A CE/SE Scheme for Flows in Porous Media and Its Applications

In this paper, a new computational scheme for solving flows in porous media was proposed. The scheme was based on an improved CE/SE method (the space-time Conservation Element and Solution Element method).We described porous flows by adopting DFB (Brinkman-Forchheimer extended Darcy) equation. The comparison between our computational results and Ghia’s confirmed the high accuracy, resolution, and efficiency of our CE/SE scheme. The proposed first-order CE/SE scheme is a new reliable way for numerical simulations of flows in porous media. After investigation of effects of Darcy number on porous flow, it shows that Darcy number has dominant influence on porous flow for the Reynolds number and porosity considered.


INTRODUCTION
Darcy effects of form-drag and boundary resistance have been studied extensively through the use of the Brinkman-Forchheimer extended Darcy model (DFB).One of the first papers to deal with these extensions was Vafai (1984).In recent years, the flow in porous media formulated by the DFB model was successfully used in a wide variety of flow conditions, including forced convection pipe flow (Alkman et al., 1998), natural convection in vertically layered porous media (Hadim, 2006), and double-diffusive natural convection (Wang et al., 2008).Numerical simulations for non-Darcy flows have improved immensely as a result of major progress in Models for flow in porous media have been developed since 1856, when Darcy postulated his well-known equation.Vafai et al. (1980) pointed out the limitations of Darcy's law.For porous media with high permeability, both the viscous effects (frictional drag at the boundary) and the inertia (form-drag) effects within the porous matrix are significant.Consequently for Newtonian fluids, the non-both computational methods and available computer facilities.
Deiber and Bortolozzi (1998) applied vorticity-stream function scheme to study natural convection in a porous annulus.Rees (2002) analyzed the onset of Darcy-Brinkman convection in a porous layer by using an asymptotic analysis method.Sman (2002) numerically solved the DFB model with 3D finite element solver (FIDAP In this presentation, a CE/SE (space-time conservation element and solution element) scheme with first-order accuracy for porous flows is established by applying new structures of CEs (conservation element) and SEs (solution element) (Zhang et al., 2001).The DFB model is used to represent the fluid transport within the porous medium.
Governing partial differential equations are transformed to algebraic ones by CE/SE, and the pressure-velocity coupling problem is solved by using artifical pressure method over structured and staggered grids.We conduct DFB-based flow simulations for single phase flow through a 2D (two-dimensional) porous medium.A benchmarking problem is simulated numerically and computational results are carefully compared with that from other literature.Influences of Darcy numbers upon porous flows are investigated.

ANALYSIS AND MODELING
The continuity equation and the momentum equation for 2D incompressible and viscous flow in porous medium can be summarized as follows (Reis et al., 2004).
Momentum balance of fluid: here all constants and variables are defined w in the nomenclature.Eq. (1a) and Eq.(1b) form the full set of equations used to model flows in porous media.Eq. (1b) contains the usual balance of forces between viscosity and pressure gradient known as Darcy's law (the 3rd and 5th terms), which is extended through the further inclusion of terms modeling in turn advective inertia (the 1st and 2nd terms), boundary effects (the 4th term: the Brinkman term) and form-drag (the 6th term: Forchheimer inertia).Quantities have been rendered dimensionless with respect to the characteristic length L, and the characteristic velocity u o , using the definitions: Eq. (1a) and Eq.(1b) take the forms 0 U (3a) where , Q E , , are vectors of primary variable, flux in x-direction, flux in y-direction and source, respectively.This set of equations describes the conservation of density Eq. ( 6) and Eq. ( 7) imply that the variables quired in computation are Q and Q y .In ting Eq. ( 6into Eq. ( 4), we obtain tegrating Eq. ( 5) on the surfaces of CE( P ) and applying source item linearization method (Wang et al., 2008) , we obtain is the time derivative of S t S . Using the ontinuity conditions at points , , E A C c and G , the derivatives of Q w respect to ith x and y are obtained ( , ) [( ) are defined as the he

