Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Turbulent jet mixing of supersonic flows in profiled flow-equalization channels

A numerical model for the calculation of gas dynamic systems with turbulent mixing of supersonic jets is proposed. The problem of designing a transitional flow-equalization channel of minimum length is solved for the viscous turbulent mixing of two parallel or mutually inclined supersonic flows. The problem is solved in two stages. In the first stage the flow-equalization channel is designed by solving the inverse problem in the ideal gas approximation. In the second stage the basic problem is solved for the channel thus obtained on the basis of the parabolized Navier-Stokes equations. Investigations have demonstrated the validity of this approach to the equalization of nonuniform flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 175–178, July–August, 1987.The authors are grateful to V. I. Kopchenov for supplying the program for solving the basic problem by a first-order Godunov method and to A. I. Kraiko and Yu. V. Kurochkin for their interest and advice.     

Stability of flows in a peristaltic transport

The stability problem related to the basic flows induced by the peristaltic waves propagating along the deformable walls is investigated numerically. The neutral stability boundary is obtained by solving the relevant Orr–Sommerfeld equation via a verified preconditioned complex-matrix solver. The critical Reynolds number becomes 577.25 when the ratio of the wave speed to the maximum speed of the basic flow (c/u_max) becomes 10.     

Propagation of acoustoelectric waves in a hollow cylinder whose physicomechanical properties have a radial symmetry axis

The general case of off-axis propagation of acoustoelectric waves in hollow cylinders made of a piezoelectric material such as a crystal of the class mm2 orthorhombic system with a two-fold radial symmetry axis is investigated. The basic system of equations for the wave problem in circular cylindrical coordinates is reduced to eight equations of the Hamiltonian type in the radial coordinate. A solution of the generalized spectral problem for harmonic waves is found by numerical methods. Special cases of the general problem are considered. The results of solving specific problems are analyzed. Taras Shevchenko National University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp 49–58, July, 1999     

Time-Periodic Solutions of the Navier-Stokes Equations in Unbounded Cylindrical Domains – Leray's Problem for Periodic Flows

Poiseuille flows in infinite cylindrical pipes, in spite of their enormous simplicity, have a main role in many theoretical and applied problems. As is well known, the Poiseuille flow is a stationary solution of the Stokes and the Navier-Stokes equations with a given constant flux. Time-periodic flows in channels and pipes have a comparable importance. However, the problem of the existence of time-periodic flows in correspondence to any given time-periodic total flux, is still an open problem. A solution is known only in some very particular cases, for instance, the Womersley flows. Our aim is to solve this problem in the general case. The above existence result opens the way to further investigations. As an example of this possibility we consider the extension of the classical Leray's problem for Poiseuille flows to arbitrary time-periodic flows. Dedicated to Louis Nirenberg on the occasion of his 80^th birthday     

水坝崩溃的计算机模拟

本文用求解化学流体力学基本方程组的方法研究了不可压自由表面问题,着重引入了流体体积分数技术,建立了水坝倒塌问题的物理模型,运用该技术和模型,计算得到了水流的速度和压力在空间的分布及其随时间的变化,与实验观测一致.     

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract

A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests are presented.     

Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance

The evolution of small perturbations in longitudinally nonuniform flows is studied with reference to the problem of the propagation of flow perturbations in a plane channel with walls of variable elasticity. Using the solution of the problem of the receptivity of the flow to local vibrations of the walls, the problem considered can be reduced to the solution of an integral equation for a single function, namely, the complex vibration amplitude of the walls. A numerical method for solving this equation on the basis of a piecewise-linear approximation of the unknown function is proposed. It is shown that the instability wave amplitude changes discontinuously at the junction of the rigid and elastic channel sections. A second method of investigating the process of propagation of perturbations in the flow considered is proposed. This method is based on laws of evolution of perturbations in nonuniform flows and an analytic solution of the problem of perturbation scattering on the junction of walls with different compliance. On the basis of this method the classical stability theory is generalized to include the case of nonuniform flows.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Towards generalization and optimization of implicit methods

A general implicit (GI) method for solving iteratively the algebraic system arising from a finite difference approximation of an elliptic partial differential equation is formulated. Under certain assumptions this method can be reduced to the already known implicit techniques. It is shown that the GI method has a very special physical meaning when solving fluid flow problems. It is shown also how this method can be optimized to achieve the maximum rate of convergence. Finally it is shown how this new strategy is applied by solving some classical numerical fluid dynamics problems.     

