A natural choice for the computation of unsteady flows would be Large-Eddy Simulation (LES), in which the large, energy-bearing eddies are resolved and only the small, isotropic eddies need modelling, thus obviating the necessity of a distinct spectral gap between flow unsteadiness and turbulence. Additionally, the underlying assumption of LES, i.e. that small-scale eddies possess a more universal character than large-scale ones, allows the use of relatively simple turbulence models. On the other hand, the required refinements in spatial and temporal resolution are quite large, especially in attached boundary layers, where the turbulent scales are very small compared to the geometrical scale. To overcome well-known deficiencies of eddy-viscosity models regarding the prediction of separated flow, while at the same time retaining their undisputed performance for attached boundary layers, hybrid RANS/LES approaches have been proposed, Detached Eddy Simulation (DES) being the most prominent example. In this non-zonal approach, attached boundary layers are treated in RANS mode while regions with separated flow are computed using LES.

The implementation of DES in a \( k\)- \(\omega\) context in TRACE follows Strelets [55] . The destruction term of the \(k\)-equation is altered as follows:

\begin{eqnarray} D_{k,DES} = \frac{\rho k^{\frac 3 2}}{\min \left(\ell, C_{DES} \Delta_{DES} \right)}, \text{with} \; \; \; \ell = \frac{\sqrt{k}}{\beta_k \omega} \end{eqnarray}

\(\Delta_{DES}\) sets the DES cell size and may be determined by different approaches. In section Determination of DES subgrid length scale you will find details about available implementations and a rough sketch of relevant functions.

The value of \(C_{DES}\) depends on the numerical scheme used.

To remedy the Modelled-Stress Depletion (MSD) problem, a variant incorporating a shielding function known as Delayed DES (DDES) has been suggested [54] . A further development merging DDES and Wall-Modelled LES (WMLES) capabilities called Improved DDES (IDDES) has also been proposed [48] . The DDES and IDDES implementations follow [14] (NB: the full version of IDDES, not the simplified one), except for the IDDES length scale, which is implemented in its original formulation [48] .

All three DES variants are available for both Wilcox and Menter SST \( k\)- \(\omega\) models.

When using the DES model, one should bear in mind the following points:

fine meshes with low aspect ratios should be used
shorter time steps should be used
it makes sense to initialize the DES model with a RANS flow field
common convergence tests will fail, as turbulent fluctuations appear in mean flow quantities
as compared to URANS, a larger number of periods should be used for averaging

Run transitional DES simulations

It is possible to run DES in combination with the \(\gamma\)-transition model. As a basis model, only Menter-SST 2003 (see Menter SST) is allowed. The \(\gamma\)-transition model is implemented in its version of Menter et al. [34] .

Determination of DES subgrid length scale

There exist a lot of different approaches to determine the DES subgrid length scale. Currently implemented in TRACE are the following ones:

Grid-based approach / baseline

\(\Delta_{\text{max}}\) - this is the original approach first mentioned in context of Detached-Eddy simulations [53] .
\[ \Delta_{\text{max}} = \text{max}(L_{\xi}, L_{\eta}, L_{\zeta}) \]

Vorticity-based approaches: Right here, the DES subgrid length scale determination also considers the orientation of vorticity. Therefore the vorticity vector is normalized as follows
\[ \vec{n}_{\omega} = \frac{\vec{\omega}}{\left|{\omega}\right|} \]

\(\Delta_{\omega, \text{Chauvet}}\) - determination of DES subgrid length scale based on Chauvet [7] .
\[ \Delta_{\omega, \text{Chauvet}} = \sqrt{n^2_{\omega, x} \Delta_y \Delta_z + n^2_{\omega, y} \Delta_x \Delta_z + n^2_{\omega, z} \Delta_x \Delta_y} \]
\(\Delta_{\omega, \text{Probst}}\) - determination of DES subgrid length scale based on Probst [41] .
\[ \Delta_{\omega, \text{Probst}} = \sqrt{\frac{1}{2} \sum_i{\left| \vec{n}_{\omega} \cdot \vec{s}_i \right |}} \]
\(\tilde\Delta_{\omega, \text{Mockett}}\) - determination of DES subgrid length scale based on Mockett [36] .
\[ \tilde\Delta_{\omega, \text{Mockett}} = \frac{1}{\sqrt{3}} \max\limits_{n,m = 8} \left| l_n - l_m \right | \]
with \( l_n = \vec{n}_{\omega} \times r_n \), where \( r_n, r_m \) states the respective vector of each vertex.
\(\tilde\Delta_{\text{Shur}}\) - determination of DES subgrid length scale based on Shur [50] .
\[ \tilde\Delta_{\text{Shur}} = \tilde\Delta_{\omega, \text{Mockett}} \; F_{KH}^{\text{lim}}(<VTM>) \]
with
\[ F_{KH}^{\text{lim}} = \begin{cases} 1.0 \; \text{if} \; f_d < (1 - \epsilon)\\ F_{KH}(<VTM>) \; \text{if} \; f_d \geq (1 - \epsilon) \end{cases} \]

\[ F_{KH}(<VTM>) = \text{max}\left\{F^\text{min}_{KH}, \text{min} \{F^\text{max}_{KH}, F^\text{min}_{KH} + \frac{F^\text{max}_{KH} - F^\text{min}_{KH}}{\alpha_2 - \alpha_1} (<\text{VTM}> - \alpha_1)\}\right\} \]

\[ \text{VTM} \equiv \frac{\sqrt{6} \left| (S \cdot \omega) \times \omega \right|}{\omega^2 \sqrt{3 tr (S^2) - [tr(S)]^2}} \; \text{max}\left\{1 , \frac{0.2 \cdot \nu}{\text{max} [ (\nu_t - \nu_{t, \infty}), (10^{-6} \cdot \nu_{t,\infty}) ]}\right\} \]
while \(\alpha_1 = 0.15\), \(\alpha_2 = 0.3\), \(F_{KH}^{\text{min}} = 0.1\) and \(F_{KH}^{\text{max}} = 1.0\). \(\nu_{t,\infty}\) is the free-stream turbulcence and \(\epsilon = 0.02\).

The baseline approach is set by default. If you'd like to activate a vorticity-based approach, see Turbulence model sub-options.

Comment on IDDES computations

For IDDES computations, some special treatment of the sub-grid length scale \(\Delta_{\text{SLS}}\) and shielding functions \(f_d\) and \(f_e\) needs to be recognized. Intuitively, one would replace all grid-based \(h_{\text{max}}\) (originally named \(\Delta_{\text{max}}\)) values by the respective sub-grid length scale approach which could also be vorticity-based. But, for the shielding functions, the geometrical information, based in \(h_{\text{max}}\), is required to shield the boundary layer appropriately and maintain WM-LES capability. For the determination of the modified IDDES sub-grid length scale (in Shur's paper labeled as \(\Delta\)) first a comparison of geometrical parameters is done ( \(\max \left[ C_w d_w, C_w h_{\text{max}}, h_{wn} \right]\)) and afterwards a comparison with the original grid- or vorticity-based length scale is done, resulting in the following expression:

\[ \Delta_{\text{<SLS approach>, IDDES}} = \min \left\{ \max \left[ C_w d_w, C_w h_{\text{max}}, h_{wn} \right], \Delta_{\text{SLS}} \right\} \]

where the originally determined \(\Delta_{\text{SLS}}\) will be limited and compared with the distance to the wall \(d_w\) and wall-normal cell size \(h_{wn}\). The constant \(C_w\) equals 0.15.