Computers & Fluids. Improving lattice Boltzmann simulation of moving particles in a viscous flow using local grid refinement

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1 Computers & Fluids 00 (204) 2 Computers & Fluids Improving lattice Boltzmann simulation o moving particles in a viscous low using local grid reinement Songying Chen a,, Cheng Peng b, Yihua Teng c, Lian-Ping Wang b,d a School o Mechanical Engineering, Shandong University, Jinan, , P.R. China b Department o Mechanical Engineering, 26 Spencer Laboratory, University o Delaware, Newark, Delaware , USA c Department o Energy and Resource Engineering, Peking University, Beijing, P.R. China d National Laboratory o Coal Combustion, Huazhong University o Science and Technology, Wuhan, P.R. China Abstract Accurate simulations o moving particles in a viscous low require an adequate grid resolution near the surace o a moving particle. Within the ramework o lattice Boltzmann approach, inadequate grid resolution could also lead to numerical instability and large luctuations o the computed hydrodynamic orce and torque. Here we eplore the use o local grid reinement around a moving particle to improve the simulation results using the multiple-relaation-time (MRT) lattice Boltzmann method (LBM). We irst re-eamine the necessary relationships, within MRT LBM, between the relaation parameters and the distribution unctions on the coarse and ine grids, in order to meet the physical requirements o the luid hydrodynamics and additional relationships as implied by the Chapman-Enskog multi-scaling analysis. Several aspects o the implementation details are discussed, including the treatment o interace buer nodes, the method to transer inormation between the the coarse domain and ine domain, and the computation o macroscopic variables including stress components. Our approach is then applied in two numerical tests to demonstrate that the local grid reinement can signiicantly improve the physical results with a high computational eiciency. We compare simulation results rom three grid conigurations: a uniormly coarse grid, a uniormly coarse grid with local reinement, and a uniormly ine grid. For the lid-driven cavity low, the local reinement essentially yields local low ield that is comparable to the use o uniormly ine grid, but with a much less computational cost. In the Couette low with a moving cylinder, the local reinement suppresses the level o orce luctuations. It was also ound that the coarse-ine grid relationships between the non-equilibrium moments o energy square and energy lues do not aect the simulation results. c 20 Published by Elsevier Ltd. Keywords: lattice Boltzmann method, local grid reinement, moving particles, orce luctuation, viscous low. Introduction As a highly eicient and capable mesoscopic computational method, the lattice Boltzmann method (LBM) [, 2, 3, 4] has been widely employed to solve a variety o luid dynamics problems. LBM describes the luid as making up by imaginative elements which can stream along a uniorm lattice grid and collide with one another only at lattice nodes. The method solves a quasi-linear collision-streaming equation or a set o distribution unctions associated Corresponding author. address: chensy66@sdu.edu.cn (Songying Chen), cpengpp@udel.edu (Chen Peng), tengyihua99@gmail.com (Yihua Teng), lwang@udel.edu (Lian-Ping Wang)

2 / Computers & Fluids 00 (204) 2 2 with discrete microscopic velocities. The macroscopic hydrodynamic variables such as pressure and velocity are obtained by taking the moments o the distribution unctions. One o the popular LBM schemes is based on the single relaation time approach (i.e., Bhatnagar-Gross-Krook collision process [5]), which is known as the LBGK model. Another popular scheme is based on the multiplerelaation-time (MRT) collision model [6, 7]. The MRT collision is perormed in the moment space with dierent moments relaing at dierent rates. By decomposing the particle relaation process into several independent relaation processes, the MRT model has been shown to not only improve the computational stability but also the accuracy [6, 7, 8]. In order to obtain accurate macroscopic quantities, such as orce and torque acting on a solid particle or the boundary, small grid spacing is needed near the solid particle or a boundary. Away rom the boundary or luid-solid interaces, the low may be more smooth so a coarser grid is adequate to resolve the low. The most eicient approach in terms o both memory and overall accuracy is thus to use a coarse grid or most o the bulk low region, combined with a local grid reinement near a luid-solid interace or wall boundary. Within the LBGK model, local grid reinement has been considered or some time to simulate incompressible viscous lows with comple geometries. Filippova and Hanel [9] was among the irst to consider patching certain regions with a ine grid in a domain mostly covered by a coarse grid, values o the distribution unctions on the coarse grid which are coming rom the ine patches are calculated on the nodes common to both grids. Filippova and Hanel [0] presented an accelerated implementation o grid reinement by using dierent molecular speeds on the coarse and ine grids. A smaller time step size was used on the ine grid while the spatial or temporal accuracy was kept. Steady-state and time-dependent problems were studied and the CPU time per time step was reduced by about 50%. Yu et al. [] proposed a multi-block technique with the BGK model in LBM. Dierent mesh sizes are used or dierent blocks that do not overlap. Macro-variables such as mass, momentum, and stress components are assumed to be continuous across the block-block interace, and this condition determines the relaation parameters in the ine domain. The cubic spline scheme was used or spatial interpolation and the three-point Lagrangian ormula was used or temporal interpolation on all nodes at the ine block boundary ater the distribution unctions are transerred rom the coarse domain to the ine domain. Yu and Girimaji [2] etended their approach to 3D using the LBGK model. Two 3D test cases, an isotropic decaying turbulence and a lid-driven cavity low, were presented to show the improved computational eiciency. Eitel-Amor et al. [3] introduced a cell-centered lattice structure to reconstruct the pre-collision distribution unctions via spatial interpolation in LBGK model. They showed that, with hierarchically reined meshes, each cell can be reined or coarsened regardless o the reinement level o neighbor cells. Lagrava [4] introduced a decimation technique to guarantee the stability o the numerical scheme especially at high low Reynolds number when the inormation is transerred rom the coarse nodes to the ine nodes. Dietzel and Sommereld [5] calculated low resistance over agglomerates with dierent morphology through LBGK local grid reinement. They slightly overlapped the coarse and ine regions and designed a method to communicate the distribution unctions between the two grids at the interace. Premnath et al. [6] presented a staggered mesh arrangement in large-eddy simulation o a comple turbulent separated low, using the MRT D3Q9 model. Subgrid scale model was employed in conjunction with the MRT to augment the relaation time scales o hydrodynamic modes that allowed better representation o the eect o subgrid scale luid motion. The irst attempt to implement local grid reinement within MRT was perormed by Peng et al. [7] using the D2Q9 model, where they related the distribution unctions and relaation parameters in the two domains based on the continuity o macro-variables at the coarse-ine interace. The method to communicate distributions unctions between the two domains was derived. They used lid-driven cavity low, steady and unsteady lows past a circular cylinder, and low over an airoil to validate their approach. As will be shown later in this paper, their implementation was not ully consistent since they ignored the relationship o energy relaation parameters in the two domains. It appears that the ull details or implementing local grid reinement within the MRT model have not been careully studied. Our irst objective is to re-eamine the details o coupling the distribution unctions and model parameters between the ine and course grids within the MRT ramework. Our second objective is to test local grid reinement in moving particle simulation. In Section 2, a brie background description o the D2Q9 MRT model is provided. Details o local grid reinement implementation are discussed in Section 3, together with the development o relationships between the course and ine grid. Necessary interpolation details at the coarse-ine interace are pre- 2

