Asymptotic methods in fluid dynamics

As~~mprotrc merhods

37.

CHERNOVKO, 1979.

127

in fluid d).namics

of mechanical systems, Lisp.Mekhan.,

F. L., Problems of optimization

38. UBTKOVSKII, A. G.. Control of systems with distributed parameters, Al~rotnal. 16-65.

U.S.S.R.

Compur.

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

Tdemehhn..

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.

0. S. Ryzhov

128 dependent turbulent

on the Prandtl number. The now so-called Dorodnitsyn

variables can be used to study

as well as laminar gas motion.

Boundary development.

lay,er theory, clearly demonstrates

the power of asymptotic

simple devices were devised for computing

and supersonic speeds. By introducing could be devised for computing

semi-empirical

methods. During its

the resistance of bodies at both subsonic

relations into the theory, effective methods

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

mathematics

as well as mechanics. By, now, solutions of boundary

various organic concepts of mathematical

physics. In particular,

that the method of matching external and internal asymptotic been given a strict proof in some comparatively As regards integration underwent

layer theory

grew; the method has

(41.

the approach to this problem naturally

drastic changes with the coming of the electronic

devices for constructing

it was from boundary expansions

simple problems

of the Prandtl equations.

into branches of

layer type have infiltrated

the velocity field were supplanted

computer,

Various approximate

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 be found in the potential

is now typical of an! asymptotic With regard to asymptotic the Computing

of the Euler and Prandtl equations theory.

methods as such for solving differential

Centre in this field had its inspiration

period of the limiting cycle of relaxation well-known

non-linear

enables the gas parameters to

and viscous flow domains with the required accuracy. A similar situation

oscillations

Van der Pol equation.

in Dorodnitsyn’s

equations.

studies of 1947 of the

[6]. These oscillations

are described by the

The idea is to find the limiting cycle by dividing it

into several overlapprng pieces in each of whtch the solution has qualitatively The main difficulty

the work of

is to find the corresponding

asymptotic

expansions

different behaviour.

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. In 1952, Dorodnitsyn characterizing

the method of matching external and internal asymptotic turned to linear second-order

the rigidity of the oscillatory

differential

equations,

system, has a singularity

expansions.

in which the coefficient.

(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 dynamics of systems with of Equations Laboratory; in a monograph published

Airy’s equation. When the Computing Centre was set up, studies in the a finite number of degrees of freedom were concentrated in the Theory the advances achieved in this field were summarized by N. N. Moiseev in 1969 [7].

Asymptotic

methods

in fluid dynamics

129

Our further discussion will be restricted to work on fluid mechanics, chiefly performed the Theory of Transport

Processes (t.t.p.)

Laboratory.

in

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.

in an inhomogeneous atmosphere

2. Wave propagation

Assume that. in the initial state. the density po. pressure po, and components particle velocity vector vu. are functions

of Cartesian space coordinatesxi

of the time r. We wish to find the asymptotic propagating

laws of damping of the weak shock waves

in this medium. The source generating

aircraft moving at supersonic

the waves may be a high-power explosion or an

speed. For plane waves in straight tubes, the problem was solved by.

Crussard in 1913. and the amplitude damping of cylindrically fronts was established in 194.5 by L. D. Landau [9]. The entire analysis in these elementary the zone of disturbed

rio of the

only, and are independent

and spherically symmetric

cases is based on the assumption

shock

that the width of

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 these quantities.

of the gas is proportronal

But when the wave travels considerable

to the cube of the variation in an> 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 amplitude.

Hence a simple rule is obtained.

Riemann wave. The non-linear unbounded

specifying the position of the discontinuity

nature of the initial Euler equations

growth of the width of the disturbed

leads in the last analysrs to

domain and to asymptotic

shock wave damping

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

consists in computing

all the gas parameters

acoustics, but assigning them, not to the coordinate but to points whose disposition of the asymptotic

is determined

from the approximation values corresponding

by non-linear

method of deformed coordinates,

b! its

in the

the

of geometrlcal

to this approximation.

processes. These are typical featur-es

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

-dxidt

=non:+r,o.

dn,

-=

dt

(n,n,-6,)

(2.1)

0. S. Ryzhov

130

Here, au is the velocity of sound in the initial atmosphere, vector II to the wave front. 6~ are the components

nr are the components

of the unit normal

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 [ 1I. 121. The application elementar!.

of this law to a volume included in an

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,-Jdenotes the initial length of the pulse. the thermodynamic gas is expressible in terms of the Poisson adiabatic exponent and 1 measures the distance along the ray’. In situations

K

coefficient

by the relation

typical of an explosion

rno for a perfect rzn= (x-l- 1) /‘2. 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 nonlinear 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.

