Computer evaluation of high order numerical schemes to solve advective transport problems

This study evaluates the performance of three representative high-order finite difference schemes to solve two sets of simple one-dimensional benchmark problems in terms of their ability to resolve spurious oscillation, numerical spreading, and peak clipping. Three models, namely QUICKEST, ULTIMATE, and ENO were constructed to represent the classical high-order schemes without a flux limiter, TVD with a flux limiter, and TVB schemes, respectively. Three sets of results generated by QUICKEST, ULTIMATE, and ENO were compared with the analytical solutions. The first set indicated that none of these high-order schemes could yield satisfactory simulations when the grid size and time-step size specified by the benchmark problems were used. The second set showed that all three numerical schemes generated excellent computations when the grid size was reduced to one-tenth and the time-step size was reduced to one-fifth of those specified by the benchmark problems. The third set demonstrated that the results obtained by these schemes deteriorated even with the reduced grid size and time-step size when 100 folds of simulation times was conducted. The ENO and ULTIMATE schemes successfully eliminated spurious oscillations for all cases as expected. The QUICKEST scheme alleviated the problem of spurious oscillations only when the reduced grid and time-step sizes were used. In terms of numerical spreading and peak clipping, none of the three schemes produced satisfactory results unless the reduced grid and time-step were used. Peak clipping poses a more severe problem for these high order schemes than numerical spreading. A common set of benchmark problems is needed for the evaluation and testing of any numerical scheme.     

Numerical prediction of unsteady vortex shedding for large leading-edge roughness

A full two-dimensional Navier-Stokes algorithm is used to investigate unsteady, incompressible viscous flow past an airfoil leading edge with surface roughness that is characteristic of ice accretion. The roughness is added to the surface through the use of a Prandtl transposition and can generate both small-scale and large-scale roughness. The focus of the study is a detailed flow analysis of the unsteady velocity fluctuations and vortex shedding induced by the surface roughness. The results of this study are compared to experimental data on roughness-induced transition for the same roughness geometry. A comparison is made between “fluctuation intensity” values from the current algorithm to experimentally determined turbulence intensity values. The effects of the roughness Reynolds number, Re_k, are investigated and compared to experimental values of the critical roughness Reynolds number. The authors speculate that there may be a possible correlation between unsteady roughness-induced vortex shedding and the onset of experimentally measured transitional flow downstream of large-scale roughness.     

High order numerical simulation of sound generated by the Kirchhoff vortex

An improved high order finite difference method for low Mach number computational aeroacoustics (CAA) is described. The improvements involve the conditioning of the Euler equations to minimize numerical cancellation errors, and the use of a stable non-dissipative sixth-order central spatial interior scheme and a third-order boundary scheme. Spurious high frequency oscillations are damped by a third-order characteristic-based filter. The objective of this paper is to apply these improvements in the simulation of sound generated by the Kirchhoff vortex. Received: 31 January 2001 / Accepted: 30 September 2001     

Accurate numerical simulations of vortex flows II: Interaction of swirling vortex rings

An accurate numerical method is applied to the investigation of the flow generated by two co-rotating vortex rings with axial flow in opposite direction at the Reynolds number Re = 2000. The rings are shown to generate a strong wake with significant axial vorticity due to the stretching induced by the rings. The rings merge and form a single ring with complex internal structure.     

一种新型涡街发生体的设计与数值仿真

涡街发生体是涡街流量计的关键部件之一,其流体力学外形决定了涡街流量计的量程、精度。设计了一种新型的涡街发生体,并通过CFD软件FLUENT对新型涡街发生体和目前常用的涡街发生体进行数值仿真、理论分析和比较。结果表明:在实际应用中,新型涡街发生体的流体力学性能优于其他涡街发生体,适合高、低流速下产生强度大且稳定的涡街。     

Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases

Two pressure-correction algorithms are studied and compared to an approximate Godunov scheme on unsteady compressible cases. The first pressure-correction algorithm sequentially solves the equations for momentum, mass and enthalpy, with sub-iterations which ensure conservativity. The algorithm also conserves the total enthalpy along a streamline, in a steady flow. The second pressure-correction algorithm sequentially solves the equations for mass, momentum and energy without sub-iteration. This scheme is conservative and ensures the discrete positivity of the density. Total enthalpy is conserved along a streamline, in a steady flow. It is numerically verified that both pressure-correction algorithms converge towards the exact solution of Riemann problems, including shock waves, rarefaction waves and contact discontinuities. To achieve this, conservativity is compulsory. The two pressure-correction algorithms and the approximate Godunov scheme are finally compared on cases with heat source terms: all schemes converge towards the same solution as the mesh is refined.     

