As~~mprotrc merhods in fluid d).namics 127

37. CHERNOVKO, F. L., Problems of optimization of mechanical systems, Lisp. Mekhan., No. 2, No. 1, 3-36, 1979.

38. UBTKOVSKII, A. G.. Control of systems with distributed parameters, Al~rotnal. Tdemehhn.. No. 11. 16-65. 1979.

U.S.S.R. Compur. Maarhs. .4larh. Phjx \‘ol. 20. No. 5. pp. 127-151 0041-5553:80,:050127-25$07.50,‘0 Printed in Great Britain 0 198 1. Pergamon Press Ltd.

ASYMPTOTlCMETHODSINFLUIDDYNAMICS*

0. S. RYZHOV

(Received 8 April 1980)

THE MAIN results obtained by asymptotic methods in various fields of fluid dynamics. in the

Laboratories of the Theory of Transport Processes of the Computing Centre of the Academy of

Sciences of the USSR. are outlined. Wave propagation in an inhomogeneous atmosphere is

considered. A unified treatment of non-linear wave processes in a radiating gas, and in chemically.

active mixtures. is given. Work on the theory, of transonic flows of both an ideal and viscous

thermally, conducting gas is described. 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. Earlier work

The application of asymptotrc methods to the solution of problems in different fields of

mechanics at the Computing Centre of the Academy of Sciences had its origins in work carried

out before the organization of the Centre. Back in 1942. Dorodnitsyn had published two papers

on the theory of the boundary, layer in a compressible gas. that have since become classical. see

[I? 21 A transformation of the independent variables was used by Dorodnitsyn whereby, at a

Prandtl number equal to unity, the equations of the laminar boundary 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.

128 0. S. Ryzhov

dependent on the Prandtl number. The now so-called Dorodnitsyn variables can be used to study

turbulent as well as laminar gas motion.

Boundary lay,er theory, clearly demonstrates the power of asymptotic methods. During its

development. simple devices were devised for computing the resistance of bodies at both subsonic

and supersonic speeds. By introducing semi-empirical relations into the theory, effective methods

could be devised for computing the friction in turbulent flows. It is in the context of this theory

that heat transfer and the heating of flight vehicle surfaces are usually calculated. The success of

boundary layer theory, was so great that its ideas and methods penetrated into branches of

mathematics as well as mechanics. By, now, solutions of boundary layer type have infiltrated

various organic concepts of mathematical physics. In particular, it was from boundary layer theory

that the method of matching external and internal asymptotic expansions grew; the method has

been given a strict proof in some comparatively simple problems (41.

As regards integration of the Prandtl equations. the approach to this problem naturally

underwent drastic changes with the coming of the electronic computer, Various approximate

devices for constructing the velocity field were supplanted by accurate numerical methods for

computing them. Having again returned in 1960 to the laminar boundary layer problem.

Dorodnitsyn described a general method of integral relations for its solution [5]. By using

smoothing functions. it was possible to write a system, approximating to high accuracy the

solution all the way up to the point of separation in the incompressible fluid. The preliminary

computatjon of potential flow past a plane body. with subsequent determination of the boundary

layer characteristics, implied in essence a synthesis of asymptotic analysis with numerical methods

for solving partial differential equations. The problem of asymptotic analysis includes simplification

of the initial Navier-Stokes equations. the simplification being performed differently in different

domains. Numerical integration of the Euler and Prandtl equations enables the gas parameters to

be found in the potential and viscous flow domains with the required accuracy. A similar situation

is now typical of an! asymptotic theory.

With regard to asymptotic methods as such for solving differential equations. the work of

the Computing Centre in this field had its inspiration in Dorodnitsyn’s studies of 1947 of the

period of the limiting cycle of relaxation oscillations [6]. These oscillations are described by the

well-known non-linear Van der Pol equation. The idea is to find the limiting cycle by dividing it

into several overlapprng pieces in each of whtch the solution has qualitatively different behaviour.

The main difficulty is to find the corresponding asymptotic expansions for the required function:

by mating these in the overlap zones. we can compute the period of the limiting cycle as a whole.

In fact. the procedure for constructing the solution precisely corresponds to what is essentially,.

in modern terminology. the method of matching external and internal asymptotic expansions.

In 1952, Dorodnitsyn turned to linear second-order differential equations, in which the coefficient.

characterizing the rigidity of the oscillatory system, has a singularity (zero, or pole, of any order).

By introducing a simpler reference equation. preserving the singularity of the initial equation, it

was possible to write a single asymptotic form of the solution throughout the interval of variation

of the independent variable. In the particular case when the singularity is a first-order zero, the

reference equation can be Airy’s equation. When the Computing Centre was set up, studies in the

dynamics of systems with a finite number of degrees of freedom were concentrated in the Theory

of Equations Laboratory; the advances achieved in this field were summarized by N. N. Moiseev

in a monograph published in 1969 [7].

Asymptotic methods in fluid dynamics 129

Our further discussion will be restricted to work on fluid mechanics, chiefly performed in

the Theory of Transport Processes (t.t.p.) Laboratory. We shall omit altogethci the interesting and

important work on wave motions in bounded domains, since a full picture of the results obtained

can be obtained from the book by Moiseev and Rumyantsev [8]. To limit the list of references.

we quote only recent or survey papers, from which a full history of the progress can be built up.

2. Wave propagation in an inhomogeneous atmosphere

Assume that. in the initial state. the density po. pressure po, and components rio of the

particle velocity vector vu. are functions of Cartesian space coordinatesxi only, and are independent

of the time r. We wish to find the asymptotic laws of damping of the weak shock waves

propagating in this medium. The source generating the waves may be a high-power explosion or an

aircraft moving at supersonic speed. For plane waves in straight tubes, the problem was solved by.

Crussard in 1913. and the amplitude damping of cylindrically and spherically symmetric shock

fronts was established in 194.5 by L. D. Landau [9].

The entire analysis in these elementary cases is based on the assumption that the width of

the zone of disturbed gas motion is much less than the distance from the shock front to the source.

On considering small wave displacements. of the order of a few wavelengths. we can assume that

the excess density,. pressure, and velocity are connected by Riemann’s relations. since the increase

of entropy with shock compression of the gas is proportronal to the cube of the variation in an>

of these quantities. But when the wave travels considerable distances. we naturally have to take

the damping of the disturbances into account. on the basis of purel!~ geometric factors (cylindrical

or spherical symmetry’ of the problem). On the whole, the procedure is in accord with the

approximation of geometric acoustics.

To allow for non-linear effects. Landau used a dependence of the disturbance propagation

velocity on the excess pressure. The shock wave moves at a speed which is also determined b! its

amplitude. Hence a simple rule is obtained. specifying the position of the discontinuity in the

Riemann wave. The non-linear nature of the initial Euler equations leads in the last analysrs to

unbounded growth of the width of the disturbed domain and to asymptotic shock wave damping

laws which differ from those predicted by sound theory. Hence the device for constructing the

solution consists in computing all the gas parameters from the approximation of geometrlcal

acoustics, but assigning them, not to the coordinate values corresponding to this approximation.

but to points whose disposition is determined by non-linear processes. These are typical featur-es

of the asymptotic method of deformed coordinates, the formal development of which is usually

linked with the names of Poincare, Lighthill. and Go [IO] .

When solving the general problem, the assumption that the zone of disturbed motion is

narrow is preserved: the wavelength is assumed small compared with the principal radii of

curvature of the shock front and with the characteristic dimension of the atmospheric

inhomogeneities. The description of the field of excess gas parameters remains as before; their

amplitude variation is found by integrating the equations of geometrical acoustics along the rays.

or bicharacteristics, defined as

dxi dn, - =non:+r,o. -= dt dt

(n,n,-6,) (2.1)

130 0. S. Ryzhov

Here, au is the velocity of sound in the initial atmosphere, nr are the components of the unit normal

vector II to the wave front. 6~ are the components of the unit tensor; repeated subscripts j, k

indicate summation from 1 to 3. In work done in I96 I- 1963 in the Mechanics of Continuous Media

(M.C.M.) Laboratory of the Computing Centre, the equations of geometrical acoustics were assigned

the form of a law expressing the conservation of sound energy when short waves of small amplitude

propagate in a moving media [ 1 I. 121. The application of this law to a volume included in an

elementar!. ray tube yields a simple expression for the excess pressure:

(2.2)

Here. u,,u is the projection of the ray velocity uo=aon+vo on the direction of the vector

n,fis the area of the wave front element inside the ray tube. and pa’, fo, poo, clco and i~,~(,

are the values of the respective quantities at the initial point.

If we now take account of non-linear factors and accurately calculate the displacement due

to them of points with pressure (2.2) along the chosen ray, we can find the asymptotic damping

law of the shock front. We shall assume for simplicity that the excess gas parameters in the sonic

pulse, bounded at the front by the weak discontinuity, have triangular profiles. Then, for the

shock wave amplitude p’* we have [ 1 I? 121

(2.3)

where X,-J denotes the initial length of the pulse. the thermodynamic coefficient rno for a perfect

gas is expressible in terms of the Poisson adiabatic exponent K by the relation rzn= (x-l- 1) /‘2.

and 1 measures the distance along the ray’. In situations typical of an explosion or supersonic flight.

the second term in the brackets on the right-hand side of (2.3) is not merely not small compared

to I, but increases without limit as I + 00.

