Finite-volume method with lattice Boltzmann flux scheme for ...

PHYSICAL REVIEW E 93, 023308 (2016)

Finite-volume method with lattice Boltzmann flux scheme for incompressible porous media flow at the representative-elementary-volume scale Yang Hu* and Decai Li† School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, People’s Republic of China

Shi Shu School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, People’s Republic of China

Xiaodong Niu College of Engineering, Shantou University, Shantou 515063, People’s Republic of China (Received 23 June 2015; revised manuscript received 2 October 2015; published 29 February 2016) Based on the Darcy-Brinkman-Forchheimer equation, a finite-volume computational model with lattice Boltzmann flux scheme is proposed for incompressible porous media flow in this paper. The fluxes across the cell interface are calculated by reconstructing the local solution of the generalized lattice Boltzmann equation for porous media flow. The time-scaled midpoint integration rule is adopted to discretize the governing equation, which makes the time step become limited by the Courant-Friedricks-Lewy condition. The force term which evaluates the effect of the porous medium is added to the discretized governing equation directly. The numerical simulations of the steady Poiseuille flow, the unsteady Womersley flow, the circular Couette flow, and the lid-driven flow are carried out to verify the present computational model. The obtained results show good agreement with the analytical, finite-difference, and/or previously published solutions. DOI: 10.1103/PhysRevE.93.023308

I. INTRODUCTION

Flow through the porous media can be found in many fields of science and engineering, ranging from oil reservoir engineering, ground water hydrodynamics, to chemical engineering. Since the pioneering work of Darcy, there has been great progress in the theoretical and experimental studies of porous media flow [1,2]. Now the flow in porous media can be considered by using four widely used models at two different space scales: the Darcy, the Brinkman extended Darcy, the Forchheimer extended Darcy, and the pore scale models. In fact, the first three models are built with the aid of the continuum hypothesis at the REV (representative-elementaryvolume) scale. Based on some semiempirical relationships, these models can describe the flow field effectively under certain conditions [3–5]. Moreover, as a new research tool, the computer simulation technique has been extended to study the porous media flow [6,7]. Recently, the lattice Boltzmann method (LBM) has been proved to be a promising computational method to simulate complex fluid systems, such as multiphase and multicomponent flows [8,9], turbulence flows [10,11], microflows [12,13], fluid-solid interactions [14–17], and thermal flows [18–22]. The LBM has been also extended to study the porous media flow. As similar as conventional computational fluid dynamics (CFD) methods, the LB scheme has been also used at two different space scales. At the pore scale, with the help of the bounce-back rule, the LBM is easy to handle with the fluid-solid interactions at the complex interface. In fact, some studies based on pore-scale models have been done by some

* †

[email protected] Corresponding author: [email protected]

2470-0045/2016/93(2)/023308(11)

scholars [23–31]. As early as 1989, Succi et al. applied the LBE model to simulate the low Reynolds number flows in a three-dimensional random medium [23]. The Darcy’s law was validated in their work. Heijs and Lowe evaluated the permeability of a random array of spheres and a clay soil based on the LB simulations [24]. Their calculated value of the permeability is in good agreement with the experimental data for the two samples. Koponen et al. presented the LB simulations of single-fluid flow in two-dimensional (2D) and three-dimensional (3D) porous media [25] and proposed a modified Kozeny-Carman law. Their numerical results also suggested that the permeability depends exponentially on porosity over a large porosity range for fluid flow through large 3D random fiber webs. Using LBM, Pan et al. proposed a modified form of Ergun’s equation to describe both low and moderate Reynolds number flows [26]. Martys and Chen applied the Shan-Chen-type lattice Boltzmann model to simulate multicomponent flow inside the porous media [27]. They calculated the relative permeability for different wetting fluid saturations of a microtomography-generated image of sandstone and compared their results and experiment data. These studies indicated that pore-scale LB simulation is an excellent method to explore new physical law governing porous flows. However, as has been pointed out by Guo and Zhao [32], the huge computational overhead and the low Mach number limit of LBM make the pore-scale LB models hard to apply widely. Alternatively, several REV-scale LB models have been developed recently. Dardis and McCloskey proposed a modified LB scheme which introduces a real numbered parameter to represent the effect of the porous structure to model the flow in porous media [33]. Spaid and Phelan presented a direct LB simulation method for Brinkman equation. In this model, the Brinkman equation can be recovered through a modification of the particle

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©2016 American Physical Society

