An immersed boundary method for incompressible flows using volume of body function

A simple and effective immersed boundary method using volume of body (VOB) function is implemented on unstructured Cartesian meshes. The flow solver is a second‐order accurate implicit pressure‐correction method for the incompressible Navier–Stokes equations. The domain inside the immersed body is viewed as being occupied by the same fluid as outside with a prescribed divergence‐free velocity field. Under this view a fluid–body interface is similar to a fluid–fluid interface encountered in the volume of fluid (VOF) method for the two‐fluid flow problems. The body can thus be identified by the VOB function similar to the VOF function. In fluid–body interface cells the velocity is obtained by a volume‐averaged mixture of body and fluid velocities. The pressure inside the immersed body satisfies the same pressure Poisson equation as outside. To enhance stability and convergence, multigrid methods are developed to solve the difference equations for both pressure and velocity. Various steady and unsteady flows with stationary and moving bodies are computed to validate and to demonstrate the capability of the current method. Copyright © 2005 John Wiley & Sons, Ltd.     

Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid–structure interaction problems

Embedded boundary methods for CFD (computational fluid dynamics) simplify a number of issues. These range from meshing the fluid domain, to designing and implementing Eulerian‐based algorithms for fluid–structure applications featuring large structural motions and/or deformations. Unfortunately, embedded boundary methods also complicate other issues such as the treatment of the wall boundary conditions in general, and fluid–structure transmission conditions in particular. This paper focuses on this aspect of the problem in the context of compressible flows, the finite volume method for the fluid, and the finite element method for the structure. First, it presents a numerical method for treating simultaneously the fluid pressure and velocity conditions on static and dynamic embedded interfaces. This method is based on the exact solution of local, one‐dimensional, fluid–structure Riemann problems. Next, it describes two consistent and conservative approaches for computing the flow‐induced loads on rigid and flexible embedded structures. The first approach reconstructs the interfaces within the CFD solver. The second one represents them as zero level sets, and works instead with surrogate fluid/structure interfaces. For example, the surrogate interfaces obtained simply by joining contiguous segments of the boundary surfaces of the fluid control volumes that are the closest to the zero level sets are explored in this work. All numerical algorithms presented in this paper are applicable with any embedding CFD mesh, whether it is structured or unstructured. Their performance is illustrated by their application to the solution of three‐dimensional fluid–structure interaction problems associated with the fields of aeronautics and underwater implosion. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

Direct numerical simulation of particle–fluid systems by combining time-driven hard-sphere model and lattice Boltzmann method

A coupled numerical method for the direct numerical simulation of particle–fluid systems is formulated and implemented, resolving an order of magnitude smaller than particle size. The particle motion is described by the time-driven hard-sphere model, while the hydrodynamic equations governing fluid flow are solved by the lattice Boltzmann method (LBM). Particle–fluid coupling is realized by an immersed boundary method (IBM), which considers the effect of boundary on surrounding fluid as a restoring force added to the governing equations of the fluid. The proposed scheme is validated in the classical flow-around-cylinder simulations, and preliminary application of this scheme to fluidization is reported, demonstrating it to be a promising computational strategy for better understanding complex behavior in particle–fluid systems.     

A LBM–DEM solver for fast discrete particle simulation of particle–fluid flows

The lattice Boltzmann method (LBM) for simulating fluid phases was coupled with the discrete element method (DEM) for studying solid phases to formulate a novel solver for fast discrete particle simulation (DPS) of particle–fluid flows. The fluid hydrodynamics was obtained by solving LBM equations instead of solving the Navier–Stokes equation by the finite volume method (FVM). Interparticle and particle–wall collisions were determined by DEM. The new DPS solver was validated by simulating a three-dimensional gas–solid bubbling fluidized bed. The new solver was found to yield results faster than its FVM–DEM counterpart, with the increase in the domain-averaged gas volume fraction. Additionally, the scalability of the LBM–DEM DPS solver was superior to that of the FVM–DEM DPS solver in parallel computing. Thus, the LBM–DEM DPS solver is highly suitable for use in simulating dilute and large-scale particle–fluid flows.     

A computational framework for fluid–porous structure interaction with large structural deformation

We study the effect of poroelasticity on fluid–structure interaction. More precisely, we analyze the role of fluid flow through a deformable porous matrix in the energy dissipation behavior of a poroelastic structure. For this purpose, we develop and use a nonlinear poroelastic computational model and apply it to the fluid–structure interaction simulations. We discretize the problem by means of the finite element method for the spatial approximation and using finite differences in time. The numerical discretization leads to a system of non-linear equations that are solved by Newton’s method. We adopt a moving mesh algorithm, based on the Arbitrary Lagrangian–Eulerian method to handle large deformations of the structure. To reduce the computational cost, the coupled problem of free fluid, porous media flow and solid mechanics is split among its components and solved using a partitioned approach. Numerical results show that the flow through the porous matrix is responsible for generating a hysteresis loop in the stress versus displacement diagrams of the poroelastic structure. The sensitivity of this effect with respect to the parameters of the problem is also analyzed.

