A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

A flux reconstruction technique is presented to perform aeroacoustic computations using implicit high-order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth-order finite-volume implicit schemes with low-dispersion and low-dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two-dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three-dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter-based Reynolds number of 4×10⁵ on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.     

High-order curvilinear mesh generation technique based on an improved radius basic function approach

A high-order curvilinear hybrid mesh generation technique is developed for high-order numerical method (eg, discontinuous Galerkin method) applications to improve the accuracy for problems with curve boundary. The grid generation technique is based on an improved radius basic function (RBF) approach by which the straight-edge mesh is converted into high-order curve mesh. Firstly, an initial straight-edge mesh is prepared by traditional grid generation software. Then, high-order interpolation points are inserted into the mesh entities such as edges, faces, and cells according to the final demand of mesh order. To preserve the original geometry, the inserted points on solid wall are then projected onto the CAD model using an open source tool “Open Cascade.” Finally, other inserted points in the field near the solid wall are moved to appropriate positions by the improved RBF approach to avoid tangled cells. If we use the original RBF approach, then the inserted points on the edge and face entities normal to the solid boundary in the region of boundary layer will move to improper positions. To overcome this problem, a weighting based on the local grid aspect ratio between normal direction and tangential direction is introduced into the baseline RBF approach. Three typical configurations are tested to validate the mesh generator. Meanwhile, a third-order solution of subsonic flow over an analytical 3D body of revolution in the second International Workshop on High-Order CFD Methods is supplied by a discontinuous Galerkin solver. These numerical tests demonstrate the potential capability of present technique for high-order simulations of complex geometries.     

Use of characteristics method with cubic interpolation for unsteady-flow computation

The specified-time-interval (STI) scheme has been used commonly in applying the method of characteristics (MOC) to unsteady open-channel flow problems. However, with the use of STI scheme, the numerical error for the simulation results can always be induced due to the interpolation used to approximate the characteristics trajectory. Hence, in order to remedy the numerical errors caused by the interpolation, one needs to seek some kind of interpolation technique with higher-order accuracy. Instead of the linear interpolation technique, which has been used very commonly and can induce serious numerical diffusion, the Holly--Preissmann two-point, method, which is a cubic interpolation technique with fourth-order of accuracy, is proposed here to integrate with the method of characteristics for the computation of one-dimensional unsteady flow in open channel. The concept of reachback and reachout in space and time directions for the characteristics is also introduced to assure the model stability. The computed results from this new model are compared with those computed by using the Preissmann four-point scheme and the multimode method of characteristics with linear interpolation.     

Parallel finite element computation of unsteady incompressible flows

A parallel semi-explicit iterative finite element computational procedure for modelling unsteady incompressible fluid flows is presented. During the procedure, element flux vectors are calculated in parallel and then assembled into global flux vectors. Equilibrium iterations which introduce some ‘local implicitness’ are performed at each time step. The number of equilibrium iterations is governed by an implicitness parameter. The present technique retains the advantages of purely explicit schemes, namely (i) the parallel speed-up is equal to the number of parallel processors if the small communication overhead associated with purely explicit schemes is ignored and (ii) the computation time as well as the core memory required is linearly proportional to the number of elements. The incompressibility condition is imposed by using the artificial compressibility technique. A pressure-averaging technique which allows the use of equal-order interpolations for both velocity and pressure, this simplifying the formulation, is employed. Using a standard Galerkin approximation, three benchmark steady and unsteady problems are solved to demonstrate the accuracy of the procedure. In all calculations the Reynolds number is less than 500. At these Reynolds numbers it was found that the physical dissipation is sufficient to stabilize the convective term with no need for additional upwind-type dissipation. © 1998 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.     

