PHYSICAL REVIEW E 87, 053306 (2013)

Sharp-interface immersed boundary lattice Boltzmann method with reduced spurious-pressureoscillations for moving boundaries

Li Chen,* Yang Yu, and Guoxiang Hou†

School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology,Wuhan 430074, People’s Republic of China

(Received 21 December 2012; revised manuscript received 20 March 2013; published 31 May 2013)

A sharp-interface immersed boundary lattice Boltzmann method (IBLBM) is developed to reduce spurious-pressure oscillations in moving boundary problems. We adopt a cut-cell-based method, i.e., the partially saturatedcomputational cell method, because the primary cause of spurious-pressure oscillations is the failure to obeythe geometric conservation law near the boundary in the sharp-interface IBLBM. We modify a sharp-interfaceIBLBM (ghost fluid method) to fit the cut-cell approach. This boundary condition can guarantee the Dirichletand Neumann boundary conditions for velocity and pressure. Some simulations are shown to test the validity ofthe method, including a circular cylinder with motions that are at rest, moving, oscillatory, and neutrally buoyant.The results illustrate that the method reduces effectively spurious pressure and can simulate moving boundaryproblems, especially when pressure field accuracy is a key concern.

DOI: 10.1103/PhysRevE.87.053306 PACS number(s): 47.11.Qr

I. INTRODUCTION

The lattice Boltzmann method (LBM) is a confirmedeffective and widely used methodology to simulate complexfluid problems because of its outstanding execution efficiency,simple implementation, and natural parallelism [1–3]. TheLBM is a mesoscopic method that simulates fluid flow throughthe collision and stream of mesoscopic particles on the fixedlattice grid. Given the nature of mesoscopic particles, the treat-ment of the boundary between the fluid and solid region can bebased naturally on an approach that describes the movement,collision, and boundary conditions of the mesoscopic particles.When mesoscopic particles move from fluid to solid nodes,these particles bounce back from the boundary walls to the fluidfield, known as the bounce-back (BB) method. The BB methodis also available in moving boundary problems with momentcorrection, which was proposed by Ladd [4]. This method hasalso been extended and improved by numerous scholars to treatcurved boundaries [5–8]. However, BB-based methods usuallylead to unreliable results when the grid node passes through theboundary after the boundary movement because of the normalfixed Cartesian lattice grid used in the LBM. We refer to thepassing node as the fresh node. The normal treatment of thefresh node is interpolated by the surrounding known fluid gridnodes or is replaced by equilibrium values [8]. Therefore,avoiding the spurious-pressure fluctuation is difficult. Thisphenomenon is generated by the violation of the geometricconservation law (GCL) [9]. The GCL was established byKamakoti et al. in their study on the body-fitted grid method.The violation of the GCL is represented in the current studyas the sudden transformation of the fresh node to a fluidnode from a solid node when passing through the movingboundary. The change in the material of the nodal cell is neithercontinuous nor smooth. This violation is the primary sourceof spurious-pressure oscillations for almost all moving gridtechniques based on the fixed grid [10].

*jasonchencl@gmail.com†Corresponding author: houguoxiang@163.com

Feng and Michaelides introduced the immersed boundary(IB) method into the LBM to remedy these problems [11]. TheIB method was introduced by Peskin, who used the methodin a fixed Cartesian grid to study biological fluid problems[12,13]. Numerous years of developing the IB method haverevealed two main types of IBs, namely, the diffused- andsharp-interface IBs. Diffused-interface IBLBMs employ a setof boundary Lagrangian nodes to denote the boundary line anduse the interpolations by the δ function between the boundarynodes and surrounding grid nodes to describe the reaction fromthe boundary to the fluid. The reaction is applied to several gridnodes, including the nodes inside and outside the boundary.Solid nodes are actually considered as fluid nodes. Therefore,no fresh nodes exist in diffused IB methods. The GCL issatisfied naturally. These kinds of IB methods are developedfor several schemes. For example, Niu et al. proposed themomentum-exchanged IB method, which computes for theforce density at the boundary [14]. The diffused IBLBMssimplify the treatment of the boundary and are very suitablefor the complex geometric boundary. However, streamlinesusually penetrate the boundary because the diffused IBLBMsblur the existence of the boundary, and the no-slip conditionis not accurately satisfied [15]. Some corrected methods areproposed for this problem, such as the Shu’s velocity correctionmethod [16,17], Kang’s iterative method [18], and Lu’smultiple relaxation-time special parameter method [19]. Thesharp-interface IBs are another option to avoid the numericalslipping phenomena. Kang proposed a sharp IB method thatuses linear and bilinear interpolation and an LBM that usesthe direct-forcing model [18]. However, the boundaries areconsidered two different situations that lead to more spuriousfluctuation and instability. Another sharp IBLBM is the ghostfluid (GF) method introduced by Tiwari and Vanka [20]from the conventional computational fluid dynamics methods[21,22]. This method is similar to that of Kang, which is aimedat obtaining the solid node (ghost node) near the boundary withthe reversed treatment of the image point in the fluid region.The GF method can guarantee the stability of the schemeand prescribe Dirichlet and Neumann boundary conditions forvelocity and pressure because the uniformed four-node bilinear

053306-11539-3755/2013/87(5)/053306(11) ©2013 American Physical Society

LI CHEN, YANG YU, AND GUOXIANG HOU PHYSICAL REVIEW E 87, 053306 (2013)

interpolation is used. Therefore, the GF is more practical thanKang’s sharp IB method. However, the sharp IBLBMs havethe same fresh node problem as that of the BBs. The freshnode has to be treated after boundary movement because theinformation on the solid nodes is unknown. Spurious-pressureoscillations also occur in the sharp IBLBMs because of thelack of the GCL.