YANG HU, DECAI LI, SHI SHU, AND XIAODONG NIU

PHYSICAL REVIEW E 93, 023308 (2016)

equilibrium distribution function [34]. Kang et al. developed a unified lattice Boltzmann method for multiscale porous media flow which can simulate flow in porous systems where multiple length scales coexist [35]. Guo and Zhao developed a LB model to solve the generalized NavierStokes equation [32]. Through including the porosity into the equilibrium distribution function and discrete forcing term, the generalized Navier-Stokes equation for incompressible flow in porous media can be derived. Chen et al. proposed a similar generalized LB model for porous media flow with considering Klinkenberg’s effect [36]. Moreover, some LB models have also been developed in simulating the porous media flow with heat transfer [37–39]. Although LBM has many advantages in simulating incompressible flow, however, compared with conventional methods, LBM still suffers some drawbacks. As pointed out by Wang et al. [40], in LBM, the time step is tied up by the grid spacing. When multiblock and adaptive mesh techniques are implemented in LBM, complicated treatment involving physical quantities at different time levels must be considered. In practical CFD applications, only boundary conditions of macroscopic physical quantities are given. It is difficult to design the boundary condition solver in LBM to satisfy the macroscopic boundary conditions. Furthermore, when adding a source term in a lattice Boltzmann equation (LBE), the discrete lattice effects must be considered [41]. Otherwise there may exist deviation terms in the corresponding macroscopic equation. Recently, to overcome these drawbacks of the LBM, Wang et al. proposed a lattice Boltzmann flux scheme based on a finite-volume method [40,42–44]. Like the flux solver in gas kinetic scheme [45], the fluxes across the cell interface are obtained by evaluating the distribution functions at the cell interface. Through Chapman-Enskog analysis, once the macroscopic physical quantities are obtained, the distribution functions at the cell interface and its surrounding points can be calculated. We notice that the viscous flux at interface usually is treated implicitly to relax the time step limit in the conventional flux scheme. However, for lattice Boltzmann flux scheme, thanks to the hyperbolic nature of the LBE, both convective and viscous fluxes are calculated explicitly. Moreover, the implementation of the boundary conditions is very simple, and the source terms can be directly added to the discrete governing equations. In this paper, we propose a computational model for incompressible porous media flow at the REV scale based on a finite-volume method with the lattice Boltzmann flux scheme. The Darcy-Brinkmann-Forchheimer equation is used to model the flow in porous medium. The fluxes across the cell interface are computed by reconstructing the local solutions of the generalized LBE which is proposed by Guo and Zhao [32]. Specifically, it is noted that in the original lattice Boltzmann flux scheme proposed by Wang et al. [40], the time step limit is stronger than the Courant-Friedricks-Lewy (CFL) condition [46]. However, the midpoint time integration rule is used in this study. As a result, the time step is only limited by the CFL condition. Moreover, the linear drag, nonlinear drag, and viscous force terms are added directly to the discrete governing equations. Some numerical experiments are simulated to validate the present computational model: the steady Poiseuille flow, the unsteady Womersley flow, the circular Couette flow,

and the lid-driven flow. It can be found that the present results show good agreement with the analytical, finite-difference, and/or previously published solutions. II. MATHEMATICAL MODELS A. Darcy-Brinkman-Forchheimer equation for incompressible porous media flow

The Forchheimer-extend Darcy equation is used to model the isotherm porous flow at the REV scale. In this model, the Darcy term, Brinkman term, and the nonlinear inertial resistance term are contained. Both steady and unsteady flows in porous media can be described. The governing equations can be expressed as follows: ∇ · u = 0,

(1)

u 1 ∂u + (u · ∇) = − ∇(p) + νe ∇ 2 u + F, ∂t  ρ

(2)

where u,p are volume-averaged velocity and pressure, respectively, ρ is the fluid density, νe is the effective viscosity, and  is the porosity. F is the forcing term including drag force inside the porous media and other external force, and can be expressed as follows: F=−

F ν u − √ |u|u + G, K K

(3)

where ν is the viscosity of the fluid. F and K are the parameters which are related to the porosity . In this study, Ergun’s relations is used: 1.75 F = √ . 150 3

(4)

It should be noted that the flow characteristic inside porous media in the real world is very complex. The above Forchheimer-extend Darcy equation (2) with Ergun’s relation (4) can only be used in some flow conditions, such as the the flow inside a packed bed or fluidized bed [4]. On the one hand, like the intermediate Reynolds number porous media flows, the viscous boundary layer effect √ cannot be neglected. As a result, a term proportional to |u|u must be included in the right-hand side of Eq. (2) [47]. On the other hand, for the case with a highly correlated microstructure, Ergun’s relations are not appropriate for estimating the permeability K. In such case, all the complexity of the microstructure must be taken into account to obtain K. B. Lattice Boltzmann equation and the corresponding macroscopic equation

