NASA / TM--2002-211694 AIAA-2002-0890

Damping of Quasi-Stationary

Two Miscible Liquids

Waiter M.B. Duval

Glem_ Research Center, Cleveland, Ohio

Waves Between

July 2002
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NASA / TM--2002-21_ 1.694 AIAA-2002-0890

Damping of Quasi-Stationary

Two Miscible Liquids

Walter M.B. Duval

Glelm Research Center, Cleveland, Ohio

Waves Between

Prepared for the

40th Aerospace Sciences Meeting and Exhibi.t

sponsored by the American Institute of Aeronautics and Astronautics

Reno, Nevada January 14-17, 2002

National Aeronau.tics and

Space Administration

Glem_ Research Center

July 2002
Acknowledgments

We thank code UG of: NASA for their support under Grant NAG3----2443.

Discussion with W. Wright is acknowledged.

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Available electronicall y at http://gltrs.grc.nasa.gov/GLTRS
DAMPING OF QUASI-STATIONARY WAVES

BETWEEN TWO MISCIBLE LIQUIDS

Walter M.B. Duval

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Two viscous miscible liquids with an initially sharp interface oriented vertically inside a cavity

become unstable against oscillatory external forcing due to Kelvin-Helmholtz instability. The instability

causes growth of quasi-stationary (q-s) waves at the interface between the two liquids. We examine

computationally the dynamics of a four-mode q-s wave, for a fixed energy input, when one of the

components of the external forcing is suddenly ceased. The external forcing consists of a steady and

oscillatory component as realizable in a microgravity environment. Results show that when there is a jump

discontinuity in the oscillatory excitation that produced the four-mode q-s wave, the interface does not

return to its equilibrium position, the structure of the q-s wave remains imbedded between the two fluids

over a long time scale. The damping characteristics of the q-s wave from the time history of the velocity

field show overdamped and critically damped response; there is no underdamped oscillation as the flow

field approaches steady state. Viscous effects serve as a dissipative mechanism to effectively damp the

system. The stability of the four-mode q-s wave is dependent on both a geometric length scale as well as the

level of background steady acceleration.

Introduction"

Quasi-stationary waves generated at the interface

between two miscible liquids are important for

understanding transport processes in a microgravity

environment. These quasi-stationary (q-s) waves,

generated via a controlled vibration source, serve as

model to understand effects of g-jitter. In the course of an

experiment one is interested in knowing how long it takes

the injected energy from the vibration source to decay

once it is ceased. The decay time interval is important to

experimentalists, since it serves as a marker to signal

when change can be made to the input amplitude and

frequency of excitation to the experiment. 1 To obtain

insight into the problem, we consider the damping

characteristic of a q-s wave due to jump discontinuity of

the oscillating component of the body force after a finite

growth time interval.

The characteristic feature of q-s waves inside a

bounded enclosure is that they are nonpropagating internal

waves of permanent form between a density interface,

generated via an oscillating parallel shear flow through

coupling with the body force. A continuous external

excitation is maintained during growth of q-s waves.

Unlike internal waves of permanent form inside

heterogeneous fluids, solitary and periodic cnoidal

waves, 26 q-s waves do not propagate; thus their phase

velocity is zero. Nonpropagating solitary waves 7'8 have

been generated inside narrow channels partially filled with

water undergoing continuous excitation, in contrast to q-s

waves there is a free surface. Another feature of q-s waves

is that they have long wavelengths similar to gravity

waves occurring on free surfaces.

