An experimental investigation of hydrodynamic coefficients for a vertical truncated rectangular cylinder due to regular and random waves

The research into hydrodynamic loading on ocean structures has concentrated mostly on circular cross-section members and relatively limited work has been carried out on wave loading on other cross-sections such as rectangular sections. These find applications in many offshore structures as columns and pontoons in semi-submersibles and tension-leg platforms. The present investigation demonstrates the behaviour of rectangular cylinders subject to wave loading and also supplies the hydrodynamic coefficients for the design of these sections.This paper presents the results of wave forces acting on a surface piercing truncated rectangular cylinder set vertically in a towing tank. The experiments are carried out in a water depth of 2.2 m with regular and random waves for low Keulegan–Carpenter number up to 6. The rectangular cylinder is of 2 m length, 0.2 m breadth and 0.4 m width with a submergence depth of 1.45 m from still water level. Based on Morison equation, the relationship between inertia and drag coefficients are evaluated and are presented as a function of KC number for various values of frequency parameter β, for two aspect ratios of cylinders, equals to 1/2 and 2/1. Drag and inertia coefficients obtained through regular wave tests are used for the random wave analysis to compute the in-line force spectrum.The results of the experiments show the drag and inertia coefficients are strongly affected by the variation in the aspect ratios of the cylinder. The drag coefficients decreases and inertia coefficients increases with increase in Keulegan–Carpenter number up to the range of KC number tested. The random wave results show a good correlation between measured and computed force spectrums. The transverse forces in both regular and random waves are found to be small compared to in-line forces.     

Hydrodynamic characteristics of a cylindrical bottom-pivoted wave energy absorber

A parametric study was carried out to investigate the hydrodynamics of a cylindrical wave energy absorber. Established methods of hydrodynamic analysis were applied to the case of a damped vertically oriented cylinder pivoted near the sea floor in intermediate depth water. The simple geometry provides a canonical reference for more complex structure shapes and configurations that may be considered for either wave energy conversion or wave energy absorption. The study makes use of the relative velocity Morison equation, with force coefficients derived from radiation and diffraction theory. Viscous effects were accounted for by including a drag term with an empirically derived coefficient, C_D. A non-linear first-order formulation was used to calculate the cylinder motion response in regular waves. It was found that the non-linear drag term, which is often neglected in studies on wave energy conversion, has a large effect on performance. Results from the study suggest a set of design criteria based on Keulegan–Carpenter (KC) number, ratio of cylinder radius to water depth (a/h), and ratio of water depth to wavelength (h/L). Respectively, these parameters account for viscous, wave radiation, and water depth effects, and optimal ranges are provided.     

Wave force coefficients for horizontally submerged rectangular cylinders

The results of wave force measurements carried out on a section of horizontally submerged rectangular cylinders, which are used as pontoons in many offshore structures, are reported in this paper. Two rectangular cylinders with aspect (depth–breadth) ratios equal to $\frac{1}{2}$ and $\frac{3}{4}$ and a square section (aspect ratio=1.0) cylinder are chosen for this study. Experiments are carried out in a wave tank at a water depth of 2.2 m at low Keulegan–Carpenter (KC) numbers to measure the horizontal and vertical wave forces acting on a 100 mm section, located at mid-length of the cylinders. For each cylinder, tests are carried out for two relative depths of submergence of 2.68 and 4.68. Measured wave forces in regular and irregular waves are then used to derive drag (C_D) and inertia coefficients (C_M). The analysis show that at very low KC numbers the inertia coefficients for all cylinders approached the potential flow values for both horizontal and vertical forces. The drag coefficients at low KC numbers exhibited large values and they decreased sharply with increase in KC number. For the square cylinder, where relatively a large KC number is obtained compared to other cylinders, inertia coefficients reached minimum values in the range of KC of about 3–4 and increased thereafter. In this range, C_M values are about 50% or so, smaller than the same at KC close to zero. The results of the experiments reveal that aspect ratio has large influence on hydrodynamic coefficients.     

Wave propagation through dense vertical cylinder arrays: Interference process and specific surface effects on damping

The purpose of this research work is to study the effect of specific surface s, the fluid–solid contact surface per volume unit, on the wave energy dissipation by porous structures consisting in dense arrays of emergent vertical cylinders. Experiments have been carried out in a 10 m long wave flume. Three cylinder diameters D are considered in order to study the effects of the specific surface while keeping the porosity constant. In a first series, the length of the porous zone is kept constant for the three cylinder diameters tested. The measurements, which include various wave steepness conditions, demonstrate the role of specific surface s on both wave attenuation and interference processes. The larger the specific surface is, the stronger the wave damping is. Damping is found to be almost proportional to 1/D when laminar, turbulent and inertial effects are of same order. Results are compared to numerical calculations based on either a constant rate of wave damping within the porous medium per unit wavelength or a quadratic damping developed using a force expression based on the work of [26]. This latter model, calibrated with drag and inertia coefficients, shows a good agreement with measurements. In a second series, both porous length and water depth are kept proportional to the cylinder diameter for the three diameters. Scale effects are then discussed and underline the importance of the flow regime within the porous medium.     