The LBM can be regarded as an alternative technique of the Navier-Stokes solver, and it is a very popular kinetic scheme in recent years. The evolution equation of the lattice Boltzmann model can be written as follows:

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fα (x + eα t,t + t) − fα (x,t)  1 = − fα (x,t) − fαeq (x,t) , τ

(5)

FINITE-VOLUME METHOD WITH LATTICE BOLTZMANN . . .

where fα (x,t) is the density distribution function for the eq discrete velocity eα , t is the time step, and fα (x,t) is the local equilibrium density distribution function, and can be computed by   eα · u (eα · u)2 u2 , (6) fαeq (x,t) = ωα ρ 1 + 2 + − cs 2cs4 2cs2 where cs is the lattice sound speed and ωα is the weight coefficient which depends on the lattice velocity model. u = (u,v)T is the fluid velocity. τ is the dimensionless relaxation time of velocity field and it can be obtained as follows: ν + 0.5, (7) τ= 2 cs t where ν is the kinematic viscosity. In this study, the D2Q9 models are used. The corresponding velocity sets are defined as ⎧ (0,0), α=0 ⎪ ⎨     π π α = 1–4 eα = cos (α − 1) 2 , sin (α − 1) 2 c, ⎪    ⎩√  π π 2 cos (2α − 1) 4 , sin (2α − 1) 4 c, α = 5–8, (8) where √ c = x/t, and x is the lattice spacing. Further, c = 3cs . The corresponding weight coefficients are given as ⎧ 4 ⎪ , α=0 ⎪ ⎨9 ωα = 19 , α = 1–4 (9) ⎪ ⎪ ⎩ 1 , α = 5–8.

PHYSICAL REVIEW E 93, 023308 (2016)

−ν[∇u + (∇u)T ] satisfy

α

ρuu + pI =

p = ρcs2 .

−ν[∇u + (∇u)T ] =

  1 f neq , eα eα 1 − 2τ α

(17) (18)

(19) (20)

the macroscopic equations (1) and (2) without force term can be recovered using the Chapman-Enskog analysis. III. NUMERICAL METHOD

We rewrite Eqs. (13) and (14) into hyperbolic conservation law form [40] ∂Qy ∂ ∂Qx + + = 0, ∂t ∂x ∂y

(21)

where

(12)

 = [ρ,ρu,ρv]T ,

(22)

Qx = [Rx ,Jxx ,Jxy ]T ,

(23)

Qy = [Ry ,Jyx ,Jyy ]T ,  Rx = eαx fαeq ,

(24)



(25)

α

Ry = Jxx

Jxy

Jyx

Jyy

(15)

where ε is the multiscale expansion parameter. eq With the definition of fα , the mass flux ρu, convective momentum flux ρuu + pI, and viscous momentum flux



  eα · u (eα · u)2 u2 − fαeq (x,t) = ωα ρ 1 + 2 + , cs 2cs4 2cs2  p = ρcs2 ,

is calculated by

fαneq = fα − fαeq ≈ εfα(1)   ∂ + eα · ∇ fαeq + O(t 2 ), = −τ t ∂t

eα eα fαeq ,

(16)

where I is the unit tensor. Substituting relationships (16), (17), and (18) into Eqs. (13) and (14), the Navier-Stokes equation can be obtained in the low Mach number limit. It is worth noting that the inviscid momentum flux in Eq. (2) is ρuu/ + pI. Through redefinition of the equilibrium distribution function and pressure as suggested in literature [32], namely,

(14) where



α

With the aid of the multiscale Chapman-Enskog procedure, the following macroscopic equations can be obtained from Eq. (5):    ∂ρ eq +∇· eα fα = 0, (13) ∂t α       ∂ρu 1 eq neq +∇· f eα eα fα + 1 − = 0, ∂t 2τ α α neq fα

eα fαeq ,

α

36

The fluid density, velocity, and pressure are given by  fα , (10) ρ= α  eα fα , (11) u= α ρ



ρu =

eαy fαeq ,

 α     1 eq neq , f = eαx eαx fα + 1 − 2τ α α      1 eq neq , f = eαx eαy fα + 1 − 2τ α α      1 fαneq , = eαy eαx fαeq + 1 − 2τ α      1 fαneq . = eαy eαy fαeq + 1 − 2τ α

(26) (27)

(28)

(29)

(30)

In the present study, the finite-volume method is used to discretize Eq. (21). The computational domain is divided into a series of control cells. Integrating Eq. (21) on a control cell

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Vi at time interval [tn−1 ,tn ], we have  n−(1/2)

t n−1 n i − i = − Qx nx + Qyn−(1/2) ny dS, Vi ∂Vi

we have ρ(x,tn−(1/2) ) =

α

(31)

ρ(x,tn−(1/2) )u(x,tn−(1/2) ) =

hmin , η  1.0, c

hmin . c

where τt is the dimensionless effective relaxation time and it is given by τt =

ν 1 2 c t 2 s

+ 0.5.