Owing to viscous dissipation between the two

fluids, damping from external excitation is very effective

for q-s waves in comparison for liquids inside containers

with a free surface. 9'1°As pointed out by Chandrasekar, 11

the problem of gravity waves which occur in an infinitely

deep ocean or the surface of a fluid with a free surface is

an analogue to the problem of oscillation of a viscous

drop, other than its spherical geometry; for gravity waves

the dispersion relation is a special condition of the more

general Rayleigh-Taylor instability problem of superposed

fluids with a jump in density and viscosity across a density

interface. Since we are interested on the effects of

microgravity, there has been experimental 12'13 and

theoretical studies 14 16on damping of viscous drops either

with a free surface or immersed in another fluid

in microgravity. These studies consider the damping

NASA/TM--2002-211694 1
characteristics ofaliquiddropwhentheexcitationforceis

suddenlyceased. Theresultshowthatoscillationsdecay

eitherperiodically, withdecreasingamplitudesimilarto

anunderdamped harmonicoscillator,or aperiodically

dependingontheparametric region.

Thedampingcharacteristicof a q-swaveis

addressed computationally. Weshowthatforabodyforce

consistingof steadyandoscillatorycomponent, theq-s

waveexhibitseitheranoverdamped orcriticallydamped

responseoncetheoscillatorycomponent ceases. This

behaviorissimilartoadampedharmonicoscillator, with

theexceptionthatnounderdamped rsponseispredicted

asfoundforviscousliquiddropsorplaneliquidsurface

insideanenclosurewithafreesurface. 9'12'13 The effect of

length scale on the characteristics of q-s wave is also

investigated. We show that the existence of a four-mode

q-s wave has both a geometrical and background

acceleration level restriction. Thus the parametric space is

not arbitrary but has to be chosen with care.

In the following, we discuss the model problem

and relevant length scale. The damping characteristic of

the q-s wave is deduced from its structure and time history

of its velocity field. We extract the geometrical and

background acceleration level for the existence of the

four-mode q-s wave from its local bifurcation as the

length scale varies as well as its background acceleration

level.

Formulation

The realization shown in Figure 1, of two

miscible liquids oriented vertically inside a cavity with an

initial jump in density across its interface, has been

achieved in a microgravity environment.1 The equilibrium

condition of a stable interface requires that the

background steady acceleration (ngo) be on the order of 1-

btg, while the unsteady component (mgo) is zero. This

implies that over a long time scale the interface obeys the

linear diffusion equation,

ac

-- = DabvZc (1)

at

When energy is injected into the system with input

amplitude (mgo), circular frequency 03, and phase angle 0,

the interface will become unstable and generate a q-s

wave. The mechanism that leads to growth of a q-s wave

is an oscillating parallel shear flow, which generates a line

vortex due to tangential discontinuity of the velocity field

near the interface. 17 The interface becomes unstable

against Kelvin-Helmholtz instability and growth of a q-s

wave occurs. We are interested on the damping

characteristics of the q-s wave when he energy input into

the system ceases. This can be modeled by a jump

discontinuity in the body force such that,

g,(t) = I rig° +mgo(COS(OOt +O)) t < tc

tngo t >_tc (2)

where to is the cut-off time of energy injection. For t < to

energy injection into the system leads to non-eqnilibrium

which must satisfy the Boussinesq momentum equation,

-DIT =-Vp+ _V217 + p_(t) (3) P-57

as well as the incompressibility condition,

V,/7 = 0 (4)

The density field is a linear function of concentration,

p=p(l+flAC) (5)

Since the flow field becomes intense, convective

acceleration must be taken into account in the species

continuity equation,

DC _DabV2 C (6)

Dt

The interface is prescribed by the following initial

condition,

{i"°

C(x,y,O)= .5

O< x< L/2

x =L/2

L/2<x<L

(7)

Y

i

I)A PB -

" H

CA C B -

i ,

i< L_

1

Figure 1 .--mnitial condition of the interface between two miscible liquids.

