Keulegan Carpenter Research Articles

Hydrodynamic coefficients on a deeply submerged heaving solid/porous disk: in analytical and experimental approaches

ABSTRACT The hydrodynamic coefficients on a deeply submerged solid/porous disk subjected to forced heave oscillation have been investigated in analytical and experimental approaches. The analytical model is developed by means of a matched eigenfunction expansion method (MEEM) using a quadratic relationship between the pressure drop and traversing fluid velocity across the porous disk. For the improvement and validation of the analytical model, a series of experiments with the solid/porous disks is conducted for various porosities, Keulegan-Carpenter (KC) numbers, and oscillation periods. In addition to the small-scale hole vortices across the porous disk, the effect of edge vortices on the hydrodynamic coefficients is empirically implemented by curve-fitting with experimental data. The inclusion of edge vortices in the analytical model results in good agreement with experimental data of the damping coefficient for a deeply submerged disk. However, there are some discrepancies in the added mass, with noticeable differences occurring in the disk with low porosity.

Numerical simulation of flow instability induced by a fixed cylinder placed near a plane wall in oscillating flow

The Honji instability of in-line forced cylinder near a plane wall is investigated using Floquet analysis and direct numerical simulation using the open-source code Nektar++. The onset of the Honji instability is analyzed over a range of the gap-to-diameter ratios (e/D) from 2.0 to 0.03125, with fixed Keulegan–Carpenter number KC = 2, Stokes number β = 150, and Re = 300. As e/D decreases from 2 to 0.375, the Honji instability is primarily on the gap side of the cylinder. For 0.25<e/D<0.375, the location of the onset of the Honji instability switches from the gap side to the top side of the cylinder. This is due to the blockage effect, which results in the two different trends in the onset of the Honji instability with decreasing e/D. For 0.125<e/D<0.25, the favorable pressure gradient suppresses the onset of instability at the gap side. The Honji instability is observed again at the top side of the cylinder at smaller gap ratio (e/D=0.0625), which is due to increased flow strength at the top side. At even small gap ratio, i.e. e/D=0.03125, the secondary vortexes and Honji instability together enhance the three-dimensional flow.

Canopy drag parameterization from field observations for modeling wave transformation across salt marshes

Understanding, quantifying, and predicting wave transformation across marsh vegetation canopies is important for assessing the efficacy of nature-based shoreline strategies. A key challenge for modeling wave dissipation in coastal marshes is accurately representing the canopy drag, particularly as previously proposed canopy drag coefficient (CD) expressions vary considerably for the same vegetation and wave properties. There is also as yet no consensus on which dimensionless parameter best explains the variation of CD with wave conditions and water depth in marshes. This study addressed these challenges using wave, elevation, and vegetation measurements collected at two natural salt marshes across two summers. The dataset was used together with a wave energy conservation model to (i) identify and test possible reasons for differences among previously published CD expressions for salt marshes; and, (ii) derive and test expressions for CD as a function of different dimensionless parameters. We found that uncertainties in measurements of vegetation properties (shoot height, diameter, density) lead to substantial uncertainties in CD expressions, which may explain some differences among previous studies. When plotted versus Keulegan Carpenter number (KC) or Reynolds number (Re), CD values vary by roughly a factor of two for large waves (KC > 100) depending on water depth. When plotted versus Cauchy number (Ca), the ratio of hydrodynamic drag force to the canopy’s elastic restoring force, CD values collapse onto a single curve. We suggest that Ca is the more appropriate parameter for representing CD under larger waves because it represents the force balance relevant for plant motion and recommend future studies to test CD(Ca) relationships for different canopy heights.

