Viscous Potential Analysis of Kelvin-Helmholtz Instability in a Channel

7/30/2019 Viscous Potential Analysis of Kelvin-Helmholtz Instability in a Channel

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J. Fluid Mech. (2001), vol. 445, pp. 263283. c 2001 Cambridge University PressDOI: 10.1017/S0022112001005572 Printed in the United Kingdom

263

Viscous potential flow analysis ofKelvinHelmholtz instability in a channel

By T. F U N A D A1 A N D D. D. J O S E P H2

1Department of Digital Engineering, Numazu College of Technology, Ooka 3600,Numazu, Shizuoka, Japan

2Department of Aerospace Engineering and Mechanics, University of Minnesota,Minneapolis, MN 55455, USA

(Received 6 November 2000 and in revised form 2 May 2001)

We study the stability of stratified gasliquid flow in a horizontal rectangular channelusing viscous potential flow. The analysis leads to an explicit dispersion relationin which the effects of surface tension and viscosity on the normal stress are notneglected but the effect of shear stresses is. Formulas for the growth rates, wavespeeds and neutral stability curve are given in general and applied to experiments inairwater flows. The effects of surface tension are always important and determinethe stability limits for the cases in which the volume fraction of gas is not too small.The stability criterion for viscous potential flow is expressed by a critical value of therelative velocity. The maximum critical value is when the viscosity ratio is equal tothe density ratio; surprisingly the neutral curve for this viscous fluid is the same asthe neutral curve for inviscid fluids. The maximum critical value of the velocity ofall viscous fluids is given by that for inviscid fluid. For air at 20 C and liquids withdensity = 1g cm3 the liquid viscosity for the critical conditions is 15 cP: the criticalvelocity for liquids with viscosities larger than 15 cP is only slightly smaller but thecritical velocity for liquids with viscosities smaller than 15 cP, like water, can be muchlower. The viscosity of the liquid has a strong effect on the growth rate. The viscouspotential flow theory fits the experimental data for air and water well when the gasfraction is greater than about 70%.

1. Introduction

It is well known that the NavierStokes equations are satisfied by potential flow;the viscous term is identically zero when the vorticity is zero but the viscous stresses

are not zero (Joseph & Liao 1994). It is not possible to satisfy the no-slip condition ata solid boundary or the continuity of the tangential component of velocity and shearstress at a fluidfluid boundary when the velocity is given by a potential. The viscousstresses enter the viscous potential flow analysis of free surface problems throughthe normal stress balance (2.9) at the interface. Viscous potential flow analysis givesgood approximations to fully viscous flows in cases where the shear from the gasflow is negligible; the RayleighPlesset bubble is a potential flow which satisfies theNavierStokes equations and all the interface conditions. Joseph, Belanger & Beavers(1999) constructed a viscous potential flow analysis of the RayleighTaylor instabilitywhich is almost indistinguishable from the exact fully viscous analysis.

The success of viscous potential flow in the analysis of RayleighTaylor instabilityhas motivated the analysis of KelvinHelmholtz (KH) theory given here. It is well


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264 T. Funada and D. D. Joseph

known that the instability that arises when surface tension and viscosity are neglected

is catastrophic: short waves with wavelengths = 2/k amplify without controllike ekt . The instability grows exponentially as the wavenumber k no matterhow small time t. This kind of catastrophic instability is called Hadamard instability(Joseph & Saut 1990). In the case of inviscid fluids this instability is regularized bysurface tension which stabilizes the short waves; surface tension is very important. Thequestion is whether viscosity, without surface tension, would regularize the Hadamardinstability of a vortex sheet on an unbounded domain. Unlike surface tension, viscositywill not cause the small waves to decay; they still grow but their growth is limited andthe growth rate Re[(k)] does not go to infinity with k as in Hadamard instability.The positive growth rate is given by

Re[+] =a

2l + l

2a

2(l + a)3(Ua Ul )2, k ,

where ,,U are respectively density, viscosity, velocity.The present paper gives a detailed report on the viscous potential flow analysis

of KH instability in a rectangular duct together with a comparison of theory andexperiment in the case of airwater flow. As we have already mentioned, potentialflow requires that we neglect the no-slip condition at solid surfaces. In a rectangularchannel the top and bottom walls are perpendicular to gravity; the bottom wall isunder the liquid and parallel to the undisturbed uniform stream; the top wall contactsgas only. The sidewalls are totally inactive; there is no motion perpendicular to thesidewalls unless it is created initially and since the two fluids slip at the walls all theconditions required in the analysis of three dimensions can be satisfied by flow in twodimensions, which is analysed here.

The viscosity in viscous potential flow enters the normal stress balance rather than

the tangential stress balance. Air over liquid induces small viscous stresses that maybe confined to the boundary layer and may be less and less important as the viscosityof the liquid increases. At a flat, free surface z = 0 with velocity components (u, w)corresponding to (x, z) the shear stress is given by

u

z+

w

x

and the normal stress is

2w

z.

The normal stress is an extensional rather than a shear stress and it is activated bywaves on the liquid; the waves are induced more by pressure than by shear. For this

reason, we could argue that the neglect of shear could be justified in wave motionsin which the viscous resistance to wave motion is not negligible; this is the situationwhich may be approximated well by viscous potential flow.

2. Formulation of the problem

A channel of rectangular cross-section with height H and width W and of lengthL is set horizontally, in which a gas layer is over a liquid layer (see figure 1): thetwo-layer Newtonian incompressible fluids are immiscible. The undisturbed interfaceis taken at z = 0 on the z-axis of Cartesian coordinates (x,y,z). We denote velocityby (u, w), pressure p, density , viscosity and acceleration due to gravity (0, g); they component is ignored herein.


