Modeling compressible turbulence for explosion applications

This thesis concerns the modeling and simulation of turbulent compressible flows for explosion applications. Part 1 considers a simple compressible turbulence model proposed in 2014, where the turbulent kinetic energy is modeled using a law associated with a uniform behavior of turbulent entropy with respect to time/space. It corresponds to a model with three balance equations for mass / momentum / energy in conservative form, associated with a thermodynamics closure law which takes into account both laminar and turbulent pressure contributions. Model properties are given, and the analysis of the one-dimensional Riemann problem allows to obtain a result of existence and uniqueness for the solution of this problem, for (almost) arbitrary initial data. A second part deals with the numerical simulation of this model using an approximate Godunov scheme, and by investigating the impact of the turbulent Mach number amplitude associated with the initial conditions. In the second chapter, we consider a compressible turbulence model proposed in 2006. In this model, the turbulent entropy can vary when crossing the shocks, with a dynamic specific constitutive law. The latter model is analyzed in detail, examining the hyperbolicity conditions, the entropy characterization and the wave structure. It is a four-equation model (for mass, momentum, total energy and turbulent entropy), integrating a source term on the shock waves for the last equation/variable. A simple Finite Volume technique, using an approximate Riemann solver, and integrating the source term in the shock waves, allows to obtain discrete approximations of the solutions. This requires in particular to define a dynamic shock detector, which is based on the entropy control and Lax conditions. A numerical convergence analysis as a function of the mesh size highlights the difficulties to simulate this model. The third chapter proposes a new model which involves four partial differential equations for mass, momentum and energy balance. The fourth PDE governs the evolution of the turbulent entropy fraction ξ_k (ratio between turbulent and total entropies), and its source term integrates a relaxation term with an associated relaxation time scale. The construction relies on a diphasic vision of the laminar/turbulence interaction. The model is again analyzed. It is hyperbolic and its wave structure is close to that of the Euler system, including a contribution in total pressure / total internal energy associated with laminar/turbulent contributions. The simulation of this model is eventually discussed, using a Finite Volume approach and Riemann solvers (approximate Godunov scheme), using variables ξ_k, ρ, U, P. (author)