U.S.S.R. Comput.!faths.fifath.Phys .,Vo1.28,No.2,pp.185-192,1988 0Q41-5553/88 $~O.OC+O.OO Printed in Great Britain 01989 Pergamon Press plc

SPATIAL PERTURBATIONS INTRODUCED BY A HARMONIC OSCILLATOR IN A BOUNDARY LAYER ON A PLATE*

O.S. RYZHOV and I.V. SAVENKOV

A linear oscillator in a boundary layer on a flat plate is considered. The fluid is assumed to be incompressible and the distances from the

leading edge of the plate are chosen to be so large that the Reynolds number may be considered as tending to infinity. The field of the perturbed motion is constructed within the framework of the theory of an interacting boundary layer, the loss of stability of which is brought about by the action of a selfinduced pressure. The solution of the linearized equations is expanded in Fourier integrals in two coordinates which lie on the surface around which the flow occurs. Numerical methods are combined with asymptotic methods when finding the inverse trans- formations.

1. Equations and boundary conditions. In experimental aerodynamics vibrating tapes, which are placed either on the surface

around which the flow occurs or at a certain distance from it, are used, as a rule, to excite unstable vibrations in the boundary layer. The waves propagating downwards through the flow are amplified, lose their two-dimensional character as they enter into the non-linear phase,

and almost-periodic inhomogeneities in the lateral direction appear in their structure. In order to follow the spatial development of the perturbations more clearly, a local small

amplitude source was used in /l/ instead of tapes. Attempts to provide a mathematical description of the essentially spatial pulsations

observed experimentally have run into considerable difficulties. A system of linearized Navier-Stokes equations was considered in /2/ on the basis of assumptions which, up to now, remain questionable. The results from /3/ are based on an even cruder approach. Hence, in order to simplify the analysis of spatial perturbations, it is advisable to assume that the Reynolds number R-cm right from the very start. Then, the Prandtl equations with a previously unknown pressure gradient /4-6/ are the asymptotic limit of the initial Navier-

Stokes equations. As applied to a non-stationary spatial boundary layer in an incompressible liquid on a flat plate, the above-mentioned equations /?, 8/ are written in the form

(l.la)

(l.lb)

(l.lc)

Here t is the time, .x,y and s are the Cartesian coordinates of the space, and u, ". and u) are the components of the velocity vector which are measured in a special dimensionless

system of units. As the joining conditions show, on the external boundary of the region under

consideration

m-y-+-4 (t, 2, 2) t w+O when y+m, (1.2)

and the selfinduced pressure p is related to the displacement thickness A by the equation

We shall assume that the perturbations in the boundary layer are introduced via a small

aperture in the plate where the vertical component of the velocity executesharmonicvibrations. In this case

u=W==o, v-Lwt, 5, Z), vO=ain (wot)~oo(x, z) when y=O, (1.4)

and the function vao differs from zero only within the circle r=(r*+Z')l"cr~ Upwards along

the flow from the source

U-+Y, w+o, p+O when r-+--m. (1.5)

The boundary value problem which has been formulated simulates the experimentalconditions in which there is a periodic blowing-in and removal of the fluid very precisely with the obvious stipulation that the measurements refer to reasonable Reynolds numbers /l, 9/. It

*Zh.vychisl.iYat.mat.Fiz.,28,4,591-602,1988 185

186

follows from this that it is only possible to make a qualitative, rather than a quantitative, comparison of the theoretical results with the experimental data. On the other hand, the first attempts to solve the problem even in the formulation which has been presented led to erroneous results in the calculation of both the two-dimensional /lo/ and spatial /ll/ per- turbations. An exhaustive analysis of the asymptotic limit as R+ml is therefore of fundamental importance.

