A Conservative Modification to the Ghost Fluid Method for Compressible Multiphase Flows

A conservative modification to the ghost fluid method (GFM) is developed for compressible multiphase flows. The motivation is to eliminate or reduce the conservation error of the GFM without affecting its performance. We track the conservative variables near the material interface and use this information to modify the numerical solution for an interfacing cell when the interface has passed the cell. The modification procedure can be used on the GFM with any base schemes. In this paper we use the fifth order finite difference WENO scheme for the spatial discretization and the third order TVD Runge-Kutta method for the time discretization. The level set method is used to capture the interface. Numerical experiments show that the method is at least mass and momentum conservative and is in general comparable in numerical resolution with the original GFM.     

A Contact SPH Method with High-Order Limiters for Simulation of Inviscid Compressible Flows

In this paper, we study a class of contact smoothed particle hydrodynamics (SPH) by introducing Riemann solvers and using high-order limiters. In particular, a promising concept of WENO interpolation as limiter is presented in the reconstruction process. The physical values relating interactional particles used as the initial values of the Riemann problem can be reconstructed by the Taylor series expansion. The contact solvers of the Riemann problem at contact points are incorporated in SPH approximations. In order to keep the fluid density at the wall rows to be consistent with that of the inner fluid wall boundaries, several lines of dummy particles are placed outside of the solid walls, which are assigned according to the initial configuration. At last, the method is applied to compressible flows with sharp discontinuities such as the collision of two strong shocks and the interaction of two blast waves and so on. The numerical results indicate that the method is capable of handling sharp discontinuity and efficiently reducing unphysical oscillations.     

A Front Tracking Method for the Simulation of Compressible Multimedium Flows

A front tracking method combined with the real ghost fluid method (RGFM) is proposed for simulations of fluid interfaces in two-dimensional compressible flows. In this paper the Riemann problem is constructed along the normal direction of interface and the corresponding Riemann solutions are used to track fluid interfaces. The interface boundary conditions are defined by the RGFM, and the fluid interfaces are explicitly tracked by several connected marker points. The Riemann solutions are also used directly to update the flow states on both sides of the interface in the RGFM. In order to validate the accuracy and capacity of the new method, extensive numerical tests including the bubble advection, the Sod tube, the shock-bubble interaction, the Richtmyer-Meshkov instability and the gas-water interface, are simulated by using the Euler equations. The computational results are also compared with earlier computational studies and it shows good agreements including the compressible gas-water system with large density differences.     

Practical Techniques in Ghost Fluid Method for Compressible Multi-Medium Flows

The modified ghost fluid method (MGFM), due to its reasonable treatment for ghost fluid state, has been shown to be robust and efficient when applied to compressible multi-medium flows. Other feasible definitions of the ghost fluid state, however, have yet to be systematically presented. By analyzing all possible wave structures and relations for a multi-medium Riemann problem, we derive all the conditions to define the ghost fluid state. Under these conditions, the solution in the real fluid region can be obtained exactly, regardless of the wave pattern in the ghost fluid region. According to the analysis herein, a practical ghost fluid method (PGFM) is proposed to simulate compressible multi-medium flows. In contrast with the MGFM where three degrees of freedom at the interface are required to define the ghost fluid state, only one degree of freedom is required in this treatment. However, when these methods proved correct in theory are used in computations for the multi-medium Riemann problem, numerical errors at the material interface may be inevitable. We show that these errors are mainly induced by the single-medium numerical scheme in essence, rather than the ghost fluid method itself. Equipped with some density-correction techniques, the PGFM is found to be able to suppress these unphysical solutions dramatically.     

A Cartesian-to-Curvilinear Coordinate Transformation in Modified Ghost Fluid Method for Compressible Multi-Material Flows

Modified ghost fluid method (MGFM) provides us an effective manner to simulate compressible multi-material flows. In most cases, the applications are limited in relatively simple geometries described by Cartesian grids. In this paper, the MGFM treatment with the level set (LS) technique is extended to curvilinear coordinate systems. The chain rule of differentiation (applicable to general curvilinear coordinates) and the orthogonal transformation (applicable to orthogonal curvilinear coordinates) are utilized to deduce the Cartesian-to-curvilinear coordinate transformation, respectively. The relationship between these two transformations for the extension of the LS/MGFM algorithm is analyzed in theory. It is shown that these two transformations are equivalent for orthogonal curvilinear grids. The extension of the LS/MGFM algorithm using the chain rule has a wider range of applications, as there is essentially no requirement for the orthogonality of the grids. Several challenging problems in two- or three-dimensions are utilized to validate the developed algorithm in curvilinear coordinates. The results indicate that this algorithm enables a simple and effective implementation for simulating interface evolutions, as in Cartesian coordinate systems. It has the potential to be applied in more complex computational domains.     

