Impact force of a surge of water and sediments mixtures against slit check dams

Science of the Total Environment 683 (2019) 351–359

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Science of the Total Environment journal homepage: www.elsevier.com/locate/scitotenv

Impact force of a surge of water and sediments mixtures against slit check dams Giulia Rossi* , Aronne Armanini University of Trento, DICAM, Trento, Italy

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Laboratory investigation on slit check dam functioning • Rational-empirical estimation of impact forces of a surge on slit check dams • Influence of a debris flow breaker positioned in the check dam opening

A R T I C L E

I N F O

Article history: Received 31 December 2018 Received in revised form 7 May 2019 Accepted 9 May 2019 Available online 17 May 2019 Keywords: Slit check dam Debris flow breaker Protection structures Dynamic impact

A B S T R A C T Slit check dams are widely used protection structures against debris flows. The role of these structures is to trap part of the debris in order to diminish the peak of the solid discharge. However, the high volume and velocity involved induce considerable impact forces. Correspondingly, an improved estimation of the impact force is fundamental to properly design the protection structures. In order to develop an analytical expression for the impact force of a debris flow surge against a slit check dam, we have adopted a rational criterion based on the principles of mass and momentum conservation. In our formulation we have introduced a proper coefficient to account for the horizontal contraction of the streamlines near the check dam slit. This coefficient is calibrated through a series of physical experiments. Furthermore, the paper addresses the influence of a debris flow breaker located within the opening of the slit check dam. The differences between the check dam with and without flow breaker are evaluated in terms of impact forces by comparing two check dams with the same slit width and the same net slit width. By comparing the force acting in the presence of the breaker or without it, we observed that the impact is more onerous in the first case, since the incoming flow is deviated from the flow breaker to the check dam wings. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The debris flows are extreme events, which affect mountainous regions. These flows consist of mixtures of water and sediments that originate in the upper part of the basins and run towards the valley with high speeds that tend to overwhelm all the obstacles * Corresponding author. E-mail addresses: [email protected] (G. Rossi), [email protected] (A. Armanini).

https://doi.org/10.1016/j.scitotenv.2019.05.124 0048-9697/© 2019 Elsevier B.V. All rights reserved.

they encounter along their path (Iverson, 1997; Takahashi, 2014). The risk associated with these events have increased considerably in recent times, mainly due to the progressive urban expansion that has affected some mountain regions, and in particular the alluvial fans, in conjunction with an exacerbation of extreme rainfall events of short duration and high intensity (Caine, 1980). The scientific community is making a substantial effort to improve knowledge of modeling these multi-phase flows (Pudasaini, 2012; Mergili et al., 2017) and enhance defence strategies (Kattel et al., 2018; Suda et al., 2009). Given the complexity and hazard

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of debris flows, effective protection strategies are of fundamental importance to avoid human losses and property damages (Hübl and Fiebiger, 2005; Jakob, 2005). The most widely used structural approach in mountainous catchments for the risk reduction is to induce the deposition of part of the sediments along the stream, to reduce peak discharges. This effect is obtained usually through stilling basins and check dams or slit check dams (Armanini and Larcher, 2001) (Piton and Recking, 2015). This kind of structures should be designed accounting for two main targets: the reduction of the peak discharge of the solid material, but also the resistance to the dynamic impact exerted by the debris flow when it reaches the dam (Hübl et al., 2017, 2009; Zanuttigh and Lamberti, 2006). The impact force is often very high, due to the velocities involved, and a reliable estimation of it is fundamental in order to properly design the structures (Kattel et al., 2018; Proske et al., 2011). In this paper we present an experimental investigation of the dynamic impact of a debris flow surge against a slit check dam, with and without the adoption of a debris flow breaker. In Section 2 we describe the experimental setup used to perform the experiments, while in Section 3 we present our rational approach to frame the dynamic impact against a slit check dam, based on the mass and momentum balances. We also present some experimental results about the slit check dam. In the fourth section, we present the experimental results and our approach, for the slit check dam in the presence of the debris flow breaker. In Section 5 we analyze the influence of the grain size on the approach adopted. Finally, in the last section, we draw some conclusions. 2. Experimental investigation The experimental investigation has been carried out in a channel of the Hydraulic Laboratory of the University of Trento. A series of tests was performed, in which surges were produced by releases of mixtures of water and sediments upstream of a removable gate placed at the upstream end of the channel. We then studied the dynamic impact against slit check dams placed at the downstream end of the channel, even in the case of slit check dams with debris flow breaker. We used a tilting open channel made of perspex, 0.25 m wide and about 3 m long (see Fig. 1), with an inclination between 15◦ and 22◦ . The removable gate was positioned in the upstream part of the channel and was used to retain debris mixture; an automatic piston system makes the gate going up instantaneously when the dam break test starts. The debris flow was simulated by a mixture of water and sediments (sand and gravel characterized by d30 = 2.0 mm, d50 = 3.5 mm and d90 = 9.0 mm), with a volume solid fraction of about 0.5.

At the end of the channel we positioned the slit check dam with or without the debris flow breaker: the impact force exerted by the mixture of sediments and water against the check dam and the debris flow breaker was measured using load cells (SP4 M by HBM, accuracy of 0.1 N), that allowed to see the evolution in time of this force. Furthermore, along the channel three ultrasonic sensor (Pepperl + Fuchs, resolution: evaluation range [mm]/4000, but >0.05 mm) were positioned at 20, 60 and 100 cm from the check dam (see Fig. 1), in order to measure the flow depth of the incoming flow and its evolution in time. Finally, all the experiments were recorded by two high speed cameras, Photron FASTCAM-1024PCI, adopting a frame rate of 500 frames per second and a resolution of 1024 × 1024 pixels. The first camera was located at the side just upstream the check dam, in order to record the impact of the flow against the dam, while the other was positioned above the channel to observe the flow front and evaluate its velocity. The velocity U0 of the front of the incoming surge, was measured by calculating the time needed for the front to cover the distance (15 cm) between two fixed points located within the channel at a distance of about 45 and 60 cm from the test wall. In order to identify the position of the two fixed points, we placed two laser pointers normally to the bed. The transit time was obtained from the analysis of the frames recorded by the camera positioned above the channel. In this way it was possible to identify the position of the front with an accuracy of about 1 mm. The two cameras, the load cell and the ultrasonic sensors were all synchronized during the experiments, through an ad hoc program, which allow us to make start the registration with all the instruments at the same time. Finally, we want to mention that we have not taken into account the impact of surges of dry granular flows, since the presence of the interstitial liquid characterizing our experiments, substantially conditions the impact mechanism (Choi et al., 2015, 2016; Ng et al., 2016; Song et al., 2018). 2.1. Experimental tests A set of experiments has been carried out in the set up described above, by inserting at the end of the channel a slit check dam. In particular, the experiments are characterized by different Froude numbers and by three different widths (Bf ) of the opening of the slit check dam (40, 60 and 100 mm). A picture of the check dam used in the experiment is shown in Fig. 2: it is a PVC structure, with a slit in the lower part. The upper part is closed and fixed to a load cell to measure the evolution of the impact force. The height of this closed part does not affect the measurements of the impact force, since, at that height from the channel bed, the jet is completely vertical and does not contribute to the force acting perpendicular to the check dam (see Armanini et al., 2019). The flow characteristics (velocity U and flow depth h) are evaluated as explained in the previous section to compute the Froude number for each test. In particular, the Froude number was varied between 4 and 6 by changing the channel slope between 17◦ and 26◦ . For each Froude number, we have analyzed the three widths of the opening, together with the case of a plain wall (Bf = 0). The range of flow depths analyzed is from 2 to 4 cm and the velocities are between 2 and 4 m/s. 3. Dynamic impact of a debris flow surge on a vertical wall with a slit

Fig. 1. Experimental apparatus, measures in mm.

According to Armanini and Scotton (1992) when a surge impacts a vertical wall two types of impacting mechanisms may occur depending on the Froude number of the incoming surge (Khattri et al., 2018). The first type consists of a reflected wave that propagates

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353

Fig. 3. Side picture of the vertical jet during the impact against the slit check dam.

Fig. 2. PVC slit check dam, being B0 the channel width and Bf the slit width.

upstream soon after the impact, while the second is characterized by a complete deviation of the flow along the vertical direction assuming a jet-like behaviour. These differences in the impact dynamics, related to the Froude number, have been observed also by other authors (Laigle and Labbe, 2017). We simulated in the laboratory flume an impact of the second type (Froude number >3). The evolution of the impact is composed by the following steps, described in detail in Armanini et al. (2019): i. the impact begins with the formation of a vertical jet (see Fig. 3); ii. the jet propagates upward, becoming wider and decelerating; iii. when the jet velocity is low enough, it breaks down falling upstream. To determine the impact of a debris flow against a rigid vertical wall in a channel, we adopt the classical approach of the propagation of a wave of finite amplitude in a channel (Henderson, 1966), under the shallow flow approximation. We apply the conservation equations of mass and momentum with respect to a suitable fixed control volume, according to the integral formulation: 

∂ qm dV = Vc ∂ t





∂ qm u dV + Vc ∂ t

qm u • n dS

(1)

Sc



 (qm u)u • n dS =

Sc

 qm f dV +

Vc

F dS

(2)

Sc

where: qm is the density of the flow; Vc the control volume; Sc the control surface; u the velocity vector; f the mass forces; and F the surface forces.

