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 |
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template<class BasicTurbulenceModel> tmp<volScalarField> SpalartAllmarasDDES<BasicTurbulenceModel>::rd ( const volScalarField& magGradU ) const { tmp<volScalarField> tr ( min ( this->nuEff() /( max ( magGradU, dimensionedScalar("SMALL", magGradU.dimensions(), SMALL) ) *sqr(this->kappa_*this->y_) ), scalar(10) ) ); tr.ref().boundaryFieldRef() == 0.0; return tr; } |
\begin{equation}
r_d \equiv \frac{\nu_t + \nu}{\sqrt{U_{i,j}U_{i,j}}\kappa^2 d^2} \tag{1} \label{eq:rd}
\end{equation}
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 \(f_{d}\):
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template<class BasicTurbulenceModel> tmp<volScalarField> SpalartAllmarasDDES<BasicTurbulenceModel>::fd ( const volScalarField& magGradU ) const { return 1 - tanh(pow3(8*rd(magGradU))); } |
\begin{equation}
f_d \equiv 1 – {\rm tanh} \left[ \left( 8r_d \right)^3 \right] \tag{2} \label{eq:fd}
\end{equation}
which is designed to be 1 in the LES region, where \(r_{d}\ \ll 1\), and 0 elsewhere.
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template<class BasicTurbulenceModel> tmp<volScalarField> SpalartAllmarasDDES<BasicTurbulenceModel>::dTilda ( const volScalarField& chi, const volScalarField& fv1, const volTensorField& gradU ) const { return max ( this->y_ - fd(mag(gradU)) *max ( this->y_ - this->CDES_*this->delta(), dimensionedScalar("zero", dimLength, 0) ), dimensionedScalar("small", dimLength, SMALL) ); } |
\begin{equation}
\tilde{d} \equiv d \; – f_d {\rm max}(0, d \; – \; C_{DES}\Delta) \tag{3} \label{eq:dTilda}
\end{equation}
| 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.