In the present paper, we propose a moving boundarymethod for the sharp IBLBM to reduce the spurious-pressurefluctuation from the fresh node. We adopt the cut-cell-basedmethod to enforce the GCL as inspired by the diffused IBmethod proposed by Noble and Torczynski, i.e., the partiallysaturated computational cell (PSC) method [23]. Noble andTorczynski used the volumetric fraction to weigh the reactionfrom the boundary to the fluid and used the BB-like methodto describe the motion of mesoscopic particles on solidboundaries. In the present study, we used the GF method todeal with the collision term on the boundary instead of theBB-like process. The GF method is adopted to avoid numericalslipping faults and ensure the Dirichlet and Neumann boundaryconditions. Considering that the PSC ensures the GCL, wecombine the advantages of the PSC with the GF method to forma sharp-interface IB boundary condition. We focus on reducingspurious-pressure oscillations. Thus, the method is tested onlyon two-dimensional problems (2D) and boundary deformationis not considered. Further studies on three-dimensional (3D)and deformable solid problems will be summarized in our nextpaper.

This paper is organized into several sections. We begin ourdiscussion by presenting the LBM theory followed by ourproposed boundary method. Numerical results are evaluatedand conclusions are finally summarized.

II. LBM

The LBM is a mesoscopic method to solve Navier-Stokes(NS) equations. The most widely used Bhatnagar-Gross-Krook model [24] evolvement equation in the LBMs can berepresented by

fi(x + ci�t,t + �t) − fi(x,t) = − 1

τ

[fi(x,t) − f

(eq)i (x,t)

],

(1)

where fi(x,t) is the distribution function at position x and timet , which is defined as the quantity of density of mesoscopicparticles, with lattice velocity ci and the ith direction of thelattice. f

(eq)i (x,t) is the equilibrium distribution function, τ is

the dimensionless relaxation time, and �t is the time step. Thedistribution function comprises two parts, i.e., equilibrium andnonequilibrium, as follows:

fi(x,t) = f(eq)i (x,t) + f

(neq)i (x,t). (2)

In this paper, the boundary method is theoretically independentfrom the dimensionality and the number of velocity. Therefore,we choose the standard D2Q9 model to perform some tests.The equilibrium distribution function can be expressed as

f(eq)i = ωiρ

[1 + ci · u

c2s

+ (ci · u)2

2c4s

− u2

2c2s

], (3)

where ρ is the fluid density, u is the fluid velocity, and ωi isthe weight factor. For the D2Q9 model, ωi and ci read as

ωi =

⎧⎪⎨⎪⎩

4/9, i = 0

1/9, i = 1,2,3,4

1/36, i = 5,6,7,8(4)

ci =

⎧⎪⎨⎪⎩

(0,0), i = 0(cos

[ (i−1)π2

], sin

[ (i−1)π2

])c, i = 1,2,3,4√

2(cos

[ (2i−1)π4

], sin

[ (2i−1)π4

])c, i = 5,6,7,8

where c = �x/�t is the lattice speed, �x is the space step, andcs = c/

√3 is the speed of lattice sound. In practical coding,

the evolution given by Eq. (1) usually divides into two steps,namely, collision and stream,

f ′i (x,t) = fi(x,t) − 1

τ[fi(x,t) − f

(eq)i (x,t)], (5)

fi(x + ci�t,t + �t) = f ′i (x,t), (6)

where f ′i (x,t) is the postcollision distribution function. Macro-

scopic physical quantities can be summated by the distributionfunction and its moments as

ρ =∑

i

fi, ρu =∑

i

cifi . (7)

Physical viscosity and pressure can be expressed as

ν =(

τ − 1

2

)2

c2s �t, p = c2

s ρ. (8)

III. A NEW SHARP-INTERFACE IMMERSEDBOUNDARY METHOD

Sharp-interface IBLBMs do not satisfy the GCL when thegrid node passes through the boundary. Therefore, we need toadopt the weighting strategy to avoid geometric discontinuity.An example of a good weighting strategy is Noble’s PSCmethod [23], which modifies evolution (1) to

fi(x + ci�t,t + �t)

= fi(x,t) − [1 − Bs(εs,τ )]1

τ

[fi(x,t) − f

(eq)i (x,t)

]+ Bs(εs,τ )s

i , (9)

where εs is the volume fraction for the solid part in each nodalcell, Bs(εs,τ ) is a weighting function related to εs , and s

i isan additional collision operator that accounts for the reactionof the solid boundary. Noble’s investigation reveals two kindsof expressions of s

i [23], i.e., the BB-like expression

si = [

fı(x,t) − f(eq)ı (ρ,u)

] − [fi(x,t) − f

(eq)i (ρ,uw)

