Sharp Treatment of Surface Tension

and Viscous Stresses in Multifluid Dynamics

R.R. Nourgaliev, T.N. Dinh and T.G. Theofanous

Center for Risk Studies and Safety, University of California, Santa Barbara

A new treatment for interfacial boundary conditions in viscous multifluid flowswith surface tension is developed and tested. In difference to the δ-formulation- andContinuum-Surface-Force-(CSF)-based methods, no smearing of material properties (vis-cosity, density, and surface tension force) is used. Instead, the interfacial jump conditionsare introduced directly into the flux terms, using sharp resolution of the interface withsubcell markers, within the framework of the level-set-based adaptive Characteristics-Based Matching (aCBM) method, incorporated into the Structured Adaptive MeshRefinement (SAMR) algorithm. This treatment exhibits significantly lower (up to threeorders of magnitude) spurious velocities at the interface compared to the smearing ap-proach, and is able to maintain sharp (within one node) jumps in pressure and viscousstresses. Performance is demonstrated on a series of test problems, including the “Os-cillation of a Non-Equilibrium-Shape Bubble”, “Single- and Multi-mode Rayleigh-TaylorInstability”, “Rising Air Bubble in Water” and “Breakup of Liquid Drop in GaseousFlow”. For Rayleigh-Taylor instability, the growth rate in the linear regime is in excel-lent agreement with the analytical solution, and, in the non-linear regime, the penetrationrate of the bubble front agrees well with available experimental data. The method isshown to be robust under large density ratios (tested up to 1,000), and large range offlow conditions, from nearly-incompressible to highly supersonic.

Introduction

THE work presented in this paper is motivatedby an interest to study fragmentation of liquid

masses subjected to gaseous flows under wide rangeof speeds, from incompressible, to highly supersonic.The primary application of interest here is relatedto atmospheric dissemination of chemical agents;though, these phenomena are also important for otherengineering applications, including spray combustionand ink-jet-printing.

It is well known that dynamics of liquid dropdeformation and breakup can be reduced to twobasic interfacial/hydrodynamic instabilities, i.e.Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH).42

The fidelity of any numerical simulation will needto rest upon an adequately sharp definition ofthe interfacial region so as to capture the truehydro-gas-dynamics interactions across the singularsurface-force-containing interface. This requirementis especially stringent when the task is to describeinterfacial breakup under highly unstable conditions,as compared for example to description of interfacialoscillations or mild evolutions in shape.

All currently in use numerical methods for inter-facial dynamics originate from Peskin’s “immersedboundary” approach,31, 32 that incorporates the effectsof an interface directly into the solution of the twocontinua (on a fixed grid) on either side of it. Knownas the “δ-formulation”, it is based on smearing mate-

rial properties, that is, density, viscosity and surfacetension, over a few grid nodes, thereby introducingnumerically continuous profiles of flow variables torepresent for the jump/discontinuity of the interfacialboundary conditions. In particular, we have the“Continuum Surface Force (CSF)” model of Brackbillet al.,6 which has been used extensively both by frontcapturing40 (FC) and Front Tracking43 (FT) methodsto produce the first breakthrough simulations ofinterfacial dynamics.

Kang et al.17 have recently questioned thisapproach, and went on to develop an approachthat implements sharply the interfacial boundaryconditions. Unfortunately, from the sample casesshown so far, both the narrow question (“when theδ-formulation can be expected to break down?”) aswell as the bigger question (“under what conditionsinterfacial simulations such as these can be consideredapriori and fundamentally correct - i.e. without needof experimental support?”) remain open. There aremany more related and interesting questions one canask, the ramifications potentially are great, and, inour opinion, the subject of using DNS to study thephysics of interfacial breakup is still wide open.

This work is a takeoff from Kang et al.17 inthat we believe that the high fidelity needed toresolve the shattering of interfaces can only beachieved by a sharp treatment (ST). In addition,we believe that it needs to be built on top of doingthe best possible job in spatial resolution, as it can

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American Institute of Aeronautics and Astronautics Paper 2005–5349

17th AIAA Computational Fluid Dynamics Conference6 - 9 June 2005, Toronto, Ontario Canada

AIAA 2005-5349

Copyright © 2005 by CRSS/UCSB. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

only be achieved by adaptive mesh refinement (AMR).

In concert with Kang’s et al.17 ST/GFM approach,our algorithm is based on the level set method.Furthermore, in constructing interface boundarycondition coupling, we also use several ideas from theFedkiw’s et al.13 “Ghost Fluid Method”.

Our method differs from the first-order-accurateChorin’s projection-based ST/GFM treatment17

in that we have built our algorithm around the“pseudocompressible (PSC) method” for incompress-ible flows.9, 25 This offers several advantages. Allflow variables are collocated, which significantlysimplifies implementation of 1) high-order-accuratediscretization (here, based on the Linear Weighted-Essentially-Non-Oscillatory schemes, LWENO, up tothe seventh-order), and 2) adaptive mesh refinement(based on Berger’s et al.3–5, 27 Structured AdaptiveMesh Refinement, SAMR, algorithm). 3) Solutionof a variable-coefficient Poisson equation is avoided,which is especially important for massive parallel sim-ulations. In conjunction with treatment of interfaceboundary conditions, the pseudocompressible methodallows for a straightforward extension of all ideasintroduced here to Godunov-based compressible flowsolvers.27 Therefore, 4) the approach can cover bothincompressible and compressible flow conditions.

Here, the interface boundary conditions are incor-porated using subcell interface markers, dynamicallycreated from the level set function, in order to modifynumerical fluxes in computational cells near the inter-face, taking into account interface coupling conditions.This approach, named “aCBM” in ref.,27 allows forrobust and sharp (within one cell) treatment ofdensity jumps and shock-interface interactions. Sincewe will focus here on incompressible flows, there is noneed to use two-fluid Riemann solvers for coupling,as utilized in ref.27 Instead, interface coupling is sim-plified, using a GFM13-inspired technique. However,for implementation on SAMR, we have eliminatedGFM’s ghost fields and extrapolations,13, 25 usinginstead the “modified-Shepard’s-based” approach34

for interpolation on irregular datasets.

By computing interface curvature and correspond-ing pressure jump conditions at subcell markers, thesurface tension force is incorporated directly into theflux terms, sharply, within one near-interface cell.Since the viscous stresses are also incorporated into thefluxes, corresponding sharp viscous stress jumps areintroduced at the subcell markers, with modificationsof stresses during the computation of numerical fluxes.

