Morison Equation Research Articles

Hydrodynamic coefficients of mussel dropper lines derived from large-scale experiments and structural dynamics

The expansion of marine aquaculture production is driven by a high market demand for marine proteins and a stagnation of wild catch of fish. Bivalve farming, i.e., the cultivation of oysters, mussels and scallops, is an important part of the ongoing market dynamics and production expansion. As marine spatial planning is considering various use purposes, available space for near-shore aquaculture is already becoming scarce; this has fueled research and development initiatives to enable production installations further offshore. The highly energetic conditions at more exposed offshore marine sites lead to increased loads on aquaculture systems and their components and it is still not sufficiently understood how the load transfer from oceanic environmental conditions onto shellfish-encrusted surfaces attached to elastic ropes may be appropriately quantified. This study data gathered large-scale data sets in a wave tank facility, which are used to validate a novel, numerical model, building on the dynamics of rope structures which allows for the determination of the hydrodynamic loads transferred to the dropper lines. The forces and hydrodynamic parameters are measured and numerically analyzed. Based on the results, drag and inertia coefficients are determined. A drag coefficient of CD=1.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\ extrm{D} = 1.1$$\\end{document} and an inertia coefficient of CM=1.7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\ extrm{M} = 1.7$$\\end{document} are recommended to model shellfish-encrusted dropper lines exposed to oscillatory flows with KC =\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=$$\\end{document} 40–90. The numerical model for the determination of wave-induced forces on mussel dropper lines is developed and validated using the experimental data. It employs a modified Morison equation, which takes into account the displacement of the mussel dropper line. The influence of varying aquaculture-related parameters is discussed by applying the numerical model. Based on the gathered insights, recommendations can be given from an engineering point of view concerning the optimal placement of mussel aquaculture within the water column.

Dynamic modeling and rotation control for flexible single-link underwater manipulator considering flowing water environment based on modified morison equation

Flexible single-link underwater manipulator (FSLUM) can be installed in autonomous underwater vehicles, it can carry out underwater tasks such as offshore oil and gas exploration, underwater cable maintenance as well as fishery farming and fishing. The design and development of the FSLUM are of great significance for ocean exploration and development. The FSLUM is affected by the underwater interference torque during underwater tasks in the flowing water environment, which will greatly affect the operational accuracy. In addition, many nonlinear factors such as the flexibility of the transmission system and the two-dimensional deformation of flexible links will aggravate the control difficulty of the FSLUM. An improved sliding mode control (SMC) strategy based on neural network identification is proposed to enhance the FSLUM's control accuracy. Neural networks are proposed to identify uncertain components including underwater interference and flexible nonlinear terms, so as to advance the control law design accuracy. Firstly, the FSLUM dynamic equations coupled with underwater interference torque considering multiple nonlinear factors are established according to the assumed mode method (AMM). A modified Morison equation is proposed to abbreviate the underwater interference torque in flowing water environments. Next, the control law of improved neural network sliding mode control (NNSMC) strategies are proposed according to the Lyapunov stability theorem, so as to ensure closed-loop stability of the FSLUM. Finally, the results of simulation and prototype tracking control experiments show that the improved NNSMC strategy has high tracking accuracy.

Numerical Study on Hydrodynamic Performance of a New Semi-Submersible Aquaculture Platform

To address the limitations of traditional aquaculture cages, such as poor performance in offshore environments, challenges in energy supply, and the inability to integrate modern farming facilities, a new semi-submersible aquaculture platform has been developed and put into operation. The hydrodynamic performance of the platform is analyzed using a fully coupled numerical model. The numerical model incorporates the main structure of the platform, the net system, and the mooring system, and is based on the linear potential flow theory and the Morison equation. The natural period of the platform is found to be far away from the common wave period. The mooring system for the platform in a specific sea area is proven to be safe. The net system significantly affects the hydrodynamic performance and alleviates the dynamic response of the platform. This study provides valuable insights for researchers and engineers in the design and optimization of this type of semi-submersible aquaculture platform in the future.