](10)

and the superpositionlike expression

si = f

(eq)i (ρ,uw) − fi(x,t)

+(

1 − �t

τ

)[fi(x,t) − f

(eq)i (ρ,u)

], (11)

where ı is the opposite direction of i, and uw is the velocity ofthe solid boundary. A nodal cell in Noble’s PSC is defined as a

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SHARP-INTERFACE IMMERSED BOUNDARY LATTICE . . . PHYSICAL REVIEW E 87, 053306 (2013)

(a)PSC strategy (b)Present strategy

FIG. 1. (Color online) Volume fraction of the PSC method and the present study. In the PSC strategy, εs is defined as the ratio of the solidportion to the whole cell. In the present strategy, εa is defined as the ratio of the small portion to the whole cell.

square cell with its center at the node point. When a nodal cellis far from the boundary, εs = 0 or 1, corresponding to purefluid or solid. When a nodal cell is near the boundary and cutby the boundary line, 0 < εs < 1. Bs(εs,τ ) can be consideredan additional factor for the nonequilibrium relaxing operator.This weighting strategy numerically diffuses the boundaryline between the fluid and solid and smoothens the changein the material of the nodal cell. The GCL is ensured by thisweighting strategy. After several benchmark tests, we foundthat the PSC method can reduce spurious-pressure oscillations.However, avoiding the faults of diffused interface IBLBMsremains difficult.

In this article, we continue to use the above weightingstrategy but replace the collision term with the GF process.The GF method is adopted to avoid numerical slipping faultsand ensure Dirichlet and Neumann boundary conditions.Therefore, the evolution equation given by Eq. (9) withoutthe additional collision term can be rewritten as

fi(x + ci�t,t + �t)

= fi(x,t) − [1 − Ba(εa,τ )]1

τ

[fi(x,t) − f

(eq)i (x,t)

], (12)

where the weighting function Ba(εa,τ ) is related to the volumefraction for the alien part in each nodal cell, εa , which isdifferent from εs . The weighting factor between the two sidesof the boundary should be the same because of the requiredsymmetry of the GF between the two sides of the boundary(Fig. 1). Therefore, εa is the ratio of the solid portion to thewhole cell if the node is outside the boundary, whereas εa isthe ratio of the fluid portion to the whole cell if the node isinside the boundary. These alien parts are always the ratio ofthe small portion to the whole cell. The ratio of εa is usedin the GF process below. Therefore, we have the followingrelationship:

εa ={

εs, εs � 0.5

1 − εs, εs > 0.5, (13)

with εa = 0 for pure fluid and solid for a nodal cell far fromthe boundary. For the nodal cell near the boundary (cut by theboundary line), 0 < εa � 0.5. Ba(εa,τ ) can be considered anadditional factor for the nonequilibrium relaxing operator. Thesimplest form is

Ba(εa,τ ) = εa (14)

that is independent from relax time τ . Noble proposed anothercomplicated form that is much closer to the physical intuitionand empirical result of Poiseuille flow [25],

Ba(εa,τ ) = εa(τ − 1/2)

(1 − εa) + (τ − 1/2). (15)

In the results section, we will compare the two weightingfunctions to obtain the boundary condition.

Calculation requires the cut-cell method, which is usuallycalculated through three methods, namely, the closed-formsolution, polygonal approximations, or the cell decompositiondiscussed by Owen et al. [26]. Owen suggested that usingpolygonal approximations to obtain εa is more effective,considering the balance of the accuracy and the cost ofcomputation. The boundary line that cuts the nodal cell can beincluded in three cases for a 2D situation, namely, cutting off atriangle, a trapezoid, or leaving a triangle (Fig. 2). These casesare differentiated by the number of edges of the polygon from5 to 3. In the 3D case, a nodal cell box cut by the boundary facecan include seven cases, which are also differentiated by thenumber of faces of the polyhedra from 10 to 4. Seo et al. [10]proposed the skilled computation of the polyhedra in 3D. Inthe present paper, we adopt the polygonal approximationsapproach. The coverage area is approximated as a polygonin 2D or a combination of polyhedra in 3D. Therefore, we canobtain εa through classical geometry, which is considered aproper compromise between accuracy and consumption.

The GF method in the LBM is proposed by Tiwariand Vanka [20]. In this paper, the method is used to treatthe boundary reactions instead of Noble’s collision term inEq. (12). The GF method, a type of sharp-interface IB,

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LI CHEN, YANG YU, AND GUOXIANG HOU PHYSICAL REVIEW E 87, 053306 (2013)

(a) (b) (c)

FIG. 2. (Color online) Polygonal approximations of the cut-cell method. The polygons in cases (a)–(c) have 5, 4, and 3 edges, respectively.

considers the solid grid node near the boundary to be a GFnode. The information of the ghost node can be solved by theimage point towards the normal boundary with informationthat can be interpolated by the bilinear scheme from thesurrounding real fluid nodes (Fig. 3).

