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Example of creating new turbulence model


    • This section deals with procedure of new turbulence model creation

       

    • We start with standard $ k-\epsilon$ model

       

    • This will be only an artificial training example

       

    • Modify model to be able to control turbulence production for $ k-\epsilon$ model

       

    • Standard $ k-\epsilon$ model has following production term:

     

     

    $\displaystyle G = \nu_{t} S^{2}$(24.3)

     

    where $ S$ represents the turbulent generation rate

     

     

    $\displaystyle S = \sqrt{2 {\mathsf {D}}_{ij}{\mathsf {D}}_{ij}}$(24.4)

     

    and velocity gradient can be decomposed into symmetric and skew-symmetric parts:

     

     

    $\displaystyle \bm{{\mathsf {L}}}=\nabla{\bm{{\mathsf {u}}}}=\left( \begin{array}{ccc} \frac{\partial {\mathsf {u}}_{1}}{\partial x_{1}} & \frac{\partial u_{1}}{\partial x_{2}} & \frac{\partial u_{1}}{\partial x_{3}} \\ [0mm] \frac{\partial {\mathsf {u}}_{2}}{\partial x_{1}} & \frac{\partial u_{2}}{\partial x_{2}} & \frac{\partial u_{2}}{\partial x_{3}} \\ [0mm] \frac{\partial {\mathsf {u}}_{3}}{\partial x_{1}} & \frac{\partial u_{3}}{\partial x_{2}} & \frac{\partial u_{3}}{\partial x_{3}} \end{array} \right) = \left( \begin{array}{ccc} u_{x} & u_{y} & u_{z} \\ v_{x} & v_{y} & v_{z} \\ w_{x} & w_{y} & w_{z} \end{array} \right) = \bm{{\mathsf {D}}} + \bm{{\mathsf {W}}}$(24.5)

     

    where $ \bm{{\mathsf {D}}}$ is rate-of-tensor (symmetric):

     

     

    $\displaystyle \bm{{\mathsf {D}}}= \frac{1}{2}\left( \nabla{\bm{{\mathsf {u}}}}+\nabla{\bm{{\mathsf {u}}}}^{T} \right)= \frac{1}{2}\left( \begin{array}{ccc} 2u_{x} & u_{y}+v_{x} & u_{z}+w_{x} \\ v_{x}+u_{y} & 2v_{y} & v_{z}+w_{y} \\ w_{x}+u_{z} & w_{y}+v_{z} & 2w_{z} \end{array} \right)$(24.6)

     

    $ \bm{{\mathsf {W}}}$ is skew-symmetric part of velocity gradient:

     

     

    $\displaystyle \bm{{\mathsf {W}}}= \frac{1}{2}\left( \nabla{\bm{{\mathsf {u}}}}-\nabla{\bm{{\mathsf {u}}}}^{T} \right)= \frac{1}{2}\left( \begin{array}{ccc} 0 & u_{y}-v_{x} & u_{z}-w_{x} \\ v_{x}-u_{y} & 0 & v_{z}-w_{y} \\ w_{x}-u_{z} & w_{y}-v_{z} & 0 \end{array} \right)$(24.7)

     

    Subsections
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