Vertical Circular Cylinder Research Articles

Numerical simulation of solitary wave scattering by a circular cylinder array

The interaction of a solitary wave with an array of surface-piercing vertical circular cylinders is investigated numerically. The wave motion is modeled by a set of generalized Boussinesq equations. The governing equations are discretized using a finite element method. The numerical model is validated against the experimental data of solitary wave reflection from a vertical wall and solitary wave scattering by a vertical circular cylinder respectively. The predicted wave surface elevation and the wave forces on the cylinder agree well with the experimental data. The numerical model is then employed to study solitary wave scattering by arrays of two circular cylinders and four circular cylinders respectively. The effect of wave direction on the wave forces and the wave runup on the cylinders is quantified.

Second Order Diffraction of Water Waves by a Bottom Mounted Vertical Circular Cylinder and Some Related Numerical Problems

A Hankel transformation is used to obtain the second order diffraction solution of vertical cylinder of circular cross section. The improper integral over the free surface is tackled carefully. The singularity at the free surface is overcome effectively using a third order nonlinear transformation. Numerical results for free surface elevations compare well with the published data.

Modelling linear wave transformation induced by dissipative structures—Random waves

The relevant theory is presented and numerical results are compared with the analytical solution for the interaction of non-breaking waves with an array of vertical porous circular cylinders on a horizontal bed. The extension to the cases of unidirectional and multidirectional waves is obtained by means of a transfer function. The influence of the mechanical properties of porous structures and wave irregularity on wave transformation is analysed. Results for unidirectional and multidirectional wave spectra are compared to those obtained for regular waves. The model presented reproduces well the analytical results and provides a tool for analysing several engineering problems.

垂直な中心線に対称なモデル物体による重力の鉛直勾配計算式と等値線図の主な特徴

We can find many symmetrical gravity anomalies such as center point symmetry in the gravity map series (1 : 200,000) published by Geological survey of Japan, AIST. To analyze these gravity anomalies, the authors reviewed the computing equations of vertical gradient of gravity caused by the model bodies with center point symmetry (vertical solid circular cylinder, circular cone, circular parabola, ellipsoid). Some characteristics are described in the maps drawn by these computing equations.

Natural convection in a stably stratified fluid along vertical plates and cylinders with temporally periodic surface temperature variations

This paper describes one-dimensional (parallel) laminar natural convection in a viscous stably stratified fluid owing to temporally periodic variations in the surface temperature of infinite vertical plates and circular cylinders. Analytical solutions of the one-dimensional (parallel) Boussinesq equations of motion and thermodynamic energy are obtained for the periodic regime for arbitrary values of ambient stratification, Prandtl number and forcing frequency. The solutions for plates and cylinders are qualitatively similar and show that (i) the flows are composed of two waves that decay exponentially with distance from the surface: a fast long wave and a slow short wave; (ii) for forcing frequencies greater than the natural frequency of the corresponding inviscid system, these two waves propagate away from the surface; and (iii) for forcing frequencies less than this natural frequency, the short wave propagates away from the surface while the long wave propagates toward the surface. This latter case provides an example of a flow for which the conventional radiation condition is not appropriate. The analytical results are complemented, for the plate problem, with three-dimensional numerical simulations of flows that start from rest and are suddenly subjected to a periodic thermal forcing at the plate. The numerical results depict the transient (start-up) stage of the flow and the approach to a periodic regime. These results confirm that the analytical solutions provide the appropriate description of the periodic regime.

Nonlinear particle kinematics of ocean waves

A fundamental relation is derived governing the Lagrangian kinematics of fluid particles on the surface of nonlinear ocean waves which may be known only stochastically. The horizontal trajectories of fluid particles on the free surface are shown to obey a pair of coupled nonlinear Ricatti-type ordinary differential equations driven by the temporal and spatial gradients of the free-surface elevation defined relative to an Eulerian frame. This equation is explicit in that it does not require the solution of a fully nonlinear potential flow free-surface problem and may be viewed as a deterministic or stochastic equation depending on the interpretation of the definition of the free-surface elevation. It is free of empirical corrections often used to estimate the particle kinematics above the calm water surface, is valid in potential flow and for waves of large steepness in two and three dimensions and in waters of all depths and may be used for the evaluation of the extreme unsteady loads exerted on surface piercing vertical circular cylinders by steep random waves.