In the procedure, the interpolation is adopted at the ghostnode after the collision at the last time step and beforethe stream at the current time step. The four-point bilinearinterpolation is given by

φ = ax + by + cxy + d. (16)

This equation requires four known nodes to construct the fourequations that solve the coefficients. The three cases in thedifferent boundary situations based on the varying number ofthe unknown node needed for interpolation are illustrated inFig. 3. The general approaches that deal with these three casesare as follows: For velocity, the interpolation can be rewrittenas

a[αjxj + (1 − αj )x ′j ] + b[αjyj + (1 − αj )y ′

j ] + c[αjxjyj

+ (1 − αj )x ′j y

′j ] + d = αj uj + (1 − αj )u′

j , (17)

where j = 1,2,3,4 is the subscript for the four known fluidnodes, and αj is the index, which is 1 for the known fluid nodeand 0 for the intersection node. For density interpolation, thezero normal gradient condition at the walls is given by

∂φ

∂n= anx + bny + c(xny + ynx) = 0, (18)

where (nx,ny) is the unit vector of normal. Here, the zeronormal gradient condition for the density is only suitable forthe constant density or the pressure in the normal gradient.However, the right side of Eq. (18) should be replaced by afinite quantity if the normal gradient is not zero physically. Ifthe zero normal gradient condition is adopted at the walls, theinterpolation equations can be given by

a[αjxj + (1 − αj )nxj ] + b[αjyj + (1 − αj )nyj ] + c[αjxjyj

+ (1 − αj )(xjnyj + yjnxj )] + αjd = αjρj . (19)

The equation meets the Neumann boundary condition byconsidering the zero normal gradient condition. We nowhave enough equations to complete the interpolation. Afterobtaining the value of velocity and density at the image point,

(a) (b) (c)

FIG. 3. GFIB. � is the ghost (solid) node, � is the fluid node, and � is the image point of the ghost node. • is the intersection of theboundary and the normal line between the ghost and image points. The quadrilateral ABCD is used to interpolate the values at image pointX. (a) G1 is the ghost node, whose image point X is set in the quadrilateral by four known fluid nodes A1, B1, C1, and D1. (b) G2 is the ghostnode. However, node D2, which should be used for interpolation, is unknown and is replaced by point D′

2, the point at the intersection of theboundary and its normal line. (c) G3 is the ghost node, and the two unknown nodes D3 and C3 are substituted with D′

3 and C ′3.

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SHARP-INTERFACE IMMERSED BOUNDARY LATTICE . . . PHYSICAL REVIEW E 87, 053306 (2013)

FIG. 4. Set of flow over a cylinder in a channel.

the information at the ghost node can be easily solved,

ughost = 2uw − uimage, ρghost = ρimage, (20)

where uw is the velocity of wall given by

uw = U + � × (x − xc), (21)

in which U is the translational velocity of the solid object,� is the angular velocity, and xc is the position of its centerof mass. We obtain the macroscopic quantities at the ghostnode to acquire the equilibrium distribution functions fromEq. (3). The nonequilibrium part can be interpolated in thesame manner as the procedure of the density computation,given by

a[αjxj + (1 − αj )nxj ] + b[αjyj + (1 − αj )nyj ] + c[αjxjyj

+ (1 − αj )(xjnyj + yjnxj )] + αjd = αjf(neq)j ,

where f(neq)j can be calculated by Eq. (2) on the known fluid

node. Tiwari’s analysis stated that this extrapolation of thenonequilibrium part is only of first-order accuracy, O(∂).However, the entire computation of the f values becomesO(∂2) because f

(neq)j = O(∂fj ) [27]. We find the distribution

functions at the ghost. However, our method introduces theweighting factor into the evolutionary equation. The weightingfactors on the image and ghost nodes should be the same.However, the image node does not usually exist. Thus, wealso need to calculate Ba directly on the ghost node usingEq. (14) or (15). This phenomenon is the reason we introducethe weighting factor εa , which is different from that ofNoble. The force evaluation at the boundary is based onthe approximate moment-exchange method. The boundarycondition to acquire directly the force by the interaction on theboundary is different from the BB method. However, the fluidnodes that surround the boundary also represent the force from

solid to fluid. Therefore, the force and torque are calculatedby

F =∑x∈B

∑ci∈ci

[f ′i (x,t) + fı(x,t + �t)]ci , (22)

T =∑x∈B

∑ci∈ci

(x − xc) × [f ′i (x,t) + fı(x,t + �t)]ci , (23)

where B is the set of the fluid node, which is near the boundaryand has at least one lattice velocity directed to the solid ghostnode, ci is the lattice velocity directed to the solid for thecurrent node, and ı is the opposite direction of i.

The present work also does not include the deformation ofthe boundary, i.e., the solid object should be rigid. The forceand torque on the object lead to the update of the boundaryline for every lattice time step using

dUdt

= FM

,d�

dt= T

I, (24)

where M and I are the mass and the moment of inertia of thesolid object, respectively. The inertia tensor is just a scalar in2D.

Finally, the fresh node xfresh that passes through theboundary when the boundary moves also needs to be treated.We only need to revert the velocity and compute for thedistribution functions because the GF method leads the GFto obtain the opposite velocity relative to the wall velocity andthe same density of image point,

fi(xfresh) = f(eq)i (u,ρ) + f

(neq)i , (25)

where u is the opposite velocity of the ghost node.The nodes in the solid region are notably insulated by the

ghost node. Therefore, whether or not collision and stream areperformed in the solid region would lead to the same resultsfor the outside fluid simulation.