(t) = ((no o + mg o cos(cot+O)) i"

NASA/TM--2002-211694 2
Theinterfacehasthevalueof0.5,whiletheleftandright

fluidhasvaluesof 1and0,respectively; thegradient at

theinterfaceliesbetween.9to0.1.Overbarintheabove

equationsdenotesaveragevalues.The boundary

conditionsat the wallsareno-slipalongthe wall

boundaries F,

/J'_=0 onF (8)

and the condition of impermeability normal _ to the

boundary,

VC'g=0 on F (9)

Equations (3,4,5) with its initial (7) and boundary

conditions (8,9) allows the prediction of the flow field

V(x,y,t) as well as its wavelength ;_ (x,y,t) as a

fimction of a parameter space A. The parameter space

may be deduced taking the curl of equation (3) and nondimensionalizing with the appropriate length and time

scale 18.The vorticity field equation in its two-dimensional

form can be expressed as,

D_ =9_V2 _ +(ng ° +mgoCOS((Ot+O))OC (10)

Dt 3x

While the continuity equation (4) is transformed

to a Poisson's equation,

V_ =-_ (11)

The velocity components and vorticity are defined as,

_3V 3V 3v 3u

u--5-f-y, v-- ax' -ax &

For a characteristic length (_ H), time (T 1/¢0),

and velocity (U_. = Ap / p • mgo H2/9-) scale,

parametric space can be expressed as

the

A, = A,(Ar, GG /Re,1/Re,AR, Sc,O ) (12)

The parametric space consisting of six parameters is the

aspect ratio CAr), the Grashof number (Grin), Reynolds

number iRe), amplitude ratio CAR), Schmidt number (Sc),

and the phase angle (0),

Ar = H GG _ Ap mgo H3 n

L ' P vz , AR m '

Re - (DH2 , Sc = g- , G G / Re - Ap mgoH

9- Dab p (09--

The ratio of Grm/Re dictates the importance of

nonlinearity in the field equations. The amplitude ratio is

also the ratio of Grashof numbers (AR Gri]Grm) based

on ngo and mgo respectively. For a Stokes-length scale

(£=8) based on diffusion of momentum (3 = x/_7-_),

the parametric space is simplified and reduces to four

parameters

A_ = A_(Res,AR,Sc,O) (13)

These parameters are essentially independent of geometric

length scale; the Stokes-Reynolds iRes) number is

defined as

Re s - Ap mg o

(,0 3/2_.. 1/2

For zero phase angle and fixed fluid properties

(Sc 1079), Sc and 0 are essentially passive parameters,

thus A based on geometric and boundary layer length

scale reduces respectively to

A, = A, (Ar, G G / Re, 1/ Re, AR)

A_ = A_ (Re s, AR)

Use of the boundary-layer length scale reduces the

parametric space to a co-dimension two bifurcation

problem. Of interest is the region of applicability of the

boundary layer versus the geometric length scale in

parametric space. This will be deduced from the

bifurcation of the wavelength of the interface,

= _(x, y, t; Re s, AR)

using a known experimental condition as reference. 1 The

velocity field can also be expressed by the same functional

relationship. Bifurcation of the wavelength of the interface

NASA/TM--2002-211694 3
is obtainedcomputationallyusingfinite difference

techniques, witha90x90gridsize,tosolveforthesetof

governingequations(6,11,12). Aflux-corrected transport

algorithmis usedto resolvethesharpgradientatthe

interface.

Discussion and Computational Results

In order to investigate the interplay of geometric

versus boundary-layer length scale; we consider fixed

injection of energy per unit volume into the system for a

reference condition (H L 5 cm), in which the input

frequency is 1 Hz, amplitude of excitation mgo 20 milli-g.

For a specific dilute system with constant (Ap / g) and

V, Res 0.3375. Variation of the parametric space is

considered by varying the length of the cavity in the range

1cm <_H <_15 cm keeping the aspect ratio (H L) fixed

to 1. Within this context the length scale issue may be

seen from the viewpoint on how the fixed input energy is

redistributed in the system as the volume increases or

decreases. For a fixed steady background acceleration ngo

(ngo 10 6go, go 980cm/sec2), AR 5x10 5, the range of

parameters is shown in Table 1.

