非圧縮性Newton流体の乱流運動エネルギ方程式

$ \frac{\overline{\rm D}k}{\overline{\rm D}t}=\frac{\bm\sigma_T}\rho:\overline{\bm d}-\varepsilon+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}\rho\right)}+\nu\bm\nabla k\right)

導出

$ \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)}

$ \implies\frac{\overline{\rm D}k}{\overline{\rm D}t}+\nu\overline{\bm d':\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{p'}{\rho}\bm I\right)}+2\nu\bm\nabla\cdot\overline{\bm v'\cdot\bm d'}

$ \because\bm\sigma'=-p'\bm I+2\rho\nu\bm d'

$ =\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla\cdot\overline{\bm v'\cdot\bm\nabla\bm v'+\bm\nabla\frac12|\bm v'|^2}

$ =\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)+\overline{\nu\bm l':\bm l'^\top+\nu\bm v'\cdot\bm\nabla(\bm\nabla\cdot\bm v')}

$ =\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)+\nu\overline{\bm l':\bm l'^\top}

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

$ \iff\frac{\overline{\rm D}k}{\overline{\rm D}t}=\frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)+\nu\overline{\bm l':(\bm l'^\top-2\bm d')}

$ = \frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)+\nu\overline{\bm l':(-\bm w'-\bm d')}

$ = \frac{\bm\sigma_T}\rho:\overline{\bm d}+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)-\nu\overline{\bm l':\bm l'}

$ \underline{= \frac{\bm\sigma_T}\rho:\overline{\bm d}-\varepsilon+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\left(\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}+\nu\bm\nabla k\right)\quad}_\blacksquare

$ \frac1\rho\frac{\overline{\rm D}\bm\sigma_T}{\overline{\rm D}t}=-2{\cal\pmb S}:\bm\sigma_T\cdot\overline{\bm l}-2{\cal\pmb S}:\rho\overline{\bm v'\bm f'}-2\overline{p'\bm d'}+\bm\nabla\cdot\left(\rho\overline{\bm v'\bm v'\bm v'}-\frac\nu\rho\bm\nabla\bm\sigma_T\right)+2{\cal\pmb S}:\bm\nabla\overline{p'\bm v'}+\mu\overline{{\bm l'}^\top\cdot\bm l'}

$ \implies\frac{\overline{\rm D}k}{\overline{\rm D}t}=\bm\sigma_T:\overline{\bm l}+\overline{\bm v'\cdot\bm f'}+\overline{p'\bm\nabla\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\frac12|\bm v'|^2}+\nu\bm\nabla^2k-\bm\nabla\cdot\overline{p'\bm v'}-\varepsilon

$ =\bm\sigma_T:\overline{\bm d}+\overline{\bm v'\cdot\bm f'}+\overline{p'\bm\nabla\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+p'\right)}+\nu\bm\nabla^2k-\varepsilon

$ =\bm\sigma_T:\overline{\bm d}+\overline{\bm v'\cdot\bm f'}+0-\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+p'\right)}+\nu\bm\nabla^2k-\varepsilon

$ \because\bm\nabla\cdot\bm v'=0

$ =\bm\sigma_T:\overline{\bm d}+\overline{\bm v'\cdot\bm f'}-\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+p'\right)}+\nu\bm\nabla^2k-\varepsilon

$ \therefore\frac{\overline{\rm D}k}{\overline{\rm D}t}=\bm\sigma_T:\overline{\bm d}-\varepsilon+\overline{\bm f'\cdot\bm v'}-\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+p'\right)}+\nu\bm\nabla^2k

項の物理的解釈

$ \underbrace{\overline{\frac{\mathrm D}{\overline{\mathrm D}t}}\left(\frac12\overline{|\bm v'|^2}\right)}_\text{乱流運動エネルギの時間変化率}+\underbrace{\overline{\bm d}:\overline{\bm v'\bm v'}}_\text{乱流エネルギ生産率}+\underbrace{\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}}_\text{拡散項}=\underbrace{\nu\bm\nabla^2\frac12\overline{|\bm v'|^2}}_\text{勾配拡散項}-\underbrace{\nu\overline{\bm l':\bm l'}}_\text{消散項}

$ \underbrace{\overline{\frac{\mathrm D k}{\overline{\mathrm D}t}}}_\text{乱流運動エネルギの時間変化率}+\underbrace{\frac1\rho\overline{\bm d}:\bm\sigma_R}_\text{乱流エネルギ生産率}+\underbrace{\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}}_\text{拡散項}=\underbrace{\nu\bm\nabla^2 k}_\text{勾配拡散項}-\underbrace{\varepsilon}_\text{消散項}

