Immersed boundary method for unsteady kinetic model equations

Predicting unsteady flows and aerodynamic forces for large displacement motion of microstructures requires transient solution of Boltzmann equation with moving boundaries. For the inclusion of moving complex boundaries for these problems, three immersed boundary method flux formulations (interpolation, relaxation, and interrelaxation) are presented. These formulations are implemented in a 2‐D finite volume method solver for ellipsoidal‐statistical (ES)‐Bhatnagar‐Gross‐Krook (BGK) equations using unstructured meshes. For the verification, a transient analytical solution for free molecular 1‐D flow is derived, and results are compared with the immersed boundary (IB)‐ES‐BGK methods. In 2‐D, methods are verified with the conformal, non‐moving finite volume method, and it is shown that the interrelaxation flux formulation gives an error less than the interpolation and relaxation methods for a given mesh size. Furthermore, formulations applied to a thermally induced flow for a heated beam near a cold substrate show that interrelaxation formulation gives more accurate solution in terms of heat flux. As a 2‐D unsteady application, IB/ES‐BGK methods are used to determine flow properties and damping forces for impulsive motion of microbeam due to high inertial forces. IB/ES‐BGK methods are compared with Navier–Stokes solution at low Knudsen numbers, and it is shown that velocity slip in the transitional rarefied regime reduces the unsteady damping force. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Combined interface boundary condition method for coupled thermal simulations

A new procedure for modeling the conjugate heat‐transfer process between fluid and structure subdomains is presented. The procedure relies on higher‐order combined interface boundary conditions (CIBC) for improved accuracy and stability. Traditionally, continuity of temperature and heat flux along interfaces is satisfied through algebraic jump conditions in a staggered fashion. More specifically, Dirichlet temperature conditions are usually imposed on the fluid side and Neumann heat‐flux conditions are imposed on the solid side for the stability of conventional sequential staggered procedure. In this type of treatment, the interface introduces additional stability constraints to the coupled thermal simulations. By utilizing the CIBC technique on the Dirichlet boundary conditions, a staggered procedure can be constructed with the same order of accuracy and stability as those of standalone computations. Using the Godunov–Ryabenkii normal‐mode analysis, a range of values of the coupling parameter is found that yields a stable and accurate interface discretization. The effectiveness of the method is investigated by presenting and discussing performance evaluation data using a 1D finite‐difference formulation for each subdomain. Copyright © 2007 John Wiley & Sons, Ltd.     

A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 John Wiley & Sons, Ltd.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order stable interpolations for immersed boundary methods

The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high‐order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two‐dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three‐dimensional flow in an air‐conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.     

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the structured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the solution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic parallel Galerkin domain decomposition procedures with grid modification for parabolic equation

Dynamic parallel Galerkin domain decomposition procedures with grid modification for semi‐linear parabolic equation are given. These procedures allow one to apply different domain decompositions, different grids, and interpolation polynomials on the sub‐domains at different time levels when necessary, in order to capture time‐changing localized phenomena, such as, propagating fronts or moving layers. They rely on an implicit Galerkin method in the sub‐domains and simple explicit flux calculation on the inter‐domain boundaries by integral mean method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the sub‐domains and across inter‐boundaries. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. Stability and convergence analysis in L²‐norm are derived for these procedures. The experimental results are presented to confirm the theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Mixed boundary elements for laminar flows

This paper presents a mixed boundary element formulation of the boundary domain integral method (BDIM) for solving diffusion–convective transport problems. The basic idea of mixed elements is the use of a continuous interpolation polynomial for conservative field function approximation and a discontinuous interpolation polynomial for its normal derivative along the boundary element. In this way, the advantages of continuous field function approximation are retained and its conservation is preserved while the normal flux values are approximated by interpolation nodal points with a uniquely defined normal direction. Due to the use of mixed boundary elements, the final discretized matrix system is overdetermined and a special solver based on the least squares method is applied. Driven cavity, natural and forced convection in a closed cavity are studied. Driven cavity results at Re=100, 400 and 1000 agree better with the benchmark solution than Finite Element Method or Finite Volume Method results for the same grid density with 21×21 degrees of freedom. The average Nusselt number values for natural convection 10³≤Ra≤10⁶ agree better than 0.1% with benchmark solutions for maximal calculated grid densities 61×61 degrees of freedom. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of free surface flows using the flux‐difference splitting scheme on the hybrid Cartesian/immersed boundary method

