A Diffuse Interface Model with Immiscibility Preservation.

A new, simple, and computationally efficient interface capturing scheme based on a diffuse interface approach is presented for simulation of compressible multiphase flows. Multi-fluid interfaces are represented using field variables (interface functions) with associated transport equations that are augmented, with respect to an established formulation, to enforce a selected interface thickness. The resulting interface region can be set just thick enough to be resolved by the underlying mesh and numerical method, yet thin enough to provide an efficient model for dynamics of well-resolved scales. A key advance in the present method is that the interface regularization is asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed. It incorporates first-order pressure and velocity non-equilibrium effects while preserving interface conditions for equilibrium flows, even within the thin diffused mixture region. We first quantify the improved convergence of this formulation in some widely used one-dimensional configurations, then show that it enables fundamentally better simulations of bubble dynamics. Demonstrations include both a spherical bubble collapse, which is shown to maintain excellent symmetry despite the Cartesian mesh, and a jetting bubble collapse adjacent a wall. Comparisons show that without the new formulation the jet is suppressed by numerical diffusion leading to qualitatively incorrect results.

Least-squares dynamic approximation method for evolution of uncertainty in initial conditions of dynamical systems.

We describe an approximation method to solve the probability density function transport equation, i.e., the Liouville equation, which is encountered in the evolution of uncertainty of the initial values of dynamical systems. A state-space based method is formulated using a least-squares technique that preserves the parabolic nature of the Liouville equation and is flexible in terms of accuracy of representation. This method is based on a global approximation in terms of analytical elementary functions with unknown parameters, whose evolution equations are determined by a global least-squares approximation. The realizability conditions of the probability density, i.e., the non-negativity and normalization conditions are enforced at all times. The method is successfully evaluated in a number of scenarios including the uncertainty evolution in a system governed by a Riccati equation and a particle moving in a fluid under the influence of Stokes drag force. The results obtained in our examples exhibit a reasonable good agreement when compared with the solution of the probability transport equation using the method of characteristics. The cost of the method is proportional to the cost of solving the deterministic system and the number of parameters used to approximate the probability density function, a feature that can make the present method very advantageous in comparison with other methods in problems involving a large number of dimensions.