Numerical examples presented here are designedto test the robustness and accuracy of this sharp

interface treatment. Using static/oscillating bubbles,we measure the level of interfacial spurious velocity,showing advantages of the ST over the CSF surfacetension model, in agreement with Kang et al.17 Inthe Rayleigh-Taylor instability test, we compare theST-based simulations to linear theory and availableexperimental data for the rate of interpenetration.We pay special attentions to a) possibility for devel-opment of spurious grid-scale disturbances, discussedby Unverdi and Tryggvason,43 and b) accuracy ofsimulation in non-linear regime in comparison toexperimental data. Next, the method is applied to arising gas bubble in water, demonstrating the abilityto robustly model high-density-ratio flows. Finally,we simulate liquid drop breakup, focusing on theimportance of grid resolution and AMR for capturinginterfacial instabilities and prediction of drop breakupregimes.

Problem Formulation

We are interested in the evolution of two-dimensional multifluid systems, consisted of two im-miscible fluids separated by infinitely thin interfaceΓ(x, t). The governing fluid dynamics equations in thebulk of each fluid are19

∂tρ + ∂xρu + ∂y ρv = −ηρv

y(1)

∂tρu + ∂x

[

ρu2 + P − Txx

]

+ ∂y

[

ρuv − Txy

]

=

η

[

ST,x

− ρuv

y

]

+ ρgx

(2)

∂tρv + ∂

x

[

ρuv − Tyx

]

+ ∂y

[

ρv2 + P − Tyy

]

=

η

[

ST,y

− ρv2

y

]

+ ρgy

(3)

and

∂tE + ∂x [u (E + βP )] + ∂y [v (E + βP )] =

−η(E + βP ) v

y(4)

where x = (x, y), t, u = (u, v), ρ, P , E = ρe, e,T

ijand g =

(

gx, g

y

)

are the position vector, time,material velocity, density, pressure, total energy, totalspecific energy, viscous stress tensor and body force,correspondingly. Heat conduction terms in eq.(4)are neglected. Viscous stress tensor is defined asT

ij= µ

(

∂iu

j+ ∂

ju

i

)

+ λ∂ku

kδ

ij, where µ, λ = − 2

3µand δ

ijare the dynamic viscosity, bulk viscosity and

Kroenekker symbol, respectively.

The switch η is zero for Cartesian (x, y) coordinates,and one for axisymmetric (x, r). In the latter case,

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American Institute of Aeronautics and Astronautics Paper 2005–5349

y = r and the stress tensor correction terms are:

ST,x

=(λ + µ) ∂

xv + µ∂

yu + v∂

xλ

y

ST,y

=(λ + 2µ)

(

∂yv − v

y

)

+ v∂yλ

y(5)

Eqs.(1)-(5) are applicable for both incompressibleand compressible fluids. In the former case, ρ = const,mass conservation equation reduces to the divergence-free condition ∇ · u = 0, and the energy equationcan be further simplified by setting switch β = 0,decoupling it from the momentum equations (2) and(3). In the compressible formulation, β = 1 andeqs.(1)-(5) must be complemented by equation of

state ρ (P, i), where i = Eρ − u2

k

2 is the specific internalenergy.

Interface jump conditions. The interface stressboundary conditions are:19

[P ] ni− [T

ik] n

k= ∂

iσ − σκn

i(6)

where nk

is the unit normal at the interface, directingfrom fluid 1 to fluid 2, and σ is the fluid surface tensioncoefficient. The local surface curvature is κ = 1

R1+ 1

R2,

where R1

and R2

are the principal radii of the inter-face curvature. By [φ] we denote the jump across theinterface, φ

1−φ

2. Since the fluids are viscous, their ve-

locities are continuous, i.e. [uk] = 0 and jump in stress

tensor is associated with discontinuity of dynamic vis-cosity, [µ] 6= 0. In general, the material density atΓ(x, t) is discontinuous, [ρ] 6= 0.

Numerical Framework

Interface. Interfaces are “captured” using the levelset method.28, 29, 38 The level set ϕ is defined as asigned distance function, which separates the compu-tational domain Ω on “positive” and “negative” sub-domains, Ω

+

and Ω−

:

x ∈ Ω+

if ϕ(x, t) > 0

x ∈ Ω−

if ϕ(x, t) < 0(7)

The interface Γ is the zero level of ϕ:

Γ = x|ϕ(x,t)=0

(8)

The motion of the interface is determined by materialvelocity field u. The evolution of ϕ is defined by thefollowing Hamilton-Jacobi equation:

∂tϕ + u · ∇ϕ = 0 (9)

We employ the Fast Local Level Set (FLLS) version ofthe method,30 with the third-order-accurate RK-TVDscheme for time discretization and the seventh-order Linear Weighted Essentially Non-Oscillatory

LWENO7 scheme for space discretization.26

Bulk fluids. We employ the following “pseudocom-pressible” equation of state:

P = (ρ − ρ0) c2

s(10)

where ρ0

is the reference density and cs

is the numer-ical sound speed, which are input parameters of thenumerical model. The numerical sound speed is setto a constant in such a way that the numerical Mach

number M =|u|max

cs≤ 0.02.

Setting β = 0 and defining total energy asE = ρ

0C

pT , where C

pand T are specific heat and

temperature, correspondingly, energy equation (4) isdecoupled from eqs.(1)-(3).

For time discretization of eqs.(1)-(4), we use thethird-order-accurate Runge-Kutta scheme. Fluxes aretreated in the following manner (here, shown for x-direction only). Defining the vectors of conservativevariables and fluxes as:

U =

ρρuρv

and F =

ρuρu2 + P − T

xx

ρuv − Tyx

(11)

numerical fluxes at cell edges are computed in theLocal-Lax-Friedrichs-like (LLF) manner:

F(i+ 1

2,j)

= 12

[

FL

(i+ 12

,j)+ F

R

(i+ 12

,j)

]

+

γλmax

(

UL

(i+ 12

,j)− U

R

(i+ 12

,j)

) (12)

where

λmax

= max

(∣

∣

∣

∣

uL

(i+ 12

,j)± c

s

∣

∣

∣

∣

,

∣

∣

∣

∣

uR

(i+ 12

,j)± c

s

∣

∣

∣

∣

)

(13)

FL

(i+ 12

,j)= F

(

UL

(i+ 12

,j), T

L

xx(i+ 1

2,j)

, TL

yx(i+ 1

2,j)

)

FR

(i+ 12

,j)= F

(

UR

(i+ 12

,j), T

R

xx(i+ 1

2,j)

, TR

yx(i+ 1

2,j)

)

(14)

and the Left and Right states of the vector of con-servative variables and viscous stress tensor are eval-uated with LWENO

Ninterpolation schemes,26 using

currently available cell-centered values U(i,j)

, Txx(i,j)

and Tyx(i,j)

, where the latter two are computed with

the 4th-order-accurate central differencing scheme. Inmost computations presented here, we employed the

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American Institute of Aeronautics and Astronautics Paper 2005–5349

Fig. 1 a) Example of a hierarchy of SAMR meshes. b) Example of a SAMR mesh with three levels ofadaptation and refinement ratio 4.

seventh-order scheme:

