NUMERICAL METHODS FOR MULTIPHASE FLOWS: FROM A SIMPLIFIED INCOMPRESSIBLE BAER-NUNZIATO MODEL TO A UNIFIED THEORY OF COMPRESSIBLE MULTIPHASE FLUID AND SOLID MECHANICS

In this thesis, I detail my contribution to the development of both the theoretical aspects of modelling compressible and incompressible multi-phase flows and the design of suitable numerical algorithms for solving such models. Currently, there is no universally accepted mathematical model to describe two-phase flows, consequently the extension to multi-phase flows (with more than two phases) is even less clear. A popular approach to describe two-phase flows is to use so-called homogenised mixture models, which are based on the diffuse interface approach. One of the most established ones is the Baer-Nunziato (BN) model. Due to its relatively simple mathematical structure, it is possible to derive reduced models, such as a model for incompressible two-phase flows, with straightforward assumptions. However, there are modelling and numerical reasons for the need to develop a mathematical model formulated in a well-defined mixture theory. Therefore, the derivation of a multi-phase and multi-material model within the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) theory and its numerical solution are of relevant interest in the work presented in this thesis. Concerning the details of my research work, I begin by addressing the problem of a reduced BN-type model for incompressible two-phase flows. In incompressible models, the pressure acts as a Lagrange multiplier which ensures the divergence free condition of the velocity field. A natural numerical approach is to treat pressure-related terms implicitly, thus in this context two efficient mass and momentum conservative semi-implicit FV schemes for different applications are developed, e.g. for complex non-hydrostatic free surface flows, or for flows interacting with moving solid obstacles. These new numerical algorithms are implemented in a distributed memory MPI-parallel Fortran code. In addition, the semi-implicit discretisation adopted for the pressure subsystem, rather than a fully implicit one for the complete Partial Differential Equation (PDE) system, leads to a small-size, symmetric positive definite discrete linear system that can be solved efficiently with a preconditioned, matrix-free conjugate gradient method. This directly results in a much higher parallel scalability compared to fully implicit schemes. A semi-implicit scheme boils down to the solution of a discrete linear system that takes into account the contributions of implicitly treated terms involved in the governing equations. As a result, understanding the numerical problems associated with the discretization of each term is more complex. Some of the numerical problems usually encountered in the numerical solution of compressible multi-phase model are related to the presence of complex interfaces described by the volume fraction of each phase. Therefore, to become more familiar with and better address these challenges, explicit schemes are considered. In particular, the numerical scheme initially adopted is an explicit second order FV method combined with the path-conservative technique of Castro and Pares for the treatment of the non-conservative products appearing in BN-type models. With the insights from this numerical scheme, the complete seven-equation BN model for compressible two phase flows is discretized. However, despite the variety of literature addressing this model, it still presents some problems, such as the fact that the model is not closed (in the sense that it requires the specification of an interface pressure and an interface velocity, the choice of which is not unique) and that it is not clear how to generalise the BN formulation to mixtures with an arbitrary number of phases. This is mainly due to the fact that the derivation procedure of the BN model is based on phenomenological and heuristic observations. To overcome these problems, a model based on a well-defined mixture theory and derived from first principles as causality and the laws of thermodynamics is considered. This model can be written in a conservative form and, as formulated, is already generalised to an arbitrary number of constituents. It takes the form of a monolithic system of first order hyperbolic PDEs that include a unified multi-phase description of fluids and solids. The origins of this model go back to the SHTC theory of mixtures firstly proposed by Romenski for the case of two fluids, and then generalized to the case of an arbitrary number of constituents. Furthermore, the Eulerian hyperelasticity equations of Godunov and Romenski (GPR) are used to introduce viscous and elastic forces. These Eulerian equations of solid mechanics are characterised by algebraic relaxation source terms that are capable of extending the applicability of the model not only to elasto-plastic solids but also to viscous and inviscid fluid flows. Within this theoretical framework, a first order hyperbolic multi-phase model is formulated, by which compressible Newtonian and non-Newtonian, inviscid and viscous fluids as well as elasto-plastic solids can be described. The question arises whether it is possible to compare the original BN model with the multi-phase model derived from SHTC theory. This can be done if the SHTC system is rewritten in terms of the equations of conservation of mass, phase momenta and phase energies. This structure can be referred to as form of Baer-Nunziato type. In addition, since the BN model is one of the most popular mathematical models for describing two-phase flow, there are many works that address it numerically. However, only a very limited number of publications exist on the mathematical and computational issues of BN models for multiphase flows describing more than two phases. Therefore, it is very interesting to numerically address the multi-phase BN-type type model derived from SHTC theory, which includes the challenges related to the GPR model of continuum mechanics. The resulting BN-type form is given by a large non-linear PDEs system, which includes highly non-linear stiff algebraic source terms as well as non-conservative products. In this thesis, a simplified version of the SHTC BN-type model is addressed numerically, neglecting the phase pressure relaxation, the temperature relaxation and assuming the absence of phase transformations. The differente challenges presented by the model are tackled by adopting a source operator splitting. The homogeneous part of the PDE system is discretized with a MUSCL-Hancock finite volume scheme using a primitive variable reconstruction and positivity preserving limiting, combined with a path-conservative technique to deal with the non-conservative products. Furthermore, the scheme employs semi-analytical time-integration methods for th e stiff source terms governing the various relaxation processes. Concerning the applicability of the models presented in this thesis for the solution of different problems, the resulting aforementioned semi-implicit algorithms for the BN models are first validated on a set of classical incompressible Navier-Stokes test problems and subsequently also by adding a fixed and moving solid phase.Most importantly, however, in this thesis I provide results for complex flows resulting from the interaction of three different phases including gases, liquids and solids. Therefore, results are shown for multiphase flows in the limit behaviour of the Newtonian inviscid and viscous fluid, as well as in the limit of nonlinear hyperelasticity for phases behaving as elastic and elasto-plastic solids. In both cases the numerical results are comparable with results obtained from established standard models, i.e. the Euler or Navier-Stokes equations for fluids, or the classical hypo-elastic model with plasticity, but, notably, everything within a unified multi-phase model of continuum mechanics.