Starting from conditions (1.2) and (1.51, we put

(p, A, a--y, V, zu)=8(p', A’, u’, v’, w’) (1.6)

and we linearize the equations of motion of the liquid with respect to the amplitude 6 of the source/sink. The next step should consist in the separating out of the harmonic depen- dence of the required solution on time in accordance with (1.4). However, it is then necessary to deal with functions which increase exponentially along the longitudinal co- ordinate. In order to ensure the selection of the unique solution in this class of functions a postulate, which is applicable to two-dimensional vibrations, was formulated in /12/ according to which the excess pressure field and the field of the components of the velocity vector must continuously depend on the frequency o0 of the pulsations of the source/sink as it passes through the critical value o.‘=2.298. The formulation of an analogous postulate in the case under consideration is less obvious as will be clearly seen subsequently. We

shall, therefore, make use of the method proposed in /13, 14/ which enables one to obtain the oscillating parameters of the fluid at any fixed point in space by passing to the limit as t-cm. A natural assumption will then be v=O when tG0, and the boundary conditions (1.4) are to be considered as being valid for instants of time n-0. Of course, the new problem on the actuating of the periodic source/sink makes it possible not only to followtheemergence of the perturbed motion onto a regime of harmonic vibrations but also to establish the formation of a forward propagating wave packet.

We shall furthernotethe limiting relationship

which is obtained from (1.2) and (1.3) by eliminating the displacement thickness. Asregards the conditions above and below with respect to the flow, they are obtained by replacing (1.5)

by u'+O, W”0, p’+O when x'+z'+m, (1.8)

since, unlike perturbations which are periodic in time, the perturbations which arise when the source/sink is actuated must degenerate as the distance becomes greater alonganydirection.

2. Integral transformations. By expanding the functions which have been introduced by means of (1.6) in Laplace

integrals with respect to time and Fourier integrals with respect to the spatial coordinates lying in the plane of the plate, we obtain

[B(w k, m), c(y;w, k,m), P(y;w, k,m),E(y;w,k,m)]=

+%t 3 dz 3 Ip’(t, x, z), u’ (tv x, y, z), v’ (4 x, y, z), 0 -m -m

L’)’ (t, x, y, z)] e-Wx+W dx,

and the limiting relationships (1.8) guarantee the existence of all of the integrals in a classical sense.

As a result of substituting the formulae which have just been written down into the linearized Eqs.Cl.1) and the boundary condition (1.4) and (1.7), a system of ordinary dif- ferential equations is obtained for the functions-image representations F, U,B and W. The integration of the above-mentioned system follows the scheme described in /15, 16/ which is based on the limiting form of the Squire's transformation as R+m. For this purpose, in the case of real k and m, we define a reduced frequency and a reduced wave number by means of the formulae

o’=(k’/k)“o, k’=sign(k) Ikl”‘(kZ+mZ)“‘, (2.1)

and, when these are taken into account, the expression for the excess pressure takes the form

842.2)

Here, &=o&(k, m)l(oo2+w*) is the image representation of the source from (1.4), the quantity F is specified with the help of the equalities

F=@(B)-Q(k’), n=o’(tk’)-~~“=o(tk)-“‘. (2.3)

Q=\k’I(ik’)%

187

and Ai denotes an Airy function which tends to zero exponentially in the sector -n/3<

argzcnl3. On equating it to zero, the denominator of the integrand in (2.2) leads to the dispersion

relationship @W-QUO, (2.4)

which relates the complex frequency 0 with the wave numbers k and m of the characteristic spatial vibrations of the boundary layer. The same dependence is obtained in the limit, as R-+=, for the frequency o' and the wave number k' of the Tollmin-Schlichting waves /17/. In the latter case, relationship (2.4) possesses a denumerable setofroots o.‘(k’)-(ik’)“G(k’) of which only the first generates unstable perturbations, sincetheinequality Reo,‘(k’)>O holds when Ik’l>k.‘=1.0005. Startingoutfromthedefinitions (2.1) for 0' andk' and (2.3) fortheinvariant 8, we concludethat, fromthewholemanifoldof spatial vibrationalmodes, only the first vibrational

mode with o,(k, m)=(ik)“Sl,(k’) can be unstable. Next, it is possible to confer the form o'=o[lf (m/k)‘]” onto the first of the formulae (2.1) with real k and m, when the estimate

IHe o,(k, m)lGJHeo,‘(k’)] follows. In the subsequent discussion, the critical frequency o.(m/ k), which determines the transverse Tollmin-Schlichting waves with an amplitude which is constant in time, plays an important role. It is expressed in terms of the critical frequency , 0. of the direct wave as o.(m/k)=o.‘[l+(m/k)*]-“.

