Stream-function Co-ordinates in Rotational Flow and an Analysis of the Flow in a Shock Layer: II. Axisymmetric flow

The stream-function co-ordinates system proposed for the analysisof inviscid rotational flow in Part I, is applied to the analysisof the flow in a shock layer around an axisymmetric blunt-nosedbody in a hypersonic stream. The problem treated in this paperis the inverse problem to be solved with a prescribed shockshape. Numerical computations after the method of characteristicsor the forward integration technique are carried out for a sphericalshock and a power-law shock, by making use of an electroniccomputer. Some remarks are then made upon the singular characterof the solution in the supersonic region of the shock layer.     

二维流面上的流动问题的速度图方法

对于一些特殊的流动,尤其是平面上的位势流动,速度图方法有其显著的优点．对于理想流体来说,流面总是存在的,在流面上,流动的速度向量总是在其切空间里．通过引入流函数和势函数,采用张量分析作为工具,给出了二维曲流面上位势流动的速度图方法,得到了流函数满足的速度图方程,为一些特殊的流动问题提供了一类分析方法．并且,对于得到的二维速度图方程,得到了相应的特征方程和特征根,从而可以对方程的类型进行分类．最后,给出了一些特殊流动的实例．     

Shock Interactions in Nonequilibrium Hypersonic Flow

A shock interaction problem is solved with finite difference methods for a hypersonic flow of air with chemical reactions. If a body has two concave corners, a secondary shock is formed in the shock layer and it meets the main shock later. As the two shocks meet, the flow becomes singular at the interaction point, and a new main shock, a contact discontinuity and an expansion wave appear as a result of interaction between the two shocks. Therefore, the problem is very complicated. Using proper combinations of implicit and explicit finite difference schemes according to the property of the equations and the boundary conditions, we compute the flow behind the interaction point successfully.     

Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid

We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.     

A Stream Function Approach to Optical Flow with Applications to Fluid Transport Dynamics

Optical flow is one of the classical problems in computer vision, but it has recently also been adapted to applications from other fields, such as fluid mechanics and dynamical systems. If the goal is to analyze the dynamics of system whose evolution is governed by a flow field that is the gradient of a potential function – which describes many flows in fluid dynamics – it is natural to approach the optical flow problem by reconstructing the potential function, also called the stream function, rather than reconstructing the components of the flow directly. This alternate approach allows one to impose scientific priors, via regularization, directly on the flow itself rather than on its components independently. We demonstrate the stream function formulation of optical flow and its application to reconstructing an oceanic fluid flow driven by satellite measurements. It is also shown how these flow fields can be used to analyze mixing and mass transport in the fluid system being imaged. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular Matched Eigenfunction Expansions for Stokes Flow around a Corner

A singular matched eigenfunction expansion method is describedfor solving Stokes flow around a corner. The flow region isdecomposed into a number of simpler rectangular subregions;this enables the stream function to be represented by meansof an expansion of Papkovich-Fadle eigenfunctions in each ofthese subregions. The coefficients in these expansions are obtainedby matching them across common interfaces in a weak sense. Theresulting solution is used in a post-processing technique todetermine the coefficients in the known locally convergent expansionof the stream function at reentrant and salient corners. A smallnumber of terms in this expansion is necessary to produce accurateapproximations.     

圆截面螺旋管道内非定常流动研究

以血液流动为背景,利用双参数摄动法研究了圆截面螺旋管内低频振荡流动,得到问题的二阶摄动解,分析了轴向速度、二次流、壁面剪应力在不同时刻的特点及随时间和Womersley数的变化情况。研究表明:挠率对圆截面曲线管道内低频振荡流动的影响不可忽略,尤其是轴向压力梯度绝对值很小时,挠率将对二次流动结构起主要影响作用。流函数的剧烈变化只发生在正、负数值发生转变的很小的时间段内,大部分时间段内变化平缓。壁面剪应力随θ的变化也很大。     

旋风分离器内流动的三维分析

本文详细阐述了旋风分离器内流动在球坐标系中的数学表述和结果,应用质量守恒定律和定常流动的运动定律,在轴对称的考虑下,用流函数方法详尽推导了流动的三个速度分量.此讨论是从三维的整体观点来全面分析流动状况的.此外,对文[1]中的一些结果作了必要的修正.     

弯曲圆管内漩涡结构分叉现象的理论研究

利用拓扑结构分析方法,分析了弯曲圆管内定常流在横截面上流线的奇点个数及分布规律,给出了二次流的漩涡数目由2个变为4个,流态结构发生分叉现象的理论判据。进而,利用Galerkin方法,得到了弯曲圆管内定常流的流函数和轴向速度的半解析表达式,给出了流态结构发生分叉现象的临界Dean数,所得结果与理论判则一致。     

Vortex generated fluid flows in multiply connected domains

A fluid flow in a multiply connected domain generated by an arbitrary number of point vortices is considered. A stream function for this flow is constructed as a limit of a certain functional sequence using the method of images. The convergence of this sequence is discussed, and the speed of convergence is determined explicitly. The presented formulas allow for an easy computation of the values of the stream function with arbitrary precision in the case of well-separated cylinders. The considered problem is important for applications such as eddy flows in oceans. Moreover, since finding the stream function of the flow is essentially identical to finding the modified Green’s function for Laplace’s equation, the presented method can be applied to a more general class of applied problems which involve solving the Dirichlet problem for Laplace’s equation.     

