### 12.1.3. Shielded Detached Eddy Simulation (SDES)

The SDES formulation is a member of the DDES model family, but offers alternatives for the shielding function and the definition of the grid scale. The impact on the turbulence model is, as usual, formulated as an additional sink term in the k-equation:

| (12–9)

The shielding function provides much stronger shielding than the corresponding function above. For this reason, the natural shielding of the model based on the mesh length definition, , can be reduced. The mesh length scale used in the SDES model is defined as follows:

| (12–10)

The first part represents the conventional LES grid length scale definition, and the second part is again based on the maximum edge length as in the DES formulation. However, the factor 0.2 ensures that for highly stretched meshes, the grid length scale is a factor of 5 smaller than for DES/DDES. Since the grid length scale enters quadratically into the definition of the eddy viscosity in LES mode, this means a reduction by a factor of 25 in such cases. It will be shown that this drastically reduces the frequently observed problem of slow ‘transition’ of DES/DDES models from RANS to LES. Note that the combination of this more ‘aggressive’ length scale with the conventional DDES shielding function would severely reduce the shielding properties of DDES and is therefore not recommended.

The SDES constant is also different from in the DES/DDES formulation, where it is calibrated based on decaying isotropic turbulence (DIT) with the goal of matching the turbulence spectrum relative to data after certain running times. However, in engineering flows, one typically has to deal with shear flows, for which a reduced Smagorinsky constant should be used. This is achieved by setting =0.4. The combination of the re-definition of the grid length scale and the modified constant leads to a reduction in the eddy viscosity by a factor of around 60 in separating shear flows on stretched grids. It will be shown later that this results in a much more rapid transition from RANS to LES.

The shielding function is formulated such that it provides essentially asymptotic shielding on any grid. In flat plate tests, the limit was pushed below .

The following test case shows the improved shielding properties of SDES/SBES models relative to DDES. The flow is a diffuser flow in an axisymmetric geometry featuring a small separation bubble. Due to the adverse pressure gradient, the boundary layer grows strongly and shielding is difficult to achieve due to the strong increase in .

The computational domain is shown in Figure 12.3: The domain and grid for the separated flow in CS0 diffuser. The length of the domain in the streamwise direction is about 7.8· [m] ( is the diameter of the cylinder and corresponds to the separation point in the experiment). For this flow, a standard RANS grid is used with steps in the streamwise and circumferential directions of and respectively (the grid step in the circumferential direction is changing in the radial direction due to the axisymmetric geometry). Here is the boundary layer thickness at the inlet section. The height of the wall cell is chosen to satisfy the condition in the entire domain and around 30 cells cover the boundary layer.

Figure 12.3: The domain and grid for the separated flow in CS0 diffuser

As seen from the contours of the SBES and DDES blending functions shown in Figure 12.4: Contours of blending functions overset by vorticity iso-lines for CS=0 diffuser for SBES and DDES models, SBES covers the entire boundary layer including the rapid growth area of the boundary layer due to the separation bubble, while DDES preserves only the portion of the domain at the inlet. This means that under adverse pressure gradient conditions, the shielding properties of DDES are substantially impaired.

Figure 12.4: Contours of blending functions overset by vorticity iso-lines for CS=0 diffuser for SBES and DDES models

Similar observations can be made for the eddy viscosity fields shown in Figure 12.5: Contours of eddy viscosity ratio for CS0 diffuser for SBES and DDES models. The eddy viscosity levels of SBES correspond to those of the SST model (not shown), while the DDES model produces much reduced levels in the adverse pressure gradient region.

Figure 12.5: Contours of eddy viscosity ratio for CS0 diffuser for SBES and DDES models

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