Nonlinear Free-Surface and Viscous-Internal Sloshing

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Daniel T. Valentine1

e-mail: [email protected] of Mechanical and Aeronautical

Engineering, Clarkson University, Potsdam, NY13699-5725

Jannette B. Frandsene-mail: [email protected]

Department of Civil and EnvironmentalEngineering, Louisiana State University, Baton

Rouge, LA 70803

Nonlinear Free-Surface andViscous-Internal SloshingThis paper examines free-surface and internal-pycnocline sloshing motions indimensional numerical wave tanks subjected to horizontal excitation. In all of thestudied, the rectangular tank of liquid has a width-to-depth ratio of 2. The first sresults are based on an inviscid, fully nonlinear finite difference free-surface modemodel equations are mapped from the physical domain onto a rectangular domainstudies at and off resonance are presented illustrating when linear theory is inadeThe next set of results are concerned with analyzing internal waves induced by sa density-stratified liquid. Nonlinear, viscous flow equations are solved. Two typbreaking are discussed. One is associated with a shear instability which causes oving on the lee side of a wave that moves towards the center of the container; this wgenerated as the dominant sloshing mode recedes from the sidewall towards thethe first sloshing cycle. The other is associated with the growth of a convective insthat initiates the formation of a lip of heavier fluid above lighter fluid behind the crethe primary wave as it moves up the sidewall. The lip grows into a bore-like structit plunges downward. It falls downward behind the primary wave as the primarymoves up the sidewall and ahead of the primary wave as this wave recedes frsidewall. This breaking event occurs near the end of the first cycle of sloshing, winitiated from a state of rest by sinusoidal forcing.fDOI: 10.1115/1.1894415g

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IntroductionThe damping of vibration can be controlled by liquids in c

tainers installed on engineered structures like tall buildings, btowers, and offshore structures to absorb the energy of vibrinduced by wind or waves; see, e.g., the relatively recent reby Ibrahim et al.f1g. If such tanks are exposed to the envirment, thermal stratification may occur and, hence, internal wamong other buoyancy effects may be important. In wcompensated-ballast fuel tanks internal interfacial sloshingoccur. Chang et al.f2g investigated the motions of the boundbetween sea water ballast and fuel oil in these kinds of taLakes sloshsor seiched when forced by wind; both surface wavand internal waves can be inducedf3g. Hence, the problem osloshing is of interest and of practical importance in a numbengineering fields. It is these kinds of waves that are explorthis paper. The experimental and theoretical studies reportedliterature on sloshing are summarized next. This is followedbrief description of the relatively simple model problems exined computationally in the present study.

One of the earlier experimental investigations of nonlinfree-surface standing waves was reported by Taylorf4g. His setupallowed him to study the crest of the wave in the centerrectangular tank. The predictions of Penney and Pricef5g for thewave of maximum amplitude, i.e., when the peak of the cresta point, described the shape of the experimental wave of mmum amplitude reasonably well. This wave height is aboutof the wave length. For waves exceeding this height Tapointed out that they are expected to be “unstable becausdownward acceleration of the free surface near the crest wexceed that of gravity.”

Hill f6g investigated theoretically the transient and asympcally time-periodic amplitudes of weakly nonlinear standwaves caused by instantaneously imposed horizontal forcingthe basin initially at rest. Hill showed, among other things, thaamplitude of the initial state of sloshing was larger than

1To whom correspondence should be addressed.Contributed by the OOAE Division for publication in the JOURNAL OF O

SHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received Ap
10, 2003. Review conducted by: K. Thiagarajan.
Journal of Offshore Mechanics and Arctic EngineeringCopyright © 200

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“steady-state” amplitude of forced waves in rectangular baAnother recent study of nonlinear sloshing by horizontal forin a rectangular tank was reported by Amundsen et al.f7g. Theyinvestigated shallow water waves modeled by a forced Kordand de VriessKdVd equation. The waves they investigatedthus, weakly nonlinear. The surface waves examined inpresent investigation are modeled by fully nonlinear equatThis study is described next.

The first set of test cases presented are for free-surface sloA comparison of results with second-order theory not only seto validate the computer code, it also illustrates when andnonlinear effects become important. The model equations foset of cases are the inviscid, irrotational flow of an incompresfluid equations of motion. The boundary conditions are pureon the side and bottom walls, and the top is the nonlinearsurface boundary condition. The inviscid equations are solveapplying an extension of the numerical approach reporteFrandsen and Borthwickf8g. The model was extended in thstudy to examine resonance sloshing motions due to horizexcitation of a two-dimensional rectangular tank.

Internal sloshing motions of a density-stratified fluid are csidered next. Mackey and Coxf9g presented a weakly nonlineanalysis of two-layered fluid sloshing near resonance in a regular tank. A forced KdV type model was derived. They foevidence of “spiked solutions switching between slowly varperiodic waves.” This mode switching is a result of cubic nonearity in the model studied by them. The theoretical study coered weak nonlinearity but not instability or breaking.