Since the decrease of the shock front amplitude along the ray is known in explicit form. we

only have to calculate its position in space. The problem amounts to numerical integration of the

system of ordinary, differential equations (2.1). The elementary area(is found from several

adjacent rays. The theory, so far developed is based on calculation of the sonic shock parameters

[ 131; it can be used up to formation of the caustic. If the ray envalope is known. the field of

disturbances in the neighbourhood of the intersection of the shock wave with it is subject to a

partial differential equation of mixed elliptic-hyperbolic type. Numerical solution of the non-

linear problem of the incidence on the caustic of a sonic pulse with triangular profile of excess

pressure, was obtained in 1978 in the Transport Process Theory (T.P.T.) Laboratory (141.

Asymptotic methods in fluid d.wamics 131

3. Non-linear waves in a radiating gas and chemically active mixtures

Non-linear acoustics was developed, for waves whose structure depends essentially on a

radiant energy flux, by joint work of the M.C.M. and T.P.T. Laboratories of the Computing

Centre in 197Z!--1975 (see [ 15. 161). Gas motions with plane, axial. and central symmetry have

been considered. The asymptotic analysis was based on the assumption that the disturbed domain

is narrow compared with the distance to the source. But the formal methods employed correspond.

not to the method of deformed coordinates. but to the ideas of the theory of short waves.

developed in 1956 b) S. A. Khristianovich [ 171. The introduction of new independent variables.

connected with the wave element. implies in essence a transformation to so-called optimal

coordinates in the method of external and internal asymptotic expansions, which ensure a unified

representation of the solution in both domains [IO]. Hence the small neighbourhood of the wave

front (possibly. smoothed bjf light quanta radiation. absorption, and scattering processes) is not

specially segregated from the zone occupied by disturbances. AU in all, the asymptotic analysis

leads to a non-linear system of integro-differential equations; but the Peierls’ equation, containing

the integral term. for the spectral density of the radiant energy, is linear. Hence it follows that the

influence of selective radiation on the flow structure in non-scattering gas is equivalent to the

action of effective pre)’ radiation. if the average of the optical thickness and of the radiant energq

density is specified according to a definite rule. In the framework of linear theory, this analog),

was pointed out b>r Vincenti and Baldwin [ 181 : its role increases especially in ordinary earth

conditions, when scattering can be neglected with high accuracy.

Further simplifications may be obtained for sonic pulses with small, or conversel). large

optical thickness. if the dependence of the radiation on the frequency is assumed to be slight.

Here. from the very beginning. the Peierls equation for a grel- gas serves as the initial equation.

Moreover. it ma) be assumed that the adiabatic and Isothermal velocities of sound in a medium

in the equilibrium state are close in value. The main result of a supplementary analysis consists in

reducing the system of integro-differential equations to a single differential equation for the

dimensionless particle velocit! 1’. For opticall! thin signals we ha\e

(3.1)

where the time t and the distance r are measured in a special dimensionless coordinate system.

moving with the wave element. while the values of the constants b and BTO are evaluated front

the rate of disturbance propagation and the thermodynamic properties of the substance. and the

parameter d depends on the symmetry of the problem. By making the substitutions

b-l/b, BT,p-- BSD=-BTOf’12. r-+-r and I;+--L , we can write at once. with the aid of

(3.1), the equation for opticall!, thick sonic pulses. These equations. with d = 1 and the time

derivatives put equal to zero, have provided the basis for computing the structure ofweak shock

waves in a radiating gas.

132 0. S. Ryzhov

A related problem arises when studying non-linear wave processes in chemically active gas

mixtures; it was examined during 197 l- 1980 in the T.P.T. Laboratory [ 19, 201. A strict

mathematical analogy was established, according to which the influence of the chemical

transformations on the quasi-equilibrium propagation of sonic pulses with shock fronts is

equivalent to the action on their structure of a longitudinal viscosity and heat conduction. In the

framework of acoustics, another statement of the analogy betweeen relaxation and viscous flows

was proposed in 1936-1937 by Leontovich and Mandel’shtam [21, 221.

When studying waves in media with any amount of reaction, it is assumed that the difference

between the equilibrium a,0 and the frozen un sound velocities is small; this assumption naturally

imposes some constraints on the equation of state of the substance. If there areI\’ relaxation

processes, there are h’ - 1 so-called intermediate sound velocities Q,,o? at which pulses of small

amplitude can travel. The quantities cr,o were introduced in accordance with the purely formal

arguments of Napolitano [12] : the strict proof of the inequalities

apL,=ao,o~a,,o~ . . . <a.,.-c. o<a.v, o(a..o

is to be found in [ 19. 201 If we characterize the difference between the p-th intermediate sound

velocity and the velocity of wave packet displacement by the contant -y(p), then we have, for the

dimensionless particle velocity 1‘.

(3.2)

In the same wa>’ as when obtaining Eq. (3. I). the time I and distance r are measured in a special

coordinate system, following the movement of the wave element, while the symbol CJ~ denotes

the sum of all possible products. made up of the eigenvalues hl: . A,y of the relaxation

matrix R, taken 1 at a time in each product. Lalrering of the disturbances in individual wave

packets is dictated b! the fact that. with i.,B , . . >i.,~ , the equilibrium state is not reached

simultaneousI!- b) all elements: conversely. completely determinate combinations of them relax

to equilibrium successively (more precisely, linear combinations of the chemical reactions densit!

vector components). Direct]) related to this process are also the sizes of the intermediate sound

velocities [ 191

Assume now thar a single reaction occurs in the mixture. Then, taking account of the

connection between the constants ~(~)=y~ and y(.VJ=y”‘=y,, Eq. (3.2) can be reduced

to a form preciseI!, the same as that of (3.1). Hence follows the complete mathematical analog)

between non-linear wave propagation in a radiating gas and in an elementar}. chemically active

system. The constatlt 7e must be associated with f3~0 for opticall}. thin signals. and with Bso for

pulses with large optical thickness. Equation (3.2) with d = I serves as the starting point when

stud>,inp the internal structure of weak shock waves [20].

Asynptotic methods in fluid dynamics 133

4. Transonic flows of ideal gas

The present subject was first actively developed in the M.C.M. Laboratory, then later in the T.P.T. Laboratory of the Computing Centre. The early studies were concerned with the flow properties close to the critical cross-section of a Lavalle nozzle, where the transition through the velocity of sound occurs. The results obtained in this direction up to 1965 were summed up in the author’s monograph [24]. The basis of the asymptotic analysis of transonic flows is the system of Fal’kovich-Kalman equations [25, 261

a1.x aL.7 atA dc, -L'x- t - I(&$L 0, -=

a.?. dr 0

r dr- ax

for the components vx, r, of the disturbed velocity. vector, which are taken in a dimensionless

system of Cartesian or cylindrical coordinates X, r. To avoid difficulties connected with the construction of the field of flow through the nozzle, the Cauchy problem was taken with the initial data

L’,=-.-l,]~(~ for s<O, L-,=A~~~ for s>O, l&=0,

(4.1)

(4.2)

preassigned along the tube axis r = 0.

It can easily be seen that Eqs. (4.1) jointly. with conditions (4.2) admit of a continuous two-parameter group of similitude transformations. in connection with which, the solution of the problem is a similarity solution:

(4.3)

The functions f and g satisfy a system of two ordinary differential equations, which are also invariant under a group of transformations. Hence it follows that the solution of the converse problem of nozzle theory reduces to the study of the integral curves of a first-order equation in the phase plane. Detailed analysis has shown that, along with the continuous (but in general, non-analytic) solutions, there exist generalized solutions, which include strong discontinuities; these represent a flow with shock waves, issuing from the channel centre, i.e. from the point of intersection of the sonic line with its axis. In plane nozzles with d = 1 three asymptotic types of flow may be realized, having no singularities on the characteristic passing through the point mentioned. The first type, with k = 1 and n = 2, was studied in detail in 194% 1946 by Frankl’

and Fal’kovich [25, 271, the corresponding velocity field being obtained analytically. The second type is obtained by solving the Cauchy problem with k = 4/3 and n = 3; in it there are discontinuities of the third derivatives of the velocity vector components with respect to the coordinates, on the characteristic issuing from the nozzle centre. The third type corresponds to values of the power exponents k = 20/ 1 I and n = 11. It is remarkable in that it points to the possibility of forming a density jump at the point of intersection of the sonic curve with the central streamline in the flow, without singularities in the supply part of the tube. All these asymptotic types of gas motion are realized in “natural” nozzles, whose walls may be as smooth as desired.

134 0. S. Rprhov

When k > 2. the solutions of the Cauchy problem yield nozzles with a straight sonic line.

They were initially studied in 1950 by Ovsyannikov [28]. In a joint work by the author and

Yu. D. Shmyglevskii, a general theorem was formulated about the properties of the surface of

transition through the velocity of sound, which is at the same time a characteristic surface of the

equations of gas dynamics [29]. It turns out that this surface has minimal area among all the

surfaces that can be stretched over the given contour.

