RANS Boundary, Region, and Initial Conditions

Non-Flow Boundaries

At wall, symmetry plane, and axis boundaries, the transported variables are defined as follows:


RANS Model	Transported Variable	Wall	Symmetry	Axis
Spalart-Allmaras models	$\tilde{ν}$	$0$	${\partial \tilde{ν} / \partial n \|}_{s y m} = 0$	${\partial \tilde{ν} / \partial n \|}_{a x i s} = 0$
K-Epsilon models	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
K-Epsilon models	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$
(Lag) Elliptic Blending model	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$
	$φ$	${\partial φ / \partial n \|}_{w a l l} = 0$	${\partial φ / \partial n \|}_{s y m} = 0$	${\partial φ / \partial n \|}_{a x i s} = 0$
	$α$	$0$	${\partial α / \partial n \|}_{s y m} = 0$	${\partial α / \partial n \|}_{a x i s} = 0$
K-Omega models	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
K-Omega models	$ω$	${\partial ω / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ω / \partial n \|}_{s y m} = 0$	${\partial ω / \partial n \|}_{a x i s} = 0$
Reynolds Stress model	$R$	${\partial R / \partial n \|}_{w a l l} = 0$ In addition, a method that is developed by Hadzic [340] is used to impose the value of production of each stress component in accordance with a wall function approach. See Strain Rate Modification for Reynolds Stress Transport Models.	See Reynolds Stress Model—Symmetry.	See Reynolds Stress Model—Axis.
Reynolds Stress model	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$

Reynolds Stress Model—Symmetry

On a symmetry boundary, the procedure for setting the Reynolds stress tensor at the face $R_{f}$ is as follows. First, the tensor at the adjacent cell center $R$ is rotated such that it is parallel to the symmetry plane to obtain a new tensor termed $R'$ . This then enables the symmetry boundary face values of the tensor in this rotated coordinate system to be explicitly set as follows:

Figure 1. EQUATION_DISPLAY

{R'}_{f} = {[\begin{matrix} R_{11}^{'} & 0 & R_{13}^{'} \\ 0 & R_{22}^{'} & 0 \\ R_{31}^{'} & 0 & R_{33}^{'} \end{matrix}]}_{c}

(1352)

The values of $R_{f}$ are then computed by rotating the ${R'}_{f}$ tensor back to the original orientation of $R$ —this is made easy by the fact that the inverse of the rotation tensor is simply its transpose, since it is an orthogonal tensor.

Reynolds Stress Model—Axis

On an axis boundary, the procedure for setting the Reynolds stress tensor at the face $R_{f}$ is similar to the symmetry plane. In this case, however, the face values of the tensor in the coordinate system parallel to the axis are as follows:

Figure 2. EQUATION_DISPLAY

{R'}_{f} = {[\begin{matrix} R_{11}^{'} & 0 & 0 \\ 0 & R_{22}^{'} & 0 \\ 0 & 0 & R_{33}^{'} \end{matrix}]}_{c}

(1353)

Flow Boundaries, Region, and Initial Conditions

When defining values for flow boundaries, region, and initial conditions, you have three choices for specifying the transported variables:

Specify the values of the transported variables directly.
Have the values of the transported variables derived from a specified turbulence intensity $I$ and length scale $L$ .
Have the values of the transported variables derived from a specified turbulence intensity $I$ and turbulent viscosity ratio $μ_{t} / μ$ .


RANS Model	Direct Specification of transported variables	Derivation of transported variables from $I$ and $L$	Derivation of transported variables from $I$ and $μ_{t} / μ$
Spalart-Allmaras model	$\tilde{ν}$	Figure 3. EQUATION_DISPLAY ${C_{μ}}^{0.25} \sqrt{\frac{3}{2}} I v L$ (1354)	Figure 4. EQUATION_DISPLAY $\frac{μ_{t}}{μ} = \frac{ρ \tilde{ν} f_{v 1}}{μ}$ (1355)
K-Epsilon model	$k$	Figure 5. EQUATION_DISPLAY $\frac{3}{2} {(I v)}^{2}$ (1356)	Figure 6. EQUATION_DISPLAY $\frac{3}{2} {(I v)}^{2}$ (1357)
K-Epsilon model	$ε$	Figure 7. EQUATION_DISPLAY $\begin{matrix} \frac{{C_{μ}}^{3 / 4} k^{3 / 2}}{L} \end{matrix}$ (1358)	Figure 8. EQUATION_DISPLAY $\frac{ρ C_{μ} k^{2}}{(μ_{t} / μ) μ}$ (1359)
(Lag) Elliptic Blending model	$k$	As for K-Epsilon model.
	$ε$	As for K-Epsilon model.
	$φ$	$φ$	$φ$
	NOTE: In porous regions, the value of $φ$ is set to $2 / 3$ .
K-Omega model	$k$	As for K-Epsilon model.
K-Omega model	$ω$	Figure 9. EQUATION_DISPLAY $\frac{\sqrt{k}}{L β^{*^{1 / 4}}}$ (1360)	Figure 10. EQUATION_DISPLAY $\frac{ρ k}{(μ_{t} / μ) μ}$ (1361)
Reynolds Stress model	$R$	Figure 11. EQUATION_DISPLAY $\frac{2}{3} {(I v)}^{2} I$ (1362)	Figure 12. EQUATION_DISPLAY ${(I v)}^{2} I$ (1363)
Reynolds Stress model	$ε$	Figure 13. EQUATION_DISPLAY $\frac{{C_{μ}}^{3 / 4} {[\frac{1}{2} t r R]}^{3 / 2}}{L}$ (1364)	Figure 14. EQUATION_DISPLAY $\frac{C_{μ} {[\frac{1}{2} t r R]}^{2}}{(μ_{t} / μ) μ}$ (1365)

where:

$v$ is the local velocity magnitude in the local frame of reference. For initial conditions, $v$ is the specified initial turbulent velocity scale.
$C_{μ}$ and $β^{*}$ are model coefficients.
$f_{v 1}$ is a Damping Function.

For the Reynolds stress model, a fourth choice allows you to enter the dissipation rate

ε

directly but have the Reynolds stresses

R

derived from the turbulent kinetic energy

k

using the relation:

Figure 15. EQUATION_DISPLAY

R = \frac{2}{3} k I

(1366)

where $I$ is the identity matrix.

Boundary Layer Initialization

When convergence is slow, or when the initialization of the boundary layer is the main consideration (as in internal flows), you can initialize flow in the near-wall region only using the local relative velocity $u_{r}$ , the wall-parallel non-dimensional velocity $u^{+}$ and the $y^{+}$ value:


RANS Model	Transported Variable	Calculation
Spalart-Allmaras model	$\tilde{ν}$	Figure 16. EQUATION_DISPLAY $\max [\frac{1}{d u^{+} / d y^{+}} - 1, 0] = \frac{ρ \tilde{ν} f_{v 1}}{μ}$ (1367)
K-Epsilon model	$k$	Figure 17. EQUATION_DISPLAY $\sqrt{(\frac{1}{d u^{+} / d y^{+}} - 1) \frac{ε κ ν}{f_{u}}}$ (1368)
K-Epsilon model	$ε$	Figure 18. EQUATION_DISPLAY $\begin{matrix} \frac{1}{\max (17, y^{+})} \frac{u_{r}^{4}}{κ ν} \end{matrix}$ (1369)
(Lag) Elliptic Blending model	$k$	As for K-Epsilon model.
	$ε$	As for K-Epsilon model.
	$φ$	As specified for region initialization.
K-Omega model	$k$	As for K-Epsilon model.
K-Omega model	$ω$	Figure 19. EQUATION_DISPLAY $\begin{matrix} \frac{ε}{β^{} f_{β^{}} k} \end{matrix}$ (1370) where $k$ and $ε$ are calculated as for K-Epsilon model.
Reynolds Stress model	$R$	Figure 20. EQUATION_DISPLAY $\frac{2}{3} k I$ (1371) where $k$ is calculated as for K-Epsilon model.
Reynolds Stress model	$ε$	As for K-Epsilon model.

where:

$κ$ is the von Karman constant.
$f_{u} = \exp (- \frac{1.0674 \cdot 10^{- 3}}{4} {y^{+}}^{3})$
$β^{*}$ is a Model Coefficient.
$f_{β^{*}}$ is the Free-Shear Modification factor.

RANS Model	Transported Variable	Wall	Symmetry	Axis
Spalart-Allmaras models	$\tilde{ν}$	$0$	${\partial \tilde{ν} / \partial n \|}_{s y m} = 0$	${\partial \tilde{ν} / \partial n \|}_{a x i s} = 0$
K-Epsilon models	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
K-Epsilon models	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$
(Lag) Elliptic Blending model	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$
	$φ$	${\partial φ / \partial n \|}_{w a l l} = 0$	${\partial φ / \partial n \|}_{s y m} = 0$	${\partial φ / \partial n \|}_{a x i s} = 0$
	$α$	$0$	${\partial α / \partial n \|}_{s y m} = 0$	${\partial α / \partial n \|}_{a x i s} = 0$
K-Omega models	$k$	${\partial k / \partial n \|}_{w a l l} = 0$	${\partial k / \partial n \|}_{s y m} = 0$	${\partial k / \partial n \|}_{a x i s} = 0$
K-Omega models	$ω$	${\partial ω / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ω / \partial n \|}_{s y m} = 0$	${\partial ω / \partial n \|}_{a x i s} = 0$
Reynolds Stress model	$R$	${\partial R / \partial n \|}_{w a l l} = 0$ In addition, a method that is developed by Hadzic [340] is used to impose the value of production of each stress component in accordance with a wall function approach. See Strain Rate Modification for Reynolds Stress Transport Models.	See Reynolds Stress Model—Symmetry.	See Reynolds Stress Model—Axis.
Reynolds Stress model	$ε$	${\partial ε / \partial n \|}_{w a l l} = 0$ In addition, the wall cell value is imposed according to the appropriate method in the wall treatment. See Wall Treatment for RANS and DES—Turbulence Quantities.	${\partial ε / \partial n \|}_{s y m} = 0$	${\partial ε / \partial n \|}_{a x i s} = 0$