Table 1 Range of Parameters, ngo=10 6go

Res=0.3375, AR=5xl0 5,, reference condition.

H Grin/Re 1/Re

(cm)

1 8.1 1.72x10 3

2 16.2 4.29x10 4

3 24.3 1.91x10 4

4 32.4 1.07x10 4

* 5 40.5 6.87x10 5

6 48.6 4.77x10 5

7 56.7 3.51x10 5

8 64.8 2.68x10 5

9 72.9 2.12x10 5

10 80.9 1.71x10 5

11 89.1 1.42x10 5

12 97.1 1.19x10 5

13 105.2 1.02x10 5

14 113.3 8.76x10 6

15 121.4 7.63x10 6

Dynamics of Interface and Flow Field

The damping characteristic of the system will be

investigated for the reference condition corresponding to

experimental results. According to the reference results, 1a

Concentration Field

1.0 ....

Streamfunction Field

t = 400 sec, Vm_ = 0.029 ore/see

t = 440 sec, Vm_ = 0.0015 ore/see

t = 800 sec, Vm_ = 0.00042 em/see

10 ....

0 2 _ _ 8 1 0 0 .. . " .0

Vo_icity Field

t = 4O0 see 10! '01

Vo_icity Field

I?1122Z_SS5 -_a_.._iiiiiii;_

7.,;i Ti :iii i

!....'ii

Figure 2.---Damping of quasi.stationary wave due to jump discontinuity in body force for t > 400 sec,

Res = 0.3375, AR = 5xl 04

four-mode q-s wave evolves from energy injection into the

system (H 5cm, Grm/Re 40.5). We consider the fate of

the four-mode q-s wave for the condition prescribed in

equation (2) for to 400 sec. Figure 2 shows the damping

characteristics of the four-mode q-s wave. The jump

discontinuity of energy input into the system causes a

global bifurcation of the flow field for which vortex

interactions at the interface from the creation term in

equation (10) becomes zero. Vortex interaction that

causes growth of the q-s wave is annihilated and the flow

field decomposes to a weak-rotating flow, which is

irrotational as shown by the vorticity field.

The strength of the rotating flow is dictated by

the magnitude of the steady background acceleration ngo.

The remarkable finding is that the interface does not

return to its equilibrium position (t 440 sec). The fourmode q-s wave structure remains imbedded into the fluid

(t 800 sec). Over a long time scale the q-s wave simply

diffuses, since there is no vorticity production, the species

continuity equation (5) is decoupled from the flow field.

The problem simply reduces to a linear diffusion problem

with the four-mode q-s wave as initial condition.

NASA/TM--2002-211694 4
Dampin_ Characteristics of the Interface

The damping characteristics of the four-mode q-s

wave are shown in Figure 3. Since the system is operating

near resonance conditions the amplitude of the velocity

field varies as a function of time. A subinterval of time

(390sec < t <410sec) for the velocity field is

shown for both the u and v components. There is an

overdamped response in the u component while the v

component is critically damped as shown by the

undershoot towards steady state. Similar characteristic is

shown in the response of the interface. Unlike systems

with free surfaces inside bounded containers or drops,

there is no underdamped response.

Viscous dissipation for internal q-s waves is very

effective at damping motion. These results are similar to

the behavior of a damped harmonic oscillator and also

u (cm/sec)

0000

0004

0002

0000

-0.002

-0.004

-0.006

-0.008

0

00010

0.0000

-0 0010

u (cm/sec) --0.0020

-0.0030

430040

--0.0050

v (cm/sec)

v (cm/sec)

390

0.08

0.04

0.02

0.00

-0.02

-0.04

-0.00

0

0008

0,004

0.002

0000

--0,002

-0.004

-0006

390

1 O0 200 300 400 500

t(sec)

395 400 405

t(sec)

1 O0 200 300 400 500

t(sec)

