# Vortex breakdown above a delta wing with sharp leading edge

## Key Fluid Physics

The flow around a delta wing with a sharp leading edge at high angle of attack is characterized by the main vortex developing above the wing. The vortex is formed as the shear layer emanating from the leading edge rolls up, starting immediately at the apex. At high Reynolds numbers, the shear layer rapidly becomes unstable and a turbulent vortex is formed. At a sufficiently high angle of attack, the vortex breaks down: the high axial velocity in the vortex core drops rapidly to a value close to zero.

To properly capture this flow, it is essential to capture the shear layer separating from the leading edge. In particular, the instability of this shear layer must be captured. It is recommended to verify these key properties using visualization of the instantaneous flow field (e.g., using the Q criterion).

## Application Uncertainties

• Use a time sample of at least 7 convective time units to compute statistical data in order to obtain statistical convergence. (Make sure the transient is not included in the time sample.)

## Computational Domain and Boundary Conditions

• Place the far field at a minimum of 3 root chords from the wing.

## Discretisation and Grid Resolution

• Use a grid with a conical structure in all directions, thus including the stream-wise direction: the mesh width should be isotropic in the main vortex and should grow together with the vortex. In other words, a much finer grid resolution is needed near the apex than further downstream. This is necessary to capture the initial development of the main vortex and the turbulence within the vortex.
• Make sure you use a low level of numerical dissipation in the LES regions.
• Use a time step that is consistent with the grid resolution of the shear layer and main vortex. This means that the convective CFL number must be of the order of one, but its precise value will depend on the numerical method employed.

## Physical Modelling

• No choice for a particular basic DES-type method (SA-DES, SST-DES, or X-LES) can be advised based on the available results.
• The standard DES approach does not suffice, but modifications are necessary to induce a more rapid development of shear-layer instabilities and full 3D turbulence.
• For the X-LES method, use the high-pass-filtered (HPF) SGS model to capture the instabilities in the shear layer emanating from the leading edge and subsequently to resolve the turbulence in the initial vortex. It is likely that this approach would be beneficial in conjunction with other DES-like methods as well.
• No advice can be given on transition modelling; all computations have assumed a fully turbulent flow.

## Recommendations for Future Work

• Investigate the use of the high-pass-filtered (HPF) SGS model in other DES-type models besides X-LES.
• Perform computations with different DES-type methods using the same solver and numerical method.
• Investigate in detail the initial development of the vortex (both numerically and experimentally) to understand the difference in pressure distribution at the first station (${\displaystyle {\left.x/c_{r}=0.2\right.}}$) between computation and experiment.
• Perform computations and experiments at different flow conditions with vortex breakdown; in particular, at higher Reynolds number.

Contributed by: J.C. Kok, H. van der Ven (National Aerospace Laboratory NLR Amsterdam, The Netherlands), E. Tangermann (Airbus Defence and Space München, Germany), S. Sanchi (Computational Fluids and Structures Engineering Lausanne, Switzerland), A. Probst and K.A. Weinman (German Aerospace Center DLR Göttingen, Germany), L. Temmerman (NUMECA International Brussels, Belgium) — '