Steady, periodic, quasi-periodic and chaotic flow regimes in toroidal pipes

Incompressible flow in a toroidal pipe was investigated by direct numerical simulation. The curvature a/c (radius of the cross section / radius of the torus) was 0.3 or 0.1 and the bulk Reynolds number ranged between 3500 and 14 700. The study revealed a rich scenario of transition to turbulence. For the higher curvature a/c = 0.3, a supercritical transition from stationary to periodic flow (Hopf bifurcation) was observed at Re=4600. The periodic flow was characterized by a travelling wave which, in the whole periodic Re range, took the form of a varicose modulation of the twin Dean vortex rings, included 8 wavelengths along the axis of the torus, and exhibited instantaneous anti-symmetry about the equatorial midplane. A further transition to quasi-periodic flow, characterized by two independent fundamental frequencies and their first few harmonics, occurred at Re=5200. The two frequencies were associated with two travelling wave systems, the first consisting of a varicose modulation of the Dean vortex rings, the second of an array of oblique near-wall vortices produced at the edge of the Dean cells, co-rotating with these latter and travelling from the inner towards the outer side, against the secondary circulation. For the lower curvature a/c=0.1, the results suggested the existence of a subcritical Hopf bifurcation at Re=5200 and of a secondary Hopf bifurcation to quasi-periodic flow at a lower Reynolds number of ~4900. Starting from zero-velocity initial conditions, the steady-state flow remained stable up to a Reynolds number of 5139, while a further increase in Re to 5208 yielded an abrupt transition to quasi-periodic flow which remained stable up to Re=6280 or larger. When a quasi-periodic solution (e.g., that obtained for Re=5658) was used as the initial condition and Re was made to decrease, the quasi-periodic regime remained stable down to values of Re well below the subcritical Hopf bifurcation at ~5200. Only a further, substantial decrease of Re to ~4108 led to the smooth disappearance of mode II and to a stable periodic solution. An abrupt transition to stationary flow was obtained when the Reynolds number decreased well below 4000 (e.g., a test case was computed for Re =3490). All periodic and quasi-periodic solutions for a/c=0.1 exhibited instantaneous symmetry about the equatorial midplane. Also the further transition from quasi-periodic to chaotic flow occurred with different mechanisms for the two curvatures. For a/c=0.3, quasi-periodic flow was obtained in the whole Reynolds number range 5270-7850. As Re increased slightly beyond this value (Re=8160), strong fluctuations, associated with random streamwise vortices, arose in the outer region. The ensuing chaotic flow regime was characterized by a broadband, almost continuous, frequency spectrum. A further increase of Re to 13180 did not modify to any appreciable extent the flow regime and the distribution of the velocity fluctuation intensity. For a/c=0.1, the convergence of the results to quasi-periodic flow became impossible to achieve as Re increased beyond ~6280, and was replaced by long and erratic transients. For Re=8160, the solution, albeit stationary in a statistical sense, was chaotic and exhibited a large number of frequencies, but the outer region remained basically stationary. Only when Re increased further, the outer region became unsteady and was characterized by the production of streamwise vortices which were then transported by the secondary flow destroying all remains of regular oscillations.