Adcock, Thomas

Dai, Saishuai

Ding, Haoyu

Tang, Tianning

Taylor, Paul H

Zang, Jun

Zhao, Guangwei

English

Oxford University Research Archive

The 39th International Workshop on Water Waves and Floating Bodies, 14-17 April 2024, St Andrews, ScotlandAn investigation of high-order harmonics in thepressure field around a vertical cylinder in steepwave conditionsHaoyu Dinga, Jun Zanga, Tianning Tangb, Paul H. Taylorc, Thomas A.A.Adcockb, Guangwei Zhaod, Saishuai Daida. Department of Architecture and Civil Engineering, University of Bath, UKb. Department of Engineering Science, University of Oxford, Oxford, UKc. Oceans Graduate School, The University of Western Australia, Crawley, Australiad. Naval Architecture, Ocean and Marine Engineering Department, University ofStrathclyde, Glasgow, UKEmail: hd484@bath.ac.uk1 IntroductionOffshore structures, encompassing foundations for offshore wind turbines, supports formarine renewable energy devices, bridge piers, and floating vessels, are consistently sub-jected to severe environmental loads. These loads often dictate the design criteria.Understanding the physics and statistics of wave-structure interaction, especially un-der non-linear loads experienced in extreme conditions, remains a complex and partiallyunresolved challenge. Notably, secondary load cycles significantly contribute to the ’ring-ing’ responses in cylindrical structures, as discussed in previous studies (e.g., Grue et al.(1993), Chaplin et al. (1997)). This paper focuses on analysing loads in focused wavegroups, representing short-term extreme wave conditions, on bottom-mounted verticalcylinders relevant to fixed offshore wind turbines. Pressure contour plots over the cylin-der’s surface were previously examined by Ghadirian & Bredmose (2020) while studyingsecondary load cycles. In this research, we adopt the phase-based harmonic separationmethod for wave forces (Fitzgerald et al. (2014)) to analyse the pressure contour plots.This method effectively isolates harmonic pressure components from the total pressures,enabling a novel exploration of the mechanisms behind secondary load cycles from theperspective of high-order harmonics on the cylinder surface.2 Numerical MethodIn this study, computational fluid dynamics (CFD) results are analysed. OpenFOAM,a well-known open-source model, has been used for this task. The waves2foam toolbox,as described in Jacobsen et al. (2012), is used for wave generation, offering accuratesimulation of the wave conditions for our study. A Reynolds-averaged Navier–Stokes(RANS) model is employed for wave-structure interaction simulations. Chen et al. (2014)conducted a series of mesh convergence tests to determine the optimal mesh size foraccurate prediction of wave-cylinder interactions. This study employs the same meshconfiguration, but with further refinement to the innermost layer around the cylinder to0.2 times the mesh size used by Chen et al. (2014). This refinement enhances the captureof spatially and temporally localised wave behaviours, such as the steep gradient of waverun-up on the cylinder and secondary load cycles.A physical experiment conducted at the Danish Hydraulic Institute (DHI) (Chen et al.(2018)) serves to validate the numerical model. In this experiment, a fixed cylinder with aFigure 1: (a) Validation of OpenFOAM predictions showing total inline force exerted onthe cylinder. (b) Validation through point pressure measurements.radius of 0.125 m was subjected to extreme focused waves in a wave tank with a depth of0.505 m. The wave parameters were a main crest amplitude of 0.15 m, a peak wave periodof 1.64 s, and a wave number of 1.98/m. Figure 1(a) and (b) present comparative resultsfor total force and point pressures, respectively. For total force, the OpenFOAM resultsshow good agreement with the experimental data. Under the impact of extreme waves,wave slamming loads are noticeable in the main peak, and distinct secondary load cyclesappear in the first trough following the main peak. These phenomena observed in thephysical experiments are also accurately predicted by OpenFOAM. In contrast to the totalforce, which is an integrated result, point pressure measurements are more sensitive andchallenging to predict. Nonetheless, OpenFOAM successfully predicts similar pressuredistributions at various stagnation points up the front of the cylinder, denoted by Z,where Z = 0 corresponds to the still water level, Z < 0 represents submersion. Thesecomparisons provide strong evidence that OpenFOAM is capable of accurately simulatingwave loading and pressure under severe wave conditions, thereby allowing subsequentinvestigations into secondary load cycles.3 ResultsA series of new experiments were carried out at the Kelvin Hydrodynamics Laboratory,University of Strathclyde, using a bottom-mounted, surface-piercing vertical cylinderwith a radius (R) of 0.2 m in a constant water depth of 1.8 m. This research providesCFD simulations for these tests. Comparisons to the experiments will be reported atthe workshop. Specifically, a focused wave group based on the JONSWAP spectrum(γ = 3.3) was selected for detailed analysis. This wave group is characterized by a maincrest amplitude (A) of 0.41 m, a peak wave period (T ) of 2.5 s, and a peak wave number(k) of 0.74/m, serving as a representative example for this section.3.1 Scattering PressureThe scattering pressure is calculated by subtracting the baseline pressure (measured inthe wave but without the cylinder) from the total pressure recorded when the cylinderis present. To facilitate this, the pressures at mesh points on the surface of a cylinderare outputted in the OpenFOAM simulation. Pressures are also taken from these samelocations