3.1. Physical picture

Much of the phenomenology of turbulent flows can be derived from the conceptual framework of ‘turbulent eddies’ that transfer energy from large to small scales through Kolmogorov's energy cascade (Richardson Reference Richardson1920; Prandtl Reference Prandtl1925; Taylor Reference Taylor1935). More recently, this physical picture was used to relate the turbulent energy spectrum to macroscopic properties of the flow, such as the vertical velocity profile and the friction coefficient (Gioia & Bombardelli Reference Gioia and Bombardelli2001; Gioia & Chakraborty Reference Gioia and Chakraborty2006; Gioia et al. Reference Gioia, Guttenberg, Goldenfeld and Chakraborty2010). If the motion of the eddies is random, they will induce a diffusion of momentum across the stream with a diffusivity, $\mathcal {D}_e$, proportional to their velocity, $V$, and length scale, $L$ ($\mathcal {D}_e \propto V L$). These eddies mix the fluid from different parts of the flow so that they can generate a flux of momentum, $F$, across the stream if the downstream velocity, $u(y,z)$, varies throughout the flow, as in figures 2 and 3. In figure 2 the bed is flat ($D' = 0$), the flow is uniform in the vertical ($\partial u / \partial z = 0$), while the depth-averaged velocity, $U(y)$, varies across the stream; in figure 3 the bed has a slope ($D' \neq 0$), the velocity, $u(z)$, varies along the vertical, while its depth average is uniform across the stream ($U' = 0$). We now consider the flux, $F$, in these two configurations, assuming that the variations of $D(y)$ and $U(y)$ across the stream are slow.

Figure 2. Momentum transfer by large eddies in a flow with a horizontal velocity gradient, $U'$, but a constant vertical velocity profile. This figure shows eddies rotating in the vertical, $y$–$z$, plane, but the same analysis applies to those rotating in the horizontal, $x$–$y$, plane. Lighter shades of blue indicate faster flow. The largest eddy has a horizontal scale, $L$, comparable to the depth, $D$, and a velocity $V$ comparable to the frictional velocity, $U^* = C_f^{1/2}U$. The downstream flow velocity, $U$, varies slowly in the transverse direction from $U(y-L/2)$ to $U(y+L/2)$ across the scale of the eddy. The eddy mixes the fluid of momentum $\rho U(y-L/2)$ with the fluid of momentum $\rho U(y+L/2)$ across a vertical slice (vertical dashed line). The flux of momentum integrated along this section is proportional to $F \sim \rho D V (U(y-L/2) - U(y+L/2))$ (3.1).

Figure 3. Momentum transfer in a flow with a vertical velocity profile, $u(z)$, and a topography gradient, $D' < 0$, but no horizontal gradient of the depth-averaged velocity, $U' = 0$. (a) Vertical profiles of velocity at two locations across the stream, $y_1$ and $y_2$. The profiles are arbitrarily assumed to be quadratic, $u(y,z) = \tfrac {3}{2} U [ 1 - z^2/D(y)^2 ]$. A horizontal velocity gradient develops from the deeper to the shallower part of the flow, inducing a corresponding flux of momentum across the stream, $F$. (b) Part of the channel corresponding to panel (a). Lighter shades of blue indicate faster flow.

Let us first consider the example of figure 2. In this case, eddies of size $L$ and velocity $V$ carry the downstream momentum across a vertical slice of the channel located at $y$, mixing the fluid from $y + L/2$ with the fluid from $y - L/2$. The parcels of fluid from $y + L/2$, therefore, carry about $\rho U(y+L/2)$ of momentum and cross the vertical slice in the negative $y$ direction, while those from $y-L/2$ carry about $\rho U(y-L/2)$ of momentum and cross the section in the positive $y$ direction. The flux, $F$, from these eddies represents the net total amount of momentum that crosses the vertical slice of height $D$ per unit time, so it is about

(3.1)\begin{equation} F \sim \rho D V [U(y-L/2) - U(y+L/2)]. \end{equation}

If the downstream flow velocity, $U$, varies slowly across the channel, we can approximate the difference in (3.1) as $U(y -L/2) - U(y+L/2) \approx -L U'$, so that the flux generated by these eddies is approximately

(3.2)\begin{equation} F \sim \rho D V L U' . \end{equation}

Empirically, in wide channels with a rough bottom, eddies reach a size, $L$, comparable to the flow depth, $D$, since eddies much larger than this are suppressed by friction at the bottom (Bouchez et al. Reference Bouchez, Lajeunesse, Gaillardet, France-Lanord, Dutra-Maia and Maurice2010) – it matters little whether these eddies rotate in the vertical, $y$–$z$, plane (as in figure 2) or in the horizontal, $x$–$y$, plane. According to Kolmogorov's theory, the largest eddies in the flow also move the fastest, so that their diffusivity, $\mathcal {D}_e \propto LV$, is the greatest. We can, therefore, expect that these eddies are responsible for carrying most of the momentum across the stream, and for setting the mixing length (Prandtl Reference Prandtl1925). The flux of momentum induced by turbulence scales with the square of the velocity fluctuations (Batchelor Reference Batchelor1967), so that the stress, which balances the flux of momentum into the channel's bottom, should scale with the square of the eddy velocity, $\tau \sim \rho V^2$. Therefore, the velocity of the largest eddies should be of the order of the frictional velocity $U^* = \sqrt {\tau /\rho }$. On the other hand, the stress is related to the mean downstream velocity, $U$, through the empirical friction coefficient, $C_f$, as $\tau = \rho C_f U^2$. Thus, the velocity, $V$, of the largest eddies, although much smaller, is proportional to the downstream velocity, $V \sim C_f^{1/2} U$. Using the scalings $L \sim D$ and $V \sim C_f^{1/2} U$ in (3.2), we can write the flux, $F$, as

(3.3)\begin{equation} F \approx- \varLambda \rho C_f^{1/2} D^2 \big(U^2\big)' , \end{equation}

where $\varLambda$ is a positive, dimensionless number of order one, and the minus sign ensures that the momentum is transferred from fast- to slow-flowing regions. The parameter $\varLambda$ controls the magnitude of the flux and we call it the dimensionless diffusion parameter.

We now consider the example of figure 3. In this case, a flux of momentum develops across the stream even though the depth-averaged velocity is uniform ($U(y) = \text {const.}$), due to a combination of a sloping bed ($D' \neq 0$) and a variable vertical velocity profile ($\partial u / \partial z \neq 0$). A non-constant vertical profile of velocity means that some eddies will generate a flux of momentum, while the bed slope, $D' \neq 0$, breaks the left–right symmetry and induces a horizontal velocity gradient, $\partial u / \partial y$ (figure 3a). If mixing is isotropic, the momentum flux across a vertical slice is proportional to this horizontal gradient, and the total flux, $F$, scales as $F \sim \rho V \int _{-D}^0 [ \partial u / \partial y ] \,\text {d}z$, where $V$ is the velocity of the dominant eddies. Since the horizontal velocity gradient is, in this case, generated by the sloping bed, to first order it scales as $\partial u / \partial y \sim U D'$, and the flux, $F$, is therefore

(3.4)\begin{equation} F \sim \rho V U D D' . \end{equation}

Using $V\sim C_f^{1/2} U$, we find an approximate expression for the flux in this configuration:

(3.5)\begin{equation} F \approx- \alpha \varLambda \rho C_f^{1/2} U^2 \big(D^2\big)' . \end{equation}

Here $\alpha$ is another dimensionless number of order one, and we keep $\varLambda$ for later convenience. This flux is proportional to the depth gradient, $D'$, so that it appears to be induced by the bed topography. For this reason, we call $\alpha$ the ‘local-shape parameter.’ The minus sign in (3.5) means that when $\alpha$ is positive, the flux is generated from the deeper to the shallower parts of the flow. If the vertical profile of velocity, $u(z)$, is top heavy, meaning that the flow is faster near the free surface than at the bottom, a horizontal velocity gradient will indeed develop from deeper to shallower parts of the flow (figure 3). Therefore, positive values of $\alpha$ indicate a top-heavy velocity profile, while $\alpha$ vanishes in the ‘plug flow’ limit, i.e. when the velocity is constant along the vertical, $u(z) = \text {const}$.

