Time Averaged Navier-Stokes Equations

NS into RANS

$$\begin{array}{c}\frac{\partial u_i}{\partial t}+\frac{\partial {\mathbf{u}}_{\mathbf{i}}{\mathbf{u}}_{\mathbf{j}}}{\partial {\mathbf{x}}_{\mathbf{j}}}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+v\frac{{\partial }^2{u}_i}{\partial x_j^2}+g_i\\ \frac{\partial {\bar{u}}_i}{\partial t}+\frac{\partial {\bar{u}}_i{\overline{\mathbf{u}}}_{\mathbf{j}}}{\partial {\mathbf{x}}_{\mathbf{j}}}+\frac{\partial \overline{{\mathbf{u}}_{\mathbf{i}}^{\prime }{\mathbf{u}}_{\mathbf{j}}^{\prime }}}{\partial {\mathbf{x}}_{\mathbf{j}}}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x_i}+v\frac{{\partial }^2{\bar{u}}_i}{\partial x_j^2}+{\bar{g}}_i\end{array}$$

Note

• The variables are split into a mean and a fluctuating component.

  $$\begin{array}{rl}{v}_i& ={\bar{v}}_i+v_i^{\prime }\\ p& =\bar{p}+p^{\prime }\end{array}$$

• These equations are substituted into the NS and then the equation is Time Averaged.

• To remember:

  $$\begin{array}{rl}& {\overline{\bar{v}}}_i{\bar{v}}_j={\bar{v}}_i{\bar{v}}_j\\ & \overline{{\bar{v}}_i{v}_j^{\prime }}={\overline{\bar{v}}}_i{\bar{v}}_j^{\prime }=0\text{ and }\overline{{\bar{v}}_j{v}_i^{\prime }}={\overline{\bar{v}}}_j{\bar{v}}_i^{\prime }=0\end{array}$$

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References

• Full derivation: Chalmers, Turbulence Modeling Pg. 82