An improved two-phase Lattice Boltzmann model for high densityratios: application to wave breaking

Amir Banari, Christian F. Janßen, and Stephan T. GrilliUniversity of Rhode Island, Department of Ocean Engineering, Narragansett, RI, USA

ABSTRACT

A new Lattice Boltzmann method (LBM) is developed to effi-ciently simulate multiphase flows with high density ratios, in or-der to study complex air-sea interaction problems, such as windwave breaking and related sea-spray generation. In this method,which builds and improves on the method proposed earlier by In-amuro et al. (2004b), the motion of (diffusive) interfaces betweenfluids is modeled by solving the convective Cahn-Hilliard equationwith the LBM. As in the latter work, we eliminate instabilities re-sulting from high density ratios by solving an additional Poissonequation for the fluid pressure. The resulting numerical schemeis computationally demanding since this equation must be solvedover the entire computational domain, which motivates imple-menting the method on the massively parallel environment of-fered by General Purpose Graphical Processing Units (GPGPU),via the nVIDIA CUDA framework. In this paper, we presentthe equations and numerical methods for the method and theinitial validation of the resulting multiphase-LBM for standardbenchmark problems such as Poiseuille flow, a rising bubble, andRayleigh-Taylor instability for two-fluid systems. A good agree-ment with the reference solutions is achieved in all cases. Finally,the method is applied to simulating an ocean breaking wave ina space periodic domain. In all the presented applications, itis observed that the GPGPU implementation leads to speed-upsof about two orders of magnitude in comparison to a single-coreCPU implementation. Although the method is only currently im-plemented in a two-dimensional (2D) framework, its extension tothree-dimensions (3D) should be straightforward, but the needfor the efficient GPGPU implementation will become even moredrastic in 3D.

KEY WORDS: Lattice Boltzmann method, Cahn-Hilliardequation, multiphase flows, GPGPU, Rayleigh-Taylor instability,ocean wave breaking, High density ratio.

INTRODUCTION

The numerical simulation of multiphase and multi-componentfluid flows is a challenging task in Computational Fluid Dynam-ics (CFD), both for conventional macroscopic and mesoscopicmethods, such as the Lattice-Boltzmann Method (LBM). In clas-sical CFD methods, multiphase flows are simulated by couplinga Navier-Stokes (NS) equation solver to an interface advection oradvection-diffusion equation (e.g., Lubin et al. (2006)). The for-mer equation is used in combination with a sharp interface model,whereas the latter is mostly used with a diffusive interface model.The interface itself is typically represented by a tracking method(such as the widely used Volume Of Fluid (VOF) method), or aninterface capturing method. Most of the interface tracking meth-ods assume a sharp interface, i.e., they consider the phase transi-tion to be clearly defined and thus the interface between two fluidsto be infinitely thin. By contrast, the interface capturing meth-ods allow for both sharp or diffusive interface representations, de-pending on the type of equations solved. An additional challenge,when using a sharp interface method, is the accurate computa-tion of the interface curvature and related surface tension forces.This has encouraged many researchers to use diffusive interfacemethods, in which interface forces can be modeled as a smoothedcontinuum by distributing them over thin but numerically resolv-able layers (Jacqmin, 1999). Such models have recently attractedmuch interest, owing to their computational advantages (Unverdiand Tryggvason, 1992; Brackbill et al., 1992). Because of thesevarious options, when developing and implementing a specific freesurface or multiphase CFD model, one has to make a priori de-cisions regarding using: (i) a sharp or diffusive interface method;(ii) an advection or advection-diffusion equation; and (iii) a track-ing or capturing method. For instance, in the two-phase modeldetailed below, we decided to model the interface motion withCahn-Hilliard (CH) interface capturing advection-diffusion equa-tion (Cahn and Hilliard, 1958), using a scalar order parameterΦα (α = 1,2) to identify each phase. The interface between thetwo phases is then defined as a smooth transition from one to the

