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 linear stability of two incompressible coaxial jets, separated by a thick duct wall, is investigated by means of both a modal and a non-modal approach within a global framework. The attention is focused on the range of unitary velocity ratios for which an alternate vortex shedding from the duct wall is known to dominate the flow. In spite of the inherent convective nature of jet flow instabilities, such behaviour is shown to originate from an unstable global mode of the dynamics linearised around the axisymmetric base flow. The corresponding wavemaker is located in the recirculating-flow region formed behind the duct wall. At the same time, the transient-growth analysis reveals that huge amplifications (up to orders of magnitude) of small flow perturbations at the nozzle exit can occur in the subcritical regime, especially for high ratios between the outer and the inner velocities

Isolated patches of turbulence in transitional straight pipes are sustained by a strong instability at their upstream front, where the production of turbulent kinetic energy (TKE) is up to five times higher than in the core. Direct numerical simulations presented in this paper show no evidence of such strong fronts if the pipe is bent. We examine the temporal and spatial evolution of puffs and slugs in a toroidal pipe with pipe-to-torus diameter ratio δ = D/d = 0.01 at several subcritical Reynolds numbers. Results show that the upstream overshoot of TKE production is at most one-and-a-half times the value in the core and that the average cross-flow fluctuations at the front are up to three times lower if compared to a straight pipe, while attaining similar values in the core. Localised turbulence can be sustained at smaller energies through a redistribution of turbulent fluctuations and vortical structures by the in-plane Dean motion of the mean flow. This asymmetry determines a strong localisation of TKE production near the outer bend, where linear and nonlinear mechanisms optimally amplify perturbations. We further observe a substantial reduction of the range of Reynolds numbers for longlived intermittent turbulence, in agreement with experimental data from the literature. Moreover, no occurrence of nucleation of spots through splitting could be detected in the range of parameters considered. Based on the present results, we argue that this mechanism gradually becomes marginal as the curvature of the pipe increases and the transition scenario approaches a dynamical switch from subcritical to supercritical.