3. Analysis of the inverse transformations. The first stage in evaluating the integrals in (2.2) consists of finding an approximate

expression for the inverse Laplace transformation. The contribution to the solution from the sum of the residues due to all the roots of the dispersion relationship, starting from the second may be estimated by analogy with the analysis of the two-dimensional perturbations

/13, 14/. Since this sum turns out to be of the order of e(t-') uniformly with respect to Z and z, it may be neglected in the calculations if the time is chosen to be sufficiently large. The contributions from the residue associated with the first root o,(k,m) and from the residues in the poles -ioO and ioO of the function a,(~, k, m)=o&(k, m)/(o”+o,‘) remain. As a result

p'=Im[l,(x,z) exp(ioot)]+Re[l,(t,2,z)], (3.1)

eimr dm 3

(ik)“* Co0 (k, m)

-m -m eikx (k* + m*)‘l*F 161, (k), k’] dk’

0

eimr dm s exp [or (k, m) t + iks] x

-_ k*F, (wl (k, m),kTml

(kg + me)“* d0 162, (k’)]/dQ dk

*

The first term in the sum (3.11, where Q~=ili.~ok-2ir c 020, has precisely the same form which the formal expansion in Fourier integrals in x and z of the solution which is periodic in time would give with the boundary conditions (1.4), which hold for all instants of time.

It would appear at first glance that it must be concluded from this that the second term tends to zero quite rapidly as t-too. This, however, is erroneous as will be seen from the

ensuing treatment. The reason is concealed in the fact that the construction of a regime of

harmonic vibrations is coupled with the need to work with fluid parameters which increase exponentially downwards through the flow. In this class, the appearance of an eigenfunction, the weight of which is determined from an analysis of the integral I?, is inevitable. More-

over, it is only necessary to take account of the contribution to the solution from the first

term in the sum (3.1) in the vicinity of the source/sink as, at large distances, it becomes unimportant compared with the contribution from the second term which, in the limit as R-m, yields the asymptotic forms of the transverse Tollmin-Schlichting waves which interact with one another.

To order to simplify the calculation of the excess pressure, we put m=kp,--m<fi<= in

the first of the integrals appearing in (3.1). Then,

(3.2)

AS far as the second integral is concerned, in addition to the transformation which has been

indicated, we shall also, following Craik /la/, make the substitution k=a(l+B’)-‘/: --o~<s<O in it. In the new variables

k’=s, o,(k, m)=o,‘(s) (l+p’)-‘“. (3.3) By taking account of (3.3), we get

J.= 5 o&, Is (1 + py-y sp (1 + pyq exp[o,‘(s)T -t is-V X lo02 + q'*(s) (1 + pa)-"a] d@ [O, (s)]/dQ ds’ -0J

188

T= (I+~)-‘9, x-(l+~“)-“‘(x+~z).

Here, the internal integral is of the same type as that which is encountered in the linear problem on the propagation of planar waves. Hence, in order to build up the structure of the spatial perturbation which is generated by the localized harmonic source, it is necessary to know the solution of the auxiliary problem on two-dimensional vibrations, which is formulated in terms of the reduced variables T and X. The well-known properties of this solution /14, 17/ enable one to simplify the calculation of 12 at any fixed point in space as t+m. Subject to this condition, the limit of J. is equal to zero, if oO(li-@')"<o.'. However, when oo(l-r~')">o.' , the required limit can be represented in terms of the residue of the integrand at a point sa=so(mo, B) in the complex plane s which is fixed by the equality o,'(s)=ioo( l+p’)” when oO>O. For this purpose it is necessary to demonstrate the analytical continuation of the quantities o,'(s) and 62,(s) occurring in the expression for 1. which were specified on the real axis. BY recalling that s=k’. we obtain the values of the quantities being considered in the whole of the complex plane s by making a cut along the positive imaginary semi-axis and writing the right-hand side of the dispersion relationship (2.4) in the more general form: Q-*k’(ik’)‘” when Re k’S0. The error which arises as a result of this is comparable with the error tolerated in deriving formula (3.1). By recalling the definition o.(~)==o.'(~+~")-" of the critical frequency for transverse Tollmin-Schlichting waves, we finally have