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.     

An analytical approach for the calculation of random wave forces on submerged tunnels

This paper deals with the random forces produced by high ocean waves on submerged horizontal circular cylinders. Arena [Arena F, Interaction between long-crested random waves and a submerged horizontal cylinder. Phys Fluids 2006;18(7):1–9 (paper 076602)] obtained the analytical solution of the random wave field for two dimensional waves by extending the classical Ogilvie solution [Ogilvie TF, First- and second-order forces on a cylinder submerged under a free surface. J Fluid Mech 1963;16:451–472; Arena F, Note on a paper by Ogilvie: The interaction between waves and a submerged horizontal cylinder. J Fluid Mech 1999;394:355–356] to the case of random waves. In this paper, the wave force acting on the cylinder is investigated and the Froude Krylov force [Sarpkaya T, Isaacson M, Mechanics of wave forces on offshore structures, Van Nostrand Reinhold Co.; 1981], on the ideal water cylinder, is calculated from the random incident wave field. Both forces represent a Gaussian random process of time. The diffraction coefficient of the wave force is obtained as quotient between the standard deviations of the force on the solid cylinder and of the Froude Krylov force. It is found that the diffraction coefficient of the horizontal force C_do is equal to the C_dv of the vertical force. Finally, it is shown that, since a very large wave force occurs on the cylinder, it may be calculated, in time domain, starting from the Froude Krylov force. It is then shown that this result is due to the fact that the frequency spectrum of the force acting on the cylinder is nearly identical to that of the Froude–Krylov force.     

Inertial wave loads on horizontal cylinders: A field experiment

A long submerged horizontal circular cylinder of .90 m diameter was assembled off the beach at Reggio Calabria where the wind waves typically have significant height ranging within 0.20 and 0.40 m and dominant period within 1.8 and 2.6 s. Three ultrasonic probes recorded the waves, and two sets of pressure transducers, the first one at the cylinder and the second one in the undisturbed wave field, enabled to compare the force amplitude on the cylinder to the force amplitude on an equivalent mass of water in the undisturbed wave field (Froude-Krylov F-K force). After ten days of measurements, the experiment was repeated with a cylinder of .45 m diameter. The Keulegan-Carpenter number was within 2.5, and the wave forces proved to be inertial. The following general features emerged: (i) the force spectrum is usually very narrow even if the wave spectrum is broad; (ii) the vertical diffraction coefficient is somewhat smaller than the horizontal diffraction coefficient; (iii) the positive extremes of F_z (vertical force referred to the buoyancy force) markedly exceed the negative extremes; (iv) the pressure fluctuations induced by the highest waves at the cylinder are very similar to the measured pressure-surface displacement covariances. In each of the 580 records obtained in the course of the experiment it was found that the propagation speed reduces to about a half at the cylinder, and the amplitude of the pressure fluctuations increases of 10–15% at the upper half of the cylinder and decreases of about the same percentage at the lower half. These phenomena fully explain why the force amplitude on the cylinder is larger than the F-K force amplitude.     

Experimental study of hydrodynamic forces and wave forces exerted on a truss structure

The drag and added mass coefficients of a truss leg of an ocean platform are obtained by using the forced-oscillation technique in a still water. Higher order forces and lift forces are also measured.The drag and inertia coefficients of the truss leg model are obtained by measuring the wave forces acting on it in regular deep waves. The moment lever of the wave force is compared with theoretical results.     

Wave-current force on horizontal cylinder

For the calculation of wave-current force on horizontal cylinder a modified Morison's equation is used. A redefined Keulegan- Carpenter number KC2 is determined for the horizontal cylinder in wave-current co-existing field. The force coefficients are well related to the redefined KC2 number. As to the comparison with the force on vertical cylinder, the characteristics of force on horizontal cylinder are quite similar to those on vertical cylinder, but the force coefficients for horizontal cylinder are larger than those for vertical cylinder. It is proved by the authors' calculation that the results of monochromatic wave can be used directly for the determination of irregular wave-current force on horizontal cylinder in time domain.     