3 / Computers & Fluids 00 (204) 2 3 sented in Section 4. In section 5, we then validate our methodology using 2D lid-driven cavity low, a Couette low with a ied or a moving cylinder. In the case o moving cylinder, the reined region also moves with the cylinder. Test results demonstrate that local grid reinement indeed improve the accuracy in moving particle simulation. We also compare CPU time and memory consumption when dierent grid arrangements (e.g., hybrid coarse / ine grid versus uniormly ine) are used. Key conclusions are summarized in Section The multiple-relaation-time lattice Boltzmann method Figure : The D2Q9 model with nine discrete velocities on two dimensional square lattice. In this section, we briely introduce the multiple-relaation-time (MRT) lattice Boltzmann method (LBM) in order to prepare or the discussions on the local grid reinement. The detailed description o MRT LBM can be ound in [6, 7]. Speciically the D2Q9 model [8, 9] is considered (Fig. ), with discrete velocities given by: e i = { (0, 0) (, 0) (0, ) (, 0) (0, ) (, ) (, ) (, ) (, ) } c. () where i = 0,, 2,..., 8, c = δ/δt. All variables are given in lattice units such that c =. The MRT LBM evolution equation [6, 7] can be written as ( + e i δt, t + δt) = (, t) M S [ m m eq]. (2) where is a vector representing a set o distribution unctions deined at a lattice node, m represents a set o independent moments, m eq is the equilibrium o m, M is an orthogonal transormation matri that transorms into m m = M, = M m. (3) The macroscopic hydrodynamic variables, including density ρ, velocity u = ( ) u, u y, and pressure p are obtained rom the moments o the mesoscopic distribution unction. We use the nearly incompressible ormulation, namely, the density is partitioned as ρ = ρ 0 + δρ with ρ 0 =, and δρ = i i, δρu = i i e i, and p = δρc 2 s, where c s = / 3 is the model speed o sound. For the D2Q9 MRT model, other moments and the transormation matri are designed as ρ e ε j m = q j y q y p p y = (4)

4 / Computers & Fluids 00 (204) 2 4 The three hydrodynamic moments, ρ, j = ρ 0 u, j y = ρ 0 u y, are locally conserved. The other si moments are not conserved, they are energy e, energy square ε, energy lu in and y directions q and q y, normal stress p, and shear stress p y. These non-conserved moments are relaed as ollows ẽ = e S e (e e eq ) ε = ε S ε (ε ε eq ) q = q S q (q q eq ) q y = q y S q (q y q eq y ) p = p S ν (p p eq ) p y = p y S ν (p y p eq y), where the symbol denotes the post-collision value. The equilibriums o non-conserved moments are designed to match the Euler and Navier-Stokes equation through the Chapman-Enskog analysis, and the results are (5) e eq = 2δρ + 3ρ 0 u 2 ε eq = δρ 3ρ 0 u 2 q eq q eq y = ρ 0 u = ρ 0 u y ( ) p eq = ρ 0 u 2 u 2 y p eq y = ρ 0 u u y, (6) where u 2 = u 2 + u 2 y. While the equilibrium ε eq plays no role in the Navier-Stokes equation and thus its orm can be leible, the speciic orm stated above leads to the standard eq = M m eq as { [ eq ei u i = w i δρ + ρ 0 + (e i u) 2 u u ]}. (7) c 2 s 2c 4 s 2c 2 s where the weighting coeicient w i is given as It ollows that the diagonal relaation matri S is w 0 = 4 9, w i = 9 (i =, 2, 3, 4), w i = (i = 5, 6, 7, 8). (8) 36 S = Diag [ 0 S e S ɛ 0 S q 0 S q S ν S ν ]. (9) The shear viscosity and bulk viscosity in the MRT model can be derived rom the Chapman-Enskog analysis as ( ) ν = c 2 s 0.5 δt S ν ( ) (0) ξ = c 2 s 0.5 δt. S e The evolution equation or can be divided into two sub-steps: collision and streaming, as (, t) = (, t) M S [ m m eq], () ( + e i δt, t + δt) = (, t). (2) In summary, the key eatures o the MRT model is that collision is perormed in the moment space, and streaming occurs in the discrete-velocity space. The two relaation times S ν and S e determine the shear and bulk viscosity, the other two relaation parameters S q and S ɛ may be viewed as ree parameters that can be used to improve the accuracy o boundary condition or to enhance numerical stability [6]. 4

