Incompressible smoothed particle hydrodynamics simulation of multifluid flows

This paper presents an incompressible SPH (ISPH) technique to simulate multifluid flows. The SPH method is a mesh‐free particle modeling approach that can treat free surfaces and multi‐interfaces in a simple and efficient manner. The ISPH method employs an incompressible hydrodynamic formulation to solve the fluid pressure that ensures a stable pressure field. Two multifluid ISPH models are proposed following different interface treatments: the coupled ISPH model does not distinguish the different fluid phases and applies the standard ISPH technique across the interface, whereas the decoupled ISPH model first treats each fluid phase separately and then couples the different phases by applying pressure and shear stress continuities across the interface. The two proposed models were used to investigate a gravity underflow with a low density ratio in a Generalized Reservoir Hydrodynamics (GRH) flume and a horizontal lock exchange flow with a high density ratio. Comparisons with data and relevant numerical error analysis indicated that the decoupled model performed well in cases of both low and high density ratios because of the accurate treatment of interface boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: Application to particulate flow

This article discusses the application of a Lagrange multiplier‐based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two‐dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright © 1999 John Wiley & Sons, Ltd.     

Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

Advection upwinding splitting method to solve a compressible two‐fluid model

In this study, the advection upwinding splitting method (AUSM) is modified for the resolution of two‐phase mixtures with interfaces. The compressible two‐fluid model proposed by Saurel and Abgrall is chosen as the model equations. Dense and dilute phases are described in terms of the volume fraction and equations of state to represent multi‐phase mixtures. Test cases involving an air–water shock tube, water faucet, and dilute particulate turbulent flows through a 90° bend are used to verify the current work. It is shown that the AUSM based on flux differences (AUSMD) contains the mechanism to correctly capture the contact discontinuity and interfaces between phases. In addition, a successful application to dilute particulate turbulence flows by the AUSMD is demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation of free surface flows using the flux‐difference splitting scheme on the hybrid Cartesian/immersed boundary method

In this study, a method is developed to simulate the interaction between free surface flows and moving or deforming boundaries using the flux‐difference splitting scheme on the hybrid Cartesian/immersed boundary method. At each physical time step, the boundary is defined by an unstructured triangular surface grid. Immersed boundary (IB) nodes are distributed inside an instantaneous fluid domain based on edges crossing the boundary. At an IB node, dependent variables are reconstructed along the local normal line to the boundary. Inviscid fluxes are computed using Roe's flux‐difference splitting scheme for immiscible and incompressible fluids. The free surface is considered as a contact discontinuity in the density field. The motion of free surface is captured without any additional treatment along the fluid interface. The developed code is validated by comparisons with other experimental and computational results for a piston‐type wave maker, impulsive motion of a submerged circular cylinder, flow around a submerged hydrofoil, and Rayleigh–Taylor instability. The developed code is applied to simulate wave generation due to a continuously deforming bed beneath the free surface. The violent motion of a free surface caused by sloshing in a spherical tank is simulated. In this case, the free surface undergoes breakup and reconnection. Copyright © 2011 John Wiley & Sons, Ltd.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Instabilities during batch sedimentation in geometries containing obstacles: A numerical and experimental study

Batch sedimentation of non‐colloidal particle suspensions is studied with nuclear magnetic resonance flow visualization and continuum‐level numerical modelling of particle migration. The experimental method gives particle volume fraction as a function of time and position, which then provides validation data for the numerical model. A finite element method is used to discretize the equations of motion, including an evolution equation for the particle volume fraction and a generalized Newtonian viscosity dependent on local particle concentration. The diffusive‐flux equation is based on the Phillips model (Phys. Fluids A 1992; 4 :30–40) and includes sedimentation terms described by Zhang and Acrivos (Int. J. Multiphase Flow 1994; 20 :579–591). The model and experiments are utilized in three distinct geometries with particles that are heavier and lighter than the suspending fluid, depending on the experiment: (1) sedimentation in a cylinder with a contraction; (2) particle flotation in a horizontal cylinder with a horizontal rod; and (3) flotation around a rectangular inclusion. Secondary flows appear in both the experiments and the simulations when a region of higher density fluid is above a lower density fluid. The secondary flows result in particle inhomogeneities, Rayleigh–Taylor‐like instabilities, and remixing, though the effect in the simulations is more pronounced than in the experiments. Published in 2007 by John Wiley & Sons, Ltd.     

A numerical model for aerated‐water wave breaking

This work presents a numerical model designed for the simulation of water‐wave impacts on a structure when aeration of the liquid phase is considered. The model is based on a multifluid Navier–Stokes approach in which all fluids are assumed compressible. The numerical method is based on a finite volume algorithm in space and a second order Runge–Kutta method in time. A validation of this model is performed. It shows a good accuracy for acoustic and shock wave propagation in a bubbly liquid and for wave breaking. Copyright © 2011 John Wiley & Sons, Ltd.     

A fluid solver based on vorticity–helical density equations with application to a natural convection in a cubic cavity

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity–helical density and velocity–Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergence‐free vorticity, and accurate kinetic energy balance in a simple and natural way. The method is applied to compute buoyancy‐driven flows in a differentially heated cubic enclosure in the Boussinesq approximation for Ra ∈ {10⁴,10⁵,10⁶}. The numerical solutions on a finer grid are of benchmark quality. The computed helical density allows quantification of the three‐dimensional nature of the flow. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite volume computation of incompressible turbulent flows in general co-ordinates on staggered grids

A brief review of the computation of incompressible turbulent flow in complex geometries is given. A 2D finite volume method for the calculation of turbulent flow in general curvilinear co-ordinates is described. This method is based on a staggered grid arrangement and the contravariant flux componets are chosen as primitive variables. Turbulence is modelled either by the standard k–ε model or by a k–ε model based on RNG theory. Convection is approximated with central differences for the mean flow quantities and a TVD-type MUSCL scheme for the turbulence equations. The sensitivity of the method to the grid properties is investigated. An application of this method to a complex turbulent flow is presented. The results of computations are compared with experimental data and other numerical solutions and are found to be satisfactory.     

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities,II

Problems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

A moving mesh BGK scheme for multi‐material flows

In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order numerical simulations of flow‐induced noise

In this paper, the flow/acoustics splitting method for predicting flow‐generated noise is further developed by introducing high‐order finite difference schemes. The splitting method consists of dividing the acoustic problem into a viscous incompressible flow part and an inviscid acoustic part. The incompressible flow equations are solved by a second‐order finite volume code EllipSys2D/3D. The acoustic field is obtained by solving a set of acoustic perturbation equations forced by flow quantities. The incompressible pressure and velocity form the input to the acoustic equations. The present work is an extension of our acoustics solver, with the introduction of high‐order schemes for spatial discretization and a Runge–Kutta scheme for time integration. To achieve low dissipation and dispersion errors, either Dispersion‐Relation‐Preserving (DRP) schemes or optimized compact finite difference schemes are used for the spatial discretizations. Applications and validations of the new acoustics solver are presented for benchmark aeroacoustic problems and for flow over an NACA 0012 airfoil. Copyright © 2010 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.