The gas flow past an infinite wing profile or past bodies of finite size in the transonic range

was examined in the period 1968- 1978 in the T.P.T. Laboratory. The first work related to

asymptotic damping laws of the three-dimensional disturbances introduced by any body into a

uniform flow with critical speed [30,31]. Frankl’ established that the principal term of the

solution for plane-parallel flows can be written in the similarity form (4.3) with n = 4/S; it gives a

velocity field with sonic line extending to infinity [32]. The results show that, in the similarity

form of the principal term? the power exponent n = 4/7, if the flow has axial symmetry [33, 341.

Attempts at a clear interpretation of the principal terms have met with no success, though the

correction terms of the relevant asymptotic sequences have a simple physical meaning: one

corresponds to the source, and another, to the lift applied to the body. These sequences can be

used to construct a channel round a half-body with generator specified by a power function [31].

Expansion of the velocity vector components in asymptotic series in ascending powers of r

was used to justify the so-called stabilization law for wing profiles and bodies of revolution

[35-371. The law was discovered by Khristianovich, Gal’perin, Gorskii, and Kovalev as a result

of systematic processing of experiments performed in 1944- 1948 at Ts AC1 [38]. The most

complete description of similar foreign experiments is to be found in Holder’s lecture, given in

memory of Reynolds and Prandtl [39]. The essence of the stabilization law consists of the fact

that, when the velocity of the subsonic incoming flow increases, the distribution of the local Mach

numbers along the body surface, as far as the density jump, deviates surprisingly little from the

limiting distribution at the critical velocity, at infinity. With regard to the actual density jump. it

moves fairly, rapidly, towards the rear edge of the surface. The sharp change in the body, resistance

is due to the motion of the shock front.

The theoretical approach to explaining these experimental laws is based on the introduction

of correction functions v’X and v’~ to the principal terms in (4.3), which represent the asymptotic

behaviour of the velocity field in the sonic flow. The corrections are written as

VI ‘=Erm-“[fl(~)+. . .], l/.l.‘=crm-l [g*(E)+. * .I

with the previous similarity variable .$. The problem is to find the powers m and the connection of

the small parameter e with the difference between the Mach number M, of the incoming flow

and unity. It has been shown that, for the wing profile of infinite dimension, in the domain ahead

of the density jump m = 8/5, while in the domain behind the shock front, m = 3/5. Similar results

for bodies of revolution lead to the values m = 8/7 and m = 2/7 respectively. Simple arguments,

based on the invariance of the system of Fal’kovich-Kalman equations under the group of

similitude transformations, enable E to be expressed in terms of M, - 1. The stabilization law

receives a quantitative statement as well as qualitative confirmation. On varying the incoming flow

velocity relative to the critical velocity, the gas parameters along the wing profile generator ahead

of the density jump differ from their limiting values for sonic flow by a quantity proportional to

I M, - 10. For a body of revolution, the gas parameter deviations in this domain from their

limiting values are of the order of I M, - 1 I,/‘. Behind the shock wave front, the pressure and local

Mach numbers vary much more strongly: along the wing profile, as I M, - 1 I%, or along a body of

revolution, as IM, - 1 1?/3. The last two estimates in fact define the growth in resistance

Asynptotic methods in fluid dynamtcr 135

experienced by flight vehicles in the transonic velocity range as the number M, increases. The

results of many computations are in excellent agreement with the conclusions of asymptotic

theory [3S-371.

A numerical study of 1978 into the finite structure of the velocity field at a wing profile of

infinite dimension led to the conclusion that the density jump closing the local supersonic zone is

generated at an interior point of the zone as a result of intersection of characteristics [40]. This

problem has given rise to long discussions in the literature, which are relected in Bers’ monograph

1411.

5. Viscous transonic flows

Systematic study of the influence of the viscosity and heat conduction of an actual gas on

the structure of transonic flows commenced in 1964 in the M.C.M. Laboratory, then was continued

in the T.P.T. Laboratory till 1978. If we apply asymptotic analysis to simplification of the system

of Navier-Stokes equations, then. following independent discussions of various authors (42-441,

we arrive at the following for the components pX, I v of the dimensionless vector of the disturbed

velocity:

dc, al_&

-Vxds + 4l,(&1)‘cli!&O, au, dvr -=

OX? -F- ax 0, r

(5.1)

It is clear from the structure of Eqs. (5.1) that it is possible to neglect the contribution of the

viscous tangential stresses, i.e. these cannot be used when investigating the boundary layer next

to the body surface.

From the exact integrals of system (5.1) we can construct the velocity field in uncalculated

operating modes of the Lavalle nozzle, when density jumps form behind its mouth, moving towards

the exhaust as the pressure at the cut falls. Another example is given by the bending of the sonic

flow round the edge of a flat plate, the expansion wave being then described by the similarity

solution (4.3) with d = 1 and n = 2/3. In both these cases the role of the dissipative factors amounts

to smearing of the weak discontinuities. carrying the jumps of velocity component derivatives, and

of the shock wave fronts [42-441.

The flow past a paraboloid of revolution is of much greater interest. The velocity field round

it is likewise subject to the similarity solution (4.3), where d = 2 and n = 2/3. It turns out that the

influence of viscosity and heat conduction on the flow formation in the case of thick paraboloids

is negligible, and the term a*v,/ax* on the left-hand side of the ftrst of Eqs. (5.1) can be omitted,

The flow past a paraboloid with average cross-sectional area is determined by the balance of

convective and dissipative processes. The role of viscosity and heat conduction becomes dominant

in the gas motion in the case of thin paraboloids, when the contribution from the non-linear term

v,&/ax on the left-hand side of the first of Eqs. (5.1) tends to zero. This conclusion remains

true when we turn to bodies of finite size: the asymptotic laws of disturbance damping, generated

in a uniform flow with critical velocity. are wholly determined by dissipation effects.

136 0. S. Ryzhov

To construct the velocity field in a viscous gas at great distances from any finite body, we

neglect the non-linear term in the first of Eqs. (5.1). The resultant system of linear equations of

quasi-elliptic type admits of an expanded (two-parameter) group of similitude transformations.

We seek the appropriate solution of the system in the similarity form [44]

With n = 2/3. relations (4.3) and (5.2) predict the same type of degeneration of the disturbances

as we move awa) without limit from the obstacle (e.g., a paraboloid of revolution). The asymptotic

behaviour of the velocit>r field at great distances from the finite body is specified by the solution

with n = 413, which corresponds to a source located in a uniform sonic flow. The solution is

completed b}. means of the relations

(5.3)

where the constant B is proportional to the source power, and * denotes the Tricomi function.

The role of viscosity and heat conduction thus appears, not only in smoothing of the discontinuities,

but also in a stronger decrease of amplitude of all the gas parameters as compared with the

amplitude obtained in [33! 341 when no account is taken of dissipative processes. In other words,

the neighbourhood of the point at infinity is a special kind of “boundary layer”. A strict proof of

relations (5.3), based on a proof of the existence of a unique solution of the problem of viscous

gas flow past a body in the context of non-linear equation (5.1), was obtained by Diesperov and

Lomakin [45,46]. The field of three-dimensional disturbances, related to creation of the body

lift. is found by Fourier expansion of the required solution with respect to the angular variable

(471.

Flows of chemically active mixtures in the transonic range represent an independent field

of study. ru’apolitano and the present author [48] pointed out the strict mathematical analogy

between quasi-equilibrium and viscous inert flows. Reactions with substantially different

characteristic times cause layering of the relaxation zones; this can be traced from the change in

the different operating modes of the Lavalle nozzle [49].

6. Non-stationary one-dimensional and similar gas motions

Work in this field started in 1968 in the T.P.T. Laboratory of the Computing Centre, with

the solution of the problem about a piston, expanding from a certain instant according to a

power law with exponent less than the value corresponding to a strong explosion (501. It was

assumed that, even before the piston was set in motion, finite energy was transmitted to the gas.

In this case the gas energy remains bounded in an infinite time interval, if the initial temperature

of the substance is zero. Hence all the required functions are obtained by linearization with respect

to the values occurring in the classical problem about a strong explosion, which was treated in

detail by Sedov [5 1. 521 and Taylor [S3].

Asynptotic methods in J7uld dynamics 137

In 1973 work was carried out on non-stationary flows with strong shock waves, which are

close to similarity flows (541. Such flows are either one-dimensional or close-to-one-dimensional.

When writing the general expressions for the mass, energy, and momentum of the material inside

the disturbed domain, the time-independent terms were isolated. To these terms correspond the

first integrals of the equations in variations, which have an extremely simple form. For instance,

assume that the basic flow has axial symmetry. We Fourier-expand the radial uy and angular V~

velocity components, the density p, and the pressurep with respect to the polar angle q, and

retain only the terms with li-th harmonic in the series. Then.

2n C, = - eLntn-1-“‘2um(i.) sin (J~cf+cr,) +. . . ,

z+1

%+I P=- z_1 ~c[g(i.)‘~t-m’2g,(~.)co~(k~i-aR)+...l.