Moreover   1 fαneq x − eα t,tn−1 2 = fαneq (x,tn−1 ) −

(34)

(35)

For the incompressible flows, the gradient terms are approximated using a central difference scheme. eq Now we consider fα (x,tn−(1/2) ). As explained above, ρ(x,tn−(1/2) ) and u(x,tn−(1/2) ) must be evaluated first. In fact,

t eα · ∇fαneq (x,tn−1 ) + O(t 2 ). 2 (41)

Taking a summation of Eq. (41) over α, we have    1 fαneq x − eα t,tn−1 = fαneq (x,tn−1 ) 2 α α (42) t  neq 2 − eα · ∇fα (x,tn−1 ) + O(t ) = 0. 2 α     1 eα fαneq x − eα t,tn−1 = eα fαneq (x,tn−1 ) 2 α α (43) t  (1) − ε ∇eα eα fα (x,tn−1 ) = 0. 2 α 

As a result, taking the zeroth-order and first-order moments of Eq. (40), we obtain that   1 (44) fαeq x − eα t,tn−1 , 2 α    1 eq ρ(x,tn−(1/2) )u(x,tn−(1/2) ) = eα fα x − eα t,tn−1 . 2 α

At the time level n − 1, once the fluid density ρ and velocity u at point x − 12 eα t are obtained, the equilibrium eq distribution function fα (x − 12 eα t,tn−1 ) can be determined. By using the Taylor expansion around the cell interface x, ρ(x − 12 eα t,tn−1 ),u(x − 12 eα t,tn−1 ) are approximated by

ρ x − 12 eα t,tn−1

= ρ(xi ,tn−1 ) + x − 12 eα t − xi · ∇ρ(xi ,tn−1 ), (36)

u x − 12 eα t,tn−1

= u(xi ,tn−1 ) + x − 12 eα t − xi · ∇u(xi ,tn−1 ). (37)

(39)

(40)

(33)

fαneq (x,tn−(1/2) )   ∂ = −τt t + eα · ∇ fαeq (x,tn−(1/2) ) ∂t    1 eq eq = −τt fα (x,tn−(1/2) ) − fα x − eα t,tn−1 + O(t 2 ), 2

eα fα (x,tn−(1/2) ).

  1 fα (x,tn−(1/2) ) = fαeq x − eα t,tn−1 2     1 1 neq f x − eα t,tn−1 . + 1− τt α 2

(32)

It will lower the computational efficiency. The key issue is to evaluate the fluxes across the cell intereq,n−(1/2) neq,n−(1/2) face. To obtain that, the values of fα and fα must be calculated. It is noticed that



(38)

Further

where hmin is the minimal grid spacing and η is the CFL number. It should be noted that in the works of Wang et al. [40,42–44], the CFL number is required to be less than 0.5, i.e., t  0.5

fα (x,tn−(1/2) ),

α

where ∂Vi ,Vi are the surface and volume of control cell Vi . n = (nx ,ny ) is the outward unit normal vector. Different from the work in the literature [40], the midpoint time integration method is used in this work. In fact, applying the midpoint rule, the time step t is determined by CFL condition t = η



ρ(x,tn−(1/2) ) =



(45) eq

neq

Once fα (x,tn−(1/2) ) and fα (x,tn−(1/2) ) are computed, the mass and momentum fluxes across the cell interface can be obtained. For the cases which consider the forcing term F, the force term is directly added to the right-hand side of Eq. (31). The force term is treated by Crank-Nicolson scheme  n−(1/2)

t ni − n−1 Qx = − nx + Qn−(1/2) ny dS y i Vi ∂Vi

t n Ri + Rn−1 + , (46) i 2

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FINITE-VOLUME METHOD WITH LATTICE BOLTZMANN . . .

where

PHYSICAL REVIEW E 93, 023308 (2016)

J . Here Re, Da, and J are defined as ⎤ √0 − u2 + v 2 u + ρGx ⎥ ⎥, R=⎣ ⎦ √ ρF √  u2 + v 2 v + ρGy − ρν v − K K ⎡

