WMLES

Spalart et al. (1997) proposed a hybrid RANS/LES methodology referred to as Detached Eddy Simulation (DES), in which a RANS closure is invoked near solid surfaces where the flow is attached, and a grid resolution dependent SGS closure is invoked for separated (detached) flow regions. The original DES concept was built around the Spalart Allmaras one-equation turbulence model, where the transition to an SGS closure was accomplished by altering the length scale that appears in the turbulent transport equation of \(\tilde{\nu}\). The base RANS model uses the wall distance, \(d\), as the turbulence length scale. The DES model replaces this length scale with the minimum of \(d\) and the DES filter width defined as

\Delta_{DES} \ = \ C_{DES} \ \Psi \ \Delta_{max} \ , \ \hspace{0.5in} \Delta_{max} \ = \ \text{max} \left[ \Delta_x \ , \ \Delta_y \ , \ \Delta_z \right]

where \(C_{DES}\) is an empirical constant, and \(\Psi\) is a low Reynolds number correction defined as

\Psi^2 = \ \text{min} \left[ 10^2 \ , \ \frac{1 \ – \ \frac{c_{b_1}}{c_{w_1} \kappa^2 f_w^*} \left[ f_{t_2} \ + \ f_{v_2} \left( 1 – f_{t_2} \right) \right]} {f_{v_1} \ \text{max} \left[ 10^{-10} \ , \ 1 – f_{t_2} \right]} \right]

The quantity \(f_w^*\) represents the equilibrium value of the function \(f_w\) present in the Spalart Allmaras closure, and takes on the value of 0.424. The remaining parameters (\(c_{b_1}\), \(c_{w_1}\), \(f_{t_2}\), \(f_{v_1}\), \(f_{v_2}\)) are as defined in the Spalart Allmaras one-equation model. The DES approach was later extended to a two-equation (Menter \(k\)–\(\omega\)) formulation by Strelets (2001). The two-equation DES formulation replaces the RANS turbulence length scale that appears in the destruction term of the turbulent kinetic energy equation, i.e.,

C_{k_d} \ \bar{\rho} \ \tilde{k} \ \tilde{\omega} \ \equiv \ C_{k_d} \ \bar{\rho} \ \frac{\tilde{k}^\frac{3}{2}}{l_{RANS}} \ , \hspace{0.25in} l_{RANS} \ \equiv \ \frac{\sqrt{\tilde{k}}}{C_{k_d} \ \tilde{\omega}}

with the DES length scale,

\begin{equation}
\hat{l} \ = \ \text{min} \left[l_{RANS} \ , \ \Delta_{DES} \right]
\label{DES_len}
\tag{1}
\end{equation}

and the quantity \(\Psi\) is taken to be 1.0 (since the Menter \(k\)–\(\omega\) models have no low Reynolds number corrections). In principle, this modification can be made to convert any two-equation RANS model to a DES formulation.

A major drawback to the original DES approach that was highlighted from the formulation’s inception appears when the grid is locally refined in multiple directions in regions not intended to be scale-resolved. This scenario often arises where grid refinement is required in some region of high geometric curvature, or near junctures where multiple solid surfaces coalesce. Given that the DES transition criteria (Eq. \ref{DES_len}) directly compares the RANS length scale to the maximum grid cell length, the eddy viscosity may be substantially reduced in the boundary layer with no mechanism to transfer the modeled turbulence energy into resolved energy. In this scenario, the flow effectively relaminarizes, corrupting the properties of the turbulent boundary layer. As a result, the early successes of the original DES formulation typically involved flow applications with relatively simple geometries where this set of circumstances can often be avoided. This can represent a severe limitation, however, for complex flow applications where local refinements are often unavoidable.