Equations (3.3) and (3.5) for the flux, $F$, have a similar form, only differing in the position of the cross-stream derivative – in both cases, the flux is proportional to $D^2$ and to $U^2$. The term $D^2$ can be interpreted as a product of the depth, $D$, of the vertical slice through which the transfer of momentum occurs and the length scale, $L$, of the dominant eddies, while the term $U^2$ can be interpreted as a product of the eddy velocity, $V$, and the downstream velocity that carries the momentum, $U$. In agreement with the discussion above, previous experimental studies suggest that the diffusion parameter $\varLambda$ is of order one. For example, Okoye (Reference Okoye1970) showed experimentally that the dimensionless eddy diffusivity, $\mathcal {D}_e / (2 C_f^{1/2} U D)$, analogous to our $\varLambda$, is of order one, varying between about $0.03$ and $1.0$. Although we also expect the local-shape parameter, $\alpha$, to be of order one, it likely vanishes as the flow becomes more and more turbulent, and its velocity profile starts to resemble a vertically uniform ‘plug’ flow. In Appendix A we estimate the values of $\varLambda$ and $\alpha$ in a standard model of turbulent diffusion, assuming a logarithmic vertical profile of velocity. There, we find $\varLambda \approx 0.03$, and $\alpha$ vanishing logarithmically with the Reynolds number.

In addition to unsteady eddies, turbulent flows often develop long-lived, coherent secondary currents (Znaien et al. Reference Znaien, Hallez, Moisy, Magnaudet, Hulin, Salin and Hinch2009; Shih, Hsieh & Goldenfeld Reference Shih, Hsieh and Goldenfeld2016). In channel flows with large aspect ratios, these secondary currents form an array of counter-rotating cells that have length scales comparable to the depth, and a transverse velocity of the order of the frictional velocity (Tominaga et al. Reference Tominaga, Nezu, Ezaki and Nakagawa1989; Blanckaert et al. Reference Blanckaert, Duarte and Schleiss2010; Chauvet Reference Chauvet2014). Therefore, the scalings we introduced previously should still hold for momentum transfer across these cells, even though the momentum transfer on the scale of an individual cell is not diffusive. Thus, if $D(y)$ and $U(y)$ vary slowly over the scale of a single cell, we expect (3.3) and (3.5) to approximately capture the average transfer of momentum smoothed over several cells, although they cannot be accurate at the scale of a single cell.

3.2. Dimensional analysis

In the previous section we estimated the flux of momentum, $F$, in two specific configurations of the flow based on physical reasoning. We now derive a generic, first-order expression for the flux, $F$, using dimensional analysis. To that end, we make the following broad assumptions about the turbulent flow.

1. (i) Inertial turbulent regime: Gravity, $g$, molecular viscosity, $\nu$, and the bed roughness, $k_s$, do not explicitly change the flux, $F$. This is certainly not exactly correct – molecular viscosity, $\nu$, and the relative roughness of the bed, $k_s/D$, both affect the friction coefficient, which can affect the momentum transfer. Nevertheless, we assume that this dependence is weak enough to remove $g$, $\nu$ and $k_s$ from the list of parameters that can affect the flux. We will later relax this assumption by allowing the model parameters to weakly depend on $g$, $\nu$ and $k_s$.

2. (ii) Single layer: The depth-averaged downstream velocity, $U$, is sufficient to characterise the vertical velocity profile of the flow. In other words, we assume that the vertical velocity profile does not change its shape across the stream, so that we do not need to keep track of higher-order moments of the velocity.

3. (iii) Locality: The flux, $F$, is locally related to the flow speed and depth. In other words, the flux $F(y)$ is a function of the local depth-averaged velocity, $U(y)$, depth, $D(y)$, and their derivatives up to some finite order, $U'$, $D'$, $U''$, $D''$, etc. However, we assume that it does not depend on the far-away parts of the flow (such as a distant wall), nor on the integrated properties of the flow, such as the total flow discharge.

4. (iv) Slow variation: Finally, we assume that the changes in $D$ and $U$ occur over large scales, so that their higher-order derivatives are much smaller than the lower-order ones, and the flux can be written as an expansion in terms of these derivatives. This is likely true in a channel with a large aspect ratio. Therefore, we neglect second- and higher-order derivatives when writing a model for $F$. In this sense, our model is a first-order correction to the shallow-water theory.

Based on the above, our cross-stream flux of momentum, $F$, can only depend on the fluid density, $\rho$, the local velocity, $U(y)$, and depth, $D(y)$, as well as their derivatives up to some finite order, $U'$, $D'$, $U''$, $D''$, etc. The most general way to construct a quantity with the units of flux ($\text {kg}\ \text {s}^{-2}$) from the quantities above is

(3.6)\begin{equation} F = \rho U^2 D \varPhi(D', DU'/U, \ldots) , \end{equation}

where $\varPhi$ is an arbitrary dimensionless function of dimensionless parameters and ‘$\cdots$’ stands for higher-order derivatives written in dimensionless form. This equation states that the dimensionless flux, $F/\rho U^2 D$, can only depend on dimensionless variables, such as $D'$ and $DU'/U$ (Barenblatt Reference Barenblatt1996). Since we are looking for the lowest-order model for $F$, we can expand the function $\varPhi$ as $\varPhi \approx c_0 + c_1 D' + c_2 DU'/U + \cdots$, for some constants $c_0$, $c_1$ and $c_2$. However, since the flux is a directed quantity, it should also change sign if we mathematically flip the orientation of the $y$ axis, $F \rightarrow -F$ as $y\rightarrow -y$. The constant term, $c_0$, does not obey this symmetry, and, therefore, must vanish. To lowest order, therefore,

(3.7)\begin{equation} F \approx- \rho C_f^{1/2} \varLambda \big[D^2 (U^2)' + \alpha U^2 (D^2)'\big] , \end{equation}

where $\varLambda$ and $\alpha$ are dimensionless numbers. We included the term $C_f^{1/2}$ in the definition of $\varLambda$, as in § 3.1, for convenience. Even though we assumed that the flux, $F$, does not depend on molecular viscosity, bed roughness or the integrated properties of the flow in order to reduce the number of parameters in the dimensional analysis, we can somewhat relax this assumption and allow the constants $C_f$, $\varLambda$ and $\alpha$ to weakly depend on these properties, as they likely do.

Equation (3.7) consists of two terms that correspond exactly to (3.3) and (3.5). Although the physical mechanisms we discussed in the previous section are representative of these terms, (3.7) is a generic first-order expression that does not depend on the detailed dynamics of the flow – only the values of the parameters $C_f$, $\varLambda$ and $\alpha$ do. If eddies and recirculation cells of a size comparable with the flow depth dominate the momentum transfer, we expect $\varLambda$ and $\alpha$ to be of order one, with $\alpha$ vanishing in the highly turbulent, ‘plug flow’ limit (§ 3.1). On the other hand, wide recirculation cells or eddies much larger than the flow depth, cannot be included in our model since they induce a fundamentally non-local flux, violating one of our key assumptions. In Appendix A we show that a standard model of turbulent diffusion satisfies (3.7). In this sense, we may also call (3.7) a diffusive approximation.

Equation (3.7) can also be written in the compact form

(3.8)\begin{equation} F =-\rho C_f^{1/2} \varLambda D^{2(1-\alpha)} ( D^{2\alpha} U^2 )' . \end{equation}

From here, we can see that the local-shape parameter $\alpha$ determines the part of $D^2$ inside the derivative. In this way, it controls the quantity that diffuses due to the flux, $F$ – if $\alpha = 0$, it is the depth-averaged velocity, $U$, while if $\alpha = 1$, it is the depth-integrated momentum, $DU$.

Wark et al. (Reference Wark, Samuels and Ervine1990) proposed a model similar to (3.7). However, they do not explicitly include the local-shape parameter, $\alpha$, but rather mention two alternative models – one in which the velocity, $U$, diffuses, and another one in which it is the depth-integrated momentum $DU$ (‘unit flow’ in their terminology). Therefore, they consider two cases corresponding to $\alpha = 0$ and $\alpha = 1$, and state that it is unclear which one is better. Here we show that these cases are not necessarily special, so that models with non-integer values of $\alpha$ may also be appropriate within the same order of approximation. Besides Wark et al. (Reference Wark, Samuels and Ervine1990), we are not aware of anyone else who explicitly considered this parameter. Moreover, to our knowledge their model and its consequences have not been carefully examined.

4.1. Connection between turbulent and laminar flows

How much does a laminar channel flow resemble a turbulent one? Here, we show that there exists a connection between these two cases, at least in the most basic, diffusive approximation we discussed above.

Devauchelle, Popović & Lajeunesse (Reference Devauchelle, Popović and Lajeunesse2022) showed that the cross-stream flux of momentum in a straight channel with a laminar flow, $F_{lam}$, is approximately

(4.8)\begin{equation} F_{lam} \approx-\tfrac{1}{3}(D^2 \tau)' . \end{equation}

This flux has the same form as the turbulent flux in (4.3) with parameters $\chi = 1/3$ and $\alpha = 1$. Therefore, despite the fact that the laminar flux is driven by molecular diffusion while in the turbulent flow, it is driven by diffusion by the largest eddies and recirculation cells, the stress on the bottom of each flow is, to first order, determined by the same (4.4). This similarity may not be all that surprising, since our approximation implicitly assumes a turbulent flux driven simply by gradients of velocity (§ 3.1). However, the above analysis shows that the equivalence between turbulent and laminar flows holds for stress rather than velocity – turbulent and laminar velocities remain different, since they scale differently with the stress.