The proceedings of the boundary treatment can be summa-rized as follows:

(1) εa is obtained for each node by the cut-cell method ofpolygonal approximations.

(2) Ba is obtained for each grid node by Eq. (14) or (15).(3) The information on the ghost node is obtained by

Eqs. (17), (19), (20), and (21).(4) Collision and stream are performed by Eq. (12).(5) Force and torque are calculated by Eqs. (22) and (23).(6) The moving boundary line is updated by Eq. (24).(7) The fresh node is treated by Eq. (25).(8) We proceed to the next time step and return to Step 1.

TABLE I. Comparison of dimensionless coefficients.

n Cd Cl L �p

20 5.3425(4.4322) 0.0125(0.0088) 0.8424(0.9314) 5.6554(5.8393)40 5.3589(4.5909) 0.0108(0.0089) 0.8512(0.8812) 5.8804(5.8047)60 5.3447(4.7041) 0.0105(0.0092) 0.8509(0.8677) 5.8853(5.8251)80 5.4012(4.9208) 0.0106(0.0099) 0.8504(0.8608) 5.8715(5.8250)PSC (n = 64) 5.1926 0.0103 0.8549 5.7915GF (n = 64) 5.6182 0.0103 0.8577 5.9897IBB (n = 80) 5.5591 0.0113 0.8402 5.8132NS 5.5700–5.5900 0.0104–0.0110 0.8420–0.8520 5.8600–5.8800

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LI CHEN, YANG YU, AND GUOXIANG HOU PHYSICAL REVIEW E 87, 053306 (2013)

X

Y

0.5 1 1.5 2 2.5 3 3.50.5

1

1.5

2

2.5

3

3.5

FIG. 5. Streamline and contour of the pressure of flow over acylinder in a channel. The result is from present B and n = 80.

IV. RESULTS

A. Stationary boundary: Flow over a cylinder in a channel

The first case is the flow over a cylinder in a channel.This benchmark problem is tested in the finite element (FEM),finite difference, and finite volume methods, as well as theBB-based LBM, and GFLBM, all of which aim to demonstratetheir consistency and basic ability to deal with the curveboundary conditions. The geometry and boundary conditionsare shown in Fig. 4. The conditions of the inlet and outletare parabolic velocity and constant pressure, respectively,using nonequilibrium extrapolation [27]. The inlet paraboliccondition is

u(0,y) = 4umaxy(H − y)

H 2, v = 0, (26)

and the constant pressure at outlet is set as initial pressure. TheReynolds number is Re = UD/ν = 20 and the Mach numberis Ma = umax/cs = 0.01, where the characteristic velocityU = 2umax/3. A different grid resolution is performed for thesimulations with n = 20,40,60,80, where n is the number ofgrid nodes on diameter D. The drag and lift coefficients of the

fluid force for the cylinder are given by

Cd = 2Fx

ρU 2D, Cl = 2Fy

ρU 2D, (27)

where ρ is the average density of the entire flow field, and Fx

and Fy are calculated by Eq. (22). These definitions are alsoavailable in the following cases.

Table I shows the results of the dimensionless coefficientsof the boundary condition compared with the PSCLBM,GFLBM [20], interpolated bounce-back (IBB) LBM [28],and incompressible NS equations. The PSCLBM is computedin the present study because it has no similar referent case.The incompressible NS equations are solved by the finitedifference, FEM, and finite volume methods [29]. The dragand lift coefficients are calculated by Eq. (27). Table I alsoshows the nondimensional length of the recirculation zone be-hind the cylinder, L = Lr/D, and nondimensional differencein the pressures between the front and back of the cylinder,�p = 2�p/(ρU 2). Values with parentheses in Table I arethe weighting factors in Ba , which are only dependent onthe geometry computed by Eq. (14) called present A. Valueswithout parentheses are the factors dependent on relaxationtime τ computed by Eq. (15), which is called present B. Thesenames are available in the following cases. Figure 5 showsthe streamline of flow over a cylinder. Streamline penetration,usually found in diffused interface IBLBMs, is not present.The contour of the pressure is very smooth.

The relaxation-time-based method present B is found to becloser to the referent results, which means that the descriptionof the relaxation-time-based Ba on Eq. (15) is closer to the realboundary layer. When the weighting factors are independentfrom relaxation time, as shown in the parentheses in Table Ifor present A, the error is larger than the former. However,very little fluctuation exists on Cd and Cl for the differentgrid resolutions. The main cause of the fluctuations is theapproximate moment-exchange method used in the forceevaluation. The relaxation-time-based results reveal that ourapproach matches very well with the referent results. Theaccuracy of our method for the basic ability of boundaryconditions is ensured.

The order of accuracy analysis on the convergence withthe grid resolution is performed by analyzing the root-mean-square norm of the error in velocity and the magnitude of the

10−2

10−1

10−4

10−3

10−2

10−1

Δx

Rel

ativ

e E

rror

and

Max

. Err

or in

Vel

ocity

Error Norm2.14Error Max.2.22

(a)Present A

10−2

10−1

10−5

10−4

10−3

10−2

Δx

Rel

ativ

e E

rror

and

Max

. Err

or in

Vel

ocity

Error Norm2.17Error Max.2.29

(b)Present B

FIG. 6. Relative root-mean-square norm error and the maximum in velocity.

