A hybrid immersed‐boundary and multi‐block lattice Boltzmann method for simulating fluid and moving‐boundaries interactions

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed‐boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving‐boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish‐like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of local DFD method to simulate unsteady flows around an oscillating circular cylinder

In this paper, the recently proposed local domain‐free discretization (DFD) method is applied to simulate incompressible flows around an oscillating circular cylinder. It is found that it is very easy for the local DFD method to handle such moving boundary flow problems. This is because it does not need to move the mesh, which is indeed needed in traditional methods. Numerical experiments show that the present numerical results agree very well with the available data in the literature, and that the local DFD method is an effective tool for the computation of moving boundary flow problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient coupling of direct forcing immersed boundary-lattice Boltzmann method and finite element method to simulate fluid-structure interactions

Fluid-structure interactions (FSI) of rigid and flexible bodies are simulated in this article. For the fluid flow, multidirect forcing immersed boundary method (IBM) is adopted to capture the moving boundary, and lattice Boltzmann method (LBM) is used to evolve the flow field. Compared with our previous no-penetration IBM, less iterations are required in this work. In addition, larger velocity in lattice units can be used and the nonphysical force oscillations are suppressed due to the C³ 6-point kernel. Multi-relaxation-time collision operator and local grid refinement are also adopted in LBM to enhance the numerical stability and efficiency. To improve the efficiency of the FSI coupling algorithm, the mesh of the deformable structure can be coarser than the Lagrangian mesh using Newton-Cotes formulas to integrate the traction on the structure surface. A variety of benchmarks, including flow around a circular cylinder with Reynold number ranging from 20 to 200, forced oscillation of a circular cylinder, vortex-induced vibration (VIV) of an elastically mounted circular cylinder in two degrees of freedom, and VIV of an elastic cantilever beam attached to a circular cylinder, are carried out to evaluate the accuracy and stability of the present coupling algorithm. For the benchmark FSI problem considered in this article, a reduction of 54% of the calculation time is achieved using coarser structure mesh. As an application of the FSI coupling algorithm, the mechanism of an elastic beam in the wake of a circular cylinder is discussed.     

Mesh adaptation for simulation of unsteady flow with moving immersed boundaries

In this work, an approach for performing mesh adaptation in the numerical simulation of two‐dimensional unsteady flow with moving immersed boundaries is presented. In each adaptation period, the mesh is refined in the regions where the solution evolves or the moving bodies pass and is unrefined in the regions where the phenomena or the bodies deviate. The flow field and the fluid–solid interface are recomputed on the adapted mesh. The adaptation indicator is defined according to the magnitude of the vorticity in the flow field. There is no lag between the adapted mesh and the computed solution, and the adaptation frequency can be controlled to reduce the errors due to the solution transferring between the old mesh and the new one. The preservation of conservation property is mandatory in long‐time scale simulations, so a P1‐conservative interpolation is used in the solution transferring. A nonboundary‐conforming method is employed to solve the flow equations. Therefore, the moving‐boundary flows can be simulated on a fixed mesh, and there is no need to update the mesh at each time step to follow the motion or the deformation of the solid boundary. To validate the present mesh adaptation method, we have simulated several unsteady flows over a circular cylinder stationary or with forced oscillation, a single self‐propelled swimming fish, and two fish swimming in the same or different directions. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive mesh refinement for simulating fluid-structure interaction using a sharp interface immersed boundary method

In this article, adaptive mesh refinement (AMR) is performed to simulate flow around both stationary and moving boundaries. The finite-difference approach is applied along with a sharp interface immersed boundary (IB) method. The Lagrangian polynomial is employed to facilitate the interpolation from a coarse to a fine grid level, while a weighted-average formula is used to transfer variables inversely. To save memory, the finest grid is only generated in the local areas close to the wall boundary, and the mesh is dynamically reconstructed based on the location of the wall boundary. The Navier-Stokes equations are numerically solved through the second-order central difference scheme in space and the third-order Runge-Kutta time integration. Flow around a circular cylinder rotating in a square domain is firstly simulated to examine the accuracy and convergence rate. Then three cases are investigated to test the validity of the present method: flow past a stationary circular cylinder at low Reynolds numbers, flow past a forced oscillating circular cylinder in the transverse direction at various frequencies, and a free circular cylinder subjected to vortex-induced vibration in two degrees of freedom. Computational results agree well with these in the literature and the flow fields are smooth around the interface of different refinement levels. The effect of refinement level has also been evaluated. In addition, a study for the computational efficiency shows that the AMR approach is helpful to reduce the total node number and speed up the time integration, which could prompt the application of the IB method when a great near-wall spatial resolution is required.     

