Mathematical and numerical modelling of rapid transients at partially lifted sluice gates

The present paper deals with the modelling of rapid transients at partially lifted sluice gates from both a mathematical and numerical perspective in the context of the Shallow water Equations (SWE). First, an improved exact solution of the dam-break problem is presented, assuming (i) the dependence of the gate contraction coefficient on the upstream flow depth, and (ii) a physically congruent definition for the submerged flow equation. It is shown that a relevant solution always exists for any set of initial conditions, but there are also initial conditions for which the solution is multiple. In the last case, a novel disambiguation criterion based on the continuous dependence of the solution on the initial conditions is used to select the physically congruent one among the alternatives. Secondly, a one (1-d) and a two-dimensional (2-d) form of a SWE Finite Volume numerical model, equipped with an approximate Riemann solver for the sluice gate treatment at cells interfaces, are presented. It is shown that the numerical implementation of classic steady state gate equations (classic equilibrium approach) leads to unsatisfactory numerical results in the case of fast transients, while a novel relaxed version of these equations (non-equilibrium approach) supplies very satisfactory results both in the 1-d and 2-d case. In particular, the 1-d numerical model is tested against (i) the proposed novel exact solutions and (ii) recent dam-break laboratory results. The 2-d model is verified by means of a test in a realistic detention basin for flood regulation, demonstrating that the novel findings can be promptly applied in real-world cases.