Variable cross-sectional effect on bi-directional blades–tower–soil–structure dynamic interaction on offshore wind turbine subject to wind–wave loads

BackgroundThis study introduces a numerical model designed to simulate interactions occurring between a wind turbine's tower and the surrounding soil, as well as between the nacelle, blades, and the surrounding environment. This simulation accounts for both fore–aft and side-to-side movements. To describe these interactions, the model leverages the Euler–Lagrange equations. It calculates wave loads utilizing the Morison equation, with wave data generated based on the JONSWAP spectrum. Furthermore, aerodynamic loads are determined using the blade element moment theory, and the wind spectrum is generated using the Von Karman turbulence model. The tower is represented as a variable cross-sectional beam, employing a two-noded Euler beam element with two degrees of freedom: transverse displacement and rotation, and utilizing Hermite polynomial shape functions.ResultsIn a comparative analysis against experimental data, this modified model demonstrates significant enhancements in accurately reproducing the dynamic behavior of wind turbines with variable cross-sectional towers, outperforming models that approximate the tower with a constant cross section. Our findings reveal that the modified model achieves a remarkable improvement of 15% in replicating the tower's dynamic response when compared to the constant cross-sectional models. As a case study, a 5 MW monopile wind turbine with a flexible foundation, specifically the one provided by the National Renewable Energy Laboratory (NREL), is employed to simulate its dynamic response.ConclusionsThis research presents a robust numerical model for simulating wind turbine behavior in various environmental conditions. The incorporation of variable cross-sectional tower representation significantly improves the model's accuracy, making it a valuable tool for assessing wind turbine dynamics. The study's findings highlight the importance of considering tower flexibility in wind turbine simulations to enhance their real-world applicability.

Beaching process of floating marine debris associated with the evolution of the nearshore wave

Floating marine debris (FMD) is a pervasive problem in marginal seas worldwide. Driven by the nearshore waves, the FMD gradually accumulates shoreward and has a large chance of being beached, posing a direct threat to the coastal environment. Thus, investigating the nearshore drifting and beaching process of the FMD is of paramount importance. In this article, the trajectories of the FMD on a sloping beach are simulated by the Smoothed Particle Hydrodynamics (SPH) method, which is pre-verified through laboratory experiments. A series of sensitivity tests are conducted numerically on the influence of attributes of FMD as well as varied wave height (H0), and wave period (T0) on its beaching process. It is found that the beaching process of the FMD can be divided into three steps: drifting in front of surf zones, surfing and leaping with plunging waves, and advancing via wave runup. The density of the FMD combined with wave steepness determines whether the FMD can enter into the surf zone where it has a large chance to beach. Finally, this article proposed a semi-analytical model with improved Morison's equations considering the second-order Lagrangian transport on a sloping beach. This semi-analytical model is much faster and is comparable in computational accuracy to the high-resolution SPH model. It has the potential to be incorporated into the existing marine models and replace the stochastic process assumed for the FMD's tracking in the nearshore, in order to achieve more accurate assessment on the stranded FMD.

Response analyses of a hybrid cable-supported and sea-crossing bridge subjected to pulse-like ground motions and hydrodynamic force

The Hybrid Cable-Supported Bridge System (HCSBS), renowned for its exceptional spanning capabilities, has become a coveted design for sea-crossing structures. Its appeal lies in the integration of the superior attributes of cable-stayed and suspension bridges. However, this system exhibits a high sensitivity and vulnerability to long-duration, pulse-like ground motions (PLGMs) and hydrodynamic forces (HFs) instigated by seawater. This study conducts an in-depth investigation into the dynamic behavior of an advanced and innovative HCSBS structure when subjected to PLGMs and HFs. A 3D Finite Element (FE) model of the Xihoumen Bridge, a sea-crossing structure with a main span length of 1488 m, is developed via the ANSYS software platform. This model aids in assessing the impact of various pulse-like effects, such as the Fling-step effect, directivity effect, pulse amplitude, and pulse period, on the response of the Hybrid Cable-Supported Sea-Crossing Bridge (HCSSCB). Additionally, two well-established techniques, the Morison Equation (ME) and Radiation Wave Theory (RWT), are employed to evaluate the influence of HFs on the response of the intricate HCSSCB, specifically in the vicinity of the pile foundation. It is worth noting that the ME may overestimate the HFs acting on the pile foundation. Furthermore, it is revealed that when impulse seismic action is considered, HFs can significantly amplify the displacement and force requirements of the HCSSCB.

