the formation of ordered vortex structures from unstable oscillations in the boundary layer

146

p,=O,pO=l. For not too small E values the algorithm turned out to be rather less efficient;

but for e-IO-" it gave approximately the same results as the preceding method.

Algorithm with grid organization (d = 3). Basic grid 36X36, auxiliary grids 12X12 and

4X4. Weights: p,=l, pp=O. For not too small e values the factor x was one order of magni-

tude less than for d = 2. But for E-IO-' the algorithm was somewhat faster. For example,

after one F-cycle along the pattern illustrated in Fig. 2, the residue decreased on the average to 1116th of the starting value and the speed of the algorithm was 50 percent higher

than when the previously shown F-cycle was used for e-10-" with a double grid.

1.

2.

3.

4.

5.

6.

7.

8.

REFERENCES

GRIFFITHS D.F., An approximately divergence-free O-node velocity element (with varia- tions) for incompressible flows , International J. Numer. Meth. Fluids , 1, 4, 324-364, 1981. TEMAM R., Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979. FEDORENKO R.P., A relaxation method for solving difference schemes of elliptic equations., Zh. Vychisl. Mat. Mat. Fiz., 1, 5, 922-927, 1961. FEDORENKO R.P., On the rate of convergence of a certain iterative process , Zh. Vychisl. Mat. Mat. Fiz., 4, 3, 559-564, 1964. ASTRAKHANTSEV G.P., On an iterative method for solving difference elliptic equations , Zh. Vychisl. Mat. Mat. Fiz., 11, 2, 439-448, 1971. MCCORMICK S.F. and RUGE J.W., Multigrid methods for variational problems , SIAM J. Numer. Anal., 19, 5, 924-929, 1982. HACKBUSCH W. (Ed.) , Multigrid Methods (Lecture Notes in Math., 960) , Springer, New York, 1982. HACKBUSCH W. (Ed.) , Multigrid Methods, II (Lecture Notes in Math., 1228), Springer, New York, 1986.

Translated by D.L.

U.S.S.R.Comput.Maths.Math.Phqls.,Vo1.30,No.6,pp.146-154,1990 0041-5553/90 $10.00+0.00 Printed in Great Britain 01992 Pergamon Press plc

THE FORMATION OF ORDERED VORTEX STRUCTURES FROM UNSTABLE OSCILLATIONS IN THE BOUNDARY LAYER*

O.S. RYZHOV

The results of an asymptotic approach to the propagation of comparatively large amplitude perturbations in a boundary layer are described. The properties of the non-linear process being considered are established using the Benjamin-On0 equation. The behaviour of the periodic solution of this equation as a function of the magnitude of the arbitrary constants is analysed. A comparison with available experimental data shows that a description of the basic regularities following from them can be achieved within the framework of the asymptotic theory. It is concluded from this that the development of Tollmin-Schlichting waves with an increasing amplitude in the boundary layer leads to the formation of ordered vortex structures of the soliton type.

1. We will first note a characteristic feature of unstable perturbations which propagate in a boundary layer, which has been recorded in numerous experiments [l-41. The amplitude of the Tollmin-Schlichting wave generated by a harmonic source increases expon- entially and, as a result of this, it enters a substantially non-linear stage in its development. However, the period of the oscillations is also maintained constant during this stage although the shape of the initial signal is distorted: long narrow "tongues" are visible in the oscillograms where the excess velocity reaches comparatively high values. Such a structure of the cycle of oscillations requires some modification of the well-known asymptotic approaches for its decription. A diagram of the cycle with a tongue occurring in it

TZh.vychisZ.Mat.mzt.Piz.,3o,l2,1804-1814,lwo

147

is represented by the dot-dash line in the upper part of Fig. 1.

-. _-

\/ I *

\.I

vq where v is a quantity of the order of unity and the small

FIG. 3 parameter E=R-/'. In accordance with (l.l), the wave-

length z*, referred to L*, is of the order ofe".The above-

mentioned normalizations of the time and distance in the direction of the propagation of the wave can serve as a basis for simplifying the system of Navier-Stokes equations. Neverthe- less, the amplitude of the perturbations still has to be specified.