J, = - n0[o, -o*(p)]exp[i(~J + .@)I X so2 (1 + B’)“4 co.1 Iso (1 + B”)“, SOS (1 + BY”1 ,

(isop a~ (52, fso (I + pyq, so} as

(3.5)

where 8 is the Heaviside unit step function. This expression can be treated as an eigen- function which is to be added to the formal expansion of the solution, which is periodic with respect to t, with the boundary conditions (1.4) which are valid at all instants of time, in Fourier integrals with respect to 5 and z. The appearance of the eigenfunction in the periodic solution at supercritical frequencies was postulated in /12/ where a rule for calculating its weight, which leads to (3.5) and is applicable to two-dimensional vibrations, is given. Hence, the extension of the above-mentioned postulate to spatial wave motions is, in fact, achieved by the introduction of the generalized critical frequency m.(B). Onpassing through its forced vibrational frequencies, ‘(00, the parameters of the fluid remaincontinuous at any specified point in the space and at an arbitrarily chosen instant of time.

4. Calculation of the inner region. If the real axis in the complex plane k is used for evaluating the internal integral

from (3.2), it is necessary to separate out the singularity which is produced by the first root of the dispersion relationship o?o.(~). This condition determines the critical longi- tudinal wave number k,(B) for transverse perturbations, which is related to the critical wave number k.‘=l.O005 of the vibrations which propagate through the flow by means of relationship k.(p)-k.'(lfB')-"@. The complex plane k is shown in Fig.1, where the broken line is the trajectory SI-Sa(m0, B) when B=O (s-k in this case). The frequency of the source/

1 O.Zi

2 +l I ““=! *

-2 4 4 -Y

-\ I

\ I

1196

\

ZJI 9.6

-0.21

’ / ’ /

_I-0.4%

Fig.1

In selecting a new integration path in the k-plane, only the part consisting of the real negative semi-axis is subject to replacement and the remaining part, which passes along the real positive semi-axis, is not deformed. Let us construct a broken line C, in such a way that it dces not touch the dispersion curves in the upper semiplane, pass the path C, under the dispersion curves in the lower semiplane and denote the positive

real semi-axis by cs. Moreover, we shall assume that C, does not have any common points with the trajectories

of the whole of the denumerable spectrum of the roots of the dispersion relationship which solely yield stable vibrations. These trajectories, which terminate, as IPI-= P at finite points in the upper semiplane, are not shown in Fig.1. In order to avoid having to separate out the singularity in the internal integral from (3.21, we shall calculate it along c,+ca when x+~z>O and along C,+C,, if z+ez<O. It is, of course, necessary to add the residue oftheintegrand, at a point whose position is fixed on one of the dispersion curves by the frequency w0 and the gradient of the wave fi, to the contour integral under consideration. Rapid convergence of the integral can be ensured by specially choosing the intensity of the source/sink in the form voO- exp[-(s'fzz)] with the image representation L=exp [i(k*+m’)/4] since this quantity was not measured in the experiments /l, 9/. Allowing for the symmetry

sink increases along so as the distance from the originincreases and, when oO=o.'-2.298, it intersects the real axis at the point k=-k.‘. The dispersion curves kso=Moo, B)-so(oo, fi)(1+ p*)-"' for the frequencies ~~-1.5, 2.298, 4.6 and‘7.3 are shown by the solid lines. The last of the curves refers to the direct Tollmin-Schlichting wave with the maxium increment in the growth of the amplitude through space. When p=O, all of the curves being considered start off at points on s0 and arrive at the origin as IpI-". The part of the dispersion curves which satisfies the condition oOCo.', intersects the abscissa at the points k--k.(l) and, hence, the amplitude of the tranverse waves can, depending on the gradient p=m/k both increase as well as decay in the downward direction through the flow.

