditions are satisfied with a very surprising flow scheme (Fig. 5). The required contour of a plane profile or body of revolution

with channel

ab causes a shock wave UCand has an in general an infinite number of break points d, dl . . which accumulate

.

on approaching

point a. The breaks cause rarefaction flows from the pencils of characteristics fdh. fLd.h,. . . . . The line of discontinuity of the Lagrange multipliers is the fall at the contour break pointsd, dl.. . , and in step-line cdc,d,, . . . : the discontinuities general. at interior points of the pencils of characteristics of type c2 on the shock wave. Singular situations are possible. when. at one of the points ci, i = 1. 2. . . no reflection of the discontinuity. from the shock wave occurs. In this case the contour has a finite number of breaks. A connection between the incoming flow velocity and the local slope of the shock wave. such that the arriving disturbances and multiplier discontinuities are extinguished. was obtained [28]

If c is a point of the ty’pe mentioned.

in

then the required contour ub has no breaks. A special

class of solutions arises. which can be found by using a check contour [8]. and which we mentioned in Section 1. The computations of [26, 271, for K = 1.4, revealed the degree of damping of the multiplier discontinuities at points d, dl. It turned out that. on reflections from these points. the discontinuities are decreased by a factor of roughly 10 000. This means that. with extremely accurate computations.

only one break at point d can be taken into account. The

solution of [26] was one of the first. in which the general method of Lagrange multipliers was used. and with the discovery, of this solution. a serious barrier was overcome to advances in the study’ of the structure of the possible solutions. The numerical realization of the necessary conditions for an extremum demands the solution of a complicated boundary value problem, including integration of Eqs. (I) and of equations for the Lagrange multipliers, in the context of an iterative process whereby all the conditions on the lines and at the points can be satisfied. Integration of the partial differential

equations

is alway,s based on the method of characteristics.

whereby high

accuracy can be achieved. The result described relates to shape optimization of part of a profile. The optimization problem for the closed profile was solved in (291. A uniform incoming supersonic flow is specified. along with the length. thickness. and angle of attack of the profile. It is assumed that the upper and lower points furthest from the chord are located on the same normal to the chord. Initially. a flow version is considered. m which the upper surface of the profile does not cause a shock wave. The resulting profiles have a quadrangular shape. The vertices of the curvilmear quadrangle are at the nose and stern points. and at the ends of a segment of the normal to the chord. which

I’ariational

determines

problems

of gas dJ,rlamics

the thickness of the profile. The necessary conditions

number of conditions

at individual

whereby all the conditions time on the BESM-6

for an extremum

include a large

points of the flow scheme. As a result. the iterative process,

are satisfied, is complicated;

computer.

123

but solution is possible in a moderate

Somewhat later [30]. the problem was solved with associated

shock waves in both the lower and upper parts of the flow.

FIG. 6

FIG. 5

Zubov’s paper [31] has a special position. In it he finds the optrmal shape of profile in an incompressible

fluid when the lift is given. A well-chosen statement

to reduce its solution to finding the extremum

of a function

obtained have large relative thickness. A supplementary! along the profile (everywhere

constraint

apart from the neighbourhood

of the problem enabled him

of several variables. All the profiles on the growth of pressure

of the end-point)

leads to profiles of

a usual shape. The problem of finding the optimal shape of a contour passing through three given points was solved in [3?]. The result is obtained for a dispoation of the given points in the incoming flow such that no head or interior shock waves arise. On the basis of the known results, it is not possible to solve three-dimensional

variational

problems of gas dynamics with the aid of a check surface. The obstacle is that the condition the stream surface pass through two contours the characteristics.

that

cannot be expressed by means of surfaces. given on

But with the general method, even this problem can be stated and solved. AI

the same time, particular

requirements

integrating

of gas dynamics

the equations

arise here on the accuracy of the numerical and the equations

technical demands on the accuracy of solving variational

methods for

for the Lagrange multipliers.

The

problems are high. At present. a 0.3%

change in nozzle thrust is looked on as extremely, important.