In this study, a method is developed to simulate the interaction between free surface flows and moving or deforming boundaries using the flux‐difference splitting scheme on the hybrid Cartesian/immersed boundary method. At each physical time step, the boundary is defined by an unstructured triangular surface grid. Immersed boundary (IB) nodes are distributed inside an instantaneous fluid domain based on edges crossing the boundary. At an IB node, dependent variables are reconstructed along the local normal line to the boundary. Inviscid fluxes are computed using Roe's flux‐difference splitting scheme for immiscible and incompressible fluids. The free surface is considered as a contact discontinuity in the density field. The motion of free surface is captured without any additional treatment along the fluid interface. The developed code is validated by comparisons with other experimental and computational results for a piston‐type wave maker, impulsive motion of a submerged circular cylinder, flow around a submerged hydrofoil, and Rayleigh–Taylor instability. The developed code is applied to simulate wave generation due to a continuously deforming bed beneath the free surface. The violent motion of a free surface caused by sloshing in a spherical tank is simulated. In this case, the free surface undergoes breakup and reconnection. Copyright © 2011 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.     

Numerical study of radial basis function interpolation for data transfer across discontinuous mesh interfaces

Allowing discontinuous or non‐matching mesh spacing across zonal interfaces within a computational domain offers many advantages, particularly in terms of easing the mesh generation process, reduction of required mesh densities, and relative motion between mesh zones. This paper presents a numerical study of a universal method for interpolating solution data across such interfaces. The method utilises radial basis functions (RBFs) for n‐dimensional volume interpolation, and treats the available solution data points simply as arbitrary clouds of points, eliminating all connectivity requirements and making it applicable to a wide range of computational problems. Properties of the developed meshless interface interpolation are investigated using analytic functions, and three issues are considered: the achievable order of spatial accuracy of the RBF interpolation alone and comparison with a variable order polynomial; the effect of a combined RBF and polynomial interpolation; and the ability of the method to recover frequency content. RBF interpolation alone is shown to achieve fourth‐order to sixth‐order spatial accuracy in one and two dimensions, and in three dimensions, using a small number of data points, third‐order and above is achievable even for a 3 : 1 discontinuous cell spacing ratio, that is a 27 : 1 volume ratio, across the interface. Hence, it is inefficient to include polynomial terms, since improving on the RBF spatial accuracy results in a significant increase in the system size and deterioration in conditioning. It is also shown that only five points per wavelength are required to capture both frequency and amplitude content of periodic solutions to less than 0.01% error.Copyright © 2013 John Wiley & Sons, Ltd.     

Computations of flow over a flexible plate using the hybrid Cartesian/immersed boundary method

A hybrid Cartesian/immersed boundary code is developed and applied to interactions between a flexible plate and a surrounding fluid. The velocities at the immersed boundary (IB) nodes are reconstructed by interpolations along local normal lines to an interface. A new criterion is suggested to distribute the IB nodes near an interface. The suggested criterion guarantees a closed fluid domain by a set of the IB nodes and it is applicable to a zero‐thickness body. To eliminate the pressure interpolation at the IB nodes, the hybrid staggered/non‐staggered grid method is adapted. The developed code is validated by comparisons with other experimental and computational results of flow around an in‐line oscillating cylinder. Good agreements are achieved for velocity profiles and vorticity and pressure contours. As applications to the fluid–structure interaction, oscillations of flexible plate in a resting fluid and flow over a flexible plate are simulated. The elastic deformations of the flexible plate are modelled based on the equations of motion for plates considering the fluid pressure as the external load on the plate. Two non‐dimensional parameters are identified and their effects on the damping of the plate motion are examined. Grid convergence tests are carried out for both cases. Copyright © 2007 John Wiley & Sons, Ltd.     

Improvements in the reliability and quality of unstructured hybrid mesh generation

This paper presents a reliable and automated approach to the generation of unstructured hybrid grids comprised of tetrahedra, prisms and pyramids for high Reynolds number viscous flow simulations. To enhance robustness, the hybrid mesh generation process starts with the formation of an isotropic tetrahedral grid. Prismatic layers are then added on no‐slip walls fully automatically by obeying user‐specified boundary conditions and three parameters: the number of the layers, an initial layer thickness normal to the walls, and a stretching factor. Topological modifications to the original isotropic tetrahedral elements are prohibited during the layer generation process. The tetrahedral elements near no‐slip walls are shifted inward and the resulting gap between the tetrahedra and the walls is filled up with prismatic elements. To enhance the quality of the prismatic layers around sharp corners, two normals are evaluated for the marching process in these regions. The addition of prismatic elements is locally stopped if negative‐volume elements are created or not enough space is left. An angle‐based smoothing method ensures that the quality of the tetrahedral elements is retained for a reasonable computational cost. The method is demonstrated for two scaled experimental supersonic airplane models designed at the National Aerospace Laboratory of Japan (NAL). Numerical results are compared with wind tunnel experimental data. Copyright © 2004 John Wiley & Sons, Ltd.     

