Scaling of wall turbulence Part I: Mean velocity
While analytical solution of mean properties of wall turbulence remains as a top challenge in classical physics, semi-empirical scaling laws are used as theoretical bases for turbulence engineering predictions. The celebrated logarithmic law describing partially the mean velocity profile (MVP), discovered by Prandtl (1925) and Von Karman (1930), has been verified for over eighty years, yet whether the Karman constant κ is universal and whether power-law is more rational than the log-law are still under vivid debating. In this series of talks (Parts I-III), I will introduce our recent studies (She, Chen & Hussain, JFM 2017; Chen, Hussain & She, JFM 2018; Chen, Hussain & She, JFM–Rapids 2019) on the scalings of canonical wall-bounded turbulence (i.e. channel, pipe and boundary layer -BL- flows), with discussions on cutting-edge issues in high Reynolds number turbulence.
In today’s talk (Part I), we present a new framework using symmetry analysis to accomplish the description of the entire MVP in three steps. First, effects of fluctuations on the mean (e.g. Reynolds shear stress) are described in terms of a series of characteristic length functions (e.g. the mixing length function). Second, a Lie-group symmetry analysis of the mean momentum equation (MME) is carried out, focusing on the dilation symmetry of the length functions, which yields to three kinds of local dilation-invariant expressions. The first kind – a power-law – characterizes viscous sublayer, buffer layer, log-layer, and a newly identified central core for channel and pipe; the second – a defect power-law – describes the bulk zone; the third kind with a generalized invariant is capable of describing scaling transition between two adjacent layers. Finally, these local invariant expressions of length function are connected to empirically observed multi-layers in simulations. The theory then reinterprets κ as a global coefficient over the entire out flow domain in wall turbulence. Evidence that κ approximately equals 0.45 in channel, pipe and TBL flows will be reported. The new approach yields a 98~99% accuracy description of recently published MVP data for Re covering over three decades, and has been applied to quantify other wall flows including rough pipes, sink flow boundary layer, compressible boundary layer flows, etc.