Approximate Theory on the Stability of Interfacial Waves between Two StreamsChanMou Tchen Citation: Journal of Applied Physics 27, 1533 (1956); doi: 10.1063/1.1722302 View online: http://dx.doi.org/10.1063/1.1722302 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stability of capillary–gravity interfacial waves between two bounded fluids Phys. Fluids 7, 3013 (1995); 10.1063/1.868678 The Born approximation for the scattering theory of elastic waves by twodimensional flaws J. Appl. Phys. 53, 4208 (1982); 10.1063/1.331245 Nonlinear stabilization of oscillating twostream instability Phys. Fluids 16, 1380 (1973); 10.1063/1.1694528 Lyapunov Stability of an Inhomogeneous TwoStream Plasma Phys. Fluids 13, 3012 (1970); 10.1063/1.1692895 Stability of a Shear Layer between Parallel Streams Phys. Fluids 6, 1391 (1963); 10.1063/1.1710959

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JOURNAL OF APPLIED PHYSICS VOLUME 27, NUMBER 12 DECEMBER, 1956

Approximate Theory on the Stability of Interfacial Waves between Two Streams* CHAN-Mou TCHEN

National Bureau of Standards, Washington, D. C. (Received June 1, 1956)

The stability of the interface of two superposed streams is studied by extending the theory of Helmholtz stability to include the viscosity and surface tension in the waves. The two uniform layers of incompressible fluids are of different densities, viscosities and stream velocities. Gravity, Taylor acceleration, and surface tension are taken into consideration. An approximate theory is developed and is valid in three ranges of wave numbers. Neutral stability, rate of growth, and rate of decay are illustrated graphically.

I. INTRODUCTION

T HE present problem consists in studying the stability of the interface of two uniform streams

of fluids of different properties. It is an extension of the problem of Helmholtz stability to include the effects of viscosity and surface tension on the waves. The pre­scribed uniform profiles of stream velocities are assumed neither affected by the waves nor by viscosity. As we know, the stability of a variable stream has been treated approximately by Heisenberg, Tollmien, etc. l Their analyses require cumbersome numerical computations, from which it is not easy to distinguish in a compre­hensive and generalizable way the essential parameters and their role in the main physical features of the flow. It is hoped that the present simple case of uniform streams may throw some light on the basic relationships among parameters governing the main physical features of the more general problem with variable streams. The knowledge thus gained may help in introducing new and useful approximations which would permit a simplification of the Heisenberg theory, when applied to the general problem of variable streams. A stability study of the latter approach will be the object of a further investigation.

II. STATEMENT OF THE PROBLEM

Consider two streams of different velocities U, 0, different densities p, {J, and different viscosities v, v. A quantity with a bar above it refers to the lower layer. Gravity g, Taylor acceleration gl, and surface tension T are to be considered. We are interested in the sta­bility of the interfacial wave, its rate of growth and rate of decay. We shall develop an approximate theory with a formula for the phase velocity which is valid simultaneously in three ranges of wave numbers. To this end, two steps are taken. First we attempt to obtain an approximate formula valid in the two extreme ranges of wave numbers. Then using the branching point which separate the periodic from the aperiodic oscillations, an extension will be made to also include an intermediate range.

* Sponsored by the Office of Naval Research. 1 C. C. Lin, The Theory of Hydrodynamic Stabilily (Cambridge

University Press, New York, 1955).

III. BASIS OF THE APPROXIMATE THEORY VALID IN THE TWO EXTREME RANGES OF WAVE NUMBERS

We shall see that the important parameters of the problem are: (a) the difference of the stream velocities, U -U; (b) the effective gravity g' (which includes gravitational acceleration g, Taylor's acceleration gI, and surface tension T); (c) an average viscosity (I') which is defined by Eq. (3b) and includes the effect of both I' and v. A remaining parameter, representing the effects of the density difference, does not present any difficulties.

