DEVELOPMENT OF PERTURBATIONS AT THE INTERFACE BETWEEN TWO LIQUIDS A. Go Kulikovskii and I~ S, Shikina

UDC 532.51.013.4

The asymptotic behavior is studied in the case of large times of initially localized, one-dimensional, small perturbations of the interface between two liquids in the presence of a tangential velocity discontinuity, taking account of surface tension and the force of gravity. The asymptotic behavior of the perturbed region is found; i.e., on the plane x, t a sector is shown with vertex at the origin of the coordinates, inside of which the perturbations tend to infinity with increase of t, and outside of which the perturbations tend to zero, and the velocities of motion of the boundaries of the perturbed region are calculated. The conditions are shown for which the instability of the tangential discontinuity will not be absolute; i.e., when they are fulfilled, flows with a tangential velocity discontinuity can occur. For the case where the effect of the force of gravity can be neglected, these conditions are independent of the magnitude of the surface tension. When investigating the stability of steady flows, which are independent of one of the coordinates (x), an approach is assumed in which an initial localized small perturbation u(x, 0) is assigned~ and its behavior with time is studied. The representation

is used, and the analytic function ~(k) is determined from the dispersion equation. By u(x, t) we shall understand the perturbation of any of the quantities describing the flow. The asymptotic behavior of u(x, t) when t § ~ determines the stability or the instability of the flow. If u(x, t) + = when t § = for any fixed value of x, then the flow is called absolutely unstable. If all exponential perturbations of the form exp(ikx-- imt) with real values of k are damped, i.e., Im ~(k) < 0, then u(x, t) + 0 uniformly with respect to x and the flow is stable. If, for any finite fixed value of x when t § ~, u(x, t) + 0, but real values of k exist for which Ira ~(k) > 0, then the instability is called convective; i.e.~ the perturbation increases and is carried out from any specified region. Criteria are known which permit the nature of the instability to be determined (see, for example, the review [I]). The concepts of absolute and convective instability are related with the choice of the system of coordinates; the problem may be posed concerning the determination of the velocities of the system of coordinates in which the instability is absolute. io Let us explain how the perturbation behaves when x § ~ and t § ~. We shall convert to a system of coordinates moving with velocity W, in which x' = x -- Wt~ t' = t, k' = k, and my = ~ _ Wk. In the new system of coordinates, the behavior of the perturbation for fixed values of x ~ when t § ~ is determined by the asymptotic of the integral

S u~(k)e~'-"~'r

t)

(1.1)

For different values of W, the behavior may be different. if, for W < W < W+ when t § =, u(x ~, t) * ~, and for W > W+, W < W the perturbation u(x', t) + 0, then in the plane x, t a sector can be distinguished inside of which the perMoscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5~ pp. 46-49~ September-October, 1977. Original article submitted February 17, 1977.

