# Spalart-Allmaras DDES model in OpenFOAM

The formulation of Spalart-Allmaras DDES (Delayed Detached Eddy Simulation) model is described in [1].

Keywords:
delayed DES, shielding function, modeled-stress depletion

 Modifications Compared to Original DES97 Model

The DES length scale $$\tilde{d}$$ is modified compared to that of DES97 model to prevent a premature switch of DES to LES mode within boundary layers. This incursion can be a cause of modeled-stress depletion (MSD) issue that the DES97 formulation suffers from.

Two functions $$r_d$$ and $$f_d$$ are introduced in new definition \eqref{eq:dTilda} and it now depends not only on the grid but also the time-dependent eddy-viscosity field as Eq. \eqref{eq:rd} shows. The new length scale \eqref{eq:dTilda} detects the boundary layer and delays the switch to LES mode until outside of it. For this reason, the method was named delayed DES.

 Implementation in OpenFOAM

r_d \equiv \frac{\nu_t + \nu}{\sqrt{U_{i,j}U_{i,j}}\kappa^2 d^2} \tag{1} \label{eq:rd}

where $$\nu_{t}$$ is the kinematic eddy viscosity, $$\nu$$ the molecular viscosity, $$U_{i,j}$$ the velocity gradients, $$\kappa$$ the Kármán constant and $$d$$ the distance to the wall.

The quantity $$r_{d}$$ is used in the shielding function $$f_{d}$$:

f_d \equiv 1 – {\rm tanh} \left[ \left( 8r_d \right)^3 \right] \tag{2} \label{eq:fd}

which is designed to be 1 in the LES region, where $$r_{d}\ \ll 1$$, and 0 elsewhere.

\tilde{d} \equiv d \; – f_d {\rm max}(0, d \; – \; C_{DES}\Delta) \tag{3} \label{eq:dTilda}

 References

[1] 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.

## Author: fumiya

CFD engineer in Japan

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