three-dimensionallatticeboltzmannfluxsolverand its ... · the density distribution functions (ddfs)...

Commun. Comput. Phys.doi: 10.4208/cicp.300514.160115a

Vol. 18, No. 3, pp. 593-620September 2015

Three-Dimensional Lattice Boltzmann Flux Solver and

Its Applications to Incompressible Isothermal and

Thermal Flows

Yan Wang1, Chang Shu1,∗, Chiang Juay Teo1, Jie Wu2 andLiming Yang2

1 Department of Mechanical Engineering, National University of Singapore, 10 KentRidge Crescent, Singapore 119260.2 Department of Aerodynamics, College of Aerospace Engineering, NanjingUniversity of Aeronautics and Astronautics, Nanjing 210016, China.

Received 30 May 2014; Accepted (in revised version) 16 January 2015

Abstract. A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presentedin this paper for the simulation of both isothermal and thermal flows. The presentsolver combines the advantages of conventional Navier-Stokes (N-S) solvers and lat-tice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) tosolve the N-S equations. Different from the conventional N-S solvers, its viscous andinviscid fluxes at the cell interface are evaluated simultaneously by local reconstruc-tion of LBE solution. As compared to the conventional LBE solvers, which apply thelattice Boltzmann method (LBM) globally in the whole computational domain, it onlyapplies LBM locally at each cell interface, and flow variables at cell centers are givenfrom the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS,the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-upof mesh spacing and time step, tedious implementation of boundary conditions, arecompletely removed. The accuracy, efficiency and stability of the proposed solver areexamined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and nat-ural convection. Numerical results show that the LBFS has a second order of accuracyin space. The efficiency of the LBFS is lower than LBM on the same grids. However,the LBFS needs very less non-uniform grids to get grid-independence results and itsefficiency can be greatly improved and even much higher than LBM. In addition, theLBFS is more stable and robust.

AMS subject classifications: 76D05

Key words: Lattice Boltzmann flux solver, lattice Boltzmann method, Navier-Stokes equation,lid-driven cavity flows, natural convection.

∗Corresponding author. Email addresses: [email protected] (Y. Wang), [email protected] (C. Shu),[email protected] (C. J. Teo), [email protected] (J. Wu), yangliming [email protected] (L. Yang)

http://www.global-sci.com/ 593 c©2015 Global-Science Press

594 Y. Wang et al. / Commun. Comput. Phys., 18 (2015), pp. 593-620

1 Introduction

In general, for simulation of incompressible fluid flows, the Navier-Stokes (N-S) solver[1–8] and the lattice Boltzmann equation (LBE) solver [9–13] are the two major solvers.Interestingly, these two kinds of solvers are established on different theoretical frame-work. The N-S solver is based on the direct discretization of the governing equationsfrom macroscopic conservation laws while the LBE solver is based on the solution of thediscrete LBE from statistical gas kinetic theory. The roots of both N-S and LBE solvers intheir respective theoretical foundations credit themselves unique and distinctive features.

Among various N-S solvers, the commonly applied methods can be roughly classifiedinto three groups: (1) the vorticity-stream-function approach (VSFA) [1–3], (2) the artifi-cial compressibility approach (ACA) [4–6] and (3) the projection approach (PA) [14–16].The VSFA takes the vorticity and stream function as primary unknowns. The govern-ing equations of the VSFA are reconstructed from the incompressible N-S equations bytaking the curl of the momentum equations and introducing the stream function into thecalculation of the vorticity. Due to its simplicity and ease in achieving high order of accu-racy, the VSFA has attained considerable popularity for simulating two-dimensional (2D)incompressible flows. Unfortunately, due to the intrinsic 2D nature of the stream func-tion, the VSFA is not efficient for three-dimensional (3D) computations. Unlike the VSFA,the ACA directly solves for velocity and pressure. This approach introduces an artificialcompressibility term into the continuity equation so that the resultant equation systemcan be solved in a consistent way. Although it is initially proposed for simulating steadyflows, extensions of the ACA for simulation of unsteady flows [6, 17] have also beenconducted successfully. Most recently, Asinari et al. [50] proposed a link-wised artificialcompressibility method (LW-ACA) by using the simple streaming-collision technologiesoriginated from LBM [11, 12]. The most popular solver in this category is perhaps thePA, which directly solves the incompressible N-S equations. The PA usually introducesa two-stage fractional step technique to resolve the coupling problem between pressureand velocity. The pressure-correction or pressure-Poisson equation is derived and nu-merically solved to guarantee the divergence-free condition. With the aid of the fractionalstep technique, the PA attains a rigorous and complete theoretical foundation and doespresent a simple and accurate solver on Cartesian grids. However, due to the slow con-vergence of the pressure-Poisson equation, the overall computational efficiency of the PAmay be degraded. In addition, as N-S equations are partial differential equations (PDEs),numerical discretization of the first and second order spatial derivatives should be care-fully conducted by applying different schemes, which may be tedious and sophisticatedfor applications on non-uniform grids.

