Turbulent flow in a square duct at extreme Reynolds number

We investigate pressure-driven turbulent flow in a square duct using wall-modelled large-eddy simulations (WMLES) of the Navier–Stokes equations. An equilibrium wall model based on the logarithmic law is employed, yielding results in close agreement with direct numerical simulations (DNS) over the range of overlapping Reynolds numbers. In particular, both the structure and intensity of the secondary motions are well captured, providing confidence in the ability of WMLES to extend to Reynolds numbers beyond the present reach of DNS, and to offer insight into the asymptotic state of turbulence. This expectation is reinforced by the good agreement with available experimental data at higher Reynolds numbers. The WMLES results confirm the validity of the classical Prandtl friction law, with the hydraulic diameter taken as the duct side, up to bulk Reynolds numbers italic Re almost equals 10 Superscript 8 Re ≈ 10 8 $\\textit{Re} \\approx 10^8$ . More interestingly, the simulations reveal an asymptotic regime in which the intensity of the secondary motions scales with the mean friction velocity. Consistent with the observations of Pullin et al. (2013, Phys. Fluids , vol. 25, 015116), this suggests that secondary motions are a finite-Reynolds-number effect, and that the asymptotic state of square-duct turbulence corresponds to a plug-like velocity profile with finite slip at the wall, when viewed in outer units.