The instability of some time periodically forced flows

• Main Author: Sagoo, Gursharan
• Other Authors: Hall, Philip
• Published: Imperial College London 2003
• Subjects: 532.05
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.406463

In this thesis the instability of two viscous incompressible flows is discussed by using numerical and analytical methods. The first problem concerns the steady streaming flow, that is contained within a hollow stationary cylinder and induced by the transverse oscillation of a solid inner cylinder. The small gap limit is taken so that a series solution in odd powers of the angular variable is possible. From the studies by Hall & Papageorgiou [37] and Watson et al. [97], it is known that the leading order equation has solutions that are steady, quasi-periodic and chaotic (period doubling). Since all the higher order equations are driven by the solution at leading order; the series solution for the steady streaming flow is investigated with an interest to determine any chaotic structures. The second problem concerns the flow in a horizontal circular pipe, that is subject to torsional oscillations about a vertical axis that passes symmetrically through the pipe. The onset of a new axisymmetric roll-type instability, as observed experimentally by Bolton & Maurer [10] for the corresponding rectangular tank problem (of small width), is sought in the high-frequency (Phi >> 1) and small-amplitude limit (alpha << 1). A perturbation of the WKBJ type is imposed upon the basic state, so that the slow angular variation of the disturbance is accounted for in the linear stability equations. Accordingly, a dispersion relation for the dimensionless frequency parameter Phi is derived. In order to identify the most dangerous disturbance, it is necessary to minimise the eigenvalue B = alpha/Phi^(1/4). The theory of Soward & Jones [82] is used to show that an acceptable solution of the governing eigenvalue problem, cannot be obtained for real values of the latitudinal variable theta; instead, the correct minimum is found in the complex theta-plane.