As compared with the N-S solvers, the LBE solvers have emerged as an alternativeand powerful algorithm for simulating incompressible flows [13, 18–23]. These solversare based on mesoscopic kinetic equations and microscopic particle models and takethe density distribution functions (DDFs) of each particle as primary unknowns. Themacroscopic flow properties are evaluated from the collective behavior of microscopic

Y. Wang et al. / Commun. Comput. Phys., 18 (2015), pp. 593-620 595

particle distributions. One approach of this kind is to solve the LBE in partial differen-tial form [24–27] for the DDFs by applying the well-established numerical schemes, suchas the finite difference method and the finite volume method. This method can be effec-tively applied on non-uniform grids and has the advantages of solving the first order par-tial differential equations. Its drawback is that much numerical dissipation is involvedand the number of governing equations depends on the lattice velocity model, whichwould need more virtual memory for 3D computations with at least 15 equations. Thedominated LBE solver is known as the lattice Boltzmann method (LBM) [9, 13, 23, 28, 29],which has two simple processes of streaming and collision. As a mesoscopic particlemethod established on the gas kinetic theory, LBM has several distinctive advantages,such as intrinsic kinetic nature, simple algebraic manipulation, ease to implement andparallel computation. Due to these attractive features, LBM has been widely applied tosimulate a variety of complex fluid flows in many different areas [10, 18, 30–36]. On theother hand, it should be indicated that LBM also has some drawbacks, such as limitationto the uniform grid and tie-up between the mesh spacing and time step. In addition, theimplementation of Newman boundary conditions in LBM is indirect and needs compli-cated transformation from the velocity and pressure to DDFs. This is not a trivial job,especially for three dimensional cases.

From the above discussion, it can be seen that both N-S solvers and LBE solvers havetheir distinctive merits and drawbacks for simulation of incompressible flows. In the lit-erature, great success has been achieved to remove their drawbacks by proposing manyimproved versions [26, 37–40, 50]. However, due to their independent developments fo-cusing on one individual theoretical framework, the improvement might be constrainedand the intrinsic drawbacks of the N-S solvers and the LBE solvers may not be com-pletely removed. Recently, to effectively combine the advantages of the N-S solvers andLBE solvers and eliminates some of their drawbacks, a 2D lattice Boltzmann flux solver(LBFS) [41–45] has been proposed. In the solver, the finite volume method is appliedto solve N-S equations. The most attractive feature of the LBFS is that its fluxes at eachinterface are evaluated by weighted summations of the discrete DDFs obtained from thelocal application of LBM. Since the DDFs include equilibrium and non-equilibrium states,both the viscous and inviscid fluxes can be computed simultaneously. Through numer-ical experiments, it has been demonstrated that LBFS not only retains advantages of theN-S solvers and LBE solvers but also avoids the tedious discretization of second-orderderivatives and the previously-discussed drawbacks of LBM.

In this series of work, we aim to develop a three-dimensional lattice Boltzmann fluxsolver (3D LBFS) for simulating both isothermal and thermal flows, which consistentlycombines the advantages of the N-S solvers and LBE solvers and further extends its ap-plications from the 2D case to the practical 3D case. Since the 3D LBFS is a newly pro-posed solver based on LBM, it is also interesting to systematically examine its accuracy,efficiency and stability in comparison with the standard LBM. Like the 2D case, the 3DLBFS applies the finite volume method to solve N-S equations on non-uniform mesh,which will give the solution for density and velocity at cell centers. The evaluation of


the macroscopic fluxes at each interface is made by local reconstruction of the thermalLBM, whose solutions are provided by double-distribution-function lattice Boltzmann(LB) model. The performance of the proposed solver will be investigated by its applica-tions to simulate the 3D plane Poiseuille flow, the lid-driven cavity flow at the Reynoldsnumbers of 102-103 and natural convection flow at the Rayleigh numbers of 102-106.

In Section 2, the standard thermal LB model is reviewed and the Chapman-Enskoganalysis is also briefly revisited to show their connections with conservation laws. Afterthat, the development of 3D LBFS is shown in Section 3. Section 4 examines the perfor-mance of the 3D LBFS in detail. Conclusions are given in Section 5.

2 Thermal lattice Boltzmann method

For incompressible thermal fluid flows, the evolution equations of the simplified thermalLBM with BGK approximation [28, 29] can be given by

fα(r+eαδt,t+δt)= fα(r,t)+f

eqα (r,t)− fα(r,t)

τv, α=0,1,··· ,N, (2.1)

gα(r+eαδt,t+δt)= gα(r,t)+g

eqα (r,t)−gα(r,t)

τc, α=0,1,··· ,M, (2.2)

where fα and gα are respectively the density distribution function and the internal energydistribution function; f

eqα and g

eqα are their corresponding equilibrium states; τv and τc are

the single relaxation parameters determined by the dynamic viscosity and the thermaldiffusivity; δt is the streaming time step and eα is the particle velocity in the α direction;M and N are the total number of discrete particles used respectively for fα and gα. In thiswork, M and N are taken the same. According to the conservation laws, the density ρ,velocity u and internal energy e are evaluated from the zeroth and first moments of fα

and gα:

ρ=N

∑α=0

fα, ρu=N

∑α=0

fαeα, ρe=M

∑α=0

gα. (2.3)

Here, e=DRT/2, D is the dimension, R is the gas constant and T represents the temper-ature.

For 3D computations, the commonly-used 15-bit lattice velocity model (D3Q15) shownin Fig. 1 is applied in this work, which is given by

eα =

0, α=0,(±1,0,0),(0,±1,0),(0,0,±1), α=1∼6,(±1,±1,±1), α=7∼14.

(2.4)

The equilibrium distribution functions of feqα and g

eqα for the density and energy distribu-


Figure 1: D3Q15 lattice velocity model.

tion functions are respectively given by

feqα (r,t)=ρwα

[

1+eα ·u

c2s

+(eα ·u)2−(cs |u|)2

2c4s

]

, (2.5)

geqα (r,t)=

−ρ

2

|u|2c2

, α=0,

ρe

18

[

1+eα ·u

c2+

9

2· (eα ·u)2

c4− 3

2· |u|

2

c2

]

, α=1∼6,

ρe

36

[

2+4· eα ·uc2

+9

2· (eα ·u)2

c4− 3

2· |u|

2

c2

]

, α=7∼14.

