|
Article information
2014 , Volume 19, ¹ 6, p.77-94
Fedotova Z.I., Khakimzyanov G.S.
The basic nonlinear-dispersive hydrodynamic model of long surface waves
Currently, the most popular nonlinear dispersive (NLD-) equations of long wave hydrodynamics, which are derived by the perturbation method, can be separated into two large groups. One of the groups is characterized by the fact that the velocity in the model is considered as the average of the horizontal velocity vector of Euler equations across the thickness of the liquid layer. In another group, the sought after velocity is the velocity of fluid flow on a surface z = z(x; y) (z is the vertical coordinate), which is either immersed in a liquid, or coincident with the boundaries of the flow (that is a free surface or bottom). This paper shows that the NLD-models of both groups can be obtained on the basis of a unified approach. Starting from the Euler equations of inviscid incompressible fluid, without the assumption of potential flow, the universal NLD-model of long-wave hydrodynamics has been derived. It considers some vector-function associated with the flow parameters. By selecting this function in a proper way, all of the most famous models of long-wave approximation can be derived. Therefore, the constructed NLD-model can be regarded as the basic model. Particular attention is attracted to the NLD model, which has the improved approximation of the dispersion relation proposed by Nwogu (1993), and generalized to the case of a movable bottom by Lynett & Liu (2002). It’s extremely cumbersome notation is a drawback. Now we have shown that under assumption of flow potentiality this equations can be obtained from the basic model. Consequently, we can write them in a compact form. Moreover, in the framework of this approach, it is possible to consider some NLD-models, based on the choice of the sought after velocity in a way which is different from the above. As an example, the derivation of the Aleshkov’s model is performed. One of the advantages of this model is a preservation of flow potentiality, if it was inherent in the original threedimensional model. The advantage of the proposed basic model is that it can be written in a quasi-conservative form, which can be reduced into the conservative form in the case of a flat bottom. In addition, the notation of the governing system of nonlinear dispersive equations is compact and physically meaningful. Therefore the base model paves the way for standardization of proven computational algorithms, which were designed previously for specific systems of the NLD-equations.
[full text]
Keywords: Long surface waves, nonlinear dispersion equations, basic model
Author(s):
Fedotova Zinaida Ivanovna
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Lavrentiev ave. 6
Phone Office: (383) 334-91-21
E-mail: zf@ict.nsc.ru
Khakimzyanov Gayaz Salimovich
Dr. , Professor
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: khak@ict.nsc.ru
SPIN-code: 3144-0877 References:
[1] Mei C.C., Le Mehaute B. Note on the equations of long waves over an uneven bottom. Geophys.Res. 1966; (71):393–400.
[2] Peregrine D.H. Long waves on a beach. J. Fluid. Mech. 1967; 27(4):815–827.
[3] Fedotova Z.I., Khakimzyanov G.S. Full nonlinear dispersion model of shallow water equations on a rotating sphere. J. Appl. Mech. and Tech. Phys. 2011; 52(6):618–638. (In Russ.)
[4] Fedotova Z.I., Khakimzyanov G.S. On analysis of conditions for derivation of nonlinear-dispersive equations. Vychisl. tehnologii. 2012; 17(5):94–108. (In Russ.)
[5] Fedotova Z. I., Khakimzyanov G. S. Nonlinear-dispersive shallow water equations on a rotating sphere and conservations laws. J. Appl. Mech. and Tech. Phys. 2014; 55(3):404–416. (In Russ.)
[6] Wu T.Y. Long waves in ocean and coastal waters. J. of Engrg. Mech. 1981; 107(3):501–522.
[7] Zheleznyak M.I., Pelinovskij E.N. Physical and mathematical models of the tsunami climbing a beach. Nakat cunami na bereg: Sb. nauchn. tr. Gor'kij: IPF AN SSSR; 1985:8–33. (In Russ.)
[8] Rygg O.B. Nonlinear refraction-diffraction of surface waves in intermediate and shallow water. Coastal Engng. 1988; (12):191–211.
[9] Dellar P.J., Salmon R. Shallow water equations with a complete Coriolis force and topography. Phys. Fluids. 2005; 17(106601):23.
[10] Fedotova Z.I., Khakimzyanov G.S. Nonlinear-dispersive shallow water equations on a movable bottom. Vychisl. tehnologii. 2008; 13(4):114–126. (In Russ.)
[11] Witting J.M. A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 1984; 56(2):203–236.
[12] Madsen P.A., Murray R., Sorensen O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engng. 1991; (15):371–388.
[13] Nwogu O. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coast. Ocean Engng. 1993; 119(6):618–638.
[14] Kennedy A.B., Kirby J.T., Chen Q., Dalrymple R.A. Boussinesq-type equations with improved nonlinear performance. Wave Motion. 2001; (33):225–243.
[15] Lynett P.J., Liu P.L.-F. A numerical study of submarine-landslide-generated waves and run-up. Proceedings of Royal Society of London. A. 2002; (458):2885–2910.
[16] Liu P.L.-F. Model equations for wave propagations from deep to shallow water. In Advances in coastal and ocean engineering (ed. P.L.-F. Liu). Singapore: World Scientific Publishing Co. 1994; (1):125–157.
[17] Wei G., Kirby J.T., Grilli, S.T., Subramanya R. The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 1995; (294):71–92.
[18] Lynett P.J., Wu T.-R., Liu P.L.-F. Modeling wave runup with depth-integrated equations characteristics. Coastal Engng. 2002; (46):89–107.
[19] Hsiao S., Liu P.L.-F., Chen Y. Nonlinear water waves propagating over a permeable bed. Proceedings of Royal Society of London. A. 2002; (458):1291–1322.
[20] Chen Q. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. J. Engrg. Mech. 2006; 132(2):220–230.
[21] Shi F., Kirby J.T., Tehranirad B., Harris J.C. A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Modelling. 2012; (43-44):36–51.
[22] Fedotova Z.I., Khakimzyanov G.S., Dutykh D. On the energy equation of approximate models in the long-wave hydrodynamics. Russ. J. Numer. Anal. Math. Modelling. 2014; 29(3):167–178.
[23] Green A.E., Naghdi P.M. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 1976; 78(2):237–246.
[24] Ertekin R.C., Webster W.C., Wehausen J.V. Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech. 1986; (169):275–292.
[25] Kim D.-H., Lynett P.J., Socolofsky S.A. A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows. Ocean Modelling. 2009; (27):198–214.
[26] Aleshkov Yu.Z. Currents and waves in the ocean. Sankt-Peterburg: Izd-vo S.-Peterburgskogo universitet; 1996: 226. (In Russ.)
[27] Gusev O.I., Shokina N.Ju., Kutergin V.A., Khakimzyanov G.S. Simulation of surface waves generated by an underwater landslide in a reservoir. Vychisl. tehnologii. 2013; 18(5):74–90. (In Russ.)
[28] Shokin Ju.I., Bejzel' S.A., Gusev O.I., Khakimzjanov G.S., Chubarov L.B., Shokina N.Ju. Numerical study of dispersive waves arising from a movement of an underwater landslide. Vestnik JuUrGU. Serija: Matematicheskoe modelirovanie i programmirovanie. 2014; 7(1):121–133. (In Russ.)
Bibliography link:
Fedotova Z.I., Khakimzyanov G.S. The basic nonlinear-dispersive hydrodynamic model of long surface waves // Computational technologies. 2014. V. 19. ¹ 6. P. 77-94
|
