It is normally written in the form

%$\nu_t = (\Delta C_s)^2 \sqrt{ S}$%

%$ S=2 S_{ij} S_{ij} $%

%$S_{ij}= {1\over 2} \left( {\partial u_i \over \partial x_j} + {\partial u_j \over \partial x_i} \right) $%

If one looks in more detail at the scalar %$S$% we have

%$S={1\over 2} \left( {\partial u_i \over \partial x_j} {\partial u_i \over \partial x_j}+ 2 {\partial u_i \over \partial x_j} {\partial u_j \over \partial x_i} + {\partial u_j \over \partial x_i} {\partial u_j \over \partial x_i} \right) $%

which ends up being

%$S=\left( {\partial u_i \over \partial x_j} {\partial u_i \over \partial x_j} + {\partial u_i \over \partial x_j} {\partial u_j \over \partial x_i} \right) $%


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Topic revision: r1 - 2008-01-16 - 17:19:52 - RichardHoward
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Computational Fluid Dynamics and Turbulence Mechanics
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