Prediction of 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 DNS and global analysis can be used, the large grids required make the computation of optimal transient and forced responses 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. In this work, we apply the One-Way Navier-Stokes (OWNS) equations to 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 high-order recursive filter. We extend the original algorithm by considering non-orthogonal body-fitted curvilinear coordinates and incorporate full compressibility with temperature-dependent fluid properties. We validate the results by comparing to DNS data for a flat plate and sharp cone, and to LST results for local disturbances on the centerline of the HIFiRE-5 elliptic cone. OWNS provides DNS-quality results for the former flows at a small fraction of the computational expense.

Resolvent analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

Accurate prediction of linear amplification of disturbances in hypersonic boundary layers is computationally challenging. While direct numerical simulations (DNS) and global analysis can be used to compute optimal (worst-case) disturbances and forced responses, their large computational expense render these tools less practical for large design parameter spaces. At the same time, parabolized stability equations can be unreliable for problems involving multimodal and non-modal interactions. To bridge this gap, we apply an approximate fast marching technique, the One-Way Navier-Stokes (OWNS) Equations, in iterative fashion to solve for optimal disturbances. OWNS approximates a rigorous parabolization of the equations of motion by removing disturbances with upstream group velocity using a higher-order recursive filter. Using OWNS, we aim to characterize disturbances of flat-plate hypersonic boundary layers over a range of Mach numbers, and find optimal disturbances under different cost functions that define corresponding receptivity problems. The calculation of optimal disturbances reveals multi-modal transition scenarios depending on the spatial support, frequency, and physical nature of the external disturbances.

In the context of transition analysis, linear input–output analysis determines the worst-case disturbances to a laminar base flow based on a generic right-hand-side volumetric/boundary forcing term. The worst-case forcing is not physically realizable, and, to our knowledge, a generic framework for posing physically realizable worst-case disturbance problems is lacking. In natural receptivity analysis, disturbances are forced by matching (typically 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 direct numerical simulations 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.

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