The transition from the laminar to the turbulent stress can be described by the change in parameters from $\chi = 1/3$ to $\chi \sim 10$ and from $\alpha = 1$ to $\alpha \approx 0$. This result clarifies the extrapolation of the small-scale, laminar rivers to real, turbulent ones (Malverti, Lajeunesse & Métivier Reference Malverti, Lajeunesse and Métivier2008; Lajeunesse et al. Reference Lajeunesse, Malverti, Lancien, Armstrong, Metivier, Coleman, Smith, Davies, Cantelli and Parker2010b).

5.1. Rectangular channel

We first consider the case of a rectangular channel with a constant depth, $D$, and a width, $W$ (figure 4). In this case, the shallow-water theory (1.1) predicts a constant stress on the flat bottom. When the cross-stream flux of momentum exists, however, some of the momentum will be transferred to the vertical side walls, thereby reducing the stress in their vicinity.

Figure 4. A rectangular channel with an aspect ratio of $W/D = 5$.

In a rectangular channel, (4.4), which assumes that the channel topography varies slowly, becomes ill-defined on the side walls where $D' \rightarrow \pm \infty$. Therefore, we cannot hope to find the distribution of the stress along them. However, the model is well defined away from the walls, so that we can still use it on the flat bottom if we take care to specify a boundary condition at the corners where the side walls meet the bottom ($\,y=\pm W/2$).

This boundary condition is often overlooked in the literature, where it is simply assumed that the velocity, $U$, vanishes at the corner. Formally, this is true – the velocity of the full Navier–Stokes equations must vanish at all boundaries, so that the proper boundary condition is always that of no slip at the side wall. However, for highly turbulent flows, this occurs within a thin viscous boundary layer where the velocity changes very rapidly. At the outer edge of this layer, where the flow becomes properly turbulent, the velocity may be significant. Since we can only hope to apply our model outside of this laminar boundary layer, it may be appropriate to consider a non-vanishing slip velocity at the side walls. Therefore, we will not simply assume the no-slip condition at the side walls, but will, instead, consider a range of possible boundary conditions.

Although we cannot determine the exact distribution of the stress along the side walls, we can find its mean value, $\langle \tau \rangle _w$, from the momentum balance, (2.1). The flux of momentum into the side wall, $F(W/2)$, must be expended by the friction at the wall, so that $F(W/2) = D \langle \tau \rangle _w$. We can, therefore, express the mean side wall stress as

(5.1)\begin{equation} \langle \tau \rangle_w = \frac{F(W/2)}{D} , \end{equation}

which, combined with (4.3) and the fact the bottom is flat ($D' = 0$), yields $\langle \tau \rangle _w = -\chi D \tau '(W/2)$. We use this average stress on the side walls to define the boundary condition at the corner. In particular, we assume that the bottom stress at the corner is some fraction, $\theta$, of the mean side wall stress,

(5.2)\begin{equation} \tau({\pm} W/2) = \theta \langle \tau \rangle_w.\end{equation}

This, therefore, defines a mixed boundary condition, wherein the stress at the corner is related to its first derivative, $\tau (W/2) = -\theta \chi D \tau '(W/2)$. We treat $\theta$ as an unknown, adjustable parameter of the model.

The parameter $\alpha$, which controls the importance of topography-induced flux of momentum, becomes irrelevant in a rectangular channel since $D' = 0$ everywhere on the bottom. Therefore, in a rectangular channel the stress is determined by parameters $\chi$ and $\theta$ rather than $\chi$ and $\alpha$.

The solution of (4.6) in a rectangular channel with the boundary condition (5.2) is

(5.3)\begin{equation} \tau(y) = \rho g S D \left( 1 - \frac{\cosh(y/\lambda)}{\cosh(W/2\lambda)\left(1+\theta \sqrt{\chi} \tanh(W/2\lambda) \right)} \right), \end{equation}

where the diffusion length scale, $\lambda$, is given by

(5.4)\begin{equation} \lambda \equiv D \sqrt{\chi}. \end{equation}

In figure 5 we show how the boundary condition, $\theta$, and the diffusion parameter, $\chi$, affect the stress distribution.

Figure 5. Solutions to (4.6) in the rectangular channel of figure 4 (aspect ratio of $W/D = 5$). Different coloured lines stand for different values of $\chi$ (the ratio of the diffusion length to channel width, $\lambda /W$, for these solutions varies between $0.11$ and $0.6$). The black dashed line stands for the shallow-water theory ($\chi =0$). (a) The stress vanishes at the corner, $\tau (y = \pm W/2) = 0$ ($\theta =0$). The dark blue line ($\chi = 1/3$ and $\theta = 0$) corresponds to the laminar flow. (b) The stress at the corner equals the mean stress on the side wall, $\tau (y = \pm W/2) = \langle \tau \rangle _w$ ($\theta =1$).

The values $\theta = 0$ (figure 5a) and $\theta = 1$ (figure 5b) have a physical interpretation. When $\theta = 0$, the stress at the corner vanishes, $\tau (\pm W/2) = 0$, implying that the depth-averaged velocity in our model also vanishes there, $U(\pm W/2) = 0$ (4.1). This is consistent with a no-slip condition at the side wall ($u(z) = 0$ along the entire side wall). On the other hand, $\theta = 1$ means that the stress at the corner equals the mean stress on the side wall. This has a simple physical interpretation in the limit of large diffusion of momentum, $\chi \rightarrow \infty$ – in this limit, $\theta = 1$ means that the stress is distributed uniformly throughout the channel, including on the side walls. For all other values, $\theta \neq 1$, the side walls are not treated equally to the flat bottom when diffusion is high. The fact that the values $\theta = 0$ and $\theta = 1$ are physically significant is the reason we defined the boundary condition through (5.2).

The length scale $\lambda$ determines how far away from the side walls the flow can feel their presence. More than several $\lambda$ away from the side walls, the stress approaches the shallow-water theory, $\tau _{sw} = \rho g S D$. The stress at the centre becomes significantly different from this value when the width is comparable to the diffusion length, $W \sim \lambda$. In other words, the effect of the side walls propagates throughout the entire channel if the aspect ratio of the channel is comparable to the square root of the diffusion parameter for the stress, $W/D \sim \sqrt {\chi }$. For large values of $\chi$ (such that $\lambda \gg W$), the stress on the bottom becomes uniform.

To appreciate more clearly the role of the boundary condition parameter, $\theta$, we now show how the stress is partitioned between the bottom and the side walls. Moreover, since this partition is often estimated in experiments, this discussion will be useful for comparison with measurements in § 6. In a rectangular channel, momentum is supplied by gravity at a rate of $\rho g S DW$, and it is transferred partly to the side walls and partly to the channel bottom. The fraction, $\varphi _w$, of the total momentum that is transferred to the side walls is, therefore,

(5.5)\begin{equation} \varphi_w = \frac{2\langle \tau \rangle_w}{\rho g S W} , \end{equation}

where $\langle \tau \rangle _w$ is the mean stress on the side walls, and the factor $2$ is due to the fact that there are two side walls. Our model predicts $\varphi _w$ to be ((5.2) and (5.3))

(5.6)\begin{equation} \varphi_w = \frac{2D \sqrt{\chi} }{W} \frac{\tanh \left( \dfrac{W}{2D\sqrt{\chi}} \right) }{1+\theta \sqrt{\chi} \tanh \left( \dfrac{W}{2D\sqrt{\chi}} \right)}. \end{equation}

In figure 6 we show the fraction, $\varphi _w$, as a function of the channel aspect ratio for various values of $\chi$ and boundary conditions defined using $\theta =0$ (figure 6a) and $\theta =1$ (figure 6b). In the shallow-water theory ($\chi \rightarrow 0$), all of the force is carried by the bottom, so $\varphi _w \rightarrow 0$. On the other hand, in the high-diffusion limit ($\chi \rightarrow \infty$), this ratio becomes

(5.7)\begin{equation} \varphi_w = \frac{1}{1+\theta\dfrac{W}{2D}},\quad \chi \rightarrow \infty. \end{equation}

As we mentioned above, when $\theta = 1$, increasing $\chi$ uniformises the stress across the entire channel (including the vertical side walls), so that the momentum is transferred to the bottom and to the side walls in proportion to their length, $\varphi _w = 2D/(2D+W)$. On the other hand, a no-slip boundary condition ($\theta = 0$) requires most of the momentum to be transferred to the side walls, so that when $\theta = 0$ and $\chi \rightarrow \infty$, the side walls carry all of the stress, $\varphi _w = 1$. In this way, the departure from $\theta = 1$ describes the asymmetry between the bottom and the side walls in the limit of high diffusion. Both $\theta = 0$ (no slip at the side wall) and $\theta = 1$ (uniform stress in the high-diffusion limit) may seem intuitive and we show in § 6 that experimental data are not unanimous about this.

