NUMERICAL PREDICTION OF FORCE ON RECTANGULAR CYLINDERS IN OSCILLATING VISCOUS FLOW

Numerical results are presented for an oscillating viscous flow past a square cylinder with square and rounded corners and a diamond cylinder with square corners at Keulegan–Carpenter numbers up to 5. This unsteady flow problem is formulated by the two-dimensional Navier–Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. Since the vorticity distribution has a mathematical singularity at a sharp corner and since the force coefficients are found in experiments to be sensitive to the corner radius of rectangular cylinders, a grid-generation technique is applied to provide an efficient mesh system for this complex flow. Local grid concentration near the sharp corners, instead of any artificial treatment of the sharp corners being introduced, is used in order to obtain high numerical resolution. The elliptic partial differential equation for stream function and vorticity in the transformed plane is solved by a multigrid iteration method. For an oscillating flow past a rectangular cylinder, vortex detachment occurs at irregular high frequency modes at KC numbers larger than 3 for a square cylinder, larger than 1 for a diamond cylinder and larger than 3 for a square cylinder with rounded corners. The calculated drag and inertia coefficients are in very good agreement with the experimental data. The calculated vortex patterns are used to explain some of the force coefficient behavior.     

Hyperbolicity and change of type in the flow of viscoelastic fluids through channels

We consider steady flow of an upper convected Maxwell fluid through a channel with wavy walls. The vorticity of this flow will change type when the velocity in the center of the channel is larger than a critical value defined by the propagation of shear waves. There is then a central region of the channel in which the vorticity equation is hyperbolic and a low speed region near the walls where the vorticity equation is elliptic. We linearize the problem for small amplitude waviness and the linearized problem is solved in detail. The characterstic nets depend on the viscoelastic “Mach” number which is the ratio (M = U/c) of the unperturbed maximum velocity U to the speed of shear waves c into the fluid at rest and the elasticity number E. There is a supercritical (hyperbolic) region around the center of the channel when M > 1. When M ? 1, the thickness of this hyperbolic region is small when E is large, and large when E is small. Regions of positive and negative vorticity are swept out along forward facing characteristics in the hyperbolic region. There is rapid damping of vorticity in the hyperbolic region away from the boundary when M ? 1 and the Weissenberg number

. (The Weissenberg number is proportional to the relaxation time of the fluid.)The rate of damping of vorticity decreases as W is increased. Flows with high M appear to be more “elastic” when W is large in the sense that the damping is suppressed as the relaxation time of the fluid is increased.     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

内管自转且公转时环管幂律流的受力分析

数值模拟了环管中内管偏心自转且公转时由轴向压力所驱动的幂律流体充分发展层流,分析了内管上的流体作用力。结果表明,内管偏心自转时流体作用力具有推动内管作和自转同向公转的效果。当只有外力矩驱动内管自转时,由于流体的作用,随内管线密度的不同,内管能达到的受力平衡态也不同:线密度较小时内管仅能在同心自转时达到受力平衡;线密度较大时内管能在作具有不变角速度和偏心率公转时达到受力平衡,且内管线密度越大,对应的受力平衡的公转的偏心率也越大。     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

A viscous vortex particle method for deforming bodies with application to biolocomotion

Bio‐inspired mechanics of locomotion generally consist of the interaction of flexible structures with the surrounding fluid to generate propulsive forces. In this work, we extend, for the first time, the viscous vortex particle method (VVPM) to continuously deforming two‐dimensional bodies. The VVPM is a high‐fidelity Navier–Stokes computational method that captures the fluid motion through evolution of vorticity‐bearing computational particles. The kinematics of the deforming body surface are accounted for via a surface integral in the Biot–Savart velocity. The spurious slip velocity in each time step is removed by computing an equivalent vortex sheet and allowing it to flux to adjacent particles; hence, no‐slip boundary conditions are enforced. Particles of both uniform and variable size are utilized, and their relative merits are considered. The placement of this method in the larger class of immersed boundary methods is explored. Validation of the method is carried out on the problem of a periodically deforming circular cylinder immersed in a stagnant fluid, for which an analytical solution exists when the deformations are small. We show that the computed vorticity and velocity of this motion are both in excellent agreement with the analytical solution. Finally, we explore the fluid dynamics of a simple fish‐like shape undergoing undulatory motion when immersed in a uniform free stream, to demonstrate the application of the method to investigations of biomorphic locomotion. Copyright © 2008 John Wiley & Sons, Ltd.     

