Reynolds Averaged Navier Stokes (RANS)

Use case

It is a method of turbulence modeling where the Time Averaged Navier-Stokes equations are used. This means that any property $f$ is split into its averaged and fluctuating components.

$$f(\vec{x},t)=\bar{f}(\vec{x})+f^{\prime }(\vec{x},t)$$

Rearranging and taking integral

$$\bar{f}(\vec{x})=\underset{\Delta T\to \infty }{\lim }\frac{1}{\Delta T}{\int }_{-\Delta T/2}^{\Delta T/2}f(\vec{x},t)dt$$

The RANS Equations are formulated taking the Time Average of the Navier Stokes Equations

• Continuity Equation

  $$\boxed{\frac{\partial {\bar{u}}_j}{\partial x_j}=0}$$

• Momentum Equations

  $$\frac{\partial {\bar{u}}_i}{\partial t}+\frac{\partial {\bar{u}}_i{\bar{u}}_j}{\partial x_j}+\frac{\partial \overline{u_i^{\prime }{u}_j^{\prime }}}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x_i}+v\frac{{\partial }^2{\bar{u}}_i}{\partial x_j^2}$$

Rearranging

$$
\boxed{
\frac{\partial \bar{u}_i}{\partial t}+\frac{\partial \bar{u}_i \bar{u}_j}{\partial x_j}=-\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}+\frac{\partial}{\partial x_j}\left[v \frac{\partial\bar u_i}{\partial x_j}+\tau_{i j}^{t u r b}\right]}
$$

where ${\tau }_{ij}^{turb}=-\overline{u_i^{\prime }{u}_j^{\prime }}$ represents the Reynolds Stresses

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References

Pg. 698 The Finite Volume Method - Moukalled