Optimization of a global seventh-order dissipative compact finite-difference scheme by a genetic algorithm

A global seventh-order dissipative compact finite-difference scheme is optimized in terms of time stability. The dissipative parameters appearing in the boundary closures are assumed to be different, resulting in an optimization problem with several parameters determined by applying a generic algorithm. The optimized schemes are analyzed carefully from the aspects of the eigenvalue distribution, the ε-pseudospectra, the short time behavior, and the Fourier analysis. Numerical experiments for the Euler equations are used to show the effectiveness of the final recommended scheme.     

Efficiency benchmarking of seventh-order tri-diagonal weighted compact nonlinear scheme on curvilinear mesh

Efficiency improvements of high-order weighted compact nonlinear scheme (WCNS) are verified using a series of benchmark cases, proposed at the International Workshop on High-Order CFD Methods. A seventh-order tri-diagonal compact one of WCNSs (WCNS-E8T7), constructed in recent years, is investigated as a basic scheme, and compared to a typical fifth-order explicit WCNS (WCNS-E6E5) and a traditional second-order TVD scheme MUSCL. Among these tests, a symmetrical conservative metric method (SCMM) is adopted to ensure the accuracy and robustness of WCNSs when solving cases in the curvilinear coordinates. The computational efficiency of schemes is evaluated based on a non-dimensional cost in achieving the same level of accuracy. Related results show that WCNS-E8T7 has a better performance than WCNS-E6E5 with the same interpolation stencils. Moreover, the opinion that high-order methods can obtain higher computational efficiency than second-order methods is demonstrated on the cases ranging from academic problems to real-life computations.     

A calibration scheme for quantitative concentration measurements using simultaneous PIV and PLIF

A new approach for calibration of planar laser-induced fluorescence (PLIF) measurements is presented. The calibration scheme is based on the fact that there is a constant concentration flux through each cross-section of a fluorescent plume in a given flow field and makes use of simultaneous measurements of particle image velocimetry (PIV) and PLIF. The following are the advantages of the current technique: (1) it is experimentally less demanding and (2) it does not require in situ calibration for generating the calibration curves. The technique can be implemented in many experimental setups (both in water and gaseous flows) provided the geometry of the time-averaged scalar field is known. Using the calibration scheme, an analysis is carried out on the measurements of concentration fields in grid turbulence to validate the proposed technique. To demonstrate the feasibility of the scheme, the distributed second-order moments (μ ₂), and concentration and velocity correlations ( á u^￠c^￠ ñ \left\langle {u^{\prime}c^{\prime}} \right\rangle and á v^￠c^￠ ñ \left\langle {v^{\prime}c^{\prime}} \right\rangle ) are computed. Good agreement is found with previous studies. In addition, a quantitative appraisal of a simple closure approximation of the moment-based transport equation is also presented using simultaneous PIV and PLIF.     

应用于计算气动声学的优化有限紧致格式

对于含间断的计算气动声学问题,数值计算的格式不仅要求低耗散低色散的设计,对短波具有较高的分辨率,还要求能捕捉激波.中心紧致格式具有高精度,具有无耗散和低色散特征,但不能捕捉间断和激波;WENO格式处理间断较为成功,而耗散和色散误差相对较大.有限紧致格式可以将紧致格式与WENO格式相结合构造成混合格式,利用光滑因子之间的关系对激波区域进行自动判断,将传统的全域求解的紧致格式划分为有限的局部紧致求解,间断点上的激波捕捉铜梁自动作为局部紧致求解的边界通量,在在光滑区域具有紧致格式的高精度低耗散性质,在激波附近不产生非物理振荡.本文利用有限紧致格式思想,构造了新的适合于气动声学问题的优化有限紧致格式,将其应用于计算气动声学一维标准测试问题,对相关格式的模拟性能进行了评估,显示该格式在宽频声波传播和含有间断的声波传播模拟方面具有优势.     

An efficient low‐dissipative WENO filter scheme

This paper presents a simple and efficient procedure developed for tracing discontinuities in flow fields. Numerical experiments are carried out to test the new sensor coupled with the associated nonlinear WENO dissipation filter developed to suppress the numerical dissipation. The tests show that, for a problem containing shocks and vortices, the implementation of the new sensor and an optimized WENO scheme can obtain a stable and highly resolved solution. The numerical experiments demonstrated that the new filter scheme performs efficiently both in parallel and serial running for one‐dimensional inviscid flow problems. Direct numerical simulation of a Mach 5 turbulent boundary layer over a flat plate was carried out to demonstrate the applicability of the scheme to the DNS practices. Copyright © 2011 John Wiley & Sons, Ltd.     

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.     

A third-order compact nonlinear scheme for compressible flow simulations

Reynolds-averaged Navier-Stokes simulations based on second-order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin-friction drag and aerodynamic heating, as well as in simulations of complex flows such as large-scale separation and transition. A remedy for this is the development of high-order schemes, by which numerically induced dissipation and dispersion errors of low-order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high-resolution nonlinear shock-capturing methods. A stencil-selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third-order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh-Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.     