(2.6)

Here, for the D3Q15 model given by Eq. (2.4), the coefficients wα and the sound speed cs

are given as w0=2/9, w1−6=1/9 and w7−14=1/72, cs =1/√

3.

The Chapman-Enskog (C-E) expansion analysis provides a solid foundation for theLB models of Eqs. (2.1)-(2.2) to simulate thermal flows, which guarantees accurate recov-ery of the macroscopic mass, momentum and energy equations. As shown in Refs. [42,43], by applying the C-E expansion analysis, Eqs. (2.1)-(2.2) can reproduce the followingequations:

∂ρ

∂t+∇·

(

N

∑α=0

eα feqα

)

=0, (2.7)

∂ρu

∂t+∇·Π=0, (2.8)

∂ρe

∂t+∇·Q=0. (2.9)


where Π and Q are the momentum flux tensor and thermal flux vector, respectively,which are defined by

Πβγ=N

∑α=0

(eα)β(eα)γ

[

feqα +

(

1− 1

2τν

)

ε f(1)α

]

, (2.10)

Q=N

∑α=0

[

eα

(

geqα +

(

1− 1

2τc

)

·εg(1)α

)]

. (2.11)

In the above equations, the non-equilibrium terms of ε f(1)α and εg

(1)α satisfy the following

relationships [12, 29, 43]:

fneqα = fα− f

eqα ≈ ε f

(1)α =−τνδt

(

∂

∂t+eα ·∇

)

feqα , (2.12)

gneqα = gα−g

eqα ≈ εg

(1)α =−τcδt

(

∂

∂t+eα ·∇

)

geqα . (2.13)

After some standard mathematical manipulations, equations (2.7)-(2.9) can recover thefollowing differential equations, which are exactly the equations governing mass, mo-mentum and energy conservations (without consideration of compression work and vis-cous heat dissipation) [12, 29, 43],

∂ρ

∂t+∇·ρu=0, (2.14)

∂ρu

∂t+∇·(ρuu)=−∇p+ν∇·

[

∇ρu+(∇ρu)T]

, (2.15)

∂

∂t(ρe)+∇·(ρue)=χ∇2(ρe), (2.16)

where ν is the kinematic viscosity and χ is the thermal diffusivity. When D3Q15 latticevelocity model is used, τν and τc can be determined by [28]:

ν= c2s

(

τυ−1

2

)

·δt, (2.17)

χ=5

3

(

τc−1

2

)

c2s ·δt. (2.18)

As can be seen, equations (2.7)-(2.9) are equivalent to equations (2.14)-(2.16). It meansthat the macroscopic flow filed can be predicted by directly solving Eqs. (2.7)-(2.9) and itsfluxes can be modeled by lattice summations of equilibrium and non-equilibrium distri-bution functions. This feature is applied in the next section to develop the 3D LBFS.

3 3D lattice Boltzmann flux solver

In this section, a consistent and complete 3D LBFS is presented for simulation of bothisothermal and thermal flows by solving Eqs. (2.7)-(2.9).


3.1 Governing equations and finite volume discretization

By including the external forcing term FE, the governing equations (2.7)-(2.9) for the flowand temperature field can be rewritten in a more general form [43]:

∂W

∂t+∇·F =FE (3.1)

where

W =

ρρu

ρe

, F=

P

Π

Q

, (3.2a)

P=N

∑α=0

eα feqα , (3.2b)

Πβγ=N

∑α=0

(eα)β(eα)γ · f∧α , (3.2c)

f∧α = feqα +

(

1− 1

2τν

)

fneqα , (3.2d)

fneqα = fα− f

eqα = ε f

(1)α =−τνδt

(

∂

∂t+eα ·∇

)

feqα , (3.2e)

Q=N

∑α=0

eαg∧, (3.2f)

g∧α = geqα +

(

1− 1

2τc

)

gneqα , (3.2g)

gneqα = gα−g

eqα = εg

(1)α =−τcδt

(

∂

∂t+eα ·∇

)

geqα . (3.2h)

In the above equations, β and γ represent the x, y and z directions. As indicated in [43],the external forcing term FE can be treated as a source term without affecting the accuracyof the flux evaluation.

The cell-centered finite volume method is applied to solve Eq. (3.1) so that the macro-scopic flow properties ρ, ρu and ρe can be updated at the cell center by marching in time.The fluxes are evaluated at the cell interface by local reconstruction of thermal LBM so-lutions. Integrating Eq. (3.1) over a control volume Ωi, we have

dW i

dt=− 1

dVi∑

k

RkdSk+FE, Rk =(n·F)k, (3.3)

where dVi is the volume of the control cell, and dSk is the area of the kth control surfaceenclosed Ωi, n=(nx,ny,nz) is the unit normal vector on the kth control surface.


Figure 2: Local reconstruction of LBM solution at an interface between two control cells.

The detailed expression for the flux Rk at a cell interface depends on the lattice veloc-ity model. For 3D simulations, the D3Q15 lattice model is applied in the present study.In this situation, the flux Rk can be given in detail as follows:

R3Dk =

nxq1+nyq2+nzq3

nxφ11+nyφ12+nzφ13



nx ϕ1+nyϕ2+nz ϕ3

k

, (3.4)

where qi, φij and ϕi are given in the appendix. It can be seen that φij is symmetric and thefinal expressions of qi, φij and ϕi are only simple algebraic combinations of the equilib-rium and non-equilibrium DDFs.

