旋转运动中弹性梁耦合振动的辛方法

研究旋转梁结构的弹性耦合振动问题。通过引入对偶体系，建立了解决该类问题的辛方法。在辛体系中描述旋转梁纵向和横向耦合振动控制方程，即哈密顿正则方程。进一步求解得到结构的固有振动频率及相应的振动模态，发现固有振动频率随转动角速度先升后降以及模态之间的某种转化规律。     

Solution of the direct problem of a laval nozzle working with a two-phase medium

A study is made of a method of numerical solution of the system of ordinary differential equations describing the flow of a two-phase medium in a Laval nozzle in the one-dimensional approximation. The paper is concerned with the direct problem, in which the law of variation of the area of the transverse section of the nozzle and functional relationships between the entrance parameters are given and the unknown parameters of the two-phase flow are found along the length of the nozzle.     

Three-dimensional axisymmetric flow in perfectly plastic metals

A method is presented for solving the three-dimensional axisymmetric field equations for a perfectly plastic material which obeys the von-Mises yield criterion and the Levy-Mises flow law. The method is used for the particular case in which a small axisymmetric perturbed flow is superposed on a uniform flow without flow reversal taking place. The method then leads to solving a fourth order differential equation for the velocity potential. The special case of a thick cylindrical shell under compressive flow is examined. The solution so obtained, being derived, from the three dimensional theory, includes a correct treatment of transverse shear distortion. A preferred mode of instability is identified having a wave-length in reasonable agreement with that obtained experimentally by other workers.     

Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.     

Non-linear vibrations of transversely isotropic rectangular'' plates

The large amplitude free flexural vibration of transversely isotropic rectangular plate, incorporating the effects of transverse shear and rotatory inertia, is studied using the von Karman field equations. A mode shape, consisting of three generalised-coordinates together with the Galerkin technique, results in a system of three non-linear simultaneous ordinary differential equations which govern the motion of the plate. These equations are integrated using a fourth-order Runge-Kutta method to obtain the period for each amplitude of vibration. The non-linear period vs amplitude behaviour is of the hardening type and it is also found that transverse shear and rotary inertia effects increase the period and that this increase is quite significant even for thin transversely isotropic plates. The results are compared with earlier results which were based on a one-term or one generalised coordinate solution and using the Berger approximation or the von Karman field equations.     

A theoretical computational model of a plate in hypersonic flow

A theoretical model of an elastic panel in hypersonic flow is derived to be used for design and analysis. The nonlinear von Kármán plate equations are coupled with 1st order Piston Theory and linearized at the nonlinear steady-state deformation due to static pressure differential and thermal loads. Eigenvalue analysis is applied to determine the system’s stability, natural frequencies and mode shapes. Numerically time marching the equations provides transient response prediction which can be used to estimate limit cycle oscillation amplitude, frequency and time to onset. The model’s predictive capability is assessed by comparison to an experiment conducted at a free stream flow of Mach 6. Good agreement is shown between the theoretical and experimental natural frequencies and mode shapes of the fluid–structure system. Stability analysis is performed using linear and nonlinear methods to plot stability, flutter and buckling zones on a free stream static pressure vs temperature differential plane.     

Aeroacoustical coupling in a ducted shallow cavity and fluid/structure effects on a steam line

A pure tone phenomenon has been observed at 460 Hz in a piping steam line. The acoustical energy has been identified to be generated in an open gate valve and to be of cavity noise type. This energy is then transmitted to the main pipe by fluid/structure coupling. The objectives here are to display the mechanism of the flow acoustic coupling in the cavity and in the duct through an aeroacoustical analysis and to understand the way of energy transfer from the fluid to the main pipe through a vibroacoustical analysis. Concerning the first objective, an experimental study by means of 2/7 scale models in air is analysed by means of numerical flow simulation. The flow acoustic phenomena are modelled by computing the Euler equations. Two different computations are carried out: in the first one, a pure Euler modelling is used, in the second one, a boundary layer obtained from experimental data is introduced in the computation in order to have a realistic flow profile upstream the cavity. The boundary layer flow profile appears to be essential to recover the experimentally observed coupling between the shear-layer instability and the acoustical transverse mode of the pipe. The numerical results confirm that the second aerodynamic mode is responsible for the oscillation. While the predicted frequency agrees about 1% with the scale model experiments, the predicted amplitude is approximately 15 dB too low. For the second objective, fluid/structure coupling in the main pipe is studied using two fully coupled methods. The first method consists in a modal analysis of the line using a fluid–structure finite element model. The second one is based on the analysis of dispersion diagrams derived from the local equations of cylindrical shells filled with fluid. The way of energy transfer in transverse acoustical waves coupled with flexion-ovalization deformations of the pipe is highlighted using both methods. The dispersion diagrams allow a fast and accurate analysis. The modal analysis using a finite-element model may complete the first one with quantitative data. The link between the fluid/acoustic and the fluid/structure analysis is then the excitation of the transverse acoustical mode of the duct.     

