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The linear and nonlinear shear instability of a fluid sheet

R. H. Rangel and W. A. Sirignano

Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 3, 2392 (1991); doi: 10.1063/1.858177

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8/12/2019 Rangel 1991


The linear and nonlinear shear instability of a fluid sheet


Department

of

Mechanical and Aerospace Engineering, University of CaEyornia, Irvine, Caltyornia 927 17

(Received 22 May 1990; accepted 18 June 1991)

A theoretical and computational investigation of the inviscid Kelvin-Helmholtz instability of a

two-dimensional fluid sheet is presented. Both linear and nonlinear analyses are performed.

The study considers the temporal dilational (symmetric) and sinuous (antisymmetric)

instability of a sheet of finite thickness, including the effect of surface tension and the density

difference between the fluid in the sheet and the surrounding fluid. Previous linear-theory

results are extended to include the complete range of density ratios and thickness-to-

wavelength ratios. It is shown that all sinuous waves are stable when the dimensionless sheet

thickness is less than a critical value that depends on the density ratio. At low density ratios,

the growth rate of the sinuous waves is larger than that of the dilational waves, in agreement

with previous results. At higher density ratios, it is shown that the dilational waves have a

higher growth rate. The nonlinear calculations indicate the existence of sinuous oscillating

modes when the density ratio is of the order of 1. Sinuous modes may result in ligaments

interspaced by half of a wavelength. Dilational modes grow monotonically and may result in

ligaments interspaced by one wavelength.

I. lNTt?ODUCTlON

II. LINEAR ANALYSIS

The distortion and breakup of a sheet of fluid under the

action of a shearing flow is of interest in a number of prob-

lems involving atomization of a fluid and the production of a

spray. In a practical atomization system, a number of differ-

ent processes contribute to the distortion and breakup.

These processes include the shearing effect of the surround-

ing medium (particularly in air-blast systems), nozzle ef-

fects, turbulence, and a few others. Of these, the shearing or

Kelvin-Helmholtz instability is recognized as one of the

main driving mechanisms. Linear analyses of the Kelvin-

Helmholtz instability on the surface of a finite-thickness liq-

uid sheet were performed by Squire,* Hagerty and Shea,2

and Taylor.3 Their analytical approach was mostly based on

the work of Lamb4 with the addition of the surface tension

effect. All of these analyses were limited to the distortion of a

liquid sheet in air, thus the ratio of the surrounding fluid

density to the fluid sheet density was very small. Dom-

browski and Hooper’ considered the effect of increasing the

air density but only by a factor of 10 or less. Range1 and

Sirignano6 considered the Kelvin-Helmholtz instability in-

cluding surface-tension effects and the complete range of

density ratios. Their nonlinear calculations were performed

using a vortex discretization method and showed the exis-

tence of a bifurcation phenomenon whereby a vorticity accu-

mulation characteristic is shifted from one region to two re-

gions as the dimensionless wave number is increased. Recent

experiments with liquid sheets have been performed by

Mansour and Chigier.7 In the following sections, the linear

theory of distortion of a finite-thickness sheet is revised to

include the complete range of density ratios. We limit our

attention to the sinuous waves (also referred to as antisym-

metric) and the dilational waves (also referred to as sym-

metric or varicose). We proceed to perform nonlinear calcu-

lations based on vortex dynamics to explore the later-time

evolution of these waves.

We consider the surface-tension-affected Kelvin-Helm-

holtz instability of a finite-thickness sheet of an inviscid fluid

in contact with two semi-infinite streams of a different fluid.

Figure 1 illustrates the configuration investigated. In both

cases, the undisturbed thickness of the sheet is 2d and the

velocity difference across each interface is AU. We consider

temporal instabilities so that periodic boundary conditions

are used at x = 0 and x = R. In our frame of reference and in

I? v-1

I

I

,

(a)

IIY

I

fb)

FIG. 1,Schematicof a sinuous ( antisymmetric) disturbance (a) and a dila-

tional (symmetric) disturbance (b).

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the absence of a disturbance, the sheet moves with velocity

-

AU /2 while the semi-infinite streams move with velocity

AU /2. Two types of instability are considered here, as illus-

trated i n Fig. 1: sinuous waves (a) and dilational waves (b).