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FIG. 7. Set of moving circular cylinder in a channel.

maximum error in velocity. The accurate solution, which isused to calculate the velocity errors and their associated ratesof convergence, is the result of the finest grid case, n = 80.The definitions of the velocity errors are as follows:

Errornorm =√∑

[(�ux)2 + (�uy)2]∑[(u∗

x)2 + (u∗y)2]

, (28)

Errormax = max[√

(�ux)2 + (�uy)2], (29)

where �ux = ux − u∗x , �uy = uy − u∗

y , in which u∗x and u∗

y

are the components of the accurate solution. The data pointsused for examination are the all-fluid grid nodes withoutthe solid nodes to isolate the convergence of the solutionat the solid grid nodes. Figure 6 shows that the rates ofconvergence of presents A and B are approximately equal to2. Therefore, the boundary conditions still comprise the sameorder of accuracy of space as the LBM. However, the rate ofpresent B is a little higher than present A. The errors of presentB are also much smaller than those of present A, which provesthat relaxation-time-based results are better.

B. Moving boundary: A moving cylinder in a channel

The second case is from Lallemand et al. [8], in which theflow passes through an impulsively started circular cylinderin a channel. Figure 7 shows that a circular cylinder withdiameter D = 24 at time t = 0 is set at an asymmetricposition C(60.3,54), namely, H = 54, in the flow grid,L × W = 1001 × 100. At t = �t , an instantaneous velocityU c = (0.04,0) is given to the cylinder. The inlet and outlet

t

drag

coef

fiec

ient

5000 6000 7000 8000

-5

0

5

restGFpresent Apresent B

t

drag

coef

fiec

ient

6000 6050 6100

-4

-2

0

2

4 restGFpresent Apresent B

(a)The drag coefficient

t

dp

5000 6000 7000 8000

-5

0

5

restGFpresent Apresent B

t

dp

6000 6050 6100

-4

-2

0

2

4 restGFpresent Apresent B

(b)The acoustic pressure

FIG. 8. (Color online) Comparison of force and acoustic pressure among the resting present B, moving GF, and present A and B methods.The acoustic pressures are measured from the measurement point. The treatment of the fresh node on GF is based on the second-orderextrapolation method [8]. The left figure is the drag. The right figure is the acoustic pressure. The drag coefficient Cd is calculated usingEq. (27). The acoustic pressure is defined as dp = 2(p − pmean)/ρU 2

w , where pmean is the time mean pressure.

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LI CHEN, YANG YU, AND GUOXIANG HOU PHYSICAL REVIEW E 87, 053306 (2013)

X

Y

3.5 4 4.5 5 5.5 6 6.50

0.2

0.4

0.6

0.8

1

(a)The streamlines

X

Y

3.5 4 4.5 5 5.5 6 6.50

0.2

0.4

0.6

0.8

1

(b)The vorticity

FIG. 9. (Color online) Stream functions and the vorticity of flow at t = 15 000, which correspond to the resting and moving cases usingpresent method B. The solid line is the rest case (the rest cylinder with moving walls). The dashed line is the moving case (the moving cylinderwith rest walls).

are the periodic boundary conditions, whereas the top andbottom are the extrapolation of the nonequilibrium proposedby Guo et al. [27]. We focus on the following two cases basedon the Galilean invariance. Case (a) allows the cylinder torest but drags the top and bottom boundaries with velocityUw = (−0.04,0) and initial flow velocity U0 = (−0.04,0).Case (b) keeps the cylinder moving with U c = (0.04,0) butcauses the top and bottom boundaries to rest with the initialflow velocity U0 = (0,0). The two cases should producesimilar results to construct a comparable group, which wouldrepresent the efficiency of the moving boundary schemes.We also set a measurement point (microphone) to view thepressure wave, which is apart from the center of the cylinder byapproximately 0.6D to the left, and this distance is maintainedwhen the cylinder moves. In the present method, methods Aand B are compared, where present A uses Eq. (14) and presentB uses Eq. (15), depending on different weighting factors.

Figure 8 shows a comparison of the force and acousticpressures. The rest cylinder using present B is the reference.We find that presents A and B have fewer fluctuations than theGF method, regardless of force or pressure. Using presentB, Fig. 9 compares the streamlines and vorticity betweenthe rest and moving cylinder. A good match is observed in

both the streamlines and the contours of vorticity. However, aphase difference exists between the results obtained using themoving and resting boundaries because of the finite speedof sound and the higher order non-Galilean effects in theviscosities and sound speed [30].

C. Moving boundary: Oscillation of a cylinder

The third case is the oscillation of a cylinder within a narrowrange, in which fresh nodes that pass through the boundary aregenerated, which was used by Seo and Mittal [10]. Figure 10shows that a circular cylinder with diameter D is set at thecenter of the square flow domain, 4D × 4D. This domainhas no-slip wall upper and lower boundaries, and Neumannboundary conditions are the left and right boundaries, namely,the zero gradient of pressure and velocity. The cylinder makesan oscillatory motion sinusoidally in the x direction,

xc(t) = xc(0) + X[1 − cos(2πf t)], uc(t) = U sin(2πf t),

(30)

where the characteristic velocity is U = 2πf X, the ampli-tude of oscillation is X, and the period of oscillation isT = 1/f . Reynolds and Strouhal numbers are defined as

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FIG. 10. Set of oscillation of a cylinder.

