Linear stability analysis of the wake of an isolated roughness
ONERA DMPE, Université de Toulouse, F-31400 Toulouse, France Jean-Philippe.Brazier@onera.fr
Keywords: Boundary layer, transition tripping, isolated roughness wake, 2D local linear stability Submitted for an oral presentation.
The effect of an isolated roughness on the laminar-turbulent transition has been the subject of numerous numerical and experimental studies for several decades. In the present work, a local linear stability analysis is performed on the wake of an isolated roughness on a flat plate. This wake is mainly characterized by a pair of strong streamwise contra-rotating vortices, exhibiting a slow variation along the streamwise coordinate X but strong variations along both wall normal coordinate Y and crosswise coordinate Z. Therefore the standard LST-1D stability analysis can no longer be applied and an extended LST-2D eigenmode ansatz must be used [1,2,3], under the general form
Following the way paved by J. Lefieux [2,3], a modal analysis has therefore been carried out with ONERA in-house LST-2D code BIGSAM in ten cross sections of a roughness wake at Mach 5 in ONERA R2Ch wind tunnel. Computations are made on one vortex only, with a symmetry or antisymmetry condition prescribed at boundary Z=0. The roughness is located at abscissa X = 100 mm on the flat plate.
Figure 1 shows the different eigenmodes found in one of the cross-sections, 145 mm downstream of the roughness. Spectra are computed first with two different grids 41×41 and 61×61 points, to identify the relevant modes, which are then tracked by varying the frequency.
Some modes are symmetric with respect to the Z=0 plane, some others are antisymmetric. These modes are closely related to the shear layer between the vortex and the external flow. The most amplified mode is symmetric, on a large frequency band. The Mack mode is also found on a narrow frequency band.
Then the amplification coefficient, also called the N factor, can be integrated over the ten X-sections. The result is plotted in figure 2. The N-factor of the Mack mode on a smooth plate in the same conditions has also been plotted for comparison. It appears that the isolated roughness induces a strong destabilization of the boundary layer and consequently an upstream shift of the transition location. Of course this result will have to be confronted to experimental data.
 Padilla Montero and F. Pinna, Analysis of the instabilities induced by an isolated roughness element in a laminar high-speed boundary layer, Journal of Fluid Mechanics, vol. 915, n° A90, 2021
 Lefieux, DNS study of roughness-induced transition in hypersonic flows, doctoral dissertation, Paris-Saclay University, February 2021
 Lefieux, E. Garnier, J.-Ph. Brazier and N.D. Sandham, Roughness-induced instabilities and transition on a generic hypersonic forebody at Mach 6, AIAA Journal, vol. 59, n°9, pp. 3529-3545, 2021