These are defined below.

Defining a velocity potential CJ~uch that u =

- Vq4, he

linearized equations for the inviscid flow inside and outside

the sheet are’*’

v24, = 0, v’q5, = 0, (1)

where the subscript 1 refers to the fluid above the sheet and

the subscript 2 refers to the fluid within the sheet. The fluid

below the sheet need not be treated explicitly if dilational or

sinuous conditions are used. For a sinuous wave, 2 = 0 at

y = 0 whereas, for a dilational wave, c@,/c$ = 0 at y = 0.

The linearized kinematic and dynamic conditions at the in-

terface are

84,

a7 1 a7

--=ar+yx,

ay

84, a7j

ay =liC

1 a77-- ---9

2

ax

(2)

(3)

(1 +p1

(

4,

F+fs

>

- (1 +p-7

X

(

a4,

1 a4,

>

w a27j

---~

at

2

ax ~~dx2’

(4)

where

w=2E l+l

b )

(AU)’ , ~2

(5)

is the dimensionless wave number, (T s the surface-tension

coefficient, /z is the disturbance wavelength, AUis the veloc-

ity difference across the interface,p, andp, are the densities

of the fluids, andp = p, /p2 is the density ratio. The dimen-

sionless variables are the time t, the parallel and normal co-

ordinates x and y, the velocity potential 4,and the location of

the interface 7. The characteristic length and time used in

the nondimensional ization are ;I and /z /AU, respectively.

The previous equations are solved by means of modal analy-

sis for the cases of sinuous and dilational waves. In both

cases, the interface initially located at y = d is perturbed ac-

cording to the relation 17, = exp[ i(wt + 27rx) 1, where w is

the dimensionless complex frequency. For the sinuous waves

(also referred to as antisymmetric ’ waves), the interface

initially located at

y= -d behaves as

vZ (x,t) =

- 7, (x + &t), whereas for the dilational or sym-

metric’

waves, the interface at y = -d behaves as

q2 (x,t) = - r], (x,t). In addition, the boundary condi tions

away from the interface require that the disturbance vanish-

esasy-+03.

A. Sinuous waves

The solution of Eqs. ( 1) that satisfies the kinematic

boundary conditions and the conditions far from the sheet

yields the velocity potentials

4,

and

42.

The dimensionless

complex frequency o is obtained from the dynamic condi-

tion, Eq. (4), as follows:

w= tanh@n-h) -P v+2T

tanh(2?rh) +p -

y{W [tanh(2nh) +p] - (1 +p) tanh(2nh) ]1’2

, .

tanh(2nh) +p

(6)

Exponentially growing waves occur when the second

term of this equation becomes imaginary. Squire’ and Ha-

gerty and Shea2 limited their analysis of Eq. (6) to the case

of small density ratios (p < 1) as corresponds to a water sheet

in air at normal conditions. Squire further limited his analy-

sis

to long

waves so

that the approximation

tanh( 2rh) = 2n-h could be used.

The dimensionless thickness h = d//z can be written as

the product of H and W, h = HW, in order to unmask the

presence of the wave number in h. The new dimensionless

thickness His given by

H=

[d(AW2/2rul

[p,pz/(p, +p,)]. (7)

Also, H may be interpreted as a Weber number based on a

characteristic densityp, defined as l/p0 = l/p, + l/p,. In

the general case of arbitrary density ratio, unstable waves

occur when the right-hand side of Eq. (6) becomes imagi-

nary. This condition occurs when the dimensionless wave

number is below a critical value ( W< WC . The critical di-

mensionless wave number is given by

WC = (1 +p)/[l +pcoth(2?rHW,)].

(8)

Equation (8) defi nes a critical dimensionl ess wave

number below which unstable waves exist. In the limit of a

very thick sheet [for practical purposes h > 0.43, since it

gives coth( 2n-h = 1 Ol ] , unstable waves occur for W < 1.