Figure 6. The fraction of the stress, $\varphi _w$, carried by the side walls of a rectangular channel (5.6) as a function of the channel's aspect ratio, $W/D$, for the case of (a) $\theta = 0$ and (b) $\theta = 1$. Different colours stand for different values of $\chi$. The dark blue line in panel (a) ($\chi = 1/3$ and $\theta = 0$) corresponds to laminar flow. The dotted lines stand for the high-diffusion limit ($\chi \rightarrow \infty$, (5.7)).

One of the most easily measured quantities is the total water discharge, $Q$, which we can find in our model as $Q = \int U D \,\text {d}y$. The velocity, $U$, that follows from (5.3) does not allow us to write this integral in a simple, closed form, so we have to solve it numerically. In figure 7 we show the discharge, $Q$, normalised by the discharge in the shallow-water theory of (1.1), $Q_{sw} = WD\sqrt {gSD/C_f}$, for $\theta = 0$ and $\theta = 1$ and various values of $\chi$ as a function of the channel's aspect ratio. The effect of increasing the diffusion parameter, $\chi$, is quite different for different boundary conditions, $\theta = 0$ and $\theta = 1$. When $\theta = 0$, increasing the diffusion parameter decreases the discharge so that in the limit $\chi \rightarrow \infty$ there is no flow (figure 7a). On the other hand, when $\theta = 1$, increasing $\chi$ brings the discharge closer and closer to the limit in which the velocity $U = \sqrt {gSR_h/C_f}$ is uniform across the entire channel. In this limit, the discharge is

(5.8)\begin{equation} \frac{Q}{Q_{sw}} = \sqrt{\frac{W}{2D+W}},\quad (\theta=1, \ \chi \rightarrow \infty) . \end{equation}

Therefore, when $\theta =1$, the discharge curves look alike for any diffusion parameter beyond $\chi \sim 1$ (figure 7b). The transfer of momentum makes the discharge, $Q$, significantly smaller than the discharge in the shallow-water theory, $Q_{sw}$, for channels with a small aspect ratio, while, for a large aspect ratio, the shallow-water theory becomes accurate.

Figure 7. The water discharge, $Q$, normalised by the shallow-water discharge, $Q_{sw} = WD\sqrt {gSD/C_f}$, in rectangular channels as a function of the channel's aspect ratio, $W/D$, in the case of (a) $\theta = 0$, and (b) $\theta = 1$. Different colours stand for different values of $\chi$. The black dashed line corresponds to the shallow-water theory ($\chi = 0$), while the black dotted line corresponds to the high-diffusion limit ($\chi \rightarrow \infty$, (5.8)).

Since $\alpha$ does not play a role in a rectangular channel, our model predicts that turbulent and laminar flows lead to the same stress distribution of (5.3) (assuming the same values of $\chi$ and $\theta$). In a laminar flow, the velocity must vanish all along the vertical side wall, so the appropriate condition is $\theta = 0$ (no slip), while the diffusion parameter is $\chi = 1/3$ (§ 4.1). The dark blue lines in figures 5(a) and 6(a) show the stress profile and the fraction of the stress carried by the side wall in a laminar flow. Since the relationship between the stress and the velocity is different for laminar and turbulent flows, the discharge, $Q$, cannot be directly compared between these two cases.

5.2. Wavy bottom

In the rectangular channel of the previous section, the stress distribution was independent of the local-shape parameter, $\alpha$ – this parameter only becomes important in channels with variable depth. Here we show that when $\alpha$ is finite, the diffusion of momentum can make the stress higher in shallower parts of the flow. In this way, momentum diffusion can invert one of the most basic predictions of the shallow-water theory, according to which the stress is always greater where the fluid is deeper.

To examine the role of $\alpha$, we consider an infinitely wide channel with a wavy bottom (figure 8a):

(5.9)\begin{equation} D = D_0 + \delta \sin(k y) .\end{equation}

Here $D_0$ is the mean depth, while $k$ and $\delta$ are the wavenumber and the magnitude of the depth perturbation. To first order in small $\delta / D_0$, the solution to our model (4.4) is

(5.10)\begin{equation} \tau = \rho g S D_0 \left[ 1 + \frac{1-2\alpha \lambda^2 k^2}{1 + \lambda^2 k^2 } \frac{\delta}{D_0} \sin(k y) \right] , \end{equation}

where $\lambda \equiv D_0 \sqrt {\chi }$ is the diffusion length scale that we already encountered in the case of rectangular channels (5.4). In figure 8 we show the solutions of (5.10) for $\alpha = 0$ and $\alpha = 1$ and several values of $\chi$.

Figure 8. (a) An infinitely wide channel with a wavy bottom (5.9), with $\delta / D_0 = 0.1$ and $D_0 k = 1/\sqrt {2}$. (b) Stress distribution of (5.10) with $\alpha = 0$. Different coloured lines stand for different values of $\chi$. The black dashed line stands for the shallow-water theory, $\tau (y) = \rho g S D(y)$ ($\chi = 0$), while the black dotted line stands for the high-diffusion limit ($\chi \rightarrow \infty$). (c) Stress distribution of (5.10) with $\alpha = 1$. The dark blue curve corresponds to the laminar flow ($\chi = 1/3$ and $\alpha = 1$). The value $\chi = 1$ is the critical value beyond which the stress distribution inverts.

The effect of momentum diffusion depends on the ratio of the diffusion length scale, $\lambda$, to the wavelength of the perturbation. When this ratio is small, $\lambda k \ll 1$, we retrieve the shallow-water theory of (1.1). If the wavelength of the perturbation is comparable to the diffusion length scale, the effect of momentum diffusion depends on the value of $\alpha$. When $\alpha = 0$ (figure 8b), momentum diffusion simply reduces the variability of the stress across the channel, so that in the limit of infinite diffusion, $\lambda k \rightarrow \infty$, the stress is uniformly spread across the channel and equal to $\rho g S D_0$. Thus, the stress for $\alpha = 0$ is greatest in the deepest parts of the flow, regardless of the strength of momentum diffusion. Conversely, when $\alpha = 1$ (figure 8c), there exists a critical bed wavelength below which the stress distribution becomes inverted – the stress becomes highest where the fluid is shallowest. This happens when

(5.11)\begin{equation} D_0 k > \frac{1}{\sqrt{2 \alpha \chi}} . \end{equation}

This inversion can occur for any $\alpha > 0$. The limit of infinite diffusion ($\lambda k \rightarrow \infty$) in this case does not correspond to a uniformly distributed stress across the channel (black dotted line in figure 8c). Instead, it becomes uniform for a finite strength of momentum diffusion, when (5.11) is an equality.

We can understand why this inversion happens by considering the flux written in a compact form, $F = -\chi D^{2(1-\alpha )} ( \tau D^{2\alpha } )'$ (4.3). The flux tends to homogenise the quantity under the derivative, $\tau D^{2 \alpha }$. In the limit of large diffusion, the quantity under the derivative becomes nearly uniform throughout the channel, $\tau D^{2 \alpha } \rightarrow \mathrm {const.}$, so that the stress scales as $\tau \propto D^{-2 \alpha }$. Therefore, when momentum diffusion is strong enough, the stress is inversely proportional to depth for positive values of $\alpha$. Relating the stress to velocity according to (4.1), this means that when $\alpha > 0$, the fluid may become faster in shallower parts of the flow. Such an inversion of the flow velocity should be readily apparent in an experiment.

The question of this stress inversion is important, since this mechanism may be responsible for setting the instability length scale for braiding rivers (Abramian, Devauchelle & Lajeunesse Reference Abramian, Devauchelle and Lajeunesse2019). This inversion is known to happen in the laminar flow (Devauchelle et al. Reference Devauchelle, Popović and Lajeunesse2022), but it is unclear whether it also happens in turbulent flows. Strictly speaking, our model is only valid when the depth varies slowly, so that its predictions become questionable when $\lambda k \sim 1$. Unfortunately, we were not able to find appropriate experiments in the literature to test this inversion.

5.3. Triangular channel

In the previous section we showed that for positive values of $\alpha$, the stress may become inverted, increasing where the flow is shallower. Here we show that the same mechanism can make the stress in our model diverge at the channel's banks, where the bed intersects the free surface. This divergence suggests that our model fails in such cases, and should therefore be used with care.

