Direct-forcing Immersed Boundary Method Research Articles

An immersed boundary method for nonuniform grids

We propose a new direct forcing immersed boundary method for simulating the flow around an arbitrarily shaped body in nonuniform grids. A new formulation of the weight function for the distribution of the immersed boundary forcing with an efficient strategy that distributes the Lagrangian points on the surface of the immersed body is developed. We validate our method for various flows induced by a translational or angular motion of a sphere by measuring the lift, drag, and moment coefficients. For an arbitrary shape particle, we test our method for flow around an ellipsoid in the uniform flow. All tests show good agreement with available references. Finally, we investigate the interaction between a tilted ellipsoid and the wall in the uniform flow and the Poiseuille flow.

Drag reduction in turbulent channel flow laden with finite-size oblate spheroids

We study suspensions of oblate rigid particles in a viscous fluid for different values of the particle volume fractions. Direct numerical simulations have been performed using a direct-forcing immersed boundary method to account for the dispersed phase, combined with a soft-sphere collision model and lubrication corrections for short-range particle–particle and particle–wall interactions. With respect to the single-phase flow, we show that in flows laden with oblate spheroids the drag is reduced and the turbulent fluctuations attenuated. In particular, the turbulence activity decreases to lower values than those obtained by accounting only for the effective suspension viscosity. To explain the observed drag reduction, we consider the particle dynamics and the interactions of the particles with the turbulent velocity field and show that the particle–wall layer, previously observed and found to be responsible for the increased dissipation in suspensions of spheres, disappears in the case of oblate particles. These rotate significantly slower than spheres near the wall and tend to stay with their major axes parallel to the wall, which leads to a decrease of the Reynolds stresses and turbulence production and so to the overall drag reduction.

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.

A non-staggered coupling of finite element and lattice Boltzmann methods via an immersed boundary scheme for fluid-structure interaction

The paper presents a numerical framework for the coupling of finite element and lattice Boltzmann methods for transient problems involving fluid-structure interaction. The solid structure is discretized with the finite element method and integrated in time with the explicit Newmark scheme. The lattice Boltzmann method is used for the simulation of single-component weakly-compressible fluid flows. The two numerical methods are coupled via a direct-forcing immersed boundary method in a non-staggered way. Without subiteration within each time-step, the proposed method can ensure the synchronization of the time integrations, and thus the strong coupling of both subdomains by resolving a linear system of coupling equations at each time-step. Hence the energy transfer at the fluid-solid interface is correct, i.e. neither energy dissipation nor energy injection will occur at the interface, which can retain the numerical stability. A well-known fluid-structure interaction test case is adopted to validate the proposed coupling method. It is shown that the stability of the used numerical schemes can be preserved and a good agreement is found with the reference results.

An implementation of the direct-forcing immersed boundary method using GPU power

ABSTRACTA graphics processing unit (GPU) is utilized to apply the direct-forcing immersed boundary method. The code running on the GPU is generated with the help of the Compute Unified Device Architecture (CUDA). The first and second spatial derivatives of the incompressible Navier-Stokes equations are discretized by the sixth-order central compact finite-difference schemes. Two flow fields are simulated. The first test case is the simulated flow around a square cylinder, with the results providing good estimations of the wake region mechanics and vortex shedding. The second test case is the simulated flow around a circular cylinder. This case was selected to better understand the effects of sharp corners on the force coefficients. It was observed that the estimation of the force coefficients did not result in any troubles in the case of a circular cylinder. Additionally, the performance values obtained for the calculation time for the solution of the Poisson equation are compared with the values for other CPUs and GPUs from the literature. Consequently, approximately 3× and 20× speedups are achieved in comparison with GPU (using CUSP library) and CPU, respectively. CUSP is an open-source library for sparse linear algebra and graph computations on CUDA.

A moving-least-squares immersed boundary method for simulating the fluid–structure interaction of elastic bodies with arbitrary thickness

