Low-mach number effects and late-time treatment of Richtmyer-Meshkov and Rayleigh-Taylor instabilities

The Richtmyer-Meshkov instability appears when the mixing between two fluids is triggered by the passage of a shock wave. It occurs in a range of different applications, such as astrophysics, inertial confinement fusion and supersonic combustion. Due to the extreme complexity of this phenomenon to be reproduced in a controlled environment, its study heavily relies on numerical methods. The presence of a shock wave as a triggering factor requires the use of compressible solvers, but once the shock has started the mixing process, the flow field freely decays and becomes incompressible. The dynamics of this instability is still to be fully understood, especially its long-time behaviour. One of the hypothesis is that the mixing layer achieves a self-similar development at some point during its evolution. However, the low-Mach flow at late-times does not always allow to push compressible simulations so far in time and when it is possible, they become extremely demanding from a computational point of view. In fact, it is known that standard compressible methods fail when the Mach number of the numerical field is low and moreover they lose time-marching efficiency. In this thesis, a new approach to the study of the very late-stage of the instability through the use of ILES is presented. The technique consists in starting the simulation by using the compressible model and to initialise the incompressible solver when the compressibility of the numerical field becomes sufficiently low. This allows to bypass the issues previously mentioned and study the very late-stage of the instability at reasonable computational costs. For this purpose, a new incompressible solver that employs high-resolution methods and which is based on the pressure-projection technique is developed. A number of different Riemann-solvers and reconstruction schemes are tested against experiments using the incompressible, impulsive version of RMI as test case. Two alternative methods are considered for triggering the mixing: velocity impulse and gravity pulse. Excellent results were obtained by using the former, whereas discrepancies were noticed when the latter was employed. Comparisons against numerical simulations in the literature allowed to identify the inviscid nature of the solver as the cause of these differences. However, this did not affect the capability of the solver to correctly compute multi-mode cases, in which viscosity is negligible. A preliminary study on the compressibility of the numerical field in time proved the feasibility of the numerical transition and a switching criterion based on the Mach number was established. The approach was therefore tested on a single-mode perturbation case and compared against compressible simulation. Very good agreement was found in the prediction of the growth of the instability and the analysis of the divergence of velocity of the numerical field proved the incompressibility of the solution generated by the hybrid solver. Finally, the approach was applied to multi-mode test cases. Excellent agreement with the theory was found. The turbulent kinetic energy presented a modified subinertial range and the growth exponent was very close to fully compressible predictions and experiments. Deeper results analysis showed against compressible simulations showed very good agreement on the flow physics. In fact, the instability settled to a self-similar regime with the same time-scale predicted by compressible analysis, but the simulated time reached by the hybrid solver was three times longer. The results obtained proved the applicability of the approach, opening to new possibilities for the study of the instability.