A multiblock Navier–Stokes algorithm using equal-order quadratic finite elements

A new multiblock pressure-based finite element algorithm has been developed. This methodology implements quadratic interpolation for both the elemental velocity and pressure fields. A direct streamline upwinding scheme previously developedby the authors is used to model the non-linear inertia effects. Details of the algorithm and its multiblock foundation are provided along with validating test cases. The results presented clearly demonstrate the accuracy of this new approach and the differences in the pressure field for an element using quadratic versus the traditional bi linear approximation of the pressure field.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.     

A parameter-free ε-adaptive algorithm for improving weighted compact nonlinear schemes

In this paper, we propose a parameter-free algorithm to calculate ε, a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth-order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high-order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high-order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high-order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth-order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders.     

EFFICIENT COMPUTATIONS USING UPWIND BIASED SCHEMES

A new multiblock unfactored implicit upwind scheme for inviscid two-dimensional flow calculations is presented. Spatial discretization is carried out by means of an upwind first-order method; an original extension to higher accuracy is also presented. The integration algorithm is constructed in a ‘δ’ form that provides a direct derivation of the scheme and leads to an efficient computational method. Fast solutions of the linear systems arising at each time step are obtained by means of the bi-conjugate gradient stabilized technique. The computational results for super/hypersonic steady state flows illustrate the efficiency and accuracy of the algorithm.     

A multiblock/multilevel mesh refinement procedure for CFD computations

A multiblock/multilevel algorithm with local refinement for general two‐ and three‐dimensional fluid flow is presented. The patched‐based local refinement procedure is presented in detail and algorithmic implementations are also presented. The multiblock implementation is essentially block‐unstructured, i.e. each block having its own local curvilinear co‐ordinate system. Refined grid patches can be put anywhere in the computational domain and can extend across block boundaries. To simplify the implementation, while still maintaining sufficient generality, the refinement is restricted to a refinement of the grid successively halving the grid size within a selected patch. The multiblock approach is implemented within the framework of the well‐known SIMPLE solution strategy. Computational experiments showing the effect of using the multilevel solution procedure are presented for a sample elliptic problem and a few benchmark problems of computational fluid dynamics (CFD). Copyright © 2001 John Wiley & Sons, Ltd.     

Multi-dimensional self-affine fractal interpolation model in tensor form

Iterated Function System (IFS) models have been explored to represent discrete sequences where the attractor of an IFS is self-affine either in R ² or R ³ (R is the set of real numbers). In this paper, the self-affine IFS model is extended from R ³ to R ⁿ (n is an integer and greater than 3), which is called the multi-dimensional self-affine fractal interpolation model. This new model is presented by introducing the defined parameter “mapping partial derivative”. A constrained inverse algorithm is given for the identification of the model parameters. The values of this new model depend continuously on all of the variables. That is, the function is determined by the coefficients of the possibly multi-dimensional affine maps. So the new model is presented as much more general and significant. Moreover, the multi-dimensional self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.     

A MULTIBLOCK IMPLEMENTATION OF A NON-ORTHOGONAL,COLLOCATED FINITE VOLUME ALGORITHM FOR COMPLEX TURBULENT FLOWS

A multiblock algorithm for general 2D and 3D turbulent flows is introduced and applied to three cases: a compressor cascade passage, a two-element high-lift aerofoil and a round-to-square transition duct. The method is a generalization of a single-block scheme which is based on a non-orthogonal, fully collocated finite volume framework, applicable to incompressible and compressible flows and incorporating a range of turbulence transport models, including second-moment closure. The multiblock implementation is essentially block-unstructured, each block having its own local co-ordinate system unrelated to those of its neighbours. Any one block may interface with more than one neighbour along any one block face. Interblock communication is handled by connectivity matrices and effected via a two-cell overlap region along block boundaries in which ‘halo data’ reside. The algorithm and the associated data communication are explained in detail, and its effectiveness is verified, with particular reference to improved numerical resolution and parallel computing.     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Aeroacoustic Modelling of Low-Speed Flows

