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

On the influence of curvature and torsion on turbulence in helically coiled pipes

Turbulent flow and heat transfer in helically coiled pipes at Retau=400 was investigated by DNS using finite volume grids with up to 2.36×10E7 nodes. Two curvatures (0.1 and 0.3) and two torsions (0 and 0.3) were considered. The flow was fully developed hydrodynamically and thermally. The central discretization scheme was adopted for diffusion and advection terms, and the second order backward Euler scheme for time advancement. The grid spacing in wall units was ~3 radially, 7.5 circumferentially and 20 axially. The time step was equal to one viscous wall unit and simulations were typically protracted for 8000 time steps, the last 4000 of which were used to compute statistics. The results showed that curvature affects the flow significantly. As it increases from 0.1 to 0.3 the friction coefficient and the Nusselt number increase and the secondary flow becomes stronger; axial velocity fluctuations decrease, but the main Reynolds shear stress increases. Torsion, at least at the moderate level tested (0.3), has only a minor effect on mean and turbulence quantities, yielding only a slight reduction of peak turbulence levels while leaving pressure drop and heat transfer almost unaffected

Numerical Simulation of Reciprocating Turbulent Flow in a Plane Channel

Direct numerical simulation results were obtained for oscillatory flow with zero time mean (reciprocating flow) in a plane channel using a finite volume method, Crank-Nicolson time stepping and central approximation of the advection terms. A pressure gradient varying co-sinusoidally in time was imposed as the forcing term, and its frequency and amplitude were made to vary so as to span a range of regimes from purely laminar to fully turbulent. For the limiting cases of reciprocating laminar flow and steady-state turbulent flow, numerical results were validated against analytical solutions and classic experimental literature data, respectively. For general reciprocating flows, predictions were in agreement with experimental or computational results in the literature. Here, the attention was focused on the flow rate-pressure gradient dependence. Interestingly, the relation between the amplitudes of the imposed pressure gradient and of the flow rate was found to be approximately linear both in the laminar and in the turbulent regime; the reasons for this peculiar behavior were investigated and discussed. The influence of oscillation frequency and pressure gradient on transition to turbulence was also studied, and a tentative flow regime chart was proposed. Finally, the effect of unsteadiness on heat transfer was investigated

Direct numerical simulation of turbulent heat transfer in curved pipes

Fully developed turbulent convective heat transfer in curved pipes was investigated by Direct Numerical Simulation for a friction velocity Reynolds number of 500, yielding bulk Reynolds numbers between 12 630 and ~17 350 according to the curvature (pipe radius/curvature radius). Three different curvatures were compared, i.e. 0 (straight pipe), 0.1 and 0.3. The Prandtl number was 0.86. The computational domain was a tract of pipe 5 diameters in length. A finite volume method was used, with multiblock structured grids of ~5.3x10E6 hexahedral volumes. Simulations were typically protracted for 20 LETOT’s starting from coarse-grid results. Results were post-processed to compute first and second order time statistics, including rms fluctuating temperature and turbulent heat fluxes, on a cross section of the pipe. In curved pipes, time-mean results exhibited Dean circulation and a strong velocity and temperature stratification in the radial direction. Turbulence and heat transfer were strongly asymmetric, with higher values near the outer pipe bend. Overall turbulence levels were lower than in a straight pip

Transition to turbulence in serpentine pipes

The geometry considered in the present work (serpentine pipe) is a sequence of U-bends of alternate curvature. It is characterized by pipe diameter, d = 2a and bend diameter, D = 2c. The repeated curvature inversion forces the secondary flow pattern, typical of all flows in curved ducts, to switch between two mirror-like configurations. This causes (i) pressure drop and heat or mass transfer characteristics much different from those occurring either in a straight pipe or in a constant-curvature pipe, and (ii) an early loss of stability of the base steady-state flow. In the present work, four values of the curvature δ = a/c (0.2, 0.3, 0.4 and 0.5) were considered. For each value of δ, the friction velocity Reynolds number Reτ = uτa/ν was made to vary in steps between 10 and 50. Fully developed flow was simulated using a three-dimensional, time-dependent finite volume method and computational grids with a number of nodes ranging from ∼1.8 to ∼4.6 × 106, according to the curvature. The computational domain included two consecutive and opposite bends and thus coincided with the minimum spatially repetitive unit. Heat transfer was also simulated for uniform wall heat flux conditions and a Prandtl number of 1. A complex scenario of transitions was predicted, leading from the base steady-state, top-down symmetric flow to turbulence through intermediate regimes which included steady-state asymmetric and time-periodic flows. For all curvatures, at the highest value of Reτ investigated (50) the flow was turbulent and exhibited top-down symmetric time averages