Let us assume that the difference between the pressure p* at a certain point in space

and the pressure p_ ’ in the approach stream is of the order of E'. Then, the domain within which the motion of the fluid is perturbed possesses the three-level structure which was first introduced when investigating supersonic detachment [7, 81 and the descent of a flow from the rear edge of a plate of finite length 19, lo]. The same structure is characteristic of each cycle of the non-linear oscillations being considered; the three sublayers which occur in it are indicated by the solid lines in Fig. 1. The normalization of their lengths is also shown in this figure and the notation 1 = 2*/L* is employed.

In the initial dimensional system of units, let t* denote the time, I* and y* the

Cartesian coordinates, U* and U* the components of the velocity vector, and p'the density.

In the lower boundary sublayer, the transition to a new independent variable and to the required functions is made in the following manner:

1 Since the period of the non-linear pulsations does

1 not change, it is determined by the frequency of the initial Tollmin-Schlichting wave. We shall assume that the perturbations develop in the boundary layer of an incompressible fluid which is adjacent to a semi-infinite planar plate and that the Reynold's number R, calculated from

the velocity U,' of the approach stream and the distance

from its edge L*, is fairly large. Then, within the framework of the linear theory of stability, the frequency Y'

of the Tollmin-Schlichting wave from the neighbourhood of the lower branch of the neutral curve is asymptotically estimated as [5, 61

Y'=&-*(Um'/L')v, (1.1)

t*=ez(L’IU,‘) t, x’=L’( 1+&$x) ( y’=l3”L’y,

U’=EU,‘U, v*=E=lJ,‘v, P’-Pm ‘=2p’U,“p.

(1.2a)

(1.2b)

The velocity field is found from the Prandtl equations:

in which the excess pressure thickness -A, by the formula

~+g=,, ap -Q z- (1.3a)

$,,~+,_E+~+$, (1.3b)

p is associated with the previously unknown displacement

m 1 aAIaX

P=; s - dX. _m x-x

(1.4)

The boundary condition on the external edge of the lower sublayer states that

u-y+A(t,s) as y-m, (1.5)

while the condition on the surface of the plate, y = 0, outside of the zone of influence of the sources of the perturbations has the usual form u = v = 0.

2. As was noted above, long narrow tongues appear as the amplitude of the pulsations in the perturbed velocity distribution increases. In order to analyse this stage of the non-linear process, we introduce an additional small parameter A, which is bounded by the

inequalities s<A<i. We assume that, in the region of the tongue, the relative excess

pressure is estimated by the quantity A2 (and not bys").Under these conditions, stratification of the lower boundary subdomain occurs to form two layers with quite different properties. The four-level structure of the wave motion of the fluid which has arisen is represented in Fig. 1 by the broken lines while the new intermediate sublayer is hatched in. Within it, the order of the frequency specified by relationship (1.1) breaks down and, as a result, the normalization of the independent variables and the required functions is realized by means of the equalities

t'=e'A-*(L'/U,')t', z'=L'(l+eLA-'z'), y'=&"AL'y,, (2.la)

u'=AU,'u,, v'=A=U,'v,, P.-P- '=AZp'U,"p,. (2.lb)

148

Since the intermediate sublayer is significantly boundary subdomain, the terms with viscous tangential tions. As a result, we have [ll, 121

au, ) 3% _ aP{ (

shorter and thicker than the initial stresses drop out of the master equa-

(2.2a)

(2.2b)

As far as the excess pressure pi is concerned, it is expressed, as before, in terms of

displacement thickness, -Ai, by means of the integral (1.4) with z replaced by X'. The

introduction of a new transverse coordinate yi also does not change the form of the boundary

condition (1.5) for the longitudinal component of the velocity on the outer edge of the inte- mediate sublayer. However, it is necessary to satisfy the simpler no-flow boundary condition Ui = Cl on the surface yi = 0. Moreover, it is necessary to join the required func-

tions occurring in (1.2) in the limit when 5 +-TO with the analogous quantities which are

determined using (2.1) with x'+Tm while simultaneously putting t-70 and t'-+F-.

