An immersed boundary-lattice Boltzmann flux solver and its applications to fluid–structure interaction problems

An immersed boundary-lattice Boltzmann flux solver (IB–LBFS) for the simulation of two-dimensional fluid–structure interaction (FSI) problems is presented in this paper. The IB–LBFS applies the fractional-step method to split the overall solution process into the predictor step and the corrector step. In the predictor step, the intermediate flow field is predicted by applying the LBFS (lattice Boltzmann flux solver) without considering the presence of immersed object. The LBFS applies the finite volume method to solve N–S (Navier–Stokes) equations for the flow variables at cell centers. At each cell interface, the LBFS evaluates its viscous and inviscid fluxes simultaneously through local reconstruction of the LBE (lattice Boltzmann equation) solutions. In the corrector step, the intermediate flow field is corrected by the implicit boundary condition-enforced immersed boundary method (IBM) so that the no-slip boundary conditions can be accurately satisfied. The IB–LBFS effectively combines the advantages of the LBFS in solving the flow field and the flexibility of the IBM in dealing with boundary conditions. Consequently, the IB–LBFS presents a much simpler and more effective approach for simulating complex FSI problems on non-uniform grids. Several test cases, including flows past one and two cylinders with prescribed motions, are firstly simulated to examine the accuracy of present solver. After that, two strongly coupled fluid–structure interaction problems, i.e., particle sedimentations and vortex-induced vibrations of a circular cylinder are investigated. Good agreements between the present results and those in literature verify the capability and flexibility of IB–LBFS for simulating FSI problems.     

An embedded boundary framework for compressible turbulent flow and fluid–structure computations on structured and unstructured grids

The finite volume method with exact two‐phase Riemann problems (FIVER) is a two‐faceted computational method for compressible multi‐material (fluid–fluid, fluid–structure, and multi‐fluid–structure) problems characterized by large density jumps, and/or highly nonlinear structural motions and deformations. For compressible multi‐phase flow problems, FIVER is a Godunov‐type discretization scheme characterized by the construction and solution at the material interfaces of local, exact, two‐phase Riemann problems. For compressible fluid–structure interaction (FSI) problems, it is an embedded boundary method for computational fluid dynamics (CFD) capable of handling large structural deformations and topological changes. Originally developed for inviscid multi‐material computations on nonbody‐fitted structured and unstructured grids, FIVER is extended in this paper to laminar and turbulent viscous flow and FSI problems. To this effect, it is equipped with carefully designed extrapolation schemes for populating the ghost fluid values needed for the construction, in the vicinity of the fluid–structure interface, of second‐order spatial approximations of the viscous fluxes and source terms associated with Reynolds averaged Navier–Stokes (RANS)‐based turbulence models and large eddy simulation (LES). Two support algorithms, which pertain to the application of any embedded boundary method for CFD to the robust, accurate, and fast solution of FSI problems, are also presented in this paper. The first one focuses on the fast computation of the time‐dependent distance to the wall because it is required by many RANS‐based turbulence models. The second algorithm addresses the robust and accurate computation of the flow‐induced forces and moments on embedded discrete surfaces, and their finite element representations when these surfaces are flexible. Equipped with these two auxiliary algorithms, the extension of FIVER to viscous flow and FSI problems is first verified with the LES of a turbulent flow past an immobile prolate spheroid, and the computation of a series of unsteady laminar flows past two counter‐rotating cylinders. Then, its potential for the solution of complex, turbulent, and flexible FSI problems is also demonstrated with the simulation, using the Spalart–Allmaras turbulence model, of the vertical tail buffeting of an F/A‐18 aircraft configuration and the comparison of the obtained numerical results with flight test data. Copyright © 2014 John Wiley & Sons, Ltd.     