For numerical simulations, the key issue in the evaluation of the flux Rk is to performan accurate evaluation of f

eqα , f∧α and g∧α at each cell interface. f

eqα and f∧α are considered

first. After that, the evaluation of g∧α is discussed.

3.2 Evaluation of feqα and f∧α at cell interface

Fig. 2 shows a typical interface between two adjacent control cells. feqα , f∧α should be

evaluated at the center of this interface. According to Eq. (3.2d), f∧α includes both theequilibrium and non-equilibrium parts f

eqα and f

neqα . In the following, the evaluations of

these two parts are illustrated respectively.

fneqα can be approximated by the equilibrium DDFs f

eqα at the cell interface and the sur-

rounding points. Discretizing Eq. (3.2e) with the second order Taylor-series expansion,we can obtain f

neqα :

fneqα (r,t)=−τν

[

feqα (r,t)− f

eqα (r−eαδt,t−δt)

]

. (3.5)


As shown in Eq. (3.5), fneqα only involves the equilibrium distribution functions f

eqα (r−

eαδt,t−δt) and feqα (r,t).

Following the conventional LBM, feqα (r−eαδt,t−δt) can be computed from the fluid

density ρ and flow velocity u at the position of (r−eαδt) by applying Eq. (2.5). As shownin Fig. 2, if ri, ri+1 and r are defined as the physical positions for the two cell centers andtheir interface, respectively, ρ and u at the location r−eαδt can be calculated by interpola-tion with given flow properties at the cell centers. One possible interpolation formulationcan be given as

φ(r−eαδt)=

φ(ri)+(r−eαδt−ri)·∇φ(r i), when r−eαδt is in the cell Ωi,φ(ri+1)+(r−eαδt−ri+1)·∇φ(ri+1), when r−eαδt is in the cell Ωi+1,

(3.6)where φ represents any macroscopic flow variable, such as ρ and u. Note that Eq. (3.6) isalso applicable to other flow variables such as e.

feqα (r,t) can also be computed from ρ(r,t) and u(r,t) by applying Eq. (2.5). ρ(r,t) and

u(r,t) are to be determined from mass and momentum conservation laws by streamingf

eqα (r−eαδt,t−δt) in a short time step:

ρ(r,t)=N

∑α=0

feqα (r−eαδt,t−δt), (3.7a)

ρ(r,t)u(r,t)=N

∑α=0

feqα (r−eαδt,t−δt)eα, (3.7b)

where feqα (r−eαδt,t−δt) has already been calculated previously.

Once feqα (r−eαδt,t−δt) and f

eqα (r,t) are obtained, f

neqα can be approximated using

Eq. (3.2e) and thus f∧α can be easily computed from Eq. (3.2d).

3.3 Evaluation of g∧α at cell interface

g∧α can be evaluated in a similar procedure to f∧α . As shown in Eq. (3.2g), g∧α includesboth equilibrium internal energy distribution functions g

eqα and non-equilibrium internal

energy distribution functions gneqα . A delicate and thorough procedure to evaluate g

eqα and

gneqα is presented as follows.

Consider gneqα first. Discretizing Eq. (3.2h) with the second order Taylor-series expan-

sion yields an expression for gneqα :

gneqα (r,t)=−τc

[

geqα (r,t)−g

eqα (r−eαδt,t−δt)

]

. (3.8)

Eq. (3.8) shows that gneqα (r,t) is only determined by the equilibrium internal energy dis-

tribution functions at the cell interface and its surrounding points. Similar to feqα , g

neqα (r,t)

and geqα (r−eαδt,t−δt) are calculated from ρ, u and e at the corresponding positions and

time according to Eq. (2.6).


As indicated previously, ρ and u at the location r−eαδt can be interpolated usingEq. (3.6). e can also be computed in the same way by applying Eq. (3.6). After ρ, u ande at (r−eαδt,t−δt) are obtained via interpolation, g

eqα (r−eαδt,t−δt) can be calculated by

using Eq. (2.6).g

eqα (r,t) can also be computed from ρ(r,t), u(r,t) and e(r,t) according to Eq. (2.6).

Note that ρ(r,t), u(r,t) have already been obtained by applying Eq. (3.7). By streamingg

eqα (r−eαδt,t−δt) in a short time step, the internal energy e(r,t) can be approximated by

energy conservation law:

ρ(r,t)·e(r,t)=N

∑α=0

geqα (r−eαδt,t−δt). (3.9)

Once e(r,t) is computed, geqα (r,t) can be calculated using Eq. (2.6). Subsequently, g∧α can

be easily computed from Eq. (3.2g).As shown above, for the evaluation of f∧ and g∧, only equilibrium distribution func-

tions feqα and g

eqα are involved. They stream to the cell interface within a short time interval

δt for the local reconstruction of f∧ and g∧. This process can be performed independentlyon each interface which makes the present solver more flexible for applications on non-uniform grids. In addition, as only the equilibrium terms are involved, there is no needto store the distribution functions at each time step during the marching process, whichsaves much virtual memory, especially for 3D computations. After local reconstructionof thermal LBM solutions at the cell interface, f

eqα , f∧α and g∧α can be easily computed.

Then the flux Rk at any interface can be effectively evaluated and Eq. (3.3) can be solvedby applying efficient time marching schemes such as four stage Runge-Kutta method.

