In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave inc...In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface ten- sion on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.展开更多
In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wav...In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.展开更多
In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rig...In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder, which is subject to a vertical oscillation.It is assumed that the fluid in the circular cylindrical vessel is inviscid,incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales without considering the effect of surface tension.It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes.The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.展开更多
The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity....The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity. Many flow patterns were found at differentforced frequencies by numerical computation. In addition, the nonlinear amplitude equation derivedin inviscid fluid was modified by adding viscous damping and the unstable regions were determined bystability analysis.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos.19772063 and 19772068) and the Doctoral Research Fund of the Ministry of Education (No.20010141024)
文摘In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface ten- sion on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.
基金Project supported by the National Natural Science Foundation of China (Nos. 19772063, 19772068)the Doctoral Research Fund of the Ministry of Education (No.20010141024)
文摘In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.
文摘In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder, which is subject to a vertical oscillation.It is assumed that the fluid in the circular cylindrical vessel is inviscid,incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales without considering the effect of surface tension.It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes.The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.
文摘The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity. Many flow patterns were found at differentforced frequencies by numerical computation. In addition, the nonlinear amplitude equation derivedin inviscid fluid was modified by adding viscous damping and the unstable regions were determined bystability analysis.
