3.1. Patterns of flow and turbulence around the bend

The patterns of flow and turbulence around the bend are investigated in the F16 experiment, where measurements are available in 13 cross-sections around the flume. Different processes induce transverse velocities near the convex bank. First, a transverse tilting of the water surface develops at the bend entrance, with transverse water surface gradients that are an order of magnitude larger than streamwise water surface gradients (figure 2). This effect leads to an adverse streamwise water surface gradient near the concave outer bank, and strongly enhanced streamwise water surface gradients near the convex inner bank from about 1 m upstream of the bend entrance to about $40^{\circ }$ into the bend (figure 2). As a result, flow decelerates at the outside of the bend and accelerates at the inside (figure 3 a). Due to mass conservation, this effect is accompanied by mass transport towards the convex bank, $v_{n}<0$ .

Figure 3.

Patterns of flow variables in the F16 experiment. (a) Normalized depth-averaged streamwise unit discharge $U_{s}h/UH$ (—), and cross-stream profiles of $U_{s}h/UH-1$ (—) in the measured cross-sections; (b) normalized near-surface transverse velocity $v_{n,surf}/U$ (—); $v_{n,surf}$ is obtained by averaging the measured velocity in the five points within the layer from 0.006 to 0.0019 m below the water surface, where the highest transverse near-surface velocities occur. (c) Normalized near-bed transverse velocity $v_{n,bed}/U$ (—); $v_{n,bed}$ is obtained by averaging the measured velocity in the five points within the layer from 0.006 to 0.0019 m above the bed, where the highest transverse near-bed velocities occur; (d) normalized depth-averaged transverse unit discharge $U_{n}h/UH$ (—); (e) normalized depth-averaged turbulent kinetic energy (tke), $\langle k\rangle /u_{\ast ,0}^{2}$ . Isoline pattern based on high-resolution measurements in the indicated cross-sections. The black line delimits the zone of convex bank flow separation at the water surface. The definition of this line is detailed in § 3.3.

Second, the interplay between the inward pressure gradient induced by the transverse tilting of the water surface and the outward centrifugal force leads to the development of a secondary flow, with outward velocities in the upper part of the water column (figure 3 b) and inward velocities in the lower part of the water column (figure 3 c). This secondary flow develops at the bend entrance, reaches a maximum magnitude between $90^{\circ }$ and $120^{\circ }$ into the bend, and weakens in the final part of the curved reach.

The transverse velocities near the convex bank resulting from these two processes affect flow separation. From the bend entrance to about $15^{\circ }$ into the bend, the secondary flow is weak, and $v_{n}<0$ over the entire water column (figure 3 b–d), which opposes flow separation. From about $15^{\circ }$ into the bend, outward transverse velocities related to the secondary flow become larger than inward transverse velocities related to the mass transport, resulting in outward transverse velocities near the water surface (figure 3 b) that favour flow separation from the convex bank. In the lower part of the water column (figure 3 c), on the contrary, the resulting transverse velocities induced by both processes are inwards over the entire length of the curved reach and oppose flow separation. Between the cross-sections at $15^{\circ }$ and $30^{\circ }$ into the curved reach, the depth-averaged transverse velocity $U_{n}$ also changes sign and becomes outward oriented (figure 3 d).

The sign reversal of $U_{n}$ near the convex bank seems to be a good indicator for the onset of flow separation, which occurs just downstream of the cross-section at $30^{\circ }$ into the bend. There, the main body of flow, including the core of highest velocities, separates from the convex bank (figure 3 a). Slower moving fluid occupies the region between the convex bank and the core of highest velocities. The velocity gradient is maximum at the abrupt lateral transition between the slow and fast moving fluids, where an internal shear layer occurs. No flow recirculation occurs in the zone of slow fluid, however, and velocities remain positive, i.e. the flow is still moving downstream. The zone of flow separation widens in the streamwise direction, but velocity gradients between the slow and fast moving fluids weaken and have almost vanished at the cross-section at $150^{\circ }$ into the bend (figure 3 a) These characteristics correspond to the first stage of flow separation (figure 1 c).

The edge of the zone of convex bank flow separation is indicated in figures 2 and 3 and all subsequent figures. The criteria for delimiting the separation zone will be discussed in § 3.3 once all relevant aspects of the flow field are presented.

Near the water surface, a streak of high outward transverse velocities induced by the convex bank flow separation is clearly distinguishable from the streak induced by the secondary flow (figure 3 b). The smallest outward transverse velocities at the water surface are observed near the shear layer at the edge of the zone of convex bank flow separation. Depth-averaged transverse velocities are small within the zone of convex bank flow separation (about $0.02U$ ), and show considerable positive values just outside this zone (maximum value of about $0.08U$ in cross-section at $90^{\circ }$ ) (figure 3 d). The pattern of inward near bed transverse velocities (figure 3 c), on the contrary, does not show a relation to the convex bank flow separation.