A new numerical algorithm for acoustic noise generation is developed. The approach involves two steps comprising an incompressible flow part and an inviscid acoustic part. The acoustic part can be started at any time of the incompressible computation. The formulation can be applied both for isentropic flows and non-isentropic flows. The model is validated for the cases of an isentropic pulsating sphere and non-isentropic flows past a circular cylinder. In the latter case the computations show that the generated acoustic field in addition to the dominant Strouhal frequency, f ₀, contains a slightly higher frequency, f ₂, and a modulating lower frequency, f ₁=f ₂−f ₀. Numerical experiments with different interpolation schemes, boundary conditions, etc., show that the appearance of these modes is not an artifact from the numerical discretization. Received 20 July 1998 and accepted 26 May 1999     

Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

The use of a colocated variable arrangement for the numerical solution of fluid flow is becoming more and more popular due to its coding simplicity. The inherent decoupling of the pressure and velocity fields in this arrangement can be handled via a special interpolation procedure for the calculation of the cell face velocity named pressure‐weighted interpolation method (PWIM) (AIAA J. 1983; 21 (11):1525–1532). In this paper a discussion on the alternatives to extend PWIM to unsteady flows is presented along with a very simple criterion to ascertain if a given interpolation practice will produce steady results that are relaxation dependent or time step dependent. Following this criterion it will be shown that some prior schemes presented as time step independent are actually not, although by using special interpolations can be readily adapted to be. A systematic way of deriving different cell face velocity expressions will be presented and new formulae free of Δt dependence will be derived. Several computational exercises will accompany the theoretical discussion to support our claims. Copyright © 2009 John Wiley & Sons, Ltd.     

抗差自适应插值粒子滤波及其在组合导航中的应用

针对粒子滤波存在的粒子退化和重要性密度函数难以选取的问题,在吸收抗差自适应滤波、二阶插值滤波和粒子滤波算法优点的基础上,提出了一种新的抗差自适应插值粒子滤波算法。该算法利用二阶插值滤波算法得到重要性密度函数,通过抗差自适应因子实时控制动力学模型误差及观测异常对导航解的影响。将该算法应用于SINS/CNS/SAR组合导航系统进行计算仿真,并与经典的粒子滤波算法进行比较分析。结果表明,提出的滤波算法得到的姿态误差控制在[-0.3′,+0.3′],速度误差控制在[-0.4 m/s,+0.4 m/s],位置误差控制在[-5 m,+5 m],性能明显优于经典的粒子滤波算法。新的滤波算法不但能够有效地抑制粒子退化,而且能够有效地控制动力学模型误差及观测异常的影响,提高了组合导航的滤波精度。     

A decoupling numerical method for fluid flow

A first-order non-conforming numerical methodology, Separation method, for fluid flow problems with a 3-point exponential interpolation scheme has been developed. The flow problem is decoupled into multiple one-dimensional subproblems and assembled to form the solutions. A fully staggered grid and a conservational domain centred at the node of interest make the decoupling scheme first-order-accurate. The discretization of each one-dimensional subproblem is based on a 3-point interpolation function and a conservational domain centred at the node of interest. The proposed scheme gives a guaranteed first-order accuracy. It is shown that the traditional upwind (or exponentially weighted upstream) scheme is less than first-order-accurate. The pressure is decoupled from the velocity field using the pressure correction method of SIMPLE. Thomas algorithm (tri-diagonal solver) is used to solve the algebraic equations iteratively. The numerical advantage of the proposed scheme is tested for laminar fluid flows in a torus and in a square-driven cavity. The convergence rates are compared with the traditional schemes for the square-driven cavity problem. Good behaviour of the proposed scheme is ascertained.     

基于信息扩散的海洋稀疏观测资料插值算法-椭圆模型

针对海洋科学研究中ARGO等观测资料零散、稀少等问题,提出了一种基于信息扩散思想的插值方法一信息扩散插值算法。该方法基于模糊映射思想,通过对稀少数据点信息进行模糊扩散和插值映射,进而实现有限数据点信息向其邻近区域点的概率插值;针对正态扩散函数在表现非对称结构的数据资料时存在的局限性,发展了一类非均匀信息扩散函数,建立了相应的非均匀信息扩散插值算法一椭圆模型,进行了试验对比和实际ARGO资料的插值应用。     