5 3. Local grid reinement implementation in MRT LBM / Computers & Fluids 00 (204) 2 5 Without loss o generality, we consider a local ine-grid domain surrounded by a coarse grid, as shown in Fig. 2. In the sketch, the boundary o the coarse domain is deined by ABCDA and that o ine grid domain by KLMNK, without any overlap. However, to acilitate implementations and inormation transer between the two grids, we add a buer layer EFGHE to the coarse domain, which is located inside the ine domain, the corresponding coarse grid nodes are reerred to as the coarse interace nodes. Likewise, a buer layer or the ine domain, which coincides with the boundary o the coarse domain are used to deine additional populations or the ine domain, which consists o ine interace nodes (those coinciding with the coarse boundary nodes) and hanging nodes (those sitting in-between the coarse boundary nodes). The buer layers are used to supply necessary inormation so that, ater streaming, all populations on the ine-grid nodes and coarse-grid nodes are available. This arrangement ollowed the work o [5]. Figure 2: Grids structure with dierent space size in coarse and ine block As a multi-block scheme [9], the ine domain has a grid spacing o δ and the coarse domain has a grid spacing o δ c. The size ratio n = δc = δtc (3) δ δt is assumed to be an integer, where the superscript c and denote the related variables located in coarse domain and ine domain. The sketch shown in Fig. 2 has n = 2. The time steps or the coarse and ine grids, δt c and δt, are deined such that the streaming in each domain goes rom a lattice grid node to a neighboring node. By this setting, the lattice velocity unit in the two domains are the same, namely, δ c /δt c = δ /δt. This also implies that the speed o sound and the transormation matri are all identical or the two domains. The irst physical requirement is that the hydrodynamic variables δρ, u, u y must be continuous across the domain interace, we must have δρ = δρ c, j = j c, and jy = j c y. All equilibrium moments deined on the two grids must be the same, namely, m eq, = m eq,c. We only need to determine the relationships between the non-equilibrium parts o the non-conservative moments. The second physical requirement is that the physical shear viscosity and bulk viscosity should be the same in the two domains, which leads to c 2 s ( ) 0.5 δt c = c 2 s S ν 0.5 δt, (4) ( ) c 2 s S e c 0.5 δt c = c 2 s 0.5 δt. (5) S e Namely, the relaation parameters or the ine domain should be related to those in the coarse domain as ( ) = n 0.5, (6) S ν 5

6 / Computers & Fluids 00 (204) 2 6 ( ) = n S e S e c 0.5. (7) The third physical requirement is that the normal and shear stress components should be the same at the domain interace. The Chapman-Enskog analysis states that τ = 6 ( 0.5S e) e neq 2 ( 0.5S ν) p neq τ yy = 6 ( 0.5S e) e neq + 2 ( 0.5S ν) p neq τ y = ( 0.5S ν ) p neq y. (8) Thereore, we demand ( 0.5S c) 6 e e neq,c 2 ( 0.5S c) 6 e e neq,c + 2 ( 0.5S c) ν p neq,c = ( ) 0.5S e e neq, ( ) 0.5S ν p neq,, (9) 6 2 ( 0.5S c) ν p neq,c = ( ) 0.5S e e neq, + ( ) 0.5S ν p neq, 6 2 ( 0.5S ν c ) p neq,c y = ( 0.5S ν, (20) ) p neq, y. (2) Solving these three equations and in view o the conditions given by Eqs. (6) and (7), we obtain the ollowing relationships between three non-equilibrium moments in the two domains, as e neq, = S e c e neq,c, p neq, ns e = S ν c ns ν p neq,c, p neq, y = S ν c p neq,c ns ν y. (22) For the three remaining non-equilibrium moments: energy square and two energy lu components, the Chapman- Enskog analysis shows that ( t δρ 3ρ0 u 2) ( ) S ε ε neq + ( ρ 0 u ) + y ρ0 u y =, (23) dt ( t ( ρ 0 u ) + + 6u 2 y 3u 2) ( ) S q q neq + y ρ0 u u y =, (24) dt ( ) ( ) ( t ρ0 u y + ρ0 u u y + y + 6u 2 3u 2) = S qq neq y. (25) dt The let hand sides involve only the hydrodynamic variables and should be the same when deined on the two grids, thereore, we set S εε c neq,c dt c = S εε neq,, (26) dt S c qq neq,c dt c S c qq neq,c y dt c = S qq neq,, (27) dt = S qq neq, y. (28) dt The two remaining relaation parameters, S ε and S q, do not enter the Navier-Stokes equation and can be treated arbitrarily. For convenience, we simple assume that these two relaation parameters are the same in the two grid systems, namely, S ε = S c ε and S q = S c q. Then Eqs. (26-28) becomes ε neq,c = nε neq,, q neq,c = nq neq,, q neq,c y = nq neq, y. (29) 6