On the other hand, as the thickness decreases, the critical

wave number decreasesbelow 1. Equation ( 8) also indicates

that there is a finite thi ckness H = H, below which all sin-

uous waves are stable. This critical thickness corresponds to

the value of H that makes WC = 0 in Eq. (8). For H < H,,

the critical wave number WC becomes negative. To deter-

mine H,, we note that, for H in a finite nei ghborhood of H,,

we must

have

HW-0,

and therefore

coth( 2?rHW) -+ 1/(27rHw) in Eq. (8). After simplifying,

one obtains

H, =p/[2~(1 +p)l.

(9)

An alternate way of determining H, is to solve Eq. (8)

for H and use L’Hopital’s rule to find the limit as WC+O.

The existence of this critical thickness was not observed in

previous works’s2 because it approaches zero in the limit of

small density ratios. Some important applications exist, in-

cluding liquid atomization in rocket engines, in which the

density ratio is not small and the critical thickness may be-

come important.

The nondimensionalization employs/z as the character-

istic length. This is an appropriate choice because it is the

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only physical length appearing in the infinitely thick case

rate, the dependence of the physical growth rate on the wave

and in the finite-thickness case, which introduces a second number is not clearly displayed. The dimensional growth

characteristic length (the sheet thickness), the former case

ratep for the sinuous waves is given by the imaginary part of

can be easily recovered. There is one drawback, however, in the complex frequency, Eq. (6), so that fi = Im{w)AU//Z,

that, by using/z in the nondimensional ization of the growth

or

“2P~ (AU)3 {W2(1 +p) tanh(2rrHW) - W3 [tanh(2rrHW) +P]}“~

tanh(2n-HW) -t-p

(10)

The result of Hagerty and Shea2 is obtained in the limit ofp

41:

P=P”~[P, (AU)‘/a] [ ( W2 - W”)/tanh(2rrHW)] “2,

(11)

while the infinitely thick behavior is obtained in the limit of

H+co:

P= [~“~/(l fp121 [p, (AU)3/tr] ( W2 - W3)“‘2. (12)

The behavior of the dimensional growth is discussed in

Sec. V.

B. Dilational waves

The solution of Eqs. ( 1) that satisfies the kinematic

boundary conditions and the conditions away from the sheet

for the dilational waves yield the velocity potentials, while

the expression for the dimensionless frequency is obtained,

again, from the dynamic condition and is

I

I

m= coth(2gh) -P n+2T

coth(2’rh) fp -

X{W[coth(2rh) +p] - (1 fp) coth(2’rh)}“’

coth(2vh) +p

(13)

Unstable waves occur for W< WC, where

WC = (1 fp)/[l +ptanh(2?rHW,)] .

Again, the critical dimensionless wave number ap-

proaches 1 as the half-thickness-to-wavelength ratio in-

creases above 0.43 for practical purposes. In contrast with

the sinuous waves, the critical wave number of the dilational

waves is greater than zero for any finite H and remains

greater than zero in the limit of an infinitely thin sheet,

WC + ( 1 + p) as H- 0. The dimensional growth rate for the

dilational waves is given by

“2 PI (AU)3Cw2(l +p) coth(2rrHW) - W3 [coth(2rHW) +P]}“~

0

coth(2IrHW) +p

(15)

As before, the result of Hagerty and Shea2 s obtained in the

limit of p < 1:

fl=p”‘[p, (AU)3/a] [ ( W2 - W3)/coth (2?rHW)]““,

(16)

while the infinitely thick behavior is the same as that of the

sinuous wave.

C. Growth-rate ratio

The ratio of the growth rate for a sinuous wave to that of

a dilational wave G, is given by

G, = coth (21rHw)

x

(

1 +p- W[i fpcoth(2rHW)l

“2

1 +p- W[l +ptanh(2rHW)]

>

x 1 fp tanh(2rHW)

1 +p coth(Zn-HW) *

(17)

In the limit of small density ratio p + 0, Eq. ( 17 ) yields

the result of Hagerty and Shea:2

G, = coth(271-HW),

(18)

which shows that, i n this limit, the growth rate of the sinuous

wave is always larger than the growth rate of the correspond-

ing dilational wave. On the other hand, in the limit of a very

2394 Phys. Fluids A, Vol. 3, No. 10, October 1991

thick sheet, the growth rates become identical, G, -+ 1 as

H-t CO, egardless of the density ratio. It can also be shown

that, for p = 1, G,(l, indicating that, in this limit, the

growth rate of the dilational wave is always larger than the

growth rate of the correspondi ng sinuous

wave.