Cavity detachment on a hydrofoil with the inclusion of surface tension effects

The non-linear problem of cavity flow past a hydrofoil is considered with taking into account fluid viscosity in the cavity closure region and surface tension, which affect the cavity detachment. The theoretical model is based on the concept of viscous–inviscid interaction between the outer inviscid cavity flow and the inner turbulent separated flow downstream of the cavity. The outer inviscid flow is solved by constructing the complex flow potential, and the wake model is based on the method of integral relationships for separated turbulent flows. The obtained numerical results are compared with experimental data.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

A domain decomposition approach to compute wave breaking (wave‐breaking flows)

A heterogeneous domain decomposition approach is followed to simulate the unsteady wavy flow generated by a body moving beneath a free surface. Attention being focused on complex free surface configurations, including wave‐breaking phenomena, a two‐fluid viscous flow model is used in the free surface region to capture the air–water interface (via a level‐set technique), while a potential flow approximation is adopted to describe the flow far from the interface. Two coupling strategies are investigated, differing in the transmission conditions. Both the adopted approaches make use of the inviscid velocity field as boundary condition in the Navier–Stokes solution. For validation purposes, two different two‐dimensional non‐breaking flows are simulated. Domain decomposition results are compared with both fully viscous and fully inviscid results, obtained by solving the corresponding equations in the whole fluid domain, and with available experimental data. Finally, the unsteady evolution of a steep breaking wave is followed and some of the physical phenomena, experimentally observed, are reproduced. Copyright © 2003 John Wiley & Sons, Ltd.     

Strongly coupled simulation of fluid–structure interaction in a Francis hydroturbine

This work simulates a complex fluid flow in fluid–structure interaction (FSI). The flow under consideration is governed by Navier–Stokes equations for incompressible viscous fluids and modeled with the finite volume method. Large eddy simulation is used to simulate the unsteady turbulent flow. The structure is represented by a finite element formulation. The present work introduces a strongly coupled partitioned approach that is applied to complex flow in fluid machinery. In this approach, the fluid and structure equations are solved separately using different solvers, but are implicitly coupled into one single module based on sensitivity analysis of the important displacement and stress modes. The applied modes and their responses are used to build up a reduced‐order model. The proposed model is used to predict the unsteady flow fields of a 3D complete passage, involving in stay, guide vanes, and runner blades, for a Francis hydro turbine and FSI is considered. The computational results show that a fairly good convergence solution is achieved by using the reduced‐order model that is based on only a few displacement and stress modes, which largely reduces the computational cost, compared with traditional approaches. At the same time, a comparison of the numerical results of the model with available experimental data validates the methodology and assesses its accuracy. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of transonic limit cycle oscillations using high-order low-diffusion schemes

This paper simulates the NLR7301 airfoil limit cycle oscillation (LCO) caused by fluid–structure interaction (FSI) using Reynolds averaged Navier–Stokes equations (RANS) coupled with Spalart–Allmaras (S–A) one-equation turbulence model. A low diffusion E-CUSP (LDE) scheme with 5th order weighted essentially nonoscillatory scheme (WENO) is employed to calculate the inviscid fluxes. A fully conservative 4th order central differencing is used for the viscous terms. A fully coupled fluid–structural interaction model is employed. For the case computed in this paper, the predicted LCO frequency, amplitudes, averaged lift and moment, all agree excellently with the experiment performed by Schewe et al. The solutions appear to have bifurcation and are dependent on the initial fields or initial perturbation. The developed computational fluid dynamics (CFD)/computational structure dynamics (CSD) simulation is able to capture the LCO with very small amplitudes measured in the experiment. This is attributed to the high order low diffusion schemes, fully coupled FSI model, and the turbulence model used. This research appears to be the first time that a numerical simulation of LCO matches the experiment. The simulation confirms several observations of the experiment.     

