Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone

Abstract

Optimal disturbances for the supersonic flow past a sharp cone are computed to assess the effects due to flow divergence. This geometry is chosen because previously published studies on compressible optimal perturbations for flat plate and sphere could not isolate the influence of divergence alone, as many factors characterized the growth of disturbances on the sphere (flow divergence, pressure gradient, centrifugal forces, and dependence of the edge parameters on the local Mach number). Flow-divergence effects result in the presence of an optimal distance from the cone tip for which the optimal gain is the largest possible, showing that divergence effects are stronger in the proximity of the cone tip. By properly rescaling the gain, wave number, and streamwise coordinate due to the fact that the boundary-layer thickness on the sharp cone is 3 p thinner than the one over the flat plate, it is found that both the gain and the wave number compare fairly well. Moreover, results for the sharp cone collapse into those for the flat plate when the initial location for the computation tends to the final one and when the azimuthal wave number is very large. Results show also that a cold wall enhances transient growth. Nomenclature A, B 0 , B 1 , B 2 , C, D, H 1 , H 2 , M,M = 5 5 matrices E = perturbation energy f = vector of perturbation unknowns G = energy ratio G E out =E in H = wall-normal characteristic length i = 1 p J = objective function L = streamwise characteristic length L = augmented functional M = Mach number m = azimuthal index m = azimuthal wave number (m m) n = nth streamwise step Pr = Prandtl number p = perturbation pressure p = vector of adjoint variables Re = Reynolds number (Re UL ) T = temperature U = base-flow streamwise velocity component u = perturbation streamwise velocity component V = base-flow wall-normal velocity component v = perturbation wall-normal velocity component w = perturbation spanwise velocity component x = streamwise coordinate y = wall-normal coordinate = spanwise wave number for flat plate = specific heat ratio x = streamwise interval (x x out x in ) = small parameter ( H ref =L ref ) = half-angle of cone tip = kinematic viscosity = density = azimuthal coordinate Subscript ad = adiabatic conditions in = inlet conditions loc = local (edge) conditions out = outlet conditions ref = reference conditions s = basic state w = wall conditions 1 = upstream conditions Superscript T = transpose

Numerical Investigation of Roughness Effects on Transition on Spherical Capsules

“…12 for both, the HLB and TAMU capsules. In accordance with previous findings in the literature, 9,11,20,27,[45][46][47] the disturbance energy gain increases with wall-cooling, whereas the effect is more pronounced in the case of the total energy norm, especially in the vicinity of the stagnation point. The disturbance energy gain based on total and kinetic energy norms is higher for the HLB capsule due to the larger body-length Reynolds number by a factor of about five.…”

Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations

“…The nonmodal analysis of compressible boundary layers was continued by Tumin & Reshotko, 16 who reformulated the temporal analysis of Hanifi et al 10 in a spatial framework, but still assuming the parallel flow approximation. The non-parallel effects were included in their subsequent publications [17][18][19][20] by solving a parabolic set of equations based on the boundary layer approximation. These authors also addressed the effects of convex surface curvature by studying optimal growth in the boundary layer over a sphere.…”

“…These studies also considered the effects of wall cooling and concluded that reducing the wall temperature increases the nonmodal growth. Zuccher et al 20 studied transient growth in the boundary layer over a sharp circular cone at zero angle of attack and freestream Mach number of 6. The basic state was based on the self-similar solution to boundary layer equations.…”

Optimal Growth in Hypersonic Boundary Layers

“…The nonmodal analysis of compressible boundary layers was continued by Tumin and Reshotko [16], who reformulated the temporal analysis of Hanifi et al [10] in a spatial framework while still assuming the parallel flow approximation. The nonparallel effects were included in subsequent publications [17][18][19][20] by solving a parabolic set of equations based on the boundary-layer approximation. These authors also addressed the effects of convex surface curvature by studying optimal growth in the boundary layer over a sphere [18,19].…”

“…Zuccher et al [20] studied transient growth in the boundary layer over a sharp circular cone at a zero angle of attack and a freestream Mach number M 6. The basic state was based on the self-similar solution to boundary-layer equations.…”