Evaluation of RBFs for volume data interpolation in CFD

A greedy method for choosing an optimum reduced set of control points is integrated with RBF interpolation and evaluated for the purpose of interpolating large‐volume data sets in CFD. Given a function defined at a set of points, the greedy method selects a small subset of these points that is sufficient to keep the interpolation error at all the remaining points below a chosen bound. This is equivalent to a type of data compression and would have useful storage, post‐processing, and computational applications in CFD. To test the method in terms of both the point selection scheme and the suitability of reduced control point volume interpolation, a trial application of the interpolation to velocity fields in CFD volume meshes is considered. To optimise the point selection process, and attempt to be able to capture multiple length scales, a variable support radius formulation has also been included. Structured and unstructured mesh cases are considered for aerofoils, a wing case and a wing‐body case. For smooth volume functions, the method is shown to work well, producing accurate velocity interpolations using a very small number of the cells in the mesh. For general complex fields including large gradients, the method is still shown to be effective, although large gradients require more interpolation points to be used.Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of ground water flow via a new approach to the local radial point interpolation meshless method

A novel approach to local radial point interpolation meshless (LRPIM) method is introduced to investigate the influence of leakage on tidal response in a coastal leaky confined aquifer system, based on a local weighted residual method with the Heaviside step function as the weighting function over a local sub-domain. The present approach is a truly meshless method based only on a number of randomly located nodes. In this approach, neither global background integration mesh nor domain integration is needed. Radial basis functions (RBFs) interpolation is employed in shape function and its derivatives construction for evaluating the local weak form integrals. Due to satisfaction of kronecker delta property in RBF interpolation, no special treatment is needed to impose the essential boundary conditions. In order to obtain the optimum parameters, shape parameters of multiquadrics (MQ)-RBF are tuned and studied. The leakage has a significant impact on the tidal behaviour of the confined aquifer. The numerical results of this research indicate that both tidal amplitude of groundwater head in the aquifer and the distance over which the aquifer can be disturbed by the tide are considerably reduced by leakage. The novelty of the approach is the use of a local Heaviside weight function in the LRPIM which does not need local domain integration and only integrations on the boundary of the local domains are needed. Therefore, in this research a new local Heaviside weight function has been proposed. Numerical results are presented and compared with the results of analytical solution. It is observed that the obtained results agreed very well with the results of analytical solution. The numerical results show that the use of a local Heaviside weight function in the LRPIM is highly accurate, fast and robust. It is also noticed that this novel meshless approach using MQ radial basis is very stable.     

Solution of time-domain acoustic wave propagation problems using a RBF interpolation model with “a priori” estimation of the free parameter

In the last decade, Meshless Methods have found widespread application in different fields of engineering and science. Beyond novelty, their mathematical simplicity and numerical accuracy have been the key of their rapid dissemination. Among meshless techniques, RBF (Radial Basis Functions) based methods can be simple and general to solve the problems related to multiple areas of applied physics and engineering. In the specific field of acoustics, there are usually two possible approaches for solving a problem: time- and frequency-domain. In this paper, the authors propose a local time-domain approach to establish an efficient methodology for the solution of large-scale acoustic wave propagation problems. For this purpose, a local interpolation scheme, based on the reproduction of the local wave field using RBFs (MultiQuadric and Gaussian), is implemented and its accuracy is verified against known closed-form solutions. An explicit time-domain marching procedure is adopted, and the quality of the numerical results is also compared with that obtained using standard space-time Finite-Difference schemes. Additionally, the RBF interpolation model is used to simulate the propagation of a Ricker pulse in two simple test cases, and applied to simulate a more complex configuration, corresponding to an underwater sound propagation problem. In this frame, the results are also compared with those computed using a fourth-order in space and second-order in time Finite-Difference scheme.     

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.     