Free-surface flows solved by means of SPH schemes with numerical diffusive terms

A novel system of equations has been defined which contains diffusive terms in both the continuity and energy equations and, at the leading order, coincides with a standard weakly-compressible SPH scheme with artificial viscosity. A proper state equation is used to associate the internal energy variation to the pressure field and to increase the speed of sound when strong deformations/compressions of the fluid occur. The increase of the sound speed is associated to the shortening of the time integration step and, therefore, allows a larger accuracy during both breaking and impact events. Moreover, the diffusive terms allows reducing the high frequency numerical acoustic noise and smoothing the pressure field. Finally, an enhanced formulation for the second-order derivatives has been defined which is consistent and convergent all over the fluid domain and, therefore, permits to correctly model the diffusive terms up to the free surface. The model has been tested using different free surface flows clearly showing to be robust, efficient and accurate. An analysis of the CPU time cost and comparisons with the standard SPH scheme is provided.     

A numerical study of an electrothermal vortex enhanced micromixer

Temperature gradients aroused from the Joule heating in a non-uniform electrical field can induce inhomogeneities of electric conductivity and permittivity of the electrolyte, thus causing an electrothermal force that generates flow motion. A 2D numerical investigation of a micromixer, utilizing electrothermal effect to enhance its mixing efficiency, is proposed in this paper. Results for temperature and velocity distributions, as well as sample concentration distribution are obtained for an electrolyte solution in a microchannel with different pairs of electrodes under AC potentials with various frequencies. Numerical solutions were first carried out for one pair of electrodes, with a length of 10 μm separated by a gap of 10 μm, on one side wall of a microchannel having a length of 200 μm and a height of 50 μm. It is found that the electrothermal flow effect, in the frequency range for which Coulomb force is predominant, induces vortex motion near the electrodes, thus stirring the flow streams and enhancing its mixing efficiency. If more than one pair of electrodes is located on the opposite walls of the microchannel, the mixing efficiency depends on the AC potential applied pattern and the electrodes arrangement pattern. The distance between two pairs of electrodes on two opposite walls is then optimized numerically. Sample mixing efficiencies, using KCl solutions as the working fluid in microchannels with different number of electrodes pairs at optimal electrodes arrangement pattern, are also investigated. If root mean squared voltages of 10 V in an AC frequency range of 0.1–10 MHz are imposed on 16 pairs of electrodes separated at an optimal distance, the numerical results show that a mixing efficiency of 98% can be achieved at the end of the microchannel having a length of 700 μm and a height of 50 μm at Re = 0.01 Pe _C = 100, and Pe _T = 0.07. However, the mixing efficiency decreases sharply at a frequency higher than 10 MHz owing to the drastically decrease in the Coulomb force.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.     

A numerical confirmation of the dual body vortex flowmeter design

In this study we demonstrate the effectiveness of adding a second bluff body as a chaos control. Numerical method involves operator splitting technique, balance tensor diffusivity (BTD), and an element-by-element conjugate gradient iterative solver. To avoid numerical instability and to satisfy the physical constrain for obtaining the lowest frequency of vortex shedding, the time step needs to be carefully selected. The simulated result shows that a second bluff body placed in the flow path can enhance the repeatability of vortex pattern and, thus, can confirm the experimental finding of Bently et al. and Okajima.     

A numerical analysis of vortex growth in a two-dimensional jet

The vortex growth in a two-dimensional jet is simulated by solving numerically full Navier-Stokes equations with primitive variables. The streakline structures visualized with marker particles are well related to the fluctuations of velocity and pressure. The mechanism of vortex growth is also explained by using the visualized structure. The predicted features will agree with the experimental fact that, for example, the dominant frequency of fluctuation exists and the longitudinal velocity fluctuation along the jet axis increases exponentially in the zone of linear growth, while in the zone of non-linear growth the increasing rate decreases.     

A numerical analysis of two-dimensional flow past a rectangular prism by a discrete vortex model

A discrete vortex model is developed to analyse the two-dimensional fully separated unsteady flow past a rectangular prism. The effects of viscous diffusion of vortices and the loss in vorticity after the stationary prism with a thickness ratio ranging from 0.5 to 2.0. The formation of Karman vortices, the vortex shedding frequency, and the fluid forces acting on the body are favourably compared with the experimental results by various research workers. The method of analysis is also shown to be applicable to the flow past a prism that is in forced vibration.     

A numerical study of the convergence properties of ENO schemes

We report numerical results obtained with finite difference ENO schemes for the model problem of the linear convection equation with periodic boundary conditions. For the test function sin(x), the spatial and temporal errors decrease at the rate expected from the order of local truncation errors as the discretization is refined. If we take sin⁴(x) as our test function, however, we find that the numerical solution does not converge uniformly and that an improved discretization can result in larger errors. This difficulty is traced back to the linear stability characteristics of the individual stencils employed by the ENO algorithm. If we modify the algorithm to prevent the use of linearly unstable stencils, the proper rate of convergence is reestablished. The way toward recovering the correct order of accuracy of ENO schemes appears to involve a combination of fixed stencils in smooth regions and ENO stencils in regions of strong gradients —a concept that is developed in detail in a companion paper by Shu (this issue, 1990).     

