Abstract
A numerical method for the optimum motion of an undulatory swimming plate is presented. The optimum problem is stated as minimizing the power input under the condition of fixed thrust. The problem is singular for the invisible modes, and therefore the commonly used Lagrange multiplier method cannot predict an optimum solution but just a saddle point. To eliminate the singularity, an additional amplitude inequality constraint is added to the problem. A numerical optimization code with a sequential quadratic programming method is used to solve the problem. The method is applied to several cases of the motion of two-dimensional and three-dimensional undulatory plates, and the optimum results are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wu, T. Y. Swimming of a waving plate. Journal of Fluid Mechanics, 10, 321–344 (1961)
Liu, P. and Bose, N. Propulsive performance from oscillating propulsors with spanwise flexibility. Proceeding of the Royal Society of London A, 453, 1763–1770 (1997)
Zhu, Q., Wolfgang, M. J., Yue, D. K. P., and Triantafyllou, M. S. Three dimensional flow structures and vorticity control in fish-like swimming. Journal of Fluid Mechanics, 468, 1–28 (2002)
Dong, G. J. and Lu, X. Y. Numerical analysis on the propulsive performance and vortex shedding of fish-like traveling wavy plate. International Journal for Numerical Methods in Fluids, 48, 1351–1373 (2005)
Lu, X. Y. and Yin, X. Z. Propulsive performance of fish-like traveling wavy wall. Acta Mechanica, 175, 197–215 (2005)
Wu, C. J. and Wang, L. Where is the rudder of a fish?—The mechanism of swimming and control of self-propelled fish school. Acta Mechanica Sinica, 26, 45–65 (2010)
Lighthill, M. J. Mathematical Biofluiddynamics, SIAM, Philadelphia (1975)
Lighthill, M. J. Note on the swimming of slender fish. Journal of Fluid Mechanics, 9, 305–320 (1960)
Lighthill, M. J. Aquatic animal propulsion of high hydromechanical efficiency. Journal of Fluid Mechanics, 44, 265–301 (1970)
Wu, T. Y. Hydromechanics of swimming propulsion, part 1, swimming of a two-dimensional flexible plate at variable forward speeds in an invisid fluid. Journal of Fluid Mechanics, 46, 337–355 (1971)
Wu, T. Y. Hydromechanics of swimming propulsion, part 2, some optimum shape problems. Journal of Fluid Mechanics, 46, 521–544 (1971)
Cheng, J. Y., Zhuang, L. X., and Tong, B. G. Analysis of swimming three-dimensional waving plates. Journal of Fluid Mechanics, 232, 341–355 (1991)
Kagemoto, H., Wolfgang, M. J., Yue, D. K. P., and Triantafyllou, M. S. Force and power estimation in fish-like locomotion using a vortex-lattice method. Journal of Fluids Engineering, 122, 239–253 (2000)
Lan, C. E. A quasi-vortex-lattice method in thin wing theory. Journal of Aircraft, 11, 518–527 (1974)
Ahmadi, A. R. and Widnall, S. E. Energetics and optimum motion of oscillating lifting surfaces of finite span. Journal of Fluid Mechanics, 162, 261–282 (1986)
Spellucci, P. An SQP method for general nonlinear programs using only equality constrained subproblems. Mathematical Programming, 82, 413–448 (1998)
Spellucci, P. A new technique for inconsistent problems in the SQP method. Mathematical Method of Operations Research, 47, 355–400 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qian, Qj., Sun, Dj. Numerical method for optimum motion of undulatory swimming plate in fluid flow. Appl. Math. Mech.-Engl. Ed. 32, 339–348 (2011). https://doi.org/10.1007/s10483-011-1419-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-011-1419-x