Abstract
A universal variational formulation for two dimensional fluid mechanics is obtained, which is subject to the so-called parameter-constrained equations (the relationship between parameters in two governing equations). By eliminating the constraints, the generalized variational principle (GVPs) can be readily derived from the formulation. The formulation can be applied to any conditions in case the governing equations can be converted into conservative forms. Some illustrative examples are given to testify the effectiveness and simplicity of the method.
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References
CHIEN Wei-zang. Generalized Variational Principle [M]. Beijing: Knowledge Press, 1985. (in Chinese)
Lin C C. Hydrodynamics of liquid helium, II: Liquid helium [A]. In: G Careri Ed. Proceedings of International School of Physics [C]. course XXI, New York: Academic Press,1963,93–146.
CHIEN Wei-zang. Method of high-order Lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals [J]. Applied Mathematics and Mechanics (English Edition), 1983,4(2):143–157.
CHIEN Wei-zang. Variational principles and generalized variational principles in hydrodynamics of viscous fluids [J]. Applied Mathematics and Mechanics (English Edition), 1984,5(3):1281–1296.
LIU Gao-lian. A systematic approach to the search and transformation for variational principles in fluid mechanics [J]. Journal of Engineering Thermophysics, 1990,11(2):136–142. (in Chinese)
LIU Gao-lian. Research development of variation principles & finite element method in fluid mechanics[J]. Shanghai Journal of Mechanics, 1989, 10(3):73–80. (in Chinese)
Herivel J W. The derivation of the equations of motion of an ideal fluid by Hamilton's principle [J]. Proc Cambridge Philos Soc,1955,51:344–349.
HE Ji-huan. On C C Lin Constraints of variational principle in fluids [A]. In: Zhuang F G Ed. Modern Mechanics and Advances in Sciences & Technology [C]. Beijing: Tsinghua University Press, 1997, 603–604.
HE Ji-huan. A new approach to establishing generalized variational principles and C C Lin's constraints[D]. Ph D Thesis. Shanghai: Shanghai University, 1997. (in Chinese).
HE Ji-huan. Generalized Variational Principles in Fluids [M]. Shanghai: Shanghai University Press, 2000. (in Chinese)
Bretherton F P. A note on Hamilton's principle for perfect fluids [J]. J Fluid Mech,1970,44(1):19–31.
Salmon R. Hamiltoinian fluid mechanics [J]. Ann Rev Fluid Mech,1988,20:225–256.
HE Ji-huan. Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics [J]. Int J Turbo & Jet-Engines,1997,14(1): 23–28.
HE Ji-huan. A family of variational principles for compressible rotational blade-to-blade flow using semi-inverse method [J]. Int J Turbo & Jet-Engines,1998,15(2):95–100.
HE Ji-huan. Hybrid problems of determining unknown shape of bladings in compressible S2-flow in mixed-flow turbomachinery via variational technique [J]. Aircraft Engineering and Aerospace Technology,1999,71(2):154–159.
HE Ji-huan. Inverse problems of determining the unknown shape of oscillating airfoils in compressible 2D unsteady flow via variational technique [J]. Aircraft Engineering and Aerospace Technology, 2000,72(1):18–24.
HE Ji-huan. Generalized Hellinger-Reissner principle [J]. ASME Journal of Applied Mechanics, 2000,67(2):326–331.
HE Ji-huan. A variational principle for thermopiezoelectricity based on Chandrasekha-raiah's generalized linear theory [J]. J University of Shanghai for Science and Technology,1999,21(4):356–365.
HE Ji-huan. Coupled variational principles of piezoelectricity [J]. Int J Engineering Science, 2000,39(3):323–341.
HE Ji-huan. A variational model for micropolar fluids in lubrication journal bearing [J]. International Journal of Nonlinear Sciences and Numerical Simulation,2000,1(2):139–141.
HE Ji-huan. Classical variational model for the micropolar elastodynamics [J]. International Journal of Nonlinear Sciences and Numerical Simulation,2000,1(2):133–138.
HE Ji-huan. On variational crisis and generalized variational principle of elasticity [J]. Journal of University of Shanghai for Sciences and Technology, 1999,21(2):127–130. (in Chinese)
HE Ji-huan. An overview of variational crises and its recent developments [J]. Journal of University of Shanghai for Sciences and Technology, 1999,21(1):29–35. (in Chinese)
HE Ji-huan. Variational crisis of elasticity and its removal [J]. Shanghai Journal of Mech, 1997, 18(4):305–310. (in Chinese)
LIU Gao-lian. A term-condensing method and generalized potential-, stream-& path functions for 3-D compressible viscous flow [A]. In: Proceeding Second Asian Congress of Fluid Mechanics[C]. Beijing: Science Press,1981,698–704.
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He, Jh. A Universal Variational Formulation for Two Dimensional Fluid Mechanics. Applied Mathematics and Mechanics 22, 989–996 (2001). https://doi.org/10.1023/A:1016310906598
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DOI: https://doi.org/10.1023/A:1016310906598