395 400 405 410

t(sec)

Figure &--Time history of velocity field showing overdamped and critically damped response,

AR = 5x10 _5, Res = 0.3375, f = 1 hz, (0.5, 0.5)

occur for certain conditions in the response of a viscous

drop. 14'16 Variation of the location (0.11,0.50) in the

cavity shows similar trends, Figure 4. The timeasymptotic behavior of the velocity field indicates that a

time interval of 40sec< t < 100sec is sufficient for

the flow field to reach steady state. The effective damping

of the q-s wave can be understood qualitatively from the

viewpoint of the decrease of energy of a wave. The energy

(E) decreases according 19to

E=cl×e -Tt (15)

y is the damping coefficient and cl is a constant. For a

wave,

7 = 29-k2 (16)

k is the wave number defined as k = 2zc/;_. For the

four-mode q-s wave, ;L 1.33 cm, k 4.72cm 1,

v- = 0.0108cm 2 / sec, the damping coefficient

becomes 7 = 0.48. For the wavelength generated by

the q-s wave the damping coefficient is significant, thus

the energy of the wave decreases quite rapidly.

Further insight is provided from the frequency

response of the flow field as shown in Figure 5. Even

though there is amplitude variation in the response of the

velocity components, the flow field oscillate with the

input frequency of 1Hz, shown by the power spectrum Pu

5.0xl 0 -4

4.0xl 0-4

3.0xl 0.-4

_ 2.ox1_ 1,0xl 0-4 y

400

0.0000

500 600 700

t_e@

800

-0.0005

_-o.oofo /

_ -0.0015

o

> -0.0020

-0.0025

-0.0030

400 500 600 700 800

t(sec)

Figure 4.--Time-asymptotic history of velocity field, AR = 5xl 0 -5, Res = 0.3375,

f=l hz, (0.11,0.50)

NASA/TM--2002-211694 5
0.006

.....

0.002

_" 0.000

_'-0.002

-0.004

-0.003

-0.008

-0.008 -0.004 0.000 O.O04

u (t)

oo6

004 [

.-. 002

000

-0.02

-0.04

-0.03

-0.06 -0.04 -0.02 -0.00 -0.02 -0.04 -0.06

v(t)

_ 00004

'_ 00002

a_ 00000 L . # . /

g.g g.5 1 .g 1.5 2.g

f (Hz)

.. 0.010

'_, 0006

_ 0004

A 0002

s-> 0 000

g.g g.5 1 .Ig 1.5 2.g

f (Hz)

0.05 0.05

ooo ;""_ 0.00

"_-0.05 -0.05

0 100 200 300 400 500 -0.10 -0.05 0.00 0.05 0.10

t (seo) _F (t)

Figure 5.---Response of system to sinusoidN body force. Res = 0.3375. AR = 5xl 0-5,1 = 1 Hz. (05. 05).

5t _~H

Viscous-region

4

Stokes-region

>.i,(

_~H

Inertial-region

>, (_ Stable

(_) Unstable

). (om:

® ® ®®

®

20 40 60 80 100 120

Gr/Re

Figure 6.--Bifurcation of mode numbers (wavelength) for nxed input energy as length scale o1 cavity increases.

Res = 0 3975. Ar = 5xl 0 -5.

and Pv. For a time interval R, the power spectrum is

defined as,

Pv(f) = ]-- R I;V(t)e-i2_f' dt 2 (17)

The effect of the amplitude variation is to produce nonchaotic attractors in phase space as shown by the pseudophase space trajectories u(t+k) and v(t+k) where k is a

time lag constant. However, the integrated value _(t)

shows a nearly constant amplitude variation with a

resulting limit cycle attractor.

