A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.     

Assessment of regularized delta functions and feedback forcing schemes for an immersed boundary method

We present an improved immersed boundary method for simulating incompressible viscous flow around an arbitrarily moving body on a fixed computational grid. To achieve a large Courant–Friedrichs–Lewy number and to transfer quantities between Eulerian and Lagrangian domains effectively, we combined the feedback forcing scheme of the virtual boundary method with Peskin's regularized delta function approach. Stability analysis of the proposed method was carried out for various types of regularized delta functions. The stability regime of the 4‐point regularized delta function was much wider than that of the 2‐point delta function. An optimum regime of the feedback forcing is suggested on the basis of the analysis of stability limits and feedback forcing gains. The proposed method was implemented in a finite‐difference and fractional‐step context. The proposed method was tested on several flow problems, including the flow past a stationary cylinder, inline oscillation of a cylinder in a quiescent fluid, and transverse oscillation of a circular cylinder in a free‐stream. The findings were in excellent agreement with previous numerical and experimental results. Copyright © 2008 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.     

An improved ghost‐cell immersed boundary method for compressible flow simulations

This study presents an improved ghost‐cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level‐set method to farther image points, incorporating a higher‐order extra/interpolation scheme for the ghost‐cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost‐cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward‐facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost‐cell method can reach the accuracy of second order in L¹ norm and higher than first order in L^∞ norm. Direct comparisons against the cut‐cell method demonstrate that the improved ghost‐cell method is almost equally accurate with better efficiency for boundary representation in high‐fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

A Voronoi‐based ALE solver for the calculation of incompressible flow on deforming unstructured meshes

An Arbitrary Lagrangian–Eulerian method for the calculation of incompressible Navier–Stokes equations in deforming geometries is described. The mesh node connectivity is defined by a Delaunay triangulation of the nodes, whereas the discretized equations are solved using finite volumes defined by the Voronoi dual of the triangulation. For prescribed boundary motion, an automatic node motion algorithm provides smooth motion of the interior nodes. Changes in the connectivity of the nodes are made through the use of local transformations to maintain the mesh as Delaunay. This allows the nodes and their associated Voronoi finite volumes to migrate through the domain in a free manner, without compromising the quality of the mesh. An MAC finite volume solver is applied on the Voronoi dual using a cell‐centred non‐staggered formulation, with cell‐face velocities being calculated by the Rhie–Chow momentum interpolation. Advective fluxes are approximated with the third‐order QUICK differencing scheme. The solver is demonstrated via its application to a driven cavity flow, and the flow about flapping aerofoil geometries. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

A new finite difference methodology is developed for the solution of computational fluid dynamics problems that do not require the use of staggered grid systems. Previous successful and robust non‐staggered methods, which used primitive variables and mass conservation in order to solve the pressure field, either interpolate cell‐face velocities or interpolate the pressure gradients in a special way, usually with an upwind‐bias to avoid the problem of odd–even coupling between the velocity and pressure fields. The new methodology presented does not detail a ‘special interpolation procedure for a primitive variable’, however, it manages to avoid the problem of odd–even coupling. The odd–even coupling is avoided by applying fourth‐order dissipation to the pressure field. It is shown that this approach can be regarded as a modified Rhie and Chow scheme. The method is implemented using a SIMPLE‐type algorithm and is applied to two test problems: laminar flow over a backward‐facing step and laminar flow in a square cavity with a driven lid. Good agreement is obtained between the numerical solutions and the corresponding benchmark solutions. The pressure dissipation term was found to successfully suppress wiggles in the pressure field. Copyright © 2000 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.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

不可压粘性流体浸入边界直接力法及软件实现

浸入边界法通过在N-S方程中施加体积力模拟不可滑移固壁边界及动边界,避免生成复杂贴体网格及动网格,极大地节省了网格建模时间及动网格计算消耗。本文提出一种新型附加体积力简化计算方法,将简化附加体积力以源项形式嵌入动量方程迭代中,通过用户自定义函数对CFD软件FLUENT二次开发,实现了浸入边界法和通用流体力学求解器的耦合计算。通过静止圆柱和动圆柱绕流数值模拟进行了验证,并探讨了插值函数对计算精度的影响。研究表明,通过引入浸入边界模型,能够提高计算效率,并实现结构网格背景下复杂边界和动边界的高效建模。     

An immersed boundary solver for inviscid compressible flows

In this paper, a simple and efficient immersed boundary (IB) method is developed for the numerical simulation of inviscid compressible Euler equations. We propose a method based on coordinate transformation to calculate the unknowns of ghost points. In the present study, the body‐grid intercept points are used to build a complete bilinear (2‐D)/trilinear (3‐D) interpolation. A third‐order weighted essentially nonoscillation scheme with a new reference smoothness indicator is proposed to improve the accuracy at the extrema and discontinuity region. The dynamic blocked structured adaptive mesh is used to enhance the computational efficiency. The parallel computation with loading balance is applied to save the computational cost for 3‐D problems. Numerical tests show that the present method has second‐order overall spatial accuracy. The double Mach reflection test indicates that the present IB method gives almost identical solution as that of the boundary‐fitted method. The accuracy of the solver is further validated by subsonic and transonic flow past NACA2012 airfoil. Finally, the present IB method with adaptive mesh is validated by simulation of transonic flow past 3‐D ONERA M6 Wing. Global agreement with experimental and other numerical results are obtained.     