Figure 3(e) shows the measured pattern of normalized depth-averaged tke. Additional strain induced by the secondary flow causes an increase of tke in the bend, notably in the central parts of the cross-sections. This process has been explained and modelled by Blanckaert (Reference Blanckaert2009). Another zone of additional depth-averaged tke is clearly related to the internal shear layer induced by the process of convex bank separation: it originates near the convex bank and moves away from the bank in the downstream direction. Depth-averaged tke reaches maximum values in the cross-section at $120^{\circ }$ into the bend that are approximately two and a half times higher than those in the straight inflow reach. In the straight outflow reach, the tke levels recover towards values for flow in the straight approach reach.

3.2. Patterns of flow and turbulence in cross-sections

Patterns of the streamwise velocity $v_{s}$ , streamwise vorticity ${\it\omega}_{s}$ and tke are examined for the F16 experiment in the cross-section at $30^{\circ }$ in the bend just upstream of the zone of convex bank flow separation, as well as in the cross-sections at $60^{\circ }$ , $90^{\circ }$ , $120^{\circ }$ and $150^{\circ }$ within the bend. Although the entire cross-sectional patterns are shown, the discussion focuses on the process of convex bank flow separation. We refer to Blanckaert (Reference Blanckaert2009) and Blanckaert et al. (Reference Blanckaert, Duarte, Chen and Schleiss2012) for a discussion of the other flow processes.

Figure 4.

Normalized streamwise velocity $v_{s}/U$ in the cross-sections at $30^{\circ },60^{\circ },90^{\circ },120^{\circ }$ and $150^{\circ }$ in the bend in the F16 experiment. A vertical profile is shown at $n=-0.53~\text{m}$ ; the horizontal line in this profile indicates $v_{s}/U=1$ . The white line delimits the zone of convex bank flow separation (§ 3.3). The dashed lines indicate the measuring grid. The same colour bar is used in all figures.

The cross-sectional patterns of the normalized streamwise velocity, $v_{s}/U$ , are illustrated in figure 4. The abrupt transverse tilting of the water surface in the region of strongly increasing curvature is accompanied by strongly enhanced streamwise water surface gradients near the convex bank (figure 2) that cause flow acceleration, and explain the occurrence of the core of highest streamwise velocities close to the convex bank in the cross-section at $30^{\circ }$ . From approximately $40^{\circ }$ into the bend, the main flow separates from the convex bank and the core of highest velocities moves farther away from the bank. A zone of convex bank flow separation with substantially lower streamwise velocities develops between the convex bank and the core of highest velocities. In the cross-sections at $60^{\circ }$ and $90^{\circ }$ into the bend (figure 4 b,c), high velocity gradients clearly demarcate the separation between the zones of slow and fast flow, and identify an internal shear layer. In these cross-sections, the difference between minimum near-surface velocities at the inner side of the shear layer and maximum near-surface velocities at the outer side of the shear layer is about 0.5  $U$ . The zone of convex-bank flow separation widens in streamwise direction, but velocity gradients at the edge of the separation zone also attenuate, as illustrated by the patterns in the cross-sections at $120^{\circ }$ and $150^{\circ }$ into the bend (figure 4 d,e). The flow separation zone is not homogenous over the water column: it narrows from the water surface towards the bed, and does not reach onto the bed. Vertical profiles of the streamwise velocity extending through the separation zone are strongly deformed, with low velocities near the water surface and maximum values near the bed underneath the separation zone (figure 4).

The cross-sectional patterns of secondary flow in the F16 experiment are illustrated in figure 5 by means of the normalized streamwise vorticity, ${\it\omega}_{s}H/U$ . The patterns reveal a centre region cell of secondary flow ( ${\it\omega}_{s}<0$ ) induced by streamline curvature, and a counter-rotating outer bank cell of secondary flow ( ${\it\omega}_{s}>0$ ). Both these secondary flow cells are described and analysed by Blanckaert (Reference Blanckaert2009) and Blanckaert et al. (Reference Blanckaert, Duarte, Chen and Schleiss2012). A zone of reversed streamwise vorticity, ${\it\omega}_{s}>0$ , occurs near the edge of the zone of convex bank flow separation in all cross-sections. It is highlighted by a thick ${\it\omega}_{s}=0$ isoline in figure 5. It is still clearly visible at $150^{\circ }$ into the bend, where the velocity gradients are already considerably attenuated (figure 4 e). The zone of reversed vorticity ${\it\omega}_{s}>0$ clearly separates a zone of negative vorticity within the zone of convex bank flow separation from the zone of negative vorticity induced by centre region cell of secondary flow.

Figure 5.

Normalized streamwise vorticity ${\it\omega}_{s}H/U$ in the cross-sections at $30^{\circ },60^{\circ },90^{\circ },120^{\circ }$ and $150^{\circ }$ in the bend in the F16 experiment. The white line delimits the zone of convex bank flow separation (§ 3.3). The black lines schematically indicate the pattern of secondary flow cells. The reversal of ${\it\omega}_{s}$ near the edge of the separation zone is highlighted by a thick ${\it\omega}_{s}=0$ isoline. The same colour bar is used in all figures.