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Viscous potential flow analysis of KH instability in a channel 265

z

y

W

L

H

g

h(x,t)

x

ha

hl

0

Liquidql, ll, Ul

Airqa, la, Ua

Figure 1. KelvinHelmholtz instability due to a discontinuity of velocity of air above liquid ina rectangular channel. The no-slip condition is not enforced in viscous potential flow so that thetwo-dimensional solution satisfies the sidewall boundary conditions.

In the undisturbed state, the gas (air) with a uniform flow ( Ua, 0) is in 0 < z < ha,and the liquid with a uniform flow (Ul , 0) is in hl < z < 0 (H = hl + ha); the pressurehas an equilibrium distribution due to the gravity. We consider KelvinHelmholtzinstability of small disturbances to the undisturbed state.

The prescription of a discontinuity in velocity across z = 0 is not compatible withthe no-slip condition of NavierStokes viscous fluid mechanics. The discontinuousprescription of data in the study of KelvinHelmholtz instability is a viscous potentialflow solution of the NavierStokes equation in which no-slip conditions at the wallsand no slip and continuity of shear stress across the gasliquid interface are neglected.

Usually the analysis of KelvinHelmholtz instability is done using potential flow foran inviscid fluid but this procedure leaves out certain effects of viscosity which canbe included with complete rigour. This kind of analysis using viscous potential flowis carried out here. An exact study of, say, air over water requires the inclusionof all of the effects of viscosity, and even the prescription of a basic flow is muchmore complicated. Problems of superposed viscous fluids have been considered, forexample, in the monograph on two-fluid mechanics by Joseph & Renardy (1991) andmore recently in the paper and references therein of Charru & Hinch (2000).

2.1. Viscous potential flow analysis

We have already noted that if the fluids are allowed to slip at the walls, then thetwo-dimensional solution will satisfy the three-dimensional equations and we may

reduce the analysis to flow between parallel plates. We found by computing thatthree-dimensional disturbances are more stable than two-dimensional ones. We nowconsider two-dimensional disturbances, for which the velocity potential (x,z,t)gives (u, w) = .

The potential is subject to the equation of continuity:

u

x+

w

z= 0

2

x2+

2

z2= 0; (2.1)

thus the potentials for the respective fluids are given by

2ax2

+2az2

= 0 in 0 < z < ha, (2.2)


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2l

x2+

2l

z2= 0 in

hl < z < 0. (2.3)

Boundary conditions at the interface (at z = h, where h h(x, t) is the interfaceelevation) are given by

h

t+ Ua

h

x= wa,

h

t+ Ul

h

x= wl , (2.4a, b)

and the conditions on the walls are given by

wa = 0 at z = ha, (2.5)

wl = 0 at z = hl . (2.6)The potential a that satisfies (2.2) and (2.5) for the air and the potential l that

satisfies (2.3) and (2.6) for the liquid are given, respectively, by

a = Aa cosh[k(z ha)] exp(t + ikx) + c.c., (2.7a)l = Al cosh[k(z + hl )] exp(t + ikx) + c.c., (2.7b)

and the interface elevation may be given by

h = A0 exp(t + ikx) + c.c., (2.7c)

where Aa, Al and A0 denote the complex amplitudes, and c.c. stands for the complexconjugate of the preceding expression; is the complex growth rate and k > 0denotes the wavenumber; i =

1. From the kinematic conditions (2.4a, b), we havethe following equations for the complex amplitudes:

( + ikUa)A0 =

kAa sinh(kha), (2.8a)

( + ikUl )A0 = kAl sinh(khl ). (2.8b)

The other boundary condition is the normal stress balance (with the normal viscousstress) at the interface:

pa + 2a waz

+ agh pl + 2l wl

z+ l gh

=

2h

x2, (2.9)

in which denotes the surface tension. Noting that the pressure is solely subject tothe Laplace equation derived from the equation of motion for small disturbances, thepressure terms in (2.9) may be eliminated using the equations of motion in which theviscous terms vanish identically when u = ; 2u = (2) 0. Thus pa may bewritten, from the equation of motion without the viscous term, as

a

uat

+ Uauax

= pa

x, (2.10a)

and with the aid of the equation of continuity, we have the expression for pa:

a

2watz

+ Ua2waxz

=

2pax2

; (2.10b)

the pressure pl may be written as

l

2wltz

+ Ul2wlxz

=

2plx2

. (2.11)


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Thus the normal stress balance is now written as

a

2watz

+ Ua2waxz

+ 2a

3wax2z

+ l

2wltz

+ Ul2wlxz

2l 3wl

x2z (l a)g

2h

x2=

4h

x4, (2.12)

hence we have the equation for , using (2.7) and (2.8):

[a( + ikUa)2 + 2ak

2( + ikUa)] coth(kha)

+[l( + ikUl)2 + 2lk

2( + ikUl)] coth(khl) + (l a)gk + k3 = 0. (2.13)2.2. Dispersion relation

From (2.13) the dispersion relation is given as

A

2

+ 2B + C = 0, (2.14)where the coefficients A, B and C are defined as

A = l coth(khl) + a coth(kha), (2.15a)

B = ik[l Ul coth(khl ) + aUa coth(kha)]

+k2[l coth(khl ) + a coth(kha)] = BR + iBI , (2.15b)

C = (l a)gk k2[lU2l coth(khl ) + aU2a coth(kha)] + k3+2ik3[l Ul coth(khl ) + aUa coth(kha)] = CR + iCI . (2.15c)

The solution may be expressed as

= BA

B2A2

CA

R + iI = BR + iBIA

DA

, (2.16)

where D is given by

D = DR + iDI = (BR + iBI )2 A(CR + iCI ), (2.17a)

DR = l a(Ua Ul )2k2 coth(khl ) coth(kha)k4[l coth(khl ) + a coth(kha)]2[l coth(khl ) + a coth(kha)][(l a)gk + k3], (2.17b)

DI = 2k3(al l a)(Ua Ul) coth(khl ) coth(kha). (2.17c)

When al = l a for which DI = 0, and if DR> 0, we have

R=

BR

DR

A , I=

BI

A .(2.18

a, b)

This is a typical case where the real and imaginary parts of can be expressed mostclearly.