7 / Computers & Fluids 00 (204) 2 7 At this point, all necessary relationships are worked out or constructing distribution unctions and model parameters on the ine grid rom those on the coarse grid, and vice versa. They can be summarized as ollows m eq,c = m eq, and m neq,c ρ ρ eq e e eq ε ε eq j j eq q q eq j y j eq y q y q eq y p p eq P y p eq Thereore, we can introduce the ollowing notations where [ T = diag [ T c = diag y c = ns e S c e n n n ns ν ns ν m neq,. (30) m neq,c = T m neq,, m neq, = T c m neq,c, (3) ns e S c e S c e ns e n n n n n n ns ν ns ν ns ν ns ν ], (32) ]. (33) We shall now eplain how to obtain the post-collision distribution unction on a ine grid in terms o the postcollision distribution unction on a coarse grid. First, the post-collision distribution unction on the ine grid is determined as = M S ( m m eq, ) = M m M S ( m m eq, ), (34) which, ater substituting Eq. (3), becomes = M m eq, + M ( I S ) T c m neq,c. (35) On the other hand, the post-collision moments in the coarse domain can be epressed as M c = m c S c (m c m eq,c ) = m eq,c + (I S c ) m neq,c. (36) So the non-equillibrium moments in the coarse block can be written in terms o its post-collision distribution unction as m neq,c = (I S c ) ( M c m eq,c). (37) Substituting Eq. (37) into Eq. (35), the post-collision distribution unction in the ine domain can be computed in terms o the post-collision distribution unction in the coarse region as where = M [ m eq,c + T c ( M c m eq,c)]. (38) T c = ( I S ) T c (I S c ) ( ) S e = diag S c e ns e ( S c e) ( S ε ) n ( S c ε) ( S q ) n ( ) S q c 7 ( S q ) n ( ) S q c ( S ν ) ns ν ( ) ( ) S ν ns ν ( S ν).(39) c

8 / Computers & Fluids 00 (204) 2 8 Similarly, the post-collision distribution unction can be transerred rom the ine domain to the coarse domain by c = M [ m eq, + T ( M m eq, )]. (40) where T = [ T c] To summarize, the distribution unctions between the coarse and ine grids can be converted either beore the collision substep or ater the collision substep. In the irst case, Eq. (30) can be used and then multiplying the converted moments by M to obtain the distribution unctions. In the second case ater the collision substep, then Eqs. (38) and (40) should be used. We have developed two versions o the code based on the two approaches, and conirm that the results are identical. 4. The computational procedure on the domain interaces Recall the grid arrangement or the coarse and ine domains shown in Fig. 2, the coarse interace nodes are inside the ine region. They provide the buer layer or the coarse-domain nodes or inormation transer rom the ine domain to the coarse domain. Basically, at the coarse interace nodes, the conversion o distribution unction rom the ine grid to the coarse grid occurs (through either Eq. (30) or Eq. (40), depending on whether the conversion was done beore or ater the collision sub-step), ollowing by streaming which eeds this converted distribution to the coarse-domain boundary nodes. Likewise, the ine interace nodes and ine hanging nodes sit on the coarse-domain boundary and provide the buer layer or the ine-domain nodes or inormation transer rom the coarse domain to the ine domain. First, the conversion o distribution unction rom the coarse grid to the ine grid is perormed or the ine interace nodes, using either Eq. (30) or Eq. (38), depending on whether the conversion was done beore or ater the collision sub-step. Second, there are no coarse nodes deined at the locations o the hanging nodes, so the ine-grid distribution unctions at the hanging nodes are obtained rom the ine-grid distribution unctions at the ine interace nodes. We employ the cubic spline interpolation at each edge o the coarse-domain boundary, namely, F i (s i ) = a i (s s i ) 3 + b i (s s i ) 2 + c i (s s i ) + d i, (4) where F i is the distribution unction being interpolated, s is the local coordinate at the edge, s i is the coordinate o the interpolated point, a i, b i, c i, d i are cubic spline coeicients which are determined by itting the known values at inite interace nodes. Furthermore, or each time step corresponding to the coarse domain, there are n time steps or the ine domain. The distribution unctions at these sub-timesteps are interpolated in time between t and t + δt. The converted ine-grid distribution unctions at the ine interace nodes and hanging nodes are then streamed onto the ine boundary nodes. I the conversion between the two grids at the buer layers are done beore the collision sub-step, the collision operation should be done on the buer layers beore perorming the steaming. The arrangement o the buer layers or the ine and coarse regions ensures each domain is ully etended, such that the distribution unctions at all nodes in the ine or coarse domain are complete ater the streaming sub-step. The two domains do not overlap, and the buer layers provide the bridges or inormation transer. Fig. 3 provides a low chart or the code when the local reinement is applied to a moving particle, namely, the grid reinement is done around a moving solid particle, with the ine domain moving with the solid particle. For this chart, it is assumed that the streaming sub-step is perormed beore the collision sub-step. One can do collision irst then streaming without changing other process in the low chart. But data transer with Eqs. (38) and (40) is replaced by the ollowing relationships, respectively. = M [ m eq,c + T c (M c m eq,c ) ], (42) c = M [ m eq, + T ( M m eq, )]. (43) 8