If the effect of the stream velocities is more im­portant than the viscous effect, we have an approxi­mate Helmholtz stability and the velocity profile can be taken as the discontinuous profile consisting of two uniform values. Hence the inviscid boundary conditions used in the stability analysis are the continuity of the vertical velocity components and the continuity of normal stresses. On the other hand, if the viscous effect is more important than the effect of stream velocities, an incorrect choice of profiles is not serious. Hence the viscous boundary conditions will be formu­lated from the continuity of horizontal velocity com­ponents and tangential stress, even though the in­correct profile made up of the two uniform velocities is used. The approximate theory presented in this paper is based on such boundary conditions with such a step pro file of velocities.

It is to be noticed that the discontinuity of motion at the interface remains a classical difficulty in this and even in the simpler problem of Helmholtz. The normal velocities and stresses are of course continuous, but the tangential velocities (U+u and U+u) are strictly speaking discontinuous, except when U = -0 is assumed at the interface, somewhat like in the Fourier representation of a step function. The usual argument is that there exists in reality a vortex-sheet at the interface which then would destroy the discontinuity of velocity profiles. This difficulty which was not con­sidered in the Helmholtz stability will also not be elaborated here.2

2 H. Lamb, Hydrodyna1!lics (Dover Publications, New York 1945), sixth edition, Sec. 231, p. 371. '

1533

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1534 CHAN-MOU TCHEN

IV. EQUATIONS OF MOTION

The perturbations of the velocity components, u, v (two-dimensional), and of the pressure, p, are assumed infinitesimal and to propagate in the x direction,

u, v, P+p(g+gl)y",(function of y)eik(x-CtJ,

for the upper layer. Similar expressions apply to the lower one. Here k is the wave number and C is the phase velocity of the disturbance. There results two stability equations which are the usual Sommerfeld-Orr equa­tions with constant stream velocities. Following a procedure similar to that given in a previous paper,3 the four boundary conditions give a characteristic equation as follows:

r-1( 1) (r) 2W*-1- r+- +2(r-1) -+1 ~* ~* ~*

x[r+~+~+l]=O, (1) ~* ~*

where ~ is an important parameter proportional to the cube of the wavelength:

~ = 'Yg' / k3J!l.

It has the structure of the Grasshof number, which is defined as the ratio of the Richardson number to the square of the Reynolds number. Further,

Tk2 g'=g+gl+-; 'Y= (p-p)/(p+p);

p-p

W*= (0 -C)/(U -C); ~*=pv/pv; p*=p/p;

[ U-CJ~ [O-CJi r= 1+ikv ; r= 1+i-,;- .

V. APPROXIMATE FORMULAS

The characteristic Eq. (1) is solved for two modes of small k, and for the creeping mode (slowly varying mode) of large k by asymptotic approximations to the first order. The two results are welded together into the following formula:

(2)

valid to the first approximation in the two extreme ranges of k. The viscous mode (rapidly varying mode) of large k, less important in practice, will be covered

3 C. M. Tchen, J. App!. Phys. 27, 760 (1956). If the stream velocities are zero, the characteristic Eg. (1) becomes much simpler and applies to the Stokes-Taylor stability with viscous effects. Such a stability problem is extended to interfacial viscoelastic waves, compare C. M. Tchen, J. App!. Phys. 27, 432 (1956) .

later in formula (4), but is left out of consideration in the present formula (2). Here

n= -ik[C-(U)J;

1-'Y2[U - UJ2 b=~o--4- k(v) ;

(3a)

~o=n'/k3(V)2;

and the average of any quantity is defined by

(cp)= (p~+pcp)/(p+p). (3b)

Formula (2) can also be derived from the secular determinental equation defined by the inviscid condi­tions only. Thus we see that, as a first approximation, the viscosities enter only through their average values (v). Formula (2) valid in two extreme ranges of k can

2.5

2.0

Unstable

I 5

1.0

N

.5

o 4

R

FIG. 1. Stable flow (.=+1): Regimes of stability in the flow domain characterized by the wave number k' and R, based on formula (4). The domain is divided by the two curves: the neutral curve N and the branching curve B. The domain above N is unstable. The stable domain below N is subdivided by the branching curve B into an aperiodic decay above B and a periodic decay below B. In the domain of the periodic decay below B, the periodically damped waves will have their frequency dispersed according to Fig. 5. There is no stable flow for .= -1. .=-1 means 'Y(g+gl) <0, according to Eg. (6).

be improved to include an intermediate range, by following closely the position of the branching point. An adjustment function M is formed by numerical computations of the complete characteristic Eq. (1) near the branching point. The numerical method can be very much simplified if the stream velocities are omitted and only the effective gravity is kept. Not many physical features are expected to be lost in this manner since we know from Eq. (2) and (3a) how the two parameters (stream velocities and effective gravity) are combined. The final result is:

n -=M[ -l±(l-b/M)t} (4) k2(v) ,

where M=1-(1-m) erf\b\-i;

with m=O.S-O.OS'Y2.

