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• W2860398082 endingPage "438" @default.
• W2860398082 startingPage "401" @default.
• W2860398082 abstract "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 $ell _{12}$ – 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 $-overline{u^{prime }v^{prime }}$ , and normal stresses $overline{u^{prime }u^{prime }}$ , $overline{v^{prime }v^{prime }}$ , $overline{w^{prime }w^{prime }}$ ) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions $ell _{11}$ , $ell _{22}$ , $ell _{33}$ (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 $ell _{12}$ and $ell _{22}$ – which have the celebrated linear scalings in the logarithmic layer, i.e. $ell _{12}approx unicode[STIX]{x1D705}y$ and $ell _{22}approx unicode[STIX]{x1D705}_{22}y$ . However, data show an invariant peak location for $overline{w^{prime }w^{prime }}$ , which theoretically leads to an anomalous scaling in $ell _{33}$ in the log layer only, namely $ell _{33}propto y^{1-unicode[STIX]{x1D6FE}}$ with $unicode[STIX]{x1D6FE}approx 0.07$ . Furthermore, another mesolayer modification of $ell _{11}$ yields the experimentally observed location and magnitude of the outer peak of $overline{u^{prime }u^{prime }}$ . The resulting $-overline{u^{prime }v^{prime }}$ , $overline{u^{prime }u^{prime }}$ , $overline{v^{prime }v^{prime }}$ and $overline{w^{prime }w^{prime }}$ 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 $y^{+}approx 12$ ; (2) the location of peak value $-overline{u^{prime }v^{prime }}_{p}$ has a scaling transition from $5.7Re_{unicode[STIX]{x1D70F}}^{1/3}$ to $1.5Re_{unicode[STIX]{x1D70F}}^{1/2}$ at $Re_{unicode[STIX]{x1D70F}}approx 3000$ , with a $1+overline{u^{prime }v^{prime }}_{p}^{+}$ scaling transition from $8.5Re_{unicode[STIX]{x1D70F}}^{-2/3}$ to $3.0Re_{unicode[STIX]{x1D70F}}^{-1/2}$ ( $Re_{unicode[STIX]{x1D70F}}$ the friction Reynolds number); (3) the peak value $overline{w^{prime }w^{prime }}_{p}^{+}approx 0.84Re_{unicode[STIX]{x1D70F}}^{0.14}(1-48/Re_{unicode[STIX]{x1D70F}})$ ; (4) the outer peak of $overline{u^{prime }u^{prime }}$ emerges above $Re_{unicode[STIX]{x1D70F}}approx 10^{4}$ with its location scaling as $1.1Re_{unicode[STIX]{x1D70F}}^{1/2}$ and its magnitude scaling as $2.8Re_{unicode[STIX]{x1D70F}}^{0.09}$ ; (5) an alternative derivation of the log law of Townsend (1976, The Structure of Turbulent Shear Flow , Cambridge University Press), namely, $overline{u^{prime }u^{prime }}^{+}approx -1.25ln y+1.63$ and $overline{w^{prime }w^{prime }}^{+}approx -0.41ln y+1.00$ in the bulk." @default.
• W2860398082 created "2018-07-19" @default.
• W2860398082 creator A5031163364 @default.
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• W2860398082 date "2018-07-05" @default.
• W2860398082 modified "2023-10-15" @default.
• W2860398082 title "Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses" @default.
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• W2860398082 doi "https://doi.org/10.1017/jfm.2018.405" @default.
• W2860398082 hasPublicationYear "2018" @default.
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