other value of Φα.Recently, the LBM has matured into a powerful alternative

to classical NS solvers, both for simulating single phase, andmultiphase/multi-component flows (Swift et al., 1995; Shan andChen, 1993; He et al., 1999). The LBM is based on the Boltz-mann equation, which governs the dynamics of molecular prob-ability distribution functions, from a microscopic scale point ofview. The LBM then discretizes the Boltzmann equation on anequidistant lattice, yielding a numerical method for computingmacroscopic distribution functions on the lattice. The macro-scopic hydrodynamic quantities, such as pressure and velocity, areobtained as low-order moments of these distribution functions.The resulting formulation can be shown to converge towards thesolution of the classical governing macroscopic equations (Chap-man and Cowling, 1970), such as NS. LBM, however, has severalsolver-specific advantages, such as a relatively easier implemen-tation, a straightforward treatment of boundary conditions, anddata and operator locality, which both yield significantly moreefficient parallel implementations than for the more traditionalsolvers. In this respect, LBM has been shown to take full advan-tage of the recent advances in General Purpose Graphical Pro-cessing Units (GPGPU) (Tolke and Krafczyk, 2008b).

While there have been numerous applications of classical CFDsolvers to multiphase flows, whose exhaustive review is outsidethe scope of this paper, over the past two decades, several note-worthy methods have been developed for simulating multiphaseflows in the context of the LBM. These are: Rothman and Keller(1988) color method, the Shan and Chen (1993) model (SC), thefree energy method of Swift et al. (1995), and the method of Heet al. (1999).

All of the above methods solved multiphase flows using vari-ous approaches but, in all of these, the maximum fluid densityratio achievable in computations was limited by the occurrenceof instabilities for high ratio values (typically larger than 10-20).Overcoming this limitation is one of the most challenging currentissues in the LBM modeling of multiphase flows and the subjectof active research. This is also the rationale for developing themethod presented in this paper, as we aim at modelling complexflows at an air-water interface, whose density ratio is about 1,000.

To this effect, we recently developed a new LBM, based on im-proving the approach proposed by (Inamuro et al., 2004b). As inthe latter work, we eliminate instabilities resulting from high den-sity ratios by solving an additional Poisson equation for the fluidpressure. In our model, however, new equilibrium functions areintroduced to retrieve NS equations, with the molecular viscos-ity being related to the LBM relaxation time similar to classicalLBMs. The new corresponding NS equation, however, does nothave the undesired terms, which appear in Inamuro et al.’s modeland contribute to triggering instabilities for very high density ra-tios. The location of diffuse interfaces between two fluids is nowtracked more accurately, by solving Cahn-Hilliard equation withthe LBM using the new equilibrium functions. Details of thesurface tension and body forces, and a detailed derivation of theequations can be found in previously published papers(Inamuroet al., 2004b,a).

The resulting numerical scheme is computationally demand-ing, as the Poisson equation must be (iteratively) solved over theentire computational domain. This is efficiently done by imple-menting the model on a latest generation GPGPU environment,via the nVIDIA CUDA framework. Such GPGPU boards (e.g.,

nVIDIA Tesla C2070) provide up to 448 cores, 6 GB of mainmemory, and a double precision computing capability. It hasbeen shown in various publications (Tolke and Krafczyk, 2008a,b;Janßen and Krafczyk, 2011) that LBM methods are especiallywell suited for such a GPGPU implementation. For all bench-mark problems presented later in this paper, the GPU implemen-tation will lead to speed-ups of about two orders of magnitude incomparison to a single-core CPU implementation.

The paper is organized as follows. First we briefly explain thediffusive interface model on the basis of the free energy concept.Then our new model for simulating multiphase flows with veryhigh density ratio is introduced. Finally, the method is validatedby comparing numerical results to reference solutions for two-fluid Poiseuille flows, a rising bubble, and the Rayleigh-Taylorinstability. Finally, we solve a case of ocean wave braking in aspace-periodic domain.

DIFFUSIVE INTERFACE MODELS

As mentioned in the introduction, numerical schemes based on asharp interface representation lead to additional problems in theirnumerical implementation, compared to diffusive interface mod-els. In particular, sharp interface models usually require a mov-ing numerical grid, whereas diffusive interface models naturallyaccommodate fixed grids (such as in the LBM). They also facedifficulties for accurately computing the interface curvature andthe related surface tension forces. This often leads to the appear-ance of “parasitic currents” in the numerical solution along theinterface. These problems are eliminated when using a diffuse-interface representation, based on the continuous variation of anorder parameter (such as fluid density or a function of density),in a way that is physically consistent with microscopic theoriesof interface processes. Three main types of diffuse-interface mod-els have thus been proposed in the literature: (i) tracking forcemodels (Unverdi and Tryggvason, 1992); (ii) continuum surfaceforce models (Brackbill et al., 1992); and (iii) phase-field models(Jacqmin, 1999).

