Wall Treatment for RANS and DES

For Reynolds-Averaged Navier Stokes (RANS) turbulence models and Detached Eddy Simulations (DES), the wall treatment provides boundary conditions to the solvers for flow and energy that are specific to turbulent boundary layers. Additionally, the wall treatment imposes special values for turbulence quantities on the centroids of the near-wall cells.

A DES solves the boundary layer by using a RANS closure model. This modeling approach allows DES to directly adopt the wall treatment from the RANS model that is used for closure.

Simcenter STAR-CCM+ provides the following types of wall treatment for RANS models:

Low- $y^{+}$
High- $y^{+}$
All- $y^{+}$
Two-layer all- $y^{+}$

Not all wall treatments are available with every RANS model or model variant. High Reynolds number models do not contain the ability to attenuate the turbulence in the viscosity-affected regions and therefore only include a high- $y^{+}$ wall treatment. Low Reynolds number models are offered only with a low- $y^{+}$ wall treatment and an all- $y^{+}$ wall treatment. The two-layer all- $y^{+}$ wall treatment is only available with the two-layer K-Epsilon and the two-layer Reynolds Stress turbulence models.

A RANS wall treatment performs the following functions:

Calculates the wall shear stress.
Calculates the wall heat flux.
Imposes values for the turbulence quantities on the centroids of the near-wall cells.

Wall Shear Stress

The wall shear stress is calculated according to Eqn. (1626), where the wall friction velocity $u_{τ}$ is calculated as:

Figure 1. EQUATION_DISPLAY

u_{τ}^{2} = \frac{u_{*}}{u^{+}} | {\hat{v}}_{tangential} |

(1630)

where:

$u_{*}$ is the velocity scale.
$u^{+}$ is the non-dimensional wall-tangential velocity component of the velocity vector.
${\hat{v}}_{tangential}$ is the RANS averaged wall-tangential velocity vector.

$u_{*}$ and $u^{+}$ are approximated using wall functions. The following table lists the wall functions that are used depending on the selected wall treatment:


Variable	Low- $y^{+}$	High- $y^{+}$	All- $y^{+}$	Two-Layer All- $y^{+}$
$u_{*}$	Eqn. (1594)	Eqn. (1596)	Eqn. (1595)	Eqn. (1595)
$u^{+}$	Eqn. (1598)	Eqn. (1603)	Eqn. (1599)	Eqn. (1599)

If the iterative $u_{*}$ method is applied with the all- $y^{+}$ wall treatment, Simcenter STAR-CCM+ calculates the wall-friction velocity $u_{τ}$ as:

Figure 2. EQUATION_DISPLAY

u_{τ} = u_{*}

(1631)

where $u_{*}$ is calculated iteratively by equating the computed value of $u^{+}$ as given by Eqn. (1585) with the wall function definition for $u^{+}$ as given by Eqn. (1599).

Wall Heat Flux

The wall heat flux is calculated according to Eqn. (1629), where the non-dimensional RANS averaged cell temperature ${\hat{T}}_{C}^{+}$ is approximated using a wall function. The wall function that is used depends on the selected wall treatment as follows:


Variable	Low- $y^{+}$	High- $y^{+}$	All- $y^{+}$	Two-Layer All- $y^{+}$
${\hat{T}}_{C}^{+}$	Eqn. (1605)	Eqn. (1609)	Eqn. (1606)	Eqn. (1606)

Turbulence Quantities

Simcenter STAR-CCM+ applies the following conditions to the transport equations for turbulence at the wall:

For the modified diffusivity $\tilde{v}$ , sets the value at the wall as $\tilde{v} = 0$ . In the near-wall cell, the transport equation for $\tilde{v}$ is solved using an imposed value for the modified diffusivity production $P_{\tilde{v}}$ .
For the turbulent kinetic energy $k$ , sets the wall-normal velocity gradient at the wall as ${\frac{\partial k}{\partial y} |}_{w} = 0$ . In the near-wall cell, the transport equation for $k$ is solved using imposed values for the turbulent kinetic energy production $P_{k}$ and the turbulent dissipation rate $ε$ .
For the turbulent dissipation rate $ε$ and the specific dissipation rate $ω$ , sets the wall normal gradients at the walls as ${\frac{\partial ε}{\partial y} |}_{w} = 0$ and ${\frac{\partial ω}{\partial y} |}_{w} = 0$ . No transport equations are solved for $ε$ and $ω$ in the near-wall cells. Instead, the values for $ε$ and $ω$ are imposed.
For the Reynolds stress tensor $R$ , sets the wall-normal velocity gradient at the wall as ${\frac{\partial R}{\partial y} |}_{w} = 0$ . In the near-wall cell, the transport equation for $R$ is solved using an imposed value for the mean strain rate tensor $S$ and the turbulent dissipation rate $ε$ .

The turbulence quantities that are imposed in the near-wall cells depend on the selected RANS Model. The value of the turbulence quantity is derived by equating the non-dimensional definition of the turbulence quantity with its algebraic approximation as provided by the applied wall function.

The following table lists the non-dimensional definitions of the imposed turbulence quantities and the wall functions that are used depending on the selected wall treatment:


RANS Model	Imposed Quantity	Non-Dimensional Definition	Low- $y^{+}$	High- $y^{+}$	All- $y^{+}$	Two-Layer All- $y^{+}$
Spalart-Allmaras	$P_{\tilde{v}}$	Eqn. (1589)	imposes no special value for $P_{\tilde{v}}$	Eqn. (1617)	Eqn. (1617)	-
K-Epsilon, Elliptic Blending	$P_{k}$	Eqn. (1591)	Eqn. (1618)	Eqn. (1620)	Eqn. (1619)	Eqn. (1619)
K-Epsilon, Elliptic Blending	$ε$	Eqn. (1592)	Eqn. (1611)	Eqn. (1613)	Eqn. (1612)	Figure 3. EQUATION_DISPLAY $ε^{+} = \frac{{(k^{+})}^{3 / 2}}{l_{ε}^{+} y^{+}}$ (1632)
K-Omega	$P_{k}$	Eqn. (1591)	Eqn. (1618)	Eqn. (1620)	Eqn. (1619)	-
K-Omega	$ω$	Eqn. (1593)	Eqn. (1614)	Eqn. (1616)	Eqn. (1615)	-
Reynolds Stress Transport (RST)	$S$	See Strain Rate Modification for Reynolds Stress Transport Models.
Reynolds Stress Transport (RST)	$ε$	Eqn. (1592)	-	Eqn. (1613)	Eqn. (1612)	Figure 4. EQUATION_DISPLAY $ε^{+} = \frac{{(k^{+})}^{3 / 2}}{l_{ε}^{+} y^{+}}$ (1633)

where:

Figure 5. EQUATION_DISPLAY

l_{ε}^{+} = \frac{l_{ε}}{y}

(1634)

and:

$k^{+}$ is given by Eqn. (1590).
$y^{+}$ is given by Eqn. (1584).
$l_{ε}$ is a length scale function that is calculated depending on the Two-Layer Model Variant.
$y$ is the wall distance.

Model Coefficients

The high-

y^{+}

, the all-

y^{+}

, and the two-layer all-

y^{+}

wall treatments use the following model coefficients:


$κ$	$E$
0.42	9