乱流運動エネルギ方程式
$ \frac{\overline{\rm D}k}{\overline{\rm D}t}+\overline{\frac{\bm\sigma'}\rho:\bm d'}=\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\cdot\left(\frac12|\bm v'|^2\bm I-\frac{\bm\sigma'}{\rho}\right)}
$ k:=\frac12\overline{|\bm v'|^2}乱流運動エネルギ $ \bm\sigma_T:=-\rho\overline{\bm v'\bm v'}Reynolds応力 $ \bm f：単位密度あたりの物体力
通常は流体に限定するが、ここでは一般の$ \rho=\rm const.な連続体で成立する式を導いている
導出
$ \rho=\rm const.とする
$ \overline{\frac{\mathrm D}{\mathrm Dt}\left(\frac12|\bm v|^2\right)}=\frac{\partial}{\partial t}\overline{\frac12|\bm v|^2}+\overline{\bm v\cdot\bm\nabla\frac12|\bm v|^2}
$ =\frac{\partial}{\partial t}\frac12|\overline{\bm v}|^2+\frac{\partial}{\partial t}\frac12\overline{|\bm v'|^2}+\overline{\bm\nabla\cdot(\bm v\frac12|\bm v|^2)}
$ \because\rho=\rm const.\implies\bm\nabla\cdot\bm v=0
$ =\frac{\partial}{\partial t}\frac12|\overline{\bm v}|^2+\frac{\partial}{\partial t}\frac12\overline{|\bm v'|^2}+\frac12\bm\nabla\cdot(\overline{\overline{\bm v}|\overline{\bm v}|^2}+2\overline{\bm v'\cdot\overline{\bm v}\ \overline{\bm v}}+\overline{\overline{\bm v}|\bm v'|^2}+\overline{\bm v'|\overline{\bm v}|^2}+2\overline{\overline{\bm v}\cdot\bm v'\bm v'}+\overline{\bm v'|\bm v'|^2})
$ =\frac{\partial}{\partial t}\frac12|\overline{\bm v}|^2+\frac{\partial}{\partial t}\frac12\overline{|\bm v'|^2}+\frac12\bm\nabla\cdot(\overline{\bm v}|\overline{\bm v}|^2+0+\overline{\bm v}\overline{|\bm v'|^2}+0+2\overline{\bm v}\cdot\overline{\bm v'\bm v'}+\overline{\bm v'|\bm v'|^2})
$ =\frac{\partial}{\partial t}\frac12|\overline{\bm v}|^2+\frac{\partial}{\partial t}\frac12\overline{|\bm v'|^2}+\overline{\bm v}\cdot\bm\nabla\frac12|\overline{\bm v}|^2+\overline{\bm v}\cdot\bm\nabla\frac12\overline{|\bm v'|^2}+\bm\nabla\cdot\left(\overline{\bm v}\cdot\overline{\bm v'\bm v'}+\overline{\bm v'\frac12|\bm v'|^2}\right)
$ = \frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12|\overline{\bm v}|^2\right)+\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12\overline{|\bm v'|^2}\right)+\bm\nabla\cdot\left(-\frac{\bm\sigma_T}{\rho}\cdot\overline{\bm v}+\overline{\bm v'\frac12|\bm v'|^2}\right)
$ \frac{\mathrm D}{\mathrm Dt}\left(\frac12|\bm v|^2\right)+\frac1\rho\bm\sigma:\bm d=\frac1\rho\bm\nabla\cdot(\bm\sigma\cdot\bm v)+\bm f\cdot\bm v
$ \implies\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12|\overline{\bm v}|^2\right)+\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12\overline{|\bm v'|^2}\right)+\bm\nabla\cdot\left(-\frac{\bm\sigma_T}{\rho}\cdot\overline{\bm v}+\overline{\bm v'\frac12|\bm v'|^2}\right)+\frac1\rho\overline{\bm\sigma}:\overline{\bm d}+\frac1\rho\overline{\bm\sigma':\bm d'}
$ =\frac1\rho\bm\nabla\cdot(\overline{\bm\sigma}\cdot\overline{\bm v})+\frac1\rho\bm\nabla\cdot\overline{\bm\sigma'\cdot\bm v'}+\overline{\bm f}\cdot\overline{\bm v}+\overline{\bm f'\cdot\bm v'}
$ \iff\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12|\overline{\bm v}|^2\right)+\frac{\overline{\bm\sigma}}\rho:\overline{\bm d}+\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12\overline{|\bm v'|^2}\right)+\overline{\frac{\bm\sigma'}\rho:\bm d'}=\bm\nabla\cdot\left(\frac{\bm\sigma_T+\overline{\bm\sigma}}{\rho}\cdot\overline{\bm v}+\overline{\frac{\bm\sigma'}\rho\cdot\bm v'}-\overline{\bm v'\frac12|\bm v'|^2}\right)+\overline{\bm f}\cdot\overline{\bm v}+\overline{\bm f'\cdot\bm v'}
$ \implies-\frac{\bm\sigma_T}\rho:\overline{\bm d}+\frac{\overline{\mathrm D}}{\overline{\mathrm D}t}\left(\frac12\overline{|\bm v'|^2}\right)+\overline{\frac{\bm\sigma'}\rho:\bm d'}=\bm\nabla\cdot\left(\overline{\frac{\bm\sigma'}\rho\cdot\bm v'}-\overline{\bm v'\frac12|\bm v'|^2}\right)+\overline{\bm f'\cdot\bm v'}
平均流の連続体の力学的エネルギ保存則$ \frac{\overline{\rm D}}{\overline{\rm D}t}\frac12|\overline{\bm v}|^2+\frac{\bm\sigma_R+\overline{\bm\sigma}}\rho:\overline{\bm d}=\bm\nabla\cdot\left(\frac{\bm\sigma_R+\overline{\bm\sigma}}{\rho}\cdot\overline{\bm v}\right)+\overline{\bm f}\cdot\overline{\bm v}を引く $ \underline{\iff\frac{\overline{\rm D}k}{\overline{\rm D}t}+\overline{\frac{\bm\sigma'}\rho:\bm d'}=\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\cdot\left(\frac12|\bm v'|^2\bm I-\frac{\bm\sigma'}{\rho}\right)}\quad}_\blacksquare