⎢− ρν u ⎢ K

ρF √  K

(47)

where Gx and Gy are the x and y components of the external force G. It should be noted that the source term Rn also contains the unknown fluid velocity un . Fortunately, similar as in Ref. [32], the above implicit equation can be solved locally, un =

u∗  , c1 + c12 + c2 |u∗ |

K νe LU , Da = 2 , J = , (51) ν L ν where L and U are the characteristic length and velocity, respectively. The dimensionless fluid velocity u, pressure p, body force G, and time t are defined as Re =

(48)

uˆ =

u , U

pˆ =

p ˆ = GL , tˆ = tU . , G ρU 2 U2 L

(52)

As a result, the dimensionless form of Eq. (2) can be expressed as   ∂ uˆ J 2 uˆ ˆ + + (uˆ · ∇) = −∇( p) ∇ uˆ ∂ tˆ  Re −

∗

 F ˆ ˆ uˆ +  G. uˆ − √ |u| Da Re Da

(53)

where u is an intermediate velocity and it is given by t u = u +  (Gn + Gn−1 ), 2 ∗

0

A. Steady Poiseuille flow between two planes

(49)

where u0 is fluid velocity without considering the forcing term. The two constant parameters c1 ,c2 are taken as c1 =

  t ν 1 1+ , 2 2 K

c2 = 

t F √ . 2 K

The Poiseuille flow through a parallel-plate channel filled with a porous medium is simulated to test the present method. The flow is driven by a constant force G = (Gx ,0). When the flow field reaches the steady state, the momentum equation can be written as νe

(50)

Now some properties of the present model are discussed as follows. First, for a conventional flux scheme, the computation of the viscous flux must adopt an implicit scheme. Therefore the conventional fluxes cannot be evaluated locally. However, the viscous flux can be expressed as the nonequilibrium part of the momentum flux in the lattice Boltzmann scheme. Like the convective flux, the viscous flux can also be calculated explicitly. We can also find that the simple collision-streaming processes of the LBM are retained. Secondly, we notice that the LBE proposed by Guo and Zhao for porous media flow can only use the uniform mesh [32]. In fact, in many cases, the velocity boundary layer may be very thin. To capture the velocity profiles near the boundary, the sufficient fine grid near the boundary must be applied. The LBM may need too much computation. Compared with the pure LBM, in the present method, the nonuniform mesh can be used. The finer mesh can be adopted near the boundary. Only the macroscopic physical quantities need be recorded. So the memory overhead in this method is much less than the LBM. Moreover, it is noted that the present model need not consider the discrete lattice effects of the boundary condition and the source term. So the present model has a great potential to simulate the practical porous media flow. IV. NUMERICAL RESULTS AND DISCUSSION

In this section, four numerical examples are simulated to validate the present approach. In fact, before the simulations, the dimensionless method is needed. This procedure is described as follows. The porous media flow can be characterized by four dimensionless parameters: the Reynolds numbers Re, the Darcy number Da, the porosity , and the viscosity ratio

F ∂ 2u ν + Gx −  u −  √ u2 = 0. ∂y 2 K K

(54)

When the nonlinear inertial effect due to the porous medium is neglected, namely, the case with F = 0, the above equation has the following analytical solution: 

   cosh λ y − H2 Gx K u(y) = , (55) 1− ν cosh λ H2  . H is the distance between the two plates. where λ = νν eK Unless specifically noted, we set νe = ν, namely, J = 1. In the present simulation, a square which has a side length of H is chosen as computational domain. The porosity  is set as 0.1. On the upper and lower walls, the nonslip boundary conditions are used. Different from the LBM, only boundary conditions of macroscopic physical quantities are implemented in this method. Here the ghost cell method is adopted to treat the no-slip boundary condition. This technique has been used in the gas kinetic scheme [48,49]. We can get ρ(xg ) = ρ(xf ),

(56)

u(xg ) = 2u(xb ) − u(xf ),

(57)

where xg , xb , and xf are ghost node, boundary node, and first layer of fluid node, respectively. At the inlet and outlet, the periodic boundary conditions are applied. The density and velocity fields are initialized to be ρ = 1.0, u = 0. The convergence criteria for velocity field is set to be un+1 − un 2  10−8 . un+1 2

(58)

The computational domain is discretized using a 40 × 40 uniform grid. The CFL number is fixed at 0.95 for all cases.

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PHYSICAL REVIEW E 93, 023308 (2016)

CPU Time (second)

1 0.8

u/u0

0.6

Analytical Da=10−1

0.4

−2

Da=10

100

−3

Da=10

0.2

−4

0.2

0.4

y/H

0.6

60 40 20 0 0

Da=10 0 0

80

0.8

LBM, 40 × 40 uniform mesh Present, 40 × 40 uniform mesh Present, 25 × 25 non−uniform mesh

20000 40000 60000 80000 100000 Iterative step

1

FIG. 1. Velocity profiles of the Poiseuille flow for different Darcy numbers when Re = 0.1.

