On Kelvin-Helmholtz instability in a rotating fluid

LaIs.

y. Scheepsbouwkundé

Tech&sche Hogeschool

353

Delit

On Kelvin-Helmholtz instabifity in a rotating fluid

By HERBERT E. HUPPERT

Institute of Geophysics and Planetary Physics,

University of California, La Jolla

(Received 16 August, 1967)

Chandrasekhar's (1961) solution to the eigenvalue equation arising from the

KelvinHelmholtz stabifity problem for a rotating fluid is shown to be incorrect. The unstable modes are correctly enumerated with the aid of Cauchy's principle of the argument. Various previously published solutions using Chandrasekliar's analysis are corrected and extended.

Introdüction

Chandrasekhar (1961, § 105) devotes_{a section of his treatise to the} investiga-tion of the effect of rotainvestiga-tion on the development of KelvinHelmholtz_instability. Two uniform, horizontally-superposed fluids of different densities are in relative horizontal motion and are rotating with uniform angular velocity about a vertical axis. Perturbing the solution with a travelling wave in the direction of streaming,

imposing the boundary.conditions at the interface, and linearizing,

Chandrase-khar obtains the eigenvalue equation for the wave speed. He then uses a graphical

method, together with the fundamental theorem of algebra, to enumerate the eigenvalues. His argument necessitates determining the singular points of the eigenvalue equation. We show that Chandrasekhar's determination of these

_/

singular points is incomplete, leading to

_{erroneous results. With the aid of-y}

Cauchy's principle of the argument, we enumerate the eigenvalues and present

a simple, sufficient condition for stability.

Alterman, in a series of papers (1961 a, b, c) dealing with the

_{KelvinHelmholtz}

stability problem under various force fields, uses Chandrasekhar's results and

generalizations of his method, in consequence of which her results are incorrect.

We present the correct results and, in_{one instance, extend our analysis to solve}

the problem for a more general flow configuration than that_{treated by Alterman.}

Chandrasekhar's analysis

Chandrasekhar considers two uniform, superposed fluids of _{densities Pi' P2} having velocities U1, U2 in the x direction and in_{a state of uniform rotation about} the z-axis with an angular velocity £1 [In what follows a subscript 1 (2) refers

to the lower (upper) fluid; Pi is greater than P2.] Imposing upon the steady-state solution a small disturbance whose dependence_{on z and t is given by}

exp [ik(xct)],

where k is the wave number and c the wave speed, linearizing the equations of

23 _FluidMech.33

ARCHIEF

J. Fluid Mech. (1968), vol. 33,part 2, pp. 353-359

354

H.E.Huppert

motion, and applying the conditions of continuity of pressure and normal

velocity at the undisturbed interface, Chandrasekhar obtains the

eigenvalue

equation

- 6)2 [1- 4l2k-2(U - c)] + a2 (U2 -

c)2 [1

- 42k-2(U2 -

C)_2J1

-g1c-'(a-a2)-kT(p1+p2)' = 0,

(2.1)

= pJ(pi+p)

(i = 1,2), (2.2a,b)

where

roots of (2.1) must be taken to have positive real parts since

the product of wave number and the inverse of

these real parts represents the rate of decay

g is the acceleration due to gravity, and T is the surface tension. The square

of the disturbance with increasing distance from the interface.

Applying the transformation

-c = g(g/k)+, 172-c = i(g/k)f, (2.3a,b)

originally due to Taylor (1931), and neglecting surface tension,Chandrasekhar reduces the problem to the simultaneous equations

(2.4)

= a1a2

g -

= V, (2.5) and (2 = 4Q2/(gk), V = (U1- U2) (k/g). (2.6 a, b) where

In order to apply the fundamental theorm of algebra, which is not directly applic-able to (2.4), Chandrasekhar introduces the following set of four equations:

- w2g2)t + cc22(1 - w2_2)l = a1 - a2, (2.7)

w2g-2)1 - a2ij2(1 - w_2)t = a1 - a2, (2.8) g2(1 _w2g-2)1 + a2172(1 _w2ìj_2)t = a1-a2, (2.9)

-a1(1 _w2g2)_a2n2(1

_w_2)l a1 a2.

(2.10)

He then states that, from the fundamental theoremof algebra, equations (2.7) to (2.10) 'have a total of exactly eight roots ...[which] are continuous functions of

the parameters of the equation[s] except at the singular points (± g0, ± w) and

(± w, ± ), ... where and i are determined by the equations

a1g2(1 _w2g-2)t = a1-a2 and a22(1

_w2_2)t = a1-a2'.

(2.11)

Using these two facts, Chandrasekhar determines the number of roots of each of (2.7)-(2.10) as follows.

The equations are solved for the particular case

V = O ( = ).

jV

is then

increased until the first singular point is reached, IVI =

g0-w

V1, say. At this

point the equations may exchnnge roots Using the enunciated theorem and the

graphs of the equations, Chandrasekhar determines thenumber of roots of each

equation at IVI = T7+. This procedure is then continued, Chandrasekhar

con-sidering the exchange of roots at each of the singular points quoted above.