Re = UD/ν and St = f D/U , respectively. We chose twosets: (a) with X = 0.05D, Re = 31, and St = 3.2, and (b)with X = 0.125D, Re = 78.5, and St = 1.27. Three differentgrid resolutions were used. The number of the grid node on thediameter D is n = 16,32,64. The initial conditions, namely,zero velocity and constant pressure, are set on each node. TheMach number is 0.1, and c = �x/�t = 1.

Figures 11 and 12 show the results of the sets (a) and (b).IBB, GF, and PSC are the same referent methods as in the firstcase. Present A is the present boundary condition by Eq. (14).

Present B is also the present work of the relaxation-timerelative method by Eq. (15). In Figure 11, the spurious-pressureoscillation is not fierce in set (a) under low Reynolds numberand small amplitude. After zooming up a part of the dragcoefficient line, IBB and GF obviously show the spuriousfluctuation, but the PSC, present A, and present B are smooth.Only present B is close to the arranged values of the IBBand GF, which further proves the physical advantage of therelaxation-time-based method. In Figure 12, set (b) shows thespurious-pressure oscillation of the high Reynolds number forthe different boundary methods. The boundary conditions arealso evidently smoother. Under the high Reynolds situation,the PSC and present A similarly deviate from the averageof IBB and GF. Present B is still the best choice. However,the methods are not completely smooth at the lower gridresolution, similar to the result obtained by Seo and Mittal [10]who reduced, instead of eliminated, the spurious numericalerror. Thus, other sources of errors apart from the non-GCLcould exist. At high grid resolution, n � 32, the methodseffectively restrain the spurious-pressure oscillation. Whenn = 64, the classic sharp-interface methods (IBB and GF) arealmost similar to the present work. Several previous studiesalso mentioned that enough fine grid resolution can suppressspurious-pressure oscillations.

D. Moving boundary: The motion of a neutrally buoyantcylinder in a linear shear flow

The last case is the simulation of the motion of a neutrallybuoyant cylinder in a linear shear flow, which has been

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2-60

-40

-20

0

20

40

60

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

20

40

60

IBBGFPSCpresent Apresent B

(a) n = 16

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2-60

-40

-20

0

20

40

60

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

20

40

60

IBBGFPSCpresent Apresent B

(b) n = 32

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2-60

-40

-20

0

20

40

60

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

20

40

60

IBBGFPSCpresent Apresent B

(c) n = 64

FIG. 11. (Color online) Oscillating cylinder of set (a) Re = 31, St = 3.2, X/D = 0.05.

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LI CHEN, YANG YU, AND GUOXIANG HOU PHYSICAL REVIEW E 87, 053306 (2013)

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2

-15

-10

-5

0

5

10

15

20

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

5

10

15

20

IBBGFPSCpresent Apresent B

(a) n = 16

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2

-15

-10

-5

0

5

10

15

20

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

5

10

15

20

IBBGFPSCpresent Apresent B

(b) n = 32

t/T

dragcoefficient

1 1.2 1.4 1.6 1.8 2

-15

-10

-5

0

5

10

15

20

IBBGFPSCpresent Apresent B

t/T

dragcoefficient

1.6 1.65 1.7 1.75 1.80

5

10

15

20

IBBGFPSCpresent Apresent B

(c) n = 64

FIG. 12. (Color online) Oscillating cylinder of set (b) Re = 78.5, St = 1.27, X/D = 0.125.

validated by numerous researchers using the moment-exchanged diffused IBLBM [14], LBM [11], and FEM [31].Figure 13 shows the set of this case. This case uses severalparameters. The height and length of the channel are H = 1and L = 25. D = 0.25H is the diameter of the circularcylinder. The top and bottom walls of the channel move atUw/2 = 1/20 in opposite directions. The inlet and outlet arethe periodic boundary conditions. The density of the cylinder isequal to that of the fluid (ρs = ρ = 1). The Reynolds number isRe = UwH/ν = 40, and the Mach number is Ma = Uw/cs =0.058. The domain of fluid simulation is on a Cartesian gridof 2000 × 80. The cylinder is set at (0.25H,0.25H) at rest nearthe bottom wall of the channel at the initial time.

The migration on the y direction of the cylinder that varieswith time using the present method is compared with theGF computed in this study, IBLBM [14], and FEM [31] inFig. 14. The migration of the cylinder is found to be close

FIG. 13. The set of a neutrally buoyant cylinder in a linear shearflow.

to the middle line when the system is at equilibrium. Goodagreement is observed between the present results and thoseof the other methods. Present method A is slightly differentfrom B. Present method B and GF also almost overlappedand have very similar abilities in predicting the migration ofan object in the fluid. Present method B and GF are closer toFEM than IBLBM. These results demonstrate the developmentof the method based on the GF method.

t*

y*

0 50 1000.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

y* IB-LBMy* FEMy* GFy* present Ay* present B

FIG. 14. (Color online) Migration on the y direction of thecylinder varying with time. The dimensionless variables are t∗ =tUw/H and y∗ = yc/H .