A versatile numerical method is presented to predict the fluid–structure interaction of bodies with arbitrary thickness immersed in an incompressible fluid, with the aim of simulating different biological engineering applications. A direct-forcing immersed boundary method is adopted, based on a moving-least-squares approach to reconstruct the solution in the vicinity of the immersed surface. A simple spring-network model is considered for describing the dynamics of deformable structures, so as to easily model and simulate different biological systems that not always may be described by simple continuum models, without affecting the computational time and simplicity of the overall method. The fluid and structures are coupled in a strong way, in order to avoid instabilities related to large accelerations of the bodies. The effectiveness of the method is validated by means of several test cases involving: rigid bodies, either falling in a quiescent fluid, fluttering or tumbling, or transported by a shear flow; infinitely thin elastic structures with mass, such as a two-dimensional flexible filament and, concerning three-dimensional cases, a flapping flag and an inverted flag in a free stream; finally, a three-dimensional model of a bio-prosthetic aortic valve opening and closing under a pulsatile flowrate. A very good agreement is obtained in all the cases, comparing with available experimental data and numerical results obtained by different methods. In particular, the method is shown to be second-order accurate by means of a mesh-refinement study. Moreover, it is able to provide results comparable with those of sharp direct-forcing approaches, and can manage high pressure differences across the surface, still obtaining very smooth hydrodynamic forces.

Effects of the Reynolds number on two-dimensional dielectrophoretic motions of a pair of particles under a uniform electric field

This paper presents two-dimensional direct numerical simulations to explore the effect of the Reynolds number on the Dielectrophoretic (DEP) motion of a pair of freely suspended particles in an unbounded viscous fluid under an external uniform electric field. Accordingly, the electric potential is obtained by solving the Maxwell’s equation with a great sudden change in the electric conductivity at the particle-fluid interface and then the Maxwell stress tensor is integrated to determine the DEP force exerted on each particle. The fluid flow and particle movement, on the other hand, are predicted by solving the continuity and Navier-Stokes equations together with the kinetic equations. Numerical simulations are carried out using a finite volume approach, composed of a sharp interface method for the electric potential and a direct-forcing immersed-boundary method for the fluid flow. Through the simulations, it is found that both particles with the same sign of the conductivity revolve and eventually align themselves in a line with the electric field. With different signs, to the contrary, they revolve in the reverse way and eventually become lined up at a right angle with the electric field. The DEP motion also depends significantly on the Reynolds number defined based on the external electric field for all the combinations of the conductivity signs. When the Reynolds number is approximately below Recr ≈ 0.1, the DEP motion becomes independent of the Reynolds number and thus can be exactly predicted by the no-inertia solver that neglects all the inertial and convective effects. With increasing Reynolds number above the critical number, on the other hand, the particles trace larger trajectories and thus take longer time during their revolution to the eventual in-line alignment.

A DF-IBM/NSCD coupling framework to simulate immersed particle interactions

Immersed granular flows are present widely in different domains under different forms (at various scales) such as in nature (rivers, muds, atmosphere, blood...), and in many industrial applications (detergents, cosmetics, etc...). Studying such flows properly requires one to represent well the physics behind their dynamics: the fluid/solid interactions (FSI), the solid/solid interactions (SSI) and the coupling mechanisms at various scales.In this work, a new coupling framework to simulate immersed granular flows has been developed. The FSI has been modeled using a direct-forcing immersed boundary method (DF-IBM) and implemented in the parallelized “PELICANS” C++ library. In this DF-IBM, all the mathematical equations, including the direct-forcing term, are discretized, both in space and time, and solved iteratively via a finite-volume and projection methods on Eulerian Grids. A sharp-edge interface, that can be smoothed, is used to represent the fluid/solid transition. The modeling of the multiple SSI at the grain’s scale is based on the Non-Smooth Contact Dynamics (NSCD) approach developed in the “LMGC90” open-source library. The coupling of the two softwares “PELICANS” and “LMGC90”, called Xper, provides an efficient framework to simulate and study dense immersed granular flows by taking into account, both advanced contact laws between grains, and hydrodynamic interactions.We address in this paper the effects of imposing a fluid-ring numerically (or fluid-mesh-cells) around two settling solid disks on modifying their dynamics. The DF-IBM approach implemented in Xper is validated, on a 2D flow over a stationary rigid cylinder benchmark, and on the settling of a rigid buoyant sphere in an incompressible laminar fluid at different Reynolds numbers. The numerical results are in good agreement with experimental and numerical data from the literature.

