Asymptotic scaling laws for periodic turbulent boundary layers and their numerical simulation up to $\textit{Re}_{\boldsymbol{\theta}}\text{ = 8300}$

We provide a rigorous analysis of the self-similar solution of the temporal turbulent boundary layer, recently proposed by Biau (2023 Comput. Fluids 254 , 105795), in which a body force is used to maintain a statistically steady turbulent boundary layer with periodic boundary conditions in the streamwise direction. We derive explicit expressions for the forcing amplitudes which can maintain such flows, and identify those which can hold either the displacement thickness or the momentum thickness equal to unity. This opens the door to the first main result of the paper, which is to prove upper bounds on skin friction for the temporal turbulent boundary layer. We use the Constantin–Doering–Hopf bounding method to show, rigorously, that the skin-friction coefficient for periodic turbulent boundary layer flows is bounded above by a uniform constant which decreases asymptotically with Reynolds number. This asymptotic behaviour is within a logarithmic correction of well-known empirical scaling laws for skin friction. This gives the first evidence, applicable at asymptotically high Reynolds numbers, to suggest that Biau’s self-similar solution of the temporal turbulent boundary layer exhibits statistical similarities with canonical, spatially evolving, boundary layers. Furthermore, we show how the identified forcing formula implies an alternative, and simpler, numerical implementation of periodic boundary layer flows. We give a detailed numerical study of this scheme presenting direct numerical simulations up to a momentum Reynolds number of $\textit{Re}_\theta = 2000$ and implicit large-eddy simulations up to $\textit{Re}_\theta = 8300$ , and show that these results compare well with data from canonical spatially evolving boundary layers at equivalent Reynolds numbers.