Direct numerical simulation of breakdown to turbulence in a Mach 6 boundary layer over a porous surface

Abstract

Transition to turbulence of a Mach 6 flat plate boundary layer over a porous surface is investigated by direct numerical simulation considering two Reynolds numbers based on the laminar boundary layer displacement thickness, namely, Reδ∗=6000 and Reδ∗=20 000. The transition was initiated by perturbing the laminar boundary layer with small random disturbances and was followed all the way to the turbulent state. The porous geometry was modeled by directly resolving the flow within the pores and the damping of the primary Mack mode of instability was verified. The presence of a porous surface was found to reduce the secondary instability growth rate by reducing the amplitude of the second mode saturation. In particular, the pores suppress the growth of the secondary wave in the near wall region, so that the secondary instability mainly happens near the critical layer. Besides the secondary instabilities Fourier analysis shows additional modes growing at the same rate as the primary instability, consistent with a model for sound waves scattering from the porous surface. The transient growth of u′, ρ′, and T′ fluctuations, in the form of streamwise streaks, appears to favor the fundamental type of secondary instability. Additional calculations revealed that an oblique first mode wave is the most amplified mode in this porous surface configuration. This wave is slightly destabilized by the pores. With the oblique first mode excited, the flow becomes turbulent due to the nonlinear interactions without the need for secondary instabilities.

Numerical Investigation of transition delay using porous walls

“…The LST analysis showed that UACs had a stabilizing effect over a wide range of frequencies and Reynolds numbers. Additional numerical analyses confirming the stabilizing effect were carried out by Brès et al, [8][9][10] Fedorov et al, 11 Egorov et al, 12 Sandham et al, 13 Fedorov 14 and de Tullio et al 15 Some experimental results for the porous coating were obtained by Rasheed et al, 16 Chokani et al, 17 Fedorov et al 18 The numerical and experimental studies were in reasonable agreement as shown in Fedorov et al 19 and Fedorov. 20 The stabilizing effect of porous walls on a Mach 6.0 boundary layer was also investigated by Hader and Fasel 21 and Hader et al 22 For the results presented in Hader and Fasel 21 a low-order immersed boundary method was employed.…”

“…As a reference the laminar skin friction coefficient and an estimate for the turbulent skin friction coefficient according to White 34 are included in figure 12(a). In addition, the skin friction evolution presented in de Tullio et al 15 was digitized and is plotted for comparison. Initially the averaged skin friction values are nearly identical for all different cases and fairly close to the estimate for the laminar value of c f .…”

Compressible turbulent channel flow with impedance boundary conditions

“…Several trials have shown that this metric is very sensitive to the integration strategy adopted when carrying out the complex indefinite integral (18). This is due to the nature of the integrand in (18) causing the accumulation of round-off error to become more severe (i.e. approaching the value of the integration result itself) as τ departs from unity, that is, for values of y departing from 0, at t = 2.…”

“…The RMS of the difference between the two solutions, evaluated over the interval −0.4 < y < 0.4 and for t = 2, decays initially with forth-order accuracy in space and second-order in time (figure 3), as expected. Several trials have shown that this metric is very sensitive to the integration strategy adopted when carrying out the complex indefinite integral (18). This is due to the nature of the integrand in (18) causing the accumulation of round-off error to become more severe (i.e.…”

“…Bres et al 17 investigated the stability properties of a two-dimensional hypersonic boundary layer over an idealized porous wall and derived a simple impedance model for companion linear stability calculations. Several high-fidelity numerical simulations have been performed 18,19 looking at the interaction between a simplified porous wall geometry and supersonic boundary layers. Similar problems have been investigated in the incompressible limit with focus on characterizing the structural alterations to wall-bounded turbulence.…”