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Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow

Published online by Cambridge University Press:  31 May 2013

Yefei Bai  and
Kwok Fai Cheung
Yefei Bai*
Affiliation:
Department of Ocean and Resources Engineering, University of Hawaii, Honolulu, HI 96822, USA
Kwok Fai Cheung
Affiliation:
Department of Ocean and Resources Engineering, University of Hawaii, Honolulu, HI 96822, USA
*
†Email address for correspondence: [email protected]
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Abstract

Non-hydrostatic multi-layer models have become a popular tool in describing wave transformation from deep water to the surf zone, but the numerical approach lacks a theoretical framework to guide implementation and assist interpretation of the results. In this paper, we formulate a non-hydrostatic model in an analytical form for the derivation and examination of dispersive and nonlinear properties. Depth integration of the dimensionless continuity and Euler equations over each layer yields the conventional multi-layer formulation. A variable transformation converts the conventional form into an integrated series form, which provides separate descriptions of flux- and dispersion-dominated processes. Substitution of the non-hydrostatic pressure and vertical velocity in the governing equations by high-order derivatives of the horizontal velocity and surface elevation provides a direct comparison with the Boussinesq equations published in the literature. Implementation of a perturbation expansion extracts the first- and second-order governing equations with respect to the nonlinear parameter. Based on that, we derive analytical solutions of the linear dispersion and the second-order super- and sub-harmonics for up to three layers and optimize the solutions in terms of the layer arrangement. In relation to the Boussinesq equations at comparable orders of expansion, the two- and three-layer models provide slightly higher errors in shallow and intermediate water in terms of dispersion and super-harmonics, but show superior performance in describing sub-harmonics in deep water.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 226 - 260
Copyright
©2013 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
Ai, C. & Jin, S. 2012 A multi-layer non-hydrostatic model for wave breaking and runup. Coast. Engng 62, 1–8.Google Scholar
Ai, C., Jin, S. & Lv, B. 2011 A new fully non-hydrostatic 3D free surface model for water wave motions. Intl J. Numer. Meth. Fluids 66, 1354–1370.Google Scholar
Bai, Y. 2012 Depth-integrated free-surface flow with non-hydrostatic formulation. PhD dissertation, University of Hawaii, Honolulu.Google Scholar
Bai, Y. & Cheung, K. F. 2012 Depth-integrated free-surface flow with a two-layer non-hydrosatic formulation. Intl J. Numer. Meth. Fluids 69 (2), 411–429.Google Scholar
Bai, Y. & Cheung, K. F. 2013 Depth-integrated free-surface flow with parameterized non-hydrostatic pressure. Intl J. Numer. Meth. Fluids 71 (4), 403–421.Google Scholar
Casulli, V. 1995 Recent developments in semi-implicit numerical methods for free surface hydrodynamics. In Proceedings of the Second International Conference on Hydroscience and Engineering, Beijing, Advances in Hydro Science and Engineering, vol. 2, Part B, pp. 2174–2181.Google Scholar
Casulli, V. 1999 A semi-implicit finite difference method for non-hydrostatic, free-surface flows. Intl J. Numer. Meth. Fluids 30, 425–440.Google Scholar
Casulli, V. & Stelling, G. S. 1998 Numerical simulation of 3D quasi-hydrostatic, free-surface flows. J. Hydraul. Engng ASCE 124, 678–686.Google Scholar
Dean, R. G. & Sharma, J. N. 1981 Simulation of wave systems due to nonlinear directional spectra. In Proceedings of the International Symposium on Hydrodynamics in Ocean Engineering, Norwegian Institute of Technology, Trondheim, vol. 2, pp. 1211–1222.Google 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, 181–210.Google Scholar
Kazolea, M., Delic, A. I., Nikolos, I. K. & Synolakis, C. E. 2012 An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations. Coast. Engng 69, 42–66.Google Scholar
Lay, T., Ammon, C. J., Yamazaki, Y., Cheung, K. F. & Hutko, A. R. 2011 The 25 October 2010 Mentawai tsunami earthquake and the tsunami hazard presented by shallow megathrust ruptures. Geophys. Res. Lett. 38, L06302.Google Scholar
Lynett, P. J. & Liu, P. L. 2004a Linear analysis of the multi-layer model. Coast. Engng 51, 439–454.Google Scholar
Lynett, P. J. & Liu, P. L. 2004b A two-layer approach to wave modelling. Proc. R. Soc. A 460, 2637–2669.Google Scholar
Ma, G., Shi, F. & Kirby, T. 2012 Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43–44, 23–35.Google 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 (2033), 1075–1104.Google Scholar