An Interface-Capturing Method for Resolving Compressible Two-Fluid Flows with General Equation of State

In this study, a stable and robust interface-capturing method is developed to resolve inviscid, compressible two-fluid flows with general equation of state (EOS). The governing equations consist of mass conservation equation for each fluid, momentum and energy equations for mixture and an advection equation for volume fraction of one fluid component. Assumption of pressure equilibrium across an interface is used to close the model system. MUSCL-Hancock scheme is extended to construct input states for Riemann problems, whose solutions are calculated using generalized HLLC approximate Riemann solver. Adaptive mesh refinement (AMR) capability is built into hydrodynamic code. The resulting method has some advantages. First, it is very stable and robust, as the advection equation is handled properly. Second, general equation of state can model more materials than simple EOSs such as ideal and stiffened gas EOSs for example. In addition, AMR enables us to properly resolve flow features at disparate scales. Finally, this method is quite simple, time-efficient and easy to implement.     

A Parallel,Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids

A reconstruction-based discontinuous Galerkin method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. In this method, an in-cell reconstruction is used to obtain a higher-order polynomial representation of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction is used to obtain a continuous polynomial solution on the union of two neighboring, interface-sharing cells. The in-cell reconstruction is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. The inter-cell reconstruction is devised to remove an interface discontinuity of the solution and its derivatives and thus to provide a simple, accurate, consistent, and robust approximation to the viscous and heat fluxes in the Navier-Stokes equations. A parallel strategy is also devised for the resulting reconstruction discontinuous Galerkin method, which is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model. The RDG method is used to compute a variety of compressible flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.     

A High Order Bound Preserving Finite Difference Linear Scheme for Incompressible Flows

We propose a high order finite difference linear scheme combined with ahigh order bound preserving maximum-principle-preserving (MPP) flux limiter tosolve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much lesscostly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP fluxlimiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guiding-center model and the incompressible Euler system. The comparison between the linearand WENO type schemes, with and without the MPP flux limiter, will demonstrate thegood performance of our proposed approach.     

A Gas Kinetic Scheme for the Simulation of Compressible Multicomponent Flows

In this paper, a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK model in [A. D. Kotelnikov and D. C. Montgomery, A Kinetic Method for Computing Inhomogeneous Fluid Behavior, J. Comput. Phys. 134 (1997) 364-388]. Different from the conventional BGK model, the collisions between different species are taken into consideration. Based on the Chapman-Enskog expansion, the corresponding macroscopic equations are derived from this two-species model. Because of the relaxation terms in the governing equations, the method of operator splitting is applied. In the hyperbolic part, the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method. Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows. The theoretical analysis on the spurious oscillations at the interface is also presented.     

A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids

A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method on non-uniform, curvilinear structured grids to simulate the compressible turbulent flows. The designed scheme utilizes two types of candidate interpolants including a higher order linear-weight polynomial as high as eleven and a THINC (Tangent of Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark broadband turbulence problem as well as real-life wall-bounded turbulent flows has been carried out to demonstrate the potential implementation of the present higher order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of compressible turbulence.     

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However, such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.     

High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

Within the projection schemes for the incompressible Navier-Stokes equations (namely "pressure-correction" method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.     

A Compact High Order Space-Time Method for Conservation Laws

This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.     

