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Trough instabilities in Boussinesq formulations for water waves

Published online by Cambridge University Press:  28 February 2020

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]
Per A. Madsen*
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, 2800 KgsLyngby, Denmark
David R. Fuhrman
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, 2800 KgsLyngby, Denmark
*
†Email address for correspondence: prm@mek.dtu.dk
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Abstract

Modern Boussinesq-type formulations for water waves typically incorporate fairly accurate linear dispersion relations and similar accuracy in nonlinear properties. This has extended their application range to higher values of $kh$ ($k$ being wavenumber and $h$ the water depth) and has allowed for a better representation of nonlinear irregular waves with a fairly large span of short waves and long waves. Unfortunately, we have often experienced a number of ‘mysterious’ breakdowns or blowups, which have perplexed us for some time. A closer inspection has revealed that short-period noise can typically evolve in the deep troughs of wave trains in cases having relatively high spatial resolution. It appears that these potential ‘trough instabilities’ have not previously been discussed in the literature. In the present work, we analyse this problem in connection with the fourth- and fifth-order Padé formulations by Agnon et al. (J. Fluid Mech., vol. 399, 1999, pp. 319–333) the one-step Padé and the two-step Taylor–Padé formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 1–30) and the multi-layer formulations by Liu et al. (J. Fluid Mech., vol. 842, 2018, pp. 323–353). For completeness, we also analyse the popular, but older, formulations by Nwogu (ASCE J. Waterway Port Coastal Ocean Engng, vol. 119, 1993, pp. 618–638) and Wei et al. (J. Fluid Mech., vol. 294, 1995, pp. 71–92). We generally conclude that trough instabilities may occur in any Boussinesq-type formulation incorporating nonlinear dispersive terms. This excludes most of the classical Boussinesq formulations, but includes all of the so-called ‘fully nonlinear’ formulations. Our instability analyses are successfully verified and confirmed by making simple numerical simulations of the same formulations implemented in one dimension on a horizontal bottom. Furthermore, a remedy is proposed and tested on the one-step and two-step formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 1–30). This demonstrates that the trough instabilities can be moved or removed by a relatively simple reformulation of the governing Boussinesq equations. Finally, we discuss the option of an implicit Taylor formulation combined with exact linear dispersion, which is the starting point for the explicit perturbation formulation by Dommermuth and Yue (J. Fluid Mech., vol. 184, 1987, pp. 267–288), i.e. the popular higher-order-spectral formulations. In this case, we find no sign of trough instabilities.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Mathematical Foundations: Computational methods Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A38
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Agnon, Y., Madsen, P. A. & Schäffer, H. A. 1999 A new approach to high order Boussinesq models. J. Fluid Mech. 399, 319–333.CrossRefGoogle Scholar
Chazel, F., Benoit, M., Ern, A. & Piperno, S. 2009 A double-layer Boussinesq-type model for highly nonlinear and dispersive waves. Proc. R. Soc. Lond. A 465, 2319–2346.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.CrossRefGoogle Scholar
Gobbi, M. F., Kirby, J. T. & Wei, G. 2000 A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O (kh 4). J. Fluid Mech. 405, 182–210.CrossRefGoogle Scholar
Kennedy, A. B., Kirby, J. T., Chen, Q. & Dalrymple, R. A. 2001 Boussinesq-type equations with improved nonlinear behaviour. Wave Motion 33, 225–243.CrossRefGoogle Scholar
Liu, Z. B. & Fang, K. Z. 2016 A new two-layer Boussinesq model for coastal waves from deep to shallow water: derivation and analysis. Wave Motion 67, 1–14.CrossRefGoogle Scholar
Liu, Z. B., Fang, K. Z. & Cheng, Y. Z. 2018 A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed. J. Fluid Mech. 842, 323–353.CrossRefGoogle Scholar
Lynett, P. & Liu, P. L.-F. 2004 A two-layer approach to wave modelling. Proc. R. Soc. Lond. A 460, 2637–2669.CrossRefGoogle Scholar
Madsen, P. A. & Agnon, Y. 2003 Accuracy and convergence of velocity formulations for water waves in the framework of Boussinesq theory. J. Fluid Mech. 477, 285–319.CrossRefGoogle Scholar
Madsen, P. A., Bingham, H. B. & Liu, H. 2002 A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1–30.CrossRefGoogle Scholar
Madsen, P. A., Bingham, H. B. & Schäffer, H. A. 2003 Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves – Derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104.CrossRefGoogle Scholar
Madsen, P. A., Murray, R. & Sørensen, O. R. 1991 A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Engng 15, 371–388.CrossRefGoogle Scholar
Madsen, P. A. & Schäffer, H. A. 1998 Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Phil. Trans. R. Soc. Lond. A 356, 3123–3181.CrossRefGoogle Scholar
Madsen, P. A. & Sørensen, O. R. 1992 A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2: A slowly varying bathymetry. Coast. Engng 18, 183–204.CrossRefGoogle Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. ASCE J. Waterway Port Coastal Ocean Engng 119, 618–638.CrossRefGoogle Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.CrossRefGoogle Scholar
Wei, G. E., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.CrossRefGoogle Scholar
Witting, J. M. 1984 A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 56, 203–236.CrossRefGoogle 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
Zhang, J., Hong, K. & Yue, D. K. P. 1993 Effects of wave length ratio on wave modelling. J. Fluid Mech. 248, 107–127.CrossRefGoogle Scholar