131

Asymptotic methods in fluid d.wamics

3. Non-linear Non-linear

waves in a radiating gas and chemically active mixtures

acoustics was developed,

for waves whose structure depends essentially

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

on a

of the Computing

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

The asymptotic

analysis was based on the assumption

that the disturbed

is narrow compared with the distance to the source. But the formal methods employed not to the method of deformed

coordinates.

developed in 1956 b) S. A. Khristianovich connected coordinates

[ 171. The introduction

in the method of external and internal of the solution in both domains

front (possibly.

smoothed

domain

correspond.

but to the ideas of the theory of short waves. of new independent

with the wave element. implies in essence a transformation

representation

have

asymptotic

variables.

to so-called optimal

expansions,

which ensure a unified

[IO]. Hence the small neighbourhood

bjf light quanta radiation.

absorption,

of the wave

and scattering processes) is not

specially segregated from the zone occupied by disturbances.

AU in all, the asymptotic

analysis

leads to a non-linear

but the Peierls’ equation,

containing

system of integro-differential

equations;

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

action of effective pre)’ radiation.

gas is equivalent

to the

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 conditions, when scattering can be neglected with high accuracy. Further simplifications

may be obtained

optical thickness. if the dependence Here. from the very beginning.

for sonic pulses with small, or conversel).

of the radiation

the Peierls equation

on the frequency

earth

large

is assumed to be slight.

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

reducing the system of integro-differential dimensionless

particle velocit!

equations

1’. For opticall!

to a single differential

analysis consists in equation

for the

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

and the

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

of the chemical

analogy was established,

transformations equivalent

according to which the influence

on the quasi-equilibrium

propagation

of sonic pulses with shock fronts is

to the action on their structure of a longitudinal

framework of acoustics, another statement was proposed in 1936-1937

viscosity and heat conduction.

of the analogy betweeen relaxation

by Leontovich

and Mandel’shtam

In the

and viscous flows

[21, 221.

When studying waves in media with any amount of reaction, it is assumed that the difference between the equilibrium imposes some constraints

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

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

processes, there are h’ - 1 so-called intermediate amplitude

can travel. The quantities

arguments of Napolitano

sound velocities Q,,o? at which pulses of small

cr,o were introduced

in accordance with the purely formal

[12] : the strict proof of the inequalities apL,=ao,o~a,,o~

is to be found in [ 19. 201 If we characterize

. . .
o
the difference between the p-th intermediate

velocity and the velocity of wave packet displacement dimensionless

naturally

sound

by the contant -y(p), then we have, for the

particle velocity 1‘.

(3.2)

In the same wa>’ as when obtaining coordinate

Eq. (3. I). the time I and distance r are measured in a special

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

connection

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

133

methods in fluid dynamics

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 -L'x-

aL.7 t -

a.?.

I(&$L

dr

0,

r

atA

dc,

-=

dr-

ax

0

(4.1)

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
L-,=A~~~ for s>O,

l&=0,

(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/ 1I 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.

0. S. Rprhov

134

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

studied in 1950 by Ovsyannikov

Yu. D. Shmyglevskii, transition equations

[28]. In a joint work by the author and

a general theorem was formulated

about the properties of the surface of

through the velocity of sound, which is at the same time a characteristic surface of the 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 was examined in the period 1968asymptotic

1978 in the T.P.T. Laboratory.

damping laws of the three-dimensional

disturbances

range

The first work related to

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

form of the principal term? the power exponent Attempts

at a clear interpretation

correction corresponds

n = 4/7, if the flow has axial symmetry

[33, 341.

of the principal terms have met with no success, though the

terms of the relevant asymptotic

sequences have a simple physical meaning:

one

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

used to construct Expansion

a channel round a half-body with generator specified by a power function of the velocity vector components

was used to justify the so-called stabilization [35-371.

[32]. The results show that, in the similarity

in asymptotic

series in ascending powers of r

law for wing profiles and bodies of revolution

The law was discovered by Khristianovich,

of systematic processing of experiments

[31].