Several modifications to the basic DES concept have been proposed by the research community to address the shortcomings of the original formulation (including the major one described above). The most recent incarnation of the DES formulation (Shur et al. 2008) is termed Improved Delayed Detached Eddy Simulation (IDDES).  The IDDES formulation incorporates two major enhancements to the original DES method. The first enhancement led to the DDES (Delayed Detached Eddy Simulation) formulation (Spalart et al. 2006) which addresses the issue described above by preventing the switch from RANS to LES mode within attached boundary layers due solely to the specifics of the grid design. This desirement was realized by altering the DES length scale to incorporate a flow dependent shielding function that maintains the RANS length scale in attached boundary layer flow regions:

\begin{equation}
\hat{l}_{DDES} \ = \ l_{RANS} \ – \ f_d \ \text{max} \left[ 0 \ , \ l_{RANS} – \Delta_{DES} \right]
\label{DDES_len}
\tag{2}
\end{equation}

where the shielding function, \(f_d\), is defined as

\begin{equation}
f_d \ = \ 1 \ – \ \tanh \left[ \left( C_{f_d} \ r_d \right)^3 \right] \ ,
\ \hspace{0.5in}
r_d \ = \ \frac{\mu \ + \ \mu_t}{\bar{\rho} \ \left( \kappa d \right)^2 \ \text{max} \left[ \sqrt{\frac{\partial{u_i}}{\partial{x_j}} \frac{\partial{u_i}}{\partial{x_j}}} \ , \ 10^{-10} \right]}
\end{equation}

and \(C_{f_d} = 8\). The argument \(r_d\) is close to unity in the sub-layer and logarithmic portions of the boundary layer, and asymptotes to zero as the edge of the boundary layer is approached. As a result, \(f_d\) is essentially zero until the defect layer is encountered and rapidly approaches unity when \(r_d \ll 1\). The second enhancement was the introduction of a wall-modeled LES functionality, Shur et al. (2008). In general, the IDDES formulation provides a wall-modeled LES response if resolved turbulent content is supplied as an inflow (or initial) condition, and resorts to a DDES response (as described above) otherwise.

The wall-modeled LES branch is intended to be active only when the inflow conditions include resolved turbulent content, and the grid is fine enough to at least resolve the largest energy containing boundary layer eddies. This branch uses the following length scale definition:

\begin{equation}
\hat{l}_{WMLES} \ = \ f_\beta \ \left( 1 + f_e \right) \ l_{RANS} \ + \ \left( 1 – f_\beta \right) \ \Delta_{DES}
\label{WMLES_len}
\tag{3}
\end{equation}

where the empirical blending function, \(f_\beta\), is formulated as:

\begin{equation}
f_\beta \ = \ \text{min} \left[ 2 \exp \left( -9 \alpha^2 \right) \ , \ 1 \right] \ , \ \hspace{0.5in}
\alpha \ = \ 0.25 \ – \ \frac{d}{\Delta_{max}}
\end{equation}

This blending function varies from 0 (LES mode) to 1 (RANS mode) and provides a rapid switch between these extremes when the wall distance is between 0.5 \(\Delta_{max}\) and \(\Delta_{max}\). The second empirical function, \(f_e\), is meant to prevent an excessive reduction of the RANS Reynolds stresses in the vicinity of the RANS/LES interface. This function was designed to address the log-layer mismatch that plagues hybrid RANS/LES models, and takes on the following form:

\begin{equation}
f_e \ = \ \text{max} \left[ \left( f_{e_1} – 1 \right) \ , \ 0 \right] \ \Psi \ f_{e_2}
\end{equation}

where

\begin{equation}
f_{e_1} \ = \ \left\{ \begin{array}{ll} 2 \ \exp \left( -11.09 \ \alpha^2 \right) & \alpha \ge 0 \\ 2 \ \exp \left( -9.0 \ \alpha^2 \right) & \alpha < 0 \end{array} \right.
\end{equation}

\begin{equation}
f_{e_2} \ = \ 1 \ – \ \text{max} \left[ \tanh \left\{ \left( c_t^2 \ r_{d_t} \right)^{3} \right\}, \ \tanh \left\{ \left( c_l^2 \ r_{d_l} \right)^{10} \right\} \right]
\end{equation}

