通常自然対流

Boussinesq近似

\begin{eqnarray} && && &\frac{\partial }{\partial X_i} U_i& &=& 0 \\ && \frac{\partial }{\partial \tau} U_i &+& &\frac{\partial }{\partial X_j} U_j U_i& &=& -&\frac{\partial }{\partial X_i} P& + Pr \frac{\partial^2 }{\partial X_j \partial X_j} U_i + Ra Pr{\delta}_{3i} \Theta \\ && \frac{\partial }{\partial \tau}\Theta &+& &\frac{\partial }{\partial X_j} U_j \Theta& &=& &\frac{\partial^2 }{\partial X_j\partial X_j} \Theta \\ \end{eqnarray}

上から連続の式、N-S方程式、エネルギー方程式。以下、無次元変数の定義と無次元数

\begin{eqnarray} t &=& \frac{L^2}{\alpha}\tau \\ u_i &=& \frac{\alpha}{L} U_i \\ x_i &=& L X_i\\ p &=& p_0 +\frac{\rho \alpha^2}{L^2}P - \rho g z\\ T &=& T_0 + \Delta T \Theta \\ Ra &=& \frac{\beta \Delta T L^3 g }{\alpha \nu}\\ Pr &=& \frac{\nu}{\alpha}\\ \end{eqnarray}