High Order Numerical Methods for the Dynamic SGS Model of Turbulent Flows with Shocks

Simulation of turbulent flows with shocks employing subgrid-scale (SGS) filtering may encounter a loss of accuracy in the vicinity of a shock. This paper addresses the accuracy improvement of LES of turbulent flows in two ways: (a) from the SGS model standpoint and (b) from the numerical method improvement standpoint. In an internal report, Kotov et al. ("High Order Numerical Methods for large eddy simulation (LES) of Turbulent Flows with Shocks", CTR Tech Brief, Oct. 2014, Stanford University), we performed a preliminary comparative study of different approaches to reduce the loss of accuracy within the framework of the dynamic Germano SGS model. The high order low dissipative method of Yee & Sjögreen (2009) using local flow sensors to control the amount of numerical dissipation where needed is used for the LES simulation. The considered improved dynamics model approaches include applying the one-sided SGS test filter of Sagaut & Germano (2005) and/or disabling the SGS terms at the shock location. For Mach 1.5 and 3 canonical shock-turbulence interaction problems, both of these approaches show a similar accuracy improvement to that of the full use of the SGS terms. The present study focuses on a five levels of grid refinement study to obtain the reference direct numerical simulation (DNS) solution for additional LES SGS comparison and approaches. One of the numerical accuracy improvements included here applies Harten's subcell resolution procedure to locate and sharpen the shock, and uses a one-sided test filter at the grid points adjacent to the exact shock location.     

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids. In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates, we develop a combination of the discontinuous Galerkin finite element (DGFE) method for the space discretization and the backward difference formulae (BDF) for the time discretization. Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step, we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step. Finally, the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.     

Extension and Comparative Study of AUSM-Family Schemes for Compressible Multiphase Flow Simulations

Several recently developed AUSM-family numerical flux functions (SLAU, SLAU2, AUSM⁺-up2, and AUSMPW+) have been successfully extended to compute compressible multiphase flows, based on the stratified flow model concept, by following two previous works: one by M.-S. Liou, C.-H. Chang, L. Nguyen, and T.G. Theofanous [AIAA J. 46:2345-2356, 2008], in which AUSM⁺-up was used entirely, and the other by C.-H. Chang, and M.-S. Liou [J. Comput. Phys. 225:840-873, 2007], in which the exact Riemann solver was combined into AUSM⁺-up at the phase interface. Through an extensive survey by comparing flux functions, the following are found: (1) AUSM⁺-up with dissipation parameters of K_p and K_u equal to 0.5 or greater, AUSMPW+, SLAU2, AUSM⁺-up2, and SLAU can be used to solve benchmark problems, including a shock/water-droplet interaction; (2) SLAU shows oscillatory behaviors [though not as catastrophic as those of AUSM⁺ (a special case of AUSM⁺-up withK_p=K_u=0)] due to insufficient dissipation arising from its ideal-gas-based dissipation term; and (3) when combined with the exact Riemann solver, AUSM⁺-up (K_p=K_u=1), SLAU2, and AUSMPW+ are applicable to more challenging problems with high pressure ratios.     

A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems

In this paper, we develop a novel approach by combining a new robust finite difference Hermite weighted essentially non-oscillatory (HWENO) method [51] with the modified ghost fluid method (MGFM) [25] to simulate the compressible two-medium flow problems. The main idea is that we first use the technique of the MGFM to transform a two-medium flow problem to two single-medium cases by defining the ghost fluids status based on the predicted interface status. Then the efficient and robust HWENO finite difference method is applied for solving the single-medium flow cases. By using immediate neighbor information to deal with both the solution and its derivatives, the fifth order finite difference HWENO scheme adopted in this paper is more compact and has higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu [14]. Furthermore, by combining the HWENO scheme with the MGFM to simulate the two-medium flow problems, less ghost point information is needed than that in using the classical WENO scheme in order to obtain the same numerical accuracy. Various one-dimensional and two-dimensional two-medium flow problems are solved to illustrate the good performances of the proposed method.     

A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.     

Fourier Collocation and Reduced Basis Methods for Fast Modeling of Compressible Flows

A projection-based reduced order model (ROM) based on the Fourier collocation method is proposed for compressible flows. The incorporation of localized artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows. Furthermore, for Euler systems, ROMs built on the conservative and the skew-symmetric forms of the governing equation are compared. To ensure efficiency, the discrete empirical interpolation method (DEIM) is employed. An alternative reduction approach, exploring the sparsity of viscosity is also investigated for the viscous terms. A number of one- and two-dimensional benchmark cases are considered to test the performance of the proposed models. Results show that stable computations for shock-dominated cases can be achieved with ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering. Under the same parameters, the skew-symmetric form shows better robustness and accuracy than its conservative counterpart, while the conservative form is superior in terms of efficiency.     

A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.