Immersed Boundary Points Research Articles

A coupled FDEM-IBM-level set method for water entry of multiple flexible objects

In order to construct a versatile technique for water entry problems with structural deformations of multiple bodies, a novel hybrid algorithm, which couples the improved immersed boundary method (IBM) based fluid solver with the finite-discrete element method (FDEM) based structural solver, is proposed. The incompressible Navier-Stokes equations added with a virtual body force are employed for the hydrodynamic behaviors in the presence of elastic bodies. Representing the boundaries of solids by a set of immersed boundary points, the direct forcing IBM is enhanced to tackle the sharply moving objects of arbitrary geometries with inherent advantages. Such treatment enables the velocity boundary condition on the interface of the fluid and solid along with the divergence-free condition to be ensured simultaneously in the implicit computation of the added body force, which give rise to a more accurate depiction for fluid-solid interactions (FSI). Described by the finite element method (FEM), the deformable objects are allowed to deform and collide with each other during water entry which is seldom reported in literature. The contact interactions are calculated according to the distance potential function based discrete element method on the outermost layer of the finite elements. Furthermore, the improved conservative level set method is adopted to deal with the violently distorted free surface during hydroelastic impact. The strongly coupled FSI system is then resolved through a partitioned procedure, where sub-iterations are required in each time step to satisfy the convergence conditions exactly. The reliability of the present solver is validated by the classical cases of slamming of a flexible wedge and a free-falling cylindrical shell, along with dam break through an elastic gate and the settlement of two interactional circular cylinders. The accuracy and convergence rates of the proposed algorithm are also studied. Finally, a self-designed case, in which water entry of multiple compliant cylindrical shells is investigated, is performed to illustrate the effectiveness of the current approach, and a three-dimensional case study of multiple rocks impacting on the calm water surface demonstrates its great potential in practical applications.

Immersed Boundary Double Layer method: An introduction of methodology on the Helmholtz equation

The Immersed Boundary (IB) method of Peskin (1977) [1] is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions.

A high-efficiency discretized immersed boundary method for moving boundaries in incompressible flows

The Immersed Boundary Method (IBM) has an advantage in simulating fluid–structure interaction, owning to its simplicity, intuitiveness, and ease of handling complex object boundaries. The interpolation function plays a vital role in IBM and it is usually computationally intensive. For moving or deforming solids, the interpolation weights of all the immersed boundary points ought to be updated every time step, which takes quite a lot CPU time. Since the interpolation procedure within all uniform structured grids is highly repetitive and very similar, we propose a simple and generalized Discretized Immersed Boundary Method (DIBM), which significantly improves efficiency by discretizing the interpolation functions onto subgrid points within each control volume and reusing a predefined universal interpolation stencil. The accuracy and performance of DIBM are analyzed using both theoretical estimation and simulation tests. The results show speedup ratios of 30–40 or even higher using DIBM when compared with conventional IBM for typical moving boundary simulations like particle-laden flows, while the error is estimated to be under 1% and can be further decreased by using finer subgrid stencils. By balancing the performance and accuracy demands, DIBM provides an efficient alternative framework for handling moving boundaries in incompressible viscous flows.

A resolved CFDEM algorithm based on the immersed boundary for the simulation of fluid-solid interaction

In order to study the movement of rigid bodies with arbitrary shapes in the fluid, a resolved CFDEM coupling algorithm is proposed. The proposed method combines the computational fluid dynamics for the fluid and the discrete element method for the rigid bodies. The behavior of the fluid phase is simulated based on the Navier-Stokes equations in the Eulerian framework, whereas the motion of the rigid bodies is governed by the Newton's second law in the Lagrangian description. The major difficulty, namely, the representation of the moving boundaries of the solids, is solved by the immersed boundary method. The interaction forces between the fluid and the solids in different frameworks are derived according to the velocity boundary conditions on the immersed boundary points. Aiming at achieving the strong interaction effects between the fluid and the solids, several iterations are carried out to solve the coupling system.

Hydraulic Simulation for Calcasieu Lake Area With Small Rivers Using an Immersed Boundary Method

Abstract A coupled one-dimensional (1D) and two-dimensional (2D) hydrodynamic model for small rivers and streams on a large background area using an immersed boundary method (IBM) was developed and implemented for a simple flooding case in a previously published study. In this study, the IBM model was applied for simulations on a real geographical area, the Calcasieu Lake and its surrounding area in Southwest Louisiana. The simulation area ranges from the city of Lake Charles on the north to Gulf of Mexico on the south, and two National Wildlife Refuges (NWR) on the east and west side of Calcasieu Lake. A small change in the natural state of Calcasieu Lake, i.e., the change in the water surface elevation or flooding, can have a drastic impact on the ecosystem of its surrounding wetlands. The flooding to the wetlands is mainly through small rivers connected to the lake. In order to study the effects of coastal flooding to the wetlands, the IBM 1D–2D coupled model was implemented in the simulation package. The main purposes of this study are to: (1) determine the appropriate number of immersed boundary points to be used for this simulation and (2) test the applicability and validate the IBM model for an actual geographical region. Due to the lack of measurement data for the small rivers, the validation was conducted by comparing simulated results from IBM to the results from a non-IBM approach (i.e., manually carved rivers). Data from NOAA and USGS were used for the boundary conditions for the 2D model.

A Moving-Least-Square Immersed Boundary Method for Rigid and Deformable Boundaries in Viscous Flow

AbstractWe present a moving-least-square immersed boundary method for solving viscous incompressible flow involving deformable and rigid boundaries on a uniform Cartesian grid. For rigid boundaries, noslip conditions at the rigid interfaces are enforced using the immersed-boundary direct-forcing method. We propose a reconstruction approach that utilizes moving least squares (MLS) method to reconstruct the velocity at the forcing points in the vicinity of the rigid boundaries. For deformable boundaries, MLS method is employed to construct the interpolation and distribution operators for the immersed boundary points in the vicinity of the rigid boundaries instead of using discrete delta functions. The MLS approach allows us to avoid distributing the Lagrangian forces into the solid domains as well as to avoid using the velocity of points inside the solid domains to compute the velocity of the deformable boundaries. The present numerical technique has been validated by several examples including a Poiseuille flow in a tube, deformations of elastic capsules in shear flow and dynamics of red-blood cell in microfluidic devices.

An extended iterative direct-forcing immersed boundary method in thermo-fluid problems with Dirichlet or Neumann boundary conditions

An iterative direct-forcing immersed boundary method is extended and used to solve convection heat transfer problems. The pressure, momentum source, and heat source at immersed boundary points are calculated simultaneously to achieve the best coupling. Solutions of convection heat transfer problems with both Dirichlet and Neumann boundary conditions are presented. Two approaches for the implementation of Neumann boundary condition, i.e. direct and indirect methods, are introduced and compared in terms of accuracy and computational efficiency. Validation test cases include forced convection on a heated cylinder in an unbounded flow field and mixed convection around a circular body in a lid-driven cavity. Furthermore, the proposed method is applied to study the mixed convection around a heated rotating cylinder in a square enclosure with both iso-heat flux and iso-thermal boundary conditions. Computational results show that the order of accuracy of the indirect method is less than the direct method. However, the indirect method takes less computational time both in terms of the implementation of the boundary condition and the post processing time required to compute the heat transfer variables such as the Nusselt number. It is concluded that the iterative direct-forcing immersed boundary method is a powerful technique for the solution of convection heat transfer problems with stationary/moving boundaries and various boundary conditions.

An Immersed Boundary Method for a Contractile Elastic Ring in a Three-Dimensional Newtonian Fluid

In this paper, we present an immersed boundary method for modeling a contractile elastic ring in a three-dimensional Newtonian fluid. The governing equations are the modified Navier---Stokes equations with an elastic force from the contractile ring. The length of the elastic ring is time dependent and the ring shrinks with time because of its elastic nature in our proposed model. We dynamically reduce the number of Lagrangian boundary points when the distance between adjacent points is too small. This point-deleting algorithm helps keep the number of immersed boundary points in a single Cartesian mesh grid from becoming too high. We perform numerical experiments with various initial configurations of the contractile elastic ring, and numerical simulations to investigate the effects of the parameters are also conducted. The numerical results show that the proposed method can model and simulate the time-dependent contractile elastic ring in a three-dimensional Newtonian fluid.

