Modal instability of the flow in a toroidal pipe

Abstract

The modal instability encountered by the incompressible flow inside a toroidal pipe is studied, for the first time, by means of linear stability analysis and direct numerical simulation (DNS). In addition to the unquestionable aesthetic appeal, the torus represents the smallest departure from the canonical straight pipe flow, at least for low curvatures. The flow is governed by only two parameters: the Reynolds number Re and the curvature of the torus δ, i.e. the ratio between pipe radius and torus radius. The absence of additional features, such as torsion in the case of a helical pipe, allows us to isolate the effect that the curvature has on the onset of the instability. Results show that the flow is linearly unstable for all curvatures investigated between 0.002 and unity, and undergoes a Hopf bifurcation at Re of about 4000. The bifurcation is followed by the onset of a periodic regime, characterised by travelling waves with wavelength O(1) pipe diameters. The neutral curve associated with the instability is traced in parameter space by means of a novel continuation algorithm. Tracking the bifurcation provides a complete description of the modal onset of instability as a function of the two governing parameters, and allows a precise calculation of the critical values of Re and δ. Several different modes are found, with differing properties and eigenfunction shapes. Some eigenmodes are observed to belong to groups with a set of common characteristics, deemed 'families', while others appear as 'isolated'. Comparison with nonlinear DNS shows excellent agreement, confirming every aspect of the linear analysis, its accuracy, and proving its significance for the nonlinear flow. Experimental data from the literature are also shown to be in considerable agreement with the present results.

The vanishing of strong turbulent fronts in bent pipes

“…Stability map of the flow in a bent pipe. The continuous black line is the neutral curve calculated using linear theory (Canton et al 2016). Red stars indicate the critical limit for transition as measured by Sreenivasan & Strykowski (1983).…”

“…We have performed an analogous analysis for a bent pipe with δ = 0.01 and present the results in figure 8. Linear optimals were calculated using the finite element code PaStA (Canton 2013;Canton et al 2016). The figure shows the optimal perturbation calculated at Re = 2870, which is characterized by an energy gain of 106 by t = 7.5 with respect to the value at t = 0.…”

“…A unifying picture for transition in bent pipes was provided by Kühnen et al (2015) who reported that subcritical transition dominates for δ 0.028, while above this value transition to turbulence occurs through a supercritical bifurcation cascade. Figure 2 reports the neutral curve for the flow in a torus (Canton et al 2016) in the δ − Re plane, and experimental data indicating the onset of turbulence taken from the literature. Subcritical transition in bent pipes was described as being "very similar as in straight pipes, where laminar and turbulent flows can coexist" (Kühnen et al 2015).…”

Critical Point for Bifurcation Cascades and Featureless Turbulence

“…Regarding the bifurcation cascade, we focus on curvature δ = 0.05, which is far enough from the critical point to guarantee no influence of the subcritical transition scenario. By means of a modal stability analysis we verified that the steady flow becomes linearly unstable at Re = 3713 and nonlinear simulations up to Re = 4000 confirmed the nature of the supercritical Hopf bifurcation [34]. Figure 2(a) depicts the state of the system at this Reynolds number: the flow is constituted by a travelling wave generated by the first supercritical Hopf bifurcation and all trajectories in the phase space eventually converge to the corresponding relative equilibrium.…”

“…Transition to turbulence is subcritical at low curvatures and qualitatively similar to the one in straight pipes [30][31][32]. For larger δ, instead, transition is initiated by a supercritical Hopf bifurcation [30,[32][33][34] and all elements point towards a bifurcation cascade [33]. The dynamics of the flow at intermediate curvatures, however, remains largely unexplored, leaving unanswered the question of whether the two transition scenarios interact, and if so how.…”