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,
$\overline{u^{\prime }u^{\prime }}^{+}\approx -1.25\ln y+1.63$
and
$\overline{w^{\prime }w^{\prime }}^{+}\approx -0.41\ln y+1.00$
in the bulk.