Bifurcations in Kolmogorov and Taylor-vortex flows

Author: Love, Philip

Year: 1999

Degree: Dissertation (Ph.D.)

Advisor: Keller, Herbert Bishop

Committee Members: Keller, Herbert Bishop; Pullin, Dale Ian; Saffman, Philip G.

Option: Applied And Computational Mathematics

DOI: 10.7907/g2f3-s507

Abstract

The bifurcation structure of Kolmogorov and Taylor-Vortex flows was computed with the aid of the Recursive Projection Method; see Schroff and Keller [32]. It was shown that RPM significantly improves the convergence of our numerical method while calculating steady state solutions. Moreover we use RPM to detect bifurcation points while continuing along solution branches, and to provide the required augmentation when continuing around a fold, or along a traveling wave branch.

The bifurcations to two and three-dimensional solutions from the shear flow solution of Kolmogorov flow are calculated both numerically, by solving an ordinary differential equation, and analytically, using an approximation method. Our results for the two-dimensional bifurcations agree with the work of Meshalkin and Sinai [26].

We also explain how the branches of Kolmogorov flows observed by Platt and Sirovich [29] are connected together, and observe that our solutions have worm like structures even at relatively low Reynolds numbers. Various statistics of our flows are calculated and compare with those from isotropic turbulence calculations.

Additionally various solution branches of the Taylor Vortex flow were computed, including spiral vortices. Furthermore, it was discovered that the Wavy Taylor Vortex branches arise from sub-critical Hopf bifurcations, and they undergo a fold close to their bifurcation point.

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