If the top and bottom are far away, hl , ha , then (2.14) gives rise to = ik(aUa + l Ul ) + k

2(a + l)

(a + l )

alk2(Ua Ul )2

(a + l)2 (l a)gk + k

3

(a + l )+

k4(a + l )2

(a + l)2

+ 2ik3(al l a)(Ua Ul)

(a + l )2

1/2,


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which is reduced, for a particular case that al = l a, to

R = k2(a + l )

(a + l )

al k2(Ua Ul )2

(a + l )2 (l a)gk + k

3

(a + l)+

k4(a + l)2

(a + l )2

1/2,

(2.19a)

I = k(aUa + l Ul )(a + l )

. (2.19b)

Here, it is easy to find that R = 0 gives a relation independent of viscosity. In otherwords, the relation holds even for inviscid fluids; this is helpful for the problem to beconsidered herein.

2.3. Growth rates and wave speeds

In terms of = R + iI , (2.14) is also written, for the real and imaginary parts, as

A(2R 2I ) + 2(BRR BI I ) + CR = 0, I = 2BI R + CI2(AR + BR) . (2.20a, b)Eliminating I from the above, we have a quartic equation for R:

a44R + a3

3R + a2

2R + a1R + a0 = 0, (2.21)

where the coefficients are given as

a4 = A3, a3 = 4A

2BR, a2 = 5AB2R + AB

2I + A

2CR, (2.22a,b,c)

a1 = 2B3R + 2BRB

2I + 2ABRCR, a0 = 14AC2I + BRBI CI + B2RCR. (2.22d, e)

The quartic equation (2.21) can be solved analytically. Neutral states for which R = 0are described in terms of the solution to the equation a0 = 0. The peak value (themaximum growth rate) m and the corresponding wavenumber km are obtained bysolving (2.21). It is usually true, but unproven, that m = 2/km will be the length ofunstable waves observed in experiments.

The complex values of are frequently expressed in terms of a complex frequency with

R + iI = = i = iR + I . (2.23)Hence

R = I , I = R. (2.24)The wave speed for the mode with wavenumber k is

CR = R/k = I /k. (2.25)

The set of wavenumbers for which unstable flows are stable is also of interest. Thewavelengths corresponding to these wavenumbers will not appear in the spectrum.Cut-off wavenumbers kC separate the unstable and stable parts of the spectrum.

2.4. Neutral curves

Neutral curves define values of the parameters for which R(k) = 0. These curves maybe obtained by putting a0 = 0:

l 2a coth(khl ) coth

2(kha) + a2l coth

2(khl) coth(kha)

[l coth(khl ) + a coth(kha)]2kV2 + (l a)g + k2 = 0,

(2.26)


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where the relative velocity V is defined by V

Ua

Ul . This equation may be solved

for V2 where

V2(k) =[l coth(khl ) + a coth(kha)]

2

l 2a coth(khl) coth2(kha) + a

2l coth

2(khl ) coth(kha)

1k

[(l a)g + k2]. (2.27)The lowest point on the neutral curve V2(k) is

V2c = mink>0

V2(k) V2(kc), (2.28)

where c = 2/kc is the wavelength that makes V2 minimum. The flow is unstable

when

V2 = (V)2 > V2c . (2.29)This criterion is symmetric with respect to V and V, depending only on the absolutevalue of the difference. This feature stems from Galilean invariance: the flow seen byan observer moving with the gas is the same as the one seen by an observer movingwith the liquid.

By recalling the results obtained by computing, it is interesting to note here thatthe three-dimensional disturbances in the sense of the viscous potential flow lead tothe relative velocity V3D, which can be expressed in terms of (2.27) as

V23D (k Ua k Ul)2

k2x=

k2

k2xV2(k), (2.30)

only if we regard the three-dimensional-wavenumber vector k = (kx, ky) as

k =

k2x + k2y , ky = 0, W,

2

W, . (2.31a, b)

It is evident in (2.30) that V23D is larger than V2(k) if ky = 0; the most dangerous

three-dimensional disturbance is two-dimensional with ky = 0.

3. KH instability of inviscid fluid

For inviscid fluids (a = l = 0), we have BR = 0 and CI = 0; thus a3 = a1 = a0 = 0and (2.21) reduces to

a44R + a2

2R = 0; (3.1)

thus we have

a42R + a2 = 0, (3.2)

and

I = BIA

= k[l Ul coth(khl ) + aUa coth(kha)]l coth(khl ) + a coth(kha)

. (3.3)

It should be noted here that the neutral curve was given by the equation a0 = 0 inthe viscous potential analysis ((2.26) and (2.27)), whereas the neutral curve in thisK-H instability is given by the equation a2 = 0. It is also noted that I is the sameas in (2.18b), though R may be different, in general, from (2.18a). But the equationR = 0 in (3.2) is the same as R = 0 in (2.18a), for the case of al = l a.


