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Master equation,design equations and runaway speed of the Kaplan turbine

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Journal of Hydrodynamics 2021年第2期33卷 282-300页

作者： Zh.Zhang ZHAW Zurch University of Applied ScienceZurichSwitzerland

To make the Kaplan turbine technology comparable to both the Pelton and the Francis turbine,the master equation for the Kaplan turbine has been established by analyses similar to that-for the Francis turbine.The analysis begins with the descriptions of free vortex flows at the runner inlet and the swirl flow at the impeller exit.By considering the Euler equation for specific work and by further evaluating the most significant shock and swirling losses,the first and the second energy equations in the form of hydraulic efficiency were formulated.The master equation is then established by combining both energy equations.In addition,three design equations and a new design parameter are presented.The master equation relates the turbine hydromechanics to the geometrical design of both the runner and the guide-vane parameters.It enables the complete hydraulic characteristics of a given Kaplan turbine to be analytically and simply computed.A computation example demonstrates the functionality and applicability of the method.With the reconstructed master equation,the runaway speed of the Kaplan turbine and its dependence on the guide-vane setting can be easily and precisely computed.For bulb turbines with guide vanes directly ahead of the turbine runner in the same tube,all computations are also applicable using another equivalent control parameter.

关键词： Kaplan turbine master equation design equations design parameter runaway speed hydraulic efficiency shock loss swirling loss bulb turbines

THE STABILITY OF THE θ-METHODS FOR DELAYDIFFERENTIAL equations

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Journal of Computational Mathematics 1999年第4期17卷 441-448页

作者： Jing-jun Zhao Ming-zhu Liu(Department of mathematics, Harbin Institute of Technology, Harbin 150001, China)Shen-shan Qiu(Department of Computer Science and Engineering, Harbin Institute of Technology, Harbin 150001, China) 哈尔滨工业大学数学系黑龙江哈尔滨 150001 哈尔滨工业大学数学系黑龙江哈尔滨哈尔滨工业大学计算机系黑龙江哈尔滨

1.Introduction1.1.Thethree0-methodsWedealwiththenumericalsolutionoftheinitialvalueproblem:Heref,no!TdenotegivenfunctionswithT(t)5t,whereasu(t)isunknown(fort>to)'Withtheso--calledone--leg0--method,linear0--methodandnew... 详细信息

关键词： Delay differential equations Numerical solution Stabilityl θ-methods.

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

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Applied Mathematics and Mechanics(English Edition) 2019年第6期40卷 889-910页

作者： Demin LIU Kaitai LI College of Mathematics and System Sciences Xinjiang UniversityUrumqi 830046China School of Mathematics and Statistics Xi’an Jiaotong UniversityXi’an 710049China

In this work, the two-dimensional convective Brinkman-Forchheimer equa- tions are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.

关键词： Brinkman-Forchheimer equations finite element method Newton iteration

The Reduction of Chazy Classes and Other Third-Order Differential equations Related to Boundary Layer Flow Models

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Chinese Physics Letters 2012年第6期29卷 5-8页

作者： K.Fakhar A.H.Kara Department of Mathematics Faculty of ScienceUniversiti Teknologi MalaysiaScience and Technology81310 UTM SkudaiJohorMalaysia Ibnu Sina Institute for Fundamental Sciences Faculty of ScienceUniversiti Teknologi MalaysiaScience and Technology81310 UTM SkudaiJohorMalaysia School of Mathematics and Centre for Differential equations Continuum Mechanics and ApplicationsUniversity of the WitwatersrandJohannesburg Wits 2050South Africa

We study the symmetries,conservation laws and reduction of third-order equations that evolve from a prior reduction of models that arise in fluid phenomena.These could be the ordinary differential equations (ODEs)that are reductions of partial differential equations (PDEs) or,alternatively,PDEs related to given ODEs.In this class,the analysis includes the well-known Blasius,Chazy,and other associated third-order ODEs.