Extensive hydro-elastic floating bridge tests: Planning, design, implementation, and numerical comparison

Extensive hydrodynamic and hydro-elastic tests for a long floating bridge have been carried out in the ocean basin of SINTEF Ocean for the Bjørnafjord floating bridge project. The complexity of the floating bridge plus cable-stayed bridge, in addition to its 52.58 m model length, mark this test campaign a major undertaking in ocean basin model testing. The design, execution and analysis of the test program has been an iterative process over 5 years. The tests provide data for validation of numerical analysis, study of hydrodynamics phenomenon that cannot be well simulated by state-of-the-art engineering tools, and verification of the design under controlled environmental loads. Wave-current-structure interaction, hydrodynamic interaction between pontoons, and their impacts on hydro-elastic global responses of floating bridge (with cable stayed bridge) are the main scope of tests. Oscillation tests under different Keulegan–Carpenter (KC) numbers were also carried out for viscous drag coefficient of a pontoon in defined direction. Correlation of test results with numerical analysis is an important part of the work scope, which provides solid basis for a robust design with reduced risk. This article tries to provide an overview of the background, design, execution and correlation work of this extensive tests. Selected examples of correlation work are also provided in the article. The tests have validated the analysis method and software, provided important input to the next phase design, and enhanced the structural reliability of the bridge concept in continuous evolution.

Modulation and hysteresis in vortex-induced vibration of a spring-mounted cylinder in a slowly varying oscillatory stream

The response of a spring-mounted circular cylinder constrained to vibrate with a single degree of freedom transverse to an oscillating stream is investigated using two-dimensional flow simulations. The Keulegan–Carpenter number was set at large values with the ratio of the oscillatory flow frequency fo to the natural frequency of the cylinder fn assuming values of fo/fn=0.01, 0.02, and 0.04, so as to investigate the effect of very-slowly-varying reduced velocity on the transient response. A maximum reduced velocity of 5, corresponding to a maximum Reynolds number of 150, was employed in all cases. Under these conditions, the cylinder underwent intermittent vortex-induced vibration (VIV) with amplitude modulations at the period of the oscillatory flow. Maximum VIV amplitudes of approximately half diameter occurred near the end of the flow acceleration stage. Pronounced hysteresis was observed in the plots of the ensemble-averaged amplitude and frequency, both being greater during the deceleration stage than during the acceleration stage at the same instantaneous reduced velocity over a significant part of the flow half period. The hysteresis was found to be related to ‘transient’ operating points that, in the space of normalized amplitude and frequency, join smoothly the distinct branches of steady-VIV where operating points do not exist (for the same system with varying the reduced velocity). The main deterministic VIV features were similar at all three frequency ratios examined but the stochastic features became more pronounced as fo/fn was increased. We also found that the cylinder motion and the driving fluid force sometimes were not synchronized around the point of maximum flow velocity, although this point corresponds to well within the synchronization range in steady-VIV.

Wave-Current Loads on a Super-Large-Diameter Pile in Deep Water: An Experimental Study

Recently, the diameters and construction water depths of the pile foundations of planned and newly built sea-crossing bridges have been increasing greatly. Hydrodynamic loads are the key control factors in the design of super-large-diameter piles. However, most of the previous studies focused on the inline force on the pile with a small diameter, and there were few cases to consider the impact of the transverse force on the hydrodynamic load of the pile under wave-current actions. In this study, to understand the hydrodynamic loads on such deep-water super-large-diameter piles, the prototype was one of the 6.3-m piles used in the Xihoumen Rail-cum-Road Bridge, and 1:60-scale model tests were carried out in an experimental tank, with the actions of regular waves and waves combined with currents used as loads. The influence of the current velocity and static wave height on the inline and transverse forces on the pile was measured and analyzed. The experimental results indicate that with increasing current velocity, the fluctuation characteristics of the wave-current-induced inline and transverse forces change significantly, and their peak values increase obviously compared to those induced by only waves. In particular, the peak transverse force increases tens of times and can become equivalent to the inline force. The modified Morison formula and Kutta–Joukowski formula are used to derive the correlations between the drag coefficient CD, inertia coefficient CM, lift coefficient CL, and redefined Keulegan–Carpenter number KC*. Under wave-current action, the transverse force contributes quite significantly to the hydrodynamic load on a super-large-diameter pile, making it easier to trigger extreme structural loads. The results presented herein are an important reference for the engineering designs of such super-large-diameter piles.

