A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

一种改进格林元方法及在渗流问题中的应用

采用格林公式和基本解推导出直接边界积分方程来求解渗流问题.边界积分方程数值离散基于格林元方法(Green element methond),改进了原方法中压力和压力导数的求解方法,命名为混合边界元方法(Mixed boundary element method).相较于格林元类方法,该方法显式考虑了求解节点的外法向流量值和压力值,并使求得的数值解在求解区域上能够连续,符合实际的物理过程,在不增加额外未知数的情况下提高了计算精度.分析了不同网格类型对模拟计算结果的影响,并对稳定渗流问题、非稳定(瞬态)渗流问题和非稳态问题进行了实例计算,结果显示改进方法提高了计算精度,并对各类渗流问题有较好的适应性.     

三维对流问题的拟协调六面体单元解法

寻找一种高精度的空间单元插值模式是数值求解三维对流问题的关键。在前人研究的基础上,探讨了一种任意空间六面体的拟协凋单元,保证节点上的物理量函数及其一阶导数连续。算例表明,该方法具有良好的计算稳定性和低数值阻尼的优点,且计算工作量大大小于协调单元法,有利于推广应用于对流扩散方程的数值求解。     

Numerical simulation of flows over 2D and 3D backward-facing inclined steps

The work deals with the numerical solution of incompressible turbulent flow in a channel with a backward-facing step having various inclination angles. Also, the inclination of upper wall is considered. The mathematical model is based on the Reynolds averaged Navier–Stokes equations. The governing equations are closed by the explicit algebraic Reynolds stress (EARSM) model according to Wallin and Johansson or by linear eddy viscosity models (SST, TNT k–ω). The numerical solution is carried out by the implicit finite-volume method based on the artificial compressibility and by the finite-element method amd both approaches compared. The numerical simulations use as reference the experimental data by Makiola and Driver and Seegmiller in large aspect ratio channels. In these cases, the results are obtained by 2D and 3D simulations. Further narrow channel PIV experimental data are used as reference for 3D simulations.     

A fast Godunov method for the water‐hammer problem

An efficient Godunov‐type numerical method with second‐order accuracy was developed to simulate the water‐hammer problem in piping. The exact solutions of the Riemann problem were analysed and illustrated on the intriguing solution diagram by properly introducing dimensionless variables within reasonably practical ranges. Based on the solution diagram, an efficient fast Riemann solver was also developed. Moreover, small perturbation analysis was performed to demonstrate the relations between the primitive variables, velocity and pressure, for the Riemann problem. The typical shock‐tube problem and the water‐hammer problem were implemented as sample ones to test the numerical method. It was shown that the present numerical method incorporated with Van Leer's flux limiter is a promising one to simulate fluid transient problem for piping in the present study. Copyright © 2002 John Wiley & Sons, Ltd.     

Fault detection for switched systems with finite-frequency specifications

This paper is concerned with the fault detection filter design problem for a class of discrete-time switched systems whose output can track a time-varying and known frequency region reference input under arbitrary switchings. Faults detection filters are designed to guarantee that the disturbance attenuation performance is satisfied for all subsystems, the reference input attenuation performance is satisfied for the fault-free case, meanwhile, the reference input sensitivity performance is satisfied for the fault cases. With the aid of virtue of the frequency of the reference input in the finite-frequency region which is known beforehand, the finite-frequency H _? performance for switched systems is firstly defined. Sufficient conditions for the fault detection filter are given in terms of solutions to a set of linear matrix inequalities, furthermore, the filter gains are characterized in terms of the solution of a convex optimization problem. A numerical example is used to demonstrate the effectiveness of the proposed design method.     

A finite element thermally coupled flow formulation for phase‐change problems

A finite element, thermally coupled incompressible flow formulation considering phase‐change effects is presented. This formulation accounts for natural convection, temperature‐dependent material properties and isothermal and non‐isothermal phase‐change models. In this context, the full Navier–Stokes equations are solved using a generalized streamline operator (GSO) technique. The highly non‐linear phase‐change effects are treated with a temperature‐based algorithm, which provides stability and convergence of the numerical solution. The Boussinesq approximation is used in order to consider the temperature‐dependent density variation. Furthermore, the numerical solution of the coupled problem is approached with a staggered incremental‐iterative solution scheme, such that the convergence criteria are written in terms of the residual vectors. Finally, this formulation is used for the solutions of solidification and melting problems validating some numerical results with other existing solutions obtained with different methodologies. Copyright © 2000 John Wiley & Sons, Ltd.     

Exact and numerical responses of a plate under a turbulent boundary layer excitation

The aim of this paper is the analysis of the predictive capabilities of the deterministic methodologies when facing the problem of a plate excited by a stochastic pressure distribution due to a turbulent boundary layer (TBL). A full analytical solution has been assembled by considering a simply supported rectangular plate wetted on one side by a TBL. This reference exact solution, developed by using a standard separable variable model, has been used as test case for comparing the approximate solutions coming from the adoption of a numerical scheme by using discrete coordinates. The numerical algorithm has been built by using a standard finite element modal approach. The approximations introduced are thoroughly discussed and analysed; they refer to the meshing condition and the transformation of the distributed stochastic load. The application of a novel numerical procedure named as Asymptotical Scaled Modal Analysis is presented too. This innovative numerical scheme allows the analysis of the structural response of a generic plane operator in the whole frequency range, which is not always amenable by exact solutions; further and equally important, it is associated to a reduction of the computational cost. The work demonstrates that some numerical advances in the prediction of the random structural responses are feasible still using standard finite element modal inputs, without increasing the computational costs.     