Horn et al.f10g and Hodges et al.f11g described the degenetion of large-scale interfacial gravity waves in two-dimensiobasins with width-to-length ratios considerably larger thantanks examined in the present study. Although shallow wtheory is more directly applicable to the examination of laboraand field observations in their papers, their work still helpprovide insight into the results presented here. The primary sing mode is the sloshing wave with wave length of twicelength of the basin containing the density-stratified liquid. Tare two mechanisms described in Ref.f10g for the degenerationthe primary sloshing mode that are important to the present s
The first mechanism is degeneration of the primary sloshing mode
MAY 2005, Vol. 127 / 1415 by ASME

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via the generation of solitons by nonlinear steepening. The semechanism is by the generation of internal bores. A computatinvestigation of the dynamics of breaking progressive wavenearly two-layered fluid interfaces was reported by FringerStreetf12g. Two of the instabilities associated with breakingscribed in their investigation are also helpful in describingbreaking sloshing waves reported in this paper.

The second set of test cases presented are for internal sloof density-stratified liquids. Internal waves induced by horiztally oscillating a closed, rectangular container are investigThe liquid is assumed to be viscous and incompressible withsity stratification caused by variations in, e.g., salinity. The ulayer is lighter than the bottom layer between which there is alayer in which the density varies from the lighter fluid toheavier fluid continuously. In ocean engineering the intermelayer is known as the pycnoclinesor, if associated with temperture, the thermoclined. The density differences are assumed sand, hence, the Boussinesq approximation is invoked. To simthe internal modes, the Navier-Stokes, continuity and convecdiffusion equations are solved by applying the ETUDE findifference methodf13g. The form of the equations solved arevorticity, stream function and the convection diffusion of the dsity anomaly equations. The density anomaly is the differbetween the local density and the mean density divided bmean density.sThe density anomaly is proportional to the conctration of salinity or other property that produces density vations in the water.d Four cases are presented to illustrate nonbring sloshing motions and sloshing that leads to breaking.types of breaking are reported.

In the next section of this paper we describe the mathemmodel and the computational method applied to solve thesurface sloshing cases. In the same section the mathemmodel and the computational method applied to solve the intsloshing cases are described. In the subsequent section resthe two sets of cases are presented and discussed. Finalprimary findings are summarized in the Conclusion.

Model Problems

Free-Surface Waves in Moving Tanks.Investigations are undertaken of two-dimensionals2Dd nonlinear motion of liquid inmoving tanks. Rectangular tanks are considered which moverespect to an inertial Cartesian coordinate systemsX,Zd with hori-zontal X axis and verticalZ axis according to lawsX=XTstd, Z=ZTstd. The Cartesian coordinatessx,zd are connected to the tanwith origin at the mean free surface at the left-hand side otank. The fluid is assumed to be incompressible, irrotationainviscid. The fluid motion is therefore described by Laplaequation

fxx + fzz= 0 s1d

wheref is the velocity potential function, and the subscriptsply differentiation with respect to the variables used assubscripts.

We assume a flat bed and that waves are generated inwith rigid walls. The fluid velocity components normal tofixed boundaries are equal to zero by definition. Hence, we

ufxux=0,b = 0,

ufzuz=−hs= 0 s2d

whereb is the length of the tank andhs denotes the still watedepth.

The dynamic free-surface boundary condition is

uftuz=z = − 12sfx

2 + fz2d − fg + ZT9stdgz − xXT9std s3d

where z is the free-surface elevation measured vertically abthe still water level,ZT9 and XT9 are the vertical and horizont
acceleration of the tank, andg denotes acceleration due to gravity.
142 / Vol. 127, MAY 2005


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The kinematic free-surface boundary condition is

uztuz=z = fz − fxzx s4d

Initially the fluid is assumed to be at rest with some initial perbation of the free surface. Thus, the initial conditions are

f = 0

z = z0sxd at t = 0 s5d

The formulation of the problem is completed by includingspecific initial conditions which depend on the desiredmovements.

Useful estimates of the natural frequencies of the free-sumodes of oscillation are given by the following formula:

Ns =1

FrÎhs

b

m

4ptanhS mp

b/hsD s6d

wherem is the number of nodes,b is the length of the tank,hs isthe depth of the water column at rest,Ns=shsvmd / s2pUcd, andUc=Îghs; see, e.g., Refs.f12–14g. Thus, for this scaling, the fresurface Froude numberFr =Uc/Îghs=1.

The computational method applied to solve the free-sucases is described next. The fully nonlinear model for idea2D waves in a numerical wave tank is solved numerically wmodifieds-transformation method. The method is used to maliquid domain onto a rectangle such that the moving free suin the physical plane becomes a fixed line in the mapped doThe details are described next.

Thes transformation was first used by Phillipsf15g in connection with weather forecasting schemes. Later thes coordinatesystem was used by Mellor and Blumbergf16g for ocean modeing to improve predictions of both surface Ekman flow and inbilities in boundary layers. More recently Chern et al.f17g used aChebyshev expansion to discretize thes-transformed potentiflow equation in their prediction of 2D nonlinear free-surfacetions. The latest model in the literature is described by Turnbal. f18g who simulate inviscid free-surface wave motions usins-transformed finite element 2D model.

The mapping has been designed so that each computationin the transformed domain is of unit size. This is why we refethis formulation as the modifieds transformation. In this modremeshing due to the moving free surface is avoided. Othevantages are that the mapping implicitly deals with the fsurface motion, and avoids the need to calculate the free-suvelocity components explicitly. Extrapolations are unnecesand so free-surface smoothing by means of a spatial filter irequired for the results presented here.

