Shock-wave diffraction over double concave cylindrical surfaces has been numerically investigated at different flow regimes by varying the incident-shock-wave Mach number from Ms=1.6 (transonic) to Ms=4.5 (supersonic regime). The purpose of this study is to better understand the dynamics of shock-wave structure and the associated wave configurations. A mesh-independent solution is obtained and the flow is assessed through different physical quantities (transition angles, triple points trajectories, wall-pressure and skin-friction distributions, velocity and shock location). It is found that the transition angles, from regular to Mach reflection, increase with the Mach number. This phenomenon remains almost the same over both concave surfaces for weak Mach numbers (up to Ms=2.5) and becomes relatively larger on the second surface for high Mach numbers. In terms of shock dynamics, it is found that by increasing the incident incident-shock-wave Mach number to Ms=4.5, unlike the first reflector, the transition from a single-triple-point (STP) wave configuration to a double-triple-point (DTP) wave configuration and back occurred on the second reflector, indicating that the flow is capable of retaining the memory of the past events over the entire process. For the shock velocity, the velocity deficit is found to be increasing with increase in Ms. A best fitting scaling law is derived, to ensure a universal decay of the shock velocity depending on the flow parameters.

The unsteady aspect of turbulent flow structures generated by a shock-wave diffraction over double cylindrical wedges, with initial diffracting angle of 75∘, are numerically investigated by means of two-dimensional high-fidelity numerical simulation. Different incident-shock-Mach numbers, ranging from transonic to supersonic regimes, are considered. Unlike previous studies where only the total vorticity production is evaluated, the current paper offers more insights into the spatio-temporal behavior of the circulation by evaluating the evolution of the instantaneous vorticity equation balance. The results show, for the first time, that the diffusion of the vorticity due to the viscous effects is quite important compared to the baroclinic term for low Mach numbers regimes, while this trend is inverted for higher Mach numbers regimes. It is also found that the stretching of the vorticity due to the compressibility effects plays an important role in the vorticity production. In terms of pressure impulses, the effect of the first concave surface on the shock strength has been quantified at both earlier and final stages of the shock diffraction process. Unlike the overpressure, the static and the dynamic pressure impulses are shown to be significantly reduced at the end of the first concave surface.

In this paper, a fully adaptive multiresolution (MR) finite difference scheme with a time-varying tolerance is developed to study compressible fluid flows containing shock waves in interaction with solid obstacles. To ensure adequate resolution near rigid bodies, the MR algorithm is combined with an immersed boundary method based on a direct-forcing approach in which the solid object is represented by a continuous solid-volume fraction. The resulting algorithm forms an efficient tool capable of solving linear and nonlinear waves on arbitrary geometries. Through a one-dimensional scalar wave equation, the accuracy of the MR computation is, as expected, seen to decrease in time when using a constant MR tolerance considering the accumulation of error. To overcome this problem, a variable tolerance formulation is proposed, which is assessed through a new quality criterion, to ensure a time-convergence solution for a suitable quality resolution. The newly developed algorithm coupled with high-resolution spatial and temporal approximations is successfully applied to shock–bluff body and shock-diffraction problems solving Euler and Navier–Stokes equations. Results show excellent agreement with the available numerical and experimental data, thereby demonstrating the efficiency and the performance of the proposed method.

The turbulent structures and long-time flow dynamics of shock diffraction over 90° convex corner associated with an incident shock Mach number Ms = 1.5 are investigated by large eddy simulation (LES). The average evolution of the core of the primary vortex is in agreement with the previous two dimensional studies. The Type-N wall shock structure is found to be in excellent agreement with the previous experimental data. The turbulent structures are well resolved and resemble those observed in the experimental findings. Subgrid scale dissipation and subgrid scale activity parameter are quantified to demonstrate the effectiveness of the LES. An analysis based on turbulent-nonturbulent interface reveals that locally incompressible regions exhibit the universal teardrop shape of the joint probability density function of the second and third invariants of the velocity gradient tensor. Stable focus stretching (SFS) structures dominate throughout the evolution in these regions. Stable node/saddle/saddle structures are found to be predominant at the early stage in locally compressed regions, and the flow structures evolve to more SFS structures at later stages. On the other hand, the locally expanded regions show a mostly unstable nature. From the turbulent kinetic energy, we found that the pressure dilatation remains important at the early stage, while turbulent diffusion becomes important at the later stage. Furthermore, the analysis of the resolved vorticity transport equation reveals that the stretching of vorticity due to compressibility and stretching of vorticity due to velocity gradients plays an important role compared to diffusion of vorticity due to viscosity as well as the baroclinic term.

The underlying sequential behavior of the multiresolution (MR) method has been exploited for parallel computing by introducing a concept of multiresolution forest structures (MFS) along with two new load-balancing algorithms. Another easy-to-implement multithreading approach has also been introduced for the multicore architectures. Tests were conducted using an Euler solver based on a fifth-order shock capturing WENO scheme and a third-order Runge–Kutta algorithm. The methods have been rigorously analyzed in terms of speedup ratio and parallel efficiency to bring forth their benefits as well as limitations. The performance yielded through these methods indicates that the MFS is a new headway for the MR method in parallel computing that has a potential to harness better scalability.

This paper concerns the use of a point-value multiresolution algorithm and its extension to three-dimensional hyperbolic conservation laws. The proposed method is applied to a high-order finite-differences discretization with an explicit time integration. The fluxes are evaluated on the adaptive grid using a fifth-order high-resolution shock capturing scheme based on a WENO solver, while the time is advanced using a third-order Runge–Kutta scheme. The multiresolution prediction operators are presented for one-, two- and three-dimensional problems. To assess the efficiency and the accuracy of the method, a new tolerance-scale diagram is introduced. This diagram enables to properly choose the adequate value of the tolerance in order to maintain an optimal multiresolution quality. Numerical examples based on advection and Euler equations are carried out to show that the proposed method yields accurate results.