各向异性交通流动力学模型的波动特性

对姜锐等人最近提出的一种新的交通流动力学模型的波动特性进行了分析.该模型中,不存在大于宏观车流速度的特征速度,从而满足交通流各向异性的要求.线性稳定性分析发现,新模型在一定条件下保持稳定,文中给出了稳定条件判定标准.分析指出,这一模型的波由一阶波和二阶波两种层次构成,既允许交通光滑波的存在,也充许交通激波的存在,但是其激波条件与其它相关模型有明显区别.正是这种区别,使得新模型在处理某些交通问题(如车辆倒退现象)上更加符合实际车流情况.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

A Lagrangian formulation for steady three-dimensional Newton-Busemann flow

A new Lagrangian formulation for steady three dimensional inviscid flow over rigid bodies is developed. First, the continuity equation is eliminated by the use of two stream functions. This is followed by a transformation to new independent variables, two of which are these stream functions and the third one is a Lagrangian time distinct from the Eulerian time. This Lagrangian formulation with the use of Lagrangian time requires only three independent variables and allows the free boundary problem of flow with shock wave to be rendered a fixed boundary one thereby making it easier to solve. In the Newtonian limit the governing equations reduce to the geodesic equations of the body surface. For flow past two-dimensional bodies, bodies of revolution, and conical bodies and wings, the problems are solved to within quadrature. All known Newtonian flow solutions are found to be special cases of the present theory.     

Singular problem for a third-order nonlinear ordinary differential equation arising in fluid dynamics

Results concerning singular Cauchy problems, smooth manifolds, and Lyapunov series are used to correctly state and analyze a singular “initial-boundary” problem for a third-order nonlinear ordinary differential equation defined on the entire real axis. This problem arises in viscous incompressible fluid dynamics and describes self-similar solutions to the boundary layer equation for the stream function with a zero pressure gradient (plane-parallel flow in a mixing layer). The analysis of the problem suggests a simple numerical method for its solution. Numerical results are presented.     

Aerodynamic Interaction of Systems of Bodies in a Supersonic Flow

Use of numerical experiment methods gives broad opportunities for studying complex flows by rather simple and inexpensive means. This work is devoted to researches on aerodynamic interference of bodies in supersonic flows. Interest to similar problems is shown in a number of areas of modern mechanics: meteoritics, movement of multiphase media, applied fields. The head shock wave before a configuration of spheres with growth of distance h monotonously breaks up to individual shock waves before spheres of forward lines. The basic conclusion of this research is the following: distortion of supersonic stream around systems of bodies as though is transferred downstream, i.e. the influence of back bodies on the flow nearby forward bodies is small. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Non-selfsimilar Riemann Problem for 2-D Zero-Pressure Flow in Gas Dynamics

The non-selfsimilar Riemann problem for two-dimensional zero-pressure flow in gas dynamics with two constant states separated by a convex curve is considered.By means of the generalized Rankine-Hugoniot relation and the generalized characteristic analysis method,the global solution involving delta shock wave and vacuum is constructed.The explicit solution for a special case is also given.     

血液动力学中血管流激波与驻波的相互作用

血液动力学问题是生物力学心血管系统中的重要研究课题.血管内斑块处,血管截面和血管壁的材质发生变化,对血液流动产生重要影响.血液流动中基本波及其相互作用对探究血液流动的规律、生理学意义及与疾病的关系有着重要的意义.本文研究血液动力学血液流动简化数学模型的基本波的相互作用.血管流模型是3×3非严格双曲型方程组.构造性地得到了初值为三段常状态时,血管流问题的解,即解决了激波与驻波的相互作用问题.特别地,给出四种后前激波与驻波的相互作用的结果.     

Study of a jet incident on a porous wall in a gravity field

The problem of an inviscid jet impacting on a porous wall ina gravity field is considered. By transforming this into a minimumproblem, existence and uniqueness results are proved. Propertiesof the flow region, of the stream function, and of the freeboundary are established. The monotonicity of the stream functionwith respect to the given velocities is also obtained. The mathematicalresults established allow one to obtain and to compare the numericalapproximations of the stream function and of the free boundaryfor the flows, with and without gravity, by using a finite-elementmethod.     

The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

The fluid flows in a variable cross-section duct are nonconservative because of the source term. Recently, the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied.In this paper, the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied. The elementary waves include rarefaction waves, shock waves,delta waves and stationary waves.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.