Dynamic Analysis of an Articulated Offshore Tower Using Stokes Fifth Order Nonlinear Wave Theory

In offshore engineering, a compliant offshore structure accommodates dynamic forces by flexibility rather than rigidly withstanding load effects. As an alternative to conventional fixed platforms for loading/mooring in deep waters, articulated offshore towers are highly appealing because of their reduced structural weight. This study presents the results of a probabilistic random wave with and without current force on a 500 m high articulated tower with two hinges. By using the Lagrangian approach, governing nonlinear equations of motion are derived. The motion equation has been resolved in the time domain with the Houbolt approach. Based on Stokes fifth order nonlinear wave theory, wave forces have been assessed. Hydrodynamic forces are determined using a modified Morison equation with stretching modifications developed by Chakrabarti. The study examined two marine states, i.e., 10 m/10 s and 15 m/15 s. The current intensity of 2 m/sec has been taken into account. The salient statistical parameters such as maximum, minimum, mean, and standard deviation are presented under both the ocean environments. The proposed model is feasible for both the sea states. When comparing sea states, those with a greater wave height and period and those with a smaller wave height and period have more energy in their PSDs for all responses.

Three-Dimensional Numerical Simulation of the Water Flow Effect on a Temporary Pier with Three Columns in Series

In order to ensure the safety of temporary piers during bridge construction, it is necessary to analyze the water flow effects on temporary piers during flood periods. Based on the practical engineering of a new bridge in Jiangxi province, a three-dimensional numerical model was established using the Reynolds Time Average Navier–Stokes equation (RANS) and SST k-ω turbulence model to simulate the flow force on three-column temporary piers in series. The stress state of the temporary pier was obtained under different flow velocities and submergence depths. The distribution characteristics of the flow field around the temporary pier and the flow force characteristics acting on the temporary pier were analyzed and compared with the monitoring measurement data and the Morison equation calculation data. The results indicate that the local flow was accelerated on both sides of the temporary pier, but the local acceleration effect along the river flow direction was weakened. The flow velocity of the water around the temporary pier decreases with the increasing of river depth. The pressure on the outer surface of the temporary pier is basically negative pressure, except for the positive pressure on the upstream side. The absolute value of the negative pressure on both sides is much greater than the positive pressure on the upstream side. As the submergence depth and flow velocity increase, the pressure in each area gradually increases. The pressure distribution of the upstream pier is basically symmetrical, while the pressure distribution of the middle and downstream piers is asymmetric due to the influence of the vortex shedding from the front pier, resulting in a change in the water flow incidence point. The turbulence intensity in each region of the flow field shows a trend of first increasing and then decreasing with the depth of the water flow. The drag coefficient and lateral force coefficient of the temporary pier are basically not affected by the inflow velocity, but the resistance and lateral force increase with the increase in submergence depth and inflow velocity.

Efficient measurement of floating breakwater vibration and controlled vibration parameters using compressive sensing

In this paper, we examine the possible usage of compressive sampling (CS) for the efficient measurement and analysis of floating breakwaters dynamics. We model the dynamics of a box-type floating breakwater in a random sea environment with the Bretschneider-Mitsuyasu spectrum. The spectral method involving FFT and IFFT routines is used for the simulation and CS reconstruction purposes of the displacement and velocity parameters. The time series of forces are calculated using the Morison equation. It is also shown that those vibration parameters can be limited and suppressed to a certain degree by a tuned-mass-damper. The advantages of using the CS for the tuned-mass-damper-controlled vibration parameter measurement and analysis are also discussed. Our findings may provide convenience for the effective and efficient strategies for the vibration-induced problems of floating platforms including wave energy converters and offshore platforms.

Hurricane Wave Loads on Spar-Type Floating Wind Turbines: A Comparison of Simulation Schemes