Introduction
Sluice gates are commonly used as regulation structures in rivers and irrigation canals (Islam et al. 2008, van Thang et al. 2010, flow measurement devices in open channels (Silva andRijo 2017, Kubrak et al. 2020), and control structures in flood detention basins (Morales-Hernández et al. 2013).
While sluice gates have been systematically studied considering steady flow conditions (Roth and Hager 1999, Lin et al. 2002, Defina and Susin 2003, Belaud et al. 2009, Habibzadeh et al. 2012, Bijankhan et al. 2012b, Castro-Orgaz et al. 2013, less attention has been paid to transient flows caused by rapid gate manoeuvres or interaction with waves propagating along the channel. There are, of course, some exceptions. De Marchi (1945) and Kubo and Shimura (1981) studied the negative wave generated upstream by the instantaneous partial uplift of a gate in a rectangular channel with water initially at rest, while Montuori and Greco (1973) studied the sudden manoeuvre (opening or closure) that caused the superposing of moving waves on preceding steady flow conditions. The exact solutions supplied by De Marchi (1945) and Kubo and Shumira (1981) were experimentally confirmed by Yamada (1992) and Reichstetter and Chanson (2013), while the exact solutions by Montuori and Greco (1973) were confirmed by their own laboratory experiments. Similar exact solutions are also present in classic textbooks such as those by Chow (1959) and Henderson (1966).
Among these studies, the work by Cozzolino et al. (2015) is particularly relevant here because a systematic analysis of the dam-break with partially lifted sluice gate in the context of the Shallow water Equations (SWE) was carried out by considering constant contraction coefficient with the adoption of the energy-momentum method by Henry (1950) for evaluating the discharge issuing under the gate in submerged flow conditions. Cozzolino et al. (2015) showed that there were initial conditions for which the dam-break problem exhibited multiple solutions and proposed a disambiguation criterion based on discharge maximization under the gate. In addition, they showed that there were initial conditions for which the dam-break problem exhibited no exact solution. The exact solutions by Cozzolino et al. (2015), which were subsequently verified for small gate opening by Monge-Gapper and Serrano-Pacheco (2021) using a smooth particle hydrodynamics model, are now a benchmark test for existing numerical models (Cui et al. 2019, Leakey et al. 2020, Delestre et al. 2023. The lack of solution to the dam-break problem with partially lifted sluice gate for certain initial conditions is due to the choice of the gate equations made in Cozzolino et al. (2015). Despite a constant value of the gate contraction coefficient is commonly used in the technical literature (Lin et al. 2002, Wu and Rajaratnam 2015, theoretical studies (Cisotti 1908, Marchi 1953, Belaud et al. 2009), numerical computations (Montes 1997, Kim 2007, Lazzarin et al. 2023, and laboratory experiments (Rajaratnam and Subramanya 1967, Rajaratnam 1977, Defina and Susin 2003, Lazzarin et al. 2023, show that the contraction coefficient depends on the gate opening and the upstream flow depth. In addition, it is well known that the energy-momentum method by Henry (1950) is unable to calculate the discharge under the gate in the transitional region between free and submerged flow, causing the formation of a non-physical discontinuity in the gate discharge equation (Bijankhan et al. 2011(Bijankhan et al. , 2012a. This issue should be corrected by continuously connecting the free and submerged flow gate equations (Cunge et al. 1980). The SWE model with sluice gate interior boundary conditions has been traditionally solved with the Method of Characteristics (Cunge et al. 1980, Islam et al. 2008, or locally coupling the Finite Volume method with the Method of Characteristics . The simultaneous solution of channel flow and gate equations with these approaches may lead to nonlinear polynomial equations with order up to twelve (Ellis 1976), for which the existence of a solution is not granted. Recently, the weak coupling of sluice gate and channel flow equations through the fluxes that the structure exchanges with the channel flow has emerged as a viable alternative in Finite Volume schemes (Zhao et al. 1994). The computation of these fluxes has been often carried out by approximating the solution of a local sluice gate Riemann problem (Morales-Hernández et al. 2013, Lacasta et al. 2014, Cozzolino et al. 2015, Cui et al. 2019, Leakey et al. 2020. Nonetheless, this approach requires that the corresponding exact solutions are known in advance for benchmarking and constructing improved approximate Riemann solvers. Except for the numerical approach by Cozzolino et al. (2015), current numerical methods do not recognize the existence of multiple Riemann solutions for certain initial conditions, and they lack a mechanism to cope with the solution multiplicity.
In the present paper, we construct novel SWE exact solutions of the dam-break at partially lifted sluice gates using variable contraction coefficient (Defina and Susin 2003) and the submerged flow gate equations by Bijankhan et al. (2012b). The novel solutions are improved with respect to those by Cozzolino et al. (2015) because the discharge gap in the transitional region between free and submerged flow is eliminated using viable experimental gate equations. We show that the dam-break solution always exists for any set of initial conditions, but there are certain initial conditions for which the solution is multiple. In this case, a criterion based on the continuous dependence of the solution on the initial conditions is used to pick up the relevant solution among the alternatives. In addition, we construct 1-and 2-d SWE Finite Volume models equipped with an approximate Riemann solver for the sluice gate treatment at cells interfaces. We show that the classic steady state gate equations lead to unsatisfactory results in the case of fast transients' numerical computation, and we propose a relaxed version of these equations, here called non-equilibrium approach, which coincides with the classic equations in the case of steady flow, and it is best suited for the construction of the approximate Riemann solver. The 1-d numerical model with the non-equilibrium approach for the gate treatment captures the novel exact dam-break solutions and the dam-break laboratory results by Lazzarin et al. (2023), while the 2-d numerical model is tested using a realistic 2-d detention basin for flood regulation.
The rest of the paper is organized as follows: in Section 2, the gate equations are presented; in Section 3, the exact solution of the dam-break problem with partially lifted sluice gate is constructed, and a novel disambiguation criterion is proposed; in Sections 4 and 5, 1-d and 2-d SWE models that incorporate the non-equilibrium numerical approach are described and tested; in Section 6, the novel disambiguation criterion is compared with the one by Lazzarin et al. (2023); finally, the paper is closed by a Conclusions section.

Sluice gate model
When fluid flows interact with a sluice gate, the corresponding regime is called orifice flow. In this case, two distinct flow conditions are possible, namely the free and the submerged flow (Henderson 1966). In free flow conditions (see Figure 1a), the supercritical jet issuing under the gate is open to the atmosphere. In contrast, in submerged flow conditions, the jet under the gate is overlaid by the downstream subcritical flow, which is characterised by intense turbulent motion (see Figure 1b).
Finally, the regime where the flow free surface does not touch the gate lip and there is no interaction with the gate is referred to as non-orifice flow ( Figure 1c).
In the present Section, the gate equations are presented, and a bifurcation phenomenon is introduced.