LWENO7

:

ΦL

(i+ 12

,j)= −Φ

(i−3,j)

140 +5Φ

(i−2,j)

84 − 101Φ(i−1,j)

420

+319Φ

(i,j)

420 +107Φ

(i+1,j)

210 − 19Φ(i+2,j)

210

+Φ

(i+3,j)

105

ΦR

(i+ 12

,j)= −Φ

(i+4,j)

140 +5Φ

(i+3,j)

84 − 101Φ(i+2,j)

420

+319Φ

(i+1,j)

420 +107Φ

(i,j)

210 − 19Φ(i−1,j)

210

+Φ

(i−2,j)

105

(15)

Parameter γ is used to control numerical diffusion.With γ = 0, we would have high-order central dif-ferencing. However, this scheme is unstable. Adding5% of LLF’s numerical diffusion (γ = 0.05) is enoughto produce stable and, at the same time, highly-accurate results. In the case of large-density-ratiomultiphase flows,

ρ1

ρ2∼ 1000, we use γ = 1 (classical

LLF flux treatment). The above-described schemewas validated on a number of benchmark tests forincompressible fluid dynamics. As an example, weprovide a comparison of this pseudocompressible(PSC) method with projection and LBE methods,in Appendix A, using the classical “doubly periodicshear layer” test.

Structured Adaptive Mesh Refinement(SAMR). Computational mesh is refined nearthe interface, using the level-set-based “tagging” cri-terium for refinement/coarsening.27 For this purpose,the above-described algorithms are incorporated intothe Structured Adaptive Mesh Refinement Appli-cation Infrastructure (SAMRAI),16, 35, 44 developed

at the Lawrence Livermore National Laboratory(LLNL). SAMRAI is based on the SAMR algo-rithm, introduced by Berger and co-authors forhyperbolic conservation laws.3–5 The SAMR grid

Ωh

L−1is composed of a sequence of nested, logically

rectangular meshes, with L levels of adaptation,h

k= h

0, h

1, ..., h

k|k=0,...,L−1, h

k+1≤ h

k, where h

k

is a grid size of the kth SAMR level. The coarsest level

Ωh0 of a hierarchy Ω

h0 ⊃ Ω

h1 ⊃ ... ⊃ Ω

hL−1

covers

the entire computational domain. Each level Ωh

k

consists of a union of Mk logically rectangular patches,G

k,m

∣

∣

m=0,...,Mk. The patches on different levels are not

nested, i.e. they may overlap and Gk,i

∩ Gk,j

6= 0∣

∣

i 6=j.

Thus, a point x of the computational domain ingeneral may exist on several grids. The solutionvector U(x) is taken from the finest level. If there areseveral equally fine grids containing the point, one canuse the value from any of them, since the solutionson the intersection of overlapping same-level grids areidentical. An example of a hierarchy and SAMR gridwith three levels of adaptation is shown in Figure 1.

Time steps on different SAMR levels are chosento have the same CFL number on all levels, i.e.∆t

0

h0

=∆t

1

h1

= ... =∆t

L−1

hL−1

and r =h

k

hk+1

=tk

tk+1

,

where r is the grid refinement ratio. With this, thesame explicit difference scheme is stable on all levels,and, at the same time, the smallest time step of thefinest level is not imposed globally. Time update of asolution on a hierarchy is organized in such a way, asto proceed sequentially from the coarser to finer grids.Thus, before updating at any level, the next-coarser

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American Institute of Aeronautics and Astronautics Paper 2005–5349

level solution must be already available, so as to allowthe complete inter-level communications needed forpopulating each patch’s ghost cells by interpolationin both time and space. More about time update ofSAMR meshes can be found in refs.3, 27, 35

For inter-level “coarse-to-fine” and “fine-to-coarse”communications, we employ the “conservative aver-age”3 and bi-cubic splines,27 correspondingly. Nearthe interface, these prolongation/restriction operatorsare modified using the level set function, in order toeliminate stability issues, as discussed in our previousstudy.27

Sharp Treatment of Interface Jump

Conditions

For interface coupling, we follow the proceduredeveloped in our previous study:27 a) compute nu-merical fluxes, neglecting the presence of interface; b)detect and tag all cells near the interface; c) for eachcomputational cell near the interface, dynamicallycreate a matrix V of the size 5×(2s − 1), whichcontains vectors of primitive variables and compo-nents of viscous stress tensor as its rows, V

(1−s),...,(s),

where V =(

P, u, v, Txx

, Txy

)Tand s is the stencil of

spatial discretization scheme. Then, d) populate thismatrix, taking into account interface pressure andstress tensor jumps. Since the density at the interfaceis discontinuous, and because it is well-known thatapplication of a standard finite difference scheme toa discontinuous function leads to large errors due tosignificant truncation errors,13 we effectively replace[ρ] by creating a numerically continuous profile ofdensity within interfacial cells, based on the principleof having continuous pressure in the absence of surfacetension†. Finally, e) using V, re-compute numericalfluxes.

More specifically, the coupling algorithm is imple-mented as follows (described here for treatment inx-direction only).

I. At the beginning of each time step, we identify allintersections of zero-level-set with grid lines (subcellmarkers). Subcell positions are characterized by θ

I,

computed as|ϕ

(i)|

|ϕ(i+1)

|+|ϕ(i)

| in 1D. Multidimensional

correction is described in ref.27 The subcell markersare utilized as reference points for coupling of numer-ical solutions across fluid-fluid interfaces.

II. We run over all (i,j) cells and compute numericalfluxes F

(i+ 12

,j)assuming that the cells are fully occu-

†The basic idea of this treatment of density jump is some-what similar to the treatment of entropy jump by one-sidedextrapolation in GFM.13

pied by the “positive” fluid, if ϕ(i,j)

≥ 0, or by the“negative” fluid, if ϕ

(i,j)< 0. Simultaneously, we tag

all cells into three groups: 1) “Internal, unaffected bysubcell markers” cells (located far from the interface);2) “Affected by subcell markers” cells (those whichare within the stencil range near subcell markers); 3)“Cut” cells (those which contain subcell markers).

III. Compute interface curvature at subcell markers.Note that each subcell marker is bracketed by two “cutcells”, (i,j) and (i+1,j). First, we compute curvature atthe center of each Cut cell, using the fourth-order-accurate central difference discretization of equations:

Unit normal: n = ∇ϕ|∇ϕ|

Curvature: κ = ∇ ·(

∇ϕ|∇ϕ|

) (16)

and then we interpolate to the subcell position as κI=

κ(i,j)

θI+ κ

(i+1,j)(1 − θ

I). Interfacial pressure jump due

to surface tension at subcell markers can be computedas:

[P ]I= (1 + η)κ

Iσ (17)

IV. Running over all “Cut” and “Affected by subcellmarkers” cells, the numerical fluxes are re-computedin the following way††.

a. First, we allocate memory for the above-discussedmatrix V.

b. Second, the matrix V is filled by values from thecorresponding fluid cells, i.e. V

1(r+s)= P

(i+r,j),

V2(r+s)

= u(i+r,j)

, V3(r+s)

= v(i+r,j)

, V4(r+s)

=

Txx(i+r,j)

and V5(r+s)

= Txy(i+r,j), r=1−s,...s.

c. Next, if there is a subcell marker to the left of thecell (i,j) (denoted here as “LM”, Left Marker), theelements of the matrix V corresponding to cells

(i+r,j) located to the left of the LM are populatedas follows.

