Quantifying wall turbulence via a symmetry approach. Part 2. 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 and pipe flows. In Part 1 (She et al. , J. Fluid Mech. , vol. 827, 2017, pp. 322–356), based on the dilation symmetry of the mean Navier–Stokes equation a four-layer formula of the Reynolds shear stress length ₁₂ – and hence also the entire mean velocity profile (MVP) – was obtained. Here, random dilations on the second-order balance equations for all the Reynolds stresses (shear stress -uv , and normal stresses uu , vv , ww ) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions ₁₁ , ₂₂ , ₃₃ (hence the three turbulence intensities) are obtained for turbulent channel and pipe flows. In particular, direct numerical simulation (DNS) data are shown to agree well with the four-layer formulae for ₁₂ and ₂₂ – which have the celebrated linear scalings in the logarithmic layer, i.e. ₁₂≈ [STIX]x1D705y and ₂₂≈ [STIX]x1D705₂₂y . However, data show an invariant peak location for ww , which theoretically leads to an anomalous scaling in ₃₃ in the log layer only, namely ₃₃∝ y^1-[STIX]x1D6FE with [STIX]x1D6FE≈ 0.07 . Furthermore, another mesolayer modification of ₁₁ yields the experimentally observed location and magnitude of the outer peak of uu . The resulting -uv , uu , vv and ww are all in good agreement with DNS and experimental data in the entire flow domain. Our additional results include: (1) the maximum turbulent production is located at y⁺≈ 12 ; (2) the location of peak value -uvₚ has a scaling transition from 5.7Re_[STIX]x1D70F1/3 to 1.5Re_[STIX]x1D70F1/2 at Re_[STIX]x1D70F≈ 3000 , with a 1+uvₚ⁺ scaling transition from <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/