Strouhal numbers in the wake of two inline cylinders

The dominant vortex frequencies f _s in the wake of two tandem circular cylinders of identical diameter d have been measured simultaneously using two hot wires placed behind each cylinder. Measurements were conducted over the Reynolds number Re (U d/, where U and are the free-stream velocity and the kinematic viscosity of fluid, respectively) =800–4.2×10⁴ and the cylinder centre-to-centre spacing L/d=1–15. The Strouhal number St (f _s d/U ) exhibits a strong dependence on L/d and Re. For L/d<(L/d)_c, which is a critical value and ranges between 3.5 and 5, there is no vortex street formed in the gap between the cylinders, and St measured behind the downstream cylinder drops rapidly for increasing L/d. For L/d>(L/d)_c, co-shedding occurs, that is, vortices are shed from the upstream as well as the downstream cylinder, and their frequencies are found to be identical. St climbs with increasing L/d, approaching a constant between 0.18 and 0.22 for L/d>10. The St–Re relationship is classified into four categories, based on their behaviours, which are associated with distinct flow physics—category 1: for 1L/d<2, the shear layers separated from the upstream cylinder roll up behind the downstream cylinder; category 2: for 2L/d3, there is a transition from the shear layer rollup behind to reattachment on the downstream cylinder; category 3: for 3<L/d5, transition from the reattachment to co-shedding regime occurs at a critical Reynolds number; and category 4: the flow for L/d>5 is characterized by co-shedding only. The present measurements reconfirm the previous observation of a bi-stable flow at the transition from the reattachment to co-shedding regime. It is found for the first time that another bi-stable flow occurs at the transition from category 1 to 2, that is, the stable reattachment co-exists with the stable rollup (behind the downstream cylinder) of shear layers separating from the upstream cylinder, resulting in two distinct vortex-shedding frequencies even at the same Re and L/d. The St behaviour is further discussed along with flow visualization using the laser-induced fluorescence technique.     

Hydrodynamic forces on dual cylinders of different diameters in steady currents

Turbulent flow past two circular cylinders of different diameters is numerically investigated. The two-dimensional Reynolds-averaged Navier–Stokes equations are solved by using a finite element method with a k–ω turbulence closure. Following a relevant numerical model validation process, effects of cylinder gap-to-diameter ratio, the angular position of the smaller cylinder and the diameter ratio of cylinders on the vortex shedding and the forces on the cylinders are investigated using the numerical model. It is found that the relative position of the small cylinder has significant effects on the hydrodynamic force and vortex shedding characteristics of the cylinders.     

Flow around two tandem square cylinders near a plane wall

An experimental study has been conducted to investigate the flow around two identical square cylinders in tandem arrangement and placed near a plane wall at a Reynolds number of 6,300. The inter-cylinder spacing ratio was varied from S ^* = 0.5 to 6, and the cylinder-to-wall gap ratio from G ^* = 0.25 to 2. Totally, 42 cases were considered to systematically examine the effects of wall proximity and the mutual interference between the two cylinders in the normalized gap–spacing (G ^*–S ^*) plane. The flow fields were captured using digital particle image velocimetry, in conjunction with measurements of the fluid forces (drag and lift) acting on the downstream cylinder using a piezoelectric load cell. The results show that the flow is highly dependent on the combined values of G ^* and S ^*. Categories relating to G ^* could be broadly classified as small-gap regime (G ^* < 0.5) at which periodic vortex shedding from the cylinders is suppressed, intermediate-gap regime (0.5 < G ^* < 1) where vortex shedding occurs but is under the influence of the wall proximity, and large-gap regime (G ^* > 1) where the wall effects become negligible. Similarly, the flow interference between the two cylinders can be divided into three basic categories as a function of S ^*, namely, shielding regime at S ^* < 1, reattachment regime at 1 < S ^* < 3, and impinging regime at S ^* > 3. Variations of force coefficients, amplitude spectra, Strouhal numbers, and Reynolds shear stress with G ^* and S ^* are presented to characterize the different flow regimes.     

