Farvre平均した連続体の運動方程式

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

$ \overline\bullet：Reynolds平均

$ \widetilde\bullet：Farvre平均

$ \bm\sigma_T:=-\overline{\rho}\widetilde{\bm v'\bm v'}：Reynolds応力

$ \bullet'：Farvre平均の変動成分

$ \frac{\widetilde{\rm D}}{\widetilde{\rm D}t}:=\frac{\partial}{\partial t}+\widetilde{\bm v}\cdot\bm\nabla

導出

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

$ \because連続体の質量保存則

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

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

$ \becauseFarvre平均の性質

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

$ =\overline\rho\frac{\widetilde{\rm D}\widetilde{\bm v}}{\widetilde{\rm D}t}+\bm\nabla\cdot\overline\rho\widetilde{\bm v'\bm v'}

$ \becauseFarvre平均した連続体の質量保存則

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

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

#2025-07-05 13:40:28

#2025-06-23 12:29:46