Reynolds Stress Model Study Comparing the Secondary Currents and Turbulent Flow Characteristics in High-Speed Narrow Open Channel and Duct Flows

In the case of LRR RSM, the following transport equation is solved to compute the Reynolds stress components (Alfonsi 2009; Hanjalić and Launder 1972; Launder et al. 1975; Speziale et al. 1991; Wilcox 2006):where $U_i$ = mean (time-averaged) velocity along the direction i; $u_i^{\prime }$ = fluctuation in velocity $U_i$; $p^{\prime }$ = pressure fluctuation; $\rho$ = density of the fluid; ${\delta }_{ik}$ = Kronecker’s delta; and $\overline{u_i^{\prime }{u}_j^{\prime }}$ = specific Reynolds stress tensor $R_{ij}$. Among the tensor terms, the most studied phenomenon is the redistribution or the pressure-strain tensor ${\mathrm{\Phi }}_{ij}$ consisting of slow ${\mathrm{\Phi }}_{ij,s}$ and rapid ${\mathrm{\Phi }}_{ij,r}$ pressure-strain parts and the corrections due to boundaries. In the LRR model, ${\mathrm{\Phi }}_{ij,s}$ and ${\mathrm{\Phi }}_{ij,r}$ are modeled as follows (Cokljat 1993; Cokljat and Younis 1995; Kadia et al. 2022a; Launder et al. 1975; Reece 1977; Rodi 1993; Rotta 1951; Wilcox 2006):where $P_{ij}=-[\overline{u_i^{\prime }{u}_k^{\prime }}(\partial U_j/\partial x_k)+\overline{u_j^{\prime }{u}_k^{\prime }}(\partial U_i/\partial x_k)]$, $D_{ij}=-[\overline{u_i^{\prime }{u}_k^{\prime }}(\partial U_k/\partial x_j)+\overline{u_j^{\prime }{u}_k^{\prime }}(\partial U_k/\partial x_i)]$, turbulent kinetic energy (TKE) $k=0.5(\overline{u_i^{\prime }{u}_i^{\prime }})$, production of TKE $P_k=-\overline{u_i^{\prime }{u}_j^{\prime }}(\partial U_i/\partial x_j)=0.5P_{kk}$, mean strain-rate tensor $S_{ij}=0.5(\partial U_i/\partial x_j+\partial U_j/\partial x_i)$, and coefficients $C_1=1.5$ and $C_2=0.4$ as utilized in previous studies by Launder et al. (1975), Cokljat (1993), Cokljat and Younis (1995), and Kadia et al. (2022a, c). At high Re values, the “Kolmogorov hypothesis of local isotropy” suggests the dissipation rate tensor ${\epsilon }_{ij}=(2/3)\epsilon {\delta }_{ij}$, in which the scalar dissipation rate of TKE $\epsilon$ is computed from the following transport equation (Hanjalić and Launder 1972; Launder et al. 1975; Wilcox 2006):where $C_{\epsilon }=0.18$, $C_{\epsilon 1}=1.45$, and $C_{\epsilon 2}=1.9$ (Cokljat 1993; Cokljat and Younis 1995; Gibson and Rodi 1989; Kadia et al. 2022a; Kang and Choi 2006). The simple gradient-diffusion model proposed by Daly and Harlow (1970) is used in OpenFOAM (OpenFOAM Foundation 2022) to define the first term in the diffusion tensor. The viscous diffusion term and the production tensor need no separate models (Alfonsi 2009; Kim 2001).

The solid boundaries and the free surface damp the normal stress component perpendicular to themselves and redistribute the same among the other directions. To account for that in the pressure strain, the wall-reflection correction models proposed by Shir (1973) and Gibson and Launder (1978) were used in combination with the nonlinear wall and free surface damping functions proposed by Naot and Rodi (1982) and then improved by Gibson and Rodi (1989), Cokljat (1993), and Cokljat and Younis (1995). Kadia et al. (2022a, b) previously incorporated the modifications required in the existing LRR model in OpenFOAM and detailed the free surface and wall corrections. Briefly, the corrections are functions of Reynolds stress, turbulent kinetic energy, energy dissipation, unit normal vector, turbulence length scale, near wall turbulence, and the average distance from the boundary. The wall and free surface corrections were combined considering that each boundary acts separately and affects its normal stress component only.

