The instability of free shear layers

3 THE INSTABILITY OF FREE SHEAR LAYERS A. MICHALKE Deutsche Forschungs- und Versuchsanstalt ffir Luft- und Raumfahrt, Institut ffir Turbulenzforschung, Berlin

Summary. This report gives a survey on our present knowledge about the instability of free shear layers which occur, for instance, in jets. The main interest is focused on the instability of incompressible free shear layers. In a first section experimental results will be discussed. Then theoretical results of the linearized theory will be considered with respect to the infuence of Reynolds number and the spatial growth of disturbances as well as to the effect of different velocity profile shapes and of three dimensional disturbances. In a further section nonlinear effects will be discussed. Finally some remarks will be given to the influence of variable basic density distribution and of Mach number. Obersicht. Dieser Bericht gibt einen 1:lberblick fiber den gegenw/irtigen Stand der Kenntnisse bezfiglich der Instabilitfit yon freien Scherschichten, wie sic in Freistrahlen auftreten. Der Hauptteil der Arbeit beschgftigt sich mit der Instabilit/it inkompressibler freier Scherschichten. Im ersten Abschnitt wird fiber experimentelle Ergebnisse berichtet. Sodann werden theoretische Ergebnisse der linearisierten Theorie diskutiert, wobei die Einflfisse der Reynoldszahl und des r/iumlichen Anwachsens der St6rungen sowie der Geschwindigkeitsprofilform und der Dreidimensionalit~itder St6rungen er6rtert werden. In einem weiteren Abschnitt werden nichtlineare Effekte behandelt. Im letzten Kapitel werden einige Hinweise auf den Einflul3 variabler Dichteverteilung und der Machzahl auf die Instabilit/it freier Scherschichten gegeben. I. I N T R O D U C T I O N The a i m o f this r e p o r t is to give a survey o n o u r p r e s e n t k n o w l e d g e a b o u t the instability o f free shear layers. Such free shear layers occur in m a n y flow configurations o f F l u i d M e c h a n i c s in c o n n e c t i o n with b o u n d a r y - l a y e r s e p a r a t i o n a n d with t h e mixing o f parallel streams. A special t y p e o f a mixing region with n e a r l y c o n s t a n t pressure exists in jets d o w n s t r e a m o f a nozzle. I f the s e p a r a t e d flow is l a m i n a r , the m i x i n g region can b e c o m e u n s t a b l e a n d m o s t l y this instability will lead to a t r a n s i t i o n f r o m l a m i n a r to t u r b u l e n t flow. Because o f the n u m e r o u s aspects o f the instability o f s e p a r a t e d flows we have to restrict t h e t o p i c o f this report. F o r this reason we here shall d e a l only with the instability o f single p l a n e free shear layers. This restriction, however, is n o t so severe as one might assume, since very often t h e realistic flow can well be a p p r o x i m a t e d b y a p l a n e shear layer. F o r large R e y n o l d s numbers, for instance, the flow in a n a x i s y m m e t r i c a l j e t is n e a r l y parallel, 213

214

A. MICHALKE

since due to boundary layer theory the velocity component normal to the flow direction is small compared with that in flow direction. With respect to a stability analysis we can therefore assume that the jet consists of a parallel flow and its velocity profile depends only on the coordinate normal to the basic flow direction. Furthermore, for large Reynolds numbers the thickness of the jet core is large compared with the jet boundary-layer thickness. Then with respect to the instability of the jet boundary layer the axial symmetry becomes unimportant. Then the jet boundary layer can be replaced by a single free shear layer and only for very small disturbance frequencies, i.e. if the disturbance wavelength is comparable with the jet diameter, the axial symmetry becomes again important. The investigation of the instability of free shear layers has essentially been initiated by Lord Rayleigh c4a,44) in 1879. But further investigations were made not before •950. In the subsequent years numerous papers have been published. Many of the results which established our knowledge about the instability of free shear layers can be found in the books of Lin, (29) Drazin and Howard "°) and Betchov and Criminale3 s) Since we are not able to repeat these results here, we shall try to give a survey on some newer contributions to this problem. The main interest will be focused on the instability properties of free shear layers for very small Mach numbers without heat conduction, so that the flow can be treated as incompressible. Finally some remarks will be given to the influence of heat conduction and Mach number.

2. THE INSTABILITY OF INCOMPRESSIBLE FREE SHEAR LAYERS A single free shear layer consists of an infinitely extended unidirectional flow in x-direction with a velocity profile U(y) where y is the coordinate normal to x. For y -+ oo the velocity U - + Uo and for y ---> -- oo U --->0 or U--~ -- Uo. Therefore the free shear layer profile U(y) is characterized by the existence of an inflexion point. The velocity profile U(y) is produced by the induction of a vorticity layer which is given by f~o(Y) = --dU/dy. At the locus of the inflexion point the vorticity distribution takes an absolute maximum. The question of the instability of a free shear layer is therefore equivalent to the question, whether the velocity induced by a small disturbance of this vorticity layer will lead to a larger displacement of the vorticity layer or not. For this reason the appropriate basic equations for stability theory are the vorticity equations. Superimposing a small two-dimensional disturbance upon the basic parallel flow, the equations for the disturbance vorticity d21(x,y,t ) and its stream function ~ 01(x,y,t) become

8~,

[

8¢1] 8~1

8~b~ [d~o

A01 = -- f/l,

E,Df21] (2)

THE INSTABILITY OF FREE SHEAR LAYERS

215

where the influence of the kinematic viscosity v on the basic flow has been neglected. E is a measure for the disturbance magnitude. If the disturbance terms in the brackets ofeq. (1) will be small compared with the corresponding basic flow terms, a neglection of the nonlinear terms is justified. Furthermore, for a wavy disturbance 41 = ~[4(Y) e'(~x-at)]

(3)

eq. (2) and the linearized eq. (1) yield the well-known Orr-Sommerfeld equation for the disturbance amplitude function 4(Y). The boundary conditions to be satisfied by the disturbance are in a free shear layer 4'(-- 00) = 4(-- 0o) = 4 ' ( + 00) ---- 4 ( + 00) ---=0.

(4)

If all quantities will be normalized with the maximum velocity Uo and a characteristic thickness L of the shear layer, the solution of the Orr-Sommerfeld equation for a given velocity profile depends on the Reynolds number R = UoL/v and on one of the parameters/3L/Uo or aL, which can be chosen freely. For real ~ the eigenvalue /3 is generally complex and (3) denotes a temporally growing or decaying disturbance. For real /3 the eigenvalue a is generally complex and (3) denotes a spatially (in x-direction) growing disturbance which depends on the Reynolds number R and on the Strouhal number S = f L / U o with /3 = 2~rf. Before discussing the theoretical aspects of the shear-layer instability some experimental results will be shown. 2.1. Experimental Results As mentioned above, for large Reynolds numbers a jet boundary layer is expected to behave like a free shear layer. The instability of plane or axisymmetric jet boundary layers can qualitatively be investigated in an experiment by the smoke technique. For this reason one has to introduce smoke into the jet boundary layer. In order to get more regularity of the flow pattern, it is useful to excite artificial disturbances with a fixed frequency in the boundary layer by means of a loudspeaker or--for a plane jet--by a small vibrating ribbon (cf. ref. 40). Illuminating the smoke by stroboscopic light of nearly the same frequency, the flow pattern can be observed as a standing or slowly varying picture. Figure 1 shows the unstable free boundary layer of a plane jet visualized by smoke. We see that the smoke bands, the envelopes of which are streaklines, roll up to smoke concentrations of meander form which move downstream. One may assume that there would be vortices formed. Finally these vortices break down and the jet boundary layer becomes turbulent. The identification of the smoke concentrations with vortices seems to be justified, since sometimes a slipping process can be observed as shown in Fig. 2 in an axisymmetric jet boundary layer. Two consecutive vortices slip around each other by the mutual induction and