Hence it follows that the accuracy

of the numerical methods must be in line with these requirements. When the available computers are used for calculating plane axisymmetric flows, the necessary accuracy may be achieved by the method of characteristics.

For accurate solution of three-dimensional

needs to be paid to improving the accuracy of the computing

problems. special attention

methods,

The advance towards a numerical scheme of high accuracy’ for computing irrotational supersonic gas flows has not been easy. References to recent progress in this field may’ be found in [33]. Some of the progress has been due to Borisov and Mikhailov [33] themselves, who first developed the tetrahedral scheme of the method of characteristics in 1970 (see [33]); then the method of characteristics. on a pattern of a different type. was developed in (341. In work carried out in 1976 and 1978 (see (331) the computational mesh is formed by sets of meridional planes (in cylindrical coordinates) and two families of characteristrcs. of surfaces passing through a line

Yu. D. Shnzyglevskii

124 of spatial type. The intersections

form, in particular,

The equations are written in a form corresponding

on each meridional

plane, a curvilinear

mesh.

to this mesh. At any point of each meridional

plane, they connect the derivatives along the directions

of the curvilinear mesh with the derivatives

with respect to the normal to the plane or with respect to the polar angle. Different approximations of the functions

with respect to the polar angle are introduced.

derivatives with respect to the polar angle to be calculated. derivatives in the meridional

planes. (In the axisymmetric

polar angle are zero, and the equations axisymmetric calculations

These approximations

The equations

case, the derivatives with respect to the

obtained are the same as the characteristic

flows.) It remains to approximate

equations

of

the equations in the planes, and to perform

similar to those when realizing the usual method of characteristics.

To approximate

the functions

with respect to the polar angle, Borisov and Mikhailov [33]

tried using central differences with respect to three or five points. trigonometric and interpolation

cubic splines. Flows were computed

critical cross-section, numerical

enable the

now contain only the

approximations,

in nozzles with a break in the surface at the

i.e. problems were solved of the type that led to development

method. The computational

accuracy was estimated

of the

from the thrust. The computations

were made in meshes of different density. The highest accuracy was always obtained by using cubic splines. The problem considered in [35,36]

consists (Fig. 6) in finding the nozzle surface cr.

maximizing the thrust and passing through two given closed contours y and r, with a plane sonic surface to which the circle y belongs. The gas flow is irrotational. The sonic velocity vector at points of y is not in general tangential to the surface u. This causes rotation of the flow (the axisymmetric

analogue of Prandtl-Meyer

characteristic

surface r. Through the output contour of the nozzle r passes a characteristic

flow) with in general non-axisymmetric

C of another farnil). The domain of determinach

is here bounded

closing surface

by surfaces u. r. Z. and has the

form of an irregular ring. The equation

of gas d!,namics is H,ritten in the form di, pyH = 0.where H is the potential

of \zelocities I’, L’= CH, and p is the density. which is a known function the no-flow condition

is YHX II= 0,where 11is the normal vector to u. The thrust T is written as

an integral over the surface u of the pressure p(VH). projection

of 'iH.On the surface u

of u onto the surface perpendicular

multiplied

to the direction

amounts to finding the surface u. giving maximum

b>, the differential

of the

of thrust computation.

The problem

T under the condition div p(‘C-H) PH = 0 in the

domain of determinacy. and with V HI )I= 0 on u. The variation of the Lagrange functional leads to necessary conditions for an extremum: connecting u, H, p.A.where ~1is the Lagrange multiplier cprresponding to the no-flow condition. while X is the multiplier corresponding to the equation of gas dynamics. The necessary conditions include the equation of gas dynamics. the equation for X in the domain of determinac). two conditions on u. a condition on Z. and a condition on r. All in all, we obtain a boundary value problem for a system of equations of hl.perbolic type, which. in the context of the iterative process. can be split into two more customar>’ problems. The first consists in finding H with known data on 7 and u. The second, similar to the first. consists in finding h with known data on S and u. The aim of the iterative process is to choose u so that the condition on u. which is not used in the two pi-oblems. can be satisfied.

l’ariariorlal

problems

01 gas d!xamics

Examples of the shapes of optimal three dimensional tetrahedral

scheme of the method of characteristics

The first results have recent&. been obtained the method of 1978 (see [33] complicated

(of 1970, see [33]). are IL)be found in (361.

j. It should be mentioned

of three-dimenstonal

nozzles using

that. on the whole, this is the most

and laborrous of all work on optimization

in gas dynamics.