Figure 6 analyses the secondary flow patterns in the zone of convex bank flow separation and near its edge in the cross-section at $90^{\circ }$ . Outside the zone of convex bank flow separation, $n>-0.35$ , the centre region cell of secondary flow induced by streamline curvature is clearly discernible (figure 6 a–d). Near the edge of the zone of convex bank flow separation, the vertical gradient of the transverse velocity, $\partial v_{n}/\partial z$ is reversed (figure 6 a), and the vertical velocity $v_{z}$ changes sign (figure 6 b). Surprisingly, downward velocities, $v_{z}<0$ , occur in a large portion of the separation zone. The vector and streamline patterns of ( $v_{n},v_{z}$ ) (figure 6 c,d) show, however, that the reversal of ${\it\omega}_{s}$ , $\partial v_{n}/\partial z$ and $v_{z}$ does not lead to an additional counter-rotating secondary flow cell, despite the distinct zones of negative streamwise vorticity (figure 5).

Figure 6.

Secondary flow in the inner part of the cross-section at $90^{\circ }$ in the bend in the F16 experiment. The black line delimits the zone of convex bank flow separation (§ 3.3). (a) Normalized transverse velocity $v_{n}/U$ (%), (b) normalized vertical velocity $v_{z}/U$ (%), (c,e,g) vector representation of ( $v_{n},v_{z}$ ) and its contributions related to the centre region cell of secondary flow ( $v_{n,CRC},v_{z,CRC}$ ) and convex-bank flow separation ( $v_{n,FS},v_{z,FS}$ ), respectively (d) streamline representation of ( $v_{n},v_{z}$ ), and (f,h) contributions to the normalized streamwise vorticity ${\it\omega}_{s}H/U$ (figure 5 c) related to the centre region cell of secondary flow, ${\it\omega}_{s,CRC}H/U$ , and convex bank flow separation ${\it\omega}_{s,FS}H/U$ , respectively.

The secondary flow patterns are further analysed by decomposing ( $v_{n},v_{z}$ ) and ${\it\omega}_{s}$ into a part corresponding to the curvature-induced centre region cell of secondary flow, ( $v_{n,CRC},v_{z,CRC}$ ) and ${\it\omega}_{s,CRC}$ , and a part related the convex bank flow separation, ( $v_{n,FS},v_{z,FS}$ ) and ${\it\omega}_{s,FS}$ . First, $v_{n,CRC}$ is obtained by parabolic interpolation between $v_{n}=0$ at the convex bank and the measured $v_{n}$ value at the outer edge of the zone of flow separation ( $n=-0.35$ in the F16 experiment), where $\partial v_{n}/\partial n=0$ is imposed. This interpolation is inspired by numerical simulations with a RANS model by Koken et al. (Reference Koken, Constantinescu and Blanckaert2013, their figure 5), which predicted a centre region cell of secondary flow that covers the entire width of the cross-section. The vertical velocity $v_{z,CRC}$ is then obtained by solving the equation for mass conservation, $\partial v_{n,CRC}/\partial n+\partial v_{z,CRC}/\partial z=0$ . The vector pattern ( $v_{n,CRC},v_{z,CRC}$ ) and corresponding vorticity ${\it\omega}_{s,CRC}$ are shown in figure 6(e,f). Subtracting these from the measured ( $v_{n},v_{z}$ ) and corresponding ${\it\omega}_{s}$ yields ( $v_{n,FS},v_{z,FS}$ ) and ${\it\omega}_{s,FS}$ (figure 6 g,h). This conceptual decomposition reveals that the convex bank flow separation is related to a secondary flow cell that co-rotates with the centre region cell, and has similar maximum vorticity levels (figures 5 c and 6 h). This conceptual decomposition also explains the different streaks of outward transverse velocities near the water surface associated with the convex bank flow separation and the centre region cell, respectively, that were observed in figure 3(b). Both the transverse gradient $\partial v_{z}/\partial n\approx \partial v_{z,FS}/\partial n$ (figure 6 b) and vertical gradient $\partial v_{n,FS}/\partial z$ (figure 6 g) contribute to the generation of a core of opposite vorticity ${\it\omega}_{s,FS}>0$ near the edge of the zone of convex bank flow separation, whereby $\partial v_{z}/\partial n$ constitutes the dominant contribution (not shown). Maximum positive values of ${\it\omega}_{s,FS}$ are comparable in magnitude to the maximum negative values of ${\it\omega}_{s}$ occurring in the zone of convex bank flow separation and in the centre region cell of secondary flow.

Van Balen et al. (Reference Van Balen, Blanckaert and Uijttewaal2010) and Koken et al. (Reference Koken, Constantinescu and Blanckaert2013) successfully simulated the secondary flow patterns in the experiment examined here with eddy resolving numerical models. RANS models, on the contrary, predicted a centre region cell of secondary flow that covered the entire cross-section. These numerical results suggest that the structure of turbulence plays an essential role in the generation of the additional secondary flow cells related to the convex bank flow separation. Accurate simulation of this additional secondary flow cell is a requisite for the accurate simulation of the process of convex bank separation, including the internal shear layer. The role of turbulence will be further investigated in § 5. Eddy resolving numerical simulations by Ottevanger (Reference Ottevanger2013, his figure 5.3) indicate the existence of a similar convex bank secondary flow cell in axisymmetric curved flow, which is defined as flow in an infinite bend of constant curvature where all streamwise gradients have vanished. Because flow separation from the convex bank is by definition impossible in axisymmetric flow, this numerical result indicates that the convex bank secondary flow cell is not a result of the convex bank flow separation, but may contribute to its onset.