Fatigue damage assessments and the influence of wave directionality

A fatigue assessment is a complex and time-consuming process with an uncertain outcome. The result is sensitive to comparatively small differences in the structure (globally or locally) or the loading environment. The predominant cause of fatigue damage for marine structures is usually wave action. It is then often asked what the beneficial effect of wave directionality is. There are two sources of wave directionality, i.e. firstly, the variable direction of uni-directional seas, and secondly, wave spreading around a given mean direction. These two effects need to be distinguished and considered separately. The influence of wave directionality is not easily predictable and its quantification normally requires extensive and detailed analysis by numerical calculations. However, under two general restrictions the influence can be rather easily determined, i.e. for a rotationally symmetric structure (a vertical circular cylinder) and for wave conditions that are of the same severity in all directions. The paper investigates this special problem and presents quantitative results for various cases.

Suppression of vortex shedding of circular cylinder in shallow water by a splitter plate

The flow behavior around a vertical circular cylinder placed in shallow water was controlled by a splitter plate inserted at various locations downstream of the cylinder. The splitter plate has a fixed length which is equal to the diameter of the cylinder, D = 50 mm. The effect of plate thickness, T , was examined using three plates with different thicknesses, T / D = 0.016 , 0.04 and 0.08. The gap between the base of the cylinder and the leading edge of the splitter plate was increased successively from 0 to 100 mm with 12.5 mm increments. Particle Image Velocimetry (PIV) was used to measure the instantaneous velocity vector field in the wake region at Reynolds number Re = 5000 based on the cylinder diameter. The mean velocity vector field, corresponding vorticity field and streamline topology and turbulence statistics were obtained using 200 instantaneous images. The water height was kept constant at 25 mm during the experiments. The splitter plate has a substantial effect on the suppression of the vortex shedding for the gap ratio ( G / D ) between 0 and 1.75 D . For G / D = 1.75 , the normalized Reynolds stress has a peak value of 0.0158, which is approximately 8.5 times smaller than the concentrations occurring for the bare cylinder. When the splitter plate was located at 2 D location, no effect of the plate was observed.

Numerical implementation of the harmonic modified mild-slope equation

A numerical solver is presented of the modified time-independent mild-slope equation, which incorporates energy dissipation. Using a second-order parabolic approximation, the following external boundary conditions are modelled: open and fully transmitting to both incoming and outgoing waves; partially reflecting, and; fully absorbing. Discretisation of the governing equation and boundary conditions is by means of a second-order accurate central difference scheme. The resulting sparse-banded matrix is solved using an inexpensive banded solver with Gaussian elimination. The numerical predictions are in excellent agreement with the analytical solution for the interaction of non-breaking waves with an array of vertical surface-piercing circular cylinders on a horizontal bed. Results are compared with those for the same array on various seabed topographies. The model is robust and can be used for wave propagation in complex geometries. It has fewer restrictions associated with wave obliqueness at boundaries than traditional models based on the mild-slope equation.

Simulation of nonlinear interactions between waves and floating bodies through a finite-element-based numerical tank

Fully nonlinear water–wave interactions with a floating structure are investigated through a numerical towing tank. A wave maker is installed on one end of the tank while a numerical beach based on a combination of damping zone and Sommerfeld condition is adopted on the other side of the tank. A floating body is placed at a location in the tank, where it will be set into motion by the waves generated by the wave maker. The simulation is based on the velocity potential theory together with the finite–element method. The mesh used follows the deformation of the free surface and the body motion. Its structure is adjusted and the distribution of elements is completely rearranged when the motion is big to avoid an over–distorted grid. Auxiliary functions are introduced to decouple the nonlinear mutual dependence between the hydrodynamic force and the body motion. Extensive numerical results are provided for vertical circular cylinders and a simplified floating production, storage and offloading, for which meshes are obtained through an efficient scheme based on a two–dimensional tri–tree method.

On diffraction and radiation problem for a cylinder over a caisson in water of finite depth

A caisson and a vertical circular cylinder floating on the surface of fluid compose a wave power device. The hydrodynamic fluid/solid interaction becomes more complex due to the caisson. In this paper, in order to investigate the effects of the caisson on the cylinder's hydrodynamic coefficients and exciting forces, the expressions for velocity potentials are derived in the presence of an incident linear wave by use of an eigenfunction expansion approach. A set of theoretical added mass, damping coefficient and exciting force expressions are obtained. Two approaches are proposed to calculate the exciting forces and they match with each other well. Finally, Analytical results of added masses, damping coefficients and exciting forces are given for the different sizes of the caisson.

