Abstract
Particle trajectories under a solitary wave are studied using the high-level Irrotational Green-Naghdi (IGN) equations. The steady IGN equations are solved using the Newton-Raphson method. It is determined that the IGN-2 model provides the converged IGN results of the velocity predictions, and the particle trajectories calculated by the IGN-2 model agree well with the available experimental data.
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Acknowledgments
The authors thank the anonymous reviewers and the associate editor for their helpful comments that improved this paper. The first and third authors’ work is supported by the National Natural Science Foundation of China (No. 11102049), the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP, No. 20112304120021), the Special Fund for Basic Scientific Research of Central Colleges, Harbin Engineering University (Harbin), and the 111 Project (B07019). This research was supported by Lloyd’s Register Foundation (LRF) through the Joint Centre involving University College London, Shanghai Jiaotong University, and Harbin Engineering University, to which the authors are most grateful. Lloyd's Register Foundation helps to protect life and property by supporting engineering-related education, public engagement, and the application of research.
References
Borluk, H., and Kalisch, H. (2012). “Particle dynamics in the KdV approximation.” Wave Motion, 49(8), 691–709.
Constantin, A. (2010). “On the particle paths in solitary water waves.” Q. Appl. Math., 68(1), 81–90.
Constantin, A., and Escher, J. (2007). “Particle trajectories in solitary water waves.” Bull. Am. Math. Soc., 44(3), 423–431.
Constantin, A., Escher, J., and Hsu, H.-C. (2011). “Pressure beneath a solitary water wave: Mathematical theory and experiments.” Arch. Ration. Mech. Anal., 201(1), 251–269.
Daily, J. W., and Stephan, S. C. (1952). “The solitary wave: Its celerity, profile, internal velocities and amplitude attenuation in a horizontal smooth channel.” Coastal Eng. Proc., 1(3), 13–30.
Ertekin, R. C. (1984). “Soliton generation by moving disturbances in shallow water: Theory, computation and experiment.” Ph.D. thesis, Univ. of California, Berkeley, CA.
Ertekin, R. C. (1988). “Nonlinear shallow water waves: The Green-Naghdi equations.” Proc., Pacific Congress on Marine Science and Technology, PACON International, Honolulu.
Ertekin, R. C., Hayatdavoodi, M., and Kim, J. W. (2014). “On some solitary and cnoidal wave diffraction solutions of the Green-Naghdi equations.” Appl. Ocean Res., 47(Aug), 125–137.
Fenton, J. (1972). “A ninth-order solution for the solitary wave.” J. Fluid Mech., 53(2), 257–271.
Hsu, H.-C., Chen, Y.-Y., and Hwung, H.-H. (2012a). “Experimental study of the particle paths in solitary water waves.” Philos. Trans. R. Soc. London, Ser. A, 370(1964), 1629–1637.
Hsu, H.-C., Chen, Y.-Y., Lin, C.-Y., and Cheng, C.-Y. (2012b). “Experimental study of the velocity field in solitary water waves.” J. Nonlinear Math. Phys., 19(S1), 1240003.
Kim, J. W., Bai, K. J., Ertekin, R. C., and Webster, W. C. (2001). “A derivation of the Green-Naghdi equations for irrotational flows.” J. Eng. Math., 40(1), 17–42.
Kim, J. W., Bai, K. J., Ertekin, R. C., and Webster, W. C. (2003). “A strongly-nonlinear model for water waves in water of variable depth—The Irrotational Green-Naghdi model.” J. Offshore Mech. Arct. Eng., 125(1), 25–32.
Kim, J. W., and Ertekin, R. C. (2000). “A numerical study of nonlinear wave interaction in regular and irregular seas: Irrotational Green–Naghdi model.” Mar. Struct., 13(4–5), 331–347.
Kim, J. W., Ertekin, R. C., and Bai, K. J. (2010). “Linear and nonlinear wave models based on Hamilton’s principle and stream-function theory: CMSE and IGN.” J. Offshore Mech. Arct. Eng., 132(2), 021102.
Korteweg, D., and de Vries, G. (1895). “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves.” Philos. Mag., 39(240), 422–443.
Laitone, E. V. (1959). “Water waves, IV: Shallow water waves.” Rep. No. 82-11, Institute of Engineering Research, Univ. of California, Berkeley, CA.
Longuet-Higgins, M. S. (1981). “Trajectories of particles at the surface of steep solitary waves.” J. Fluid Mech., 110(Sep), 239–247.
McCowan, J. (1891). “VII. On the solitary wave.” Philos. Mag., 32(194), 45–58.
Munk, W. H. (1949). “The solitary wave theory and its application to surf problems.” Ann. N.Y. Acad. Sci., 51(3), 376–424.
Rayleigh, L. (1876). “XXXII. On waves.” Philos. Mag., 1(4), 257–279.
Umeyama, M. (2013). “Investigation of single and multiple solitary waves using superresolution PIV.” J. Waterway, Port, Coastal, Ocean Eng., 304–313.
Zhao, B. B., Ertekin, R. C., Duan, W. Y., and Hayatdavoodi, M. (2014). “On the steady solitary-wave solution of the Green–Naghdi equations of different levels.” Wave Motion, in press.
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© 2014 American Society of Civil Engineers.
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Received: Mar 26, 2014
Accepted: Aug 29, 2014
Published online: Sep 25, 2014
Published in print: May 1, 2015
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