In the current work, we use the latter approach, in which thetotal free energy F of a two-fluid system is specified to be mini-mum for the equilibrium interface profile φ(ζ). Here, φ denotes acontinuously varying order parameter (with values φ1 and φ2 re-ferring to fluid 1 and 2 on either side of the interface, respectively;and φ2 > φ1, φ ∈ [φ1,φ2]), and ζ is a coordinate normal to theinterface. Following Jacqmin (1999), the motion of the diffusiveinterface is modeled, as a function of the order parameter, withthe extended Cahn-Hilliard (CH) equation (Cahn and Hilliard,1958), which includes convection terms,

∂φ

∂t+∇· (φuuu) =M∇2Γφ, (1)

where the left hand side represents the interface advection, andthe right hand side represents the interface diffusion, expressed asa function of a mobility coefficient M and the chemical potentialΓφ,

Γφ = βdΨdφ−k∇2φ (2)

with the bulk free-energy density Ψ(φ) = (φ−φ2)2(φ−φ1)2. Thecoefficients k and β are related to the surface tension coefficient

σ12 and interface thickness W as,

W = 4φ1−φ2

√k

2β (3)

σ12 = (φ1−φ2)3

6√

2kβ (4)

A more detailed derivation of the model can be found in (Banariet al., 2012).

LATTICE BOLTZMANN MODEL

Two-dimensional multiphase (2D) flows are simulated by solvingtwo sets of equations: (i) the NS equations, which provide theflow fields, based on the conservation of mass and momentum;(ii) the extended Cahn-Hilliard Eq. (1), which describes the in-terface motion. We solve these equations using a new LatticeBoltzmann method (LBM), based on two sets of Lattice Boltz-mann particle distribution functions, one for each equation (i)and (ii), for which we find mesoscopic equilibrium distributionfunctions, which reproduce the desired macroscopic equations.To discretize the LBM equations, we use the D2Q9 set of particlevelocities (Qian et al., 1992), which introduces 9 discrete particlevelocities in directions ei (Fig. 1), with,

eee0 = 0;eeei = c(cos((i−1)π/4) ,sin((i−1)π/4)) , i=1,3,5,7

eeei =√

2c(cos((i−1)π/4) ,sin((i−1)π/4)) , i=2,4,6,8 (5)

where c = ∆x/∆t is the propagation speed on the lattice, takenas c = 1 in this work, with grid spacing ∆x and time step ∆t.The speed of sound in the D2Q9 lattice is cs = c/

√3 (Qian et al.,

1992), and the weighting factors wi are given as,

w0 = 49 ,w1,3,5,7 = 1

9 ,w2,4,6,8 = 136 (6)

1

2 3 4

7

e2e3

e1

e4

e5

e6e7 e8

5

6 8

1f

2f3f

8f7f6f

5f

4f

Figure 1: Discrete velocity vectors for the D2Q9 model (5)

LBM of Navier-Stokes equations

A set of particle distribution functions gi (xxx,t) is specified for cal-culating the flow velocities and pressure, whose time evolution is

computed as (assuming a single relaxation time (SRT) formula-tion, Yu et al. (2003))(i= 0, ...,8),

gi(xxx+eeei∆t, t+ ∆t) = gi(xxx,t)−∆tτg

(gi(xxx,t)−g(eq)i (xxx,t))

