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.
\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 \(f_{d}\):
\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.
\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.
DES (Detached Eddy Simulation) turbulence model is pioneered in 1997 by Spalart et al. based on the Spalart-Allmaras RANS model [1]. It is referred to as DES97 model in the CFD (Computational Fluid Dynamics) community and several improved versions (DDES and IDDES) have been developed so far that have continually widened the areas of application. They have been implemented in both commercial and open-source CFD softwares and used on a regular basis in both industry and academia.
In this blog post, I will try to describe in detail DES97 model that serves as a base for the subsequent models:
The motivation and aim of DES models is to decrease the computational cost of massively separated turbulent flows compared to LES (Large Eddy Simulation). The reduction in the cost is achieved by modeling the boundary layers using unsteady RANS (Reynolds-Averaged Navier-Stokes) model [2].
Detached-Eddy Simulation (DES) is a hybrid technique first proposed by Spalart et al. (1997) for prediction of turbulent flows at high Reynolds numbers (see also Spalart 2000). Development of the technique was motivated by estimates which indicate that the computational costs of applying Large-Eddy Simulation (LES) to complete configurations such as an airplane, submarine, or road vehicle are prohibitive. The high cost of LES when applied to complete configurations at high Reynolds numbers arises because of the resolution required in the boundary layers, an issue that remains even with fully successful wall-layer modeling.
In Detached-Eddy Simulation (DES), the aim is to combine the most favorable aspects of the two techniques, i.e., application of RANS models for predicting the attached boundary layers and LES for resolution of time-dependent, three-dimensional large eddies. The cost scaling of the method is then favorable since LES is not applied to resolution of the relatively smaller-structures that populate the boundary layer.
Fig. 1 Sketch of a Detached-Eddy Simulation [7]The DES97 model accomplishes the switching between RANS and LES modes by changing the length scale as will be described later in this post.
The model is available in OpenFOAM. I will detail its implementations in various OpenFOAM versions.
Mathematical Expression
\begin{align}
&\;\frac{\partial (\rho \tilde{\nu})}{\partial t} + \frac{\partial (\rho u_j \tilde{\nu})}{\partial x_j} \\
-&\;\frac{1}{\sigma} \left[ \frac{\partial}{\partial x_j} \left( \rho \left( \tilde{\nu} + \nu \right) \frac{\partial \tilde{\nu}}{\partial x_j} \right) + C_{b2} \rho \frac{\partial \tilde{\nu}}{\partial x_i} \frac{\partial \tilde{\nu}}{\partial x_i} \right] \tag{1} \label{eq:sa4} \\
=&\;C_{b1} \rho \tilde{S} \tilde{\nu}\;-\;C_{w1} \rho f_{w} \left( \frac{\tilde{\nu}}{\tilde{d}} \right)^2
\end{align}
In the case of incompressible fluid flow, the density \(\rho = 1\) in Eq. \eqref{eq:sa4}. The third term in the l.h.s. of Eq. \eqref{eq:sa4} is the diffusion term. In the r.h.s, the terms starting from the left represent the production and destruction terms for the eddy viscosity \(\tilde{\nu}\). The destruction term is changed from the Spalart-Allmaras RANS model by replacing the distance to the wall \(d\) with a new length scale \(\tilde{d}\) [2]
\begin{equation}
\tilde{d} = {\rm min} \left( l_{RANS}, l_{LES} \right) = {\rm min} \left( d, C_{DES}\Delta \right) \tag{2} \label{eq:dTilda}
\end{equation}
where \(\Delta\) is the chosen measure of grid spacing and \(C_{DES}\) is a calibration constant and its default value is 0.65.
When balanced with the production term, this term adjusts the eddy viscosity to scale with the local deformation rate \(S\) and \(d\): \(\tilde{\nu} \propto Sd^2\). Subgrid-scale (SGS) eddy viscosities scale with \(S\) and the grid spacing \(\Delta\), i.e., \(\nu_{SGS} \propto S\Delta^2\). A subgrid-scale model within the S-A formulation can then be obtained by replacing \(d\) with a length scale \(\Delta\) directly proportional to the grid spacing.
