This study is concerned with the development of extended fluid-dynamic models for the prediction of micro-scale flows. When compared to classical fluid descriptions, such models must remain valid on scales where traditional techniques fail. Also, knowing that solution to these equations will be sought by numerical methods, the nature of the extended models must also be such that they are amenable to solution using computational techniques. Moment closures of kinetic theory offer the promise of satisfying both of these requirements. It is shown that the hyperbolic nature of moment equations imbue them with several numerical advantages including an extra order of spacial accuracy for a given reconstuction when compared to the Navier-Stokes equations and a reduced sensitivity to grid irregularities. In addition to this, the expanded set of parameters governed by the moment closures allow them to accurately model many strong non-equilibrium effects that are typical of micro-scale flows. Unfortunately, traditional moment models have suffered from various closure breakdowns, and robust models that offer a treatment for non-equilibrium viscous heat-conducting gas flows have been elusive. To address these issues, a regularized 10-moment closure is first proposed herein based on the maximum-entropy Gaussian moment closure. This mathematically well-behaved model avoids closure breakdown through a strictly hyperbolic treatment for viscous effects and an elliptic formulation that accounts for non-equilibrium thermal diffusion. Moreover, steps toward the development of fully hyperbolic moment closures for the prediction of non-equilibrium viscous gas flow are made via two novel approaches. A thorough study of each of the proposed techniques is made through numerical solution of many classical flow problems.

This study is concerned with the development of extended fluid-dynamic models for the prediction of micro-scale flows. When compared to classical fluid descriptions, such models must remain valid on scales where traditional techniques fail. Also, knowing that solution to these equations will be sought by numerical methods, the nature of the extended models must also be such that they are amenable to solution using computational techniques. Moment closures of kinetic theory offer the promise of satisfying both of these requirements. It is shown that the hyperbolic nature of moment equations imbue them with several numerical advantages including an extra order of spacial accuracy for a given reconstuction when compared to the Navier-Stokes equations and a reduced sensitivity to grid irregularities. In addition to this, the expanded set of parameters governed by the moment closures allow them to accurately model many strong non-equilibrium effects that are typical of micro-scale flows. Unfortunately, traditional moment models have suffered from various closure breakdowns, and robust models that offer a treatment for non-equilibrium viscous heat-conducting gas flows have been elusive. To address these issues, a regularized 10-moment closure is first proposed herein based on the maximum-entropy Gaussian moment closure. This mathematically well-behaved model avoids closure breakdown through a strictly hyperbolic treatment for viscous effects and an elliptic formulation that accounts for non-equilibrium thermal diffusion. Moreover, steps toward the development of fully hyperbolic moment closures for the prediction of non-equilibrium viscous gas flow are made via two novel approaches. A thorough study of each of the proposed techniques is made through numerical solution of many classical flow problems.

This dissertation presents a study of hyperbolic turbulence modelling for the Gaussian ten-moment equations. In gaskinetic theory, moment closures offer the possibility of deriving a series of gas-dynamic governing equations from the Boltzmann equation. One typical example, the Gaussian ten-moment model, which takes the form of hyperbolic-relaxation equations, is considered as a competitive model for viscous gas flow when heat transfer effects are negligible. The hyperbolic nature of this model gives it several numerical advantages, compared to the Navier-Stokes equations. However, until this study, the application of the ten-moment equations has been limited to laminar flows, due to the lack of appropriate turbulence models. In this work, the ten-moment equations are, for the first time, Reynolds-averaged. The resulting equations inherit the hyperbolic balance-law form from the original equations with new unknowns, which require approximation by turbulence models. Most of the traditional turbulence models for the Reynolds-averaged Navier-Stokes equations are not perfectly well-suited for the Reynolds-averaged ten-moment equations, because the second-order derivatives presented in these models can break the pure hyperbolic nature of the original model. The relaxation methods are therefore proposed in this project to reform the existing turbulence models. Two relaxation methods, the Chen-Levermore-Liu p-system and Cattaneo-Vernotte models, are used to hyperbolize the Prandtl’s one-equation model, standard k-ε model and Wilcox k-ω model. The hyperbolic versions of these turbulence models are first shown to be equivalent to their original forms. They are then coupled to the Reynolds-averaged ten-moment equations to build the overall hyperbolic governing equations for turbulence flows. An axisymmetric version of Reynolds-averaged ten-moment equations is also derived. A dispersion analysis is conducted for the resulting governing equations, which shows the corresponding dispersive behaviour and stability. The effect of the relaxation parameters is investigated through several numerical tests. All derived turbulence models are applied to solve canonical validation test problems, including two-dimensional planar mixing-layer, free-jet and circular free-jet. The numerical evaluations are analysed and compared against existing experimental measurements.