Floating wind turbines are sensitive to hurricane events. Since the turbine rotors are parked and the blades are feathered during hurricanes, the aerodynamic loads due to boundary-layer winds are relatively small compared to the hydrodynamic loads due to sea surface elevations. Hence, accurate modeling of the hurricane wave loads is crucial to ensure the safety of floating wind turbines. During a hurricane, large wave heights with severe flow separation make it inaccurate to use either linear panel method-based models (without nonlinear consideration associated with fluid viscosity) or Morison equation-based models (without unsteady consideration associated with fluid memory). Efforts have been made to advance simulation schemes of hurricane wave loads on spar-type floating wind turbines. This study systematically compares and assesses the efficacy of six hydrodynamic models available in the literature along with a newly proposed model. The ability of these seven hydrodynamic models to capture nonlinear and/or unsteady effects is investigated. As a demonstration example, the wave loads on a spar-type wind turbine are calculated using these seven models to highlight the underlying role of each simulation scheme in accurately acquiring the dynamic responses of this type of offshore floating structure in severe hurricane seas. It is found that the nonlinear viscous term in the Morison equation and hybrid model serves as an important nonlinear damping mechanism. The reduction of the low-frequency wave load and added mass in the modified hybrid model collectively leads to larger displacements compared to those based on the hybrid model. While the displacements based on the stretching method and Rainey’s equation are similarly larger than those based on the Morison equation, their nonlinear wave loads are much smaller than those in FNV theory.

A semi-analytical hydrodynamic model for floating offshore wind turbines (FOWT) with application to a FOWT heave plate tuned mass damper

This paper proposes a semi-analytical MacCamy–Fuchs–Morison (MFM) hybrid hydrodynamic model for very efficient simulations of FOWT responses without resorting to potential flow solvers. First, closed-form expressions of the 6-by-6 added mass matrix and 6-dimensional complex wave load transfer function are derived based on the Morison equation. Comparison of the wave load models show that the pure Morison model over predicts high-frequency wave loads above the diffraction limit, while the pure MacCamy–Fuchs model performed poorly at low frequencies due to the existence of heave plates. The MFM hybrid model resolves both issues and gives good agreement with the potential flow theory at all frequencies. The proposed MFM hybrid model is then employed to a representative application case, i.e. a semi-submersible FOWT installed with heave plate tuned mass dampers (HPTMDs). It is shown that implementing a deep installation of the HPTMDs effectively suppresses foundation roll, heave responses and tower side–side vibrations.

Dynamic analysis of offshore triceratops supporting wind turbine: Preliminary studies

Offshore triceratops, the recent innovation in deep water compliant platforms, is primarily designed to withstand lateral forces through its geometric shape. The uniqueness of the platform is the presence of the ball joints between the legs and deck, which partially restrain the transfer of rotations from the legs to the deck and vice-versa. However, displacements such as surge, sway, and heave motions are transferred, ensuring a rigid connection between the legs and the deck. Efficient operations of offshore wind turbines are more dependent on the support systems on which they are mounted. Increased stability and reduction in stress concentration in rigid connections are desirable. Nonlinear dynamic response analysis is carried out in FAST by coupling the frequency response of the platform obtained in ANSYS AQWA with that of the HydroDyn module of FAST. The current study investigates a fully coupled three-dimensional hydro-aerodynamic model of triceratops mounted with a horizontal axis wind turbine. Unsteady Blade Element Momentum theory (BEM) is used to estimate the aerodynamic loads, which encompasses the effect of wind shear using a power law and spatially coherent turbulence. In contrast, Morison equations are used to estimate the hydrodynamic loads on the platform. After the preliminary proportioning of the platform, Response Amplitude Operator (RAO) plots are drawn to illustrate the partial motion transfer between the deck and the buoyant legs. Based on the preliminary studies, it is seen that the environmental loads do not impose instability, reinforcing the dynamic stability of the platform. Frequency responses for operating and parked conditions illustrate the coupling between the degrees of freedom and the influence of the rotor motion of the wind turbine on the platform deck. Tether tension variation is assessed in all three legs for the operational sea states to check the safety standards for a compliant system to avoid tether pull-out. The presented study is prima facie to encourage the suitability of triceratops as floaters to support the wind turbine under moderate sea states. Highlights This study is focused on new-generation offshore triceratops as support system for wind turbine Prelimiary dynamic anlaysis of coupled action of the supprting system and wind turbine are presented Use of ball joints help partial isolation of the deck and restrain transfer of moment from the turbine shaft to the supporting system, which is a novelty Infleunce of rotor motion of the triceratops is illustarted to highlight the advantage of complinacy of trirceratops This study disucsses only the performance assessment and not the design perspectives

Development of an efficient calculation technique for dynamics of mooring lines by using discrete forms of rotation