Interaction Between a Slow Magnetohydrodynamic Shock Wave and a Tangential Discontinuity

The self-similar problem of the oblique interaction between a slow MHD shock wave and a tangential discontinuity is solved within the framework of the ideal magnetohydrodynamic model. The constraints on the initial parameters necessary for the existence of a regular solution are found. Various feasible wave flow patterns are found in the steady-state coordinate system moving with the line of intersection of the discontinuities. As distinct from the problems of interaction between fast shock waves and other discontinuities, when the incident shock wave is slow the state ahead of it cannot be given and must to be determined in the process of solving the problem. As an example, a flow in which the slow shock wave incident on the tangential discontinuity is generated by an ideally conducting wedge located in the flow is considered. The basic features of the developing flows are determined.     

Full and reduced model solutions of steady axi-symmetric ice sheet flow over small and large bed topography slopes

Two numerical methods for solving the full steady ice-sheet equations in axi-symmetric flow are described. The free-boundary problem is treated by transforming the problem to a fixed domain using either an orthogonal co-ordinate transformation or a variant of a transformation proposed by Landau, and difficulties with the former, more sophisticated, method are demonstrated. The simpler Reduced Model is also presented, and accurate solutions for flows over bed topography with moderate to large slopes are generated by an inverse method for comparison with the numerical solutions of the full equations. The reduced model is not valid for such bed slopes, and the comparisons demonstrate the extent and nature of the errors arising from the use of the simpler model. Received September 10, 1999     

A staggered grid‐based explicit jump immersed interface method for two‐dimensional Stokes flows

In this paper the explicit jump immersed interface method (EJIIM) is applied to stationary Stokes flows. The boundary value problem in a general, non‐grid aligned domain is reduced by the EJIIM to a sequence of problems in a rectangular domain, where staggered grid‐based finite differences for velocity and pressure variables are used. Each of these subproblems is solved by the fast Stokes solver, consisting of the pressure equation (known also as conjugate gradient Uzawa) method and a fast Fourier transform‐based Poisson solver. This results in an effective algorithm with second‐order convergence for the velocity and first order for the pressure. In contrast to the earlier versions of the EJIIM, the Dirichlét boundary value problem is solved very efficiently also in the case when the computational domain is not simply connected. Copyright © 2007 John Wiley & Sons, Ltd.     

Contact interaction of an elastic punch and an elastic half-space with initial (residual) stresses

The contact interaction of an elastic punch of arbitrary cross-section and an elastic semi-space with initial (residual) stresses is studied. A general method to solve the problem is proposed. It allows solving contact problems for bodies with initial (residual) stresses when the solution of the corresponding elastic problem is known __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 28–40, December 2007.     

Dynamic analysis of nonlinear systems by modal synthesis techniques

Different kinds of modal synthesis method have been used widely in dynamic analysis of linear structure systems, but, in general, they are not suitable for nonlinear systems.In this paper, a kind of modal synthesis techniques is extended to dynamic analysis of nonlinear systems. The procedure is based upon the method suggested in [20],[21], which is applicable to vibration analysis for complex structure systems with coupling attachments but with simplified forms of linear springs and dampers. In fact, these attachments have nonlinear characteristics as those generally known to the cases of nonlinear elasticity and nonlinear damping, e.g., piecewise-linear springs, softening or hardening springs. Coulomb damping,elas-ioplastic hysteresis damping, etc. So long as the components of structure are still linear systems, we can get a set of independent free-interface normal mode information hut only keep the lower-order for each component. This can be done by computations or experiments or both. The global equations of linear vibration are set up by assembling of the component equations of motion with nonlinear coupling forces of attachments. Then the problem is reduced to less degrees of freedom for solving nonlinear equations. Thus considerable saving in computer storage and execution time can be expected. In the case of a very high-order system, if sufficient degrees of freedom are reduced, then it may be possible for the problem to be solved by the aid of a computer of ordinary grade.As the general nonlinear vibration of multiple degrees of freedom systems is quite involved, in general, the exact solution of a nonlinear system equations is not easy to find, so the numerical method can be adopted for solving the reduced nonlinear equations to obtain the transient response of system for arbitrary excitations.     