Direct numerical simulation of a liquid sheet in a compressible gas stream in axisymmetric and planar configurations

A thin liquid sheet present in the shear layer of a compressible gas jet is investigated using an Eulerian approach with mixed-fluid treatment for the governing equations describing the gas–liquid two-phase flow system, where the gas is treated as fully compressible and the liquid as incompressible. The effects of different topological configurations, surface tension, gas pressure and liquid sheet thickness on the flow development of the gas–liquid two-phase flow system have been examined by direct solution of the compressible Navier–Stokes equations using highly accurate numerical schemes. The interface dynamics are captured using volume of fluid and continuum surface force models. The simulations show that the dispersion of the liquid sheet is dominated by vortical structures formed at the jet shear layer due to the Kelvin–Helmholtz instability. The axisymmetric case is less vortical than its planar counterpart that exhibits formation of larger vortical structures and larger liquid dispersion. It has been identified that the vorticity development and the liquid dispersion in a planar configuration are increased at the absence of surface tension, which when present, tends to oppose the development of the Kelvin–Helmholtz instability. An opposite trend was observed for an axisymmetric configuration where surface tension tends to promote the development of vorticity. An increase in vorticity development and liquid dispersion was observed for increased liquid sheet thickness, while a decreasing trend was observed for higher gas pressure. Therefore surface tension, liquid sheet thickness and gas pressure factors all affect the flow vorticity which consequently affects the dispersion of the liquid.      

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of Isolation Effectiveness of Shear Waves by a Row of Hollow Pipe Piles in Saturated Soils

Wave numbers about the three types of waves in saturated soils are firstly given in this paper. The lengths of the pipe piles are much larger than their diameters, so the isolation problem about SV waves by discontinuous barriers composed of a row of pipe piles can be simplified as a two-dimensional scattering problem. The expansion method of wave functions is adopted, the stresses and displacements at the boundaries between the pipe piles and adjacent soils are considered as continuous and the inner sides of the pipe piles are free, and then the theoretical solutions are obtained about this two-dimensional scattering problem. Normalized displacements are introduced, which are the displacements behind the barriers caused by both the incident and scattered waves to those only by the incident SV waves, contours and curves of the normalized displacements are drawn, and the influences of wall thickness of pipe piles, modulus ratio of pipe piles to soils, spacing distance between the pipe piles and pipe pile numbers on the isolation effectiveness are analyzed.     

The viability of a simple procedure for determining the criterion for the instability of circumferential growth of a through-wall crack in a stainless steel piping system, Part 2: The applied displacements produce bending and tensile deformation

Against the background of the technological problem of integranular stress corrosion cracking of Type 304 stainless steel Boiling Water Reactor piping systems, the stability of a circumferential through-wall crack is examined, for the case where the cracked section is at a position where the pipe enters a larger component. The paper presents a general methodology for determining the instability criterion for fixed displacement loadings, which are appropriate for an accident condition, the criterion being expressed in terms of the applied and material tearing moduli, or equivalently in terms of an effective pipe length. The methodology is applied to models which simulate bends in a piping system, and general conclusions are drawn with regards to the effect of pipe bends on crack instability.     

Motion of a deformed contour in a flow of an ideal incompressible liquid with a constant vorticity

The article gives a solution to the plane problem of the motion of a deformed contour in a flow of an ideal incompressible liquid with a constant vorticity. An explicit expression is obtained for the hydrodynamic force when the velocity of the external flow depends linearly on the coordinates. In the case of a contour of small dimensions, this expression is valid also for an arbitrary external flow.     

Projection conditions on the vorticity in viscous incompressible flows

The problem of establishing appropriate conditions for the vorticity transport equation is considered. It is shown that, in viscous incompressible flows, the boundary conditions on the velocity imply conditions of an integral type on the vorticity. These conditions determine a projection of the vorticity field on the linear manifold of the harmonic vector fields. Some computational consequences of the above result in two-dimensional calculations by means of the nonprimitive variables, stream function and vorticity, are examined. As an example of the application of the discrete analogue of the projection conditions, numerical solutions of the driven cavity problem are reported.     

Self-sustained amplification of disturbances in pipe Poiseuille flow

The nonlinear development of disturbances in pipe Poiseuille flow is studied with a low-dimensional model. The basic system from which the model is derived governs disturbances closely related to the radial velocity and radial vorticity disturbances. The analysis is restricted to the interaction of the two first harmonics of streamwise elongated disturbances since they are the most transiently amplified ones in linear theory. In the resulting dynamical system a nonlinear feedback from the normal vorticity disturbance (which is transiently amplified according to linear theory) to the radial velocity disturbance is present. Above a threshold of the initial amplitude, the feedback leads to a self-sustained amplification of the disturbances continuing for all times.     