应用有限差分法的轴向流作用下叠层板的稳定性和模态分析

分析了轴向流作用下两端简支和固支叠层板的稳定性。基于势流理论建立轴向流作用下叠层板的流固耦合系统连续型运动方程,基于有限差分法建立了流场网格和结构网格统一的离散化格式,流场势函数用板的横向振动位移变量来表示,得到关于叠层板的横向振动位移变量的控制方程。求解控制方程的广义特征值,计算分析结果表明,两端简支和两端固支模型发生屈曲失稳,且得到了屈曲失稳临界速度与叠层板的层数和无量纲板间距的关系。此外,轴向流作用下叠层板的一阶模态并不是叠层板的同相弯曲模态。     

Spectral algorithm for the analysis of flows in grooved channels

A grid‐less, spectrally accurate algorithm for the analysis of flows in grooved channels is presented. The algorithm is based on the immersed boundary conditions concept, where the boundary conditions are submerged inside the computational domain and are treated as internal constraints. When grooves' ridges are orthogonal to the flow direction (transverse grooves), the flow remains two‐dimensional. As the grooves rotate away from this direction, the flow becomes three‐dimensional. An auxiliary coordinate system is defined in such a way that one of its axes is aligned with the grooves. It is shown that the governing equations expressed in this system decouple into a two‐dimensional flow across the grooves and a flow in the direction along the grooves, resulting in improved solution efficiencies. Fourier series are used for discretization in the direction transverse to the grooves and Chebyshev expansions for the direction across the channel. Special solvers that take advantage of the matrix structure have been implemented providing a significant acceleration of computation and reduction of memory requirements. Various tests have been conducted in order to illustrate the performance of the algorithm, to show its spectral accuracy and to characterize the effects of various numerical and physical parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Scaling Transformation for the Effect of Temperature-Dependent Fluid Viscosity with Thermophoresis Particle Deposition on MHD-Free Convective Heat and Mass Transfer Over a Porous Stretching Surface

This article concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants, a third-order ordinary differential equation corresponding to the momentum equation, and two second-order ordinary differential equations corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet, the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. Impact of thermophoresis particle deposition in the presence of temperature-dependent fluid viscosity plays an important role on the concentration boundary layer. The results, thus, obtained are presented graphically and discussed.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

Numerical study of magnetohydrodynamic laminar flow separation in a channel with smooth expansion

The flow of an electrically conducting incompressible viscous fluid in a plane channel with smooth expansion in the presence of a uniform transverse magnetic field has been analysed. A solution technique for the governing magnetohydrodynamic equations in primitive variable formulation has been developed. A co‐ordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. The governing equations are discretized using finite‐difference approximations in staggered grid. Pressure Poisson equation and pressure correction formulae are derived and solved numerically. It is found that with increase in the magnetic field, the size of the flow separation zone diminishes and for sufficiently large magnetic field, the separation zone disappears completely. The peak u‐velocity decreases with increase in the magnetic field. It is also found that the asymmetric flow in a symmetric geometry, which occurs at moderate Reynolds numbers, becomes symmetric with sufficient increase in the transverse magnetic field. Thus, a transverse magnetic field of suitable strength has a stabilizing effect in controlling flow separation, as also in delaying the transition to turbulence. Copyright © 2008 John Wiley & Sons, Ltd.     

