Keh-Ming Shyue

Three-Dimensional Front Tracking

We describe a three-dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two-dimensional computations, we expect three-dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability.

A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions

We present a simple volume-of-fluid approach to interface tracking for inviscid compressible multicomponent flow problems in two space dimensions. The algorithm uses a uniform Cartesian grid with some grid cells subdivided by tracked interfaces, approximately aligned with the material interfaces in the flow field. A standard volume-moving procedure that consists of two basic steps: (1) the update of a discrete set of volume fractions from the current time to the next and (2) the reconstruction of interfaces from the resulting volume fractions, is employed to find the new location of the tracked interfaces in piecewise linear form at the end of a time step. As in the previous work by LeVeque and the author to front tracking based on a surface-moving procedure (R.J. LeVeque, K.-M. Shyue, Two-dimensional front tracking based on high-resolution wave propagation methods, J. Comput. Phys. 123 (1996) 354-368), a conservative high-resolution wave propagation method is applied on the resulting slightly non-uniform grid to update all the cell values, while the stability of the method is maintained by using a large time step idea even in the presence of small cells and the use of a time step with respect to the uniform grid cells. We validate our algorithm by performing the simulation of a Mach 1.22 shock wave in air over a circular R22 gas bubble, where sensible agreement of some key flow features of the computed solutions are observed when direct comparison of our results are made with the existing experimental and numerical ones that appear in the literature. Other computations are also presented that show the feasibility of the algorithm together with a mixture type of the model equations developed by the author (K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Gruneisen equation of state, J. Comput. Phys. 171 (2001) 678-707) for practical multicomponent problems with general compressible materials characterized by a Mie-Gruneisen form of the equation of state.

One-dimensional front tracking based on high resolution wave propagation methods

A simple approach to shock tracking is presented in conjunction with conservative high resolution shock-capturing methods in one space dimension. An underlying uniform grid is used with additional grid interfaces introduced at appropriate points for tracked shocks. Conservative high resolution methods based on the large time step wave propagation approach are used on the resulting nonuniform grid. This method is stable even if some of the small cells created by the tracked interface are orders of magnitude smaller than the regular cells used to determine the time step. A fractional step method is used to handle source terms. Several calculations are presented to demonstrate the effectiveness of the method, including an unstable detonation wave calculation where mesh refinement in the reaction zone is required in addition to shock tracking. Stability and accuracy results of the method are also shown for some sample problems. The basic ideas described here can be extended to two space dimensions, as will be...

A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves

We model liquid-gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel-Petitpas-Berry (Saurel et al., 2009) 9]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxation terms to model heat and mass transfer and hence liquid-vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid-vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics.

Time evolution of cosmic-ray modified plane shocks

We have developed a novel computer code designed to follow the evolution of cosmic-ray modified shocks, including the full momentum dependence of the particles for a realistic diffusion coefficient model. In this form the problem is technically very difficult because one needs to cover a wide range of diffusive scales, beginning with those slightly larger than the physical shock thickness. With most finite difference schemes for Eulers equations, the numerical shock thickness is at least one zone across, so this provides a lower bound on the physical scale for diffusive transport computation. Our code uses subzone shock tracking and multilevel adaptive mesh refinement to provide enhanced spatial resolution around shocks at a modest cost compared to the coarse grid and vastly improved cost effectiveness compared to a uniform, highly refined grid. We present and discuss the implications from our initial results.

An Eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach

Article history: We describe a novel interface-sharpening approach for efficient numerical resolution of a compressible homogeneous two-phase flow governed by a quasi-conservative five-equation model of Allaire et al. (2001) (1). The algorithm uses a semi-discrete wave propagation method to find approximate solution of this model numerically. In the algorithm, in regions near the interfaces where two different fluid components are present within a cell, the THINC (Tangent of Hyperbola for INterface Capturing) scheme is used as a basis for the reconstruction of a sub-grid discontinuity of volume fractions at each cell edge, and it is complemented by a homogeneous-equilibrium-consistent technique that is derived to ensure a consistent modeling of the other interpolated physical variables in the model. In regions away from the interfaces where the flow is single phase, standard reconstruction scheme such as MUSCL or WENO can be used for obtaining high-order interpolated states. These reconstructions are then used as the initial data for Riemann problems, and the resulting fluctuations form the basis for the spatial discretization. Time integration of the algorithm is done by employing a strong stability-preserving Runge-Kutta method. Numerical results are shown for sample problems with the Mie-Gruneisen equation of state for characterizing the materials of interests in both one and two space dimensions that demonstrate the feasibility of the proposed method for interface-sharpening of compressible two-phase flow. To demonstrate the competitiveness of our approach, we have also included results obtained using the anti-diffusion interface sharpening method.

A Volume-of-fluid Type Algorithm for Compressible Two-phase Flows

We present a simple approach to the computation of a simplified two-phase flow model involving gases and liquids separated by interfaces in multiple space dimensions. In contrast to the many popular techniques which are mainly concerned with the incompressible flow, we consider a compressible version of the model equations without the effect of surface tension and viscosity. We use the iy-law and the Tait equation of states for approximating the material property of the gas and the liquid in a respective manner. The algorithm uses a volume-of-fluid formulation of the equations together with a stiffened gas equation of state that is derived to give an approximate model for the mixture of more than one phase of the fluids within grid cells. A standard high-resolution shock capturing method based on a wave-propagation view-point is employed to solve the proposed model. We show results of some preliminary calculations that illustrate the viability of the method to practical application without the occurrence of any spurious oscillation in the pressure near the interfaces. This includes results of a planar shock wave in water over a bubble of air.

A high-resolution mapped grid algorithm for compressible multiphase flow problems

We describe a simple mapped-grid approach for the efficient numerical simulation of compressible multiphase flow in general multi-dimensional geometries. The algorithm uses a curvilinear coordinate formulation of the equations that is derived for the Euler equations with the stiffened gas equation of state to ensure the correct fluid mixing when approximating the equations numerically with material interfaces. A @c-based and a @a-based model have been described that is an easy extension of the Cartesian coordinates counterpart devised previously by the author [30]. A standard high-resolution mapped grid method in wave-propagation form is employed to solve the proposed multiphase models, giving the natural generalization of the previous one from single-phase to multiphase flow problems. We validate our algorithm by performing numerical tests in two and three dimensions that show second order accurate results for smooth flow problems and also free of spurious oscillations in the pressure for problems with interfaces. This includes also some tests where our quadrilateral-grid results in two dimensions are in direct comparisons with those obtained using a wave-propagation based Cartesian grid embedded boundary method.

A Moving-Boundary Tracking Algorithm for Inviscid Compressible Flow

This paper is concerned with the development of a Cartesian-grid approach for the numerical simulation of general (single or multicomponent) compressible flow problems with complex moving geometries. As a preliminary, in this work, we are interested in a class of moving objects that undergo solely rigid-body motion with the propagation speeds determined by either a given function of time or the Newton’s second law of motion. We use the two-dimensional Euler equations of gas dynamics as a model system in which the principal motion of the single-component flow is governed by

An Eulerian Interface-Sharpening Algorithm for Compressible Gas Dynamics

We describe a novel Eulerian interface-sharpening approach for the efficient numerical resolution of contact discontinuities arising from inviscid compressible flow in more than one space dimension. The algorithm uses the single-phase compressible Euler equations as the model system, and introduces auxiliary differential terms to the model so as to neutralize numerical diffusion that is inevitable when the original Euler system is solved by a diffused interface method. A standard fractional-step method is employed to solve the proposed model equations in two steps, yielding an easy implementation of the algorithm. Preliminary results obtained using an anti-diffusion based model system are shown to demonstrate the feasibility of the algorithm for practical problems.