Numerical study of second-order wave forces using a Rankine panel method

Influence of second-order wave force and focusing position on dynamic responses of tension leg platform under a freak wave

Dynamic analysis of bridge structures under combined earthquakes and wave loadings based on a simplified nonlinear Morison equation considering limit wave steepness

In this paper, the second-order long period wave forces acting on a two-dimensional fixed obstacle of arbitrary shape in bichromatic waves in a finite depth water are investigated. In the evaluation, the second-order effect of a wave maker is included to satisfy the precise boundary condition in an experimental tank. The solution of the second-order diffraction problem is obtained by using hybrid type boundary integral equation method. One of the merit of this method is that the complicated second-order free surface condition can be treated easily by using the analytical form of second-order velocity potential which was derived in 1st report.Computations are done for a half immersed circular cylinder and the calculated results are compared with the results obtained from Green's second identity. These results are also compared with experimental values and the validity of the present method is discussed.

The second-order wave force is analysed for diffraction of monochromatic water waves by a vertical cylinder. The force is evaluated directly from pressure integration over the cylinder, and the second-order potential is derived by Weber transformation of the corresponding forcing function on the free surface. This forcing function is reduced to a form which involves a simple factor inversely proportional to the radial coordinate plus an oscillatory function which decays more rapidly in the far field. This feature alleviates the slow rate of convergence involved in capturing the far-field effect. Benchmark computations are obtained and compared with other works. Asymptotic approximations are derived for long and short wavelengths. The analysis and results are primarily for the case of infinite fluid depth, but the finite-depth case is also considered to facilitate comparison with other computations and to illustrate the importance of finite-depth effects in the long-wavelength asymptotic regime.

Standard analysis models applied for motions of moored floaters are based on potential flow perturbation methods with wave frequency response governed by first order wave forces and low-frequency response governed by second-order difference frequency wave forces. These models have been shown to have limitations in extreme sea states where nonlinear wave excitation and viscous drag forces above still water level may dominate. This effect is particularly visible for the low frequency excitation since the potential flow contribution goes to zero for long waves. In the present study non-linear wave excitation and viscous drag contributions on a semi-submersible is modelled by Morison’s load formula since the columns and pontoons are slender elements. A numerical simulation model is developed using SIMO [6], in which viscous forces and damping are included by the drag term of Morison equation and with drag coefficients recommended from DNV-RP-C205 [1]. Low frequency surge responses calculated by the combined potential flow drift forces and viscous drag from Morison load model are compared with model tests for waves only and for combined wave and current conditions. A simplified formula for current and viscous effects on wave drift force, generalized to non-collinear conditions is presented and compared with model test results.

Motion responses of a moored barge in shallow water

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves was put forward. The flow field was divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain was surrounded by a ship-wetted surface and a matching surface as well as part of the free surface. The free-surface condition for the inner domain was formulated by perturbation about the double-body (DB) flow assumption. The outer domain was surrounded by a matching surface and the rest had a free surface as well as an infinite far-field radiation boundary. The free-surface condition for the outer domain was formulated by perturbation of the uniform incoming flow. The simple Green function and transient free-surface Green function were used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. The Taylor expansion boundary element method (TEBEM) was adopted to solve the DB flow and inner-domain and outer-domain unsteady flow BIE. Matching conditions for the inner-domain flow and outer-domain flow were enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on the ship-wetted surface was applied to obtain the first- and second-order wave forces. The numerical prediction on the displacement, acceleration and added resistance of the 14000-TEU container ship at different forward speeds were investigated by the proposed TEBEM method. Reynolds-Averaged Navier–Stokes (RANS) equations based on Computational Fluid Dynamics (CFD) method were adopted to compare with TEBEM method. The physical tank experiment results also validated the accuracy of the numerical tank results. Compared with the experimental solutions, TEBEM obtained good agreement with the RANS CFD method. TEBEM, however, was much more efficient and robust.

A complete semi-analytical solution is given for second-order diffraction of monochromatic waves by a truncated vertical circular cylinder in water of uniform finite depth. The methodology presented in detail elsewhere (Eatock Taylor & Huang 1996) is adopted to find a particular solution which exactly satisfies the governing equation, the inhomogeneous free-surface condition and the seabed condition. In order to satisfy the boundary condition on the cylinder bottom, the fluid domain around the cylinder is divided into two regions. First- and second-order velocity potentials are described separately in the two regions and matched on the interface by the pressure and normal-velocity continuity conditions. Based on the formulation, the second-order wave field in the vicinity of the cylinder and the corresponding wave forces and overturning moments on the cylinder are studied in detail. Numerical results for the double frequency forces obtained by using the present semi-analytical approach are compared with those computed with a higher-order boundary element method (Eatock Taylor & Chau 1992). As well as the exact solution, an approximate solution is also given for the second-order potential and the corresponding forces. Numerical results show that the approximate solution possesses excellent accuracy for the total second-order heave force over a wide range of conditions. When kb > 1.2 (where k, b are the incident wavenumber and the draught of the cylinder respectively), the accuracy for total second-order surge force and pitch moment is also satisfactory. These results could lead to the development of very efficient solutions and corresponding algorithms for the analysis of second-order wave diffraction by more complicated structures such as tension leg platforms. Numerical results based on the present solution show that in many cases, both the first- and the second-order-free surface elevation in the vicinity of a truncated cylinder is very close to that of a bottom-seated cylinder. For waves with larger amplitudes, the maximum free-surface elevation around a vertical cylinder predicted with the second-order theory can significantly exceed that given by linear theory. There is also a considerable difference in the location of the maximum elevation predicted by the linear and nonlinear theories.