A comparative study of Brinkman penalization and direct-forcing immersed boundary methods for compressible viscous flows

This paper deals with the comparison between two methods to treat immersed boundary conditions: on the one hand, the Brinkman penalization method (BPM); on the other hand, the direct-forcing method (DFM). The penalty method treats the solid as a porous medium with a very low permeability. It provides a simple and efficient approach for solving the Navier–Stokes equations in complex geometries with fixed boundaries or in the presence of moving objects. A new approach for the penalty-operator integration is proposed, based on a Strang splitting between the penalization terms and the convection-diffusion terms. Doing so, the penalization term can be computed exactly. The momentum term can then be computed first and then introduced into the continuity equation in an implicit manner. The direct-forcing method however uses ghost-cells to reconstruct the values inside the solid boundaries by projection of the image points from the interface. This method is comparatively hard to implement in 3D cases and for moving boundaries. In the present paper, the performance of both methods is assessed through a variety of test problems. The application concerns the unsteady transonic and supersonic fluid flows. Examples include a normal shock reflection off a solid wall, transonic shock/boundary layer in a viscous shock tube, supersonic shock/cylinder interaction, and supersonic turbulent channel flow. The obtained results are validated against either analytical or reference solutions. The numerical comparison shows that, with sufficient mesh resolution, the BPM and the DFM methods yield qualitatively similar results. In all considered cases, the BPM is found to be a suitable and a possibly competitive method for viscous-IBM in terms of predictive performance, accuracy and computational cost. However, despite its simplicity, the method suffers from a lack of regularity in the very near-wall pressure fluctuations, especially for the turbulent case. This is attributed to the fact that the method requires no specific pressure condition at the fluid/solid interface.

An analysis and an affordable regularization technique for the spurious force oscillations in the context of direct-forcing immersed boundary methods

The framework of this paper is the improvement of direct-forcing immersed boundary methods in presence of moving obstacles. In particular, motivations for the use of the Direct Forcing (DF) method can be found in the advantage of a fixed computational mesh for fluid–structure interaction problems. Unfortunately, the direct forcing approach suffers a serious drawback in case of moving obstacles: the well known spurious force oscillations (SFOs). In this paper, we strengthen previous analyses of the origin of the SFO through a rigorous numerical evaluation based on Taylor expansions. We propose a remedy through an easy-to-implement regularization process (regularized DF). Formally, this regularization is related to the blending of the Navier–Stokes solver with the interpolation, but no modification of the numerical scheme is needed. This approach significantly cuts off the SFOs without increasing the computational cost. The accuracy and the space convergence order of the standard DF method are conserved. This is illustrated on numerical and physical validation test cases ranging from the Taylor–Couette problem to a cylinder with an imposed sinusoidal motion subjected to a cross-flow.

A suspension balance direct-forcing immersed boundary model for wet granular flows over obstacles

This paper addresses a new suspension balance direct-forcing immersed boundary model (SB-DF-IBM), for wet granular flows over obstacles. We mean by “wet granular” term all non-Newtonian flows that are made of non-Brownian particles (rigid spheres) immersed in a Newtonian fluid. This new SB-DF-IBM couples the Suspension Balance Model (SBM) to a Direct-Forcing Immersed Boundary Method (DF-IBM). The mathematical formulation in this contribution is based on a mixed Eulerian–Lagrangian approach.A novel feature of this new formulation is that many Lagrangian particles of the same size of the suspended particles, can be introduced into the continuum medium. This extrapolates the physics from a macroscopic to a microscopic scale, making the SB-DF-IBM behaves as a Dynamic-Scale Model (DSM).This SB-DF-IBM is implemented, as a new solver, in the OpenFOAM® open-source Computational Fluid Dynamics (CFD) software. Then, a numerical study is conducted on isodense mono-dispersed suspension flows over a stationary cylinder, in both wide and micro-channels, to test the functionality and performances of this SB-DF-IBM. The pressure difference, drag and lift forces are addressed for different initial bulk concentrations between 0% and 50%.Moreover, the effect of a cylindrical obstacle on the suspension flow separation in a microchannel is analyzed. The numerical results are presented and compared to recent experimental measurements from the literature addressing the non-Newtonian response of a suspension flow.As a conclusion, this SB-DF-IBM is a promising tool that can be introduced into different CFD packages for simulating wet granular non-Newtonian flows over obstacles.

Reduction of Numerical Oscillations in Simulating Moving-Boundary Problems by the Local DFD Method

AbstractIn this work, the hybrid solution reconstruction formulation proposed by Luo et al. [H. Luo, H. Dai, P. F. de Sousa and B. Yin, On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries, Computers & Fluids, 56 (2012), pp. 61–76] for the finite-difference discretization on Cartesian meshes is implemented in the finite-element framework of the local domain-free discretization (DFD) method to reduce the numerical oscillations in the simulation of moving-boundary flows. The reconstruction formulation is applied at fluid nodes in the immediate vicinity of the immersed boundary, which combines weightly the local DFD solution with the specific values obtained via an approximation of quadratic polynomial in the normal direction to the wall. The quadratic approximation is associated with the no-slip boundary condition and the local simplified momentum equation. The weighted factor suitable for unstructured triangular and tetrahedral meshes is constructed, which is related to the local mesh intervals near the immersed boundary and the distances from exterior dependent nodes to the boundary. Therefore, the reconstructed solution can account for the smooth movement of the immersed boundary. Several numerical experiments have been conducted for two- and three-dimensional moving-boundary flows. It is shown that the hybrid reconstruction approach can work well in the finite-element context and effectively reduce the numerical oscillations with little additional computational cost, and the spatial accuracy of the original local DFD method can also be preserved.

