### 12.2.2. Locally Unstable Flows

#### 12.2.2.1. Flow Physics

The expression “locally unstable flows” is not easily defined, because every turbulent flow is unstable by nature. It is meant to characterize flows that also produce new turbulence, typically downstream of a geometry change, but where the flow instability producing this turbulence is significantly weaker than for globally unstable flows.

Consider the computation of a mixing layer starting from two wall boundary layers with different freestream velocities in RANS mode (see Figure 12.24). As the flat plate ends, the two boundary layers form a turbulent mixing layer, which quickly becomes independent of the turbulence of the two boundary layers on the flat plate (yellow circles). The mixing layer instability (red) provides for a de-coupling of the boundary layer and the mixing layer turbulence. For this reason, you can neglect the boundary layer turbulence downstream of the trailing edge (the dashed yellow boundary layer turbulence sketched in Figure 12.24) and concentrate on using SRS mode to resolve the mixing layer turbulence, which will quickly dominate the flow.

Figure 12.24: Schematic of locally unstable flow: Mixing layer originating from a flat plate with two boundary layers of different freestream velocity. Full yellow circles are boundary layer turbulence. Dashed yellow circles are remains of the boundary layer turbulence. Red arrows are new mixing layer turbulence

Examples of locally unstable flows:

- Generic Flows

  - All equilibrium free-shear flows emanating from walls (jets, wakes, mixing layers)

  - Backward-facing step flow

  - Weakly interacting equilibrium flows

  - Flows with weak swirl

#### 12.2.2.2. Modeling

The goal in SRS is to cover the boundary layer turbulence (solid yellow circles in Figure 12.24) in RANS and the mixing layer turbulence (red in Figure 12.24) in resolved mode. This can only be achieved if the impact of the RANS turbulence model is significantly reduced downstream of the trailing edge; otherwise the formation of unsteady structures would be suppressed.

The SAS model will typically not switch to SRS mode in such situations, independent of the mesh provided, as the eddy-viscosity produced in the mixing layer will be too large for the flow instability at hand. From a pure turbulence modeling standpoint, this is often acceptable, as such flows are typically covered with reasonable accuracy by using RANS models (mixing layers, wakes, back step, and so on). However, in cases where unsteady information is required for other reasons (for example, acoustics), the SAS model will likely not be suitable, unless an interface is used that converts modeled turbulence energy into resolved energy (see Section 12.1.7).

The DDES, SDES and SBES models allow SRS behavior because the shielding function is turned off past the trailing edge of the plate and, assuming a fine (LES) grid is provided downstream of the plate, the eddy-viscosity is reduced. These models then switch to LES mode in the wake, and the mixing-layer instability is strong enough to generate resolved turbulence relative quickly (within a few boundary layer thicknesses). It is important to point out that the ability of the DDES/SDES/SBES models to generate unsteady structures in the mixing layer depends on the grid provided in that area. Assuming an overly coarse grid (for example, in the spanwise direction), the grid limiter would not engage and the model would stay in RANS mode, which will not enable the formation of resolved structures. Remember that, for the DES model, the length scale is defined as:

| (12–29)

where is the largest edge length for each cell. For this case, assume that the grid in the plane shown in Figure 12.24 is very fine (of LES quality), and is the grid resolution in the spanwise () direction. Conversely, if is very coarse, the DES limiter would always select the RANS length scale and the model would remain in RANS mode in the wake region. No unsteady structures would develop as the RANS model will damp them out. As the grid in the -direction is refined, the DES limiter will be activated at some location downstream of the trailing edge where (note that grows as the mixing layer becomes thicker). With further grid refinement, the location of the implicit RANS-LES interface would move closer to the trailing edge. Eventually, the entire mixing layer would be covered by LES. This behavior of (D)DES is both a disadvantage and an advantage. The disadvantage and the danger lie in the strong grid sensitivity introduced explicitly into the turbulence model. As a result, when using (D)DES (as well as SDES/SBES), you must be very careful to provide a suitable grid for a given application when using DDES. The advantage is that the model can be applied to locally unstable flows without the definition of an explicit RANS-LES interface. However, the grid sensitivity can be reduced by employing an interface which converts modeled turbulence to resolved turbulence using the DDES/SDES/SBES model upstream and downstream of the interface (see Section 12.1.7).