Sluice gate equations
In steady free flow conditions, the unit-width discharge qF under the gate depends on the gate opening a and the upstream depth hu, which is measured at a distance from the gate sufficient to re-establish gradually varied flow (Figure 1a). The cross-section where the depth of the supercritical jet issuing under the gate is minimum and the flow is gradually varied is called vena contracta.
At the generic cross-section, the energetic content of the flow is measured by the total head ( ) where h is the flow depth and q is the unit-width discharge. If the energy loss through the gate is neglected and steady state conditions are assumed, the invariance of total head and unit-width discharge between the upstream cross-section and the vena contracta implies  (Rouse 1946, Henderson 1966, Defina and Susin 2003 ( In the present paper, the parametric formulation by Defina and Susin (2003) ( In steady submerged flow conditions (Figure 1b), the unit-width discharge qS under the gate also depends on the tailwater depth ht, i.e., the depth of the downstream subcritical flow measured at a distance sufficient to re-establish the gradually varied flow (Henry 1950, Rajaratnam and Subramanya 1967, Lozano et al. 2009  hh of the tailwater depth represents the limit condition between free and submerged flow (Rajaratnam and Subramanya 1967, Lin et al. 2002, Habibzadeh et al. 2011).
Since the term inside the parentheses to the right-hand side of Eq. (5) is minor than one, it follows that [Insert Figure 1 about here]

Free flow multiple solutions with variable contraction coefficient
Before considering the dam-break problem solution with partially lifted sluice gate, it is instructive to contrast the effect of variable and constant Cc on the discharge calculated with Eq. (2). In addition to the experimental Cc of Eq. (3) by Defina and Susin (2003), the theoretical formulation by Marchi (1953), which is based on the assumption of irrotational flow with gravity effects included, will be considered in the present Section.
In Figure 2a, the experimental contraction coefficient Cc by Defina and Susin (2003) (Lin et al. 2002, Cozzolino et al. 2015. The inspection of the figure shows that the theoretical expression of Cc by Marchi (1953) is a convex function of a/hu that exhibits a minimum in a/hu = 0.29 and satisfies the condition Cc = 1.0 for a/hu = 1.0. The last condition expresses the fact that no flow contraction is expected when the gate lip barely trims the flow free surface. Similarly, the experimental Cc by Defina and Susin (2003) is convex with a minimum in a/hu = 0.48 and satisfies the no flow-contraction condition for a/hu = 1.0. Trivially, this requirement cannot be met by a constant value of Cc.  Lazzarin et al. (2023) have associated this behaviour with flow instability phenomena at gates with high relative opening.
In the following, it will be shown that the shape of the Cc curve has a dramatic influence on the existence and uniqueness of the dam-break problem solution, even in the simplest case of dambreak on dry bed.
[Insert Figure 2 about here]

Exact solution to the dam-break problem at partially lifted gates
Under the assumptions of a constant-width rectangular channel with a horizontal frictionless bed, the  The Riemann problem is the initial value problem where Eq. (6) is solved with the discontinuous initial conditions In the dam-break problem, the flow velocity is initially null, and the initial states reduce to . This corresponds to the situation where two reservoirs with water initially at rest are separated by a sluice gate in x = 0 that is suddenly lifted leaving an opening of height a. The dam-break solution is trivial when the gate is lifted enough to avoid subsequent interaction between the flow free-surface and the gate lip because the corresponding non-orifice regime is equivalent to the case of gate complete removal already discussed in the literature (Stoker 1957, Toro 2001, LeVeque 2002. In the present section, we generalize the dam break to consider the case where the flow interacts with the gate because the device is only partially lifted.
Without loss of generality, we will assume in the following that hL  hR, implying that the flow moves from the left to the right under the gate during the transient caused by the gate lifting. It follows that, in free flow conditions, the state 2 u coincides with the vena contracta state , where qF is calculated using Eq.
(2) with hu = h1. In submerged flow conditions, the state 2 u is such that , where qS is calculated using Eq. (5) with ht = h2 and hu = h1.
From the discussion related to Eq. (5) (Section 2.1), it follows that the limit tailwater state u2 in submerged flow conditions is

Preliminaries: dam-break on a dry bed and multiple solutions
To introduce the issue of multiple solutions to the dam-break problem at partially lifted gates, first we explore the solution of the dam-break on dry bed (hR = 0 m), considering the parameters a = 0.47 m and hL = 1 m (Test E1 of Table 1) (2) where hu = hF. It follows that the flow depth F h corresponding to F u satisfies the equation Depending on the initial relative opening a/hL, Eq. (12) admits one, two, or no solutions.
Correspondingly, one, two, or no states u1 = uF can be associated to orifice free flow conditions. When Eq. (12) exhibits two solutions, a subscript l [h] characterizes the lower [higher] value hF,l [hF,h] of the flow depth solution and the corresponding state uF,l [uF,h], determining a lower [higher] solution. In the example with a = 0.47 m and hL = 1 m (Test E1 of Table 1) From the preceding discussion, it follows that three distinct dam-break solutions on dry bed are possible for the example considered, namely one non-orifice and two orifice flow solutions ( Figure 3). If the same procedure is repeated for different values of a and hL, the diagram of the relative depth h1/hL as a function of the initial relative opening a/hL is obtained (see Figure 4). The inspection of the figure shows that the dam-break problem on dry bed admits a single orifice free flow solution for low initial relative openings ( ).
In the next Sections, we will construct the general solution to the dam-break problem at partially lifted gates for the case with hR > 0 and we will provide a criterion for the disambiguation of multiple solutions.
[Insert Figure 3 about here] [Insert Figure 4 about here] [Insert Table 1 about here]

