Shielded Detached Eddy Simulation (SDES)

Shielded Detached Eddy Simulation (SDES) is a hybrid RANS-LES turbulence model family. Starting from the DDES-SST model (see DES with the BSL or SST k-ω Model (p. 95) for details), it provides substantial improvements:

• a shielding function that protects the RANS wall boundary layer region against influences from the LES model

• improved definition of the grid length scale, for faster “transition” from RANS to LES in separating shear layers

Shielded Detached Eddy Simulation (SDES)

The improved shielding function offers nearly perfect boundary layer shielding against a premature switch to the LES model. Such premature switching could otherwise cause strong deterioration of the RANS capabilities of the model in attached boundary layers [446] (p. 783).

Note

No special calibration has been performed for the combination of SDES with Transition SST and the Intermittency Transition model.

Note that the activation of the LES term in the freestream in any hybrid RANS-LES model can affect the decay of the freestream turbulence, which in turn affects the transition location.

For an example demonstrating the effects of the SDES model, see SDES and SBES Example (p. 101).

4.12.1. Shielding Function

The starting point for the development of the Shielded Detached Eddy Simulation (SDES) shielding function is the existing Delayed Detached Eddy Simulation (DDES) models (see Detached Eddy Simulation (DES) (p. 93)). In DDES, the primary goal is to model the attached and mildly separated boundary layers in RANS mode and then switch to LES mode for separated (detached) shear layers. Starting from a given RANS model, a DDES formulation is achieved in two-equation models by a modification of the sink term in the -equation. In the -equation of the SST model, the modification is reformulated as follows:

(4.254) where

(4.255)

In the previous equation, is the turbulence length scale, is the maximum edge length of the local cell, is the DDES shielding function and is a coefficient. When equals zero, the RANS model is recovered. This happens when one or both of the following conditions are satisfied:

(4.256) and/or

(4.257)

The function is designed to protect attached boundary layers from the switch to the LES regime.

The limit at which the boundary layer becomes affected depends on the ratio of the grid spacing to the boundary layer thickness (for example, = / ). Without shielding (that is, when =0) the limit is approximately =1, so that at <1 the solution in the boundary layer becomes affected by the DES sink term. With the shielding function as given in Detached Eddy Simulation (DES) (p. 93), the limit for is reduced to 0.2-0.3, depending on the pressure gradient. Violating these limits has severe consequences, as the RANS solution will become compromised, and you might not be aware of it. DES / DDES methods were originally developed for aircraft simulations, where the boundary layer is typically thin relative to the dimensions of the aircraft / wing. Under those conditions (and with diligent mesh generation practices), the previously described limits for can typically be ensured. However, it is harder to ensure these limits for general industrial applications, due to lower Reynolds numbers, less control over the grid spacing, and more complex interactions between components.

Turbulence

The first goal of the development of the SDES method was therefore the reformulation of the function to achieve more reliable shielding, meaning the ability to maintain RANS boundary layers even under much more severe mesh refinement in attached boundary layers than is possible for DDES. For this purpose, a new shielding function called was developed, which provides substantially safer shielding than the functions currently used. The resulting sink term is equivalent to the one from the DDES model:

(4.258) where

(4.259)

Figure 4.7: Eddy Viscosity Profiles (p. 99) shows the eddy viscosities for self-similar zero-pressure-gradient boundary layer simulations under mesh refinement, where = / (note the different range of the refinement ratio for the two models). The SDES shielding (on the right) is clearly superior to the DDES shielding, and maintains the RANS boundary layer even under severe mesh refinement.

Figure 4.7: Eddy Viscosity Profiles

4.12.2. LES Mode of SDES

After the SDES model switches to LES mode, the additional source term is activated and reduces the eddy viscosity to a level comparable to a conventional LES model. The level of the achieved eddy vis-cosity can be estimated by imposing equilibrium on the source terms of the underlying RANS model (in this case, the SST model). This means that the convection and diffusion terms in both the - and

-equations are neglected, and the source and sink terms are equated. The result is:

(4.260)

where is the strain rate, and and are the constants of the source and sink terms in the -equation, respectively. represents the constants and used in the different models, and represents the definition of the grid spacing used in the different models. The formulation is obviously equivalent

Shielded Detached Eddy Simulation (SDES)

to the classical Smagorinsky model (see Subgrid-Scale Models (p. 103)). The combination of ( / )^3/4 is equivalent to the constant in the Smagorinsky model. Using the outer constants from the SST model that are relevant for the LES region ( =0.44, =0.083, =0.61) produces an equivalent =0.175 for the DES / DDES models. This is close to the value of =0.18, which is recommended for the

Smagorinsky model for Decaying Isotropic Turbulence (DIT) simulations. This is not surprising, as the LES portion of DES / DDES was calibrated for DIT. However, it is known that the Smagorinsky model requires different calibration constants for DIT and shear turbulence. The value for shear flows is closer to =0.11. Since shear flows are much more relevant for engineering simulations than DIT, it was decided to use a value of =0.4 in the SDES model, which results in the desired value for the equivalent constant.

The definition of the LES length scale is = in the DES / DDES models. This definition is problem-atic, as it can result in overly high levels of eddy viscosity in separating shear layers, where the mesh aspect ratio is typically high (for example, separating flow from a backward facing step, where the spanwise grid spacing is much larger than the grid spacing in the other two directions). This can lead to a slow 'transition' from RANS to LES mode. In order to avoid this issue, the SDES model uses the following formulation for the LES length scale:

(4.261) The first part is the classical length scale based on the volume of the cell, and the second part ensures a viable limit for very high aspect ratios. This formulation is smaller by a factor of five for high aspect ratio meshes compared to the DES definition.

It is important to note that both the constant / and the mesh definitions enter the equivalent Smagorinsky model quadratically. For highly stretched meshes, as is typical for separating shear layers, the combined effect of a lower constant and a smaller grid spacing definition results in a reduction by a factor 60 in the eddy viscosity level for SDES compared to DES / DDES.

It should also be noted that both modifications could equally be applied to the DES / DDES model formulations. However, such changes would severely reduce the shielding properties of the DES / DDES formulation, which are based on the combination of . Lowering these values alone or in combination would impair the shielding properties of these models.