The scope of the present

of developing the numerical method. of

the iterative processes. and of finding ways to obtain the optimal shapes.

Informatron maximum

nozzles. obtained by using the

on optimization

paper dots not allow us to discuss all the difficulties constructing

125

ma! be found in [Z] about obtaining

momentum

the shapes of composite nozzles with

in a given time interval. about optimization

in the case of two-phase and

two-la),er flows. and abijut solving problems in the case of twisted flows. along with man? other results obtained

b! Kraiko and his associates on the basis of the general method.

Finall!,. we hope that the present surge! has given some idea of the work of the Laborator!. of h!echanics of Cont inuous hledta of the Computmg

Centre of the Academ), of Sciences of the

USSR. in the field of variational

with non-linear

problems

connecred

partial differential

equations.

It is concerned

MItI1 the area where optimization of s!‘stems with distributed parameters merges wtth numerical methods: though irxx~mplete. our material VIiii be found to supplement the sune!‘s of 19?9 [.37. 381.

REFERENCES 1. OKHOTSIlrlSKI1. D. L.. On the theor!. of 2

KRAIKO. A. h’.. I’ariurrorlal probfcms Sfosco\r

:ocker motion. f,ihI warern. mekharz.. 10. No. 2. 251- 272. 1946

0.: ,?a8 JJ r~amics

(Variatsionn! e zadachi eazovoi dmamikir. X’auka.

1979.

3. r
ol rhcofcricol

work on aerod).namics

(Sb. reor. rabor po aerodinamike). Oborongiz.

Moscow. 195?.

4. GL’DFRLEY. K. G.: and HANTSCH. F Beste Formen fur achsensymmetrtsche Z Fhr,n~iss.. 3, No. 9. 305-315.

Uberschallschubduscn

1955.

5. TERENT’EV. E. D.. and SHhlYGLEVSKII. Yu. D.. Complete system of dtvergence equations of ideal gas dytamics. Z/r ~,.vchrsl..Ilar. ma{. FK.. 15. No. 6. 1535- 1544. 19!5.

6. TERENT’EV, E. D.. and SHMYGLEVSKII. Yu. D.. Complete system of divergenceequations of electromagnetic

d! namics of an ideal ~3s. Z/I. t:~chlsi Alcr mcr Fi-

I. IBRAGIItO\‘.

N. KH.. Law of conswation 1307-1309.1973.

8. SHMYGLEVSKII. Yu. D.. Sovtc I.ariariona/ problems

ofgas

Sazovoi dinamikt). VTs Akad. Kauk SSSR. Moscou

9. KRAIKO. A. Ir; . I’ariarior~al problems

Dokl. .4X&

in h!drod!namics.

of supersomi

dl.tlamics

16, No 3. 725-737. .“rbok SSSR

1976

210. ~‘0. 6.

(Nekotor! e variatsionn!c

zadacht

1963.

$o~~s 01 .gas wr/j an> thermod~.,?amic

propcutcs

(Variatsionnye zadachi sverkhzvukovykh techenii gaza s proizvol’n),mi termodinamicheskiml svoistvami). VTs Akad. h’auk SSSR. Moscow. 1963.

10. RAO, C. V. R.. Exhaust nozzle contours for optimum thrust. Jr/ Prolwlsiort, 28, No. 6. 373-382.

1958.