Experimental and numerical modelling of scour at vertical wall abutment under combined wave-current flow in low KC regime

The present study investigates the scour at vertical-wall abutments under a strong current-dominated combined wave-current and current-only environment using experiments and computational fluid dynamics (CFD) modeling. The CFD modelling is performed using an open-source three-phase numerical model (REEF3D), which solves Reynolds-Averaged Navier Stokes (RANS) equations with k-ω turbulence closure. The CFD model is first validated using the experiments and then used for further investigation. The Exner equation was used to compute alterations in bed elevation, while the Level-Set method recorded the free surface and bed topography in a realistic manner. The experimental findings demonstrate that the initial phase of scour development is more rapid under combined wave-current flow, but the equilibrium scour depth is higher in current-only conditions. The increase in Keulegan Carpenter (KC) number increases the scour depth at vertical-wall abutments for a specific relative flow velocity (Ucw). In contrast, a mild increase in vertical-wall abutment scour depth is observed with an increase in Ucw. To the best of the authors’ knowledge, the present work is the first of a kind of experimental and three-dimensional CFD study that estimates abutment scour under strong current-dominated combined wave-current flow.

Optimization of the drag coefficient in wave attenuation by submerged rigid and flexible vegetation based on experimental and numerical studies

This study investigates wave attenuation due to submerged rigid and flexible vegetation through a combination of experimental and numerical approaches. Accurate reproduction of physical measurements was achieved by adjusting the drag coefficients within the numerical model to account for the influence of wave conditions and vegetation characteristics on wave attenuation. The decay of wave height in both rigid and flexible vegetation fields is closely linked to wave nonlinearity, which can be quantified using the Ursell number. Wave nonlinearity has a more pronounced impact on wave attenuation when rigid vegetation is present compared to flexible vegetation. According to the relationship between the drag coefficient and the Keulegan–Carpenter number, a new drag coefficient formula was proposed based on a revised Keulegan–Carpenter number that considers wave nonlinearity. The deformation of flexible vegetation in response to waves was characterized by introducing the effective vegetation height, which can be correlated with the wave Cauchy number and the stem height ratio. By identifying the transition between rigid and flexible vegetation, an empirical expression encompassing multiple vegetation types was developed by incorporating vegetation flexibility. The derived empirical equation can be implemented in wave simulations with applicability to wave attenuation by vegetation and optimal flood response strategies.

Suppression of vortex shedding in the wake of a circular cylinder through high-frequency in-line oscillation

This paper presents a new flow control approach to suppress the vortex shedding in the wake of a circular cylinder through high-frequency oscillation. The circular cylinder is forced to oscillate in the streamwise direction at high-frequency and low amplitude, corresponding to a high Stokes number (β = 100–1000) and low Keulegan–Carpenter number (KC = 0.001–4). Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations of an oscillating circular cylinder in steady current have been carried out in the parameter space of KC, Rec, and β. Our numerical results show that when the flow remains in the two-dimensional vortex shedding regime, the cylinder wake sequentially experiences transitions from the vortex shedding regime to the suppression of the vortex shedding regime and finally to the symmetry breaking regime, with increasing KC. Corresponding wake characteristics and variations of hydrodynamic forces over the three wake regimes are explored. Three quantities that represent shear-layer characteristics, including the length of separating shear layers, the circulation of shear layers and wake recirculation length, reach maxima at the onset of suppression. The physical mechanisms for the suppression of vortex shedding and occurrence of symmetry breaking are also explained. Once the flow becomes 3-D, vortex shedding from the cylinder cannot be suppressed, primarily because the outer shear layers induced by the steady approaching flow are enhanced in 3-D flows. The cylinder oscillation over the frequency range investigated in the present study delays wake transition to 3-D. The cylinder oscillation alters the 3-D vortical structure and its spanwise wavelength significantly.