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From (3.2) with a2 < 0, the growth rate R is expressed as

R=

l ak2 coth(khl ) coth(kha)V2[l coth(khl )+a coth(kha)][(l a)gk +k3]l coth(khl) + a coth(kha)

.

(3.4)At the neutral state R = 0 for which a2 = 0, we have

l ak coth(khl ) coth(kha)

l coth(khl) + a coth(kha)V2 [(l a)g + k2] = 0. (3.5)

Instability arises if

V2 >

tanh (khl )

l+

tanh (kha)

a

1

k

(l a) g + k2

V2i (k). (3.6)In the stable case for which a2 > 0, the wave velocity CR is given by

kCR = I = BIA

B2IA2

+CRA

. (3.7)

4. Dimensionless form of the dispersion equation

The dimensionless variables are designated with a hat and are

k = kH, ha =haH

, hl = hlH

= 1 ha, = al

, =al

, =

l gH2,

Ua

=Ua

Q, U

l=

Ul

Q, V = U

a U

l, =

H

Q, =

l

l HQ,

where

Q =

(1 )gH

1/2.

The dimensionless form of (2.14) is given by

[coth(khl ) + coth(kha)]2

+ 2{ik[Ul coth(khl ) + Ua coth(kha)] + k2[coth(khl ) + coth(kha)]} k2[U2l coth(khl ) + U2a coth(kha)] + 2ik3[Ul coth(khl ) + Ua coth(kha)]

+ k

1 + k

2

(1 )

= 0. (4.1)

The expression (2.27) for the neutral curve R(k) = 0 is written in dimensionlessvariables as

V2 =[tanh(kha) + tanh(khl)]

2

tanh(kha) + (2/) tanh(khl)

1

k

1 +

k2

(1 )

. (4.2)

Notice that the growth rate parameter = l/(l HQ), which depends linearly on thekinematic viscosity l = l /l of the liquid, does not appear in (4.2). Note also thatthe value of (1 ) is close to unity, since = 0.0012 for airwater.


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The neutral curves for an inviscid fluid (3.5) can be obtained by putting = or

l /l = a/a. This gives from (4.2) the following expression:

V2 = [tanh(kha) + tanh(khl)]1

k

1 +

k2

1

(4.3)

which is the dimensionless form of (3.6). Though this reduction is immediate it issurprising.

Evaluating (4.2) for = 0, we get

V2 = tanh(kha)1

k

1 +

k2

1

. (4.4)

Evaluating (4.2) for = we getV2 = tanh(khl )

1

k

1 +

k2

1

. (4.5)

It is easy to verify that V2 is maximum at = , for inviscid fluids. Viscosity inviscous potential flow is destabilizing; however, large viscosities are less destabilizingthan small viscosities.

Since = 0.0012, which is very small, the variation in the stability is large when varies between and , and is very small when varies between and zero. Thevalue = 0.018 > = 0.0012, and is in the interval in which V2 is rapidly varying(see figure 4).

In the literature on gasliquid flows a long-wave approximation is often made toobtain stability limits. For long waves k 0 and tanh(kh) kh and (4.2) reduces to

V2 =(ha + hl )

2

ha + (2/)hl

1 +

k2

1

. (4.6)

The effect of surface tension disappears in this limit but the effects of viscosity areimportant. To obtain the long-wave limit in the inviscid case put = .

The regularization of short waves by surface tension is an important physical effect.

For short waves, k , tanh(kh) 1 and

V2 =( + 1)2

1 +

2

/

1

k1 +

k2

(1 ) . (4.7)

To obtain the short-wave limit in the inviscid case put = .The effects of surface tension may be computed from (4.6) and (4.7). The stability

limit for long waves k 0 is independent of . For short waves (4.6) has a minimumat k =

(1 )/ with a value there given by

V2 =2( + 1)2

1 + 2/

1 . (4.8)

Equation (4.8) shows that short waves are stabilized by increasing . For small longwaves are unstable.


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2000

1500

1000

500

0102 101 100 101 102 103

k(cm1)

V(cms

1)

0.99

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

= 0.01

Figure 2. Neutral curves for air and water ( = 0.018, see table 1 and figure 4); = ha is the gasfraction. As in the usual manner, the disturbances will grow above the neutral curve, but decaybelow it. For larger than about 0.2, the critical velocity Vc arises, below which all the disturbanceswill decay.

5. The effect of liquid viscosity and surface tension on growth rates and

neutral curves

For air and water at 20 C

a = 0.0012gcm3, l = 1 g cm3, = a/l = 0.0012,

a = 0.00018 P, l = 0.01 P, = a/l = 0.018.

The surface tension of air and pure water is = 72.8dyncm1. Usually the wateris slightly contaminated and = 60dyncm1 is more probable for the waterairtension in experiments. For all kinds of organic liquids = 30dyncm1 is a goodappproximation.

Neutral curves for = 0.018 (air/water) and = = 0.0012 (inviscid flow) and = 3.6 106 (l = 50P) with = 60dyncm1 are selected here; the former twoare shown in figures 2 and 3. The liquid viscosities l = l a/a corresponding tothese three cases are l = 0.01P, 0.15 P and 50 P. The neutral curves for > arenearly identical. The neutral curves for = 0.018 (air/water) are to be compared

with experiments. We have identified the minimum values of (4.2) over k > 0 in theair/water case, and in tables 1, 2 and 3 the critical velocity Vc = V(kc), the criticalwavenumber kc (and wavelength c = 2/kc) and associated wave speeds CRc = CR(kc)are listed. In the tables, Vs and CRs denote the values taken at k = 10

3 cm1, whichmay be representative of values in the limit of long waves, k 0. The variation ofthe critical velocity with the viscosity ratio = a/l for a representative gas fraction = 0.5 is shown in figure 4. The vertical line = identifies the stability limit for


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2000

1500

1000

500

0102 101 100 101 102 103

k(cm1)

V(cms

1)

0.99

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

= 0.01

Figure 3. Neutral curves for inviscid fluids ( = = 0.0012) for different gas fractions = ha.This neutral curve arose for the special case = = 0.0012 = a/l with a = 0.00018P; hencel = 0 .15 P. Surprisingly it is identical to the case a = l = 0 (see table 2 and figure 4). The neutralcurves for viscous fluids with l > 15 cP are essentially the same as these (cf. table 2 and 3).

inviscid fluids. Points to the left of this line have high liquid viscosities l > 0.15 P,and for points to the right, l < 0.15 P.