The cross-sectional patterns of tke in the F16 experiment are illustrated in figure 7. The centre region cell of secondary flow (figure 5) is clearly the dominant process with respect to the generation of additional tke (figure 7) in the bend. The most pronounced increase in tke occurs near the centre of the centre region cell, as best illustrated in the cross-sections at $30^{\circ }$ and $60^{\circ }$ into the bend. Convex bank flow separation enhances tke in two ways. First, an increase in tke occurs near the edge of the zone of convex bank flow separation. It is best discernible in the cross-sections at $60^{\circ }$ and $90^{\circ }$ in the bend, weakens considerably in the cross-section at $120^{\circ }$ , and has vanished at $150^{\circ }$ . Figure 8(a) shows that the streamwise turbulent normal stress $\overline{v_{s}^{\prime 2}}$ is the main contribution to the increased tke near the edge of the zone of convex bank flow separation. Second, the zone of convex bank flow separation confines the streamwise flow (figure 4) and the core of maximum vorticity associated with the centre region cell of secondary flow (figure 5) to the lower part of the water column, below the zone of convex bank flow separation. This confinement leads to enhanced values of streamwise velocity, vorticity and associated tke. Figure 8(b) shows that the transverse turbulent normal stress $\overline{v_{n}^{\prime 2}}$ is the main contribution to the increased tke below the zone of convex bank flow separation. It is also the main contribution to the core of maximum tke related to the centre region cell of secondary flow. The vertical turbulent normal stress $\overline{v_{z}^{\prime 2}}$ (figure 8 c) is smaller than the other two components, but still of a dominant order of magnitude.

Figure 7.

Normalized tke $k/u_{\ast ,0}^{2}$ in the cross-sections at $30^{\circ },60^{\circ },90^{\circ },120^{\circ }$ and $150^{\circ }$ in the bend in the F16 experiment. The white line delimits the zone of convex bank flow separation (§ 3.3). The same colour bar is used in all figures.

Figure 8.

Normalized turbulent normal stresses and shear stresses in the cross-section at $90^{\circ }$ in the bend in the F16 experiment. (a) $\overline{v_{s}^{\prime 2}}/u_{\ast ,0}^{2}$ , (b) $\overline{v_{n}^{\prime 2}}/u_{\ast ,0}^{2}$ , (c)  $\overline{v_{z}^{\prime 2}}/u_{\ast ,0}^{2}$ , (d) $\overline{v_{s}^{\prime }v_{n}^{\prime }}/u_{\ast ,0}^{2}$ , (e) $\overline{v_{s}^{2}v_{z}^{\prime }}/u_{\ast ,0}^{2}$ , (f) $\overline{v_{n}^{\prime }v_{z}^{\prime }}/u_{\ast ,0}^{2}$ . The white line delimits the zone of convex bank flow separation (§ 3.3).

Contrary to straight uniform open channel flow where $\overline{v_{s}^{\prime }v_{n}^{\prime }},\overline{v_{n}^{\prime }v_{z}^{\prime }}\ll \overline{v_{s}^{\prime }v_{z}^{\prime }}$ (Sukhodolov, Thiele & Bungartz Reference Sukhodolov, Thiele and Bungartz1998), all three turbulent shear stresses are of similar magnitude in curved open channel flow (figure 8). Especially the shear stresses $\overline{v_{s}^{\prime }v_{n}^{\prime }}$ and $\overline{v_{s}^{\prime }v_{z}^{\prime }}$ , which represent the shear induced by cross-sectional gradients in the streamwise velocity, reach high values near the shear layer at the edge of the zone of convex bank flow separation. The shear layer is characterized by pronounced positive values of the turbulent shear stress $\overline{v_{s}^{\prime }v_{z}^{\prime }}$ (figure 8 e). The pattern of $\overline{v_{n}^{\prime }v_{z}^{\prime }}$ (figure 8 f) relates to the secondary flow cells (figure 5 c). This shear stress shows a sign reversal in the shear layer, which coincides with the region of opposite vorticity (cf. figure 5 c). The pattern of $\overline{v_{s}^{\prime }v_{n}^{\prime }}$ (figure 8 d) shows pronounced variations in the zone of convex bank flow separation.

3.3. Dependence of convex bank flow separation on flow depth

Figures 9–11 show patterns of the most relevant flow variables in the cross-section at $135^{\circ }$ into the bend in the F11 experiment and at $75^{\circ }$ into the bend in the F21 experiment. A change in flow depth $H$ causes different values of the non-dimensional parameters $H/R$ , $C_{f}^{-1}H/B$ and Fr, which are all known to be important control parameters in curved open channel flow.

Figure 9.