Wave forces on a large, horizontal submerged cylinder

A numerical boundary integral equation method combined with a non-linear time stepping procedure is used for the calculation of wave forces on a large, submerged, horizontal circular cylinder. As the method is based on potential theory, all computations are performed in the inertia dominated domain, that is, for small Keulegan-Carpenter numbers. Computations are carried out for the Eulerian mean current under wave trough level equal to zero. When the cylinder is moved towards the sea bed the computations show that the inertia coefficients increase significantly, which is associated with a blockage effect. Furthermore, the effect of the wave steepness is reduced when the submergence of the cylinder is increased. In the vicinity of the free water surface the vertical inertia coefficient is highly dependent upon the wave steepness, which tends to reduce it, whereas the horizontal inertia coefficient is only slightly dependent on the wave steepness. Computations are also carried out for cylinder diameters comparable with the wave length. Finally, inertia coefficients computed by the present method are compared with some analytical results by Ogilvie [(1963), First and second order forces on a cylinder submerged under a free surface. J. Fluid Mech. 16, 451–472]. As long as the assumptions leading to Ogilvie's theory are fulfilled (cylinder radius small compared to the wave length), the results are quite similar.     

Forces on a smooth submarine pipeline in random waves — A comparative study

Forces on a smooth submarine pipeline, fixed horizontally near a plane boundary, have been investigated under random wave conditions. The submarine pipeline was subjected to Pierson-Moskowitz spectrum (P-M spectrum) at various energy levels. The water particle kinematics were computed based on the linear random wave model and the Morison equation was chosen as the wave force predictor model. The inline hydrodynamic coefficients of drag and inertia were evaluated using two different methods, one in the frequency domain and the other in the time domain. Five mathematical formulations were considered for the analysis of transverse wave forces and these were compared in terms of the correlation coefficient. The transverse force was also analyzed in terms of the transverse root mean square (rms) coefficient. The inline hydrodynamic coefficients of drag and inertia and the transverse rms coefficient were correlated with the Keulegan-Carpenter number or period parameter, the relative clearance of the pipeline from the bed and the depth parameter. Finally, the results of the random wave tests were compared with those of regular waves under similar pipeline conditions.     

Wave forces on submarine pipelines near a plane boundary

Forces induced by regular waves on submarine pipelines resting on as well as near a plane boundary and aligned parallel to wave fronts of the oncoming waves are investigated experimentally. The inline hydrodynamic coefficients of drag and inertia are evaluated through the use of Morison equation and the least squares method. The transverse force is analysed in terms of maximum transverse force and transverse root mean square (r.m.s.) coefficients. The resulting inline and transverse hydrodynamic coefficients are correlated with the period parameter or Keulegan-Carpenter number and relative clearance of the pipeline from the plane boundary. The effect of depth parameter on these coefficients and the correlation between maximum transverse force and transverse r.m.s. coefficients are also reported.     

Wave forces on, and water-surface fluctuations around a vertical cylinder encircled by a perforated square caisson

The wave forces and moments on and the water surface fluctuations around a vertical circular cylinder encircled by a perforated square caisson were experimentally investigated. The porosity of the outer square caisson was varied from 4.24 to 14.58%. The in-line wave forces on the inner vertical cylinder are influenced by changing the porosity of the outer caisson, whereas the variations in the water surface fluctuations are less influenced in this porosity range. The in-line moment on the vertical cylinder is relatively less sensitive when the porosity is increased from 4.24 to 8.75%, but varies substantially when it is increased from 8.75 to 14.58%. The force and moment ratio (i.e. the ratio of the force or moment on the vertical cylinder, when it is encircled by the perforated caisson to the force or moment on the cylinder without any protection around it) reduces with increased wave height, H, and wave length, L, whereas the wave height ratio (ratio of the wave height at a point in the vicinity of the structure to the incident wave height) is less sensitive for the varying H and L. A new non-dimensional parameter, p^1.5 (D/L)/(H/d), is introduced to predict the in-line force and moment on the inner vertical cylinder, where d is local water depth, D is the diameter of the inner cylinder and p is the porosity of the outer caisson in percentage. Simple predictive equations for forces, moments and water surface fluctuations are provided.     

The virtual mass of a rectangular flat plate of finite aspect ratio

The skin friction of a two-dimensional planing flat plate is made up of two opposing components; a drag force from the flow aft of the stagnation line and an opposing thrust force from the jet flow. This paper is concerned only with the drag term and the wake velocity defect which it causes in the water behind the transom.It is concluded that the skin friction is less than would be expected from a flat plate at ambient static pressure (D_fo say) and is approximately equal to D_fo (1 − C_R), where C_R is the normal force coefficient based on wetted area. The wake velocity decrement due to this drag is found to be significant, particularly for surface piercing propellers.     

WAVE-CURRENT FORCE ON VERTICAL PILE

The drag and the interia coefficients Cd,CM of wave-current force on vertical pile are well related to the redefined Keulegan-Carpenter number N' Kc by the test data of the authors. The relation could be also used for irregular wave-current force to calculate in time domain. A simplified method for the calculation of cumulative probabilistic distribution of peak value of irregular wave-current force is also recommended in this paper. These methods were justified by the model test of the authors.     