The \(f_{t2}\) laminar suppression term is included in com versions (v1606+ – v1812) as against v4. The subscript \(t\) stands for “trip”. The production term is multiplied by \(1-f_{t2}\) so that \(\tilde{\nu} = 0\) is a stable solution when solving a transitional flow where both laminar and turbulent regions exist in one computational domain. The destruction term is also altered in accordance with the change in the production term in order to balance the energy budget near the wall [3]. For fully turbulent flows at high Reynolds numbers, the \(f_{t2}\) term takes small values and makes essentially no difference between the above two equations.
In addition to this difference, the low-Reynolds number correction option [4] is available only in com versions (v1606+ – v1812):
If the low-Re number correction option is deactivated (lowReCorrection false;) and the value of \(C_{t3}\) is set to 0 in v1606+ – v1812, the same calculation can be done as v4.
Low-Reynolds Number Term Correction \(\Psi\) Function
The low-Reynolds number terms in the S-A RANS model were designed to account for wall proximity, “measured” by the ratio of the eddy and molecular viscosities \(\nu_t/\nu\) (an expedient to avoid using the friction velocity). Most RANS models have similar terms, also based on various turbulent Reynolds numbers. However, in the LES mode of DES, the subgrid eddy viscosity (normalized by the molecular viscosity) decreases with grid refinement and decrease of the flow Reynolds number. At some point, standard DES will mis-interpret the situation as “wall proximity” and lower the eddy viscosity excessively, relative to the ambient velocity and length scales, through the \(f_v\) and \(f_t\) functions of the S-A model. Although this deficiency shows up only at relatively low Reynolds numbers and/or with very fine grids, it is still desirable to eliminate it, and to do so without zonal instructions from the user.
The DES97 model is based on the assumption that the wall-parallel grid spacing near the wall \(\Delta_{||}\) is in excess of the boundary layer thickness \(\delta\) by a good margin so that the entire boundary layer is handled by RANS model. As we can see from Eq. \eqref{eq:dTilda}, the new length scale of DES97 model \(\tilde{d}\) depends only on the geometric features of computational mesh and not on the computed flow fields at all.
If we choose the following measure of grid spacing that is maxDeltaxyz option in OpenFOAM
\begin{equation}
\Delta \equiv {\rm max} \left( \Delta x, \Delta y, \Delta z \right), \tag{6} \label{eq:delta}
\end{equation}
the wall-parallel grid spacing \(\Delta_{||}\) should meet the following inequality to satisfy the above mesh requirement
\begin{equation}
\Delta_{||} \approx \Delta \geq \frac{d}{C_{DES}} \geq \frac{\delta}{C_{DES}}, \tag{7} \label{eq:deltaInequality}
\end{equation}
where we assume that \(y\) is the wall normal direction and \(\Delta_{||} \approx \Delta_x \approx \Delta_z\).
There are several problems that DES97 model suffers from.
Grey area
Modeled-Stress Depletion (MSD)
Delayed DES and Zonal DES are the solutions to MSD.
Grid-induced Separation (GIS)
MSD is the root cause of GIS.
Laminar Suppression Term ft2
In the excellent Turbulence Modeling Resource page [5] of NASA Langley Research Center, the various deformations of the Spalart-Allmaras turbulence (RANS) model are clearly documented.
According to the naming method, the “Standard” Spalart-Allmaras One-Equation Model (SA) is implemented in OpenFOAM v1606+ – v1812 and the Spalart-Allmaras One-Equation Model without \(f_{t2}\) Term (SA-noft2) is implemented in v4.
Spalart-Allmaras One-Equation Model without \(f_{t2}\) Term (SA-noft2) Many implementations of Spalart-Allmaras ignore the \(f_{t2}\) term, which was a numerical fix in the original model in order to make zero a stable solution to the equation with a small basin of attraction (thus slightly delaying transition so that the trip term could be activated appropriately). It is argued that if the trip is not used, then \(f_{t2}\) is not necessary. The equations are the same as for the “standard” version (SA), except that the term \(f_{t2}\) does not appear at all (i.e., ct3=0 instead of 1.2).