The most general approach to the flows discussed here is the use of the embedded or zonal RANS-LES methods, where the boundary layers are covered by a RANS model and the mixing layer by a LES model. The models are explicitly switched from RANS to LES at a pre-defined interface upstream or at the trailing edge. In order to obtain a proper LES solution, a grid with LES resolution is required in the mixing layer. Frequently a non-conformal interface between the RANS and the LES part is used to reduce the grid resolution in the upstream RANS region. For a fully consistent simulation, one must introduce synthetic turbulence at the RANS-LES interface. By such injection of synthetic turbulence, the balance between RANS and LES turbulence across the interface is preserved (that is, the yellow dashed circles in Figure 12.24 are accounted for).

The recommendation for flows with local instabilities is to use ELES/ZLES models if the geometry and the application enable the definition of well-defined interfaces (for example, internal flows, like pipe flows). Synthetic turbulence should be introduced at these interfaces in order to preserve the balance between the RANS and LES turbulence content. Should the geometry/application be complex such that the definition of explicit RANS and LES zones is not easily possible (for example, turbomachinery flows, external flows), apply the DDES/SDES/SBES model. However, ensure careful tailoring of the grid with sufficient resolution on the LES region to avoid undefined model behavior somewhere between RANS and LES mode. It is advisable to refrain from using conventional DES in flows with extensive boundary layers, because the danger of affecting the boundary layers is too high. For such flows, the newest members of the family, namely the SBES models, are recommended.

It is very important to understand that for locally unstable flows, failure to capture the instability of the Separating Shear Layer (SSL) can have a pronounced effect on the solution downstream. The turbulence field is a result of this initial instability and missing it can severely limit the resolved content of the simulation and contaminate an expensive SRS solution. This danger is much reduced with ELES/ZLES models, (relative to DDES) because the flow enters the SSL with a prescribed synthetic turbulent content from the RANS-LES interface.

#### 12.2.2.3. Meshing Requirements

In order to generalize the concepts discussed for the mixing layer example (Figure 12.24), we introduce the terminology of a Separating Shear Layer (SSL). It refers to the shear layer that starts at the point of separation from the body and moves into a free shear flow (we are not considering small separation bubbles embedded within the boundary layer). In Figure 12.24 the SSC would be the mixing layer forming downstream of the plate. In other flows it can be a separating boundary layer from a corner. In the case of locally unstable flows, the spacing should be sufficiently small to enable resolution of the initial flow instability of the SSL. The main quantity of relevance is the ratio of RANS to grid length scale:

| (12–30)

It is important to emphasize that this quantity should be evaluated based on a precursor RANS solution. This implies that such a solution exists and is meaningful. If the precursor solution is not available, then you can estimate the ratio based on the thickness of SSL. For equilibrium mixing layers, the following ratio is approximately correct:

| (12–31)

where is the thickness of the mixing layer. The value of RL should be:

| (12–32)

where 0.2 should be considered an extreme lower limit of resolution and 0.1 the desirable lower limit. Again, higher grid resolution should be used if computing power permits. The value of corresponds to a resolution of 15 cells across the mixing layer. This amount of cells is not a very fine grid resolution, but equal resolution should ideally be provided in all three space dimensions. In addition, the SSL can be thin relative to the body dimensions, resulting in very high computational costs. The initial SSL instability is akin to a Helmholtz instability and is initially two-dimensional. A doubled grid spacing in the spanwise direction is therefore acceptable.