9 / Computers & Fluids 00 (204) 2 9 Computational procedure Set initial values in coarse and ine blocks Transer data rom ine to coarse at t Transer data rom coarse to ine at t Streaming in coarse block Spatial interpolation to get hanging nodes value in ine interace at t Calculate macroscopic variables in coarse block Collision in coarse block Streaming in ine block t (t + 0.5δt) Transer data rom coarse to ine at t + δt Spatial interpolation to get hanging nodes value in ine interace at t + δt Calculate macroscopic variables in ine block Temporal interpolation to get ine interace value at t + 0.5δt Collision in ine block Streaming in ine block (t + 0.5δt) ( t + δt) Calculate macroscopic variables in ine block Collision in ine block No Converge Yes End Figure 3: The low chart o the code when the streaming sub-step is done beore the collision sub-step. This low chart assumes n = 2. It should be stressed that the macroscopic variables should be processed ater the steaming to be consistent with the LBM evolution. 5. Results rom numerical simulations and discussions In order to validate the approach and to highlight the beneits o local grid reinement in improving computational accuracy using the MRT LBM model, we apply the approach to solve two low problems. The irst is the lid-driven cavity low and we implement local grid reinement on the upper-let corner, to show that the local grid reinement can provide more accurate results or this region where large velocity gradients eit. The second is a 2D Couette low over a ied or moving cylinder. Here local grid reinement is applied to region near the cylinder to demonstrate that grid reinement can suppresses luctuations in the hydrodynamic orce acting on the moving cylinder. 9

10 / Computers & Fluids 00 (204) 2 0 Table : Parameters used in the simulations o two dimensional square cavity low under three dierent grid conigurations. UCG UCG-L UFG U w H ν Re N N y Total node points Lid driven cavity low The lid-driven cavity low has been etensively used as a benchmark case to test a numerical method [2, 2]. In this low, the two corners under the moving lid are singular points, and higher grid resolution is desired in order to obtain more accurate stress distribution near the corner points. We apply local grid reinement to the top-let corner (Fig. 4). The simulations are carried out using three dierent grid conigurations: a uniorm coarse grid (UCG), a uniorm coarse grid with local grid reinement (UCG-L) at the top-let corner, and a uniorm ine grid (UFG). The physical and simulation parameters are listed in Table. For all three grid resolutions, the low Reynolds number Re = HU w /ν is ied to 000, where H is the width o the square cavity, U w is the lid velocity, and ν is the kinematic viscosity. The bulk viscosity ξ was set to be equal to ν, and these lead to S c e =.8868, =.8868 in the coarse region where δt c = and δ c =. The two remaining relaation parameters in the coarse region are S c ε =.54 and S c q =.9. For the UCG-L case, a reined grid with d = 0.5 is applied to a region near the top-let corner o the size These lead to the setting that S e = 0.64, S ν = 0.64, S ε =.54,and S q =.9. u w 0.26N y 0.26N N y N Figure 4: Local grid reinement block layout or a 2D cavity low The low is initially at rest. Ater a suiciently long time (over 00,000 coarse-grid time steps), the low reaches to a steady state. The two velocity components along a vertical line at = 3 and a horizontal line at y = 86 are shown in Fig. 5 and 6, respectively. Both lines cut though the ine domain. Clearly, the proiles are continuous at the ine-coarse boundary (marked by the vertical line). Second, the results rom the three grid conigurations essentially overlap, but the UCG-L proiles match better the UFG results, when compared to the UCG results. This shows that the local grid reinement improves the results. 0

11 / Computers & Fluids 00 (204) 2 Figure 5: Comparison o normalized velocity along vertical line = 3 computed with three grid conigurations, let: u ; right: u y. The vertical line marks the boundary between the coarse domain and the ine domain. Figure 6: Comparison o normalized velocity proiles along horizontal line y = 86 computed via three grid conigurations, let: u ; right: u y. The vertical line marks the boundary between the coarse domain and the ine domain. Near the upper lid and let wall, there eists an area o highly velocity gradients. In Fig. 7, we plot the viscous stress proiles. The viscous stress components are computed using Eq. (8). The continuity o the stress proiles at the ine-coarse boundary (marked by the black vertical line) validates our implementation. Such stress proiles are rarely shown in published literature. Interestingly, there are some oscillations in the proiles o normal viscous stresses that could be caused by undamped acoustic waves or simply due to inadequate grid resolution. In Fig. 8, we compare the shear stress (τ y ) proiles obtained rom the three grid conigurations. This igure demonstrates a great beneit o local grid reinement: without the reinement, the results rom UCG show unphysical oscillations. On the other hand, the UCG-L and UFG results are almost identical. Fig. 9, 0 and shows the vorticity contours obtained rom the three grid conigurations. Clearly, the local grid reinement makes a huge dierence in the accuracy o the results when compared to the UFG run, namely, the vorticity contours are as smooth as the UFG run. The UCG run did not resolve the vorticity distribution in the upper let corner and lead to strong unphysical noises.