Local Bifurcation of the Interface

The local bifurcation of the interface to

determine the region of applicability of various length

scales, as well as the stability of the four-mode q-s wave is

shown in Figure 6. The Stokes-length scale (3is applicable

when inertia balances viscous effects. This occurs for

36< Gr/Re< 65, the reference experiment 1 for the

stable four-mode q-s wave corresponds to Grm/Re 40.5

(H 5cm). The four-mode q-s wave loses stability to twomodes for Grm/Re < 36. As the cavity size increases H--+

8cm the wavelength increases for a fixed mode number.

The four-mode loses stability for Gr > 65. In the region

65 < Gr_ / Re < 102 the eight-mode is unstable,

however it gains stability for 102<Grm/Re<122. The

criterion for the use of the boundary layer scale is based

on the parametric range for which the four-mode is stable.

In terms of designing a test-cell, this means that

4.5cm< H< 8cm, for the energy injection to yield the

stable four-mode q-s wave.

As the cavity size (H<4.5cm) decreases viscous

forces become important and the number of modes

decrease. On the other hand when the cavity size increases

for H > 8cm (Grm/Re > 65), inertia dominates and the

number of modes increases. The eight-mode gain stability

through a supercritical bifurcation in the neighborhood of

Grm/Re 102. The loss of stability from higher to lower

mode numbers occurs through a subcritical bifurcation in

which the wavelength increases (Grm/Re 102,

Grm/Re 36). In contrast the gain of stability from lower to

higher mode number occurs through a supercritical

bifurcation. In the inertial region, buoyancy forces

overwhelms viscous forces, thus energy injected into the

system is much more effectively transmitted against

viscous dissipation.

The effect of length scale on the magnitude of the

velocity field is shown in Figure 7. The magnitude of the

velocity decreases in the Stokes-region. This trend can be

understood from the characteristic velocity scale. Since

(U_. = Ap / -p •mg o/V- •_C2 ), as the length scale (Z;)

transition from geometry (H) to boundary layer

(_ = _ (O)in which (H>>8), there is a decrease in

the characteristic velocity. The velocity reaches a

minimum in the transition region from four to eight

modes, and increases for Grm/Re>105 in the inertial

region.

Effect of Background Acceleration

For a fixed energy input as well a geometrical

length, the effect of the background acceleration (ngo) on

the stability of the four-mode q-s wave is investigated

(Figure 8). Note that the parametric space (Grm/Re 41)

NASA/TM--2002-211694 6
0 0 20 40 60 80 100 120

Gr/Re

FigureT--Magnitude of velocityfield as cavi_ sizeirlcreasesforfixed energyirlputinto sy_enl,

Res = 03375, AI= 5x10 -5.

5O

4O

20

10

0

10 --7

(_) Stable

z: . -a

.........,_____,___,__,_,_,,,_, .........,_____,___,__,__,_,_,_,,______,__,'__,,__,,,,,,,,,, .........,____,___,__,__,_,_,_,,__,

10--_ 1_-5 10-4 10-3

AR

Figure &--Stability region of fouH_1ode quasi-stationary wave, Res=0.3375_ Gr/Re=41.

corresponds to the Stokes-region in which the boundary

layer scale applies (;_ 8). For the reference condition of

lgg (AR 5x10 5), the region of stability of the four-mode

q-s wave is 3x104 < AR < 7x10 5 or in terms of g-level

(6.0x10 7 go < ngo > 1.4x10 6 go). For AR > 7x10 5, the

intensity of the background mean flow destabilizes the

higher order modes. The four-mode q-s wave undergoes a

supercritical bifurcation to two modes. Beyond AR O

(10 3) q-s waves no longer exist, buoyancy effects

dominate in the limit and the interface overturns to a

stably stratified configuration.