Shallow-water mooring chains of floating offshore wind turbines pose challenges such as nonlinear tension characteristics, high cost and prominent fatigue issues, which seriously hinder the large-scale use of floating wind power. Based on the hydrological and meteorological data of an offshore wind farm in the South China Sea, a sensitivity analysis of the mooring chain fatigue of a semi-submersible floating offshore wind turbine in shallow water was conducted with different environmental loads and mooring chain design parameters. It was found that mooring fatigue is mainly caused by the wave condition within diagonal distribution of the wave dispersion diagram. The mooring chain fatigue calculated using the main monochromatic wave was more conservative than that calculated using the wave scatter diagram with a difference of about 28%. The mooring fatigue calculated using turbulent wind spectrum was greater than that calculated using steady wind. In contrast, waves are the main source of mooring fatigue, and the second-order wave force has an undeniable impact on the mooring fatigue of a semi-submersible floating offshore wind turbine, which accounts for approximately 26% of wave-induced mooring fatigue. In addition, the current-induced mooring fatigue should not be neglected, although its effect is not as large as that caused by the wave or wind loads. An increase in the cross-sectional diameter and the length of a mooring chain, as well as a reasonable clump weight arranged, all contribute to improving the fatigue life of mooring chains in shallow water, which provide useful technical references for the optimization of mooring lines fatigue in shallow-water areas.

The fully coupled aerodynamic-hydrodynamic-mooring numerical model of floating offshore wind turbine(FOWT) has been established. The complex conditions such as second-order wave force, full-field turbulent wind, and mooring break have been taking into account. Using OrcaFlex and FAST to calculate the dynamic response of FOWT under the combined action of wind, wave and current. The time-domain response of floating foundation, mooring tension, blade root contact force, aerodynamic power, and wind rotor thrust are obtained. And the effect of the turbulent wind and mooring break at different positions on the FOWT are studied. The results show that compared with the platform motion and mooring tension, the aerodynamic power and rotor thrust are more sensitive to the turbulent wind, and there are fluctuations in amplitude and multiple peaks in the frequency spectrum. Compared with the leeward side, the mooring break on the windward side is a more dangerous situation, and it is easy to cause the collapse of platform and the break of the remaining mooring. The mooring break on the leeward side is relatively safe, and it will produce a drift in the sway direction, and have little impact on the safety of the wind turbine.

Research on Second-Order Low Frequency Wave Forces on Floating Circular Cylinder in Shallow Water

Evaluation is performed of the data used for T2015 test cases for the KCS captive added resistance σAR and ONRT free running course keeping/speed loss in head and oblique waves. For KCS calm water resistance, the individual facility N-order level testing uncertainty is UXi = 1%D and the multiple facility standard deviation based on three institutes using 2 model sizes (L = 7.3 and 6.1 m) is SD = 0.74%D, such that the individual facility MxN-order level testing uncertainty is UDi = 1.75%D. UXi was not reported for sinkage and trim; however, the SD = 4 and 7%D, respectively. For KCS head waves, the analysis was based on three institutes with model sizes L = 6.1, 3.2 and 2.7 m. For the 6.1 m model, FORCE provided UXi = 8, 4 and 4%D, respectively, for σAR and first harmonic heave z1/ζ1 and pitch θ1/ζ1k amplitudes, whereas for the 2.7 m model, FORCE/IIHR provided UXi = 7/18, 8/2 and 9/5%D, respectively. In consideration of the differences in model sizes and rigid vs. surge free mounts the agreement between the facilities is reasonable: for the primary variables, SD = 3, 20 and 10%D for z1/ζ1 and θ1/ζ1k amplitudes and σAR, respectively; and for secondary variable SD = 66%D for first harmonic resistance CT1. For KCS oblique waves, the data is only available from IIHR for tests in 2015 and 2016. The agreement is good, i.e., SD values are comparable to the corresponding values for head waves: for primary variables, SD = 6 and 63%D for z1/ζ1 and θ1/ζ1k amplitudes and σAR, respectively; and for secondary variable SD = 13%D for CT1. For ONRT self-propulsion and head and oblique waves, the data is only available from IIHR and only preliminary uncertainty analysis is available. The evaluation showed reasonably reliable data for both KCS and ONRT. However, clearly it is desirable to have data from more facilities including uncertainty analysis for assessment of facility biases, which will provide more robust data and uncertainty analysis for CFD validation.

