Spalart-Allmaras Model

The Spalart-Allmaras turbulence model solves a transport equation for the modified diffusivity $\tilde{ν}$ in order to determine the turbulent eddy viscosity.

This approach is in contrast to many of the early one-equation models that solved an equation for the transport of turbulent kinetic energy and required an algebraic prescription of a length scale. (See [302].)

The original model was developed primarily for the aerospace industry, and has the advantage of being readily implemented in an unstructured CFD solver (unlike the more traditional aerospace models, such as those of Baldwin-Lomax [297] and Johnson-King [299]). This advantage has resulted in its popularity increasing as the use of unstructured CFD methods has grown more widespread in the aerospace industry.

The authors of the original Spalart-Allmaras turbulence model presented results for attached boundary layers and flows with mild separation (such as flow past a wing). It is reasonable to expect that these cases are the types of flows for which the model yields the best results. Wilcox [303] presents free-shear flow spreading rates for the model. While acceptable results are obtained for wake, mixing layer and radial jet flows, the predicted spreading rates for plane and round jets are inaccurate. Therefore, Wilcox concludes that the model is not suited to applications involving jet-like free-shear regions. It is also likely to be less suited to flows involving complex recirculation and body forces (such as buoyancy) than two-equation models such as K-Epsilon and K-Omega or Reynolds Stress Transport.

In its standard form, the Spalart-Allmaras model is a low-Reynolds number model, meaning that it is applied without wall functions. According to the formulation of the model, the entire turbulent boundary layer, including the viscous sublayer, can be accurately resolved and the model can be applied on fine meshes (small values of $y^{+}$ ). In Simcenter STAR-CCM+, this model is also available with an all- $y^{+}$ wall treatment. To account for anisotropy of turbulence, Simcenter STAR-CCM+ offers a quadratic constitutive relation.

Relation for Turbulent Viscosity

The turbulent eddy viscosity $μ_{t}$ is calculated as:

Figure 1. EQUATION_DISPLAY

μ_{t} = ρ f_{v 1} \tilde{ν}

(1148)

where:

$ρ$ is the density.
$f_{v 1}$ is a Damping Function.

Transport Equation

The transport equation for the modified diffusivity $\tilde{ν}$ is:

Figure 2. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \tilde{ν}) + \nabla\cdot (ρ \tilde{ν} \bar{v}) = \begin{matrix} \frac{1}{σ_{\tilde{ν}}} \nabla\cdot [(μ + ρ \tilde{ν}) \nabla \tilde{ν}] + P_{\tilde{ν}} {+ S}_{\tilde{ν}} \end{matrix}

(1149)

where:

$\bar{v}$ is the mean velocity.
$σ_{\tilde{ν}}$ is a Model Coefficient.
$μ$ is the dynamic viscosity.
$P_{\tilde{ν}}$ is the Production Term.
$S_{\tilde{ν}}$ is the user-specified source term.

Production Term

The production term $P_{\tilde{ν}}$ is defined as:

Figure 3. EQUATION_DISPLAY

P_{\tilde{ν}} = D_{\tilde{ν}} + G_{\tilde{ν}} + G_{nl} - ϒ_{\tilde{ν}}

(1150)

where the contributions to the production term are:


	Description	Formulation	where:
$D_{\tilde{ν}}$	Non-conservative diffusion	Figure 4. EQUATION_DISPLAY $\frac{C_{b 2}}{σ_{\tilde{ν}}} ρ (\nabla \tilde{v} \cdot \nabla \tilde{v})$ (1151)	$C_{b 2}$ is a Model Coefficient.
$G_{\tilde{ν}}$	Turbulent production	Figure 5. EQUATION_DISPLAY ${ρ (1 - f_{t 2}) C}_{b 1} f_{r 1} \tilde{S} \tilde{ν}$ (1152)	$f_{t 2}$ is a Damping Function. $C_{b 1}$ is a Model Coefficient. $f_{r 1}$ is the rotation function given by Eqn. (1286). $\tilde{S}$ is the Deformation Parameter.
$G_{nl}$	"Non-linear" production	Figure 6. EQUATION_DISPLAY $(T_{RANS, NL}) : \nabla \bar{v}$ (1153)	$T_{RANS, NL}$ is a non-linear Constitutive Relation.
$ϒ_{\tilde{ν}}$	Turbulent dissipation	Figure 7. EQUATION_DISPLAY $ρ (C_{w 1} f_{w} - \frac{C_{b 1}}{κ^{2}} f_{t 2}) {(\frac{\tilde{ν}}{d})}^{2}$ (1154)	$C_{w 1} = \begin{matrix} \frac{C_{b 1}}{κ^{2}} + \frac{1 + C_{b 2}}{σ_{\tilde{ν}}} \end{matrix}$ $f_{w} = g {(\frac{1 + C_{w 3}^{6}}{g^{6} + C_{w 3}^{6}})}^{1 / 6}$ where: $g = r + C_{w 2} (r^{6} - r)$ $r = \min (\frac{\tilde{ν}}{\tilde{S} κ^{2} d^{2}}, 10)$ $C_{w 2}$ and $C_{w 3}$ are Model Coefficients. $κ$ is the von Karman constant, see Model Coefficients. $d$ is the distance to the wall.

Deformation Parameter

The deformation parameter $\tilde{S}$ is calculated as:

Figure 8. EQUATION_DISPLAY

\tilde{S} = \hat{S} + \frac{\tilde{ν}}{κ^{2} d^{2}} f_{v 2}

(1155)

where:

$f_{v 2}$ is a Damping Function.

Two alternatives are provided for evaluation of the scalar deformation $\hat{S}$ . The original model uses the magnitude of the vorticity:

Figure 9. EQUATION_DISPLAY

\hat{S} = W

(1156)

where:

$W$ is given by Eqn. (1131).

An alternative suggested in Dacles-Mariani and others [298] combines the strain rate and vorticity tensor magnitudes as follows:

Figure 10. EQUATION_DISPLAY

\hat{S} = W + C_{prod} min [0, S - W]

(1157)

where:

$C_{prod}$ is a Model Coefficient.
$S$ is given by Eqn. (1129).

Damping Functions

For turbulence models that resolve the viscous- and buffer-layer, damping functions mimic the decrease of turbulent mixing near the walls.


$f_{v 1}$	$f_{v 2}$	$f_{t 2}$
Figure 11. EQUATION_DISPLAY $\frac{χ^{3}}{χ^{3} + C_{v 1}^{3}}$ (1158)	Figure 12. EQUATION_DISPLAY $1 - \frac{χ}{1 + χ f_{ν 1}}$ (1159)	Figure 13. EQUATION_DISPLAY $1.1 exp (- 2 χ^{2})$ (1160)

where:

$χ = \frac{\tilde{ν}}{ν}$
$C_{ν 1}$ is a Model Coefficient.

Model Coefficients


$C_{b 1}$	$C_{b 2}$	$C_{w 2}$	$C_{w 3}$
0.1355	0.622	0.3	2


$κ$	$σ_{\tilde{ν}}$	$C_{v 1}$	$C_{prod}$
0.41	2/3	7.1	2