Taylor–least-squares finite element for two-dimensional advection-dominated unsteady advection–diffusion problems

A new finite element method, the Taylor–least-squares, is proposed to approximate the advection-dominated unsteady advection–diffusion equation. The new scheme is a direct generalization of the Taylor–Galerkin and least-squares finite element methods. Higher-order spatial derivatives in the new formulation necessitate higher-degree polynomials. Hermite cubic shape functions are used. Extensive comparisons with other methods in one dimension proved that the new scheme is a step forward in modelling this difficult problem. The method offers straightforward generalizations to higher dimensions without losing the accuracy demonstrated in one dimension, i.e. the method preserves the important property of the Taylor–Galerkin scheme of being free of numerical crosswind diffusion. Several numerical experiments were made in two dimensions and excellent results were obtained from the representative experiments.     

CFD结合降阶模型预测阵风响应

传统的阵风响应主要在频域内进行分析,气动载荷基于线性方法计算,不能考虑黏性和跨声速流动影响. 飞机设计需考虑不同频率和不同形状阵风的响应,基于CFD的阵风响应预测由于计算工况太多,工作量巨大. 本文发展了一种CFD结合非定常气动力ARMA(autoregressive-moving-averagemodel)降阶模型的阵风响应分析方法,CFD只要针对给定频率和形状的一种阵风响应进行计算,对获得的气动力时间历程运用线性最小二乘法参数辨识ARMA降阶模型的系数,则对任意频率和形状的阵风,代入降阶模型即可确定该阵风的响应,大大提高了计算效率. 为效验发展的方法,先计算NACA0012翼型在低马赫数0.11的阵风响应,通过对比CFD、ARMA降阶模型及早期发展的不可压阵风响应预测方法的结果,验证了方法的有效性. 再对比CFD、ARMA在跨声速马赫数0.8的阵风响应预测结果,证实所发展的方法对跨声速阵风响应预测亦是有效的.      

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

On a finite element CFD algorithm for compressible,viscous and turbulent aerodynamic flows

This paper develops and analyses individual construction aspects of an efficient and accurate finite element algorithm for prediction of viscous and turbulent flow fields of impact in aerodynamics. The theoretical construction employs a Taylor weak statement (TWS) for coincident embedding of stability mechanisms within a classic Galerkin finite element formulation of semi-discrete approximation error orthogonalization. A wide variety of the stabilizing mechanisms of independently derived CFD algorithms are contained within the TWS theory. An implicit construction that meets the requirement of efficient convergence to steady state is developed. The theoretical asymptotic error estimates of the TWS finite element algorithm for supersonic and viscous boundary layer flows are verified. Application to a three-dimensional turbulent flow is cited.     

A modified conservation principles theory leading to an optimal Galerkin CFD algorithm

A modified conservation principles theory in one, then multi‐dimensions, admits the prediction of an optimally accurate algorithm construction for the unsteady incompressible Navier–Stokes (INS) equations. Via a time Taylor series (TS) operation, followed by a pseudo‐limit process, the theory generates a modified, but still analytical, INS system parameterized by a set of coefficients constrained only by a convexity requirement. A spatially discretized finite element implementation of a Galerkin weak statement on this modified INS system, termed the ‘Taylor weak statement (TWS), ’ generates a parameterized CFD algorithm for analysis. TWS algorithm phase velocity and amplification factor error functions are derived for linear and bilinear basis implementations assembled at the generic node. A subsequent TS expansion in wave number space admits analytical identification of parameter set options affecting lowest order error terms. The results of definitive verification‐ and validation‐class computational experiments for a range of published CFD algorithms belonging to the TWS class, reported herein, clearly confirm theoretical prediction of the optimal TWS algorithm for INS thermal/fluid transport applications. Copyright © 2007 John Wiley & Sons, Ltd.     

Group analysis for the effects of the temperature-dependent fluid viscosity and chemical reaction on free convective heat and mass transfer

A group analysis has been carried out to study heat and mass transfer characteristics of an incompressible Newtonian fluid having a temperature-dependent viscosity over a vertical stretching surface in the presence of thermal radiation and a chemical reaction. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The vertical surface is assumed to be permeable so as to allow for possible wall suction or injection. The governing differential equations are derived and transformed using the Lie group analysis. The transformed equations are solved numerically by applying the Runge—Kutta—Gill scheme with the shooting technique. Favorable comparisons with previously published works on various special cases of the problem are obtained     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

用拟协调元直接构造平面任意四边形单元——进入有限元的禁区

利用拟协调元方法,在直角坐标系下直接构造了一族平面任意四边形单元,对其收敛性进行了分析,并与平面等参元进行了对比研究.结果证明平面任意四边形单元可采用多项式基函数直接列式,并可以保障单元的收敛性;拟协调元列式可以使平面问题的有限元方法得到统一.与平面等参元相比,单元列式简单,性能稳定,具有显式的刚度阵,计算量小,这说明对于有限元平面问题拟协调元是一个更正确、有效的做法.      