The steady state uniform flow simulations were performed using OpenFOAM: version dev (OpenFOAM Foundation 2022), full LRR pressure-strain model, and simpleFoam solver, which attains the velocity-pressure coupling using the SIMPLE algorithm proposed by Caretto et al. (1973). Orthogonal hexahedra mesh configurations were generated using the blockMeshDict dictionary. For DF simulations, a uniform mesh of 0.75 mm cell size was used, which conforms to the log-law criteria at the first cell center (see Table 1). In the case of the OCFs, the near wall cells were of size 0.75 mm (also conforming to the log-law solution as found from Table 1) and increased toward the free surface and the channel center up to $\approx 3\mathrm{mm}$ as detailed by Kadia et al. (2022a), who already performed a grid convergence test. Figs. 2(a and b) compare the used grid arrangements for $a_r=2$ cases and explain the domain boundaries. Furthermore, Table 1 provides the grid arrangements used for the tested six simulations. A total of 3,844–7,626 cells were generated for OCFs and 17,956–35,912 cells were created for DFs. Along the flow direction, one cell layer was required whose normal faces, i.e., the inlet and outlet boundaries, were neighbourPatch to each other and assigned cyclic or periodic boundary conditions to achieve the steady state converged solutions. A momentum source was introduced to all the cells using the fvOptions dictionary, which adjusts the velocity and the pressure gradient so that the volumetric average velocity reaches the target ($\overline{U}$, 0, 0) and produces a dynamically adjusted longitudinal pressure gradient to drive the flow [see Talebpour (2016), Talebpour and Liu (2019), and Kadia et al. (2021, 2022a, c) for further details].

The solid walls were no slip boundaries with fixedFluxExtrapolatedPressure condition used for pressure $p$, kqRWallFunction used for $R_{ij}$, epsilonWallFunction used for $\epsilon$, and nutUWallFunction used for eddy viscosity ${\nu }_t$ (OpenFOAM Foundation 2022). The local equilibrium condition near the wall, i.e., $P_k=\epsilon$, is satisfied using such $\epsilon$ wall function. Only one half of the domain was simulated considering a symmetry plane condition at channel or duct mid width ($y=0$) to minimize the computational cost, and such an assumption did not influence the results. In OCF simulations, a symmetry plane condition was applied at the free surface for all variables except $\epsilon$, for which the following condition proposed by Naot and Rodi (1982) was implemented in OpenFOAM using the codedFixedValue function (Kadia et al. 2022a):where coefficient $C_{\mu }=0.09$, von Kármán constant $\kappa =0.41$, $k_{fs}=0.5{(\overline{u_i^{\prime }{u}_i^{\prime }})}_{fs}$, $z^{\prime }=0.07h$, and $z^{*}$ = average distance of a face center from the nearest wall calculated following Naot and Rodi (1982), Cokljat (1993), and Kadia et al. (2022a). See the source file provided by Kadia et al. (2022a, b) for further details.