To compare with analytical solutions, the case with F = 0 is first considered. As shown in Fig. 1, the velocity profiles u/u0 for different Darcy numbers are presented when Re = 0.1. Here u0 is the maximum velocity along the centerline   Gx K 1 . u0 = (59) 1− ν cosh λ H2 Obviously, the present results show excellent agreement with the analytical solutions. This indicates that the present numerical method is reliable. Moreover, it can be observed that the thickness of the velocity boundary layer decreases with decreasing Darcy number. To test the accuracy of the present method, a set of simulations on different meshes are also carried out. Four different uniform meshes with mesh spacing of x = H /20,H /40,H /80,H /160 are used. The L2 error for velocity field is measured by  (un − ua )2 , (60) L2 − error = N

FIG. 3. Comparison of CPU time costs between the generalized LBM (Guo and Zhao [32]) and the present method.

where un and ua are the numerical solution and analytical solution, respectively. N is the number of nodes. The numerical results are obtained at Re = 0.1, Da = 10−2 . As shown in Fig. 2, second-order accuracy of the present method can be confirmed clearly. Moreover, it is interesting to compare the computational efficiency between the present finite-volume (FV) scheme and LBM [32]. On the 40 × 40 uniform mesh, the CPU time costs of two methods versus iterative step are presented in Fig. 3. The simulations are done on a desktop (CPU 3.4 GHz and 8 G RAM). It can be found that the LBM is about 2.28 times faster than the present method. Even so, the present FV method can be applied on nonuniform grids conveniently. A 25 × 25 nonuniform mesh is also used. The following transformation is used to generate the nonuniform mesh xi = (ξ + 1) yj =

H , 2

(61)

H [tanh(c) + tanh(cηj )], 2 tanh(c)

(62)

1.4 −2

10

1.2

Present, Slope=1.992 Slope=2.0

1

−3

0

u/u

L2−error

10

0.8 0.6

−4

10

0.4 0.2

−5

10

−2

10

0 0

−1

Δx

FIG. 2. The L2 errors of u at different grid sizes.

Analytial 40 × 40 uniform mesh, LBM 25 × 25 nonuniform mesh, Present 0.1

0.2

0.3

y/H

10

FIG. 4. Comparison of velocity profiles obtained from the LBM, present method, and analytical method. 023308-6

FINITE-VOLUME METHOD WITH LATTICE BOLTZMANN . . .

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1.2

x 10 Analytical t=T/4 3 t=T/2 t=3T/4 2 t=T

/u0

0.6

1 u

max

0.8

u

1

0.4 0.2 −2 10

0 Present Guo and Zhao,LBM

−1

0

2

10 Log(Re)

10

−2 0

FIG. 5. Peak velocities of the Poiseuille flow with nonlinear inertial effect for different Reynolds numbers when Da = 10−5 .

where c = 1.5. The points (ξi ,ηj ) are uniformly distributed on the computational domain [−1,1] × [−1,1]. It can be found clearly that the computational efficiency of the present method on the 25 × 25 nonuniform mesh is higher than that of LBM on the 40 × 40 uniform mesh. In Fig. 4, the velocity profiles at Re = 0.1, Da = 10−4 obtained from the LBM, present method, and analytical method are displayed. Obviously, compared with the LBM, the present results are close to the exact solutions. This is because the mesh can be locally refined near the boundary in the present finite-volume scheme. An additional advantage of the present approach is that it uses less memory. In fact, only macroscopic physical quantities are recorded and updated during the simulation. The distribution functions at the cell interface are reconstructed locally using the macroscopic physical quantities. Then the case with nonlinear inertial effect is investigated. Figure 5 shows the peak velocities for different Reynolds numbers when Da = 10−5 . As had been pointed out in Ref. [32], as the Reynolds number increases, the effect

0.2

0.4

0.6

0.8

1

y/H FIG. 7. Comparison of the present velocity profiles with the analytical solutions at K = 10−2 .

of the porous drag becomes more significant. Furthermore, good agreement between the present results and the LBM results can be observed. To be more informative, the velocity profiles of the Poiseuille flow with nonlinear inertial effect for different Darcy numbers when Re = 0.1 are plotted in Fig. 6. Excellent agreement between the present FVM results and LBM solutions is presented. B. Unsteady Womersley flow between two planes

In this section, the unsteady Womersley flow inside the porous media is simulated. The geometric configuration and boundary conditions in this problem are the same as those in the the Poiseuille flow problem. The difference is that the Womersley flow is driven by a periodic force Gx = A cos ωt,

(63)

where A is the amplitude and ω is the frequency. −4

x 10 Analytical 16 t=T/4 t=T/2 12 t=3T/4 8 t=T

1

u

0.6

u/u

0

0.8

LBM Present, Da=10−2

0.4

Present, Da=10−3

0

−4

0.2 0 0

4

Present, Da=10

0.2

0.4

0.6

−4 0.8

−8 0

1

0.2

0.4

0.6

0.8

1

y/H

y/H FIG. 6. Velocity profiles of the Poiseuille flow with nonlinear inertial effect for different Darcy numbers when Re = 0.1.

