Analytical solution of oscillating inviscid flow over oblate spheroids with spheres and flat disks as special cases

The hydrodynamic forces acting on an oblate spheroid placed in an oscillating free stream are obtained for the special case of inviscid flow. The flow is assumed incompressible and axisymmetric and the free stream oscillations are harmonic. The flow direction is always along the spheroid's axis of symmetry. Analytical expressions are obtained for the potential and stream functions as well as the surface pressure distribution and the hydrodynamic force coefficient. The analysis is based on the solution of the unsteady equations of motion and continuity in oblate spheroidal coordinate system. The parameters involved are the major to minor axes ratio and the Strouhal number. The solutions for the two limiting cases of oscillating flows over disks and spheres can be easily obtained from the presented analytical solution.     

Oscillating viscous flow over an inclined elliptic cylinder

The problem of oscillating viscous flow over a stationary cylinder of elliptic cross-section is investigated. The free stream direction is horizontal and is always normal to the cylinder axis and oscillations are only allowed in the magnitude of the stream velocity. The flow is incompressible and two-dimensional and the free steam oscillations are harmonic. The elliptic cylinder considered has a major-minor axis ratio of 0.6 and is inclined to the free stream direction at angles of incidence of either 30 or 60°. The investigation is based on the solution of the time-dependent Navier-Stokes equations together with the mass conservation equation. The parameters involved are the cylinder axis ratio, angle of incidence, Reynolds number, and Strouhal number. The Reynolds number ranges between 10² and 10³ and the Strouhal number ranges between π/4 and π/2. The time variation of the flow field is presented in the form of streamline patterns as well as surface vorticity distribution. The surface pressure distribution and the time variation of the in-line and transverse force coefficients are also presented and compared with an inviscid flow solution for the same problem. The comparison between viscous and inviscid flow results shows a better agreement for higher values of Reynolds and Strouhal numbers.     

A model equation for the transverse forces on cylinders in oscillatory flows

A quasi-steady model is presented to predict the transverse force on cylinders in waves and oscillating flows. The model assumes that the Strouhal number, based on the instantaneous flow velocity, is constant, taking a value of 0.2. It is also assumed that the lift coefficient, based on the instantaneous dynamic pressure of the flow, is constant over a half cycle of the flow. The predictions of the model are compared with measurements taken on a circular cylinder in planar oscillatory flow over the Keulegan Carpenter number, KC, range from 5 to 53. The agreement between predicted and measured transverse forces is good at high KC but deteriorates at low KC. For high KC, it is shown that the model can be further improved if additional variables are introduced into the model equation.     

Reflection of surface waves by submerged cylinders

Reflection from submerged cylinders are studied by means of integral equations. By expressing the solution as a distribution of vortices, the integral equations become non-singular for closed contours. It is shown that the method gives a short and easy proof for the classical result that no reflection occurs for the circular cylinder. The reflection power for the elliptic contour and the flat plate are studied when the bodies are situated deeply below the surface.     

Forces on partly buried, tandem twin cylinders in waves at low Keulegan-Carpenter numbers

This study deals with the forces on the circular cylinder, laid on, or partly buried in the bed with a parallel twin dummy cylinder nearby and without it. They were determined by measuring the pressure distribution on the cylinder in the case of wave at low KC numbers. The forces on the cylinder were calculated by the integration of the measured pressures determined by pressure transducers on the surface of the cylinder. Force coefficients were obtained for the low KC numbers (KC<5), for the burial-depth-to-the diameter ratio = 0-0.7. The distance between axis of the measurement and dummy cylinders to diameter ratio (x/D) was 2, 1.5 and 1. The dummy cylinder was replaced downstream and upstream of the measurement cylinder.     

五圆柱体结构群绕流特性及互扰效应研究

基于Fluent流体计算平台,运用大涡模拟方法对亚临界雷诺数Re=3900下“X”形排列五圆柱体结构群三维绕流特性进行研究,主要分析来流攻角α与间距比L/D两个关键参数对五圆柱体结构群的尾流区三维涡结构演化与流体力系数的影响,并揭示其内在流动互扰机理。研究表明:来流攻角和间距比的变化对五圆柱体结构群流动控制及互扰效应的影响显著。在小间距比工况下,观察到柱体群间隙区域内流体高速流动的现象,导致五圆柱体之间的互扰作用十分强烈。间隙流对中间圆柱体和下游圆柱体有较强的冲击作用,对其表面的流体力分布特性有显著的影响。另外,大间距比工况下,当α=0°与L/D≥5.0工况时,柱体群尾流效应强于其间隙流效应。当α=22.5°与L/D=7.0时,位于下游与中间处的圆柱体流体绕流特性存在较大差异。而当α=45°与L/D≥6.0时,位于上游与中间处的圆柱体尾流区均会产生正负交替的漩涡结构。     