The mappings from the physicalsx,z,td domain to the transformed sX,s ,td domain are given by

x ↔ X, X = m1 +sm2 − m1d

bx

z↔ s, s = n1 +sn2 − n1dsz+ hsd

h

t ↔ T, T = t s7d

whereh=z+hs; the wave amplitude isz, the still water depth ishs,andb is the tank width. We designate the grid size to spanm1 to m2 in the horizontalx direction andn1 to n2 in the verticaz direction.

The derivatives of the potential functionfsx,z,td are transformed with respect tox, z, and t into derivatives ofFsX,s ,Td.The first derivatives of the velocity potential,f, are

fx =sm2 − m1dSFX +

aFsD
b h
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fz =sn2 − n1d

hFs

ft = FT +g

hFs s8d

wherea=−ss−n1dzX andg=−ss−n1dzT. Similarly, Laplace’s Eqs1d can be rewritten as

FXX +1

hFaX −

2a

hhXGFs + 2

a

hFsX + Fa2

h2 +b2sn2 − n1d2

h2sm2 − m1d2GFss

= 0 s9d

The fixed vertical wall boundary condition onX=m1,m2, and theflat bed boundary conditions2d on s=n1 become

FX = −a

has

sn2 − n1dh

Fs = 0 s10d

The dynamic free-surface boundary conditions3d on s=n2 be-comes

FT =sn2 − n1d

hzTFs −

1

2H sm2 − m1d2

b2 FFX −sn2 − n1d

hzXFsG2

+sn2 − n1d2

h2 sFsd2J − sg + ZT9dz − S X − m1

m2 − m1DbXT9 s11d

whereZT9 andXT9 are the vertical and horizontal acceleration oftank.

The kinematic free-surface boundary conditions4d on s=n2becomes

zT =sn2 − n1d

hFsF1 +

sm2 − m1d2

b2 szXd2G −sm2 − m1d2

b2 zXFX

s12d

Equationss8d–s12d are then discretized using the second oAdams-Bashforth scheme and solved in the transformed doiteratively using successive over-relaxation. More details onmethod can be found in Frandsenf19,20g.

Internal Waves in Moving Tanks. Investigations are undetaken next of 2D nonlinear, internal viscous motions of a liqthat is density stratified. The equations solved are the NaStokes, continuity, and convection-diffusion equations.boundary conditions are no-slip walls. The Boussinesq appmation is invokedsthis means that the density difference betwthe upper and lower layers of liquid is less than about fivecentd. The equations are

zt + suzdx + swzdz = −FstdFV

2 uz −1

Frd2ux +

1

Reszxx + zzzd

ut + suudx + swudz =1

Re Scsuxx + uzzd

cxx + czz= z s13d

The functionFstd is the temporal variation of the horizontal aceleration of the container. The parameterFV denotes the sloshinparameter. The coefficient 1/FV

2 is a dimensionless parameter tcan be written alternatively asg0AFVF

2, whereAF=ah/hs is theamplitude of the horizontal excitation andVF=vhhs/U is theforcing frequency. In this study we appliedFstd=−sinfh. If pe-riodic excitation of the tank is set at a single frequency, thefh= lpt+fhs, wherefhs is a specified phase angle. The dimens
less frequency is, thus,VF=pl. The dimensionless time ist
Journal of Offshore Mechanics and Arctic Engineering


rin

is

r-ei-

nr-

t

-

= t*U /hs, wheret* is time in seconds. Thus, the parameterFV isrelated to the amplitude of the horizontal acceleration of thesit is described in more detail laterd. The y component of thvorticity is z=uz−wx, whereu andw are the horizontal and vetical components of the velocity vector, respectively. The dendifference ratio isu=sr−rld / sru−rld, whererl is the density othe lower layersand, hence, atz=0, u=0d andru is the density othe upper layersand, hence, atz=1, u=1d. The continuity equation is ux+wz=0; this equation allows us to define the strefunction, c, in terms of the velocity field as follows:u=cz andw=−cx. The Reynolds number, Re=Uhs/n, is based on the dephs, of the water column in the container and the charactespeed of a gravity wave along an infinitely thin pycnocline,

U=Îsgg0h1h2d / sh1+h2d, where the reference value of the denanomalyg0=srl −rud /rl, andh1 andh2 are the depths of the uppand lower layers of the liquid, respectively. The Schmidt numSc=n /D, wheren is the kinematic viscosity for the liquidse.g.,waterd andD is the diffusivity of a solutese.g., saltd. The densimetric Froude number squared is Frd

2=U2/ sgug0uhsd, whereg is thegravitational acceleration. Gravity acts in the negativez direction.Thus, Frd

2=sh1h2d /hs, wherehs=h1+h2.The horizontal-oscillation parameter,FV, can be written as

product of the densimetric Froude number and the ratio ogravitational acceleration divided by the amplitude of the horital acceleration,f, as follows:

FV2 = Frd

2g

fs14d

Hence, e.g., iff =1g, then FV2 =Frd

2=sh1h2d /hs2. If FV=1, then f

=Frd2g. The parameterf depends, of course, on the case of inte

to the investigator.Frandsenf19g, for free surface sloshing, used the follow

dimensionless parameter as a measure of nonlinearitykh

=ahvh2/g, whereah and vh are the amplitude and frequency

horizontal excitation. The equivalent parameter for internal wis described next. Multiplying and dividing the right-hand sidthis formula byg0, we can write this parameter in terms ofcharacteristic parameters that describe internal waves. Thukh

=g0AFFrd2VF

2. Recalling Eq.s13d and the relationship betweenFVand the parameters in the formula forkh, we can rewrite this as

kh =Frd

2

FV2 s15d

Finally, a useful formula for estimating the natural frequenof the internal sloshing modes is the formula for a two-layfluid system, i.e.

