Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-22T17:31:56.036Z Has data issue: false hasContentIssue false
  • English
  • Français

Depth-integrated wave–current models. Part 1. Two-dimensional formulation and applications

Published online by Cambridge University Press:  20 November 2019

Z. T. Yang Open the ORCID record for Z. T. Yang [Opens in a new window]  and
P. L.-F. Liu Open the ORCID record for P. L.-F. Liu [Opens in a new window]
Z. T. Yang
Affiliation:
Department of Civil and Environmental Engineering, National University of Singapore, 117576, Singapore
P. L.-F. Liu*
Affiliation:
Department of Civil and Environmental Engineering, National University of Singapore, 117576, Singapore School of Civil and Environmental Engineering, Cornell University, Ithaca, NY14850, USA Institute of Hydrological and Oceanic Sciences, National Central University, Jhongli,Taoyuan, 320, Taiwan
*
†Email address for correspondence: Philip.Liu@nus.edu.sg
Get access Rights & Permissions [Opens in a new window]

Abstract

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler’s equations in the $\unicode[STIX]{x1D70E}$-coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of $\unicode[STIX]{x1D70E}$-coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of $\unicode[STIX]{x1D70E}$ and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models’ linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green–Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.

JFM classification

Geophysical and Geological Flows: Coastal engineering Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A4
Copyright
© 2019 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation.Google Scholar
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
Babuska, I., Práger, M. & Vitásek, E. 1966 Numerical Processes in Differential Equations. Sntl.Google Scholar
Bai, Y. & Cheung, K. F. 2013 Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow. J. Fluid Mech. 726, 226–260.CrossRefGoogle Scholar
Barthélemy, E. 2004 Nonlinear shallow water theories for coastal waves. Surv. Geophys. 25, 315–337.CrossRefGoogle Scholar
Beji, S. & Battjes, J. A. 1994 Numerical simulation of nonlinear wave propagation over a bar. Coast. Engng 23, 1–16.CrossRefGoogle Scholar
Belibassakis, K. & Touboul, J. 2019 A nonlinear coupled-mode model for waves propagating in vertically sheared currents in variable bathymetry—collinear waves and currents. Fluids 4, 61.CrossRefGoogle Scholar
Bellman, R. E. & Roth, R. S. 1986 Methods in Approximation: Techniques for Mathematical Modelling. Springer Netherlands.CrossRefGoogle Scholar
Biezeno, C. B. 1924 Graphical and numerical methods for solving stress problems. In Proceedings of the First International Congress for Applied Mechanics, Waltman, Delft, pp. 3–17.Google Scholar
Buldakov, E., Stagonas, D. & Simons, R. 2017 Extreme wave groups in a wave flume: controlled generation and breaking onset. Coast. Engng 128, 75–83.CrossRefGoogle Scholar
Castro, A. & Lannes, D. 2014 Fully nonlinear long-wave models in the presence of vorticity. J. Fluid Mech. 759, 642–675.CrossRefGoogle Scholar
Chen, L. F., Stagonas, D., Santo, H., Buldakov, E. V., Simons, R. R., Taylor, P. H. & Zang, J. 2019 Numerical modelling of interactions of waves and sheared currents with a surface piercing vertical cylinder. Coast. Engng 145, 65–83.CrossRefGoogle Scholar
Chen, Q., Madsen, P. A., Schäffer, H. A. & Basco, D. R. 1998 Wave-current interaction based on an enhanced Boussinesq-type approach. Coast. Engng 33, 11–40.CrossRefGoogle Scholar
Chen, Y. & Liu, P. L.-F. 1995 Modified Boussinesq equations and associated parabolic models for water wave propagation. J. Fluid Mech. 288, 351–381.CrossRefGoogle Scholar
Demirbilek, Z. & Webster, W. C.1992 Application of the Green–Naghdi theory of fluid sheets to shallow-water wave problems. Report 1. Model development Tech. Rep. Coastal Engineering Research Center Vicksburg MS.CrossRefGoogle Scholar