An immersed boundary method for unstructured meshes in depth averaged shallow water models

SummaryThe representation of geometries as buildings, flood barriers or dikes in free surface flow models implies tedious and time‐consuming operations in order to define accurately the shape of these objects when using a body fitted numerical mesh. The immersed boundary method is an alternative way to define solid bodies inside the computational domain without the need of fitting the mesh boundaries to the shape of the object. In the direct forcing immersed boundary method, a solid body is represented by a grid of Lagrangian markers, which define its shape and which are independent from the fluid Eulerian mesh. This paper presents a new implementation of the immersed boundary method in an unstructured finite volume solver for the 2D shallow water equations. Moving least‐squares is used to transmit information between the grid of Lagrangian markers and the fluid Eulerian mesh. The performance of the proposed implementation is analysed in three test cases involving different flow conditions: the flow around a spur dike, a dam break flow with an isolated obstacle and the flow around an array of obstacles. A very good agreement between the classic body fitted approach and the immersed boundary method was found. The differences between the results obtained with both methods are less relevant than the errors because of the intrinsic shallow water assumptions. Copyright © 2015 John Wiley & Sons, Ltd.

Lattice Boltzmann simulation of two cold particles settling in Newtonian fluid with thermal convection

The knowledge on the sedimentation of double particles is important in multiphase flow research. However, the available studies almost all are limited on the sedimentation of isothermal particles where there is no thermal convection between particles and fluid. In order to reveal the effects of thermal convection, in the present work the behavior of two cold particles freely settling in vertical channel is investigated using the lattice Boltzmann method (LBM) with direct-forcing immersed boundary method (IBM). By changing the initial separation distance and relative angle between two cold particles, the effects of these initial parameters on the interaction of cold particles settling in Newtonian fluid are also investigated. With different initial separation distance and relative angle between the two cold particles, we find that there are three regimes, namely repulsion, attraction and transition, during the sedimentation. The effects of thermal convection on the interactions between the two cold particles are revealed comprehensively for the first time by making comparisons between sedimentation of the two cold particles and their isothermal counterparts. The results reveal that thermal convection significantly influences the interactions between two cold particles during the sedimentation. The repulsive process will be enhanced due to thermal convection especially when the two cold particles approach each other very close. In addition, we observe the “drafting, kissing and tumbling” (DKT) phenomenon occurs earlier due to thermal convection. Owing to thermal convection, the attraction and repulsion process in the transition regime are enhanced. Moreover, thermal convection enhances the oscillations of trajectories of the two cold particles.

A simple and efficient direct forcing immersed boundary method combined with a high order compact scheme for simulating flows with moving rigid boundaries

A non-boundary conforming formulation for simulating complex flows with moving solid boundaries on fixed Cartesian grids is proposed. The direct forcing immersed boundary method (IBM) is implemented in a direct numerical simulation (DNS) code (called Incompact3d) based on high-order compact schemes for incompressible flows.To satisfy the boundary conditions on the immersed interface, the velocity field at the grid points near the interface is reconstructed via momentum forcing on a Cartesian grid by means of interpolation at forcing points in the fluid domain. A novel interpolation scheme which is applicable to boundaries of arbitrary shape is introduced and compared to a bi-linear model. A variance of this method utilizes a more compact stencil which allows compatibility with two-dimensional domain decomposition of the DNS code.Local force distributions and velocity fields were compared to identify which interpolation scheme best represents the solid boundaries by computing flow induced by a transversely oscillating cylinder.The accuracy and efficiency of the present technique are examined by simulating two-dimensional flow over a traveling wavy foil and comparing against numerical reference data. Finally, we present results from a three-dimensional simulation of a lamprey-like body undulating with prescribed experimental kinematics of anguilliform type, in order to demonstrate the ability of the present implementation in computing flows around moving solid objects with non-trivial geometries.

Direct-Forcing Immersed Boundary Method for Mixed Heat Transfer

AbstractA direct-forcing immersed boundary method (DFIB) with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.

