CFD Theory

Basic CFD Theory

CFD-Equations

Navier-Stokes (N-S) Equation

$$\begin{align} & \frac{\partial u_{i}}{\partial t}+ u_{j}\frac{\partial u_{i}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu \frac{\partial^2 u_{i}}{\partial x_{j}\partial x_{j}}+g_{i} \tag{1.1}\\ & \frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1.2} \end{align}$$ $$\begin{align} & \frac{\partial u_{i}}{\partial t}+ u_{j}\frac{\partial u_{i}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu \frac{\partial^2 u_{i}}{\partial x_{j}\partial x_{j}}+g_{i} \\ & \frac{\partial u_{i}}{\partial x_{i}}=0 \end{align}$$

where $u_i$ represents the air velocity, p is the pressure, $g_i$ is the gravitational acceleration.

Energy Equation

$$\begin{aligned} &\rho \frac{\partial\left(c_{p} T\right)}{\partial t}+\rho \frac{\partial\left(c_{p} u_{j} T\right)}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[k \frac{\partial T}{\partial x_{j}}\right]+\Phi+S_{T}\\ &\Phi = \mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) \frac{\partial u_{i}}{\partial x_{j}} \end{aligned}$$

DEM

$$\begin{aligned} & m_{p} \frac{d u_{i}^{p}}{d t}=\sum F_{i}+\left(m_{p}-m_{f}\right) g_{i}\\ & F_{i}^{D}=\frac{\pi}{8} \rho_{f} d_{p}^{2} C_{D}\left(u_{i}^{f}-u_{i}^{p}\right)\left|u_{i}^{f}-u_{i}^{p}\right|\\ & C_{D}=\frac{24}{\operatorname{Re}_{p}}+3.6 \operatorname{Re}_{p}^{-0.313} \end{aligned}$$ $$C_{D}=\left\{\begin{array} {c} \frac{24\left(1+0.15 R e^{0.687}\right)}{\operatorname{Re}} \text {Re} \leq 1000 \\ 0.44 \mathrm{Re}>1000 \end{array}\right.$$ $$\begin{aligned} C_{\mathrm{D}}=\left\{ \begin{array}{cc} \frac{24}{\operatorname{Re}_{\mathrm{p}}} & \text { if } \operatorname{Re}_{\mathrm{p}} \leq 10^{3} \\ \frac{24}{\mathrm{Re}_{\mathrm{p}}}\left(1+0.15 \mathrm{Re}_{\mathrm{p}}^{0.678}\right) & \text { if } \quad \operatorname{Re}_{\mathrm{p}}>10^{3} \end{array}\right. \end{aligned}$$ $$m_{p} \frac{d u_{i}^{p}}{d t}=\sum F_{i}+\left(m_{p}-m_{f}\right) g_{i}\\$$ $$F_{p} = \frac{C_D Re_{p}}{24}\\$$ $$C_D = \left\{ \begin{array} {**lr**} \frac{24(1+0.15Re_{p}^{0.687})}{Re_{p}} \qquad &Re_{p}\leq 10^{3} \\ 0.44Re_{p}\qquad &Re_{p}<10^{3} \end{array} \right.$$ $$Re_{p}=\frac{\rho\left|\vec{u}-\vec{u}_{p}\right| d_{p}}{\mu_{p}}$$