     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Simulation of self‐propelled anguilliform swimming by local domain‐free discretization method

The local domain‐free discretization method is extended in this work to simulate fluid–structure interaction problems, the class of which is exemplified by the self‐propelled anguilliform swimming of deforming bodies in a fluid medium. Given the deformation of the fish body in its own reference frame, the translational and rotational motions of the body governed by Newton's Law are solved together with the surrounding flow field governed by Navier–Stokes equations. When the body is deforming and moving, no mesh regeneration is required in the computation. The loose coupling strategy is employed to simulate the fluid–structure interaction involved in the self‐propelled swimming. The local domain‐free discretization method and an efficient algorithm for classifying the Eulerian mesh points are described in brief. To validate the fluid–structure interaction solver, we simulate the ‘lock‐in’ phenomena associated with the vortex‐induced vibrations of an elastically mounted cylinder. Finally, we demonstrate applications of the method to two‐dimensional and three‐dimensional anguilliform‐swimming fish. The kinematics and dynamics associated with the center of mass are shown and the rotational movement is also presented via the angular position of the body axis. The wake structure is visualized in terms of vorticity contours. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

Numerical study of Saffman–Taylor instability in immiscible nonlinear viscoelastic flows

In this paper, a numerical solution for Saffman–Taylor instability of immiscible nonlinear viscoelastic-Newtonian displacement in a Hele–Shaw cell is presented. Here, a nonlinear viscoelastic fluid pushes a Newtonian fluid and the volume of fluid method is applied to predict the formation of two phases. The Giesekus model is considered as the constitutive equation to describe the nonlinear viscoelastic behavior. The simulation is performed by a parallelized finite volume method (FVM) using second order in both the spatial and the temporal discretization. The effect of rheological properties and surface tension on the immiscible Saffman–Taylor instability are studied in detail. The destabilizing effect of shear-thinning behavior of nonlinear viscoelastic fluid on the instability is studied by changing the mobility factor of Giesekus model. Results indicate that the fluid elasticity and capillary number decrease the intensity of Saffman–Taylor instability.     

3-D inviscid self-excited vibrations of a blade row in the last stage turbine

Presented here is a three-dimensional (3-D) nonlinear time-marching method for the aeroelastic behaviour of an oscillating turbine blade row. The approach has been based in the solution of a coupled fluid–structure problem where the aerodynamic and structural dynamic equations are integrated simultaneously in time. This provides the correct formulation of a coupled problem, as the interblade phase angle (IBPA) at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity in the entire annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonic second-order accurate Godunov–Kolgan finite-volume scheme and a moving hybrid H–O (or H–H) grid. The fluid and the structural equations are solved using the modal superposition method. An aeroelasticity prediction of a turbine blade of 0.765 m is presented. The natural frequencies and modal shapes of the blade were calculated by using 3-D finite element models. The instability regions for five mode shapes and the distribution of the aerodamping coefficient along the blade length were shown for harmonic oscillations with an assumed IBPA. The coupled fluid–structure oscillations in which the IBPA is part of the solution are shown.     

Numerical stabilities of loosely coupled methods for robust modeling of lightweight and flexible structures in incompressible and viscous flows

The growing interest to examine the hydroelastic dynamics and stabilities of lightweight and flexible materials requires robust and accurate fluid–structure interaction(FSI)models. Classically, partitioned fluid and structure solvers are easier to implement compared to monolithic methods;however, partitioned FSI models are vulnerable to numerical("virtual added mass") instabilities for cases when the solid to fluid density ratio is low and if the flow is incompressible.As a partitioned method, the loosely hybrid coupled(LHC)method, which was introduced and validated in Young et al.(Acta Mech. Sin. 28:1030–1041, 2012), has been successfully used to efficiently and stably model lightweight and flexible structures. The LHC method achieves its numerical stability by, in addition to the viscous fluid forces, embedding potential flow approximations of the fluid induced forces to transform the partitioned FSI model into a semi-implicit scheme. The objective of this work is to derive and validate the numerical stability boundary of the LHC. The results show that the stability boundary of the LHC is much wider than traditional loosely coupled methods for a variety of numerical integration schemes. The results also show that inclusion of an estimate of the fluid inertial forces is the most critical to ensure the numerical stability when solving for fluid–structure interaction problems involving cases with a solid to fluid-added mass ratio less than one.     

Development of a hybrid model based on mesh and meshfree methods and its application to fluid–elastic structure interaction for free surface waves

In this paper, a hybrid scheme, Fluid–Fluid–Elastic Structure (FFES) model was developed in the time domain to address the wave breaking impact on the structure. The model is developed based on the partitioned approach with different governing equations that describe various regions of the model domain. The fluid–fluid model denotes that two different fluid models were used to describe fluid in the actual physical domain. The method is a physics-based approximation to reduce the computational time, i.e. in the far-field inviscid fluid (fully nonlinear potential flow theory model), and near to the structure, viscous fluid (Navier Stokes model) is used. The coupled model then interacts with the elastic structure (based on Euler–Bernoulli beam theory). The system of equations is strongly coupled both in space and time. The Fluid–Fluid coupling uses an implicit predictor–corrector scheme, and the fluid–structure coupling works based on an iterative scheme. This approach makes the method more robust and for future extension. Three different possibilities for introducing the coupling was identified and implemented. The model was validated against results from the analytical solution and literature. The method proposed is a reliable, robust, and efficient alternative for simulating fluid–structure interaction problems.     