189

of the perturbation field, the inner region was defined by the inequalities -2GxG6, OG64. The computed points on the surface of the plate were chosen with a step size Ax=Az=O.2 and the modulus Ml of the integration step size in the complex plane k was varied from 0.04 to 0.11. Almost all of the computer time was taken up in evaluating the double integral (3.2).

5. Analysis of the far region. When x=6, the contribution to the solution, due to the above-mentioned integral,

becomes small (-1%) and it may therefore be neglected in the remote region with x>6. The analysis of the wave motion is greatly simplified here since only the second termremains in the same (3.1) and this term, by virtue of (3.4) and (3.5), reduces to a single integral. As far as the estimates are concerned, it is convenient to work with the initial variables k and m to obtain them. Rewriting Iz in the form

eimJk (m; t, x) dm,

0

Jk y s exp [o, (k, m) t + ikx] x

-ca cook%,, (k, m) dk

[coo* + cola (k, m)] (k* + ma)‘U-D [Q, (k’)]/dn ’

we represent JI in terms of the residue of the integrand at the point k,s- kmo(uo, m) in the complex plane k which is specified by the equality ol(k,m)=ioo. As a result

II=-a-031 [ml-m.(oa)] exp [i(oot+ %ox)]k,,%oo(k mo. m){(ik,o)“(k,o”+ ma)“W[Q,(k,,), k’(k,o, m)]/ak}-‘. (5.2) The interpretation of expression (5.2) is not so obvious as that discussed in connection

with (3.5), although it is completely analogous to it. The role of the critical frequency

o*(B) in the case of the transverse Tollmin-Schlichting waves in the variables s=k',p here belongs to the critical lateral wave number ?a.(~&). The simple relationship between the two above-mentioned critical quantities is: m.(oo)=k.‘[o.‘/oo-(oo/o.‘)J]“. Its values have been calculated in /16/. Nevertheless, the motion in the remote region, which is established by means of (5.1), can be considered as a superposition of all possible internal supercritical perturbations.

Fig.2

x InI

0.2

0.1 !Lc w

0 5 10

Fig.4

t0 w

. . A. \

‘-.A*\. ‘\ '.0.3 1.1. '\

5 Ll .,;_l_.> :

.-a.?.. ./

'Y?, .._. .-.

&99 ' m

0 2 4

Fig.3

Fig.5

Let us denote the increment in the growth of the amplitude of the transverse Tollmin- Schlichting waves in the direction of the x.-axis by x,0==---Im k,,(o,,m). Graphs of the functions xrnO for the frequencies roe-1.5, 2.298, 4.6 and 7.3 when ma0 are shown in Fig.2. In accordance with the results of the preceding paragraph, critical values of m.(o,) are only obtained subject to the condition that oo<o.'--2.298. In the range O<(uo<7.3, apositive maxx,o=x,o(m., 00)-%.(~0) is attained /16/ at a certain finite m=m,(oo) and then, when

