The Kelvin Waves in Vortex Dynamics and Their Stability
Abstract
The rotating mwaves of Kelvin form a class K_{m}, m ⩾ 2, of mfold symmetric regions D of constant vorticity ω_{0} ≡ 1 which are uniformly rotating with an angular velocity (bifurcation parameter) Ω in the range Ω _{m}1 < Ω ⩽ Ω _{m}, Ω _{m} ≡ {(m 1)}/{2m}. The class K_{2} corresponds to the rotating ellipse of Kirchoff. We present a numerical method for the determination of the stabilitycharacteristics of the class K_{m}. The method, based on Burbea's theory of the stability of vortexmotions, uses conformal mappings to construct the spectrum of a certain crucial "stabilityoperator" B. Numerical results show the existence of a critical value Ω _{cr}(m) = {(3Ω _{m} + Ω _{m1}) }/{4} with the following property: Given any Ω ɛ ( Ω _{m1}, Ω _{m}], there exists a steady state D ɛ K_{m}, unique up to rotation, magnification and reflection, whose angular velocity is Ω. Moreover, this state is (secularly) stable if and only if Ω _{cr} (m) < Ω ⩽ Ω _{m}. Other stabilitycharacteristics of the class K_{m} are determined. The entire results of Love for the particular class K_{2} are obtained here as special cases.
 Publication:

Journal of Computational Physics
 Pub Date:
 January 1982
 DOI:
 10.1016/00219991(82)901061
 Bibcode:
 1982JCoPh..45..127B
 Keywords:

 Computational Fluid Dynamics;
 Flow Stability;
 Vorticity Equations;
 Algorithms;
 Conformal Mapping;
 Eigenvalues;
 Flow Charts;
 Nonlinear Equations;
 Wave Propagation;
 Fluid Mechanics and Heat Transfer