# Hybrid Simulations

In recent years, hybrid methods have increasingly been employed for the simulation of unsteady turbulent flows.

These models form a bridge between Large Eddy Simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) by utilizing RANS for attached boundary layer calculations and LES for the separated regions. In general, hybrid simulations need spatial filtering processes to determine the local sub grid turbulent viscosity. Compared to LES and Direct Numerical Simulation (DNS), hybrid simulations are extremely robust since the numerical requirement is less severe than the two other approaches. Hybrid simulations include Detached Eddy Simulation (DES), Delayed Detached Eddy Simulation (DDES) and Improved Delayed Detached Eddy Simulation (IDDES).

## Detached Eddy Simulations

Detached Eddy Simulation (DES) uses one of the one or two equation Reynolds-averaged Navier-Stokes (RANS) turbulence models to define the turbulence length scale or distance from the wall. The distance is then used to determine the region for which the RANS or Large Eddy Simulation (LES) models will be used.

DES using the Spalart-Allmaras (SA) model employs a modified distance function to replace the distance (d) in the SA model’s dissipation term (Spalart et al., 1997).

where d is the distance to the closest wall, ${C}_{DES}$ is a constant and $\Delta $ is a metric of the local element size, often chosen as the largest grid spacing in all three directions ( $\text{\Delta}=\text{max}\left[\text{\Delta}x,\text{\Delta}y,\text{\Delta}z\right]$ ).

The SA model is activated in a boundary layer region where d < ∆, while a Smagorinsky-like LES model is used in regions where ∆ < d. Similarly, Strelets (2001) adopted the SST model as the baseline model for DES, which computes a turbulence length scale and compares it with a grid length size for a switch between LES and RANS. However, both approaches need careful determinations of local grid sizes because these grid sizes act as the switch for the activation of LES and RANS.

## Delayed Detached Eddy Simulations

Delayed Detached Eddy Simulation is an improved version of DES that avoids the undesired activation of Large Eddy Simulation (LES) in boundary layer regions when the maximum grid size is less than the distance to the closest wall (∆ < d).

In this model (Spalart et al., 2006), the length scale is redefined as:

where ${f}_{d}=1-tanh\left({\left(8h\right)}^{3}\right)$ , $h=\frac{\tilde{\nu}}{\sqrt{{\overline{u}}_{ij}{\overline{u}}_{ij}}{\kappa}^{2}{d}^{2}}$ , $\kappa =0.41$ .

## Improved Delayed Detached Eddy Simulations (DDES)

Shur et al. (2008) proposed an improved DDES to ensure that most of the turbulence is resolved when the model is operating as a wall modeled Large Eddy Simulation (LES).

This is achieved by introducing a blending function of length scales defined as:

- $\tilde{{f}_{d}}=\mathrm{max}\left(\left[1-{f}_{d}\right],{f}_{b}\right)$ ,
- ${f}_{b}=\mathrm{min}\left(2{e}^{-9{\alpha}^{2}},1.0\right)$ ,
- $\alpha =0.25-\frac{d}{\text{\Delta}}$ ,
- ${f}_{e}=\mathrm{max}\left(\left[{f}_{e1}-1\right],1.0\right){f}_{e2}$ ,
- ${f}_{e1}=\{\begin{array}{c}2{e}^{-11.09{\alpha}^{2}}\frac{\text{\Delta}}{d},if\alpha \ge 0\\ 2{e}^{-9{\alpha}^{2}}\frac{\text{\Delta}}{d},if\alpha 0\end{array}$ ,
- ${f}_{e2}=1.0-\mathrm{max}\left(\mathrm{tanh}{\left(({C}_{t}^{2}{h}_{t}\right)}^{3}),\mathrm{tanh}{\left(({C}_{l}^{2}{h}_{l}\right)}^{10}\right))$ ,
- ${h}_{t}=\frac{{\nu}_{t}}{\sqrt{{\overline{u}}_{ij}{\overline{u}}_{ij}}{\kappa}^{2}{d}^{2}}$ ,
- ${h}_{l}=\frac{\nu}{\sqrt{{\overline{u}}_{ij}{\overline{u}}_{ij}}{\kappa}^{2}{d}^{2}}$ .