Vortex breakdown above a delta wing with sharp leading edge

Application Challenge AC1-09 © copyright ERCOFTAC 2024

Solution Strategy

Detached Eddy Simulations (DES) have been performed by a number of partners in the EU-project ATAAC. DES (Spalart, 2009) is a hybrid RANS--LES approach that was originally based on the Spalart--Allmaras (SA) model, but that has also been extended to $k$--$\omega$ models. In DES switching between LES and RANS is effectively achieved by defining the turbulence length scale employed in the turbulence model as the minimum of the LES and RANS length scales. The LES length scale is the filter width, which is defined as the maximum of the mesh size in all three computational directions at each grid point. The RANS length scale depends on the RANS model employed: essentially the wall distance for the SA model and $\sqrt{k}/\omega$ for the $k$--$\omega$ model. Note that for the SA model the RANS length scale is static and therefore the RANS--LES interface is fixed, whereas for the $k$--$\omega$ model the RANS--LES interface is dynamic.

Computations have been performed with different DES-type methods as listed in Table \ref{ref-models}. For the underlying RANS model, the SA model (SA-DES; Spalart \emph{et al.}, 1997), the SST $k$--$\omega$ model (SST-DES; Travin \emph{et al.}, 2002), and the TNT $k$--$\omega$ model (X-LES; Kok \emph{et al.} 2004) have been employed. The main difference between SST-DES and X-LES is that in SST-DES the blended turbulent length scale is only used to define the dissipation term in the $k$-equation, whereas in X-LES it is used to define the dissipation term as well as the eddy-viscosity coefficient. The delayed approach of Spalart \emph{et al.} (2006), shielding attached boundary layers against inadvertently switching to LES (so-called shear-stress depletion), is used in all DES and X-LES computations (denoted as DDES and DX-LES) but one. Finally, CFSE has employed the improved variant IDDES (Shur \emph{et al.}, 2008). Table \ref{ref-models} also lists the type of solvers and grids.

One of the difficulties of DES consists of the development of resolved turbulence as the method switches from RANS to LES. In particular, the development of instabilities in free shear layers, such as those separating from the leading edges in the present case, may be significantly delayed. Two approaches that improve the prediction of free shear layers have been considered for X-LES: a stochastic SGS model (Kok and Van der Ven, 2009) and a high-pass-filtered (HPF) SGS model (Kok and Van der Ven, 2012). For the present case, the effect of the HPF model was significant, but the effect of the stochastic model was limited when used in combination with the HPF model. Therefore, results are mainly presented with only the HPF model. However, some of the sensitivity studies were performed with the stochastic model. Note that both approaches can be combined with other DES variants as well.

Some numerical details of the different computations are summarized in Table \ref{ref-methods}. It gives the spatial and temporal order of accuracy of the different numerical methods as well as the chosen time-step size and the time sample used to compute statistical data. The latter two are expressed in convective time units (CTU), i.e., they are made dimensionless with the free-stream velocity and the mean aerodynamic chord. NLR used a fourth-order symmetry-preserving low-dispersion finite-volume scheme with sixth-order artificial dissipation (Kok, 2009). DLR, CASS, and NUM used a standard second-order JST-type finite-volume scheme with fourth-order artificial dissipation (Jameson \emph{et al.}, 1981), but with a strongly reduced level of artificial dissipation. CFSE used a fourth-order central finite-volume scheme with JST-type artificial dissipation.

For time integration, all partners used the second-order backward implicit scheme. A time step of $3.75\cdot 10^{-4}$ CTU and a mesh width of $0.003 c_{\text{mac}}$ (see grid characteristics below) imply a convective CFL number of approximately $1/8$ based on the free-stream velocity. The experiment reports a dominant frequency of $St = f c_\text{mac}/u_{\infty} = 2$ implying 1,333 time steps per period. Schiavetta \emph{et al.} (2007) also report higher relevant frequencies, e.g., for shear layer instabilities $St = 8$ -- $10$. These frequencies are also well resolved by the time step. Note that a time sample of 10 CTU implies that twenty periods of the dominant frequency have been captured.

Computational Domain

The geometries of the wing and the sting are given analytically by Chu and Luckring (1996). In the computational domain, the sting is simplified: the shape of the sting upstream of the wing trailing edge conforms to the analytic definition, but it is extended as a straight cylinder from the trailing edge of the wing down to the far-field boundary (whereas the analytical definition of the sting diameter increases at some distance from the trailing edge).

NLR has generated a common multi-block structured grid, consisting of 22 blocks and 6.3 million grid cells. The grid has a conical structure over a large part of the wing: the grid covering the main vortex is essentially isotropic at each chord-wise station (outside the boundary layer) and the mesh width grows in all directions (including the stream-wise direction) together with the main vortex, going from approximately $0.003 c_{\text{mac}}$ to $0.011 c_{\text{mac}}$. In other words, the grid resolution relative to the main vortex is kept constant. Only in a small region near the apex, the conical structure is not fully maintained, avoiding a grid singularity. The first grid cell at the wing surface has a height of $y^+ = O(1)$. The far-field boundary is located at three root chord lengths from the wing in all directions, i.e., given that the apex is at the origin, the extent of the computational domain is given by $-3 c_r \leq x \leq 4 c_r$, $-3.5 c_r \leq y \leq 3.5 c_r$, and $-3 c_r \leq z \leq 3 c_r$. To study grid sensitivity, also a finer grid with the mesh width reduced by a factor 2/3 in all directions (21.4 million grid cells) as well as a modified grid with the far field located at 15 root chord lengths (8.7 million grid cells) have been generated. Additionally, CASS has generated a fine, unstructured grid (17.4 million grid cells, far field at 15 root chord lengths) that also essentially has a conical structure. An impression of the common grid is given in Figure \ref{fig-grid-struc}.