圆柱形弹体Taylor撞击方法研究

研究了圆柱形弹体垂直撞击刚性靶体的Taylor撞击问题,提出了弹体撞击过程中未发生变形部分的速度变化规律,即二次非线性变减速运动,并通过弹体的运动方程对Taylor撞击进行了理论分析;同时利用再生核质点法(Reproducing kernel particle method,RKPM)对Taylor撞击过程进行了数值分析。利用该理论对五种具体材料进行分析,结果表明,解析结果与试验结果及数值分析结果吻合较好。     

On a generalized taylor theorem: A rational proof of the validity of the homotopy analysis method

IntroductionIn 1 7thcenturyIsaacNewton[1]gavesuchabinomialexpressionforfractionalandnegativeexponents(1 +t) α,i.e.,(1 +t) α =1 + +∞k=1α(α-1 ) (α-2 )… (α -k+ 1 )k !tk　　 (α≠ 0 ,1 ,2 ,… ) ,(1 )whoseconvergenceradiusisone.Furthermore ,theclassicalTaylorseries (seeRef.[2 ] )limm→+∞ mk=0f(k) (z0 )k !(z-z0 ) k (2 )ofacomplexfunctionf(z)atz=z0 isvalidmostlyinarestrictedconvergenceregion｜z-z0 ｜相似文献   

Efficient retrieval of the thermo‐acoustic flame transfer function from a linearized CFD simulation of a turbulent flame

The stability of thermo‐acoustic pressure oscillations in a lean premixed methane‐fired generic gas turbine combustor is investigated. A key element in predicting the acoustically unstable operating conditions of the combustor is the flame transfer function. This function represents the dynamic relationship between a fluctuation in the combustor inlet conditions and the flame's acoustic response. A transient numerical experiment involving spectral analysis in computational fluid dynamics (CFD) is usually conducted to predict the flame transfer function. An important drawback of this spectral method application to numerical simulations is the required computational effort. A much faster and more accurate method to calculate the transfer function is derived in this paper by using a most important basic assumption: the fluctuations must be small enough for the system to behave linear. This alternative method, which is called the linear coefficient method, uses a linear representation of the unsteady equations describing the CFD problem. This linearization is applied around a steady‐state solution of the problem, where it can consequently describe the dynamics of the system. Finally, the flame transfer function can be calculated from this linear representation. The advantage of this approach is that one only needs a steady‐state solution and linearization of the unsteady equations for calculating a dynamic transfer function, i.e. no time‐consuming transient simulations are necessary anymore. Nevertheless, as a consequence of the large number of degrees of freedom in a CFD problem, an extra order reduction step needs to be performed prior to calculating the transfer function from the linear representation. Still, the linear coefficient method shows a significant gain in both speed and accuracy when calculating the transfer function from the linear representation as compared to a spectral analysis‐based calculation. Hence, this method gives a major improvement to the application of the flame transfer function as a thermo‐acoustic design tool. Copyright © 2007 John Wiley & Sons, Ltd.     

Gravitational instability of thin gas layer between two thick liquid layers

We consider the problem of gravitational instability (Rayleigh–Taylor instability) of a horizontal thin gas layer between two liquid half-spaces (or thick layers), where the light liquid overlies the heavy one. This study is motivated by the phenomenon of boiling at the surface of direct contact between two immiscible liquids, where the rate of the “break-away” of the vapor layer growing at the contact interface due to development of the Rayleigh–Taylor instability on the upper liquid–gas interface is of interest. The problem is solved analytically under the assumptions of inviscid liquids and viscous weightless vapor. These assumptions correspond well to the processes in real systems, e.g., they are relevant for the case of interfacial boiling in the system water-n-heptane. In order to verify the results, the limiting cases of infinitely thin and infinitely thick gas layers were considered, for which the results can be obviously deduced from the classical problem of the Rayleigh–Taylor instability. These limiting cases are completely identical to the well-studied cases of gravity waves at the liquidliquid and liquid–gas interfaces. When the horizontal extent of the system is long enough, the wavenumber of perturbations is not limited from below, and the system is always unstable. The wavelength of the most dangerous perturbations and the rate of their exponential growth are derived as a function of the layer thickness. The dependence of the exponential growth rate on the gas layer thickness is cubic.     