0015-4628/77/1205-0679507.50

9 1978 Plenum Publishing Corporation

679

turbation increases, but outside of which it is damped. ( G e n e r a l l y speaking, there may be several of these sectors.) We shall call W+ the velocity of the leading edge, and W the velocity of the trailing edge of the perturbed region. In order to find the asymptotic of the integral (i), we shall use the method of steepest descent [2], according to which when t § ~, u(x', t) ~ t -- ~ exp(t Im ~'). Here Im ~' is Calculated at the saddle point, through which we shall draw the integration contour in Eq. (i.I); ~ > 0 and is determined by the order of the first nonvanishing derivative of the function m'(k) at this point. At the saddle point dm'/dk = O, i.e., Re (d~/dk) =W, Im (d~/dk) = 0. The lines lm(d~/dk) = 0 in the plane k will be called "crests" (although some of them may be "gullies"). If, at a certain point of the crest Im m' = Im m-- W, Im k < O, then for the corresponding value of W we obtain u(x', t) § 0 when t § =, and if Im m' > 0 then u(x', t) + =. Thus, according to the sign of Im m' for values of k lying on crests, the behavior of the perturbation u(x, t) can be determined along any ray in the plane x, t* The following statements are valid in relation to the crests. The lines Im (d~/dk) = 0 in the complex plane k can start and finish at the branching points of the even order of the function f(k) = dm/dk, or they can go out to infinity. The crests may be intersected at points where d2~/dk 2 = 0. The crests may also have points of self-intersection of the pole higher than the first order of the function f(k), but the case of the presence of a singularity of this type in the dispersion equation is not being considered here. If the crest is closed, then there is necessarily a singular point of the function f(k) inside it. In fact, let us suppose that the branching point k* of the function f(k) lies on a crest. In the vicinity of it, f(k) -- f(k*) = %(k -- k*) I/n, where n is a whole number. With even values of n on the crest in the vicinity of the point k*, the equation arg (k -- k*) = 2~p -- n arg % holds, where p = 0, i, 2, ... and, consequently, the crest emerges from the point k* in one direction. With odd values of n on the crest in the vicinity of the branching point, arg (k -- k*) = ~p -- n arg % and, consequently, the crest leaves in two directions, having a common tangent. Thus, the point k* is the final point of the crest if n is an even number. Suppose that at the vector point ko the crest f'(ko) = 0. Then along the crest in the vicinity of kon!, Im f(k) = I m [fln)(ko)(k-- ko) n] = 0 (n > 2). Since during rotation of Ak = k -- ko by the angle ~/n in the complex plane k, (Ak)n changes sign, then at the point ko we have n intersecting crests. The presence of a singular point inside the closed line Im (d~/dk) = 0 follows from the analyticity of the function d~/dk # const [2]. 2. We shall use the reasoning given in order to explain how an initial localized perturbation of the surface of separation between two ideal incompressible liquids develops with time. Suppose that in the unperturbed state the upper liquid is moving with constant velocity vl and the lower liquid is moving with velocity v2, 01 and 02 are the densities of the upper and lower liquids, respectively, and ~ is the coefficient of surface tension. The initial perturbation of the surface ~(x, 0) and its velocity ~t(x, 0) will be assumed to be small. In order to determine the velocity potentials of the perturbed motions of both liquids and the equations of the perturbed surface of separation ~(x, t), we shall use the Laplace equation and the linearized boundary conditions. The dispersion equation for K with Re K > 0 has the form [3, 4]

(2.1) f~ =

(oe

Iv , - v ~ l ~ ( p , + p ~ )

K --

'

k(~

G =

Fv,-v~i~(p,+o~l '

pi ,+p2'

g(~

Iv,-v~l'(p,+p,)

pi-t-p~.

In view of t h e c o n d i t l o n of damping of the perturbations of the velocity potentials with z = • for K with Re K < 0, K must be replaced by --K in the dispersion equation (2); then the analytic function g(K) will be determined in the whole complex plane K with a branch cut along the imaginary axis.

680

if s = G(8= -- 8~)(818a) -= > ~/~, then ~ is real for all real values of K and the flow is stable. If e < ~ / ~ then fo~ real values of K there exist values of ~ with Im ~ > 0, the exponential perturbations increase and a Kelvin--Helmholtz instability occurs [3, 4]. Let us investigate how the initial perturbation is developed in this case. First of all, we note one general property of the system of equations for small perturbations. If u(x, z, t) = u(z) exp (ikx -- i~t) is the solution of this system, then there is always also a complex-conjugate solution ~(x, z, t) = u(z) exp (--ikx + imt). This means that for values of K, located symmetrically relative to the imaginary axis K, on the complex plane ~ the roots of the dispersion equation also are symmetrical relative to the axis Im ~. Therefore, in the plane K, the crests Im(d~/dK) = (d Im ~/d Re K) = 0 lie symmetrically relative to the axis Im K, and the values Re

d~ d Re ~ ....

dK

dReK

W

Im ~ ' = I m Q --

Iv~--v~] '

W

Iv~-v~]

Im K

for K with Re K < 0 coincide with their values for K with Re K > 0. This confirmation follows also directly from the dispersion equation (2) and the remark about the cut of the function ~(K). Taking into account the symmetry of the crests, all further considerations ried through adequately in the right-hand half-plane of Re K > O.