‘7.3<ooC13.0, it is shifted to m=m,(oo)-0 (we recall that 00-7.3 corresponds to the direct Tollmin-Schlichting wave with the greatest growth in amplitude through the space). Forced vibrations with oo>13.0 are not considered further. Since

k,a--oo/m+e-‘“l’oo’“/mZ+ . as (rR(-+=, the inequality

x,,-(00/2)v'/ma+I.. >O as ImJ--

190

is satisfied for all 00. The isolines of the function LO in the rn, o0 plane are drawn in Fig.3 using broken curves for the stable region and dotted and dashed curves for the un- stable region. The neutral isoline with x,,,~=O, yielding the above-mentioned relationship m=m.(oo), and which separates the regions is marked with the number 1. The number 2 refers to the curve m.(o0) with the maximum growth increment of xm,(oO). The nature of the change in the latter as o0 is varied can be understood from Fig.4 where the maximum value is on the direct Tollmin-Schlichting wave with 00=7.3. Hence, over the whole range O<oO<m

x,,(oa)>O and XIII,-+0 only when W+O,~. This means that spatial perturbations which are harmonic in time always generate instability in the boundary layer while two-dimensional waves increase downwards through the flow only in the case of frequencies exceedingacritical frequency 0: /13, 14/.

Taking account of the fact that the quantities o. and m in (5.1) and (5.2) are real, we shall write the simplest estimate of I, as:

Il*(t,I,Z) I=GCexp (xmo(m., 00)x), C=const. (5.3)

Hence, the increment in the growth of any periodic perturbations in the direction of

the main flow of the fluid does not exceed Xmo(ms,Oo). Let us show that, in spite of its apparent crudeness, the estimate which has been obtained is in good agreement withtheprecise estimate for certain paths z=ztga, a=Ta,. originating from the origin.

Let cp(m;oo,tga)-k,O(oO, m)+m tga. In order to find the asymptotic forms of 1, as X,Z-+~, we apply the method of steepest descent to expression (5.1), (5.2). Foranarbitrary fixed a, the coordinates of the saddle points in the complex plane m satisfy the equation

dcpldm=dk,,ldm+tg a=O. It is only for certain values of this parameter that the saddle points can lie on the real axis which coincides with the initial integration contour. In the latter case the problem is simplified since there is no need now for an analysis of the singularities

of the integrand which, as a rule, is very tedious /17/. The equalities

*+tga=O, -=-_ dImkmo ds,, _. dm dm

hold for real m. Three values satisfy the second of these equalities when 0<00<7.3: ml=0 and mr.s=*m.(oo). Substitution of these values into the first equality of (5.4) enables one to determine the corresponding angles, a,=0 and ar,s=Fa,,of inclinationoftherequired paths in the x,z-plane and, moreover, al=-al by virtue of the fact that k,o(oo,-m)-k,o(oo,

m). In the case of frequencies 7.3<00<13.0, in accordance with the displacement of mar xnro to the point m- m.(%)=O, we obtain the unique value a=a.=O.

We note that Imcp(m;oo,tga)=Imk,,(oo,m) for real m and, in practice, thisisindependent of a. Subject to the condition that 0<00<7.3, the function --Imcp(m;oo, lga) attains its positive maximum xmo(m,, 4 at the points m=m.(%) while, in the case of m=O, we have -Imcp(O;oo,O)< x,o(m.,oo). The results of the calculations of a. are shown in Fig.5.

In accordance with what has been described above, a.=0 over the range 7.3G0~Gi3.0. It is clear thatallthe conditions which guarantee the applicability of the method of

steepest descent are satisfied in the case of the saddle points ml,s=fm,(oO) and the angles

corresponding to them a1,3=Ta. when 0<oa<7.3 and, in the case of the point m=m.(w)=O

with the angle a=a.=O when 7.3alhG 13.0 . In view of the symmetry of the solution about

the r-axis, we shall only consider the upper half plane z>o. Then, in order to describe

the perturbation field, it is possible to confine oneself either to the saddle point ms=

-m.(o0), if 0C00<7.3 or to the saddle point m=m.(oo)=O, when 7.3<'00G13.0. Let k.=

kmo(oa,-m,) for any forced vibrational frequency 00 from the two ranges under consideration and the quantity ma=m,(oo) corresponding to it. According to the method of steepestdescent, we have on the path z=xtg'a.-

W,,O~i -me) 1~“~ k,“P,, (k,, - me) (ik,) 18 (keg + m.‘)‘jr

x

[ aF[80(kJ7 k’(ke9 -mJl

8k 3 -lexp(~(o,t+cp(-m ‘0 tga)s+r]) S?01 e

(5.5)

where r is the angle between the positive direction of the m axis and the tangent to the line Recp=const at the point of steepest descent. Since -Im cp(-m.;o,, tga.)=x,Jm,,W), the asymptotic forms of (5.5) are in good accord with the crude estimate (5.3) although they contain the additional factor z-lb.