Pressure. First, we add a correction term, de-signed to effectively eliminate density jump [ρ],i.e., V

1(r+s)+ = ∆P

(r), where ∆P

(r):

0 if

ϕ(i+r,j)

≥ 0 & ϕ(i,j)

≥ 0or

ϕ(i+r,j)

< 0 & ϕ(i,j)

< 0

∆Pρ if ϕ(i+r,j)

< 0 and ϕ(i,j)

≥ 0−∆P

ρotherwise

(18)

The density jump correction ∆Pρ

is computed us-ing equations of state (10):

∆Pρ

=(

ρ+− ρ

0

)

c2+−

(

ρ−− ρ

0

)

c2−

(19)

††Since we operate here in the incompressible limit M → 0,we do not need two-fluid Riemann solvers for interface couplingat “Cut” cells, employed in our previous study27 for highly com-pressible multiphase flows. Therefore, treatment of “Cut” and“Affected by subcell markers” cells is identical.

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American Institute of Aeronautics and Astronautics Paper 2005–5349

where ρ±

and c±

are the density and numericalsound speed for the “positive”/“negative” fluids,correspondingly. With this treatment, in the ab-sence of surface tension, the pressure is effectivelycontinuous across the interface. To account forthe surface tension, we further correct V1(r+s)

as:

V1(r+s)+ =

[P ]I

if ϕ(i+r,j)

< 0 & ϕ(i,j)

≥ 0

− [P ]I

if ϕ(i+r,j)

≥ 0 & ϕ(i,j)

< 0(20)

In the case of gravity, we also include gravitycorrection terms, to represent local hydraulichead changes due to the presence of anotherfluid, on a subcell level.

Velocity. Since the velocity is continuous acrossthe interface, [u] = 0, no additional correctionsare necessary.

Stresses. Jumps in viscous stresses are incorpo-rated as

V4(r+s)= Txx(iL,j)

V5(r+s)

= Txy(iL,j)

(21)

where (iL,j) is the index of the cell to the leftof the marker LM. Since the velocity is treatedas a continuous variable, with eq.(21), jumps inviscous stresses are ensured through the jumpsin dynamic viscosity, [µ], since T

xx(i,j)and T

xy(i,j)

are computed using local viscosity: µ(i,j)

= µ+

ifϕ

(i,j)≥ 0 and µ

(i,j)= µ

−otherwise.

d. If there is a subcell marker to the right of the cell

(i,j) (denoted as “RM”, Right Marker), the ele-ments of the matrix V corresponding to the cells

(i+r,j) located to the right of the RM are corrected,using similar arguments as given above for step c.

Finally, using V, numerical fluxes are re-computedas described in previous section, eqs.(11)-(15).

V. Since the whole algorithm is Cartesian-grid-based,there are always a number of cells which changetheir “signs” (fluid) due to motion of the level setat a particular time step. We denote these cellsas “degenerate”. The solution at this time step for“degenerate” cells is not appropriate, since it wasobtained assuming that these cells are occupied byanother fluid. In the GFM,13 this situation is takencare of by (i) keeping two sets of flow variables for eachmodeled fluid; and (ii) using extrapolation techniquesto populate ghost nodes. As a ghost node is uncovered- the extrapolated solution is always available. Sincewe do not use ghost fields and extrapolation here, the“degenerate” cells are corrected using interpolation

Fig. 2 Pressure jump for a spherical bubble atequilibrium: CSF model (top) vs. ST (bottom).Uniform grid with resolution of 20 computationalnodes per diameter.

Fig. 3 Spurious velocity for a spherical bubble atequilibrium: CSF model (top) vs. ST (bottom).Uniform grid with resolution of 20 computationalnodes per diameter.

on irregular datasets, as described in our previouspaper.27 Here, instead of the “Natural NeighborInterpolation (NNI)” used in ref.,27 we employ the“modified Shepard’s” approach,34 since it can beeasily extended to three dimensions.

The above-described algorithm can be straightfor-

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American Institute of Aeronautics and Astronautics Paper 2005–5349

Fig. 4 Dynamics of the interface for three grid resolutions using the ST method. Red, solid lines: 20nodes/S; Blue, dash lines: 40 nodes/S; Green, dash-dot lines: 80 nodes/S.

Fig. 5 Dynamics of the effective radius for theoscillating bubble test.

wardly modified for highly-compressible flows withshock waves. In this case, the treatment shouldinclude two-fluid Riemann solutions at subcell mark-ers, as described in our previous study.27 Finally,even though all numerical examples shown here aretwo-dimensional, extension to 3D is trivial.

Numerical Examples

In this section, the performance of our sharp inter-facial treatment is tested on five numerical examples.In example I, we consider a spherical bubble atequilibrium, with a particular focus placed on thesharpness of the pressure jump and the level of spuri-ous velocity, comparing the result to the implementedin our code surface-tension-smearing CSF approach.In example II, we investigate the performance of oursurface tension treatment on a dynamic problem withoscillation of non-equilibrium-shape bubble. Of par-ticular interest here the convergence to non-decayingbehavior, with refinement of the computationalgrid. In example III, we concentrate on ability toaccurately describe the Rayleigh-Taylor instability,one of the two most important interfacial instabilitymechanisms. Investigation of the Kelvin-Helmholtzinstability will be performed in future. In exampleIV, we test our method for large-density-ratio flow,using the case of a rising air bubble in water. Finally,in example V, we apply our method for a liquiddrop breakup under external gaseous flow, in a first

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American Institute of Aeronautics and Astronautics Paper 2005–5349

Fig. 6 The dependence of the disturbance

growth rate, α (measured in units of

g2

ν

1/3

),

on wavenumber, κ (measured in units of

g2

ν2

1/3

)

and surface tension, σ (measured in units of (ρh +

ρl)gν4

1/3

). Comparison with analytical solution(Chandrasekhar,7 1981). Computational resultsare obtained on uniform grid with resolution of 64nodes/W.

attempt to properly capture the basic regimes of thedrop breakup dynamics.