General solution to the dam-break problem
The In the following, we will separately consider the cases of low ( ).

Low initial relative opening
For low initial relative openings ( 49  L ah ), the state uL cannot be connected by the rarefaction R1(uL) to a non-orifice state u1 characterized by h1 < a (see Section 3.1), there is only one solution to Eq. (12) (see Section 3.1). For this reason, it is possible to enumerate the three distinct wave configurations represented in Figure 5, corresponding to the exact solutions of the dam-break tests from E2 to E4 contained in Table 1, which are characterized by a/hL = 0.2 and increasing values of hR.
In the wave configurations of Figures 5a and 5b (Tests E2 and E3 of Table 1,  In the wave configuration of Figure 5c (Test E4 of

High initial relative opening
For high initial relative openings ( 0.495  L ah ), the orifice free flow is forbidden because Eq. (12) has no solution (see Section 3.1). In this case, it is possible to enumerate the two wave configurations represented in Figure 7, which correspond to dam-break examples with a/hL = 0.6 and increasing values of hR.
In the wave configuration of Figure 7a (Test E5 of Table 1 -the locus of the states u2 along TC.
From Figure 8a, we observe that the L-M curve corresponding to the dam-break with high relative opening is continuous and strictly decreasing, proving that the solution of the dam-break problem always exists and it is unique in this case. In Figures 8b and 8c, we consider the graphical solution of the dam-break tests E5 and E6 (Table 1), respectively, characterized by high initial relative opening a/hL = 0.6. In the case of Test E5 (Figure 8b

Intermediate initial relative opening
For The first L-M curve considered is the one where the state u1 in the orifice free-flow solutions coincides with uF,h. In Figure 9, the corresponding flow depth exact solution at t = 5 s for the dambreak tests from E7 to E9 (Table 1)  The inspection of Figure 9 shows that the dam-break wave configurations connected to the higher solution uF,h coincide with the wave configurations obtained for low values of a/hL ( Figure 5).
Following the methods of Section 3.2.1, it is possible to construct the corresponding L-M curve ( Figure 10a). This curve is continuous and strictly decreasing, which proves that the corresponding dam-break solution always exists and it is unique. In Figures 10b,c,d, the graphical solution of the dam-break tests from E7 to E9 (Table 1)  To shed light on the origin of the gap on these L-M curves, we observe that the states u2 along the TC curve satisfy the equalities h2u2 = qR,1 and h2u2 = qS, which is possible only when qR,1  qF (see comments to Eq. [5]). In Figure 12, the unit-width discharges qR,1 and qF are plotted as a function and to the non-orifice solution of the dam-break on-dry bed (Figure 12c).
[Insert Figure 11 about here] [Insert Figure 12 about here]

A disambiguation criterion for multiple solutions
The discussion of Section 3.2 shows that the exact solution to the dam-break at partially lifted gate can be obtained by intersecting the L-M curve, which depends on the state uL and the opening a, with the ( ) , while non-orifice flow regime is obtained for 0.495  L ah .

One-dimensional numerical modelling
If the friction is added to Eq. (6) is used to calculate the vector 1 + n i u of the conserved variables at the time level n + 1 (Cozzolino et al. 2012). In Eqs. (16) and (17), t is the time step, x is the length of the cell, 12 is the contribution to the cell Ci across the interface i+1/2 between Ci and Ci+1, while 12 is the contribution to the cell

Ci+1.
It is assumed that gates are located at cell interfaces, and a distinction is made between ordinary and gate interfaces. When the gate is absent, the interface is ordinary and the two contributes at gate interfaces, where the device is located. Two different procedures, called equilibrium and non-equilibrium approach, respectively, are described in the following.