It is not always possible to achieve such resolution directly from the onset of the separating shear layer, especially if this layer is very thin relative to the body dimensions. This inability is not necessarily a problem as, typically, the thickness of the SSL increases strongly downstream of the separation point/line. Therefore is decreasing relatively quickly and reaches sufficiently low values to provide the required resolution. It is important to note that for cases where the small scales play a significant role, such as in acoustics simulations, the delay of the initial instability can result in a loss of spectral information at high wave numbers (small scales). It is advisable to visually inspect the displayed results for the presence of the unsteady turbulent structures at the intended locations.

Of special concern are geometries with high aspect ratios, meaning a large domain size in the direction perpendicular to the SSL (long cylinders in crossflow, stalled wings of high aspect ratios, and so on). In such situations, it is not always possible to sufficiently resolve the third direction. It might then be necessary to solve only a portion of the real flow domain in SRS mode, either by using suitable boundary conditions (for example, periodicity in the spanwise direction), or by restricting the SRS to a limited portion of the domain.

#### 12.2.2.4. Numerical Settings

The general numerical settings described in Section 12.3 should be applied. In addition, locally unstable flows can be very sensitive with respect to numerics. For the application of the DDES/SDES/SBES models, the recommended choice for the advection terms is the Bounded Central Difference (BCD) in the entire domain. The PRESTO pressure interpolation should be avoided in such simulations because it has been observed that this option can suppress the initial formation of resolved turbulence.

Experience suggests that the BCD scheme is also the most suitable choice when using ELES/ZLES methods. In some applications with high demands on accuracy and where a high quality isotropic mesh can be provided in the LES region, the application of the CD scheme in the LES zone might be advantageous.

#### 12.2.2.5. Examples

##### 12.2.2.5.1. Mixing Layer

The single-stream mixing layer flow is experimentally investigated in Morris and Foss, 2003 [21]. The experimental section consists of a flat plate which suddenly terminates at a corner step (Figure 12.25: The domain and grid for the single stream mixing layer flow), which induces the formation of the mixing layer from the separation point. The experiment is carried out at a Reynolds number of based on the momentum thickness of the incoming boundary layer and on the free-stream velocity .

The inlet of the computational domain shown in Figure 12.25: The domain and grid for the single stream mixing layer flow is located at , while the outlet boundary is located at ( corresponds to the step location). The size of the domain in the wall normal direction is upstream of the step and downstream of it. Finally, in the spanwise direction the domain is wide.

Two computational grids are considered (Figure 12.25: The domain and grid for the single stream mixing layer flow). Mesh 1 consists of about hexahedral cells with the maximum grid step of downstream of the step. Mesh 2 consists of hexahedral cells and has a maximum grid step of in the mixing layer region (spanwise direction). It should be noted that both meshes have the same node distribution upstream of the step with the grid steps in the wall normal direction chosen to satisfy in the entire domain.

The boundary conditions are specified as follows (Figure 12.25: The domain and grid for the single stream mixing layer flow). A constant velocity is specified at the flat plate inlet. Following the experimental setup, a velocity of is set at the bottom inlet below the mixing layer to provide the correct entrainment rate for a shear layer at zero pressure gradient and to ensure that the freestream velocity gradient in the streamwise direction is approximately zero. A constant pressure is set at the outlet boundary. No-slip conditions are utilized at solid walls. Finally, periodic conditions are employed in the spanwise direction.

Figure 12.25: The domain and grid for the single stream mixing layer flow

As seen from the instantaneous structures visualized with the use of isosurfaces of the Q-criterion (Figure 12.26: Isosurfaces of the Q-criterion colored with the velocity magnitude), SBES and SDES provide a swift development of three-dimensional turbulence downstream of the step, while DDES yields almost two-dimensional structures in the entire domain regardless of the used grid. For SDES, this rapid transition relative to DDES is based on the alternative definition of the grid length scale and the reduced constant. For SBES, it is based on the activation of the low eddy viscosity levels from the WALE model in the LES domain.