We now solve (4.6) in a triangular channel (figure 9a). Although there exist analytical solutions to our model in this configuration, the algebra is too cumbersome to show here. We leave this derivation for Appendix C, and, here, we only discuss the results for $\alpha = 0$ and $\alpha = 1$.

Figure 9. (a) A triangular channel with a bank angle of $\phi \approx 27^\circ$. (b,c) Dimensionless stress ($\tilde {\tau }$) profiles in our model for different values of $\chi$ (different coloured lines) and $\alpha$. The black dashed line is the shallow-water theory ($\chi = 0$). Panel (b) corresponds to $\alpha = 0$, while panel (c) corresponds to $\alpha = 1$. The dark blue line in panel (c) ($\chi = 1/3$ and $\alpha = 1$) corresponds to the laminar flow in our model, while the red dotted line corresponds to the unapproximated laminar flow. The largest value of $\chi$ shown ($\chi = 20/9$) corresponds to the highest value for which the stress in our model does not diverge in the corners of the above channel when $\alpha = 1$ (5.12).

The dimensionless stress profiles, $\tilde {\tau }(\tilde {y})$, are shown in figure 9(b,c). The solutions for $\alpha = 0$ and $\alpha = 1$ look quite different. When $\alpha = 0$, the stress is smooth and reaches a maximum at the centre of the channel. As $\chi$ increases, it simply becomes more evenly distributed across the channel. On the other hand, when $\alpha = 1$, the stress reaches a maximum somewhere off-centre and has a kink in the centre. As $\chi$ increases in this case, the maximum of the stress moves more and more towards the corners where the flow depth vanishes. When $\chi$ reaches a critical value, the stress reaches its maximum at the corners, decreasing linearly towards the centre (light blue line in figure 9c). For even greater values of $\chi$, the stress diverges at the corners, and the theory, therefore, fails. In Appendix C we show that this happens when

(5.12)\begin{equation} \left(\frac{D_{max}}{W}\right)^2 > \frac{ 1 + \sqrt{1 + (4\alpha \chi)^2} }{ 2 (4\alpha \chi)^2} , \end{equation}

where $D_{max}$ is the channel depth at its centre and $W$ is the total channel width. This criterion is analogous to (5.11), and the divergence occurs for the same reason as the stress inversion we discussed in § 5.2 – when $\alpha > 0$, a strong diffusion of momentum will make the stress diverge with depth as $\tau \sim D^{-2\alpha }$. Equation (5.12) shows that for any $\alpha > 0$, a steep enough bank angle will cause our theory to fail. Since this divergence does not happen for vanishing $\alpha$, setting $\alpha = 0$ may be a useful practical choice for modelling purposes.

Laminar flow offers a benchmark for our model since it can be solved without approximation by solving the Poisson equation, $\nabla ^2 u = - g S / \nu$ (C.1), and analogies with this equation were sometimes used to estimate the stress in turbulent flows (figure 7.6 in Chow Reference Chow1959). The divergence of the stress at the banks we discussed does not exist in the Poisson equation. Still, in the triangular channel of figure 9, the solution to the Poisson equation closely matches our model with $\chi = 1/3$ and $\alpha =1$, except near the sharp corner at the centre (red dotted line in figure 9c), where the symmetries of the Poisson equation require the stress to rapidly vanish (as in all sharp corners). Although our model is not well suited to treat the flow around such sharp corners, it nonetheless makes a sharp dip at the centre, correctly capturing the trend of the exact solution. We can conclude that even when the bed topography varies quickly, our model predictions should not be disregarded altogether, although one should be careful when using them.

5.4. Variable roughness

So far, we assumed that the friction coefficient, $C_f$, is constant. However, when the roughness of the bed varies across the stream, the diffusion of momentum induces a phenomenon that does not exist in the shallow-water theory – the stress concentrates on the rough parts of the channel (figure 10a).

Figure 10. Solutions to (5.13) on a flat bed with variable roughness – the left half ($\tilde {y} < 0$) is smooth, while the right half ($\tilde {y} > 0$) is rough (brown line in panel a). We are assuming that $C_f$ adjusts instantaneously to bed roughness, so that it is constant in two halves of the channel, with $C_{f, {rough}}/C_{f, {smooth}} = 5$. We are assuming a constant diffusion parameter, $\varLambda$. Different blue lines correspond to different values of $\varLambda$ (the diffusion length scale on the rough side varies between $\lambda / D \approx 0.25$ for the dark blue curve and $\lambda / D \approx 2.5$ for the light blue curve). The black dashed line corresponds to the shallow-water theory. (a) Dimensionless stress, $\tau / \rho g S D$. (b) Dimensionless velocity, $U/\sqrt {gSD}$.

In the shallow-water theory (1.1), the stress at each point simply balances the input of momentum by gravity in the overlying fluid column, so that variations of the channel roughness do not affect the stress. On the other hand, the roughness affects the velocity through the friction coefficient, $C_f$, so that the flow is slower over the rough parts of the channel (4.1). Therefore, a gradient of velocity develops from the smooth to the rough parts, even in the shallow-water theory. A flux of momentum, $F$, will thus develop from the smooth to the rough parts of the channel, increasing the stress over the rough and decreasing it over the smooth parts.

Although the friction coefficient, $C_f$, is in reality determined by the details of the flow, for simplicity, here we assume that it is simply a function of the local bed roughness. Accordingly, when the roughness changes discontinuously, the stress on the bottom can also become discontinuous. However, the velocity, $U$, will remain continuous. To emphasise the fact that we are looking for a continuous velocity when the roughness varies across the stream, we solve our model in terms of $U$,

(5.13) \begin{equation} \big( \varLambda C_f^{1/2} \big( D^2 (U^2)' + \alpha U^2 (D^2)' \big) \big)' - C_f U^2 (1+D'^2)^{1/2} + gSD = 0 , \end{equation}

where we keep the diffusion parameter, $\varLambda$ (3.7), inside the derivative, since it may also vary. We note that, when the friction coefficient, $C_f$, varies across the stream, it is not possible to collect $\varLambda$ and $C_f$ into a single parameter, $\chi$, by rewriting the momentum balance in terms of the stress, $\tau$ (even if $\varLambda$ is constant).

In figure 10 we show a solution to (5.13) on a flat bed on which the roughness suddenly changes (assuming a constant diffusion parameter, $\varLambda$). The stress develops a discontinuity at the point where the smooth and the rough parts meet (figure 10a). Over the rough part, there is an excess of stress compared with the shallow-water theory, while over the smooth part there is a deficit of stress. In both regions, away from the discontinuity, the stress relaxes exponentially to the shallow-water theory, $\tau _{sw} = \rho g S D$, over the diffusion length scale $\lambda = D\sqrt {\varLambda C_f^{-1/2}}$ (the same one as in (5.4)). The magnitude of the discontinuity does not depend on the value of the diffusion parameter, $\varLambda$ – around the junction, any amount of momentum diffusion will uniformise the velocity, thereby creating a jump in the stress on the rough part and a dip in the smooth part. The ratio of the stresses on two sides of the junction equals the ratio of the friction coefficients, $\tau _{rough}/\tau _{smooth} = C_{f, {rough}}/C_{f, {smooth}}$. The excess of stress in the rough region creates an excess of force, $\Delta \mathcal {F} = \int _0^{\infty } (\tau - \tau _{sw})\,\text {d}y =\Delta \tau \lambda$, where $\Delta \tau = \tau _{rough} - \tau _{sw}$ is the deviation from the shallow-water theory at the junction point. This excess of force is proportional to the diffusion length scale, $\lambda$. Strong cross-stream diffusion of momentum, therefore, makes the flow pull significantly more on the rough parts of the channel, an effect that the shallow-water theory cannot account for.

The velocity, $U$, continuously decreases from the smooth to the rough region over the same length scale, $\lambda$ (figure 10b). The gradient in velocity, therefore, creates a flux of momentum from the smooth region to the rough one. In the shallow-water theory, this decrease in $U$ is discontinuous.

6.1. The Seine river

We begin by comparing (5.3) with velocity measurements of Chauvet et al. (Reference Chauvet, Devauchelle, Métivier, Lajeunesse and Limare2014) in the Seine river in Paris, France (figure 11a). In order to make the river navigable throughout the year, the banks of the Seine were paved into vertical walls where it crosses Paris. The river flows over bedrock, which is virtually flat. Therefore, the channel of the Seine is nearly rectangular with an aspect ratio of about $23.9$. We give the details about the geometry of the Seine at the time and the location of the measurements in Appendix D.1.