The convection term was discretized using second order schemes: bounded Gauss limitedLinearV for $U_i$ and bounded Gauss limitedLinear for $R_{ij}$ and $\epsilon$. A single gradient limiter is applied to all the components while using the $V$ scheme based on the most rapidly changing gradient, and such a scheme helps to stabilize the solution (Almeland et al. 2021; Greenshields and Weller 2022). Furthermore, the used cellLimited Gauss linear scheme for the velocity gradient improved the near wall solutions. The used default fvSchemes are Gauss linear for the gradient, Gauss linear orthogonal for the Laplacian term, and orthogonal for the surface-normal gradient. The used matrix solvers are the preconditioned (bi-)conjugate gradient (PBiCG) for $U_i$, $R_{ij}$, and $\epsilon$ and the geometric-algebraic multigrid (GAMG) for $p$. The converged OCF solutions were obtained when the equation residuals for the variables $U_i$, $p$, $R_{ij}$, and $\epsilon$ reached ${\le 7.5\times 10}^{-5}$ during the iterations. In DFs, those were set at ${\le 5\times 10}^{-5}$, which provided improved bulging of the contour lines of the longitudinal velocity and longitudinal turbulence intensity. A relaxation factor of 0.7 was used for $U_i$, $R_{ij}$, and $\epsilon$, and the same was 0.3 for $p$, which were suggested for the steady state simulations (Greenshields 2022; Greenshields and Weller 2022) and were used previously by Talebpour (2016) and Kadia et al. (2022a). To converge, the OCFs required between 8,490 and 9,714 iterations and the DFs required between 8,575 and 14,512 iterations, which were increased with an increase in the flow depth and number of cells (see Table 1). The simulations are computationally economical and took (only) between 8.7 and 22.6 min and between 31.0 and 124.3 min to achieve the converged solutions for OCFs and DFs, respectively (see Table 1). The increase in the number of cells impacted the simulation execution time, as expected. The resulting postprocessing was performed using ParaView: version 5.9.1 and MATLAB: version R2021a.

The longitudinal and lateral velocity distributions are symmetric, and the vertical velocity distributions are antisymmetric about the mid depth $z/h=0.5$ for DFs, and $U_{\max }$ values are located at the center of the flow area [see Figs. 3, 4(a and b), and 5(a–c)]. However, the locations of $U_{\max }$ for OCFs shift toward the free surface as the free slip top boundary condition and its damping and redistribution effects differ from those imposed by a no slip wall (Naot and Rodi 1982). At the mid width, such values are situated at $z/h=0.68$, 0.61, and 0.625 for $a_r=2.0$, 1.25, and 1.0, respectively, which correspond well to the results obtained by Nezu and Nakagawa (1993), Auel et al. (2014), and Demiral et al. (2020). However, these locations of $U_{\max }$ are not a monotonically increasing function of $a_r$, possibly because of the developing or developed intermediate vortices at lower $a_r$. Guo (2014) proposed a model that shows exponential shifting of the velocity-dip location from the free surface to the mid depth with decreasing $a_r$. However, the model was validated in detail for $a_r\ge 2$.

DFs produce 1.0%–2.3% higher $U_{\max }$ values than OCFs (see Table 2). These $U_{\max }$ values are 14.4%–14.9% higher than $\overline{U}$ for OCFs and 15.5%–17.4% higher than $\overline{U}$ for DFs [see also Fig. 4(a)]. Such incremental values and the relative positions of the velocity-dip can be useful to obtain approximate solution for $U_{\max }$ using the discharge and $h$ values only. Marginally higher maximum normalized mean vertical velocity $W_{\max }/U_{\max }$ for DFs than for OCFs but significantly lower maximum absolute normalized mean lateral velocity $|V_{\max }/U_{\max }|$ and maximum normalized mean resultant secondary velocity $U_{WV,\max }/U_{\max }$ (where the mean resultant secondary velocity $U_{WV}=\sqrt{W^2+V^2}$) for DFs than for OCFs (see Table 2) signify the stronger free surface vortices in OCFs than weaker corner vortices in DFs. Interestingly, the deviations reduce with a reduction of $a_r$ as the free surface influence reduces. At the mid width, considerably higher deviations between the OCF and DF profiles, especially in the outer flow region $z/h>0.2$, are observed for the vertical velocity as seen from Fig. 4(b). Broadly, the secondary velocity values provided in Table 2 are comparable to previous studies on OCFs (Albayrak and Lemmin 2011; Broglia et al. 2003; Kadia et al. 2022a, c; Naot and Rodi 1982; Nezu and Nakagawa 1986, 1993; Nezu and Rodi 1985; Tominaga et al. 1989) and on DFs (Melling and Whitelaw 1976; Naot and Rodi 1982; Nezu and Nakagawa 1993). However, Nikitin (2021) reported higher $U_{WV,\max }$ (up to 5% of $\overline{U}$) in a DNS study conducted for an OCF with $a_r=4.0$.