(6.1)

P= z p,~‘nt’:n-!’ h [ (i.)~~t-“‘2h,(3.)co~(k~Saa)+. . .I.

where the functionsf, g, and h of the first approximation depend only on the similarity

combination X = r/(bt)n of time t and the cylindrical coordinate I; the small parameter E is

introduced for convenience as a coefficient in the correction terms; po denotes the initial density.

K the Poisson adiabatic index, and b and ak are arbitrary constants.

We direct the z axis along the vector of total momentum communicated to the gas. We

consider the volume V between the shock wave surface X2 and a surface h = const. The contribution

to the momentum I(h2, A) is given by the integral. containing only the 1:). particle velocity

component: in relations (6.1) it is sufficient to put k = 1. ok = 0. All in all,

This expression is time-independent if m = 2(3/r - 1). The derivative of the momentum is found

from the relation

dI (L, i.)

dt = . [ !,l’, (.Ycr,) -pnj]da, 4

I

(6.3)

which takes account asymptotically of the Rankin-Hugoniot condition on the strong shock wave

front. Here,nY denotes the projection onto the), axis of the unit normal n to element da;N, gives

the rate of displacement of this element along the normal n, and v, is the normal component of

the particle velocity. With m = 2(3n- 1). in accordance with relation (6.2). the derivative

dZ(h,, A)ldl=cl. From (6.3) we have the relatron

138 0. s. Ryzhov

(6.4

which connects the functions ,fm, g,, h, and u,. It is easily shown that (6.4) is the first

integral of the equations in variations.

Similar arguments apply to the momentum transfer in waves with near-spherical shape. In

three-dimensional flows, all the gas parameters are expanded in series in spherical functions I’$,

whose arguments are the angles 9 and 9 of the spherical coordinate system. In this wasy Terent’ev

found the solution of the explosion problem, characterized by transmission of momentum as well

as energy’ to a finite volume of gas [55]. Construction of the first integrals of the equations in

variations, corresponding to the laws of conservation of mass and energy of a substance, is a

simpler problem, since the gas motion remains symmetric in both the first and the second

approximations. Divergence forms of the partial differential equations have found wide

application when solving very diversified problems of gas dynamics by asymptotic methods [56].

Intermediate stages of gas motions, generated by a source communicating to the particles a

certain initial velocity, were studied analytically and numerically during 197 l- 1979. The starting.

point for this work was the study of an explosion at the boundary of two media with different

densities [57]. It was found that the domain close to the shock wave, travelling through the gas

with higher density, can be fairly accurately described in a considerable time interval by the well-

known solution of the short shock problem [58,59]. A similar conclusion was reached by

Derzhavina, when considering the energy separation in the internal and kinetic forms in a gas

with uniformly distributed initial density (601. Elucidation of the role of the initial particle

velocity in the formation of non-stationary gas motion led to the construction of cylindrical and

spherical analogues of the solution of the short shock problem [61]. The envelope of the

characteristic curves m the actual flow is cut off by the front of a secondary shock wave issuing

from the centre. Parkhomenko considered the departure of the flow to the asymptotic form in

waves converging to the axis or centre of symmetry; these waves arise from the separation of the

internal and kinetic energy in the peripheral domain [6?].

7. Hypersonic flows

The work in this field at the T.P.T. Laboratory of the Computing Centre started in 1969

with the study of uniform viscous gas flow past a half-body at Mach number equal to infinity [63],

So-called strong interaction was also considered, when the pressure in the gas is primarily

determined by the growth of the boundary layer thickness, while body shape variations only

contribute small disturbances. Under these conditions, the flow parameters in the non-viscous

domain can be expanded in series in which the principal terms correspond to the pressure induced

by the boundary layer on a flat plate.

With supersonic flow past a blunt body, a density jump is formed at some distance ahead of

the nose section. At hypersonic incoming flow speeds, the pressure behind the front of the direct

shock wave increases strongly. Hence the profile of a wing of infinite dimension or a body of

revolution is washed by filaments of high entropy. The properties of this special type of thin layer,

in an ideal gas (i.e. discounting dissipative processes), were first studied by Cheng (641 and Sychev

(651. During 1969-1980, the theory) of the high-entropy layer has been developed in the

As),mptotic methods in fluid dynamics 139

Transport Process Theory Laboratory in the framework of the method of matching external and

internal asymptotic expansions [66, 671. In particular, the simple relation

was found for the generator rb of the body contour in the converse problem with density jump

corresponding to a strong explosion:

r2=c5~ (??“)* (7.2)

In (7.1) ho is the value of the function h of Section 6 at h = 0. Notice that the incoming flow

velocity U, is simply, the ratio of the coordinate x to the time I, in obvious analogy between non-

stationary gas motions and hypersonic flows in space c with one fewer dimensions [68]. It is now

easily shown that relations (7.1) define a partrcle trajectory initiated by an explosion wave; the

entropy of the particle corresponds exactly to the gas compression in stationary hypersonic flow

at a direct density jump. Hence we have the following extremely simple rule: the results of strong

explosion theory due to Sedov and Taylor [5 I-531 can be used without modification in the

entire domain between the body (7.1) and the shock wave (7.2).

Subsequent reflection of disturbances from the shock front and their interaction with the

high-entropy layer adjacent to the lateral surface of a blunted wedge, were studied in detail by

Manuilovich and Terent’ev [67], basing their analysis on the earlier results of Chernyi [68]. It was

found that the Line of transition through the velocity, of sound does not necessarily join the

density jump with the body; it may deviate at its ends to infinity, parallel to the wedge faces. In

three-dimensional hypersonic flows past blunt bodies. a singular eddy layer forms at the bodl,

surface, if the entropy does not reach an extremal value on the critical (branched) steam line [69].

Calculation of the flow past a supporting body,. acted on by lift as well as resistance, is one

of the basic tasks of aerodynamics. Chemyi’s results [68] show that, in the context of the above-

mentioned analogy between non-stationary gas motions and hypersonic flows, the velocity field

induced by the body resistance can be obtained from a solution of the strong explosion problem.

Studies were made in 1974-1975 at the T.P.T. Laboratory, of the disturbances generated by the

lift remote from the profile of a wing of infinite dimension, or from any bounded body [70, 7 11.

In the latter case, the longitudinal vX ) radial Y,. and angular v~, velocity components, along with

the density p and pressure p, can be expanded in Fourier series with respect to the polar angle q.

The terms with the first harmonic are

140 0. S. Ryihov

(Cont’d)

x-1 p= ~p_(~t:,J~)+DY~--[ln~~i?(E)+p13(~!lcO~(F+...}?

1

p= 2(211)2 p-_ c-,? -+*:(e)+b ,x-“‘[ln~p,?(~)Sp,~(LS) lcosqf.. .},

where the functions IJ,,~, pil and iIll of the first approximation depend only on the

similarity, combination j-r / (bs) “I , and are found by solving the strong explosion problem;

the small parameter by is proportional to the body lift, while U,, p_, and p_ denote respectively

the velocity, density. and pressure in the incoming flow.

In the systems of the second and third approximations, the equations for the disturbances

vX 12 and vX 13 of the longitudinal velocity component separate out from the rest, and can be

integrated after finding all the other parameters uri2,. . . , p12 and u,~~,. . . , pis. The latter

two groups of parameters are subject to systems of equations, arising when studying the second

and third approximations in the theory of non-stationary two-dimensional gas motions. The

functions of the second approximation can be found in explicit form, by using the invariance of

the solution of the strong explosion problem under displacement along they axis. It is this

property that stipulates the inclusion of the logarithmic terms in expansion (7.3): they also have

to be supplemented by relations (6.1) with II = ‘/i and m = I. The lift Fy is expressible in terms

of the functions of the third approximation:

It is clear from the results of Section 6 that they, must be connected by the first integral of the

appropriate system of ordinary differential equations. In fact, the connection is given by relation

(6.4) if, in the latter, we make the replacements J+v,,,, g-p,,, h-+p,, and fm-+~r,3,

gm+pi3, 12 m’,P13, wTl--tV,i? , and we add to the right-hand side of the resulting relation

the quantity -2vrll,,ll. Now. the hypersonic flow parameters can be computed in any plane

x = const for any body to which both resistance and lift forces are applied, from the solution of

the strong cord explosion problem, when momentum perpendicular to the cord (along they axis),

as well as energy. is communicated to the gas. According to calculations, 12 = 0.2775.

8. Laminar trail

The stationary uniform flow of a viscous incompressible fluid past a finite body is an

extremely rare case in which we can prove an existence theorem for the solution and obtain exact

estimates of the order of decrease of the disturbance velocity vector along any direction. Following

publication of Ladyzhenskaya’s book [72], the strongest results here were obtained by Babenko

[73]. In 1975-1976, the T.P.T. Laboratory studied the velocity field at great distances from the

profile of a wing of infinite dimension and from a body of revolution, using matching of the

external and internal asymptotic expansions [74]. The external expansion describes the domain

of potential flow; the trail structure is established by means of the asymptotic expressions of the

internal expansion. Of course. there may be more terms in the asymptotic expansions than when

the problem is investigated strictly; but in all approximations that admit of comparison. both

approaches lead to the same results.