Madsen, P. A., Fuhrman, D. R. & Wang, B. 2006 A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Engng 53, 487–504.CrossRefGoogle Scholar
Madsen, P. A., Murray, R. & Sørensen, O. R. 1991 A new form of Boussinesq equations with improved linear dispersion characteristics. Coast. Engng 15, 371–388.Google Scholar
Mahadevan, A., Oliger, J. & Street, R. L. 1996a A non-hydrostatic mesoscale ocean model, Part I: Well-posedness and scaling. J. Phys. Oceanogr. 269, 1868–1880.Google Scholar
Mahadevan, A., Oliger, J. & Street, R. L. 1996b A non-hydrostatic mesoscale ocean model, Part II: Numerical implementation. J. Phys. Oceanogr. 269, 1881–1900.Google Scholar
Marshall, J., Hill, C., Perelman, L. & Adcroft, A. 1997 Hydrostatic, quasi-hydrostatic, and non-hydrostatic ocean modelling. J. Geophys. Res. 102, 5733–5752.Google Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway Port Coast. Ocean Engng 119, 618–638.Google Scholar
Ottesen-Hansen, N. E. 1978 Long period waves in natural wave train. Progress Report no. 46, Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, pp. 13–24.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Roeber, V. & Cheung, K. F. 2012 Boussinesq-type model for energetic breaking waves in fringing reef environments. Coast. Engng 70, 1–20.Google Scholar
Roeber, V., Yamazaki, Y. & Cheung, K. F. 2010 Resonance and impact of the 2009 Samoa tsunami around Tutuila, American Samoa. Geophys. Res. Lett. 37, L21604.CrossRefGoogle Scholar
Sand, S. E. & Mansard, E. P. D. 1986 Reproducetion of higher harmonics in irregular waves. Ocean Engng 13, 57–83.CrossRefGoogle Scholar
Schäffer, H. A., Madsen, P. A. & Deigaard, R. A. 1993 A Boussinesq model for wave breaking in shallow water. Coast. Engng 20, 185–202.Google Scholar
Shi, F., Kirby, J. T., Harris, J. C., Geiman, J. D. & Grilli, S. T. 2012 A high-order adaptive time stepping TVD solver for Boussinesq modelling of breaking waves and coastal inundation. Ocean Model. 43–44, 36–51.Google Scholar
Stansby, P. K. & Zhou, J. G. 1998 Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems. Intl J. Numer. Meth. Fluids 28, 541–563.Google Scholar
Stelling, G. S. & Duinmeijer, S. P. A. 2003 A staggered conservative scheme for every Froude number in rapidly varied shallow water flows. Intl J. Numer. Meth. Fluids 43, 1329–1354.Google Scholar
Stelling, G. S. & Zijlema, M. 2003 An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. Intl J. Numer. Meth. Fluids 43, 1–23.Google Scholar
Tonelli, M. & Petti, M. 2012 Shock-capturing Boussinesq model for irregular wave propagation. Coast. Engng 61, 8–19.CrossRefGoogle Scholar
Wei, G. & Kirby, J. T. 1995 Time-dependent numerical code for extended Boussinesq equations. J. Waterway Port Coast. Ocean Engng 121 (5), 251–261.CrossRefGoogle Scholar
Wei, G., Kirby, J. T. & Grilli, S. T. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.CrossRefGoogle Scholar
Wu, C. H., Young, C. C., Chen, Q. & Lynett, P. J. 2010 Efficient non-hydrostatic modelling of surface waves from deep to shallow water. J. Waterway Port Coast. Ocean Engng 136 (2), 104–118.Google Scholar
Yamazaki, Y. & Cheung, K. F. 2011 Shelf resonance and impact of near-field tsunami generated by the 2010 Chile earthquake. Geophys. Res. Lett. 38, L12605.Google Scholar
Yamazaki, Y., Cheung, K. F. & Kowalik, Z. 2011a Depth-integrated, non-hydrostatic model with grid nesting for tsunami generation, propagation and runup. Intl J. Numer. Meth. Fluids 67, 2081–2107.Google Scholar
Yamazaki, Y., Kowalik, Z. & Cheung, K. F. 2009 Depth-integrated, non-hydrostatic model for wave breaking and runup. Intl J. Numer. Meth. Fluids 61, 473–497.Google Scholar
Yamazaki, Y., Lay, T., Cheung, K. F., Yue, H. & Kanamori, H. 2011b Modelling near-field tsunami observations to improve finite-fault slip models for the 11 March 2011 Tohoku earthquake. Geophys. Res. Lett. 38, L00G15.Google Scholar
Yuan, H. & Wu, C. H. 2004 A two-dimensional vertical non-hydrostatic sigma model with an implicit method for free-surface flows. Intl J. Numer. Meth. Fluids 44, 811–835.Google Scholar
Zelt, J. A. 1991 The runup of non-breaking and breaking solitary waves. Coast. Engng 15, 205–246.Google Scholar
Zhou, J. G. & Stansby, P. K. 1999 An arbitary Lagrangian–Eulerian $\sigma $ (ALES) model with non-hydrostatic pressure for shallow water flows. Comput. Meth. Appl. Mech. Engng 178, 199–214.Google Scholar
Zijlema, M. & Stelling, G. S. 2005 Further experiences with computing non-hydrostatic free-surface flows involving water waves. Intl J. Numer. Meth. Fluids 48, 169–197.Google Scholar
Zijlema, M. & Stelling, G. S. 2008 Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure. Intl J. Numer. Meth. Fluids 55, 780–790.Google Scholar
Zijlema, M., Stelling, G. S. & Smit, P. 2011 SWASH: an operational public domain code for simulating wave fields and rapidly varied flows in coastal waters. Coast. Engng 58, 992–1012.Google Scholar