The longitudinal, vertical, and lateral turbulence intensities are computed from the converged solutions of the specific Reynolds (normal) stresses as
,
, and
, respectively [Figs.
7 and
8(a–c)]. The shear velocity values at the bed and sidewall were obtained from the log-law (
von Kármán 1930;
Prandtl 1932) using an integral constant of 5.29 used previously by Nezu and Rodi (
1986), Nezu and Nakagawa (
1993), Auel et al. (
2014), and Demiral et al. (
2020) and following Kadia et al. (
2022a). The bulging of
contour lines is stronger than that of
contour lines [comparing Figs.
3(a–c) with Figs.
7(a–c)], which agree well with previous findings (
Auel et al. 2014;
Melling and Whitelaw 1976;
Nezu and Nakagawa 1993). However, the trends are opposite; i.e., the higher
values are found toward the no slip boundaries where
reduces rapidly and produce steeper velocity gradient in the normal direction while the lower
values are obtained toward the center of the flow area and around the velocity dips where
reduces mildly and produces flatter gradient. The diagonal flows (combining a pair of corner vortices for DFs and combining bottom vortex, free surface vortex, and or intermediate vortex for OCFs) toward the solid corners push
contour lines diagonally. Furthermore, the secondary flows around the mid depth and mid width for the
and 1.0 cases bulge the
contour lines toward the center of the flow area. In the case of OCF_2 and DF_2,
contour lines are pushed away from the solid boundaries around
and
, respectively, but not up to the mid width due to the restricted growth of the bottom vortices. Additionally, the inner secondary vortices modify
contour lines at the mixed corners. Figs.
7(a and b) and
9(a) indicate that
contour lines in the central area below the mid depth for
and 1.25 are more closely spaced for DFs than those found for OCFs. This is apparently influenced by the steeper
found in such zones for DFs than those for OCFs [see Figs.
3(a and b) and
4(a)]. Similar variations are also found for
,
, and TKE
[see Figs.
7(d and e),
8(a and b), and
9(b–d)] for
and 1.25. Significantly higher deviations (but similar trends) are observed in the central area above the mid depth for
,
, and
possibly due to the stronger bulging of
(steeper
) found in such zone for DFs than the same for OCFs. However, the distributions of
and primary specific Reynolds shear stress
diverge due to dissimilar top boundaries and their influences. In uniform OCFs,
reduces toward the free surface and attains the minimum value near the free surface (
Auel et al. 2014;
Komori et al. 1982;
Nezu and Nakagawa 1986,
1993). Such observed trends are comparable to those reported in uniform supercritical flows by Nezu and Nakagawa (
1986,
1993), Auel et al. (
2014), and Jing et al. (
2019). However, the trends can alter in nonuniform flows (
Kironoto and Graf 1995;
Song and Chiew 2001), while further deviations are observed in decelerating supercritical flows with higher Froude numbers (
Demiral et al. 2020) experiencing surface perturbation and surface undulations as reported by Auel et al. (
2014) and discussed by Kadia et al. (
2022a), which apparently restrict the damping of
. In contrast,
increases in the central area above the mid depth for DFs as found in Figs.
7(d–f) and
9(b). However,
does not change significantly very close to the bed and top wall due to its damping and redistribution. Interestingly, the distributions of the gap between
and
(represented by the turbulence anisotropy stress
) observed in the central area for DFs differ insignificantly from the same found for OCFs as seen from Figs.
8(d–f) and
9(f). Besides, the patterns of
near the sidewalls for DFs are comparable to those for OCFs, where the maximum
values [and the maximum negative turbulence anisotropy shown in Figs.
8(d–f)] are observed for both flow types [Figs.
7(d–f)] due to the comparable damping and redistribution of
toward the sidewalls [see Figs.
8(a–c)]. Similarly, the damping and redistribution of
toward the bed and top wall or the free surface increases
toward such boundaries [see Figs.
7(d–f),
8(a–c), and
9(b and c)]. As a result, the maximum positive turbulence anisotropy values are obtained near such boundaries [Figs.
8(d–f)]. Overall, the distributions of
,
, and
in the bottom half depth of DFs are comparable to the same observed in OCFs, especially for lower
values 1.25 and 1.0. In addition, the magnitudes of the turbulence intensities presented in Fig.