In all numerical tests, we denote AMR simula-

tions as SAMRA/B

C, where A is the grid refinement

ratio between levels of adaptation in a hierarchy ofSAMR grids, B is the total number of levels in ahierarchy, and C is the effective grid resolution ifcorresponding uniform grids were applied. To monitormass conservation, we compute Ω

−

as describedin a companion paper26 of the same proceedings.Unless otherwise specified, we use the seventh orderLWENO

7scheme for spatial discretization of fluid

dynamics solver, eq.(15).

Example 1: Laplace Law

A spherical (axisymmetrical) bubble of radius 0.2 ispositioned at the center of the computational domainof size 1×1. Initial pressure field was set to satisfy theLaplace law:

∆P =2σ

R(22)

At equilibrium, the solution should preserve sharppressure jump at the interface. This is achievedwith our and Kang et al.17’s ST treatment of surfacetension, Fig.2. The CSF’s pressure is smeared over 3-4nodes, consistently with the chosen diameter (3∆h) ofthe tube for smearing surface tension. Furthermore,in the CSF, there are non-zero velocity streams nearthe interface. Spurious velocity field exists also forthe ST models (both ours and ST/GFM), due tonumerical discretization errors in computing curvatureat positions of subcell markers, κI . However, thesevelocities are negligible - three orders of magnitudelower than those of the CSF method.

Example 2: Oscillation of Non-Equilibrium-ShapeBubble

A gas bubble (density ratioρliq

ρgas= 10) is placed in

the center of the computational domain of size 1×1,under uniform pressure conditions. Initial shape ofthe bubble is square (S = 0.4 × 0.4), with cornersrounded as shown in Fig.4 (Rc = 1/20). Simulationsare performed on a sequence of grids: uniform 502 (20

nodes per S), SAMR2/2

40/Sand SAMR

2/2

80/S.

Due to the effects of the surface tension, the bubblebegins to oscillate, Fig.4, with a period of oscillationapproximately 0.7. Since there are no any dissipativephysical mechanisms in the system, the bubble shouldoscillate indefinitely.

Computational results are presented in Figs.4 and5. Due to numerical diffusion in under-resolvedconfigurations with resolutions of 20 and 40 nodes/S,the amplitudes are seen to decay. With a sufficientgrid resolution of corners (10 nodes/Rc, correspondingto 80 nodes/S), the oscillations of bubble becomeessentially non-decaying, for both ST and CSF sur-face tension treatments (in Fig.5, grid convergence isshown only for the ST method). However, the periodof oscillations of the CSF method is slightly higherthan the one of the ST method.

Example 3: Rayleigh-Taylor Instability

A heavy fluid (density ρh) is placed atop of a layer

of a lighter fluid (density ρl) in the gravitational field.

Small perturbations are added to the interface, gener-

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Fig. 7 Dynamics of the interface and velocity field for a single-mode Rayleigh-Taylor instability test.

SAMR2/3

256/W. At=0.5. σ = 37.5(ρh + ρl)

gν4

h

1/3. Reh = 1000 and Rel = 2000.

ating the level set as

Single-mode: ϕ(x) = y −[

H0 + ǫ cos(

2πW x

)]

Multi-mode: ϕ(x) = y −[

H0 + ǫ7∑

i=0

cos(πω

i

W x)

]

(23)

where W is the width of the computational domain,H0 is the position of the interface, ǫ is the amplitudeof initial disturbances (0.02 for a single-mode and0.000125 for a multi-mode) and ω

iare the wavenum-

bers (4,14,23,28,33,42,51,59). Each simulation isstarted from the motionless configuration with theinitial pressure field prescribed by the local hydraulichead. Simulations are performed on a sequence

of grids: uniform 64×256 (64/W), SAMR2/2

128/W,

SAMR2/3

256/Wand SAMR

2/4

512/W.

Single-mode. Computational results for a single-mode RT instability are presented in Figs.6 and 7.

Atwood numbers, defined as At =ρh−ρ

l

ρh+ρ

l

, were 0.5

and 0.875. Viscosity was varied to achieve the desired

wavenumber, measured in units of(

g2

ν2

)1/3

. First,

we analyze the dependence of the disturbance growthrate on the wavenumber and surface tension. Theanalytical solution is available from Chandrasekhar.7

As one can see from Fig.6, the computed by the

ST model growth rates, α =ln h(t)

ǫ

t , are in excellentagreement with the theory, accurately describing boththe maximum growth rate and the critical stabilitylimits, due to surface tension. Importantly, settingamplitudes of initial disturbances below the resolvedgrid scales, we found no spurious grid-scale (noise)instabilities, in a wide range of tested wavenumbers,surface tensions and disturbance amplitudes. Further-more, as shown by grid convergence study in Table 1,our interface treatment is high-order-accurate.

Multi-mode. Fig.8 illustrates the dynamics ofthe interface for a multi-mode RT instability. Initialconditions for this test are similar to those of Bell andMarcus,2 i.e. density ratio

ρliq

ρgas= 7 and At = 0.75.

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Fig. 8 Dynamics of the interface for a multi-mode Rayleigh-Taylor instability test. SAMR2/4

512/W. At=0.75.

σ = 0.0875(ρh + ρl)gν4

h

1/3

.

Table 1 Grid convergence study for the growthrate in the single-mode Rayleigh-Taylor instability.At=0.5. σ = 1, κ = 0.6 (in dimensionless units, seecaption to Fig.6). The exact solution is αexact =9.711 · 10−2.

Grid resolution α Error Rate32 nodes/W 0.103824 6.71·10−3

1.6864 nodes/W 9.92 · 10−2 2.093·10−3

5.05128 nodes/W 9.70216 · 10−2 8.837·10−5

Kinematic viscosities for heavy and light fluids arechosen to get Re

h= 1000 and Re

l= 2000, where

the Reynolds numbers are defined as Rek

=√

WgWν

k

.

To test the performance of our sharp surface ten-sion model, surface tension coefficient was set to

σ = 0.0875(ρh

+ ρl)(

gν4h

)1/3.

There are eight distinct waves growing from theinitially perturbed interface. It can be seen that, atthe early stage, the waves grow nearly independent ofeach other. Eventually, strong interactions develop,causing significant turbulent-like mixing. Similarly tothe simulation by Bell and Marcus,2 there are vortexdipoles with length scales significantly larger than theinitial perturbation.

The effect of grid resolution on interface structurein the non-linear regime is shown in Fig.9. Onthe grid 256/W, the grid-scale filaments/drops areunder-resolved, compared to the simulation with 512nodes/W. Nevertheless, due to high-order-accuratetreatment of the level set equation, mass conservationis maintained within 0.2%, see Fig.10.

The bubble acceleration constant αb

providesthe most basic characterization of the Rayleigh-

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Fig. 9 On grid convergence for a multi-modeRayleigh-Taylor instability test. t = 2.3.