Gal’perin, Gorskii, and Kovalev as a result

performed in 1944- 1948 at Ts AC1 [38]. The most

complete description

of similar foreign experiments

memory of Reynolds

and Prandtl

is to be found in Holder’s lecture, given in

[39]. The essence of the stabilization

that, when the velocity of the subsonic incoming

law consists of the fact

flow increases, the distribution

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

of the local Mach 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 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,

of

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, - 1I,/‘. 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

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

experienced

results of many computations

of asymptotic

are in excellent agreement with the conclusions

theory [3S-371. A numerical infinite

dimension

generated

study of 1978 into the finite structure led to the conclusion

of the velocity field at a wing profile of

that the density jump closing the local supersonic

at an interior point of the zone as a result of intersection

problem has given rise to long discussions

in the literature,

of characteristics

zone is

[40]. This

which are relected in Bers’ monograph

1411.

5. Viscous transonic Systematic the structure

study of the influence

of transonic

in the T.P.T. Laboratory of Navier-Stokes

of the viscosity and heat conduction

flows commenced

then. following independent

we arrive at the following for the components velocity:

+

-Vxds It is clear from the structure viscous tangential

of an actual gas on

in 1964 in the M.C.M. Laboratory,

till 1978. If we apply asymptotic

equations,

dc,

flows

analysis to simplification

of the system

discussions of various authors

pX, vI of the dimensionless

al_& 4l,(&1)‘cli!&O, r

then was continued

OX?

au, -= dvr -F- ax

0,

(5.1)

of Eqs. (5.1) that it is possible to neglect the contribution

stresses, i.e. these cannot be used when investigating

(42-441,

vector of the disturbed

the boundary

of the 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

in the gas motion in the case of thin paraboloids,

when the contribution

becomes dominant

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 quasi-elliptic

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

type admits of an expanded (two-parameter)

We seek the appropriate

group of similitude

transformations.

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

of the disturbances

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 but also in a stronger decrease of amplitude amplitude

obtained

the neighbourhood

thus appears, not only in smoothing of all the gas parameters

of the discontinuities,

as compared with the

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

[45,46].

The field of three-dimensional

lift. is found by Fourier expansion

equation

(5.1), was obtained by Diesperov and

disturbances,

related to creation of the body

of the required solution with respect to the angular variable

(471. Flows of chemically

active mixtures in the transonic

range represent an independent

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

field

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

In 1973 work was carried out on non-stationary close to similarity

flows with strong shock waves, which are

flows (541. Such flows are either one-dimensional

When writing the general expressions the disturbed

137

methods in J7uld dynamics

for the mass, energy, and momentum

domain, the time-independent

first integrals of the equations

in variations,

which have an extremely We Fourier-expand

the density p, and the pressurep

retain only the terms with li-th harmonic

2n

C, = z+1

of the material inside

terms were isolated. To these terms correspond

assume that the basic flow has axial symmetry. velocity components,

or close-to-one-dimensional. the

simple form. For instance,

the radial uy and angular V~

with respect to the polar angle q, and

in the series. Then.

eLntn-1-“‘2um(i.)

sin (J~cf+cr,) +. . . ,

(6.1)

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

z

where the functionsf, combination introduced K

X = r/(bt)n

[ h (i.)~~t-“‘2h,(3.)co~(k~Saa)+.

p,~‘nt’:n-!’

g, and h of the first approximation of time t and the cylindrical

for convenience

as a coefficient

depend only on the similarity

coordinate

in the correction

the Poisson adiabatic index, and b and ak are arbitrary

. .I.

I; the small parameter E is

terms; po denotes the initial density.

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 component:

I(h2, A) is given by the integral. containing

in relations (6.1) it is sufficient

This expression

is time-independent

only the 1:). particle velocity

to put k = 1. ok = 0. All in all,

if m = 2(3/r - 1). The derivative of the momentum

is found

from the relation

dI (L, i.) dt

4

= . [ !,l’, (.Ycr,)

-pnj]da,

(6.3)

I

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 the particle velocity. With m = 2(3n- 1). in accordance with relation (6.2). the derivative From (6.3) we have the relatron dZ(h,, A)ldl=cl.

of

138

0.

s. Ryzhov

(6.4

which connects the functions integral of the equations

,fm, g,, h,

and

Similar arguments apply to the momentum three-dimensional whose arguments

u,.