9 indicate that the major contributor to
is the longitudinal component, followed by lateral and vertical components. However, the maximum
and
values, where
= bed shear velocity at the mid width, obtained near the bed are comparatively lower than the theoretical (
Nezu and Nakagawa 1993) and experimental (
Auel et al. 2014;
Demiral et al. 2020) values, which is a limitation of the used RSM, as also noticed by Cokljat (
1993).
The primary specific Reynolds shear stress
is obtained from the converged solutions of the Reynolds shear stress component
. Figs.
9(e) and
10(a–c) indicate that
extrema values are attained close to the no slip walls for both OCFs and DFs, which is attributed to the steeper gradient
there. Although the distributions of
in the bottom half flow depth are mostly comparable, significant differences are observed above the mid depth. First, the weaker bulging of
contour lines above the mixed corner bisectors (i.e., weaker velocity dips and flatter
) for OCFs [see Figs.
3(a–c)] produces more distantly spaced contour lines of negative
there than those obtained for DFs. It further generates absolute minimum
values around 0.2 for OCFs, which are consistent with previous experimental data (
Nezu and Nakagawa 1993;
Nezu and Rodi 1985) and numerical results (
Broglia et al. 2003;
Kadia et al. 2022a,
c;
Kang and Choi 2006;
Shi et al. 1999) but are significantly lower than DFs as seen in Figs.
10(a–c). Second, with a decrease in
, the intermediate vortex develops and modifies
isovels, and thus, the contour line of
around the channel quarters above the mid depth for OCFs is pushed toward the free surface and follows the patterns as in DFs. Such an interesting feature is clearly seen in Figs.
10(b–c) at
to
and
around 0.55 to 0.6 for OCF_1.25, and more clearly for OCF_1 at
around
and
around 0.5 to 0.6 [comparable to the findings of Broglia et al. (
2003)]. Lastly, the positive
values observed at
around 0.95 for OCFs are attributed to the inward lateral components of the small-scale inner secondary vortices, which create
there [see Figs.
3(a–c) and
10(a–c)].
The antisymmetric nature of the turbulence anisotropy stress
at the solid and mixed corners and its steeper gradients toward the corners are the major contributors to the generation of counterrotating secondary flows at the corners, which are reflected from the vorticity equation provided by Einstein and Li (
1958), Demuren and Rodi (
1984), Nezu and Nakagawa (
1984,
1993), Kang and Choi (
2006), and Nikora and Roy (
2012). Interestingly,
contour lines for DFs are comparable to those for OCFs [Figs.
8(d–f)], especially for
and 1.0 and more specifically for the bottom half flow depth. The differences between the free surface and solid top wall effects on the Reynolds stresses produce some dissimilarity in
contour lines above the mid depth as seen in Figs.
8(d–f). Such a difference can produce different turbulence anisotropy gradients (toward the corners), which are relevant to the vorticity production. Eventually, the secondary current patterns for DFs obtained from the mean secondary velocity vector plots (Fig.
6) and mean longitudinal vorticity
contours [around the mid depth in Figs.
10(d–f)] differ from those for OCFs. It is apparently impacted by the similar influences (on the damping and redistribution of the Reynolds stresses) of the sidewalls and top and bottom walls in DFs, but dissimilar influence of free surface and solid boundaries in OCFs. Furthermore, the spatial domination of the free surface vortices in OCFs produces small-scale inner secondary vortices (smaller than the sidewall corner vortices in DFs) and restricts the development of intermediate vortices at higher
[see Figs.
6 and
10(d–f)]. The turbulence anisotropy distribution observed for OCF_2 [Fig.
8(d)] agrees well with previous studies performed for similar
(
Cokljat 1993;
Cokljat and Younis 1995;
Shi et al. 1999). In addition, two pairs of counterrotating corner vortices are found in one half of the ducts [see Figs.
10(d–f)], which are antisymmetric about the mid depth, as also visible from Figs.
6(a–c). The sidewall corner vortices swell, and the top and bottom corner vortices shrink with a reduction of
. The vorticity plots indicate that the intermediate vortex is a part of the free surface vortex that separates with a decrease in
for OCFs. Although a fully developed intermediate vortex is found for OCF_1, a part of the free surface vortex still reaches the bottom solid corner before going up toward the free surface while covering the intermediate vortex. In the lower flow region, the longitudinal vorticity distribution for OCF_1 is comparable to the same for DF_1.