Secondary flow in isothermal and thermal square-duct turbulence

Mean secondary flows in isothermal and thermal square-duct turbulence at low Reynolds numbers are characterized in terms of coherent structures and buoyancy-driven convection.

Firstly we examine the mean flow structure of isothermal turbulence in a numerically-simulated square duct at marginal Reynolds number. The secondary flow in a marginal state exhibits a four-vortex pattern alternating in time, which is very different from the usual eight-vortex secondary flow in a fully turbulent state. It is shown that at the marginal Reynolds number only one pair of opposite walls can accommodate buffer-layer coherent structures, i.e., streamwise vortices, leading to the four-vortex mean secondary flow. We discuss the relevance of this observation with the four-vortex traveling-wave recently found by Okino, Nagata, Wedin and Bottaro (2010).

Next we move on to the higher-Reynolds-number regime in the thermal turbulent square-duct flow. It is found that at higher Reynolds numbers all the walls accommodate a pair of counter-rotating instantaneous streamwise vortices, so that the usual eight-vortex secondary flow appears as their statistical footprint. We can say that the buffer-layer coherent structures play a crucial role in the appearance of secondary flow of Prandtl's second kind. We also present the eight-vortex traveling wave which provides a theoretical support of a direct link between the coherent structures and the turbulence-driven secondary flow.

Lastly we discuss turbulence- and buoyancy-driven secondary flow in a numerically-simulated horizontal square duct heated from below. At moderate Richardson numbers the mean secondary flow is observed to be represented by a single large-scale thermal convection roll and four turbulence-driven corner vortices of the opposite sense of rotation to the roll, as contrasted with the isothermal secondary flow. This remarkable structural difference in the corner regions can be interpreted in terms of combined effects, on instantaneous streamwise vortices, of the large-scale circulation and of the geometrical constraint by the duct corner.