# Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

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## Abstract

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.

## Keywords

Lid-driven cavity flow Porous medium CE/SE method CO_{2}storage Forchheimer

## List of symbols

**u**Dimensionless volume-averaged velocity vector [–]

*P*Dimensionless pressure [–]

*α = μ*_{e}/*μ*Viscosity ratio [–]

*μ*Shear viscosity of the fluid [ML

^{−1}T^{−1}]*μ*_{e}Effective viscosity [ML

^{−1}T^{−1}]- Re =
*LU/μ* Reynolds number [–]

- \({\varphi}\)
Porosity [–]

- Da
Darcy number [–]

*t*Dimensionless time [–]

*L*Characteristic length [L]

*U*Characteristic velocity [LT

^{−1}]*b*Depth ratio [–]

*F*_{b}Forchheimer parameter (Geometric function) [L

^{−1}]*K*Permeability [L

^{2}]*u*Dimensionless horizontal velocity [–]

*v*Dimensionless vertical velocity [–]

*τ*Dimensionless virtual time [–]

*C*^{2}Artificial compressibility coefficient [–]

*ρ*Dimensionless density [–]

- (
*j*,*k*,*n*) Space-time mesh points

**Q**Vectors of primary variable

**E**Flux in

*x*-direction**F**Flux in

*y*-direction**S**Source term vector

*S*_{m}Components of the source term vector

*S*(*V*)Boundary of an arbitrary space-time region

*V*Space-time region

- dσ
Area of a surface element

**n**Outward unit normal of a surface element

*P*′(*j*,*k*,*n*)Spatial point

**H**_{m}Space-time flux vector

*Q*_{m}Components of vector

**Q***E*_{m}Components of vector

**E***F*_{m}Components of vector

**F**

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