A minor

discussion of the scaling parameters should be made here.

Our choice of characteristic density p. clearly demons trates

thep “* dependency of the growth rate. If one had usedp, as

the characteristic density, the critical thickness for the sin-

uous case would be H, = 1/27r, while the infinitely thick

limit would yield WC = p/( 1 + p) for both symmetries,

and, for the dilational case, WC +p as h --t 0.

III. NONLINEA R ANALYSIS

The linear results are valid only during the initial evolu-

tion of the disturbance. For a growing disturbance, the non-

linear effects become dominant and distortion of the inter-

face dominates over exponential growth. The basis of our

nonlinear analysis is the fact that, in the two-dimensional ,

inviscid flow under consideration, the only nonzero compo-

nent of vorticity w, is convected by the flow field according

to Dw,/Dt = 0. The solution for the case in which vorticity

is confined to an infinitesimally thin sheet of arbitrary shape


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(Batchelor’ ) yields the velocity components at any point

CGYV) s

u&y) = -$

f

yqi

y(s)ds,

r

u(x,y) =$

x

r2

y(s)&

(20)

where y(s) is the circulation per unit length (strength) of

the

vortex sheet at the

point

W,Y’),

r ’ = (x - x’ )’ + (y - y’)‘, and s is a coordinate running

along the vortex sheet. The integration is along the interface

from - co to + co. As we are interested in disturbances

that are periodic in the horizontal direction, we need only

consider the integration along one wavelength of the distur-

bance. Using complex notation with z = x + iy, Eqs. (19)

and (20) are replaced by (van de Vooren’ )

u(z) -

iv(z) =$ y(s) cot[n-(z-z’)]ds,

‘S

(21)

where the integration is over one period of the disturbance

(one unit of length). Separating Eq. (21) into its real and

imaginary parts results in

1 ’

a= --

I

y sinh 2~(y - y’)

2

o cash 27r(y - y’) - cos 27r(x - x’)

ds,

(22)

y sin 2~-(x - x’)

cash 2n(y -y’) - cos 2n(x - x’)

ds. (23)

In the absence of a density discontinuity across the in-

terface or in the absence of surface tension, the circulation

associated with each segment of the vortex sheet remains

invariant. An equation governing the evolution of the circu-

lation when there is a density discontinuity and surface ten-

sion can be derived using the Kelvin theorem, the momen-

tum equations for each fluid, and the interface force balance

(Range1 and Sirignano6 ) :

*=

dt

(24)

where AT = yA.s s the circulation associated with a segment

of the interface of length As, u is the velocity of the vortex

element, and K is the curvature of the interface. The Atwood

number A = (p2 -p, )/(p, + p, ) is related to the density

ratio through the expression A = (1 -p)/( 1 +p), while

the Weber number We = (p, + p2 )R ( AlJ)‘/a is related to

the dimensionless wave number and the density ratio

through the expression We = 27r( 1 + p)‘/(p2 w). The fac-

torA missing in the second term of the right-hand side of Eq.

( 13) in Ref. 6 is a typographical error.

IV. NUMERICAL SOLUTION

Following Range1 and Sirignano,6 we employ the vor-

tex-discretization approach, whereby the continuous vortex

sheet representing the interface i s discretized into a finite

number of vortices n, replacing the integrals in Eqs. (22)

and (23) with the summations

u= 2

2

Arj

sinh 2r(y - y,)

2

j=1cosh2rr(y-yyi) -cos2rr(x-xxi) ’

(25)

‘2

j=-

AJYj

in 27r(x - xi)

2

j=l cosh2n(y-yj) -coS2T(X-Xj) *

(26)

Van de Vooren9 presents a discussion of the vortex-dis-

cretization method. In Eqs. (25) and (26), the summations

are taken over all the vortices in both interfaces.

Equations (25) and (26) provide the velocity compo-

nents in a reference frame moving with the interface velocity.