Asymptotic methods in f7uid dynamics 141

For the three-dimensional trail behind a supporting body of limited size in a transonic flow.

the solution of Landau and Lifshits for flows of incompressible fluid [75] proved suitable [47].

The only difference is linked to the density variations due to gas compressibility; these are found

independently by integration of the classical equation of heat conduction. In the near-by zone of

a viscous laminar trail, there is a rolling of the stream surface similar to the twisting of a vortex

sheet along its lateral sides when the sheet converges from a wing of finite size in an ideal fluid

(where there are no dissipative processes). In the central part of the far trail the flow tends

asymptotically to plane-parallel flow, in which connection the stream lines deviate to infinity

logarithmically, away from the side on which the lifting force acts [76].

Matching of the external and internal asymptotic expansions was used in 1974-1978 to find

the structure of the eddy trail behind a body in the hypersonic flow of viscous heat-conducting gas, in joint work of the present author and Terent’ev [70, 771. The flow field in the outer domain is

subject to the system of Euler equations; its construction is described in Section 7. Comparing the

relative size of the convective terms appearing in the initial Navier-Stokes equations, and the terms

due to heat transfer, Sychev concluded that, in order to continue the gas parameters into the inner

domain of the laminar trail, we need to use. instead of the similarity combination g=r / (bs) “’ , the new variable % =r / b”>z VW+*) (see [78] ). This conclusion remains true for finite bodies,

to which lift is applied in addition to resistance. For the domain occupied by the trail, the

asymptotic expansions have the form

+b,x-k~[~,?,(5)cos(k,ln 2~)+~.~~~(5)sin(k~lnz) lcosq+. . .},

1

x+1 -l/(X+iJ {pzi (t;)

+b,x-k6[pdf;) cos(k,lnx)+pPl(~)sin(k3 In~)]cos@-. . .},

1 P

= 2( %-+l)z pmu,2 +{p*i (L) +x-y/(x+i)P22 (5)

(8.1)

+b,x- ~~~x+~~-k~[p~c(~)~o~(k3lns)+p~s(l;)sin(kgln~)]coscp+. . .},

2-X k., = k, =

2(X+-l).

142 0. S. Ryzhov

One result of substituting relations (8.1) into the initial Navier-Stokes equations is the

equivalence principle, according to which the trail characteristics in any x = const plane can be

evaluated (regardless of their values in the other planes) from the solution of the directed explosion

problem with total momentum having non-zeroy-component. Further, in the axisymmetric flow,

the viscous stresses become significant when finding the longitudinal velocity component, whereas

the field of the remaining parameters can be constructed by taking account of heat transfer only.

On the contrary, in the asymmetric disturbances due to the body lift, the fields both of the

velocity vector and the thermodynamic quantities equally depend on the viscosity and the heat

conduction of the gas.

The limiting conditions as { + m for the trail functions are determined by the asymptotic

properties of the gas motion on approaching the inner boundary of the outer domain. Here, inertia

forces are balanced only by the pressure forces, in such a way that the distributions of the gas

parameters take on oscillatory properties [7 I]. In fact, the asymptotic expansions for asymmetric

disturbances as 5’0 contains terms with cos(k In E) and sin(kln E), k=[ (~-X)/(X-I)]‘“.

Arising at the trail boundary,, the oscillaiions are then transmitted throughout its length; in

connection with this. terms appear in relations (8.1) with cos(k, In 3) and sin(k, In 5).

In view of the equation k,=k(z-1)/2(x+1). the variations of the frequency of

oscillation along and across the trail are different.

9. Boundary layer

We mentioned above that, next to the surface of a blunt body, in three-dimensional hyper-

sonic flow, there is a singular vortex layer. unless the entropy reaches its maximum value on the

critical streamline 1691. Formatron of such a singular layer is also possible when an incompressible

fluid flows past an obstacle. pro\,ided that the total pressure is different on different streamlines

1791. In this lay,er. as we approach the body, surface, the normal derivative of the velocity increases

without limit. In turn. the fact that the velocity derivative has a singular-it! in the exterior flow

domain plays a fundamental role when choosing the asymptotic expansion for the boundary layer.

As the Rey,nolds number tends to infinity. singular terms appear in this expansion, which are

absent in the higher approximations of classical Prclndtl theor), [69. 791,

The Prandtl theory takes on a whole series of entirely new features when we study the

so-called free interaction of the boundary lay,er with the exterior (non-viscous) flow. Even to a

first approximation. the pressure gradient here is evaluated, not from the solution of the problem

of ideal fluid flow, past a body. but on the assumption that it is determined by the growth in the

displacement thickness of the filaments lying close to the rigid surface. Non-linear perturbation

theory. describing the effect of free displacement, was formulated b), Neiland (80, 811, and

jointly by Stewartson and Williams [82: 831. In the framework of this theory we can explain the

propagation of disturbances up-flow at supersonic particle velocities at infinity, and obtain a

picture of the laminar break-away which is accompanied by the appearance of recirculation zones.

In the T.P.T. Laboratory of the Computing Centre, study of free interaction of a boundary layer

began in 1977 and is being actively pursued today: the basic idea is that the gas motion in the

break-away, zone is in general non-stationar!,: hence in the differential equations of the

mathematical model. we have to retain the principal time-derivatives.

Asymptotic methods in fluid dynamics 143

Asymptotic analysis of the system of Navier-Stokes equations was undertaken

independently by different authors from several different stand-points; the common result is to

obtain the Prandtl equations [84-861

a vr , au, P-J----_

ax ay 0,

z+ azv,

at &%+L”du,=_!$+_,

dX aY dY2

ap -= SY

0,

which contain the self-induced pressure

dA --1

ax if MS=-1,

P-

1 -

ia, aAjax dX

r[ s ~ 1 x-x if M,<l,

_m

(9.1)

(9.2)

the function A (t, x) being found during solution of the problem. In Eqs. (9.1) and (9.3). both the

independent variables and the required gas characteristics are referred to a special system of

measurement units. The boundary conditions on a plane platey = 0 are obvious: yX = 1’). = 0.

The remaining boundary conditions in free interaction theory are posed as limiting conditions.

In fact. as x + - ~0, we have z:,+ y. p+O. while asy + 00. we have 11~ - 1’ + A. Equations

(9.1) and (9.7) define the structure of the velocity, field in a narrow sublayer immediately

adjacent to the body. The two other domatns are occupied by the main boundary layer and b)

the exterior irrotational flow: the gas motion in them is quasi-stationary, wirh the result that the

equations of the first approximation do not contam time-derivatives. Conversely, in transonic

particle velocities at infmity,, the time-dependence of the required functions plays an important

role in the external potential part of the flow. Here disturbances are propagated, to which 11 IS

possible instantaneously to adjust the flow in the viscous layer next to the wall [87]. The statement

of the boundary value problems for three-dimensional flows is rather more complicated [88. 891.

To obtain an idea of the non-stationary process of free interaction at M, > 1. system (9.1)

was linearized. If we write the solution as

p=a exp (olikx),

r,=y---a exp(wlSX )% “ dy ’

(9.3)

t’,=ak CSj’(W-F;s\f(,!/)

and neglect in all the relations the terms proportional to the amplitude a squared, the function f

has to satisfy an ordinary, third-order differential equation. In the boundary conditions, stated

above for a flat plate, there are no sources of excitation of oscillations, so that the eigenvalue

problem is posed for the differential equation; in general. this problem contains two unattached

parameters. The general properties of the dispersion relation

(9.4)

144 0. S. Ryzhov

where the variable z=o/k*“+k’“y, and Ai is Airy’s function, were studied jointly by the present author and Zhuk [90] . A complete solution was given there for the eigenvalue problem.

It was shown that, for a fixed complex wave number k, there is an entire (discrete) spectrum of eigenfrequencies. A similar conclusion holds for a given complex frequency o, when the wave number is required. The distribution of the eigenvalues in the “tails” of the spectra is established by means of asymptotic methods; some of the first eigenvalues may be found by numerical solution of Eq. (9.4).

On substituting the eigenfunctions of the boundary value problem into the right-hand sides of (9.3) we can construct the field of gas flows in the boundary layer. They may be treated as internal waves, resulting from the joint action of the self-induced pressure and the viscous tangential stresses. If the internal wave is a travelling wave, then, for futed k, its rate of displacement up-flow is uniquely determined. For travelling waves carried down-stream, the dispersion equation has an infinite set of solutions; close to the wall, high-frequency vibrations with respect to the transverse coordinate make their appearance in these waves.

For M, > 1, Terent’ev solved the boundary value problem of the small harmonic oscillations of an oscilIator located at a distance from the edge of a fmed flat plate [9 I]. The disturbances radiated by the oscillator propagate against the flow as internal waves, uniquely determined by the

eigenvalue k. The gas motion down-stream from the source includes an infinite system of internal waves with different k. The length of each wave depends only on the oscillator frequency. Asymptotic analysis of the solution reveals the disturbance damping laws at fairly large distances

from their point of generation. For high frequencies of the oscillator, the pressure in the boundary layer proves to be close to the pressure found by solving the external supersonic flow problem for a vibrating obstacle (with an ideal (non-viscous) gas).