Classic equilibrium approach
The name "equilibrium approach" refers to the fact that the numerical fluxes 12 − + i f and 12 at the gate are computed by imposing the gate equations in their original equilibrium form of Section 2 (steady state conditions). This approach seems well justified, since the states u1 and u2 immediately upstream and downstream of the gate, respectively, are constant for t > 0 (local steady state conditions) in the exact Riemann solution.

Algorithm structure
We assume, without loss of generality, that the flow depth to the left of the gate is greater than the flow depth to the right, i.e.,

Dam-break on dry bed
To test the capability of the equilibrium approach, the solution of the dam-break problem on dry bed (Test E1 of Table 1)  where the time-graphs of numerical unit-width discharge qF (Figure 14a), he numerical upstream flow depth hu (Figure 14b), and numerical relative opening a/hu (Figure 14c), are plotted. To interpret the figure, one must recall from Section 2.2 that qF is an increasing function of hu for a/hu < (a/hu)lim = 0.86, while it is a decreasing function of hu for a/hu > (a/hu)lim. The inspection of Figure 14a shows that qF rapidly increases from 0 to 1.05 m 2 /s at the beginning of the transient, when the upstream flow depth is maximum, then it starts to decrease because hu decreases due to the channel emptying ( Figure   14b). Interestingly, a/hu increases rapidly and crosses the threshold (a/hu)lim at time t = 0.08 s ( Figure   14c). This causes the discharge qF to stop its descent and start to increase again, accelerating the channel emptying and causing the final detachment of the flow from the gate lip at t = 0.69 s, after numerous oscillations caused by the alternating passage from orifice to non-orifice flow regime and vice versa. From the preceding analysis, it seems that the equilibrium approach overestimates qF during the initial phase of the dam-break transient, causing a too rapid decrease of hu.
To Hh . This explains why the discharge qF under the gate is overestimated at the beginning of the transient. A similar phenomenon will occur during the subsequent time steps because the head in the cell Ci immediately upstream of the gate will generally differ from the required head associated to the discharge qF computed with Eq. (2).
The observations above suggest that the gate equation should be modified to take into account strong transients. A heuristic approach able to cope with this issue will be considered in the next Section.
[Insert Figure 13 about here] [Insert Figure 14 about here]

Non-equilibrium approach
If we assume that the total head is invariant through the gate and relax the assumption of discharge invariance, we obtain the free flow equation (see Appendix A) (2) when the discharge invariance is attained during steady state conditions. Note that Eq. (23) should be regarded as a numerical relaxation approach with a physical justification, and not as a novel physics equation.
The steps that constitute the "non-equilibrium" numerical approach coincide with the steps of the equilibrium-approach described in Section 4.1.1, with the only difference that the Eq. (23)  Congruently, the qF of Eq. (23) is also used to compute the limit tailwater depth # c h of Eq. (4) and the submerged flow discharge qS of Eq. (5).

Numerical tests with exact solution
The solution of the dam-break problem on dry bed (Test E1 of Table 1) is approximated with the nonequilibrium approach, using the same numerical parameters of Section 4.1.2. The corresponding flow depth at time t = 5 is compared with the exact solution in Figure 15, where the computational results are represented with dots (only one in five dots is represented). The inspection of the figure shows that the non-equilibrium approach approximates the relevant free flow solution. In particular, the flow depth jump through the gate is nicely captured, together with the strength and celerity of the moving waves. The inspection of Figure 16a shows that the non-equilibrium approach reaches the goal of reducing the overestimation of qF during the initial part of the transient. This allows to limit the decrease of hu (Figure 16b) and the increase of a/hu (Figure 16c), ensuring that the orifice flow regime is kept during the entire simulation.
[Insert Figure 15 about here] [Insert Figure 16 about here] The remaining tests of Table 1 are tackled with the non-equilibrium approach and computation parameters of Section 4.1.2, and the corresponding numerical results (flow depth at time t = 5 s) are compared with the exact solutions in Figure 17 (tests from E2 to E4), Figure 18 (tests E5 and E6), and Figure 19 (tests from E7 to E9). The inspection of the figures shows that the non-equilibrium numerical approach nicely approximates the exact solution, independent on the initial relative opening a/hL.
[Insert Figure 17 about here] [Insert Figure 18 about here] [Insert Figure 19 about here]