基于RBF插值技术的CFD/CSD非线性耦合分析方法研究

基于非结构混合网格的N-S方程求解器和结构柔度影响系数法,发展了一种考虑气动、结构非线性的基于RBF插值技术CFD/CSD耦合分析方法,适用于解决现代大展弦比飞机的非线性静气动弹性问题。该方法采用时间相关法（即求解非定常方程组,用长时间的渐近解趋于定常状态）求解静气弹分析时的定常流动。考虑大展弦比飞机结构变形问题为大变形小应力问题,在利用柔度系数法求解结构方程时,假设每次求解结构方程时应力与应变为线性关系,整体静气弹分析过程为非线性关系,因此每次求解结构方程时要更新柔度影响系数矩阵。在非定常N-S方程每求解一个时间步耦合一次结构有限元分析,由于结构有限元分析的时间相对于气动分析时间是很短的,所以这种方法实际上近似使用了一次求解非定常气动力的时间完成了整个静气动弹性分析的过程。对于气动网格与结构有限元网格不一致性,本文采用径向基函数（RBF）插值方法中的TPS方法进行结构弹性变形和气动载荷插值,采用虚功原理完成气动载荷数据交换。为了节省气弹分析时间,采用动网格方法对气动网格进行更新,本文基于RBF插值方法发展一种适用于混合网格（四面体、三棱柱、金字塔和六面体）变形的动网格方法,可以保证附面层网格的质量与分布从而准确模拟其流动。利用该方法对M6机翼、DLR-F6翼身组合体和某大型客机机翼进行了静气动弹性特性分析,结果验证了本文开发的非线性CFD/CSD耦合分析方法的可行性、精确性和高效性。     

A NEW FLUX-CONSERVING NEWTON SCHEME FOR THE NAVIER-STOKES EQUATIONS

SUMMARY

A new numerical method is developed for the two-dimensional, steady Navier-Stokes equations. Using local polynomial expansions to represent the discrete primitive variables on each cell, we construct a scheme which has the following properties: First, the local discrete primitive variables are functional solutions of both the integral and differential forms of the Navier-Stokes equations. Second, fluxes are balanced across cell interfaces using exact functional expressions (to the order of accuracy of the local expansions). No interpolation, flux models, or flux limiters are required. Third, local and global conservation of mass, momentum, and energy are explicitly provided for. Finally, the discrete primitive variables and their derivatives are treated in a unified and consistent manner. All are treated as unknowns to be solved together for simulating the local and global flux conservation.

A general third-order formulation for the steady, compressible Navier-Stokes equations is presented. As a special case, the formulation is applied to incompressible flow, and a Newton's method scheme is developed for the solution of laminar channel flow. H is shown that, at Reynolds numbers of 100, 1000, and 2000, the developing channel flow boundary layer can be accurately resolved using as few as six to ten cells per channel width.     

A novel finite point method for flow simulation

A novel finite point method is developed to simulate flow problems. The mashes in the traditional numerical methods are supplanted by the distribution of points in the calculation domain. A local interpolation based on the properties of Taylor series expansion is used to construct an approximation for unknown functions and their derivatives. An upwind‐dominated scheme is proposed to efficiently handle the non‐linear convection. Comparison with the finite difference solutions for the two‐dimensional driven cavity flow and the experimental results for flow around a cylinder shows that the present method is capable of satisfactorily predicting the flow separation characteristic. The present algorithm is simple and flexible for complex geometric boundary. The influence of the point distribution on computation time and accuracy of results is included. Copyright © 2002 John Wiley & Sons, Ltd.     

Flux vector splitting solutions for coupling hydraulic transient of gas-liquid-solid three-phase flow in pipelines

The gas-liquid-solid three-phase mixed flow is the most general in multiphase mixed transportation. It is significant to exactly solve the coupling hydraulic transient problems of this type of multiphase mixed flow in pipelines. Presently, the method of characteristics is widely used to solve classical hydraulic transient problems. However, when it is used to solve coupling hydraulic transient problems, excessive interpolation errors may be introduced into the results due to unavoidable multiwave interpolated calculations. To deal with the problem, a finite difference scheme based on the Steger-Warming flux vector splitting is proposed. A flux vector splitting scheme is established for the coupling hydraulic transient model of gas-liquid-solid three-phase mixed flow in the pipelines. The flux subvectors are then discretized by the Lax-Wendroff central difference scheme and the Warming-Beam upwind difference scheme with second-order precision in both time and space. Under the Rankine-Hugoniot conditions and the corresponding boundary conditions, an effective solution to those points located at the boundaries is developed, which can avoid the problem beyond the calculation region directly induced by the second-order discrete technique. Numerical and experimental verifications indicate that the proposed scheme has several desirable advantages including high calculation precision, excellent shock wave capture capability without false numerical oscillation, low sensitivity to the Courant number, and good stability.     