高阶紧致格式分区并行算法

针对超声速多尺度复杂流动问题,发展了一种高精度并行算法。计算格式采用五阶迎风紧致格式,用特征型通量限制方法抑制非物理振荡。在对接边界处采用五阶WENO格式,以保证整个计算域内计算精度一致。通过网格分区和数据交换,在MPI平台实现了并行计算。通过超声速算例对算法进行了验证,并对并行效率和加速比进行了分析。最后,将算法应用于超声速转捩、湍流问题的数值模拟。计算结果表明,提出的算法具有较高的精度和分辨率,对接边界光滑连续,并且并行效率较高,在高超声速湍流流动数值模拟中取得了较好的应用效果。     

Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.     

An improved parallel compact scheme for domain-decoupled simulation of turbulence

An improved domain-decoupled compact scheme for first and second spatial derivatives is proposed for domain-decomposition-based parallel computational fluid dynamics. The method improves the accuracy of previously developed decoupled schemes and preserves the accuracy and bandwidth properties of fully coupled compact schemes, even for a very large degree of parallelism, and enables the Navier-Stokes equations to be solved independently on each processor. The scheme is analysed using Fourier analysis and error analysis, and tested on one-dimensional wave-packet propagation, a two-dimensional vortex convection problem, and in the direct numerical simulation of the three-dimensional Taylor-Green vortex problem and turbulent channel flow. Our results demonstrate the scheme's effectiveness in performing direct numerical simulation of turbulence in terms of accuracy and scalability.     

The three-point fifth-order accurate generalized compact scheme and its applications

A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics. The project supported by the National Natural Science Foundation of China (19972038) and Foundation of the National CFD Laboratory of China     

Numerical approximation of viscous terms in finite volume models for shallow waters

Two methods for the numerical treatment of viscous terms in shallow water equations are studied and computational details are given for structured grids. It is demonstrated that the first scheme, which is widely used, may lead to spurious oscillations arising from computational modes. In fact, the shortest resolvable waves of wave length 2Δx are invisible to this method. The second method, although more expensive, is free of computational modes and it presents a more accurate approximation of viscous terms. The dispersion relation of the second method is closer to the analytical case and it has a smaller truncation error, which is due to the fact that it uses a more localized control volume. Numerical experiments are also presented that support the study. Copyright © 2009 John Wiley & Sons, Ltd.     

Implicit,non-oscillatory containing no free parameters and dissipative (INND) scheme

An implicit,non-oscillatory.containing no free parameter and dissipative(INND)scheme solving.Navier Stokes equations is developed.This scheme is one of total variationdiminishing(TVD)algorithms.The results show that this scheme is applicable for solvingNavier Stokes equations,and that the shock-capturing capabilbility and the convergencerate are satisfactory.     

数值求解Navier—Stokes方程的迎风紧致差分格式

从迎风紧致逼近^[1]出发,提出数值求解可压Navier-Stokes方程的一种高精度的数值方法。利用Steger-Warming的通量分裂技术^[2]将守恒型方程中的流通向量分裂成两部分,在此基础上据风向构造逼近于无粘项的三阶迎风紧致有限差分格式。对方程中的粘性部分采用通常的二阶差分逼近。所建立的差分格式被用来数值求解了三维粘性绕流问题。     

Formulation of thermoelastic dissipative material behavior using GENERIC

We show that the coupled balance equations for a large class of dissipative materials can be cast in the form of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). In dissipative solids (generalized standard materials), the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows for an efficient derivation of thermodynamically consistent coupled field equations, while revealing additional underlying physical structures, like the role of the free energy as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects. Applications to large and small-strain thermoplasticity are given. Moreover, for the quasistatic case, where the deformation can be statically eliminated, we derive a generalized gradient structure for the internal variable and the temperature with a reduced entropy as driving functional.     

Damping efficiency of the Tchamwa–Wielgosz explicit dissipative scheme under instantaneous loading conditions

To deal with dynamic and wave propagation problems, dissipative methods are often used to reduce the effects of the spurious oscillations induced by the spatial and time discretization procedures. Among the many dissipative methods available, the Tchamwa–Wielgosz (TW) explicit scheme is particularly useful because it damps out the spurious oscillations occurring in the highest frequency domain. The theoretical study performed here shows that the TW scheme is decentered to the right, and that the damping can be attributed to a nodal displacement perturbation. The FEM study carried out using instantaneous 1-D and 3-D compression loads shows that it is useful to display the damping versus the number of time steps in order to obtain a constant damping efficiency whatever the size of element used for the regular meshing. A study on the responses obtained with irregular meshes shows that the TW scheme is only slightly sensitive to the spatial discretization procedure used. To cite this article: L. Mahéo et al., C. R. Mecanique 337 (2009).     

A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations

We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity form, can be approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low-to-medium Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re ? 7500 point-SOR iteration is used and the convergence is fast.     

污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。     

Discrete non-linear adaptive data driven control based upon simultaneous perturbation stochastic approximation

A novel adaptive data driven control strategy is proposed for general discrete non-linear systems. The controller is designed based upon the Simultaneous Perturbation Stochastic Approximation (SPSA) method, and is constructed through use of a Function Approximator (FA), which is fixed as a neural network here. In this novel control strategy, the parametric estimation is designed to be adaptive with convergence analysis, and the control ability has been greatly improved. The proposed control method is finally applied into the non-linear tracking problems, as well as near-optimal control problems for discrete-time non-linear systems. Simulation comparison tests were conducted on typical non-linear plants, through which, the convergence and feasibility of the proposed adaptive data driven control strategy are well demonstrated.