Small Keulegan-Carpenter Numbers Research Articles

Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts

A first-order analytical theory on a truncated and surface-piercing vertical circular cylinder with a circular plate mounted at the bottom of the cylinder was generalized to solve the linear wave diffraction and radiation problems based on potential flow and the hypothesis of small wave and motion amplitudes. The domain was decomposed, and the linearized velocity potentials were derived in each subdomain and matched on each subdomain interface employing pressure and normal velocity continuity. The linear hydrodynamic loads obtained with the analytical method were compared with the results of linear boundary element method (BEM) solvers. Dedicated experimental campaigns were performed on fixed models with regular waves (diffraction) and then on models oscillating in surge, heave, and pitch motions without incident waves (radiation). The analytical method describes well the wave excitation loads from the experiments for small wave steepness (H/λ=0.02). The predictions of added mass and damping are shown to be applicable to the surge motion, only for small Keulegan-Carpenter numbers (KC<1). On the other hand, the analytically predicted radiated waves demonstrate satisfactory agreement with experiments for all motions.

Large-scale experimental study on scour around offshore wind monopiles under irregular waves

The Keulegan-Carpenter (KC) number is the main dimensionless parameter that affects the local scour of offshore wind power monopile foundations. This study conducted large-scale (1:13) physical model tests to study the local scour shape, equilibrium scour depth, and local scour volume of offshore wind power monopiles under the action of irregular waves with different KC numbers. Systematic experiments were carried out with the KC number ranging from 1.0 to 13.0. With a small KC number (KC < 6), and especially when the KC number was less than 4, the scour mainly occurred on both cross-flow sides of the monopile with a low scour depth. When the KC number exceeded 4, the shape of the scour hole changed from a fan to an ellipse, and the maximum scour depth increased significantly with KC. With a large KC number (KC > 6), the proposed method better predicted the equilibrium scour depth when the wave broke. In addition, according to the results of three-dimensional terrain scanning, the relationship between the local equilibrium scour volume of a single offshore wind power monopile and the KC number was derived. This provided a rational method for estimation of the riprap redundancy for monopile protection against scour.

Hydrodynamic damping of an oscillating cylinder at small Keulegan–Carpenter numbers

Direct numerical simulations (DNS) of oscillatory flow around a cylinder show that the Stokes–Wang (S–W) solution agrees exceptionally well with DNS results over a much larger parameter space than the constraints of $\beta K^2\ll 1$ and $\beta \gg 1$ specified by the S–W solution, where $K$ is the Keulegan–Carpenter number and $\beta$ is the Stokes number. The ratio of drag coefficients predicted by DNS and the S–W solution, $\varLambda _K$ , mapped out in the $K\text {--}\beta$ space, shows that $\varLambda _K &lt; 1.05$ for $K\leq {\sim }0.8$ and $1 \leq \beta \leq 10^6$ , which contradicts its counterpart based on experimental results. The large $\varLambda _K$ values are primarily induced by the flow separation on the cylinder surface, rather than the development of three-dimensional (Honji) instabilities. The difference between two-dimensional and three-dimensional DNS results is less than 2 % for $K$ smaller than the corresponding $K$ values on the iso-line of $\varLambda _K = 1.1$ with $\beta = 200\text {--}20\,950$ . The flow separation actually occurs over the parameter space where $\varLambda _K\approx 1.0$ . It is the spatio-temporal extent of flow separation rather than separation itself that causes large $\varLambda _K$ values. The proposed measure for the spatio-temporal extent, which is more sensitive to $K$ than $\beta$ , correlates extremely well with $\varLambda _K$ . The conventional Morison equation with a quadratic drag component is fundamentally incorrect at small $K$ where the drag component is linearly proportional to the incoming velocity with a phase difference of ${\rm \pi} /4$ . A general form of the Morison equation is proposed by considering both viscous and form drag components and demonstrated to be superior to the conventional equation for $K &lt; {\sim }2.0$ .