In the non-linear process of interaction of the weak shock wave with the boundary layer, the excess pressure 1921 is

p=OH(X)-g, H(x)= 0, X<G,

1, x>o, where 0 measures the amplitude of the disturbances. If the boundary layer is next to a moving plate,

then, in the boundary conditions for y =O and the limiting conditions as x + - m, and y +w, we have to introduce suitable modifications to allow for the plate velocity [93]. Numerical solution of

the problem shows that the characteristic dimension of the interaction domain decreases when the

the shock wave intensity is kept constant, while the wall velocity is increased. Recently, the Moore-Rothe-Sirs criterion [94]. originally postulated in 1956-1958, has become popular in the analysis of the structure of the zones of recirculatory gas motion. However, the data of a large number of computations refute this criterion; they reveal, as a typical feature of a separation with moving surfaces, the presence of two recirculatory zones with filaments separating them.

We must specially mention the deep connection between the free interaction of the boundary layer and its stability. Since, for an incompressible fluid (M, = 0) the self-induced pressure is expressible in terms of an improper integral with infinite limits, the real part of the wave number k in relations (9.3) must be equal to zero. In this case, the left-hand side of dispersion relation (9.4) remains as before, while the right-hand side has to be replaced by Tik’/3, where the choice of sign depends on the sign of Im k. Reduction to precisely the same form is possible for the secular equation in the classical Orr-Sommerfeld problem [95] provided that the critical layer for the relevant long-wave oscillations is immediately adjacent to the plate. An eigenfrequency w with zero real part gives the value of the wave number with tile aid of which we can immediately

Asymptotic methods in J7uki dynamics 145

write the asymptotic relation for the lower branch of the neutral stability curve as the Reynolds number tends to infinity, since the normalization of the variables used in the present section

includes powers of this number which are multiplies of l/S. It is worth recalling here that, according to the idea put forward jointly by Dorodnitsyn and Loitsyanskii back in 1945, the transition from laminar to turbulent boundary layer occurs as a result of local non-stationary recirculatory zones [96].

10. Kinetic processes in gas mixtures

The state of a neutral gas can be fairly accurately described by the non-linear Boltzmann integro-differential equation for a single-frequency distribution function. For a gas mixture, we

have to introduce an entire set of distribution functions fLIy, where the subscript ~1 = I, 2. . . . , m

corresponds to the chemical type of molecules. while subscript v = 1,2, . . . , n, refers to the

quantum levels of their degrees of freedom. Each such function depends on time, the coordinates, and the components of the microscopic particle velocity vector t. In a mixture of gases with

different properties, there can be several relaxation times of both elastic and inelastic processes. For simplicity, we assume that relaxation times T’~’ and T”) are uniquely defined for the two processes, while @=rcE)/r’R’. Then. the system of Boltzmann equations referred to dimensionless variables, has the form

(10.1)

where the Knudsen number, denoted by Kn, plays the role of a small parameter. The expressions for the integrals of elastic I(n,, and inelastic $f)collisions may be found in the literature of kinetic theory (971.

It is well known that, in an asymptotic analysis of the properties of the Boltzmann equation, the Chapman-Enskog method is mainly used; this method was originally used to derive the system of Navier-Stokes equations for a gas consisting of structureless particles. Extension of the

method to a mixture of substances when bimolecular reactions are present is relatively simple. provided that the reactions are extremely rapid @ m l), or conversely, are very slow (6 + Kn). A similar situation occurs for the excitation of the internal degrees of freedom of the molecules. In system of equations (10.1) for a mixture with arbitrary speeds of chemical reactions and

excitation of internal degrees of freedom, the size of the parameter /l can vary widely. It is much

more difficult to extend the Chapman-Enskog method in this case. A general approach was proposed by Alekseev in 1969, when working at the M.C.M. Laboratory of the Computing Centre [98]. Further development of the mathematical apparatus was started five years later by Galkin, Kogan, and Makashev [99, 1001. To them is due the derivation of the Navier-Stokes equations

with the associated equations for the chemical reactions, in conditions typical of external aerodynamical problems.

A vital step is the expansion of the solution of system (10.1) in the asymptotic series

!U,=!Py) (1’Kn h,,+. . .) (10.7)

146 0. S. Ryzhov

with respect to the small parameter Kn, about the locally Maxwellian distribution functionsf,,@)

for any 0. The functionsf,,(o) themselves satisfy the Boltzmann equations in the limit for Kn = 0.

Substitution of asymptotic expansions (10.2) into system (10.1) leads to a system of linear integral

equations for the required disturbances h,,. The derivation of these is accompanied by the

elimination from the left-hand sides of (10.1) of the total time-derivatives of the mixture macro-

parameters by means of conservation equations, containing only first-order terms in the Knudsen

number. The equations of conservation for the numberical densities nfiV are

D 8 -=- Dt dt

+(nv), (10.3)

where bv U we mean the macroscopic flow velocity vector. The two integral terms Q,,!? [j,,l”’ ]

and Q? [h,,] in (10.3) are in general of the same order O(Kn). This means that the

reaction speeds are established, not only by Maxwellian particle distributions, but also, to an

equal degree, by, the non-equilibrium corrections h,,. If we neglect the terms Q!*“,) [A,,,] in

Eqs. (10.3). we arrive at integral operators, the number of eigenfunctions of which is less than

the number of independent macroparameters. since all the eigenfunctions are the same as the

invariants of the inelastic collisions. In this version of the extended Chapman-Enskog method,

it is not possible to treat each macroparameter of the mixture in terms of integrals which can be

evaluated in terms of the zero approximation of the distribution functions; part of the macro-

parameters appear in asymptotic series with respect to the Knudsen number [99, 1001.

Another version of the extended Chapman-Enskog method, developed in the T.P.T.

Laboratory. is extremely attractive from the point of view of the physical interpretation of the

quantities appearing in it. hlapuk and Rykov proposed that, when eliminating the total time-

derivatives, we retain in the conservation equations all the first-order terms in the Knudsen

number 1101, 1031 . For instance, in the right-hand sides of Eqs. (10.3) along with the term

Qu?’ [I,,:.” ] , the term Q’R’ [h,vl is also retained. As a result of considerable modification

of the mathematical formalis:, integral operators are obtained in a vector Hibert space with as

many eigenfunctions as there are macroparameters of the gas mixture. Apart from the eigenfunctions

which can be identified with invariants of the ineleastic collisions, the integral operators have

supplementary eigenfunctions, which transform, when the reactions are “frozen,” into inva:iants

of the elastic collisions, though they are not in general identical with them. It is important that no

macroparametersof the mixture need be distributed in the series with respect to the Knudsen

number; though in the expressions for the chemical reaction speeds, account is taken of the

contribution from the disturbances of the Maxwellian distribution functions in the system of

Euler equations. The presence of the supplementary eigenfunctions ensures that the present

approach becomes identical with the classical Chapman-Enskog method if the reactions, of the

internal degrees of freedom of the molecules, are “frozen.” In the context of the present

asymptotic theory, it is also possible to derive multi-temperature equations of a continuous

medium, when the reacting gases consist of particles with substantially different masses.

Translated by D. E. Brown.

Asymptotic methods in fluid dynamics 147

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14. EREMENKO, V. A., and RYZHOV, 0. S., On fiow structure in the neighbourhood of the point of intersection of a shock wave with a caustic, Dokl. Akad. Nauk SSSR, 238, No. 3, 541-544, 1978.

15. ALEKSANDROV, V. V., and RYZHOV, 0. S., On the non-linear acoustics of a radiating gas, III, Exponential approximation, Zh. rvj;chisl. Mat. mat. Fiz., 14, No. 3,717-727, 1974.

16. ALEXANDROV, V. V., and RYZHOV, 0. S., Sur la decroissance des ondes courtes dans un gaz rayonnant, J. Mecanique, 14,No. 1, 135-159, 1975.

17. KHRISTIANOVICH, S. A., Shock wave at a great distance from a point of explosion, PrikL matem. mekhan., 20, No. 5,599-605, 1956.

18. VINCENT], W. C., and BALDWIN, B. S.. Jr.. Effect of thermal radiation on the propagation of plane acoustic waves, J. Fluid Mech., 12, No. 3,449-477, 1962.

19. NI, A. L., and RYZHOV, 0. S., On the speeds of sound in multi-component chemically active gas mixtures, Vestrr LGU, Marem., mekhan., astron., 13, No. 3. 117-128, 1976.

20. Nl, A. L., and RYZHOV, 0. S., On the structure of relaxation zones behing a shock front in chemically active gas mixtures, PrikL Mekhan. tekhn fiz., No. 1, 33-45, 1980.

21. LEONTOVICH, M. A., Notes on the theory of sound absorption in gases, Zh. eksperim. teor. fiz., 6, No. 6, 561-576,1936.