It is easily shown that (6.4) is the first

in variations. transfer in waves with near-spherical

are the angles 9 and 9 of the spherical coordinate

found the solution of the explosion problem, characterized as energy’ to a finite volume of gas [55]. Construction variations,

corresponding

to the laws of conservation

application

system. In this wasy Terent’ev

by transmission

of momentum

as well in

of mass and energy of a substance, is a

Divergence forms of the partial differential

in both the first and the second equations

have found wide

when solving very diversified problems of gas dynamics by asymptotic

Intermediate

I’$,

of the first integrals of the equations

simpler problem, since the gas motion remains symmetric approximations.

shape. In

flows, all the gas parameters are expanded in series in spherical functions

stages of gas motions, generated by a source communicating

methods

[56].

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 known solution of the short shock problem

[58,59].

time interval by the well-

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

with the study of uniform viscous gas flow past a half-body So-called strong interaction

was also considered,

Centre started in 1969

at Mach number equal to infinity

[63],

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

Transport internal

Process Theory Laboratory asymptotic

expansions

methods

139

in fluid dynamics

in the framework of the method of matching external and

[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~

In (7.1) ho is the value of the function

(??“)*

(7.2)

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 nonstationary

gas motions and hypersonic

entropy

of the particle corresponds

initiated

exactly to the gas compression

at a direct density jump. Hence we have the following extremely explosion

[68]. It is now

flows in space c with one fewer dimensions

easily shown that relations (7.1) define a partrcle trajectory

theory due to Sedov and Taylor

[5 I-531

by an explosion wave; the in stationary

hypersonic

flow

simple rule: the results of strong

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

of the basic tasks of aerodynamics. mentioned

body,. acted on by lift as well as resistance, is one

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

analogy between non-stationary

gas motions and hypersonic

induced by the body resistance can be obtained

Studies were made in 1974-1975 at the T.P.T. Laboratory, lift remote from the profile of a wing of infinite dimension, In the latter case, the longitudinal

flows, the velocity field

from a solution of the strong explosion

problem.

of the disturbances generated by the or from any bounded body [70, 7 11.

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

where the functions

IJ,,~, pil

similarity, combination

,x-“‘[ln~p,?(~)Sp,~(LS)

p-_c-,? -+*:(e)+b

j-r

lcosqf..

.},

and iIll of the first approximation depend only on the / (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

vX12 and vX13 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 property that stipulates the inclusion to be supplemented of the functions

of the logarithmic

by relations (6.1) with II = ‘/i and

along they

terms in expansion

(7.3): they also have

m = I. The lift Fy is expressible in terms

of the third approximation:

It is clear from the results of Section 6 that they, must be connected appropriate

axis. It is this

system of ordinary differential

equations.

(6.4) if, in the latter, we make the replacements

by the first integral of the

In fact, the connection

J+v,,,,

g-p,,,

h-+p,,

is given by relation and

fm-+~r,3,

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

For the three-dimensional the solution

trail behind a supporting

body of limited size in a transonic

of Landau and Lifshits for flows of incompressible

The only difference independently

141

is linked to the density variations due to gas compressibility;

by integration

of the classical equation

flow.

fluid [75] proved suitable [47].

of heat conduction.

these are found

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 logarithmically,

to plane-parallel flow, in which connection the stream lines deviate to infinity 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 due to heat transfer, Sychev concluded

that, in order to continue

equations,

the gas parameters

and the terms into the inner

domain of the laminar trail, we need to use. instead of the similarity

combination g=r / (bs) “’ , %=r / b”>zVW+*) (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 the new variable asymptotic

expansions

have the form

+b,x-k~[~,?,(5)cos(k,ln

2~)+~.~~~(5)sin(k~lnz)

lcosq+.

. .},

1

(8.1)

x+1

-l/(X+iJ

+b,x-k6[pdf;)

{pzi

(t;)

cos(k,lnx)+pPl(~)sin(k3

In~)]cos@-.

. .},

1

pmu,2 +{p*i (L) +x-y/(x+i)P22 (5) = 2( %-+l)z +b,x- ~~~x+~~-k~[p~c(~)~o~(k3lns)+p~s(l;)sin(kgln~)]coscp+.

P

k., =

k, =

2-X 2(X+-l).

. .},

0. S. Ryzhov

142

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

the viscous stresses become significant when finding the longitudinal the field of the remaining parameters On the contrary,

in the asymmetric

can be constructed disturbances

velocity vector and the thermodynamic conduction

velocity component,

flow, whereas

by taking account of heat transfer only.

due to the body lift, the fields both of the

quantities

equally depend on the viscosity and the heat

of the gas.