Figure 11. Comparison with measurements in rectangular channels. The red dots represent measurements. The blue lines represent our model with the default values of model parameters from table 2 ($\chi = \varLambda C_f^{-1/2}$ with $\varLambda = 0.3$; since the bottom is flat, the value of the local-shape parameter, $\alpha$, is irrelevant). The blue shaded area corresponds to the uncertainty in $\varLambda$ and $\theta$ from table 2. The black dashed lines represent the shallow-water theory. (a) Measurements of the depth-averaged downstream velocity, $U$, in the Seine river by Chauvet et al. (Reference Chauvet, Devauchelle, Métivier, Lajeunesse and Limare2014). We show our solution with $C_f =4\times 10^{-3}$ and the default $\theta = 0$ (table 2). (b) Measurements by Knight et al. (Reference Knight, Demetriou and Hamed1984) of the stress, $\tau\!$, in a rectangular flume with an aspect ratio of $7.73$. We estimate $C_f \approx 2.8\times 10^{-3}$ from measurements of channel-averaged velocity and stress, and use the default $\theta = 0.8$ (table 2).

In order to find the velocity, $U$, in our model we need to specify three parameters – the friction coefficient, $C_f$, the diffusion parameter for the stress, $\chi$, and the boundary condition, $\theta$. Allowing all these parameters to adjust, we find the best fit for $C_f \approx 4.0\pm 0.2\times 10^{-3}$, $\chi = 6.6 \pm 2.5$ ($\varLambda = 0.42\pm 0.15$) and $\theta \approx -0.03 \pm 0.03$ (we estimate the uncertainty by bootstrapping). Our model with these best-fit parameters matches closely the measurements – the largest error is about $3\,\%$ of the mean velocity. In figure 11(a) we show that our model with the default parameters of table 2 ($\chi = \varLambda C_f^{-1/2}$, $C_f \approx 4\times 10^{-3}$, $\varLambda = 0.3$ and $\theta = 0$) also reproduces the data well – maximum error is about $11\,\%$ of the mean velocity.

Consistent with our discussion above, the measurements on the Seine suggest a $\varLambda$ of order one. The best-fit value of $C_f$ is also reasonable – Chézy (Reference Chézy1775) estimated it to be about $C_f \approx 5\times 10^{-3}$, while the Colebrook equation (4.2) yields our best-fit value assuming an equivalent sand roughness, $k_s$, of the order of several centimetres, a sensible value for the Seine. The measurements above suggest a no-slip condition at the side walls, which, in our model, translates to a vanishing stress at the corners ($\theta \approx 0$).

6.2. Rectangular flumes

Next, we turn our attention to the experiments of Knight et al. (Reference Knight, Demetriou and Hamed1984) who measured the distribution of the stress across smooth rectangular channels with varying aspect ratios and Reynolds numbers. We provide more details about their experiments in Appendix D.2.

In figure 11(b) we show an example of their measurements of the stress, $\tau\!$, in a single channel, and compare them to our model. Since the stress rather than velocity is measured, we do not need to specify the friction coefficient, $C_f$ (§ 4) – our model for $\tau$ in a rectangular channel is fully determined by specifying $\chi$ and $\theta$. Nevertheless, in order to use our scaling, $\chi = \varLambda C_f^{-1/2}$, we estimate $C_f$ using the reported measurements of the channel-averaged velocity, $\langle U \rangle$, and the mean stress, $\langle \tau \rangle$, as $C_f = \langle \tau \rangle / \rho \langle U \rangle ^2$. For the experiment in figure 11(b), this yields $C_f \approx 2.8\times 10^{-3}$, a value consistent with an estimate based on the Colebrook equation (4.2) assuming a smooth channel with a Reynolds number ${Re} = \langle U \rangle D / \nu$. Allowing $\varLambda$ and $\theta$ to adjust, we find the best fit with $\varLambda = 0.22 \pm 0.05$ and $\theta = 0.73 \pm 0.1$ (for other experiments reported in Knight et al. (Reference Knight, Demetriou and Hamed1984), the best-fit $\varLambda$ ranges from $0.07 \pm 0.01$ to $0.22 \pm 0.05$, and the best-fit $\theta$ from $0.7\pm 0.08$ to $0.85\pm 0.15$). These best-fit solutions reproduce the measurements very well – the error is at most $9\,\%$ of the average stress (for other experiments, it is between $5\,\%$ and $9\,\%$). In figure 11(b) we show that our default solution ($\varLambda = 0.3$ and $\theta = 0.8$) also captures the data well (maximum error is about $16\,\%$ of the mean stress).

Knight et al. (Reference Knight, Demetriou and Hamed1984) reported the fraction of momentum transferred to the side walls, $\varphi _w$, for each of their 25 experiments and compared it with 35 other experiments performed in smooth rectangular flumes in studies by Cruff (Reference Cruff1965), Ghosh & Roy (Reference Ghosh and Roy1970), Kartha & Leutheusser (Reference Kartha and Leutheusser1970), Knight & Macdonald (Reference Knight and Macdonald1979), Myers (Reference Myers1978) and Noutsopoulos & Hadjipanos (Reference Noutsopoulos and Hadjipanos1982). The fraction $\varphi _w$ for all of these studies approximately falls onto a single curve as a function of the channel's aspect ratio (figure 12a). Allowing $\varLambda$ and $\theta$ to vary, we find the best fit with $\varLambda \approx 0.14 \pm 0.07$ and $\theta \approx 0.88 \pm 0.1$, which reproduces the data well (the mean error is about $2\,\%$). The blue line in figure 12(a) shows that our model with default parameters ($\varLambda = 0.3$ and $\theta = 0.8$) also falls close to the measurements (the mean error is about $4\,\%$). Moreover, the limit of uniform stress across the channel ($\chi \rightarrow \infty$ and $\theta = 1$, (5.7)) also explains the data well (black dotted line in figure 12a). In fact, the data for $\varphi _w$ are consistent with our model for any $\theta \sim 1$, and $\chi$ greater than about $\chi \sim 1$.

Figure 12. (a) The fraction of momentum transferred to channel side walls, $\varphi _w$ (5.6), and (b) the normalised water discharge, $Q / Q_{sw}$, as a function of the aspect ratio. We assume the friction coefficient $C_f = 2.8\times 10^{-3}$, as in figure 11(b). The blue lines correspond to our model with default parameters of table 2 ($\varLambda = 0.3$ and $\theta = 0.8$), while the blue shaded area corresponds to the uncertainty in these parameters. The red dots in panel (a) represent a collection of 60 experiments by various authors reported in Knight et al. (Reference Knight, Demetriou and Hamed1984), while the red dots in panel (b) represent 25 experiments performed by Knight et al. (Reference Knight, Demetriou and Hamed1984) themselves. The black dotted lines correspond to a uniform stress across the entire channel ($\chi \rightarrow \infty$ and $\theta =1$, (5.7) and (5.8)). The horizontal black dashed line in panel (b) corresponds to the shallow-water prediction, $Q=Q_{sw}$.

Although we find good agreement between predicted and measured stress in all of the experiments, measuring stress directly is difficult. For this reason, we also test our model against the measurements of water discharge, a quantity that is easy to measure. In figure 12(b) we show that normalising the water discharge, $Q$, by its value in the shallow-water theory, $Q_{sw} = U_{sw} DW$, where $U_{sw} = \sqrt {gSD/C_f}$, collapses all of the data points of Knight et al. (Reference Knight, Demetriou and Hamed1984) onto a single curve as a function of the channel aspect ratio. Here, we assumed that $C_f = 2.8\times 10^{-3}$, the same as in the experiment of figure 11(b). Allowing $\varLambda$ and $\theta$ to vary, we find the best fit with $\varLambda = 0.06 \pm 0.03$ and $\theta = 0.8 \pm 0.1$ that reproduces the data with at most $7\,\%$ error. Figure 12(b) shows that our default model ($\varLambda = 0.3$ and $\theta = 0.8$) also reproduces the data well (maximum error is $11\,\%$). Again, the data are consistent with the limit of uniform stress ($\chi \rightarrow \infty$ and $\theta = 1$, (5.8)), so that our model with any $\theta \sim 1$ and $\chi \gtrsim 1$ is compatible with the measurements. Unlike $\varphi _w$, which is a statement about the stress, the water discharge is a statement about velocities. Figure 12(b), therefore, suggests that the velocity is close to uniform across the channel.

Although $\varLambda$ in these experiments is smaller than in the Seine (the best fits range from $0.07$ to $0.22$), it is still of order one, and our default value ($\varLambda = 0.3$) still explains the data well. However, all of the experiments above strongly suggest a boundary condition specified through $\theta$ of order one, in stark contrast with the no-slip condition we found for the Seine (§ 6.1). At this point, it is unclear why this difference exists.