Demirbilek, Z. & Webster, W. C. 1999 The Green–Naghdi theory of fluid sheets for shallow-water waves. In Dev. Offshore Eng., pp. 1–54. Gulf Professional Publishing.Google Scholar
Dingemans, M.1994 Comparison of computations with Boussinesq-like models and laboratory measurements. Mast-G8M note, H1684. Delft Hydraulics.Google Scholar
Ellingsen, S. A. & Li, Y. 2017 Approximate dispersion relations for waves on arbitrary shear flows. J. Geophys. Res. Ocean. 122, 9889–9905.CrossRefGoogle Scholar
Ertekin, R. C., Webster, W. C. & Wehausen, J. V. 1986 Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech. 169, 275–292.CrossRefGoogle Scholar
Finlayson, B. A. 2013 The Method of Weighted Residuals and Variational Principles, vol. 73. SIAM.CrossRefGoogle Scholar
Galerkin, B. G. 1915 Series solution of some problems of elastic equilibrium of rods and plates. Vestn. Inzh. Tekh 19, 897–908.Google Scholar
Gobbi, M. F. & Kirby, J. T. 1999 Wave evolution over submerged sills: tests of a high-order Boussinesq model. Coast. Engng 37, 57–96.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, 181–210.CrossRefGoogle Scholar
Green, A. E., Laws, N. & Naghdi, P. M. 1974 On the theory of water waves. Proc. R. Soc. Lond. A 338, 43–55.CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976a A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976b Directed fluid sheets. Proc. R. Soc. Lond. A 347, 447–473.CrossRefGoogle Scholar
Hwung, H. H., Chiang, W. S. & Hsiao, S. C. 2007 Observations on the evolution of wave modulation. Proc. R. Soc. A 463, 85–112.CrossRefGoogle Scholar
Kantorovich, L. V. & Krylov, V. I. 1958 Approximate Methods of Higher Analysis. P. Noordhoff Ltd.Google Scholar
Kemp, P. H. & Simons, R. R. 1982 The interaction between waves and a turbulent current: waves propagating with the current. J. Fluid Mech. 116, 227–250.CrossRefGoogle Scholar
Kemp, P. H. & Simons, R. R. 1983 The interaction of waves and a turbulent current: waves propagating against the current. J. Fluid Mech. 130, 73–89.CrossRefGoogle Scholar
Kim, D. H., Lynett, P. J. & Socolofsky, S. A. 2009 A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows. Ocean Model. 27, 198–214.CrossRefGoogle Scholar
Kirby, J. T. & Chen, T. M. 1989 Surface waves on vertically sheared flows: approximate dispersion relations. J. Geophys. Res. 94, 1013–1027.CrossRefGoogle Scholar
Kway, J. H. L., Loh, Y.-S. & Chan, E.-S. 1998 Laboratory study of deep-water breaking waves. Ocean Engng 25, 657–676.CrossRefGoogle Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.CrossRefGoogle Scholar
Lanczos, C. 1938 Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17, 123–199.CrossRefGoogle Scholar
Lannes, D. & Bonneton, P. 2009 Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601.CrossRefGoogle Scholar
Levich, V. G. & Krylov, V. S. 1969 Surface-tension-driven phenomena. Annu. Rev. Fluid Mech. 1, 293–316.CrossRefGoogle Scholar
Liang, B., Wu, G., Liu, F., Fan, H. & Li, H. 2015 Numerical study of wave transmission over double submerged breakwaters using non-hydrostatic wave model. Oceanologia 57, 308–317.CrossRefGoogle Scholar
Liu, P. L.-F. 1995 Model equations for wave propagations from deep to shallow water. In Adv. Coast. Ocean Eng. (ed. Liu, P. L.-F.), vol. 1, pp. 125–157. World Scientific Publishing.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. 2004a A two-layer approach to wave modelling. Proc. R. Soc. Lond. A 460, 2637–2669.CrossRefGoogle Scholar
Lynett, P. J. & Liu, P. L.-F. 2004b Linear analysis of the multi-layer model. Coast. Engng 51, 439–454.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. & 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. 1991 A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Engng 15, 371–388.CrossRefGoogle Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves, vol. 1. World Scientific Publishing.Google Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway Port Coastal Ocean Engng 119, 618–638.CrossRefGoogle Scholar
