Two dimensional shallow-water flow model with immersed boundary method

This study demonstrates an immersed boundary (IB) method which integrates a depth-averaged two dimensional flow model is proposed to tackle a typical fluid-solid phase problem in fluid dynamics field. The finite-difference scheme with curvilinear coordinate system is employed to discretize the shallow-water flow equations. Lagrangian markers and Eulerian grid are applied to portray the geometric contour of interior boundary and discretize the flow domain, respectively. The Dirac delta function is accordingly conducted to link both Lagrangian and Eulerian coordinate systems. The numerical simulations of single pier are performed and compared to examine the effect of marker’s mesh width, grid size, and the various Dirac delta functions. Experimental data from literatures are compared with numerical results to justify the validity of the proposed IB model. To further demonstrate the model capability, the model is applied to the hypothetical cases of piers in parallel, and compared with theoretical results.     

Modelling and simulation of porous immersed boundaries

The immersed boundary method has been used to simulate a wide range of fluid–structure interaction problems from biology and engineering, wherein flexible solid structures deform in response to a surrounding incompressible fluid flow. We generalize the IB method to handle porous membranes by incorporating an additional transmembrane flux that obeys Darcy’s law. An approximate analytical solution is derived that clearly illustrates the effect of porosity on the immersed boundary motion. Numerical simulations in two dimensions are used to validate the analytical results and to illustrate the motion of more general porous membrane dynamics.     

An immersed boundary method based on the lattice Boltzmann approach in three dimensions,with application

The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier–Stokes (N–S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N–S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N–S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) within the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are: (1) the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; (2) the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow.     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.     

Drag of a flexible fiber in a 2D moving viscous fluid

This paper reports the numerical study of the drag of a flexible elastic fiber immersed in a two-dimensional viscous flow using the immersed boundary (IB) method. We found drag reduction of a flexible fiber compared to a stiff one and the drag coefficient decreases with respect to the dimensionless fiber length within a certain range. The results are a starting point for the understanding of the role of flexibility in biological organisms in fluid flows.     

Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow

Coupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and moving boundaries. To investigate the possibility for future IB-LB coupled simulations of the heart flow dynamics, an IB-LB coupling scheme suitable for rapid boundary motion and large pressure gradient FSI is proposed, and the mitral valve jet flow considering the interaction of leaflets and fluid is simulated. After analyzing the respective concepts, formulae and advantages of the IB and LB methods, we first explain the coupling strategy and detailed implementation procedures, and then verify the effectiveness and second-order accuracy of the scheme by simulating a benchmark case, the relaxation of a stretched membrane immersed in fluid. After that, the diastolic filling jet flow between mitral leaflets in a simplified 2D left heart model is simulated. The model consists of the simplified transmitral passage of the heart and two curvilinear leaflets. In the simulation, the atrial and ventricular pressure histories of normal human are specified as boundary conditions, and the leaflets are treated as fibers that interact with the fluid to define their deformations and movements. The resulting opening and closing movements of the leaflets and the flow patterns of the filling jet are qualitatively reasonable and compare well with existing numerical and measured data. It is shown that this IB-LB coupling method is feasible for treating flexible boundary FSI problems with rapid boundary motion and large pressure gradient, the results of the mitral leaflet flow are valuable for understanding the transmitral FSI dynamics, and it is possible to simulate the more realistic 3D heart flow by the scheme in the future.     

A collocated finite volume embedding method for simulation of flow past stationary and moving body

A simple and conservative numerical scheme is introduced in this paper to simulate unsteady flow around stationary and moving body. Based on the embedding method (immersed boundary (IB) + volume of fluid (VOF)) implemented in the finite-volume framework, flow past the arbitrarily complex geometry can be readily computed on any existing mesh system. Flow variables stored at cell centers, including those residing within the immersed body, are computed where the induced effect on the flow due to the immersed body is realised via a simple acceleration term (forcing function) derived based on the VOF value. In the current work, an identical VOF value is used for all momentum equations, in contrast to that of the pre-existing method, whereby numerical interpolation is required. The method is verified with a number of flow cases, including flow in a 2D square cavity, flow past a stationary and oscillating cylinder and flow induced by a flapping ellipse in an enclosure.     

A finite element immersed boundary method for fluid flow around moving objects

This paper presents the extension of a recently proposed immersed boundary method to the solution of the flow around moving objects. Solving the flow around objects with complex shapes may involve extensive meshing work that has to be repeated each time a change in the geometry is needed. Mesh generation and solution interpolation between successive grids may be costly and introduce errors if the geometry changes significantly during the course of the computation. These drawbacks are avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. This work presents an extension of our recently developed finite element Immersed Boundary (IB) method to transient applications involving the movement of immersed fluid/solid interfaces. As for the fixed solid boundary case, the method produces solutions of the flow satisfying accurately boundary conditions imposed on the surface of immersed bodies. The proposed algorithm enriches the finite element discretization of interface elements with additional degrees of freedom, the latter being eliminated at element level. The boundary of immersed objects is defined using a time dependent level-set function. Solutions are shown for various flow problems and the accuracy of the present approach is measured with respect to solutions on body-conforming meshes.     