Simulation of Propulsive Dynamics of an Organism in a Viscous Fluid Using an Immersed Boundary Finite Volume Method

Inspired by the propulsion of organisms in a viscous fluid, we develop a two-dimensional computational model to study the propulsive and fluid dynamic features of an organism modeled as an elastic filament in viscous fluid using immersed boundary (IB) finite volume method. The elastic filament is modeled using discrete number of IB points. The elastic forces are computed based on an elastic energy function. The Navier-Stokes equations governing the fluid flow are solved on a staggered Cartesian grid system using the fractional step based finite volume method. The computational model is validated by comparing the numerical simulation results pertinent to the swimming of an infinite with that of the existing analytical results. The interplay of propulsive and fluiddynamic features of the organism in the viscous fluid is well captured using the developed model.

An immersed boundary method for simulation of inviscid compressible flows

SUMMARYIn this paper, an immersed boundary method for simulating inviscid compressible flows governed by Euler equations is presented. All the mesh points are classified as interior computed points, immersed boundary points (interior points closest to the solid boundary), and exterior points that are blanked out of computation. The flow variables at an immersed boundary point are determined via the approximate form of solution in the direction normal to the wall boundary. The normal velocity is evaluated by applying the no‐penetration boundary condition, and therefore, the influence of solid wall in the inviscid flow is taken into account. The pressure is computed with the local simplified momentum equation, and the density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively. With a local coordinate system, the present method has been extended easily to the three‐dimensional case. The present work is the first endeavor to extend the idea of hybrid Cartesian/immersed boundary approach to compressible inviscid flows. The tedious task of handling multi‐valued points can be eliminated, and the overshoot resulting from the extrapolation for the evaluation of flow variables at exterior points can also be avoided. In order to validate the present method, inviscid compressible flows over fixed and moving bodies have been simulated. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical study on bacterial flagellar bundling and tumbling in a viscous fluid using an immersed boundary method

The bundling and tumbling behavior of bacterial flagella in a viscous fluid has got immense significance in the field of biological fluid dynamics. In this paper we investigate the hydrodynamic interaction among two and more than two flagella in a viscous fluid based on an immersed boundary method. We model each helical flagellum by a number of triangular cross-sections with three immersed boundary (IB) points on each cross-section. Three types of elastic links are generated from each IB point to create an elastic network model of the flagellum and the first cross-section is modeled as the flagellar motor. The elastic forces are computed based on the elastic energy approach and the motor forces are obtained from the applied angular frequency of rotation of the motor. The Stokes equations governing the flow are solved on a staggered Cartesian grid system using a fractional-step based finite-volume method. It is observed that when two left-handed helical flagella rotate in the counter-clockwise direction, the resulting hydrodynamic interaction leads to bundling. When one of the flagella reverses the direction of rotation to clockwise the hydrodynamic interaction results in tumbling. During the bundling, the flagella wrap and intertwine each other, whereas during the tumbling they separate in an erratic way. There exists an exact combination of the handedness and rotational direction of the flagella to achieve the bundling. The bundling-to-tumbling behavior of the flagella is studied and it is concluded that the tumbling occurs faster than the bundling. Further, the hydrodynamic interaction among three flagella in a viscous fluid is studied for the cases of rotation in the same direction and in different directions. The bundling and tumbling behavior is well captured even for the case of multiple (more than two) flagella using the developed model.

A Multigrid Method for a Model of the Implicit Immersed Boundary Equations

AbstractExplicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for the boundary force, with a time step 100 times larger than the explicit method, the implicit method is about 15-100 times more efficient than the explicit method. A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.