22. MANDEL’SHTAM, L. I., and LEONTOVICH, M. A., On the theory of sound absorption in fluids, Zh. eksperim. teor. fiz., 7, No. 3,438-449, 1937.

23. NAPOLITANO, L. G., Generalized velocity potential equation for pluri-reacting mixtures, Arch. mech. stosowanej, 16, No. 2, 373-390, 1964.

24. RYZHOV, 0. S., Study of transonic flows in Lavalle nozzles (Issledovanie transvukovykh techenii v soplakh Lavalya), VTs Akad. Nauk SSSR, Moscow, 1965.

148 0. s Ryzhov

25. FAL’KOVICH, S. V., On the theory of the Lavalle nozzle, Prikl. matem. mekhan., 10, No, 4,503-512, 1946.

26. VON KARMAN, TH., The similarity law of transonic flow, J. Math. Phys, 26, No. 3, 182-190, 1947.

27. FRANKL’, F. I., On the theory of Lavalle nozzles, Izr: Akad. Nauk SSSR, Ser. Matem, 9, No. 5, 387-422, 1945.

28. OVSYANNIKOV’, L. V., Study of gas flows in a straight sonic line, Tr. LKVVIA. No. 33, 1950.

29. RYZHOV, 0. S., and SHMYGLEVSKII, Yu. D., A property of transonic gas flows, Prikl. matem mekhan., 25, No. 3,453-455, 1961.

30. DIESPEROV, V. N., and RYZHOV, 0. S., Threedimcnsional sonic ideal gas flow past a body, Rikl. matem. mekhan., 32, No. 2, 285-290, 1968.

31. LIFSHITZ, Yu. B., and RYZHOV, 0. S.. Sonic ideal gas flow past a half-body, Zh. vLchirl. Mat. mat. Fiz., 9, No. 2, 389-396, 1969.

32. FRANKL’, F. I., Study of the theory of a wing of infinite dimension moving at the velocity of sound, Dokl. Akad. Nauk SSSR, 57, No. 7, 661-664, 1947.

33. GUDERLEY, K. G., and JOSHIHARA, H., An axial-symmetric transonic flow pattern, Quart. Appl. Mafh., 8, No. 4, 333-339, 1951.

34. FAL’KOVICH, S. V., and CHERNOV, I. A., Sonic gas flow past a body of revolution, Prikl matem mekhan., 28, No. 2,280-284, 1964.

35. DIESPEROV, V. I., LIFSHITZ, Yu. B., and RYZHOV, 0. S., Proof of the stabilization law for wing profiles, Uch. zap. TsAGI, 5, No. 5, 30-38, 1974.

36. DIESPEROV, V. N., LIFSHITZ, Yu. B.. and RYZHOV, 0. S., Stabilization law for transonic flows past a body of revolution, Izv. Akad. Nauk SSSR, Mekhan. zhidkostigaza, No. 5,49-54, 1974.

37. DIESPEROV, V. N., LIFSHITZ, Yu. B., and RYZHOV, 0. S., Stabilization law and drag in the transonic range of velocities, in: Froc. Sy,mp. Transsonicum II, Springer, Berlin, 1976.

38. CAL’PERIN, V. G.. er al., Physical foundations of transonic aerodynamics, Uch. zap. TsAGI, 5, No. 5, l-29, 1974.

39. HOLDER, D. W., The transonic flow past twodimensional aerofoils, J. Roy,. Aeronaut. Sot., 68, No. 644, 501-516, 1964.

40. EREMENKO, V. A., and RYZHOV, 0. S., On flow in a local supersonic zone at the profile of a wing of infinite dimensions, Dokl. Akad. Nauk SSSR, 240, No. 3,560-563, 1978.

41. BERS, L., Mathematical aspects of subsonic and transonic gas dynamics, Wiley, New York, 1958.

42. RYZHOV, 0. S., On viscosity and heat conduction effects in compressible fluid dynamics, in: Fluid d.ynamics transactions, Vol. 4, Polish Scient. Pubs., Warsaw, 1969.

43. SICHEL, M., Two-dimensional shock structure in transonic and hypersonic flow, in: Advances Appl. Mech., Vol. 11, Acad. Press, 197 1.

44. RYZHOV, 0. S., Viscous transonic flows, in: Ann. Ret. Fluid Mech., Vol. 10, hn, Revs. Inc., Palo Alto, Calif., 1978.

45. DIESPFROV. V. N., and LOM:2KIN, L. A., A boundary value problem for the linearized axisymmetric VT equation, Zh. $chisl. Mat. mat. Fiz., 14, No. 5, 1244-1260, 1974.

46. DIESPCROV, V. N., and LOMAKIN, L. A., On the asymptotic properties of the solution of a boundary value problem for the viscous transonic equation, Zh. v_Fchisl. Mat. mat. Fiz., 16, No. 2,470-481,1976.

47. RYZHOV, 0. S., and TERENT’EV, E. D., On the disturbances involved in creating lift on a body in transonic dissipative gas flow. Prikl. matem. mekhan., 31, No. 6, 1035-1049, 1967.

48. NAPOLITANO, L. J., and RYZHOV, 0. S., The analogy between non-equilibrium and viscous Inert flows at transonic speeds, Zh. I1.Fchisl Mat. mat. Fiz., 11, No. 5, 1229-1261, 1971.

49. NI, A. L., and RYZHOV, 0. S., Two-dimensional flows of a relaxing mixture and the structure of weak shock waves. Prikl. matem. mekhan.. 39, No. 1, 66-79, 1975.

As),mptoric methods in fluid d.snamics 149

50. LIFSHITZ, Yu. B.. and RYZHOV, 0. S., On onedimensional non-stationary motions, of gas expelled by a piston, Prik/. marem mekhan., 32, No. 6, 1005-1013, 1968.

51. SEDOV, L. I., Air motion in a strong explosion, DokL Akad. Nauk SSSR, 52, No. 1, 17-20, 1946.

52. SEDOV. L. I., Propagation of strong explosion waves, PrikL mafem mekhan., 10, No. 2, 241-250, 1946.

53. TAYLOR. J. 1.. The formation of a blast wave by a very intense explosion, 11, The atomic explosion of 1945,Proc. RoJ,. Sot.. Ser. A, 201. No. 1065, 159-186, 1950.

54. RY’ZHOV, 0. S.. and TERENT’EV, E. D., On the general theory of nonstationary flows close to similarity flows, Priki. mui~‘m. mekahn., 37, E;o. 1, 65-74, 1973.

55. TERENT’EV. E. D.. On disturbances connected with finite momentum in the strong point explosion probIem,Prlkl. marem. mekhan.. 43, No. I, 51-56, 1979.

56. RYZHOV, 0. S.. and TERENT’EV, E. D., On the application cf conservation laws to the solution of gas- dynamics problems by asymptotic methods, in: Aeromechanics and gas dynamics (Aeromekhan. i gazovaya dinamika). Nauka, !vloscow, 1976.

57. VLASOV, 1.0.. DERZHAVINA, A. I., and RYZHOV. 0. S., On an explosion at the interface of two media, Zh. v_ichisl. Mat. mat. Fiz., 14, No. 6, 1544-1552, 1974.

58. M-EIZSACKER, C. F., Gen&herte Darstellung starker instationarer Stosswellen durch Homologie-Losungen, Z. .Vaturforsch 9a, No. 4. 269-275. 1954.

59. ZEL’DOVICH, Ya. B., Gas motion under the action of short-term pressure (shock), Akusr. rh., 2, No. 1, 28-38, 1956.

60. DERZHAVINA. A. l., On the asymptotic behaviour of transient gas motion under pulse action, frikl. mafem. mekhan.. 40, No. 1. 185-189. 1976.

61. PARKHO!viENKO, V. P.. POPOV’, S. P., and RYZHOV, 0. S.. Influence of initial particle velocity on transient sphericall!, sy,mmetric gas motions, Zk. v?_chisl. Mar. mar. Fiz., 17, No. 5, 1325- 1329. 1977.

62. PARPHOMENKO! V. P.. Gas motion close to a centre (axis) of symmetry on separation of internal and kinetic energ! at the peripher!, Ix. Akad. .Vauh SSSR, Mekhan. Zhidkostigaza. No. 1. 194-198, 1979.

63. PETROVA. L. 1.. and RY’ZHOV. 0. S.. On hy-personic viscous gas flow past a half-body. Iir. Akad :I.alth SSSR. .11ekhan.. rhidkosti guru. No. 1. 64-68. 1969.

64. CHENG. H. K.. Simrlitude of h! personic real gas flops over slender bodies with blunted noses. J. .4erona~r. SCL, 26, No. 9.575-585. 1959.

65. SYCHEV, V. V’.. The method of smalI perturbations in problems of hypersonic gas flow past slender blunted bodies, Prikl. mekhan. tekhtz. fi:., No. 6. 50-59. 1962.

66. RSZHOV, 0. S.. and TERENT’EV, E. D.. On the theor! of the high-entropy layer in hypersonic flon, Zh. v_ichisl. mat. mat. Fiz., 11, No. 2, 462-480. 1971.

