One Equation Eddy Viscosity Models

One equation Reynolds-averaged Navier-Stokes (RANS) models solve a single scalar transport equation to compute the eddy viscosity.

Common one equation models include the Spalart-Allmaras (SA) model and the Nut-92 model. The SA model is discussed in this manual due to its application in general purpose CFD codes and its popularity for the simulation of external flows and internal flows. The details of the Nut-92 model can be found in Shur et al. (1995).

Spalart-Allmaras (SA) Model

The SA model uses a transport equation to solve for a modified kinematic eddy viscosity, $ˆ υ$ $\hat{υ}$ , as a function of the kinematic eddy viscosity ( $ν_{t}$ $ν_{t}$ ) (Spalart and Allmaras, 1992).

In this model, a length scale (d) in a dissipation term of the modified kinematic eddy viscosity transport equation is specified to determine the dissipation rate. This model has an advantage of having economic solutions for attached flows and moderately separated flows, but it is not recommended for massively separated flows, free shear flows and decaying turbulence.

Transport Equations

(1)

\frac{\partial (ρ ˆ υ)}{\partial t} + \frac{\partial (ρ ˆ υ ¯ ¯¯ ¯ u_{j})}{\partial x_{j}} = \frac{1}{σ} (\frac{\partial}{\partial x_{j}} [(μ + ρ ˆ υ) \frac{\partial ˆ υ}{\partial x_{j}}] + ρ C_{b 2} \frac{\partial ˆ υ}{\partial x_{j}} \frac{\partial ˆ υ}{\partial x_{j}}) + P + D

$\frac{\partial (ρ \hat{υ})}{\partial t} + \frac{\partial (ρ \hat{υ} \bar{u_{j}})}{\partial x_{j}} = \frac{1}{σ} (\frac{\partial}{\partial x_{j}} [(μ + ρ \hat{υ}) \frac{\partial \hat{υ}}{\partial x_{j}}] + ρ C_{b 2} \frac{\partial \hat{υ}}{\partial x_{j}} \frac{\partial \hat{υ}}{\partial x_{j}}) + P + D$

where $σ$ $σ$ and $C_{b 2}$ $C_{b 2}$ are constants and $μ$ $μ$ is the fluid dynamic viscosity. P and D are the production term and destruction term of the modified turbulent viscosity, respectively.

Production of $ˆ υ$ $\hat{υ}$

(2)

P = ρ C_{b 1} ˆ S ˆ υ

$P = ρ C_{b 1} \hat{S} \hat{υ}$

where

$ˆ S = \sqrt{2 Ω_{i j} Ω_{i j}} + \frac{ˆ υ}{κ^{2} d^{2}} f_{v 2}$ $\hat{S} = \sqrt{2 Ω_{i j} Ω_{i j}} + \frac{\hat{υ}}{κ^{2} d^{2}} f_{v 2}$ ,
$Ω_{i j} = \frac{1}{2} (\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} - \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}})$ $Ω_{i j} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} - \frac{\partial \bar{u_{j}}}{\partial x_{i}})$ is the rotation tensor,
$f_{v 2} = 1 - \frac{χ}{1 + χ f_{v 1}}$ $f_{v 2} = 1 - \frac{χ}{1 + χ f_{v 1}}$ ,
$d$ $d$ is the distance from the nearest wall,
$κ$ $κ$ is the Von Kármán constant,
$C_{b 1}$ $C_{b 1}$ is a constant.

Destruction of $ˆ υ$ $\hat{υ}$

(3)

D = - ρ C_{w 1} f_{w} {(\frac{ˆ υ}{d})}^{2}

$D = - ρ C_{w 1} f_{w} {(\frac{\hat{υ}}{d})}^{2}$

where

$f_{w} = g {(\frac{1 + C_{w 3}^{6}}{g^{6} + C_{w 3}^{6}})}^{1 / 6}$ $f_{w} = g {(\frac{1 + C_{w 3}^{6}}{g^{6} + C_{w 3}^{6}})}^{1 / 6}$
$f_{t 2} = C_{t 3} e x p (- C_{t 4} χ^{2})$ $f_{t 2} = C_{t 3} e x p (- C_{t 4} χ^{2})$
$g = r + C_{w 2} (r^{6} - r)$ $g = r + C_{w 2} (r^{6} - r)$
$r = \frac{ˆ υ}{ˆ S κ^{2} d^{2}}$ $r = \frac{\hat{υ}}{\hat{S} κ^{2} d^{2}}$
$C_{w 1}$ $C_{w 1}$ is a constant.

Modeling of Turbulent Viscosity $μ_{t}$ $μ_{t}$

The kinematic eddy viscosity for the Spalart-Allmaras model is computed using the following relationship(4)

μ_{t} = ρ ˆ υ f_{v 1}

$μ_{t} = ρ \hat{υ} f_{v 1}$

where

$f_{v 1} = \frac{χ^{3}}{χ^{3} + C_{v 1}^{3}}$ $f_{v 1} = \frac{χ^{3}}{χ^{3} + C_{v 1}^{3}}$ is the viscous damping function.
$χ = \frac{ˆ υ}{ν}$ $χ = \frac{\hat{υ}}{ν}$ .

Model Coefficients

$C_{w 1} = \frac{C_{b 1}}{κ^{2}} + \frac{1 + C_{b 1}}{σ^{2}}$ $C_{w 1} = \frac{C_{b 1}}{κ^{2}} + \frac{1 + C_{b 1}}{σ^{2}}$ , $C_{w 2} = 0.3$ $C_{w 2} = 0.3$ , $C_{w 3} = 2.0$ $C_{w 3} = 2.0$ , $C_{b 1} = 0.1355$ $C_{b 1} = 0.1355$ , $C_{b 2} = 0.622$ $C_{b 2} = 0.622$ , $C_{v 1} = 7.1$ $C_{v 1} = 7.1$ $σ$ $σ$ = $\frac{2}{3}$ $\frac{2}{3}$ , $κ = 0.41$ $κ = 0.41$ .

Spalart-Allmaras (SA) Model with Rotation/Curvature Correction

The effects of system rotation and streamline curvature are present in turbomachinery components. Some examples include axial turbines, radial turbines, axial fans, compressors and centrifugal impellors.

This is where most linear eddy viscosity models fail. In order to incorporate the rotational and curvature effects for the SA model, Shur et al. (2000) introduced a version with the modified production term of the transport equation by multiplying the rotation function $f_{r 1}$ $f_{r 1}$ .

Modified Production of $ˆ υ$ $\hat{υ}$

(5)

P = ρ f_{r 1} C_{b 1} ˆ S ˆ υ

$P = ρ f_{r 1} C_{b 1} \hat{S} \hat{υ}$

where

$f_{r 1} = (1 + C_{r 1}) \frac{2 r *}{1 + r *} (1 - C_{r 3} t a n^{- 1} [C_{r 2} ˆ r]) - C_{r 1}$ $f_{r 1} = (1 + C_{r 1}) \frac{2 r *}{1 + r *} (1 - C_{r 3} t a n^{- 1} [C_{r 2} \hat{r}]) - C_{r 1}$
$r * = \frac{S}{ˆ ω}$ $r * = \frac{S}{\hat{ω}}$
$S = \sqrt{2 S_{i j} S_{i j}}$ $S = \sqrt{2 S_{i j} S_{i j}}$ is the strain rate magnitude.
$S_{i j} = \frac{1}{2} (\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} + \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}})$ $S_{i j} = \frac{1}{2} (\frac{\partial \bar{u_{i}}}{\partial x_{j}} + \frac{\partial \bar{u_{j}}}{\partial x_{i}})$ is the strain rate tensor.
$ˆ ω = \sqrt{2 {ˆ ω}_{i j} {ˆ ω}_{i j}}$ $\hat{ω} = \sqrt{2 {\hat{ω}}_{i j} {\hat{ω}}_{i j}}$ is the modified vorticity magnitude.
${ˆ ω}_{i j} = \frac{1}{2} ([\frac{\partial ¯ ¯ ¯ u_{i}}{\partial x_{j}} - \frac{\partial ¯ ¯¯ ¯ u_{j}}{\partial x_{i}}] + 2 ε_{m j l} {Ω'}_{m})$ ${\hat{ω}}_{i j} = \frac{1}{2} ([\frac{\partial \bar{u_{i}}}{\partial x_{j}} - \frac{\partial \bar{u_{j}}}{\partial x_{i}}] + 2 ε_{m j l} {Ω'}_{m})$ is the rotation tensor.
$ˆ r = \frac{2 ω_{i k} S_{j k}}{D^{4}} (\frac{\partial S_{i j}}{\partial t} + ¯ ¯¯ ¯ u_{i j} \frac{\partial S_{i j}}{\partial x_{j}} + [ε_{i m n} S_{j n} + ε_{j m n} S_{i n}] {Ω'}_{m})$ $\hat{r} = \frac{2 ω_{i k} S_{j k}}{D^{4}} (\frac{\partial S_{i j}}{\partial t} + \bar{u_{i j}} \frac{\partial S_{i j}}{\partial x_{j}} + [ε_{i m n} S_{j n} + ε_{j m n} S_{i n}] {Ω'}_{m})$
$D = \sqrt{\frac{1}{2} (S^{2} + {\hat{ω}}^{2})}$
$Ω'$ is the rotation rate.

Model Coefficients

$C_{r 1}$ =1.0, $C_{r 2}$ =12.0, $C_{r 3}$ =1.0

One Equation Eddy Viscosity Models

Spalart-Allmaras (SA) Model

Transport Equations

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Destruction of ˆυˆυ<math xmlns="http://www.w3.org/1998/Math/MathML"> <mover accent="true"> <mi>υ</mi> <mo>^</mo> </mover> </math>

Modeling of Turbulent Viscosity μtμt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Model Coefficients

Spalart-Allmaras (SA) Model with Rotation/Curvature Correction

Modified Production of ˆυˆυ<math xmlns="http://www.w3.org/1998/Math/MathML"> <mover accent="true"> <mi>υ</mi> <mo>^</mo> </mover> </math>

Model Coefficients

Production of $ˆ υ$ $\hat{υ}$

Destruction of $ˆ υ$ $\hat{υ}$

Modeling of Turbulent Viscosity $μ_{t}$ $μ_{t}$

Modified Production of $ˆ υ$ $\hat{υ}$