Ne =1

FrdÎ m

4p

hs

bFcothSmphs

b

h1

hsD + cothSmphs

b

h2

hsDG−1/2

s16d

wherem is the number of nodes,b is the width of the tank andhsis the depth of the liquid in the tank as well as the height oftank; see, e.g., Ref.f21g.

The method applied to solve the system of Eqs.s13d is theETUDE finite difference method described in detail by Valenf13g. It was successfully applied to solve related problems, ewas applied by Valentine and Sipcicf22g to examine internal soltary waves, and by Jahnke et al.f23g to examine cellular flowinduced by thermal effects in square containers. In additionmerical simulations of the internal waves observed by Kao ef24g in the laboratory were computed with this method. Thedictions were successfully compared by Valentine et al.f25g andSaffarinia and Kaof26g. The grid size in the present study is fithan what was used in previous studies because of the incrcomputational power of the computer used. The grid select
the present study is 2313131 in thex and z directions, respec-
MAY 2005, Vol. 127 / 143


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tively. The grid is on a domain with two units in thex directionand one unit in thez direction. For clarity only a part of the gris illustrated in Fig. 1. The grid is finer near the walls to resothe boundary layers. The grid is square in the center of the tathe center the size of the grid issDx,Dzd=s0.01,0.01d. The timestep used in the simulations isDt=0.0001. This is an ordermagnitude smaller than what was used in previous, succestudies. Hence, the spatial and temporal phenomena are rably well resolved. This conclusion is based on the full assessof this second order finite-difference method reportedValentinef13g.

Case Studies

Free-Surface Sloshing Motion.Investigations of forced slosing of liquid in a rectangular 2D tank subjected to horizontal bexcitation is undertaken. The only change to the governings8d–s12d is the dynamical free-surface boundary conditions12d inwhich ZT9=0. A linear solution for fluid motions with surface tesion in a horizontally excited tank was first obtained by Faltinf27g. In the present study surface tension is assumed to begible. We prescribed the forced motion ofXTstd=ah cossvhtd,whereah denotes the horizontal forcing amplitude,t is time, andvh is the angular frequency of forced horizontal motion. Thetial conditions arefn

s1,2ds0d=0, zns1,2ds0d=0, corresponding to th

fluid being at rest. The coordinate system selected is withxaxis fixed on the undisturbed water surface and thez axis fixed onthe left-hand wall of the tank. The numerical predictions offree surface motions in the horizontally excited tanks are cpared with analytical results from second order potential the

zsx,td = ahFon=0

`

cossknxdZns1dsvhtd

+ Sahvh2

gDo

n=0

`

cossknxdZns2dsvhtdG s17d

and

fsx,z,td =ahg

vhHo

n=0

`coshfknsz+ hsdg

coshsknhsdcossknxdfn

s1dsvhtd

+ Sahvh2

gDo

n=0

`coshfknsz+ hsdg

coshsknhsdcossknxdFn

s2dsvhtdJs18d

where the equations for the time evolution of the second-oFourier components of the nondimensional surface elevationZn

s2d

and velocity potentialFns2d are

Zns2d8std − Vn

2Fns2dstd =

12hCnfKn

2Fns1d,Zn

s1dg − SnfKnFns1d,KnZn

s1dgj

Fig. 1 Lower-left corner details of grid for internal wave study

B

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fulon-nt

y

es.

nli-

-

e-

er

Fns2d8std + f1 + knV9stdgZn

s2dstd = −1

2B2SnfKnFns1d,KnFn

s1dg

−1

2CnfVn

2Fns1d,Vn

2Fns1dg

− CnfVn2Fn

s1d8,Zns1dg s19d

whereKn=pn are nondimensional wave numbers andB=bvh2/g

denotes the ratio of the tank width to wavelength correspondivh. The dimensionaless free-sloshing frequency is defined aVn=vn/vh, where the linear dispersion relation vn

=ÎsgKn/bdtanhsKnhs/bd, andSn, Cn contain Fourier coefficienand Fourier series. The vertical forcing parameterskn andV9 arezero in the case study presented. Further details on the apmate model can be found in Frandsenf19g. The free-surface motions are numerically examined off and at resonance, wherenance occurs when the external horizontal forcing frequencysvhdis equal to the natural sloshing frequencysvnd of the liquid. Thefree-surface behavior is investigated by varying the externalthrough the parameterkh=ahvh

2/g which is a measure of nonliearity. The results presented are for a tank of aspect ratiob/hs=2. Also, b=2 m andhs=1 m is used when dimensional valuare presented later.

Figures 2sad–2sdd show the free-surface elevation at thewall in an off-resonance condition withvh/v1=1.3 for small horizontal forcing amplitude wherekh=0.0036sad, and for large horizontal forcing amplitude wherekh=0.072sdd. The time historieof the forced sloshing analyses are nondimensionalized witfirst natural sloshing frequency. A grid size of 40340 and 40380, respectively, were prescribed for small and large forfrequency. A time step of 0.003 s was used in both cases.corresponds to a dimensionless time stepDt / tc=0.001 wheretc=hs/Uc andUc=Îghs. As shown in the small amplitude spectrscd, an additionalsthirdd natural sloshing frequency is presenthe solution of this particular free-surface problem. Althoughv3has a low energy content it contributes to the lower numepredicted peaks compared to the second order solution. Theciated wave phase diagramsbd displays irregular peaks atroughs in bounded orbits.