Unsteady characteristics of low-Re flow past two tandem cylinders

This study investigated the two-dimensional flow past two tandem circular or square cylinders at Re = 100 and D / d = 4–10, where D is the center-to-center distance and d is the cylinder diameter. Numerical simulation was performed to comparably study the effect of cylinder geometry and spacing on the aerodynamic characteristics, unsteady flow patterns, time-averaged flow characteristics and flow unsteadiness. We also provided the first global linear stability analysis and sensitivity analysis on the physical problem for the potential application of flow control. The objective of this work is to quantitatively identify the effect of the cylinder geometry and spacing on the characteristic quantities. Numerical results reveal that there is wake flow transition for both geometries depending on the spacing. The characteristic quantities, including the time-averaged and fluctuating streamwise velocity and pressure coefficient, are quite similar to that of the single cylinder case for the upstream cylinder, while an entirely different variation pattern is observed for the downstream cylinder. The global linear stability analysis shows that the spatial structure of perturbation is mainly observed in the wake of the downstream cylinder for small spacing, while moves upstream with reduced size and is also observed after the upstream cylinder for large spacing. The sensitivity analysis reflects that the temporal growth rate of perturbation is the most sensitive to the near-wake flow of downstream cylinder for small spacing and upstream cylinder for large spacing.     

Force coefficients and Strouhal numbers of three circular cylinders subjected to a cross-flow

In this paper, wind tunnel experiments were conducted to measure the mean force coefficients and Strouhal numbers for three circular cylinders of equal diameters in an equilateral-triangular arrangement when subjected to a cross-flow. These experiments were carried out at five subcritical Reynolds numbers ranging from 1.26 × 10⁴ to 6.08 × 10⁴. The pressure distributions on the surface of the cylinders were measured using pressure transducers. Furthermore, the hot-wire anemometer was employed to measure the vortex shedding frequencies behind each cylinder. Six spacing ratios (l/d) varying from 1.5 to 4 were investigated. It is observed that for l/d > 2, the upstream cylinder experiences a lower mean drag coefficient compared with the downstream cylinders. The minimum values of the drag coefficient for the downstream cylinders occur at l/d = 1.5 and l/d = 2, because there is no vortex shedding from the foregoing cylinders. Also, the value of the pressure coefficient behind the upstream cylinder reduces by increasing l/d. Moreover, by decreasing the value of l/d, the Strouhal number for the upstream cylinder increases. It can be concluded that the flow pattern and aerodynamic coefficients are basically dependent on l/d; in other words, decreasing l/d results in an increase in the effects of the flow interference between the cylinders.     

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     

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.     

Numerical computation of the flow around two square cylinders arranged side-by-side

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Vortex induced vibration and vortex shedding characteristics of two side-by-side circular cylinders of different diameters in close proximity in steady flow

This paper presents results of a numerical study of vortex-induced vibrations of two side-by-side circular cylinders of different diameters in steady incompressible flow. The two-dimensional Reynolds-averaged Navier–Stokes equations with a SST k–ω turbulence model are solved using the Petrov–Galerkin finite element method and the Arbitrary-Lagrangian–Eulerian scheme. The diameter ratio of the two cylinders is fixed at 0.1 and the mass ratio of both cylinders is 5.0. Both cylinders are constrained to oscillate in the transverse direction only. The Reynolds number based on the large cylinder diameter and free stream velocity is fixed at 5000. The effects of the reduced velocities of the cylinders on the vibration amplitude and vortex shedding regimes are investigated. It is found that for the range of parameters considered, collision between the two cylinders is dependent on the difference of the reduced velocities of the cylinders. Presence of the small cylinder in the proximity of the large one appears to have significant effects on the vortex shedding regime and vibration amplitude of the large cylinder.     