A polycrystal plasticity model based on the mechanical threshold

A temperature and rate-dependent viscoplastic polycrystalmodel is presented.It uses a single crystal constitutive response that is based on the isotropic Mechanical Threshold Stress continuum model. This combination gives us theability to relate the constitutive model parameters between the polycrystaland continuum models. The individual crystal response is used to obtain themacroscopic response through the extended Taylor hypothesis. A Newton-Raphsonalgorithm is used to solve the set of fully implicit nonlinear equations for each crystal. The analysis also uses a novel state variable integration method which renders the analysis time step size independent for constant strain rate simulations. Material parameter estimates are obtained through an identification study, where the error between experimental and computed stress response is minimized. The BFGS method, which is used to solve theidentification problem, requires first-order gradients. These gradients arecomputed efficiently via the direct method of design sensitivity analysis.Texture augmentation is performed in a second identification study by changing crystal weights (volume fractions).     

Interferometric measurement of density in nonstationary shock wave reflection flow and comparison with CFD

We present density measurements from the application of interferometry and Fourier transform fringe analysis to the problem of nonstationary shock wave reflection over a semicircular cylinder and compare our experimental measurements to theoretical results from a CFD simulation of the same problem. The experimental results demonstrate our ability to resolve detailed structure in this complex shock wave reflection problem, allowing visualization of multiple shocks in the vicinity of the triple point, plus visualization of the shear layer and an associated vortical structure. Comparison between CFD and experiment show significant discrepancies with experiment producing a double Mach Reflection when CFD predicts a transitional Mach reflection.Received: 12 November 2003, Accepted: 21 October 2004, Published online: 31 March 2005[/PUBLISHED]PACS: 47.40.-x, 42.40.Kw     

基于CFD的气动力建模及其在气动弹性中的应用

CFD技术为带有气动力非线性的气动弹性分析提供了一种研究途径,但是基于CFD的气动弹性直接数值模拟方法的计算量很大,不便开展定性分析和参数设计.基于CFD的非定常气动力模型的降阶技术为缓解计算效率与计算精度之间的矛盾以及系统的复杂性与易分析、易设计性之间的矛盾提供了行之有效的技术途径.综述了近年来发展的两类基于CFD技术的非定常气动力降阶技术和一种非线性气动力的谐波平衡方法,以及这些方法在非线性气动弹性研究中的运用.对比了各种方法的优越性并作了进一步的展望.      

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

Interferometry and reconstruction of strongly refracting fields in two-dimensional boundary layer flow

 Optical methods like interferometry as non-intrusive experimental techniques are used for fine analysis of flowfields. The accuracy of the measuring method is very important for the applicability of its results to CFD validation. A common evaluation method to reconstruct interferograms is based on the assumption, that the object ray propagates along a straight line. But the strong bending of rays that occurs, e.g. in supersonic boundary layer flow, cannot be neglected without losses in reconstruction quality. Since the reconstruction of a two-dimensional boundary layer flow can be considered as an one-dimensional problem, the phase difference of the object and reference ray at the interferogram can be related analytically to the refractive-index distribution using a Taylor series expansion. The resulting interferometric equation is an ordinary non-linear second-order differential equation, which can be integrated by numerical methods. By application of this interferometric equation on the one hand, the error in the “classical” interferometry resulting from the ray bending neglection can be estimated. On the other hand, the accuracy in evaluation of interferograms of two-dimensional boundary layer flow can be improved by solving this equation. Received: 23 May 1996 / Accepted: 21 September 1996     

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.     

Stochastic methods in dispersion theory

In this study we show how methods from the theory of stochastic processes can be applied to problems in dispersion theory.First, we show that Taylor dispersion with adsorbing boundaries is easily transformed into a new Taylor dispersion problem without adsorbing boundaries. The transformed problem can then be solved using any of the traditional methods used for Taylor dispersion.Secondly, we consider the dispersion of particles in a channel (between parallel plates) with one partially adsorbing surface and one perfectly reflecting boundary. We determine the exact law of the position of adsorption for an arbitrary channel flow in terms of an infinite series of iterated integrals of the flow field, which is assumed to be a function of the cross-channel coordinate only. We also consider the case of shear flow over an adsorbing plane, by taking the limit where one of the boundaries is taken to infinity