An adaptive multiresolution semi‐intrusive scheme for UQ in compressible fluid problems

This paper deals with the introduction of a multiresolution strategy into the semi‐intrusive scheme, recently introduced by the authors, aiming to propagate uncertainties in unsteady compressible fluid applications. The mathematical framework of the multiresolution setting is presented for the cell‐average case, and the coupling with the semi‐intrusive scheme is described from both the theoretical and algorithmic point‐of‐view. Some reference test cases are performed to demonstrate the convergence properties and the efficiency of the overall scheme: the linear advection problem for both smooth and discontinuous initial conditions, the inviscid Burgers equation, and an uncertain shock tube problem obtained by modifying the well‐known Sod shock problem. For all the cases, the convergence curves are computed with respect to semi‐analytical (exact) solutions. In the case of the shock tube problem, an original technique to obtain a reference highly‐accurate numerical stochastic solution has also been developed. Copyright © 2015 John Wiley & Sons, Ltd.     

A robust numerical method for flow through a pipe driven by an oscillating pressure gradient

The problem of periodic flow of an incompressible fluid through a pipe, which is driven by an oscillating pressure gradient (e.g. a reciprocating piston), is investigated in the case of a large Reynolds number. This process is described by a singularly perturbed parabolic equation with a periodic right‐hand side, where the singular perturbation parameter is the viscosity ν. The periodic solution of this problem is a solution of the Navier–Stokes equations with cylindrical symmetry. We are interested in constructing a parameter‐robust numerical method for this problem, i.e. a numerical method generating numerical approximations that converge uniformly with respect to the parameter ν and require a bounded time, independent of the value of ν, for their computation. Our method comprises a standard monotone discretization of the problem on non‐standard piecewise uniform meshes condensing in a neighbourhood of the boundary layer. The transition point between segments of the mesh with different step sizes is chosen in accordance with the behaviour of the analytic solution in the boundary layer region. In this paper we construct the numerical method and discuss the results of extensive numerical experiments, which show experimentally that the method is parameter‐robust. Copyright © 2006 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.     

The characteristic‐based split (CBS) meshfree method for free surface flow problems in ALE formulation

A semi‐implicit characteristic‐based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Error estimate and adaptive refinement for incompressible Navier‐Stokes equations using the discrete least squares meshless method

In this paper, an adaptive refinement strategy based on a node‐moving technique is proposed and used for the efficient solution of the steady‐state incompressible Navier–Stokes equations. The value of a least squares functional of the residual of the governing differential equation and its boundary conditions at nodal points is regarded as a measure of error and used to predict the areas of poor solutions. A node‐moving technique is then used to move the nodal points to the zones of higher numerical errors. The problem is then resolved on the refined distribution of nodes for higher accuracy. A spring analogy is used for the node‐moving methodology in which nodal points are connected to their neighbors by virtual springs. The stiffness of each spring is assumed to be proportional to the errors of its two end points and its initial length. The new positions of the nodal points are found such that the spring system attains its equilibrium state. Some numerical examples are used to illustrate the ability of the proposed scheme for the adaptive solution of the steady‐state incompressible Navier–Stokes equations. The results demonstrate a considerable improvement of the results with a reasonable computational effort by using the proposed adaptive strategy. Copyright © 2011 John Wiley & Sons, Ltd.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Plane-parallel flow around a gas cavity

The stationary motion of a gas cavity in an ideal incompressible fluid is studied taking account of surface tension by using a variational equation. Approximate analytical dependences of the dimensionless parameters on the degree of cavity deformation are obtained. It is shown that the variational equation admits of an exact analytical solution. The stability of motion corresponding to the exact solution is proved relative to arbitrary perturbations in the cavity shape. A solution is given for the problem of stationary motion of an elliptical cavity in a gravity viscous fluid and the stability problem is investigated. Dependences are found for the velocity of cavity rise, the Reynolds number, and the Froude number as a function of the cavity size.     

Electromagnetic field generated in air by a nonstationary gamma-ray source in an ideally conducting plane

We consider a model problem on the generation of a radio signal by a nonstationary gamma-ray source. The problem is essentially two-dimensional in space but is reduced to a number of one-dimensional nonstationary problems. The results of a numerical solution of the problem are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 18–26, May–June, 1975.In conclusion, the authors wish to thank A. A. Milyutin and I. E. Dinaburg for working out the numerical methods used for the solution of this problem, and also thank I. N. Mikhailov and G. M. Gandel'man for their participation in the development of the problem.     

基于边界值方法的微分动力系统数值计算方法

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。     

Influence of the velocity gradient on the stagnation point heating in hypersonic flow

In a number of experimental and numerical publications a deviation has been found between the measured or computed stagnation point heat flux and that given by the theory of Fay and Riddell. Since the formula of Fay and Riddell is used in many applications to yield a reference heat flux for experiments performed in wind tunnels, for flight testing and numerical simulations, it is important that this reference heat flux is as accurate as possible. There are some shortcomings in experiments and numerical simulations which are responsible in some part for the deviations observed. But, as will be shown in the present paper, there is also a shortcoming on the theoretical side which plays a major role in the deviation between the theoretical and experimental/numerical stagnation point heat fluxes. This is caused by the method used so far to determine the tangential velocity gradient at the stagnation point. This value is important for the stagnation point heat flux, which so far has been determined by a simple Newtonian flow model. In the present paper a new expression for the tangential velocity gradient is derived, which is based on a more realistic flow model. An integral method is used to solve the conservation equations and, for the stagnation point, yields an explicit solution of the tangential velocity gradient. The solution achieved is also valid for high temperature flows with real gas effects. A comparison of numerical and experimental results shows good agreement with the stagnation point heat flux according to the theory of Fay and Riddell, if the tangential velocity gradient is determined by the new theory presented in this paper.This article was processed by the author using theLAT_EX style filepljour2 from Springer-Verlag.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.