Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions

Homogeneous-heterogeneous reactions in boundary-layer flow of a nanofluid near the forward stagnation point of a cylinder

A mathematical model describing the homogeneous-heterogeneous reactions in the vicinity of the forward stagnation point of a cylinder immerged in a nanofluid is established. We assume that the homogeneous reaction is given by isothermal cubic autocatalator kinetics, while the heterogeneous reaction is chosen as first-order kinetics. The existence of multiple solutions through hysteresis bifurcations is discussed in detail for the various diffusion coefficients of reactant and autocatalyst

LH-moment estimation for statistical analysis on the wave crest distributions of a deepwater spar platform model test

The design of fixed and compliant offshore platforms requires the reliable estimation of extreme values with small probabilities of exceedance based on an appropriate probability distribution. The Weibull distribution is commonly utilised for the statistical analysis of wave crests, including near-field wave run-ups. The parameters are estimated empirically from experimental or onsite measurements. In this paper, the data set of wave crests from a Spar model test was statistically analysed by using the method of LH-moments for parameter estimation of the Weibull distribution. The root-mean-square errors (RMSEs) and the error of LH-kurtosis were used to examine the goodness-of-fit. The results for the first four LH-moments, the estimated parameters, and the probability distributions showed that the level of the LH-moments has a significant influence. At higher levels, the estimation results gave a more focused representation of the upper part of the wave crest distributions, which indicates consistency with the intention of the method of LH-moments. The low tail RMSE values of less than 2.5% demonstrated that a Weibull distribution model estimated by using high-level LH-moments can accurately represent the probability distribution of large extreme wave crests for incident waves, wave run-ups, and moon pool waves. Goodness-of-fit test on the basis of comparison of sampling LH-kurtosis and theoretical LH-kurtosis was recommended as a procedure for selecting an optimum level

Time-domain analysis of substructure of a floating offshore wind turbine in waves

This paper aims to analyze the dynamic response of a floating offshore wind turbine (FOWT) in waves. Instead of modeling the incident random wave by the traditional wave spectrum and superposition theory, an impulse response function method was used to simulate the incident wave. The incident wave kinematics were evaluated by a convolution of the wave elevation at the original point and the impulse response function in the domain. To check the validity of current wave simulation method, the calculated incident wave velocities were compared with analytical solutions; they showed good agreement. The developed method was then used for the hydrodynamic analysis of the substructure of the FOWT. A direct time-domain method was used to calculate the wave-rigid body interaction problem. The proposed numerical scheme offers an effective way of modeling the incident wave by an arbitrary time series

Hydrodynamic coefficients of oscillating flat plates at 0.15 ⩽ KC⩽ 3.15

This article presents an experimental investigation on the hydrodynamic coefficients of oscillating flat plates. The plates are forced to oscillate harmonically in still water. The range of Keulegan–Carpenter number (KC= 2 πa/ D, where a is the single amplitude of oscillation and D is the equivalent diameter of the plate) is 0.15 ⩽ KC⩽ 3.15. The hydrodynamic forces acting on the plates are measured and the hydrodynamic coefficients including added mass and damping coefficients are calculated using the Morison’s equation. The influences of the thickness ratio, shape, edge corner radius, perforation ratio and hole size on the hydrodynamic coefficients of a single plate are analyzed and presented. For the twin- and triplet-plate configurations, the spacing effects are also evaluated

Flow around an oscillating circular disk at low to moderate Reynolds numbers

Direct numerical simulations of the flow induced by a circular disk oscillating sinusoidally along its axis are performed. The aspect ratio of the disk is 10. The Reynolds number , based on the maximum speed and the diameter of the disk, is in the range of . The Keulegan-Carpenter number is in the range of . Five flow regimes are observed in the considered-space: (I) axisymmetric flow (AS), (II) planar symmetric flow in the low-region (PSL), (III) azimuthally rotating flow in the low-region (ARL), (IV) planar symmetric flow in the high-region (PSH) and (V) azimuthally rotating flow in the high-region (ARH). The critical boundaries between different flow regimes are identified based on the evolutions of the magnitude and direction of transverse force acting on the disk. For the non-axisymmetric flow regimes, the flow is one-sided with respect to the axis of the disk and is associated with a non-zero mean value of the transverse force acting on the disk

Performance characteristics of a conceptual ring-shaped spar-type VLFS with double-layered perforated-wall breakwater