Steady flow past a vertical surface-piercing circular cylinder

This paper describes experiments in which a vertical surface-piercing circular cylinder with a large draught was towed at steady speeds through water initially at rest. The cylinder diameter d was 210 mm, and measurements were made of pressures around its circumference at elevations between 2.4 d below still water level to 0.7 d above, at Froude numbers (based on d) up to 1.67. The tests were carried out at a constant ratio of Reynolds number to Froude number of 2.79×10 5. The total resistance coefficient reached a maximum at a Froude number of about 1, when that part of the loading that can be attributed to the presence of the free surface was equivalent to the submerged form drag on a length of cylinder of about 0.9 d. Measurements are also presented of the run-up on the front of the cylinder and of the depth of the depression at the back. Previous measurements by Hay (Flow about Semi-submerged Cylinders of Finite Length. Princeton University Report, Princeton, NJ, 1947) for the case of a cylinder with a submerged free end, and by Hsieh (Proc. Am. Soc. Civil Eng. 90 (1964) 161) of forces on cylinders standing on the floor of an open channel, are reanalysed. In most respects these results are found to be compatible with the present data for a cylinder of large draught.

Anomalous sedimentation of a small Brownian sphere in a vertical circular cylinder of periodically varying radius

The mean velocity of a Brownian sphere sedimenting under the influence of gravity through a viscous fluid confined within a vertical circular cylinder whose radius changes periodically along its effectively infinite length is investigated for circumstances where, owing to the small radius of the sphere relative to the minimum cylinder radius, conventional hydrodynamic wall effects are absent. Using generalized Taylor dispersion theory, explicit results are presented for the case where the periodic changes in cylinder radius occur abruptly, representing the computationally and conceptually simplest configuration. Despite the apparent absence of wall effects, the (mean) settling velocity of the sphere is always less than its Stokes law value, and depends nonlinearly upon the gravity force. Moreover, the sphere’s dispersivity fails to obey a Stokes–Einstein relation. This anomalous behavior may be exploited to enhance the force-driven separation of physicochemically disparate Brownian particles.

On scattering and radiation problem for a cylinder in water of finite depth

Wave loadings due to scattering and radiation for a floating vertical circular cylinder in water of finite depth are derived. These are derived from the total velocity potential which can be decomposed as four velocity potentials; one due to scattering in the presence of an incident wave on fixed structure (diffraction problem), and the other three due to radiation respectively by surge, heave and pitch motion on calm water (radiation problem). For each case, the velocity potential is derived by considering two regions, namely, interior region and exterior region. The complex matrix equations can be solved numerically to determine the unknown coefficients to compute the wave loads. Numerical results can be obtained for different depth to radius and draft to radius ratios.

On four highly nonlinear phenomena in wave theory and marine hydrodynamics

Some recent developments in the formation of extreme waves, kinematics of steep waves, the phenomenon of ringing and currents in the ocean induced by internal waves are reviewed. Formation of extreme waves are simulated by means of a rapid fully nonlinear model. A large wave event taking place in a wave group is characterized by an elevation being significantly larger than the initial amplitude of the group. Recurrence occurs. PIV measurements of Stokes waves exhibit an exponential velocity profile all the way up to the surface elevation (wave slope up to 0.16). The computed velocity profile under crest of an extreme wave corresponds also to an exponential profile. Experimental results of the horizontal force on a vertical circular cylinder in long and steep waves exhibit a secondary cycle of high frequency in the force history. This typically occurs for waves longer than about 10 times the cylinder diameter and a Froude number ωη m / gD larger than about 0.4, ω the wave frequency, η m the maximal elevation, g the acceleration of gravity, D the cylinder diameter. Properties of internal solitons and the induced fluid velocities are described in terms of weakly and fully nonlinear models supported by PIV measurements. A rapid scheme for fully nonlinear interfacial waves in three dimensions is derived, complementing the rapid model of free surface waves.