−3wi∆teiασviscαβ∂

∂xβ

(1ρ

)+ 3wieiαBα∆t

(7)

with particle velocities ei at point xxx and the time t, relaxationtime τg, that relates to fluid kinematic viscosity (ν)as τg = ν/c2

s+12 ∆t, viscous stress tensor σviscαβ = νρ

(∂uα∂xβ

+ ∂uβ

∂xα

), gravitational

body force Bα, density ρ, and velocity u. We denote by g(eq)i ,

the equilibrium state of particle distribution functions,

geqi = wi

{1 + eiαuα

c2s

+ (eiαuα)2

2c4s− |u

uu|2

2c2s

}+wi

k

ρGαβeiαeiβ

−vik

2ρ |∇φ|2 (8)

where implicit summations are performed on repeated indices αand β (but not on i). The last two terms in the above equationexpress the surface tension forces, where vi is defined as,

v0 =−53 ,vi = 3wi(i= 1,2, ...,8) (9)

and

Gαβ(φ) = 92∂φ

∂xα

∂φ

∂xβ− 9

4∂φ

∂xγ

∂φ

∂xγδαβ (10)

Note that, compared to the classical LBM models, the fluiddensity ρ no longer appears in the equilibrium distribution func-tions g(eq)

i . This eliminates potential instabilities caused by highdensity differences across interfaces and is equivalent to removingthe hydrodynamic pressure gradient from the field. As a result,the velocity resulting from this set of equations is not divergence-free and thus has to be corrected. The predicted velocity uuu∗ iscomputed as the first order moment of gi’s as,

uuu∗ =8∑i=0

gieeei (11)

and is corrected by ∆uuu, to satisfy the complete momentumequation. Following Inamuro et al. (2004b), we state,

∆uuu'−∆t∇pρ

; uuu= uuu∗+ ∆uuu (12)

Thus, for the corrected velocity field to satisfy continuity equation∇.uuu= 0, uuu∗ must satisfy the following Poisson equation,

∇.uuu∗ =∇.(∆t∇pρ

) (13)

Note, in practice, the collision time τg can be considered as anelementary time of collision, so that ∆t can be replaced by τg inEq. (13) (see (Sukop and Throne, 2005)). The Poisson equation(13) can be discretized by various methods. Here, we also solveit in the LBM framework, which yields the following evolution

equation for a new set of particle distribution functions hi (i =0, ...,8),

hi(xxx+eeei∆t, t+ ∆t) = hi(xxx,t)−∆tτh

(hi−h(eq)i ) +wi(∇.uuu∗) (14)

New equilibrium distribution functions are defined as well, as,

heqi = wipn(xxx,t), (15)

where n denotes the n-th iteration in the Poisson equation solu-tion. The relaxation time τh is related to that of the NS LBMansatz,

τh = τg

c2s ρ

+ 12∆t, (16)

and the pressure is calculated as the zero-th order moment of theparticle distribution functions,

pn+1 =8∑i=0

hi. (17)

This scheme is iteratively run at a given time t until the pressurefield converges.

Hence, in this new method, the two previously derived LBMschemes solve: (i) the (pressureless) NS equations for high den-sity ratios, with surface tension forces included in the equilibriumdistribution functions; and (ii) a Poisson equation for the correc-tion of velocity fields to account for pressure gradients. By con-trast with sharp interface methods, the calculation of interfacecurvature is not necessary and only the gradients of the phasefield parameter φ have to be calculated. Applying the Chapman-Enskog expansion (Chapman and Cowling, 1970) to Eq. 7 withthe equilibrium distribution function in Eq. 8, these schemes canbe shown to converge to the NS equation (Banari et al., 2012),

∂t(uα) +∂β(uαuβ

)= k

ρ∂β

(∂φ

∂xk

∂φ

∂xkδαβ−

∂φ

∂xα

∂φ

∂xβ

)+∂β

[ν(∂αuβ +∂βuα

)]+Bα (18)

Lattice Boltzmann scheme for solving Cahn-HilliardequationThe diffusive interface motion is modeled by the convective Cahn-Hiliard equation,

∂t(φ) +∂α (φuα) =M∇2Γφ (19)

To solve this equation, we introduce a third probability distri-bution function, fi(x,t), whose evolution is again governed by astandard LBM scheme,

fi(xxx+eeei∆t, t+ ∆t) = fi(xxx,t)−∆tτf

(fi(xxx,t)−f(eq)i (xxx,t)) (20)

with corresponding equilibrium distribution functions f (eq)i ,

feqi =Hiφ+viΓφ+φwi

{1 + eiαuα

c2s

+ (eiαuα)2

2c4s− |u

uu|2

2c2s

}, (21)

wi and vi according to (6) and (9),

H0 = 1 and H1,2,...,8 = 0 (22)