3.4 Computational sequence

Overall, the basic solution procedure of 3D LBFS can be summarized below:

(1) At first, we have to specify a streaming time step δt. The choice of δt should satisfythe constraint that the location of (r−eαδt) must be within either the cell Ωi or thecell Ωi+1. Once δt is chosen, τv and τc can be calculated by Eqs. (2.17)-(2.18).

(2) For the considered interface position r, identify its surrounding positions (r−eαδt),and then use Eq. (3.6) to compute the macroscopic flow variables at those positions.

(3) Use Eqs. (2.5)-(2.6) to calculate the equilibrium density distribution function feqα (r−

eαδt,t−δt) and geqα (r−eαδt,t−δt).

(4) Compute the flow variables at the cell interface by using Eqs. (3.7) and (3.9), andfurther calculate f

eqα (r,t) and g

eqα (r,t) by applying Eqs. (2.5)-(2.6).

(5) Calculate fneqα (r,t), f∧α (r,t), g

neqα (r,t) and g∧α (r,t) by using Eqs. (3.2d), (3.2g), (3.5)

and (3.8).

(6) Compute the fluxes Rk at the cell interface by Eq. (3.4) and further calculate theexternal forcing term FE if available.


(7) Once fluxes at all cell interfaces are obtained, solve ordinary differential equations(3.3) by using 4-stage Runge-Kutta scheme.

(8) Repeat steps (2)-(7) until convergence criterion is satisfied.

From the above solution process, it can be seen that present LBFS is different from theconventional LBE solvers. It only applies the LBE solutions locally to construct the fluxesat each interface. As a finite volume method, the LBFS is also completely different fromthe recently proposed LW-ACM [50], which applies perfect shift model in its solutionprocess and is intrinsically a finite difference scheme.

4 Numerical results and discussion

In this section, the accuracy, efficiency and stability of the 3D LBFS is validated by nu-merical simulations of the plane Poiseuille flow, lid-driven cavity flow and the thermalnatural convection flow in a square cube. Numerical results will be compared with bothnumerical and analytical solutions. Note that for all test cases, D3Q15 lattice velocitymodel is adopted and the energy equation given in Eq. (3.1) is not solved for isothermalcases since it has no effect on the flow field.

4.1 Plane Poiseuille flows between two parallel plates

The numerical accuracy of the 3D LBFS is examined by simulating the plane Poisseuilleflow. This flow takes place in a rectangular domain with a size of −L/46 x,y6 L/4 and−L/26 z6 L/2. No-slip boundary condition is applied on upper and lower boundariesat z=±L/2 while periodic boundary conditions are set on boundaries in both x and ydirections. An external force FE =(Fx,0,0) is imposed on the fluid in this domain. Withthese conditions, the solution of this flow can be given analytically as:

u(x,y,z)=Umax

([

1−( z

L

)2])

, v(x,y,z)=0, w(x,y,z)=0, (4.1)

where Umax=FxL2/8ν.Four different uniform grids of 6×6×11, 11×11×21, 21×21×41 and 41×41×81 are

applied to simulate this flow at the Reynolds number of Re=UmaxL/ν=40. The obtainedresults are measured by the L2 norm, which is defined as:

L2(u)=

√

√

√

√

1

Ntotal

Ntotal

∑k = 1

(

unumerical−uexact

uexact

)2

, (4.2)

where Ntotal is the total number of grid points. Fig. 3 shows the present results of theu-velocity profile along the centerline of the flow domain. It can be seen that the presentresults agree well with the analytical solutions. Fig. 4 shows the L2 norm of u versus mesh


Figure 3: u-velocity along the vertical centerline for the plane Poiseuille flow at Re=40.

Figure 4: L2 norm of relative error u versus mesh spacing h for the plane Poiseuille flow at Re=40.

spacing h in the log scale. The slope of the line is 1.995, which shows that the accuracy ofthe 3D LBFS is the second in space. The velocity components in the y and z directions arein the order of 10−16, which also agree well with the analytical solutions. It is worth topoint out that, in the numerical computations, the iterative time step can be set as largeas 2.2h when the four-stage Runge-Kutta method is applied. This phenomenon indicatesthat the temporal stability of the present solver can be enhanced if more stable temporalschemes are applied.

4.2 3D lid-driven cavity flows

After validating the accuracy of the LBFS, the 3D lid-driven cavity flow illustrated inFig. 5 is studied to examine the performance of the LBFS on both uniform and non-


Figure 5: Illustration of the 3D lid driven cavity flows (a) and a typical non-uniform computational grid (b).

uniform grids. Particular attention will be paid on its accuracy, efficiency and stability,which will be compared in detail with the standard LBM.

First, to validate the present solver for this flow problem, numerical simulations arecarried out on a very fine non-uniform grid of 81×81×81. Three different Reynoldsnumbers of Re= ρU0L/µ= 100, 400 and 1000 are considered, where U is the velocity ofthe top lid and L is the length of the cavity. No-slip boundary conditions are applied at allwalls. The parameters are set as U0=0.1, L=1 and ρ=1. Initially, the flow field is at restand the density in the domain is set as 1. Fig. 6 shows the u- and v- velocity componentsalong the vertical centerlines of the cubic cavity. The numerical results of Ku et al. [46] andDing et al. [47] are also included for comparison. Obviously, good agreements have beenachieved, which successfully validates the reliability of the present solver in simulating3D incompressible flows on non-uniform grids. To show the flow patterns of the lid-driven cavity flows, three middle planes of the cube located at x=0.5, y=0.5 and z=0.5respectively are chosen. On each of these planes, the streamlines at Re=100, 400 and 1000are displayed respectively in Figs. 7-9. The effects of the Reynolds numbers on the flowpatterns in different planes can be clearly observed. As shown in Fig. 7, the flow patternon the z=0.5 plane demonstrates that, when the Reynolds number is low (Re=100), theprimary vortex is generated in the upper half region and very small secondary vorticesappear at the bottom corners. As Re is increased from 100 to 1000, the axis of this primaryvortex gradually moves from the upper half region to the cavity center and the strengthof the secondary vortices at the bottom corners is enhanced. From the flow patterns onthe x= 0.5 and y= 0.5 planes shown in Figs. 8-9, it can be observed that the strength ofsecondary vortices is also gradually enhanced as the Reynolds number increases. Thisalso shows a strong 3D effect. All these observations were also founded and reported byKu et al. [46] and Ding et al. [47].


