Optimal Receptivity and the Generalization of the One-Way Navier-Stokes (OWNS) Equations to Complex High-Speed Boundary Layers and Jets

Prediction of the linear amplification of disturbances in hypersonic boundary layers is challenging due to the presence and interactions of discrete modes (e.g. Tollmien-Schlichting and Mack) and continuous modes (entropic, vortical, and acoustic). While direct numerical simulations (DNS) and global analysis can be used, the large grids required make the stability calculations expensive, particularly when a large parameter space is required. At the same time, parabolized stability equations are non-convergent and unreliable for problems involving multi-modal and non-modal interactions. We therefore apply the One-Way Navier-Stokes (OWNS) Equations to study transitional hypersonic boundary layers. OWNS is based on a rigorous, approximate parabolization of the equations of motion that removes disturbances with upstream group velocity using a higher-order recursive filter. We extend the original algorithm by considering non-orthogonal curvilinear coordinates and incorporate full compressibility with temperature-dependent fluid properties. The generalized OWNS methodology is validated by comparing to DNS data for flat plates and a sharp cone, and to linear stability theory results for local disturbances on the centerline of the Mach 6 HIFiRE-5 elliptic cone. OWNS provides DNS-quality results for the former flows at a small fraction of the computational expense. We further demonstrate the capability of OWNS to track fully 3D instabilities by applying the algorithm to a complex Mach 6 finned-cone geometry as well as a 3D Mach 1.5 turbulent jet. It is often desirable, especially for design purposes, to compute worst-case disturbances, i.e. solving the inverse problem, otherwise known as resolvent or input-output analysis. While DNS and global analysis can be used to compute optimal forced responses, their large computational expense render these tools less practical for large design parameter spaces. We address this issue by modifying the original OWNS framework to find the optimal forcing and responses using Lagrangian multipliers via an iterative, adjoint-based, space-marching technique that appreciably reduces the computational burden compared to the global approach that uses singular value decomposition without sacrificing accuracy. The input-output OWNS model is validated against optimal forcings and responses of a Mach 4.5 flat-plate boundary layer from literature and a Mach 1.5 turbulent jet. We then apply these equations to study worst-case disturbances on the centerline of the Mach 6 HIFiRE-5 elliptic cone and on a highly cooled Mach 6 flat-plate boundary layer. Although the worst-case forcings are theoretically informative, they are not physically realizable. In natural receptivity analysis, disturbances are forced by matching local solutions within the boundary layer to outer solutions consisting of free-stream vortical, entropic, and acoustic disturbances. We pose a scattering formalism to restrict the input forcing to a set of realizable disturbances associated with plane-wave solutions of the outer problem. The formulation is validated by comparing with DNS of a Mach 4.5 flat-plate boundary layer. We show that the method provides insight into transition mechanisms by identifying those linear combinations of plane-wave disturbances that maximize energy amplification over a range of frequencies. We also discuss how the framework can be extended to accommodate scattering from shocks and in shock layers for supersonic flow.