Turbulence Models in OpenFOAM – Hybrid Methods


Large Eddy Simulation (LES) can be used to accurately simulate unsteady flow behaviors but its numerical cost is higher compared to the conventional Reynolds Averaged Navier-Stokes (RANS) approach.

Several methods have been developed in order to save computational time and cost for unsteady flow computations compared to LES.

Keywords
Detached-Eddy Simulation (DES), Scale Adaptive Simulation (SAS), Grey area problem

Detached Eddy Simulation (DES)

Detached-Eddy Simulation (DES) is a hybrid Reynolds Averaged Navier-Stokes/Large-Eddy Simulation model. The DES model was first proposed in 1997 by Spalart et al. [1] based on the Spalart-Allmaras RANS model and it is commonly referred to as DES97.

Spalart-Allmaras Based DES Formulation (DES97)

Its formulation is briefly described in [2]:

The driving length scale of the RANS S-A model is the distance to the closest wall, \(d\). This makes a modification to this model for DES mode quite straightforward (exactly for this reason it was used as a basis of DES in the first publication [1]). The modification consists in substituting for \(d\), everywhere in the equations, the new DES length scale, \(\tilde{d}\). This length is also based on the grid spacing \(\Delta\) and is defined as:
\begin{equation}
\tilde{d} = {\rm min}\left( d, C_{DES}\Delta \right), \tag{1} \label{eq:dTilda}
\end{equation}
where \(C_{DES}\) is the only new adjustable model constant, and \(\Delta\) is based on the largest dimension of the local grid cell
\begin{equation}
\Delta = {\rm max}\left( \delta_{x}, \delta_{y}, \delta_{z} \right). \tag{2} \label{eq:delta}
\end{equation}
Here we assume for simplicity that the grid is structured and that the coordinates \(\left(x, y, z\right)\) are aligned with the grid cell, but the generalizations are obvious.

For wall-bounded separated flows, the above formulation results in a bybrid model that functions as the standard RANS S-A model inside the whole attached boundary layer, and as its subgrid-scale version in the rest of the flow including the separated regions and near wake. Indeed, in the attached boundary layer, due to the significant grid anisotropy \(\left(\delta_{x} \approx \delta_{z} \gg \delta_{y}\right)\) typical of this flow region, in accordance with \eqref{eq:dTilda}, \(\tilde{d} = d\), and the model reduces to the standard S-A RANS model. Otherwise, once a field point is far enough from walls \(\left( d > C_{DES}\Delta \right)\), the length scale of the model becomes grid-dependent, i.e., the model performs as a subgrid-scale version of the S-A model. Note that at “equilibrium” (meaning a balance of production and destruction terms) this model reduces to an algebraic miximg-length Smagorinski-like subgrid model.

The “DES limiter” defined by Eq. \eqref{eq:dTilda} that is used in DES97 switches between the RANS length scale and LES length scale so that the model behaves in RANS-like and LES-like manners as illustrated in Figure 1. The length scale \eqref{eq:dTilda} depends only on the grid used in the simulation and it is solution-independent.

Figure 1: Schematic illustration of S-A based DES model.

Menter’s SST Based DES Formulation

  • k-\(\omega\) SST DES model
  • k-\(\omega\) SST DDES model
  • k-\(\omega\) SST IDDES model

Other Models

  • \(v^2-f\) DES model

There is a project called DESider(Detached Eddy Simulation for Industrial Aerodynamics) and many other DES models have been developed based on different RANS models, including RSM ones.

desider

We can check the RANS and LES regions using the DESModelRegions function objects in OpenFOAM.

Scale Adaptive Simulation (SAS)
  • k-\(\omega\) SST SAS model
Embedded LES (ELES)
Useful Links
  • FLOMANIA(Flow Physics Modelling An Integrated Approach) Project: 2002-2004
  • DESider(Detached Eddy Simulation for Industrial Aerodynamics) Project: 2004-2007
  • ATAAC(Advanced Turbulence Simulation for Aerodynamic Application Challenges) Project
References

[1] P. R. Spalart, W.-H. Jou, M. Strelets and S. R. Allmaras, Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, 1997, Ruston, LA. In “Advances in DNS/LES”, C. Liu and Z. Liu Eds., Greyden Press, Columbus, OH.
[2] M. Strelets, Detached Eddy Simulation of Massively Separated Flows. AIAA, 2001-0879.
[3] P. R. Spalart, S. Deck, M. L. Shur, K. D. Squires, M. Kh. Strelets, and A. Travin, A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoretical and Computational Fluid Dynamics, 20(3), 181-195, 2006.
[4] M. L. Shur, P. R. Spalart, M. Kh. Strelets, and A. Travin, A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. International Journal of Heat and Fluid Flow, 29, 1638–1648, 2008.
[5] Detached eddy simulation (DES). Available at: http://www.cfd-online.com/Wiki/Detached_eddy_simulation_(DES) [Accessed: 16 October 2016].
[6] OpenFOAM® v3.0+: New Solver and Physical Modelling Functionality. Available at: http://www.openfoam.com/version-v3.0+/solvers-and-physics.php#physics-kOmegaSSTDES [Accessed: 16 October 2016].
[7] C. M. Winkler, A. J. Dorgan and M. Mani, Scale Adaptive Simulations of Turbulent Flows on Unstructured Grids. AIAA, 2011-3559.

Author: fumiya

CFD engineer in Japan