Armanini et al. (2019) applied the above equations to calculate the dynamic impact of a rectangular surge of height ho and velocity Uo that impacts a plain wall normal to the bottom, by forming a jetlike impact. These authors adopted a control volume fixed with respect to the wall at the instant when the jet begins to form, having experimentally observed that this is the instant in which the impact force against the wall assumes the maximum. In this condition, the control volume contains the whole zone of the flow characterized by a curvature: this entails that at the inflow section all the streamlines are parallel to the bed and at the outflow section all the streamlines are parallel to the vertical wall. Furthermore, they observed that the width of the output section (i.e. of the front of the jet) is, with good approximation, equal to the flow depth of the incoming surge (this statement is also a derivable from the energy balance), so that the mass and momentum time variations within the control volume (first terms of Eqs. (1) and (2)) are negligible. In the case of a jet-like flow against a slit check dam, we apply an analogous procedure. The hypotheses we assumed are: – negligible variation of the solid fraction concentration, since the control volume is very small and we did not observe any deposit of material in the time interval considered; – constant density qm = qs C + qw (1 − C) of the mixture and no variation of the fluid density qw and solid material density qs , since the phenomenon is characterized by velocity far lower than the sound velocity. C is the solid phase concentration; – same velocities of the two phases (solid particles and interstitial fluid, water); – nearly uniform velocity distribution of the incoming surge, and consequently we set equal to 1 the momentum correction coefficients; – negligible contribution of the longitudinal weight component of the fluid and negligible bed shear stresses, given that we assumed the length of the base of the control volume to be small enough and that the two forces are opposite. With respect to the control volume defined as above, represented in Fig. 4 (dashed red line), and according to the previous

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(a) longitudinal section crossing the wings

(b) longitudinal section crossing the slit

Fig. 4. (a) at the moment of impact the flow near to the walls of the channel reaches the wings of the check-dam giving rise to a jet-like impact; (b) at the center of the channel the flow continue across the slit. U0 and Uf are the depth averaged velocities of the incoming front and of the mixture crossing the slit respectively, h0 and hf the flow depth of the incoming flow and of the flow across the slit, p is the pressure, to is the tangential stress.

assumptions, the momentum balance in the longitudinal direction can be written as:

S=

1 1 qm gh2o cos a B0 − qm gh2f cos a Bf +qm Uo2 ho B0 −qm Uf2 hf Bf (3) 2 2

In Eq. (3) ho and Uo are the depth and velocity of the incoming surge when entering the control volume and hf and Uf are the depth and velocity of the flow crossing the slit, B0 and Bf are respectively the channel and slit width. The first and the second right-hand terms represent the hydrostatic force on the entering section and on the slit section of the control volume respectively. The last two terms are the momentum across the same sections. The impact force is then due to the difference between the hydrostatic force and the momentum of the incoming flow and the hydrostatic force and the momentum corresponding to the slit. Eq. (3) can be made dimensionless with respect to the hydrostatic force of the incoming flow:

S˜ =

S 1 qm gh20 cos a B0 2

    Fr02 Bf Bf 1− = 1− +2 B0 cos a B0

(4)

Thus, we obtain a direct relation between the dimensionless impact  force and the Froude number, defined as Fr = U0 / gh0 . In Fig. 5 we compare the experimental data with this formula. In general Eq. (4) overestimates the impact force with respect to the experimental values, except for R = 0 which corresponds to the case of a plain wall. As expected, in this case Eq. (4) reduces to S˜ =  2 √ 1 + 2 Fr0 / cos a , which coincides with the results obtained by Armanini et al. (2019) for the plain wall. In order to understand this discrepancy, we analyzed accurately the laboratory experiments. We observed that at the moment of impact the incoming surge is divided into three parts. A central part that passes through the slit and two lateral parts (one right and one left) that instead form two jets rising along the wings. The recordings from the camera positioned above the flow showed that the part of the flow passing through the slit was wider with respect to the part of the incoming mixture flowing in the strip large as the slit width (grey area in Fig. 6). That is, a non-negligible part of the surge (more than 20%), which should hit the wings of the check dam, before the impact is diverted towards the opening (see magenta streamlines in Fig. 6). Considering the mass conservation crossing the slit, we may write:

Qf = Uf hf Bf = U0 h0 Bf c

Fig. 5. Behaviour of the dimensionless impact force S˜ in function of the Froude number Fr, for different values of the ratio R = Bf /B0 between the width of the slit Bf and the width of the channel B0 . The symbols represent the experimental data, while the curves the dimensionless impact force calculated according to Eq. (4).

Fig. 6. Scheme of the streamlines path.

(5)

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355

where Qf is the flow rate passing through the slit and c represents the relative increase of the outgoing flow rate with respect to the opening width, experimentally observed and described above. Let us now make the further simplification hypothesis that the flow depth in the output section, which we remember is at the beginning of the slit, is the same as the depth in the input section. This hypothesis is certainly valid in the supercritical conditions in which we carried out the experiments (Fr0 >∼ 3). We can then assume hf = h0 , obtaining that c = Uf /U0 . According to this assumption, we may rewrite Eq. (3) in the following form:

S=

1 qm gh20 cos a(B0 − Bf ) + qm U02 h0 B0 − qm U02 2



Uf2 U02

h0 Bf

(6)

By making dimensionless Eq. (6) with respect to the hydrostatic force of the incident front S0 = 12 qm gh20 cos aB0 , we may write:  2

  2 Uf Bf ˜S = 1 − Bf + 2 Fr0 1− B0 cos a U02 B0

(7)

From Eq. (5), introducing the coefficient c, we obtain:     Fr02 Bf Bf S˜ = 1 − 1 − c2 +2 B0 cos a B0

(8)

with the coefficient c to be experimentally determined. In Eq. (8) we may observe that the ratio Bf /B0 accounts for the presence of the slit, while the coefficient c is necessary to take into account the deviation of the streamlines near the check dam slit. We may call this coefficient contraction coefficient. 3.1. Definition of the coefficient c Starting from the observation of the results of Fig. 5, we may notice that the difference between the data calculated by neglecting the streamlines divergence (i.e. c = 1) and the experimental values increases at increasing slit width. For this reason, we assumed that in Eq. (8) the coefficient c varied as a function of the ratio R = Bf /B0 . Fig. 7 shows the trend of the c coefficient that gives the best fitting between the data and Eq. (8), as a function of ratio R = Bf /B0 . As expected, at increasing the relative influence of the opening, the effect of the lateral deviation of the streamlines increases. The best fitting of the relation between c and R is given by: c = 0.8929 R2 − 1.7556 R + 1.8625

Fig. 7. Evolution of the coefficient c, as a function of the ratio R.

(9)

Fig. 8. Comparison between experimental data and the semi-empirical approach, obtained by combining Eqs. (8) and (9).

This regression curve is characterized by R2 = 0.9996, which indicates a very good agreement between the experimental data and Eq. (9). Combining Eqs. (8) and (9), we obtain the curves reported in Fig. 8, which are in good agreement with the experimental points. In order to verify the general validity of this approach, we performed two further series of tests characterized by different ratios R (0.32 and 0.5), reported in Fig. 8. We represent a further comparison in Fig. 9, which shows the behaviour of the dimensionless impact force S˜ as a function of the ratio R: S˜ = (1 − R) + 2Fr02 (1 − c R)

(10)

where c is computed according to Eq. (9). In this figure each curve is obtained for a constant value of the Froude number. Even though the model has been calibrated on the values of R equal to 0.16, 0.24 and 0.4, Figs. 8 and 9 show a very good agreement between theory and experiments also in the further two tests with R = 0.32 and R = 0.5. 3.2. Final formulation By relating directly the dimensionless impact force S˜ to a practical design value, we may simply apply the following relation:

S=

1 qm g h20 cos a B0 2

  1−

2  

 Bf Bf Fr0 +2 √ 1 − c2 B0 B0 cos a (11)

that is by multiplying the dimensionless impact force for the hydrostatic force of the incident front, we obtain the design value. It is

Fig. 9. Comparison between experimental data and the theory, obtained by combining Eqs. (8) and (9).

