串列双圆柱绕流问题的数值模拟

本文运用有限体积方法,对绕串列放置的双圆柱的二维不可压缩流动进行了数值计算。为研究两圆柱不同间距对圆柱相互作用和尾流特征的影响,选取间距比Ｌ／Ｄ（Ｌ为两圆柱中心间的距离,Ｄ为圆柱直径）在１．５～５．０之间每隔０．５共八个有代表性的间距进行了计算模拟。计算均在Ｒｅ＝２００条件下进行。计算结果表明：对该绕流问题,流动特征在很大程度上取决于间距的大小。且间距存在一临界值,间距比从小于临界值变化到大于临界     

Control of vortex shedding behind circular cylinder for flows at low Reynolds numbers

It has been observed by researchers in the past that vortex shedding behind circular cylinders can be altered, and in some cases suppressed, over a limited range of Reynolds numbers by proper placement of a second, much smaller, ‘control’ cylinder in the near wake of the main cylinder. Results are presented for numerical computations of some such situations. A stabilized finite element method is employed to solve the incompressible Navier–Stokes equations in the primitive variables formulation. At low Reynolds numbers, for certain relative positions of the main and control cylinder, the vortex shedding from the main cylinder is completely suppressed. Excellent agreement is observed between the present computations and experimental findings of other researchers. In an effort to explain the mechanism of control of vortex shedding, the streamwise variation of the pressure coefficient close to the shear layer of the main cylinder is compared for various cases, with and without the control cylinder. In the cases where the vortex shedding is suppressed, it is observed that the control cylinder provides a local favorable pressure gradient in the wake region, thereby stabilizing the shear layer locally. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical predictions of low Reynolds number flows over two tandem circular cylinders

Flows over two tandem cylinders were analysed using the newly developed collocated unstructured computational fluid dynamics (CUCFD) code, which is capable of handling complex geometries. A Reynolds number of 100, based on cylinder diameter, was used to ensure that the flow remained laminar. The validity of the code was tested through comparisons with benchmark solutions for flow in a lid‐friven cavity and flow around a single cylinder. For the tandem cylinder flow, also mesh convergence was demonstrated, to within a couple of percent for the RMS lift coefficient. The mean and fluctuating lift and drag coefficients were recorded for centre‐to‐centre cylinder spacings between 2 and 10 diameters. A critical cylinder spacing was found between 3.75 and 4 diameters. The fluctuating forces jumped appreciably at the critical spacing. It was found that there exists only one reattachment and one separation point on the downstream cylinder for spacings greater than the critical spacing. The mean and the fluctuating surface pressure distributions were compared as a function of the cylinder spacing. The mean and the fluctuating pressures were significantly different between the upstream and the downstream cylinders. These pressures also differed with the cylinder spacing. Copyright © 2004 John Wiley & Sons, Ltd.     

Interference effect of two equal‐sized square cylinders in tandem arrangement: With planar shear flow

Numerical simulations have been performed for flow past two equal‐sized square cylinders in tandem arrangement subjected to incoming planar shear flow. Effect of L/d ratio and the shear parameter has been studied. The range of L/d ratio (ratio of center‐to‐center distance (L) to cylinder width (d)) is varied from 2 to 7 and the non‐dimensional shear parameter (K) is varied from 0.0 to 0.4 in steps of 0.1. For all the cases the Reynolds number (Re) based on centerline velocity and cylinder width is fixed at 100. The results are compared with that of isolated square cylinder with uniform flow. Strouhal number decreases with increasing shear parameter. There are more than one shedding frequency at high shear parameters and L/d ratios. The mean drag coefficient is decreased with shear parameter and lesser than that of the single cylinder. The root mean square (RMS) value of both lift and drag coefficients is higher for the downstream cylinder for all values of shear parameter. With increasing L/d ratio, for both lift and drag, the RMS value increases and then decreases for upstream cylinder, whereas it continuously increases for the downstream cylinder. The stagnation point is moved towards the top leading edge with increasing shear. The critical L/d ratio, which is defined as the distance between two cylinders, beyond which the vortex shedding from the upstream cylinder occurs, decreases with increasing shear parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Unsteady RANS simulation of oscillating mould flows

Mould flow oscillations are of major importance for the performance of the continuous casting process. They are suspected to promote entrainment of slag and other unwanted secondary phases into the melt pool. These oscillating turbulent flows are investigated by means of numerical simulations. The numerical model is based on the equation of continuity and the unsteady Reynolds averaged Navier–Stokes equations. The system of flow equations is closed by a Reynolds stress turbulence model in combination with non‐equilibrium wall functions. The unsteady simulation resolves low‐frequency oscillations of the flow field. These frequencies and numerically resolved mean values are in agreement with results of corresponding model experiments. The proposed model should be advantageous in order to investigate the mechanisms of the oscillations and the process of slag entrainment in more detail. Copyright © 2006 John Wiley & Sons, Ltd.     

