Mixed convection in metallic foams modeled with Darcy's law under local thermal non-equilibrium conditions is investigated, where the solid phase thermal conductivity is assumed to be infinitely larger than its fluid phase counterpart. A linear and modal stability analysis was employed to evaluate the convective and absolute instability thresholds as well as their respective cell patterns. This analysis indicates that local thermal non-equilibrium always has a stabilizing effect and the spanwise uniform mode is always the most unstable. At the onset of convective instability, however, the number of equally unstable cell patterns increases with both the aspect ratio and the local thermal non-equilibrium strength.

The linear stability of a thermally stratified fluid layer between horizontal walls, where non-Brownian thermal particles are injected continuously at one boundary and extracted at the other – a system known as particulate Rayleigh–Bénard (pRB) – is studied. For a fixed volumetric particle flux and minimal thermal coupling, reducing the injection velocity stabilises the system when heavy particles are introduced from above, but destabilises it when light particles are injected from below. For very light particles (bubbles), low injection velocities can shift the onset of convection to negative Rayleigh numbers, i.e. heating from above. Particles accumulate non-uniformly near the extraction wall and in regions of strong vertical flow, aligning with either wall-impinging or wall-detaching zones depending on whether injection is at sub- or super-terminal velocity. The increase of the volumetric particle flux always enhances these effects.

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp.

A novel implementation of weighted essentially non-oscillatory (WENO) schemes is presented to solve eigenvalue problems resulting from numerical discretization of the compressible linearized Navier–Stokes equations within a matrix-forming framework. In boundary layer profiles featuring a discontinuity, agreement of the eigenvalues delivered by the proposed method and those found in the literature, calculated using the linearized Rankine–Hugoniot boundary condition, is excellent at the physically most significant conditions of peak amplification. Moreover, the present WENO approach delivers accurate eigenfunctions, drastically reducing oscillations that appear in the amplitude functions of linear perturbations when spatial discretization is performed using standard finite-difference methods. In the absence of discontinuities, accurate recovery of convective instabilities is demonstrated in open and closed shear flows, at the nominal theoretical convergence rate of the WENO method.