Fluid–solid interaction problems with thermal convection using the immersed element‐free Galerkin method

In this work, the immersed element‐free Galerkin method (IEFGM) is proposed for the solution of fluid–structure interaction (FSI) problems. In this technique, the FSI is represented as a volumetric force in the momentum equations. In IEFGM, a Lagrangian solid domain moves on top of an Eulerian fluid domain that spans over the entire computational region. The fluid domain is modeled using the finite element method and the solid domain is modeled using the element‐free Galerkin method. The continuity between the solid and fluid domains is satisfied by means of a local approximation, in the vicinity of the solid domain, of the velocity field and the FSI force. Such an approximation is achieved using the moving least‐squares technique. The method was applied to simulate the motion of a deformable disk moving in a viscous fluid due to the action of the gravitational force and the thermal convection of the fluid. An analysis of the main factors affecting the shape and trajectory of the solid body is presented. The method shows a distinct advantage for simulating FSI problems with highly deformable solids. Copyright © 2009 John Wiley & Sons, Ltd.     

Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method

Mould filling processes, in which a material flow front advances through a mould, are typical examples of moving boundary problems. The moving boundary is accompanied by a moving contact line at the mould walls causing, from a macroscopic modelling viewpoint, a stress singularity. In order to be able to simulate such processes, the moving boundary and moving contact line problem must be overcome. A numerical model for both two- and three-dimensional mould filling simulations has been developed. It employs a pseudo-concentration method in order to avoid elaborate three-dimensional remeshing, and has been implemented in a finite element program. The moving contact line problem has been overcome by employing a Robin boundary condition at the mould walls, which can be turned into a Dirichlet (no-slip) or a Neumann (free-slip) boundary condition depending on the local pseudo-concentration. Simulation results for two-dimensional test cases demonstrate the model's ability to deal with flow phenomena such as fountain flow and flow in bifurcations. The method is by no means limited to two-dimensional flows, as is shown by a pilot simulation for a simple three-dimensional mould. The reverse problem of mould filling is the displacement of a viscous fluid in a tube by a less viscous fluid, which has had considerable attention since the 1960's. Simulation results for this problem are in good agreement with results from the literature. © 1998 John Wiley & Sons, Ltd.     

Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

A numerical method is presented for the analysis of interactions of inviscid and compressible flows with arbitrarily shaped stationary or moving rigid solids. The fluid equations are solved on a fixed rectangular Cartesian grid by using a higher‐order finite difference method based on the fifth‐order WENO scheme. A constrained moving least‐squares sharp interface method is proposed to enforce the Neumann‐type boundary conditions on the fluid‐solid interface by using a penalty term, while the Dirichlet boundary conditions are directly enforced. The solution of the fluid flow and the solid motion equations is advanced in time by staggerly using, respectively, the third‐order Runge‐Kutta and the implicit Newmark integration schemes. The stability and the robustness of the proposed method have been demonstrated by analyzing 5 challenging problems. For these problems, the numerical results have been found to agree well with their analytical and numerical solutions available in the literature. Effects of the support domain size and values assigned to the penalty parameter on the stability and the accuracy of the present method are also discussed.     

Forces on oscillating bodies in viscous fluid

This paper describes a method for determining the fluid forces on oscillating bodies in viscous fluid when the corresponding flow problem has been solved using the finite element method. These forces are characterized by the concept of added mass, added damping and added force. Numerical results are obtained for several example body shapes. Comparison is made with exact analytical results and other finite element results for the limiting cases of Stoke's flow and inviscid flow, and good agreement is obtained. The results for finite values of the body amplitude parameter β show the appearance of added force from the steady streaming component of the flow for asymmetric bodies. Results are also obtained for the associated flow where the fluid remote from a fixed body is oscillating.     