can be car-

We designate Ui

U~

Let us first consider the case when the effect of the gravity field cannot be taken into account, when ~i = K/K-- B182 and d~i/dK = (3/2K-- BIB2)(K-- B:82) -~/2. We construct the lines Im (d~i/dk) = 0 (crests). For real values of K the half-interval [8182, =) is a crest. For complex values of K, a circle with center at the point of branching K = ~i~2 and radius r = I / ~ I B a serves as the crest, as on it arg (K-- 2 / 3 8 1 ~ 2 ) = 1/2 arg ( K - ~i~2). Both crests intersect at the point Ko = ~/3B~82, and at this point d 2 ~ / d K 2 = 0. There are no other crests. Let us examine how Re (d~/dk) =U andIm ~,' = Im ~ -- U Im K vary along the crests. In [~2, =)IuI there is a minimum at the point Ko. During motion from the point Ko around the circle U, it varies monotonically, and in the contrary case there should be one further point of intersection of the crests on the circle, which is impossible because of the absence of other crests. On the upper semicircle, U > 0 and on the lower semicircle, U < 0. On the crest [8~:, = ] I m ~ ' = 0. Since, along any crest d~: '/dK =--K dU/dK, then on the circle 0n both semicircles IUI decreases from U = ~381B2 at the point Ko also d Im ~!~ ' = --Im KdU to zero at the point K = 2/a~,82; Consequently, Im ~a' increases from Im ~ ' = 0 at the point Ko to Im ~,~ = 2(~/38182) a/= > 0 for K = ~/aB~8~. (Along the circle, lying in the ! second sheet of the complex plane K, Im ~: < 0.) Thus, the inequality Im ~ ' > 0 can be satisfied only when IUI < ~ and when I U ~ 3r Im ~,' = 0. In the plane x, t inside the sector bounded by the rays W+ = B,v~ + 8~v~ + 3 ~ , 8 = I v ~ v= I and W = ~ v ~ + 8=v~ -- 38/~-~82[v~ -- v~ I, the perturbations of the velocity potentials and the perturbation of the surface of separation ~(x -- Wt, t) tend to ~ for t § ~, but outside the sector they tend to zero. Consequently, with large values of t the leading edge of the perturbed region is moving with a velocity W+ and the trailing edge with a velocity W_. If I ~ v ~ + ~av~| < 3~/~]~B=Iv~ - v, I, then this sector envelops the axis t and the flow being considered is absolutely unstable. If l~,v2 + 8~v=I > 38~,8=Iv, -- v=i, then the instability is convective and the growing perturbation is communicated to the liquid. We note that as the velocities of the boundaries of the perturbed region W+ and W_ are independent of the coefficient of surface tension o, then a convective instability is possible also for ~ + 0. 3. In the discussion developed, the effect of the force of gravity has been neglected completely. This is possible if ~s I << i. Taking account of the force of gravity

3

2

2

2

681

As the exponential perturbations increase when e < ~/k, then d~,/dK has, in addition to K = 0, two real points of branching (K+ > 0 and K_ < K+). If Pl < p2, then ~ > 0 and K_ > 0. The crests lying on the real axis will be ( O K ] and [K+, ~). When e < 0 in the half-plane Re K > 0, there will be only one crest on the real axis [K+, ~) as K < 0. On (OK ], d2~,/dK 2 # 0. On [K+, ~) a point Ko > K+ exists, at which d 2 ~ / d K 2 = 0~ and therefor~ there is also a closed crest enveloping the point K+. It intersects the real axis at the points K s and Ko, where K s > 0 is the major root of the equation d~I/dK = 0 and Ks < K+. Carrying on the same reasonings as in Sec. 2 for the change of U and Im ~I' along a closed crest, we show that the inequality Im (~I -- KU) > 0 is satisfied only for IUl < U(Ko) and Im (QI -- KU) = 0 for IUI Z U(Ko). Consequently, when t + ~, the perturbations of the surface of separation and perturbations of the velocity potentials increase infinitely for W t < x < W+t where W+ = 8,vi + 82v2 + u(no) Iv, - v~I, and W = B1v: + B=v= -- U(Ko)Ivl ----v21. If |el is small, then in linear approximation we obtain U(Ko) = 3B~182(I-- "/ke). 4. Similarly, we can consider the limiting case where v, = v2 = v and p, > p~, and we can investigate how the initial perturbation develops in the case of Rayleigh-Taylor instability. It is found that this instability will be absolute if v ~ < Aog(pl -- p2)(p: + p2) -= and A = 3~3(i + ~ ) ~ / 4 ( 2 + ~ ) = 2.1 ~. In the contrary case the instability is convective. Thus, the velocities of propagation W+ and W have been found for the boundaries of the region in which an increase of the initial loc~lized perturbations occurs. If the velocitiesW+and W_ have identical signs, then the instability of the discontinuity will be convective, the initial perturbations in the course of time will leave any part of the discontinuity considered, and in the absence of newly arriving perturbations it will exist. It is possible that the achievement of the tangential discontinuities observed in actual flows is explained by this. LITERATURE CITED i. 2. 3. 4.

682

A . I. Akhiezer and R. V. Polovin, "Criteria for the increase of waves," Usp. Fiz. Nauk, !O4, No. 2 (1971). M . A . Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1965). L . D . Landau and E. M. Lifshits, Mechanics of Continuous Media [in Russian], Gostekhizdat, Moscow (1954). S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford

(1961).