Large-amplitude wavetrains and solitary waves in vortices

Abstract

Large-amplitude axisymmetric waves on columnar vortices, thought to be related to flow structures observed in vortex breakdown, are found as static bifurcations of the Bragg–Hawthorne equation. Solutions of this equation satisfy the steady, axisymmetric, Euler equations. Non-trivial solution branches bifurcate as the swirl ratio (the ratio of azimuthal to axial velocity) changes, and are followed into strongly nonlinear regimes using a numerical continuation method. Four types of solutions are found: multiple columnar solutions, corresponding to Benjamin's ‘conjugate flows’, with subcritical–supercritical pairing of wave characteristics; solitary waves, extending previously known weakly nonlinear solutions to amplitudes large enough to produce flow reversals similar to the breakdown transition; periodic wavetrains; and solitary waves superimposed on the conjugate flow that emerge from the periodic wavetrain as the wavelength or amplitude becomes sufficiently large. Weakly nonlinear soliton solutions are found to be accurate even when the perturbations they cause are fairly strong.

The role of non-uniqueness in the development of vortex breakdown in tubes

“…This equation, attributed by Leibovich & Kribus (1990) to Bragg & Hawthorne (1950), will be referred to as the BRE and is given by (51) The circulation, r (defined in (5a)), is solely a function ofljf in inviscid flow. The total head, H, is also a function of ljf only: (52) The explicit dependence of the right-hand side of (51) on ljf is obtained through the specification of the flow state at a particular axial station.…”

Solitary waves with recirculation zones in axisymmetric rotating flows

“…In turn, for narrow jetlike profiles of the axial velocity, the solitary waves with a recirculation zone exist only for sufficiently large swirls (to the right of curve (ii), figure 15). This is a typical case for the experiments of Leibovich (1991) where the 'bubble' type of breakdown was not observed for small swirls. In figure 16, we show that wider profiles of the swirl results in a smaller maximum amplitude of the recirculation zone.…”

“…It is not our intention here to discuss the mechanisms leading to these various kinds of breakdown. However, we note that our model can explain the observation (see Leibovich 1991) that for a jetlike profile of axial velocity, the 'bubble' type of breakdown occurs only for sufficiently large inflow swirls. The margin between swirls such that the 'bubble' does or does not exist is quite sharp.…”

“…It is pertinent to compare our results with those of others, such as Leibovich and collaborators (see, for instance, Leibovich 1970Leibovich , 1991Leibovich & Randall 1973;Leibovich & Kribus 1990), who have used long-wave theories to study vortex breakdown. We note here two crucial differences.…”

“…However, in experiments, the observed finite size of the recirculation zone may be due to a combination of inertial effects and viscous diffusion at low Reynolds numbers, or to complicated instabilities inside and around the breakdown bubble at high Reynolds numbers. Theoretical models of this phenomenon have been proposed by Leibovich and collaborators (see, for instance, the review by Leibovich 1991). Thus, Leibovich & Kribus (1990) established the existence of four types of solution: multiple columnar solutions corresponding to the conjugate states of Benjamin (1962Benjamin ( , 1967, solitary waves, periodic wavetrains and solitary waves superimposed on the conjugate flow that emerges from the wavetrain.…”