6. Results. Taking account of the relationships which have been obtained above, the form p'=P(x,

z)cos[@6t+$(Ga)l may be given to the expression for the excess pressure, where P, obviously, has the meaning of an amplitude and q is the phase of the perturbations at a given point 2,~ on the surface of the plate. The numerical calculation of both of these quantities in the inner region -2Gx<6 was augmented with data from an asymptotic analysis in the remote region 6&&12 and the change in the transverse coordinate was constrained by the inequalities OGz44. In the calculations, the frequency o0 of the forced vibrations was chosen to be equal to the critical frequency a.'-2.298 of the direct neutral Tollmin- Schlichting wave. In this case, the greatest increment xno (m., oJ=O.1127 in the increase in the amplitude was attained on a path inclined to the x-axis at an angle a.--7.15" and the maximum in the amplitude, P.-3.84, in the calculated region was located at the point

191

z=O.8,z=O. The oscillations in the pressure only start to exceed this maximum beyond the limits of the calculated region.

The isolines of the relative amplitude PIP, are shown in Fig.6; its values are given by the numbers and the point where P/P,=1 is indicated by a small circle. The broken line is the path z-xtga. with the angle a., found by solving the asymptotic system (5.4) and the calculated positions of the maxima in the distribution of the relative amplitude with respect to s for different fixed z are marked with small crosses. The small deviations are explained by the error which is incurred when formula (5.5) is used, if the values of s-10. Control calculations with x=30 showed that the error which is tolerable in the determination of the integral Zz on passing from (5.1), (5.2) to (5.5) does not exceed 1%. It can be seen from Fig.6 that the perturbations upwards through the flow die off very rapidly after attain- ing a local maximum P."'=LiZ when x=-l.& z=o. In the calculated region and beyond, they also decay strictly downwards through the flow behind the source/sink. In accordance with the conclusions from the asymptotic analysis, the points at which the amplitude of the vibrations attains its greatest value, depart from the axis z-0 (starting from x=8.6).

Fig.6

I

5 8.5 I.? s

Fig.7

Fig.8 Fig.9

Figure 7 provides a more accurate representation of the nature of the distribution of the amplitude of the perturbations with respect to x along various lines z=const . The confluence of the pulsations along the axis of symmetry on the neutral frequency of the direct Tollmin-Schlichting wave is most noteworthy. The displacement of the points with the greatest vibrational amplitudes from the axis of symmetry, when s=const+8.6, is illustrated in Fig.8. By the time x=12 the maximum of PIP. on the path z=xtga. is extremely pronounced. Finally, the isolines 4 of the phase of the perturbations are shown in Fig.9 with a step of n/2 . Qualitative agreement with experiment in the distribution both of the amplitude and ofthephase of the forced vibrations is obtained which is completely satisfac- tory and the value a,-7.15" , which is predicted by the asymptotic theory, is extremely close to the value of a.=7.5" for the same angle measured experimentally /l/. Such a small dif- ference in the data for a, can obviously be explained by the fact that the angles in the plane of the plate are not normalized on changing to the variables used in the analysis. Actually,

asymptFti.c the experiments were carried out at a Reynolds number R--1.54x10 . The

parameter e==R-'I*-0.22 from /4-8/, calculated using it, is sufficiently small to ensure a direct quantitative comparison of the theoretical and experimental results.

192

REFERENCES

1. GILEV V.M., KACHANOV YU.S. and KOZLOV V.V., Development of a spatial wave packet in a boundary layer, Preprint 34, Inst. Tekhn. i Prikl. Mat., Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1981.

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Translated by E.L.S.