Next, the horizontal forcing parameter was increased tkh=0.072 and the free-surface elevation simulated, as shown i2sdd. The increase ofkh introduces nonlinearity in the solutiresulting in discrepancy in amplitudes between the fully nonlimodel and the second-order solution. As time evolves the pbetween the numerical model and the approximate solutionates, the numerical model having a longer period. This is dthe two additional secondary frequenciessvh±v1d, as shown inspectrumsfd, which are generated by nonlinear interactiontween modes. For this reason the wave phase-planesed displaysmore irregular patterns compared to the small forcing frequcasesbd.

Figure 3 shows the free-surface elevation at the left waresonance,vh=v1=3.76 rad/s sor Ns=0.191 in dimensionlesformd, for sad smallkh=0.0014 andscd largekh=0.014 horizontaforcing amplitudes. For the small amplitude case there isagreement between the approximate solution and the nummodel. For the large amplitude case, at the initial statestv1,20d while the amplitude is still small, the numerical solutcoincides with both the linear and second-order solutions. Etually, as the amplitude increases, the nonlinear effects begplay a considerable role leading to higher peaks and smtroughs in the surface elevation. The second-order solutioncapture these nonlinear features but discrepancy in amplitudphase compared to the fully nonlinear model is evident. Thiscess can be observed even more clearly on the wave phasewhen the spiral trajectory of the linear solutionsbd deforms gradu
ally from cycle to cycle in the nonlinear casesdd, as the center of


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the trajectory gradually moves towards higher amplitudes.maximum steepness shown for the resonant solution in Figscdcan be estimated to be approximately 0.25, and as shownumerical solution begins to deviate from the linear one assteepness reaches about 0.1.

Internal Sloshing Motions. In this section we examine wavinside an enclosed rectangular tank with no-slip rigid top, botand sidewalls. The waves are support by density stratificaThey are generated by horizontal excitation of the containerformulations13d includes viscous effects and density stratificaeffects.

The initial condition of density stratification for the internsloshing simulations is illustrated in Fig. 4. A lighter fluid istop of a heavier fluid separated by an intermediate layer of d

Fig. 2 Free-surface elevation at the left wall in hori„a… kh =0.0036 and „d… kh =0.072; dashed line secoThe corresponding wave phase plane and spectra of„f… nonlinear solution.

less than 0.1 of the depth of the total water column. Shown is th



e

thee

,n.e

th

density field at timet=0 at which time the fluid in the rectangucontainer is at rest. The characteristic depths of the uppelower layers are measured from the center of the pycnoclinestart of the computation for the purpose of computing the dmetric Froude number. The container is set into a horizontacillatory motion in thex direction of thesx,zd coordinates att=0+. The evolution of sloshing in the container is computedthe dimensionless range of time from 0 to 10. The results ofsimulations are discussed in this section. They are for a conset into back-and-forth horizontal motion withl =2, 0.85, 0.75and 0.5, respectively; thus, the dimensionless forcing frequecorrespond toNF=VF / s2pd=1, 0.425, 0.375, and 0.25 cyclesunit dimensionless time for the four internal wave simulatirespectively. The phase angle for all of these cases isfhs=0. For

tally excited tank off resonance; vh /v1=1.3;-order solution; solid line, numerical solution.numerical model: „b…, „c… linear solution; „e…,

zonndthe

ethe two parameters associated with vorticity generation ins13d we

MAY 2005, Vol. 127 / 145


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selected Frd=0.433 andFV=1.09. To model laboratory scalesmotion we selected Re=10,000 and Sc=833. The parameteconsistent with the internal waves investigated in the laboraby Kao et al.f24g.

To help understand the nature of the internal oscillationsduced by horizontal excitation in the four internal-waves cawe computed estimates of the natural frequencies of the syste

Fig. 3 Free-surface elevation at the left wall in horizkh =0.0014 and „d… kh =0.014; dash-dot line, linear sline, numerical solution. The corresponding phase-pl„d… nonlinear solution.

Fig. 4 Initial condition of the density-difference ratio field forthe internal waves simulations. The isopycnals shown are,
from bottom to top of the pycnocline, 0.1 <uÏ0.9, Du=0.1.
146 / Vol. 127, MAY 2005


arery

-s,by

applying s16d. As pointed out by Wüest and Lorkef3g, Eq. s16dfor Ne provides reasonable estimates of the seiche frequeobserved in many lakes with nearly two-layer density strucThis is the type of density stratification used as the initial cotion for u illustrated in Fig. 4. The estimated internal-sloshfrequencies for the natural modes of oscillation are, form=1, 2,and 3 nodes,Ne=0.23, 0.41, and 0.54, respectively.

Figure 5sad illustrates the temporal history of the stream fution for the NF=1 case as measured at two points withincontainer and Fig. 5sbd illustrates the associated spectra. Theillustrate the fact that the forcing frequency is the dominant mof oscillation. In addition, a long period oscillation associawith the primary sloshing mode due to gravity is observed.results show that the dominant mode is indeedN=1, i.e., theforced horizontal mode. In addition, the next significant frequethat appears isN=0.22. This is the natural frequency of the pmary gravity mode of the system.

