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.     

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.     

A moving mesh BGK scheme for multi‐material flows

In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

CFD simulation of two‐ and three‐dimensional free‐surface flow

The paper describes the implementation of moving‐mesh and free‐surface capabilities within a 3‐d finite‐volume Reynolds‐averaged‐Navier–Stokes solver, using surface‐conforming multi‐block structured meshes. The free‐surface kinematic condition can be applied in two ways: enforcing zero net mass flux or solving the kinematic equation by a finite‐difference method. The free surface is best defined by intermediate control points rather than the mesh vertices. Application of the dynamic boundary condition to the piezometric pressure at these points provides a hydrostatic restoring force which helps to eliminate any unnatural free‐surface undulations. The implementation of time‐marching methods on moving grids are described in some detail and it is shown that a second‐order scheme must be applied in both scalar‐transport and free‐surface equations if flows driven by free‐surface height variations are to be computed without significant wave attenuation using a modest number of time steps. Computations of five flows of theoretical and practical interest—forced motion in a pump, linear waves in a tank, quasi‐1d flow over a ramp, solitary wave interaction with a submerged obstacle and 3‐d flow about a surface‐penetrating cylinder—are described to illustrate the capabilities of our code and methods. Copyright © 2003 John Wiley & Sons, Ltd.     

A comparative study of direct‐forcing immersed boundary‐lattice Boltzmann methods for stationary complex boundaries

In this study, we assess several interface schemes for stationary complex boundary flows under the direct‐forcing immersed boundary‐lattice Boltzmann methods (IB‐LBM) based on a split‐forcing lattice Boltzmann equation (LBE). Our strategy is to couple various interface schemes, which were adopted in the previous direct‐forcing immersed boundary methods (IBM), with the split‐forcing LBE, which enables us to directly use the direct‐forcing concept in the lattice Boltzmann calculation algorithm with a second‐order accuracy without involving the Navier–Stokes equation. In this study, we investigate not only common diffuse interface schemes but also a sharp interface scheme. For the diffuse interface scheme, we consider explicit and implicit interface schemes. In the calculation of velocity interpolation and force distribution, we use the 2‐ and 4‐point discrete delta functions, which give the second‐order approximation. For the sharp interface scheme, we deal with the exterior sharp interface scheme, where we impose the force density on exterior (solid) nodes nearest to the boundary. All tested schemes show a second‐order overall accuracy when the simulation results of the Taylor–Green decaying vortex are compared with the analytical solutions. It is also confirmed that for stationary complex boundary flows, the sharper the interface scheme, the more accurate the results are. In the simulation of flows past a circular cylinder, the results from each interface scheme are comparable to those from other corresponding numerical schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Based on the Bhatnagar–Gross–Krook (BGK) Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non‐linear source terms. Based on the unsteady time‐splitting technique and the non‐oscillatory, containing no free parameters, and dissipative (NND) finite‐difference method, the gas kinetic finite‐difference second‐order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm for the gas dynamical problems from various flow regimes is developed. To test the reliability of the present numerical method, the one‐dimensional shock‐tube problems and the flows past two‐dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related flows indicate that both high resolution of the flow fields and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

An evaluation of a 3D free‐energy‐based lattice Boltzmann model for multiphase flows with large density ratio

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free‐energy‐based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56 :1653–1671) is applied to solve the 3D C–H equation. To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long‐time simulation of a bubble rising. How to handle the inlet/outlet and moving‐wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation. This model's performance on two‐phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high‐Re cases, the mass conservation seems not so good as the low‐Re case. Copyright © 2009 John Wiley & Sons, Ltd.     

A multi‐energy‐level lattice Boltzmann model for the compressible Navier–Stokes equations

In this paper, we propose a new lattice Boltzmann model for the compressible Navier–Stokes equations. The new model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. As the 25‐bit model, we obtained the equilibrium distribution functions and the compressible Navier–Stokes equations with the second accuracy of the truncation errors. The numerical examples show that the model can be used to simulate the shock waves, contact discontinuities and supersonic flows around circular cylinder. The numerical results are compared with those obtained by traditional method. Copyright © 2007 John Wiley & Sons, Ltd.     

