Scaling of wall turbulence Part II: Reynolds stresses
We present new scaling expressions, including high-Reynolds-number (Re) predictions, for all Reynolds stress components in the entire flow domain of turbulent channel, pipe and boundary layer flows (Chen, Hussain & She, JFM 2018). In Part I, based on the dilation symmetry of the mean momentum equation a multi-layer formula of the Reynolds shear stress length l12 – and hence also the entire mean velocity profile – was obtained. Here (Part II), random dilations on the second-order balance equations for all the Reynolds stresses (shear stress – , and normal stresses , , ) are analysed layer by layer, and similar multi-layer formulae of the corresponding stress length functions l11, l22, l33 (hence the three turbulence intensities) are obtained, enabling us to answer some major longstanding questions. Compared to the channel/pipe ﬂow, TBL differs by the absence of a centerline mirror symmetry, which results in a small but ﬁnite wall-normal velocity V. The latter leads to a higher total shear stress and fully explains previously observed larger Reynolds shear stress, higher wake strength, larger turbulent production and hence higher turbulence intensities in the bulk of TBL. Our Reynolds stresses profiles are validated by a large set of flow data for Reynolds number ( ) covering four decades (from 102 to 106), including atmospheric boundary layer data measured at Great Salt Lake Desert (USA, 2007-2012) and at the Qingtu Lake (China, 2016). A series of scaling laws are obtained, including the Townsend’s log profiles of and , and a newly identified non-universal scaling transition for the Reynolds shear stress peak in channel/pipe and TBL flows (Chen, Hussain & She, JFM Rapids 2019).