Orbital flow past a cylinder: A numerical approach

Orbital flow past a cylinder is relevant to offshore structures. The numerical scheme presented here is based on a finite-difference solution of the Navier–Stokes equations. Alternating-directional-implicit (ADI) and successive-over-relaxation (SOR) techniques are used to solve the vorticity-transport and stream-function equations. Theoretical simulations to low Reynolds number flows (up to 1000) are discussed for cases involving uniform flow past stationary and rotating cylinders and orbital flow past a cylinder. The separation points for cylinders that are rotating or immersed in an orbital flow are deduced from velocity profiles through the boundary layer using a hybrid mesh scheme. During the initial development of orbital flow surface vorticity on the impulsively started cylinder dominates the flow. A vortex then detaches from behind the cylinder and establishes the flow pattern of the orbit. After some time a collection of vortices circles the orbit and distorts its shape a great deal. These vortices gradually spiral outward as others detach from the cylinder and join the orbital path.     

A new modification of the immersed‐boundary method for simulating flows with complex moving boundaries

In this paper, a new immersed‐boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite‐difference approach on a staggered mesh together with a fractional‐step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no‐slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed‐boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method. Copyright © 2006 John Wiley & Sons, Ltd.     

On the flows produced by sudden application of a constant pressure gradient or by impulsive motion of a boundary

Some properties of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Five illustrative examples are given. They are: unsteady flow in a circular cylinder moving parallel to its length, starting flow in a circular pipe, unsteady flow in a rotating cylinder, starting flow in a rectangular channel moving parallel to its length and unsteady flow in a channel of rectangular cross-section. It is found that the expressions of the quantities such as velocity, flux and skin friction are in series forms which may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa. It is shown that if their expressions can be found for one of large values of the time or small values of the time, these expressions can be used for the other.     

分区Lagrangian涡方法及瞬时起动圆柱初期流动的模拟

本文提出一种分区Lagrangian涡方法:将附着流动和分离流动分开处理,在附着区解边界层方层,只在分离区用涡方法解N-S方程。由于将尺度不同的区域分开了,求解分离区流动的涡方法中,每一时间步上物面引出的涡数在较小程度上依赖于Re数。这样,求解高Re数流动时,流场内的涡数,因而计算机内存和时间得以大大减小。用该方法计算了瞬时起动圆柱的初期流动,与实验结果比较相符很好。     

An improved Rhie–Chow interpolation scheme for the smoothed‐interface immersed boundary method

Rhie–Chow interpolation is a commonly used method in CFD calculations on a co‐located mesh in order to suppress non‐physical pressure oscillations arising from chequerboard effects. A fully parallelized smoothed‐interface immersed boundary method on a co‐located grid is described in this paper. We discuss the necessity of modifications to the original Rhie–Chow interpolation in order to deal with a locally refined mesh. Numerical simulation with the modified scheme of Choi shows that numerical dissipation due to Rhie–Chow interpolation introduces significant errors at the immersed boundary. To address this issue, we develop an improved Rhie–Chow interpolation scheme that is shown to increase the accuracy in resolving the flow near the immersed boundary. We compare our improved scheme with the modified scheme of Choi by parallel simulations of benchmark flows: (i) flow past a stationary cylinder; (ii) flow past an oscillating cylinder; and (iii) flow past a stationary elliptical cylinder, where Reynolds numbers are tested in the range 10–200. Our improved scheme is significantly more accurate and compares favourably with a staggered grid algorithm. We also develop a scheme to compute the boundary force for the direct‐forcing immersed boundary method efficiently. Copyright © 2016 John Wiley & Sons, Ltd.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Unsteady natural convective flow past a moving vertical cylinder with heat and mass transfer

Transient natural convection boundary layer flow of an incompressible viscous fluid past an impulsively moving semi- infinite vertical cylinder is considered. The temperature and concentration of the cylinder surface are taken to be uniform. The unsteady, nonlinear and coupled governing equations of the flow are solved using an implicit finite difference scheme. The finite difference scheme is unconditionally stable and accurate. Numerical results are presented with various sets of parameters for both air and water. Transient effects of velocity, temperature and concentration profiles are analyzed. Local and average skin friction, rates of heat and mass transfer are shown graphically. Received on 1 November 1999     

Computation of high Reynolds number flow around a circular cylinder with surface roughness

Incompressible high-Reynolds-number flows around a circular cylinder are analyzed by direct integration of the Navier-Stokes equations using finite-difference method. A generalized coordinate system is used so that a sufficient number of grid points are distributed in the boundary layer and the wake. A numerical scheme which suppresses non-linear instability for calculations of high-Reynolds-number flows is developed. The computation of an impulsively started flow at Re = 1200 is compared with corresponding experimental observations, and excellent agreements are obtained.A series of computations are carried out on the flow around a circular cylinder with surface roughness. The height of the roughness in these computations is 0.5% of the diameter. The range of Reynolds numbers is from 10³ to 10⁵; no turbulence model is employed. Sharp reduction of drag coefficient is observed near Re = 2 × 10⁴, which indicates that the critical Reynolds number is captured in the present computation.     

Simulation of dynamic fluid–solid interactions with an improved direct-forcing immersed boundary method

Dynamic fluid–solid interactions are widely found in chemical engineering, such as in particle-laden flows, which usually contain complex moving boundaries. The immersed boundary method (IBM) is a convenient approach to handle fluid–solid interactions with complex geometries. In this work, Uhlmann's direct-forcing IBM is improved and implemented on a supercomputer with CPU–GPU hybrid architecture. The direct-forcing IBM is modified as follows: the Poisson's equation for pressure is solved before evaluation of the body force, and the force is only distributed to the Cartesian grids inside the immersed boundary. A multidirect forcing scheme is used to evaluate the body force. These modifications result in a divergence-free flow field in the fluid domain and the no-slip boundary condition at the immersed boundary simultaneously. This method is implemented in an explicit finite-difference fractional-step scheme, and validated by 2D simulations of lid-driven cavity flow, Couette flow between two concentric cylinders and flow over a circular cylinder. Finally, the method is used to simulate the sedimentation of two circular particles in a channel. The results agree very well with previous experimental and numerical data, and are more accurate than the conventional direct-forcing method, especially in the vicinity of a moving boundary.     

Coupling methods between finite element–based Boussinesq‐type wave and particle‐based free‐surface flow models

We develop one‐way coupling methods between a Boussinesq‐type wave model based on the discontinuous Galerkin finite element method and a free‐surface flow model based on a mesh‐free particle method to strike a balance between accuracy and computational cost. In our proposed model, computation of the wave model in the global domain is conducted first, and the nonconstant velocity profiles in the vertical direction are reproduced by using its results. Computation of the free‐surface flow is performed in a local domain included within the global domain with interface boundaries that move along the reproduced velocity field in a Lagrangian fashion. To represent the moving interfaces, we used a polygon wall boundary model for mesh‐free particle methods. Verification and validation tests of our proposed model are performed, and results obtained by the model are compared with theoretical values and experimental results to show its accuracy and applicability.     

An extended finite element method for the simulation of particulate viscoelastic flows

We present an extended finite element method (XFEM) for the direct numerical simulation of the flow of viscoelastic fluids with suspended particles. For moving particle problems, we devise a temporary arbitrary Lagrangian–Eulerian (ALE) scheme which defines the mapping of field variables at previous time levels onto the computational mesh at the current time level. In this method, a regular mesh is used for the whole computational domain including both fluid and particles. A temporary ALE mesh is constructed separately and the computational mesh is kept unchanged throughout the whole computations. Particles are moving on a fixed Eulerian mesh without any need of re-meshing. For mesh refinements around the interface, we combine XFEM with the grid deformation method, in which nodal points are redistributed close to the interface while preserving the mesh topology. Our method is verified by comparing with the results of boundary fitted mesh problems combined with the conventional ALE scheme. The proposed method shows similar accuracy compared with boundary fitted mesh problems and superior accuracy compared with the fictitious domain method. If the grid deformation method is combined with XFEM, the required computational time is reduced significantly compared to uniform mesh refinements, while providing mesh convergent solutions. We apply the proposed method to the particle migration in rotating Couette flow of a Giesekus fluid. We investigate the effect of initial particle positions, the Weissenberg number, the mobility parameter of the Giesekus model and the particle size on the particle migration. We also show two-particle interactions in confined shear flow of a viscoelastic fluid. We find three different regimes of particle motions according to initial separations of particles.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.