This is defined as the average of the velocities above and

below the interface. The location of the interface is found by

solving the vectorial equation

where u = ui + uj is the vortex-induced velocity vector

whose components are given by Eqs. (25) and (26) and

up =

(AU/2)i

is a uniform potential velocity field added in

order to have a reference frame movi ng with the average

velocity of the semi-infinite stream and the sheet, as indicat-

ed in Sec. II.

The evolution of an initially sinuous disturbance is de-

termined by integrating Eq. (27) with the aid of Eqs. (25)

and (26) for the velocity field. The tangential acceleration

appearing in the first term of the right-hand side of Eq. (24)

is obtained by differentiating Eqs. (25) and (26) with re-

spect to time and iterating at each time step until conver-

gence is achieved. To eliminate the problem of accumulation

and separation of discrete vortex elements, the interface is

rediscretized after each integration step by means of linear

interpolation, and a new vortex is introduced whenever the

vortex separation i ncreases by more than 2.5% of the initial

separation. After rediscretization of the interface, the indi-

vidual vortex strengths are recalculated by interpolating on

the functional variation of the vortex strength versus the

interface coordinates.6 Equation (27) needs to be solved for

the elements of one interface only since the dilational or sin-

uous conditions are used to determine the location of the

second interface. All the nonlinear calculations presented

are started with 40 vortex elements per wavelength. Differ-

ences n the amplitude of less than 2% are observed when 50

vortex elements per wavelength are used at t = 0. Other de-

tails of the numerical procedure are given elsewhere.6

V. RESULTS

A. Linear theory

The critical wave number WCas a function of the dimen-

sionless thickness H for several values of the density ratiop is

plotted in Fig. 2. The solid lines correspond to the sinuous

waves and the broken lines correspond to the dilational

waves. The values of H at the intersection of the solid lines

with the WC = 0 axis represent the critical thickness for the

various density ratios. For large H, all curves approach

WC = 1 asymptotically. For a given density ratio and thick-

ness, the critical dimensionless wave number is lower for a

sinuous wave than it is for a dilational wave. The implication

is that the dilational wave becomes unstable at a shorter

wavelength.

2395


R. H. Rangel and W. A. Sirignano 2395

s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

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H

FIG. 2. The critical wave number for linear instabiiity as a function of the

FIG. 4. The dimensionless rowth rateD’ asa function of wavenumber W

dimensionless hickness for sinuous waves (solid lines) and dilational

for p = 0.25 and for different dimensionless hicknesses.Sinuous waves

waves broken ines), for various density ratios.

(solid lines) and dilational wav es broken lines).

Figures 3-5 show the dimensionless growth rate

P’ = [o/p, (iW3]8

(28)

for the sinuous waves (solid lines) and for the dilational

waves (broken lines) as a function of the dimensionless wave

number W with the dimensionless thickness H as a param-

eter for density ratios of 0.01 (Fig. 3), 0.25 (Fig. 4), and 1

(Fig. 5). The first of these corresponds to the same qualita-

tive situation investigated by Squire’ and by Hagerty and

Shea’ of a very l ow density ratio. In this case, the sinuous

waves exhibit larger growth rates than the dilational waves

except for W just below its critical value for the sinuous

FIG. 3. The dimensionless rowth ratep’ asa function ofwave number W

FIG. 5. The dimensionless rowth rate@’ asa function of wavenumber W

for p = 0.01 and for different dimensionless hicknesses.Sinuous waves

(solid Iines) and dilational waves broken lines).

forp = 1 and for different dimensionlesshicknesses. inuouswaves solid

lines) and dilational waves broken lines).

W

wave. For Wabove WC,or for H below H, ( = 0.001 58)) the

sinuous wave is stable and therefore has zero growth rate,

while the dilational wave still exists. At a density ratio of 0.25

(Fig. 4), the difference in growth rates for dilational and

sinuous waves is not so pronounced unless the dimensionless

thickness is just above the critical value (H, = 0.03 18)) in

which case the sinuous wave is barely unstable. Note that,

for H = 0.5, the growth rates are almost identical for

0.75 < WC 1, indicating the infinitely thick limit. The situa-

tion is completely reversed for the case of equal densities

(Fig. 5), as the dilational waves exhibit a larger growth rate

for any value of W. As noted earlier, the dilational waves are

unstable for a larger range of dimensionless wave numbers.