The nearly symmetrical $U/U_{\max }$ contours observed about the bottom corner bisectors for OCFs, especially for lower $a_r$ values, are comparable to the DF results with same bottom corner conditions [see Figs. 3(a–c)]. Additionally, $U/U_{\max }$ contour patterns in the bottom half depth for OCFs match fairly with the same found for DFs. In the case of DFs, the diagonal components of the corner vortices push the high-momentum fluids and isovels of $U/U_{\max }$ toward the corners, whereas the inward lateral components of the sidewall corner vortices [up to $y/h\approx \pm 0.8$ for $a_r=2.0,y/h\approx \pm 0.35$ for $a_r=1.25$, and $y/h\approx \pm 0.13$ for $a_r=1.0$ as seen in Figs. 5 and 6(a–c)] and the inward vertical flows of the bottom and top wall corner vortices push the low-momentum fluids and isovels of $U/U_{\max }$ toward the center of the flow area as found in Figs. 3(a–c) [see also Figs. 3(d–f), 5, and 6(a–c)]. The inward lateral components of the sidewall corner vortices are restricted up to such $y/h$ values (around which a zone of negligible secondary velocity is formed surrounded by corner vortices) due to the outward lateral components of the bottom and the top wall corner vortices [see Figs. 6(a–c)], which dominate the sidewall corner vortices for $a_r>1.0$. The restriction relaxes with a reduction of $a_r$, which enlarges the sidewall corner vortices in comparison to the bottom and top wall corner vortices. Therefore, the inward bulging of the isovels of $U/U_{\max }$ is more pronounced for DF_1 than the same observed for higher $a_r$ of 1.25 and 2.0 as found in Figs. 3(a–c). Interestingly, the zone of negligible secondary velocity is also found for OCF with $a_r=1.0$ at $z/h\approx 0.6$ and $y/h\approx \pm 0.24$ as found in Figs. 6(c–d). In OCFs, fully developed intermediate vortices are observed for $a_r=1.0$ [comparable to Broglia et al. (2003) and Kadia et al. (2022a)], which are developing for $a_r=1.25$ [see Fig. 6(b)] and absent for $a_r=2.0$ [see Fig. 6(a)]. The inward lateral components of the developing or the developed intermediate vortices at $z/h$ around 0.5–0.6 [see Figs. 5(b–c)] push the low-momentum fluids inward and result in bulging of the isovels of $U/U_{\max }$ toward the channel center. In both OCFs and DFs, the reduction of $a_r$ eases the lateral transfer of low-momentum fluids toward the center of the flow area and results in narrowing the area of higher velocity fluids as found in Figs. 3(a–c). Although the bottom wall corner vortices are prominent up to the mid width for DFs, the similar (i.e., the bottom vortices in OCFs) could not reach the mid width for wider channels, especially for the OCF_2 case, as observed from Figs. 3(d–f) and 6(a–c). As a result, the upward bulging of the isovels of $U/U_{\max }$ away from the bed for the OCF_1 and OCF_1.25 cases is similar to that observed for the DF_1 and DF_1.25 cases, respectively, but differs noticeably for $a_r=2.0$ as observed from Figs. 3(a–c). In the case of OCFs, the diagonal components of the bottom vortices and free surface vortices (additionally intermediate vortex components when they exist) toward the bottom corner bisectors bulge the isovels of $U/U_{\max }$ toward the corners [see Figs. 3(a–c) and 6].

Unlike the lower half flow depth, significant differences between the OCF velocity fields and DF velocity fields are observed in the upper half flow depth due to the differences in top boundary conditions. The no slip top wall in DFs produces steeper velocity gradients $\partial U/\partial z$ and enhanced velocity-dip phenomena compared to those obtained in OCFs with free surface [see Figs. 3(a–c) and 4(a)] although the OCFs produce stronger secondary velocities than DFs as discussed earlier [relate Figs. 3(d–f), 4(b), 5(a–c), and 6 and Table 2]. Unlike in DFs, where the flow conditions are mirrored about the mid depth, in OCFs, the free surface vortices spatially dominate the other vortices as shown in Figs. 6(a–c), especially for higher $a_r$ values. The inward lateral components near the free surface [see Figs. 5(a–c) and 6] and the downward components near the channel center [see Figs. 3(d–f) and 6] of the free surface vortices convey the low-momentum fluids toward the mid width from the sidewalls, which bulges the isovels of $U/U_{\max }$ toward the center of the flow area and results in velocity dips as found in Figs. 3(a–c). For OCF_2, the outward lateral components of the free surface vortices at $z/h\approx 0.6$ transfer the high-momentum fluids toward the sidewalls and push the isovels of $U/U_{\max }$ outward. Such transfer of fluids relaxes and even opposes with the decrease in $a_r$ (and the development of intermediate vortices) as found in Figs. 3(b and c). In addition, Fig. 6 shows that the inner secondary vortices formed at the mixed corners in OCFs are smaller than the free surface vortices [also reported previously by Grega et al. (1995, 2002), Broglia et al. (2003), Kang and Choi (2006), Nikitin (2021), and Kadia et al. (2022a, 2024)] and as compared to the corner vortices in DFs. However, such small-scale vortices could still alter the isovels of $U/U_{\max }$ at the top mixed corners as seen in Figs. 3(a–c).

The longitudinal, vertical, and lateral turbulence intensities are computed from the converged solutions of the specific Reynolds (normal) stresses as $u_{\mathrm{rms}}=\sqrt{R_{uu}}$, $w_{\mathrm{rms}}=\sqrt{R_{ww}}$, and $v_{\mathrm{rms}}=\sqrt{R_{vv}}$, 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 $u_{\mathrm{rms}}/U_{*l}$ contour lines is stronger than that of $U/U_{\max }$ 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 $u_{\mathrm{rms}}/U_{*l}$ values are found toward the no slip boundaries where $U/U_{\max }$ reduces rapidly and produce steeper velocity gradient in the normal direction while the lower $u_{\mathrm{rms}}/U_{*l}$ values are obtained toward the center of the flow area and around the velocity dips where $U/U_{\max }$ 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 $u_{\mathrm{rms}}/U_{*l}$ contour lines diagonally. Furthermore, the secondary flows around the mid depth and mid width for the $a_r=1.25$ and 1.0 cases bulge the $u_{\mathrm{rms}}/U_{*l}$ contour lines toward the center of the flow area. In the case of OCF_2 and DF_2, $u_{\mathrm{rms}}/U_{*l}$ contour lines are pushed away from the solid boundaries around $y/h\approx \pm 0.5$ and $y/h\approx \pm 0.4$, respectively, but not up to the mid width due to the restricted growth of the bottom vortices. Additionally, the inner secondary vortices modify $u_{\mathrm{rms}}/U_{*l}$ contour lines at the mixed corners. Figs. 7(a and b) and 9(a) indicate that $u_{\mathrm{rms}}/U_{*l}$ contour lines in the central area below the mid depth for $a_r=2.0$ and 1.25 are more closely spaced for DFs than those found for OCFs. This is apparently influenced by the steeper $\partial U/\partial z$ 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 $w_{\mathrm{rms}}$, $v_{\mathrm{rms}}$, and TKE $k$ [see Figs. 7(d and e), 8(a and b), and 9(b–d)] for $a_r=2.0$ and 1.25. Significantly higher deviations (but similar trends) are observed in the central area above the mid depth for $u_{\mathrm{rms}}$, $v_{\mathrm{rms}}$, and $k$ possibly due to the stronger bulging of $U$ (steeper $\partial U/\partial z$) found in such zone for DFs than the same for OCFs. However, the distributions of $w_{\mathrm{rms}}$ and primary specific Reynolds shear stress $-\overline{u^{\prime }{w}^{\prime }}$ diverge due to dissimilar top boundaries and their influences. In uniform OCFs, $w_{\mathrm{rms}}$ 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 $w_{\mathrm{rms}}$. In contrast, $w_{\mathrm{rms}}$ increases in the central area above the mid depth for DFs as found in Figs. 7(d–f) and 9(b). However, $w_{\mathrm{rms}}$ 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 $v_{\mathrm{rms}}$ and $w_{\mathrm{rms}}$ (represented by the turbulence anisotropy stress $R_{\mathrm{aniso}}=R_{vv}-R_{ww}$) 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 $w_{\mathrm{rms}}/U_{*l}$ near the sidewalls for DFs are comparable to those for OCFs, where the maximum $w_{\mathrm{rms}}/U_{*l}$ 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 $v_{\mathrm{rms}}$ toward the sidewalls [see Figs. 8(a–c)]. Similarly, the damping and redistribution of $w_{\mathrm{rms}}$ toward the bed and top wall or the free surface increases $v_{\mathrm{rms}}$ 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 $u_{\mathrm{rms}}/U_{*l}$, $v_{\mathrm{rms}}/U_{*l}$, and $R_{\mathrm{aniso}}$ in the bottom half depth of DFs are comparable to the same observed in OCFs, especially for lower $a_r$ values 1.25 and 1.0. In addition, the magnitudes of the turbulence intensities presented in Fig. 9 indicate that the major contributor to $k$ is the longitudinal component, followed by lateral and vertical components. However, the maximum $u_{\mathrm{rms}}/u_{*l}$ and $w_{\mathrm{rms}}/u_{*l}$ values, where $u_{*l}$ = 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 $-\overline{u^{\prime }{w}^{\prime }}$ is obtained from the converged solutions of the Reynolds shear stress component $R_{uw}$. Figs. 9(e) and 10(a–c) indicate that $-\overline{u^{\prime }{w}^{\prime }}$ extrema values are attained close to the no slip walls for both OCFs and DFs, which is attributed to the steeper gradient $\partial U/\partial z$ there. Although the distributions of $-\overline{u^{\prime }{w}^{\prime }}/U_{*l}^2$ in the bottom half flow depth are mostly comparable, significant differences are observed above the mid depth. First, the weaker bulging of $U$ contour lines above the mixed corner bisectors (i.e., weaker velocity dips and flatter $\partial U/\partial z$) for OCFs [see Figs. 3(a–c)] produces more distantly spaced contour lines of negative $-\overline{u^{\prime }{w}^{\prime }}/U_{*l}^2$ there than those obtained for DFs. It further generates absolute minimum $-\overline{u^{\prime }{w}^{\prime }}/U_{*l}^2$ 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 $a_r$, the intermediate vortex develops and modifies $U$ isovels, and thus, the contour line of $-\overline{u^{\prime }{w}^{\prime }}/U_{*l}^2$ 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 $y/h\approx -0.4$ to $-0.45$ and $z/h$ around 0.55 to 0.6 for OCF_1.25, and more clearly for OCF_1 at $y/h$ around $-0.35$ and $z/h$ around 0.5 to 0.6 [comparable to the findings of Broglia et al. (2003)]. Lastly, the positive $-\overline{u^{\prime }{w}^{\prime }}/U_{*l}^2$ values observed at $z/h$ around 0.95 for OCFs are attributed to the inward lateral components of the small-scale inner secondary vortices, which create $\partial U/\partial z>0$ there [see Figs. 3(a–c) and 10(a–c)].

The antisymmetric nature of the turbulence anisotropy stress $R_{\mathrm{aniso}}$ 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, $R_{\mathrm{aniso}}/U_{*l}^2$ contour lines for DFs are comparable to those for OCFs [Figs. 8(d–f)], especially for $a_r=1.25$ 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 $R_{\mathrm{aniso}}/U_{*l}^2$ 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 ${\omega }_x=\partial V/\partial z-\partial W/\partial y$ 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 $a_r$ [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 $a_r$ (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 $a_r$. The vorticity plots indicate that the intermediate vortex is a part of the free surface vortex that separates with a decrease in $a_r$ 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.

Computation of the cross-sectional distribution of the boundary shear stress is crucial for better estimation of sediment transport and safer design of channels from erosion and of SBTs from hydro-abrasion, which can play a significant role in defining the capital and maintenance costs of such designs. The difference in the secondary currents between OCFs and DFs and the influence of $a_r$ on such flow structures impacts the boundary shear stress. The obtained $U_{*l}$, $u_{*l}$, laterally averaged bed shear stress ${\overline{\tau }}_b$, and depth averaged wall shear stress ${\overline{\tau }}_w$ values for OCFs and DFs with a constant bulk velocity are compared in Table 2. Interestingly, higher $U_{*l}$ (between 4.9% and 5.5%), i.e., higher ${\overline{\tau }}_b$ (between 9.8% and 11.4%) values are found for DFs as compared to OCFs. The trend is the same across the entire channel width as reflected from Fig. 11(a) and is related to the higher near wall gradient $\partial U/\partial z$ observed for DFs than OCFs as found at the mid width (as an example) from the slopes of $U/\overline{U}$ profiles provided in Fig. 4(a). However, the increments of ${\overline{\tau }}_w$ (between 3.1% and 3.9%) are comparatively lower than those observed for ${\overline{\tau }}_b$. Interestingly, the increments of ${\overline{\tau }}_b$ and ${\overline{\tau }}_w$ decrease marginally (for ${\overline{\tau }}_b$, from being 11.4% for $a_r=2.0$ to 9.8% for $a_r=1.0$ and for ${\overline{\tau }}_w$, from being 3.9% for $a_r=2.0$ to 3.1% for $a_r=1.0$) with an increase in the flow depth or reduction of $a_r$ (i.e., reduction of the free surface contribution as compared to the flow depth). The decrease in $a_r$ reduces both ${\overline{\tau }}_b$ and ${\overline{\tau }}_w$ for OCFs as well as for DFs. However, such reductions are greater for ${\overline{\tau }}_b$ than the same for ${\overline{\tau }}_w$. Additionally, the reduction of $a_r$ increases the deviation between ${\overline{\tau }}_w$ and ${\overline{\tau }}_b$ for OCFs but decreases the same for DFs as found in Table 2. Eventually, the quantities become equal for a square duct, i.e., both being ${8.63\mathrm{N}/\mathrm{m}}^2$ for the tested conditions. Remarkably, such value is very close to the ${\overline{\tau }}_b$ (${8.61\mathrm{N}/\mathrm{m}}^2$) and ${\overline{\tau }}_w$ (${8.71\mathrm{N}/\mathrm{m}}^2$) values obtained for OCF_2. The restricted lateral growth of the bottom vortices undulates ${\tau }_b$ distributions for $a_r=2.0$ [see Figs. 3(d–f), 4(b), 6, and 11]. However, the distributions become more regular for lower $a_r$ values 1.25 and 1.0 as the bottom vortices reach the mid width. In all cases, the downflows of the bottom vortices toward the sidewall push the isovels of $U$ toward the bed and increase ${\tau }_b$, whereas the upward flows toward the mid width for $a_r$ 1.25 and 1.0 and around the channel quarter for $a_r=2.0$ bulge the isovels of $U$ away from the bed, which reduces ${\tau }_b$. Additionally, the bed shear increases toward the mid width for $a_r=2.0$. The obtained ${\tau }_b/{\overline{\tau }}_b$ undulations for OCF_2 and DF_1 are comparable to the results reported previously by Nezu and Rodi (1985) and Nezu and Nakagawa (1986, 1993) for similar $a_r$ values. However, those studies compared the OCF results obtained for $a_r\approx 2.0$ with the DF results obtained for $a_r=1.0$ (although their consideration of $h$ = half flow depth, using horizontal symmetry plane, for DF makes $b/h=2.0$) and observed a significant difference between the ${\tau }_b/{\overline{\tau }}_b$ profiles as also found in Fig. 11. Such a difference is attributed to the development of bottom vortices whose lateral extends reach up to the mid width for DF_1 but are limited up to the channel quarter for OCF_2 (see Fig. 6). Therefore, the bed shear stress distribution of a square duct cannot be compared to that of an OCF with $a_r=2.0$ even though ${\overline{\tau }}_b$ values are close (see Table 2). For $a_r=1.25$ and 1.0, the lateral distributions of ${\tau }_b/{\overline{\tau }}_b$ for OCFs differ insignificantly from those obtained for DFs as seen in Fig. 11(b).

