Secondary flow in turbulent ducts with increasing aspect ratio

Direct numerical simulations of turbulent duct flows with aspect ratios 1, 3, 5, 7, 10, and 14.4 at a center-plane friction Reynolds number Re-tau,Re- c similar or equal to 180, and aspect ratios 1 and 3 at Re-tau,Re- c similar or equal to 360, were carried out with the spectral-element code NEK5000. The aim of these simulations is to gain insight into the kinematics and dynamics of Prandtl's secondary flow of the second kind and its impact on the flow physics of wall-bounded turbulence. The secondary flow is characterized in terms of the cross-plane component of the mean kinetic energy, and its variation in the spanwise direction of the flow. Our results show that averaging times of around 3000 convective time units (based on duct half-height h) are required to reach a converged state of the secondary flow, which extends up to a spanwise distance of around similar or equal to 5h measured from the side walls. We also show that if the duct is not wide enough to accommodate the whole extent of the secondary flow, then its structure is modified as reflected through a different spanwise distribution of energy. Another confirmation of the extent of the secondary flow is the decay rate of kinetic energy of any remnant secondary motions for z(c)/h > 5 (where z(c) is the spanwise distance from the corner) in aspect ratios 7, 10, and 14.4, which exhibits a decreasing level of energy with increasing averaging time t(a), and in its rapid rate of decay given by similar to t(a)(-1). This is the same rate of decay observed in a spanwise-periodic channel simulation, which suggests that at the core, the kinetic energy of the secondary flow integrated over the cross-sectional area, < K >(yz), behaves as a random variable with zero mean, with rate of decay consistent with central limit theorem. Long-time averages of statistics in a region of rectangular ducts extending about the width of a well-designed channel simulation (i.e., extending about similar or equal to 3h on each side of the center plane) indicate that ducts or experimental facilities with aspect ratios larger than 10 may, if properly designed, exhibit good agreement with results obtained from spanwise-periodic channel computations.