An effective approach for modeling fluid flow in heterogeneous media using numerical manifold method

A major challenge of modeling fluid flow in heterogeneous media is to model the material interfaces, which may be arbitrarily oriented or intersected with Dirichlet, Neumann, or other boundaries, making it difficult to mesh and accurately satisfy the boundary constraints. In order to solve these problems, we derived a new continuous approach in the numerical manifold method (NMM). NMM is an ideal method to handle boundaries, considering its flexibility and efficiency with fixed mathematical mesh and its integration precision. With the two‐cover‐meshing system, we construct physical covers containing gradient jump terms defined as extended degrees of freedom to realize the refraction law across material interfaces. In the global equilibrium equations, the jump terms are naturally considered with the energy‐work seepage model. In this approach, high accuracy is expected from the newly constructed jump function together with simplex integration. Moreover, high mesh efficiency is realized by fixed triangular mathematical mesh with algorithms fully considering interfaces intersecting with Dirichlet, Neumann, or other boundaries and simplex integration on elements in arbitrary shapes. The new approach was coded into our NMM fluid flow model. We calculated examples involving fluid flow through a domain including (1) a single interface, (2) an idealized fault represented by multiple material interfaces, (3) intersected interfaces, and (4) an octagonal inclusion. We compared the simulated results to analytical solutions or results with denser mesh to test precision and efficiency and thereby proved that the new approach is accurate, efficient, and flexible, especially when considering intense geometric change or intersections. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical investigation of ice breaking by a high-pressure bubble based on a coupled BEM-PD model

In this paper, by combining the boundary element method (BEM) and peridynamics (PD), a bubble-ice interaction model is established, which can investigate the dynamic interactions between a high-pressure bubble and an ice plate with particular focus on the mechanical behaviors of ice breaking. The bubble dynamics are solved by BEM based on the potential flow theory. Ice cracks initiation and propagation are simulated by the bond-based peridynamics which is validated by a three-point bending test. The fluid–structure interaction (FSI) is achieved by matching the normal velocity and hydrodynamic loads at the fluid–structure interface. To validate the proposed FSI model, an experiment is carried out in which an oscillating bubble is generated under an ice plate by underwater discharge system. The whole interaction process is captured by a Phantom V711 high-speed camera. Qualitative agreements are achieved between the numerical and experimental results. The underlying mechanism of cracks initiation, propagation, branching, and coalescence of the ice plate is found to highly depend on three parameters, i.e., bubble–ice distance, ice thickness and bubble size. The present study is expected to provide further assists in the understanding of ice breaking problems.     

Dynamic stall modelling on airfoils based on strong viscous–inviscid interaction coupling

The prediction of the aerodynamic performance of pitching airfoils in stall conditions is considered in the context of strong viscous–inviscid interaction modelling. The aim of the work is to demonstrate the capabilities of a low‐cost dynamic stall model well suited for engineering applications. The model is formulated on the basis of a standard panel method combined with a vortex blob approximation of the wake. The development of the boundary layer over the airfoil and the evolution of the shear layer in the wake are taken into account by means of strong viscous–inviscid interaction coupling. To this end a transpiration layer is added to the inviscid formulation which represents the displacement effect viscosity results in the flow while the non‐linear coupled equations are solved simultaneously. Separation is modelled by introducing a second wake originating from the separation point (‘double‐wake’ concept) which is provided as part of the boundary layer solution. The theoretical presentation of the model is supported with favourable comparisons to four sets of wind tunnel measurements. Copyright © 2007 John Wiley & Sons, Ltd.     

Fluid–structure interaction of turbulent pulsatile flow within a flexible wall axisymmetric aortic aneurysm model

Pulsatile turbulent flow characteristics in an axisymmetric aortic aneurysm (AA) model were analyzed numerically using a simulated physiological waveform. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. A fully-coupled fluid–structure interaction (FSI) analysis was utilized in this work. We investigated the effects of turbulent flow characteristics on the distribution of wall stress and flow patterns in AA models. Wall stress distributions were calculated by computational solid stress (CSS) model, which ignores the effect of the blood flow, and the FSI model that takes into account flow and solid mechanics. Our results showed that peak wall stress and peak deformation were found to occur shortly after peak systolic flow in the FSI model and at the peak luminal pressure condition in the CSS model. Further, CSS model underestimated wall stress calculations when compared to the FSI model. There were also significant differences in the structure of flow fields between the flexible and rigid wall aneurysm models. Contour plots of kinetic energy dissipation and the application of the Kolmogorov microscale suggest that the conditions that result in red blood cell damage and platelet activation most likely occur in the near-wall region of AA during turbulent flow.     