Figures 6sad and 6sbd provide information about the spatvariation of theNF=1 simulation. At t=2 the contours of thstream function are shown in Fig. 6sad. Three isopycnals arelustrated in Fig. 6sbd at t=8.75. Both figures illustrate the impotance of the boundary layers in the motions induced insidetank. The no-slip boundary conditions on the side walls causboundary-layer oscillations to induce horizontal pulses of wthat emanate from the boundaries and interact in the middlecontainer. In addition, the vertical fluctuations in the motion ofisopycnals are confined to the pycnocline region. This is antration of the fact that density stratification tends to enhancezontal motion and suppress vertical motion.

The amplitude of forcing in theNF=1 case is apparently suf

tally excited tank at resonance; vh /v1=1; „a…tion; dashed line, second-order solution; solidof the numerical model: „b… linear solution;

onoluane

ciently small that the gravity waves and the forcing frequencies



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are not strongly coupled via the nonlinear terms in the NaStokes equations. The two vorticity-production terms, the firstterms on the right hand side of the first Eq.s13d, are summelinearly. Any acceleration imposed in thex direction, howevesmall, produces vorticity because the initial condition is suchuz is finite. If the initial acceleration is in the negativex directionsas it is in the present simulationsd, the bottom fluid first moves uthe x=0 wall and down thex=b wall. Once the fluid particlemove vertically away from their initial equilibrium configuratiwith respect to gravity, gradients in thex direction are createHence, the finite values ofux produce vorticity via the seconterm on the right-hand side of the first Eq.s13d. The first term onthe right-hand side induces motion with frequencies assocwith the forcing frequency. The second term induces motionsociated with the natural sloshing modes associated with grwaves. The latter are longer waves with frequencies commrate with the tank width. In theNF=1 case the forcing frequenis larger than the natural sloshing frequency. In addition, foFV=1.09, the amplitude of the acceleration af horizontal forcinf =0.1578g.

The simulations forNF=0.425, 0.375, and 0.25 simulationsexamined next. For these casesl =0.85, 0.75, and 0.5, respetively. By changing the frequency of forcing in this way, we

Fig. 5 Internal sloshing for NF=1 case. „a… Time histories ofthe streamfunction at „x ,z…= „1.45,0.25… „the large amplitude …

and at „x ,z…= „0.05,0.25… „the small amplitude …. „b… The corre-sponding spectra.

not change the amplitude of acceleration of the sinusoida



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horizontal forcing. Thus,kh, as defined bys15d, is constant fothese cases. Figures 7–9 illustrate the onset of breakininternal-sloshing waves predicted forNF=0.425, 0.375, and 0.2respectively. At the initiation of motion the tank moves innegative horizontal direction first. The primary wave doesbreak when it first rises and subsequently falls along thex=0 wall.The breaking occurs when the heavier liquid rises for the firstalong thex=b wall. This is the situation for all three cases. Tdistinct types of breaking are observed. For theNF=0.425 casbreaking is initiated after the wave recedes from thex=b wall. Itemits a smaller wave of elevation that breaks on the leewardas this wave propagates in the negativex direction. Figure 7 illustrates the degeneration of the primary sloshing mode by theeration of a solitary wavesor solitond that, in this case, suffeleeward side breaking. Figure 7sad illustrates the fact that in thcase the pycnocline is moving downward in such a way thapycnocline in the lee of the solitary wave overturns. The crethis solitary wave is located atx=1.2 att=4, as illustrated in Fig7sad. This emission of a solitary wave is one of the degeneraprocesses discussed by Horn et al.f10g. The breaking processan overturning associated with a shear instability similar to“type B” regime described by Fringer and Streetf12g.

Fig. 6 Spatial variations of internal sloshing for NF=1. „a… Thestreamline pattern at t =2; −0.004ÏcÏ0.004, Dc=0.00038. Notethat c is positive in the center of the tank. „b… Isopycnal con-tours at t =8.75; for the three isopycnals shown, from lower tothe upper, u=0.3, 0.5, and 0.7, respectively.

l- For the two cases with smaller values ofNF the wave breaking

MAY 2005, Vol. 127 / 147


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is initiated by the formation of a lip of heavier fluid extendbehind the flattened crest as the heavier liquid moves upx=b wall. Figure 8 illustrates this distinctly different degeneraprocess. The pycnocline adjacent to thex=b wall is near its highest vertical location in this figure. At the instant of time showthe figure, the crest is just beginning to move downward.quite flat in horizontal extent. The instantaneous flow alongcrest is away from the wall. Just above the crest the flow iwards the wall. Hence, the crest experiences a high shearpoint in time. This is analogous with the bore-like regimescribed by Horn et al.f10g. The heavier fluid that was convectto form an overhanging lip in the present case causes mwhen it falls and, thus, transfers energy to smaller scales oftion. This lip is the growth of a convective instability similarthe “type C” regime described by Fringer and Streetf12g. Theheavier fluid in the lip formed at the onset of breaking is ablighter fluid. The lip descends due to gravity as breaking conti