Figure 12.26: Isosurfaces of the Q-criterion colored with the velocity magnitude

Due to the much faster transition from RANS to LES, both, the SDES and the SBES model show superior agreement with the experiments in terms of the velocity profiles compared to DDES (see Figure 12.27: Profiles of the mean velocity at different sections (x/θ=19.3, 29.6, 40.6, and 54.2) for mixing layer test case). Similar performance could be obtained with DDES only under severe grid refinement – meaning that SDES/SBES can produce more realistic solutions on much coarser grids than required for DDES.

Figure 12.27: Profiles of the mean velocity at different sections (x/θ=19.3, 29.6, 40.6, and 54.2) for mixing layer test case

##### 12.2.2.5.2. Backward-Facing Step I

This flow has been experimentally studied by Vogel and Eaton, 1985 [35] for ( is the bulk velocity at the inlet section, is the step height, is the constant density, and is the constant dynamic viscosity). The expansion ratio relative to the upstream plane channel is 1.25.

The computational domain (Figure 12.28: The domain and grid for the backstep test case) extends from at the inlet to at the outlet ( corresponds to the step location), while in the spanwise direction it has the length of . The computational grid shown in the figure consists of hexahedral cells with maximum steps in the streamwise and spanwise directions of and respectively, which corresponds to and to in the wall units. The steps in the wall normal direction correspond to along all walls in the domain.

Figure 12.28: The domain and grid for the backstep test case

The boundary conditions are specified as shown in Figure 12.28: The domain and grid for the backstep test case. The inflow profiles are obtained from the precursor SST-RANS simulation of the developing plane channel flow up to the experimental boundary layer thickness of . Periodic conditions are applied in the spanwise direction. No-slip conditions are specified on the solid walls. At the outlet boundary, a constant pressure is specified, while the other transported quantities are extrapolated from the domain interior.

A non-dimensional time step of ensures that the CFL number is less than one in the entire domain. The instantaneous flow fields are averaged over 5000 time steps in order to achieve a sufficient time sample for the unsteady statistics.

As seen from iso-surfaces of the Q-criterion shown in Figure 12.29: Isosurfaces of the Q-criterion colored with the velocity magnitude for backstep test case, SBES and SDES provide visually similar turbulent structures downstream of the step, while relatively larger structures are observed for DDES. It can also be seen that the blending functions of SBES and SDES, relative to that of DDES, cover a smaller portion of the flow near the bottom wall (Figure 12.30: Contours of the blending function with the vorticity iso-lines for backstep test case), while, in the channel center, the SDES and SBES models work in the RANS mode. DDES also remains in RANS mode there, but this is not visible due to the lack of grid information in the DDES shielding function. DDES provides a larger eddy viscosity level in the mixing zone relative to those of SBES and SDES, as can be seen from Figure 12.31: Contours of the eddy viscosity ratio for backstep test case. Again, this explains the larger turbulent structures observed in this region with DDES.

Figure 12.29: Isosurfaces of the Q-criterion colored with the velocity magnitude for backstep test case

Figure 12.30: Contours of the blending function with the vorticity iso-lines for backstep test case

Figure 12.31: Contours of the eddy viscosity ratio for backstep test case

However, despite the above differences, the skin friction coefficient distribution depicted in Figure 12.32: Distribution of the skin friction coefficient for backstep test case along the bottom wall shows that the results of all the models are in good agreement with the experimental data. However, the DDES model shows a slightly overly large separation zone and a slight over-prediction of -recovery downstream of the reattachment point. Finally, the mean and RMS velocity profiles for all the considered models agree fairly well with the experimental data. The better prediction for the backstep as compared to the mixing layer above is not surprising because the mesh for the backstep has been designed for the DDES model. It features lower grid spacing (relative to the upstream boundary layer thickness) in spanwise direction compared to the mixing layer case.

Figure 12.32: Distribution of the skin friction coefficient for backstep test case

Figure 12.33: Profiles of the mean and RMS velocity at different sections (x/H=3.2, 4.55, 5.87, and 7.2) for backstep test case

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