216

A. M I C H A L K E

coalesce finally to a single one. This phenomenon has been observed by Wille, ~57,ss) Wehrmann and Wille, t55) Michalke and Wille, t41) Freymuth t13~ and recently by Becker and Massaro. t2) The instability and the rolling-up process of the free boundary layer seems to be essentially two-dimensional, which was especially confirmed by B r o w a n d F ~ For quantitative measurements of the velocity and its fluctuations in the disturbed jet boundary layer the hot-wire technique is very useful. Such hotwire investigations were made by Fabian, t12) Sato, ~45-47) Wehrmann and Wille, t55) Wehrmann, t56) Michalke and Wille, ~41) Freymuth tla) and Browand. ~7) It was, however, found that the flow properties in a jet boundary layer are naturally not quite periodical. The frequency of these natural disturbances varies slightly about a mean value. Only, if artificial disturbances will be excited, the flow is strictly periodical. This can be seen from the hot-wire signals in Fig. 3 showing the oscillogram of the velocity fluctuations and the sound pressure without and with excitation by a loudspeaker at various distances downstream of an axisymmetric nozzle. 3* is the displacement boundary-layer thickness in the final cross-section of the special nozzle. One can easily see that for excitation the flow frequency is identical with the sound frequency with the exception of the last oscillogram where the main frequency component of the fluctuation is the half of the sound frequency indicating the fusion of two vortices to a single one. Furthermore, by the excitation the development of the disturbances is shifted closer to the nozzle, since the initial disturbance magnitude is increased with increasing sound intensity. The mean value of the frequency of natural disturbances is found by Michalke and Wille t41) to befS*/Uo = 0.023 for this special nozzle independent of the Reynolds number based on the nozzle diameter in the range of 104-105. A similar result was found by Sato ~4s-47~ for a plane jet and for a separated boundary layer. If a hot wire is continuously moved across the jet boundary layer at a position before the rolling-up takes place, then a record is obtained as shown in Fig. 4. It was made by Freymuth ~3) who used artificial disturbances excited by a loudspeaker. At the bottom we see the profile of the total velocity with fluctuations and above the amplitude distribution of the fundamental component which is essentially the u'-fluctuation. There are two maxima in the amplitude distribution denoted by ?~ and 72 with a remarkable phase reversal in between at small values of the basic velocity. This typical amplitude distribution was also found by Sato, ~46'47) Wehrmann and Wille, ~sS) Wehrmann ts6) and B r o w a n d F ) The amplitude is proportional to the initial disturbance magnitude, i.e. for acoustic excitation proportional to the ratio of sound pressure to dynamic pressure of the jet as was shown by Freymuth.O 3) The amplitude of the disturbance grows in basic flow direction x as shown in Fig. 5. For fixed Strouhal number--the characteristic length L is here the

I

y

Uo= 8rnls~ D= 7.5era x = 1.5cm

: f =/,16Hz

FIG. 4. Total velocity and amplitude distribution in an axisymmetric jet boundary layer; ref. 13.

FIG. | 5. Smoke distribution in a jet boundary layer during rolling-up; ref. 38.

-~,= 15

t

s~gnal ----,--

x 5.=5o

"~,:30

"~,: 50

with sound 225cps. 95db

~.°30

sound f

L~

X ~-.= 80

~.-" 80

FIG. 3. Oscillograms of hot-wire signals in an axisymmetric jet boundary layer and sound pressure without and with excitation at various distances downstream of the nozzle; ref. 41.

SOUD~

hot-w~re

without

FIG. 1. Two-dimensional jet boundary layer visualized by smoke; nozzle width 4 cm, jet velocity 2 m/sec, frequency 30 c/sec; ref. 41.

FIG. 2. Axisymmetric jet boundary layer visualized by smoke; nozzle diameter 7.5 cm, jet velocity 3 m/sec, frequency 95 c/sec; ref. 13.

THE INSTABILITYOF FREE SHEARLAYERS

217

momentum boundary-layer thickness 0--both peaks 71 and ~2 of the amplitude distribution grow exponentially in x-direction in the beginning, while further downstream a saturation occurs. Furthermore, we see in Fig. 5 that for large Reynolds numbers R which were also based on 0, the growth of the amplitude is nearly independent of the Reynolds number. It follows that for small amplitudes and large Reynolds numbers the instability of a free shear layer should approximately be described by the inviscid linearized stability theory with spatially growing disturbances. ~2

-+÷~* ~

l

+

I,

.

"~

"'"

_

_ _

Q02 Q01

i

~:7:

~2,,,__I~:" + . I. -":. +__ t EO0~ ~---- .-~"~---- ~ "''I | QO02 '

i

o

22

~I

••

+w, ;73

~

33z.

x

+ I +

236

QO01 20

l.O

60

811

100

120

FIG. 5. Growth of the maxima di and d2 of the amplitude distribution in downstream direction in an axisymmetricjet boundary layer; ref. 13. From the measured growth of the disturbances one can evaluate the spatial growth rate --ai which should agree with the imaginary part of a calculated theoretically. Furthermore, the disturbance wavelength A can be measured by hot-wire technique as well as the amplitude distribution. Therefore many flow quantities can be obtained from the experiments which enable us to examine the theoretical results. 2.2. Theoretical Results of the Linearized Theory In the past temporally growing disturbances mostly were assumed in theoretical investigations due to the classical stability theory. It was found by Lessen, <2+) Curie, m) Esch, °1) Tatsumi and Gotoh, (Sa) Betchov and Szewczyk(6) and Tatsumi, Gotoh and Ayukawa (s+) that in fact for large Reynolds numbers the instability properties of a free shear layer were nearly independent on the Reynolds number as shown in Fig. 6 for the hyperbolic-tangent velocity profile

U(y) = tanh y

(5)

and temporally growing disturbances. For small Reynolds numbers the free

218

A. MICHALKE

shear layer does not become stable, i.e. a critical Reynolds number does not exist. But it is generally accepted that for very low Reynolds numbers the assumption of a parallel basic flow is somewhat doubtful, since the variation of the basic velocity profile in x-direction caused by the viscosity may be very important. Therefore Ko and Lessen(21) took into account the growth of the boundary-layer thickness approximately and then found a critical Reynolds number.

ci

/

'

_

.,//

=-0/,

---03 -

-O2 -0.1

.._.----

- -

r

0.1

---02

- - - - 0./, --0.5

o?O.~ -08 ogl.o

"'

10"

1

lO

1 ctRe

FIG. 6. Lines of constant temporal amplification ct in the (~, aR)-plane for the tanh-velocity profile; ref. 54.