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STABILITY OF INTERFACIAL WAVES 1535

Formula (4) is valid in three regions of ~l (intermediate and two extreme h. It differs from formula (2) only by the factor M. The adjustment function M almost always has the value j, except that for h very large, M is unity.

VI. GRAPHICAL ILLUSTRATIONS

In order to reduce the number of physical quantities, we group them into some essential nondimensional parameters so that the plotting of curves is condensed as much as possible. The dimensionless parameters introduced in the figures are as follows:

.6

n' .4

n' = n I I' (g+gl) I-I(v)l,

k'=k[\'Y(g+gl) \ (V)-2J-t,

1-1'2 _ R=-(U - U)2[\'Y(g+gl) \ (v)]-l,

4

1+1' T'= T-[I'Y(g+gl) I {V)4J-t.

2{>

):;---;.5~--;I';;-O ---,';I'5;-----cz'""o;;-----;zf;'5°-----c;3c';:'O,..-----.J31.5

k'

(5)

FIG. 2. Unstable flow: The variation of the rate of growth n' with wave-number k' in the case of E= -1 according to Eq. (4). R is defined by Eq. (5) and represents the relative velocity of the streams. Each curve gives a maximum and an asymptotic rate of growth to be shown in Fig. 4. Circles are the exact results of numerical computations from the complete characteristic equation (1) using (v) as viscosity and taking R=O.

We have plotted the approximate results obtained from formula (4), for various values R, and some exact results are obtained from the numerical computation of the complete Eq. (1) for R=O.

VII. GROWTH OF DISTURBANCES

Figure 1 shows the neutral curve N and the branch­ing curve B for the case of a light fluid over a heavier one. The region above N is unstable and the region below it is stable. The stable region can be divided into aperiodic decay (above B), and periodic decay (below B). For simplicity of discussions we take the reference axes moving with (U) and g+gl>O. On the other hand, if g+gl <0, the reasonings which follow will still be valid by interchanging the densities.

4

.3

.2

o 1.0 1.5 2.0 25 3.0 3.5

k'

FIG. 3. Unstable flow: The variation of the rate of growth n' with wave-number k' in the case of E= + 1. The curves are based on formula (4). The rate of growth has an asymptotic maximum to be shown in Fig. 4, and vanishes at neutral wave numbers as illustrated in Fig. 1.

The N curve shows that, for a given stream condition, waves of large wave-numbers are more unstable, in agree­ment with observations.

Figure 2 shows the rate of growth for the case of a heavy fluid over a lighter one. The relative velocity of streams (represented by R) destabilizes the disturbance.

Figure 3 shows the rate of growth for the case of a light fluid over a heavier one. The relative velocity of streams again destabilizes the disturbance in such a way that a stable situation can become unstable by increasing the relative velocity of streams.

Figure 4 shows the maximum and asymptotic mini­mum rates of growth for the case E= -1, and the asymptotic maximum for the case E=+1. (Compare Figs. 2 and 3.) E is a unit operator introduced, for the sake of abbreviation, to represent the conditions of densities and accelerations. It is defined as follows:

-1, for I' (g+gl) <0; /

E= (6)

'" +1, for 'Y(g+gl»O.

2.0

Om. 1.5

n~.,",

1.0

"~ (£--1)

.5

o 2 3 4

R

FIG. 4. Unstable flow: Dependence on R of maximum and asymptotic rate of growth (respectively n'm and n' •• ym) based on formula (4). The curve n'm is valid in the case of E = -1 only, and is absent in the case of E= + 1. The curve n' •• ym applies to both cases. In the case of .=+1, n'a.yrn serves as an asymptotic maximum. Compare Fig. 2 and 3.

