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On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves

Published online by Cambridge University Press:  19 April 2021

Mathias Klahn Open the ORCID record for Mathias Klahn [Opens in a new window] ,
Per A. Madsen Open the ORCID record for Per A. Madsen [Opens in a new window]  and
David R. Fuhrman Open the ORCID record for David R. Fuhrman [Opens in a new window]
Mathias Klahn*
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs.Lyngby, Denmark
Per A. Madsen
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs.Lyngby, Denmark
David R. Fuhrman
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs.Lyngby, Denmark
*
†Email address for correspondence: matkla@mek.dtu.dk
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Abstract

We consider nonlinear, directionally spread irregular wave fields in deep water and study the statistical properties of the total inline force that would be induced by the waves on a vertical circular cylinder. Starting from the two-dimensional Morison equation, we specifically investigate the effect of wave steepness and directionality on the probability density functions (PDFs) of the inertia and drag forces. To do so, we derive new analytical expressions for the PDFs of these forces based on first-order theory and compare them with the results of fully nonlinear numerical simulations. We show that the inertia force for the main direction ($x$) of the wave field is in general unaffected by nonlinear effects, while the inertia force for the direction perpendicular to the main direction ($y$) is subject to substantial third-order effects when the steepness is appreciable and the wave field becomes relatively long crested. Moreover, we show that the drag force for the $x$-direction is in general subject to substantial second-order effects. The drag force for the $y$-direction is also affected by second-order effects, but to a much smaller degree than the $x$-direction. It is, however, strongly affected by third-order effects under the same conditions as the inertia force in the $y$-direction. We conclude that the total force can be accurately approximated by first-order theory when the ratio $k_p D/\varepsilon$ is large (with $k_p$ the peak wavenumber, $D$ the cylinder's diameter and $\varepsilon$ the wave steepness), while first-order theory underestimates the probability of large forces considerably when this ratio is small.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A59
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abramowitz, M. & Stegun, I. 1972 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards.Google Scholar
Annenkov, S.Y. & Shrira, V.I. 2009 ‘Fast’ nonlinear evolutions in wave turbulence. Phys. Rev. Lett. 102, 024502.CrossRefGoogle Scholar
Barratt, D., Bingham, H.B. & Adcock, T.A.A. 2020 Nonlinear evolution of a steep, focusing wave group in deep water simulated with OCEANWAVE3D. J. Offshore Mech. Arctic Engng 142, 021201.CrossRefGoogle Scholar
Berg, C. 2013 Complex Analysis. Department of Mathematical Sciences, University of Copenhagen.Google Scholar
Borgman, L.E. 1958 Computation of the ocean-wave forces on inclined cylinders. EOS Trans. AGU 39, 885–888.CrossRefGoogle Scholar
Borgman, L.E. 1967 Random hydrodynamic forces on objects. Ann. Math. Statist. 38, 37–51.CrossRefGoogle Scholar
Borgman, L.E. 1972 Statistical models for ocean waves and wave forces. Adv. Hydrosci. 8, 139–181.CrossRefGoogle Scholar
Clamond, D. & Grue, J. 2001 A fast method for fully nonlienar water-wave computations. J. Fluid Mech. 447, 337–355.CrossRefGoogle Scholar
Dommermuth, D. 2000 The initialization of nonlinear waves using an adjustment scheme. Wave Motion 32 (4), 307–317.CrossRefGoogle Scholar
Dommermuth, D.G. & Yue, D.K.P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.Google Scholar
Dysthe, K.B., Trulsen, K., Krogstad, H.E. & Socquet-Juglard, H. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J. Fluid Mech. 478, 1–10.CrossRefGoogle Scholar
Fructus, D., Clamond, D., Grue, J. & Kristiansen, Ø. 2005 An efficient model for three-dimensional surface wave simulations. Part 1. Free space problems. J. Comput. Phys. 205, 665–685.CrossRefGoogle Scholar