A numerical investigation of the flow over a pair of identical square cylinders in a tandem arrangement

This paper describes a numerical study of the two‐dimensional and three‐dimensional unsteady flow over two square cylinders arranged in an in‐line configuration for Reynolds numbers from 40 to 1000 and a gap spacing of 4D, where D is the cross‐sectional dimension of the cylinders. The effect of the cylinder spacing, in the range G = 0.3D to 12D, was also studied for selected Reynolds numbers, that is, Re = 130, 150 and 500. An incompressible finite volume code with a collocated grid arrangement was employed to carry out the flow simulations. Instantaneous and time‐averaged and spanwise‐averaged vorticity, pressure, and streamlines are computed and compared for different Reynolds numbers and gap spacings. The time averaged global quantities such as the Strouhal number, the mean and the RMS values of the drag force, the base suction pressure, the lift force and the pressure coefficient are also calculated and compared with the results of a single cylinder. Three major regimes are distinguished according to the normalized gap spacing between cylinders, that is, the single slender‐body regime (G < 0.5), the reattach regime (G < 4) and co‐shedding or binary vortex regime (G ≥4). Hysteresis with different vortex patterns is observed in a certain range of the gap spacings and also for the onset of the vortex shedding. Copyright © 2011 John Wiley & Sons, Ltd.     

Flow-induced oscillation of two circular cylinders in tandem arrangement at low Re

Results are presented for flow-induced vibrations of a pair of equal-sized circular cylinders of low nondimensional mass (m^*=10) in a tandem arrangement. The cylinders are free to oscillate both in streamwise and transverse directions. The Reynolds number, based on the free-stream speed and the diameter of the cylinders, D is 100 and the centre-to-centre distance between the cylinders is 5.5D. The computations are carried out for reduced velocities in the range 2≤U^*≤15. The structural damping is set to zero for enabling maximum amplitudes of oscillation. A stabilized finite element method is utilized to carry out the computations in two dimensions. Even though the response of the upstream cylinder is found to be qualitatively similar to that of an isolated cylinder, the presence of a downstream cylinder is found to have significant effect on the behaviour of the upstream cylinder. The downstream cylinder undergoes very large amplitude of oscillations in both transverse and streamwise directions. The maximum amplitude of transverse response of the downstream cylinder is quite similar to that of a single cylinder at higher Re beyond the laminar regime. Lock-in and hysteresis are observed for both upstream and downstream cylinders. The downstream cylinder undergoes large amplitude oscillations even beyond the lock-in state. The phase between transverse oscillations and lift force suffers a 180 jump for both the cylinders almost in the middle of the synchronization regime. The phase between the transverse response of the two cylinders is also studied. Complex flow patterns are observed in the wake of the freely vibrating cylinders. Based on the phase difference and the flow patterns, the entire flow range is divided into five sub-regions.     

Three-dimensional simulations of flow past two circular cylinders in side-by-side arrangements at right and oblique attacks

The flow past two identical circular cylinders in side-by-side arrangements at right and oblique attack angles is numerically investigated by solving the three-dimensional Navier–Stokes equations using the Petrov–Galerkin finite element method. The study is focused on the effect of flow attack angle and gap ratio between the two cylinders on the vortex shedding flow and the hydrodynamic forces of the cylinders. For an oblique flow attack angle, the Reynolds number based on the velocity component perpendicular to the cylinder span is defined as the normal Reynolds number Re_N and that based on the total velocity is defined as the total Reynolds number Re_T. Simulations are conducted for two Reynolds numbers of Re_N=500 and Re_T=500, two flow attack angles of α=0° and 45° and four gap ratios of G/D=0.5, 1, 3 and 5. The biased gap flow for G/D=0.5 and 1 and the flip-flopping bistable gap flow for G/D=1 are observed for both α=0° and 45°. For a constant normal Reynolds number of Re_N=500, the mean drag and lift coefficients at α=0° are very close to those at α=45°. The difference between the root mean square (RMS) lift coefficient at α=0° and that at α=45° is about 20% for large gap ratios of 3 and 5. From small gap ratios of 0.5 and 1, the RMS lift coefficients at α=0° and 45° are similar to each other. The present simulations show that the agreement in the force coefficients between the 0° and 45° flow attack angles for a constant normal Reynolds number is better than that for a constant total Reynolds number. This indicates that the normal Reynolds number should be used in the implementation of the independence principle (i.e., the independence of the force coefficients on the flow attack angle). The effect of Reynolds number on the bistable gap flow is investigated by simulating the flow for Re_N=100–600, α=0° and 45° and G/D=1. Flow for G/D=1 is found to be two-dimensional at Re_N=100 and weak three-dimensional at Re_N=200. While well defined biased flow can be identified for Re_N=300–600, the gap flow for Re_N=100 and 200 changes its biased direction too frequently to allow stable biased flow to develop.     