Fig. 7 Internal sloshing motion for NF=0.425. The onset of thefirst type of breaking is illustrated. „a… At t =4 the streamlines„solid lines … correspond to −0.12 ÏcÏ0.2, Dc=0.01 with nega-tive values near the center of the tank. The three isopycnals„dashed lines … correspond to, from bottom to top, u=0.3, 0.5,and 0.7, respectively. „b… At t =4.4 the streamlines „solid lines …

correspond to −0.056 ÏcÏ0.016, Dc=0.005 with negative val-ues near the center of the tank. The three isopycnals „dashedlines … correspond to, from bottom to top, u=0.3, 0.5, and 0.7,respectively.

with the fluid in the lip falling towardsz=0; it descends ahead of

148 / Vol. 127, MAY 2005


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the primary wave as the wave profile descends and begimove towards thex=0 wall. Interestingly, for theNF=0.375 casethe onset of breakingsi.e., the formation of the lipd occurs at thtime the heavier liquid just reaches its highest level on thex=bwall. At its highest elevation the amplitude of the isopycnal acenter of the pycnocline is aa/h1<0.8.

A more energetic breaking of the second type is illustrateFig. 9. This is for theNF=0.25 case. The sloshing amplituderelatively large and, hence, the formation of an overhanging lthe leeward side of the crest occurs as the primary wave movthex=b wall. In addition, the heavier fluid in the lip collapses ia bore-like structure before the sloshing wave reaches itsmum amplitude on thex=b wall. In both theNF=0.375 and 0.2cases, the overhanging lip is formed as the primary wave mup thex=b wall. Since the lip forms just as the wave reachepeak amplitude in theNF=0.375 case, this value ofNF is near thetransition from one breaking regime to another.

In summary, we have identified two types of breaking in innal sloshing. The first is caused by the degeneration of the prsloshing mode into a solitary wave that breaks. The solitary wsor solitond breaks due to the growth of a shear instabilityleads to lee-side overturning. The second type of breaking obecause of the growth of a convective instability that deveinto a lip of heavier fluid above light fluid. This second typebreaking is initiated by the formation of an overhanging lip onleeward side of the primary sloshing mode as it moved upx=b wall during the first sloshing cycle.

ConclusionsThe nonlinear free surface behavior reveals changes in

and interaction of modes that alter the temporal fluctuationfree-surface sloshing. Nonlinear effects become evident fortively large forcing frequencies. The forced horizontal motiotanks can also trigger resonance behavior. This was demonsin a case study, showing growing free-surface amplitudes. Idition, the free-surface mapping method is proving to be a rosimple and accurate method for solving nonoverturning ssloshing waves on relatively coarse grids.

When density stratification in rectangular basins is importhe major action of internal waves is confined to a region invicinity of the pycnocline. This is because the density grad

Fig. 8 Internal sloshing motions for NF=0.375. The onset ofbreaking of the second type is illustrated. At t =3.4 the stream-lines „solid lines … correspond to −0.045 ÏcÏ0.015, Dc=0.003.The isopycnals „dashed lines … correspond to, from bottom totop, u=0.3, 0.5, and 0.7, respectively.

are largest in this region and, hence, it is this region that supports



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waves. The simulations illustrate the impact of no-slip wallsinternal waves. For weak forcing with frequency higher than rnance shorter wave length, spatial phenomena are generatthe waves propagate horizontally from the boundary layerwards the center of the tank. For strong forcing with frequencloser to resonance, two distinct regimes of breaking were idfied. They illustrate how the primary sloshing mode degeneinto smaller wavelength phenomena; thus, they describe twoof energy dissipation by mixing in liquid systems in rectangtanks that support internal waves. The two types of breakinassociated with the growth of two types of instabilities. One isdegeneration of the primary wave into a solitary wave that bron its leeward side due to a shear instability; the other isgrowth of an overhanging lip that collapses due to a conveinstability. The two types of breaking are similar to the two tydescribed by Horn et al.f10g in their investigation of seichinglong-shallow tanks, and by Fringer and Streetf12g in their inves-tigation of breaking progressive waves.

Fig. 9 Internal sloshing motion for NF=0.25. The onset of thesecond type breaking of the primary sloshing mode is shown.„a… At t =3.4 the streamlines „solid lines … correspond to 0 ÏcÏ0.3, Dc=0.015. The isopycnals „dashed lines … correspond to,from bottom to top, u=0.3, 0.5, and 0.7, respectively. „b… At t=3.8 the streamlines „solid lines … correspond to −0.02 ÏcÏ0.2,Dc=0.01 with positive values near the center of the tank. Theisopycnals „dashed lines … correspond to, from bottom to top,u=0.3, 0.5, and 0.7, respectively.

This study addressed some aspects of sloshing motion and



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computational modeling of both free-surface and internal waThe present work has yet to address interaction effects bethe free-surface and the internal waves. On this topic a rpaper by Hill f28g discusses an analytic solution describinweakly nonlinear interaction between surface waves and incial waves in a two-layer fluid system. His investigation shthat internal waves can be generated by surface waves throsubharmonic resonance. His work provides theoretical resulvalidating future computational studies.