Obviously, a similar joining of the solutions must be carrried out in the case of the two upper sublayers which occur both in the three-level structure of the perturbed motion of the fluid and in the shorter tongue zone. However, in spite of the difference in the characteristic dimensions which determine the scales of the longitudinal coordinate in the three-level structure and in the tongue zone, the weakly perturbed potential flow and also the main thickness of the boundary layer obey the same equations over the whole of their extension and the difference reduces solely to a renormalization of the velocity field and the excess pressure in terms of the small parameters e and A. Actually, the retention of the relationship (1.4) between the excess pressure and the displacement thickness and the boundary condition (1.5) when an additional sublayer is introduced into the model of the perturbed fluid is explained by this. The joining process is discussed in greater detail in 111, 121.

It now just remains to consider the lower boundary sublayer in the zone of the tongue. The definition of the transverse coordinate

y'=e'A-'L'y, (2.3)

suggests that it is significantly thicker than the corresponding sublayer from the region with a three-level structure. The required functions which are specified by the relation- ships

u'=AU,'ut, v*=e'AU,'v,, p'-p_'=A=p'U,'=p~ (2.4)

satisfy the Prandtl equations in which the pressure pz is expressed in terms of the known

displacement thickness, -Al = -Ai(t’, ~‘1, by means of formulae (1.4). In accordance with

this, the boundary condition on the external edge of the lower boundary sublayer has the form

ul-+u,(t', x', O)=U'(t',z') as Yl+m, (2.5)

and the sticking conditions 1.4~ = vz = 0 are satisfied on the surface of the plate yz = 0.

Hence, the fundamental propositions of classical boundary layer theory are accurately repeated in the formulation of the boundary value problem where the pressure is found from a preliminary analysis of the external flow of an ideal (non-viscous) fluid.

Let us now pass to the limit when yl-m which is conventional within the framework

of the classical theory by substituting relationship (2.5) for the longitudinal component of the velocity and expression (1.4) for the pressure into the Prandtl equations. As a

result, we conclude that the function uO(t',s') must satisfy the following condition:

.W

(2.6)

3. Already at this stage of the analysis, the possibility arises of stating certain conclusions of a qualitative nature regarding the tongues which arise during the course of the non-linear process of the amplification of Tollmin-Schlichting waves. Of course, it has to be kept in view that the theory of bidimensional oscillations must be used with caution since the non-linear effects lead to a substantially spatial picture of the propagation of the perturbations. Nevertheless, during the earlier stage of the development of the non-linear pulsations, a spatial character is apparently not one of the predominant properties which determine their regularities [3, 41. The conclusions of the bidimensional theory are confined to this phase.

A comparison of different normalizations of the transverse coordinate which are specified by formulae (1.2), (2.1) and (2.3) shows the stratification of the lower boundary sublayer which is contained in the structure of the Tollmin-Schlichting waves does not take

149

place for the condition e=A. It is readily concluded from this that the tongues are formed

from initially harmonic oscillations and that the viscous boundary sublayer serves as the site of their generation. By virtue of the same formulae (1.2) and (2.1) and the asymptotic representations (2.4), the amplitude of both components of the velocity vector and of the perturbed pressure is the same in all of the subdomains of the flow being considered,

if e=A. However, amplification of the perturbations, which is characterized by the inequal-

ity e<A is accompanied by the formation of vortex structures, that is, tongues, which are

narrower in their extent. Since the new intermediate sublayer which is formed is thicker than the initial lower boundary subdomain of the Tollmin-Schlichting waves, the tongues come to the surface, shifting, as h increases, in the direction of the outer edge of the boundary layer.

