Optimal Growth in Hypersonic Boundary Layers

Abstract

The linear form of the parabolized linear stability equations is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic Mach number regime without any high-enthalpy effects. The influence of wall cooling is studied, with particular emphasis on the role of the initial disturbance location and the value of the spanwise wave number that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary-layer equations, mean flow solutions based on the full Navier-Stokes equations are used in select cases to help account for the viscousinviscid interaction near the leading edge of the plate and for the weak shock wave emanating from that region. Using the full Navier-Stokes mean flow is shown to result in further reduction with Mach number in the magnitude of optimal growth relative to the predictions based on the self-similar approximation to the base flow. Nomenclature A, B, C, D = linear matrix operators c = normalization coefficient E = total energy norm G = energy gain h 1 = streamwise metric factor h 3 = spanwise metric factor J = objective function K = kinetic energy norm K = bilinear concomitant L = flat-plate characteristic length L = Lagrangian function M = Mach number M E = energy weight matrix N = logarithmic amplitude ratio relative to neutral stability location of modal instability N η = number of discretization points along the wall-normal direction q = vector of base flow variables q = vector of perturbation variableŝ q = vector of amplitude variables R = local Reynolds number Re = Reynolds number T = temperature T ad = adiabatic wall temperature T w = wall temperature u; v; w = streamwise, wall-normal, and spanwise velocity components ν = kinematic viscosity x; y; z = Cartesian coordinates α = streamwise wave number β = spanwise wave number γ = heat capacity ratio δ = local similarity length scale of boundary layer δ 0.995 = 0.995 boundary-layer thickness ξ; η; ζ = streamwise, wall-normal, and spanwise coordinates in body-fitted coordinate system ρ = density Ω = domain of integration ω = angular frequency Subscripts ad = adiabatic wall condition r = reference value w = wall condition 0 = initial disturbance location 1 = final optimization location Superscripts H = conjugate transpose T = transpose = dimensional value † = adjoint

Blunt-Body Paradox and Improved Application of Transient-Growth Framework

“…[18,19]. The only significant difference from the theoretical framework employed by Paredes et al 19 is that the spanwise wavelength of the disturbance is allowed to be an arbitrary but prescribed function of the location along the leeward symmetry plane. For completeness, a brief summary of the optimal growth framework is presented in this section.…”

“…The iterative procedure is terminated when the value of J has converged up to a specified tolerance, which was set to a relative error of 10 −4 in the current computations. Additional details of the current implementation are found in Paredes et al 17,19 Nonuniform, stable, high-order, finite-difference schemes (FD-q) 26,27 of sixth order are used for discretization of the PSE along the wall-normal coordinate. The discretized PSE are integrated along the streamwise coordinate by using second-order backward differentiation.…”

“…Verification of the present optimal-growth framework against available results from the literature is presented in Refs. [19,28].…”

Numerical Investigation of Roughness Effects on Transition on Spherical Capsules

“…The optimal transient-growth analysis is performed using the framework of linear parabolized stability equations (PSE) as elucidated in the literature. [19][20][21][22] The method is outlined here for the sake of completeness.…”

“…The optimal-growth framework developed by NASA is used for the computations for the TAMU ACE capsule configuration and has been extensively verified. 21 On the other hand, a newly developed optimal transient-growth code by DLR is employed for characterizing the non-modal growth properties of the boundary layer on the HLB capsule. A more detailed overview of the two different capsule configuration is given in Sec.…”