However, (±

,

0) and (0, ± ) are also singular points,

in consequence of

which Chnndrasekhar's results are incorrect.tFurthermore, the inclusion of these

Prof. Chandrasekhar informs me that he is aware of this error and will present an

355 singular points renders the method inapplicable, as it is not possible, by Chan-drasekhar's method, to determine the_{exchange of Ñots between (2.7) and(2.8)}

at (±

,O) and between (2.7) and (2.9) at (O,_{± 21o).}

In the next section

_{we use Cauchy's principle of the argument to yield the}

correct enumeration of the eigenvalues of (2.1).

3 Cauchy's principle of the

_argument

Introducing the transformations

c=(Uj+U2)+V, V=(U1U2),

_(3.la,b)

= oct,

ic = kV2/g, z = 2Q/(kV), = k2T/(p1g), (3.2a,b,c,d)

we write the eigenvalue equation as

E(a) 0T 1 +ox(a+ 1)2[1 z2(a+ 1)_2]+K(8-

_{1)2 [1 z2(a 1)2j1}

_{= 0, (3.3)}

where the square roots are to have positive _{real parts. (From here on weassume}

V, and hence z, to be positive; the results are not altered if V is negative.)

'w

{s}

FIGuRE 1. The contour in the 8-plane to which_{Cauchy's principle of the argument}

is applied. D may be to the right of E and the branch cuts overlap.

To render E(a) analytic, we introduce branch cuts in the 8-plane extending along the real axis from

_{- 1x to 1+ z and from 1x to 1+ x.}

In order to determine the solutions of (3.3) we apply Cauchy's principle of the

argument: for any function f(z), analytic within and on a closed contour ', the number of zeros minus the number of_{poles of ¡(z) within ' is (1/2ir) times the}

increase of argf(z) as z traverses _{once in an auticlockwise direction (see Copson} 1962, § 6.2). We apply this principle to F(s) and take Sf to be the real axis of the

8-plane, indented above the branch cuts, and_{a semi-circle whose radius tends to}

infinity in the upper halfplane. A typical contour is shown in figure 1.

To construct the map of ' onto the F-plane (the OauchyNyquist _diagram),

we define x(o-) as that value of x for which

= o-[2(s+ 1)2x2]{(8+1) [lx2(s+ 1)-2]*}_1

±[2(s l)2x2]{(s_ 1) [1x2(& 1)_2]}1

_{= Ø (34)}

356

H.E.Huppert

has a double root in

i ±x <8 < i-x (or, equivalently, that value

of z for

which F"(s) = O in this range of a). x1(o-), which isdetermined numerically, is

shown in figure 2. For sin the range (-1+x,1-x),

F(s) has one maximum if

o.

FIGm 2. The curves x1(o), x2(o).

z x1(o) and two maxima and oneminimum if z <x1(o). We label the valuesof

s, at the maxima a

and b, E(a)

E(b)

F, and at the minimum m,

F(m)

We also define x2(ci), determinednumerically and shown in figure 2, by

for

x=x2(o).

It can be shown that z

x2(o) implies E F1. In addition, the stipulation that

the square roots of (3.3) havepositive real parts requires that forall z

[1_x2(s+1)_2Il

j[x2(s1)_2_i]lsgn(8+i)

(son BD), (3.5)

[i x2(s 1)_2j = ix2(s

1)2_ 1]sgn(s 1)

(s on EG). (3.6)

The construction of the Cauchy-Nyqiiist diagrams, examples ofwhich are shown in figure 3, is now straightforward.

To enumerate thé complex eigenvalues, we determine the number oftimes the Cauchy-Nyquist diagram encircles the origin [from Cauchy's principle and the absence of poles of F(s) within , this is the number of pairs of complex

conjugate eigenvalues]. The positiòn of the origin in figure 3 isdependent upon the particular values of X, o, K, r. Exploring all possibilities (there are

approxi-mately thirty), we determine the number of complex eigenvalues as shown in table i [F_1

F( - 1) therein]. From the table, we see that: there cannot be

(e)

{F

(f)

FIOuRE 3. The CauchyNyquist diagrams. (a) x <z1, X2. (b) z2 <z <z1. (e) z1 <x< z2.

(d) z1,z2 <z < 1. (e) 1.< z < 2. (f) z> 2.

(a) (b)

(c) _(d)

358

H.E.Huppert

instability for x 2; a sufficiönt condition for stabffity in z < 2 is

TABLE 1. The distribution of complex éigenvalues. N is, the number of pairs of complex

conjugate eigenvalues in the indicated range of n

For max (z1, x2)

z

2 this is also a necessary condition. For z < max (z1,z2), we see from lines (iii)and (iv) of table 1 that there may be a region of stabifity for which (3 7)is not satisfied Returning to physicalvariables, we find that a

suffi-cient condition for stabifity is.

