A direct‐forcing pressure‐based lattice Boltzmann method for solving fluid–particle interaction problems

A direct‐forcing immersed boundary‐lattice Boltzmann method (IB–LBM) is developed to simulate fluid–particle interaction problems. This method uses the pressure‐based LBM to solve the incompressible flow field and the immersed boundary method to handle the fluid–particle interactions. The pressure‐based LBM uses the pressure distribution functions instead of the density distribution functions as the independent dynamic variables. The main idea is to explicitly eliminate the compressible effect due to the density fluctuation. In the IB method, a direct‐forcing method is introduced to capture the particle motion. It directly computes an IB force density at each lattice grid from the differences between the pressure distribution functions obtained by the LBM and the equilibrium pressure distribution functions computed from the particle velocity. By applying this direct‐forcing method, the IB–LBM becomes a purely LBM version. Also, by applying the Gauss theorem, the formulas for computing the force and the torque acting on the particle from the flows are derived from the volume integrals over the particle volume instead of from the surface integrals over the particle surface. The order of accuracy of the IB–LBM is demonstrated on the errors of velocity field, wall stress, and gradients of velocity and pressure. As a demonstration of the efficiency and capabilities of the new method, sedimentation of a large number of spherical particles in an enclosure is simulated. Copyright © 2010 John Wiley & Sons, Ltd.     

A pressure correction method for fluid–particle interaction flow: Direct‐forcing method and sedimentation flow

A direct‐forcing pressure correction method is developed to simulate fluid–particle interaction problems. In this paper, the sedimentation flow is investigated. This method uses a pressure correction method to solve incompressible flow fields. A direct‐forcing method is introduced to capture the particle motions. It is found that the direct‐forcing method can also be served as a wall‐boundary condition. By applying Gauss's divergence theorem, the formulas for computing the hydrodynamic force and torque acting on the particle from flows are derived from the volume integral of the particle instead of the particle surface. The order of accuracy of the present method is demonstrated by the errors of velocity, pressure, and wall stress. To demonstrate the efficiency and capability of the present method, sedimentations of many spherical particles in an enclosure are simulated. Copyright © 2010 John Wiley & Sons, Ltd.     

Analytical and numerical studies of the boundary slip in the immersed boundary‐thermal lattice Boltzmann method

We analytically and numerically investigate the boundary slip, including the velocity slip and the temperature jump, in immersed boundary‐thermal lattice Boltzmann methods (IB‐TLBMs) with the two‐relaxation‐time collision operator. We derive the theoretical equation for the relaxation parameters considering the effect of the advection velocity on the temperature jump of the IB‐TLBMs. The analytical and numerical solutions demonstrate that the proposed iterative correction methods without the computational cost of the sparse matrix solver reduce the boundary slip and boundary‐value deviation as effectively as the implicit correction method for any relaxation time. Because the commonly used multi‐direct forcing method does not consider the contributions of the body force to the momentum flux, it cannot completely eliminate the boundary slip because of the numerical instability for a long relaxation time. Both types of proposed iterative correction methods are more numerically stable than the implicit correction method. In simulations of flow past a circular cylinder and of natural convection, the present iterative correction methods yield adequate results without the errors of the velocity slip, the temperature jump, and the boundary‐value deviation for any relaxation time parameters and for any number of Lagrangian points per length. The combination of the present methods and the two‐relaxation‐time collision operator is suitable for simulating fluid flow with thermal convection in the multiblock method in which the relaxation time increases in inverse proportion to the grid size.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

A Q2Q1 finite element/level‐set method for simulating two‐phase flows with surface tension

A Q2Q1 (quadratic velocity/linear pressure) finite element/level‐set method was proposed for simulating incompressible two‐phase flows with surface tension. The Navier–Stokes equations were solved using the Q2Q1 integrated FEM, and the level‐set variable was linearly interpolated using a ‘pseudo’ Q2Q1 finite element when calculating the density and viscosity of a fluid to avoid an unbounded density/viscosity. The advection of the level‐set function was calculated through the Taylor–Galerkin method, and the direct approach method is employed for reinitialization. The proposed method was tested by solving several benchmark problems including rising bubbles exhibiting a large density difference and the surface tension effect. The numerical results of the rising bubbles were compared with the existing results to validate the benchmark quantities such as the centroid, circularity, and rising velocity. Furthermore, we focused our attention mainly on mass conservation and time‐step. We observed that the present method represented a convergence rate between 1.0 and 1.5 orders in terms of mass conservation and provided more stable solutions even when using a larger time‐step than the critical time‐step that was imposed because of the explicit treatment of surface tension. Copyright © 2011 John Wiley & Sons, Ltd.     