Development of efficient calculation techniques is of key importance for design of deepwater moored floating structures consisting of many subsystems, such as a floating vessel, risers and mooring lines. In particular, this study focuses on a method for simulating dynamic behavior of the mooring line, since it includes a lot of degrees of freedom and geometric nonlinear effects. A modified Morison equation based on the linear potential flow theory is employed to derive the hydrodynamic forces acting on the lines. The present approach describes the mooring lines (chain) as multi-rigid bodies based on the multibody dynamics approach. In addition, a discrete form of rotation called “the rotation update equation” is introduced. This technique approximates the rotation by using incremental components of rotation in terms of the angular velocity. This approach can bypass the parameterization of rotation in the equations of motion. Clearly, it contributes to reduction of computational costs. Furthermore, a preconditioning method is applied to the equations of motions in order to prevent the ill-conditioning which affects convergence of the numerical integration procedures for the set of equations with configuration level constraints. Finally, a comparison of results by the present method and the conventional method are presented in order to show the validity of the method.

Wave–current-structure blockage: Validation of one-way hydro-structural coupling with large-scale model tests of a stiffness-similar jack-up in elevated condition

This paper documents extensive comparison between one-way hydro-structural coupling model (OF-ABAQUS) and key findings from the physical model tests of a jack-up performed in TCOMS Ocean Basin. A generic jack-up model in elevated condition is considered at 1:30 scale. A series of model tests involving waves and current is performed on one of the jack-up legs for calibration of hydrodynamic coefficients, as well as on the whole jack-up for characterisation of its dynamic response. With the novel use of polycarbonate rods for the jack-up legs, stiffness similarity can be maintained at model-scale which allows direct strain measurements on the legs and meaningful connection force measurements at the spudcan modules and leg-to-hull connection modules. Using the OF-ABAQUS coupling model, the jack-up legs are first numerically represented as porous blocks and simulated in OpenFOAM to derive the hydrodynamic forces and additional fluid damping following Santo et al. (2018a), where static simulations are performed. The forces and damping are subsequently mapped into a Finite Element (FE) model of the jack-up in ABAQUS as a one-way transfer to derive its dynamic response following Li et al. (2022), where dynamic simulations are performed. The novelty of this paper is on the demonstration of the coupling model in capturing dynamic structural modes and responses in multiple degree-of-freedoms (DOFs) which is more realistic, unlike the earlier work by Santo et al. (2018a) who demonstrated the coupling model for an idealised single DOF system.Comparisons with model test results yield reasonably good agreement for all the cases considered, covering not just bending moment at one of the spudcan modules but also strain and leg-to-hull connection force of one of the legs. From the dynamic response spectra, the total system damping at the surge resonance frequency is adequately represented using the coupling model, which accounts for the additional fluid damping contributed by the effect of wave–current-structure blockage, or more precisely by the relative velocity. Most importantly, the use of single and consistent set of hydrodynamic coefficients for the derivation of hydrodynamic forces and additional fluid damping is demonstrated throughout all the cases. In contrast, the use of standard Morison equation will require tuning of the hydrodynamic coefficients and arbitrary increase in the hydrodynamic damping to match the measured peak response of each wave condition, resulting in huge uncertainties when evaluating the structural reliability in larger and unseen sea-states. This paper demonstrates that, once the hydrodynamic inputs are adequately represented, the corresponding dynamic structural responses can be derived accurately for any sea-states. The proposed approach can be regarded as a step improvement in accurate prediction of hydrodynamic forces and dynamic responses of space-frame structures, which has its important applications covering new-builds, re-assessment of old platforms, and structural health monitoring or digital twinning of offshore assets.

Numerical study on the forces and moments on a bottom-mounted cylinder by internal solitary wave

The forces and moments on a bottom-mounted cylinder by internal solitary wave are investigated with numerical simulation. The solution of Dubreil–Jacotin–Long equation is used to initialize internal solitary wave in the computational domain, which overcomes the deficiencies of internal solitary wave generated by gravity collapse method. The continuous density stratification is taken into account in this study and numerical results show that the shape of pycnocline could influence the forces and moments exerted on cylinder by internal solitary wave. The forces and moments on different cylinder sections are calculated separately, the derived forces exhibit similar patterns observed in previous laboratory measurements. The results show that the loads by internal solitary wave on cylinder are quite complex as the wave induced velocity field is unsteady and the directions are opposite between upper and lower layers. The performance of Morison equation is examined and three coefficient determination methods are tested, force based global coefficients, moment based global coefficients and local coefficients. The analysis shows that the global coefficients are sensitive to the data adopted, which explains why universal coefficients are not available from previous studies.