关键词： equations ordinary evolve

Parameter Optimisation of Stress-strain Constitutive equations Using Genetic Algorithms

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Journal of Materials Science & Technology 2003年第Z1期19卷 5-8页

作者： Y. Y. Yang M. Mahfouf D.A.Linkens Department of Automatic Control and Systems Engineering University of Sheffield Mappin St. Sheffield S1 3JD UK Department of Automatic Control and Systems Engineering University of Sheffield Mappin St. Sheffield S1 3JD UK Department of Automatic Control and Systems Engineering University of Sheffield Mappin St. Sheffield S1 3JD UK

The accuracy of numerical simulations and many other material design calculations, such as the rolling force, rollingtorque, etc., depends on the description of stress-strain relationship of the deformed materials. One common methodof describing the stress-strain relationship is using constitutive equations, with the unknown parameters fitted byexperimental data obtained via plane strain compression (PSC). Due to the highly nonlinear behaviour of the constitutive equations and the noise included in the PSC data, determination of the model parameters is difficult. Inthis paper, genetic algorithms were exploited to optimise parameters for the constitutive equations based on thePSC data. The original PSC data were processed to generate the stress-strain data, and data pre-processing wascarried out to remove the noise contained in the original PSC data. Several genetic optimisation schemes have beeninvestigated, with different coding schemes and different genetic operators for selection, crossover and mutation.It was found that the real value coded genetic algorithms converged much faster and were more efficient for theparameter optimisation problem.

关键词： Genetic algorithms, Parameter optimisa tion, Aluminium alloy, Flow stress, Constitutive equations

A New Method to Solve the Spheroidal Wave equations

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Chinese Physics Letters 2010年第3期27卷 40-43页

作者：田贵花 School of Science Beijing University of Posts and Telecommunications Beijing 100876

The perturbation method in supersymmetric quantum mechanics is used to study the spheroidal wave functions＇ eigenvalue problem. The super-potential are solved in series of the parameter α, and the general form of all its terms is obtained. This means that the spheroidal problem is solved completely in the way for the ground eigen-value problem. The shape invariance property is proved retained for the super-potential and subsequently all the excited eigen-value problem could be solved. The results show that the spheroidal wave equations are integrable.

关键词： Hive equipment Method equations

Symmetry Groups and Gauss Kernels of the Schr(o)dinger equations with Potential Functions

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Chinese Physics Letters 2012年第7期29卷 1-5页

作者：康静屈长征 Department of Mathematics Northwest UniversityXi’an 710069 Department of Mathematics Ningbo UniversityNingbo 315211

We study the Gauss kernels for a class of (2+1)-dimensional linear Schr(o)dinger equations with potential functions.The relationship between the Lie point symmetries and Gauss kernels for the Schr(o)dinger equations is established.It is shown that a classical integral transformation of the Gauss kernel can be generated by a proper Lie point symmetry admitted by the equation.Then we can recover the Gauss kernels for the Schr(o)dinger equations by performing the inverse integral transformation.

关键词： functions equation equations

A Method to Construct the Nonlocal Symmetries of Nonlinear Evolution equations

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Chinese Physics Letters 2013年第10期30卷 5-8页

作者：辛祥鹏陈勇 Shanghai Key Laboratory of Trustworthy Computing East China Normal UniversityShanghai 200062

A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations.The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation,the coupled Korteweg-de Vries system,the Kadomtsev–Petviashvili equation,the Ablowitz–Kaup–Newell–Segur equation and the potential Korteweg-de Vries equation.The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.

关键词： equations equation equation

A unified approach to the large deviations for small perturbations of random evolution equations

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Science China Mathematics 1997年第7期40卷 697-706页

作者：胡亦钧 Department of Mathematics Wuhan UniversityWuhan 430072China

Let be the processes governed by the following stochastic differential equations:where v (t) is a random process independent of the Brownian motion B(·).Some large deviation (LD) properties of are proved.For a particular case,an explicit representation of the rate function is also given,which solves a problem posed by Eizenberg and Freidlin.In the meantime,an abstract LD theorem is obtained.

关键词： large deviations random evolution equations small perturbations.