## Filter Width – maxDeltaxyz

Several options are available in OpenFOAM for calculation of the filter width used in the large eddy simulation (LES) and detached eddy simulation (DES). In this blog post, the maxDeltaxyz option is covered in some detail.

OpenFOAM Version: OpenFOAM-dev, OpenFOAM v1612+

 Implementation in OpenFOAM

The maxDeltaxyz option calculates the filter width of the $$i-$$th cell $$\Delta_i$$ by taking the maximum distance between the cell center $$P_i$$ and each face center $$F_j$$
\begin{equation}
\Delta_i = {\rm deltaCoeff} \times \max_{1 \le j \le n_i} \left\{ \overline{P_iF_j} \right\}, \tag{1} \label{eq:deltaxyz}
\end{equation}
where $${\rm deltaCoeff}$$ is a constant of proportion (user input) and $$n_i$$ is the number of the faces of the $$i-$$th cell.

For a regular hexahedral cell shown in Fig. 1, the computed $$\Delta$$ using Eq. \eqref{eq:deltaxyz} equals one-half of the maximum cell width $$\Delta x$$, so the deltaCoeff coefficient should be set to 2 in the turbulenceProperties file as shown below.

The filter width is calculated in the following function.

 Implementation in OpenFOAM v1612+

In OpenFOAM v1612+, the face normal vectors are considered in order to take into account the mesh non-orthogonality.

This definition \eqref{eq:deltaxyz} is often used in the detached eddy simulation (DES) where some anisotropic grid cells such as “book”, “pencil” and “ribbon”-shaped cells exist in a boundary layer mesh. The position where switching between the RANS and LES modes occurs in the DES97 model  depends on how to calculate the filter width $$\Delta$$
\begin{equation}
\tilde{d} \equiv {\rm min}\left( d, C_{DES}\Delta \right), \tag{2} \label{eq:dTilda}
\end{equation}
where $$d$$ is the distance to the closest wall and $$C_{DES}$$ is a calibration constant.
Several modified length scales have been developed to prevent a delay of the Kelvin-Helmholtz instability in free and separated shear layers .

 References

 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.
 P. R. Spalart, Detached-Eddy Simulation. Annu. Rev. Fluid Mech. 41, 181-202, 2009.

## Spalart-Allmaras DES model in OpenFOAM

Last Updated: June 23, 2019

DES (Detached Eddy Simulation) turbulence model is pioneered in 1997 by Spalart et al. based on the Spalart-Allmaras RANS model . 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:

##### Motivation and Aim of DES

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 .

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 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.

OpenFOAM versions

* OpenFOAM v4
* OpenFOAM v1606+
* OpenFOAM v1612+
* OpenFOAM v1706
* OpenFOAM v1712
* OpenFOAM v1806
* OpenFOAM v1812

##### Governing Equation in v4

Source Code: src/TurbulenceModels/turbulenceModels/
LES/SpalartAllmarasDES/SpalartAllmarasDES.C

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 $$\tilde{d}$$ 
\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.

##### Governing Equation in v1606+ – v1812

Source Code: src/TurbulenceModels/turbulenceModels/
DES/SpalartAllmarasDES/SpalartAllmarasDES.C

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{3} \label{eq:sa1606} \\
=&\;C_{b1} \rho \tilde{S} \tilde{\nu} \left( 1 – \color{#dd4b39}{f_{t2}} \right)\;-\;\rho \left( C_{w1} f_{w} – \frac{C_{b1}}{\kappa^2} \color{#dd4b39}{f_{t2}} \right) \left( \frac{\tilde{\nu}}{\tilde{d}} \right)^2
\end{align}

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 . 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  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.

\begin{equation}
\tilde{d} = {\rm min} \left( d, \Psi C_{DES}\Delta \right) \tag{4} \label{eq:dTildaPsi}
\end{equation}
\begin{equation}
\Psi^2 = {\rm min} \left[ 10^2, \frac{1-\frac{C_{b1}}{C_{w1}\kappa^2 f_w^*} \left( f_{t2} + \left( 1 – f_{t2} \right) f_{v2} \right)}{f_{v1} {\rm max}\left[ 10^{-10}, 1-f_{t2} \right]} \right] \tag{5} \label{eq:Psi}
\end{equation}

##### Requirement for Computational Mesh

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  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).

##### Common Definitions between the Two Versions

\begin{equation}
\chi = \frac{\tilde{\nu}}{\nu} \tag{8} \label{eq:chi}
\end{equation}

\begin{equation}
f_{v1} = \frac{\chi^3}{\chi^3 + C_{v1}^3} \tag{9} \label{eq:fv1}
\end{equation}

\begin{equation}
f_{v2} = 1\;-\;\frac{\chi}{1+\chi f_{v1}} \tag{10} \label{eq:fv2}
\end{equation}

\begin{align}
W_{ij} &= \frac{1}{2}\left( \nabla \boldsymbol{u} – (\nabla \boldsymbol{u})^T \right) = \frac{1}{2}\left( \frac{\partial u_j}{\partial x_i} – \frac{\partial u_i}{\partial x_j} \right) \tag{11a} \\
\Omega &= \sqrt{2 W_{ij} \; W_{ij}} \tag{11b} \label{eq:Omega}
\end{align}

\begin{equation}
\tilde{S} = {\rm max} \left[ \Omega + \frac{\tilde{\nu}}{\kappa^2 \tilde{d}^2} f_{v2}, C_{s} \Omega \right] \tag{12} \label{eq:Stilda}
\end{equation}

\begin{equation}
r = {\rm min} \left[ \frac{\tilde{\nu}}{\Omega \kappa^2 \tilde{d}^2}, 10 \right] \tag{13} \label{eq:r}
\end{equation}

\begin{align}
g &= r + C_{w2} \left( r^6 – r \right) \tag{14a} \\
f_{w} &= g \left[ \frac{1 + C_{w3}^6}{g^6 + C_{w3}^6} \right] ^{\frac{1}{6}} \tag{14b} \label{eq:fw}
\end{align}

##### Different Definitions between the Two Versions

\begin{equation}
f_{t2} = C_{t3} e^{-C_{t4} \chi^2} \tag{15} \label{eq:ft2}
\end{equation}

##### References

 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.
 K. D. Squires, DETACHED-EDDY SIMULATION: CURRENT STATUS AND PERSPECTIVES.
 P. R. Spalart and S. R. Allmaras, A one-equation turbulence model for aerodynamic flows. La Rech. Aerospatiale 1, 5-21, 1994.
 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.
 Turbulence Modeling Resource. Available at: https://turbmodels.larc.nasa.gov/index.html [Accessed: 10 October 2016].
 P. R. Spalart and C. L. Rumsey, Effective inflow conditions for turbulence models in aerodynamic calculations. AIAA Journal 45(10), 2544-2553, 2007.
 P. R. Spalart et al., THE USES OF DES: NATURAL, EXTENDED, AND IMPROPER. DESider, July 14-15 2005, Stockholm.