Simulation of dynamic fluid–solid interactions with an improved direct-forcing immersed boundary method

Dynamic fluid–solid interactions are widely found in chemical engineering, such as in particle-laden flows, which usually contain complex moving boundaries. The immersed boundary method (IBM) is a convenient approach to handle fluid–solid interactions with complex geometries. In this work, Uhlmann's direct-forcing IBM is improved and implemented on a supercomputer with CPU–GPU hybrid architecture. The direct-forcing IBM is modified as follows: the Poisson's equation for pressure is solved before evaluation of the body force, and the force is only distributed to the Cartesian grids inside the immersed boundary. A multidirect forcing scheme is used to evaluate the body force. These modifications result in a divergence-free flow field in the fluid domain and the no-slip boundary condition at the immersed boundary simultaneously. This method is implemented in an explicit finite-difference fractional-step scheme, and validated by 2D simulations of lid-driven cavity flow, Couette flow between two concentric cylinders and flow over a circular cylinder. Finally, the method is used to simulate the sedimentation of two circular particles in a channel. The results agree very well with previous experimental and numerical data, and are more accurate than the conventional direct-forcing method, especially in the vicinity of a moving boundary.     

Direct-forcing immersed boundary lattice Boltzmann simulation of particle/fluid interactions for spherical and non-spherical particles

The lattice Boltzmann method (LBM) is a useful technique for simulating multiphase flows and modeling complex physics. Specifically, we use LBM combined with a direct-forcing (DF) immersed boundary (IB) method to simulate fluid–particle interactions in two-phase particulate flows. Two grids are used in the simulation: a fixed uniform Eulerian grid for the fluid phase and a Lagrangian grid that is attached to and moves with the immersed particles. Forces are calculated at each Lagrangian point. To exchange numerical information between the two grids, discrete delta functions are used. The resulting DF IB-LBM approach is then successfully applied to a variety of reference flows, namely the sedimentation of one and two circular particles in a vertical channel, the sedimentation of one or two spheres in an enclosure, and a neutrally buoyant prolate spheroid in a Couette flow. This last application proves that the developed approach can be used also for non-spherical particles. The three forcing schemes and the different factors affecting the simulation (added mass effect, corrected radius) are also discussed.     

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.     

Modeling particle sedimentation in viscous fluids with a coupled immersed boundary method and discrete element method

Numerical techniques have increasingly been used to model fluid–particle two-phase flows. Coupling the immersed boundary method (IBM) and discrete element method (DEM) is one promising approach for modeling particulate flows. In this study, IBM was coupled with DEM to improve the reliability and accuracy of IBM for determining the positions of particles during the sedimentation process within viscous fluids. The required ratio of the particle diameter to the grid size (D/dx) was determined by comparing the simulation results with the analytical solution and experimental data. A dynamic mesh refinement model was utilised in the IBM model to refine the computational fluid dynamics grid near the particles. In addition, an optimum coupling interval between the IBM and DEM models was determined based on the experimental results of a single particle sedimentation within silicon oil at a Reynolds number of 1.5. The experimental results and the analytical solution were then utilised to validate the IBM–DEM model at Reynolds numbers of 4.1, 11.6, and 31.9. Finally, the validated model was utilised to investigate the sedimentation process for more than one particle by modeling the drafting-kissing-tumbling process and the Boycott phenomenon. Benchmark tests showed that the IBM–DEM technique preserves the advantages of DEM for tracking a group of particles, while the IBM provides a reliable and accurate approach for modeling the particle–fluid interaction.     

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.     

浸入边界法及其应用

浸入边界法主要用于模拟存在复杂外形结构的流场的运动情况和处理各种动边界问题,目前已广泛应用于计算流体力学领域.浸入边界法既是数学建模方法又是数值离散方法,它将物体边界与流体的相互作用通过在流体运动方程中加体积力项来体现,并在数值计算中采用简单的笛卡尔网格,避免了按照物体边界形状生成贴体网格时所遇到的各种问题.浸入边界法分为连续力法和离散力法:连续力法主要用于处理弹性边界问题,它的力源项满足特定的力学关系式;离散力法主要用于处理固体界面问题,它的力源项由边界条件推导得到.着重阐述了浸入边界法的基本原理和数学构造,对目前已有的几种不同的浸入边界法做了简单地介绍,并给出了一些应用实例,最后提出了浸入边界法未来的发展方向.     

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.     

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 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.     

Numerical modeling of flow past a volumeless and thin rigid body using direct forcing immersed boundary method

The new capability has been added as the numerical method for modeling volumeless and thin rigid bodies to the direct forcing immersed boundary (DFIB) method. The DFIB approach is based on adding a virtual force to the Navier–Stokes equations of incompressible flow to account for the interaction between the fluid and structures. The volume of a solid function (VOS) identifies the stationary or moving solid structures in a given fluid domain. A new VOS-based algorithm was developed to identify thin, rigid structure boundary points in fluid flow and ensure that the fluid cannot cross through the boundary of a thin rigid structure while moving or stationary. The DFIB method was first validated in a three-dimensional (3D) turbulent flow over a circular cylinder. The large-eddy simulation simulated the turbulent flow scales. The proposed algorithm was tested using a 3D turbulent flow past a stationary and rotating Savonius wind turbine that functions as a thin, rigid body. The validation results showed that the selected DFIB approach, combined with the novel algorithm, could simulate a thin, volumeless, rigid structure that is stationary and rotating in incompressible turbulent flows. The current method is also applicable for two-way fluid-structure interaction problems.     