FIG. 8. Comparison of the present velocity profiles with the analytical solutions at K = 10−3 .

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PHYSICAL REVIEW E 93, 023308 (2016)

When the nonlinear force due to the porous media is neglected, we have the following analytical solution:   !

cos λ y − H2 A u(y,t) = Re − 1 eiωt , iω +  Kν cos λ H2 (64) where " λ=

−

iω +  Kν , νe

(65)

In this simulation, the computational domain is set as [0,1] × [0,1]. A 40 × 40 grid is used. The CFL number is set to be η = 0.95. The flow field is at rest initially. The amplitude of the force is fixed at A = 1.0 × 10−4 and the frequency ω is 2π/100.  = 0.1, νe = ν = 10−3 are adopted for all cases. Figures 7 and 8 present the comparison of velocity profiles for different values of permeability K between the present numerical results and analytical solutions. As expected, the results at four different times t = T /4,T /2,3T /4,T for different K show good agreement with the analytical solutions. It can be concluded that the present method shows a good performance in simulating the unsteady flow.

and Re represents the operator to calculate the real component of a complex number and i is imaginary unit.

C. Circular Couette flow

The standard LBM can only be implemented on the regular grids. One advantage of the finite-volume method is to apply the irregular grids conveniently. As shown in Fig. 9(a), an annulus with internal radius Ri and external radius Ro is filled with porous medium. The inner cylinder rotates with a constant velocity u0 and the outer cylinder remains stationary. Figure 9(b) presents an O-type grid of 60 × 250 which is used in this simulation. Here the radii Ri ,Ro are set to be 1,2, respectively. The porosity  is fixed at 0.9. 1

Da=10−1, Present −2 Da=10 , Present −3 Da=10 , Present Finite−difference solutions

0.6

θ

u /u0

0.8

0.4 0.2

(a)

0 1

1.2

1.4

1.6

1.8

2

r (a)Re = 10

1

Re=0.1, Present Re=100, Present Re=1000, Present Finite−difference solutions

0.6

θ

u /u0

0.8

0.4 0.2 0 1

1.2

1.4

1.6

1.8

2

r

(b)

(b)Da = 10-1

FIG. 9. Schematic diagram and computational mesh for the circular Couette porous media flow.

FIG. 10. The azimuthal velocity profiles of the circular Couette flow for different Darcy numbers (a) and Reynolds numbers (b).

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When the flow reaches the steady state, the azimuthal velocity uθ satisfies     ∂uθ uθ 1 ∂ ν F νe r − 2 − uθ − √ u2θ = 0, (66) r ∂r ∂r r K K

as pointed out in Ref. [32], the ratio of the linear and the nonlinear drags can be measured by √ √ F |u|/ K ∼ F Da Re. (67) ν/K

where r is the radial coordinate. The Reynolds number is defined as Re = u0 (Ro − Ri )/ν. It is difficult to derive the analytical solution of the above equation. The finite-difference scheme with a uniform mesh of size 2000 is used to solve Eq. (66). The obtained nonlinear system is solved by the Newton-Raphson method. The finite-difference solutions are used to compare with the present finite-volume results. In Fig. 10(a), the azimuthal velocity profiles for different Darcy numbers when Re = 10 are presented. As Da decreases, the velocity boundary layer becomes thinner. Good agreement between the present results and finite-difference solutions can be observed. Furthermore, Fig. 10(b) demonstrates variation of the azimuthal velocity profile versus Re when Da = 0.1. As expected, the results are satisfactory. It should be noted that Eqs. (1) and (2) can describe the flows in both the Darcy regime and the Forchheimer regime. We also study the effect of the nonlinear inertial term. In fact,

It indicates that the effect of the nonlinear drag force is more significant at high Da or Re. In Fig. 11(a), the dimensionless azimuthal velocity uθ /u0 at radial position r = (Ri + Ro )/2 as a function of Re when Da = 10−1 is presented. Obviously, as Re increases, the nonlinear drag force must be considered. In addition, as shown in Fig. 11(b), it can be found that the nonlinear inertial effect becomes more significant as Da increases. The results of numerical simulation are consistent with the theoretical analysis. D. Lid-driven flow

The lid-driven flow in a cavity has been widely studied as a benchmark testing in computational fluid dynamics [50–52]. The geometric configuration of this problem is very simple. The upper wall moves with a constant velocity u0 tangent to the side and the other walls are stationary. When flow reaches

0.2

1

0.16

0.8

0.12

0.6 y/H

uθ((Ri+Ro)/2)/u0

FINITE-VOLUME METHOD WITH LATTICE BOLTZMANN . . .