Calculation of principal characteristics of turbulent streams in a state of structural equilibrium

A review of articles on the study of turbulent streams having transverse displacement, in which a turbulent energy balance equation is used, is contained in [1]. Levin [2] proposed a certain development of Rotta's method [3] making it possible to determine the characteristics of the average flow and the radial distribution of pulsation magnitudes. However, in this article the scale of the turbulence (the quantityl) was given as an empirical function of the coordinates. At the same time it is clear that the distribution of the turbulence scale depends on the conditions of the problem. A special differential equation proposed in [4,5] describing the variation in time and space of the quantityl has the drawback that in deriving this equation it is necessary to invoke additional hypotheses which are difficult to test experimentally. In the present article, along with the velocity of the average flow, the pressure, and the pulsation magnitudes, the scale of the turbulence is considered as an important characteristic of the stream, determined by the reference system which consists of the Reynolds equations, continuity equations, and equations for the component of the Reynolds stress tensor. Rotta's approximate semiempirical relations and an experimental relation for the single-point correlation coefficient between the turbulent pulsations in velocity are used for closure of the system obtained. An approximate calculation is given for the principal average and pulsation characteristics of the flow for the region of the stream where the turbulence is in a state of structural equilibrium [6]. A comparison of the calculated and experimental data is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–99, January–February, 1973.     

Propagation of wave packets in the boundary layer on a curved surface

The development of wave packets excited in a boundary layer by means of a local deformation of the surface in the longitudinal-transverse interaction regime is considered. A solution of the linearized system of equations of interaction theory is constructed using a Laplace transformation with respect to time and a Fourier transformation with respect to the space variables. Two problems are separately examined. In the first, the disturbances are induced by a surface deformation sinusoidal in the transverse direction. It is shown that the center of the wave packet with the greatest oscillation amplitude moves in a direction opposite to that of the flow in the boundary layer. At the same time the wave packet expands, so that in the course of time any fixed point will enter the region of growing oscillations. In the second problem the source of the disturbances is isolated. In this case the wave packet acquires a horseshoe shape. Expanding, it carries the disturbances away from the source in all directions, including upstream relative to the flow in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 59–68, March–April, 1990.     

Scaling group transformation for the effect of temperature-dependent nanofluid viscosity on an mhd boundary layer past a porous stretching surface

This paper deals with a steady two-dimensional flow of an electrically conducting incompressible fluid over a porous vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. The partial differential equations governing the problem under consideration are transformed by a special form of Lie group transformations, namely, scaling group of transformations, into a system of ordinary differential equations, which are solved numerically using the Runge-Kutta-Gill algorithm and the shooting method. The conclusion is drawn that the flow field and temperature profiles are significantly influenced by the Lewis number, Brownian motion number, and thermophoresis number.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The steady state solution of the system of equations consisting of the full Navier-Stokes equations and two turbulence equations has been obtained using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time-stepping scheme with a stability-bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positivity. Low-Reynolds-number modifications to the original two-equation model are incorporated in a manner which results in well-behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved, initializing all quantities with uniform freestream values. Rapid and uniform convergence rates for the flow and turbulence equations are observed.     

A finite element approximation of steady flow through a rotating non-aligned straight tube

A finite element solution is developed for a penalty function formulation of the equations which govern the steady motion of a Newtonian fluid through a pipe that rotates about an axis not parallel to its own. The motion in this system is driven by the Coriolis acceleration, which has components in the axial direction as well as in the transverse plane of the pipe. The relative magnitudes of these components significantly affect the qualitative and quantitative nature of the primary and secondary flow field. The present results compare favourably with those of previously reported experimental and theoretical studies over a wide range of flow regimes.     

Viscous laminar flow in smooth coil tubes

A mathematical model of incompressible viscous laminar flow in smooth coil tubes is proposed and the results of its numerical realization in a nonorthogonal helical coordinate system are presented. This coordinate system is free of singularities in the domain of definition of the unknown functions, that is, the pressure and the velocity components, which makes it possible to refine the existing distributions of the axial component and the secondary crossflows obtained using the well-known orthogonal coordinate system having a singularity at the center of the coil channel. The momentum transport equation is written in the projections on the axes of the natural basis of the coordinate system, which makes it possible to subdivide the system of equations into two alternately solved subsystems. The distributions of the axial and two transverse components show that at the center of coil tubes the transverse components are comparable with the axial velocity (the transverse components can be as high as half the mean-flow-rate velocity and one third of it at the center of the channel).