Numerical study on the rotation of an elastic rod in a viscous fluid using an immersed boundary method

We present a three-dimensional computational model based on an immersed boundary (IB) method to study the hydrodynamic features of a solid flexible cylindrical rod in a viscous fluid driven at one side by a tiny motor. The elastic rod is modelled by a number of circular cross-sections with twelve IB points on each cross-section. Three types of elastic links are created from each IB point to obtain an elastic network model of the rod and the first cross-section is modelled as the motor part. The elastic forces are computed based on an elastic energy approach and the motor forces are obtained from the applied angular frequency of rotation of the motor. The Stokes equations governing the fluid are solved on a staggered Cartesian grid system using the fractional-step based finite-volume method. Numerical simulations are performed to demonstrate the three dynamical stages of rod motion-twirling, whirling and overwhirling for different rotational frequency of the motor. It is revealed that for low rotational frequencies, the rod undergoes stable rigid body motion known as twirling. For high rotational frequencies of the motor, it is observed that the rod initially undergoes whirling motion and attains an unstable helical shape. Further, it is noticed that a discontinuous shape transition occurs for the rod and it folds back on itself. This unstable motion is referred to as overwhirling. It is also found that there exists a critical value of angular frequency of rotation of the motor below which the rod is subjected to twirling motion and above which it undergoes overwhirling motion.

Numerical simulation of interaction between laminar flow and elastic sheet

A numerical simulation of the interaction between laminar flow with low Reynolds number and a highly flexible elastic sheet is presented. The mathematical model for the simulation includes a three-dimensional finitevolume based fluid solver for incompressible viscous flow and a combined finite-discrete element method for the three-dimensional deformation of solid. An immersed boundary method is used to couple the simulation of fluid and solid. It is implemented through a set of immersed boundary points scattered on the solid surface. These points provide a deformable solid wall boundary for the fluid by adding body force to Navier-Stokes equations. The force from the fluid is also obtained for each point and then applied on the boundary nodes of the solid. The vortex-induced vibration of the highly flexible elastic sheet is simulated with the established mathematical model. The simulated results for both swing pattern and oscillation frequency of the elastic sheet in low Reynolds number flow agree well with experimental data.

Numerical study on the propulsion of a bacterial flagellum in a viscous fluid using an immersed boundary method

The propulsion of a bacterial flagellum in a viscous fluid has gained much attention in the field of biological fluid dynamics. Inspired by the bacterial propulsion, we present a three-dimensional computational model based on an immersed boundary (IB) method to study the propulsive and fluid dynamic features of a solid flexible flagellum in a viscous fluid driven at one side by an external torque. The helical flagellum is modelled by a series of triangular cross-sections with three IB points on each cross-section. Three types of elastic links are created to connect the IB points of one cross-section to the IB points of the next cross-section to obtain an elastic network model of the flagellum. An external torque is applied at the center of the first cross-section modelled as the flagellum motor. The elastic forces are computed based on elastic energy approach and the motor forces are obtained from the magnitude and direction of the applied torque. The Stokes equations governing the flow are solved on a staggered Cartesian grid system using the fractional-step based finite-volume method. The computational model is validated by comparing the swimming speed of the flagellum obtained from the numerical results with that of the existing numerical results. The interplay of propulsive and hydrodynamic features of a left handed helical flagellum driven by an external torque is well captured using the developed model. The effect of elasticity of various types of links on the swimming speed of the flagellum is investigated. It is revealed that among the three types of elastic links, the diagonal links have dominant role in the propulsion behavior of the flagellum. The effect of pitch on the propulsion speed of the flagellum is analyzed by computing its optimum value for a constant flagellum length based on the maximum swimming speed. Numerical simulations are performed to demonstrate the forward to backward swimming behavior of the flagellum and it is concluded that the backward swimming speed is higher than the forward swimming speed. The effect of the channel wall on the forward and backward propulsion of the flagellum is examined. It is observed that the flagellum swims faster in a channel than that in an unbounded fluid domain. Further, it is found that the swimming speed of the flagellum near to a channel wall is considerably higher than that away from the walls.

