Spalart-Allmaras DES

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

The DES formulation ([362], [359]) is obtained by replacing the distance to the nearest wall $d$ in the turbulent dissipation term $ϒ_{\tilde{ν}}$ (see Eqn. (1154)) by $\tilde{d}$ . The definition of $\tilde{d}$ depends on the DES approach used.

DDES Formulation

To prevent a premature onset of the LES mode within the boundary layer of ambiguous meshes, Spalart and others [363] have introduced a correction to the definition of the dissipation length scale $\tilde{d}$ that depends on the value of the turbulent eddy viscosity and velocity gradient. The new approach has been named Delayed DES (DDES).

For DDES, the length scale $\tilde{d}$ is computed as:

Figure 1. EQUATION_DISPLAY

\tilde{d} = d - f_{d} max (0, d - {ψ C}_{D E S} Δ)

(1408)

and:

$C_{D E S}$ is a Model Coefficient.
$Δ$ is the largest distance between the cell center under consideration and the cell centers of the neighboring cells.

The function $f_{d}$ is [363]:

Figure 2. EQUATION_DISPLAY

f_{d} = 1 - \tanh [{(8 r_{d})}^{3}]

(1409)

where:

Figure 3. EQUATION_DISPLAY

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

(1410)

and $κ$ is the von Karman constant. See Spalart-Allmaras Model—Model Coefficients.

The optional low-Reynolds number correction function $ψ$ is designed to prevent activation of the low-Reynolds number terms when in “LES” mode [363]. It is defined as:

Figure 4. EQUATION_DISPLAY

ψ^{2} = min {100, \frac{1 - C_{b 1} ([f_{t 2} + (1 - f_{t 2}) f_{v 2}] / {{f_{w}^{*} κ}^{2} C}_{w 1})}{(1 - f_{t 2}) f_{v 1}}}

(1411)

where:

$C_{b 1}$ and $C_{w 1}$ are given in Spalart-Allmaras Model—Model Coefficients.
$f_{t 2}$ is a damping function defined as appropriate for the DES model. See DES Damping Function.
$f_{v 1}$ and $f_{v 2}$ are given in Spalart-Allmaras Model—Damping Functions.

The coefficient $f_{w}^{*}$ is a computed iteratively as a nonlinear function of the other model coefficients, as described in [363]. It has a value of 0.424 when the standard model coefficients are used.

IDDES Formulation

For the IDDES formulation of Shur and others [361], two more functions are introduced to the calculation of the modified wall distance $\tilde{d}$ to add Wall-Modeled LES (WMLES) capability: a blending function $f_{B}$ and the so-called “elevating” function $f_{e}$ .

The length scale $\tilde{d}$ is computed as:

Figure 5. EQUATION_DISPLAY

\tilde{d} = \tilde{f_{d}} (1 + f_{e}) d + (1 - \tilde{f_{d}}) {ψ C}_{D E S} Δ_{I D D E S}

(1412)

where:

Figure 6. EQUATION_DISPLAY

f_{B} = min [2 exp (- 9 α^{2}), 1]

(1413)

Figure 7. EQUATION_DISPLAY

α = 0.25 - \frac{d}{Δ}

(1414)

Figure 8. EQUATION_DISPLAY

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

(1415)

Figure 9. 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}

(1416)

Figure 10. EQUATION_DISPLAY

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

(1417)

Figure 11. EQUATION_DISPLAY

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

(1418)

Figure 12. EQUATION_DISPLAY

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

(1419)

Figure 13. EQUATION_DISPLAY

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

(1420)

Figure 14. EQUATION_DISPLAY

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

(1421)

where $C_{t}$ and $C_{l}$ are Model Coefficients.

The introduction of the low-Reynolds number correction function $ψ$ in the elevating function $f_{e}$ 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 15. EQUATION_DISPLAY

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

(1422)

Figure 16. EQUATION_DISPLAY

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

(1423)

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

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

Figure 17. EQUATION_DISPLAY

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

(1424)

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

DES Damping Function

The DDES and IDDES versions of the Spalart-Allmaras model contain a modification to the

f_{t 2}

damping function (see Spalart-Allmaras Model—Damping Functions).

$f_{t 2}$ is defined as recommended by Squires and others [364] as:

Figure 18. EQUATION_DISPLAY

f_{t 2} = C_{t 3} exp (- C_{t 4} χ^{2})

(1425)

This modification does not affect the predictions of fully turbulent flows, but prevents spurious upstream propagation of the eddy viscosity into attached laminar regions.

DES Model Coefficients


$C_{d t}$	$C_{D E S}$	$C_{t}$	$C_{l}$
8	0.65	1.63	3.55