The computation of multiphase flows presents a subtle energetic equilibrium between potential (i.e., surface) and kinetic energies. The use of traditional interface-capturing schemes provides no control over such a dynamic balance. In the spirit of the wellknown symmetry-preserving and mimetic schemes, whose physics-compatible discretizations rely upon preserving the underlying mathematical structures of the space, we identify the corresponding structure and propose a new discretization strategy for curvature. The new scheme ensures conservation of mechanical energy (i.e., surface plus kinetic) up to temporal integration. Inviscid numerical simulations are performed to show the robustness of such a method.

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The appearence of unphysical velocities in highly distorted meshes is a common problem in many simulations. In collocated meshes, this problem arises from the interpolation of the pressure gradient from faces to cells. Using an algebraic form for the classical incompressible Navier-Stokes equations, this problem is adressed. Starting from the work of F. X. Trias et. al. [FX.Trias et al. JCP 258: 246-267, 2014], a new approach for studying the Poisson equation obtained using the Fractional Step Method is found, such as a new interpolator is proposed in order to found a stable solution, which avoid the appearence of these unpleasant velocities. The stability provided by the interpolator is formally proved for cartesian meshes and its rotations, using fully-explicit time discretizations. The construction of the Poisson equation is supported on mimicking the symmetry properties of the differential operators and the Fractional Step Method. Then it is reinterpreted using a recursive application of the Fractional Step Method in order to study the system as an stationary iterative solver. Furthermore, a numerical analysis for unstructured mesh is also provided.

The essence of turbulence are the smallest scales of motion. They result from a subtle balance between convective transport and diffusive dissipation. Mathematically, these terms are governed by two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and positive-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows around complex configurations. With this in mind, a fully-conservative discretization method for general unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: it exactly preserves the symmetries of the underlying differential operators on a collocated mesh. However, any pressure-correction method on collocated grids suffer from the same drawbacks: the cell-centered velocity field is not exactly incompressible and some artificial dissipation is inevitable introduced. On the other hand, for staggered velocity fields, the projection onto a divergence-free space is a well-posed problem: given a velocity field, it can be uniquely decomposed into a solenoidal vector and the gradient of a scalar (pressure) field. This can be easily done without introducing any dissipation as it should be from a physical point-of-view. In this work, we explore the possibility to build up staggered formulations based on collocated discrete operators.

The numerical simulation of multiphase flows presents several challenges, namely the transport of different phases within de domain and the inclusion of capillary effects. Here, these are approached by enforcing a discrete physics-compatible solution. Extending our previous work on the discretization of surface tension [N. Valle, F. X. Trias, and J. Castro, "An energy-preserving level set method for multiphase flows," J. Comput. Phys., vol. 400, p. 108991, 2020] with a consistent mass and momentum transfer a fully energy-preserving multiphase flow method is presented. This numerical technique is showcased within the simulation of a falling film under several working conditions related to the normal operation of LiBr absorption chillers.