Detached eddy simulation of flow around two wall-mounted cubes in tandem

We investigate the performance of unsteady Reynolds-averaged Navier–Stokes (URANS) computation and various versions of detached eddy simulation (DES) in resolving coherent structures in turbulent flow around two cubes mounted in tandem on a flat plate at Reynolds number (Re) of 22,000 and for a thin incoming boundary layer. Calculations are carried out using four different coherent structure resolving turbulence models: (1) URANS with the Spalart–Allmaras model; (2) the standard DES [Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S.R., 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: Liu, C., Liu, Z., (Eds.), Advances in DNS/LES. Greyden Press, Columbus, OH]; (3) the Delayed DES (DDES); and (4) the DES with a low-Re modification (DES-LR) [Spalart, P., Deck, S., Shur, M., Squires, K., Strelets, M., Travin, A., 2006. A new version of detached eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn. 20 (3), 181–195]. The grid sensitivity of the computed solutions is examined by carrying out simulations on two successively refined grids. The computed results for all cases are compared with the experimental measurements of Martinuzzi and Havel [Martinuzzi, R., Havel, B., 2000. Turbulent flow around two interfering surface-mounted cubic obstacles in tandem arrangement. ASME J. Fluids Eng. 122, 24–31] for two different cube spacings. All turbulence models reproduce essentially identical separation of the approach thin boundary layer and yield an unsteady horseshoe vortex system consisting of multiple vortices in the leading edge region of the upstream cube. Significant discrepancies between the URANS and all DES solutions are observed, however, in other regions of interest such as the shear layers emanating from the cubes, the inter-cube gap and the downstream wake. Regardless of the grid refinement, URANS fails to capture key features of the mean flow, including the second horseshoe vortex in the upstream junction and recirculating flow on the top surface of the downstream cube for the large cube spacing, and underestimates significantly turbulence statistics in most regions of the flow for both cases. On the coarse mesh, all three DES approaches appear to yield very similar results and fail to reproduce the second horseshoe vortex. The standard DES and DDES solutions obtained on the fine meshes are essentially identical and both suffer from premature switching to unresolved DNS, due to the mis-interpretation of grid refinement as wall proximity, which leads to spurious vortices in the inter-cube region. Numerical solutions show that the low-Re modification (DES-LR) is critical prerequisite in DES on the ambiguously fine – not fine enough for full LES – mesh to prevent excessive nonlinear drop of the subgrid eddy viscosity in low cell-Re regions like in the inter-obstacle gap. Mean flow quantities and turbulence statistics obtained with DES-LR on the fine mesh are in good overall agreement with the measurements in most regions of interest for both cases.     

Numerical investigation on flow around circular cylinders in tandem arrangement at a subcritical Reynolds number

Three-dimensional fluid computations have been performed to investigate the flows around two circular cylinders in tandem arrangements at a subcritical Reynolds number, Re=2.2×10⁴. The center-to-center space between the cylinders was varied from twice the cylinder diameter to five times that, and the flows and fluid-dynamic forces obtained from the simulations are compared with the experimental results reported in the literature. Special attention is paid to the characteristics of the vortices shed from the upstream cylinder such as the convection, the impingement onto the downstream cylinder and the interaction with the vortices from the downstream cylinder. The effects of the vortices from the upstream cylinder on the fluid-dynamic forces acting on the downstream cylinder are discussed.     

Effect of acoustic resonance on the dynamic lift forces acting on two tandem cylinders in cross-flow

Direct measurements of the dynamic lift force acting on two tandem cylinders in cross-flow are performed in the presence and absence of acoustic resonance. The dynamic lift force is measured because it represents the integrated effect of the unsteady wake and therefore it is directly related to the dipole sound source generated by vortex shedding from the cylinder. Three spacing ratios inside the proximity interference region, L/D=1.75, 2.5 and 3 are considered. During the tests, the first transverse acoustic mode of the duct housing the cylinders is self-excited. In the absence of acoustic resonance, the measured dynamic lift coefficients agree with those reported in the literature. When the acoustic resonance is initiated, a drastic increase in the dynamic lift coefficient is observed, especially for the downstream cylinder. This can be associated with abrupt changes in the phase between the lift forces and the acoustic pressure. The dynamic lift forces on both cylinders are also decomposed into in-phase and out-of-phase components, with respect to the resonant sound pressure. The lift force components for the downstream cylinder are found to be dominant. Moreover, the out-of-phase component of the lift force on the downstream cylinder is found to become negative over two different ranges of flow velocity and to virtually vanish between these two ranges. Acoustic resonance of the first mode is therefore excited over two ranges of flow velocity separated by a non-resonant range near the velocity of frequency coincidence. It is therefore concluded that the occurrence of acoustic resonance is controlled by the out-of-phase lift component of the downstream cylinder, whereas the effect of the in-phase lift component is confined to causing small changes in the acoustic resonance frequency.