The difference between the velocity c.* of the perturbations propagating in it and the velocity c* with which the waves move In the whole of the viscous boundary subdomain serves as a second consequence of the formation of an intermediate sublayer. Actually, according to (2.1), the order of magnitude of the first of the named velocities is estimated

using c,'-AU,' while the normalizations of the Tollmin-Schlichting wave (1.2) have to be

used for the estimation of the second velocity whereupon it follows that c'-eU,'. It has

been pointed out above that the motion of the fluid in the sublayer adjacent to the wall in the zone of a tongue obeys the classical Prandtl equations with a well-known pressure distribution. This sublayer plays a passive role in the propagation of vortex perturbations and it cannot serve as a site for their generation. Hence, an increase in the amplitude of the oscillations is accompanied not only by their stratification in space but also by an increase in the velocity of the displacement of the tongue with respect to the pulsations in the viscous boundary subdomain, the velocity of which does not change in order of magnitude under the action of non-linear effects.

4. Let us continue with the analysis of the intermediate sublayer in the tongue zone. It is readily seen using a simple substitution that the solution of Eqs. (2.2) which satisfies the limiting condition (1.5) has the form

u,=y,fA,, 84 aA, a-4, ap.

V+=-~~/C-~--A,~-~, (4.1)

and that the expression for vi is identical to the asymptotic representation of this

function when y,+m (see [ill). By satisfying the no-flow condition ri = 0 when yi = 0,

we see that the displacement thickness must be determined from the Benjamin-On0 integro- differential equation [13, 141

aA, 1 i a2A,lax2dx. 'as'=;__ X-x'

(4.2)

Since uO=A, by virtue of (4.1), the previously derived relationship (2.6) reduces to (4.2).

The Benjamin-On0 equation belongs to a number of remarkable non-linear wave equations in mathematical physics, the solutions of which possess soliton properties which are the subject of extremely intensive study at the present time. The aims put forward in our paper do not require the invocation of the general theory of solitons and it suffices to consider the simplest periodic integral

k, 2k,(i-A,,")"

"--z+ (G-A,')'" - i-Accost' g=k,x’-v,t‘, (4.3)

which depends on three arbitrary constants, vi, ki and Ai. In order to establish the funda-

mental properties of the tongues which are formed in each cycle of non-linear oscillations, let us elucidate how the behaviour of (4.3) changes when these constants are varied. The analysis set out below is carried out with the aim of facilitating a comparison of the results which flow from it and the data from the latest experimental investigations [3, 41. However, the comparison itself will be made later.

It is well known that the asymptotic behaviour of the first root of the dispersion

relationship which relates the frequency v to a real positive number k+m in Tollmin- Schlichting waves has the form

v=ka+2”(l+i)/2+. . . . (4.4)

The leading term occurring here is also obtained from the integral (4.3) if it is used to describe the perturbations which are propagating through a null background with an amplitude coefficient AC-to. It is readily seen that this is a common property of any small oscilla-

tions, the evolution of which obeys the Benjamin-On0 equation. One can arrive at the assertion which has been made in another way by taking account of the renormalization of

the frequencies v=(A/e)'v, in (4.4) which follows from (1.2) and (2.1) and the wavenumbers

k= (A/e)k, as applied to the analysis of the structure of the intermediate sublayer. Hence,

150

the neutral character of the small oscillations in the sublayer under consideration is a

result of passing to the limit with respect to e/A+O.

Let us now make use (although this is unimportant in the subsequent discussion) of

the relationship Y,= k:, by assuming that it is satisfied in non-linear processes. Then,

the integral (4.3) can be written as

Ai=k. [ l+(l-A,z)--lb - 2(1-A<‘)“s 1 l-A,cos 5 .

Here, let ApI. Denoting the points where the function Ai vanishes by E,, we have

The maximum values

while the minimum value

(4.5)

go=* [6(i-Ai)]“. (4.6)

of expression (4.5) are attained at the ends of the interval g=Fn

is attained at its centre E=O. These values are

respectively and their ratio Aimin/Aimax = -3.

The limiting dependences clearly demonstrate that, when A,+i, the asymmetry in the

distribution of the positive and negative phases with respect to each oscillatory cycle

increases sharply. The length of the interval g, carrying the negative values of Ai,

contracts (although slowly) down to zero. The absolute magnitudes of both the positive and the negative extrema of Ai in the intermediate sublayer increase without limit. However,

the growth of the negative extremum is completed more rapidly than that of the positive extremum.