PDE融合方法的两种离散方案的比较研究

基于Pop.S等提出的基于偏微分方程（PDE）的连续图像融合方程,提出了一种新的离散化方案,将原来的先计算梯度再求散度的离散化方案变化为直接计算拉普拉斯的离散化方案,对方程的正则化项进行相应的调整。将这两种离散化方案分别对一些一维信号和二维信号进行融合对比实验,实验结果显示：对所有的测试信号,新方案都能进行有效的融合;对某些测试的一维和二维信号,新方案能够加速信号的收敛过程并且使得到的融合图像在质量指标上比原方案有提高;而对某些测试信号,新方案与原方案在输出信号的收敛速度上和输出图像的质量指标上也是十分接近的。     

Global communication schemes for the numerical solution of high-dimensional PDEs

The numerical treatment of high-dimensional partial differential equations is among the most compute-hungry problems and in urgent need for current and future high-performance computing (HPC) systems. It is thus also facing the grand challenges of exascale computing such as the requirement to reduce global communication. To cope with high dimensionalities we employ a hierarchical discretization scheme, the sparse grid combination technique. Based on an extrapolation scheme, the combination technique additionally mitigates the need for global communication: multiple and much smaller problems can be computed independently for each time step, and the global communication shrinks to a reduce/broadcast step in between. Here, we focus on this remaining synchronization step of the combination technique and present two communication schemes designed to either minimize the number of communication rounds or the total communication volume. Experiments on two different supercomputers show that either of the schemes outperforms the other depending on the size of the problem. Furthermore, we present a communication model based on the system’s latency and bandwidth and validate the model with the experiments. The model can be used to predict the runtime of the reduce/broadcast step for dimensionalities that are yet out of scope on current supercomputers.     

A numerical analysis of vortex growth in a bounded rectangular jet

A low apsect ratio bounded rectangular jet flow with an artificial disturbance is numerically analyzed by solving three-dimensional Navier-Stokes equations with primitive variables. The streakline structures are related to the velocity and pressure fields. To explain the mechanism initiating a secondary flow in the initial field where the flow shows a well ordered structure, the streakline distortions are discussed by the velocity fluctuations in the flow field. The results advocate the existing relations between the secondary flow and the vortex growth.     

Impact of computational physics on multi-scale CFD and related numerical algorithms

Development of accurate computational methods for the constitutive relation that plays a role as the bridge between microscopic and macroscopic physics becomes a key issue in a continuum approach for describing rarefied and micro-scale gas flows. The mathematical form of the constitutive relation dictates the resulting computational methods and related algorithms. It is, therefore, vital to develop proper computational models on the basis of a correct understanding of the multi-scale physics inherent in non-equilibrium gases. In this study the computational issue is discussed by considering two benchmark multi-scale problems: the compression-dominated shock structure and velocity shear-dominated gas flows. Special emphasis is placed on efficient CFD algorithms within the finite volume formulation. In addition, the verification and validation issue of the multi-scale methods is discussed and a simple verification method based on basic physical laws is proposed.     

数值型敏感属性的近邻泄露保护方法研究*

针对在发布数值型敏感属性数据时,因同一分组中个体的敏感属性值之间过小的差异而导致攻击者可以较高的概率以及较小的误差推导出目标个体的敏感信息,从而出现近邻泄露问题,提出了一种有效的防止近邻泄露的模型:(εp,l)-anonymity。该模型根据不同的敏感属性值区间设置不同的阈值εi(1≤i≤p)控制敏感属性值之间的相似度,并采用有损链接的方法对隐私数据进行保护。实验结果表明,该方法可以明显减少近邻泄露,提高信息可用性,增强数据发布的安全性。     

Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids

Freestream and vortex preservation properties of a weighted essentially nonoscillatory scheme (WENO) and a weighted compact nonlinear scheme (WCNS) on curvilinear grids are investigated. While the numerical technique used for the compact difference scheme can be applied to WCNS, applying it to WENO is difficult. This difference is caused by difference in the formulation of numerical fluxes. WENO computed in the generalized coordinate system does not work well for either freestream or vortex preservation, whereas WENO computed in the Cartesian coordinate system works well for both freestream and vortex preservation, but its resolution is lower than that of WCNS. In addition, WENO in the Cartesian coordinate system costs three times as much as WENO or WCNS in the generalized coordinate system. Therefore, WENO in the Cartesian coordinate system is not suitable for solving Euler equations on a curvilinear grid. On the other hand, WCNS computed in the generalized coordinate system works well for freestream and vortex preservation when used with the numerical technique proposed for the compact difference scheme. The results show that WCNS with this numerical technique can be used for an arbitrary grid system. In this paper, the excellent freestream and vortex preservation properties of WCNS when used with the numerical technique, compared with those of WENO, are shown for the first time.