For half-thicknesses above 0.5, the two growth rates become

very similar as the two interfaces start to act independentl y

and the dilational or sinuous character is lost.

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B. Nonlinear calculations

The nonlinear results are presented for the same three

characteristic density ratios used in the linear calculations,

namely p = 0.01, 0.25, and 1. For each density ratio, we

investigated the behavior of the sheet for a range of thick-

nesses going from a very thin sheet to the infinitely thick

case. The infinitely thick case was reported in our previous

work.6 In all cases, the ratio of the initial amplitude to the

wavelength is chosen equal to 0.025 to ensure the proper

linear behavior for small time. Figure 6 shows the evolution

of the sinuous mode when the density ratio is 0.01, the di-

mensionless wave number Wis 0.5, and the half-thickness to

wavelength ratio h is 0.05 (H = 0.1). This figure and all of

its kind show the dimensionless time at the upper left corner

of each frame. Also shown is the number of discrete vortex

elements n used to define the i nterface. This case approxi-

mately corresponds to the maximum growth rate for this

value of H, as can be seen n Fig. 3. Figure 6 indicates that the

sinuous character remains for a substantial period of time.

This result agrees with the fact that, for small density ratios,

vorticity concentrations occur near the crests of each inter-

face; a result that can be predicted by linear theory and that

was explored in our infinitely thick calculati ons.6 This figure

also indicates that the sheet breaks up into ligaments at each

half-wavelength. The behavior of the sheet at other thi ck-

ness-to-wavel ength ratios is qualitatively similar to that

,- .60-42

lnrl

f - 3.20 l¶=S.?

t=

.90 n=66

I

I = 2.40 IL=45

FIG. 6. Time evolution of the sinuous mode for p = 0.01, W= 0.5, and

h=0.05(H=O.l).

shown in Fig. 6. The important difference occurs in the rate

of growth of the disturbance. At small density ratios such as

this, the growth of the Kelvin-Helmholtz instability is very

slow because of the reduced inertia of the surroundi ng fluid.

Figure 7 summarizes the results for a density ratio of

0.01 in terms of the amplitude of the disturbance as a func-

tion of time. In the nonlinear calcul ations, the dimension less

amplitude is defined by (y,,, - ymin / ( 2~)) where y,,, and

Y

m,n are the maximum and minimum displacements of one

interface and E s the initial amplitude [E = (y,,, - Ymin /2

at t = 01. Figure 7 also shows the corresponding amplitude

as predicted by the linear theory for three different values of

h. In each case, the dimensionless wave number for maxi-

mum growth predicted by the linear theory is used. In all

cases, he nonlinear growth rate is less han the linear predic-

tion, as expected.

The next set of results is for the case of a density ratio of

0.25. As Fig. 4 indicates, the growth rate of the dilational

waves at this density ratio is comparabl e to that of the sin-

uous waves. In fact, Fig. 4 indicates that, for a value of

H = 0.04, the dilational wave grows faster than the shifted-

dilational one for Wgreater than about 0.2. For a value of the

dimensionless wave number W= 0.5 and a dimensionless

half-thickness h = 0.05 (H = 0.1)) Fig. 4 indicates that the

sinuous wave grows faster than the dilational one. The non-

linear behavior for these two cases s illustrated in Figs. 8 and

9 for the dilational and sinuous waves, respectively. The dila-

tional wave behavior is interesting because t is similar to the

axidilational behavior of the round jet.” As seen in Fig. 8,

the planar sheet under a dilational disturbance woul d pro-

duce ligaments interspaced at one wavelength.

The behavior of the faster-growing sinuous mode has

some interesting features illustrated in Fig. 9. As seen n this

figure, the sheet undergoes an oscillatory distortion during

which the amplitude grows and decays without evidence of

sheet breakup. The onset of this phenomenon had been ob-

served in our previous work for the case of fluids of the same

0

I I I 1

0

DimensionlessTime

FIG. 7. Dimensionlessdisturbanceamplitude for sinuouswavesas a func-

tion of time for p = 0.01 and various dimensionless hicknesses.Nonlinear

calculations (solid lines) and linear predictions (broken lines).

