SST K-Omega DES

The SST K-Omega Detached Eddy model combines features of the SST K-Omega RANS model in the boundary layers with a large eddy simulation (LES) in unsteady separated regions.

The DES formulation of the SST K-Omega model is obtained by modifying the dissipation term in the transport equation for the turbulent kinetic energy.

DDES Formulation

Based on the work by Menter and Kuntz [360], the specific dissipation rate $ω$ in Eqn. (1215) is replaced by $\tilde{ω}$ , where:

Figure 1. EQUATION_DISPLAY

\tilde{ω} = ω ϕ

(1429)

and $ϕ$ is defined as:

Figure 2. EQUATION_DISPLAY

ϕ = max (\begin{matrix} l_{ratio} F, 1 \end{matrix})

(1430)

The length scale ratio $l_{ratio}$ is calculated as:

Figure 3. EQUATION_DISPLAY

l_{ratio} = \frac{l_{RANS}}{l_{LES}}

(1431)

Figure 4. EQUATION_DISPLAY

l_{RANS} = \frac{\sqrt{k}}{f_{β^{*}} β^{*} ω}

(1432)

Figure 5. EQUATION_DISPLAY

l_{LES} = C_{DES} Δ

(1433)

where:

$k$ is the turbulent kinetic energy.
$f_{β^{*}}$ is the free-shear modification factor.
$β^{*}$ is a K-Omega Model—Model Coefficient.
$Δ$ is the largest distance between the cell center under consideration and the cell centers of the neighboring cells.

$F$ is defined as:

Figure 6. EQUATION_DISPLAY

F = 1 - F_{2}

(1434)

where $F_{2}$ is the blending function given by Eqn. (1214).

For $ϕ = 1$ , the RANS solution is recovered, and for $ϕ > 1$ , the solution tends towards the LES definition.

The model coefficient $C_{D E S}$ , which blends the values obtained from independent calibration of the K-Epsilon and K-Omega branches of the K-Omega SST model [365], is evaluated as follows:

Figure 7. EQUATION_DISPLAY

C_{D E S} = {C_{D E S, k - ω} F}_{1} + C_{D E S, k - ε} (1 - F_{1})

(1435)

where:

$C_{D E S, k - ω}$ and $C_{D E S, k - ε}$ are Model Coefficients.
$F_{1}$ is defined in Eqn. (1230).

IDDES Formulation

For the IDDES formulation of Shur et al. [361], the specific dissipation rate $ω$ in Eqn. (1215) is replaced by $\tilde{ω}$ defined as:

Figure 8. EQUATION_DISPLAY

\tilde{ω} = \frac{\sqrt{k}}{l_{HYBRID} f_{β^{*}} β^{*}}

(1436)

where:

$f_{β^{*}}$ is the free-shear modification factor.
$β^{*}$ is given in K-Omega Model—Model Coefficients.

and:

Figure 9. EQUATION_DISPLAY

l_{HYBRID} = {\tilde{f}}_{d} (1 + f_{e}) l_{RANS} + (1 - {\tilde{f}}_{d}) C_{D E S} Δ_{IDDES}

(1437)

Two more functions are introduced to the length scale calculation to add Wall-Modeled LES (WMLES) capability, a blending function $f_{B}$ and the so-called “elevating” function $f_{e}$ :

Figure 10. EQUATION_DISPLAY

f_{B} = min [2 exp (- 9 α^{2}), 1]

(1438)

Figure 11. EQUATION_DISPLAY

α = 0.25 - \frac{d}{Δ}

(1439)

Figure 12. EQUATION_DISPLAY

f_{e} = max [(f_{e 1} - 1), 0] ψ f_{e 2}

(1440)

Figure 13. EQUATION_DISPLAY

f_{e 1} = {(\begin{matrix} 2 exp (- 11.09 α^{2}) if α \geq 0 \\ 2 exp (- 9 α^{2}) if α < 0 \end{matrix})

(1441)

Figure 14. EQUATION_DISPLAY

f_{e 2} = 1 - max (f_{t}, f_{l})

(1442)

Figure 15. EQUATION_DISPLAY

f_{t} = \tanh [{(C_{t}^{2} r_{d t})}^{3}]

(1443)

Figure 16. EQUATION_DISPLAY

f_{l} = \tanh [{(C_{l}^{2} r_{d l})}^{10}]

(1444)

Figure 17. EQUATION_DISPLAY

r_{d t} = \frac{ν_{t}}{\sqrt{\nabla v {: \nabla v}^{T}} κ^{2} d^{2}}

(1445)

Figure 18. EQUATION_DISPLAY

r_{d l} = \frac{ν}{\sqrt{\nabla v {: \nabla v}^{T}} κ^{2} d^{2}}

(1446)

where:

$C_{t}$ and $C_{l}$ are Model Coefficients.
$ν$ is the kinematic viscosity.
$ν_{t} = \frac{μ_{t}}{ρ}$ , where $μ_{t}$ is the turbulent eddy viscosity.
$κ$ is the von Karman constant. See K-Omega Model—Model Coefficients.
$d$ is the distance to the wall.

The introduction of the low-Reynolds number correction function $ψ$ in the formulation of $f_{e}$ is unrelated to the low-Reynolds number correction role of this function in the LES mode of DDES, and is purely empirical.

The WMLES and DDES branches of the model are combined using a modified version of the DDES $f_{d}$ function as follows:

Figure 19. EQUATION_DISPLAY

{\tilde{f}}_{d} = max ((1 - f_{d t}), f_{B})

(1447)

where:

Figure 20. EQUATION_DISPLAY

f_{d t} = 1 - \tanh [{(C_{d t} r_{d t})}^{3}]

(1448)

and $C_{d t}$ is a Model Coefficient.

The IDDES model also uses an altered version of the mesh length scale $Δ_{IDDES}$ , computed as:

Figure 21. EQUATION_DISPLAY

Δ_{IDDES} = min (max (0.15 d, 0.15 Δ, Δ_{min}), Δ)

(1449)

where $Δ_{min}$ is the smallest distance between the cell center under consideration and the cell centers of the neighboring cells.

DES Model Coefficients


$C_{D E S, k - ω}$	$C_{D E S, k - ε}$	$C_{d t}$	$C_{l}$	$C_{t}$
0.78	0.61	20	5	1.87