There is considerable interest in the spectral characteristics of the non-linear oscillations which are being studied. In order to determine these characteristics, let us expand the function

2k,(l-A,‘)‘”

1-A< CCIS E =4k,++ ~@‘cos@z~)]. (4.8)

n-1

Q= 1-(1-A,‘)“*

A, ’

in a Fourier series. It can immediately be seen from this that the coefficients of the series which give

the amplitudes of the successive harmonics form a geometrical progression with a ratio q > 0

and that g-+1 when As-l. Hence, as A. increases, the contribution from the higher harmonics

to the sum of the series becomes ever%more important. The spectral properties of the integral (4.5) of the Benjamin-On0 equation beautifully express the process of the non-linear distortion of the periodic oscillations and the formation, in accordance with (4.6), of narrow tongues with negative values of Ai. A second consequence of the spectral decompo-

sition (4.8) can be formulated as a sychronization of the phases of the successive harmonics. In fact, since the Fourier series is sign constant by virtue of the fact that q > 0, there is no shift in the phases of all the modes of oscillation which are being considered.

5. We will now compare the conclusions drawn from the asymptotic analysis which has been described and the results of calculations of the non-linear wave motions of a fluid and data from the latest experimental investigations. Up to now, the asymptotic theory had as its aim the explanation of the structure of each individual cycle of pulsations: as far as their time evolution is concerned, this naturally can only be established by numerical integration of the system of partial differential equations. As the initial equations, it is preferable to work with Eqs. (1.3) and (1.41, while obeying their limiting condition

(1.5) when y+m and the boundary condition u = u = 0 when y = 0. As the magnitude of the

required functions becomes larger, the term 13*u/~Yy' makes a relatively small contribution

to the solution at a certain distance from the plate around which the flow occurs, which leads to the formation of the intermediate sublayer with the master equations (2.2).

A program for carrying out such calculations has been written in [15]. A formulation of the problem of perturbations initiated by a source which is harmonic with respect to time must include boundary conditions on a boundary which closes the computational domain from below with respect to the flow. The techniques which have been developed up to the present time for simulating these conditions are various and, generally speaking, they are not equivalent to one another. The above-mentioned difficulty is automatically overcome if, instead of periodic oscillations, one considers a wave packet from a source operating under pulsed conditions.

The typical structure of a wave packet in the non-linear stage of its development is shown in Fig. 2. The influence of non-linear effects manifests itself primarily in the

151

5 70 i5 20 25 3

FIG. 2

FIG. 3 fact that the central parts of the cycles with negative Ai values are narrower than

those to which the values of Ai are positive. Moreover, the peak to peak amplitude of the

oscillations in the negative phase is appreciably (approximately 1.7 times) greater than in the positive phase. These results reproduce the distinctive properties of the integral (4.5) of the Benjamin-On0 equation which have been discussed above. Here, however, it is necessary to bear in mind that the spectrum of the integral being considered consists of discrete lines while the wave packet possesses a continuous spectrum. The spectrum of the wavenumbers is shown in Fig. 3; the segment around k = 2.5 associated with the pulsations which are growing most rapidly in the linear phase predominate in this spectrum. A second maximum, as the non-linear theory predicts, is formed close to the doubled value of k = 5.0. Similar properties have also been observed in other calculations and they are indirec- tly confirmed by the results from [16, 171.

The available experimental data provide even more definite evidence in favour of the conclusions drawn from the asymptotic analysis if one confines oneself to the start of the non-linear stage of the propagation of perturbations when the velocity field has still not acquired a pronounced spatial character. It is likely that this is explained by the fact that sources of oscillations which are periodic in time were used in the experiments [3, 41.