(U1 - U2)2 2Q2k-2 + k-2{44 + k2pj2[o1 - P2) +k2T]2}. (3.8)

A table enumerating the real eigenvalues can

be easily obtained, but is of

excessive length and hence is omitted. It can be simply seen from the Cauchy-Nyquist diagrams, however, that the numberofreal eigenvalues varies between zero and four. We eau also show that the total number of eigenvalues is never

more than four Finally, we

note from the Cauchy-Nyquist diagrams

that if

<o

1?(1 +x) (3 3) has no solution whatever, in which case the original linearized perturbation equation does not admit adiscrete spectrum solution.

Generalizing Chandrasekhar's argumeat to include the effect of surface

ten-sion, Alterman (1961 a) asserts thatthe system is unstable to long waves. From (3.8) we see that the' actual conditionfor stability is

(U1_U2)2<42k-2+O(1)

(k-i-0), (3.9)

and hence there is stability to a long wave length disturbance.

In a later publication, Alterman (1961 b) considers the problem of two hetero-geneous fluids, with horizontal velocities U1, U2inthe same direction, separated by a horizontal interface at z = 0,the densities being given by

p

=p1e_ßZ (z < 0), p = P2e (z> 0). (3.10 a, b) Invoking the Boussinesq approximatiôn, she shows that the eigenvalue equation governing stabffity is' formally equivalent to (2.1) if 2 is replaced by ßg. The correct sufficient condition for stability is hence given by (3.8) once this

replace-By stability here we mean stability to an exponentially growing disturbance; the

as-sumed form of the disturbance riles out any possibility of investigating algebraic instability,

for which an initial value approach is required.

In Chandìasekhar's notation, this becomes <0after setting r = 0.

1-cT+'r ? 4,C[1_(x2/4)]

(z <2).

F

N (i)

[2, )

.. O (ii) (0,2) F_1 <0 0 (iii) (0,x2) Fa O,Fi < O O (iv) (0,x1) Fb>_0,Fm<O O (y) [1,2) F1>O 2 (vi) (x2,1)

Fj>Q,Fa<0

2 (vii) Otherwise i

359

mént has been made. Generalizing Alterman's model such that the fluid densities

are given by

p = p1e-ftic (z < O), _{p = p2eft2t (z> O), (3.11a,b)}

we obtain the eigenvalue equation

- e)2 [1 -fi1gk-2(U1 - c)-2]t + 2(U2 - e)2 [1 -fi2gk-2(U2

--glc'(c--c2)-kT(p1+p2)-1

= 0, (3.12) and the eigenvaJues can be enumerated in the same manner as before. For the sake of brevity it suffices to saythat (3.8)_{is a sufficient condition for stabifity if}

is replaced by (g/4) min (fi1, fi2).

In anotherpaper, Alterman (1961 c) obtains_{the eigenvalue equation pertinent}

to the firnd system ongmally considered by

_{(Jhandrasekhar, with the added} condition that the fluid be a perfect conductor_{under the influence of a uniform,} horizontal magnetic field of intensity H. In the limit as the wave lenSth of the disturbance tends to iuithiity, Alterman obtains the equation (3.12) with

_fig

replaced by 4û2 + (pH21c2/2n'p) (i _{= 1,2). Her sufficient condition for stabifity} should be replaced by

(U1 - 172)2 g2-2 + (pR2/47rp1)_{+ k-2{4[L2 + (uHSkS/87rp1)]2 + k2pj2 {g(p1}

- P2) +k2TJ2}i (3.13)

4. Conclusion

We conclude that an eigenvalue equation of the form (2.1) arises in various Kelvin-Helmholtz stability problems and that applying Cauchy's principle of

the argument is a simple and efficient method of enumerating the eigenvaiues.

The author wishes to thank Dr J. W. Miles for his oft-given advice and the University of Sydney for the award of a University of Sydney Post-Graduate

Travelling Fellowship. This research was sponsored in part by National Science

Foundation grant GP-2414.

REFERENCES

AVrsuÇAN, Z _{1961 a Effect of surface tension on the Kelvin-Hehnholtz instábility of}

two rotating fluids. Proc. NaU Acad. Sci. U.S.47, 224-7.

AL'rFrRMAN, Z. 1961 b Kelvin-Ee]mholtz instábility _{in media of variable density. Phy8.}

Fluids, 4, 1177-9.

Azw, Z. 1961 c Effect of magnetic field_{and rotatiòn on Kelvin-Héhujholtz insta.} bility. Phys.. Fluids, 4, 1207-10.

CHA1WBASEXLuAB, S. 1961 Hydrodynamic and _{Hydromagnetic Stability. Oxford: Claren.}

don Press.

COPSON, E. T. 1962 The Theory of Functions of a Complex Variable. Oxford: Clarendon

Press.

TALOn, G. I. 1931 Effect of variation in density on the stability of superposed streams