Referencesf1g Ibrahim, R. A., Pilipchuk, V. N., and Ikeda, T., 2001, “Recent Advance

Liquid Sloshing Dynamics,” Appl. Mech. Rev.,54, pp. 133–199.f2g Chang, P. A., Percival, S., and Hill, B., 1996, “Computations of Two-F

Flows through Two Compartments of a Compensated Fuel/Ballast TankHy-dromechanics Directorate Research and Development ReportCRDKNSWC/HD-1370-01.

f3g Wüest, A., and Lorke, A., 2003, “Small-Scale Hydrodynamics of LakAnnu. Rev. Fluid Mech.,35, pp. 373–412.

f4g Taylor, G. I., 1953, “An Experimental Study of Standing Waves,” ProcSoc. London, Ser. A,218, pp. 44–59.

f5g Penney, W. G., and Price, A. T., 1952, Philos. Trans. R. Soc. London, S244, p. 254.

f6g Hill, D., 2003, “Transient and Steady-State Amplitudes of Forced WavRectangular Basins,” Phys. Fluids,15, pp. 1576–1587.

f7g Amundsen, D. E., Cox, E. A., Mortell, M. O., and Reck, S., 2001, “Evoluof Nonlinear Sloshing in a Tank Near Hall the Fundamental Resonance,”Appl. Math., 107, pp. 103–125.

f8g Frandsen, J. B., and Borthwick, A. G. L., 2002, “Free and Forced SloMotions in a 2-D Numerical Wave Tank,”Proceedings of the 21st Interntional Conference on Offshore Mechanics and Arctic Engineering, PapeOMAE2002-28105, Oslo, Norway.

f9g Mackey, D., and Cox, E. A., 2003, “Dynamics of a Two-Layer Fluid SlosProblem,” IMA J. Appl. Math., 68, pp. 665–686.

f10g Horn, D. A., Imberger, J., and Ivey, G. N., 2001, “The Degeneration of LaScale Interfacial Gravity Waves,” J. Fluid Mech.,434, pp. 181–207.

f11g Hodges, B. R., et al., 2000, “Modeling the Hydrodynamics of StratLakes,” Hydroinformatics 2000 Conference, Iowa Institute of Hydraulics Research, Iowa.

f12g Fringer, O. B., and Street, R. L., 2003, “The Dynamics of Breaking Prosive Interfacial Waves,” J. Fluid Mech.,494, pp. 319–353.

f13g Valentine, D. T., 1995, “Decay of Confined, Two-Dimensional, SpatiPeriodic Arrays of Vortices: A Numerical Investigation,” Int. J. Numer. Mods Fluids, 21, pp. 155–180.

f14g Kinsman, B., 1965,Wind Waves, Prentice-Hall, Englewood Cliffs, NJ.f15g Phillips, N. A., 1957, “A Coordinate System Having Some Special Advan

for Numerical Forcasting,” J. Meteorol.,14, pp. 184–185.f16g Mellor, G. L., and Blumberg, A. F., 1985, “Modelling Vertical and Horizon

Diffusivities with the Sigma Transform System,” Appl. Ocean. Res.,113, pp.1379–1383.

f17g Chern, M. J., et al., 1999, “A Pseudospectrals-Transformation Model of 2-DNonlinear Waves,” J. Fluids Struct.,13, pp. 607–630.

f18g Turnbull, M. S., et al., 2003, “Numerical Wave Tank Based os-Transformed Finite Element Inviscid Flow Solver,” Int. J. Numer. MethFluids, 42, pp. 641–663.

f19g Frandsen, J. B., 2002, “Sloshing Effects in Periodically and Seismicallycited Tanks,”Proceedings of the Fifth World Congress on Computationalchanics, Vienna.

f20g Frandsen, J. B., 2004, “Sloshing Motions in Excited Tanks,” J. Comp. P196, 53–87.

f21g Yih, C.-S., 1972, “Wave Motion in Stratified Fluids,” inNonlinear Waves,edited by S. Leibovish and A. R. Seebass, Cornell University Press,Chap. X.

f22g Valentine, D. T., and Sipcic, R., 2002, “Nonlinear Internal Solitary WavePycnoclines,” ASME J. Offshore Mech. Arct. Eng.,124, pp. 120–124.

f23g Jahnke, C. C., Subramanayam, V., and Valentine, D. T., 1998, “On thevection in an Enclosed Container with Unstable Side Wall Temperaturetributions,” Int. J. Heat Mass Transfer,41, pp. 2307–2320.

f24g Kao, T. W., et al., 1985, “Internal Solitons on the Pycnocline: GeneraPropagation and Shoaling on Breaking Over a Slope,” J. Fluid Mech.,159, pp.19–53.

f25g Valentine, D. T., Barr, B., and Kao, T. W., 1999, “Large Amplitude SoliWave on a Pycnocline and Its Instability,” inFluid Dynamics at Interface,edited by W. Shyy, Cambridge University Press, Cambridge, Chap. 1227–239.

f26g Saffarinia, J., and Kao, T. W., 1996, “A Numerical Study of the Breaking oInternal Soliton and its Interaction with a Slope,” Dyn. Atmos. Oceans,23, pp.379–391.

f27g Faltinsen, O. M., 1978, “A Numerical Nonlinear Method of Sloshing in Tawith Two Dimensional Flow,” J. Ship Res.,18, pp. 224–241.

f28g Hill, D. F., 2004, “Weakly Nonlinear Cubic Interactions Between SurWaves and Interfacial Waves: An Analytic Solution,” Phys. Fluids,16, pp.

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Nonlinear Free-Surface and Viscous-Internal Sloshing