Figure 6: u and v velocity profiles along the centerlines of cubic cavity for 3D lid-driven cavity flow at Reynoldsnumbers of 100, 400 and 1000.

After validation, the accuracy and efficiency of the LBFS are examined by comparingwith the standard LBM on both uniform and non-uniform grids. Consider the numericalexperiment on the uniform mesh of 81×81×81 first. The computation stops when the


Figure 7: Streamlines on the plane of z=0.5 for the 3D lid-driven cavity flow.

Figure 8: Streamlines on the plane of x=0.5 for the 3D lid-driven cavity flow.

Figure 9: Streamlines on the plane of y=0.5 for the 3D lid-driven cavity flow.

convergence criterion satisfies

∑ij

∣

∣

∣

∣

(√u2+v2+w2

)n+1−(√

u2+v2+w2)n∣

∣

∣

∣

∑ij

(√u2+v2

)n+1610−6. (4.3)

Fig. 10 compares the LBFS and LBM results of u-x and v-y on the centerlines for the lid-


Table 1: Comparison of the efficiency of LBFS and LBM on the same mesh for lid-driven cavity flow at Re=1000.

Virtual Memory Maximum Computational

Grid Size Methods (Megabytes) iterative steps time (s)

LBFS 111.6 27616 28732

81×81×81 LBM 190.8 27081 11048

LBFS/LBM 0.585 1.02 2.60

Table 2: Comparison of the efficiency of LBFS and LBM with grid-independence results for lid-driven cavityflow at Re=1000.

Required Virtual Memory Maximum Computational

Methods grids (Megabytes) iterative steps time (s)

LBFS 61×61×31 24.2 22705 4471

LBM 101×101×101 361.8 32529 33527

LBFS/LBM 11.20% 6.69% 68.8% 13.33%

driven cavity flow at Re=1000. It can be seen that good agreements have been achieved.This implies that, with the same grid resolutions, the LBFS and LBM have the same ac-curacy for the velocity fields in this case. Table 1 compares the efficiency and requiredvirtual memory of these two methods. It is shown that the LBFS takes about 2.6 timesof computational time of the LBM to get the final solutions on the same grids. This isbecause the LBFS needs additional time to do interpolations at each cell interface. Then,we investigate the efficiency of the LBFS and LBM when the grid-independence (con-verged) numerical results are achieved. Fig. 11 and Fig. 12 present the LBFS results ofv-component and pressure, and the LBM results of the pressure on the centerline of thecavity at Re=1000 on three different grids. Here, the grids applied by LBFS are non-uniform. As shown in these two figures, the LBFS need a coarse mesh of 61×61×31 to getgrid-independence results while the LBM requires a much finer mesh of 101×101×101.The efficiency of the LBFS and LBM on these two grids is compared in Table 2. It canbe seen that, when grid-independence results are obtained, the LBFS takes only 13.33%computational time of the LBM. The required virtual memory by the LBFS is reducedsignificantly to 6.69% as compared with the LBM.

The pure stability of the LBFS is further examined without considering the accuracyof the numerical results. Table 3 shows the minimum computational mesh required bythe LBFS for the lid-drive cavity flows at Re=400 and 1000. The requirements of the LBMare also included for comparison. As can be seen, the LBFS requires much less grids to getstable solutions. In addition, the stability of the LBFS can also be verified by comparingthe grid-independence results. Fig. 13 compares the pressure contours of the LBFS andthe LBM on z= 0.5 at Re= 1000. The pressure contours obtained by the LBFS are verysmooth while those obtained by the LBM have unphysical oscillations in the upper twocorners. Therefore, the stability and robustness of the LBFS are superior to the LBM.


Figure 10: Comparison of the LBFS and LBM results of u-x and v-y on the centerlines for the 3D lid-drivencavity flows at Re=1000 on the same uniform grids of 81×81×81.

Figure 11: The velocity profile u-x on different non-uniform grids for the 3D lid-driven cavity flows at Re=1000.

Table 3: Comparison of minimum required mesh size of the LBFS and LBM for the lid-driven cavity flows atdifferent Reynolds numbers.

Re LBFS LBM

400 4×4×4 16×16×16

1000 4×4×4 31×31×31

4.3 3D natural convection in a cubic cavity

The natural convection in a cubic cavity is studied here to further examine the perfor-mance of the present solver for 3D thermal flows. The schematic diagram of this flowproblem is depicted in Fig. 14. As can be seen, the fluid is filled in a cubic cavity, whichis constituted by two vertical walls with different temperatures and four other adiabatic


Figure 12: Comparison of pressure obtained by the LBFS and the LBM on the horizontal centerline for the 3Dlid-driven cavity flows at Re=1000.