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1536 CHAN-MOU TCHEN

.7

.6 R'O

. 5

ni 4

.3

o .2 .4 1.4

FIG. 5. Dispersion of decaying waves: The dependence of the imaginary part n', (angular frequency of oscillation) on the wave­number k'. The curves are based on formula (4). n', vanishes at the branching points corresponding to curve B of Fig. 1. The decaying waves are dispersive. Circles are the exact results of numerical computations from the complete characteristic equa­tion (1) using (v> as viscosity and taking R=O.

VIII. DECAY OF DISTURBANCES

While all amplified disturbances are aperiodic, the damped waves are periodic for small k, and aperiodic for large k, as shown already by Fig. 1. The frequency of the periodically damped waves is shown in Fig. 5. The relative velocity of streams tends to destroy the periodicity.

Figure 6 shows the rate of decay (- nr') for the case of a light fluid over a heavier one. The relative velocity of streams destabilizes the disturbances as should be expected. The corresponding unstable case has been shown in Fig. 3.

IX. COMPARISON OF RESULTS

In order to note the difference between the two approximate formulas (2) and (4), a comparison with the exact numerical solutions of the complete charac­teristic equation was made for the case of zero stream velocity. It was found that the approximate formula (4), valid in three ranges of k, is substantially in better agreement with the exact numerical solutions. For intermediate wave numbers, formula (2) has a poor validity in all cases, with both damped and amplified modes. For large wave numbers, formula (2) is poor in determining the rate of decay of rapidly damped modes, and in evaluating the frequencies of damped waves. Therefore only the results of formula (4) instead of formula (2) are plotted. Formula (4) reduces to the known formula of inviscid stability of Helmholtz, if the viscosities vanish. On the other hand, if the stream velocities vanish, formula (4) determines the viscous effect of the Taylor-Stokes stability, as should be expected.3

The effects of streams, as formulated in formula (4) and illustrated in Figs. 1 to 6, can be discussed (for gl>O) in physical terms as follows:

(a) As far as the criterion of stability is concerned (see Fig. 1), the relative velocity of streams has the effect of destabilization. For a given relative velocity of streams, short waves grow faster than long ones .

(b) With respect to the growth of waves, the relative velocity of streams helps to accelerate the growth in the case of a heavy fluid over a light one (see Fig. 2). For sufficiently short waves, the relative velocity of streams changes the stable flow into unstable flow in the case of a light fluid over a heavy one (see Fig. 3). As the wave number increases indefinitely, the rate of growth attains some asymptotic values, determined by the balance between the dissipative effects of viscosity, the stabilizing (fJ> p) or destabilizing (fJ<p) effects of gravity on the one hand and destabilizing effects of rel­ative velocity of streams on the other hand. The relative velocity of streams increases the asymptotic value of the rate of growth (see Fig. 4).

(c) The amplified modes are aperiodic, while the decaying modes are periodic in the range of small wave numbers and aperiodic in the range of large wave numbers (see Fig. 1). The periodic waves are dispersed

1.0

.9

.8

On; .6

.5

4

.3

.2

.1 ~---..

o .2 .4 .6 8 1.0 1.2 1.4 1.6 1.8 2.0 k'

FIG. 6. Stable flow (.=+1): The rate of decay, -n'" as a function of wave-number k'. The curves represent the real part of n' given by formula (4). Each curve consists of three parts: a rising part corresponding to a periodic decay, a part of slow creeping, and finally, a part of rapid and aperiodic decay. Circles are the exact results of numerical computations from the com­plete characteristic equation (1) using (v> as viscosity and taking R=O.

by gravity and viscosity, in such a way that there is a maximum attainable frequency corresponding to each value of the relative velocity of streams (see Fig. 5). The effect of the relative velocity of streams is to decrease the maximum attainable frequency and to shrink the range of wave numbers which excite periodic oscillations. This is understandable from its desta­bilizing effect which converts the periodic decaying wave motion into an unstable aperiodic motion through a stable aperiodic motion (see Fig. 1).

(d) The destabilizing effect of the relative velocity of streams is to shrink the range of wave numbers for which the decay of disturbances persists (Fig. 6).

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