Direct numerical simulations are carried out on boundary-layer flow at Mach 6 over a porous surface, in which a Mack mode of instability is excited. The pores are resolved rather than modeled, allowing an evaluation to be made of the accuracy of simplified analytical models used in previous investigations based on linear stability theory. It is shown that the stabilizing effect of porosity is stronger in the simulations than in the corresponding theory for both two-and three-dimensional pores. From comparisons of spanwise grooves, streamwise slots, and square pores, it appears that the detailed surface structure is not as important as the overall porosity, and the hydraulic diameter is able to collapse the results for different pore shapes to good accuracy. When the porous surface consists of fewer larger pores, the flow is noisier, with sound waves generated at the pore edges. Nomenclatureenergy f, g = forcing terms J 0 , J 2 = Bessel functions of the first kind k v , k t = arguments of Bessel functions L x;y;z = domain dimensions M = Mach number N g = number of cells in the pore n = porosity n p = number of pores Pr = Prandtl number q = heat flux R = gas constant Re = Reynolds number based on boundary-layer displacement thickness r = pore radius T = temperature t = time u = velocity x, y, z = coordinates Y 1 = shunt admittance Z 1 = series impedance , = wave numbers = ratio of specific heats = nondimensional wall-normal coordinate m = coordinate of pore bottom = thermal conductivity = viscosity = density = stress tensor ! = complex frequency Subscripts i, j = directions (1, 2, 3) rms = root mean square w, aw = wall, adiabatic wall Superscripts = base flow quantitŷ = eigenfunction

I. IntroductionAMINAR-turbulent transition in hypersonic boundary layers remains a challenging subject. This is especially true of the hypervelocity regime, in which an intriguing phenomenon is the possible damping of second-mode disturbances by chemical and vibrational nonequilibrium processes 1,2 . To generate flows with sufficiently high enthalpy to investigate such effects, the use of shock-tunnel facilities is necessary; furthermore, it is now generally accepted that direct measurements of the instability mechanisms active within the boundary layer, together with a characterization of the free-stream disturbance environment, are required, as simple measurements of transition locations can lead to ambiguous conclusions 3,4 . However, as difficult as the accurate measurement of instability waves in conventional hypersonic facilities can be, in shock tunnels it is appreciably more so. For identical unit Reynolds numbers, the higher stagnation temperature in a shock tunnel means that the dominant second-mode disturbances lie at even higher frequencies (typically hundreds of kHz or higher); moreover, because of the destructive testing environment, hot-wire techniques, a staple for instability measurements in conventional tunnels, cannot be used. Fast-response pressure transducers are an obvious alternative, but recent experiments 5 have highlighted the challenging nature of interpreting data from mechanically-sensitive sensors in the high-noise L

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Hypersonic boundary-layer-transition is dominated by so-called Mack modes, second mode instabilities that can be damped passively by acoustic absorptive coatings. Those coatings are realised in practise by porous walls. In the present paper a second mode stability analysis is performed for a boundary layer flow at Mach 6 over a smooth wall and various porous walls. The influence of the porosity, the radius and the thickness of the pores is investigated. For this study three different codes are used: The linear stability code SLST of the University of Southampton in comparison with the DLR linear stability code NOLOT and finally direct numerical simulations including the modelling of the pores through the wall. Good agreement for a smooth wall as well as for different porous wall cases is demonstrated.