Laboratory dam-break tests
In the present Section, the numerical scheme of Eqs.  present for a short time until detachment was completed at times t * = 0.5 s and t * = 2 s, respectively (see Table 2). Finally, Experiment L6 was characterized by stable orifice flow conditions. For the sake of comparison, the dam-break exact solutions obtained with the methods of Section 3.1 and the corresponding Finite Volume solution without friction are reported in Table 3.
The inspection of Table 3 confirms that the numerical model without friction is able to nicely capture the corresponding exact solutions, as already deduced in Section 4.2.1. More interestingly, the comparison with Table 2 shows that, contrarily to the numerical results with friction, the frictionless exact and numerical solutions do not reproduce all the experimental flow regimes. It can be deduced that the friction has a decisive influence in determining the numerical simulation results in the case of rapid transients with partially lifted sluice gates. The discussion of the friction influence on the laboratory dam-break solutions will be tackled in Section 6.
[Insert Table 2 about here] [Insert Figure 20 about here] [Insert Table 3 about here]

Two-dimensional framework
The 2-d SWE model with uneven bed elevation and friction can be written as (Audusse and Bristeau 2005) (24) where the conserved variable vector U, and the flux vectors F and G along x and y, respectively, are , In the following, it will be shown how the 1-d gate model of Section 2 can be adjusted for implementation in 2-d SWE models, and a 2-d example application will be presented.

Gate model for 2-d flows
Consider the plan-view of a sluice gate as represented in Figure 21, where the gate is aligned with the y-axis of the fixed reference Oxy (Figure 21a). The flow particle approaching the gate has velocity components uu and vu along x and y, respectively. After the passage under the gate, the components of the velocity become ud and vd, respectively. Mass conservation at the gate implies that qg = huuu = hdud, where qg is the unit-width discharge under the gate, while hu and hd are the flow depths upstream and downstream, respectively. If the gate is frictionless, it exerts no action on the flow particle along the y-axis, implying that the flow particle has no acceleration along y while passing under the gate.
This supplies vu = vd (invariance of the transverse velocity, Figure 21a) and qgvu = qgvd (invariance of the transverse momentum flux).
Consider now a moving reference O'xy' that translates along y with uniform velocity vu and such that ' =− u y y v t . In this moving reference, the particle passes perpendicularly under the gate and has no velocity component along y' (Figure 21b) [Insert Figure 21 about here]

Two-dimensional numerical modelling
The solution of the 2-d SWE of Eq. (24) is approximated by means of a first-order Finite Volume scheme on unstructured triangular grid, where a time splitting approach is adopted to separately treat the advective and the friction part of the mathematical model (Toro 2001 while the implicit friction step is subsequently used to calculate the vector 1 + n i U of the conserved variables at the time level n + 1. In Eqs. (28) The matrix Rij allows the passage from the global reference framework Oxy to a local reference whose axes are aligned with the interface between Ci and Cj. In Eq. (28)

Computation of ordinary interface contributions
At ordinary interfaces, where the gate is not present, a simplified HLLC approximate Riemann solver (Toro 2001) is used to solve the local SWE plane Riemann problem while the hydrostatic reconstruction approach by Audusse et al. (2004) is adopted to cope with the source term S0(U). For this reason, it is possible to write where the HLLC numerical flux HLLC F corresponding to the SWE plane Riemann problem is calculated using the projections ˆn

Computation of gate interface contributions
For the sake of simplicity, it is assumed that the bed is horizontal under gates and that the opening is a. When

Two-dimensional idealized detention basin
The 2-d numerical model described above is applied to simulate the filling and emptying of a detention basin (see Figure 22)