保留する

$ \rho\frac{\mathrm D}{\mathrm Dt}\left(\frac12|\bm v|^2\right)+\bm\sigma:\bm d=\bm\nabla\cdot(\bm\sigma\cdot\bm v)+\bm f\cdot\bm v

$ \iff\rho\frac{\mathrm D}{\mathrm Dt}\left(\frac12|\bm v|^2\right)=(\bm\nabla\cdot\bm\sigma)\cdot\bm v+\bm f\cdot\bm v

$ \frac{\mathrm D}{\mathrm Dt}\left(\frac12|\bm v|^2\right)=\bm v\cdot\bm\nabla\cdot(-\frac p\rho\bm I+\frac\lambda\rho(\bm\nabla\cdot\bm v)\bm I+2\nu\bm d)

$ =-\frac1\rho\bm v\cdot\bm\nabla p+\frac\lambda\rho\bm v\cdot\bm\nabla(\bm\nabla\cdot\bm v)+\nu\bm v\cdot\bm\nabla^2\bm v+\nu\bm v\cdot\bm\nabla(\bm\nabla\cdot\bm v)

$ =-\frac1\rho\bm\nabla\cdot(p\bm v)+\nu\bm\nabla^2\frac12|\bm v|^2-\nu\bm\nabla\bm v:\bm\nabla\bm v

$ \bm v\cdot\bm\nabla^2\bm v=\bm\nabla\cdot((\bm\nabla\bm v)\cdot\bm v)-\bm\nabla\bm v:\bm\nabla\bm v

$ = \bm\nabla^2\frac12|\bm v|^2-\bm\nabla\bm v:\bm\nabla\bm v

$ \iff\overline{\frac{\mathrm D}{\mathrm Dt}}\left(\frac12|\overline{\bm v}|^2\right)+\overline{\frac{\mathrm D}{\mathrm Dt}}\left(\frac12\overline{|\bm v'|^2}\right)+\bm\nabla\cdot(\overline{\bm v}\cdot\overline{\bm v'\bm v'}+\frac12\overline{\bm v'\bm v'\cdot\bm v'})=-\frac1\rho\bm\nabla\cdot\overline{(p\bm v)}+\nu\overline{\bm v\cdot\bm\nabla^2\bm v}

$ \iff\overline{\frac{\mathrm D}{\mathrm Dt}}\left(\frac12|\overline{\bm v}|^2\right)+\overline{\frac{\mathrm D}{\mathrm Dt}}\left(\frac12\overline{|\bm v'|^2}\right)+\bm\nabla\cdot(\overline{\bm v}\cdot\overline{\bm v'\bm v'}+\frac12\overline{\bm v'\bm v'\cdot\bm v'})=-\frac1\rho\bm\nabla\cdot\overline{p'\bm v'}-\frac1\rho\overline{\bm v}\cdot\bm\nabla\overline{p}+\nu\overline{\bm v}\cdot\bm\nabla^2\overline{\bm v}+\nu\overline{\bm v'\cdot\bm\nabla^2\bm v'}

$ \overline{|\bm v|^2}=|\overline{\bm v}|^2+2\overline{\bm v}\cdot\overline{\bm v'}+\overline{|\bm v'|^2}

$ =|\overline{\bm v}|^2+\overline{|\bm v'|^2}

$ \overline{\bm v|\bm v|^2}=\overline{\overline{\bm v}|\overline{\bm v}|^2+2\overline{\bm v}\ \overline{\bm v}\cdot\bm v'+\overline{\bm v}|\bm v'|^2}+\overline{\bm v'|\overline{\bm v}|^2+2\bm v'\overline{\bm v}\cdot\bm v'+\bm v'|\bm v'|^2}

$ =\overline{\bm v}(|\overline{\bm v}|^2+|\bm v'|^2)+2\overline{\bm v}\cdot\overline{\bm v'\bm v'}+\overline{\bm v'\bm v'\cdot\bm v'}

$ \overline{p\bm v}=\overline{p}\ \overline{\bm v}+\overline{p'\bm v'}

$ \bm\nabla\cdot\overline{\bm v}=0

$ \overline{\frac{\mathrm D}{\mathrm Dt}}\frac12|\overline{\bm v}|^2=-\frac1\rho\overline{\bm v}\cdot\bm\nabla\overline{p}+\nu\overline{\bm v}\cdot\bm\nabla^2\overline{\bm v}-\overline{\bm v}\cdot\bm{\nabla}\cdot\overline{\bm{v}'\bm{v}'}

$ \therefore\overline{\frac{\mathrm D}{\mathrm Dt}}\left(\frac12\overline{|\bm v'|^2}\right)+\underbrace{\bm\nabla\overline{\bm v}:\overline{\bm v'\bm v'}}_\text{乱流エネルギー生産率}+\underbrace{\bm\nabla\cdot\overline{\bm v'\left(\frac12|\bm v'|^2+\frac{p'}{\rho}\right)}}_\text{拡散項}=\underbrace{\nu\bm\nabla^2\frac12\overline{|\bm v'|^2}}_\text{勾配拡散項}-\underbrace{\nu\overline{\bm\nabla\bm v':\bm\nabla\bm v'}}_\text{消散項}

$ \rho\bm\nabla\overline{\bm v}:\overline{\bm v'\bm v'}=\overline{\bm d}:\bm\sigma_R

解説詳しい

$ I_2^{\bm l}=\frac12\left((\mathrm{tr}\bm l)^2-\mathrm{tr}\left(\bm l^2\right)\right)

$ =\frac12\left((\mathrm{tr}\bm d)^2-\mathrm{tr}\left(\bm d^2+\bm d\cdot\bm w+\bm w\cdot\bm d+\bm w^2\right)\right)

$ =\frac12\left((\mathrm{tr}\bm d)^2-\mathrm{tr}\left(\bm d^2\right)-\mathrm{tr}\left(\bm w^2\right)-\mathrm{tr}\left(\bm d\cdot\bm w-(\bm d\cdot\bm w)^\top\right)\right)

$ =\frac12\left((\mathrm{tr}\bm d)^2-\mathrm{tr}\left(\bm d^2\right)-\mathrm{tr}\left(\bm w^2\right)\right)

反対称tensorのtraceは0

$ \iff\frac{\mathrm D\bm l}{\mathrm Dt}=\bm\nabla\frac{\mathrm D\bm v}{\mathrm Dt}-\bm l^2

$ = \bm\nabla(-\frac1\rho\bm\nabla p+\nu\bm\nabla^2\bm v)-\bm l^2

$ = -\frac1\rho\bm\nabla\bm\nabla p+\nu\bm\nabla^2\bm l-\bm l^2

$ = -\bm l^2-\frac1\rho\bm\nabla\bm\nabla p+\nu\bm\nabla^2\bm l

$ \frac12(\bm l^2+{\bm l^2}^\top)=\frac12((\bm d+\bm w)^2+(\bm d-\bm w)^2)

$ = \bm d^2+\bm w^2

$ \therefore\frac{\mathrm D\bm d}{\mathrm Dt}=-\bm d^2-\bm w^2-\frac1\rho\bm\nabla\bm\nabla p+\nu\bm\nabla^2\bm d

$ \frac12(\bm l^2-{\bm l^2}^\top)=\frac12((\bm d+\bm w)^2-(\bm d-\bm w)^2)

$ =\frac12(\bm d\cdot\bm w+\bm w\cdot\bm d+\bm d\cdot\bm w+\bm w\cdot\bm d)

$ = \bm d\cdot\bm w-(\bm d\cdot\bm w)^\top

$ \therefore\frac{\mathrm D\bm w}{\mathrm Dt}=\bm d\cdot\bm w-(\bm d\cdot\bm w)^\top+\nu\bm\nabla^2\bm w

$ \frac{\overline{\rm D}}{\overline{\rm D}t}\frac12|\overline{\bm v}|^2+\frac{\bm\sigma_R}\rho:\overline{\bm d}=\bm\nabla\cdot\left(\frac{\bm\sigma_R}{\rho}\cdot\overline{\bm v}\right)+\overline{\bm v}\cdot\bm\nabla\cdot\frac{\bm\sigma}\rho+\overline{\bm f}\cdot\overline{\bm v}

$ =\bm\nabla\cdot\left(\frac{\bm\sigma_R-\overline p\bm I}{\rho}\cdot\overline{\bm v}+\nu\bm\nabla\frac12|\overline{\bm v}|^2\right)-\nu\overline{\bm l}:\overline{\bm l}+\overline{\bm f}\cdot\overline{\bm v}

定常流れ：統計量の時間微分が0

$ 0=\frac{\bm\sigma_R}\rho:\overline{\bm d}-\varepsilon+0

$ \iff\frac{\bm\sigma_R}\rho:\overline{\bm d}=\varepsilon

一様でない乱流でも、オーダーがほぼ同じになるらしい

連続体力学での導出