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2

r-o.4

It=41

FIG. 8. Time evolution of the dilational mode or p = 0.25, IV= 0.5, and

h=0.05 (H=O.l).

FIG. 9. Time evolution of the sinuousmode or p = 0.25, W= 0.5, and

L=0.05

(N=O.l).

density and an infinitely thick sheet. For a fixed physical

situation, the surface tension is fixed and the system will

select the fastest growing modes. Several waveleng ths would

be present as prescribed by the solution of the di spersion

equations. There are longer waves with smaller grow th rates

that are characterized by dimensionless wave numbers W

smaller than the optimum value. Thes e onger wa ves (lower

W) are less affected by surface tension than the fastest grow-

ing wave. This behavior was studied in Ref. 6. The oscilla-

tory behavior is cause d by the relatively stronger surface

tension typically existing in the faster-growing modes.6

Note that the period of the distortion is not constant but

decreasesduring the second cycle. The last frame of Fig. 9

hints at the possibility of breakup occurring at each half-

wavelength, thus resulting in ligaments of approximately

half of the mass of the ligaments formed in the dilational

mode. Note, however, that the larger ligaments of the dila-

tional case are formed in one-sixth or less of the time re-

quired to form a ligament with a sinuous wave, thus the

former may undergo secondary breakup sooner. Figure 10

shows the disturbance amplitude for these two cases ogeth-

er with the correspond ing linear prediction. The oscillatory

behavior of the sinuous disturbance is apparent in this figure.

Figures 11 and 12 llustrate the effect of decreasing the

dimensionless wave number W to 0.05 while maintaining the

half-thickness-to-wavelength ratio h at 0.05. This results in a

dimensionle ss half-thickness Hof 1. Physically, this could be

achieved by red ucing the surface tension by a factor of 10 or

by increasing both the wavelength and the sheet hickness by

a factor of 10, while keepin g other quantities as in the pre-

vious case. Figure 11 shows the development of the dila-

tional wa ve which is qualitatively similar to that of Fig. 8.

The main difference is that the disturbance grows faster in

dimensionle ss time units. Note, however, that, if the wave-

10

-2

a

”

5

8

5

-E

I

;;

.-

cl

0

Dimensionless ime

FIG. IO. Dimensionless isturbanceamplitude as a function of time for

p = 0.25, W = 0.5, and h = 0.05 (H = 0.1 . Nonlinear calculations solid

lines) and inear predictions broken ines).

2398

Phys. Fluids A, Vol. 3, No. IO, October 1991

R. H. Rangel and W . A. Sirignano

2398

s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 205 208 105 215 On: Tue 24 Jun 2014 23:05:25

8/12/2019 Rangel 1991


0.5 I

t-0

It=41

t-6

-45

L--4-4

FIG. 1 . Time evolutionof the dilational mode orp = 0.25, W= 0.05,and

h=0.05(H=O.l).

p0.75 n=52

I

1-I n-65

FIG. 12.Time evolution

of the sinuous

mode or

p = 0.25, W = 0.05, and

FIG. 13. Time evolution of the sinuousmode for p = 1, W= 0.67, and

h=0.05(H=O.l). h = 0.25 (H = 0.373).

2399


length is the quantity

beingvaried (increased o reduceW,

the real time is increased proportionally. This issue was dis-

cuss ed elsewhere.6 The reduced s urface tension is manifest

in Fig. 11 through the incipient formation of satellite liga-

ments. The sinuous disturbance for this reduced dimension-

less wave number is qualitatively different from that of the

previous case. This can be observe d in Fig. 12, where it is

evident that the disturbance is monotonically increasing

with time. This is a result of the reduc ed relative effect of

surface tension. Realize that, if it is the wavelength that has

been varied as explained abo ve, the last frame of Fig. 12

(t = 1.23) corresponds to t = 12.3 n Fig. 9. Therefore there

is no evidence of oscillatory motion in Fig. 12 up to an equiv-

alent time roughly equal to twice the time after which two

cycles occurred in Fig. 9.