Taylor instability growth in the non-linear regime,h

b(t) = α

bAt gt2, where h

b(t) = (y

b(t) − H

0) is

the position of the bubble front. Experimental datatogether with available computational results aresummarized in Table 2. Our α

bis consistent with

both experiments and the most recent simulations bythe advanced front-tracking approach.14

Example 4: Rising Gas Bubble in Liquid

A gas bubble of radius 0.5 is centered in thecomputational domain of size 10×10, filled by liquid.Density and dynamic viscosity ratios are set to 1000.Kinematic viscosities and gravity are chosen a waysuch that the scaling parameters are Bo = 9.99 andRe = 1000, where the Bond and Reynolds numbers

are defined as Bo =g[ρ]D2

b

σ and Re =

√D

bgD

b

νl

,

respectively. The simulation was started from themotionless configuration and pressure field defined bythe local hydraulic head.

For stability reasons, the numerical diffusion coef-

Fig. 10 On mass conservation for a multi-modeRayleigh-Taylor instability test.

Fig. 11 History of the position and velocity of thebubble’s center of inertia and effective diameter.

Table 2 Summary of the experimental and simu-lation data for αb

ExperimentsRead33 1984 α

b∼ 0.063 − 0.077

Kucherenko et al.18 1991 αb∼ 0.07

Snider&Andrews39 1994 αb∼ 0.07

Schneider et al.36 1998 αb

> 0.054Dimonte&Schneider12 2000 α

b∼ 0.05

SimulationsYoungs45, 46 1984 α

b∼ 0.04 − 0.05

He at al.15 1999 αb∼ 0.04

Glimm at al.14 2001 αb∼ 0.07

Present study 2005 αb∼ 0.06 − 0.07

ficient γ used for computation of the numerical fluxin eq.(12) is increased to 1. For spatial interpolation,the third-order-accurate LWENO

3scheme is applied

for fluid dynamics solver.

Computational results are presented in Figs.11 and

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Fig. 12 Dynamics of the interface and velocity field for the “rising gas bubble” test. SAMR2/4

40/D. Time

is measured in unitsp

Db/g.

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Fig. 13 Dynamics of the interface and velocity field for the cylindrical drop deformation. We=5.

SAMR2/5

160/D. For velocity, 20 isolines are plotted, uniformly distributed from 0 to 2.11 (measured in units

U∝).

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Fig. 14 Dynamics of the interface and velocity field for the cylindrical drop deformation. We=50.

SAMR2/5

160/Dmesh is shown for t = 3.7. For velocity, 20 isolines are plotted, uniformly distributed from 0

to 2.01 (measured in units U∝).

12. Dynamics of the bubble rise can be divided intothree major stages. During the first 0.7 time units, thebubble accelerates. Then, the bubble cap has beganto form, with a liquid tongue penetrating into thegas, 0.7 < t < 1.8. At this stage, the bubble slightlydecelerates, as shown in Fig.11. For the chosen Bondnumber, the surface tension wins, preventing furtherdevelopment of the cap. Finally, for t > 2, the bubblebegins to flatten, leading to further deceleration andrapid increase of its effective diameter. Our approachis able to maintain mass conservation very well, evenwith very coarse grid (40/D) used here: by the endof the simulation, t = 3.3, only 0.6% of the total gasmass has been lost.

Example 5: Breakup of Liquid Drops

A drop of radius 0.5 is positioned at x = (4; 5) inthe computational domain of size 10×10. At zeromoment, we set uniform pressure in the whole domain

and turn on the inflow boundary conditions at theleft boundary. All the rest boundaries are kept atfree outflow boundary conditions. In all simulations,we use relatively low density ratio

ρl

ρg= 10. The

free-stream velocity U∝

and material properties arechosen so that the scaling parameters are Re = 1000and We is varied from 5 to 50, where the Reynolds

and Weber numbers are defined as Re =U

∝D

d

νgand

We =ρgU2

∝

Dd

σ , correspondingly. The ratio of dynamicviscosities is set to one.

We use SAMR2/5

160/D. The actual number of compu-

tational cells is varied from 4.5 · 104 to 8.5 · 104. If auniform grid with the same resolution were applied,it would require approximately 2.5 · 106 cells. Eachrun took approximately 100 hrs on four INTEL-Xeon2.8GHz CPUs.

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Fig. 15 Dynamics of the interface and velocity field for the axisymmetrical drop deformation. We=50.

SAMR2/5

160/D. For velocity, 20 isolines are plotted, uniformly distributed from 0 to 1.76 (measured in units

U∝).

Figs.13-15 present dynamics of interface and ab-solute value of fluid velocity for time up to t = 4,where the time is given in units D

U∝

, for different

We numbers. In the range of We numbers from 5to 50, we observed two different regimes of the dropbreakup. Under relatively low We numbers . 10, thedrop passes through three stages: first, it is subjectedto shearing, forming a “forward bag”, t . 2; thenit begins to flatten, t = 2.5; and, finally, it forms a“backward bag”, due to the penetration of Taylorwaves, t = 4; see Fig.13. Under larger We numbers& 20, the drop is always sheared, forming a “forwardbag”, Figs.14 and 15. This kind of behavior is qual-itatively consistent with experimental observationsof drop breakups discussed by Theofanous et al.42

Direct comparison with experiment is unfeasible dueto different flow conditions.

The effect of the We number is demonstratedin Fig.16, which shows the shapes of cylindricaldrops for time t = 3.7. On the top, we presentedthe “backward-bag-forming” simulations, . 10, andat the bottom - the “forward-bag-forming” ones, & 20.

In all numerical simulations presented here, themass is conserved very well, with losses below 0.2%.This is attributed to high resolution due to AMR(an example of SAMR mesh for t = 3.7 of We = 50simulation is shown in Fig.14) and the seventh-order-accurate LWENO

7scheme used for discretization of

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Fig. 16 Effect of We number on the shape of the drop, for t = 3.7. SAMR2/5

160/D. The velocities are plotted

with 20 isolines, uniformly distributed from 0 to 2 (measured in units U∝).

the level set equation.26

Concluding Remarks

• Compared to treatments based on δ- and CSF-formulations, the sharp treatment of interface isshown to possess advantageous performance, i.e.,minimal spurious velocity and interfacial pressurejump in one node. This confirms the work byKang et al.17 We believe that such features areessential to the accuracy of DNS of realistic mul-tiphase flow configurations, when description ofnear-interfacial flows is not hampered by spuriouseffect of interfacial stress treatment.

• In comparison to Kang et al.,17 the present imple-mentation of sharp interface is done for collocatedgrids and pseudocompressible solver, that renderthe method simple for AMR implementation andextendable to compressible systems. These fea-tures are instrumental in dealing with interfacial

breakup problems of practical interest.

• Our method is accurate for description ofRayleigh-Taylor instability growth rates, in bothlinear and non-linear regimes. The performanceof the method for very high values of At numberand for the Kelvin-Helmholtz instability will beexamined in future.

• As demonstrated on the example of rising gasbubble, the method is stable for large-density-ratio flows.