12 / Computers & Fluids 00 (204) 2 2 Figure 7: Stress proiles computed with local grid reinement, let: along = 3 ; right: along y = 86. The vertical line marks the boundary between the coarse domain and the ine domain. Figure 8: Comparison o τ y along vertical line = 3 computed via three grid conigurations, let: whole domain view; right: the zoom-in plot. The vertical line marks the boundary between the coarse domain and the ine domain. 2

13 / Computers & Fluids 00 (204) y Figure 9: Voticity contours in the cavity: the UCG case y Figure 0: Voticity contours in the cavity: the UCG-L case. 3

14 / Computers & Fluids 00 (204) 2 4 Table 2: Computer CPU and memory occupied by three grid structures with cavity low. UCG UCG-L UFG CPU (ms) Memory (kb) y Figure : Vorticity contours in the cavity: the UFG case. Finally, in Table 2, we compare the CPU per time step used and memory required or the three cases. While the UFG run uses 4 times CPU and about 3.5 times memory when compared to the UCG run, the UCG-L only uses 2.26 times CPU and 29% more memory An asymmetrically placed cylinder in a 2D Couette low Net, we consider the same low studied in [20], namely, an asymmetrically placed cylinder in a 2D Couette low (Fig. 2). The low can be simulated in two rames o reerence to study the accuracy o moving particle simulation. In the irst (or the ied cylinder case) case, the cylinder particle is ied relative to the lattice grid and the upper and lower channel boundaries move in opposite direction with a same constant velocity (U b ). In the second case (the moving cylinder case), the cylinder moves at a velocity u 0, with the top wall and bottom wall moving at velocity U b = U b + u 0 and U b2 = U b + u 0, respectively. Physically, the two cases are identical. Numerically, the second case is much more diicult due to the need to treat the curved moving luid-cylinder surace. We implemented local grid reinement in both cases. For the moving cylinder case, the ine domain shits by one lattice grid every time the center o the cylinder is moved by one lattice grid. The geometric parameters or this problem include the channel width L y and length L, the diameter o the cylinder D, and the cylinder center at the initial time (X c0, Y c0 ). Periodic boundary condition is used in the direction, and the no-slip condition is assumed at the top and bottom channel walls as well as on the cylinder surace. Again, we consider three grid conigurations: uniorm coarse grid (UCG), uniorm coarse grid with local grid reinement around the cylinder (UCG-F), and uniorm ine grid (UFG). The kinematic viscosity is /9, which yields S ν =.2 in the 4

15 / Computers & Fluids 00 (204) 2 5 Table 3: Parameters setting o three grid structures in lattice Boltzmann space. Parameter UCG UCG-L UFG N N y D (X c0, Y c0 ) (30, 54) (30, 54) (60, 08) u U b coarse domain. Other relaation parameters in the coarse domain are set to S ε =.4 and S e = S q =.5 (i.e., the bulk viscosity is /8). The other parameters in the moving cylinder simulations are set in Table 3. Note that the parameters or the ied cylinder case are the same, ecept u 0 = 0. In the ine domain with n = 2, the resulting relaation parameters are S ν = 6/7, S e =.2, S q =.5, S ε =.4. Figure 2: Sketch o 2D Couette low containing a cylinder The ied cylinder case Figure 3: The hydrodynamic orce F acting on the ied cylinder, let: the whole time interval; right: zoom-in plot. In this case, u 0 = 0. The size o the ine domain is a square o size equal to 36. The region covers 2 < < 48 and 36 < y < 72. The no-slip condition on the cylinder surace was handled by a quadratic interpolation scheme [20, 22]. Figs. 3, 4 and 5 show the drag orce F, lit orce F y, and torque as unctions o time acting on the particle, 5

16 / Computers & Fluids 00 (204) 2 6 respectively. The orce and torque are computed by the Galilean invariant momentum echange method [22, 23]. Overall, the results rom the three grid conigurations are in ecellent agreement. The zoom-in plots or 4000 < t < 400 show a very minor dierence, typical 0.05% relative dierence or less. This is clearly negligible. Thereore, each o these ied cylinder results can be used as a benchmark to eamine results or the moving cylinder case Moving particle low simulation When the cylinder is moving at u 0, the ine domain is also a squared but with width equal to 32, initially covering 4 < < 46 and 38 < y < 70. It is more or less placed with the cylinder near the center. Every time the cylinder moves by one lattice unit, the ine domain is shited in the same direction. When the cylinder moves relative to the grid, a solid node may become a luid node and the distribution unctions at such new luid node need to be illed. The reilling scheme is based on a newly developed velocity-constrained etrapolation scheme [22]. In Figs. 6, 7, 8, we show the drag orce F, lit orce F y, and torque as unctions o time acting on the particle, respectively. Note that due to the improved scheme, the level o orce luctuations in Figs. 6 and 7 is signiicantly less than the level o orce luctuations shown in Fig. 5 o [20]. The zoom-in view shows that the UCG run has larger magnitude o orce luctuations when compared to that o the UFG run. The results rom the UCG-L run are more similar to the UFG run than to the UCG run, showing the beneit o local grid reinement. In order to compare the level o orce luctuations quantitatively, we use the data rom the ied cylinder case as the benchmark and compute the L 2 norm o the dierence as [20], (t) = F (t) F 0 (t) 2 F0 (t) 2 (44) Where F (t) and F 0 (t) are the orce values o the later part simulated with moving particle and ied particle, respectively. The results are listed in Table 4. The local grid reinement reduces the level o unphysical orce luctuations by roughly a actor o 2. Figure 4: The hydrodynamic orce F y acting on the ied cylinder, let: the whole time interval; right: zoom-in plot. 6