In all cases the critical velocity is influenced by surface tension; the critical velocityis given by long waves only when is small (small air gaps). For larger values of (say > 0.2), the most dangerous neutrally unstable wave is identified by a sharpminimum determined by surface tension, which is identified in table 1 (cf. equation(4.8)).

The growth rates depend strongly on the liquid viscosity, unlike the neutral curves.The most dangerous linear wave is the one whose growth rate R is maximum atk = km,

Rm = R(km) = maxk>0

R(k) (5.1)

with an associated wavelength m = 2/km and wave speed CRm = CR(km). Typicalgrowth rate curves are shown in figure 5. Maximum growth rate parameters for = 0.018 (figure 5), = = 0.0012, l = 15cP and = 3.6 106(l = 50P) arelisted for V = 1500 and 900 cm s1 in table 4.

6. Comparison of theory and experiments in rectangular ducts

Kordyban & Ranov (1970) and Wallis & Dobson (1973) are the only authors toreport the results of experiments in rectangular ducts. Many other experiments havebeen carried out in round pipes; the results of these experiments are not perfectlymatched to the analysis done here or elsewhere, and will be discussed later. All


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1000

600

400

0

106 101 100 101

l

V(cms

1)

102103104105

Water

200

0.0180.0012

800

Figure 4. Critical velocity V vs. for = 0.5. The critical velocity is the minimum value on theneutral curve. The vertical line is = = 0 .0012 and the horizontal line at V = 635.9c m s1 is thecritical value for inviscid fluids. The vertical dashed line at = 0.018 is for air and water. Typicalvalues for a high-viscosity liquid are given in table 3.

ha Vs (cms1) CRs (cms1) kc (cm1) c (cm) Vc (cms1) CRc (cms1)

0.01 76.04 198.6 0.649 9.676 72.92 155.90.1 285.6 43.220.2 478.5 20.820.3 643.4 12.50 0.692 9.076 651.3 9.432

3.893 1.614 572.5 5.5100.4 788.8 8.150 4.020 1.563 573.9 5.4840.5 919.4 5.481 4.052 1.551 574.1 5.4810.6 1039 3.676 4.052 1.551 574.1 5.4790.7 1149 2.373 4.052 1.551 574.3 5.4590.8 1252 1.389 4.117 1.526 575.7 5.3190.9 1348 0.619 4.354 1.443 585.3 4.4150.99 1430 0.056 4.150 1.514 628.0 0.585

Table 1. Typical values of the neutral curves in figure 2 for airwater with a = 0.0012gcm3,

a

= 0 .00018P, l

= 1.0 g c m

3, l

= 0.01 P, g = 980.0c m s

2, = 60 .0dyncm

1, H = 2.54 cm. (Thistable was based upon the results of computation that the neutral curves with = 0.1 and 0.2 infigure 2 increase monotonically from the values Vs cm s

1 at k = 103 cm1; the curve with = 0 .3in figure 2 increases from the value Vs cm s

1 at k = 103 cm1, takes a maximum V = 651.3c m s1at k = 0.692 cm1, and then takes a minimum Vc = 572.5c m s1 (the critical) at kc = 3.893cm1;for the other values of, the corresponding curves give the critical Vc at kc.)

experimenters were motivated to understand the transition from stratified flow witha flat smooth interface to slug flow. They note that in many cases the first transition,which is studied here, is from smooth stratified flow to small-amplitude sinusoidalwaves, called capillary waves by Wallis & Dobson (1973). The data given by theseauthors are framed as a transition to slug flow, though the criteria given are for the


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ha Vs (cms1) CRs (cms1) kc (cm1) c (cm) Vc (cms1) CRc (cms1)

0.01 144.0 0.11040.1 455.2 0.01000.2 643.7 0.0045 2.990 2.101 619.4 0.00120.3 788.4 0.0026 3.924 1.601 634.1 0.00110.4 910.4 0.0017 4.020 1.563 635.4 0.00110.5 1018 0.0011 4.052 1.551 635.5 0.00110.6 1115 0.0007 4.052 1.551 635.5 0.00110.7 1204 0.0005 4.052 1.551 635.5 0.00110.8 1287 0.0003 4.052 1.551 635.5 0.00110.9 1366 0.0001 4.052 1.551 635.5 0.00090.99 1432 1.1 105 4.052 1.551 635.5 0.0001

Table 3. Typical values of the neutral curves for airhigh-viscosity liquid with a = 0 .0012gcm3,

a = 0 .00018P, l = 1 .0gc m3, l = 50 .0 P, g = 980.0c m s2, = 60 .0dyncm1, H = 2.54 cm;thus = 3 .6 106. This corresponds to a high viscosity case in figure 4. (The curves with ha = 0 .5through 0.8 take almost the same minimum value at k = kc, though the values at k = 10

3 cm1change as Vs = 10181287 cm s

1 and CRs = 0 .00110.0003 cm s1). (See table 4 for the maximumgrowth rate.)

air fluxes less than those predicted by their empirical formula, j < 0.53/2. All thissuggests that we may be looking at something akin to subcritical bifurcation withmultiple solutions and hysteresis.