Patterns of flow variables in the cross-section at $135^{\circ }$ in the bend in the F11 experiment (a) normalized streamwise velocity $v_{s}/U$ ; a vertical profile is shown at $n=-0.53~\text{m}$ ; the horizontal line in this profile indicates $v_{s}/U\!=\!1$ , (b) normalized vertical velocity $v_{z}/U$ (%), (c) normalized streamwise vorticity ${\it\omega}_{s}H/U$ , (d) normalized tke $k/u_{\ast ,0}^{2}$ , (e) normalized turbulent normal stress $\overline{v_{s}^{\prime 2}}/u_{\ast ,0}^{2}$ and (f) normalized turbulent shear stress $\overline{v_{s}^{\prime }v_{z}^{\prime }}/u_{\ast ,0}^{2}$ . The measuring grid is indicated in (a). The white line delimits the zone of convex bank flow separation (§ 3.3). Same colour bars as in corresponding figures 4, 5, 7, 8 and 10.

Figure 10.

Patterns of flow variables in the cross-section at $75^{\circ }$ in the bend in the F21 experiment (a) normalized streamwise velocity $v_{s}/U$ ; a vertical profile is shown at $n=-0.53~\text{m}$ ; the horizontal line in this profile indicates $v_{s}/U\!=\!1$ , (b) normalized vertical velocity $v_{z}/U$ (%), (c) normalized streamwise vorticity ${\it\omega}_{s}H/U$ , (d) normalized tke $k/u_{\ast ,0}^{2}$ , (e) normalized turbulent normal stress $\overline{v_{s}^{\prime 2}}/u_{\ast ,0}^{2}$ and (f) normalized turbulent shear stress $\overline{v_{s}^{\prime }\,v_{z}^{\prime }}/u_{\ast ,0}^{2}$ . The measuring grid is indicated in (a). The white line delimits the zone of convex bank flow separation (§ 3.3). Same colour bars as in corresponding figures 4, 5, 7–9.

Figure 11.

Secondary flow in the inner part of the cross-section. (a,c,e) at $135^{\circ }$ in the bend in the F11 experiment, and (b,d,f) at $75^{\circ }$ in the bend in the F21 experiment. (a,b) streamwise vorticity ${\it\omega}_{s,CRC}H/U$ related to the centre region cell of secondary flow, (c,d) streamwise vorticity ${\it\omega}_{s,FS}H/U$ related to the convex bank flow separation, (e,f) vector representation ( $v_{n,FS},v_{z,FS}$ ) of secondary flow related to the convex bank flow separation. The black line delimits the zone of convex bank flow separation (§ 3.3). Same colour bar as in figure 6(f,h).

The zone of convex bank flow separation has a width at the water surface of $1H$ at the cross-section at $75^{\circ }$ into the bend in the F21 experiment (figure 10), and of $0.7H$ and $1.5H$ in the cross-sections at $60^{\circ }$ and $90^{\circ }$ into the bend, respectively, in the F16 experiment (figure 4). The width of the separation zone in the F11 experiment at $135^{\circ }$ into the bend is $2.4H$ (figure 9), and it increases from $2.4H$ to $2.8H$ in the F16 experiment from $120^{\circ }$ to $150^{\circ }$ into the bend (figure 4). These observations indicate that the width of the zone of convex bank flow separation primarily scales with the flow depth. In the three experiments, the flow separation zone narrows from the water surface towards the bed (figures 4, 9, 10) and does not extend to the bed.

Vertical profiles of the streamwise velocity extending through the separation zone are strongly deformed (figures 4, 9 a, 10 a), with low velocities near the water surface and maximum values near the bed underneath the separation zone. In the three experiments, defining characteristics at the edge of the zone of convex bank flow separation are the occurrence of a shear layer, high velocity gradients and inflexion points in the velocity profiles (figures 4, 9, 10), a zone of reversed vorticity ${\it\omega}_{s}>0$ (figures 5, 9, 10), reversal of the vertical velocities (figures 6, 9, 10), enhanced tke that is mainly due to the streamwise turbulent normal stress $\overline{v_{s}^{\prime 2}}$ (figures 8, 10) and high positive values of the shear stress $\overline{v_{s}^{\prime }v_{z}^{\prime }}$ (figures 8–10). The streamwise turbulent normal stress $\overline{v_{s}^{\prime 2}}$ and tke recover fastest to their background values, and do not show a clear signature of the convex bank flow separation in the cross-section at $150^{\circ }$ in the F16 experiment (figure 7), and in the cross-section at $135^{\circ }$ in the F11 experiment (figure 9). On the contrary, the streamwise vorticity ${\it\omega}_{s}$ (figures 5 and 9), vertical velocity $v_{z}$ (figure 9) and shear stress $\overline{v_{s}^{\prime }v_{z}^{\prime }}$ (figure 9) still clearly show the signature of the convex bank flow separation in these cross-sections. The position of the edge of the zone of convex bank flow separation has been estimated from these characteristics and indicated in all relevant figures. This estimation involves some ‘expert’ judgment, because the defining characteristics of flow separation associated with the different flow variables do not occur at exactly the same location. In the F21 experiment, for example, the inflexion point in the $v_{s}$ profiles, the sign reversal in $v_{z}$ and the zone of reversed ${\it\omega}_{s}$ are slightly offset (figure 10 a–c). Therefore the position of the edge of the separation zone should be considered approximate.