Developing a new mesh deformation technique based on support vector machine

Mesh deformation technique is widely applied in numerical simulations involving moving boundaries, and the deforming capability and efficiency is the key of it. In this paper, we present a new point-by-point mesh deformation method based on the support vector machine. This proposed method, to certain extent, is similar to the radial basis function (RBF) interpolation method with data reduction, but the new approach selects key boundary points automatically without specifying an initial set, and the function coefficients are obtained by solving a simple quadratic programming problem. Therefore, it is more efficient than the RBF method. Typical 2D/3D applications and realistic unsteady flow over a pitching airfoil are simulated to demonstrate the capability of the new method. With proper setting, the quality of the deformed meshes after using the new method is comparable to that of the RBF method, and the performance is improved by up to 7 ×.     

A further assessment of interpolation schemes for window deformation in PIV

We have evaluated the performances of the following seven interpolation schemes used for window deformation in particle image velocimetry (PIV): the linear, quadratic, B-spline, cubic, sinc, Lagrange, and Gaussian interpolations. Artificially generated images comprised particles of diameter in a range 1.1 ≤ d _p ≤ 10.0 pixel were investigated. Three particle diameters were selected for detailed evaluation: d _p = 2.2, 3.3, and 4.4 pixel with a constant particle concentration 0.02 particle/pixel². Two flow patterns were considered: uniform and shear flow. The mean and random errors, and the computation times of the interpolation schemes were determined and compared.     

无网格算法在多段翼型流动计算中的应用

研究了一种求解欧拉方程的无网格算法，发展出了一套布点及点云自动生成的方法；在点云离散的基础上，采用最小二乘法求解矛盾方程的方法来求取空间导数，进而获得数值通量；采用四步龙格-库塔方法进行时间推进，并引入当地时间步长和残值光顺等加速收敛措施。通过对NA-CA0012翼型的跨音速流动和多段翼型复杂绕流的数值模拟，验证了上述无网格算法的正确性和实用性。     

一种基于增量径向基函数插值的流场重构方法

由于流场参数重构中, 用于重构的基网格单元的物理参数波动量相对于均值较小, 径向基函数（RBF） 直接插值方法重构会产生较大的数值振荡, 论文提出了一种增量RBF 插值方法, 并用于有限体积的流场重构步, 明显改善了插值格式的收敛性和稳定性. 算例首先通过简单的一维模型说明该方法的有效性, 当目标函数波动量相对于均值为小量时, 增量RBF 插值能够抑制数值振荡; 进一步通过二维亚音速、跨音速定常无黏算例、静止圆柱绕流非定常算例以及超音速前台阶算例来说明该方法在典型流场数值求解中的通用性和有效性. 研究表明增量RBF 重构方法可陡峭地捕捉激波间断, 可有效改善流场求解的收敛性和稳定性, 数值耗散小, 计算效率高.      

A RECONSTRUCTION METHOD FOR FINITE VOLUME FLOW FIELD SOLVING BASED ON INCREMENTAL RADIAL BASIS FUNCTIONS

由于流场参数重构中, 用于重构的基网格单元的物理参数波动量相对于均值较小, 径向基函数（RBF） 直接插值方法重构会产生较大的数值振荡, 论文提出了一种增量RBF 插值方法, 并用于有限体积的流场重构步, 明显改善了插值格式的收敛性和稳定性. 算例首先通过简单的一维模型说明该方法的有效性, 当目标函数波动量相对于均值为小量时, 增量RBF 插值能够抑制数值振荡; 进一步通过二维亚音速、跨音速定常无黏算例、静止圆柱绕流非定常算例以及超音速前台阶算例来说明该方法在典型流场数值求解中的通用性和有效性. 研究表明增量RBF 重构方法可陡峭地捕捉激波间断, 可有效改善流场求解的收敛性和稳定性, 数值耗散小, 计算效率高.     

附面层修正应用于无网格算法的研究

研究了无网格算法中的附面层修正方法,在一种布置点云方法的基础上,发展一种曲面拟合的重构方式构造流场物理量;找出了无网格算法与网格算法之间的联系,成功将AUSM＋-up格式移植到无网格算法当中,并应用于计算欧拉方程的数值通量;计算中采用了一种改进的隐式时间推进,并引入当地时间步长和残值光顺等加速收敛措施,成功的将附面层修...     

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.