The normalized wall shear stress ${\tau }_w/{\overline{\tau }}_w$ increases up to $z/h\approx 0.2–0.25$ but decreases above such $z/h$ and up to the mid depth for DFs (see Fig. 12) due to the outward and inward components of the sidewall corner vortices, respectively, which alter $U$ isovels and near wall gradient $\partial U/\partial y$. In contrast, ${\tau }_w/{\overline{\tau }}_w$ increases around the mid depth for OCF_2 as the outward flow of the free surface vortices push $U$ contour lines toward the sidewall. Such increments relax and ${\tau }_w/{\overline{\tau }}_w$ values remain roughly constant with the developing (for OCF_1.25) and developed (for OCF_1) intermediate vortices, which push the low-momentum fluids toward the mid width (see Figs. 6 and 12). In addition, the rapid rise in ${\tau }_w$ toward the free surface is attributed to the outward bulging of $U$ contour lines due to the small-scale inner secondary vortices. Similarly, the inward bulging of $U$ contour lines due to such vortices around $z/h\approx 0.8–0.9$ reduces the near wall $\partial U/\partial y$ and eventually the wall shear. Such influence of the inner secondary vortices was reported previously by Kang and Choi (2006) for a subcritical flow with $a_r=2$. However, the Speziale et al. (1991) or SSG RSM used by Kang and Choi (2006) simulated weaker bulging of the $U$ contour lines toward the bottom solid corner and, thus, observed lower ${\tau }_w/{\overline{\tau }}_w$ near the bed as seen in Fig. 12. It is challenging to experimentally determine the impact of inner secondary vortices because of the difficulties in measuring the velocity data close to the boundaries at the mixed corner. Therefore, Nezu and Rodi (1985) could not provide the near free surface undulation of ${\tau }_w/{\overline{\tau }}_w$. The vertical distribution of ${\tau }_w/{\overline{\tau }}_w$ can be useful in the design of sidewall or bank erosion protections.

The simulated OCF conditions are comparable to the recent experimental studies on high-speed flows, e.g., Auel et al. (2014, 2017a, b) and Demiral et al. (2020, 2022), and the studied Re and $h$ values (see Table 1) satisfy the recommended minimum values of ${10}^5$ (Boes and Hager 2003) and 0.04 m (Heller 2011), respectively, to limit the scale effects in high-speed flows. Therefore, the results can be scaled up for geometric, kinematic, and dynamic similarities to compare the prototype designs using Froude similitude [see Heller (2011) and Pummer and Schüttrumpf (2018)]. For example, the scale factor for Solis SBT = existing channel width divided by model $\mathrm{width}=4.4/0.2=22$. Furthermore, Demiral-Yüzügüllü (2021) compared the hydro-abrasion results from the scaled flow, sediment, and invert material conditions with the field observations and observed no or negligible scale effects. In addition, the tested DFs have significantly higher Re than fully turbulent DF cases tested by Pirozzoli et al. (2018) and Nikitin (2021) using DNS, where it was found that the development of the corner vortices and their effect on the flow properties depended on Re when it was comparatively low. The results obtained for the DFs can be scaled up using the geometric scale and Euler number similarity (Aberle et al. 2020). However, further research at project scale shall enlighten the existing knowledge.