6.3. Floodplain

The experiments presented above did not allow us to constrain the local-shape parameter, $\alpha$. To do this, we now consider the flow in a channel with variable depth across the stream. We are unaware of any experiments in the literature that were performed in channels with slowly varying bottom topography and no sharp corners, which would conform with our model's assumptions, and would, therefore, be ideal for this study. However, many previous studies considered the flow in the so-called floodplain geometry (figure 13a). Such a configuration was designed to mimic the flow in rivers during flooding, and consists of a deep and relatively narrow main channel joined with a wide and shallow ‘floodplain’. Although many models were developed that accurately estimate the stage-discharge relations in this geometry, i.e. how the water level varies with total water discharge (Bousmar & Zech Reference Bousmar and Zech1999; Martin-Vide & Moreta Reference Martin-Vide and Moreta2008; Proust et al. Reference Proust, Bousmar, Riviere, Paquier and Zech2009; Kaddi et al. Reference Kaddi, Cierco, Faure and Proust2022), they typically paid little attention to the detailed distribution of the stress across the banks between the main channel and the floodplain. Yet, in this geometry, our parameter $\alpha$ only plays a direct role on these banks, so, unlike most previous studies, we pay special attention to this transitional region between the main channel and the floodplain.

Figure 13. (a) Cross-section of one half of the symmetric ‘floodplain’ channel used by Shiono & Knight (Reference Shiono and Knight1991) with the depths of the main channel and the floodplain $D_{mc} \approx 19.8\ \text {cm}$ and $D_{fp} \approx 4.8\ \text{cm}$. (b) Measurements (red circles) of the stress, $\tau\!$, in the channel above. The solid blue line is the solution to (5.13) assuming $\alpha = 0$, $\varLambda = 0.3$ and $C_f$ estimated from the Colebrook equation using an approximated Reynolds number ${Re}(y) = U_{sw}(y) D(y) / \nu$, where $U_{sw}(y) = \sqrt {g S D(y) / C_{f, {def}}}$ and $C_{f, {def}} = 2\times 10^{-3}$. The blue shading corresponds to uncertainty in $\varLambda$ from table 2. The blue dotted line corresponds to $\alpha = 0.3$ and $\varLambda = 0.3$. The black dashed line is the shallow-water prediction. (c) The depth-averaged velocity, $U$, with the same notation as in panel (b). (d) Estimates of the friction coefficient, $C_f$. The red circles are estimates using the stress and velocity measurements, $C_f(y) = \tau (y)/\rho U^2(y)$. The black dotted line is the estimate based on the Colebrook equation (4.2) assuming a smooth channel and a local Reynolds number, ${Re}(y) = U(y)D(y)/\nu$. This equation predicts a higher friction coefficient in the floodplain, where the Reynolds number is lower, and a divergence near the right corner of the channel where the Reynolds number vanishes.

Here we consider the experiments of Shiono & Knight (Reference Shiono and Knight1991) in a symmetric floodplain channel (figure 13a). The water level changed from one experiment to the next, so that the flow depth in the main channel, $D_{mc}$, and in the floodplain, $D_{fp}$, varied. The main channel and the floodplain were connected by banks of unit slope ($D' = \pm 1$), and the floodplain was connected to the water surface by banks of the same slope. In this channel, Shiono & Knight (Reference Shiono and Knight1991) measured both the stress, $\tau (y)$ (figure 13b), and the depth-averaged velocity, $U(y)$ (figure 13c), using independent methods (Appendix D.3). We note that bank slopes of $D' = \pm 1$ are not really small so that we consider these experiments only because we lack a more appropriate alternative.

We start by considering the friction coefficient, $C_f$. Since the Reynolds number is larger in the deep main channel than in the shallow floodplain, $C_f$ is lower in the main channel than in the floodplain. We can estimate the friction coefficient at each point across the stream directly from the measurements of the stress and velocity as $C_f(y) = \tau (y)/\rho U^2(y)$ (figure 13d). We find that $C_f \approx 2\times 10^{-3}$ in the main channel for all experiments, while it varies in the floodplain from one experiment to the other. This variability of $C_f$ across the stream is well captured by the Colebrook equation (4.2) assuming a smooth channel ($k_s/D = 0$), and a local Reynolds number, ${Re}(y) = U(y)D(y)/\nu$ (black dotted line in figure 13d).

To compute the stress, $\tau\!$, and the velocity, $U$, from our model when the friction coefficient varies across the stream, we need to solve (5.13). We supply our model with the friction coefficient $C_f(y)$ estimated using the Colebrook equation with a local Reynolds number ${Re}(y) = U_{sw}(y) D(y) / \nu$, where $U_{sw}(y) = \sqrt {g S D(y) / C_{f, {def}}}$ is the shallow-water velocity and $C_{f, {def}} = 2\times 10^{-3}$ is a default friction coefficient that we take to be that of the main channel. This rough estimate of the Reynolds number is sufficient, since $C_f$ in the Colebrook equation depends only weakly on the Reynolds number – using twice the value of $C_{f, {def}}$, the Colebrook equation yields only about $9\,\%$ higher $C_f$ in the floodplain, which corresponds to a floodplain velocity only about $4\,\%$ smaller (and similarly in the main channel). Finally, we assume that the parameters $\varLambda$ and $\alpha$ are constant. Therefore, besides the depth profile, $D(y)$, our model in this case requires three parameters – $C_{f, {def}}$, $\varLambda$ and $\alpha$.

The solutions to our model here are not as accurate as in the rectangular channels of the previous two sections. For this reason, it is not clear how to define the best fit to the data – we may accurately reproduce the data in the main channel, in the floodplain or in the transition region, but not all three at the same time (focusing only on the main channel yields the best-fit $\varLambda$ from $\varLambda \approx 0.48$ to $\varLambda \approx 1.05$ depending on the experiment, regardless of $\alpha$). Therefore, we do not attempt to show the best-fit solutions, and, instead, we simply compare our model with the default diffusion parameter ($\varLambda = 0.3$) and two different values of $\alpha$: $\alpha = 0$ and $\alpha = 0.3$ (figure 13b,c).

In the main channel and the floodplain, the solutions with $\alpha = 0$ and $\alpha = 0.3$ behave very similarly and reproduce the data equally well. The two solutions differ significantly only in the transition region. They both show a peak in the stress that is not observed in the data, but this peak is more pronounced for $\alpha = 0.3$. When $\alpha = 0$, this peak is caused by the rapidly changing friction coefficient in the transition region, while for $\alpha > 0$, it is caused by the bed topography itself, and would remain even for a constant friction coefficient. Using a negative $\alpha$ could remove this peak of the stress, but would, at the same time, make the stress change more abruptly from the main channel to the floodplain, unlike what we see in the data. The solution with $\alpha = 0$ shows no peak in the velocity, consistent with the measurements, while $\alpha = 0.3$ still shows a small peak. For these reasons, our model conforms with the data only for small (positive or negative) values of $\alpha$, so, for simplicity, we set $\alpha = 0$ as our default parameter.

In the floodplain, our solution quickly converges to the shallow-water prediction, while the measured stress and velocity remain about $20\,\%$ above it – a modest difference that our model cannot account for. This discrepancy can either be the result of a slight bias in the measurements, or, as suggested by Shiono & Knight (Reference Shiono and Knight1991), caused by a real feature of the flow such as a wide recirculation cell that spans the floodplain or large eddies that exist in the horizontal $x$–$z$ plane. These features, which are normally suppressed by the bottom friction in rivers, may play a significant role in smooth channels such as this one.

6.4. Variable roughness

Finally, we test the predictions of § 5.4 for a bed with variable roughness by comparing our model with the experiments of Blanckaert et al. (Reference Blanckaert, Duarte and Schleiss2010). They performed high-precision measurements of velocity in half-trapezoidal channels (figure 14a) in which the bottom of the channel had a fixed roughness while the right, sloping bank was either smoother, as rough, or rougher than the bottom. They report $C_f \approx 5.3 \times 10^{-3}$ for the bottom based on direct measurements of stress and velocity, and we use the Colebrook equation to estimate $C_f \approx 2.4\times 10^{-3}$ for the deepest part of the smooth bank and $C_f \approx 1.9\times 10^{-2}$ for the deepest part of the rough bank (on the sloping bank, $C_f$ increases as the bank becomes shallower). We give the details of these experiments in Appendix D.4.

Figure 14. (a) A channel of Blanckaert et al. (Reference Blanckaert, Duarte and Schleiss2010) with a bank angle of $30^\circ$. Banks were coated with different material so that the bank roughness was less than, equal to or greater than the roughness of the flat bottom in different experiments. The friction coefficient is $C_f \approx 5.3\times 10^{-3}$ on the bottom, while it varies on the banks. (b) Measurements (red circles) of the stress, $\tau\!$, in the channel above with smooth banks. The blue line is the solution to (5.13) assuming default parameters $\varLambda = 0.3$, $\alpha = 0$ and $\theta = 0.8$ on the left vertical side wall, and $C_f$ estimated using the Colebrook equation. The blue shading corresponds to the uncertainty in $\varLambda$ and $\theta$ (table 2). The black dashed line corresponds to the shallow-water prediction. The downstream slope is $S \approx 5.5\times 10^{-4}$. (b) A rough bank experiment. The notation is the same as above. The downstream slope is $S \approx 7.6\times 10^{-4}$.

