Reynolds Averaged Simulation (RAS)

OpenFOAM includes Reynolds Averaged Simulation turbulence closures based on linear and non-linear eddy viscosity models, and Reynolds stress transport models.

RAS is selected by setting the `simulationType`

entry

simulationType RAS; RAS { \\ Model input parameters }

Suitable for:

- 1-D, 2-D and 3-D cases
- steady-state or transient

Care should be taken to provide an adequate initialisation of turbulence fields. For open systems, the flow can evolve via the inlets. However, note that the initial field values should be approximate for the flow medium, where non-physical levels will result in non-physical turbulence viscosity predictions and case instability.

Appropriate formulae and/or methods are provided in the documentation for each model.

Reynolds decomposition of the velocity into its mean and fluctuating contributions takes the form

\[ \u(\vec{x}, t) = \av{\u}(\vec{x}, t) + \u'(\vec{x}, t) \]

where the mean of the fluctuating component is defined as zero:

\[ \av{\u'} = 0 \]

Applied to the Navier Stokes equations, this leads to equations for the mean velocity and pressure:

\[ \ddt{\rho} + \div \left(\rho \av{\u} \right) = 0 \]

\[ \ddt{\rho \av{\u}} + \div \left( \rho \av{\u} \otimes \av{\u} \right) = \vec{g} + \div \av{\tensor(\tau)} - \div \left( \rho \tensor{R} \right) \]

where the averaged stress tensor, \( \av{\tau} \), for Newtonian fluids is given by:

\[ \av{\tensor{\tau}} = - \left(p + \frac{2}{3}\mu \div \av{\u} \right) \tensor{I} + \mu \left(\grad \av{\u} + \left(\grad \av{\u} \right)^T \right) \]

Using the relationship:

\[ \div \av{\u} = \mathrm{tr} \left( \grad \av{\u} \right) = \mathrm{tr} \left( \left( \grad \av{\u} \right)^T \right) \]

the stress tensor becomes:

\[ \av{\tensor{\tau}} = - p \tensor{I} + \mu \left[\grad \av{\u} + \left( \grad \av{\u} \right)^T - \frac{2}{3} \mathrm{tr} \left( \left( \grad \av{\u} \right)^T \right) \tensor{I} \right] = - p \tensor{I} + \mu \left[\grad \av{\u} + \mathrm{dev2} \left( \left( \grad \av{\u} \right)^T \right) \right] \]

and the \( \mathrm{dev2} \) operator is defined as:

\[ \mathrm{dev2}\left(\phi\right) = \phi - \frac{2}{3} \mathrm{tr} \left(\phi\right) \tensor{I} \]

\( \tensor{R} \), is the *Reynolds stress tensor*

\[ \tensor{R} = \av{\u' \otimes \u'} \]

The Reynolds stress tensor is further divided into isotropic and deviatoric anisotropic contributions:

\[ \tensor{R} = \av{\u' \otimes \u'} = \underbrace{\frac{2}{3}k \tensor{I}}_{\mathrm{isotropic}} + \underbrace{\av{\u' \otimes \u'} - \frac{2}{3}k \tensor{I}}_{\mathrm{deviatoric}} \]

where \( k \) is the turbulent kinetic energy, defined by

\[ k = \frac{1}{2} \av{\u' \dprod \u'} = \frac{1}{2} \mathrm{tr}\left( \tensor{R} \right) \]

Only the anisotropic contribution of the Reynolds stress tensor transports momentum, whereby the isotropic contribution can be added to the mean pressure, leading to the full form:

\[ \ddt{\rho \av{\u}} + \div \left( \rho \av{\u} \otimes \av{\u} \right) = \vec{g} - \grad \av{p}' + \div \left( \mu \grad \av{\u} \right) + \div \left[ \mu \, \mathrm{dev2}\left(\left(\grad \av{\u}\right)^T \right) \right] - \div \left( \rho \tensor{R}_\mathit{dev} \right) \]

where

\[ \tensor{R}_\mathit{dev} = \av{\u' \otimes \u'} - \frac{2}{3}k \tensor{I} \]

and

\[ \av{p}' = \av{p} + \frac{2}{3} \rho k \]

The RAS family of turbulence models provide methods to approximate the deviatoric anisotropic stress contribution due to \( \tensor{R}_\mathit{dev} \).

- See: Wall modelling

Source code:

- $FOAM_SRC/TurbulenceModels/turbulenceModels/RAS
- $FOAM_SRC/TurbulenceModels/turbulenceModels/derivedFvPatchFields/wallFunctions

Related:

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