We consider a generic homogenized model of two-phase flows, obtained by averaging of balance laws for single phases. The resulting system of equations is non-strictly hyperbolic and non-conservative, i.e. it cannot be written in divergence form. This poses serious difficulties for the theoretical investigation of this system, as well as for its numerical solution.
We use a physically motivated principle in order to obtain a discretization of the non-conservative terms in the generic model and propose a numerical method on its basis. We show its accuracy and robustness on a number of test problems, and through the comparison with experimental data.
Further, we study several submodels of the generic model, characterized by the different choice of the interface parameters. In particular, we consider the model for a deflagration-to-detonation transition (DDT) in gas-permeable, reactive granular materials. For the homogeneous part of the model, we study a simple initial-value problem, the Riemann problem. We note that the non-conservative terms act only along one wave in the solution to the Riemann problem. This alows us to define a weak solution to the Riemann problem as a composition of weak solutions to conservation laws in sectors. Then, we give a physical interpretation to the situations when some waves in the solution to the Riemann problem coincide. Also, we construct the exact solution to Riemann problems and implement in a software package. With its help, we propose a number of test problems, which are intended to assess the perfomance of numerical methods for a certain type of non-conservative systems.
It appears that the solution across one wave in the Riemann problem is not unique. To deal with this, we adopt a physically motivated criterion, the evolutionarity condition. We argue that the well-known discontinuities arising in the solution of conservation laws must be evolutionary. For the classical case of strictly hyperbolic conservation laws, we show that the usual conditions on these discontinuities are equivalent to the evolutionarity criterion.
Under certain assumptions, the generic model of two-phase flows reveals the well-known Euler equations in a duct of variable cross-section. Since this system is much simpler than the generic model, its study provides deeper insight into the structure of the generic model. For the Euler equations in a duct, we show that the solution to the Riemann problem is not unique. We study the conditions, which lead to the non-uniqueness, as well as the conditions for a unique solution. In order to decide, which solution is physically relevant, we carry out 2D computations in a duct of corresponding geometry. Then, we compare the 1D solution to the Riemann problem with the averaged 2D computations. It appears that the 1D solution, picked out by the 2D computations, satisfies a kind of entropy rate admissibility criterion.
The system of the Euler equations in a duct belongs to the class of resonant non-strictly hyperbolic systems. Such systems have been studied in the literature, and it is known that one wave in the solution to the Riemann problem is not unique. To deal with it, an admissibility criterion has been proposed. We show that this criterion is actually a particular case of the evolutionarity criterion.
Finally, we solve the Riemann problem for the Euler equations in a duct exactly and propose a Godunov-type method on its basis. It employs the idea that the non-conservative terms act only along cell boundaries, so we are left with a conservation law inside a cell. The numerical experiments show excellent accuracy of the scheme.