Wake-induced vibration of tandem and staggered cylinders with two degrees of freedom

The wake-induced vibration (WIV) of two staggered cylinder with two degrees of freedom (2-dof) has been investigated by experiments in a water channel for Reynolds number between 2000 and 25 000. The streamwise separation was fixed to 4 diameters and the lateral separation varied between 0 and 3 diameters for tandem and staggered configurations. Results are presented in the form of trajectories of motion and dynamic response curves of displacements, frequencies and force coefficients. Excitation caused by the WIV mechanism is found to get weaker as the initial position of the downstream cylinder is increased from the centreline of the wake (tandem arrangement) towards the sides. For a lateral separation of 3 diameters wake interference was already found to be negligible. Evidence of a type of wake-stiffness concept is also observed to occur for 2-dof WIV in tandem arrangement, especially for higher reduced velocities. A similar mechanism may also be occurring for staggered arrangements around the centreline.     

虚拟边界法研究正交双圆柱及串列双圆球绕流

把Goldstein等人提出的虚拟边界法推广到三维情况,研究了 Re=150时不同间距下正交双圆柱绕流,和Re=250时不同间距下串列双 圆球绕流流场. 对于正交双圆柱绕流,当间距比大于3,下游圆柱对上游圆柱尾流的影响只 限定在下游圆柱的尾流所扫过的范围之内;当间距比小于等于3,下游圆柱对上游圆柱尾流 的影响扩大,下游圆柱尾流扫过区上下出现两排三维流向二次涡结构. 对于串列圆球绕流, 研究发现,在小间距比(L/D≈ 1.5)的情况下,由于上下游圆球尾流区的相互抑 制消除了压力不稳定性,整个流场呈现稳 态轴对称特征;间距比为2.0时,周向压力梯度诱发出流体的周向输运,流场呈现稳态非对 称性,但流场中存在特定的对称面;间距比增大到2.5后,绕流场开始周期振荡,原有的对 称面依旧存在;在间距比3.5时下游圆球下表面的涡结构强度有所减弱,导致占优频率发生 交替;间距比增至7.0时,整个流场恢复稳态特征,两圆球尾部同时出现双线涡,这时流场 对称面的位置发生了变动.     

Flow past a sphere in density-stratified fluid

Stratified flow past a three-dimensional obstacle such as a sphere has been a long-lasting subject of geophysical, environmental and engineering fluid dynamics. In order to investigate the effect of the stratification on the near wake, in particular, the unsteady vortex formation behind a sphere, numerical simulations of stratified flows past a sphere are conducted. The time-dependent Navier–Stokes equations are solved using a three-dimensional finite element method and a modified explicit time integration scheme. Laminar flow regime is considered, and linear stratification of density is assumed under Boussinesq approximation. The effects of stratification is implemented by density transport without diffusion. The computed results include the characteristics of the near wake as well as the effects of stratification on the separation angle. Under increased stratification, the separation on the sphere is suppressed and the wake structure behind the sphere becomes planar, resembling that behind a vertical cylinder. With further increase in stratification, the wake becomes unsteady, and consists of planar vortex shedding similar to von Karman vortex streets.     

