Reynolds平均した連続体の運動方程式

$ \frac{\overline{\rm D}\overline{\bm v}}{\overline{\rm D}t}=\frac1\rho\bm\nabla\cdot(\bm\sigma_T+\overline{\bm\sigma})+\overline{\bm f}

導出

$ \overline{\frac{\mathrm D\bm v}{\mathrm Dt}}=\frac{\partial\overline{\bm v}}{\partial t}+\overline{\bm v\cdot\bm\nabla\bm v}

$ =\frac{\partial\overline{\bm v}}{\partial t}+\overline{\bm\nabla\cdot(\bm v\bm v)}

$ \because\rho=\rm const.\implies\bm\nabla\cdot\bm v=0

$ =\frac{\partial\overline{\bm v}}{\partial t}+\bm\nabla\cdot(\overline{\overline{\bm v}\ \overline{\bm v}}+\overline{\overline{\bm v}\bm v'}+\overline{\bm v'\overline{\bm v}}+\overline{\bm v'\bm v'})

$ =\frac{\partial\overline{\bm v}}{\partial t}+\bm\nabla\cdot(\overline{\bm v}\ \overline{\bm v}+0+0+\overline{\bm v'\bm v'})

$ =\frac{\partial\overline{\bm v}}{\partial t}+\overline{\bm v}\cdot\bm\nabla\overline{\bm v}+\bm\nabla\cdot\overline{\bm v'\bm v'}

$ ={\frac{\overline{\mathrm D}\overline{\bm v}}{\overline{\mathrm D}t}}-\frac{\bm\sigma_T}{\rho}

$ \frac{\mathrm{D}\bm{v}}{\mathrm{D}t}=\frac1\rho\bm{\nabla}\cdot\bm{\sigma}+\bm f

$ \implies{\frac{\overline{\mathrm D}\overline{\bm v}}{\overline{\mathrm D}t}}-\frac{\bm\sigma_T}{\rho}=\frac1\rho\bm\nabla\cdot\overline{\bm\sigma}+\overline{\bm f}

$ \underline{\iff{\frac{\overline{\mathrm D}\overline{\bm v}}{\overline{\mathrm D}t}}=\frac1\rho\bm\nabla\cdot(\bm\sigma_T+\overline{\bm\sigma})+\overline{\bm f}\quad}_\blacksquare