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V. CONCLUSIONS

In this study, we proposed a coupled GF and PSC approachto reduce the spurious-pressure oscillations of moving prob-lems with the sharp-interface IBLBM. The PSC method canenforce the GCL on the boundaries by adopting the cut-cellstrategy. The GF is a kind of sharp-interface IBLBM withthe assurance of the nonslip boundary conditions, Dirichletand Neumann. We made proper modifications in the GFprocess to introduce the weighting strategy of the PSC intothe GF. The effectiveness of the present method is tested usingsome benchmark problems, including a stationary boundarytest (the flow over a cylinder) and three moving boundarytests (a moving cylinder in a channel, the oscillation of acylinder, and the motion of a neutrally buoyant cylinder ina linear shear flow as stationary and moving cases). Thebasic abilities of the boundary condition are also tested.The results prove that the boundary condition with therelaxation-time-based weighting factors is not only accurate

for static boundaries, but also for moving boundaries. Thisrelaxation-time-based method can reduce spurious-pressureeffectively and is more compatible in simulating movingboundary problems when the accuracy of the pressure fieldis the major consideration. We also found that the weightingfactors are important in our method. The relaxation-time-based weighting factor is more accurate and closer to realitycompared with the relaxation-time-independent factor. Furthervalidations and verifications will be performed in a futurestudy.

ACKNOWLEDGMENTS

This study is supported by the National Natural ScienceFoundation of China (Grant No. 50975103) and the NationalNatural Science Funds for Distinguished Young Scholars(Grant No. 51006039). We gratefully acknowledge the con-tributions of the Palabos open-source project.

[1] C. K. Aidun and J. R. Clausen, Annu. Rev. Fluid Mech. 42, 439(2010).

[2] D. Yu, R. Meia, L.-S. Luo, and W. Shyy, Prog. Aerospace Sci.39, 329 (2003).

[3] S. Chen and G. D. Doolen, Annu. Rev. Fluid Mech. 30, 329(1998).

[4] A. J. C. Ladd, J. Fluid Mech. 271, 285 (1994).[5] O. Filippova and D. H Nel, J. Comput. Phys. 147, 219 (1998).[6] R. Mei, L.-S. Luo, and W. Shyy, J. Comput. Phys. 155, 307

(1999).[7] M. Bouzidi, M. Firdaouss, and P. Lallemand, Phys. Fluids 13,

3452 (2001).[8] P. Lallemand and L. S. Luo, J. Comput. Phys. 184, 406

(2003).[9] R. Kamakoti and W. Shyy, Int. J. Num. Meth. Heat Fluid Flow

14, 851 (2004).[10] J. H. Seo and R. Mittal, J. Comput. Phys. 230, 7347 (2011).[11] Z. G. Feng and E. E. Michaelides, J. Comput. Phys. 195, 602

(2004).[12] C. S. Peskin, J. Comput. Phys. 25, 220 (1977).[13] C. S. Peskin, Acta Numerica 11, 479 (2003).[14] X. D. Niu, C. Shu, Y. T. Chew, and Y. Peng, Phys. Lett. A 354,

173 (2006).[15] G. Le and J. Zhang, Phys. Rev. E 79, 026701 (2009).[16] C. Shu, N. Liu, and Y. T. Chew, J. Comput. Phys. 226, 1607

(2007).

[17] J. Wu and C. Shu, J. Comput. Phys. 228, 1963 (2009).[18] S. K. Kang and Y. A. Hassan, Int. J. Num. Meth. Fluids 66, 1132

(2010).[19] J. Lu, H. Han, B. Shi, and Z. Guo, Phys. Rev. E 85, 016711

(2012).[20] A. Tiwari and S. P. Vanka, Int. J. Num. Meth. Fluids 69, 481

(2011).[21] R. Mittal, H. Dong, M. Bozkurttas, F. M. Najjar, A. Vargas, and

A. Von Loebbecke, J. Comput. Phys. 227, 4825 (2008).[22] Y. H. Tseng and J. H. Ferziger, J. Comput. Phys. 192, 593 (2003).[23] D. R. Noble and J. R. Torczynski, Int. J. Mod. Phys. C 9, 1189

(1998).[24] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511

(1954).[25] D. J. Holdych, Ph.D. thesis, University of Illinois at Urbana-

Champaign, 2003.[26] D. R. J. Owen, C. R. Leonardi, and Y. T. Feng, Int. J. Num.

Meth. Eng. 87, 66 (2011).[27] Z. Guo, C. Zheng, and B. Shi, Phys. Fluids 14, 2007 (2002).[28] A. Mussa, P. Asinari, and L. S. Luo, J. Comput. Phys. 228, 983

(2009).[29] M. Schafer, S. Turek, F. Durst, E. Krause, and R. Rannacher,

Notes Num. Fluid Mech. 52, 547 (1996).[30] P. Lallemand and L. S. Luo, Phys. Rev. E 61, 6546 (2000).[31] J. Feng, H. H. Hu, and D. D. Joseph, J. Fluid Mech. 277, 271

(1994).

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