Large-scale experimental study on scour around offshore monopile under combined wave and current condition

This work conducts large scale (1:13) experiments to study local scour around a monopile under combined irregular wave and current condition. The Keulegan–Carpenter (KC) numbers range from 1.1 to 6.73, and the relative current strength parameter Ucw numbers range from 0.17 to 0.77. The large-scale experiment indicates the 3D shape of scour holes is always an inverted cone under wave-current condition which has nothing to do with wave irregularity and Ucw. The changes of maximum scour location are associated with KC and Ucw numbers. The location of maximum scour depth is shifted from upstream to downstream of the monopile with large KC number under current-dominated condition. The characteristic scour timescale is closely related to KC, Ucw and Shields parameter θw when Ucw<0.3 and may be only related to θw when Ucw>0.7. The equilibrium scour depth is a function of vortex scale parameter referring to KC and Ucw numbers. A change in the definition of KC can improve the accuracy of existing formula under large-scale experimental condition, and scour volume formulae are further proposed from the relation with maximum scour depth. Finally, a scour volume formula of KC and Ucw numbers is suggested to estimate the protection material amount around monopile.

CFD-assisted linearized frequency-domain analysis of motion and structural loads for floating structures with damping plates

Damping plates are widely used on floating offshore structures to reduce the dynamic responses in waves. Linearization of the nonlinear drag loads are needed to solve the floater motions by frequency-domain methods, which, to date, still dominate the analysis of wave-frequency responses of floating structures in engineering practice. While existing studies focus on damping effects on heave motions, there is a lack of understanding on how the drag loads on damping plates should be modeled to predict the angular motions and structural loads of the floaters. Two different configurations of Morison elements are investigated in this paper. The first considers the entire damping plate as a whole, while the second locates Morison elements at the estimated ‘points of acting’ of drag loads for different parts of the damping plate. Computational Fluid Dynamics (CFD) analysis is utilized to obtain the required drag coefficients and the ‘points of acting’ as function of Keulegan–Carpenter number. In comparison to full CFD analysis of the same floater in regular waves, a hybrid approach is shown to give satisfactory wave-frequency results, not only for motion responses, but also bending moments and shear forces on the damping plate. However, non-negligible second-harmonic structural loads that cannot be properly accounted for by the existing Morison-drag formulation are also identified in CFD analysis, calling for special attention to structural assessment under extreme wave conditions. This study also demonstrates that locating the Morison elements at ‘points of acting’ improves the prediction of motion responses and structural loads at resonance.

Oscillatory flow around a vertical circular cylinder placed in an open channel: coherent structures, sediment entrainment potential and drag forces

