A Preliminary Calculation of Three-Dimensional Unsteady Underwater Cavitating Flows Near Incompressible Limit

Recently, cavitated flows over underwater submerged bodies have attracted researchers to simulate large scale cavitation. Comparatively Computational Fluid Dynamics (CFD) approaches have been used widely and successfully to model developed cavitation. However, it is still a great challenge to accurately predict cavitated flow phenomena associated with interface capturing, viscous effects, unsteadiness and three-dimensionality. In this study, we consider the preconditioned three-dimensional multiphase Navier-Stokes equations comprised of the mixture density, mixture momentum and constituent volume fraction equations. A dual-time implicit formulation with LU Decomposition is employed to accommodate the inherently unsteady physics. Also, we adopt the Roe flux splitting method to deal with flux discretization in space. Moreover, time-derivative preconditioning is used to ensure well-conditioned eigenvalues of the high density ratio two-phase flow system to achieve computational efficiency. Validation cases include an unsteady 3-D cylindrical headform cavitated flow and an 2-D convergent-divergent nozzle channel cavity-problem.     

An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution approach has been applied to a high order spectral volume (SV) method for unstructured tetrahedral grids. The LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with a LU decomposition. The compact feature of SV reconstruction facilitates the efficient solution algorithm even for high order discretizations. The developed implicit solver has shown more than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme for typical inviscid and viscous problems. A convergence to a high order solution for high Reynolds number transonic flow over a 3D wing with a one equation turbulence model is also indicated.     

Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

In this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D'Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.     

An Efficient Parallel/Unstructured-Multigrid Implicit Method for Simulating 3D Fluid-Structure Interaction

A finite volume (FV) method for simulating 3D Fluid-Structure Interaction (FSI) is presented in this paper. The fluid flow is simulated using a parallel unstructured multigrid preconditioned implicit compressible solver, whist a 3D matrix-free implicit unstructured multigrid finite volume solver is employed for the structural dynamics. The two modules are then coupled using a so-called immersed membrane method (IMM). Large-Eddy Simulation (LES) is employed to predict turbulence. Results from several moving boundary and FSI problems are presented to validate proposed methods and demonstrate their efficiency.     

Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit,Nesterov Accelerated Schemes

We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.     

Performance of Low-Dissipation Euler Fluxes and Preconditioned LU-SGS at Low Speeds

In low speed flow computations, compressible finite-volume solvers are known to a) fail to converge in acceptable time and b) reach unphysical solutions. These problems are known to be cured by A) preconditioning on the time-derivative term, and B) control of numerical dissipation, respectively. There have been several methods of A) and B) proposed separately. However, it is unclear which combination is the most accurate, robust, and efficient for low speed flows. We carried out a comparative study of several well-known or recently-developed low-dissipation Euler fluxes coupled with a preconditioned LU-SGS (Lower-Upper Symmetric Gauss-Seidel) implicit time integration scheme to compute steady flows. Through a series of numerical experiments, accurate, efficient, and robust methods are suggested for low speed flow computations.     

Adaptive Fully Implicit Simulator with Multilevel Schwarz Methods for Gas Reservoir Flows in Fractured Porous Media

Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of the gas reservoir flow in fractured porous media. We develop and study a parallel and highly scalable reservoir simulator based on an adaptive fully implicit scheme and an inexact Newton type method to solve this dual-continuum mathematical model. In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully implicit discretization, which is second-order and L-stable. And then we focus on the family of Newton–Krylov methods for the solution of a large sparse nonlinear system of equations arising at each time step. To accelerate the convergence and improve the scalability of the solver, a class of multilevel monolithic additive Schwarz methods is employed for preconditioning. Numerical results on a set of ideal as well as realistic flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors are also carried out to show that the proposed reservoir simulator is highly scalable.     

Unified Gas-Kinetic Wave-Particle Methods VI: Disperse Dilute Gas-Particle Multiphase Flow

A coupled gas-kinetic scheme (GKS) and unified gas-kinetic wave-particle (UGKWP) method for the disperse dilute gas-particle multiphase flow is proposed. In the two-phase flow, the gas phase is always in the hydrodynamic regime and is followed by GKS for the Navier-Stokes solution. The particle phase is solved by UGKWP in all regimes from particle trajectory crossing to the hydrodynamic wave interaction with the variation of particle’s Knudsen number. In the intensive particle collision regime, the UGKWP gives a hydrodynamic wave representation for the particle phase and the GKS-UGKWP for the two-phase flow reduces to the two-fluid Eulerian-Eulerian (EE) model. In the rarefied regime, the UGKWP tracks individual particle and the GKS-UGKWP goes back to the Eulerian-Lagrangian (EL) formulation. In the transition regime for the solid particle, the GKS-UGKWP takes an optimal choice for the wave and particle decomposition for the solid particle phase and connects the EE and EL methods seamlessly. The GKS-UGKWP method will be tested in all flow regimes with a large variation of Knudsen number for the solid particle transport and Stokes number for the two-phase interaction. It is confirmed that GKS-UGKWP is an efficient and accurate multiscale method for the gas-particle two-phase flow.     

