Linear stability analysis of the wake of an isolated roughness

Jean-Philippe Brazier

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.

Abstract:

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

where

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.

Figure 1: Local wave numbers and growth rates of different eigenmodes in a cross-stream section: vortex symmetric (light blue) and antisymmetric (orange) modes, Mack mode (violet)

Figure 2: N factor values at different frequencies for the most amplified vortex symmetric mode downstream of the roughness, compared with the Mack mode on a smooth plate (dashed line)

References :

[1] 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

[2] Lefieux, DNS study of roughness-induced transition in hypersonic flows, doctoral dissertation, Paris-Saclay University, February 2021

[3] 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