Instability of an Oscillating Cylinder in a Circulation Flow of Ideal Fluid

The problem of the stability of a circular cylinder in a circulation flow is considered under the condition that the cylinder can perform both free (free cylinder) and forced oscillations (cylinder on a spring). It is shown that this simple system can be unstable in the presence of flow vorticity. Particular cases of vorticity distributions which make it possible to obtain an analytic solution are considered. The case of weak monotonically decreasing vorticity of an arbitrary form is analyzed for an arbitrary relation between the densities of the cylinder and the fluid. It turns out that the instability can develop only for a cylinder whose density is greater than that of the fluid. An approximate method of solving this problem based on consideration of the energy balance in the system is constructed. This makes it possible to obtain an expression for the growth rates and explain the physical mechanism realizing the instability, which is associated with the possibility of energy transfer from perturbations in the critical layer to the cylinder oscillations.     

点火条件对密闭管道内预混氢气/空气燃爆特性的影响

基于流体动力学软件Fluent,开展数值模拟,研究点火位置（距管左端壁面100、200和500 mm）、点火温度（1 000、1 500和2 000 K）和点火面积（管左端壁面处半径为50、35和20 mm的点火域）等点火条件对1 000 mm密闭管道中预混氢气/空气（H₂/air）燃爆特性的影响。研究表明:点火位置距管左端壁面越远,中间节点处温度越高,温升越快;不同点火温度下管内最高温升速率基本同步,且提高点火温度,使得燃烧反应更剧烈,能提高管内气体温升速率,但却降低管内的压力峰值;点火面积越小,预混H₂/air燃烧前期温升越快。当采用半径为35 mm的点火域和点火位置距管左端壁面100 mm的点火方式时,预混H₂/air燃爆的各项参数相对较高。不同点火条件对密闭管内气体的动能和内能的影响规律类似于其对管内气体的流速和温度的影响规律,而对涡量的影响不明显。     

Baroclinic vortex sheet production by shocks and expansion waves

Vortex sheet production by shocks and expansion waves refracting at a density discontinuity was examined and compared using an analytical solution and numerical simulations. The analytical solution showed that with a small exception, vortex sheet strength is generally stronger in fast/slow shock refractions. In contrast, expansion waves generated a stronger vortex sheet in slow/fast refractions. This difference results in larger vorticity deposited by shocks in fast/slow refractions and by expansion waves in slow/fast refractions. Shock refractions become irregular and the analytical solution fails when either incident, transmitted or reflected shock, exceeded the angle limit for an attached shock. To investigate vortex sheet production outside the range of analytical solutions and to verify the applicability of the planar-interface analytical solution to a curved interface, shock refraction through a sinusoidal interface was numerically simulated in the shock frame of reference. It is found that variation in the local incidence angle along the curved interface creates pressure waves that affect the level of deposited vorticity. This contributes to the difference between predictions from local analysis and numerical computation. Furthermore, an interesting behavior of the shock and expansion wave-deposited vorticity in supersonic ramp flow was discovered. When the high- and low-density streams were swapped, while keeping the incident flow Mach numbers constant, a vortex sheet of equal magnitude but of opposite sign was generated.     

A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids

The solution of the Gromeko problem [1] on unsteady flow of a viscous fluid in a long circular pipe is among the few exact solutions of the Navier-Stokes equations. Its effective solution is obtained only when the longitudinal pressure gradient is given as an arbitrary time function. However, in practice we encounter cases when the flow rate is a known time function. This sort of problem arises, in particular, in rheological experiments using viscometers with a given flow rate. In this case the determination of the pressure gradient from the given flow rate leads in the general case to a very unwieldy expression. Below we present an effective solution of this problem for viscous and elasticoviscous media using the method of solving the inlet flow problem for a steady flow of a viscous fluid in a semi-infinite pipe. It is shown that for the case of a viscous fluid these two problems are actually equivalent.     

The propagation of vorticity in a viscoelastic fluid

A layer of constant vorticity exists in an infinite space of incompressible isotropic viscoelastic fluid for which the shear stress for rectilinear shearing flows depends linearly on the history of the velocity gradient. At some instant of time the forces maintaining this flow are removed. The subsequent time-dependent vorticity field is calculated explicitly in the particular cases when the fluid is Newtonian and when it is Maxwellian. The limiting case in which the vorticity layer becomes a vortex sheet is also calculated.     

Magnetohydrodynamic pipe flow with reverse motion

An exact solution of the Magnetohydrodynamic pipe flow equations is found in terms of elementary functions. As the Hartmann number increases from zero reverse motion in the pipe occurs and eventually separates from the boundary. With further increase in the Hartmann number the vorticity on the boundary continuously changes sign and the flow is analogous to laminar separation in field free hydrodynamics.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.