• The method enables effective simulations of defor-mations and breakups of light liquid drops (den-sity ratio ∼ 10). Computational results presentedhere seem to agree with expectation, includingformation of the “backward bag” under We ∼ 10conditions.

Acknowledgment. This work was supported by the

Lawrence Livermore National Laboratory (“ALPHA”, “MIX”

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Fig. 17 Dynamics of the vorticity field for inviscid doubly-periodic shear layer. LWENO7 scheme with3 AMR levels and refinement ratio 4, corresponding to effective grid resolution 10242. Numerical Machnumber M=0.02.

and “ASOS” projects), the National Ground Intelligence

Center (NGIC) and the US Army. Support of Drs. Frank

Handler and Glen Nakafuji at LLNL, Dr. Rick Babarsky at

NGIC, Ed Conley, JBCCOM at RDEC, and Mr. John Pace

and Mr. Charles Fromer of the JSTO at DTRA is gratefully

acknowledged.

References1Bell, J.B., Colella, P., and Glaz, H.M., A Second-Order Pro-

jection Method for the Incompressible Navier-Stokes Equation,Journal of Computational Physics, 85, 257-283, 1989.

2Bell, J.B., and Marcus, D.L., A Second-Order ProjectionMethod for Variable-Density Flows, Journal of ComputationalPhysics, 101, 334-348 (1992).

3Berger, M.J., and Colella, P., Local Adaptive Mesh Re-finement for Shock Hydrodynamics, Journal of ComputationalPhysics, 82, 67-84 (1989).

4Berger, M.J., and Oliger, J., Adaptive Mesh Refinement forHyperbolic Partial Differential Equations, Journal of Compu-tational Physics, 53, 482-512 (1984).

5Berger, M.J., and Rigoutsos, I., An Algorithm for PointClustering and Grid Generation, IEEE Transactions on Sys-tems, Man, and Cybernetics, 21(5), 1278-1286 (1991).

6Brackbill, J.U., Kothe, D.B., and Zemach, C., A COntinuumMethod for Modeling Surface Tension, Journal of Computa-tional Physics, 100, 335-354 (1992).

7Chandrasekhar, S., Hydrodynamic and Hydromagnetic Sta-bility, Dover Publications Inc., New York, pp.444-449.

8Chen, S., and Doolen, G. D., Lattice Boltzmann Method forFluid Flows, Annu. Rev. Fluid Mech., 30 (4), 329-364, 1998.

9Chorin, A.J., A Numerical Method for Solving Incompress-ible Viscous Flow Problems, Journal of Computational Physics,2, 12, 1967 (Recent edition: 135, pp.118-125, 1997).

10Chorin, A.J., Numerical Solution of the Navier-StokesEquations, Mathematics of Computation, 22, 745-762, 1968.

11Dellar, P.J., Incompressible Limits of Lattice BoltzmannEquations using Multiple Relaxation Times, Journal of Com-putational Physics, 190, 351-370, 2003.

12Dimonte, G., and Schneider, M.B., Density Ratio Depen-dence of Rayleigh-Taylor Mixing for Sustained and ImpulsiveAcceleration Histories, Physics of Fluids, 12, 304 (2000).

13Fedkiw, R.P., Aslam, T., Merriman, B., and Osher, S., ANon-oscillatory Eulerian Approach to Interfaces in Multimate-rial Flows (the Ghost Fluid Method), Journal of ComputationalPhysics, 152, 457-492 (1999).

14Glimm, J., Grove, J.W., Li, X.L., Oh, W., and Sharp, D.H.,A Critical Analysis of Rayleigh-Taylor Growth Rates, Journalof Computational Physics, 169, 652-677 (2001).

15He, X., Chen, S., and Zhang, R., A Lattice BoltzmannScheme for Incompressible Multiphase Flow and Its Applica-tion in Simulation of Rayleigh-Taylor Instability, Journal ofComputational Physics, 152, 642-663 (1999).

16Hornung, R.D., and Kohn, S.R. Managing ApplicationComplexity in the SAMRAI Object-Oriented Framework, inConcurrency and Computation: Practice and Experience (Spe-cial Issue), 14, 347-368 (2002).

17Kang, M., Fedkiw, R.P., and Liu, X.-D., A Boundary Con-dition Capturing Method for Multiphase Incompressible Flow,Journal of Scientific Computing, 15(2), 323-360 (2000).

17 of 20

American Institute of Aeronautics and Astronautics Paper 2005–5349

Fig. 18 Convergence study for inviscid doubly-periodic shear layer. LWENO7-based pseudocom-pressible approach.

18Kucherenko, Ya.I., Shibarshev, L.I., Chitaikin, V.I., Bal-abin, S.A., and Palaev, A.P., Experimental Investigation intothe Self-Similar Mode of the Gravitational Turbulent Mixing, inProceedings of the 3rd International Workshop on the Physicsof Compressible Turbulent Mixing, (1991).

19Landau, L.D., and Lifschitz, E.M., Theoretical Physics, v.1:Mechanics, Nauka, Moscow, 1988 (Fourth Edition).

20Liu, X.-D., Fedkiw, R., and Kang, M., A Boundary Con-dition Capturing Method for Poisson’s Equation on IrregularDomains, Journal of Computational Physics, 160, 151-178(2000).

21Minion, M.L., and Brown, D.L., Performance of Under-Resolved Two-Dimensional Incompressible Flow Simulations, II,

Journal of Computational Physics, 138, 734-765, 1997.22Nourgaliev, R.R., Dinh, T.N., Theofanous, T.G., and

Joseph, D., The Lattice Boltzmann Equation Method: Theoret-ical Interpretation, Numerics, and Implications, Intern. Journalof Multiphase Flow, 29(1), 117-169 (2003).

23Nourgaliev, R.R., Dinh, T.N., and Theofanous, T.G.,Godunov-Bi-Grid-Interpolation-Based Projection Method,UCSB-CRSS Research Report, 07/04-1, 33p., July 17, 2003.

24Nourgaliev, R.R., Dinh, T.N., and Theofanous, T.G.,Comparison of Pseudocompressible Methods for Incompress-ible Fluid Flows: Incompressibility Limits, Convergence, Accu-racy and Efficiency, CRSS-Research Report 06/01/04-1, June1, 2004.

25Nourgaliev, R.R., Dinh, T.N., and Theofanous, T.G., APseudo-Compressibility Method for the Numerical Simulation ofIncompressible Multifluid Flows, Intern. Journal of MultiphaseFlow, 30(7-8), 901-937 (2004).

26Nourgaliev, R.R., Wiri, S., Dinh, T.N., and Theofanous,T.G., Adaptive Strategies for Mass Conservation in Level SetTreatment, Proceedings of the 17th AIAA Computational FluidDynamics Conference, AIAA Paper 2005-5348, June 6-9, 2005,Toronto, Canada.