Nevertheless, for large Reynolds numbers it is evident that contrary to wallboundary layers for a free shear layer the inviscid stability theory is applicable. Then the influence of the Reynolds number drops out and the Strouhal number is the only parameter affecting the instability of an incompressible free shear layer. The instability of inviscid free shear flows has been exhaustively investigated by many authors and it is here only referred to Drazin and Howard (1°) and Betchov and Criminale. (s) Lin(29) explained that the instability of free shear layers is caused by the mutual induction of the vorticity distributed in the shear layer. Because of the inflexion point of the velocity profile a free shear layer is always unstable with respect to certain wavy disturbances. The results obtained by means of the inviscid stability theory, however, were mostly restricted to temporally growing disturbances. It was found that a free shear layer is always unstable for small frequencies. For a finite thick shear layer a maximum of temporal amplification exists, but for higher frequencies beyond the neutral value the shear layer becomes stable. In order to compare theoretical results with experimental ones, the temporal growth rates were usually transformed by means of the disturbance phase velocity into spatial growth rates as used by Sato, (47) Schade and Michalke(s°) and

THE INSTABILITY OF FREE SHEAR LAYERS

219

Michalke and Wille.(41) The agreement between theory and experiment was relatively good with respect to wave numbers and growth rates, but with respect to the finer structure of the disturbed flow the agreement was unsatisfactory. For instance, contrary to the experiments, no phase reversal of the z/-fluctuation was found theoretically for temporally growing disturbances. From theoretical considerations Gastero”) came to the conclusion that the growth rates obtained from a stability calculation for temporally growing disturbances cannot be transformed by means of the phase velocity into spatial growth rates. Gaster 05) stated that a stability calculation with complex a and real frequency /3 should be carried out for large amplification as present in free shear layers. Furthermore, GasteP) was able to show that, when a small local disturbance will be artificially created in a boundary layer,

FIG. 7. Comparison

of measured and calculated amplitude distribution Strouhal number S = 0.008; ref. 13.

for a

the disturbance downstream will grow spatially and the corresponding eigen-mode of the linearized stability problem will finally dominate. But in using spatially growing disturbances the difficulty is that most of the theorems of the classical stability theory were not applicable for complex a such as the Squire theorem or the Rayleigh inflexion point theorem. Nevertheless, some stability calculations for complex a have been performed by Michalke,(34) Betchov and Criminale.(4) Lessen and Ko(‘~) and Ko and Lessen.(21) The surprising result was that concerning the free shear layer many essential features of the instability properties observed in the experiments were confirmed by the results of the theory of spatially growing disturbances. For instance, the distribution of the u’-fluctuation with its two maxima c”Iand E2 and the phase reversal shown in Fig. 7 as well as their dependence on the Strouhal number was in good agreement with the experimental results (cf. ref.

220

A. MICHALKE

13). The basic velocity profile used for the theoretical results of Fig. 7 was the hyperbolic-tangent profile

U(y) =

0.5 [1 + tanh y].

(6)

The spatial growth rates --cq0 as shown in Fig. 8 were also in good agreement with the measured values, at least for small Strouhal numbers. But here another problem becomes evident, namely that agreement with the measured -a~o

]

i

0.08

I

/ ~ t~ i

0.0/,

axisyrnmetricjet O Uo:8 rn/s Q'~ane jet o Uo:8 m/s

1i"

0

0.01

Tfi" ~ r y

0.02

(th-profite)

0.03

0.04

FIG. 8 Measured and theoretical spatial growth rate vs. Strouhal number; refs. 13, 35.

values is only found in the frequency region I. For higher Strouhal numbers in a region II the measured values were smaller than the theoretical ones and, finally, in the region III it was impossible for Freymuth (z3) to excite artificial disturbances in the experiment at all. The experimental results therefore imply that for a realistic free boundary layer the neutral disturbance is physically insignificant. Furthermore, the mean value of the natural frequency, i.e. the frequency of the disturbances which exist without artificial excitation, is

1.U(y) o

U(y) =1-[1+ rne2mfy] -~ with f

_;l-z

1

dz

j

oo 1 e =/°°u (1-U)dY= 2 ~

- ~ - - ~ - T -~

sf~/

0. /

-~

"

i "

oo

i

~

--y ~ ....

FIG. 9. The generalized tanh-velocity profiles; ref. 35.

THE INSTABILITY OF FREE SHEAR LAYERS

221

found to agree with the frequency at the border of the frequency region I, but not with the theoretical frequency of maximum amplification. We have, however, to realize that in the theory a parallel flow was assumed which was infinitely extended in basic flow direction x, while in the experiment the flow was not strictly parallel. The basic velocity profile changed continuously from a wall-boundary-layer profile at the nozzle to a free boundarylayer-type profile with inflexion point downstream of the nozzle. It seems to be quite reasonable to suppose that all these local velocity profiles will generally have different instability properties. Then a disturbance moving from the nozzle downstream will meet different conditions for growth. In order to estimate the influence of the variation of the basic velocity profile, the instability properties of infinitely extended parallel flows with different velocity profile shapes were compared by Michalke. (3s) His "generalized hyperbolic-tangent profiles" are shown in Fig. 9. For the profile 2.5! - O q

m=oo

t 2.0

1.5

1.0

I i ]I~

J ~=6

0.5

/

,

I

0.2

0.4

._.,.. p 016

FIG. 10. The spatial growth rates of the generalizedtanh-profiles; ref. 35. parameter m ----- 1 we have the original tanh-profile (6), while for m = oo we have a wall-boundary-layer-type profile, but all profiles have the same momentum boundary-layer thickness 0. The spatial growth rates of these profiles are shown in Fig. 10. We can discern three frequency regions which agree with those used in Fig. 8. For small frequencies the growth rates are

222

A. MICHALKE

nearly independent on the profile type, while especially in the frequency region III the differences become large. This confirms a statement of Esch °1) that only for higher frequencies the special profile shape becomes important. The comparison of these theoretical results with the experimental ones of Fig. 8 seems to indicate that the growth of a disturbance in downstream direction is only possible if the local instability properties do not change very much, i.e. for small Strouhal numbers. A more general treatment of the non-parallel flow problem was proposed by Lanchon and Eckhaus. (22) Up to now we discussed only two-dimensional disturbances. For temporally growing disturbances it is known that the growth rate of three-dimensional disturbances cannot exceed that of two-dimensional ones. For spatially growing disturbances, however, this fact has not yet been proved*, but

T (1.2

Cr=[31Ctr100~.i~'5'=0

1o3 \

01

0.2

03

0 t,

0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Fz~. 11. Phase velocity and spatial growth rate of the tanh-profile for various spanwise wave numbers; ref. 36. calculations for the tanh-profile (6) by Michalke (36) seem to confirm that it is true even for spatially growing disturbances. In Fig. 11 the phase velocity e, and the spatial growth rate --at are plotted as function of the frequency/3 for various spanwise wave-numbers 7 in z-direction. We see that the growth rates for 7 =~ 0 are always smaller than for 7 = 0. Therefore the three-dimensional disturbances are more stable than two-dimensional disturbances, at least in this case. It should, however, be noted that for a three-dimensional neutral disturbance the u'- and w'-fluctuations become singular due to the linearized theory at the critical layer where U" ---- 0 as was found by Benney. (3) He supposed that the singular behaviour was due to the neglection of viscosity. This assumption seems to be rather unsatisfactory. If the idea of an induction instability is adopted, it is difficult to see why the neglection of viscosity * Note added in proof: In the meantime this has been proved by M. GASTER,3. Fluid Mech. 43, 837-9 (1970).