A lattice Boltzmann based implicit immersed boundary method for fluid–structure interaction

The immersed boundary method is a practical and effective method for fluid–structure interaction problems. It has been applied to a variety of problems. Most of the time-stepping schemes used in the method are explicit, which suffer a drawback in terms of stability and restriction on the time step. We propose a lattice Boltzmann based implicit immersed boundary method where the immersed boundary force is computed at the unknown configuration of the structure at each time step. The fully nonlinear algebraic system resulting from discretizations is solved by an Inexact Newton–Krylov method in a Jacobian-free manner. The test problem of a flexible filament in a flowing viscous fluid is considered. Numerical results show that the proposed implicit immersed boundary method is much more stable with larger time steps and significantly outperforms the explicit version in terms of computational cost.     

A Mass Conservative Scheme for Fluid–Structure Interaction Problems by the Staggered Discontinuous Galerkin Method

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method.     

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151–72] is extended for 3D unsteady flow simulation with fluid–structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier–Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17–20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on “Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425–36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V₀ can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood–leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.     

On optimal node spacing for immersed boundary–lattice Boltzmann method in 2D and 3D

A computational study on optimal spacing of Lagrangian nodes discretizing a rigid and immobile immersed body boundary in 2D and 3D is presented in order to show how the density of the Lagrangian points affects the numerical results of the Immersed Boundary–Lattice Boltzmann Method (IB–LBM). The study is based on the implicit velocity correction-based IB–LBM proposed by Wu and Shu (2009, 2010) that allows computing the fluid–body interaction force. However, the (original) method fails for densely spaced Lagrangian points due to ill-conditioned or even singular linear systems that arise from the derivation of the method. We propose a modification that improves the solvability of the linear systems and compare the performance of both methods using several benchmark problems. The results show how the spacing of the Lagrangian points affects the numerical results, mainly the permeability of the discretized body boundary in applications to fluid flows over rigid obstacles and blood flows in arteries in 2D and 3D.     

On Boundary Condition Capturing for Multiphase Interfaces

This review paper begins with an overview of the boundary condition capturing approach to solving problems with interfaces. Although the authors’ original motivation was to extend the ghost fluid method from compressible to incompressible flow, the elliptic nature of incompressible flow quickly quenched the idea that ghost cells could be defined and used in the usual manner. Instead the boundary conditions had to be implicitly captured by the matrix formulation itself, leading to the novel approach. We first review the work on the variable coefficient Poisson equation, noting that the simplicity of the method allowed for an elegant convergence proof. Simplicity and robustness also allowed for a quick extension to three-dimensional two-phase incompressible flows including the effects of viscosity and surface tension, which is discussed subsequently. The method has enjoyed popularity in both computational physics and computer graphics, and we show some comparisons with the traditional delta function approach for the visual simulation of bubbles. Finally, we discuss extensions to problems where the velocity is discontinuous as well, as is the case for premixed flames, and show an example of multiple interacting liquids that includes all of the aforementioned phenomena.     

A novel immersed boundary procedure for flow and heat simulations with moving boundary

This article describes a novel immersed boundary procedure for computing the flow and heat transfer problems with moving and complex boundary. Although the immersed boundary techniques have been successfully implemented to these flow and heat simulations, a frequently encountered drawback of this method is the relatively low accuracy proximate to the boundary due to the spreading of forcing function or the interpolation scheme. In this study, we propose a moving-grid process under the arbitrary Lagrangian-Eulerian framework to reduce the numerical diffusion near the immersed boundary. The incompressible Navier-Stokes equations are discretized spatially using unstructured finite element method, and advanced temporally by an operator-splitting scheme. The methodology is validated by the simulations of flow induced by an oscillating cylinder in a free stream. The capability of the proposed method is further demonstrated by good predictions of flow passing the rotating fan in a channel and also flow driven by two independent rotating fans in a circular cavity.     

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.     

Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces

We propose a fast, explicit numerical method for computing approximations for the immersed boundary problem in which the boundaries that separate the fluid into two regions are stiff. In the numerical computations of such problems, one frequently has to contend with numerical instability, as the stiff immersed boundaries exert large forces on the local fluid. When the boundary forces are treated explicitly, prohibitively small time-steps may be required to maintain numerical stability. On the other hand, when the boundary forces are treated implicitly, the restriction on the time-step size is reduced, but the solution of a large system of coupled non-linear equations may be required. In this work, we develop an efficient method that combines an integral equation approach with the immersed interface method. The present method treats the boundary forces explicitly. To reduce computational costs, the method uses an operator-splitting approach: large time-steps are used to update the non-stiff advection terms, and smaller substeps are used to advance the stiff boundary. At each substep, an integral equation is computed to yield fluid velocity local to the boundary; those velocity values are then used to update the boundary configuration. Fluid variables are computed over the entire domain, using the immersed interface method, only at the end of the large advection time-steps. Numerical results suggest that the present method compares favorably with an implementation of the immersed interface method that employs an explicit time-stepping and no fractional stepping.     

Peristaltic pumping of solid particles

We simulate solid particle transport by peristalsis in a two-dimensional channel with sinusoidal waves. The fluid is regarded as viscous and incompressible, and the walls of the channel and the particle as neutrally buoyant elastic boundaries immersed in this fluid. Using the immersed boundary technique, we are able to computationally model fluid-particle interaction. To demonstrate the validity of our computational approach, we first present results of computations without solid particles and compare them with existing theory and computations. We then examine how the transport of the solid particle depends upon its diameter, the Reynolds number and its initial placement in the channel. We also present evidence that the mean transport speed of a particle increases geometrically as its diameter approaches the channel width.     

A unified operator splitting approach for multi-scale fluid–particle coupling in the lattice Boltzmann method

A unified framework to derive discrete time-marching schemes for the coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales.