A phase field-immersed boundary-lattice Boltzmann coupling method for fluid-structure interaction analysis

Body-fitted and Cartesian grid methods are two typical types of numerical approaches used for modelling fluid–structure interaction problems. Despite their extensive applications, there is a lack of comparing the performance of these two types of approaches. In order to do this, the present paper presents benchmark numerical solutions for two two-dimensional fluid–structure interaction problems: flow-induced vibration of a highly flexible plate in an axial flow and a pitching flexible plate. The solutions are obtained by using two partitioned fluid–structure interaction methods including the deforming-spatial-domain/stabilized space–time fluid–structure interaction solver and the immersed boundary–lattice Boltzmann method. The deforming-spatial-domain/stabilized space–time fluid–structure interaction solver employs the body-fitted-grid deforming-spatial-domain/stabilized space–time method for the fluid motions and the finite-difference method for the structure vibrations. A new mesh update strategy is developed to prevent severe mesh distortion in cases where the boundary does not oscillate periodically or needs a long time to establish a periodic motion. The immersed boundary–lattice Boltzmann method uses lattice Boltzmann method as fluid solver and the same finite-difference method as structure solver. In addition, immersed boundary method is used in the immersed boundary–lattice Boltzmann solver to handle the fluid–structure interaction coupling. Results for the characteristic force coefficients, tail position, plate deformation pattern and the vorticity fields are presented and discussed. The present results will be useful for evaluating the performance and accuracy of existing and new numerical methodologies for fluid–structure interaction.

Fluid-structure interaction (FSI) is a very common physical phenomenon which extensively exists in nature, human daily life and many engineering applications. The lattice Boltzmann method (LBM) is an alternative of solving Navier–Stokes equations to obtain complex fluid dynamics. Since the proposal of lattice Bhatnagar–Gross–Krookmodel, the LBM has been improved and applied to various complex flows ranging from laminar flow to turbulent flow and Newtonian flow to non-Newtonian flow. To handle the associated FSI, the bounce-back scheme was proposed and then the immersed boundary method (IBM) was incorporated into the LBM for the simplicity and robustness of IBM in handling complex and moving geometries and the significant efficiency of LBM in solving fluid dynamics. In recent years, the combined frameworks of LBM, that is, bounce-back–lattice Boltzmann method and immersed boundary–lattice Boltzmann method have witnessed a significant development and the successful applications can be found in a range of physical problems such as flow around bluff bodies, flapping wings, wind turbines, aeroacoustics, sea wave modelling and et al. In this paper, the recent progress of LBM and some typical applications involving FSI are reviewed.

We present implicit time integration schemes suitable for modeling free surface Stokes flow dynamics with marker in cell (MIC) based spatial discretization. Our target is for example thermal convection surrounded by deformable surface boundaries to simulate the long term planetary formation process. The numerical system becomes stiff when the dynamical balancing time scale for the increasing/decreasing load by surface deformation is very short compared with the time scale associated with thermal convection. Any explicit time integration scheme will require very small time steps; otherwise, serious numerical oscillation (spurious solutions) will occur. The implicit time integration scheme possesses a wider stability region than the explicit method; therefore, it is suitable for stiff problems. To investigate an efficient solution method for the stiff Stokes flow system, we apply first (backward Euler (BE)) and second order (trapezoidal method (TR) and trapezoidal rule—backward difference formula (TR-BDF2)) accurate implicit methods for the MIC solution scheme. The introduction of implicit time integration schemes results in nonlinear systems of equations. We utilize a Jacobian free Newton Krylov (JFNK) based Newton framework to solve the resulting nonlinear equations. In this work we also investigate two efficient implicit solution strategies to reduce the computational cost when solving stiff nonlinear systems. The two methods differ in how the advective term in the material transport evolution equation is treated. We refer to the method that employs Lagrangian update as “fully implicit” (Imp), whilst the method that employs Eulerian update is referred to as “semi-implicit” (SImp). Using a finite difference (FD) method, we have performed a series of numerical experiments which clarify the accuracy of solutions and trade-off between the computational cost associated with the nonlinear solver and time step size. In comparison with the general explicit Euler method, the second order accurate Imp methods reduce total computational cost successfully through the utilization of a large time step without sacrificing accuracy and stability. Moreover, the proposed SImp method is effective in reducing the computational cost associated with evaluating the nonlinear residual while obtaining a solution similar to the Imp method.

Stability of semidiscrete formulations for parabolic problems at small time steps

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

A strong-coupled method combined finite element method and lattice Boltzmann method via an implicit immersed boundary scheme for fluid structure interaction

The Svenning, E. Mark, A. Edelvik, F. Fluid structure interaction Immersed boundary methods Sloshing tank FSI benchmark aim of this paper is to present and validate a modeling framework that can be used for simulation of industrial applications involving fluid structure interaction with large deformations. Fluid structure interaction phenomena involving elastic structures frequently occur in industrial applications such as rubber bushings filled with oil, the filling of liquid in a paperboard package or a fiber suspension flowing through a paper machine. Simulations of such phenomena are challenging due to the strong coupling between the fluid and the elastic structure. In the literature, this coupling is often achieved with an Arbitrary Lagrangian Eulerian framework or with smooth particle hydrodynamics methods. In the present work, an immersed boundary method is used to couple a finite volume based Navier-Stokes solver with a finite element based structural mechanics solver for large deformations. The benchmark of an elastic rubber beam in a rolling tank partially filled with oil is simulated. The simulations are compared to experimental data as well as numerical simulations published in the literature. 2D simulations performed in the present work agree well with previously published data. Our 3D simulations capture effects neglected in the 2D case, showing excellent agreement with previously published experiments. The good agreement with experimental data shows that the developed framework is suitable for simulation of industrial applications involving fluid structure interaction. If the structure is made of a highly elastic material, e.g. rubber, the simulation framework must be able to handle the large deformations that may occur. Immersed boundary methods are well suited for such applications, since they can efficiently handle moving objects without the need of a body-fitted mesh. Combining them with a structural mechanics solver for large deformations allows complex fluid structure interaction problems to be studied.

In this paper, a diffuse-interface immersed boundary method (IBM) is proposed to treat three different thermal boundary conditions (Dirichlet, Neumann, Robin) in thermal flow problems. The novel IBM is implemented combining with the lattice Boltzmann method (LBM). The present algorithm enforces the three types of thermal boundary conditions at the boundary points. Concretely speaking, the IBM for the Dirichlet boundary condition is implemented using an iterative method, and its main feature is to accurately satisfy the given temperature on the boundary. The Neumann and Robin boundary conditions are implemented in IBM by distributing the jump of the heat flux on the boundary to surrounding Eulerian points, and the jump is obtained by applying the jump interface conditions in the normal and tangential directions. A simple analysis of the computational accuracy of IBM is developed. The analysis indicates that the Taylor-Green vortices problem which was used in many previous studies is not an appropriate accuracy test example. The capacity of the present thermal immersed boundary method is validated using four numerical experiments: (1) Natural convection in a cavity with a circular cylinder in the center; (2) Flows over a heated cylinder; (3) Natural convection in a concentric horizontal cylindrical annulus; (4) Sedimentation of a single isothermal cold particle in a vertical channel. The numerical results show good agreements with the data in the previous literatures.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Convective heat transfer coefficient for a rod-like particle in a uniform flow

Aeroelastic study of flexible flapping wings by a coupled lattice Boltzmann-finite element approach with immersed boundary method

The objective of this brief review is to showcase recent applications of sharp-interface immersed boundary (IB) method to computationally challenging problems in fluid–structure interaction (FSI). The sharp-interface IB method is briefly reviewed and a scheme to strongly couple the flow and structural solver is discussed. The following applications have been discussed. First, the vortex-induced vibration of a cylinder has been presented. Second, FSI benchmarks of an elastic and a viscoelastic splitter attached on a cylinder involving large-scale flow-induced deformation have been presented. Third, numerical simulations of a FSI benchmark in a heated channel, demonstrating augmentation in convective heat transfer, have been reviewed. Finally, the flow-induced deformation of a thin elastic plate due to blast loading has been presented. In all of these FSI problems, the IB method has been successfully employed to simulate the coupled fluid and structural dynamics.

Fluid-structure interaction using lattice Boltzmann method: Moving boundary treatment and discussion of compressible effect

A surface flux correction-based immersed boundary-multiphase lattice Boltzmann flux solver applied to multiphase fluids–structure interaction

A new and efficient direct numerical method with second-order convergence accuracy was developed for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method combines the state-of-the-art immersed boundary method (IBM), the multi-direct forcing method, and the lattice Boltzmann method (LBM). First, the multi-direct forcing method is adopted in the improved IBM to better approximate the no-slip/no-penetration (ns/np) condition on the surface of particles. Second, a slight retraction of the Lagrangian grid from the surface towards the interior of particles with a fraction of the Eulerian grid spacing helps increase the convergence accuracy of the method. An over-relaxation technique in the procedure of multi-direct forcing method and the classical fourth order Runge-Kutta scheme in the coupled fluid-particle interaction were applied. The use of the classical fourth order Runge-Kutta scheme helps the overall IB-LBM achieve the second order accuracy and provides more accurate predictions of the translational and rotational motion of particles. The preexistent code with the first-order convergence rate is updated so that the updated new code can resolve the translational and rotational motion of particles with the second-order convergence rate. The updated code has been validated with several benchmark applications. The efficiency of IBM and thus the efficiency of IB-LBM were improved by reducing the number of the Lagragian markers on particles by using a new formula for the number of Lagrangian markers on particle surfaces. The immersed boundary-lattice Boltzmann method (IBLBM) has been shown to predict correctly the angular velocity of a particle. Prior to examining drag force exerted on a cluster of particles, the updated IB-LBM code along with the new formula for the number of Lagrangian markers has been further validated by solving several theoretical problems. Moreover, the unsteadiness of the drag force is examined when a fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. The simulation results agree well with the theories for the short- and long-time behavior of the drag force. Flows through non-rotational and rotational spheres in simple cubic arrays and random arrays are simulated over the entire range of packing fractions, and both low and moderate particle Reynolds numbers to compare the simulated results with the literature results and develop a new drag force formula, a new lift force formula, and a new torque formula. Random arrays of solid particles in fluids are generated with Monte Carlo procedure and Zinchenko's method to avoid crystallization of solid particles over high solid volume fractions. A new drag force formula was developed with extensive simulated results to be closely applicable to real processes over the entire range of packing fractions and both low and moderate particle Reynolds numbers. The simulation results indicate that the drag force is barely affected by rotational Reynolds numbers. Drag force is basically unchanged as the angle of the rotating axis varies.