THE INSTABILITYOF FREE SHEARLAYERS

223

should be responsible for the singular behaviour of the solution. Another possible reason for the singularity may be the linearization of the equations. This seems to be probable as was mentioned by Stuart (52) and will be discussed later. We note at this point that the linearization of the disturbance equation can lead to some trouble with the solutions. Therefore it seems to be worthwhile to discuss the properties of the nonlinear equations in order to obtain criteria for the validity of the linearized solutions. 2.3. The Influence of Nonlinear Terms The influence of the nonlinear terms will become important when a small amplified disturbance in a free shear layer has reached a certain magnitude. In order to estimate what must be expected in this situation, let us discuss the inviscid two-dimensional vorticity equation. From this non-linear equation it follows that the vorticity is fixed at the fluid particles and remains constant, when the particles move along their paths. Furthermore, if in the beginning there was somewhere an absolute maximum of the vorticity, then this maximum value cannot be exceeded in space and time. This means that new or higher-valued vorticity cannot be created in the inviscid flow. If the influence of the viscosity would be taken into account, it follows that for unbounded flows the viscosity has only diffusive character. Hence the maximum value of the vorticity can only become smaller in space and time under the influence of viscosity. An inviscid free shear layer is equivalent to a vorticity layer which has one maximum as mentioned above. Hence due to the nonlinear inviscid vorticity equation this vorticity layer will be deformed by an amplified two-dimensional disturbance in such a way that the maximum of the vorticity layer will be preserved. Furthermore, any line of constant vorticity must correspond to a fluid line, or more precisely, to a streakline for spatially growing disturbances. Contrary to this, solutions of the linearized vorticity equation will violate this law of conservation of vorticity. Therefore we must expect that for a linearized solution the vorticity can grow in space and time beyond its original maximum value. This is, in fact, true as can be seen in Fig. 12. Here the lines of constant vorticity denoted by f2 of the disturbed flow are shown in the (x,y)-plane at a fixed time. The curves were calculated for the most strongly amplified disturbance of the tanh-profile (6) due to the inviscid linearized theory for complex a by Michalke334) Here ~ is the wavelength of the disturbance and g(x) ---=, e - ' , x (7) is the exponentially growing local disturbance magnitude which is very small at x = 0 (E = 5 × 10-4). The maximum of the undisturbed vorticity is f~ = --0.5. We can see that with increasing x new extremum values of the

224

A. MICHALKE

vorticity appear. It is quite clear that this phenomenon must disappear, if nonlinear terms of the solution would be included in the calculation as mentioned above. #

~Y

:l

-0.2

'

I

o!

LI.0 -1'5{g.OO005

~.~.o176 lslt k

~=~oo3o I 10

0

o.~Jk

20

~._~io~3 : 2s

"~s,Bs 30

1.~x

2x

FIG. 12. Lines of constant vorticity of the disturbed tanh-profile at maximum amplification; ref. 34. On the other hand, due to the nonlinear vorticity equation the lines of constant vorticity must be identical with streaklines. Therefore we can compare both lines in order to estimate the validity of the linearized solution. This is shown in Fig. 13. The dashed curves are the lines of constant vorticity as shown in Fig. 12, while the curves with points are the corresponding streaklines. We see that outside the critical layer (y = 0) the agreement between the streaklines and the lines of constant vorticity is good, until the local disturbance magnitude ~ becomes nearly 0.1 at x ~ 23. For larger local disturbance magnitude g differences between both curves appear indicating that nonlinear terms should be taken into account. But in the neighbourhood of the critical layer at y = 0 we find the disagreement even for small g as it was stated by Lin. (3°)

,.o' . . . . . . . o~.. . . .

!

/

-1.5

'

,

!

I

!

I

!

t-15T

-2,0J

. . . . .

6 ......

5-

~ 10 O.5X

' ~-

15; k

~

20

__

I 25 1.5k

30 : - x 2),

FIG. 13. Comparison of streaklines and lines of constant vorticity for the disturbed tanh-profile at maximum amplification; ref. 34.

THE INSTABILITY OF FREE SHEAR LAYERS

225

The vorticity is, however, proportional to the derivatives of the velocity, while the streaklines were calculated by integration of the velocity. Therefore it is believed that the range of validity of the linearized solution with respect to the streaklines is much greater than with respect to the lines of constant vorticity. Hence the streakline pattern calculated by means of the linearized solution should give a correct impression of the vorticity distribution even for relatively large local disturbance magnitudes. By means of the calculated streaklines the smoke pictures of the experiment can also be interpreted. A detail of the calculated streakline pattern of Fig. 13 is shown in Fig. 14. We see that the streaklines roll up in a complicated manner with a coincidental folding. If smoke would be introduced at x : 0 in that part of the shear layer which contains most of the vorticity, the hatched

FIG. 14. A detail of streakline pattern during rolling-up; ref. 3 8.

area would become visible. On the other hand, the hatched area would then contain most of the vorticity. Therefore the smoke distribution gives an impression of the vorticity distribution. It follows from Fig. 14 that we have to expect vorticity concentrations in between of which the vorticity layer becomes very thin. A similar smoke distribution is found in the experiment as shown in Fig. 15 at the left. We see also here the further stage of the rolling-up process. The smoke distribution suggests a vorticity concentration which can surely be interpreted as a discrete vortex with respect to its effect on the flow. One can also clearly see the diffusive effect of the viscosity on the very thin vorticity layer which is the connection to the preceding vortex. But this stage of the rolling-up process is evidently far beyond the validity of the linearized theory and nonlinear terms have to be included in the theoretical analysis. In order to obtain a solution for the disturbed free shear layer which is valid even for larger disturbance magnitude, one can expand the solution of the nonlinear equations in powers of the disturbance magnitude ~. Then

226

A. MICHALKE

the first-order term corresponds to the solution of the linearized equation and the higher-order terms can be calculated recursively. Since the first-order term grows exponentially in time or space for amplified disturbances, it is doubtful whether the series expansion will converge or not. It is, however, known from the experiments that the disturbance velocities remain finite in time and space. Therefore Stuart (sl) introduced a modified amplitude time function A(t) for temporally growing disturbances. Stuart assumed that this function A(t) has to satisfy a nonlinear differential equation in t dA

d t = A[ao + at IA[ 2 + ....]