A ring-shaped spar-type Very Large Floating Structure (VLFS) is proposed as an intermediate base for supporting deepwater resource exploitation far away from the coast line. The proposed VLFS is composed of eight rigidly connected deep-draft spar-type modules and an inside harbor. A double-layered perforated-wall breakwater is vertically attached to the VLFS and pierces through the water surface to attenuate the wave energy in the inside harbor. The hydrodynamic performance characteristics of the ring-shaped VLFS was experimentally evaluated in the present study, focusing on the motion responses, wave elevations, and wave run-ups. The natural periods of the motions in vertical plane were determined to be larger than 40s, which is much larger than common wave periods. This enhanced the motion performance in vertical plane and afforded favorable habitation and operation condition on the VLFS. A large surge damping was induced by the vertical breakwater, which tended to significantly affect the surge and pitch motions, but had a negligible effect on the heave motion. The component frequencies of the wave elevations in the inside harbor and the wave run-ups were identical with those of the incident waves. The wave attenuation was frequency-dependent and effective for the common wave frequencies, with a smaller loss coefficient observed in higher sea state. The wave attenuation and wave run-ups tended to improve in the absence of the leeward walls

Energy transformation on flow-induced motions of multiple cylindrical structures with various corner shapes

A comprehensive numerical study on flow-induced motions (FIMs) of a deep-draft semi-submersible, a typical multiple cylindrical structure in offshore engineering, was carried out to investigate the energy transformation of the vortex shedding process. In addition, the corner shape effect on the flow characteristics, the hydrodynamic forces, and the FIM responses are presented for a multiple cylindrical structure with various corner shapes (sharp, rounded, and chamfered) under 45○ current incidence. Different energy transformations, hydrodynamic characteristics, and FIM responses were observed due to the slight variation of the corner shape. The galloping at 45○ incidence for a square section shape column was observed when the corner shape modified as a chamfered corner. A “re-attached vortex shedding” phenomenon is discovered when the “lock-in” happened for a chamfered corner design. Further insights of the fluid physics into the flow characteristics due to the difference of the corner shape are revealed. In addition, the energy transformation and the mechanism for reducing the hydrodynamic forces and the FIM responses are analyzed

Fatigue assessment of flange connections in offshore wind turbines under the initial flatness divergence

Bolted ring flange connections are widely utilized in offshore wind turbines to connect steel tubular segments. After the massive production and installation of offshore wind turbines in the past decade, flatness divergence is regarded as one of the most important initial imperfections for the fatigue design of flange connections. Offshore wind turbines are subjected to wind, wave, and current loads. This initial imperfection may alter the structural response and accelerate the fatigue crack growth. This paper aims to analyse the impact of the initial flatness divergence on the structural response of flange connections and evaluate its consequences on fatigue damage. Two different offshore wind turbines with fixed foundations and floating foundations are modelled to simulate their global responses to environmental loads. Based on a superposition method, local finite-element models of flange connections are established with three types of flatness divergence. Using the same bolt pretension and external loads from global modelling, the impact of these geometric imperfections is further examined by comparing the structural responses of flanges under different radial and peripheral opening lengths. Then, the fatigue assessments on flange connections in both fixed wind turbines and floating wind turbines are conducted, and the impacts of initial flatness divergence on these two different wind turbines are analysed

Vortex-induced-vibration of jack-ups with cylindrical legs in multiple modes

A simple mathematical model was developed based on the single-degree-of-freedom analogy and principle of conservation of energy evaluating various modes of Vortex-Induced-Vibration (VIV) of a jack-up with cylindrical legs in steady flow. Mass ratio, damping ratio and mode factor were found to be the important parameters controlling the inline and cross flow VIV and radius of gyration for the yaw VIV. Criteria for the initiation of the three VIV modes were developed for the cases of a single 2D cylinder, four rigidly coupled 2D cylinders in rectangular configuration and a jack-up experiencing uniform flow. The model tests demonstrated that the jack-up with cylindrical legs experienced cross flow and yaw VIV in uniform flows, with amplitude ratios greater than 0.1D. Further, there was considerable overlap of the lock-in ranges and coupling at higher current speeds of the aforementioned modes making the jack-up practically redundant throughout the operating currents. The analysis of the mean inline responses of the model revealed drag amplification due to the VIV. The test results validated the developed VIV model, VIV criteria and the importance of mass ratio in suppressing VIV. The mathematical method will enable practising engineers to consider the effect of VIV in jack-up designs