Finite element computations for unsteady fluid and elastic membrane interaction problems

The interaction between the hydrodynamic forces of a flow field and the elastic forces of adjacent deformable boundaries is described by elastohydrodynamics, a coupled fluid–elastic membrane problem. Direct numerical solution of the unsteady, highly non-linear equations requires that the dynamic evolution of both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. This paper describes a numerical algorithm based on the deformable spatial domain space–time (DSD/ST) finite element method for the unsteady motion of an incompressible, viscous fluid with elastic membrane interaction. The unsteady Navier–Stoke and elastic membrane equations are solved separately using an iterative procedure by the GMRES technique with an incomplete lower-upper (ILU) decomposition at every time instant. One-dimensional, two-dimensional and deformable domain model problems are used to demonstrate the capabilities and accuracy of the present algorithm. Both steady state and transient problems are studied. © 1997 John Wiley & Sons, Ltd.     

Fluid structure interaction of cantilever micro and nanotubes conveying magnetic fluid with small size effects under a transverse magnetic field

Micro and nanotubes have found major application in fluidic systems as channels for conveying fluid. In some micro and nanofluidic applications such as drug delivery, a transverse magnetic field can be used to guide the fluid flow by generating an axial force in the flow direction. An important issue in the design of micro and nanofluidic systems is the structural vibration caused by the fluid flow. In the current study, we investigate the effect of transverse magnetic field on the vibration of cantilever micro and nanotubes conveying fluid by considering the small size effects. We couple the nonlocal Euler–Bernoulli beam model with Navier–Stokes theory to determine a fluid structure interaction (FSI) model for the vibration analysis of the system. We modify the FSI governing equation by driving a velocity correction factor to consider the effect of transverse magnetic field on the fluid flow’s pattern through the tube. Then, we use the Galerkin’s method to obtain the frequency diagrams for the instability analysis of the system. We show that the transverse magnetic field can have a substantial effect on the dynamics of tube conveying fluid by increasing the system’s natural frequencies and critical flow velocity which contributes to the flutter instability. We also discover that although the transverse magnetic field plays a crucial role on dynamics of microstructures, its effect on the dynamics of nanotubes is not significant and can be ignored.     

Comparison between computational fluid dynamics,fluid–structure interaction and computational structural dynamics predictions of flow-induced wall mechanics in an anatomically realistic cerebral aneurysm model

Haemodynamically induced stress plays an important role in the progression and rupture of cerebral aneurysms. The current work describes computational fluid dynamics (CFD), fluid–structure interaction (FSI) and computational structural dynamics (CSD) simulations in an anatomically realistic model of a carotid artery with two saccular cerebral aneurysms in the ophthalmic region. The model was obtained from three-dimensional (3D) rotational angiographic imaging data. CFD and FSI were studied under a physiologically representative waveform of inflow. The arterial wall was assumed elastic or hyperelastic, as a 3D solid or as a shell depending on the type of modelling used. The flow was assumed to be laminar, non-Newtonian and incompressible. The CFD, FSI and CSD models were solved with the finite elements package ADINA. Predictions of velocity field and wall shear stress (WSS) on the aneurysms made using CFD and FSI were compared. The CSD model of the aneurysms using complete geometry was compared with isolated aneurysm models. Additionally, the effects of hypertensive pressure on CSD aneurysm models are also reported. The vortex structure, WSS, effective stress, strain and displacement of the aneurysm walls showed differences, depending on the type of modelling used.