# Darcy-Brinkman-Forchheimer方程的无量纲化

$$\nabla\cdot\vec{u}’=0$$

$$U\rho\frac{\partial \vec{u}’}{\partial t}+U^2\nabla\cdot\left(\frac{\rho\vec{u}’\vec{u}’}{\epsilon}\right)=-(\rho U^2)\nabla(\epsilon p’)+\mu U\nabla^2\vec{u}’-\frac{U\mu\epsilon}{K}\vec{u}’-\frac{\rho U^2\epsilon C_F|\vec{u}’|}{\sqrt{K}}\vec{u}’$$

$$\frac{1}{U}\frac{\partial \vec{u}’}{\partial t}+\nabla\cdot\left(\frac{\vec{u}’\vec{u}’}{\epsilon}\right)=-\nabla(\epsilon p’)+\frac{\mu}{\rho U}\nabla^2\vec{u}’-\frac{\mu\epsilon}{\rho UK}\vec{u}’-\frac{\epsilon C_F|\vec{u}’|}{\sqrt{K}}\vec{u}’$$

$$\frac{D}{U}\frac{\partial \vec{u}’}{\partial t}+D\nabla\cdot\left(\frac{\vec{u}’\vec{u}’}{\epsilon}\right)=-D\nabla(\epsilon p’)+\frac{D^2\mu}{\rho UD}\nabla^2\vec{u}’-\frac{\mu\epsilon D^2}{\rho UDK}\vec{u}’-\frac{D\epsilon C_F|\vec{u}’|}{\sqrt{K}}\vec{u}’$$

$$\frac{D}{U}\frac{\partial \vec{u}’}{\partial t}+D\nabla\cdot\left(\frac{\vec{u}’\vec{u}’}{\epsilon}\right)=-D\nabla(\epsilon p’)+\frac{D^2}{Re}\nabla^2\vec{u}’-\frac{1}{Re}\frac{\epsilon}{Da}\vec{u}’-\frac{\epsilon C_F|\vec{u}’|}{\sqrt{Da}}\vec{u}’$$

$$\frac{\partial \vec{u}’}{\partial t’}+\nabla’\cdot\left(\frac{\vec{u}’\vec{u}’}{\epsilon}\right)=-\nabla'(\epsilon p’)+\frac{1}{Re}\nabla’^2\vec{u}’-\frac{1}{Re}\frac{\epsilon}{Da}\vec{u}’-\frac{\epsilon C_F|\vec{u}’|}{\sqrt{Da}}\vec{u}’$$

$$\nabla’\cdot\vec{u}’=0$$

## 参考文献

[chen2008] Chen X, Yu P, Winoto S H, et al. Numerical analysis for the flow past a porous square cylinder based on the stress-jump interfacial-conditions[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2008, 18(5): 635-655.

[nithiarasu2002] Nithiarasu P, Seetharamu K N, Sundararajan T. Finite element modelling of flow, heat and mass transfer in fluid saturated porous media[J]. Archives of Computational Methods in Engineering, 2002, 9(1): 3.