11. BORISOV. V. M.: and SHSlYGLEVSKII. Yu. D.. On the statement of variational problems of gas d!,namics Prikl. matem. mekhau.. 27, Ir;o. 1. 183- 185. 1963. 12. KATSEOVA, 0. h’.. et ai., Experieuct. 111compuri~~,~ plow aud axis?‘mmrrnc st~p~‘rson~ ,eas jlows bl rhc merhod o.t characrerisrics (0~) t rascheta ploskikh i osesimmetrichnykh sverkhzwkovykh techenii gaza metodom kharaktrrisrik). VTs .4kad. Nauk SSSR. Moscou. 1961. 13. STERKIN. 1. E.. On the boundar! h’auk SSSR.

of the domam oi ciirtence

139. No. 2, 335- 336. 196

I

of shockless optimal nozzles. DoXi. 1Xod.

YIC.D. Shmyglevskii

126

14. KRAIKO, A. N., NAUMOVA, I. N., and SHMYGLEVSKII, Yu. D., On the construction shape in supersonic flo\s, Prikl. matem. mekhan, 28, No. 1, 178- 182, 1964. 15. SHIPILIN. A. V.. Domain of discontinuous mekhan.. 27, No. 2, 342, 1963.

of bodies of optimal

solutions of variational problems of gas dynamics, Prikl. matem.

16. SHIPILIN, A. V., On bodies with minimum wave resistance in non-uniform incoming gas flow, Prikl. matem. mekhan.. 28, No. 3,543-547, 1964. 17. BORISOV. V. M., On the optimal shape of bodies in supersonic gas flow, Zh. @his/. No. 4, 788-793, 1963. 18. SHMYCLE\SEIl. 604-608.

Yu. D. On ma\irnum resistance in supersonic flow, Prikl. matem. mekhan., 1965.

19. SHMYGLEVSKII. Yu. D., On minimum resistance of the stern parts of body, Ix zhidkostigaza, No. 5, 102-104. 1966. 20.

Mar. mat. Fiz., 3,

29, No. 3,

Akad. Nauk SSSR, Mekhan.

N’AUMOVA. 1. N., and SHMYGLEVSKII, Yu. D., Increase of nozzle thrust by rotation of flow, Izr. Al&. Nauk SSSR, hiekhan. :hidkosti gaia, NO. 1, 34- 31, 1967.

21. GUDERLLY! K. G., and ARMITAGE, J. V., A general method for the determination of best supersonic rocket nozzles, S.rmposiurn 011extremal problems in aerod_vnamics, Boeing Scient. Res. Lab., Seattle, Washington, Dec. 3-4, 1962. 22.

SIRAZETDINOV. T. K.. Optimal problems of gas d! namics. Ariatsionnaj,a

rekhn., No. 2, 11-21,

1963

23. KRAIKO. A. N., Variational problems of gas dynamics of non-equilibrium and equilibrium flows, Prikl. matem. mekhan., 28, so. 2, 285-295. 1964. 24. FEDOROV. A. V., The Legendre condition in optimal problems of gas dynamics, Prikt. matem 39. No. 6, 1032-1042. 25.

BORISOV, V. \I. and SHIPILIN. A. V.. On nozzles of maximum thrust with arbitrar! isoperimetric conditions. Prikl. matem. mekhan.. 28, No. 1, 182-183, 1964.

26. SHIPILIN’. A. V.. Optimal shapes of bodies with associated shock waves, Ix zhrdkosti 27.

mekhan.,

1975.

,caza. No 4. 9-

SHlPlLIN, A. V.. V’ariationai problems of theoretical

Akad. Nauk SSSR, Mekhan,

IS. 1966.

k,orX ori /),I drorm chanics

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pas

MOSCO\\I 1970. 28. CHERNY’I. G. G.. Gas j7ows ,$,ith high supersonic skorost’!uj. Fizmatgiz, Moscow. 1959.

speed (Techeniga gaza s bol’shoi sverkhzvukovoi

29. ZUBOV, V. I., On optimal supersonic profile of a given bulge, Ix. Akad. .Vauk SSSR. Mekhan. zhidkosti gara. No. 1, 89-96, 1976. 30. ZUBOV, V. I.. Optimal profiles at small angles of attack in supersonic gas flow, Dep. at VINITI. No. 174977. 1977; DEP (RZh .1l(Xh, No. 8. 1977). 31. ZUBOV. V. I., On optimal u ing profile in ideal incompressible fluid flow, Z/r. rFc111sl.Mar. mar. Fiz., 20, No. 1,241-245.1980. 32. BORISOV. V. M.. On s! stem of bodiesuith 1965.

minimum wave resistance.Irrzh.