As was noted in Section 3, an increase in the amplitude of the pulsations gives rise to their stratification, with the soliton part rising to the surface and shifting in the direction of the outer edge of the boundary layer. As previously, the other part of the pulsations is concentrated in a narrow boundary sublayer, the order of magnitude of the height of which is estimated by the corresponding characteristics dimension of the Tollmin-Schlichting waves. The rate of propagation of the soliton part of the perturbations which are associated with the tongues in the oscillograms is greater than the rate of displacement of the waves in the boundary layer which are generated by the secondary instability of the fluid motion. Direct quantitative measurements lead to similar conclusions [3, 41.

The asymmetry of each individual oscillatory cycle both as regards the extent (duration) of the positive and negative phases and the amplitude of these phases has already been discussed above in relation to the results of the numerical modelling of wave packets. It only remains to add to what has already been said that the very existence of tongues had already been observed in earlier experiments [l, 21 but the proposal regarding their soliton nature has only recently been put forward in [3, 4, 151.

The most convincing confirmation of this hypothesis follows from the spectral properties of the periodic solution (4.5) of the Benjamin-On0 equation. According to (4.8), the

152

amplitudes of the higher harmonics occurring in the frequency spectrum form a geometrical progression while the phases of all of the oscillatory modes are identical. The above-mentioned strict hierarchy of amplitudes of successive harmonics during the stage of the direct formation and appearance of tongues (at a distance of 430-440 mm from the leading edge of the plate) has been reliably established in experiments. Direct measurements have also revealed the sychronization of the phases of the recorded oscillatory modes. Taken as a starting point, these data enable us to reverse the arguments since they encompass about 20 harmonics 13, 41. The partial sum constituted from them is a good approximation to the sum of the Fourier series from the right side of (4.8) which converges absolutely and uni- formly by virtue of the rapid decay of its coefficients when 4 < 0. It is clear from this that the partial sum being considered is equal, with high accuracy, to the function from the left side of (4.8) which only differs from the integral (4.5) of the Benjamin-On0 equation by a constant term.

Using the results of the experiments, let us quantitatively estimate the magnitude of the amplitude coefficient Ai, the extend of the negative and positive phases of each oscilla-

tory cycle and the amplitude of these phases. At the beginning of the non-linear stage of the development of the pulsations, the ratio q of the geometrical progresssion can be put, according to the measurements, equal to "0.6-0.65. By taking the mean value q = 0.625, we

have Ai = 0.9. The extension of the negative phase is found from the equality cos~,=O.81,

whence E0=36". The ratio of the amplitude of the oscillations in the negative and positive

phases is independent of the wavenumber ki; by virtue of (4.51, this ratio Aimin/Aimax=

-2.2 which is greater than the analogous value found by the numerical calculation of the structure of a wave packet.

The estimates presented characterize the tendencies in the non-linear distortion of an initially harmonic signal which lead to the formation of tongues. However, even the value of the limiting ratio Aimin/Aimax = -3 which follows from (4.7) is smaller than that

which is recorded in experiments. Nevertheless, fully developed tongues are observed in essentially three-dimensional oscillations [l-4] and it is therefore impossible to describe them within the framework of the asymptotic analysis which has been given.

6. The results which have been obtained are of far-reaching importance. At the present time, the problem of the correlation between order and chaos in various phenomena is being widely discussed in physics. As applied to hydromechanics the problem involves the loss of stability by an initial laminar flow of a viscous fluid, a subsequent increase in the complexity of the pulsations which are generated and the development of a turbulent velocity field from them. However, there are ordered formations even in a turbulent flow which are referred to as coherent structures [18, 191. The natural question concerning at what stage large-scale ordered vortex structures arise has become of great interest. The approach which has been developed enables one to assert that this takes place as a result of the non-linear amplification of Tollmin-Schlichting waves and the separation from them of the parts of the perturbations with a comparatively large amplitude which come to the surface while shifting in the direction of the upper edge of the boundary layer. Actually, the cycle of fluid oscillations includes a narrow and long tongue here which is of a soliton nature and, at the earlier stage, is described using the simplest solution (4.5) of the Benjamin-On0 equation.