0.4

0.08 0.04 0

Present −2 Da=10 , LBM −3 Da=10 , LBM Da=10−4, LBM

0.2

Without nonlinear term With nonlinear term 0

0 −0.2

0

0.2

2

10

10 Re (a)Da = 0.1

0.08

0 −0.05

Present Da=10−2, LBM Da=10−3, LBM Da=10−4, LBM

−0.15

0.04 −0.2 −2

1.2

0.05

−0.1

10 Da

1

0.1

Without nonlinear term With nonlinear term

0.12

0 −3 10

0.8

0.15

v(H/2)

uθ((Ri+Ro)/2)/u0

0.16

0.4 0.6 u(H/2) (a)

0

0.2

0.4

0.6

0.8

1

x/H

−1

(b)

10

(b)Re = 100 FIG. 11. The azimuthal velocities at radial position r = (Ri + Ro )/2 for different Reynolds numbers (a) and Darcy numbers (b).

FIG. 12. Velocity profiles of the lid-driven flow through the cavity center. (a) The u-velocity profiles along the vertical line passing through the center of the cavity. (b) The v-velocity profiles along the horizontal line passing through the center of the cavity.

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PHYSICAL REVIEW E 93, 023308 (2016) TABLE I. The locations of the center of the dominant eddy for different Darcy numbers. Da

0.01

0.25

106

Present (0.6597,0.6934) (0.5662,0.5998) (0.5493,0.5870) Yang et al. [53] (0.65,0.69) (0.56,0.61) (0.55,0.59)

the steady state, the counter-rotating vortices appear at the corners of the cavity. First, the case with  = 0.1, Re = 10 is simulated. The results are obtained based on a 150 × 150 grid. The CFL number is fixed at 0.95. The initial flow field is at rest. As shown in Fig. 12, the horizontal and vertical velocity profiles with Da = 10−2 ,10−3 ,10−4 through the cavity center are presented. Obviously, it confirms that the present results show good agreement with the LBM results [32]. As pointed out in Ref. [32], when the Darcy number decreases, the velocity boundary layer near the upper becomes thinner due to strong porous drag effect. Second, the simulations with corresponding parameters  = 0.2, Re = 100 are carried out. The same as in Ref. [53], the Darcy number varies from 0.01 to 106 . Figure 13 depicts the streamline patterns for Da = 0.01,0.25,106 . When Da = 0.01, the penetrating capacity of the porous layer is very weak. The center of dominant eddy deviates significantly from the center of the cavity. When the Darcy number increases to 0.25, the secondary vortices appear in the lower left corner of the cavity. And the size of the vortices in the lower right corner of the cavity become larger. When Da further increases to 106 , the drag force due to the porous media is very small, and the center of the dominant eddy becomes close to the center of the cavity. For comparison of the flow details, the locations of the dominant eddy for different cases are listed in Table I. As expected, the present results agree well with that in Ref. [53].

(a)Da = 0.01

(b)Da = 0.25

V. CONCLUSIONS

In this paper, a finite-volume method with lattice Boltzmann flux scheme is developed to simulate the incompressible flow in porous media at the REV scale. Through Chapman-Enskog analysis, the Darcy-Brinkman-Forchheimer equation can be expressed as a hyperbolic system with a nonlinear source term in conservation law form. Then the convective and viscous fluxes across the cell interface can be reconstructed locally by the solutions of the generalized LBE. The boundary conditions can be implemented without any special treatment. Numerical simulations of several 2D porous media flow problems are carried out to validate the present model. The present results are in good agreement with the analytical, finitedifference, and/or previously published solutions. ACKNOWLEDGMENTS

(c)Da = 106 FIG. 13. Streamlines for different values of Darcy number when Re = 100,  = 0.2.

This research was supported by the National Natural Science Foundation of China (Grant No. 51375039), Beijing Natural Science Foundation (Grant No. 4142046), and Program for Changjiang Scholars and Innovative Research Team (Grant No. IRT13046).

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