The existence of q-s waves occurs for AR

< 5x10 4. As the level of ngo decreases, there exists a

region 1.2x10 5<AR<3xl0 5, for which the eigthmode surprisingly becomes stable, this corresponds to

2.4xl 0 7go<ngo>6.0xl 0 7go. In this region the background

flow field becomes weaker thus resulting in an increase in

the number of modes. However, for AR<I.2xl0 5 the

eigth-mode loses stability. This means that the weak flow

field created by the steady component of the body force

(ngo) is important in terms of stabilizing the eight mode qs wave. Further decrease of the g-level for AR<lxl0 7

(ngo<2Xl0 9go) is no longer effective. This shows that a

finite g-level of the background acceleration is necessary

for stabilizing q-s waves, it is not necessary to have

absolute zero.

Summary and Conclusions

For a parametric space in which a four-mode q-s

wave exists, when the injected energy into the system

suddenly ceases by a jump discontinuity, the damping

characteristic of the q-s wave is either overdamped or

critically damped as the flow field approaches steady

state. The characteristic damping time is on the order of

60 sec. The stability of the four-mode q-s wave is

dependent on both the length scale and the background

acceleration. The four mode q-s wave is stable for

4.5cm< H< 8cm, and g-level in the range of6.0xl0 7

go<ngo > 1.4x10 6 go.

The dependence on length scale shows that the

Stokes-region is bounded by the viscous and inertial

region. The four-mode q-s wave loses stability

supercritically to the eight-mode in the inertial region and

subcritically to the second-mode in the viscous region.

The reverse bifurcation trend occurs for the dependence of

g-level on the stability of four-modes. This study shows

that the four-mode q-s wave is stable over a narrow

parametric space. For a given energy input, the existence

of the four-mode q-s wave has both a geometric length

scale as well as a background g-level restriction. For the

reference experimental configuration, the system is well

parametrized by the Stokes-length scale.

References

1) Duval,W.M.B., Tryggvason, B.V., "Effects of G-jitter

on Interfacial Dynamics of Two Miscible Liquids:

Application of MIM," NASA/TM--2000-209789, March

2000.

2) Brooke Benjamin, T., "Internal Waves of Finite and

Pelmanent Form," J. Fluid Mech., Vol. 25, part 2,

pp. 241 270, 1966.

3) Brooke Benjamin, T., "Internal Waves of Permanent

Form in Fluids of Great Depth," J. Fluid Mech., Vol. 29,

part 3, pp. 559 592, 1967.

4) Davis, R.E., Acrivos, A., "Solitary Internal Waves in

Deep Water," J. Fluid Mech., Vol. 29, part 3, pp. 593

607, 1967.

5) Davis, R.E., Acrivos, A., "The Stability of Oscillatory

Internal Waves," J. Fluid Mech., Vol. 30, part 4, pp. 723

736, 1967.

6) Terez, E.D., Knio, M.O., "Numerical Simulation of

Large-Amplitude Internal Solitary Waves," J. Fluid

Mech., Vol. 362, pp. 53 82, 1998.

7) Wu, J., Keolian, R., Rudnick, I., "Observation of a

Nonpropagating Soliton," Phys. Rev. Lett., Vol. 52,

No. 16, pp. 1421 1424, 1984.

NASA/TM--2002-211694 7
8)Denardo, B.,Wright,W.,Putterman, S.,"Observation

ofaKinkSolitonontheSurfaceofaLiquid,"Phys.Rev.

Lett.,Vol.64,No.13,pp.15181521,1990.

9)Howell,R.D.,Buhrow,B.,Heath,T.,McKenna, C.,

Schatz, F.M.,"Measurements ofSurface-Wave Damping

inaContainer," Phys.ofFluids,Vol.12,No.2,pp.322

326,2000.

10)Martel,C.,Nicolas,J.A.,Vega,J.M.,"Surface-Wave

DampinginaBrimfulCircularCylinder,"J.FluidMech.,

Vol.360,pp.213228,1998.

11)Chandrasekhar, S.,Hydrodynamic andHydromagneticStability,DoverPublications, Inc.,pp.451477,

1981.