The quantity \(\Psi\) is an optional low Reynolds number correction, and \(r_{d_t}\) and \(r_{d_l}\) are the turbulent and laminar portions of the \(r_d\) parameter defined previously. The constants \(c_t\) and \(c_l\) are meant to ensure that the function \(f_{e_2}\) is virtually zero when either \(r_{d_t}\) or \(r_{d_l}\) is close to 1.0. The appropriate value for these constants are dependent on the specific RANS model utilized. Based on fully developed channel flow simulations, Shur et al. (2008) determined that \(c_t\)=1.63 and \(c_l\)=3.55 worked well with the Spalart Allmaras model, while \(c_t\)=1.87 and \(c_l\)=5.0 worked well with the Menter SST model. The quantity \(\Psi\) is the same low Reynolds number correction introduced in the definition of \(\Delta_{DES}\) when the Spalart model is used as the parent RANS model, and is set to unity for RANS models that do not invoke low Reynolds number corrections.

The DDES (Eq. \ref{DDES_len}) and WMLES (Eq. \ref{WMLES_len}) length scales as written above are not easily blended in a manner that ensures the desired branch is realized depending on whether resolved turbulent content is present within the boundary layer. However, this desirement is possible with a modified DDES length scale expression of the form:

\hat{l}_{DDES} \ = \ \tilde{f}_d \ l_{RANS} \ + \ \left( 1 – \tilde{f}_d\right) \ \Delta_{DES}

where

\tilde{f}_d \ = \ \text{max} \left[ 1 – f_{d_t} \ , \ f_\beta \right] \ , \ \hspace{0.5in} f_{d_t} \ = \ 1 \ – \ \tanh \left[ \left( C_{f_d} \ r_{d_t} \right)^3 \right]

This essentially equivalent functional replacement for the DDES length scale allows the blending of the DDES and WMLES length scale definitions to be realized via the following expression:

\begin{equation}
\hat{l} \ = \ \tilde{f}_d \ \left( 1 + f_e \right) \ l_{RANS} \ + \ \left( 1 – \tilde{f}_d \right) \ \Delta_{DES}
\label{IDDES_len}
\tag{4}
\end{equation}

The hybrid length scale given by Eq. \ref{DES_len} provides the desired WMLES behavior for simulations that contain resolved turbulent content within the boundary layer (since \(r_{d_t} \ll 1 \Rightarrow f_{d_t} \approx 1\) so that \(\tilde{f}_d = f_\beta\)). Conversely, a simulation without resolved turbulence within the boundary layer leads to \(f_e \approx 0\) and \(f_{d_t} \approx 0 \Rightarrow \tilde{f}_d = 1\).

The DES filter width was also modified in the IDDES formulation to explicitly include the wall distance (\(d\)),

\Delta_{DES} \ = \ \text{min} \left\{ \ \text{max} \left[ C_w \ d \ , \ C_w \ \Delta_{max} \ , \Delta_n \right] \ , \ \Delta_{max} \ \right\}

The quantity \(\Delta_n\) is the grid spacing in the wall normal direction, which is somewhat ambiguous for complex geometries with multiple walls. One method to evaluate this quantity involves averaging the cell center values of the wall distance to grid nodes, allowing an effective “wall-normal” grid spacing to be computed as the difference between the maximum and minimum nodal values for each grid cell. The value of \(C_w\) was arrived at through simulations of fully developed channel flow and taken to be 0.15.

References

1. Spalart, P. R., et al. Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach. 1st AFOSR International Conference on DNS/LES (invited).
2. Strelets, M. “Detached Eddy Simulation of Massively Separated Flows.” 39th Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics, 2001. Crossref, doi:10.2514/6.2001-879.
3. Shur, Mikhail L., et al. “A Hybrid RANS-LES Approach with Delayed-DES and Wall-Modelled LES Capabilities.” International Journal of Heat and Fluid Flow, Elsevier BV, Dec. 2008, pp. 1638–49. Crossref, doi:10.1016/j.ijheatfluidflow.2008.07.001.
4. Spalart, P. R., et al. “A New Version of Detached-Eddy Simulation, Resistant to Ambiguous Grid Densities.” Theoretical and Computational Fluid Dynamics, Springer Science and Business Media LLC, May 2006, pp. 181–95. Crossref, doi:10.1007/s00162-006-0015-0.