Figure 13: Comparison of pressure contours obtained by the LBFS and the LBM for the 3D lid-driven cavityflows at Re=1000.

walls. The vertical wall at x= 0 is cooled with a lower temperature of T = 0 while thatat x = 1 is heated with a higher temperature of T = 1 respectively. Other four walls areassumed to be adiabatic. No-slip boundary conditions are applied on all walls. The flowpattern of this problem is governed by the Prandtl number Pr and the Rayleigh numberRa, which are defined as:

Pr=ν

χ, Ra=

gβ·∆T ·L3

ν·χ =V2

c ·L2

ν·χ , (4.4)


Figure 14: Schematic diagram of the 3D natural convection in a cubic cavity.

where L is the characteristic length of the cavity and Vc =√

gβL·∆T is the characteristicthermal velocity which is constrained by the low Mach number limit. With the abovedefinitions and the Boussinesq approximation, the buoyancy force in the z direction canbe defined as Fb =−ρgβ(T−Tm), where Tm is the average temperature of the cold andhot walls. Then, the external forcing term FE in Eqs. (3.1) and (3.3) can be given by FE =(0,0,0,−Fb,0). In this study, the Prandtl number is fixed at Pr= 0.71 while the Rayleighnumber is varied from 103 to 106. A grid size of 81×81 is used for simulations.

To quantitatively examine the heat transfer rate of this problem, the mean Nusseltnumber Numean in the y direction and the overall Nusselt number Nuoverall at the isother-mal walls are computed, which can be respectively defined as follows [47]:

Numean (y)=

1∫

0

Nulocaldz=

1∫

0

∂T(y,z)

∂x

∣

∣

∣

∣

x=0 or x=1

dz, (4.5)

Nuoverall =

1∫

0

Numean(y)dy. (4.6)

Table 4 shows the present results of the overall Nusselt number Nuoverall on the heatedwall (x=1) at different Rayleigh numbers of 103, 104, 105 and 106. Also included in thistable are the numerical solutions of Fusegi et al. [48] and Ha et al. [49]. Note that theresults of Fusegi et al. [48] were obtained by the third order QUICK scheme with highresolutions and can be taken as the benchmark solutions. It can be seen from the tablethat, for all the cases considered, the relative errors between the present results and thoseof Fusegi et al. [48] are within 1.3%. Obviously, good agreements have been achieved,which verify the reliability of the present solver. In addition, it can also be seen that, with


Table 4: Comparison of overall Nusselt number Nuoverall at the heated wall (x=1) for 3D natural convectionin a cavity.

Ra 103 104 105 106

Present 1.071 2.062 4.344 8.684

Nuoverall Fusegi et al. [48] 1.085 2.085 4.361 8.770

Ha et al. [49] 1.072 2.07 4.464 -

Relative Error to Ref. [48] 1.29% 1.10% 0.38% 0.98%

the increase of Ra, the overall Nusselt number is gradually increased, which indicatesthe enhancement of the heat transfer rate all over the flow domain. To further explorethe details of the heat transfer characteristic, the local Nusselt number distribution on theheated wall (x= 1) is displayed in Fig. 15. It can be observed that, the local Nu number

Figure 15: Local Nusselt number distributions at the heated wall (x= 1) for the 3D natural convection in acubic cavity.


Table 5: Comparison of representative field properties in the symmetric plane at y = 0.5 for the 3D naturalconvection.

Ra 103 104 105 106

umax Present 0.132 0.200 0.142 0.082

Peng et al. [28] 0.132 0.206 0.149 -

Fusegi et al. [48] 0.131 0.201 0.147 0.084

z Present 0.187 0.176 0.147 0.146

Peng et al. [28] 0.188 0.163 0.136 -

Fusegi et al. [48] 0.200 0.183 0.145 0.144

wmax Present 0.133 0.221 0.244 0.253

Peng et al. [28] 0.133 0.221 0.240 -

Fusegi et al. [48] 0.132 0.225 0.247 0.259

x Present 0.829 0.885 0.932 0.968

Peng et al. [28] 0.826 0.887 0.935 -

Fusegi et al. [48] 0.833 0.883 0.935 0.967

Numean Present 1.092 2.289 4.622 8.921

Peng et al. [28] 1.097 2.304 4.658 -

Fusegi et al. [48] 1.105 2.302 4.646 9.012

near the lower wall is higher than that close to the upper wall and its distribution issymmetric about the plane of y=0.5. As Ra is increased, the local Nusselt number at anyphysical point also increases. For the cases at high Rayleigh numbers of Ra=105 and 106,the contours of Nu near y=0.5 are almost parallel to the y axis and have roughly uniformdistribution in the z direction.

Table 5 shows the representative quantities at the symmetric plane of y= 0.5 for allconsidered cases of Ra = 103, 104, 105 and 106. These properties include the maximumhorizontal speed umax on the vertical central line and its position z, the maximum verticalvelocity wmax on the horizontal central line and its position and the mean Nusselt numberon the wall in the plane of y=0.5. Also included in this table are the solutions of Fusegi etal. [48] and Peng at al. [28] for comparison. It can be seen that the quantitative agreementsbetween the present results and those published data are satisfactory. To more clearlydemonstrate flow field information, Fig. 16 depicts the velocity profiles on the horizontaland vertical central lines in the symmetric plane (y=0.5). The results of Fusegi et al. [48]at Ra=105 and 106 and those of Ha et al. [49] at Ra=105 are also included in this figurefor comparison. Once again, good agreements are achieved. The temperature fields atthis symmetric plane for all cases considered are depicted in Fig. 17. As can be seen, the


Figure 16: Velocity profiles in the symmetric plane at y=0.5 for 3D natural convection.

thermal boundary layers near the heated and cooled walls are becoming thinner as Rais increased. In addition, the overall patterns at different Ra are very similar to thosein 2D cases [43]. Fig. 18 further shows the 3D temperature field in a cubic cavity. Fivedifferent iso-surfaces at T=0.15, 0.25, 0.5, 0.75, and 0.85 are depicted in each figure. Theseiso-surfaces are concentrated to the region near upper-cooled and lower-heated walls asRa is increased. This phenomenon is in good accordance with the temperature contoursin 2D views as shown in Fig. 17. The enhancement of the heat transfer rate at higherRayleigh numbers is outstanding and the 3D effects on the temperature filed can also beobserved.