Re
Re .
Using the time operator splitting method (Jue, 2000) to split pressure item in momentum Eq. ( 13), we obtain is the visual time, and is the oefficient effecting the numerical stability.ti (14a) Substitu ng the visual time derivative of velocities into Eq.(1b), we obtain We interpret Eq. (14a) as yielding an intermediate value of , denoted by Here, the cript indicates the tim is alue at time step ulated by CE/SE , for , Eq. ( 12) is consistent to Eq. (1b).Applying the pressure splitting method, Eq. ( 12 where marching solutions and internal iteration ) .

Numerical Validation
Here, we focus on the 2 ark problem D n cavity is a we wn enc for numerical me e simple geom co square, lid-driven ll-kno thods for etry and cavity filled with fluid-saturated porous medium.The drive b hm laminar flows due to th mplicated flow behaviors.We apply the improved CE/SE method to the fluid flow in a square cavity filled with a porous medium.The flow condition is the same as that of liddriven flow problem (Guo et al., 2002) as shown in Fig. 2(a).At the left, right and down boundaries, no-slip condition holds at the wall, respectively.While, at the upper boundary, the horizontal velocity is specified and held fixed.Hence, in Fig. 2(a), 0 . 1 u .The computational domain covered 202 202 grid used for the calculations.We set Da = 10 4 , = 0.9 and Re = 100 and 1000.In Fig. 2 and Fig. 3, the velocity profiles through the cavity center are plotted.The benc lutions (Ghia et al., 1982)

SUMMARY
and 10 diminish a lead to sin vortex lid by the reduction of scheme has been si curacy of the new CE/SE scheme is validated by comparing the numerical results of lid-driven cavity flow with the corresponding results by Ghia et al. (1982) It is observed for the Reynolds number and porosity considered, as the Darcy number is reduced the primary vortex becomes weaker and tends to move towards the lid.Secondary vortices formed at higher Darcy numbers (10 -1 scheme is a new reliable way for numerical simulation of flows in porous medium.Further work will be required to improve CE/SE computational efficiency by using multi-grid method. authors also greatly appreciate Prof. Chuenreviewers r their constructive comments, which have m ith a Porous aterial.Int.J. Heat Mass Transfer . 41: llan, F.M. and Hamdan, M.H. (2002) means of timemethod, Eq. (11) and Eq.(14b) yield n 0 )

Fig. 1 .
Fig. 1.Mesh construction of the improved CE/SE method.(a) Mesh points projection on xy plane, (b) Conservation element CE(P (c) Solution element SE(P orthogonal grid (as shown in Fig.1(a) which eliminates the possibility of a checkerboard pressure pattern.

Fig. 3 .Fig. 2 .
Fig.3, the velocity profiles through the cavity center are plotted.The benc lutions(Ghia et al., 1982) are also included for comparison.It is seen that the CE/ hmark

Fig. 5 .
Fig. 5. Velocity profiles through the cavity center for different Darcy number ( = 10 -1 , 10 -2 , 10 -3 and 10 -4 ).(a) v com vertical line through the cavity center, (b) u component along the horizontal line through the cavity center ( Da ponent along the Zhang et al. (2001)ensional CE/SE method(Chang et al.,  1999)is complicated as the special design of CEs and SEs.Zhang et al. (2001)proposed an improved CE/SE method by adopting general hexahedrons mesh to construct CEs and SEs, as shown in Fig.1(b) and Fig.1(c).The structure of CEs and SEs simplifies the process of scheme derivation.In this study, we deduce the two-dimensional CE/SE scheme with first-order accuracy for flows in porous medium.Let denote a set of space- Field Model for Natural Convection in a Porous Annulus at High Rayleigh Numbers.Chem.Eng.Sci.53: 1505-1516.hia, U., Ghia, K.N. and Shin, C.T. (1982).High-Re Solutions for Incompre Jinn Tsai, as well as the anonymous fo uch improved this manuscript.