The Flow of A Falling Ellipse: Numerical Method and Classification

AbstractA modified direct-forcing immersed-boundary (IB) pressure correction method is developed to simulate the flows of a falling ellipse. The pressure correct method is used to solve the solutions of the two dimensional Navier-Stokes equations and a direct-forcing IB method is used to handle the interaction between the flow and falling ellipse. For a fixed aspect ratio of an ellipse, the types of the behavior of the falling ellipse can be classified as three pure motions: Steady falling, fluttering, tumbling, and two transition motions: Chaos, transition between steady falling and fluttering. Based on two dimensionless parameters, Reynolds number and the dimensionless moment of inertia, a Reynolds number-inertia moment phase diagram is established. The behaviors and characters of five falling regimes are described in detailed.

A robust direct-forcing immersed boundary method with enhanced stability for moving body problems in curvilinear coordinates

A robust immersed boundary method for semi-implicit discretizations of the Navier–Stokes equations on curvilinear grids is presented. No-slip conditions are enforced via momentum forcing, and mass conservation at the immersed boundary is satisfied via a mass source term developed for moving bodies. The errors associated with an explicit evaluation of the momentum forcing are analysed, and their influence on the stability of the underlying Navier–Stokes solver is examined. An iterative approach to compute the forcing term implicitly is proposed, which reduces the errors at the boundary and retains the stability guarantees of the original semi-implicit discretization of the Navier–Stokes equations. The implementation in generalized curvilinear coordinates and the treatment of moving boundaries are presented, followed by a number of test cases. The tests include stationary and moving boundaries and curvilinear grid problems (decaying vortex problem, stationary cylinder, flow in 90° bend in circular duct and oscillating cylinder in fluid at rest).

A non-iterative direct forcing immersed boundary method for strongly-coupled fluid–solid interactions

A non-iterative direct forcing immersed boundary method is presented for the strongly-coupled simulations of fluid–solid interactions. While it retains many advantages of the immersed boundary framework by Yang and Stern (2012) [30], especially the simplified field extension strategy for moving boundary treatment and the pointwise integration of hydrodynamic force using the momentum forcing term, the present approach improves upon the previous method in several aspects including optimized computational cost for a strong coupling scheme, reduced algorithm complexity for a straightforward implementation, and enhanced numerical stability for low density ratio problems. Central to these improvements is a simple intermediate step in which the velocity fields around solid bodies are predicted on temporary non-inertial reference frames attached to moving solid bodies in a one-to-one manner. This step enables the explicit inverse of the implicit equations for rigid body dynamics, thus rendering unnecessary the previous predictor–corrector scheme for iteratively adjusting the displacements and velocities of the immersed bodies until reaching a convergence. In addition, a simple, generalized procedure is developed to obtain the interpolation coefficients in a local reconstruction stencil explicitly from the geometric relationship. For verification and validation, the vortex-induced vibration of a circular cylinder and the rotational galloping of a rectangular body are considered first; then several particulate flow problems, including settling and buoyant particles of low density ratios, a settling particle in a small container, and the kissing–drafting–tumbling problem of two settling particles, are studied. The agreement between the present results and the reference data in the literature is excellent. An overall second-order accuracy of the algorithm is verified in two systematic grid convergence tests. The present idea can be easily applied to similar methods for achieving a strong coupling scheme on top of a weak one with a nominal increase in the computational cost. Details of the algorithm are provided to facilitate its implementation in other solvers using non-boundary-conforming grids.

A Comparison of Fluid-Cell and Ghost-Cell Direct Forcing Immersed Boundary Method for Incompressible Flows with Heat Transfer

In direct-forcing immersed boundary methods, the forcing terms are generally applied to forcing points on either fluid side (fluid-cell forcing approach) or body side (ghost-cell forcing approach) of the immersed boundary. These direct forcing terms are added to the discretized equations over forcing points in order to implicitly enforce proper boundary conditions on the immersed boundary; hence they dictate the development of the computed flow field. Generally, different forcing approaches adapt different reconstruction models with different stencil supports to compute forcing terms. In this article, a general second-order accurate reconstruction model for solving incompressible Navier-Stokes equations with heat transfer is applied to both the fluid-cell forcing approach and the ghost-cell forcing approach. This allows a meaningful comparison of the solution accuracy and convergence between the two forcing approaches under various boundary conditions. In particular, a simple remedy to reduce the spurious oscillations in pressure force calculation over moving bodies for the ghost-cell forcing approach is devised and verified.