The order parameter φ is computed from the first moment ofdistribution function,

φ=8∑i=0

fi (23)

It can be shown that, with this definition of f (eq)i , the scheme

solves the diffusive Cahn-Hilliard equation, for a relaxation timeτf = 1

2 ∆t+M (Banari et al., 2012).Once the order parameter is computed, the space distribution

of fluid properties (density ρ, kinematic viscosity ν and dynamicviscosity µ) can be calculated. These properties are assumed tovary smoothly along the phase interface as a function of the orderparameter φ,

ρ(φ) = φ−φ2φ1−φ2

(ρ1−ρ2) +ρ2 (24)

ν(φ) = φ−φ2φ1−φ2

(ν1−ν2) +ν2 (25)

µ(φ) = φ−φ2φ1−φ2

(µ1−µ2) +µ2 (26)

VALIDATION

In this section, the new LBM multiphase method is validatedby comparing the numerical results to analytical and experimen-tal reference solutions, for three different test cases: a two-fluidPoiseuille flows, rising bubbles, and the Rayleigh-Taylor instabil-ity. The initial analysis of the multiphase- LBM results shows agood agreement with reference solutions for all three test cases.Finally, the method is applied to simulating a breaking oceanwave, in order to demonstrate the applicability of the method toair-sea interaction problems.

Figure 2: A schematic velocity profile of a two-fluid Poiseuilleflow between infinite plates

Two-component Poiseuille flowThe two-fluid Poiseuille flow between two infinite plates is a goodtest case for validating the method for high viscosity and densityratios. Two immiscible fluids are accelerated by a body force (inthe positive x1 direction) in a rectangular channel and sloweddown by viscous drag along the plate surface Fig. 2. At thephase interface, the continuity of fluid velocity and stresses hasto be satisfied. Since the interface is planar, its curvature andsurface tensions forces are zero. The analytical solution of theNavier-Stokes equations for the velocities u1 and u2 in fluid 1and fluid 2, respectively, is given by,

u1 = gh2 (ρ1 +ρ2)2 (µ1 +µ2) −

x2 gh (µ1 ρ2−µ2 ρ1)2µ1 (µ1 +µ2) − gρ1x

22

2µ1(27)

u2 = gh2 (ρ1 +ρ2)2 (µ1 +µ2) −

x2 gh (µ1 ρ2−µ2 ρ1)2µ2 (µ1 +µ2) − gρ2x

22

2µ2(28)

with fluid densities ρi, dynamic viscosities µi, gravity g and chan-nel height 2h. The simulations were started from a state of rest,with zero flow velocities in the whole computational domain. Aperiodic boundary condition in the flow direction and no-slipboundary conditions at the plate surfaces were used. The in-terface thickness is set to W = 4 lattice units. The Reynoldsnumber is Re = 100, based on half of the channel width, the vis-cosity of the first fluid phase, and the maximum velocity in thechannel. The LB Mach number is fixed to Ma = 0.01, and τf isset to 1; ρ1/ρ2 = 100 and the viscosity ratio is ν1/ν2 = 0.1. Thesimulations were stopped as soon as a steady state was reached.To check the accuracy of the numerical scheme, the followingrelative L2-norm error is used,

L2 =

√√√√∫ 2h0[u(x2)−u1,2 (x2)

]2 dx2∫ 2h0 u2

1,2 (x2) dx2(29)

where u1,2 denotes the analytical solution in fluid 1 or 2, respec-tively, depending on the value of x2.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y/h

u x

(a) h = 100

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y/h

u x

(b) h = 300Figure 3: Non-dimensional velocity profiles (ux = u/umax) fordifferent grid resolutions (Re = 100, ρ1/ρ2 = 100, ν1/ν2 = 0.1,y=x2), solid line is the analytical solution, and doted line is thenumerical result

Fig. 3 shows steady-state velocity profiles computed for differ-ent two grid sizes, as compared to the analytical solution. In pre-vious studies of two-phase Poiseuille flows with LBM multiphasemodels (Gross et al., 2011; Freudiger, 2009), local oscillations ofthe fluid velocity near the phase interface were observed, evenfor low density ratios. Our new LBM multiphase method doesnot show these oscillations, and accurately reproduces the slopediscontinuity in velocity profile. Fig. 4 shows the L2 errors fordifferent grid configurations; first-order convergence can clearlybe observed with decreasing grid size.

