User:MasterHD/Turbulent Energy Dissipation Rate

A two-equation k-${\displaystyle \epsilon }$ turbulence model commonly used in modern engineering applications uses the following two additional differential equations to close the RANS equation,

${\displaystyle {\frac {\partial K}{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial K}{\partial x_{j}}}={\mathcal {P}}-\epsilon +\nu \nabla ^{2}K-{\frac {\partial }{\partial x_{i}}}\left({\frac {\overline {p\upsilon _{i}^{\prime }}}{\rho }}+{\overline {\upsilon _{i}^{\prime }(\upsilon _{j}^{\prime \,2}/2)}}\right)}$

${\displaystyle {\frac {\partial \epsilon }{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial \epsilon }{\partial x_{j}}}=P_{\epsilon }^{1}+P_{\epsilon }^{2}+P_{\epsilon }^{3}+P_{\epsilon }^{4}+\Pi _{\epsilon }+T_{\epsilon }+D_{\epsilon }-\Upsilon _{\epsilon }}$