At a later stage the picture of the perturbed fluid flow becomes spatial and along the

coordinate z'=s'A-'fiz' which is measured off in a lateral direction, the transverse

component of the velocity is determined by means of w’=AU,*wr. Recalling relationship

(2.11, we obtain the system of equations [201

(6.la)

(6.lb)

(6.1~)

where the excess pressure

(6.2)

and, as before, the function -Ai specifies the instantaneous displacement of the flow lines.

The boundary conditions for the system of equations (6.1) when y‘+m are formulated in the following manner:

153

if the perturbations decay at infinity upwards through the flow. As is usual, we have a no-flow condition Vi = 0 on the surface of the plate yi = 0. In the case of a two-dimen-

- + j dX, [A,@‘, X,, z’) ap’(;;fi’ “) $

-m

x1

+ dA,(f,Xi,Z’) s * aP,(t’,x2,z’) dX +

3% 8Z' _s

+ j' a~~~(f,x~,~‘) dX,], at’ ad

_w

(6.3a)

(6.3b)

sional velocity field which is independent of the z' coordinate, the relationship (6.2) between pi and Ai transforms into the improper integral (1.4).

The occurrence of terms on the right-hand side of (6.3) which algebraically tend to zero on the outer edge of the intermediate sublayer is explained by the growth of the pulsations of the transverse component of the velocity vector in this sublayer on account of oscillations of the self-induced pressure in a lateral direction. In fact, these oscillations lead, in the final analysis, to the breakdown of the two-dimensional flow pattern and the formation of spatial solitons [3, 41. Nevertheless, the asymptotic expression (6.3) together with the representations connected with it [20]

for the transverse component of the velocity vector do not satisfy the system of equations (6.1). Hence, the question of the possibility of the derivation of a master equation for the functions Ai in the general case being considered still remains unresolved up to the

present time.

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Translated by E.L.S.

U.S.S.R.Comput.Maths.Math.Phys.,Vo1.30,No.6,pp.154-162,1990 0041-5553/90 $10.00+0.00 Printed in Great Britain 01992 Pergamon Press plc

ABLATION OF LARGE METEORS DURING RADIATIVE HEATING"

P.I. CHUSHKIN and A.K. SHARIPOV

A problem in radiative gas dynamics concerning the removal of mass and the change in shape of a large cosmic body caused by heating due to the emission of the shock layer on entering the earth's atmosphere is solved. The equations of motion of an ablating body are calculated simultaneously with the determination of the aerodynamic forces and the radiative thermal fluxes. The spectral emission is treated in the diffusion approximation and under the assumption of a locally one-dimensional optically planar layer. The results of calculations for three natural cosmic objects belonging to different classes (stony and iron meteorites and a snow-ice comet body) are presented.

1. Introduction The problem of the motion of cosmic objects at hypersonic velocities in the atmosphere

of a planet is of considerable interest both in the case of space craft and probes and in the case of natural meteoritic bodies. A cosmic body decelerates on entering the dense layers of the atmosphere and its surface experiences intense heating which leads to intense ablation. A particularly large removal of mass occurs during the flight of meteoroids, the speeds of which, on entering the earth's atmosphere, lie within a range from 11.2 to 73.2 km/s. During this intense ablation, the shape of the body, its aerodynamic characteristics and the thermal fluxes acting on its surface undergo pronounced changes. These changes have to be taken into account in calculating the trajectories of cosmic objects.

When a relatively large cosmic body moves in the atmosphere at a high hypersonic speed, radiation from the gas in the high-temperature shock layer plays a dominant role in the heating and break up of its surface. Estimates have shown that, under these conditions, heat transfer to the body around which the flow occurs due to other factors (convection, heat conduction and diffusion) turns out to be secondary compared with heating by the radiative fluxes.

In the rigorous calculation of the trajectory of a cosmic body its equations of motion have to be calculated simultaneously with the solution of the problem in radiative gas dynamics regarding the circumfluence and heating of the body, the mass and shape of which are changing during flight. The execution of such calculations is extremely complex and time-consuming and various simplified approaches are therefore employed. A widely used approach in the approximate physical theory of meteors 111 is that in which the retardation

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the formation of ordered vortex structures from unstable oscillations in the boundary layer

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