We show the estimates of Blanckaert et al. (Reference Blanckaert, Duarte and Schleiss2010) for the stress in figure 14(b,c). As predicted in § 5.4, the stress develops a discontinuity at the junction between the bed and the banks. If the bank is smoother than the flat bottom, the stress on the bank suddenly decreases, while a rough bank makes the stress exceptionally high on the bank, but, also, exceptionally low on the bottom near the junction point. We solve (5.13) using the default parameters ($\varLambda = 0.3$, $\alpha = 0$ and $\theta = 0.8$ on the left vertical side wall), and using the Colebrook equation (4.2) to estimate $C_f(y)$ as a function of the local Reynolds number approximated as ${Re}(y) = U_{sw}D(y)/\nu$ (where $U_{sw} = \sqrt {gSD(y)/C_{f, {def}}}$ and $C_{f, {def}} = 5.3 \times 10^{-3}$), and the relative roughness, $k_s(y)/R_h$, where $k_s(y)$ assumes different values on the bottom and the banks. This solution reproduces the measurements qualitatively well – it captures both the drop in the stress in the case of the smooth bank and its jump in the case of the rough bank. The value of $\varLambda$ does not affect the magnitude of the jump in the stress – it only affects the shape of the stress curves. We find the best fit with $\varLambda \approx 0.18$ for the smooth bank experiment and $\varLambda \approx 0.31$ for the rough bank experiment, close to our default value.

Even though our model reproduces the measurements qualitatively well, there are still significant deviations. The measurements show strong variability around our solutions on scales comparable to the channel depth. This variability is of the same order as the deviation of our theory from the shallow-water prediction, and is, therefore, not negligible. It is likely that this is due to strong recirculation cells. This is an effect that we cannot hope to capture with our model on the scale of individual cells – our model can only give the stress smoothed out over several such cells.

6.5. Summary of model parameters

In summary, we find that our model reproduces most of the observations. The friction coefficient, $C_f$, is well captured by the Colebrook equation (4.2). The best-fit $\varLambda$ varied from one experiment to another but was always of order one. The lowest best-fit value we found was $\varLambda = 0.07$ for the rectangular flume of Knight et al. (Reference Knight, Demetriou and Hamed1984) with the smallest aspect ratio ($W/D \approx 2$), while the largest was $\varLambda \approx 1.05$ in the main channel of the floodplain experiment of Shiono & Knight (Reference Shiono and Knight1991) with the smallest flow depth ($D_{mc} \approx 17\ \text {cm}$). We chose our default value ($\varLambda = 0.3$) to be between these two extremes. Together with our estimates of $C_f$, this yields the diffusion parameter for the stress, $\chi = \varLambda C_f^{-1/2}$, between $\chi \approx 1.3$ and $\chi \approx 23.4$. The values of $\varLambda$ we found here are compatible with previous estimates – Okoye (Reference Okoye1970) tracked tracer dispersion in various experiments and field studies, and estimated the effective eddy diffusivity, $\mathcal {D}_e$, to be between $\mathcal {D}_e \approx 0.07 U^*D$ and $\mathcal {D}_e \approx 2.0 U^*D$. These measurements imply that our parameter $\varLambda = \mathcal {D}_e / 2 U^*D$ lies in the range between $[0.03, 1.0]$, consistent with our present estimates.

Experimental evidence suggests that $\alpha$ is small – significant positive values of $\alpha$ lead to peaks in the stress that are not observed in the floodplain experiments of Shiono & Knight (Reference Shiono and Knight1991), while significant negative values lead to an abrupt transition of the stress from the main channel to the floodplain which is, again, inconsistent with measurements. Theoretical arguments also suggest a small $\alpha$ for highly turbulent flows – the model of turbulent diffusion of Appendix A suggests that $\alpha$ decays logarithmically with the Reynolds number, as the velocity profile starts to resemble a vertically uniform ‘plug’ flow. For these reasons, we suggest the default value $\alpha = 0$ with a tentative uncertainty range around zero, $\alpha \in (-0.1,0.1)$. Nevertheless, we note that care should be taken when dealing with flows at lower Reynolds numbers that do not resemble a ‘plug’ flow – approaching the laminar regime, the values of $\alpha$ should approach $\alpha = 1$.

In rectangular channels, it is necessary to prescribe a boundary condition, $\theta$, at the vertical side wall. Measurements on the Seine suggest a no-slip boundary condition ($\theta \approx 0$) and do not support $\theta$ greater than $\theta \sim 0.1$. On the other hand, rectangular flume experiments suggest $\theta$ of order one. Depending on the experiment, the best-fit values range from $\theta \approx 0.6$ to $\theta \approx 1$, and we take the mid-value ($\theta = 0.8$) to be our default. We currently do not understand what causes this significant difference between the natural channel of the Seine and the experimental flumes.

Using ${Re} = 10^5$ and $C_f = 10^{-3}$ in the model of turbulent diffusion of Appendix A, we find $\varLambda \approx 0.03$ and $\alpha \approx 0.17$ (A11a,b). The value of $\varLambda$ in this model is lower than our observed values, while the value of $\alpha$ is too large to be supported by the data. This suggests that, although the model of turbulent diffusion captures many qualitative features of turbulent momentum transfer, it is not quantitatively accurate. We note that $\varLambda \approx 0.03$ corresponds to the lowest value measured by Okoye (Reference Okoye1970), who hypothesised that this is the theoretical lower bound.

6.6. Validity of model assumptions in the experiments

We derived our model under four assumptions: (i) the transfer of momentum across the stream does not depend on the gravity, $g$, viscosity, $\nu$, and the bed roughness, $k_s$; (ii) the vertical profile of velocity does not change shape across the stream; (iii) the flux of momentum, $F$, is locally related to the flow depth and velocity; and (iv) the depth and velocity vary slowly across the stream. None of these assumptions were exactly satisfied in the experiments we considered.

The flux in our model depends on the Reynolds number and the bed roughness through the friction coefficient, although it does so only weakly. This violates the first assumption of our model, and we partly took it into account by allowing $C_f$ to vary according to the Colebrook equation.

Chauvet (Reference Chauvet2014) measured the vertical profiles of velocity at different locations in the Seine. Based on their measurements, we estimate that the contribution to the flux, $F$, due to the changing shape of the vertical velocity profile is likely more than an order of magnitude smaller than the contribution due to the velocity gradient, $U \tilde {\sigma }_U' / U' \sim 0.04$, where $\tilde {\sigma }_U = \sigma _U / U$ is the dimensionless standard deviation of the velocity profile that characterises its shape. This is consistent with our second assumption above, but we note that variations of the profile's shape likely become significant near the corners where the bed intersects the free surface, especially when $\alpha > 0$ and our model diverges (§ 5.3).

The non-local contributions to the flux generated two notable features in the data that we could not reproduce: (i) the large fluctuations of the stress in the experiments of Blanckaert et al. (Reference Blanckaert, Duarte and Schleiss2010) on the scale of the flow depth, probably related to recirculation cells; and (ii) the slight but consistent underestimate of the stress in the floodplain region in the experiments of Shiono & Knight (Reference Shiono and Knight1991), possibly caused by large, horizontal eddies or recirculation cells. These features contributed to about $20\,\%$ error in our model, but they may become weaker in natural, irregular channels with a rough bed (Blanckaert et al. Reference Blanckaert, Duarte and Schleiss2010).

Finally, none of the experiments we considered really had a slowly varying topography throughout the channel – they all contained sharp corners or steep banks. The fact that the experiments of Shiono & Knight (Reference Shiono and Knight1991) had banks with $D' = 1$, means that the higher-order terms in the flux (for example, those proportional to $D'^2$) are of a similar order of magnitude to the one we considered. This could explain why our model produced a small peak of the stress between the main channel and the floodplain, which was not observed by Shiono & Knight (Reference Shiono and Knight1991).

Despite these inconsistencies, most of the data fell within the uncertainty bounds of our model, while our default parameters (table 2) reproduced the data to within about $30\,\%$. Our model always provided a significant improvement to the shallow-water approximation. The data we considered cover about three orders of magnitude in Reynolds number (from about $10^4$ for some rectangular flumes to about $7\times 10^6$ for the Seine), and include channels of varying shape with a smooth or rough bed, as well as those with variable roughness. These reasons suggest that the model we presented here is reasonable.