Turning next to linearized theory Wallis & Dobson (1973) do an inviscid analysisand state that . . . if waves are long (khL 1, khG 1) and surface tension can beneglected, the predicted instability condition is

(vG vL)2 > (L G) g

hGG

+hLL

. (6.1)

If G L and L G they may be simplified further to giveG

2G > g(L G)hG (6.2)

which is the same as

jG > 3/2 (6.3)

. . . Here = hG/H and

jG =G

G

gH(L G)> 3/2.

Their criterion (6.1) is identical to our (4.6) for the long-wave inviscid case = andk 0. They compare their criterion (6.3) with transition observations that they callslugging and note that empirically the stability limit is described well by

jG > 0.53/2,

rather than (6.3).In figure 6 we plot j vs. showing jG = 3/2 and 0.53/2; we give the results from

our viscous potential flow theory for the inviscid case in table 2 and the airwatercase in table 1 and we show the experimental results presented by Wallis & Dobson(1973) and Kordyban & Ranov (1970). Our theory fits the data better than Wallis &


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V (cms1) ha km (cm1) m (cm) Rm (s1) CRm (cms1)

0.018 1500 0.01 29.90 0.2101 1448 3.0440.10.9 29.66 0.2118 872.5 2.049

0.99 32.13 0.1955 706.2 1.454

900 0.01 15.40 0.408 615.3 3.0460.1 10.00 0.628 167.7 1.1830.2 10.24 0.613 164.2 1.175

0.30.8 10.24 0.613 164.2 1.1740.9 10.33 0.609 163.3 1.1640.99 11.36 0.553 84.66 0.367

0.0012 1500 0.01 26.95 0.233 1340 3.0220.10.9 27.17 0.231 768 1.798

0.99 30.14 0.209 584.7 1.159

900 0.01 14.45 0.435 585.2 3.0640.1 9.456 0.665 155.1 1.097

0.20.7 9.685 0.649 151.0 1.0790.8 9.763 0.644 151.0 1.0790.9 9.841 0.638 149.9 1.0640.99 10.66 0.589 69.59 0.285

3.6 106 1500 0.01 1.821 3.450 295.1 24.550.1 0.916 6.861 60.04 4.4950.2 0.845 7.432 34.43 2.0490.3 3.087 2.035 21.96 0.086

0.40.6 4.44.5 1.421.40 21.89 0.0450.040.7 4.679 1.343 21.85 0.0400.8 5.360 1.172 21.61 0.032

0.9 7.743 0.812 20.21 0.0170.99 20.54 0.306 6.801 0.003

900 0.01 1.323 4.750 145.9 19.640.1 0.676 9.297 24.80 3.0170.2 0.581 10.82 10.48 1.1990.3 0.984 6.385 4.294 0.135

0.40.6 4.024.08 1.561.51 4.86 0.0100.7 4.150 1.514 4.840 0.0090.8 4.460 1.409 4.735 0.0080.9 5.534 1.135 4.100 0.0050.99 7.994 0.786 0.741 0.001

Table 4. Wavenumber, wavelength and wave speed for the maximum growth rate (5.1).

Dobsons (1973) j = 3/2 curve; it still overpredicts the data for small but fits thelarge data quite well; we have good agreement when the water layer is small.

The predicted wavelength and wave speed in table 1 can be compared withexperiments in principle, but in practice this comparison is obscured by the focuson the formation of slugs. For example, Wallis & Dobson (1973) remark that at acertain minimum air velocity, ripples appeared at the air entry end, and slowly spreaddown the channel. These waves were about 2-in. (0.05 m) long and were made up oflong wave crests, with three or four capillary waves riding on the troughs. The longwaves traveled faster than the capillary waves. The speed of these long waves wasreported by Wallis & Dobson (1973) to be less than 0.3 m s1 in all cases. Theoreticalresults from table 1 show that the wavelength c increases with the water depth (as in


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0.5

0.4

0.3

0.2

0.01

0.02

0.03

0.04

0.05

0.08

0.1

0.1 0.2 0.3 0.4 0.5 0.8 1.0

Kordyban &Ranov (1970)

j*= 0.53/2

Lin & Hanratty (1986)j* vs. 1h/D

0.0254 m,L/D = 600

0.0953 m,L/D = 260

j

*

=

3/2

Wallis & Dobson (1973)j* vs. 1h/H

f= 1.1f= 1.2f= 1.3f= 1.4f= 1.5f= 1.6f= 1.7f= 1.9

j*=gH(wa

)

V

a

Figure 6. j vs. is for marginal stability of air and water in a frame in which the water velocityis zero. The heavy line through = airwater, our result with = 60 dynes/cm from table 1; =inviscid fluid from table 2. j = 3/2 is the long-wave criterion for an inviscid fluid put forwardby Wallis & Dobson (1973). j = 0.53/2 was proposed by them as best fit to the experimentsf1.1 through f1.9 described in their paper. The shaded region is from experiments by Kordyban &

Ranov (1970). Also shown are experimental data in rectangular conduits j vs. 1 h/H = and inround pipes j vs. 1 h/D = (Lin & Hanratty 1986, figure 4).

the experiment) and the wave speed varies from 0.1 m s1 to 0.04m s1. The predictedspacing of the waves on average is about 1.5 cm s1. The predicted wavelength andwave speed from viscous potential flow are apparently in good agreement with thewaves Wallis & Dobson (1973) call capillary waves.

