Summary
The linearized stability analysis is applied to investigate the wave behavior in a water vortex produced in a cylindrical tank with a flat disk rotating at the bottom. Two flow cases are considered herein. The first case deals with waves developed on the free surface of a hollow vortex, while the second with waves generated in the core of a Rankine vortex. It is evident from the analysis that the experimental dispersion velocity approaches the calculated one when the wave amplitude becomes smaller. The latter is consistent with the small perturbation assumption that is inherent in the theory. For the case where the core is flooded, the presence of a cylindrical wall is shown to enhance the wave speed. A hypothesis as to how the core develops in the mixed state regions is proposed. The graphical simulations appear to predict reasonably well the main features of the observations.
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Abbreviations
- a :
-
radius of the liquid surface whenz=0
- h 0 :
-
initial liquid height
- m :
-
slope of the tangential velocity in [o,r c]
- P :
-
static pressure
- n :
-
wave number
- r, φ,z, t :
-
radial, azimuthal, axial and time coordinates
- r c :
-
core radius
- r i :
-
cylindrical tank radius
- r e, ze :
-
radius and elevation of the free surface
- u, v :
-
velocity components (inr and φ directions respectively)
- χ:
-
vortex strength of the base flow
- v :
-
kinematic viscosity of the fluid
- ϑ:
-
liquid density
- σ:
-
angular frequency
- Φ:
-
velocity potential
- Φ :
-
disk speed
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Vatistas, G.H., Wang, J. & Lin, S. Recent findings on Kelvin's equilibria. Acta Mechanica 103, 89–102 (1994). https://doi.org/10.1007/BF01180220
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DOI: https://doi.org/10.1007/BF01180220