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This work explores the different shapes that can be realized by the one-particle velocity distribution functions (VDFs) associated with the fourth-order maximum-entropy moment method. These distributions take the form of an exponential of a polynomial of the particle velocity, with terms up to the fourth-order. The 14- and 21-moment approximations are investigated. Various non-equilibrium gas states are probed throughout moment space. The resulting maximum-entropy distributions deviate strongly from the equilibrium VDF, and show a number of lobes and branches. The Maxwellian and the anisotropic Gaussian distributions are recovered as special cases. The eigenvalues associated with the maximum-entropy system of transport equations are also illustrated for some selected gas states. Anisotropic and/or asymmetric non-equilibrium states are seen to be associated with a non-uniform spacial propagation of perturbations.

Maximum-entropy moment methods allow for the modelling of gases from the continuum regime to strongly rarefied conditions. The development of approximated solutions to the entropy maximization problem has made these methods computationally affordable. In this work, we apply a fourth-order maximum-entropy moment method to the study of supersonic rarefied flows. For such conditions, we compare the maximum-entropy solutions to results obtained from the kinetic theory of gases at different Knudsen numbers. The analysis is performed for both a simplified model of a gas with a single translational degree of freedom (5-moment system) and for a typical gas with three degrees of freedom (14-moment system). The maximum-entropy method is applied to the study of the Sod shock-tube problem at various rarefaction levels, and to the simulation of two-dimensional low-collisional crossed supersonic jets. We show that, in rarefied supersonic conditions, it is important to employ accurate estimates of the wave speeds. Since analytical expressions are not presently available, we propose an approximation, valid for the 14-moment system. In these conditions, the solution of the maximum-entropy system is shown to realize large degrees of non-equilibrium and to approach the Junk subspace, yet provides a good overall accuracy and agreement with the kinetic theory. Numerical procedures for reaching second-order accurate discretizations are discussed, as well as the implementation of the 14-moment solver on Graphics Processing Units (GPUs).

We address the problem of shock induced ignition and transition to detonation in a reactive medium in the presence of mechanically induced fluctuations by a moving oscillating piston. For the inert problem prior to ignition, we provide a closed form model for the generation of the train of compression and expansions, their steepening into a train of N-shock waves and their reflection on the lead shock, as well as the distribution energy dissipation rate in the induction zone. The model is found in excellent agreement with numerics. Reactive calculations were performed for hydrogen and ethylene fuels using a novel high-fidelity numerical Lagrangian scheme. Different regimes of ignition and transition to detonation, controlled by the time scale of the forcing and the two time scales of the chemistry: the induction and reaction times. Two novel hot spot cascade mechanisms were identified. The first relies on the coherence between the sequence of hot spot formation set by the piston forcing and forward wave interaction with the lead shock, generalizing the classic runaway in fast flames. The second hot spot cascade is triggered by the feedback between the pressure pulse generated by the first generation hot spot cascade and the shock. For slow forcing, the sensitization is through a modification to the classic run-away process, while the high frequency regime leads to very localized sub-critical hot-spot formation controlled by the cumulative energy dissipation of the first generation shocks at a distance comparable to the shock formation location.

The 10 and 14-moment maximum-entropy methods are applied to the study of high-Mach-number non-reacting crossflows past a flat plate at large degrees of rarefaction. The moment solutions are compared to particle-based kinetic solutions, showing a varying degree of accuracy. At a Knudsen number of 0.1, the 10-moment method is able to reproduce the shock layer, while it fails to predict the low-density wake region, due to the lack of a heat flux. Conversely, the 14-moment method results in accurate predictions of both regions. At a Knudsen number of 1, the 10-moment method produces unphysical results in both the shock layer and in the wake. The 14-moment method also shows a reduced accuracy, but manages to predict a reasonable shock region, free of unphysical sub-shocks, and in qualitative agreement with the kinetic solution. Accuracy is partially lost in the wake, where the 14-moment method predicts a thin unphysical high-density layer, concentrated on the centreline. An analysis of the velocity distribution functions (VDF) indicates strongly non-Maxwellian shapes, and the presence of distinct particle populations, in the wake, crossing each other at the centreline. The particle-based and the 14-moment method VDFs are in qualitative agreement.

This work introduces a limitation on the minimum value that can be assumed by the energy of a relativistic gas in the presence of a non-zero heat flux. Such a limitation arises from the non-negativity of the particle distribution function, and is found by solving the Hamburger moment problem. The resulting limitation is seen to recover the Taub inequality in the case of a zero heat flux, but is more strict if a non-zero heat flux is considered. These results imply that, in order for the distribution function to be non-negative, (i) the energy of a gas must be larger than a minimum threshold; (ii) the heat flux, on the other hand, has a maximum value determined by the energy and the pressure tensor; and (iii) there exists an upper limit for the the adiabatic index $\Gamma$ of the relativistic equation of state, and that limit decreases in the presence of a heat flux and pressure anisotropy, asymptoting to a value $\Gamma = 1$. The latter point implies that the Synge equation of state is formally incompatible with a relativistic gas showing a heat flux, except in certain gas states.

Ions in Hall effect thrusters are often characterized by a low collisionality. In the presence of acceleration fields and azimuthal electric field waves, this results in strong deviations from thermodynamic equilibrium, introducing kinetic effects. This work investigates the application of the 14-moment maximum-entropy model to this problem. This method consists in a set of 14 PDEs for the density, momentum, pressure tensor components, heat flux vector and fourth-order moment associated to the particle velocity distribution function. The model is applied to the study of collisionless ion dynamics in a Hall thruster-like configuration, and its accuracy is assessed against different models, including the Vlasov kinetic equation. Three test cases are considered: a purely axial acceleration problem, the problem of ion-wave trapping and finally the evolution of ions in the axial-azimuthal plane. Most of this work considers ions only, and the coupling with electrons is removed by prescribing reasonable values of the electric field. This allows us to obtain a direct comparison among different ion models. However, the possibility to run self-consistent plasma simulations is also briefly discussed, considering quasi-neutral or multi-fluid models. The maximum-entropy system appears to be a robust and accurate option for the considered test cases. The accuracy is improved over the simpler pressureless gas model (cold ions) and the Euler equations for gas dynamics, while the computational cost shows to remain much lower than direct kinetic simulations.

A 14-moment maximum-entropy system of equations is applied to the description of non-equilibrium electrons in crossed electric and magnetic fields and in the presence of low collisionality, characteristic of low-temperature plasma devices. The flexibility of this formulation is analyzed through comparison with analytical results for steady-state non-equilibrium velocity distribution functions and against particle-based solutions of the time-dependent kinetic equation. Electric and magnetic source terms are derived for the 14-moment equations, starting from kinetic theory. A simplified BGK-like collision term is formulated to describe the collision of electrons with background neutrals, accounting for the large mass disparity and for energy exchange. An approximated expression is proposed for the collision frequency, to include the effect of the electrons drift velocity, showing good accuracy in the considered conditions. The capabilities of the proposed 14-moment closure to capture accurately non-equilibrium behaviour of electrons for space homogeneous problems under conditions representative of those found in Hall thrusters is demonstrated.