Flow and turbulent structures generated by the interaction of forced incoming oscillatory flow with a circular, vertical cylinder placed in an open channel with a horizontal bed are investigated using eddy-resolving simulations. Validation simulations performed with a Keulegan–Carpenter (KC) number of 20 and multiple-mode forcing corresponding to the laboratory experiment of Sumeret al.(J. Fluid Mech., vol. 332, 1997, pp. 41–70) show that detached eddy simulation (DES) predicts more accurately the amplification of the bed shear stress beneath the horseshoe vortex system (upstream side of the cylinder) and the maximum magnitude of the bed shear stress at the downstream (wake) side of the cylinder compared with unsteady Reynolds-averaged Navier–Stokes simulations. High-Reynolds-number DES simulations are then conducted with 1.5 ≤ KC ≤ 30.8 and one-mode sinusoidal forcing of the streamwise velocity in the approaching flow to investigate the changes in the wake vortex-flow regimes, the coherence of the horseshoe vortices and the generation of other near-bed coherent structures in the wake during the oscillatory cycle. The flow is periodic, no horseshoe vortices form and no vortices are shed in the wake forKC = 1.5. By contrast, forKC ≥ 8 horseshoe vortices are present over part of the oscillatory cycle and up to three wake vortices are shed over each half-cycle asKCis increased to 30.8. For an intermediate range ofKCnumbers, one (KC = 15.4) or two (KC = 8) of the vortices forming at the back of the cylinder during each half-cycle are washed around it when the flow reverses. The main horseshoe vortex and other horizontal near-bed vortices have a large capacity to amplify the bed shear stresses when the incoming velocity magnitude is significantly less than its peak value. Assuming the depth-averaged velocity in the incoming (undisturbed) oscillatory flow is the same in simulations conducted with differentKCnumbers, the peak values of the sediment entrainment potential measured by the mean (cycle-averaged) volumetric flux of sediment entrained from the bed over one oscillatory cycle occur for 8 ≤ KC ≤ 15.4. For allKCnumbers, the in-line force variation over the oscillatory cycle is fairly well approximated by the Morison equation. ForKC = 1.5, the in-line force is only due to inertia effects. ForKC = 30.8, the maximum and minimum values of the phase-averaged in-line force are approximatively in phase with those of the incoming flow velocity. ForKC ≥ 15, the phase-averaged in-line force coefficients vary between 0.8 and 1.1 during most of the oscillatory cycle (e.g. when the incoming flow velocity is not close to zero). This is different from cases withKC ≤ 8 where the in-line force coefficient is equal to zero twice during the oscillatory cycle as the in-line force becomes equal to zero for non-zero values of the incoming velocity. The largest cycle-to-cycle variations of the in-line force coefficient and in-line force are observed aroundKC = 8. ForKC = 8 and 15.4, the cylinder is subject to relatively large phase-averaged spanwise drag forces that are comparable to the peak phase-averaged streamwise drag forces. AsKCis increased to 30.8, the phase-averaged spanwise drag force becomes zero over the whole oscillatory cycle but the cylinder is still subject to large instantaneous spanwise forces over part of the oscillatory cycle.

Interaction of two cylinders immersed in a viscous fluid. On the effect of moderate Keulegan–Carpenter numbers on the fluid forces

This work deals with the hydrodynamic interaction of two parallel circular cylinders, with identical radii, immersed in a viscous fluid initially at rest. One cylinder is stationary while the other one is imposed a harmonic motion with a moderate amplitude of vibration. The direction of motion is parallel to the line joining the centers of the two cylinders. The two dimensional fluid–structure problem is numerically solved by the Arbitrary Lagrangian–Eulerian method implemented in the open-source CFD code TrioCFD. First, we show that the fluid forces on the two cylinders are aligned with the direction of the imposed motion. Second, we show that the moderate oscillations of the moving cylinder create nonlinear effects in the fluid that strongly affect the characteristics (Fourier harmonics) of the hydrodynamic force acting on the stationary cylinder. The fluid force on the moving cylinder is shown to be poorly affected by the nonlinear effects, which makes it possible to extend the linear concept of self-added mass and damping coefficients. First, we show that the self-added coefficients decrease as Sk−1/2, with Sk the Stokes number (dimensionless number constructed from the imposed vibration frequency). Second, we show that the self-added mass (resp. damping) decreases (resp. increases) as −KC3 (resp. +KC3), with KC the Keulegan–Carpenter number (ratio between the imposed amplitude vibration and the separation distance between the cylinders). These variations are included in new power laws derived from nonlinear regressions of the numerical results. These new power laws for the self-added coefficients combine the effect of both Sk and KC, covering the viscous (Sk≥500) and weakly nonlinear (KC≤0.3) regimes.