Mitigating deepwater jacket offshore platform vibration under wave and earthquake loadings utilizing nonlinear energy sinks

The random vibrations of jacket offshore platforms under wave and earthquake actions can reduce the service life and exacerbate fatigue damage of the offshore platforms. Aiming to mitigate those unwanted vibrations, this paper introduces the nonlinear energy sinks (NES) to the deep-water jacket offshore platform as a passive control device. Its coupling dynamic model is built with assumed parameters of wave and earthquake loads, and the wave forces of the platform structure are calculated by Morison's equation. Further, the optimal parameters of the NES system are obtained by an exhaustive search algorithm under the objective function. The results show that the NES system can effectively contribute to mitigating the maximum, minimum, and root mean squares (RMS) of deck-level responses of the platform structure. What's more, despite various mass or stiffness of platform structure, the NES system not only greatly reduces the response peaks, but also demonstrates a more steady phenomenon, in which the robustness is presented clearly. In this way, it strongly proves the feasibility and robustness of the NES system under wave and earthquake loadings.

Study on the hydrodynamic added mass of rectangular pier under sine wave actions

As for deep-water bridges in river, sea, and reservoirs, the water-structure interaction is non-negligible when investigating its seismic performance. Morison equation and radiation wave theory are two main methods for calculation of hydrodynamic pressure resulted from water-structure interaction, which cannot truly simulate the complicated interaction between water and structures. This study conducts a series of underwater shaking table tests on seven rectangular steel tubes to predict the hydrodynamic added mass of rectangular piers considering water-pier interaction. The influence of action amplitude, action frequency, water depth, depth of cross section, and width of upstream face of the pier on hydrodynamic added mass is investigated. The results show that the hydrodynamic added mass of pier first decreases and then increases with action frequencies. The hydrodynamic added mass is proportional to a-th (a > 1) power of water depth due to the influence of structures on the movement of water. The hydrodynamic added mass increases with depth of cross section and width of upstream face of pier, and changes slightly with the action amplitude with maximum percentage of 10% and thus can be neglected. Therefore, except for the water depth, depth of cross section, and width of upstream face, the frequency characteristics of ground motions should be considered when analyzing the water-pier interaction under earthquake actions.

Analysis of hydrodynamic forces on freestanding risers with buoyancy cans

Free-standing hybrid risers are vertical slender steel pipes that are commonly used in the oil and gas industry. These slender structures are subjected to complex hydrodynamic loads that results in fatigue damage. The tidal forces that develop on Freestanding Risers (FSR) with buoyancy cans (BC) because of varying outer pipe dimensions are investigated in this work. The study considered riser diameters of 0.2, 0.3, 0.5 and 0.6 meters as well as buoyancy can diameters ranging from 2, 3, 5 and 6 meters. The Morison equation was used in this study to investigate the wave hydrodynamic forces. Separate assessments of the lift force are also carried out consider a water depth (d) of 30 m. The findings reveal that the pipe diameter, wave heights and the wave periods have significant impacts on the hydrodynamic forces on the riser.

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.

Forces of fully nonlinear interfacial periodic waves on a cylindrical pile in a two-layer fluid with free-surface boundary conditions

In the frame of fully nonlinear potential flow theory, series solutions of interfacial periodic gravity waves in a two-layer fluid with free surface are obtained by the homotopy analysis method (HAM), and the related wave forces on a vertical cylinder are analyzed. The solution procedure of the HAM for the interfacial wave model with rigid upper surface is further developed to consider the free surface boundary. And forces of nonlinear interfacial periodic waves are estimated by both the classical and modified Morison equations. It is found that the estimated wave forces by the classical Morison equation are more conservative than those by the modified Morison’s formula, and the relative error between the total inertial forces calculated by these two kinds of Morison’s formulae remains over 25% for most cases unless the upper and lower layer depths are both large enough. It demonstrates that the convective acceleration neglected in the classical Morison equation is rather important for inertial force exerted by not only internal solitary waves but also interfacial periodic waves. All of these should further deepen our understanding of internal periodic wave forces on a vertical marine riser.