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 l(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 stressess (shear stress - (u'v') over bar, and normal stresses (u'u') over bar, (v'v') over bar, (w'w') over bar) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions l(11), l(22), l(33 ) (hence the three turbulence intensities) are obtained for turbulent channel and pipe flows. In particular, direct numerical simulation (DNS) data arc shown to agree well with the four-layer formulae for l(12) and l(22) - which have the celebrated linear scalings in the logarithmic layer, i.e. (hence the three turbulence intensities) are obtained for turbulent channel and pipe flows. In particular, direct numerical simulation (DNS) data arc shown to agree well with the four-layer formulae for l(12) and l(22) - which have the celebrated linear scalings in the logarithmic layer, i.e. l(12) approximate to kappa y and l(22) approximate to kappa(22)y. However, data show an invariant peak location for (w'w') over bar, which theoretically leads to an anomalous scaling in l(33) in the log layer only, namely l(33) alpha y(1-gamma) with gamma approximate to 0.07. Furthermore, another mesolayer modification of l(11) yields the experimentally observed location and magnitude of the outer peak of (u'u') over bar. The resulting (-u'v') over bar, (u'u') over bar, (u'v') over bar and (w'w') over bar 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(+) approximate to 12; (2) the location of peak value (-u'v') over bar (p) has a scaling transition from 5.7Re(tau)(1/3) to 1.5Re(tau)(-1/2) Re-tau approximate to 3000, with a 1 + (-u'v') over bar p(+ ) scaling transition from 8.5Re(tau)(-2/3) to 3.0Re(tau)(-1/2) (Re(tau )the friction Reynolds number); (3) the peak value (w'w') over bar p(+) approximate to 0.84Re(tau)(0.14 )(1 - 48/Re tau); (4) the outer peak of (u'u') over bar emerges above Re-tau approximate to 10(4 )with its location scaling as 1.1Re(tau)(1/2 )and its magnitude scaling as 2.8Re(tau)(0.09); (5) an alternative derivation of the log law of Townsend (1976, The Structure of Turbulent Shear flow, Cambridge University Press), (u'u') over bar (+) approximate to -1.25 ln y + 1.63 (w'w') over bar (+) approximate to ln y + 1.00 in the bulk.