3D CFD Study of Scour in Combined Wave–Current Flows around Rectangular Piles with Varying Aspect Ratios

This study utilizes three-dimensional simulations to investigate scour in combined wave–current flows around rectangular piles with various aspect ratios. The simulation model solves the Reynolds-averaged Navier–Stokes (RANS) equations using the k–ω turbulence model, and couples the Exner equation to compute bed elevation changes. The model also employs the level-set approach to realistically capture the free surface, and couples a hydrodynamic module with a morphological module to simulate the scour process. The morphological module employs a modified critical bed shear stress formula on a sloping bed and a sand-slide algorithm for erosion and deposition calculations in the sediment bed. To validate the numerical model, simulations are conducted in a truncated numerical wave tank with the Dirichlet boundary condition and active wave absorption method. After validation, the numerical model is used to investigate the effect of aspect ratio and the Keulegan–Carpenter (KC) number on scour depth in a combined wave–current environment. The study finds that the normalized scour depth is highest for a rectangular pile with an aspect ratio of 2:1 and lowest for an aspect ratio of 1:2. The maximum normalized scour depth (S/D) for aspect ratios of 2:1 are 0.151, 0.218, and 0.323 for KC numbers 3.9, 5.75, and 10, respectively, whereas the minimum normalized scour depth (S/D) for aspect ratios of 1:2 are 0.132, 0.172, and 0.279. Additionally, the research demonstrates that the normalized scour depth increases with an increase in the KC number for a fixed wave–current parameter (Ucw).

Numerical investigation of submerged flexible vegetation dynamics and wave attenuation under combined waves and following currents

Wave attenuation by flexible vegetation is an increasingly significant research area in coastal and ocean engineering. Given the numerical research on flexible vegetation along with the combined waves and currents conditions have received insufficient emphasis, this study seeks to extend the application range of established numerical models to conditions with combined waves and currents, and further quantitatively reveal the effects of colinear following currents on submerged flexible vegetation dynamics and wave attenuation. By comparing with measurement data obtained from previous experimental studies, the reliability of the applied numerical methods in simulating flexible vegetation motions and wave attenuation under combined waves and following currents is confirmed. Scenario simulations illustrate that the Keulegan–Carpenter number and velocity ratio exert substantial nonmonotonic effects on the maximum horizontal force, stem tip excursion, and maximum horizontal stem tip displacement of flexible vegetation. And the presence of the following currents can both enhance and suppress the wave attenuation by flexible vegetation depending on the velocity ratio and water depth. Disparate regularities are revealed as the velocity ratio is higher or less than 3. This investigation can broaden the application of the flexible vegetation model and further advance the understanding of wave attenuation by flexible vegetation.

Computer vision-based measurement of wave force on the rectangular structure

Measurement of the wave force acting on the structure is normally achieved by the traditional load cells. In this study, a non-contact computer vision (CV)-based method is proposed, which can conveniently acquire the wave pressure and force without installing pressure sensors or load cell mounted on the structure. Based on the velocity field measured by the Optical Flow technique, the pressure field is successfully computed by solving the Poisson pressure equation, and the wave force is further determined by integrating the pressure over the surfaces of the structure. The proposed method is verified by conducting a series of laboratory experiments of wave interacting with the rectangular structure. The accuracy of the method is validated by comparing with the results measured by pressure sensors. It indicates that the CV obtained pressure and wave forces generally agree well with the sensor measured results, including the time history of wave force and the peak wave force, and the reasons for the deviation and error are also discussed. The effects of wave height and wave period on the pressure field and wave force are studied, and it is found that the Keulegan–Carpenter number is an important parameter determining the peak wave force. Finally, the limitations existed in this method are noted, including the two-dimensional simplification and air bubble effects, which need to be addressed in the future.

A laboratory study of wave-induced local scour at an emergent pile breakwater

A pile breakwater is a row of closely-spaced piles, placed in a side-by-side arrangement, to dissipate wave energy and prevent coastal erosion. It is important to understand the local scour at a pile breakwater for its foundation design. This paper reports a set of wave-flume tests on the local scour around the piles in regular waves. A laser scanner was used to measure the change in the three-dimensional bed elevation. Five values of Keulegan–Carpenter number were examined in this study. Scour holes near the pile breakwater were developed due to the oscillatory jet flow through the gaps between adjacent piles, and sand ripples were observed on both sides of the structure. The migration of the sand ripples was either onshore or offshore, depending on the wave conditions. The maximum scour-hole depth had an exponential growth with the duration of wave attack, and the equilibrium depth was found to be deeper than that for a standalone pile. Both the scour-hole depth and length were found to be scaled by the pile diameter. The ripple length and steepness were in general agreement with recent empirical formulas in the literature in spite of the presence of a pile breakwater.

