相容拉格朗日-欧拉法求解黏性流体中弹性圆柱壳的振动
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摘要
采用相容拉格朗日-欧拉法,研究圆柱壳表面不间断振动时黏性流体的速度以及壳体的推进速度。根据黏性流体分子的黏附条件简化接触面条件,变形后的变量通过变形前各量的泰勒级数展开式近似表示。假设雷诺数Re<0.1,纳维-斯托克斯方程采用斯托克斯近似,考虑弹性圆柱壳表面发生横向振动与纵向振动。结果表明,弹性圆柱壳半径增大时,流体的速度趋近于薄板振动时的结果,纯横向振动时壳的位移方向与波的方向相反,纯纵向振动时二者方向相同,两种形式的振动均有,则方向可能相同,也可能相反。
Here, when a cylindrical shell vibrated continuously in viscous fluid, velocity of viscous fluid and the cylindrical shell's propulsion speed were studied with the united Lagrangian-Eulerian method. The contact surface conditions were simplified according to adhesion conditions of viscous fluid molecules. Variables after deformation were approximately expressed with their Taylor series expansions before deformation. When Reynolds number Re < 0.1, Navier-Stokes equations were replaced by Stokes ones to study transverse and longitudinal vibrations of the elastic cylindrical shell. The results showed that fluid velocity distribution approaches a thin plate vibration results with increase in cylindrical shell's radius; the shell's displacement direction is opposite to the wave's one during shell having transverse vibration, but their directions are the same during shell having longitudinal vibration; their directions may be the same or opposite during shell having both transverse and longitudinal vibrations.
Here, when a cylindrical shell vibrated continuously in viscous fluid, velocity of viscous fluid and the cylindrical shell's propulsion speed were studied with the united Lagrangian-Eulerian method. The contact surface conditions were simplified according to adhesion conditions of viscous fluid molecules. Variables after deformation were approximately expressed with their Taylor series expansions before deformation. When Reynolds number Re < 0.1, Navier-Stokes equations were replaced by Stokes ones to study transverse and longitudinal vibrations of the elastic cylindrical shell. The results showed that fluid velocity distribution approaches a thin plate vibration results with increase in cylindrical shell's radius; the shell's displacement direction is opposite to the wave's one during shell having transverse vibration, but their directions are the same during shell having longitudinal vibration; their directions may be the same or opposite during shell having both transverse and longitudinal vibrations.
引文
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[2] 徐晓锋.仿生鱼游动与运动控制的数值分析技术[D].上海:上海交通大学,2011.
[3] LIGHTHILL J.Mathematical Biofluiddynamics[M].SIAM,1975.
[4] TAYLOR G.Analysis of the swimming of long narrow animals[J].Proc.R.Soc.Lond.A,1952,214:158-183.
[5] LIGHTHILL J.Note on the swimming of slender fish[J].Journal of Fluid Mechanics,1960(9):305-317.
[6] WU T Y.Swimming of a waving plate[J].Journal of Fluid Mechanics,1961,10:321-344.
[7] CHOPRA M G.Hydromechanics of lunate-tail swimming propulsion[J].Journal of Fluid Mechanics,1974,64:375-391.
[8] TRIANTAFYLLOU M S.Hydrodynamics of fishlike swimming[J].Annu Rev Fluid Mech,2000,32(1):33-53.
[9] TIAN F B,LU X Y,LUO H.Propulsive performance of a body with a traveling-wave surface[J].Physical Review E Statistical Nonlinear & Soft Matter Physics,2012,86:3-5.
[10] 严宗毅.低雷诺数流理论[M].北京:北京大学出版社,2002.
[11] ИЛЬГАМОВ М А.Введение в нелинейную гидроупругость[M].Москва:Изд.Наука,1991.
[12] 郝亚娟,石运会.势流中可渗透圆柱壳的变形与压力分析[J].力学季刊,2016,37(1):176-183.HAO Yajuan,SHI Yunhui.Analysis of deformation and pressure of porous circular cylinder in potential flow[J].Chinese Quarterly of Mechanics,2016,37 (1):176-183.
[13] 郝亚娟,平畔畔.势流中弹性薄板的变形与应力分析[J].力学季刊,2014,35(4):651-657.HAO Yajuan,PING Panpan.Analysis of deformation and stress of elastic thin plate in potential flow[J].Chinese Quarterly of Mechanics,2014,35(4):651-657.
[14] 陈懋章.黏性流体动力学基础[M].北京:高等教育出版社,2001.
[15] 上海交通大学船舶制造系编.流体力学[M].北京:科学教育出版社,1961.
[16] 王竹溪,郭敦仁.特殊函数类[M].北京:科学出版社,1979.
[17] BLAKE J R.Infinite models for ciliary propulsion[J].Journal of Fluid Mechanics,1971,49(2):209-222.