The paper provides a brief introduction to the near-wall problem of LES and how it can be solved through modeling of the near-wall turbulence. The distinctions and key differences between different approaches are emphasized, both in terms of fidelity (LES, wall-modeled LES, and DES) and in terms of different wall-modeled LES approaches (hybrid LES/RANS and wall-stress-models). The focus is on approaches that model the wall-stress directly, i.e., methods for which the LES equations are formally solved all the way down to the wall. Progress over the last decade is reviewed, and the most important and promising directions for future research are discussed.

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The interaction between isotropic turbulence and a normal shock wave is investigated through a series of direct numerical simulations at different Reynolds numbers and mean and turbulent Mach numbers. The computed data are compared to experiments and linear theory, showing that the amplification of turbulence kinetic energy across a shock wave is described well using linearized dynamics. The post-shock anisotropy of the turbulence, however, is qualitatively different from that predicted by linear analysis. The jumps in mean density and pressure are lower than the non-turbulent Rankine-Hugoniot results by a factor of the square of the turbulence intensity. It is shown that the dissipative scales of turbulence return to isotropy within about 10 convected Kolmogorov time scales, a distance that becomes very small at high Reynolds numbers. Special attention is paid to the 'broken shock' regime of intense turbulence, where the shock can be locally replaced by smooth compressions. Grid convergence of the probability density function of the shock jumps proves that this effect is physical, and not an artefact of the numerical scheme.

We present wall-modelled large-eddy simulations (WLES) of oblique shock waves interacting with the turbulent boundary layers (TBLs) (nominal $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\delta _{99}=5.4\ \mathrm{mm}$ and ${\mathit{Re}}_{\theta }\approx 1.4\times 10^4$) developed inside a duct with an almost-square cross-section ($45\ \mathrm{mm}\times 47.5\ \mathrm{mm}$) to investigate three-dimensional effects imposed by the lateral confinement of the flow. Three increasing strengths of the incident shock are considered, for a constant Mach number of the incoming air stream $M\approx 2$, by varying the height (1.1, 3 and 5 mm) of a compression wedge located at a constant streamwise location that spans the top wall of the duct at a 20° angle. Simulation results are first validated with particle image velocimetry (PIV) experimental data obtained at several vertical planes (one near the centre of the duct and three near one of the sidewalls) for the 1.1 and 3 mm-high wedge cases. The instantaneous and time-averaged structure of the flow for the stronger-interaction case (5 mm-high wedge), which shows mean flow reversal, is then investigated. Additional spanwise-periodic simulations are performed to elucidate the influence of the sidewalls, and it is found that the structure and location of the shock system, as well as the size of the separation bubble, are significantly modified by the lateral confinement. A Mach stem at the first reflected interaction is present in the simulation with sidewalls, whereas a regular shock intersection results for the spanwise-periodic case. Low-frequency unsteadiness is observed in all interactions, being stronger for the secondary shock reflections of the shock train developed inside the duct. The downstream evolution of secondary turbulent flows developed near the corners of the duct as they traverse the shock system is also studied.