A Comparative Study of Nonlinear Convection in a Confined Fluid Overlying a Porous Layer

Abstract

The alternating direction implicit method (ADI method) has been used to investigate a comparative study between the Darcy (DM), Brinkman-extended Darcy (BM) and Brinkman–Forchheimer-extended Darcy models (BFM) of free convection in a cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The two-dimensional enclosure is heated from below and cooled from above, while the other two vertical sides are adiabatic. The Beavers–Joseph empirical boundary condition at the fluid/porous layer interface is employed, while the Darcy model is used to simulate momentum transfer in a porous medium. In case of the BM and BFM models, the two regions are coupled by matching the velocity and stress components at the interface. Numerical simulations have revealed that all three models have yielded almost same results for Darcy numbers up to about $10^{-4}$. For higher Darcy numbers, the BFM model can be used to simulate momentum transfer as it accounts for the effects of inertia.

Appendices

Appendix

Mathematical Modeling of Governing Equations in Vector Form

Following paper of Kim and Choi [16].

For fluid region:

Conservation equation of mass, momentum, and energy equation for fluid region

$$\begin{aligned} \nabla \cdot \overrightarrow{q}= & {} 0, \end{aligned}$$

(27)

$$\begin{aligned} \overrightarrow{q}\cdot \nabla \overrightarrow{q}= & {} - \frac{1}{\rho _{f}}\nabla p+\nu _{f}\nabla ^{2}\overrightarrow{q}+\overrightarrow{g}\end{aligned}$$

(28)

$$\begin{aligned} \overrightarrow{q}\cdot \nabla T= & {} \alpha _{f}\nabla ^{2} T \end{aligned}$$

(29)

where $\overrightarrow{q}$ represents velocity similarly T, temperature, p, pressure, and all other symbols are defined above.

For porous region:

Brinkman–Forchheimer-extended Darcy model

$$\begin{aligned} \nabla \cdot \overrightarrow{q}= & {} 0, \end{aligned}$$

(30)

$$\begin{aligned} \overrightarrow{q}\cdot \nabla \overrightarrow{q}= & {} -\frac{1}{\rho _{f}}\nabla p\nu _{eff}\nabla ^{2}\overrightarrow{q} -\frac{\nu _{f}}{K}\overrightarrow{q}-\frac{F}{\sqrt{K} } \left| {\overrightarrow{q}} \right| \overrightarrow{q}+g \end{aligned}$$

(31)

$$\begin{aligned} \overrightarrow{q}\cdot \nabla T= & {} \alpha _{eff}\nabla ^{2} T \end{aligned}$$

(32)

where the symbols used defined above, $\alpha _{eff}=k_{eff}\nabla ^{2}T$ is effective thermal diffusivity.

Following [22]

Darcy model: It is enough to show only momentum equation (other two are same)

$$\begin{aligned} \overrightarrow{q}=-\frac{K}{\mu }\nabla p \end{aligned}$$

(33)

Brinkman’s Model:

$$\begin{aligned} \nabla p=-\frac{\mu }{K}\overrightarrow{q}+\tilde{\mu }\nabla ^{2}\overrightarrow{q}, \end{aligned}$$

(34)

where $\mu$ is the dynamic viscosity of the fluid, $\tilde{\mu }$ is effective viscosity, K is intrinsic permeability of porous medium.