(8)

For small IA[ the solution A(t) of the linearized eq. (8) is exponential with ao : --i/3, but for t--> ~ the solution of the nonlinear eq. (8) can reach an equilibrium amplitude ]Ae I- In the third-order approximation this equilibrium amplitude is determined by the so-called Landau constant al. Expanding the solution of the nonlinear vorticity equation in a Fourier series in x-direction and its coefficient functions in a power series in A(t), a system of ordinary differential equations for the y-dependent amplitude functions is obtained. The first-order equation is identical with the homogeneous Orr-Sommerfeld equation, while the following differential equations are i nhomogeneous and can be solved recursively. The Landau constant al appears in the inhomogeneous part of a differential equation which belongs to a thirdorder term of the fundamental Fourier component. The homogeneous part of that differential equation is identical with the Orr-Sommerfeld equation, if the value of t3 in the latter equation is replaced by/3 + 2i/31. Since/3 is an eigenvalue of the Orr-Sommerfeld equation, the, inhomogeneous differential equation has solutions satisfying the boundary conditions for every value al for amplified disturbances with /31 • 0 And the Landau constant remains undetermined. Contrary to this, if/3~ : 0, that inhomogeneous differential equation has a solution satisfying the boundary conditions only for a special value of al. Therefore only in the neutral case the Landau constant can be calculated by the usual methods of the theory of differential equations. Hence it was supposed that the Landau constant calculated for the neutral case is also significant for amplified disturbances. For the tanh-velocity profile (5) Schade (4s,¢9) succeeded in calculating the Landau constant al for the neutral case. But he had some trouble, since his solution became singular at the critical layer. As usual he introduced viscosity in order to rule out this singularity. Nevertheless, the physical interpretation of Schade's solution is somewhat unsatisfactory. The equilibrium amplitude IAe I is zero in the neutral case, and the disturbance magnitude of a finite neutral disturbance, which remains constant due to the linearized theory, vanishes with increasing time due to the nonlinear theory. Thus the nonlinear terms have apparently a damping influence that is unsatisfactory in the inviscid

THE INSTABILITY OF FREE SHEAR LAYERS

227

limit. Furthermore, Schade assumed in his calculation that the basic velocity profile should not be modified by the nonlinear terms. This assumption is doubtful, since for any small amplification this modification occurs. Therefore it is believed that the singular behaviour of Schade's solution is due to his incorrect assumptions. Recently Gotoh (xT) showed that the Landau constant a x as calculated by Schade (4s) is not correct and he obtained a Landau constant ax which is proportional to (/3~/a)-s in the inviscid limit. Surely, this result of Gotoh is also not correct, since the Landau constant cannot depend on/3~ as mentioned above. Gotoh apparently evaluated formulas, which were derived for the neutral case only, also for the case of amplified disturbances which clearly violates his assumptions. Stuart <52) tried to obtain a regular equilibrium solution for the tanh profile by expanding the neutral solution for wavenumbers ct which were amplified due to the linearized theory. But the question arises whether such an expansion is allowed in the inviscid case, since according to the linearized theory the disturbed vorticity distribution (9) does not converge uniformly, if fl~ tends to zero as it was remarked by MichalkeJ s2) The reason f o r this behaviour becomes evident, if we realize that due to the nonlinear vorticity equation an amplified disturbance is created by a lateral displacement of the basic vorticity layer as mentioned above, while by the neutral disturbance the basic vorticity layer is modified only by peaks and valleys without a lateral displacement. This behaviour in the neutral case can be found quite Clearly from the linearized solution (cf. ref. 32) as well as from an exact nonlinear solution of Stuart (s2) which is given by the stream-function ~band the vorticity f] as follows-

~(x,y)

---- ln[cosh y + ~ cos x],

(lO)

f~(x,y)

= -- (1 -- e2)[cosh y + ~ cos x] -2.

(II)

This solution is regular for 0 < E < 1. For E = 0 the solution (10) represents the tanh-profile (5) and for e = 1 the flow of a one-row street of potential vortices. Equation (11) corresponds to a vorticity distribution with absolute maxima at y = 0 periodic in x-direction. Expanding (10) in powers of E < 1 and introducing Fourier components, the solution (10) agrees with the series expansion obtained by Schade (4s) up to the second-order terms with the exception of a term independent on x which modifies the basic flow (5). This term was arbitrarily assumed by Schade to be zero. It is easy to show that without this assumption a series expansion can be obtained which

228

A. MICHALKE

agrees with that of Stuart's solution (10). From this solution, however, it can be seen that all Landau constants are zero. It follows that the same must be expected for amplified disturbances. This result indicates either that for amplified disturbances an equilibrium amplitude does not exist or that Stuart's statement (8) is not successful. For amplified disturbances the lateral displacement of the lines of constant vorticity was found by Michalke (aa) and shown in Fig. 16. Here the lines of constant vorticity were calculated in a third-order approximation of Stuart's series for the most strongly amplified disturbance of the tanh-profile (6)

lST7

-15r

I

--

thtd cooer a~oxu~gat,o~

]

2 2 i 2 3 ~ 5 6 7 8 9 ~ I I ~

FiG. 16. Lines of constant vorticity of the disturbed tanh-profile for maximum amplification due to Stuart's series; ref. 33. ([Ai = 0.1.)

for temporal amplification. The Landau constant al used was due to Schade. (48) The dashed lines correspond to the first-order approximation. It is assumed that the disturbance magnitude A(t) has grown to a value of IAI = 0.1. We see that the maximum value of the vorticity is even in the third-order approximation something larger than the maximum value of the undisturbed vorticity. Therefore we can conclude that for this disturbance magnitude the third-order approximation does not suffice, and the convergency of the series expansion is not very good in the inviscid case. Surely, putting the Landau constant to zero, which will also influence the vorticity distribution, some minor improvement may be expected. Nevertheless, we can predict that for amplified disturbances the lines of constant vorticity should roll up yielding broad local concentrations of vorticity with small ridges of vorticity in between. In the presence of viscosity these ridges of vorticity will be smoothed because of their large vorticity gradient changes. This is confirmed by the computation of Amsden and Harlow <1) who calculated the nonlinear viscous instability of a free shear layer numerically. Their computed lines of constant vorticity with the concentra-

229

THE INSTABILITY OF FREE SHEAR LAYERS

tion of vorticity and the smoothed ridges are shown in Fig. 17. These concentrations of vorticity may surely be interpreted as vortices with regard to their effect on the flow as it was stated by Domm. (9)

Y

I •i i iiiiiiiiiiii!ili!iii!ii iiiiiiiiii i~::iiiii~:~:::i.::.~.iii)i!i::i.ii.2".q.:.:...., :..:::. ".'.....:::::)!.:;

X FiG. 17. Lines of constant vorticity of a disturbed mixing layer (viscous, nonlinear, timewise theory); ref. 1.

In the inviscid two-dimensional case these results suggest a redistribution of vorticity for amplified disturbances which is of the type

D.(x,y,t) -= f~o(Y + ,f(x,y,t))

(12)

where f~o(Y) is the undisturbed vorticity distribution. From the vorticity equation it follows that the displacement f of the vorticity has to satisfy the nonlinear equation +U~x

+v

1+~

=0.

