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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

Stability properties of vortex structure over slender bodies at high angles of attack and instability mechanism are the subjects of many experimental and computational studies, most of which are observed in [1,2]. However, the reasons and mechanism of spontaneous symmetric §ow breaking are not established up to now. In the ¦rst part of the present paper, a criterion for asymmetry origin for slender conical bodies is obtained using slender body theory [3] and catastrophe theory [4]. The theoretical results are veri¦ed by comparison with experimental data [5 8]; they helped to explain most of experimental observations and numerical simulations. In the second part, based on the obtained criterion and numerical solutions of boundary layer equations evaluations are made to estimate an e¨ectiveness of global §ow structure control method using local volumetric or surface gas heating. Qualitative con¦rmation of these estimations was done in experiments [7,8] with gas heating by plasma discharge. STABILITY ANALYSISA §ow over a slender pointed body of self-similar cross-section shape r = F (θ)a(X) where a(0) = 0; X is scaled with the body length l; r and θ are the polar radius and angle of the cylindrical coordinates (X, r, θ) scaled with maximal transversal size δl and maximal relative width δ, respectively, is considered.It is assumed that the body is symmetric with respect to the plane Z = 0 where (X, Y, Z) are the Cartesian coordinates; Y and Z are scaled with δl. The freestream velocity is U ∞ ; the vertical freestream velocity component is Progress in Flight Physics 5 (2013) 155-168

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