17 / Computers & Fluids 00 (204) 2 7 Figure 5: The hydrodynamic torque acting on the ied cylinder, let: the whole time interval; right: zoom-in plot. Figure 6: The hydrodynamic orce F acting on the moving cylinder, let: the whole time interval; right: zoom-in plot. Figure 7: The hydrodynamic orce F y acting on the moving cylinder, let: the whole time interval; right: zoom-in plot. 7

18 / Computers & Fluids 00 (204) 2 8 Table 4: L 2 error norm or three grid conigurations: the moving cylinder case. Force UCG UCG-L UFG (t) y (t) Figure 8: The hydrodynamic torque acting on the moving cylinder, let: the whole time interval; right: zoom-in plot. In Fig.9 we show the normal stress components τ, τ yy and shear stress τ y at the end o the simulation t = The two inner and two outer vertical lines mark the edge o the cylinder and the ine-coarse boundary, respectively. O importance is that all stress proiles show a consistency at ine-coarse grid interace, namely, both the value and slope at the ine-coarse boundary are continuous. This again validates our implementation o the local grid reinement. Figure 9: Stress distribution along y = 5. The thin vertical lines mark the ine-coarse domain boundaries and locations o solid surace. The CPU time per running step and computer memory requirements are compared in Table 5. We ind that the computing resources needed or UCG-L run are very similar to UCG, while these or UFG are much larger. 8

19 / Computers & Fluids 00 (204) 2 9 Table 5: Computer CPU and memory occupied by three grid structures UCG UCG-L UFG CPU (ms) Memory (kb) Fig. 20 shows the eect o ine-domain size in the computed F, or size ranging rom 32 to 48. For this range, the level o orce luctuations are independent o the ine-domain size. The minimum ine-domain size in this case is which is 3. S min = D + 6, (45) Figure 20: Comparison o F computed with dierent sizes o the ine domain Simpliied treatment or the ghost moments From the Chapman-Enskog analysis, energy square ε and energy lues q and q y are ghost moments. They do not aect the hydrodynamic variables including the stress components. We use Eqs. (23-25) to relate the non-equilibrium parts o these moments. I these relations are ignored and we simply make each the same in the coarse domain and ine domain, namely, by setting the diagonal conversion matri to [ T = diag ns e S c e ns ν ns ν ]. (46) The results with this simple treatment are shown in Fig. 2. It is ound that how we relate the non-equilibrium parts o these ghost moments does not aect the resulting hydrodynamic orces and their luctuations. This is anticipated as the non-equilibrium parts o these moments do not enter the Navier-Stokes equation. Here, the treatment o ghost moments is the same as Peng et al. [7], but the authors also neglected the second term in T shown in Eq (46). 9

20 / Computers & Fluids 00 (204) 2 20 Figure 2: The computed hydrodynamic orces using dierent treatments o the conversion matri, let: F ; right: F y. 6. Summary and conclusions This paper was motivated by the desire to improve LBM simulation o the interaction o a moving particle with the carrier viscous low. The use o uniorm grid in LBM is not the ideal choice or resolving the viscous boundary layer near the surace o a solid particle. As one option, local grid reinement near the surace o the solid particle can be used to improve the simulation results. We irst re-eamined the necessary relationships, with the MRT LBM approach, between the relaation parameters and the distribution unctions on the coarse and ine grids, in order to meet the physical requirements o the luid hydrodynamics (continuity in pressure, velocity, and stress) and additional relationships (i.e., the non-equilibrium epressions o the non-conservative moments in terms o hydrodynamic variables) as implied by the Chapman-Enskog multi-scaling analysis. The details o grid arrangement, speciically the inormation transer on interace (or buer) layers have been presented. We pointed out that the conversion between the two domains can be perormed either beore or ater the collision sub-step, but the conversion relations are dierent. Both orms o the conversion relations have been developed here. Although not shown in the paper, we ound that the results rom these two alternatives o inormation transer between the two domains are identical provided that the hydrodynamic variables including the stress components are computed ater the streaming sub-step. Our approach is general in that multiple levels o grid reinement could be implemented. The boundary between the coarse domain and ine domain can be arbitrary and move with the solid particle. Our approach was then applied to two numerical test cases to demonstrate that the local grid reinement can signiicantly improve the physical results with a high computational eiciency. Simulations rom three grid conigurations were compared: a uniormly coarse grid, a uniormly coarse grid with local reinement, and a uniormly ine grid. In addition to velocity proiles, stress proiles were careully eamined in these tests. For the lid-driven cavity low, the local reinement essentially yields local low ield that is comparable to the use o uniormly ine grid, but with a much less computational cost. In the Couette low with a moving cylinder, the local reinement suppresses the level o orce luctuations. Results rom the moving particle test case show that even grid reinement in a small region surrounding the solid particle can signiicantly improve the simulation results, implying a great potential or the local grid reinement strategy in the lattice Boltzmann method or moving particle problems. The numerical tests showed that the approach does not compromise numerical stability. It was ound that the coarse-ine grid relationships between the non-equilibrium moments o energy square and energy lues do not aect the simulation results. This observation implies that there is some leibility at the domain interace which may be used to urther optimize the numerical stability. We are in the process o applying our approach to a reely moving particle suspended in a turbulent low. 20