Interaction of linear waves with an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth

The linear water wave scattering and radiation by an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth is investigated by use of the multipole expansion method. The diffracted and radiated potentials are expressed as a linear combination of infinite multipoles placed at the centre of each cylinder with unknown coefficients to be determined by the cylinder boundary conditions. Analytical expressions for wave forces, hydrodynamic coefficients, reflection and transmission coefficients and energies are derived. Comparisons are made between the present analytical results and those obtained by the boundary element method, and some examples are presented to illustrate the hydrodynamic behavior of multiple horizontal circular cylinders in a two-layer fluid. It is found that for two submerged circular cylinders the influence of the fluid density ratio on internal-mode wave forces is more appreciable than surface-mode wave forces, and the periodic oscillations of hydrodynamic results occur with the increase of the distance between two cylinders; for four submerged circular cylinders the influence of adding two cylinders on the wave forces of the former cylinders is small in low and high wave frequencies, but the influence is appreciable in intermediate wave frequencies.     

Scattering of surface waves by submerged cylinders

A train of surface waves is normally incident upon a cylinder that is totally submerged in a body of deep water. Details are given for the cases of circular or elliptic cross-sections, with estimates for the transmission and reflection constants when the cylinder is many wavelengths below the surface. Corresponding results are suggested for arbitrary smooth cylinders.     

基于粘性流模型的筒型基础防波堤波浪力数值分析

筒型基础防波堤是一种新型港口海岸工程结构,其基础上部是由连续排列的圆筒构成的直立防浪墙.采用粘性流数值模型,研究连续圆筒防波堤上波浪力竖向分布、水平(沿圆筒环向)分布和波浪力合力特性,并对粘性流数值模型计算的平面直墙波浪力与海港水文规范方法计算结果;粘性流数值模型计算的连续圆筒墙面波浪力与平面直墙波浪力;无限长连续圆筒墙面波浪力与有限长连续圆筒墙面波浪力进行比较分析.针对所选工程算例,建议按<海港水文规范>中平面直墙波浪力计算方法确定连续圆筒防波堤上的波浪力时,波峰时考虑0.90左右的折减系数,波谷时考虑0.95左右的折减系数.     

Wave interaction with multiple horizontal cylinders

The interaction of waves with submerged two-dimensional circular cylinder groups is investigated. Linear wave theory is used in the analysis and the viscous effects are neglected. The boundary value problem for the wave potential is based on Green's theorem, and the resulting integral equation is solved numerically. The added mass and damping coefficients in sway, heave and roll of oscillating twin-cylinders and the total wave forces on a fixed cylinder in a group are examined. The effects of the free surface, and particularly, the spacing of the cylinders are shown. The numerical results are tested against known solutions. Results for a two-cylinder configuration at different orientations are presented. It is found that the interaction between closely spaced cylinders is large. The spacings at which the interaction effect is important are shown graphically.     

Vibration of two elastically mounted cylinders of different diameters in oscillatory flow

Vibration of two elastically mounted cylinders in an oscillatory flow at a Keulegan-Carpenter number of 10 is simulated numerically. The two cylinders are rigidly connected with each other and are allowed to vibrate in the cross-flow direction only. The aim of this paper is to identify the effects of the orientation of the cylinders and the gap between the cylinders on the vibration. The two-dimensional Reynolds-Averaged Navier-Stokes equations are solved to predict the flow and the cylinder vibration is predicted using the equation of motion. When the two cylinders are in a tandem arrangement, a combined single pair flow regime and attached pair flow regime are observed as reduced velocity exceeds 10 and this combined regime and the single pair regime occurs intermittently. Periodic vibration is found when the two cylinders are in a staggered arrangement with a 45° flow attack angle. When the two cylinders are in a side-by-side arrangement, a new single vortex regime is observed. This single vortex remains attached to the cylinder surface and rotates around the cylinder. The intermittent switch between this single vortex regime and the single pair regime are observed.     

