Recent progress has uncovered a small but significant error in the paper (Ricco & Hahn Reference Ricco and Hahn2013), which propagates into the paper (Wise & Ricco Reference Wise and Ricco2014). The caption of figure 9 in Ricco & Hahn (Reference Ricco and Hahn2013) should read:
(a) Wall-normal profiles of r.m.s. of
$\boldsymbol{u}_{\boldsymbol{d}}$
components and of
$\langle u_{d}v_{d}\rangle ^{+}$
(the latter multiplied by a factor of
$-\mathbf{6}$
); the disc-flow boundary layer thickness
$\unicode[STIX]{x1D6FF}$
, defined in § 3.4, is shown. (b) Wall-normal profiles of r.m.s. of velocity components and Reynolds stresses, where
$u$
,
$v$
,
$w$
are indicated in the legend.
The only difference is the minus sign in front of 6. The Reynolds stresses
$\langle u_{d}v_{d}\rangle ^{+}$
in the left graph were plotted with the opposite sign to show clearly that their maximum amplitude occurs where the
$u_{d,rms}^{+}$
also peaks. Although the results and the conclusions in the text of Ricco & Hahn (Reference Ricco and Hahn2013) are unaltered, the sign error is also found in the analysis in Wise & Ricco (Reference Wise and Ricco2014). The effect of this error on the results and conclusions in Wise & Ricco (Reference Wise and Ricco2014) is addressed below.
Figure 11(b) in Wise & Ricco (Reference Wise and Ricco2014) shows that
$\widehat{u_{d}v_{d}}^{+}$
is positive, which implies that it contributes favourably to the drag reduction
$\mathscr{R}$
, as elucidated by the analysis based on the modified Fukagata–Iwamoto–Kasagi identity (see § 4.3 of Wise & Ricco (Reference Wise and Ricco2014)). However, figure 11(b) in Wise & Ricco (Reference Wise and Ricco2014) should be corrected as in figure 1(b) below because the profiles should instead be multiplied by
$-6$
.
The isosurfaces of
$\langle u_{d}v_{d}\rangle ^{+}$
for
$u_{d}<0$
,
$v_{d}>0$
displayed in figure 11(a) of Wise & Ricco (Reference Wise and Ricco2014) are corrected in figure 1(a) below. Thanks to the modified Fukagata–Iwamoto–Kasagi identity, it emerges that the structures contribute negatively to the total drag reduction.
Figure 1. (a) Isosurfaces of
$\langle u_{d}v_{d}\rangle ^{+}$
observed from the
$y$
–
$z$
plane at
$x^{+}=0$
,
$x^{+}=160$
,
$x^{+}=320$
(from top to bottom). The plots show only
$\langle u_{d}v_{d}\rangle ^{+}$
for
$u_{d}<0$
,
$v_{d}>0$
as within the contour range the contributions from other contributions of
$u_{d}$
and
$v_{d}$
are negligible. (b) Wall-normal profiles of
$u_{d,rms}^{+}$
(solid lines) and
$\widehat{u_{d}v_{d}}^{+}$
(dashed lines). Profiles are shown for phases from the first half of the disc oscillation.
This is further shown in the diagram of figure 2, where figure 14 on p. 557 of Wise & Ricco (Reference Wise and Ricco2014) has been corrected: the arrows of the interdisc structures now point upstream and upward as they contribute directly to strengthen the Reynolds stresses.
Figure 2. Reproduction of figure 14 on p. 557 of Wise & Ricco (Reference Wise and Ricco2014). Note that the arrows of the interdisc structures now point upstream and upward in the wall-normal direction.
Figure 15 in Wise & Ricco (Reference Wise and Ricco2014) is recreated in figure 3 below with the corrected values of
$\mathscr{R}_{t}$
and
$\mathscr{R}_{d}$
. The drag reduction contribution from the modification of the turbulent Reynolds stresses,
$\mathscr{R}_{t}$
still scales linearly with the penetration depth of the disc-flow boundary layer,
$\unicode[STIX]{x1D6FF}^{+}$
.
$\mathscr{R}_{d}$
also still scales linearly with the same simple combination of the disc-flow parameters (i.e.
$W^{m}T^{n}$
) and the exponents remain unchanged, i.e.
$(m,n)=(2,0.3)$
.
Figure 3. (a)
$\mathscr{R}_{t}$
, the contribution to drag reduction due to the modification to the turbulent Reynolds stresses, versus
$\unicode[STIX]{x1D6FF}^{+}$
, the penetration depth of the disc-flow boundary layer. (b)
$\mathscr{R}_{d}$
, contribution to drag reduction due to the disc-flow Reynolds stresses, versus
$W^{2}T^{0.3}$
. White circles:
$W^{+}=3$
, light grey:
$W^{+}=6$
, dark grey:
$W^{+}=9$
.
Acknowledgement
We would like to thank one of the referees for pointing out the missing ‘
$g$
’ subscript of the square parenthesis in equation (4.2) on p. 551 of Wise & Ricco (Reference Wise and Ricco2014).