102

10−1

Number of lattice nodes

ER

RO

R (

L2

Nor

m)

Figure 4: L2-norm error plotted versus grid resolution, in steady-state two-fluid Poiseuille flow

Rising bubbleThe transient behaviour of bubbles under the influence of gravityhas always been a major and demanding test case for validatingnumerical schemes for two-phase flows. Although the simula-tion setup in terms of grid initialization and boundary conditionsis straightforward, the flow structure simultaneously illustratesthe significant effects of viscosity, buoyancy, and surface tensionforces. Several experimental studies on the rising and deforma-tion of single bubbles in a quiescent liquid have been reported inthe literature (Clift et al., 1978; Bhaga and Weber, 1981). Thebubble shapes vary greatly in different flow regimes, as a func-tion of several non-dimensional parameters. Typically the Bondnumber, the Reynolds number, and the Morton number are usedto describe the test case. These are defined as,

Bo = g∆ρD2

σ(30)

Mo = g∆µ41

σ3ρ21

(31)

Re = UtD

ν1(32)

where D is the bubble diameter, ∆ρ= ρ1−ρ2 the density differ-ence between the two fluids, ρ1 the density of the heavier fluid, ρ2the density of the lighter fluid, g the gravitational acceleration, Utthe terminal velocity of the bubble, and ν1 and µ1 are kinematicand dynamic viscosity, respectively, of the heavier fluid.

A sphere with a diameter of D = 40 lattice nodes is placedin a computational domain of 128× 256 lattice nodes. Periodic

Table 1: Simulation parameters for the rising bubble test caseCase ν1 k β g Bo Mo(a) 2.85×10−3 0.1 0.4 1.5×10−8 0.1 10−3

(b) 2.85×10−3 0.3 1.2 5×10−9 10 10−1

(c) 1.4×10−2 0.06 0.26 1×10−7 100 103

boundary conditions are used on the lateral sides of the domain,a bounce-back scheme boundary condition is applied as on thetop and bottom of the domain. The remaining flow parametersare set to ρ1 = 600 and ρh/ρl = 100, τf = 1, φ1 = 0.4, φ2 = 0.1.We run the simulation for three different test cases (a) - (c),with different Mo and Bo; the specific simulation parameters aregiven in Table 1. Fig. 5 shows the computed terminal shape ofthe bubble and velocity fields, compared to the numerical resultsof Sun and Tao (2010), who used a NS-VOF Level set method.

Fig. 6 shows the time evolution of the bubble shape duringrising, for a density ratio of 1000. Here, the domain size is128× 512 lattice nodes, and the dimensionless parameters areset to Bo = 100 and Mo = 0.001. During the early stages of thesimulation, buoyancy forces are dominant. Hence, the bubble isaccelerated and starts rising, causing its shape to change. Even-tually, the terminal shape of the bubble is formed, when buoyancy, surface tension and viscous forces are balanced.

Rayleigh-Taylor instability

The classic Rayleigh-Taylor instability is used to demonstrate theaccuracy of our current model to solve more complicated flows.Here, two immiscible fluids of different densities are placed in achannel, the heavier fluid on top, the lighter one on the bottom.Under the influence of gravity, the heavier fluid will graduallysink into the lighter fluid, which is displaced upwards. The di-mensionless numbers that are important in this test case are theAtwood number and the Reynolds number, which are defined as:

A= ρh−ρlρh+ρl

and Re =√WgW

ν, (33)

where W is the width of the channel, and ρh and ρl are densitiesof heavy and light fluids, respectively. We set-up this simulationfollowing He et al. (1999) and use no-slip boundary conditionsat the top and bottom boundaries, and periodic boundary condi-tions at the lateral sides; gravity is chosen to satisfy

√Wg= 0.04.