An Experimental and Numerical Study of Added Mass and Damping for Side-by-Side Plates in Oscillating Flow

Abstract Forced harmonic oscillations of nine configurations consisting of horizontal side-by-side plate elements are performed experimentally and numerically. The configurations are oscillated in vertical direction and represent generalized mudmats of subsea structures. The tests are performed for Keulegan–Carpenter (KC) numbers relevant for force estimation during lifting operations. Hydrodynamic added mass and damping coefficients are presented. The coefficients are found to be amplitude dependent for all tested configurations. The interaction effects between the plates increase with increasing amplitude and decreasing distance between the plates. For small oscillation amplitudes, compared with the gap between the plates, the plates behave approximately like individual plates. A study of the relation between the damping force and the added mass force for the tested structures illustrates the importance of applying representative damping coefficients in numerical analysis of marine operations. Numerical results are obtained using a potential flow solver (BEM) and a viscous flow solver (CFD). Low-KC added mass coefficients predicted with the BEM are in accordance with the experiments. There is acceptable agreement between the CFD and the experiments. Best agreement is obtained for small KC numbers. As the KC numbers increase, the differences are, in general, larger. This is possibly due to the CFD being based on the two-dimensional laminar flow.

Drag and added mass coefficients of a flexible pipe undergoing vortex-induced vibration in an oscillatory flow

The drag and added mass coefficients of a flexible pipe undergoing vortex-induced vibration (VIV) in an oscillatory flow are investigated experimentally with maximum reduced velocities ranging from 4 to 7.9 and Keulegan-Carpenter (KC) numbers ranging from 10 to 178. The strain responses are measured by fiber bragg grating sensors. By using displacement reconstruction and inverse analysis methods, displacement response and hydrodynamic force are identified. Then, through least square method, drag and added mass coefficients are extracted. The results show that drag coefficient varies with both the maximum reduced velocity and KC number. It is stable in cases with large KC numbers but significantly amplified in the cases of small KC numbers. The maximum value of the drag coefficient can reach 3.5 at kC < 30. A comparison with the results of a stationary cylinder without VIV in an oscillatory flow demonstrates that VIV can amplify the drag coefficient and reduce the sensitivity of the hydrodynamic coefficients to the Stokes number. Based on the experimental results, an empirical formula for predicting the drag coefficients in an oscillatory flow is proposed that accounts for the VIV response and KC number, and its applicability is effectively verified via a comparison between the predicted and experimental results.

Magnification of hydrodynamic coefficients on a flexible pipe fitted with helical strakes in oscillatory flows

The features of hydrodynamic coefficients including drag and added mass coefficients on a flexible pipe fitted with helical strakes in an oscillatory flow are studied experimentally. The experiment of a flexible straked pipe was conducted in an oscillatory flow with the Keulegan-Carpenter (KC) number varying from 9 to 165 and the maximum reduced velocities ranging from 4 to 8. Strain response in both in-line and cross-flow directions are measured by fiber bragg grating sensors. Using displacement reconstruction and inverse analysis methods, displacement response and hydrodynamic force of the straked pipe are identified. Then, through the least squares method, corresponding drag and added mass coefficients are extracted. The asymmetric features at the acceleration and deceleration stages of hydrodynamic force on a flexible pipe in the in-line direction under oscillatory flow are first observed. Compared with a bare pipe, helical strakes can effectively reduce the higher frequency fluctuating force and enhance the wake effects. The hydrodynamic coefficients on the straked pipe are significantly magnified in the case of a relatively small KC number. The maximum mean drag coefficients can reach approximately 10. The presented work suggests that the risk of helical strakes application should be fully evaluated through flexible pipe experiments in both steady flow and unsteady flow.

Hydrodynamic damping of a circular cylinder at low KC: Experiments and an associated model

This paper investigates the hydrodynamic damping of a smooth circular cylinder undergoing forced oscillations at Keulegan-Carpenter (KC) numbers smaller than 5 and Reynolds (Re) numbers from 103–105 with and without background steady currents. A series of experiments are conducted with a circular cylinder oscillating in still water, in-line currents and cross currents. The measured drag coefficients of the smooth cylinder in the still water condition match with the well-published results and the theoretical solution of Stokes and Wang at very small KC numbers. The hydrodynamic damping increases with the in-line steady current whereas it remains almost constant at small transverse velocities and increases notably when the latter becomes large. To predict the hydrodynamic damping in in-line steady currents, the performance of the Morison equation based on relative velocity and independent velocity is explored, respectively. The latter model, by separating the drag into two independent parts, leads to a better fit of the drag force than the former, which is not surprising. However, the former is still a preferable option for engineering design due to its simplicity. The experimental data suggest that the existing design guidelines such as ISO-19902 or DNVGL-RP-C205 should be used with caution for KC < 5.

Experimental study on a yawed square cylinder in oscillatory flows

The vortical structures around a yawed square cylinder oscillating in quiescent water are investigated using the particle image velocimetry technique. Following a previous study on the hydrodynamics (Lou et al., 2017), the present experiments are performed at different yaw angles (α) and Keulegan-Carpenter (KC) numbers to correlate the independence principle (IP) to vortical flow structures. At KC = 6, the vortex pair shows no shedding. Similar vortex patterns at different yaw angles result in similar hydrodynamic behaviors, which validates the IP at small KC numbers. The single and double pairs of vortex shedding regimes are observed for α= 0° at KC = 11 and 19, respectively, and the vortex shedding process is determined by the movement of the cylinder as well as the interaction between vortices and shear layers. As α increases to 45°, the shear layers are stretched and show attachment to the upper and lower sides of the cylinder for most of the time within one oscillating cycle. The shedding is only observed at the end of each half cycle and the vortices are found to reattach to the cylinder body. The subsequent drag force behavior of the yawed cylinder displays significant differences from that at α= 0° and hence the IP is no longer applicable at KC = 11 and 19. When KC increases to 25, a three pairs of vortex shedding regime can be observed at both α= 0° and 45°. A similar flow feature, characterized by the shear layer attachment when the cylinder is at the neutral position and the vortex shedding at the end of each half cycle, has been found for both α= 0° and 45°. This result indicates that the IP becomes valid for the yawed square cylinder when the KC is sufficiently large until it is analogous to the steady flow.

Vortex-induced vibration of flexible pipe fitted with helical strakes in oscillatory flow

Helical strakes are widely used to suppress vortex-induced vibration (VIV) in offshore engineering. However, the VIV response and suppression efficiency of the straked pipe remain unclear in an oscillatory flow. In this paper, an experimental study of VIV for a flexible pipe fitted with helical strakes was conducted in an oscillatory flow with the Keulegan-Carpenter (KC) number varying from 21 to 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on the VIV response, Strouhal number, suppression efficiency and fatigue damage in an oscillatory flow were investigated. The results show that the suppression efficiency and fatigue damage reduction ratio are not as ideal in oscillatory flow as those in steady flow. Moreover, the VIV dominant frequency is obviously magnified and a large Strouhal number is observed, reaching 0.439 under a lower reduced velocity of 4. As the reduced velocity further increases to 6, 8, 12, two branches of Strouhal numbers are clearly seen. The dominant frequency decreases at the first branch with a small KC number and increases at the second branch with a large KC number.

Numerical Investigation on Vortex-Induced Vibration caused by Vessel Motion for a Free Hanging Riser Under Small Keulegan-Carpenter Numbers

A free-hanging riser (FHR) is a typical riser configuration seen in the disconnected drilling riser, the water-intake riser, and the deep-sea mining riser. In offshore productions, these marine risers will move back and forth in water and further generate an equivalent oscillatory current around themselves, due to the vessel motions. Both in full-scale marine operations and model tests, it has been reported that such oscillatory current leads to riser vortex-induced vibration (VIV) and therefore causes structural fatigue damage. Recently, there have been some attempts to numerically predict vessel motion-induced VIV on the compliant production risers, with emphasize on relatively large Keulegan–Carpenter (KC) numbers. In the real marine operations, the risers experience small KC number scenarios during most of their service life. Therefore, the investigation of vessel motion-induced VIV under small KC number is of great significance, especially considering its contribution to the fatigue damage. In this paper, numerical investigation of VIV of a FHR attached to a floating vessel is carried out. A new response frequency model for vessel motion-induced VIV under small KC numbers is proposed and implemented in vivana. Validation of the proposed numerical methodology is performed against the published experimental results, where a good agreement is achieved.

ДИНАМИКА ЖИДКОСТИ В ПОДВИЖНОМ СОСУДЕ С НАКЛОННЫМИ РЕШЕТКАМИ

Forced oscillations of a rectangular tank p artially filled with liquid are considered in this work. Two inclined screens are located in the tank as inserts. The system of Reynolds–averaged Navier–Stokes equations is solved numerically with the SST k - ω turbulence model. The volume of fluid (VOF) method is taken for simulation of gas–liquid interface. The problem is solved in two-dimensional statement. Unstructured triangular mesh converted to a fine structured rectangular mesh is used near the screens. A comparison of two turbulence models with experimental data is carried out, and the optimal model for the pressure loss coefficient calculation is chosen. The effect of the inserts inclination angle on the pressure loss coefficient is investigated at different values of the oscillation amplitude of the tank and different heights of the screen slats. It is obtained that the pressure loss coefficient dependence on the inclination angle at small Keulegan–Carpenter number has the local maximum depending on the oscillation amplitude and the geometric parameters of the screens. Comparison with the known analytical dependence of the pressure loss coefficient dependence on the inclination angle that obtained for the stationary flow through the inclined screen is conducted. It is found that this dependence needs to be verified for the case of oscillation tank at big screen inclination angle and small Keulegan–Carpenter number.

A Nonlinear Computational Model of Floating Wind Turbines

The dynamic motion of floating wind turbines is studied using numerical simulations. The full three-dimensional Navier–Stokes equations are solved on a regular structured grid using a level set method for the free surface and an immersed boundary method for the turbine platform. The tethers, the tower, the nacelle, and the rotor weight are included using reduced-order dynamic models, resulting in an efficient numerical approach that can handle nearly all the nonlinear hydrodynamic forces on the platform, while imposing no limitation on the platform motion. Wind speed is assumed constant, and rotor gyroscopic effects are accounted for. Other aerodynamic loadings and aeroelastic effects are not considered. Several tests, including comparison with other numerical, experimental, and grid study tests, have been done to validate and verify the numerical approach. The response of a tension leg platform (TLP) to different amplitude waves is examined, and for large waves, a nonlinear trend is seen. The nonlinearity limits the motion and shows that the linear assumption will lead to overprediction of the TLP response. Studying the flow field behind the TLP for moderate amplitude waves shows vortices during the transient response of the platform but not at the steady state, probably due to the small Keulegan–Carpenter number. The effects of changing the platform shape are considered, and finally, the nonlinear response of the platform to a large amplitude wave leading to slacking of the tethers is simulated.

HYDRODYNAMIC DAMPING AND QUASI-COHERENT STRUCTURES AT LARGE STOKES NUMBERS

Unsteady flow about solid and perforated cylinders at large Stokes numbers and very small Keulegan–Carpenter numbers has not been sufficiently understood to predict the hydroelastic response of compliant structures subjected to high frequency excitation. It is impossible to compute, difficult to quantify, and extremely difficult to visualize. Here, following a brief review of the previous research, new results for both forced and pluck-induced oscillations are presented. It is shown that: (a) the measured drag coefficients are larger (about double for smooth cylinders) than those predicted from the Stokes–Wang analysis (in the region of its applicability); (b) there is an unstable flow regime in which the oscillatory boundary layer develops quasicoherent structures (QCS) over a range of K values before giving rise to Honji-type coherent structures (HTCS); (c) the drag coefficients for perforated cylinders are much larger than those for solid cylinders and do not follow the Stokes–Wang prediction even at small β. As expected, their inertia coefficients are considerably smaller than those for solid cylinders.

HYDRODYNAMIC DAMPING OF A CYLINDER AT β≈106

This paper describes delicate, but large-scale, experiments aimed at measuring the hydrodynamic damping of a circular cylinder oscillating in still water and transversely in a current. Attention is concentrated on the regime of very small Keulegan–Carpenter numbers, in which the drag coefficient is inversely proportional to the Keulegan–Carpenter number. Measurements in still water at β=650 000 and 1250 000 point to drag coefficients about twice those appropriate to two-dimensional laminar flow, in common with earlier measurements at β≈105. In the presence of a slowly varying transverse current (generated by placing the cylinder at the node of standing waves of long period), the damping increased with the reduced velocity of the ambient flow at a rate that increased with the Reynolds number.

Wave forces on a large, horizontal submerged cylinder

Abstract 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.

Inertia dominated forces on oblique horizontal cylinders in waves near a plane boundary

The hydrodynamic forces exerted on horizontal cylinders placed obliquely in plane waves in a laboratory flume have been measured for small Keulegan-Carpenter numbers at which inertia forces dominate. Three different gaps between the cylinder and the bed of the channel have been considered for two oblique angles of the incident waves. The linear diffraction theory is numerically solved using the boundary element method to determine the wave forces. It is found that the computed results compare favourably with the experimental results pertaining to the in-line forces. The lift forces evaluated through linear theory agree with the empirical results when a correction is made to the pressures taking account of the velocity field around the cylinder, especially when it is close to the bed of the channel.

Oscillating flows over periodic ripples of finite slope

The analysis of a laminar oscillating flow over a wavy surface is an important first step toward understanding the complicated dynamics over sand ripples along sea coasts. In a previous paper, two separate asymptotic theories for modestly large Reynolds numbers were presented, one for finite ripple slope but small Keulegan–Carpenter number, the other for finite Keulegan–Carpenter number but small ripple slope, where the Reynolds number is defined with the ambient oscillation velocity and the ripple amplitude, and the Keulegan–Carpenter number is the ratio of the oscillation amplitude to the ripple wavelength. Different types of vortex motion were found above the ripples. Here, a seminumerical solution of the Navier–Stokes equations in power series of the ripple slope, with emphasis on the intermediate regime where both parameters are finite and the Reynolds number is from order unity to moderately large is presented. For moderately large Reynolds number, the present results are similar to the second asymptotic theory in which there are two boudary layers with an oscillating vortex high above the rippled bed. For relatively small Reynolds numbers of order unity, the Stokes and the outer layers merge. In addition, the presence of bed ripples is found to increase the rate of energy dissipation by as much as 40%.

A numerical study of oscillating flow around a circular cylinder

This paper presents results obtained from a numerical solution to a stream function–vorticity formulation of the Navier–Stokes equations for the flow around a circular cylinder in planar oscillating flow at small Keulegan–Carpenter numbers (KC) in the subcritical Reynolds number (Re) range. The equations are solved by finite-difference methods. For very small KC ([les ] 1), the numerical results coincide with analytical solutions. As KC is increased, the incipient separation and instability leading to an asymmetrical flow with vortex shedding are predicted. Computed flow fields at small KC values are compared to flow visualizations, and good agreement is found for moderate β-values (≈ 250). The well-documented flow regimes with the transverse vortex street, single-, double- and three-pair shedding, are predicted by the model. Although the flow is not fully resolved for the highest Re values, comparisons of calculated drag and inertia coefficients with experimental data for three different values of the frequency parameter β in the range 196–1035 for 0 < KC < 26 show good agreement.