Figures 13-l 5 summarize the results for a density ratio

of 1. Figure 5 indicates that t he dilational waves grow faster

than the sinuous ones or all values of Wan d H in the linear

regime. We consider the effect of the wave type and the sheet

thickness in the vicinity of or at the dimensionless wave

number corresponding to maximum growth rate. The effect

of Wwas investigated earlier.6 F igure 13 shows the develop-

ment of a sinuous wave for p = 1, W = 0.67, and h = 0.25

(H = 0.373). Oscillatory behavior is observe d without any

evidenc e of breakup to the extent of the calculations. As in

I=8

II=68

t-6

n=68

I


2399

s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

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8/12/2019 Rangel 1991


0.5

(=a

n=41

FIG. 14.Time evolution of the dilational mode or p = 1, IV= 0.67, and

h = 0.25 (H = 0.373).

the case of lower density ratio, the restoring effect of the

surface tension force acts against the inertia forces that t end

to destabilize the sheet. The decay and reappearanceand the

sinuous behavior in Fig. 13 can be compared with a nonlin-

ear potential-kinetic energy system. IIere, surface tension

acts as he restoring f orce while the inertia of the fluid carries

the kinetic energy. The dilational mod e is show n in Fig. 14

and has the same features observed at p = 0.25, mainly the

indication of breakup in ligaments interspaced by one wave-

length. Figure 15 shows the dimensionless amplitude as a

function of time for these two cas es.

VI. CONCLUSIONS

The linear an d nonlinear instability of a fluid sheet of

finite thickness under relative shearing motion (Kelvin-

E-lelmholtz) has been analy zed including the effect of inter-

facial tension, density ratio, and thickness-to-wavelength ra-

tio. Both sinuous and dilational waves have been considered.

The linear theory analysis encompasses he complete range

of density ratios and thickness-to-wavelength ratios. The lin-

ear theory is revised and two significant contributions are

made. The first on e is the existence of a critical thickness

Oc’

t0

Dimensionless ime

FIG. 15. Dimensionless i sturbance mplitude as a function of time for

p = 1, W= 0.67,andh = 0.25 (H = 0.373 . Nonlinear calculations solid

lines) and inear predictions broken ines).

below which all sinuous waves are stable. The limited results

of the previous linear theory implied that sinuous waves

were always more unstable. For low density ratios, the

growth rate of the sinuo us wave s is larger t han that of the

dilational waves, in agreement with previous results. The

seco nd finding is that dilational wave s are more unstable

than sinuous ones when the density ratio approaches unity.

This result has mportant implications in near critical liquid

rocket atomization. The nonlinear calculations indicate the

existence of si nuous oscillating modes when the density ratio

is of the order of 1. The sinu ous distortion may result in

ligaments interspaced by half of a wavelength, whereas the

dilational distortion, which grows monotonically, may re-

sult in ligaments interspaced by one wavelength, thus being

larger than those produced by the sinuous mode.

ACKNOWLEDGMENT

This work has been supported in part by Air Force Of-

lice of Scientific Resear ch Grant No. 86-00 16D and by Il. C.

Irvine Committee on Research Grant No. 90/9 l-20.

’ H. B. Squire, rit. J. AppLPhys.4, 167 1953).

* W. W. Hager& a nd J. F. Shea, . Appl. Mech. 22, 509 ( t955).

‘G. I. Taylor, Proc. R. Sot. Lond onSer. A 253,296 ( 1959).

4H. Lamb,

Hydrodynamics

(CambridgeUP., Cambridge, 1932).

’ N. Dombrowski and P. C. Hooper,Chem. Eng. Sci. 17,2 91 ( 1962),

bR. H. Ran ge1 nd W. A. Sirignano,Phys. Fluids 31, 1845 1988).

‘A. Mansourand N. Chigier, Phys. Fluids A 2,706 ( 1990).

“G. K. Batchelor,An Introduction to Fluid D ynamics (CambridgeU.P.,

Cambridge, 1970).

9A. I. vande Vooren, Proc. R. Sot. Londo nSer. A 373,67 ( 1980).

‘%3.P. Lin, Phys. Fluids 30,200O 1987).

2400 Phys. Fluids A, Vol. 3, No. 10, October 1991

R. H. Rangei and W. A. Sirignano

2400


Rangel 1991

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