Fuhrman, D.R. & Bingham, H.B. 2004 Numerical solutions of fully non-linear and highly dispersive Boussinesq equations in two horizontal dimensions. Intl J. Numer. Meth. Fluids 44, 231–255.CrossRefGoogle Scholar
Holmes, M.H. 2013 Introduction to Perturbation Methods. Springer.CrossRefGoogle Scholar
Klahn, M., Madsen, P.A. & Fuhrman, D.R. 2021 a Simulation of three-dimensional nonlinear water waves using a pseudospectral volumetric method with an artificial boundary condition. Int. J. Numer. Meth. Fluids https://doi.org/10.1002/fld.4956.CrossRefGoogle Scholar
Klahn, M., Madsen, P.A. & Fuhrman, D.R. 2021 b On the statistical properties of surface elevation, velocities and accelerations in multi-directional irregular water waves. J. Fluid Mech. 910, A23.Google Scholar
Klahn, M., Madsen, P.A. & Fuhrman, D.R. 2021 c Data related to ‘On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves’. doi:10.11583/DTU.13359089.CrossRefGoogle Scholar
Kopriva, D.A. 2009 Implementing Spectral Methods for Partial Differential Equations. Springer.CrossRefGoogle Scholar
Lighthill, J. 1986 Fundamentals concerning wave loading on offshore structures. J. Fluid Mech. 173, 667–681.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1963 The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid Mech. 17, 459–480.CrossRefGoogle Scholar
Manners, W. & Rainey, R.C.T. 1992 Hydrodynamic forces on fixed submerged cylinders. Proc. R. Soc. Lond. A 450, 391–416.Google Scholar
Morison, J.R., O'Brien, M.P., Johnson, J.W. & Schaaf, S.A. 1950 The force exerted by surface waves on piles. J. Petrol. Tech. 2 (5), 149–154.CrossRefGoogle Scholar
Nicholls, D.P. 2011 Efficient enforcement of far-field boundary conditions in the transformed field expansions method. J. Comput. Phys. 230, 8290–8303.CrossRefGoogle Scholar
Onorato, M., et al. 2009 Statistical properties of mechanically generated surface gravity waves: a lobaratory experiment in a three-dimensional wave basin. J. Fluid Mech. 627, 235–257.CrossRefGoogle Scholar
Pierson, W.J. & Holmes, P. 1965 Irregular wave forces on a pile. J. Waterways Harbors Div. ASCE 91 (WW4), 1–10.CrossRefGoogle Scholar
Rainey, R.C.T. 1989 A new equation for calculating wave loads on offshore structures. J. Fluid Mech. 204, 295–324.CrossRefGoogle Scholar
Rainey, R.C.T. 1995 Slender-body expressions for the wave load on offshore structures. Proc. R. Soc. Lond. A 450, 391–416.Google Scholar
Saad, Y. & Schultz, M.H. 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.Google Scholar
Spiegel, M.R. 1965 Theory and Problems of Laplace Transforms. McGraw-Hill.Google Scholar
Song, J.-B. & Wu, Y.-H. 2000 Statistical distribution of water-particle velocity below the surface layer for finite water depth. Coast. Engng 40, 1–19.CrossRefGoogle Scholar
Song, J.-B., Wu, Y.-H. & Wiwatanapataphee, B. 2000 Probability distribution of random wave forces in weakly nonlinear random waves. Coast. Engng 27, 1391–1405.Google Scholar
Tanaka, M. 2001 Verification of Hasselmann's energy transfer among surface gravity waves by direct numerical simulations. J. Fluid Mech. 444, 199–221.CrossRefGoogle Scholar
Tickell, R.G. 1977 Continuous random wave loading on structural members. Struct. Engng 55 (5), 209–222.Google Scholar
Toffoli, A., Gramstad, O., Truelsen, K., Monbaliu, J., Bitner-Gregersen, E. & Onorato, M. 2010 Evolution of wekly nonlinear random directional waves: laboratory experiments and numerical simulations. J. Fluid Mech. 664, 313–336.CrossRefGoogle Scholar
Tucker, M.J., Challenor, P.G. & Carter, D.J.T. 1984 Numerical simulation of a random sea: a common error and its effect upon wave group statistics. Appl. Ocean Res. 6 (2), 118–122.CrossRefGoogle Scholar
West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M. & Milton, R.L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res. 92 (C11), 11803–11824.CrossRefGoogle Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D.K.P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357–392.Google Scholar
Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar