2.1. Criterion for laminar flow

The criterion adopted here to verify the laminar flow follows the empirical method proposed by Darby (Reference Darby1986) and subsequently applied by Ng & Mei (Reference Ng and Mei1994). The flow is considered laminar if the wide-channel Reynolds number, defined as

(2.7) \begin{equation} \textit{Re} = \frac {\rho V^{2-n} Y^{n}}{K}, \end{equation}

where $\rho$ is the density of the fluid, remaining below the critical threshold

(2.8) \begin{equation} \textit{Re}_c = 0.125 \left (\frac {3n+1}{2n}\right )^n \left [2100 + 875 (1-n)\right ]. \end{equation}

3.1. Analytical solution based on mass and momentum balance

In the geometry presented in § 2 where the channel axis undergoes an abrupt deflection, the flow dynamics can be described by a system of three governing equations (Levi Reference Levi1965). The first is the continuity equation, written per unit width and projected normal to the wavefront, reading as

(3.1) \begin{equation} \int _0^{Y_1}u_1(z)\sin (\beta )\,\text{d}z=\int _0^{Y_2}u_2(z)\sin (\beta -\theta )\,\text{d}z. \end{equation}

The second and third equations derive from the momentum balance expressed along normal and parallel directions to the wavefront, respectively, assuming a hydrostatic pressure distribution $p=\rho g (Y-z)$ ; these two projections can be written as

(3.2) \begin{equation} \int _0^{Y_2}\rho u_2^2(z)\sin ^2(\beta -\theta )\,\text{d}z-\int _0^{Y_1}\rho u_1^2(z)\sin ^2(\beta )\,\text{d}z=\int _0^{Y_1}p_1\,\text{d}z-\int _0^{Y_2}p_2\,\text{d}z, \end{equation}

(3.3) \begin{equation} \int _0^{Y_1}u_1(z)\frac {1}{Y_1}\cos (\beta )\,\text{d}z=\int _0^{Y_2}u_2(z)\frac {1}{Y_2}\cos (\beta -\theta )\,\text{d}z. \end{equation}

It is convenient to define the following dimensionless ratios: $\zeta =V_2/V_1$ and $\eta =Y_2/Y_1$ . By integrating the velocity profile given (2.2) the system of (3.1)–(3.3) can be solved. This yields a transcendental system of three equations that determines the unknown set $(Y_2,V_2,\beta )$ , given the geometric and rheological parameters of the flow:

(3.4) \begin{equation} \left \{ \begin{array}{ll}\!\! \displaystyle \zeta \eta =\frac {\sin (\beta )}{\sin (\beta -\theta )},\\[12pt] \!\! \displaystyle \zeta =\frac {\cos (\beta )}{\cos (\beta -\theta )},\\[12pt] \!\! \displaystyle \frac {1-\eta ^2}{2\textit{Fr}_1^2C_{\!M}}=\zeta ^2 \eta \sin ^2(\beta -\theta )-\sin ^2(\beta ). \end{array} \right . \end{equation}

In Appendix A, the system (3.4) is derived following the standard approach used in gas dynamics for the analysis of oblique shock waves.

Figure 2.

Mach angle $\beta$ as a function of depth ratio $\eta$ for various values of $F_{*1}$ .

From system (3.4) the following equation can be derived:

(3.5) \begin{equation} \sin ^2(\beta )=\frac {\eta ^2+\eta }{2\textit{Fr}_1^2C_{\!M}}=\frac {\eta ^2+\eta }{2F_{*1}^2}. \end{equation}

The plot of the relationship $\eta -\beta$ in figure 2 reveals that at high values of $F_{*1}$ , the depth ratio $\eta$ increases rapidly with the Mach angle $\beta$ . In contrast, for values of $F_{*1}$ closer to unity, the increase in $\eta$ with $\beta$ is more gradual.

Equation (3.5) is a quadratic expression in $\eta$ , which is solved by considering only the positive root, since $\eta$ must be non-negative, i.e.

(3.6) \begin{equation} \eta =\frac {1}{2}\sqrt {1+8\textit{Fr}_1^2C_{\!M}\sin ^2(\beta )}-\frac {1}{2}. \end{equation}

Furthermore, (3.1) and (3.3) can be combined as follows:

(3.7) \begin{equation} \eta =\frac {\tan (\beta )}{\tan (\beta -\theta )}. \end{equation}

Figure 3.

Mach angle $\beta$ as a function of deflection angle $\theta$ for various values of $F_{*1}$ ; the dashed line at $F_{*2}=1$ delineates the boundary between tranquil (subcritical) and rapid (supercritical) flow downstream; the dot-dashed line connects the maxima $\theta _{\textit{max}}$ and identifies the point of detachment.

Substituting (3.6) into (3.7) yields a valuable relationship between the Mach angle $\beta$ and the known flow conditions, channel geometry and fluid rheology:

(3.8) \begin{equation} \tan (\theta )=\frac {\tan (\beta )\left (3-\sqrt {1+8\textit{Fr}_1^2C_{\!M}\sin ^2(\beta )}\right )}{1-2\tan ^2(\beta )-\sqrt {1+8\textit{Fr}_1^2C_{\!M}\sin ^2(\beta )}}. \end{equation}

Figure 3 illustrates the relationship between the deflection angle  $\theta$ and the Mach angle $\beta$ for various values of $F_{*1}$ upstream; the curve of $\beta$ versus $\theta$ for given $F_{*1}$ increases, reaches a maximum and decreases again; physically, this means that for known deflection angle $\theta$ and Froude number upstream, there are two possible values of $\beta$ , one corresponding to a weak shock, the other to a strong shock. Note that by combining the continuity equation with the momentum equation projected normal to the shock (i.e. the first and third equations in (3.4)), and rearranging the resulting relations, one can express them in terms of the downstream Froude number $\textit{Fr}_2$ as

(3.9) \begin{equation} \sin ^2(\beta -\theta )=\frac {1+\eta }{2\eta ^2\textit{Fr}_2^2C_{\!M}}, \end{equation}

which provides a convenient framework for analysing the downstream flow regime and to plot the critical curve $\textit{Fr}_2 = \textit{Fr}_c$ in figure 3. As shown in the figure, for a given deflection angle $\theta$ there are generally two possible values of $\beta$ : one corresponds to a supercritical flow downstream and the other to a subcritical flow downstream; the coexistence of the latter with the supercritical flow upstream gives rise to a hydraulic jump.