A comparative study of direct‐forcing immersed boundary‐lattice Boltzmann methods for stationary complex boundaries

In this study, we assess several interface schemes for stationary complex boundary flows under the direct‐forcing immersed boundary‐lattice Boltzmann methods (IB‐LBM) based on a split‐forcing lattice Boltzmann equation (LBE). Our strategy is to couple various interface schemes, which were adopted in the previous direct‐forcing immersed boundary methods (IBM), with the split‐forcing LBE, which enables us to directly use the direct‐forcing concept in the lattice Boltzmann calculation algorithm with a second‐order accuracy without involving the Navier–Stokes equation. In this study, we investigate not only common diffuse interface schemes but also a sharp interface scheme. For the diffuse interface scheme, we consider explicit and implicit interface schemes. In the calculation of velocity interpolation and force distribution, we use the 2‐ and 4‐point discrete delta functions, which give the second‐order approximation. For the sharp interface scheme, we deal with the exterior sharp interface scheme, where we impose the force density on exterior (solid) nodes nearest to the boundary. All tested schemes show a second‐order overall accuracy when the simulation results of the Taylor–Green decaying vortex are compared with the analytical solutions. It is also confirmed that for stationary complex boundary flows, the sharper the interface scheme, the more accurate the results are. In the simulation of flows past a circular cylinder, the results from each interface scheme are comparable to those from other corresponding numerical schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid immersed‐boundary and multi‐block lattice Boltzmann method for simulating fluid and moving‐boundaries interactions

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed‐boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving‐boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish‐like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.     

A discrete forcing method dedicated to moving bodies in two‐phase flow

The numerical simulation of interaction between structures and two‐phase flows is a major concern for many industrial applications. To address this challenge, the motion of structures has to be tracked accurately. In this work, a discrete forcing method based on a porous medium approach is proposed to follow a nondeformable rigid body with an imposed velocity by using a finite‐volume Navier‐Stokes solver code dedicated to multiphase flows and based on a two‐fluid approach. To deal with the action reaction principle at the solid wall interfaces in a conservative way, a porosity is introduced allowing to locate the solid and insuring no diffusion of the fluid‐structure interface. The volumetric fraction equilibrium is adapted to this novelty. Mass and momentum balance equations are formulated on a fixed Cartesian grid. Interface tracking is addressed in detail going from the definition of the porosity to the changes in the discretization of the momentum balance equation. This so‐called time‐ and space‐dependent porosity method is then validated by using analytical and elementary test cases.     

Explicit characteristic‐based finite volume element method for level set equation

Accurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two‐phase flow problems. This paper presents an explicit characteristic‐based finite volume element method for solving the two‐dimensional level set equation. The method is applicable for the case of non‐divergence‐free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time. Copyright © 2010 John Wiley & Sons, Ltd.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

An immersed smoothed point interpolation method (IS‐PIM) for fluid‐structure interaction problems

An immersed smoothed point interpolation method using 3‐node triangular background cells is proposed to solve 2D fluid‐structure interaction problems for solids with large deformation/displacement placed in incompressible viscous fluid. In the framework of immersed‐type method, the governing equations can be decomposed into 3 parts on the basis of the fictitious fluid assumption. The incompressible Navier‐Stokes equations are solved using the semi‐implicit characteristic‐based split scheme, and solids are simulated using the newly developed edge‐based smoothed point interpolation method. The fictitious fluid domain can be used to calculate the coupling force. The numerical results show that immersed smoothed point interpolation method can avoid remeshing for moving solid based on immersed operation and simulate the contact phenomenon without an additional treatment between the solid and the fluid boundary. The influence from information transfer between solid domain and fluid domain on fluid‐structure interaction problems has been investigated. The numerical results show that the proposed interpolation schemes will generally improve the accuracy for simulating both fluid flows and solid structures.     

High‐order stable interpolations for immersed boundary methods

The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high‐order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two‐dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three‐dimensional flow in an air‐conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.     

A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids

This paper is concerned with the development of a new high‐order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non‐uniform grids. Specifically, both a high‐order fluxes integration and the implicit deconvolution of the volume‐averaged field are considered. This way, the numerical solution effectively stands for a fourth‐order approximation of the point‐wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high‐order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large‐eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two‐dimensional time‐evolving mixing layer is simulated, with and without sub‐grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

A new modification of the immersed‐boundary method for simulating flows with complex moving boundaries

In this paper, a new immersed‐boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite‐difference approach on a staggered mesh together with a fractional‐step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no‐slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed‐boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupling methods between finite element–based Boussinesq‐type wave and particle‐based free‐surface flow models

We develop one‐way coupling methods between a Boussinesq‐type wave model based on the discontinuous Galerkin finite element method and a free‐surface flow model based on a mesh‐free particle method to strike a balance between accuracy and computational cost. In our proposed model, computation of the wave model in the global domain is conducted first, and the nonconstant velocity profiles in the vertical direction are reproduced by using its results. Computation of the free‐surface flow is performed in a local domain included within the global domain with interface boundaries that move along the reproduced velocity field in a Lagrangian fashion. To represent the moving interfaces, we used a polygon wall boundary model for mesh‐free particle methods. Verification and validation tests of our proposed model are performed, and results obtained by the model are compared with theoretical values and experimental results to show its accuracy and applicability.     

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.     

Development of a local MQ‐DQ‐based stencil adaptive method and its application to solve incompressible Navier–Stokes equations

In this paper, a local stencil adaptive method is presented, which is designed for solving computational fluid dynamics (CFD) problems with curved boundaries accurately. A local multiquadric‐differential quadrature (MQ‐DQ) method is used to discretize the governing equations, taking advantage of its meshless nature. The present method bears the properties of both local MQ‐DQ method and local stencil adaptive method and is thus named the local MQ‐DQ‐based stencil adaptive method. Two test problems with curved boundaries are solved to investigate the performance of this solution‐adaptive method. The numerical results indicate that the proposed method is effective and efficient by combining the advantages of meshless property for complex geometries and local adaptation for accuracy improvement. Copyright © 2007 John Wiley & Sons, Ltd.