(13)

Here the velocity components u and v are solutions of the Poisson equation, i.e.

l ff u(x,y,t) = -- 2-~ v(x,y,t) = ~1 f f

0, -- ~)f2(~:,~/,t)

(x -- ~:)2 + (y _ .,7)2

d e d~/,

f ) f 2+( f , (y ~ , t _) 7)2 d~:d~. (x (--x - - ~:)2

(14)

(15)

For ~ = 0, (14) and (15) yield u = U(y), v = 0, i.e. the basic flow. Since due to (12) lf~(x,y,t)[ < max If2o(y)l, it follows from (14) and (15) that also for > 0 the velocity has to remain finite. Hence we have to expect an equilibrium amplitude with respect to the velocity components u and v in the inviscid case, but not necessarily with respect to the derivatives of u, v and f. An indication of this phenomenon can be found on the last oscillogram of Fig. 3 which shows very large temporal derivatives of the hot-wire signal at certain

230

A. MICHALKE

instants. This is surely not compatible with Stuart's assumption (8). Linearizing f2o(~ + ~f(~,~,t)) in the integrals of (14) and (15) as well as the eq. (13) one obtains to 0(~)

f(x,y,t ) --- ~ U~-~-~)/~ e "~x-Bt )

(16)

where ~(y) is identical with the quantity used in eq. (3). It is quite clear that with the first approximation (16) of the displacement the disturbed vorticity distribution (12) avoids the vorticity source character which is found in its linearized form (9). Returning now to the three-dimensional neutral disturbance, Stuart ~52) was able to show by means of his nonlinear two-dimensional solution (10) that the velocity components of the three-dimensional disturbed flow (5) have to be u = a 2 ~y (In g) + 72 F(g),

v = -- tg--x(In g), w = ay

(17)

(In g) -- F(g

,

with g = c o s h y + Ecos ~;

~=~x+Tz

(18)

where 7 is the spanwise wave number with ~: + 72 -----1. The arbitrary function F(g) has to satisfy the following conditions: (i) For y --* ± oo the component u must tend to tanh y and w to 0 . (ii) The function F(g) has to be real and regular, especially for y = 0. (iii) For E = 0 the solution must become u = tanh y and v = w = 0. A possible solution satisfying these conditions is surely

F(g) = q- ~/[g2 _ (1 -- e)2]/g

(19)

which can be written in the form F = sinh__y J ( 1 + ~ 2 ( c o s h y c o s ~ - 1 - 1 ) - - E s i n 2 ~) . g sinh 2 y

(20)

It is easy to see that for ~ < 1 eq. (20) is indeed regular and real even at y ---- 0. But introducing (20) in (17) and expanding the nonlinear solution in powers of ~, one obtains an invalid expansion which is singular at y = 0. The reason is that f o r y ~ 0 the function F(g) is of 0(a/[~(cos ~ + 1)]). Hence the singu-

THE INSTABILITYOF FREE SHEARLAYERS

231

larity appearing in the linearized theory can be explained as a result of the neglection of the nonlinear terms as mentioned above. Another aspect of the nonlinear development in a free shear layer is the appearance of a secondary instability. From the experiments (see Fig. 2) it is known that the vortices formed in the disturbed free shear layer can interact by the mutual induction. Thus a slipping m o t i o n of two consecutive vortices around each other and their fusion to a single, larger vortex is possible. This phenomenon is connected with the occurrence of a subharmonic fluctuation component found in the hot-wire output as shown in Fig. 3. Evidently, this secondary instability can only occur, if the concentration process of the vorticity would have reached a certain stage, i.e. if the amplitude of the fundamental disturbance would have grown to a critical value. Kelly ~2°) was able to show theoretically that this secondary instability can in fact occur in free shear layers and is due to a parametric resonance of the secondary and the fundamental disturbance component. The critical amplitude of the fundamental component calculated by Kelly was in good agreement with that observed by Browand ~7) in the experiment. The final stage of the laminar-turbulent transition in a free shear layer, however, seems to be quite unclear up till now. From smoke experiments we only know that the smoke concentrations suddenly seem to explode three-dimensionally. In the framework of the inviscid theory we can therefore suppose that the vorticity distribution in the vortices becomes highly unstable to three-dimensional disturbances with growth rates very large compared with that of the primary disturbance in the shear layer. A first approach of the vortex instability in free shear layers was made by Michalke and Timme. ~a9) But the very simplified vorticity distributions and the restrictions which must be applied in this analysis make it difficult to draw general conclusions from these special results. On the other hand, the experimental investigation of the vortex breakdown in free shear layers is very difficult to handle and no essential contributions have been made up till now. Hence the breakdown of the vortices in a free shear layer seems to be an unsolved problem.

3. THE INSTABILITY OF COMPRESSIBLE FREE SHEAR LAYERS Finally some remarks about the influence of the compressibility on the instability of free shear layers will be given. The compressibility leads to density fluctuations which can occur (i) for larger Much numbers and (ii) for a non-zero density gradient of the basic flow. Then four new parameters will, generally, influence the instability problem, namely the Much number, the temperature ratio of the heat conduction, the Prandtl number and the ratio of the heat capacity coefficients.

232

A. MICHALKE

We can discern two types of influences. Firstly, the basic velocity-, temperature- and density profiles, which are again assumed to be only functions of y, depend generally on all these parameters. Thus the instability of the free shear layer is also influenced implicitly by these parameters via the basic flow. Secondly, the disturbance itself is now compressible and the Mach number, the Prandtl number and the heat capacity ratio appear explicitly in the disturbance equation via the density and temperature fluctuations. For large Reynolds numbers and a Prandtl number of 0(1) the disturbance motion can again be assumed as inviscid and non-conducting. Then the heat conduction affects the instability only via the basic flow. For very small Mach numbers the influence of the Mach number becomes unimportant, but the temperature ratio of heat conduction can influence the instability properties essentially, since the basic density distribution varies due to this parameter. Even in the absence of gravity g, or more precisely, for large

~iL

1.5|Mode

~p•0,5

0,2

0,1f

FIG. 18. Temporal growth rate of the linear shear layer of thickness L with density variation for various density ratios p2/p~; ref. 37.

Froude numbers Fr = U,~/gL, the inhomogeneity of tile inertia can be responsible for a remarkable variation of the eigenvalues and produce additional disturbance modes. A simple example is the linear shear layer of thickness L with a discontinuous basic density variation p(y) which was treated by Michalke. <37) The temporal growth rate ~tL/U1 vs. wave-number aL is shown in Fig. 18. U1 is the maximum velocity of the shear layer and px the corresponding density, pz and po the density inside and outside of the shear layer. Here the density ratio po/px is 1.2 and the parameter of the curves is the density ratio p2/pl. We see that for values of p2/pl < 1.2 two or three different disturbance modes can exist. Following a criterion of Howard °9~ the number of the possible

THE INSTABILITY OF FREE SHEAR LAYERS

233

disturbance modes seems to be determined by the number of zeros of the expression d

K(y) = #-t ~ [ao(y)#Cv)].

(21)

Using a continuous velocity and temperature profile Gropengiesser (is) found for spatially growing disturbances at a Mach number M = 0 that a cooling of the free shear layer acts destabilizing. This is shown in Fig. 19 where the spatial growth rate --a s is plotted vs. frequency /3. For a fixed velocity profile with a Mach number M = 0 the maximum amplification becomes larger with decreasing temperature ratio T2, i.e. for increasing density outside the shear layer.