Another feature, already pointed out by Ippen (Reference Ippen1943) for inviscid flows and clearly illustrated in figure 3, is that for each value of $\textit{Fr}_1$ upstream the $\theta$ -curve possesses a maximum $\theta _{\textit{max}}$ . When, for a given $\textit{Fr}_1$ , the deflection angle $\theta$ exceeds this maximum, the shock is no longer attached to the point of wall deflection but propagates upstream, producing what is referred to as a detached shock. In this case, the disturbance takes the form of a curved perturbation wave. The location of these maxima can be determined by setting the derivative of $\theta$ with respect to $\beta$ to zero. To this end, (3.8) can be rewritten as

(3.10) \begin{equation} \theta =\beta -\arctan \left [\frac {1+\sqrt {1+8\textit{Fr}_1^2C_{\!M}\sin ^2(\beta )}}{2\textit{Fr}_1^2C_{\!M}\sin (2\beta )}\right ]\!. \end{equation}

Differentiating this relation with respect to $\beta$ yields the value $\beta _{\textit{max}}$ corresponding to the maximum deflection angle $\theta _{\textit{max}}$ for a given $\textit{Fr}_1$ . When $\theta$ exceeds this value, shock detachment occurs. The resulting expression leads to a quartic equation in $\sin ^2 (\beta _{\textit{max}} )$ :

(3.11) \begin{align} 16\textit{Fr}_1^6C_{\!M}^3\sin ^8(\beta _{\textit{max}})-&\big (32\textit{Fr}_1^6C_{\!M}^3+6\textit{Fr}_1^4C_{\!M}^2\big)\sin ^6(\beta _{\textit{max}})\nonumber \\ &\,+\big(16\textit{Fr}_1^6C_{\!M}^3-12\textit{Fr}_1^4C_{\!M}^2\big)\sin ^4(\beta _{\textit{max}}) \nonumber \\ &\,+\big (10\textit{Fr}_1^4C_{\!M}^2+5\textit{Fr}_1^2C_{\!M}\big )\sin ^2(\beta _{\textit{max}})+1+2\textit{Fr}_1^2C_{\!M}=0. \end{align}

The above equation allows us, for a given value of the upstream Froude number $\textit{Fr}_1$ , to implicitly calculate $\beta _{\textit{max}}$ and determine whether the deflection is such as to induce the detachment. The curve connecting the maxima $\theta _{\textit{max}}$ , obtained by plotting $\theta _{\textit{max}}$ as a function of $\beta _{\textit{max}}$ according to (3.10), is depicted in figure 3, and can be considered as the criterion for the onset of detachment.

Furthermore, the momentum equation in the streamwise direction (3.7) can be rearranged to express it explicitly in terms of both Froude numbers as

(3.12) \begin{equation} \frac {\textit{Fr}_2^2}{\textit{Fr}_1^2}=\frac {\tan (\beta -\theta )}{\tan (\beta )} \left [\frac {1+\tan ^2(\beta -\theta )}{1+\tan ^2(\beta )}\right ]\!. \end{equation}

By imposing $\textit{Fr}_2 = \textit{Fr}_c = 1/\sqrt {C_{\!M}}$ and replacing $\theta$ using the (3.10), the resulting expression can be manipulated to yield following the cubic equation in $\sin ^2(\beta _{\textit{crit}})$ :

(3.13) \begin{align} 4\textit{Fr}_1^6C_{\!M}^3\sin ^6(\beta _{\textit{crit}})-\big (8\textit{Fr}_1^6C_{\!M}^3+16\textit{Fr}_1^4C_{\!M}^2\big)\sin ^4(\beta _{\textit{crit}})\nonumber \\ +\big (8\textit{Fr}_1^4C_{\!M}^2+6\textit{Fr}_1^2C_{\!M}+4\textit{Fr}_1^6C_{\!M}^3\big )\sin ^2(\beta _{\textit{crit}})+\textit{Fr}_1^2C_{\!M}+1=0. \end{align}

Solving this equation yields the value of $\beta _{\textit{crit}}$ associated with the critical flow condition. It is worth noting that, setting $C_{\!M}=1$ in (3.11) and (3.13), the expressions reduce to those derived by Gray & Cui (Reference Gray and Cui2007) for a granular stream. Substituting $\beta _{\textit{crit}}$ into (3.8) directly yields the critical condition, indicating whether the flow passes from rapid to tranquil and produces a hydraulic jump.

In summary, the general validity of the system (3.4) enables the solution for a power-law fluid to be obtained by analogy with the classical case of a vertically uniform velocity profile (Ippen Reference Ippen1936), i.e. $C_{\!M}=1$ , provided that the Froude number is scaled by its corresponding critical value (the $F_{*1}$ ratio).

Consequently, the analogy between oblique shocks in rapid free-surface flows, generated by an abrupt channel deflection, and those in gas dynamics, produced when a uniform gas stream encounters an inclined surface, extends naturally to power-law currents (Gray & Cui Reference Gray and Cui2007). Based on the preceding analysis, three types of shock may arise in a steady rapid current of a power-law fluid when the channel is suddenly deflected. For sufficiently large upstream Froude numbers and small deflection angles $\theta$ , two shock angles $\beta$ are possible (see figure 3): the larger angle corresponds to a strong shock, in which the downstream flow becomes subcritical ( $\textit{Fr}_2\lt \textit{Fr}_c$ ) and a hydraulic jump forms, while the smaller angle corresponds to a weak shock, in which the downstream flow remains supercritical ( $\textit{Fr}_2\gt \textit{Fr}_c$ ) and only a wavefront connects two distinct rapid-flow states. The onset of the more energetically dissipative strong shock is determined by downstream boundary conditions, which force the flow to become subcritical.

Furthermore, when the deflection angle exceeds the maximum value $\theta _{\textit{max}}$ identified above, or when the upstream Froude number is not sufficiently large, an oblique shock solution to system (3.4) ceases to exist. Under these conditions, the shock detaches and takes a curved form (Ippen Reference Ippen1943). In turbulent flows of clear-water or granular free-surface flows, the occurrence of weak, strong and detached shocks depends solely on $\textit{Fr}_1$ and $\theta$ . By contrast, for laminar power-law fluids, rheological properties also influence the type of shock, since the governing parameters are the pair $(F_{*1}, \theta )$ , where $F_{*1}$ depends on the Boussinesq coefficient $C_{\!M}$ and, consequently, on the fluid behaviour index $n$ .

3.2. Head loss

The specific head $(E)$ at any cross-section is defined as the sum of the flow depth $(Y)$ and a kinetic energy term corrected for the non-uniform velocity distribution by the Coriolis coefficient $\alpha$ , also known as the kinetic energy correction factor. For a power-law fluid, $\alpha$ is given by

(3.14) \begin{equation} \alpha = \frac {\displaystyle \int _A u^3 \, \mathrm{d}A}{A \left ( \frac {1}{A} \int _A u \, \mathrm{d}A \right )^3} = \frac {6(2n+1)^2}{(3n+2)(4n+3)}, \end{equation}

where $u$ is the velocity, $A$ is the cross-sectional area and $n$ is the flow behaviour index.

Using this coefficient, the specific head is expressed as

(3.15) \begin{equation} E = Y + \alpha \frac {V^2}{2g}. \end{equation}

Hydraulic jump is a dissipative phenomenon characterized by localized energy loss, which can be quantified as the difference in the specific head between the upstream and downstream sections (Ugarelli & Di Federico Reference Ugarelli and Di Federico2007). The specific head loss $\Delta E$ (equal to the hydraulic head loss $\Delta H$ for a horizontal channel) is thus

(3.16) \begin{equation} \Delta E = E_1 - E_2 = Y_1 - Y_2 + \alpha \frac {V_1^2}{2g} - \alpha \frac {V_2^2}{2g}, \end{equation}

where subscripts 1 and 2 denote the upstream and downstream conditions, respectively.

Substituting the downstream velocity $V_2$ from the continuity equation and using the relationship given in (3.5), $\Delta E$ can be rewritten as

(3.17) \begin{equation} \Delta E = Y_1 (1 - \eta ) + \alpha Y_1 \textit{Fr}_1^2 \left [ 1 - \frac {\eta + 1}{2 \eta \textit{Fr}_1^2 C_{\!M}} \frac {1}{\sin ^2 \left ( \arcsin \sqrt {\frac {\eta ^2 + \eta }{2 \textit{Fr}_1^2 C_{\!M}}} - \theta \right )} \right ]\!. \end{equation}

After some algebraic manipulations, (3.17) can be reformulated in a different form by eliminating the arcsine function to yield

(3.18) \begin{equation} \Delta E=Y_1(1-\eta )+\alpha Y_1 \textit{Fr}_1^2 \left [ 1-\frac {\eta +1}{2\eta \textit{Fr}_1^2C_{\!M}}\left (\cos (\theta )\sqrt {\frac {\eta ^2+\eta }{2\textit{Fr}_1^2C_{\!M}}}-\sin (\theta ) \sqrt {1-\frac {\eta ^2+\eta }{2\textit{Fr}_1^2C_{\!M}}} \right )^{-2}\right ]\!. \end{equation}

Therefore, it is evident that head loss cannot be determined solely from upstream conditions; the downstream flow state also plays a role. Specifically, head loss $\Delta E$ depends on the upstream Froude number $\textit{Fr}_1$ and flow depth $Y_1$ , the fluid rheology characterized by the Boussinesq and Coriolis coefficients $C_{\!M}$ and $\alpha$ , as well as on the downstream flow condition expressed by the fluid depth ratio $\eta$ . Moreover, the geometry of the channel influences the loss of head through the deflection angle $\theta$ .

In § 4, (3.18) will be used to calculate the head loss with experimental parameters. It will be evident that the case of a strong shock is more energetically dissipative than the weak one.

4.1. Results of the experiments

Table 1 provides an overview of the 44 experiments performed.

Table 1.

Parameters of the experimental tests. The symbols $n$ , $K$ and $\rho$ denote the rheological index, consistency index and density of the fluid; $\theta$ is for the channel deflection angle; $\textit{Fr}_1$ and $\textit{Re}_1$ are for the upstream Froude and Reynolds numbers; ${F}_{*1}$ and ${F}_{*2, th}$ are for the upstream and (theoretical) downstream normalized Froude number; $\textit{Re}_{2,{th}}$ is for the theoretical downstream Reynolds number; $\beta _{{exp}}$ and $\beta _{\textit{th}}$ are for the experimental and theoretical shock angles; and $\eta _{{exp}}$ and $\eta _{\textit{th}}$ are for the experimental and theoretical ratios between downstream and upstream flow depth. The critical Reynolds number is $\textit{Re}_c=488$ for shear-thinning fluids, $525$ for Newtonian fluids and $526$ for shear-thickening fluids. The channel bed slope was $3.4^\circ$ for shear-thinning and Newtonian fluids, and $15^\circ$ for shear-thickening fluids. The gate opening height was $h_G=5\,\mathrm{mm}$ for shear-thinning and Newtonian fluids, and $h_G=3\,\mathrm{mm}$ for shear-thickening fluids.

Table 2.

Theoretical head loss $\Delta E$ for weak (letter a) and strong (letter b) shocks, calculated via (3.18) using the experimental parameters from cases 5, 26 and 34 from table 1.

Three types of fluids were considered: (i) a strongly shear-thinning fluid with flow behaviour index $n\lt 1$ ; (ii) a Newtonian fluid with $n=1$ ; (iii) a shear-thickening fluid with $n\gt 1$ . Preliminary rheological tests on the shear-thinning mixtures indicated that their parameters remained nearly constant over time, confirming the good repeatability of the experiments. For Newtonian and shear-thickening fluids, two channel-deflection angles, $\theta =15^\circ$ and $25^\circ$ , were tested, while for shear-thinning fluids four angles were used, ranging from $15^\circ$ to $34^\circ$ . Changing the flow rate $Q$ , the upstream Froude number $\textit{Fr}_1$ was established between 1.72 and 9.15, corresponding to the upstream Reynolds numbers $\textit{Re}_1$ (calculated from (2.8) in § 2.1) in the range of 12–374. Figure 6 shows, for each fluid and for the two angles $\theta =15^\circ$ and $25^\circ$ , the experiment with the largest upstream Froude number.

An interesting application of (3.18) for the head loss is presented in table 2. For some selected experimental scenarios – only the weak shock case was experimentally obtained – the theoretical values of $\eta$ and $\beta$ for both weak and strong shock were calculated; obviously, the Mach angle and the downstream depth would have been higher in the latter case. As expected, the value of $\Delta E$ is much higher – almost an order of magnitude – for strong than for weak shocks; in fact, only in the latter case a proper hydraulic jump, a strongly dissipative phenomenon, takes place.

Figure 6.

Snapshots from six representative experiments performed with three different fluids at wall deflection angles of $15^\circ$ and $25^\circ$ , illustrating the resulting hydraulic jump configurations. (a) Test 1, shear-thinning fluid, $\textit{Fr}_1=6.50$ ; (b) Test 12, shear-thinning fluid, $\textit{Fr}_1=9.15$ ; (c) Test 22, Newtonian fluid, $\textit{Fr}_1=6.29$ ; (d) Test 28, Newtonian fluid, $\textit{Fr}_1=5.23$ ; (e) Test 30, shear-thickening fluid, $\textit{Fr}_1=5.14$ ; ( f) Test 37, shear-thickening fluid, $\textit{Fr}_1=5.18$ . The dashed lines represent the intersection between the bottom of the channel and the walls.

The uncertainty in the rheological parameters $ n$ and $ K$ arises mainly from three sources: the non-viscometric character of the flow conditions during rheometric testing; the inherent approximation involved in fitting the fluid behaviour to a power-law model; fluctuations in temperature throughout the measurement process. These combined factors produce relative uncertainties estimated at $\Delta n / n \leqslant 4\,\%$ and $\Delta K / K \leqslant 8\,\%$ . Fluid density measurements were conducted with high precision; however, minor variations in temperature contribute to a conservative uncertainty estimate of $\Delta \rho / \rho \leqslant 0.2\,\%$ .

The free-surface depth measurements upstream ( $Y_1$ ) and downstream ( $Y_2$ ) of the hydraulic jump were subject to experimental errors arising from measurement resolution and surface disturbances, with relative uncertainties quantified as $\Delta Y_1 / Y_1 \leqslant 7\,\%$ and $\Delta Y_2 / Y_2 \leqslant 3\,\%$ , respectively. These combine to yield an uncertainty on the experimentally determined depth ratio $\eta _{\textit{exp}} = Y_2 / Y_1$ of up to $\Delta \eta _{\textit{exp}} / \eta _{\textit{exp}} \leqslant 10\,\%$ .

Discharge $Q$ was measured using a calibrated flow meter, resulting in a relative uncertainty $\Delta Q / Q \leqslant 1\,\%$ . An additional systematic error is associated with the channel width $b$ , estimated to be within $\pm 1\, \mathrm{mm}$ (that is, $\Delta b / b \leqslant 1\,\%$ ), which propagates into the calculation of the upstream Froude number $\textit{Fr}_1$ , contributing to the uncertainty $\Delta \textit{Fr}_1 / \textit{Fr}_1 \leqslant 13\,\%$ . The propagation of uncertainty for a function of several parameters and variables is obtained through a first-order Taylor series expansion, in which the variances of all independent arguments are linearly combined. Further details can be found in Longo et al. (Reference Longo, Di Federico, Archetti, Chiapponi, Ciriello and Ungarish2013).

To estimate the Mach angle $\beta _{\textit{exp}}$ , the images were processed to extract the pixels corresponding to the jump front using a colour-based thresholding approach. A straight line was then fitted to these pixels, and the procedure was repeated over several tens of frames, with the resulting angles subsequently averaged. This method yields a result comparable to drawing a best-fit line by eye. Altering the colour threshold modifies the detected front primarily through translation and, to a much lesser extent, rotation. The uncertainty derived from this procedure therefore reflects the intrinsic variability within the ensemble of measured shock angles. The estimated uncertainty of $\Delta \beta _{\textit{exp}} / \beta _{\textit{exp}} \leqslant 5\,\%$ , accounts for the uncertainty described previously and for the resolution limits of the images.

To estimate the uncertainty in the theoretical depth ratio $\eta _{\textit{th}}$ , a Monte Carlo simulation was performed. In this approach, the input parameters – namely the upstream Froude number $\textit{Fr}_1$ , the power-law exponent $n$ and the channel deflection angle $\theta$ – were treated as independent random variables, each assumed to follow a normal distribution centred on its nominal value and with a standard deviation corresponding to its respective uncertainty. For the deflection angle, a standard deviation of $\Delta \theta = 0.1^{\circ }$ was assumed. By propagating these uncertainties through the governing equations, it was found that the resulting relative uncertainty in $\eta _{\textit{th}}$ is $\Delta \eta _{\textit{th}} / \eta _{\textit{th}} \leqslant 11\,\%$ .

In addition, the uncertainty in the Reynolds number $\textit{Re}_1$ was evaluated, which yielded a relative error estimate of $\Delta \textit{Re}_1 / \textit{Re}_1 \leqslant 21\,\%$ .

Figure 7 compares the theoretical and experimental Mach angles in varying Froude numbers for shear-thinning, Newtonian and shear-thickening fluids. The overall agreement is satisfactory, with the experimental error bars generally encompassing the theoretical predictions. Nevertheless, for Newtonian fluids – and, to a lesser degree, shear-thickening fluids – the theoretical Mach angles consistently overpredict the values observed experimentally.

Figure 7.

Comparison of the experimental Mach angle. (a) Shear-thinning fluid, (b) Newtonian fluid and (c) Shear-thickening fluid. Symbols are the experiments, curves are the theoretical values. Error bars refer to $\pm 1$ standard deviation.

Figure 8 presents a comparison between the experimental and theoretical values of $\eta$ for the cases where the downstream depth $Y_2$ was measurable. The agreement is particularly good at higher $\eta$ values, corresponding to larger $\textit{Fr}_1$ . At lower $\textit{Fr}_1$ , accurate measurement of $Y_2$ is challenging due to the surface undulations associated with the hydraulic jump. Other techniques could be used to measure $Y_2$ , which will form part of our future activities on this topic. Given that the experimental activity was planned with a balance of resources and objectives in mind, and with the measurement of angle $\beta$ in view, our focus was on achieving this objective.

Discrepancies between the theoretical and experimental values of $\eta$ become pronounced again at very high Froude numbers, where the flow forms a jet impinging on the channel wall. This generates a highly curved flow structure that deviates significantly from the hydrostatic pressure distribution assumed in the theoretical model (see figure 9). Extending the length of the inclined wall and increasing the channel width may facilitate the development of a flow field that is more consistent with the theoretical assumptions, potentially improving the agreement between predictions and observations. The range of parameter combinations explored in the experiments listed in table 1 is considered adequate for validating the proposed models. Some limitations on the maximum attainable Froude number arise from the nature of the fluids used. In particular, shear-thinning fluids experience rapid degradation of the polymer chains responsible for their pseudoplastic behaviour when subjected to very high shear rates, which in our tests correspond to large Froude numbers. We acknowledge, however, that drawing more conclusive insights into the characteristics of the shocks would have benefited from a broader range of test conditions. Nevertheless, the selected experimental conditions span a sufficiently broad range of rheological behaviours and flow regimes – from subcritical to strongly supercritical – thus ensuring a meaningful validation of the models under diverse hydraulic conditions.

Figure 8.

Comparison between experimental measurements and theoretical predictions of the depth ratio $\eta$ . Error bars refer to $\pm 1$ standard deviation.

Figure 9.

The jump evolves in a jet-like manner at very high Froude numbers. Test 17, shear-thinning fluid with $\textit{Fr}_1=6.88$ . Panels (a) and (b) are taken 1/8 of a second apart.

5.1. Governing equations and numerical set-up

We consider for generality the two-dimensional (2-D) depth-averaged mass and momentum conservation equations for a power-law fluid flowing over a gently inclined ( $x,y$ ) plane, reading, respectively, as

(5.1) \begin{equation} \frac {\partial Y}{\partial t}+ \frac {\partial \left (V_x Y\right ) }{\partial x}+ \frac {\partial \left (V_y Y\right ) }{\partial y} \;=\;0, \end{equation}

(5.2) \begin{equation} \frac {\partial \left (V_x Y\right )}{\partial t}+ \frac {\partial \left (C_{\!M}\;V_x V_x Y\right ) }{\partial x}+ \frac {\partial \left (C_{\!M}\;V_x V_y Y \right )}{\partial y}+ \frac {\partial }{\partial x}\left (\frac { g Y^2}{ 2}\right )\;=\; S_{x}, \end{equation}

(5.3) \begin{equation} \frac {\partial \left (V_y Y\right )}{\partial t}+ \frac {\partial \left (C_{\!M}\; V_x V_y Y\right ) }{\partial x}+ \frac {\partial \left (C_{\!M}\; V_y V_y Y\right ) }{\partial y}+ \frac {\partial }{\partial y}\left (\frac {g Y^2}{ 2}\right )=\; S_{y}, \end{equation}

where $t$ denotes time; $V_x$ and $V_y$ are the velocity components averaged by depth in the $x$ and $y$ directions, respectively; $C_{\!M}$ is the Boussinesq coefficient defined in (2.4). The source terms $S_x$ and $S_y$ incorporate the effects of the bottom slope and the basal shear stress, and are given by

(5.4) \begin{equation} S_{x}\;=g\;Y\; S_{0x} -\;\frac {K}{\rho }\left (\frac {2n+1}{n}\frac {1}{Y}\right )^n \left (V_x^2+V_y^2\right )^{\frac {n-1}{2}} V_x, \end{equation}

(5.5) \begin{equation} S_{y}\;= g\;Y\; S_{0y}-\frac {K}{\rho }\left (\frac {2n+1}{n}\frac {1}{Y}\right )^n \left (V_x^2+V_y^2\right )^{\frac {n-1}{2}} V_y, \end{equation}

with $S_{0x}$ and $S_{0y}$ representing the bottom slopes along the $x$ and $y$ directions, respectively.

Equations (5.1)–(5.3) extend the 1-D model of Ng & Mei (Reference Ng and Mei1994) to two dimensions (Greco et al. Reference Greco, Di Cristo, Iervolino and Vacca2019; Yu & Chu Reference Yu and Chu2022), assuming vertical accelerations to be negligible and adopting a hydrostatic distribution of the pressure (Hogg & Pritchard Reference Hogg and Pritchard2004).

The flow model (5.1)–(5.5) belongs to a class of models in the literature often referred to as the Saint–Venant approach, boundary-layer or lubrication approximation, and similar frameworks (Ancey Reference Ancey2007). The primary advantage of the Saint–Venant approach is that it enables the analysis of the flowing layer dynamics without requiring detailed knowledge of the internal flow structure. However, this simplification comes at the cost of losing some information about the finer flow dynamics (Forterre & Pouliquen Reference Forterre and Pouliquen2003). More rigorously derived models, though mathematically more complex, have been proposed particularly to better characterize the stability conditions of sheet flows. For example, focusing on the power-law model and applying the weighted-residual technique (Ruyer-Quil & Manneville Reference Ruyer-Quil and Manneville1998, Reference Ruyer-Quil and Manneville2000), Amaouche, Djema & Bourdache (Reference Amaouche, Djema and Bourdache2009) and Fernandez-Nieto, Noble & Vila (Reference Fernandez-Nieto, Noble and Vila2010) corrected the averaged momentum equation originally derived by Ng & Mei (Reference Ng and Mei1994) and formulated two-equation models consistent up to first order in the shallow-water parameter. Second-order models have been developed by Ruyer-Quil et al. (Reference Ruyer-Quil, Chakraborty and Dandapat2012) and Noble & Vila (Reference Noble and Vila2013). A comprehensive overview of existing models incorporating yield stress, such as those for Herschel–Bulkley fluids, can be found in Denisenko (Reference Denisenko2024) and Muchiri, Monnier & Sellier (Reference Muchiri, Monnier and Sellier2025).

Governing equations (5.1)–(5.3), together with (5.4)–(5.5), are solved numerically using a standard cell-centred finite-volume method. The scheme is first-order accurate in both space and time. Details of the numerical implementation are provided in the Supplementary material.

The experimental tests described in § 4 have been numerically reproduced using a simplified square computational domain. This domain extends from the first measurement point upstream of the jump to the second measurement point downstream, resulting in a length of $L_x = 0.16\,\mathrm{m}$ in the $x$ -direction. The channel deviation begins at $x_d = 0.034\,\mathrm{m}$ . A schematic representation of the computational domain for a deviation angle of $\theta = 15^\circ$ is reported in the Supplementary material.

The bed channel slope values are set as follows: $S_{0x} = 0.06$ , $S_{0y} = 0$ for both shear-thinning and Newtonian fluids; and $S_{0x} = 0.26$ , $S_{0y} = 0$ for shear-thickening fluids.

The boundary conditions were designed to replicate the experimental set-up. At the left boundary of the computational domain, $x=0$ , the supercritical inflow conditions measured, namely the flow depth $Y_0$ and the flow rate per unit width $q_x = q_0=Q/b$ , were uniformly prescribed for all $y$ . Consequently, the Froude number $\textit{Fr}_1$ , as reported in table 1, is imposed along the inflow boundary for all $y$ .

An impermeability condition was applied along the sidewall and the wedge-shaped boundary corresponding to the channel deviation. At the right-hand and top boundaries, no explicit boundary conditions were enforced, assuming the flow remains supercritical as in the experiments in table 1. Therefore, only experimental cases that fulfilled this assumption were simulated.

The number of triangular cells varies with the deviation angle, ranging between 62 783 and 72 788. The time step was fixed at $10^{-4}$ s, which satisfies the Courant–Friedrichs–Lewy stability condition. To assess the quality of the numerical results, preliminary simulations neglecting the source terms were performed for all tests, and the numerical outcomes were compared with the theoretical predictions. The maximum relative differences were found to be 0.9 % and 0.6 % for $\beta$ and $\eta$ , respectively. When including the source terms, computations with a doubled mesh resolution demonstrated that the numerical results are mesh-independent.

5.2. Two-dimensional simulation of selected experimental tests

In this section, we analyse the numerical predictions of the shallow-water model formulated in (5.1)–(5.3), which incorporates bed friction and channel slope – factors present in the experiments but neglected in the simplified 1-D analytical model.

Detailed findings are presented for representative tests from table 1, one for each fluid rheology: shear-thinning, Newtonian and shear-thickening.

The results are presented in dimensionless form using the gate opening height $h_G$ as reference length and $u_R=\sqrt {g h_G}$ as velocity scale, with $h_G=5\,\mathrm{mm}$ for the shear-thinning and Newtonian cases and $h_G=3\,\mathrm{mm}$ for the shear-thickening fluid. Dimensionless quantities are denoted with a superscript hat.

Figure 10.

Contour plots of the $F_*=\textit{Fr}/\textit{Fr}_c$ ratio with streamlines superimposed. Black dashed line, shock front $\hat {y}_{F}^{1-\text{D}}$ ; red dashed line, shock front $\hat {y}_{F}^{1-\text{D}-S}$ . (a) Test 4: shear-thinning fluid ( $\textit{Fr}_1=4.04$ ); (b) Test 26: Newtonian fluid ( $\textit{Fr}_1=3.48$ ); (c) Test 30: shear-thickening fluid ( $\textit{Fr}_1=5.14$ ).

Figures 10(a), 10(b) and 10(c) show, in a colour map, the ratio $F_*$ of the local Froude number to its critical value with streamlines superimposed for Test 4 ( $n=0.39$ , $\textit{Fr}_1=4.04$ ), Test 26 ( $n=1.00$ , $\textit{Fr}_1=4.21$ ) and Test 30 ( $n=1.09$ , $\textit{Fr}_1=5.14$ ) in table 1, respectively. For comparison, figure 10(a–c) include the straight shock front with slope $\beta _{th}$ (see table 1) predicted by the 1-D model, shown as $\hat {y}_{F}^{1-\text{D}}$ (black straight dashed line).

Regardless of the power-law exponent and consistent with the theoretical and experimental findings, figure 10(a–c) show that an oblique shock develops in all tests. Moreover, the flow remains supercritical ( $F_*\gt 1$ ) downstream of the jump, i.e. a weak shock takes place.

However, the numerical shock front exhibits a slight concavity, with a local slope consistently greater than $\beta _{th}$ for all $ x \geqslant x_d$ . Moreover, the streamline pattern downstream of the shock front indicates that the flow becomes approximately parallel to the deviated sidewall, particularly in the case of the shear-thinning fluid. A similar behaviour was reported by Gray & Cui (Reference Gray and Cui2007), who numerically investigated weak oblique shocks in granular flows on the free-surface generated by an abrupt channel deviation. Gray & Cui (Reference Gray and Cui2007) attributed the deformation of the shock front to the bottom friction and the slope of the channel, which determine the development of a gradually varied flow-depth profile along the channel and, consequently, the spatial variability of the Froude number upstream of the shock. This variability in the incoming Froude number along the shock front modifies the local Mach angle at the deviation edge, causing the shock to bend.

Figure 10(a–c) show that, as in granular flows, the Froude number $\textit{Fr}$ decreases markedly in the streamwise ( $x$ ) direction upstream of the shock, indicating that a profile of depth of flow decelerated in the $x$ direction occurs. In contrast, $\textit{Fr}$ remains essentially uniform in the transverse direction ( $y$ ), as a consequence of the boundary conditions imposed at $x=0$ (constant inflow) and at the sidewall $y=0$ (impermeable boundary).

5.3. One-dimensional model with source terms and comparison with the 2-D model

This section introduces a 1-D model, derived from the 2-D formulation, which accounts for the influence of bed slope and friction. The analysis begins with an investigation of the mechanisms responsible for the curvature of the shock line observed in figure 10(a–c) and in some experimental tests. Following the approach of Gray & Cui (Reference Gray and Cui2007), it is assumed that the flow downstream of the shock remains aligned with the deviated sidewall. Furthermore, the source-term contributions (bed slope and friction) to the tangential and normal momentum equations across the shock are neglected. Under these assumptions, condition (3.8) is applied locally along the curved shock front $y_F = y_F(x)$ .