5 Conclusions

In this paper, a 3D lattice Boltzmann flux solver (LBFS) is proposed for the simulationof both isothermal and thermal flows. The 3D LBFS applies the finite volume method(FVM) to solve the N-S equations for the macroscopic conservative variables at cell cen-


Figure 17: Temperature contours in the symmetric plane at y= 0.5 for the 3D natural convection in a cubiccavity.

ters. The fluxes are evaluated at each cell interface by local reconstruction of the LBMsolutions. In this step, the relationships between the expressions of fluxes in the N-S equations and density distribution functions (DDFs) in the LBE, obtained from themulti-scale Chapman-Enskog expansion analysis, are applied. Since equilibrium andnon-equilibrium terms are included in the DDFs, both the viscous and inviscid fluxesare computed simultaneously. Due to the local application of the LBM and the FVM, the3D LBFS successfully combines the advantages of the N-S solvers and LBE solvers. Thedrawbacks of the conventional LBM, such as lattice uniformity, tie-up between time stepand mesh spacing and tedious implementation of boundary conditions, can be removedin the present solver. In addition, the drawbacks of the incompressible N-S solvers, i.e.,the velocity-pressure coupling and the requirement of staggered grid, are removed aswell.

The performance of the LBFS is examined systematically by simulating the planePoiseuille flows, lid-driven cavity flows and natural convection in a cube. First, it hasbeen demonstrated that the accuracy of the LBFS is the second order in space. With


Figure 18: Temperature field for 3D natural convection a cubic cavity (Iso-surface levels in the positive xdirection: 0.15, 0.25, 0.5, 0.75 and 0.85).

the same computational grids, the numerical results of the LBFS and the standard LBMare also similar in quantity. Second, in terms of efficiency, the LBFS is about 2.6 timesslower than the LBM due to additional interpolations at each cell interface for the fluxcomputations. However, the computational efficiency of the LBFS is greatly improvedwhen non-uniform is applied to get grid-independence solutions. For instance, for thelid-driven cavity flow at Re=1000, the LBFS takes only about 13.33% computational timeand 6.69% virtual memory of the LBM to get converged results. In addition, the LBFS isalso more stable than the LBM. The LBFS solutions of the pressure are very smooth whilethe pressure filed obtained by LBM has non-physical oscillations in the upper corners.Moreover, the capability of the 3D LBFS in simulating the natural convection in a widerange of Rayleigh numbers from 102 to 106 is also demonstrated. Given the performancementioned above, it is believed that the LBFS has a high potential for solving various 3Disothermal and thermal flow problems in practice.


Appendix

By applying the D3Q15 lattice velocity model, the flux R3Dk can be written as follows:

R3Dk =

nxq1+nyq2+nzq3




nx ϕ1+ny ϕ2+nz ϕ3

k

, (A.1)

where

q1= feq1 − f

eq2 + f

eq7 − f

eq8 + f

eq9 − f

eq10+ f

eq11− f

eq12+ f

eq13− f

eq14 , (A.2a)

q2= feq3 − f

eq4 + f

eq7 − f

eq8 + f

eq9 − f

eq10− f

eq11+ f

eq12− f

eq13+ f

eq14 , (A.2b)

q3= feq5 − f

eq6 + f

eq7 − f

eq8 − f

eq9 + f

eq10+ f

eq11− f

eq12− f

eq13+ f

eq14 , (A.2c)

φ11= f∧1 + f∧2 + f∧7 + f∧8 + f∧9 + f∧10+ f∧11+ f∧12+ f∧13+ f∧14, (A.2d)

φ12=φ21=(

f∧7 + f∧8 + f∧9 + f∧10

)

−(

f∧11+ f∧12+ f∧13+ f∧14

)

, (A.2e)

φ13=φ31= f∧7 + f∧8 − f∧9 − f∧10+ f∧11+ f∧12− f∧13− f∧14, (A.2f)

φ22= f∧3 + f∧4 + f∧7 + f∧8 + f∧9 + f∧10+ f∧11+ f∧12+ f∧13+ f∧14, (A.2g)

φ23=φ32= f∧7 + f∧8 − f∧9 − f∧10− f∧11− f∧12+ f∧13+ f∧14, (A.2h)

φ33= f∧5 + f∧6 + f∧7 + f∧8 + f∧9 + f∧10+ f∧11+ f∧12+ f∧13+ f∧14, (A.2i)

ϕ1= g∧1 −g∧2 +g∧7 −g∧8 +g∧9 −g∧10+g∧11−g∧12+g∧13−g∧14, (A.2j)

ϕ2= g∧3 −g∧4 +g∧7 −g∧8 +g∧9 −g∧10−g∧11+g∧12−g∧13+g∧14, (A.2k)

ϕ3= g∧5 −g∧6 +g∧7 −g∧8 −g∧9 +g∧10+g∧11−g∧12−g∧13+g∧14. (A.2l)

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