12)Trinh,E.,Wang,G.T.,"Large-Amplitude Freand

Driven Drop-ShapeOscillations:Experimental

Observations," J.FluidMech.,Vol. 122,pp.315338,

1982.

13)Trinh,E.,Zwern,A.,Wang,G.T.,"AnExperimental

Studyof Small-AmplitudeDrop Oscillationsin

ImmiscibleLiquidSystems," J.FluidMech.,Vol.115,

pp.453474,1982.

14)Miller,A.C.,Scriven,E.L.,"TheOscillationof a

FluidDropletImmersedinanotherFluid,"J.FluidMech.,

Vol.32,pp.417435,1968.

15)Lundgren,S.T.,Mansour,N.N.,"Oscillationsof

Dropsin ZeroGravitywithWeakViscousEffects,"J.

FluidMech.,Vol.194,pp.479510,1988.

16)Basaran, A.O.," NonlinearOscillationsofViscous

LiquidDrops,"J.FluidMech.,Vol.241,pp.169198,

1992.

17)Duval,W.M.B.,"Sensitivityof aWaveStructureto

InitialConditions," 38thAerospaceSciencesMeeting&

Exhibit,AIA_20001051Paper, Jan.1013,2000.

18)Duval,W.M.B.,"TheKinematicsof Buoyancy

InducedMixing,"Proceedings of theVIIIthEuropean

Symposiumon Materialsand Fluid Sciencesin

Microgravity, UniversiteLibredeBruxelles, Belgium,

April1216,pp.855861,1992.

19)Landau,D.L.,Lifshitz,M.E.,FluidMechanics,

Butterworth-Heineman, pp.9294,2000.

NASA/TM--2002-211694 8
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

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July 2002 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Damping of Quasi-Stationary Waves Between Two Miscible Liquids

& AUTHOR(S)

Walter M.B. Duva]

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135 - 3191

WU-101-53-00-00

8. PERFORMING ORGANIZATION

REPORTNUMBER

E----13425

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

Washington, DC 20546-0(101 NASA TM--2002-211694

AIAA-2002-0890

11. SUPPLEMENTARY NOTES

Prepared for the 40th Aerospace Sciences Meeting and Exhibit sponsored by the American Institute of Aeronautics and

Astronautics, Reno, Nevada January 14-17, 2002. Responsible person, Walter M.B. Duval, organization code 6712,

216-433-5023.

12a, DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Categories: 29 and 34 Distribution: Nonstandard

Available electronically at Ntp://gltrs.grc.n;_sa.gov/GIil'RS

"l-his publication is available from the NASA Center for AeroSpace In_brmation, 301-621-0390.

12b. DISTRNBUTION CODE

13. ABSTRACT (Maximum 200 words)

Two viscous raise] ble liquids with an initially sharp interface oriented vertically inside a cavity become unstable against

oscillatory external tbrcing due to Kelvin-tielmholtz instability. The instability causes growth of quasi-stationary q-s)

waves at the interface between the two liquids. We examine computationally the dynamics of a tbur-mode q-s wave, tier a

fixed energy input, when one of the components of the external fbrcing is suddenly ceased. The external forcing consists of

a steady and oscillaiory component as realizable in a microgruvity enviromnent. Results show that when there is a jump

discontinuity in the oscillatory excitation that produced the four-mode q-s wave, the interface does not return to its equilibrium position, the sm_cture of the q-s wave remains imbedded between the two fluids over a long time scale. The damping

characmristics of the q-s wave from the time history of the velocity field show overdamped and critically damped response;

there is no underdamped oscillation as the flow field approaches steady state. Viscous effects serve as a dissipative mechanism to effectively damp the system. The stability of the fou>mode q-s wave is dependent on both a geometric length scale

as well as the level of background steady acceleration.

14. SUBJECT TERMS

Fluid dynamics; Interfacial instability; Microgravity; Quasi-stationary waves;

M] scible liquid s

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OF REPORT

Unclassified

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14

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