Three-Dimensional Lattice Boltzmann Simulation of Two-Phase Flow Containing a Deformable Body with a Viscoelastic Membrane

The lattice Boltzmann method (LBM) with an elastic model is applied to the simulation of two-phase flows containing a deformable body with a viscoelastic membrane. The numerical method is based on the LBM for incompressible two-phase fluid flows with the same density. The body has an internal fluid covered by a viscoelastic membrane of a finite thickness. An elastic model is introduced to the LBM in order to determine the elastic forces acting on the viscoelastic membrane of the body. In the present method, we take account of changes in surface area of the membrane and in total volume of the body as well as shear deformation of the membrane. By using this method, we calculate two problems, the behavior of an initially spherical body under shear flow and the motion of a body with initially spherical or biconcave discoidal shape in square pipe flow. Calculated deformations of the body (the Taylor shape parameter) for various shear rates are in good agreement with other numerical results. Moreover, tank-treading motion, which is a characteristic motion of viscoelastic bodies in shear flows, is simulated by the present method.     

Vectorial Kinetic Relaxation Model with Central Velocity. Application to Implicit Relaxations Schemes

We apply flux vector splitting (FVS) strategy to the implicit kinetic schemes for hyperbolic systems. It enables to increase the accuracy of the method compared to classical kinetic schemes while still using large time steps compared to the characteristic speeds of the problem. The method also allows to tackle multi-scale problems, such as the low Mach number limit, for which wave speeds with large ratio are involved. We present several possible kinetic relaxation schemes based on FVS and compare them on one-dimensional test-cases. We discuss stability issues for this kind of method.     

An Efficient,Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.     

Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

The purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks, T. J. R. Hughes, "An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves", Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.     

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is O(h⁴) for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an O(h⁴+k⁴) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one- and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.     

A Parallel Domain Decomposition Algorithm for Simulating Blood Flow with Incompressible Navier-Stokes Equations with Resistive Boundary Condition

We introduce and study a parallel domain decomposition algorithm for the simulation of blood flow in compliant arteries using a fully-coupled system of nonlinear partial differential equations consisting of a linear elasticity equation and the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition plays an interesting role in the accuracy of the blood flow simulation and we provide a numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition method for solving the fully coupled nonlinear system on a supercomputer with a few hundred processors.     

Three-Dimensional Simulation of Balloon Dynamics by the Immersed Boundary Method Coupled to the Multiple-Relaxation-Time Lattice Boltzmann Method

The immersed boundary method (IBM) has been popular in simulating fluid structure interaction (FSI) problems involving flexible structures, and the recent introduction of the lattice Boltzmann method (LBM) into the IBM makes the method more versatile. In order to test the coupling characteristics of the IBM with the multiple-relaxation-time LBM (MRT-LBM), the three-dimensional (3D) balloon dynamics, including inflation, release and breach processes, are simulated. In this paper, some key issues in the coupling scheme, including the discretization of 3D boundary surfaces, the calculation of boundary force density, and the introduction of external force into the LBM, are described. The good volume conservation and pressure retention properties are verified by two 3D cases. Finally, the three FSI processes of a 3D balloon dynamics are simulated. The large boundary deformation and oscillation, obvious elastic wave propagation, sudden stress release at free edge, and recoil phenomena are all observed. It is evident that the coupling scheme of the IBM and MRT-LBM can handle complicated 3D FSI problems involving large deformation and large pressure gradients with very good accuracy and stability.     

Time Implicit Unified Gas Kinetic Scheme for 3D Multi-Group Neutron Transport Simulation

In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D multi-group neutron transport equation with delayed neutron is developed. The explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport, the microscopic angular flux and the macroscopic scalar flux are fully coupled in an implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step. Mathematically, the current scheme has the asymptotic preserving (AP) property in recovering automatically the diffusion solution in the continuum regime. Since the explicit scanning along neutron traveling direction within the computational domain is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other schemes, such as the explicit UGKS, the commonly used finite difference, and finite volume methods. This study shows that the IUGKS can be used faithfully to study neutron transport in practical engineering applications.     

New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.     

Development of a Volume of Fluid Method for Computing Interfacial Incompressible Fluid Flows

This study is aimed to develop a volume of fluid (VOF) method to capture the free surface flow. The incompressible two-phase flow is computed by second-order Adams-Bashforth algorithm with a uniform staggered Cartesian grid arrangement. The tangent of hyperbola for interface capturing (THINC) scheme and weighted linear interface calculation (WLIC) based geometrical reconstruction procedure have been implemented in the operator-splitting method for the VOF method. The proposed VOF method preserves mass well, and the interface normal vector can be easily estimated from the level set (LS) function. The LS function, which is a continuous signed distance function around the interface, is represented by solving the re-initialization equation. Numerical results using the present scheme are compared with experimental data and other numerical results in the Rayleigh-Taylor instability, dam-break flow, travelling solitary wave, Kelvin-Helmholtz instability, rising bubble and merging bubble problems. We also present numerical results in detail between computations made with the proposed VOF method and computations made with the conventional LS method.     

Lattice Boltzmann Simulation of Nucleate Pool Boiling in Saturated Liquid

A thermal lattice Boltzmann method (LBM) for two-phase fluid flows in nucleate pool boiling process is proposed. In the present method, a new function for heat transfer is introduced to the isothermal LBM for two-phase immiscible fluids with large density differences. The calculated temperature is substituted into the pressure tensor, which is used for the calculation of an order parameter representing two phases so that bubbles can be formed by nucleate boiling. By using this method, two-dimensional simulations of nucleate pool boiling by a heat source on a solid wall are carried out with the boundary condition for a constant heat flux. The flow characteristics and temperature distribution in the nucleate pool boiling process are obtained. It is seen that a bubble nucleation is formed at first and then the bubble grows and leaves the wall, finally going up with deformation by the buoyant effect. In addition, the effects of the gravity and the surface wettability on the bubble diameter at departure are numerically investigated. The calculated results are in qualitative agreement with other theoretical predictions with available experimental data.     

An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi-implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.