0.10

-ct i

N=O

T

-12 -8

-l,

0

6

-1Z -0

-6

I 0

4

f - - \ Tz:O6

/

0O8

/ / 006

,/ ,

/

/ / /

0.0/, /

,/

,

i"

\\-.

/

0.02

x

\

¢!

\

O.Ot,

006

012

016

12

\

\

0 20

0

\

0.26

029

FIG. 19. Spatial growth rates of a shear layer for vanishing Mach number and various temperature ratios; ref. 18. For Mach number M > 0 the free shear layer becomes more stable with increasing M. This is already known from the compressible vortex sheet which was investigated by Landau, (23) Pai (42) and Miles.(31) A continuous shear layer velocity profile was investigated by Lin, (2s) Lessen, Fox and Zien (25'26) and recently by Gropengiesser "s) who used spatially growing disturbances and included heat conduction in his analysis. The spatial growth rates --ai of two-dimensional disturbances (0 = 0) are shown in Fig. 20 in a three-dimensional plot as function of the disturbance frequency/3 and the Mach number M for a heated free shear layer, i.e. for a temperature ratio ~r2 = 2. We see that the growth rates decrease with increasing Mach number. Above a Mach number of 1.8 disturbances with supersonic phase velocity exist indicated by the sonic lines c, = a2 and al -- cr = at, respectively. The effect of cooling the shear layer, i.e. a temperature ratio T2 = 0.6, is shown in Fig. 21, We ~ee that the maximum of the growth rates is much larger for

234

A. MICHALKE -cti

~= 0.10 TZ= 2 0 =0"

t~ x

Ul "Cr:°l

\M

~-2a

a

FIG. 20. Spatial growth rates of two-dimensional disturbances in a compressible free shear layer vs. frequency and Mach number; temperature ratio T2 = 2; ref. 18.

0.12

0.10

0.0~

0.0~

0.0~

0.0;

[

FIG. 21. Spatial growth rates of two-dimensional disturbances in a compressible free shear layer vs. frequency and Mach number; temperature ratio ~2 = 0.6; ref. 18.

THE INSTABILITY OF FREE SHEAR LAYERS

235

small Mach numbers compared with that of Fig. 20, while the range of unstable frequencies is smaller. About a Mach number of 1.8 a remarkable change of the curves occurs which corresponds to the appearance of a second disturbance mode and to an overlapping of the characteristics of these two modes which is shown in Fig. 22 in more detail.

II,I,IIII\

o,oo,

0

I

0 0 2 ~ 0.02

0.0~

0.0&

005

0.06

0.07

0,~1

0.09

FIG. 22. Spatial growth rates vs. frequency for various Mach numbers, when the second mode appears; temperature ratio ~2 = 0.6; ref. 18.

A further aspect of the instability of a compressible free shear layer is that three-dimensional disturbances can be more unstable than two-dimensional ones. An example is shown in Fig. 23 for the Mach number M = 2 and a temperature ratio T2 = 0.6 and various wave angles 0 = arccos (~,/V'[~ 2, + ~,2]). We see that the m a x i m u m amplification occurs for a wave angle 0 ~ 60 °. For O = 30 ° the second mode is included which has two supersonic neutral disturbances.

0.03 / f \ ',?O=fi0*

//

0.02

'~\ 4r

rz:a6

"'~F\\ 75' \ \

0.01

~)~-~', --\\ 30' II.Model

//

\\

0.02

0.04

0.06

0.08

0.10

FIG. 23. Spatial growth rates vs. frequency of three-dimensional disturbances for various wave angles at M = 2, T2 = 0.6; ref. 18.

236

A. MICHALKE

It should, however, be noted that in the neutral case the temperature and density fluctuations become singular at the critical layer for subsonic as well as for supersonic disturbances. The same is true for the u'- and the w'-fluctuation of three-dimensional neutral disturbances, while for two-dimensional disturbances the u'-fluctuation becomes singular only for supersonic neutral disturbances. It is, however, supposed that this singular behaviour may also be a consequence of the linearization of the equation like that found for three-dimensional disturbances in the incompressible case and mentioned above. A comparison of the theoretical results with experiments is not yet possible up to now, since experimental investigations of the instability of free shear layers for high subsonic as well as for supersonic flows have apparently not been performed in the past. Therefore this lack of experimental results should warn us of using the theoretical results unreservedly.

4. O U T L O O K

AND

ACKNOWLEDGEMENTS

This survey on the present knowledge of the instability of free shear layers leads to some suggestions for future investigations. The main open questions of the subject in hand seem to be as follows: (i) The influence of a non-parallel basic flow, i.e. of the variation of the velocity profile from a wall-boundary layer- to a free boundary layertype profile, should be investigated theoretically. (ii) The nonlinear development of spatially growing disturbances in a free shear layer is another point of interest. The main attention should be focused to amplified disturbances instead of to neutral ones. (iii) The experimental investigation of the vortex breakdown in free shear layers seems to be very desirable. (iv) The influence of heat conduction and higher Mach numbers on the instability of free shear layers should be investigated experimentally. It is hoped that these suggestions will encourage further investigations of the instability and transition in free shear layers. This report was made at the Institut ftir Turbulenzforschung of the Deutsche Forschungs- und Versuchsanstalt ftir Luft- und Raumfahrt e.V. at Berlin. The author wishes to express his gratitude to Professor Dr.-Ing. R. Wille, the Director of the Institute. This survey is based on a lecture "Instability in Mixing Regions", given at the NATO Advanced Study Institute, "Transition from Laminar to Turbulent Flow", in 1968 at Imperial College, London. The author thanks Professor J. T. Stuart and the Organizing Committee very much for inviting him to give this lecture.

