A Second Order Fully-discrete Linear Energy Stable Scheme for a Binary Compressible Viscous Fluid Model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flow mixtures derived from the generalized Onsager Principle. The hydrodynamic model not only possesses the variational structure, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model in an equivalent form using the energy quadratization method and then discretize the reformulated model to obtain a semi-discrete partial differential equation system using the Crank-Nicolson method in time. The numerical scheme so derived preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using the 2nd order finite difference method, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and two numerical examples on phase separation due to the spinodal decomposition in two polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications

On two-dimensional finite amplitude electro-convection in a dielectric liquid induced by a strong unipolar injection

The hydrodynamic stability of a dielectric liquid subjected to strong unipolar injection is numerically investigated. We determined the linear criterion Tc (T being the electric Rayleigh number) and finite amplitude one Tf over a wide range of the mobility parameter M. A noticeable discrepancy is shown for Tf between our numerical prediction and the value predicted by stability analysis, which is due to the velocity field used in stability analysis. Recent studies revealed a transition of the flow structure from one cell to two with an increase in T. We demonstrate that this transition results in a new subcritical bifurcationMinisterio de Ciencia y Tecnología FIS2011-25161Junta de Andalucía P10-FQM-5735Junta de Andalucía P09-FQM-458

Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library

In this paper we employ two implementations of the fictitious domain (FD) method to simulate water-entry and water-exit problems and demonstrate their ability to simulate practical marine engineering problems. In FD methods, the fluid momentum equation is extended within the solid domain using an additional body force that constrains the structure velocity to be that of a rigid body. Using this formulation, a single set of equations is solved over the entire computational domain. The constraint force is calculated in two distinct ways: one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method and another using a fully-Eulerian approach of the Brinkman penalization (BP) method. Both FSI strategies use the same multiphase flow algorithm that solves the discrete incompressible Navier-Stokes system in conservative form. A consistent transport scheme is employed to advect mass and momentum in the domain, which ensures numerical stability of high density ratio multiphase flows involved in practical marine engineering applications. Example cases of a free falling wedge (straight and inclined) and cylinder are simulated, and the numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some parts of it for the reader's convenienc

Entropy-Production-Rate-Preserving Algorithms for Thermodynamically Consistent Nonisothermal Models of Incompressible Binary Fluids

We derive a thermodynamically consistent, non-isothermal, hydrodynamic model for incompressible binary fluids following the generalized Onsager principle and Boussinesq approximation. This model preserves not only the volume of each fluid phase but also the positive entropy production rate under thermodynamically consistent boundary conditions. Guided by the thermodynamical consistency of the model, a set of second order structure-preserving numerical algorithms are devised to solve the governing partial differential equations along with consistent boundary conditions in the model, which preserve the entropy production rate as well as the volume of each fluid phase at the discrete level. Several numerical simulations are carried out using an efficient adaptive time-stepping strategy based on one of the structure-preserving schemes to simulate the Rayleigh-B\'{e}nard convection in the binary fluid and interfacial dynamics between two immiscible fluids under competing effects of the temperature gradient, gravity, and interfacial forces. Roll cell patterns and thermally induced mixing of binary fluids are observed in a rectangular region with insulated lateral boundaries and vertical ones with imposed temperature difference. Long time simulations of interfacial dynamics are performed demonstrating robust results of new structure-preserving schemes