Higher-harmonic wave forces and ringing of vertical cylinders

A pronounced secondary load cycle may occur in the force on a vertical circular cylinder exposed to moderately steep transient water wave events. The timing is about one quarter wave period later than the main peak of the force. The load cycle contributes locally to a pronounced higher harmonic wave force. Experimental observations of the phenomenon in small and moderate scale are characterized in terms of the local wave slope of the incoming individual waves, kη m ( k the wavenumber, η m the local maximal wave elevation), and the wavenumber relative to the cylinder radius, kR ( R the cylinder radius). The height of the secondary load cycle is typically up to about 0.1–0.15 times the peak to peak force on the cylinder during a wave event. In the small scale experiments the secondary load cycle occurs when 0.3< kη m and kR<0.33 (registrations are made for 0.1< kR). In the experiments of moderate scale the secondary load cycle occurs for a smaller wave slope than observed in the small scale experiments. This is true in the long wave regime. The secondary load cycle is then observed for a nondimensional wavenumber kR in the range 0.1–0.33 and a Froude number (Fr=ωη m / gD ; ω, the wave frequency; g, the acceleration of gravity; D=2 R) exceeding about 0.4. The important difference between the experiments in moderate and small scale is explained by the effect of flow separation. This effect is weaker in moderate than in small scale. Registrations of pronounced resonant high-frequency ‘ringing’ motions of vertical cylinders and models of offshore structures of moderate scale are compared with the force recordings. Large motion occurs for the same wave parameters as for the pronounced secondary load cycle in the moderate scale experiments. The additional load cycle contributes to a build-up of resonant body responses if the resonance frequency is close to about four times the wave frequency.

Gravity flow of granular materials in contracting cylinders and tapered tubes

This paper presents two applications of the double-shearing theory for flow of granular materials under gravitational forces for axially symmetric flow. In the first the material is contained in a vertical circular cylinder which compresses the material by contracting radially. Exact solutions for the stress and velocity fields are derived, under the boundary conditions that the cylinder wall is either `perfectly rough' or subject to Coulomb friction. In the second problem the material flows under gravity through a tube with circular cross-section of radius that decreases slowly with depth. For this problem an approximate solution is derived that is accurate to first order in the slope of the tube wall relative to the vertical. It is also shown that an alternative theory, in which it is postulated that the principal axes of the stress and rate-of-strain tensors are coincident, leads to the prediction of unacceptable singularities in the flow field in the interior of the material.

Wave Generation in Open Channels by Vortex Shedding from Channel Obstructions

Transverse surface waves were produced in a rectangular open channel in the region where the otherwise steady, uniform, flow passed through a cluster of vertical circular cylinders. The cylinders could represent either naturally occurring obstacles such as vegetation or man-made structures such as a group of piles driven into the bed of a river. The waves were produced from the periodic forces created by the vortex shedding from the cylinders. These forces generated and then amplified transverse waves in the channel. In some experiments the waves had amplitudes of 35% of the mean flow depth. These resonance-generated waves produce a seiching that can occur in hydraulic models as well as prototype systems. Both laboratory experiments and a theory-based analysis were used to determine the relationship between the amplitude of the vortex-generated wave, average velocity of the approach flow, viscosity of the fluid, width of the rectangular channel, depth of flow, cylinder diameter, and cylinder placement configuration.

Gravitational attraction of solids of revolution: Part 1: Vertical circular cylinder with radial variation of density

Integral expressions are derived for the vertical component of gravitational attraction arising from vertical circular cylinders and horizontal circular disks with radial variation of density. The expressions contain a single elliptic integral of either the first or second kind and are evaluated easily by numerical integration. Various closed-form solutions are obtained for special cases. For constant density, the expressions reduce to well-known solutions as well as approximations that are based on the solid angle and line element, thus providing a unified theme for the derivations of those solutions and approximations. Numerical implementation is discussed in terms of practical application.

Wave Diffraction by a Vertical Cylinder with a Porous Ring Plate

Linearized potential wave theory is applied to investigate the phenomenon of wave diffraction by a vertical circular cylinder with a thin ring plate in water of finite depth. By means of the eigenfunction expansion method, harmonic expressions for the velocity potential are obtained. The numerical results for the wave loads and the wave height surrounding the body are discussed. It is found that the presence of a thin ring plate causes the focusing of wave energy near the rear edge of the cylinder for small dimensionless porous-effect parameters. A porous ring plate behaves as a wave absorber, which leads to a decrease of the wave height until setdown occurs. The ring plate can decrease not only the horizontal wave force but also the moment on the cylinder. In general, the mechanism of decreasing the wave loads on the cylinder by a porous ring plate is different from that by an impermeable ring plate. An impermeable ring plate attached to the cylinder causes wave focusing near the rear edge of the cylinder so that the difference between wave heights at the front edge and at the rear edge of the cylinder becomes small; hence the wave loads on the cylinder are decreased. A porous ring plate behaves as a wave absorber, which decreases the wave height at the front edge of the cylinder, and thus leads to a decrease of wave loads on the cylinder.