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.     

Numerical simulation of polygonal particles moving in incompressible viscous fluids

A two-dimensional coupled lattice Boltzmann immersed boundary discrete element method is introduced for the simulation of polygonal particles moving in incompressible viscous fluids. A collision model of polygonal particles is used in the discrete element method. Instead of a collision model of circular particles, the collision model used in our method can deal with particles of more complex shape and efficiently simulate the effects of shape on particle–particle and particle–wall interactions. For two particles falling under gravity, because of the edges and corners, different collision patterns for circular and polygonal particles are found in our simulations. The complex vortexes generated near the corners of polygonal particles affect the flow field and lead to a difference in particle motions between circular and polygonal particles. For multiple particles falling under gravity, the polygonal particles easily become stuck owing to their corners and edges, while circular particles slip along contact areas. The present method provides an efficient approach for understanding the effects of particle shape on the dynamics of non-circular particles in fluids.     

Migration of a solid particle in the vicinity of a plane fluid–fluid interface

The slow migration of a small and solid particle in the vicinity of a gas–liquid, fluid–fluid or solid–fluid plane boundary when subject to a gravity or an external flow field is addressed. By contrast with previous works, the advocated approach holds for arbitrarily shaped particles and arbitrary external Stokes flow fields complying with the conditions on the boundary. It appeals to a few theoretically established and numerically solved boundary-integral equations on the particle’s surface. This integral formulation of the problem allows us to provide asymptotic approximations for a distant boundary and also, implementing a boundary element technique, accurate numerical results for arbitrary locations of the boundary. The results obtained for spheroids, both settling or immersed in external pure shear and straining flows, reveal that the rigid-body motion experienced by a particle deeply depends upon its shape and also upon the boundary location and properties.     

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.     

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.     

On the approximate zeroth and first‐order consistency in the presence of 2‐D irregular boundaries in SPH obtained by the virtual boundary particle methods

In this paper, a new method to impose 2‐D solid wall boundary conditions in smoothed particle hydrodynamics is presented. The wall is discretised by means of a set of virtual particles and is simulated by a local point symmetry approach. The extension of a previously published modified virtual boundary particle (MVBP) method guarantees that arbitrarily complex domains can be readily discretised guaranteeing approximate zeroth and first‐order consistency. To achieve this, three important new modifications are introduced: (i) the complete support is ensured not only for particles within one smoothing length distance, h, from the boundary but also for particles located at a distance greater than h but still within the support of the kernel; (ii) for a non‐uniform fluid particle distribution, the fictitious particles are generated with a uniform stencil (unlike the previous algorithms) that can maintain a uniform shear stress on a particle‐moving parallel to the wall in a steady flow; and (iii) the particle properties (density, mass and velocity) are defined using a local point of symmetry to satisfy the hydrostatic conditions and the Cauchy boundary condition for pressure. The extended MVBP model is demonstrated for cases including hydrostatic conditions for still water in a tank with a wedge and for curved boundaries, where significant improved behaviour is obtained in comparison with the conventional boundary techniques. Finally, the capability of the numerical scheme to simulate a dam break simulation is also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

Towards simulation of flapping wings using immersed boundary method

In this work the immersed boundary method is applied to simulate incompressible turbulent flows around stationary and moving objects. The goal is to demonstrate that the immersed boundary technique along with a large eddy simulation approach is capable of simulating the effect of the so‐called leading edge vortex (LEV), which can be found in flapping wing aerodynamics. A Lagrangian method is used to approximate the solutions in the freshly cleared cells that lay within solid objects at one time step and emerge into fluid domain at the next time step. Flow around a stationary cylinder at Re_D = 20, 40, and 3900 (based on cylinder diameter D) is first studied to validate the immersed boundary solver based on the finite volume scheme using a staggered grid. Then, a harmonically oscillating cylinder at Re_D = 10 000 is considered to test the solver after the Lagrangian method is implemented to interpolate the solution in the freshly cleared cells. Finally, this approach is used to study flows around a stationary flat‐plate at several angles of attack and fast pitching flat‐plate. The rapidly pitching plate creates a dynamic LEV that can be used to improve the efficiency of flapping wings of micro air vehicle and to determine the optimum flapping frequency. Copyright © 2012 John Wiley & Sons, Ltd.     

Direct numerical simulation of a particle-laden low Reynolds number turbulent round jet

Direct numerical simulation method is used for the investigating of particle-laden turbulent flows in a spatially evolution of low Reynolds number axisymmetric jet, and the Eulerian–Lagrangian point-particle approach is employed in the simulation. The simulation uses an explicit coupling scheme between particles and the fluid, which considers two-way coupling between the particle and the fluid. The DNS results are compared well with experimental data with equal Reynolds number (R_e = 1700). Our objects are: (i) to investigate the correlation between the particle number density and the fluctuating of fluid streamwise velocity; (ii) to examine whether the three-dimensional vortex structures in the particle-laden jet are the same as that in the free-air jet and how the particles modulate the thee-dimensional vortex structures and turbulence properties with different Stokes number particles; (iii) to discover the particle circumferential dispersion with different Stokes number particles. Our findings: (i) all the particles, regardless of their particle size, tend to preferentially accumulate in the region with large-than-mean fluid streamwise velocity; (ii) the small Stokes number particles take an important part in the modulation of three-dimensional vortex structures, but for the intermediate and larger sized particles, this modulation effect seems not so apparent; (iii) the particle circumferential dispersion is more effective for the smaller and intermediate sized particles, especially for the intermediate sized particles.     

Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

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.