The limiting conditions

as { + m for the trail functions

properties of the gas motion on approaching

are determined

the inner boundary

by the asymptotic

of the outer domain. Here, inertia

forces are balanced only by the pressure forces, in such a way that the distributions parameters take on oscillatory disturbances

as 5’0

properties

[7 I]. In fact, the asymptotic

contains terms with cos(k

Arising at the trail boundary,, the oscillaiions

In E) and sin(kln

are then transmitted

expansions E), k=[

throughout

of the gas for asymmetric

(~-X)/(X-I)]‘“. its length; in

connection

with this. terms appear in relations (8.1) with cos(k, In 3) and sin(k, In 5). the variations of the frequency of In view of the equation k,=k(z-1)/2(x+1). 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 hypersonic 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 displacement

theory. describing the effect of free displacement, jointly

that it is determined

by the growth in the

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

by Stewartson

was formulated

perturbation

b), Neiland (80, 811, and

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

equations of the

Asymptotic methods in fluid dynamics

Asymptotic independently

analysis of the system of Navier-Stokes

by different

avr

,

au,

the common

result is to

0,

ay

ax

z+

azv, &%+L”du,=_!$+_,

at

dX

ap -= the self-induced

(9.1)

dY2

aY

0,

SY

pressure dA --1

P-

1

if

ax

ia,aAjax

s~

MS=-1, (9.2)

dX if

1

r[ _m x-x the function

was undertaken

stand-points;

[84-861

P-J----_

which contain

equations

authors from several different

obtain the Prandtl equations

143

M,
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 the exterior irrotational equations

flow: the gas motion in them is quasi-stationary,

of the first approximation

do not contam time-derivatives.

particle velocities at infmity,, the time-dependence role in the external potential possible instantaneously of the boundary

Conversely,

of the required functions

part of the flow. Here disturbances

layer and b)

wirh the result that the in transonic

plays an important

are propagated,

to which 11IS

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

value problems for three-dimensional

To obtain an idea of the non-stationary was linearized. If we write the solution as p=a

process of free interaction

[88. 891.

at M, > 1. system (9.1)

exp (olikx),

r,=y---a t’,=ak

flows is rather more complicated

exp(wlSX

“

)% dy ’

(9.3)

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
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)

0. S. Ryzhov

146

with respect to the small parameter Kn, about the locally Maxwellian distribution for any 0. The functionsf,,(o) Substitution

of asymptotic

themselves satisfy the Boltzmann expansions

equations

functionsf,,@)

in the limit for Kn = 0.

(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 macroparameters by means of conservation number. The equations

D -=Dt

8 dt

containing

equations,

of conservation

for the numberical

only first-order terms in the Knudsen densities nfiV are

(10.3)

+(nv),

where bv U we mean the macroscopic

flow velocity vector. The two integral terms

Q,,!? [j,,l”’ ]

in (10.3) are in general of the same order O(Kn). This means that the and Q? [h,,] reaction speeds are established, not only by Maxwellian particle distributions, but also, to an equal degree, by, the non-equilibrium

corrections

Eqs. (10.3). we arrive at integral operators, the number of independent invariants

since all the eigenfunctions

are the same as the

macroparameters.

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

evaluated in terms of the zero approximation appear in asymptotic

Laboratory.

is extremely

attractive

in

method,

of the mixture in terms of integrals which can be of the distribution

functions;

part of the macro-

series with respect to the Knudsen number

Another version of the extended quantities

Q!*“,) [A,,,]

If we neglect the terms

of which is less than

it is not possible to treat each macroparameter parameters

h,,.

the number of eigenfunctions

Chapman-Enskog

[99, 1001.

method, developed in the T.P.T.

from the point of view of the physical interpretation

appearing in it. hlapuk and Rykov proposed that, when eliminating

of the

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

147

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U.S.S.R.

Cornput. Marhs. Marh. Phys. Vol. 20. No. 5. pp.

151 -168

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

Printed in Great Britain

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 and the radiation have no prospects of being solved analytically. methods for this field was begun in the Computing USSR at the beginning

The development

of the motion of numerical

Centre of the Academy of Sciences of the

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. Onedimensional

problems have already been extensively

The numerical parts: calculation

investigation

of the transport

the gas-dynamic equations. or by time integration. The multigroup

investigated.

of the flows of a radiating gas includes three interconnecting along a ray. calculation

within a solid angle. and integration

of

They have to be dealt with separately either within the iterative process

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 simplification 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.