The kinematic viscosity for both fluids is the same, and the At-wood and Reynolds numbers are 0.5 and 256, respectively (withaccordingly ρh/ρl = 3). The simulation is carried out on a gridof 256× 1024 lattice nodes. Fig. 7 compares the results of ourpresent method to the numerical results of He et al. (1999) forfour selected time steps; a very good agreement can be seen be-tween both methods.

Breaking wave

Previous work (Longuet-Higgins and Cokelet, 1976; Grilli et al.,1989) showed that a periodic sinusoidal wave of large amplitude,with the initial velocities being calculated from linear theory, isnot stable and rapidly breaks, since the initial velocity field isnot in equilibrium with the initial wave profile. To limit compu-tational time, as in earlier work, the simulation is assumed to be

(a)

(b)

50 100

(c)Figure 5: Terminal shape and velocity vectors for a bubble risingunder buoyancy; comparison of present LBM (red) and Sun andTao (2010) (black) results, for three cases: (a) Bo = 0.1, Mo =0.001; (b) Bo = 10, Mo = 0.1; (c) Bo = 100, Mo = 1000.

periodic in the flow direction. This characteristic makes such pe-riodic sinusoidal waves a convenient and efficient way of studyingwave breaking (Lubin et al., 2006).

The initial wave velocity and interface shape, obtained fromlinear theory are,

η = H

2 cos(kx) (34)

u= H

2 σcoshk(h+z)

sinh(kh) cos(kx)

v = H

2 σsinhk(h+z)

sinh(kh) sin(kx)

where σ is the wave angular frequency and other wave parame-ters are shown in Fig. 8.

Fig. 9 shows the time evolution of a high amplitude sinusoidalwave with H/L= 0.13 in depth h/L= 0.25. We see that the waveit is not stable and rapidly breaks, after traveling one wavelengthfrom initialization.

(a) t = 0 (b) t = 10,000 (c) t = 20,000 (d) t = 40,000

Figure 6: Time evolution (at selected time steps) of rising bubbleshape during the bubble acceleration phase, for ρ1/ρ2 = 1,000,Bo = 100, and Mo = 0.001.

The number of grid nodes used in this simulation is 512×256.The nodes update per second (NUPS) for this case is 5× 105,which is two orders of magnitude faster compare to a single-coreCPU implementation.

CONCLUSIONS AND OUTLOOK

We presented the concepts, equations, numerical methods, andinitial validation and application of a 2D Lattice-Boltzmannmodel for the simulation of two-phase flows with high fluid den-sity ratios. Very efficient simulations on conventional GPGPUhardware, where speed-ups of up to two orders of magnitude canbe achieved, as compared to computations on a single CPU. Inthe future work, we will focus on developing a 3D model and im-plementing turbulence models for simulating more realistic oceanwaves.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge support for this research fromthe nVIDIA Academic Partnership Program (APP) as well asGrant OCE-09-27014 from the US National Science Foundation.

REFERENCES

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Bhaga, D. and Weber, M. (1981). Bubbles in viscous liquids:shapes, wakes and velocities. Fluid Mech, 105:61–85.

(a) t′ = 0.0 (b) t′ = 1.5

(c) t′ = 3.0 (d) t′ = 4.5

Figure 7: Rayleigh-Taylor instability problem for ρh/ρl = 3, A=0.5, Re = 256. Time evolution of the two-fluid interface for fourdimensionless times t′ = t/

√W/g; leftward panels: results of He

et al. (1999); rightward panels: present results.

Figure 8: Definition sketch for initial interface profile of largeamplitude sinusoidal wave.

Brackbill, J., Kothe, D. B., and Zemach, C. (1992). A contin-uum method for modeling surface tension. J. Comput. Phys,100:335–354.

Cahn, J. and Hilliard, J. (1958). Free energy of a nonuniformsystem. J. Chem. Phys, 28:258–267.

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(a) t′ = 0.0 (b) t′ = 0.33

(c) t′ = 1.33 (d) t′ = 1.66

(e) t′ = 2.0 (f) t′ = 2.33

(g) t′ = 2.66 (h) t′ = 3.0

Figure 9: Space-periodic wave breaking (Fig. 8). Time evolutionof an overturning breaking wave with H/L= 0.13 and h/L= 0.25(dimensionless time t′ = t/

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