CFD Theory

本文最后更新于：2020年11月23日上午

Basic CFD Theory

\begin{aligned} (1.1) & \frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x_{i}} + ν \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}} + g_{i} \\ (1.2) & \frac{\partial u_{i}}{\partial x_{i}} = 0 \end{aligned}

\begin{aligned} \frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x_{i}} + ν \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}} + g_{i} \\ \frac{\partial u_{i}}{\partial x_{i}} = 0 \end{aligned}

where $u_{i}$ represents the air velocity, p is the pressure, $g_{i}$ is the gravitational acceleration.

\begin{aligned} ρ \frac{\partial (c_{p} T)}{\partial t} + ρ \frac{\partial (c_{p} u_{j} T)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [k \frac{\partial T}{\partial x_{j}}] + Φ + S_{T} \\ Φ = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) \frac{\partial u_{i}}{\partial x_{j}} \end{aligned}

\begin{aligned} m_{p} \frac{d u_{i}^{p}}{d t} = \sum F_{i} + (m_{p} - m_{f}) g_{i} \\ F_{i}^{D} = \frac{π}{8} ρ_{f} d_{p}^{2} C_{D} (u_{i}^{f} - u_{i}^{p}) | u_{i}^{f} - u_{i}^{p} | \\ C_{D} = \frac{24}{{Re}_{p}} + 3.6 {Re}_{p}^{- 0.313} \end{aligned}

C_{D} = {\begin{matrix} \frac{24 (1 + 0.15 R e^{0.687})}{Re} Re \leq 1000 \\ 0.44 Re > 1000 \end{matrix}

\begin{array}{r} C_{D} = {\begin{array}{cc} \frac{24}{{Re}_{p}} & if {Re}_{p} \leq 10^{3} \\ \frac{24}{{Re}_{p}} (1 + 0.15 {Re}_{p}^{0.678}) & if {Re}_{p} > 10^{3} \end{array} \end{array}

m_{p} \frac{d u_{i}^{p}}{d t} = \sum F_{i} + (m_{p} - m_{f}) g_{i}

F_{p} = \frac{C_{D} R e_{p}}{24}

C_{D} = {\begin{array}{lr} \frac{24 (1 + 0.15 R e_{p}^{0.687})}{R e_{p}} & R e_{p} \leq 10^{3} \\ 0.44 R e_{p} & R e_{p} < 10^{3} \end{array}

R e_{p} = \frac{ρ | \vec{u} - {\vec{u}}_{p} | d_{p}}{μ_{p}}

CFD

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