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.     

Numerical simulations of fluid–structure interaction based on Cartesian grids with two boundary velocities

This work proposes an innovative numerical method for simulating the interaction of fluid with irregularly shaped stationary structures based on Cartesian grids. Instead of prescribing an artificial force to enforce the no‐slip boundary condition at the solid–fluid interface, this work imposes two boundary velocities, referred to as the solid and mass‐conserving boundary velocities, to satisfy the no‐slip boundary condition and mass conservation in the ghost cells around the immersed solid boundary. Both the traditional level set method [41] and the hybrid particle level set method [45] were used to represent the solid boundary and the complex free‐surface evolution, respectively. Consequently, the boundary velocities close to the immersed solid boundary can be determined in terms of the level set function and the neighboring fluid velocity. The projection method is further modified to incorporate the solid and mass‐conserving boundary velocities into the solution algorithm. A series of numerical experiments were conducted to demonstrate the feasibility of the proposed method. They involved uniform flow past a stationary circular cylinder and the propagation of water waves over a submerged trapezoidal breakwater. Comparisons between the numerical results and experimental data showed very good agreement in all cases of interest. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved hybrid Cartesian/immersed boundary method for fluid–solid flows

An improved hybrid Cartesian/immersed boundary method is proposed based on ghost point treatment. A second‐order Taylor series expansion is used to evaluate the values at the ghost points, and an inverse distance weighting method to interpolate the values due to its properties of preserving local extrema and smooth reconstruction. The present method effectively eliminates numerical instabilities caused by matrix inversion and flexibly adopts the interpolation in the vicinity of the boundary. Some typical fluid–solid flows, including viscous flow past a circular cylinder, a sphere, two cylinders in a side‐by‐side arrangement, and an array of 18 staggered cylinders, are examined. These benchmark simulations reasonably indicate the reliability and capability of the present method. Copyright © 2007 John Wiley & Sons, Ltd.     

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

In this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Employment of the second‐moment turbulence closure on arbitrary unstructured grids

The paper presents a finite‐volume calculation procedure using a second‐moment turbulence closure. The proposed method is based on a collocated variable arrangement and especially adopted for unstructured grids consisting of ‘polyhedral’ calculation volumes. An inclusion of 23k in the pressure is analysed and the impact of such an approach on the employment of the constant static pressure boundary is addressed. It is shown that this approach allows a removal of a standard but cumbersome velocity–pressure –Reynolds stress coupling procedure known as an extension of Rhie‐Chow method (AIAA J. 1983; 21 : 1525–1532) for the Reynolds stresses. A novel wall treatment for the Reynolds‐stress equations and ‘polyhedral’ calculation volumes is presented. Important issues related to treatments of diffusion terms in momentum and Reynolds‐stress equations are also discussed and a new approach is proposed. Special interpolation practices implemented in a deferred‐correction fashion and related to all equations, are explained in detail. Computational results are compared with available experimental data for four very different applications: the flow in a two‐dimensional 180o turned U‐bend, the vortex shedding flow around a square cylinder, the flow around Ahmed Body and in‐cylinder engine flow. Additionally, the performance of the methodology is assessed by applying it to different computational grids. For all test cases, predictions with the second‐moment closure are compared to those of the k–εmodel. The second‐moment turbulence closure always achieves closer agreement with the measurements. A moderate increase in computing time is required for the calculations with the second‐moment closure. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.     

Improvement of mass source/sink for an immersed boundary method

An improved immersed boundary method using a mass source/sink as well as momentum forcing is developed for simulating flows over or inside complex geometries. The present method is based on the Navier–Stokes solver adopting the fractional step method and a staggered Cartesian grid system. A more accurate formulation of the mass source/sink is derived by considering mass conservation of the virtual cells in the fluid crossed by the immersed boundary. Two flow problems (the decaying vortex problem and uniform flow past a circular cylinder) are used to validate the proposed formulation. The results indicate that the accuracy near the immersed boundary is improved by introducing the accurate mass source/sink. Copyright © 2006 John Wiley & Sons, Ltd.