Clear water scour around a finite array of cylinders

This experimental study presents clear-water scour and deposition patterns around hexagonal arrays of circular cylinders in steady flow conditions. Understanding the scour processes around such configurations could facilitate the design of several hydraulic and marine engineering structures, such as bridge piers and piles. The flow alteration caused by the examined porous obstacles depends on the solid volume fraction of the obstacles and on the angle of attack of the incoming flow, due to the limited number of cylinders constituting the array. Flume experiments with erodible bed were carried out for four array densities (solid volume fractions: 0.14, 0.20, 0.32 and 0.56) under three different orientations (regular, angled and staggered configurations). The scour/deposition characteristics were obtained by means of laser scanner and the results were compared to solid cylinders of equal circumambient diameter. Different angles of attack of the incoming flow lead to different blockage ratios, which have direct impact on the scour characteristics and deposition patterns. The arrays with the higher solid volume fraction generated scour/deposition patterns similar to solid cylinder, while in the arrays with the lower solid volume fractions, local scour around the individual small cylinders became evident. Finally, considering that the load bearing capacity of a pier basically depends on the area of its cross-section, a comparison of the maximum induced scour depth and volume by the cylinder arrays and the solid cylinder with equal solid cross-sectional area is presented, in order to introduce an alternative pier configuration that induces less scour. The results showed that the array of cylinders could generate 27% less scour volume and 22% less scour depth compared to its single solid cylinder counterpart.     

In-line force on a cylinder translating in oscillatory flow

Experiments were conducted with smooth and sand-roughened cylinders moving with constant velocity in a sinusoidally oscillating flow to determine the drag and inertia coefficients and to examine the effect of wake biasing on the modified Morison equation. The various flow parameters such as the relative cylinder velocity. Reynolds number, and the Keulegan-Carpenter number were varied systematically and the in-line force measured simultaneously. The principal results, equally valid for both smooth and rough cylinders, are as follows: the drag coefficient decreases with increasing relative current for a given Reynolds number and Keulegan-Carpenter number; the effect of wake biasing on the drag and inertia coefficients is most pronounced in the drag-inertia dominated regime; and the two-term Morison equation with force coefficients obtained under no-current conditions is not applicable to the prediction of wave and current induced loads on circular cylinders.     

Flow around circular and noncircular cylinders by a low-order panel method

Time-dependent cross-flow was studied around cylinders with circular and noncircular cross-sections. The numerical approach for the analysis was a low-order panel method based on constant source and dipole values along each panel. The method was previously used successfully for several applications, such as calculation of the added mass and damping coefficients. In simulating the viscous time-dependent flow around the cylinder, the time-dependent wake feature of the code was used. For the circular and D-cylinders, the results agreed well with the experiments. Suggestions for improving the results for T-cylinders with angle of attack are included.     

On splitter plate coverage for suppression of vortex-induced vibrations of flexible cylinders

Experimental results show how vortex-induced vibration (VIV) amplitudes of flexible cylinders can be reduced up to a 90% by covering less than half of the length of the cylinder with splitter plates elastically mounted to the surface of the cylinder.The VIV amplitude reduction takes place along with drag coefficient reductions of up to a 50% for the reduced velocities investigated.     

Water wave interaction with an array of bottom-mounted surface-piercing porous cylinders

The interaction of water waves with arrays of bottom-mounted, surface-piercing circular cylinders is investigated theoretically. The sidewall of each cylinder is porous and thin. Under the assumptions of potential flow and linear wave theory, a semi-analytical solution is obtained by an eigenfunction expansion approach first proposed for impermeable cylinders by Spring and Monkmeyer (1974), and later simplified by Linton and Evans (1990). Analytical expressions are developed for the wave motion in the exterior and all interior fluid regions. Numerical results are presented which illustrate the effects of various wave and structural parameters on the hydrodynamic loads and the diffracted wave field. It is found that the porosity of the structures may result in a significant reduction in both the hydrodynamic loads experienced by the cylinders and the associated wave runup.     