27Nourgaliev, R.R., Dinh, T.N., and Theofanous, T.G.Adap-tive Characteristics-Based Method for Compressible MultifluidDynamics, Submitted to the Journal of Computational Physics.

28Osher, S., and Sethian, J.A., Fronts Propagating withCurvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, Journal of Computational Physics, 79,12-49 (1988).

29Osher, S., and Fedkiw, R., Level Set Methods and Dy-namic Implicit Surfaces, Applied Mathematical Sciences, 153,Springer-Verlag, New York, Berlin, Heidelberg (2003).

30Peng, D.P., Merriman, B., Osher, S., Zhao, H., and Kang,M., A PDE-Based Fast Local Level Set Method, Journal ofComputational Physics, 155, 410-438 (1999).

31Peskin, C., Numerical Analysis of Blood Flow in the Heart,Journal of Computational Physics, 25, 220-252 (1977).

32Peskin, C., and Printz, B., Improved Volume Conservationin the Computation of Flows with Immersed Elastic Boundaries,Journal of Computational Physics, 105, 33-46 (1993).

33Read, K.I., Experimental Investigation of Turbulent Mix-ing by Rayleigh-Taylor Instability, Physica D, 12, 45 (1984).

34Renka, R.J., Multivariate Interpolation of Large Sets ofScattered Data, ACM Transactions on Mathematical Software,14(2), 139-148 (1988).

35SAMRAI: Structured Adaptive Mesh Refinement Applica-tion Infrastructure, http://www.llnl.gov/CASC/SAMRAI.

36Schneider, M.B., Dimonte, G., and Remington, B., Largeand Small Scale Structure in Rayleigh-Taylor Instability, Phys-ical Review Letters, 80(16), 3507 (1998).

37Serna, S., and Marquina, A., Power ENO Methods: aFifth-Order Accurate Weighted Power ENO Method, Journalof Computational Physics, 194, 632-658 (2004).

38Sethian, J.A., Level Set Methods and Fast Marching Meth-ods, Cambridge University Press, 1999.

39Snider, D.M., and Andrews, M.J., Rayleigh-Taylor andShear Driven Mixing with an Unstable Thermal Stratification,Physics of Fluids, 6(10), 3324 (1994).

40Sussman, M., Smereka, P., and Osher, S., A Level SetMethod for Computing Solutions to Incompressible Two-PhaseFlow, Journal of Computational Physics, 119, 146 (1994).

41Sussman, M., Almgren, A.S., Bell, J.B., Colella, P., Howell.L.H., and Welcome, M.L., An Adaptive Level Set Approach forIncompressible Two-Phase Flows, Journal of ComputationalPhysics, 148, 81-124 (1999).

42Theofanous, T.G., Nourgaliev, R.R., Li, G.J., and Dinh,T.N., Compressible Multi-Hydrodynamics (CMH): Breakup,Mixing, and Dispersal, of Liquids/Solids in High Speed Flows,

18 of 20

American Institute of Aeronautics and Astronautics Paper 2005–5349

Proceedings of the IUTAM Symposium on “Computational Ap-proaches to Disperse Multiphase Flow”, ANL, Argonne, Illinois,4-7 October, 2004, Springer Verlag Heidelberg, 2005.

43Unverdi, S.O., and Tryggvason, G., A Front-TrackingMethod for Viscous, Incompressible, Multi-Fluid Flows, Journalof Computational Physics, 100, 25 (1992).

44Wissink, A., Hysom, D., and Hornung, R., Enhancing Scal-ability of Parallel Structured AMR Calculations, in Proceedingsof the 17th ACM International Conference on Supercomputing(ICS03), San Francisco, CA, June 23-26, 2003, pp. 336-347.

45Youngs, D.L., Numerical Simulation of Turbulent Mixingby Rayleigh-Taylor Instability, Physica D, 12, 32-44 (1984).

46Youngs, D.L., Three-Dimensional Numerical Simulationof Turbulent Mixing by Rayleigh-Taylor Instability, Physics ofFluids A, 3, 1312 (1991).

Appendix A: Doubly-Periodic Shear

Layer

In this appendix, we demonstrate the performanceof the LWENO-based pseudocompressible (PSC)solver for single-phase incompressible fluid flows, com-paring it with the Exact Projection (EP) and Lat-tice Boltzmann Equation (LBE) methods, using the“doubly-periodic shear layer”. This test was originallyintroduced by Bell, Colella and Glaz,1 and examinedmore recently by Minion and Brown.21 The problemconsists of two jets in a doubly periodic geometry towhich a sinusoidal perturbation perpendicular to theorientation of the shear layers is imposed at the lowestwavenumber supported by the computational mesh.In the absence of any additional perturbations, eachof the shear layers rolls up in a single vortex as theflow evolves, Fig.17. To set the problem, the followingvelocity fields are initially generated in the periodicaltwo-dimensional computational domain of size [1 × 1]:

u =

tanh (ϑ (y − 0.25)) , for y ≤ 0.5tanh (ϑ (0.75 − y)) , for y > 0.5

v = γ sin (2π (x + 0.25))(24)

where ϑ is the shear layer width parameter and γis the strength of initial perturbation. Here, wedemonstrate the results for “thin” inviscid shearlayer, i.e. γ = 0.05, ϑ = 80 and ν = 0, as it is morechallenging due to the thinner shear layer formed.

The exact projection method used here is basedon the fourth-order-accurate discrete divergence andgradient operators, the third-order Runge-Kuttatime discretization and the fifth-order power WENOscheme37 for space discretization (see refs.23, 24 fordetails). For the LBE, we use the MRT11 (“MultipleRelaxation Time”) scheme, as it is found to be stableeven in the limiting case of inviscid flow used here, indifference to more commonly used LBGK8, 22 scheme.

Computational results are presented in Figs.17-19.First, the PSC approach is found to be free of “spu-rious vorticities”, found in the exact and approximateprojection methods (see Minion and Brown21 and

Fig.19). Second, the PSC method is grid-convergent,which is demonstrated in Fig.18. The effects of under-resolution on coarse grids revealed in thicker shearlayers and under-revolved vortices. The artificial com-pressibility in M = 0.2 simulations also results inslightly under-revolved vortex (compare Figs.18c vs.18d). Finally, due to the high-order LWENO schemesused for spatial discretization, our PSC approach ismore accurate than both the EP and the LBE-MRTmethods, as the stretched shear layers appear to bethinner under the same grid resolutions, Fig.19.

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Fig. 19 Comparison of the “4th-order exact projection” (top), “Multiple-Relaxation-Time LBE” (center)and “Pseudocompressible (PSC)” (bottom) methods. Numerical Mach number were M=0.02 (for MRT-LBE and PSC methods). WpENO5 and LWENO7 schemes are used for the EP and PSC methods,correspondingly.

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