Nwogu, O. G. 2009 Interaction of finite-amplitude waves with vertically sheared current fields. J. Fluid Mech. 627, 179–213.CrossRefGoogle Scholar
Ohyama, T., Kioka, W. & Tada, A. 1995 Applicability of numerical models to nonlinear dispersive waves. Coast. Engng 24, 297–313.CrossRefGoogle Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.CrossRefGoogle Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.CrossRefGoogle Scholar
Picone, M. 1928 Sul metodo delle minime potenze ponderate e sul metodo di Ritz per il calcolo approssimato nei problemi della fisica-matematica. Rend. del Circ. Mat. di Palermo 52, 225–253.CrossRefGoogle Scholar
Pike, J. & Roe, P. L. 1985 Accelerated convergence of Jameson’s finite-volume Euler scheme using Van Der Houwen integrators. Comput. Fluids 13 (2), 223–236.CrossRefGoogle Scholar
Schäffer, H. A. & Madsen, P. A. 1995 Further enhancements of Boussinesq-type equations. Coast. Engng 26, 1–14.CrossRefGoogle Scholar
Serre, F. 1953 Contribution à l’étude des écoulements permanents et variables dans les canaux. Houille Blanche 6, 830–872.CrossRefGoogle Scholar
Shaprio, R. 1970 Smoothing, filtering and boundary effects. Rev. Geophys. 8, 359–387.CrossRefGoogle Scholar
Shields, J. J. & Webster, W. C. 1988 On direct methods in water wave theory. J. Fluid Mech. 197, 171–199.CrossRefGoogle Scholar
Skjelbreia, L. & Hendrickson, J. 1960 Fifth-order gravity wave theory. In Proceedings of the 7th International Conference on Coastal Engineering, pp. 184–196.Google Scholar
Son, S. & Lynett, P. J. 2014 Interaction of dispersive water waves with weakly sheared currents of arbitrary profile. Coast. Engng 90, 64–84.CrossRefGoogle Scholar
Stagonas, D., Buldakov, E. & Simons, R. 2018 Experimental generation of focusing wave groups on following and adverse-sheared currents in a wave–current flume. J. Hydraul. Engng ASCE 144 (5), 04018016.CrossRefGoogle Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539.CrossRefGoogle Scholar
Swan, C., Cummins, I. P. & James, R. L. 2001 An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Part 1. Regular waves. J. Fluid Mech. 428, 273–304.CrossRefGoogle Scholar
Thomas, G. P. 1981 Wave–current interactions: an experimental and numerical study. Part 1. Linear waves. J. Fluid Mech. 110, 457–474.CrossRefGoogle Scholar
Thomas, G. P. 1990 Wave–current interactions: an experimental and numerical study. Part 2. Nonlinear waves. J. Fluid Mech. 216, 505–536.CrossRefGoogle Scholar
Thompson, P. D. 1949 The propagation of small surface disturbances through rotational flow. Ann. N.Y. Acad. Lond. Sci. 51, 463–474.CrossRefGoogle Scholar
Waseda, T. & Tulin, M. P. 1999 Experimental study of the stability of deep-water wave trains including wind effects. J. Fluid Mech. 401, 55–84.CrossRefGoogle Scholar
Webster, W. C., Duan, W. & Zhao, B. 2011 Green–Naghdi theory, part A: Green–Naghdi (GN) equations for shallow water waves. J. Mar. Sci. Appl. 10, 253–258.CrossRefGoogle Scholar
Wei, G., 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
Yoon, S. B. & Liu, P. L.-F. 1989 Interactions of currents and weakly nonlinear water waves in shallow water. J. Fluid Mech. 205, 397–419.CrossRefGoogle Scholar
Zhang, Y., Kennedy, A. B., Panda, N., Dawson, C. & Westerink, J. J. 2013 Boussinesq–Green–Naghdi rotational water wave theory. Coast. Engng 73, 13–27.CrossRefGoogle Scholar
Zhao, B. B., Duan, W. Y. & Ertekin, R. C. 2014 Application of higher-level GN theory to some wave transformation problems. Coast. Engng 83, 177–189.CrossRefGoogle Scholar
Supplementary material: File

Yang and Liu supplementary material

Yang and Liu supplementary material

Download Yang and Liu supplementary material(File)
File 1.1 MB