In a similar manner to the F16 experiment (figure 6), the secondary flow patterns in the F11 and F21 experiments can also be decomposed in a curvature-induced centre region cell and a co-rotating turbulence-induced cell related to the convex bank flow separation (figure 11). A zone of opposed vorticity, ${\it\omega}_{s,FS}>0$ , also occurs near the edge of the zone of convex bank flow separation. The maximum magnitude of the negative values of ${\it\omega}_{s,FS}$ in the core of the secondary flow cell related to the convex bank flow separation, and the maximum magnitude of the positive values of ${\it\omega}_{s,FS}$ near the edge of the separation zone are smallest in the F11 experiment, and similar in the F16 and F21 experiments (figures 6 and 11). The increasing value of the curvature ratio $H/R$ , and the different locations around the bend of the investigated cross-sections might be responsible for these different maximum vorticity magnitudes.

These results show that the main characteristics of the flow and turbulence in the zone of convex bank flow separation are similar for the three investigated flow depths. The F11 experiment is not appropriate for further analysis of convex bank flow separation because the signature of the flow separation on some turbulence characteristics has already weakened too much in the investigated cross-section at $135^{\circ }$ into the bend. Analysis of the physics underlying convex bank flow separation has yielded similar results for the F16 and F21 experiments; therefore only an analysis of the F16 experiment will be reported in the forthcoming sections.

4.1. Momentum equations

The depth-averaged momentum equations can be written in curvilinear coordinates in the following form (Jin & Steffler Reference Jin and Steffler1993):

Here $\partial /\tilde{\partial }s=(1+n/R)^{-1}\partial /\partial s$ and $\partial /\tilde{\partial }n=\partial /\partial n$ , ${\it\delta}$ is the Kronecker symbol, $\langle \,\rangle$ is the depth-averaging operator and the Einstein summation convention is applied. The elevation of the water surface is given by $z_{s}$ and $h$ indicates the local flow depth. In (4.3), the local flow velocity $v_{i}~(i=s,n)$ has been decomposed into time- and depth-averaged values, $\langle v_{i}\rangle =U_{i}$ , local time-averaged spatial deviations, $v_{i}^{\ast }$ and temporal fluctuations, $v_{i}^{\prime }$ :

The examined experiments concern steady flow, but the unsteady term is retained in (4.1) to facilitate interpretation: positive/negative terms on the right-hand side tend to increase/decrease $U_{i}$ . The term GRAV in (4.1) represents the gravitational forcing of the flow, which occurs through water surface gradients in open channel flow. The bed shear stress ${\it\tau}_{b}$ opposes the flow. The term INERT mainly represents inertial effects related to the depth-averaged flow field. The term VERT represents the effect of the vertical structure of the flow, including momentum redistribution by the secondary flow and the term TURB represents turbulence effects. The terms in (4.3) multiplied by the Kronecker symbol appear due to the transformation from Cartesian to curvilinear coordinates. They include the centrifugal force in the term $\text{INERT}_{n}$ . The term $O({\it\nu})$ represents viscous dissipation, and is known to be small.

The terms GRAV, INERT, VERT and TURB can be estimated from the experimental data, which would in theory allow estimation of the magnitude of the bed shear stress from (4.1). Error propagation, however, prevents a reliable quantitative estimation of the bed shear stress. The streamwise and transverse components of the bed shear stress, ${\it\tau}_{bi}~(i=s,n)$ are of order of magnitude 1 ${\it\tau}_{bs,bend}/{\it\rho}$ , where ${\it\tau}_{bs,bend}$ is the bend-averaged bed shear stress. According to the experimental data investigated here, the terms GRAV, INERT and VERT are of order of magnitude 10 ${\it\tau}_{bs,bend}/{\it\rho}$ , and TURB is of order of magnitude 1 ${\it\tau}_{bs,bend}/{\it\rho}$ . Blanckaert & de Vriend (Reference Blanckaert and de Vriend2004) estimated the experimental uncertainty in the term GRAV at about 2 %, and at about 20 % in the terms INERT, VERT and TURB. In absolute values, this means that the experimental uncertainty in the terms INERT and VERT is of order of magnitude 2 ${\it\tau}_{bs,bend}/{\it\rho}$ , and the corresponding uncertainty in their sum of order of magnitude $\text{sqrt}(2)\times 2~{\it\tau}_{bs,bend}/{\it\rho}$ , which is considerably larger than the magnitude of the bed shear stress components. Hence, the magnitude of the bed shear stress will not be estimated from (4.1) but from near bed flow measurements performed with high spatial resolution. A term-by-term analysis of (4.1) based on the experimental data does provide insight in the relative importance of the different flow processes governing the velocity redistribution and convex bank flow separation.

4.2. Streamwise bed shear stress

The bed shear stress is the most important flow parameter with respect to frictional energy losses and the morphologic development in loose boundary configurations. It is often represented by means of a Chézy-type dimensionless friction factor, which relates the bed shear stress to the depth-averaged velocity:

Figure 12.