Observations similar to those of Wallis & Dobson (1973) were made by Andritsos,Williams & Hanratty (1989) who note that for high-viscosity liquid (80 cP) a regionof regular two-dimensional waves barely exists. The first disturbances observedwith increasing gas velocity are small-amplitude, small-wavelength, rather regular 2Dwaves. With a slight increase in gas velocity, these give way to a few large-amplitude

waves with steep fronts and smooth troughs, and with spacing that can vary from afew centimeters to a meter.

7. Critical viscosity and density ratios

The most interesting aspect of our potential flow analysis is the surprising im-portance of the viscosity ratio = a/l and density ratio = a/l; when = equation (4.2) for marginal stability is identical to the equation for the neutral stabilityof an inviscid fluid even though = in no way implies that the fluids are inviscid.Moreover, the critical velocity is a maximum at = ; hence the critical velocity issmaller for all viscous fluids such that = and is smaller than the critical velocityfor inviscid fluids. All this may be understood by inspection of figure 4, which shows


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0.05

0.09

0.03

0.01

0.002

0.004

0.006

0.0009

0.0001

0.0003

0.0005

1 3 5 7 11 15 25

Ugs(m s1)

Uls(ms

1)

2.52 cm, 1cP

2.52 cm, 16cP

2.52 cm, 70cP9.53 cm, 1cP9.53 cm, 12cP9.53 cm, 80cP

Figure 7. (After Andritsos & Hanratty 1987.) The borders between smooth stratified flow anddisturbed flow observed in experiment for various heights of the liquid and viscosities. The waterairdata are well below the cluster of high-viscosity data that are bunched together.

that = is a distinguished value that can be said to divide high-viscosity liquidswith < from low-viscosity liquids. The stability limits of high-viscosity liquids can

hardly be distinguished from each other while the critical velocity decreases sharplyfor low-viscosity fluids. This result may be framed in terms of the kinematic viscosity = / with high viscosities l > a. The condition a = l can be written as

l = ala

. (7.1)

For air and water

l = 0.15 P. (7.2)

Hence l > 0.15 P is a high-viscosity liquid and l < 0.15 P is a low-viscosity liquidprovided that l 1 g c m3.

Other authors have noted unusual relations between viscous and inviscid fluids.Barnea & Taitel (1993) note that the neutral stability lines obtained from the viscous

Kelvin-Helmholtz analysis and the inviscid analysis are quite different for the caseof low liquid viscosities, whereas they are quite similar for high viscosity, contrary towhat one would expect. Their analysis starts from a two-fluid model and it leads todifferent dispersion relations; they do not obtain the critical condition = . Earlier,Andritsos et al. (1989) noted a surprising result that the inviscid theory becomesmore accurate as the liquid viscosity increases.

Andritsos & Hanratty (1987) have presented flow regime maps for flows in 2.52 cmand 9.53 cm pipes for fluids of different viscosity ranging from 1 cP to 80 cP. Thesefigures present flow boundaries; the boundaries of interest to us are those that separatesmooth flow from disturbed flow. Liquid holdups (essentially ) are not specified inthese experiments. We extracted the smooth flow boundaries from figures in Andritsos& Hanratty (1987) and collected them in our figure 7. It appears from this figure that


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channels. The common practice for round pipes is to express results in terms of h/D

where D is the pipe diameter and h is the height of liquid above the bottom of thepipe; h/H is the liquid fraction in rectangular pipes and = 1 h/H is the gasfraction, but h/D is not the liquid fraction in round pipes and 1 h/D is not the gasfraction in round pipes. Lin & Hanratty (1986) presented experimental results for thinliquid films in round pipes giving (their figure 4) h/D vs. j; we converted their resultsto j vs. 1h/D and compared them in figure 6 with the results for rectangular pipes.The experimental results for round pipes are much lower than those for rectangularpipes. All this points to the necessity of taking care when comparing results betweenround and rectangular pipes and interpreting results of analysis for one experimentto another.

In general we do not expect viscous potential flow analysis to work well in two-liquid problems; we get good results only when one of the fluids is a gas so that

retarding effects of the second liquid can be neglected. However, the case of Holmboewaves studied by Pouliquen, Chomaz & Huerre (1994) may have a bearing on the two-liquid case. These waves appear only near our critical condition of equal kinematicviscosity. They account for viscosity by replacing the vortex sheet with layers ofconstant vorticity across which no-slip conditions and the continuity of shear stressare enforced for the basic flow but the disturbance is inviscid. They did not considerthe notion that an inviscid analysis is just what would emerge from the condition ofequal kinematic viscosity for viscous potential flow.

9. Nonlinear effects

None of the theories agree with experiments. Attempts to represent the effects ofviscosity are only partial, as in our theory of viscous potential flow, or they require

empirical data on wall and interfacial friction, which are not known exactly and maybe adjusted to fit the data. Some choices for empirical inputs underpredict and othersoverpredict the experimental data.

It is widely acknowledged that nonlinear effects at play in the transition fromstratified to slug flow are not well understood. The well-known criterion of Taitel& Dukler (1976), based on a heuristic adjustment of the linear inviscid long-wavetheory for nonlinear effects, is possibly the most accurate predictor of experiments.Their criterion replaces j = 3/2 with j = 5/2. We can obtain the same heuristicadjustment for nonlinear effects on viscous potential flow by multiplying the criticalvalue of the velocity in table 1 by . Plots of j = 3/2, j = 5/2 and the heuristicadjustment of viscous potential flow, together with the experimental values of Wallis& Dobson (1973) and Kordyban & Ranov (1970) are shown in figure 8. The good

agreement there lacks a convincing foundation.

