Abstract
Two-dimensional, two-layer steady interfacial flow about a point vortex is studied in a uniform stream for each layer. The upper layer is of finite depth with a rigid lid on the upper surface, and the depth of the lower layer is assumed infinite. The point vortex is located in lower-layer fluid. We study this problem using not only a linear analytical method but also a nonlinear numerical method. A linear solution is derived in terms of a complex exponential integral function. The fully nonlinear problem is formulated by an integro-differential equation system. The equation system is solved using Newton’s method to determine the unknown steady interfacial surface. The numerical results of the downstream wave are provided by a linear solution and fully nonlinear solution. A comparison between linear solutions and nonlinear solutions shows that the nonlinear effect is apparent when the vortex strength increases. The effects of point vortex strengths, Froude numbers, and density ratios on the amplitudes of the downstream waves are studied. We analyze the effects of point vortex strengths, Froude numbers, and density ratios on the wavelengths of the downstream waves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baines PG (1995) Topographic effect in stratified flows. Cambridge University Press, Cambridge
Vanden-Broeck JM (2010) Gravity-capillary free surface flows. Cambridge University Press, Cambridge
Melville WK, Helfrich Karl R (1987) Transcritical two-layer flow over topography. J Fluid Mech 178:31–52
Wehansen JV, Laitone EV (1960) Surface waves. In: Flugge S (ed) Handbuch der Physik, vol 9. Springer, Berlin
Kochin NE, Kibel IA, Roze NV (1964) Theoretical hydromechanics. Wiley Interscience, New York
Forbes LK (1985) On the effects of non-linearity in free-surface flow about a submerged point vortex. J Eng Math 19:139–155
Dias F, Vanden-Broeck JM (2004) Two-layer hydraulic falls over an obstacle. Eur J Mech B 23(6):879–898
Forbes LK (1989) Two-layer critical flow over a semi-circular obstruction. J Eng Math 23:325–342
Sha H, Vanden-Broeck JM (1993) Two layer flows past a semicircular obstruction. Phys Fluids A 5(11):2661–2668
Belward SR, Forbes LK (1993) Fully non-linear two-layer flow over arbitrary topography. J Eng Math 27(4):419–432
Forbes LK, Schwartz LW (1982) Free-surface flow over a semicircular obstruction. J Fluid Mech 114:299–314
Liang H, Zong Z, Sun L, Zou L, Zhou L, Zhao YJ, Ren ZR (2013) Generalized Weissinger’s L-method for prediction of curved wings operating above a free surface in subsonic flow. J Eng Math 83:109–129
Faltinsen OM, Timokha AN (2009) Sloshing. Cambridge University Press, New York
Abramowitz M, Stegun I (1964) Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover, New York
Giesing JP, Smith AMO (1967) Potential flow about two-dimensional hydrofoils. J Fluid Mech 28:113–129
Newman JN (1987) Evaluation of the wave-resistance Green function: part 1—the double integral. J Ship Res 31:79–90
An S, Faltinsen OM (2012) Linear free-surface effects on a horizontally submerged and perforated 2D thin plate in finite and infinite water depths. Appl Ocean Res 37:220–234
Acknowledgments
This work is supported by National Science Foundation of China (51579040, 51379033, 50921001), the Fundamental Research Funds for the Central Universities (DUT2015LK34, DUT2015LK45), and National Key Basic Research Project of China (2013CB036101, 2010CB32700). Many thanks for the anonymous reviewers’ valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Z., Zou, L., Liang, H. et al. Nonlinear steady two-layer interfacial flow about a submerged point vortex. J Eng Math 103, 39–53 (2017). https://doi.org/10.1007/s10665-016-9859-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-016-9859-5