Friction coefficient 1000 $c_{f,bed}$ based on high-resolution measurements in the indicated cross-sections. (a) F16 experiment. The black line approximately delimits the zone of convex bank flow separation at the water surface (§ 3.3); (b) experiments according to label in Figure; the horizontal lines of constant value represent the flume-averaged value of 1000 $\tilde{C}_{f}$ (table 2).

Figure 12 shows the distribution around the bend of $c_{f,bed}$ , estimated from the velocity measurements. The estimation of $U_{s}$ and $U_{n}$ from the measured velocity patterns (figures 4, 9, 10) is straightforward. The estimation of ${\it\tau}_{bs}$ from the velocity measurements is more difficult. In straight uniform open channel flows, Nezu & Nakagawa (Reference Nezu and Nakagawa1993) have proposed a semi-theoretical logarithmic profile for the mean velocity, semi-theoretical exponential profiles for the turbulent normal stresses and the tke and a semi-theoretical linear profile for the streamwise vertical turbulent shear stress $\overline{v_{s}^{\prime }v_{z}^{\prime }}$ , which all scale with the bed shear stress. Fitting of the measured vertical profiles to these semi-theoretical similarity solutions allows estimation of the bed shear stress in straight uniform flow. Nezu & Nakagawa (Reference Nezu and Nakagawa1993) report considerable differences between estimates based on the different flow quantities in straight uniform laboratory flows. Wilcock (Reference Wilcock1996), Biron et al. (Reference Biron, Lane, Roy, Bradbrook and Richards1998), Kim et al. (Reference Kim, Friedrichs, Maa and Wright2000), Biron et al. (Reference Biron, Robson, Lapointe and Gaskin2004) and Sime, Ferguson & Church (Reference Sime, Ferguson and Church2007) have compared and analysed the same methods in various configurations of quasi-straight, shallow open channel flow. Their findings indicate that the estimation based on the near bed tke is most reliable. For the estimation based on the logarithmic velocity profile, these investigations show that the inclusion of near bed measurements is crucial. Jamieson et al. (Reference Jamieson, Post and Rennie2010), Termini & Piraino (Reference Termini and Piraino2011) and Sukhodolov (Reference Sukhodolov2012) (for configurations with mobile bed topography) and Abad & Garcia (Reference Abad and Garcia2009) (for cases with flat immobile bed topography) have shown that the vertical profiles of the turbulent normal stresses, the turbulent shear stresses and the tke differ fundamentally in a curved open channel flow from their counterparts in straight open channel flow. This is confirmed by figures 7–10. These discrepancies invalidate Nezu & Nakagawa’s (Reference Nezu and Nakagawa1993) approach based on semi-theoretical similarity profiles for turbulent flow quantities in sharply curved open channel flows. Therefore, it can be inferred that the best estimation in such flows is obtained by fitting a logarithmic profile to the measured profile of the streamwise velocity in the inner 20 % of the boundary layer close to bed. The profiling capacity of the ADVP allowed measuring with high spatial resolution in the lower 20 % of the water column and resulted in a bed shear stress estimation with an uncertainty of approximately 20 % (Duarte Reference Duarte2008).

Integration of the depth-averaged momentum equation (4.1) around the curved reach, indicated by the subscript bend, approximately yields:

where $R_{h,bend}$ is the bend-averaged hydraulic radius, and $E_{s,bend}$ is the energy slope in the bend obtained by a linear fit to the cross-sectionally-averaged energy levels $E=H+U^{2}/(2g)$ along the curved reach. In the F16 experiment, the bend-averaged bed shear stress ${\it\tau}_{bs,bend}/{\it\rho}$ estimated from the velocity patterns, ${\it\tau}_{bs,bend}/{\it\rho}=1.08\times 10^{-3}~\text{m}^{2}~\text{s}^{-2}$ , is nearly identical to the value estimated from $R_{h,bend}$ and $E_{s,bend}$ of $-gR_{h,bend},E_{sbend}=1.07\times 10^{-3}~\text{m}^{2}~\text{s}^{-2}$ , which provides support for the general reliability of the estimated patterns of the friction factor $c_{f,bed}$ shown in figure 12.

The friction factor $c_{f,bed}$ increases around the bend in the F16 experiment (figure 12 a), which can mainly be attributed to the gradual deformation of the vertical profiles of the velocity. The maximum velocities are situated at the water surface near the bend entry, and then gradually move towards the lower part of the water column (figure 4). This deformation of the velocity profile increases the near bed velocity gradients and the bed shear stress without modifying the depth-averaged velocity, and therefore leads to an increase of $c_{f,bed}$ (4.5).

The major feature in the patterns of $c_{f,bed}$ is, however, its pronounced increase beneath the zone of convex bank flow separation, with values that are enhanced by up to 50 %. This increase is most pronounced in the initial part of the zone of convex bank flow separation, and weakens in the downstream direction. Similar increased values of $c_{f,bed}$ are found beneath the zone of convex bank flow separation in the F11 and F21 experiments (figure 12 b). The velocity patterns shown in figures 4, 9 and 10 confirm that this increase is mainly due to the deformation of the profiles of the streamwise velocity, which have low values within the zone of convex bank flow separation situated in the upper part of the water column, and high values below the zone of convex bank flow separation near the bottom.