一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法

相对于有限元法,边界单元法在求解断裂问题上有着独特的优势,现有的边界单元法中主要有子区域法和双边界积分方程法.采用一种改进的双边界积分方程法求解二维、三维断裂问题的应力强度因子,对非裂纹边界采用传统的位移边界积分方程,只需对裂纹面中的一面采用面力边界积分方程,并以裂纹间断位移为未知量直接用于计算应力强度因子.采用一种高阶奇异积分的直接法计算面力边界积分方程中的超强奇异积分;对于裂纹尖端单元,提供了三种不同形式的间断位移插值函数,采用两点公式计算应力强度因子.给出了多个具体的算例,与现存的精确解或参考解对比,可得到高精度的计算结果.      

A comparative study of lattice Boltzmann methods using bounce‐back schemes and immersed boundary ones for flow acoustic problems

In order to find applicable treatments of moving boundary conditions based on the lattice Boltzmann method in flow acoustic problems, three bounce‐back (BB) methods and four kinds of immersed boundary (IB) methods are compared. We focused on fluid–solid boundary conditions for flow acoustic problems especially the simulations of sound waves from moving boundaries. BB methods include link bounce‐back, interpolation bounce‐back and unified interpolation bounce‐back methods. Five IB methods are explicit and implicit direct‐forcing (Explicit‐IB and Implicit‐IB), two kinds of partially saturated computational methods and ghost fluid method. In order to reduce the spurious pressure generated by the fresh grid node changing from solid domain to fluid domain for BB methods and sharp IB methods, we proposed two new kinds of treatments and compared them with two existing ones. Simulations of the benchmark problems prove that the local evolutionary iteration (LI) is the best one in treatments of the fresh nodes. In addition, for standing boundary problems, although BB methods have a little higher accuracy, all the methods have similar accuracy. However, for moving boundary problems, IB methods are more appropriate than BB methods, because IB methods' smooth interpolation of pressure eld produces less disturbing spurious pressure waves. With improved treatments of fresh nodes, BB methods are also acceptable for moving boundary acoustic problems. In comparative tests in respective type, unified interpolation bounce‐back with LI, Implicit‐IB, and ghost fluid with LI are the best choices. Copyright © 2013 John Wiley & Sons, Ltd.     

Implicit boundary method for determination of effective properties of composite microstructures

A method that uses a structured grid to perform micromechanical analysis for determining effective properties of a composite microstructure is presented. This approach eliminates the need for constructing a mesh that has nodes along the interfaces between constituent materials of the composite. Implicit boundary method is used to ensure that interface conditions are satisfied at the material boundaries. In this method, solution structures for test and trial functions are constructed using approximate step functions such that the interface conditions are satisfied, even if there are no nodes on the material interface boundary. Since a structured grid does not conform to the geometry of the analysis domain, the geometry of the microstructure is defined independently using equations of the interface boundary curves/surfaces. Structured grids that overlap the geometry are easy to generate, and the elements in the grid are regular shaped and undistorted. A numerical example is presented to demonstrate that the proposed solution structure accurately models the solution across material interface, and convergence analysis is performed to show that the method converges as the grid density is increased. Fiber reinforced microstructures are analyzed to compute the effective elastic properties using both 2D and 3D models to show that the results match closely with the ones available in the literature.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

An accurate moving grid Eulerian Lagrangian localized adjoint method for solving the one‐dimensional variable‐coefficient ADE

An accurate finite‐volume Eulerian Lagrangian localized adjoint method (ELLAM) is presented for solving the one‐dimensional variable coefficients advection dispersion equation that governs transport of solute in porous medium. The method uses a moving grid to define the solution and test functions. Consequently, the need for spatial interpolation, or equivalently numerical integration, which is a major issue in conventional ELLAM formulations, is avoided. After reviewing the one‐dimensional method of ELLAM, we present our strategy and detailed calculations for both saturated and unsaturated porous medium. Numerical results for a constant‐coefficient problem and a variable‐coefficient problem are very close to analytical and fine‐grid solutions, respectively. The strength of the developed method is shown for a large range of CFL and grid Peclet numbers. Copyright 2004 John Wiley & Sons, Ltd.