21 / Computers & Fluids 00 (204) Acknowledgements This work is supported by the National Natural Science Foundation o China (57602) and granted by China Scholarship Council ( ). The work is also support by the U.S. National Science Foundation (NSF) under grants CBET and AGS and by Air Force Oice o Scientiic Research under grant FA , LPW also acknowledges support rom the Ministry o Education o P.R. China and Huazhong University o Science and Technology through Chang Jiang Scholar Visiting Proessorship. Reerences [] S. Chen and G. Doolen, Lattice Boltzmann method or luid lows, Annu. Rev. Fluid Mech. 30 (998) [2] D. Yu, R. Mei, L.-S. Luo, W. Shyy, Viscous low computations with the method o lattice Boltzmann equation, Progr. Aerospace Sci.39 (2003) [3] D. Raabe, Overview o the lattice Boltzmann method or nano- and microscale luid dynamics in materials science and engineering, Model. Simul. Mater. Sci. Eng. 2 (2004) R3-R46. [4] S. Succi, The Lattice Boltzmann Equation or Fluid Dynamics and Beyond, Clarendon Press, Oord, 200. [5] P. L. Bhatnagar, E. P. Gross, M. Krook, A model or collision processes in gases. I. Small amplitude processes in charged and neutral one-component system, Physical Review 94 (954) [6] P. Lallemand and L.-S. Luo, Theory o the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E 6 (2000) [7] D. Humieres, I. Ginzburg, M. Kraczyk, P. Lallemand, L.-S. Luo, Multiple relaation-time lattice Boltzmann models in three dimensions, Philos. Trans. R. Soc. London A 360 (2002) [8] R. Mei, L.-S. Luo, P. Lallemand, D. d Humieres, Consistent initial conditions or lattice Boltzmann simulation, Computer & Fluids, 35 (2006) [9] O. Filippova and D. Hanel, Grid reinement or lattice-bgk models, J. Comput. Phys. 47 (998) [0] O. Filippova and D. Hanel, Acceleration o lattice-bgk schemes with grid reinement, J. Comput. Phys. 65 (2000) [] D. Yu, R. Mei, W. Shyy, A multi-block lattice Boltzmann method or viscous luid lows, Int. J. Numer. Methods Fluids 39 (2002) [2] D. Yu and S. S. Girimaji, Multi-block Lattice Boltzmann method: Etension to 3D and validation in turbulence, Physica A 362 (2006) [3] G. Eitel-Amor, M. Meinke, W. Schroder, A lattice-boltzmann method with hierarchically reined meshes, Computers and Fluids 75 (203) [4] D. Lagrava, O. Malaspinas, J. Latt, B. Chopard. Advances in multi-domain lattice Boltzmann grid reinement, Journal o Computational Physics 23 (202) [5] M. Dietzel and M. Sommereld, Numerical calculation o low resistance or agglomerates with dierent morphology by the LatticeBoltzmann Method, Powder Technology, 250 (203) [6] K. N. Premnath, M. J. Pattison, S. Banerjee, An Investigation o the Lattice Boltzmann Method or Large Eddy Simulation o comple Turbulent Separated Flow, Journal o Fluids Engineering 35 (203) [7] Y. Peng, C. Shu, Y. T. Chew, X. D. Niu, X. Y. Lu, Application o multi-block approach in the immersed boundary lattice Boltzmann method or viscous luid lows, J. Comput. Phys. 28 (2006) [8] B. Chopard, M. Droz, Cellular Automata Modeling o Physical Systems, Cambridge University Press, 998. [9] X. Shan, X. F. Yuan, H. Chen, Kinetic theory representation o hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid Mech. 550 (2006) [20] P. Lallemand, L. S. Luo, Lattice Boltzmann method or moving boundaries, J. Comput. Phys. 84 (2003) [2] U. Ghia, K. N. Ghia, C. T. Shin, High resolution or incompressible low using the Navier-Stokes equations and a multi-grid method, J. Comput. Phys. 48 (982) [22] C. Peng, Y. Teng, B. Hwang, Z. Guo, L.-P. Wang, Implementation issues and benchmarking o moving particle simulations in a viscous low, submitted to the ICMMES204 special issue. [23] B. Wen, C. Zhang, Y. Yu, et al, Galilean invariant luid-solid interacial dynamics in lattice Boltzmann simulations, J Comp. Phys. 266 (204)