THE INSTABILITY OF FREE SHEAR LAYERS

237

5. R E F E R E N C E S 1. A. A. AMSDENand F. H. HARLOW, Slip instability. Physics of Fluids 7 (3), 327-34 (1964). 2. H. A. BECKERand T. A. MASSARO,Vortex evolution in a round jet. J. Fluid Mech. 31, pt. 3,435-48 (1968). 3. O. J. BENNEY, A non-linear theory for oscillations in a parallel flow. J. Fluid Mech. 10, pt. 2, 209-36 (1961). 4. R. BETCHOVand W. O. CRIMINALE,Spatial instability of the inviscid jet and wake. Physics of Fluids 9, 359-62 (1966). 5. R. BETCHOVand W. O. CRIMINALE,Stability of Parallel Flows, New York-London: Academic Press, 1967. 6. R. BETCHOVand A. SZEWCZYK, Stability of a shear layer between parallel streams. Physics of Fluids 6 (10), 1391-6 (1963). 7. F. K. BROWAND,An experimental investigation of the instability of an incompressible separated shear layer. J. FluidMech. 26, 281-307 (1966). 8. N. CURLE,On hydrodynamic stability of unlimited antisymmetrical velocity profiles. Great Britain, Aeronautical Research Council, unpublished Rept. 18564 (1956). 9. U. DOMM, /fiber eine Hypothese, die den Mechanismus der Turbulenz-Entstehung betrifft, Deutsche Versuchsanstalt fi~r Luftfahrt E. V., DVL-Bericht No. 23 (1956). 10. P. G. DRAZIN and L. N. HOWARD, Hydrodynamic stability of parallel flow of inviscid fluid. Advances in Applied Mechanics 991-89 (1966). I I. R. E. EscH, The instability of a shear layer between two parallel streams. J. Fluid Mech. 3, 289-303 (1957). 12. H. FAnIAN, Experimentelle Untersuchungen der Geschwindigkeitsschwankungen in der Mischungszone eines Freistrahles nahe der D~senm~ndung, Deutsche Versuchsanstalt f/~r Luftfahrt E.V. DVL-Bericht No. 122 (1960). 13. P. FREYMUTH,On transition in a separated laminar boundary layer. J. Fluid Mech. 26, pt. 4, 683-704 (1966). 14. M. GASTER, A note on the relation between temporally-increasing and spatiallyincreasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, pt. 2, 222-4 (1962). 15. M. GASTER,The role of spatially growing waves in the theory of hydrodynamic stability. Progress in Aeronautical Sciences 6, 251-70 (1965). 16. M. GASTER, On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22 pt. 3,433-41 (1965). 17. K. GoTorl, The equilibrium state of the finite disturbance in free flows. J. Phys. Soc. Japan 24 (5), 1137-46 (1968). 18. H. GROPENGIESSER, Beitrag zur Stabilitdt freier Grenzschichten in kompressiblen Medien, DLR Forsch.-Ber. 69-25, 1969, DVL-Bericht No. 867. 19. L. N. HOWARD, The number of unstable modes in hydrodynamic stability problems. J. M~canique 3 (4), 433-43 (1964). 20. R. E. KELLY, On the stability of an inviscid shear layer which is periodic in space and time. J. FluidMech. 27, 657-89 (1967). 21. S. H. Ko and M. LESSEN,LOW Reynolds number instability of an incompressible halfjet. Physics of Fluids 12 (2), 404-7 (1969). 22. H. LANCHONand W. ECK~AVS, Sur l'analyse de la stabilit6 des ~coulements faiblement divergents. J. Mdcanique 3 (4), 445-59 (1964). 23. L. D. LANDAU,Stability of tangential discontinuities in compressible fluid. Dokl. Akad. Nauk SSSR 44, 139-41 (1944). 24. M. LESSEN,On the Stability of Free Laminar Boundary Layer Between Parallel Streams, NACA Rep. No. 979 (1950). 25. M. LESSEN,J. A. FOX and H. M. ZIEN, On the inviscid stability of the laminar mixing of two parallel streams of a compressible fluid. J. Fluid Mech. 23, 355-67 (1965). 26. M. LESSEN,J. A. FOX and H. M. ZIEN, Stability of the laminar mixing of two parallel streams with respect to supersonic disturbances. J. Fluid Mech. 25, 737-42 (1966). 27. M. LESSENand S. H. Ko, Viscous instability of an incompressible fluid half-jet flow. Physics of Fluids 9 (6), 1179-83 (1966).

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A. MICHALKE

28. C. C. LIN, On the stability of the laminar mixing region between two parallel streams of gas. NACA Tech. Note No. 2887 (1953). 29. C. C. LIN, The Theory of Hydrodynamic Stability, London-New York: Cambridge University Press, 1955. 30. C. C. LIN, On the instability of laminar flow and its transition to turbulence. In: Boundary Layer Research, pp. 144-57, Ed. H. G6RTLER, Berlin: Springer, 1958. 31. J. W. MILES, On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, pt. 5, 538-52 (1958). 32. A. MICHALKE, On the inviscid instability of the hyperbolic-tangent velocity profile. J. FluidMech. 19, pt. 4, 543-56 (1964). 33. A. MICHALKE,Vortex formation in a free boundary layer according to stability theory. J. FluidMech. 22, pt. 2, 371-83 (1965). 34. A. MICHALKE, On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, pt. 3,521-44 (1965). 35. A. MICHALKE,The influence of the vorticity distribution on the inviscid instability of a free shear layer. Fluid Dynamics Transactions, Polish Acad. Sci., Warsaw, 4, 751-60 (1969). 36. A. MICHALKE, A note on spatially growing three-dimensional disturbances in a free shear layer. J. FluidMech. 38, pt. 4, 765-7 (1969). 37. A. MXCHALKE,Der EinfluB variabler Dichte auf die Instabilit/it einer freien Scherschicht. lng.-Arch. 40, 29-39 (1971). 38. A. MICHALKEand P. FREYMUTH,The instability and the formation of vortices in a free boundary layer. A G A R D Conference Proc. No. 4, Separated Flows (1966), pt. II, pp. 575-95. 39. A. MICHALKE and A. TIMME, On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, pt. 4, 647-66 (1967). 40. A. MICHALKE and O. WEHRMANN, Akustische Beeinflussung von Freistrahl-Grenzschichten. Proc. Intern. Counc. Aeron. Sci., 3rd Congress Stockholm, 1962, pp. 773-85, Washington: Spartan Books, Inc., 1964. 41. A. MICHALKEand R. WILLE, Str6mungsvorg~inge im laminar-turbulenten ~bergangsbereich yon Freistrahlgrenzschichten. In: Applied Mechanics, Proc. l l th Intern. Congr. Appl. Mech., Munich, 1964, pp. 962-72, Ed. H. GORTLER, Berlin: Springer, 1966. 42. S. I. PAl, On the stability of a vortex sheet in an inviscid compressible fluid. J. Aeronautical Sciences 21 (5), 325-8 (1954). 43. LORD RAYLEIGH,On the instability of jets. Proc. London Math. Soc. X, 4-13 (1879). 44. LORD RAYLEIGH,On the stability, or instability, of certain fluid motions. Proc. London Math. Soc. XI, 57-70 (1880). 45. H. SATO, Experimental investigation on the transition of laminar separated layer. J. Phys. Soc. Japan 11 (6), 702-9 (1956). 46. H. SATO, Further investigation on the transition of two-dimensional separated layer at subsonic speeds. J. Phys. Soc. Japan 14 (12), 1797-1810 (1959). 47. H. SATO, The stability and transition of a two-dimensional jet. J. Fluid Mech. 7, 53-80 (1960). 48. H. SCHADE, Contribution to the nonlinear stability theory of inviscid shear layers. Physics of Fluids 7 (5), 623-8 (1964). 49. H. SCHADE,Einf~hrung in die nichtlineare hydrodynamische Stabilitdtstheorie, DLR Forsch.-Ber. 64-35, 1964-DVL:Bericht No. 379. 50. n . SCHADEand A. MICHALKE,Zur Entstehung von Wirbeln in einer freien Grenzschicht. Zeits. Flugwissen. 10, 147-54 (1962). 51. J. T. STUARX,On three-dimensional non-linear effects in the stability of parallel flows. Advances in ,4eronautical Sciences 3, 121-42 (1961). 52. J. T. STUART,On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, pt. 3,417-40 (1967). 53. T. TATSUr~Iand K. GoToH, The stability of free boundary layers between two uniform streams. J. FluidMech. 7, 433-41 (1960). 54. T. TATSUMI,K. GOTOH and K. AYUKAWA,The stability of a free boundary layer at large Reynolds numbers. J. Phys. Soc. Japan 19 (10), 1966-80 (1964).

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