Bilge keel loads and hull pressures created by bilge keels fitted to a rotating cylinder

This paper presents bilge keel loads and hull pressure measurements carried out on a rotating cylinder in a free surface water basin. A flat plate bilge keel and one more complex shaped bilge keel were studied to investigate the geometry effect. The draft of the cylinder was varied to study the effect of the vicinity of the free surface on the bilge keel loads and hull pressures. The rotation axis of the cylinder was fixed to define a pure roll experiment (one degree of freedom).The cylinder was subject to forced oscillations of varying amplitude leading to a KC range of 0.3–16. Using Fourier analysis the first three harmonic coefficients representing the normal bilge keel load were derived. The first harmonic drag and inertia coefficients are in good agreement to existing experimental data obtained for wall bounded flat plates fitted in a U-shaped water tunnel as reported by Sarpkaya and O’Keefe (1996). New insight is gained by the fact that the addition of higher harmonic contributions is essential to capture the time varying bilge keel normal force.The pressure measurements next to the bilge keel are compared to measurements reported by Ikeda et al. (1979). Similar findings are obtained, showing that the pressure on the hull in front of the moving bilge keel is KC independent while the vortex system in the wake of the bilge keel leads to KC dependent hull pressure distributions. The hull pressure jump over the bilge keel correlates well to the force coefficient on the bilge keel. The complex nature of the vortex induced hull pressures is manifested. The empirically derived hull pressure distribution by Ikeda et al. (1979) for the time instant of maximum velocity is shown to correlate reasonably well to the measured data with some conservatism in the absolute value.Although a cylinder is very different from a ship-shaped section, the experiments provide essential insight into the physics associated with roll damping and into the factors that should be included in a roll damping prediction method.     

Pressures and forces on inclined cylinders due to regular waves

The variation of the dynamic pressures around a circular cylinder due to regular waves is studied in a wave basin of constant water depth of 3 m. The measuring segment consisted of 12 pressure transducers placed at an elevation of 0.8 m below the still water level. The tests were conducted with the cylinder axis inclined with respect to the vertical plane along and against wave direction. The results on the variation of dimensionless pressures with the non-dimensional input wave parameters are reported for different angles of orientation of the cylinder. The sectional normal force obtained by integrating the pressures is also presented as a function of wave steepness and the effect of angle of orientation of the cylinder is also reported.     

Flow induced vibrations of a sharp edge square cylinder in the wake of a circular cylinder

The response of an oscillating circular cylinder at the wake of an upstream fixed circular cylinder was classified by different researchers as galloping, wake induced galloping or wake induced vibration. Furthermore it is already known that a sharp edge square cylinder would undergo galloping if it is subjected to uniform flow. In this study the influence of the wake of a fixed circular cylinder on the response of a downstream square cylinder at different spacing ratios (S/D = 4, 8, 11) is experimentally investigated. The subject appears not to have received previous attention. The lateral displacements, lift forces and the pressure data from gauges mounted in the wake of the oscillating cylinder are recorded and analyzed. The single degree of freedom vibrating system has a low mass-damping parameter and the Reynolds number ranges from 7.7 × 10² to 3.7 × 10⁴.In contrast to that for two circular cylinders in tandem arrangement, the freely mounted downstream square cylinder displays a VIV type of response at all spacing ratios tested. There is no sign of galloping or wake induced galloping with the square cylinder. With increase at the spacing ratio the cross-flow oscillations decrease. It is shown that the vortices arriving from the upstream fixed circular cylinder play a major role on the shedding mechanism behind the downstream square cylinder and cause the square cylinder to shed vortices with frequencies above Strouhal frequency of the fixed square cylinder (St = 0.13). The VIV type of oscillations in the downstream square cylinder is most probably caused by the vortices newly generated behind the square cylinder.     

Three dimensional wave scattering by arrays of elliptical and circular cylinders

Recently, Chatjigeorgiou and Mavrakos (2009, 2010a) provided an analytic solution for the three dimensional wave scattering by arrays of elliptical cylinders. The present paper extends the contents of the existing study to tackle the problem of the hydrodynamic interactions between elliptical and circular cylinders. The main task is to derive an analytic solution for the total velocity potential for an arbitrary body of the array and accordingly, to express the hydrodynamic pressure, the exciting forces and the wave elevation in compact analytic closed-forms.The solution method is rather complicated as it considers the circular cylinders as different geometries and not special cases of elliptical cylinders with zero elliptic eccentricity. Nevertheless, the adopted procedure enhances the mathematical reconstruction of the physical subject as it requires the derivation and the employment of addition theorems that transform expressions from elliptic to polar coordinate systems in all four possible combinations.