in a simulation with the cylinder removed and replaced by water to gauge thescattering pressure contributions from incident waves and hydrostatic pressure alone.Figure 2: Contour plot of inline scattering force per unit length against cylinder heightand time, with overlaying time histories of total force (red line). Also depicted are thetotal free surface elevations along the cylinder’s central line, from the front (white dashedline) to the rear (purple dash-dot line), and undisturbed wave elevations at the centrepoint in the absence of the cylinder (blue dotted line). The scattering wave field aroundthe cylinder is shown at three time points: (a) 15.30 s, (b) 15.45 s, and (c) 15.60 s.The scattering wave field, depicted in Figure 2, represents the difference between thetotal waves and the undisturbed waves, and is derived using the same methodology.The inline scattering force per unit length vertically is found from the circumferen-tial integral of the local scattering pressure resolved along the inline direction. This isvisualised in a contour plot against time and the vertical elevation in Figure 2. Accompa-nying the contour is the total force. A secondary load cycle is apparent in the total forcetrace around 15.5 seconds. Regarding the scattering force per unit length contour plot, anegative ’chamber’ emerges below the positive region at Z = 0.2m around 15 s, markingthe onset of the secondary load cycle. A positive region, termed a ’positive insertion’,transiently inserts into the negative zone as a minor crest appears in the trough of thetotal force, followed by a substantial negative region around Z = −0.2m. Three criticalpoints during the force trough are labeled as a, b, and c in Figure 2. The scatteringfields at these points, along with the central line free surface elevations, illustrate the freesurface behaviours during the secondary load cycle. The significant wave run-up at pointa and the wave set-down at point c at the cylinder’s front stagnation point are observed,which may contribute to the variations in force. These variations in the scattering forceper unit length highlight the pressure field’s sensitivity around the cylinder to secondaryload cycles, suggesting that abrupt shifts from positive to negative pressures could havesignificant implications for structural fatigue.3.2 Pressure DecompositionThe four-phase harmonic separation method proposed by Fitzgerald et al. (2014) hasbeen demonstrated as effective in decomposing individual force harmonics when the hy-Figure 3: Contour plots of inline scattering force per unit length for (a) linear harmonic,(b) second harmonic, (c) third harmonic, (d) fourth harmonic, (e) fifth harmonic and (f)beyond fifth harmonicdrodynamic force in focused waves possesses a Stokes-like structure. In this research,this method is also applied to the point pressures on the cylinder surface, facilitating theintegration of contour plots for inline scattering force per unit length across individualharmonics. Evidence supporting the method’s applicability to pressure measurementswill be presented at the workshop. Figure 3 displays the results following this decompo-sition on the force per unit length below the still water level. The oscillation betweenpositive and negative regions in these contour plots accurately reflects the relationshipbetween harmonics from first to fifth order, with their magnitudes generally diminishingwith higher harmonic. Notably, harmonics beyond the fifth harmonic still show signifi-cant magnitudes with negative regions (-224 N/m) exceeding the linear scattering forceper unit length (-178 N/m). The pronounced values of these high harmonics, particularlyevident around 15.5 s, correspond with the ’positive insertion’ identified in Figure 2 andmay also influence secondary load cycles. This decomposition underscores the criticalinfluence of higher-order harmonics on the scattering force amid severe wave conditionsand suggests that components beyond the fifth harmonic have substantial implicationsfor secondary load cycles.ReferencesChaplin, J. R., Rainey, R. C. T. & Yemm, R. W. (1997), ‘Ringing of a vertical cylinder in waves’, J.Fluid Mech. pp. 119–147.Chen, L. F., Zang, J., Hillis, A. J., Morgan, G. C. & Plummer, A. R. (2014), ‘Numerical investigationof wave-structure interaction using OpenFOAM’, Ocean Eng. 88, 91–109.Chen, L., Zang, J., Taylor, P. H., Sun, L., Morgan, G., Grice, J., Orszaghova, J. & Ruiz, M. T. (2018), ‘Anexperimental decomposition of nonlinear forces on a surface-piercing column: Stokes-type expansionsof the force harmonics’, J. Fluid Mech. 848, 42–77.Fitzgerald, C. J., Taylor, P. H., Eatock Taylor, R., Grice, J. & Zang, J. (2014), ‘Phase manipula-tion and the harmonic components of ringing forces on a surface-piercing column’, Proc. R. Soc. A470(2168), 20130847.Ghadirian, A. & Bredmose, H. (2020), ‘Detailed force modelling of the secondary load cycle’, J. FluidMech. 889, 1–30.Grue, J., Bjørshol, G. & Strand, Ø. (1993), ‘Higher harmonic wave exciting forces on a vertical cylinder’,Preprint series. Mechanics and Applied Mathematics http://urn. nb. no/URN: NBN: no-23418 .Jacobsen, N. G., Fuhrman, D. R. & Fredsøe, J. (2012), ‘A wave generation toolbox for the open-sourceCFD library: OpenFoam®’, Int. J. Numer. Methods Fluids 70, 1073–1088.

An investigation of high-order harmonics in the pressure field around a vertical cylinder in steep wave conditions

https://ora.ox.ac.uk/objects/uuid:5be0d48b-a1df-4767-a551-780b6eee1c64/files/s4q77ft044

An investigation of high-order harmonics in the pressure field around a vertical cylinder in steep wave conditions

Abstract

Similar works

Full text

Available Versions

Oxford University Research Archive