The pronounced transverse variation of $c_{f,bed}$ has important consequences for numerical modelling by means of depth-averaged flow models, which are typically used for long-time and large-scale morphological simulations. These models are inherently unable to resolve the three-dimensional nature of the zone of convex bank flow separation. They could be able, however, to resolve the deceleration of the depth-averaged streamwise velocities $U_{s}$ adjacent to the convex bank (figure 3 a) and the related outward depth-averaged transverse velocities $U_{n}$ (figure 3 d). Such models require the prescription of $c_{f,bed}$ as input parameter. The common choice of a constant value of $c_{f,bed}$ will lead to an underestimation of flow friction near the convex bank, and hence an underestimation of the deceleration of $U_{s}$ and the outward $U_{n}$ , which will hamper the accurate simulation of the flow near the convex bank.

4.3. Transverse momentum balance

As mentioned in § 3.1, the sign reversal of $U_{n}$ near the convex bank (figure 3 d) seems to be a good indicator for the onset of flow separation. The term-by-term analysis of the transverse momentum equation (4.1) provides insight into the relative importance of the flow processes governing the distribution of $U_{n}$ . Figure 13 shows the different terms in the transverse momentum equation (4.1) normalized by the bend-averaged streamwise bed shear stress ${\it\tau}_{bs,bend}/{\it\rho}$ (4.6). The transverse bed shear stress has been estimated from the near bed velocity measurements as:

The second term represents the deviation between the direction of the depth-averaged velocity and the bed shear stress, due to the secondary flow. This deviation is estimated from the maximum value of $v_{n}^{\ast }/v_{s}$ near the bed.

Figure 13.

Normalized terms in the transport equation for transverse momentum (4.1) based on high-resolution measurements in the indicated cross-sections in the F16 experiment. (a) $({\it\tau}_{bn}/{\it\rho})/({\it\tau}_{bs,bend}/{\it\rho})$ , (b) $\text{VERT}_{n}/({\it\tau}_{bs,bend}/{\it\rho})$ , (c) $\text{TURB}_{n}/({\it\tau}_{bs,bend}/{\it\rho})$ , (d) $(\text{GRAV}_{n}+\text{INERT}_{n})/({\it\tau}_{bs,bend}/{\it\rho})$ . The black line delimits the zone of convex bank flow separation at the water surface (§ 3.3). ${\it\tau}_{bs,bend}/{\it\rho}$ is the bend-averaged streamwise bed shear stress (4.6).

The evolution of ${\it\tau}_{bn}$ around the curved reach (figure 13 a) is similar to the evolution of the near bed transverse velocity (figure 3 c). Maximum values of ${\it\tau}_{bn}$ are larger than the bend-averaged streamwise bed shear stress ${\it\tau}_{bs,bend}/{\it\rho}$ and occur within and just outside the zone of convex bank flow separation. The term $\text{VERT}_{n}$ (figure 13 b) is considerably larger than ${\it\tau}_{bn}$ . The positive values outside the zone of convex bank flow separation represent outward transport of momentum induced by the centre region cell of secondary flow. This process has been explained and modelled by Blanckaert & de Vriend (Reference Blanckaert and de Vriend2004). The term $\text{VERT}_{n}$ becomes negative near the convex bank. This change in sign can be attributed to the additional secondary flow cell related to the convex bank flow separation (figure 6 g). The term $\text{TURB}_{n}$ (figure 13 c) is small and does not play an important role in the transverse momentum balance.

The negative term $\text{GRAV}_{n}$ and positive term $\text{INERT}_{n}$ are dominant in the transverse momentum balance; they attain maximum magnitudes larger than 20 ${\it\tau}_{bs,bend}/{\it\rho}$ . Due to inertia, flow tends to move along a straight path at the bend entry, as represented by $\text{INERT}_{n}$ in the transverse momentum equation written in curvilinear coordinates. Mass accumulation at the outer side of the bend leads to the generation of a transverse tilting of the water surface, and an inward pressure gradient that causes the flow to follow the bend, represented by $\text{GRAV}_{n}$ . Figure 13(d) shows the resulting effect $\text{GRAV}_{n}+\text{INERT}_{n}$ , The magnitude of $\text{GRAV}_{n}+\text{INERT}_{n}$ increases just downstream of the bend entry, attains maximum values between $30^{\circ }$ and $90^{\circ }$ into the bend and subsequently decreases. A core of high positive values of $\text{GRAV}_{n}+\text{INERT}_{n}$ at the convex bank between $30^{\circ }$ and $60^{\circ }$ into the bend seems to be the major cause of the flow separation. Further downstream, values of $\text{GRAV}_{n}+\text{INERT}_{n}$ remain positive just outside the zone of convex bank flow separation, and are close to zero within the separation zone. In the outer half of the cross-section, $\text{GRAV}_{n}+\text{INERT}_{n}$ is negative. These results highlight and quantify the inertial lag between centrifugal and pressure gradient forces that causes the flow to separate from the convex bank in sharp bends. In the investigated experiments, the flow separation is in the first stage (figure 1).