Conversely, when the Froude number is defined approaching the shock front, the contributions of the source terms are retained. In other words, the present 1-D formulation partially accounts for the effects of bed slope and friction. The resulting problem to be solved can thus be written as

(5.6) \begin{equation} \frac {\textrm {d}y_F}{\textrm {d}x}=\tan {\beta (x)}, \end{equation}

with the local Mach angle $\beta (x )$ satisfying the following algebraic equation adapted from (3.8):

(5.7) \begin{equation} \tan {\theta }=\frac {\tan (\beta \left (x\right ))\left (3-\sqrt {1+8\textit{Fr}_u^2\;C_{\!M}\sin ^2(\beta \left (x\right ))}\right )}{1-2\tan ^2(\beta \left (x\right ))-\sqrt {1+8\textit{Fr}_u^2\;C_{\!M}\sin ^2(\beta \left (x\right ))}}, \end{equation}

where $\textit{Fr}_u= \textit{Fr} (x,y_F )$ denotes the Froude number upstream of the shock at the point with coordinates $ (x,y_F )$ .

Being independent of the inflow boundary condition $y$ , the variation of the upstream Froude number in the $x$ direction is obtained by integrating the steady 1-D counterpart (along $x$ ) of system (5.1)–(5.5), which can be expressed solely in terms of flow depth as follows:

(5.8) \begin{equation} \frac {\textrm {d} Y}{\textrm {d}x}=\frac {S_{0x}}{Y^{2n-2}}\frac {Y^{2n+1}-Y_{\!N}^{2n+1}}{Y^3-C_{\!M} \textit{Fr}_{N}^2 Y_{\!N}^3}, \end{equation}

with the initial condition

(5.9) \begin{equation} Y\left (x=0\right )=Y_0. \end{equation}

In (5.8), $Y_{\!N}$ denotes the normal, i.e. uniform, flow depth,

(5.10) \begin{equation} Y_{\!N}=\left [\frac {k}{\rho }\frac {1}{g S_{0x}}\left (\frac {2n+1}{n} q_0\right )^n\right ]^{\frac {1}{2n+1}}\!, \end{equation}

and $\textit{Fr}_{N}$ is the normal Froude number, defined as $\textit{Fr}_{N}=q_0/ (g Y_{\!N}^3 )^{1/2}$ .

Following the procedure suggested by Di Cristo, Iervolino & Vacca (Reference Di Cristo, Iervolino and Vacca2018), the analytical solution of the nonlinear ordinary differential (5.8), subject to the initial condition (5.9), has been derived (see Appendix B) and is given by

(for $n\neq 1/2$ )

(5.11) \begin{align} \begin{aligned} \displaystyle x=& -\frac {Y_{\!N}} {S_{0x}} \left [\frac {1}{2n+2} \left (\frac {Y}{Y_{\!N}}\right )^{2n+2} {\mathcal F}\left (\frac {2n+2}{2n+1}, 1;\frac {4n+3}{2n+1};\left (\frac {Y}{Y_{\!N}}\right )^{2n+1} \right ) \right . \\ &\,-\displaystyle \left .C_{\!M} \textit{Fr}_{N}^{2 \frac {1}{2n-1}} \left (\frac {Y}{Y_{\!N}}\right )^{2n-1} {\mathcal F}\left (\frac {2n-1}{2n+1}, 1;\frac {4n}{2n+1};\left (\frac {Y}{Y_{\!N}}\right )^{2n+1} \right ) \right . \\ &\,-\displaystyle \left .\frac {1}{2n+2} \left (\frac {Y_0}{Y_{\!N}}\right )^{2n+2} {\mathcal F}\left (\frac {2n+2}{2n+1}, 1;\frac {4n+3}{2n+1};\left (\frac {Y_0}{Y_{\!N}}\right )^{2n+1} \right ) \right . \\ &\,+\displaystyle \left .C_{\!M} \textit{Fr}_{N}^{2 \frac {1}{2n-1}} \left (\frac {Y_0}{Y_{\!N}}\right )^{2n-1} {\mathcal F}\left (\frac {2n-1}{2n+1}, 1;\frac {4n}{2n+1};\left (\frac {Y_0}{Y_{\!N}}\right )^{2n+1} \right ) \right ]\!; \end{aligned} \end{align}

(for $n = 1/2$ )

(5.12) \begin{align} \begin{aligned} \displaystyle x=& \frac {Y_{\!N}} {S_{0x}} \left [\frac {Y}{Y_{\!N}} +\frac {1}{2}\ln {\frac {Y-Y_{\!N}}{Y+Y_{\!N}}}-\frac {C_{\!M} \textit{Fr}_{N}^2}{2}\ln {\frac {Y^2-Y_{\!N}^2}{Y^2}} \right . \\ &\,-\displaystyle \left . \frac {Y_0}{Y_{\!N}} -\frac {1}{2}\ln {\frac {Y_0-Y_{\!N}}{Y_0+Y_{\!N}}}+\frac {C_{\!M} \textit{Fr}_{N}^2}{2}\ln {\frac {Y_0^2-Y_{\!N}^2}{Y_0^2}} \right ]\!. \end{aligned} \end{align}

In (5.11), ${\mathcal F} (a,b;c;z )$ denotes the Gauss hypergeometric function, often indicated as $_2F_1 (a,b;c;z )$ .

The numerical solution of system (5.6)–(5.7) is obtained via a standard explicit fourth-order Runge–Kutta scheme, initiated at the point ( $x_d,0$ ). At each $x\geqslant x_d$ and at every Runge–Kutta stage, the flow depth upstream of the shock front $Y_u=Y (x,y_F )$ is determined iteratively by solving (5.11) (or (5.12) when $n=1/2$ ). Consequently, the local Froude number is evaluated as $\textit{Fr}_u=q_0/ (g Y_u^3 )^{1/2}$ . The local Mach angle $\beta (x )$ is then computed by iteratively solving (5.7).

The shock front computed with the 1-D formulation, which incorporates the source terms, is shown in figure 10(a–c) (denoted by $\hat {y}_{F}^{1-\text{D}-S}$ , red dashed line). The close agreement between the shock front locations predicted by the 1-D and 2-D models, as observed in figures 10(a) and 10(b), suggests that the local deceleration of the flow upstream of the front is the primary cause of the reduction of the Mach angle at the deviation edge and of the resultant curvature of the shock front. In contrast, figure 10(c) reveals a somewhat poorer correspondence between the predictions of the 2-D and 1-D models. Taking into account the relatively larger bottom slope in this test ( $15^\circ$ ), compared with the others ( $3.4^\circ$ ), the observed discrepancy is attributed to this significant slope. Specifically, the bottom slope determines a non-negligible weight component in the momentum balance across the shock and additionally reduces the flow parallelism with the deviated sidewall downstream of the shock.