Hydrodynamic behavior of a straight floating pipe under wave conditions

This paper examines the hydrodynamic behavior of a floating straight pipe under wave conditions. The main problem in calculating the forces acting on a small-sized floating structure is obtaining the correct force coefficients C_n and C_t, which differ from a submerged structure. For a floating straight pipe of small size, we simplify it into a 2D problem, where the pipe is set symmetrically under wave conditions. The force equations were deduced under wave conditions and a specific method proposed to resolve the wave forces acting on a straight floating pipe. Results of the numerical method were compared to those from model tests and the effects of C_n and C_t on numerical results studied. Suggestions for the selection of correct C_n and C_t values in calculating wave forces on a straight floating pipe are given. The results are valuable for research into the hydrodynamic behavior of the gravity cage system.     

Drag and inertia coefficients for a circular cylinder in steady plus low-amplitude oscillatory flows

Computer simulations of steady plus low-amplitude oscillatory flow about a circular cylinder are reported at a fixed Reynolds number of 150 based on the steady component. The conventional Keleugan–Carpenter number based on the oscillatory component is fixed at π/5. The oscillation frequency is varied so as to study a wide spectrum of flows where inertial forces dominate at one end and viscous drag forces at the other as a function of the modified Keleugan–Carpenter number. The hydrodynamic force on the cylinder in-line with the flow direction is represented by Morison's equation and an extended version with three terms. The drag and inertia coefficients in Morison's equation are determined by least-squares fits to data directly computed from integration of skin friction and pressure distributions around the periphery of the cylinder. The root-mean-square value of the residue of reconstructed minus directly-computed forces varies between 2 and 41% depending on the flow parameters. Comparable results can be obtained with a semi-theoretical approach using inviscid inertia and quasi-steady viscous drag terms. Physical explanations for the variation of the force coefficients are provided and implications for pertinent flow–structure interactions are discussed.     

WAVE-CURRENT FORCE COEFFICIENTS FOR ISOLATED PILE

Based on the linear wave theory and model experiment results, the wave-current force coefficients for isolated pile are investigated, using Morison's formula to calculate the wave-current force. A formula is'presented for determining the relative coefficients of the drag, in which the feature value of the wave-current field is. proposed and used as an important parameter. According to the maximum wave-current force measured in the experiment, the coefficients of the drag force and inertia force are determined by statistical method of two-variable regression so that both of them are fitted in optimum.     

Drag and Sediment Dispersion Over Sand Waves

Estimates of the drag coefficient over sand waves during calm weather in the southern North Sea have been obtained from measurements of the water slope and currents at different heights (z) above the sea-bed using the log profile and momentum balance methods. An observed phase difference between principal terms in the momentum balance equation is examined theoretically. Drag coefficient estimates are found to agree broadly with previous studies. Owing to bedform asymmetry, average drag coefficient values obtained atz=1 m (C₁₀₀) are found to be 0·0021 and 0·0029 for flood and ebb tides, respectively. Systematic changes in bed roughness are not detected. Using a momentum balance approach, the average drag coefficient value (C_d) atz=10 m is found to be 0·0056. Changes in 10-min averageC_dvalues over sand waves during the tidal cycle are found to be small with bedform asymmetry having no detectable effect. Correlation betweenC_dandC₁₀₀is found to be poor and separation of skin friction and form drag terms is not possible with existing measurements. The inclusion of form drag inC₁₀₀values at the present site leads to over-estimation of the bed shear stress ({q) available to mobilize and transport sediment. Mobile sediment, detected through the use of tracers and a transmissometer, was not found to have any measurable effect on eitherC_dorC₁₀₀in calm weather conditions.     

A Prediction Method of Internal Solitary Wave Loads on the Semi- Submersible Platform

An investigation into the prediction method for internal solitary waves (ISWs) loads on the columns and caissons of the semi-submersible platform found on three kinds of internal solitary wave theories and the modified Morison Equation is described. The characteristics of loads exerted on the semi-submersible platform model caused by the ISWs have been observed experimentally, and the inertial and drag coefficients in Morison Equation are determined by analyzing the forces of experiments. From the results, it is of interest to find that Reynolds number, KC number and layer thickness ratio have a considerable influence on the coefficients. The direction of incoming waves, however, is almost devoid of effects on the coefficients. The drag coefficient of columns varies as an exponential function of Reynolds number, and inertia coefficient of columns is a power function related to KC number. Meanwhile, the drag coefficient of caissons is approximately constant in terms of regression analysis of experimental data. The results from different experimental conditions reveal that the inertia coefficient of caissons appears to be exponential correlated with upper layer depths.