This study is concerned with the development and implementation of a preconditioner for a set of hyperbolic partial differential equations resulting from a new 5-moment closure for the prediction of gas flows both in and out of local equilibrium. This new 5-moment closure offers a robust and efficient system of first-order hyperbolic partial differential equations that has proven to provide an accurate treatment of one-dimensional gases, both in and for significant departures from local thermodynamic equilibrium. However, numerical computations using this model have proven to be difficult as a result of a singularity in the closing flux of the system. This also causes infinitely large wavespeeds in the system. The main goal of this work is to mitigate these numerical issues. Since the solution of a hyperbolic system is characterized by the waves of the system, one could suggest to scale these wavespeeds to remove the arbitrarily large speeds without altering the solution of the system. To accomplish this, this work starts with a detailed study of the behaviour of the system’s wavespeeds, given by the eigenvalues of the flux Jacobian of the system. Since, it is not possible to solve for these eigenvalues explicitly, it is suggested to approximate them by interpolation between the few states at which these waves can be solved for explicitly. With an estimate for the wavespeeds, the nature of the singularity in the system can be analyzed mathematically. The results of this mathematical analysis are used to develop a preconditioner matrix to remove the singularity from the model. To implement the proposed preconditioned model numerically, a centred-difference scheme with artificial dissipation is proposed. A dual-time-stepping strategy is developed and implemented with implicit Euler time marching for both physical and pseudo time iteration. This dual-time treatment allows the preconditioned system to remain applicable to time-accurate problems and is found to greatly increase the robustness of the solution of the steady-state problems. Solutions to several canonical problems for both continuum and non-equilibrium flow are computed and comparisons are made to classical models.

In this study, the behavior of pressure in the Rapid Distortion Limit, along with its concomitant modeling, are addressed. In the first part of the work, the role of pressure in the initiation, propagation and suppression of flow instabilities for quadratic flows is analyzed. The paradigm of analysis considers the Reynolds stress transport equations to govern the evolution of a dynamical system, in a state space composed of the Reynolds stress tensor components. This dynamical system is scrutinized via the identification of the invariant sets and the bifurcation analysis. The changing role of pressure in quadratic flows, viz. hyperbolic, shear and elliptic, is established mathematically and the underlying physics is explained. Along the maxim of "understanding before prediction", this allows for a deeper insight into the behavior of pressure, thus aiding in its modeling. The second part of this work deals with Rapid Pressure Strain Correlation modeling in earnest. Based on the comprehension developed in the preceding section, the classical pressure strain correlation modeling approaches are revisited. Their shortcomings, along with their successes, are articulated and explained, mathematically and from the viewpoint of the governing physics. Some of the salient issues addressed include, but are not limited to, the requisite nature of the model, viz. a linear or a nonlinear structure, the success of the extant models for hyperbolic flows, their inability to capture elliptic flows and the use of RDT simulations to validate models. Through this analysis, the schism between mathematical and physical guidelines and the engineering approach, at present, is substantiated. Subsequently, a model is developed that adheres to the classical modeling framework and shows excellent agreement with the RDT simulations. The performance of this model is compared to that of other nominations prevalent in engineering simulations. The work concludes with a summary, pertinent observations and recommendations for future research in the germane field.