Development of new finite volume schemes on unstructured triangular grid for simulating the gas–liquid two‐phase flow

This paper presents a coupled finite volume inner doubly iterative efficient algorithm for linked equations (IDEAL) with level set method to simulate the incompressible gas–liquid two‐phase flows with moving interfaces on unstructured triangular grid. The finite volume IDEAL method on a collocated grid is employed to solve the incompressible two‐phase Navier–Stokes equations, and the level set method is used to capture the moving interfaces. For the sake of mass conservation, an effective second‐order accurate finite volume scheme is developed to solve the level set equation on triangular grid, which can be implemented much easier than the classical high‐order level set solvers. In this scheme, the value of level set function on the boundary of control volume is approximated using a linear combination of a high‐order Larangian interpolation and a second‐order upwind interpolation. By the rotating slotted disk and stretching and shrinking of a circular fluid element benchmark cases, the mass conservation and accuracy of the new scheme is verified. Then the coupled method is applied to two‐phase flows, including a 2D bubble rising problem and a 2D dam breaking problem. The computational results agree well with those reported in literatures and experimental data. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐dimensional implicit time‐dependent calculations on adaptive unstructured meshes with time evolving boundaries

An implicit multigrid‐driven algorithm for two‐dimensional incompressible laminar viscous flows has been coupled with a solution adaptation method and a mesh movement method for boundary movement. Time‐dependent calculations are performed implicitly by regarding each time step as a steady‐state problem in pseudo‐time. The method of artificial compressibility is used to solve the flow equations. The solution mesh adaptation method performs local mesh refinement using an incremental Delaunay algorithm and mesh coarsening by means of edge collapse. Mesh movement is achieved by modeling the computational domain as an elastic solid and solving the equilibrium equations for the stress field. The solution adaptation method has been validated by comparison with experimental results and other computational results for low Reynolds number flow over a shedding circular cylinder. Preliminary validation of the mesh movement method has been demonstrated by a comparison with experimental results of an oscillating airfoil and with computational results for an oscillating cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

Unified gas kinetic scheme combined with Cartesian grid method for intermediate Mach numbers

We develop a method to seamlessly simulate flows over a wide range of Knudsen numbers past arbitrarily shaped immersed boundaries. To achieve seamless computation, ie, not use any zone division to distinguish between continuum and non‐continuum regions, we use the unified gas kinetic scheme (UGKS), which is based on the Bhatnagar‐Groos‐Krook (BGK) approximation of the Boltzmann equation. We combine UGKS with an appropriately designed Cartesian grid method (CGM) to allow us to compute flows past arbitrary boundaries. The CGM we use here satisfies boundary conditions at the wall by using a constrained least square interpolation procedure. However, it differs from the usual, continuum CGMs in 2 ways. Firstly, to allow us capture non‐continuum effects at the boundaries, the CGM used herein interpolates the microscopic velocity distribution function in addition to the macroscopic variables. Secondly, even for the macroscopic variables, we use a gas kinetic method–based density interpolation procedure at the boundaries that allows the CGM to interface well with the UGKS method. We demonstrate the robustness and efficacy of the method by testing it on stationary immersed boundaries at various Knudsen numbers ranging from continuum to transition regimes.     

Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

A parallel unstructured dynamic mesh adaptation algorithm for 3‐D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three‐dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell‐centred finite‐volume method and the Roe's flux‐difference splitting. Second‐order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time‐dependent solutions was enhanced by adopting an h‐refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed‐up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock‐tube problem. The flows around oscillating NACA0012 wing and F‐5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

On improvements of simplified and highly stable lattice Boltzmann method: Formulations,boundary treatment,and stability analysis

In this paper, we present a detailed report on a revised form of simplified and highly stable lattice Boltzmann method (SHSLBM) and its boundary treatment as well as stability analysis. The SHSLBM is a recently developed scheme within lattice Boltzmann framework, which utilizes lattice properties and relationships given by Chapman‐Enskog expansion analysis to reconstruct solutions of macroscopic governing equations recovered from lattice Boltzmann equation and resolved in a predictor‐corrector scheme. Formulations of original SHSLBM are slightly adjusted in the present work to facilitate implementation on body‐fitted mesh. The boundary treatment proposed in this paper offers an analytical approach to interpret no‐slip boundary condition, and the stability analysis in this paper fixes flaws in previous works and reveals a very nice stability characteristic in high Reynolds number scenarios. Several benchmark tests are conducted for comprehensive evaluation of the boundary treatment and numerical validation of stability analysis. It turns out that by adopting the modifications suggested in this work, lower numerical error can be expected.     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.