An extended unresolved CFD-DEM coupling method for simulation of fluid and non-spherical particles

This study develops an extended unresolved CFD-DEM coupling method for simulation of the fluid–solid flow with non-spherical particles. The limitation of fluid grid size is discussed, by simulating the settling of a cylinder in a Newtonian fluid based on the resolved and unresolved CFD-DEM coupling method. Then, the calculation of porosity and the fluid–particle relative velocity based on the particle shape enlargement method for simulation of non-spherical particles is proposed. The availability of the particle shape enlargement method for the simulation of non-spherical particles with different sphericity is discussed in this work, by comparing it with the results from the equivalent diameter enlargement method. The limitation of the equivalent diameter enlargement method for non-spherical particles is revealed from the simulation results. Several typical cases are employed to elaborate and verify the extended unresolved CFD-DEM method based on particle shape enlargement method, by presenting a good consistency with the experimental results. It proves that the extended unresolved CFD-DEM method is suitable for different CFD grid size ratios, and consolidates that it is a universal calculation method for CFD-DEM coupling simulation.     

An analytical homogenization method for heterogeneous porous materials

The objective of this work is to develop an analytical homogenization method to estimate the effective mechanical properties of fluid-filled porous media with periodic microstructure. The method is based on the equivalent inclusion concept of homogenization applied earlier for solid–solid mixture. It is assumed that porous media are described by the poroelastic constitutive law developed by Biot where porosity is a material parameter. By solving the governing equations of poroelasticity in Fourier transformed domain, the relation between periodic strain and eigenstrain in porous media is established. This relation is subsequently used in an average consistency condition involving both solid and fluid phase stresses and strains. The geometry of the porous microstructure is captured in the g-integral. This homogenization method can also be applied to estimate the equivalent properties of solid–fluid mixture where a pure solid and fluid can be modeled by assuming very low and high porosity, respectively. Several examples are considered to establish this new method by comparing with other existing analytical and numerical methods of homogenization. As an application, poroelastic properties of cortical bone fibril are estimated and compared with previously computed values.     

Free vibration analysis of a finite circular cylindrical shell in contact with unbounded external fluid

The free flexural vibration of a finite cylindrical shell in contact with external fluid is investigated. The fluid is assumed to be inviscid and irrotational. The cylindrical shell is modeled by using the Rayleigh–Ritz method based on the Donnell–Mushtari shell theory. The fluid is modeled based on the baffled shell model, which is applied to fluid–structure interaction problems. The kinetic energy of the fluid is derived by solving the boundary-value problem. The natural vibration characteristics of the submerged cylindrical shell are discussed with respect to the added virtual mass approach. In this study, the nondimensionalized added virtual mass incremental factor for the submerged finite shell is derived. This factor can be readily used to estimate the change in the natural frequency of the shell due to the presence of the external fluid. Numerical results showed the efficacy of the proposed method, and comparison with previous results showed the validity of the theoretical results.     

A solution adaptive simulation of compressible multi‐fluid flows with general equation of state

The unstructured quadrilateral mesh‐based solution adaptive method is proposed in this article for simulation of compressible multi‐fluid flows with a general form of equation of state (EOS). The five equation model (J. Comput. Phys. 2002; 118 :577–616) is employed to describe the compressible multi‐fluid flows. To preserve the oscillation‐free property of velocity and pressure across the interface, the non‐conservative transport equation is discretized in a compatible way of the HLLC scheme for the conservative Euler equations on the unstructured quadrilateral cell‐based adaptive mesh. Five numerical examples, including an interface translation problem, a shock tube problem with two fluids, a solid impact problem, a two‐dimensional Riemann problem and a bubble explosion under free surface, are used to examine its performance in solving the various compressible multi‐fluid flow problems with either the same types of EOS or different types of EOS. The results are compared with those calculated by the following methods: the method with ROE scheme (J. Comput. Phys. 2002; 118 :577–616), the seven equation model (J. Comput. Phys. 1999; 150 :425–467), Shyue's fluid‐mixture model (J. Comput. Phys. 2001; 171 :678–707) or the method in Liu et al. (Comp. Fluids 2001; 30 :315–337). The comparisons for the test problems show that the proposed method seems to be more accurate than the method in Allaire et al. (J. Comput. Phys. 2002; 118 :577–616) or the seven‐equation model (J. Comput. Phys. 1999; 150 :425–467). They also show that it can adaptively and accurately solve these compressible multi‐fluid problems and preserve the oscillation‐free property of pressure and velocity across the material interface. Copyright © 2010 John Wiley & Sons, Ltd.