格子玻尔兹曼法多块网格的交界结构优化研究

基于格子玻尔兹曼方法LBM(Lattice Boltzmann Method)对多块网格方法(Multi-Block)的粗细网格交界结构进行了研究,提出了一种新的优化处理方案。解决了原有网格交界结构存在的三个问题,即两套插值运算造成的程序结构复杂的问题,存储前几个时间步的节点流场数据以备插值运算造成内存浪费的问题和基于时间插值结果进行空间插值计算造成插值误差积累的问题。用一次多点二维空间插值的方式,将原方法的空间和时间双插值,简并成一次空间插值。通过对经典的非定常的圆柱绕流算例和定常的标准顶盖方腔驱动流算例的仿真模拟,验证了交界面处质量、动量及应力的连续性以及网格交界面数据过渡的流畅度,最终验证了改进方法的正确性。数值模拟结果表明,改进后多块算法可实现局部网格细化,进一步推动LBM方法在实际工程问题中的应用。     

基于单元相交的混合网格精确守恒插值方法

基于网格切割思想,发展了二维/三维混合网格条件下的单元相交算法,可精确计算任意两个多边形/多面体的交集。在此基础上,实现了基于单元相交(CIB/DC)的精确守恒插值算法。二维和三维验证算例表明,该方法能够保证插值过程中计算域内物理量的严格守恒,且具有比常规二阶插值更高的精度。     

On the use of characteristic‐based split meshfree method for solving flow problems

This study presents characteristic‐based split (CBS) algorithm in the meshfree context. This algorithm is the extension of general CBS method which was initially introduced in finite element framework. In this work, the general equations of flow have been represented in the meshfree context. A new finite element and MFree code is developed for solving flow problems. This computational code is capable of solving both time‐dependent and steady‐state flow problems. Numerical simulation of some known benchmark flow problems has been studied. Computational results of MFree method have been compared to those of finite element method. The results obtained have been verified by known numerical, analytical and experimental data in the literature. A number of shape functions are used for field variable interpolation. The performance of each interpolation method is discussed. It is concluded that the MFree method is more accurate than FEM if the same numbers of nodes are used for each solver. Meshfree CBS algorithm is completely stable even at high Reynolds numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

基于分形的捷联惯组历次测试数据混合插值算法

针对捷联惯组历次测试数据小样本、非等间隔、非线性的问题,提出了一种基于分形插值的三次混合插值算法.通过第一次分形插值保证原始测试数据的变化趋势;通过第二次样条函数插值保证了插值的准确性,实现原始测试数据等间隔化;通过第三次分段线性插值,扩大样本容量,同时保证了原有测试序列的统计特性不变.实例分析表明,该算法很好的实现了对捷联惯组历次测试数据的等间隔处理和样本容量扩大,为捷联惯组历次测试数据小样本建模分析提供良好的基础.     

Fractional Discrete-Time Signal Processing: Scale Conversion and Linear Prediction

A generalization of the linear prediction for fractional steps isreviewed, widening well-known concepts and results. This prediction isused to derive a causal interpolation algorithm. A reconstructionalgorithm for the situation where averages are observed is alsopresented. Scale conversion of discrete-time signals is studied takingas base the fractional discrete-time system theory. Some simulationresults to illustrate the behaviour of the algorithms will be presented.A new algorithm for performing the zoom transform is also described.     

An operator splitting algorithm for the three-dimensional advection–diffusion equation

Operator splitting algorithms are frequently used for solving the advection–diffusion equation, especially to deal with advection dominated transport problems. In this paper an operator splitting algorithm for the three-dimensional advection–diffusion equation is presented. The algorithm represents a second-order-accurate adaptation of the Holly and Preissmann scheme for three-dimensional problems. The governing equation is split into an advection equation and a diffusion equation, and they are solved by a backward method of characteristics and a finite element method, respectively. The Hermite interpolation function is used for interpolation of concentration in the advection step. The spatial gradients of concentration in the Hermite interpolation are obtained by solving equations for concentration gradients in the advection step. To make the composite algorithm efficient, only three equations for first-order concentration derivatives are solved in the diffusion step of computation. The higher-order spatial concentration gradients, necessary to advance the solution in a computational cycle, are obtained by numerical differentiations based on the available information. The simulation characteristics and accuracy of the proposed algorithm are demonstrated by several advection dominated transport problems. © 1998 John Wiley & Sons, Ltd.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.