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33. BORISOV, V M., and MIKH.AILOV. 1 IL.. Method of characteristics for computing three-dimensional supersonic irrotational flows. Z/I. r?_chrsl. Afar. mat. Fiz., 18, No. 5, 1243-1252, 1978. 34. AKHMETOV. G. B.. and MIKHAILOV. 1. F.., On a scheme of a numerical method of threedimensional characteristics for computing urotati~inai gas flows. Ix. Akad. nhuk Ka:SSR, Ser. fiz.-matem., No. 3, 29-35, 35.

1975.

BORISOV. \‘. ?I.. and \1IKH4ILO\. 1. L.. Optimal shape of a nozzle for threedimensional in. FluidDi~rraniic~s t/.atls, Vol. 4. Polish Scient. Publs.. 149-153. Warsaw, 1970.

36. MIKHAILO\‘. I. E.. Shape of a supersonic threedimensional Mar. mat. PI:. 13. ho. 1. 257-262. 1973.

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37.

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F. L., Problems of optimization

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U.S.S.R.

Compur.

No. 2, No. 1, 3-36,

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No.

11.

1979.

Maarhs. .4larh. Phjx

\‘ol. 20. No. 5. pp. 127-151

0041-5553:80,:050127-25$07.50,‘0 0 198 1. Pergamon Press Ltd.

Printed in Great Britain

ASYMPTOTlCMETHODSINFLUIDDYNAMICS* 0. S. RYZHOV

(Received

THE MAIN results obtained Laboratories

by asymptotic

of the Theory of Transport

Sciences of the USSR. are outlined. considered.

A unified treatment

active mixtures.

8 April 1980)

methods in various fields of fluid dynamics.

Processes of the Computing

Wave propagation

of non-linear

gas is described.

Centre of the Academy of

in an inhomogeneous

atmosphere

is

wave processes in a radiating gas, and in chemically.

is given. Work on the theory, of transonic

thermally, conducting

in the

flows of both an ideal and viscous

Relevant to the study of almost one-dimensional

non-

stationary flows, the first Integrals of the equations in variations are obtained: they characterize the conservation of mass. momentum. and energ! of matter. One of these integrals provides the basis of studies of stattonar)

hy,personic flow round supporting

bodies. The velocity field in the

interior domain is constructed by solving the problem of the laminar eddy. wake stretching behind the body. Non-stationary processes in a boundary, layer, freely interacting with the external potential flow, are discussed. Finally. the dertvation from Boltzmann’s equation of the sy’stem of hydrody,namic equations. for mixtures in which chemical transiormations occur, IS examined.

1. Introduction. The application mechanics

of asymptotrc

at the Computing

out before the organization

Earlier work

methods to the solution of problems in different

Centre of the Academy of Sciences had its origins in work carried of the Centre. Back in 1942. Dorodnitsyn

on the theory of the boundary, layer in a compressible [I? 21 A transformation

fields of

of the independent

had published two papers

gas. that have since become classical. see

variables was used by Dorodnitsyn

Prandtl number equal to unity, the equations

of the laminar boundary

whereby, at a

layer in the gas reduce to

the form that they take for incompressible fluid flows. If the new variables are used. the methods for computing the velocity field, developed for the boundary, layer in an incompressible fluid. extend automatically to the motion of a compressible gas. In particular, to construct the solution of the problem of the flow past a flat plate. it is sufficient

to take the well-known

Blasius

formulae and to find the corresponding compressible flow stream lines. In 1948. Dorodnitsyn extended his boundary layer analysis to supersonic flows with arbitrary. Prantdl number [3]. This extension

was of a fundamental

kind. since the heat fluxes to the hod! surface are strong]!

*Dr. r.%his/. Mat. mar. Fiz.. 20,5, 1221-1248.

1980.