An immersed boundary method for simulating a single axisymmetric cell growth and division

For an animal cell, cytokinesis is the process by which a cell divides its cytoplasm to produce two daughter cells. We propose a new mathematical model for simulating cytokinesis. The proposed model is robust and realistic in deciding the position of the cleavage furrow and in defining the contractile force leading to cell division. We use an immersed boundary method to track the morphology of cell membrane during cytokinesis. For accurate calculation, we adaptively add and delete the immersed boundary points. We perform numerical simulations on the axisymmetric domain to have sufficient resolution and to incorporate three-dimensional effects such as anisotropic surface tension. Finally, we investigate the effects of each model parameter and compare a numerical result with the experimental data to demonstrate the efficiency and accuracy of our proposed method.

A fast, robust, and non-stiff Immersed Boundary Method

We propose a fast and non-stiff approach for the solutions of the Immersed Boundary Method, for Newtonian, incompressible flows in two or three dimensions. The proposed methodology is built on a robust semi-implicit discretization introduced by Peskin in the late 70s which is solved efficiently through the novel use of a fast, treecode strategy to compute flow-structure interactions. Optimal multipole-type expansions are performed numerically by solving a least squares problem with a new, fast iterative algorithm. The new Immersed Boundary Method is particularly well suited for three-dimensional applications and/or for problems where the number of immersed boundary points is large. We demonstrate the efficacy and superiority of the method over existing approaches with two simple but illustrative examples in 3D.

Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers

We present an immersed-boundary algorithm for incompressible flows with complex boundaries, suitable for Cartesian or curvilinear grid system. The key stages of any immersed-boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary (a line in 2D, a surface in 3D), and the spreading of a force field from the immersed boundary to the neighboring mesh points, to enforce the desired boundary conditions on the immersed-boundary points. We propose a technique that uses the Reproducing Kernel Particle Method [W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids 20(8) (1995) 1081–1106] for the interpolation and spreading. Unlike other methods presented in the literature, the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved, independent of the grid topology (uniform or non-uniform, Cartesian or curvilinear). The technique is easy to implement, and is able to maintain the order of the original underlying spatial discretization. Applications to two- and three-dimensional flows in Cartesian and non-Cartesian grid system, with uniform and non-uniform meshes are presented.

Unstructured grid immersed boundary method for numerical simulation of fluid structure interaction

This paper presents an improved unstructured grid immersed boundary method. The advantages of both immersed boundary method and body fitted grids which are generated by unstructured grid technology are used to enhance the computation efficiency of fluid structure interaction in complex domain. The Navier-Stokes equation was discretized spacially with collocated finite volume method and Euler implicit method in time domain. The rigid body motion was simulated by immersed boundary method in which the fluid and rigid body interface interaction was dealt with VOS (volume of solid) method. A new VOS calculation method based on graph was presented in which both immersed boundary points and cross points were collected in arbitrary order to form a graph. The method is verified with flow past oscillating cylinder.

An immersed boundary-thermal lattice Boltzmann method using an equilibrium internal energy density approach for the simulation of flows with heat transfer

In present paper, a novel immersed boundary-thermal lattice Boltzmann method by the name of “an equilibrium internal energy density approach” is proposed to simulate the flows around bluff bodies with the heat transfer. The main idea is to combine the immersed boundary method (IBM) with the thermal lattice Boltzmann method (TLBM) based on the double population approach. The equilibrium internal energy density approach based on the equilibrium velocity approach [X. Shan, H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E 47 (1993) 1815] is used to combine IBM with TLBM. The idea of the equilibrium internal energy density approach is that the satisfaction of the energy balance between heat source on the immersed boundary point and the amount of change of the internal energy density according to time ensures the temperature boundary condition on the immersed boundary. The advantages of this approach are the simple concept, easy implementation and the utilization of original governing equation without modification. The simulation of natural convection in a square cavity with various body shapes for different Rayleigh numbers has been conducted to validate the capability and the accuracy of present method on solving heat transfer problems. Consequently, the present results are found to be in good agreement with those of previous studies.