An analytical approach of finding out the equilibrium scour depth at a cylindrical pier when the current is making an angle with the wave

The present work describes the development of a mathematical model for the evaluation of the equilibrium scour depth around a single cylindrical pier under the influence of co-existing waves and current. The presented equation includes the effect of the Keulegan–Carpenter number and Froude number on the formation of scour hole. The principle of energy balance is used to form a nonlinear equation of scour, where the energy of incoming flow when collided with the pier is equated with the weight of the sediments moving away from the vicinity of the pier to establish an equilibrium scour depth. An effort has been made to capture the nature of the flow along x-, y-, and z-directions when the current is inclined at an angle θ with the direction of progress of the wave. Further, the scour profile chosen can incorporate any shape and size of the scour hole. The model has been tested against published experimental data pertaining to two cases, namely when the waves are following the current and when the waves are traveling normal to the current. Here, only linear wave is considered while deriving the equation.

Flow-mediated interaction between a forced-oscillating cylinder and an elastically mounted cylinder in less regular regimes

This study investigates the interaction between an actively oscillating cylinder and a passive cylinder elastically mounted with a damper. Both cylinders are rigid, immersed in a viscous fluid, of the same diameter and constrained to move along the two cylinders' centerline. This problem is simulated by an in-house finite-element solver. Six non-dimensional groups are chosen as input: the active cylinder's frequency f1/fn=0.05−3.2 and amplitude A1/D=0.159−1.432, the passive cylinder's damping ratio ζ=0, 0.02 and mass ratio m*=2, the Reynolds number Rem=35−315, and gap distance G/D=2.5. The resulting Keulegan–Carpenter and the Stokes numbers are KC=1−9 and β=35. In total, 2176 combinations are studied in this parametric space. An increase in KC leads to higher irregularity and larger vibration amplitude of the passive cylinder. In regime C, the passive cylinder vibrates in a pulse-beating pattern due to the periodic switching of the streaming direction. In regime E, the passive cylinder responds with intermittent irregularity. In regime F, the flow structure switches intermittently among unrecognizable irregularities and three regular patterns resembling those observed in regimes C and E. In regime G, the flow is highly irregular and circular, where vortices shed from consecutive cycles can merge, forming a much larger one.

Hydrodynamic coefficients of a forced oscillating flexible pipe with energy competition model

The objective of this study is to reveal the characteristics of hydrodynamic force and coefficients undergoing vortex-induced vibration (VIV) of a flexible pipe in an oscillatory flow. The flexible pipe is forced to oscillate with different amplitudes and periods in the still water to simulate the equivalent oscillatory flow. Different from the traditional model, the hydrodynamic forces for the flexible pipe under the new energy competition force model in both in-line (IL) and cross-flow (CF) directions are divided into drag, excitation and added mass force. The identification methods of the hydrodynamic coefficients for the energy competition force model are then established. The detailed features of the hydrodynamic coefficients under the new force model in an oscillatory are illustrated. The distribution comparison of the two forces, i.e., drag force plays a role in energy dissipation and excitation force acts as energy input, explains the differences of the VIV response in magnitudes under different oscillatory flows. By summarizing the hydrodynamic coefficients, the drag and excitation coefficients are found to be inverse proportion to the Keulegan-Carpenter (KC) number. The independence of the added mass coefficients and KC number is revealed. On the basis of the observation, the empirical models for hydrodynamic coefficients are preliminarily proposed. The effects of the oscillatory flow on the hydrodynamics are established to be determined by the KC numbers under the same reduced velocity. The present investigations will provide a better guideline for developing flow-induced vibration prediction methods in future work.