A boundary-integral method is developed for computing first-order and mean second-order wave forces on floating bodies with small forward speed in three dimensions. The method is based on applying Green's theorem and linearizing the Green function and velocity potential in the forward speed. The velocity potential on the wetted body surface is then given as the solution of two sets of integral equations with unknowns only on the body. The equations contain no water-line integral, and the free-surface integral decays rapidly. The Timman-Newman symmetry relations for the added mass and damping coefficients are extended to the case when the double-body flow around the body is included in the free-surface condition. The linear wave exciting forces are found both by pressure integration and by a generalized far-field form of the Haskind relations. The mean drift force is found by far-field analysis. All the derivations are made for an arbitrary wave heading. A boundary-element program utilizing the new method has been developed. Numerical results and convergence tests are presented for several body geometries. It is found that the wave exciting forces and the mean drift forces are most influenced by a small forward speed. Values of the wave drift damping coefficient are computed. It is found that interference phenomena may lead to negative wave drift damping for bodies of complicated shape.

\n This thesis describes the development of METiS, a numerical tool for the seakeeping analysis of floating offshore wind turbines (FOWT). It is based on a slender-body approximation for evaluating the first- and second-order wave loads acting on a floating structure comprised of slender cylinders that combines Raineys equation, which can be seen as an extension of the inertial part of Morisons equation to include nonlinear terms, with Pinksters formulation for the low-frequency second-order loads on floating bodies. This combination is followed to allow the evaluation of the forces considering the mean body position, in such a way that an Inverse Fast Fourier Transform algorithm can be used to efficiently compute the time series of wave kinematics and second-order wave loads in a real sea condition directly in time domain, as opposed to the most common procedure of solving the second-order radiation/diffraction problem in frequency domain and importing the results to time-domain solvers. As a drawback of the approximation, end effects due to the extremities of the cylinder and effects due to wave scattering and radiation are lost, which is acceptable as long as the diameters of the cylinders that compose the structure are small in face of their draft and in face of the length of the incoming waves. These conditions may be too restrictive to modern oil & gas spars and semi-submersibles, which have large diameter columns, but it is satisfied by FOWTs in many significant wave conditions. The slender-body approximation is first verified by analyzing the simple case of a single surface piercing cylinder under the action of bichromatic waves, both bottom mounted and floating, for different combinations of the dimensionless parameters that describe the problem. The results are compared with the ones obtained with diffraction theory and Newmans approximation, evidencing an interesting complementarity between the two approximations and the conditions in which each of them performs the better. The relevance of the second-order terms from Raineys formulation is shown, demonstrating that the common practice of analyzing second-order loads by simply applying Morisons equation with second-order wave kinematics is not strictly correct. The method is then applied to the analysis of a semi-submersible FOWT model, moored by three caternary lines, that was tested at the wave basin of the Numerical Offshore Tank of the University of São Paulo. Three sets of tests are presented: free decays of the moored model; forced oscillations of the hull; and motions under the action of waves (bichromatic, JONSWAP and white-noise) and wind. The results obtained with METiS are compared with the experiments and with WAMIT and OpenFAST, illustrating the capabilities and limitations of each software.\n

In this paper, we investigate to what extent the far-wake ‘signature’ of the near-wake vortex dynamics of a nominally two-dimensional bluff body is affected by the character of the free-stream noise. We confirm the existence of an oblique wave resonance (at frequency, fK–fT), which is caused by nonlinear ‘quadratic’ interactions between primary oblique shedding waves (fK) and secondary two-dimensional waves (fT), which are amplified from free-stream disturbances. In this work, oblique wave resonance is induced by acoustic forcing of two-dimensional waves. The use of acoustic forcing reveals a set of higher-order oblique wave resonances corresponding to frequencies (fK–nfT), where n is an integer. We find from visualization that, even when the secondary two-dimensional waves have the same frequency as the oblique waves, it is the oblique waves that are preferentially amplified. Oblique wave angles up to 74° have been observed. The response of the wake to a large range of forcing frequencies shows a broad region of peak response, centred around F = (fT/fK) = 0.55, and is in reasonable agreement with predictions from linear stability analysis. A similar broad response is found for each of the higher-order oblique wave modes. Simple equations for the oblique waves yield approximate conditions for maximum wake response, with a frequency for peak response given by Fmax = 1/2n = 1/2, 1/4, 1/6,…, and an oblique wave angle given by θmax = 2θK, where θK is the angle of oblique vortex shedding. An increase in forcing amplitude has the effect of bringing the nonlinear wave interactions, leading to oblique wave resonance, further upstream. Paradoxically, the effect of an increase in amplitude (A) of the two-dimensional wave forcing is to further amplify the oblique waves in preference to the two-dimensional waves and, under some conditions, to inhibit the appearance of prominent two-dimensional waves where they would otherwise appear. With a variation in forcing amplitude, the amplitude of oblique wave response is found to be closely proportional to A½. In summary, this investigation confirms the surprising result that it is only through the existence of noise in the free stream that the far wake is ‘connected’ to the near wake.