67. MANLILOVICH, S. V., and TERENT’EV, E. D., On asymptotic behaviour of supersonic perfect gas Boa past a blunted wedge. L‘ch. :ap. 7r4GI. 11. No. 5, 1980.

68. CHERNYI, G. G.. Gas .flo~. a? high supersonic speeds (Techeniy a gaza s bol’shoi sverkhzvukovoi skorost’! u,), Fizmatgiz. ~loscow. 1959.

69. RYZHOV, 0. S., and SHIFRIN, E. G.. Dependence of aerodynamrc forces on Reynolds number in three. dimensional viscous flou- past a hod! ~ I:I,. .4kad. .Vaub SSSR, Mekhan :hidkostiga:a’. No. 2. 90-96, 1974.

70. RYZHOV, 0. S., and TtREhT’EV. E. D.. On h! personic flou past a supporting profile, frikl. marem. mekhan., 38. Ko. 1, 92-104, 1974.

71. RYZHOV, 0. S., and TERENT’EV, E. D., On threedtmensional hypersonic flows, Prikl. mart-m mekhan.. 39, No. 3, 458-465. 1975.

72. LADYZHENSKAY A, 0. A.. Mathematical aspects of rhe d,vnamics oja viscous incompressible &id (Mathematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti). Nauka. Xiosco\s, 1970.

73. BABENKO. K. 1.. On stationar! solutions of viscous incompressible fluid flow past a body. Marem. sb (New series). 91, No. 1. 3-26. 1973.

150 0. S. Ryzhov

74. LOMAKIN, L. A., and RYZHOV, 0. S., On VISCOUS incompressible fluid flow past a body of revolution, II, Third approximation, Uch. zap. TsAGI, 7, No. 2, 17-24, 1976.

75. LANDAU, L. D., and LIFSHITZ, E. M., Mechanics of continuous media (Mekhanika sploshykh sred), Fizmatgiz, hioscow, 1954.

76. RYZHOV, 0. S., and TERENT’EV, E. D., On the trail behind a supporting body in a viscous fluid, Prikl. mekhan. tekhn. fiz., No. 5, 1980.

77. RYZHOV, 0. S., and TERENT’EV, E. D., Laminar hypersonic trail behind a supporting body, Prikl. matem. mekhan., 42, No. 2,277-288, 1978.

78. SYCHEV, V. V., On the flow in the laminar hypersonic trail behind a body, in: Fluid d_vnamics tram_, Vol. 3, Polish scient. Pubs., Warsaw, 1966.

79. RYZHOV, 0. S., and SHIFRIN, E. G., Singular eddy layer on a body surface in three-dimensional flow, Dokl. Akad. Nauk SSSR, 215, No. 2,297-300, 1974.

80. NEILAND, V. Ya., On the theory of break-away of a laminar boundary layer in supersonic flow, Izv. Akad. fiauk SSSR, Mekhan. zhidkostiga:a, No. 4,53-57, 1969.

81. NEILAND, V. Ya., Asymptotic problems of viscous supersonic flow theory, Tr. TsAGI, No. 1529, 1974.

82. STEWARTSON, K. and WILLIAMS, P. G., Self-induced separation, Proc. Roy. Sot., Ser. A., 312, No. 1509, 181-206, 1969.

83. STEWARTSON, K., Multistructured boundary layers on flat plates and related bodies, in: Advances Appl. Mech., Acad. Press, New York, Vol. 14, 1974.

84. SCHNEIDER, W., Upstream propagation of unsteady disturbances in supersonic boundary layers, J. Fluid Mech., 63, No. 3,465-485, 1974.

85. BROWN, S. N., and DANIELS, P. G., On the viscous flow about the trailing edge of a rapidly oscilIatine plate, J. FluidMech.. 67, No. 4, 743-761, 1975.

86. RYZHOV, 0. S., and TERENT’EV, E. D., On a non-stationary boundary layer with self-induced pressure, Prikl. matem. mekhan, 41, No. 6, 1007-1023, 1977.

87. RYZHOV, 0. S., On a non-stationary boundary layer with self-induced pressure at transomc speeds of external tIow,Dokl. Akad. Nauk SSSR, 236, No. 5, 1091-1094, 1977.

88. SMITH, F. T., SYKES, R. I.. and BRIGHTON, P. H’. M., A two-dimensional boundary lay,er encountering a threedimensional hump,J. FluidMech., 83. No. 1, 163-176, 1977.

89. RYZHOV, 0. S., On non-stationary three-dimensional boundary layer freely interacting with external flow. Prikl. matem mekhan., 44, No. 5, 1980.

90. RYZHOV, 0. S., and ZHLJK, V. 1.. Internal waves in the boundary layer with the self-induced pressure, J. Mechanique, 19, No. 2, 1980.

91. TERENT’EV E. D., Calculation of the pressure in the linear problem of an oscillator in a supersonic boundary layer, Prikl. marem. mekhan., 43, No. 6, 1014-1028, 1979.

92. NEILAND, V. Ya., On the asymptotic theory of supersonic flow interaction with a boundary layer, ~zI,. Akad. Nauk SSSR, Mekhan. zhidkosrigaza, No. 4,41-47, 1971.

93. ZHUK, V. I., and RYZHOV, 0. S., On the formation of recirculation zones in a boundary layer on a moving surface, lx. Akad. Nauk SSSR, Mekhan. zhidkostigaza, No. 5, 1980.

94. WILLIAMS, J. C., III. Incompressible boundary layer separation, in: Ann. rev f7uid mech.. Vol. 9, Ann. revs. Inc., Palo Alto, Calif., 1977.

95. LIN, CHIA CH’LAO, Hydrodynamic srabiliry, Cambridge UP., 1968.

96. DORODNITSYN, A. A., and LOITSYANSKII, 1. G., On the theory of the transition of laminar layer to a turbulent layer, Prikl. matem. mekharr., 9, No. 4, 269-285, 1945.

97. HIRSCHFELDER, J. O., et al., Molecular theor), ofgases and liquids, Wiley, 1964

98. ALEKSEEV, 8. V., On the theory of the generalized Enskog method, Teor. eksperim. khimi).a, 5, No. 4, 541-546,1969.

Numerical methods in radiorive gas dsnamics 151

99. GALKIN, V. S., KOGAN, M. N., and MAKASHEV, N. K., Generalized Chapman-Enskog method, Part 1, Equations of non-equilibrium gas dynamics, Uch. zap. TsAGI, 5, No. 5, 66-76, 1974.

100. GALKIN, V. S., KOGAN, M. N., and MAKASHEV, N. K., The generalized Chapman-Enskog method, Part 11, Equation of a multi-velocity multi-temperature gas mixture, Cch. zap. TsAGI, 6, No. 1, 15-26, 1975.

101. MATSUK, V. A., and RYKOV, V. A., On the Chapman-Enskog method for a gas mhture,Dokl. Akad. Nauk SSSR, 233, No. 1,49951, 1977.

102. MATSUK, V. A., and RYKOV, V. A., On the Chapman-Enskop method for a multi-velocity multi-temper- ature reacting gas mixture, Zh. r.vchisl. Mat. mar. FE., 18, No. 5, 1230-1242, 1978.

U.S.S.R. Cornput. Marhs. Marh. Phys. Vol. 20. No. 5. 151 -168 Printed in Great Britain

pp. 0041-5553/80/050151-18$07.50/O 0 1981. Pergamon Press Ltd.

NUMERICAL METHODS IN RADIATIVE GAS DYNAMICS*

A. A. CHARAKHCH’YAN and Yu. D. SHMYGLEVSKH

MOSCOH

(Received ‘2 Junuar~ 1980)

A SURVEY of papers on numerical methods for radiative gas dynamics compiled rn the Laborator!

of the Mechanics of Continuous Media of the Computing Centre of the Academy of Sciences of

the USSR from 1970 is given.

Problems of the dynamics of a spectrally radiaring gas with strong interaction of the motion

and the radiation have no prospects of being solved analytically. The development of numerical

methods for this field was begun in the Computing Centre of the Academy of Sciences of the

USSR at the beginning of the seventies. The aim of the work was to solve problems with spherical

and axial symmetry. At the present time approaches already exist enabling such calculations to be

performed in principle. but they require an unrealistically large amount of computer time. One-

dimensional problems have already been extensively investigated.

The numerical investigation of the flows of a radiating gas includes three interconnecting

parts: calculation of the transport along a ray. calculation within a solid angle. and integration of

the gas-dynamic equations. They have to be dealt with separately either within the iterative process

or by time integration.

The multigroup method of integrating the transport equation is well known. It is laborious

and does not enable the role of the spectral lines to be taken jnto account in detail. To reduce the

volume of calculations Nemchinov proposed averaging of the transport equation [I]. To obtain

the averaged coefficients it is necessary from time to time to calculate the radiative transport. In

[?I the integral of the transport [3] is used. and after the introduction of an appropriate simplifi-

cation the computing formulas appear. This approach is of limited accuracy and is less economical

than methods using the original equation.

+Zh. v?chisl. Mat. mat. Fiz., 20, 5, 1249-1265, 1980.