10. Conclusion

We studied KelvinHelmholtz stability of superposed uniform streams in rectan-gular ducts using viscous potential flow analysis. Viscous potential flows satisfy theNavierStokes equations. Because the no-slip condition cannot be satisfied the effectsof shear stresses are neglected, but the effects of extensional stresses at the interfaceon the normal stresses are fully represented. The solution presented is complete andmathematically rigorous. The effects of shear stresses are neglected at the outset;after that no empirical inputs are introduced. The main result of the analysis is theemergence of a critical value of velocity, discussed in 7. The main consequence of


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0.5

0.4

0.3

0.2

0.05

0.1

0.1 0.2 0.3 0.4 0.5 0.6 1.0

Viscous potential flow:

table 1 withcritical values

reduced by factor(TD)

j*= 3/2

WD theory

j*

0.7

j*= 0.53/2

WD best fit

j*= 5/2

(TD)

X

X

X

X

xx

x

x

xx

x

Figure 8. Nonlinear effects. The TaitelDukler (1976) (TD) correction (multiply by ).WD denotes Wallis & Dobson (1973).

this result is that for airliquid systems the critical values of velocity for liquids with

viscosities greater than 15 cP are essentially independent of viscosity and the same asfor an inviscid fluid, but for liquids with viscosities less than 15 cP the stability limitsare much lower. The criterion for stability of stratified flow given by viscous potentialflow analysis is in good agreement with experiments when the liquid layer is thin,but it overpredicts the data when the liquid layer is thick. Though viscous potentialflow neglects the effects of shear the qualitative prediction of the unusual effects ofliquid viscosity have been obtained by other authors using other methods of analysisin which shear is not neglected.

A quite accurate predictor of experimental results is given by applying the nonlinearcorrection factor to account for the effect of finite-amplitude waves on the results ofviscous potential flow.

This work was supported by the Engineering Research Program of the Office ofBasic Energy Sciences at the DOE, by the NSF/CTS-0076648, by an ARO grantDA/DAAH04 and by INTEVEP S.A. A longer (unreferred) version of this paper canbe found at http://www.aem.umn.edu/people/faculty/joseph/.

REFERENCES

Andritsos, N. & Hanratty, T. J. 1987 Interfacial instabilities for horizontal gas-liquid flows inpipelines. Intl J. Multiphase Flow 13, 583603.

Andritsos, N., Williams, L. & Hanratty, T. J. 1989 Effect of liquid viscosity on the stratified-slugtransition in horizontal pipe flow. Intl J. Multiphase Flow 15, 877892.

Barnea, D. 1991 On the effect of viscosity on stability of stratified gas-liquid flow application toflow pattern transition at various pipe inclinations. Chem. Engng Sci. 46, 21232131.


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Barnea, D. & Taitel, Y. 1993 Kelvin-Helmholtz stability criteria for stratified flow: viscous versus

non-viscous (inviscid) approaches. Intl J. Multiphase Flow 19, 639649.Charru, F. & Hinch, E. J. 2000 Phase diagram of interfacial instabilities in a two-layer Couette

flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195224.

Crowley, C. J., Wallis, G. B. & Barry, J. J. 1992 Validation of a one-dimensional wave modelfor the stratified-to-slug flow regime transition, with consequences for wave growth and slugfrequency. Intl J. Multiphase Flow 18, 249271.

Francis, J. R. D. 1951 The aerodynamic drag of a free water surface. Proc. R. Soc. Lond. A 206,387406.

Joseph, D. D., Belanger, J. & Beavers, G. S. 1999 Breakup of a liquid drop suddenly exposed toa high-speed airstream. Intl J. Multiphase Flow 25, 12631303.

Joseph, D. D. & Liao, T. 1994 Potential flows of viscous and viscoelastic fluids. J. Fluid Mech. 265,123.

Joseph, D. D. & Renardy, Y. Y. 1991 Fundamentals of Two-Fluid Dynamics, Chap. IVVIII,Springer.

Joseph, D. D. & Saut, J.-C. 1990 Short wave instabilities and ill-posed problems. Theor. Comput.Fluid Mech. 1(4), 191228.

Kordyban, E. S. & Ranov, T. 1970 Mechanism of slug formation in horizontal two-phase flow.Trans. ASME: J. Basic Engng 92, 857864.

Lin, P. Y. & Hanratty, T. J. 1986 Prediction of the initiation of slugs with linear stability theory.Intl J. Multiphase Flow 12, 7998.

Mishima, K. & Ishii, M. 1980 Theoretical prediction of onset of horizontal slug flow. Trans. ASME:J. Fluids Engng 102, 441445.

Pouliquen, O., Chomaz, J. M. & Huerre, P. 1994 Propagating Holmboe waves at the interfacebetween two immiscible fluids. J. Fluid Mech. 226, 277302.

Taitel, Y. & Dukler, A. E. 1976 A model for predicting flow regime transitions in horizontal andnear horizontal gas-liquid flow. AIChE J. 22, 4755.

Wallis, G. B. 1969 One-dimensional Two-phase Flow. McGraw-Hill.Wallis, G. B. & Dobson, J. E. 1973 The onset of slugging in horizontal stratified air-water flow.

Intl J. Multiphase Flow 1, 173193.

Wu, H. L., Pots, B. F. M., Hollenberg, J. F. & Meerhoff, R. 1987 Flow pattern transitions intwo-phase gas/condensate flow at high pressure in an 8-inch horizontal pipe. Third Intl Conf.on Multiphase Flow, The Hague, The Netherlands, Paper A2, pp. 1321.

Viscous Potential Analysis of Kelvin-Helmholtz Instability in a Channel

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