Application of a 3-D FEM/FDM Overlapping Scheme to Heat and Mass Transfer and Moving-Boundary Problems

A 3-D FEM/FDM overlapping scheme for viscous, incompressible flow problems is presented that combines the finite element method, which is best suited to analyze flow in any arbitrarily shaped flow geometry, with the finite difference method, which is advantageous in both computing time and computer storage. The combination of both methods enables large-scale viscous flow to be analyzed, which is crucial both for detailed analysis of 3-D flows and for solving flow problems around moving bodies, A modified ABM AC method is used as the basic algorithm, to which a sophisticated time integration scheme, proposed by the present authors, has been applied. In this paper, some numerical results including 3-D heat and mass transfer problem and moving-boundary problems are presented.     

Generation of large-scale structures and vortex systems in numerical experiments for rotating annular channels

Methods for solving shallow-water equations that describe flows in rotating annular channels are considered and the results of numerical calculations are analyzed for the possible generation of global large-scale flows, narrow jets, and numerous small-scale vortices in laboratory experiments. External effects in fluids are induced using a mass source–sink and the MHD-method of interaction of radial electric current with the magnetic field generated by the field of permanent magnets. A central–upwind scheme modified to suit the specific aspects of geophysical hydrodynamics. Initially, this method was used to solve shallow-water equations only in hydraulic problems, such as for flows in dam breaks, channels, rivers, and lakes. Geophysical hydrodynamics (in addition to free surface and topography) requires a rotation of the system as a whole, which is accompanied by the appearance of a complex system of vortices, jets, and turbulence (these should be taken into account in the formulation of the problem). Accordingly, the basic features of the central–upwind method should be changed. The modifications should ensure that the scheme is well-balanced and choose interpolation methods for desired variables. The main result of this modification is the control over numerical viscosity affecting the fluid motion variety. The active dynamics of a large number of vortices transformed into jets or generating large-scale streams is the general result of modifications suitable for geophysical hydrodynamics. Because there are technical difficulties in the creation of an appropriate laboratory setup for modeling of geophysical flows with the help of numerous source–sinks, it will be appropriate to use numerical experiments for studying the motions generated by this method. Unlike this method, the MHD-method can be rather easily used in laboratory conditions to generate a large variety of flows and vortex currents in the channel by a relatively small number of permanent magnets. Specifically, this method made it possible to obtain large-scale circular flows over the entire channel area, jets, and systems of interacting vortices. For the purpose of experiments, the distributions of source–sinks and systems of permanent magnets over the bottom of annular channels are determined.     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

Flow of a viscous fluid in a cylindrical vessel with a rotating cover

The problem considered arises in solving various technical problems associated with flows of a viscous fluid in a closed space near rotating plane surfaces, turbomachine disks, thrust bearings, rotational viscosimeters, etc. The approximate solution of the problem on the basis of a simplified flow scheme was first obtained by Schultz-Grunow [1], The most complete investigation has been made recently by Grohne [2], who outlined a program for solving the problem by joining several partial solutions on the basis of definite hypotheses concerning the flow core.With the development of electronic digital computers and the necessary numerical methods, the most effective means of solving the considered problem is the use of the grid methods for solving partial differential equations. The present paper is devoted to presenting the results of the solution of the problem using the grid method on a digital computer.