Discussion
The solution disambiguation criterion proposed in Section 3.3 is based on the concept of existence and uniqueness of frictionless dam-break exact solutions for general initial conditions. On the other hand, the disambiguation criterion proposed by Lazzarin et al. (2023), which has been validated by means of the laboratory dam-break experiments of Table 2, is based on the stability of the sluice gate equations of Section 2 only, without regard for the existence and the uniqueness of the dam-break solutions for general initial conditions. The last criterion requires that the ratio a/hu between the gate opening a and the flow depth hu immediately upstream of the gate satisfies the condition To compare the two disambiguation criteria, the flow depth h1 corresponding to the state u1 in the exact solution to the dam-break problem on dry bed for different values of a and hL is considered. In Figure 26a, which is obtained from Figure 4  (see Section 3.3). In Figure 26b, a similar diagram is plotted after the application of the disambiguation criterion by Lazzarin et al. (2023). In the case of the dam-break problem, this criterion is equivalent to the condition ( ) , where the laboratory experiment L5 of Table 2 falls. For this experiment, the criterion by Lazzarin et al. (2023) predicts non-orifice flow regime, which is confirmed by the laboratory results, while the disambiguation criterion of Section 3.3 predicts orifice flow regime (see Table 3, fifth column).
Although the last observation is apparently negative for its credibility, we notice that the disambiguation criterion of Section 3.3 is expressly formulated for exact and numerical solutions without friction. For this reason, its direct application is inappropriate in real world cases while it comes useful in the construction of numerical models based on the local solution of a Riemann problem. This is demonstrated by the 1-d numerical model of Section 4.2, which satisfies the disambiguation criterion of Section 3.3 for very idealistic cases without friction and nicely reproduces the L5 laboratory results (sixth column of Table 2)  [Insert Figure 28 about here]

Conclusions
In the present paper, an improved solution of the dam-break problem at partially lifted sluice gates has been presented. This novel solution assumes not only the dependence of the gate contraction coefficient on the upstream flow depth (Defina and Susin 2003), but also recent developments for the definition of a physically congruent submerged flow equation (Bijankhan et al. 2012b). The improvement of the Riemann problem physical representation amends the limitations of the preceding work by Cozzolino et al. (2015), namely the lack of solution existence for certain values of the initial downstream flow depth.
As common for the Riemann problem of the Shallow water Equations at geometric discontinuities and hydraulic structures, there are initial conditions for which the solution is multiple, and a disambiguation criterion must be introduced to pick up a physically congruent choice among the alternatives. In the present work, a disambiguation criterion based on the continuous dependence of the solution on the initial conditions allows to single out a well-posed solution. Interestingly, this criterion supplies results that slightly differ from the ones obtained with the disambiguation criterion by Lazzarin et al. (2023), and the corresponding discrepancies are discussed.  Lazzarin et al. (2023). It follows that the numerical scheme can distinguish between the dam-break initial conditions that either lead to orifice flow under the gate or to a flow that is detached from the gate lip in the flume experiments. Interestingly, the comparison between the numerical simulations with and without friction shows that the friction may have a role in the inception of the instability phenomena that lead to the detachment of the flow from the gate lip, and this has a consequence in the interpretation of laboratory experiments.
Finally, a 2-d Finite Volume scheme based on the non-equilibrium approach is used to simulate the filling and emptying of a detention basin with complicate topography and a sluice gate located at its downstream end, demonstrating how the novel findings can be promptly used in realworld applications.
along the particle trajectory under the gate from the upstream position A to the position B at the vena contracta ( Figure 29). In Eq. (A.1), s is the local abscissa along the trajectory, v is the particle velocity modulus, vs is the component of the velocity along the trajectory, z is the elevation of the particle above the datum, p is the local pressure, and  is the fluid specific weight. If we integrate in space from A to B and assume that the local acceleration is negligible, Eq. [Insert Figure 29 about here] Tables List9 Table 1. Initial conditions of the dam-break problems with exact solution. friction. An asterisk denotes the passage from orifice free flow to non-orifice flow regime.       Table 1 with a/hL = 0.2: Test E2 (a); Test E3 (b); Test E4 (c).   Table 1 with a/hL = 0.6: Test E5 (a); Test E6 (b).   Table 1 with a/hL = 0.47: Test E7 (a); Test E8 (b); Test E9 (c).         Table 1 with a/hL = 0.2. Flow depth at t = 5 s: Test E2 (a); Test E3 (b); Test E4 (c). Figure 18. Comparison between exact (thin black line) and numerical solution with the nonequilibrium approach (dots, one in five is represented to enhance the clarity of the plot) for the dam-break tests of Table 1 with a/hL = 0.6. Flow depth at t = 5 s: Test E5 (a); Test E6 (b). Figure 19. Comparison between exact (thin black line) and numerical solution with the nonequilibrium approach (dots, one in five is represented to enhance the clarity of the plot) for the dam-break tests of Table 1 with a/hL = 0.47. Flow depth at t = 5 s: Test E7 (a); Test E8 (b); Test E9 (c).  Table 2: Test L1 (a); Test L2 (b); Test L3 (c); Test L4 (d); Test L5 (e); Test L6 (f).
The position of the gate lip is represented with a dashed line.