Table 3 reports the value of the Froude number computed at the edge of the deviation $\textit{Fr}_{d}$ , obtained from (5.11), together with the inlet Froude number ( $\textit{Fr}_{1}$ ). In all cases, table 3 clearly shows a reduction in the Froude number at the edge of the deviation compared with the inlet value, the effect being particularly pronounced in tests involving shear-thickening fluids.

Table 3.

Inlet Froude number $\textit{Fr}_{1}$ and Froude number $\textit{Fr}_{d}$ at the deviation edge $ x_d$ , for all simulated tests.

Figure 11.

Contour plots of the $F_*= \textit{Fr}/\textit{Fr}_c$ ratio with streamlines superimposed. Black dashed line, shock front $\hat {y}_{F}^{1-\text{D}}$ ; red dashed line, shock front $\hat {y}_{F}^{1-\text{D}-S}$ . (a) Test 42, shear-thickening fluid ( $\textit{Fr}_1=2.50$ ); (b) Test 27, Newtonian fluid ( $\textit{Fr}_1=2.59$ ).

The reduced value of the Froude number at the deviation edge may lead to the formation of a detached shock. This phenomenon was observed in Tests 34–36 and 40–42, all involving shear-thickening fluids.

As a representative example of this condition, figure 11(a) depicts the colour map of the ratio $F_*$ with superimposed streamlines for Test 42 ( $n=1.11$ , $\textit{Fr}_1=2.50$ ). Figure 11(a) clearly illustrates that the shock is detached and curved. For this case, the maximum deviation angle, calculated from (3.11) and (3.10) using the Froude number value $\textit{Fr}_d=1.40$ (see table 3), is $\theta _{\textit{max}} \sim 11^\circ$ . Since the deviation angle in this test ( $\theta =25^\circ$ ) exceeds $\theta _{\textit{max}}$ the shock must detach. At the detachment point, the shock is normal, i.e. $\beta =90^\circ$ . Moreover, as in the case of detached oblique-shock waves in gas dynamics (see, e.g. Anderson Reference Anderson2019), figure 11(a) shows that subcritical conditions ( $F_*\lt 1$ ) occur immediately downstream of the shock (strong shock). Farther downstream, however, supercritical conditions are re-established ( $F_*\gt 1$ ), indicating that the shock is weak.

To investigate in depth the magnitude of the shock front curvature, the transversal coordinate of the front at the abscissa of the downstream measurement point ( $x_M$ ) predicted by the 2-D model ( $y_{F,x_M}^{2-\text{D}-S}$ ) is compared with the corresponding predictions of the 1-D formulations, both with ( $y_{F,x_M}^{1-\text{D}-S}$ ) and without ( $y_{F,x_M}^{1-\text{D}}$ ) source terms. Tests exhibiting detached shock have been excluded from this comparison. Table 4 reports, in dimensionless form, the computed values of $y_{F,x_M}^{2-\text{D}-S}$ , $y_{F,x_M}^{1-\text{D}-S}$ and $y_{F,x_M}^{1-\text{D}}$ , together with the estimated uncertainty in determining $y_{F,x_M}^{2-\text{D}-S}$ .

Table 4.

Depth of the current at downstream measurement point $x_M$ : results from 2-D model $(\hat {y}_{F,x_M}^{2-\text{D}-S})$ and from the 1-D formulation including $(\hat {y}_{F,x_M}^{1-\text{D}-S})$ and neglecting $(\hat {y}_{F,x_M}^{1-\text{D}})$ the source terms.

Figure 12.

Depth of the current at downstream measurement point $x_M$ predicted by the 2-D model, $\hat {y}_{F,x_M}^{2-\text{D}-S}$ , compared with the results of the 1-D formulation (a) without the source terms, $\hat {y}_{F,x_M}^{1-\text{D}}$ , and (b) including the source terms, $\hat {y}_{F,x_M}^{1-\text{D}-S}$ .

Tests 21, 27 and 32–39 are not included in table 4, since, as in the cases with a detached shock, the reduction of $\textit{Fr}_u$ along the shock front becomes so pronounced that (5.7) no longer admits a solution. Consequently, the integration of (5.6)–(5.7) cannot be continued.

As an illustration of these conditions, figure 11(b) shows the colour map of the ratio $F_*$ , together with the streamlines for Test 27 ( $n=1.0$ , $\textit{Fr}_1=2.59$ ). Superimposed on the colour map are the shock fronts predicted by the 1-D formulation with source terms (red dashed line) and without source terms (black dashed line). Figure 11(b) highlights that the 1-D model with source terms reproduces the position of the shock front with reasonable accuracy only up to $\hat x^*=21.5$ ; for $\hat x\gt \hat x^*$ , it does not capture the correct shock position.

To aid readability and interpretation of table 4, figure 12 compares the dimensionless transverse coordinates of the shock front from the 2-D model, $\hat {y}_{F,x_M}^{2-\text{D}-S}$ , with those predicted by the 1-D model without source terms, $\hat {y}_{F,x_M}^{1-\text{D}}$ (figure 12 a) and with source terms $\hat {y}_{F,x_M}^{1-\text{D}-S}$ (figure 12 b).

Figure 12 shows that for all tests with shear-thinning and Newtonian fluids (bed slope $3.4^\circ$ ) the reduction of the Froude number upstream of the shock, due to the bed slope and bottom friction, explains the difference observed between the predictions of the 1-D and 2-D models. This conclusion does not hold for tests 30 and 31 (shear-thickening fluid, bed slope $15^\circ$ ) in which the gravity-related source term plays a non-negligible role in the momentum balance and in determining the flow direction downstream the shock.

5.4. Effect of bottom stress in the 2-D shallow-water model

A careful inspection of the results of the 2-D shallow-water model shows that including shear stress leads to a concave deformation of the planar front predicted by the corresponding inviscid solution. This behaviour can be explained by noting that with increasing distance from the deflected sidewall, the length of the decelerated upstream flow profiles grows, causing a local reduction in the Froude number upstream of the jump. As illustrated in figure 3, a decrease in the upstream Froude number results in an increase of the Mach angle along the hypercritical branch of the curve. Consequently, the shock front shifts slightly farther away from the sidewall, in qualitative agreement with the larger Mach angles observed in some of the experiments (see figure 7).