Three-dimensional computation for flow-induced vibrations of an upstream circular cylinder in two tandem circular cylinders

It is well known from a lot of experimental data that fluid forces acting on two tandem circular cylinders are quite different from those acting on a single circular cylinder. Therefore, we first present numerical results for fluid forces acting on two tandem circular cylinders, which are mounted at various spacings in a smooth flow, and second we present numerical results for flow-induced vibrations of the upstream circular cylinder in the tandem arrangement. The two circular cylinders are arranged at close spacing in a flow field. The upstream circular cylinder is elastically placed by damper-spring systems and moves in both the in-line and cross-flow directions. In such models, each circular cylinder is assumed as a rigid body. On the other hand, we do not introduce a turbulent model such as the Large Eddy Simulation (LES) or Reynolds Averaged Navier-Stokes (RANS) models into the numerical scheme to compute the fluid flow. Our numerical procedure to capture the flow-induced vibration phenomena of the upstream circular cylinder is treated as a fluid-structure interaction problem in which the ideas of weak coupling is taken into consideration.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Vortex shedding and acoustic resonance of single and tandem finned cylinders

The effect of fins on vortex shedding and acoustic resonance is investigated for isolated and two tandem cylinders exposed to cross-flow in a rectangular duct. Three spacing ratios between the tandem cylinders (S/D_e=1.5, 2 and 3) are tested for a Reynolds number range from 1.6×10⁴ to 1.1×10⁵. Measurements of sound pressure as well as mean and fluctuating velocities are performed for bare and finned cylinders with three different fin densities. The effect of fins on the sound pressure generated before the onset of acoustic resonance as well as during the pre-coincidence and coincidence resonance is found to be rather complex and depends on the spacing ratio between cylinders, the fin density and the nature of the flow-sound interaction mechanism.For isolated cylinders, the fins reduce the strength of vortex shedding only slightly, but strongly attenuate the radiated sound before and during the occurrence of acoustic resonance. This suggests that the influence of the fins on correlation length is stronger than on velocity fluctuations. In contrast to isolated cylinders, the fins in the tandem cylinder case enhance the vortex shedding process at off-resonant conditions, except for the large spacing case which exhibits a reversed effect at high Reynolds numbers. Regarding the acoustic resonance of the tandem cylinders, the fins promote the onset of the coincidence resonance, but increasing the fin density drastically weakens the intensity of this resonance. The fins are also found to suppress the pre-coincidence resonance for the tandem cylinders with small spacing ratios (S/D_e=1.5, 2 and 2), but for the largest spacing case (S/D_e=3), they are found to have minor effects on the sound pressure and the lock-in range of the pre-coincidence resonance.     

Vortex-induced vibration of two elastically coupled cylinders in side-by-side arrangement

Vortex-induced vibration (VIV) of two elastically coupled circular cylinders in side-by-side arrangement is investigated numerically. The Reynolds-averaged Navier–Stokes equations are solved by the finite element method for simulating the flow and the equation of motion is solved for calculating the vibration. The mass ratio (the ratio of the mass of the cylinder to the displaced fluid mass) is 2 and the Reynolds number is 5000 in the simulations. Simulations are carried out for one symmetric configuration (referred to be Case A) and one asymmetric configuration (referred to be Case B). In both Case A and Case B, the primary response frequencies of the two cylinders are found to be the same both inside and outside the lock-in regimes. Five response regimes are found in both cases and they are the first-mode lock-in regime, the second-mode lock-in regime, the sum-frequency lock-in regime and two transition regimes. When the vibration is transiting from the first- to the second-mode lock-in regimes, the vibration of each cylinder contains both first- and the second-mode natural frequencies, and the vibrations are usually irregular. In the transition regime between the second-mode lock-in and the sum-frequency lock-in regimes, the response frequencies of both cylinders increases with an increase in the reduced velocity until they are close to the sum of the two natural frequencies. In both cases, the lower boundary reduced velocity of the total lock-in regime (the sum of the five lock-in regimes) is about 3 and the upper boundary reduced velocity is about 11 times the first-to-second-mode natural frequency ratio.     

Numerical simulations of the vortex-shedding in flows around rectangular cylinders

Flows around rectangular cylinders with a series of width-to-height ratios are calculated by means of the Improved Finite Analytic Method (IFAM) and the formation, development and shedding of vortices from the cylinders are simulated successfully. According to these results of time-dependent processes the physical phenomena in the flows are investigated in detail, and the discontinuity of Strouhal number is explained in the case of the width-to-height ratio equal to 3.0. The numerical solutions here show good agreement with the experimental results. In addition, based on several hundreds of the calculated flow patterns a moving picture is made by the computer image processing technology and recorded on a video tape, and then the vivid pictures of the physical process of vortex-shedding can be replayed later and analysed in detail. The project supported by the National Natural Science Foundation of China     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.