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necessary to know the solid fraction concentration, the flow depth and velocity or flow rate of the incident front. These values may be obtained from numerical simulations, by applying suitable twophase models (Pudasaini, 2012; Mergili et al., 2017; Armanini et al., 2009). Then by inserting these values in Eq. (11) the value of the impact force is obtained. We may notice that in order to determine the impact force some authors (Hübl et al., 2009; Lichtenhahn, 1973; Scheidl et al., 2013) use the product between a coefficient a s and the hydrostatic force ˜ which as of the incident front. According to our approach as = S, expected depends on the Froude number. In addition, the same formula may be written as a function of the   momentum, S = ad qm B0 U02 h0 ; in this case the formulation is:  S=

1 cos a 2 Fr02

Bf 1− B0



Bf + 1−c B0 2

   qm B0 U02 h0

(12)

These two approaches are equivalent. 4. The role of the debris flow breaker In order to reduce the dynamic impact against slit check dams, it is diffused in the professional practice the use of debris flow breakers, with the purpose to absorb a large part of the impact. In the literature and in the field three main types of debris flow breakers may be identified: i. grid type: this structure, called flat-board debris flow breaker, consists of an horizontal grid which according to the proposers has the aim of separating the sediments from the water and reducing the volume involved in the events (Watanabe et al., 1980; Mizuyama, 2008; Kim et al., 2012). The grid allows the water to remove from the bulk of the debris flow, so that the sediments stop; ii. prismatic flow-breakers: they consist in prismatic blocks of rectangular or square section, positioned perpendicular to the flow or turned of 45◦ with respect to the flow direction. These structures decelerate the flow without retain material (Zanuttigh and Lamberti, 2006). iii. prismatic flow breaker within a check dam (Fig. 10): prismatic blocks, usually inclined of 45◦ with respect to the flow direction, are situated in the check dam opening. According to Hübl and Suda (2008) and Suda et al. (2010) this structure is aimed at the retention of part of the sediments (due to the check dam) and at the energy dissipation due to the interaction of the debris flow with the prismatic debris flow breakers.

Fig. 10. Example of a debris flow breaker of the III type, in Rio Dona. By courtesy of Bacini Montani, Provincia di Trento.

A load cell was positioned on the check dam and a second one was located on the debris flow breaker in order to distinguish the impact force acting on each of two components of the structure. In the following, we analyze the dynamic impact forces exerted by the mixture on the structure and we compare the results with those obtained for the simple slit-check dam, without debris flow breaker. In particular, we did the comparison in two cases. In the first case, the impact force registered on the slit check dam is compared with the impact force acting on a slit check dam with the same opening width, but in presence of the debris flow breaker within the opening (Fig. 11). In particular, we compared three slits with openings of

In this section we intend to study the third kind of debris flow breaker, which is the most widespread in field applications and would seem to be the one that offers the greatest effectiveness in the absorption of dynamic impact. We plan to analyze its effect on the slit check dam and compare the impact on the dam in the presence and in the absence of the debris flow breaker. These structures are often adopted to trap at the same time the woody material; however this may change the velocity field and the efficiency of the structure. For these reasons we do not consider wood transport in the present study and we suggest to use different structures to trap the woods. 4.1. Experimental evidence In this section we present some results regarding the check dam equipped with a debris flow breaker of the third type, located within the opening of the check dam, as in Fig. 11.

Fig. 11. Scheme of the debris flow breaker inserted in the check dam opening.

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(a) Case I: same slit width of the check dam

357

(b) Case II: same net slit of the check dam

Fig. 12. Comparison between the impact against the slit check dam in presence or without the debris flow breaker.

125, 100 and 80 mm with the same check dams equipped with a flow breaker, so that the net width of the opening was respectively 100, 80 and 60 mm. In the second case, we compared the impact force acting on a check dam of a certain opening width, with another slit check dam characterized by a larger opening, but equipped with a flow breaker so that the net slit was the same. In this case we compared the impact force on check dams with openings of 100, 80 and 60 mm and wider slits equipped by debris flow breaker, so that the net slits were the same of the check dams without the flow breaker. In Fig. 12a we compare the dimensionless impact force for the check dam opening of 125 mm, with (black circles) and without (pink triangles) the debris flow breaker. In this case the ratio R is 0.5 for the check dam without breaker but 0.4 in presence of the debris flow breaker. The main difference is that in the presence of the breaker the impact force on the check dam increases, so that this solution seems to be more onerous for the check dam. We also plotted two curves relative to Eq. (8), considering the coefficient c = 1 (dashed black), that is neglecting the curvature of the streamlines, and with c = 1.2 (dash-dotted magenta) as computed previously through the experimental calibration. The points of the test with the breaker are in between the two curves: indeed, the

behaviour of the streamlines, represented in Fig. 13, is a combination of the two cases. They deviate towards the opening center, but the breaker makes them turn back towards the check dam wings. Fig. 12b shows the comparison between the impact force recorded on the slit check dam with an opening of 100 mm and the case of the opening of 125 mm, but with a breaker of 25 mm (same net slit). Here the ratio R is 0.4 both with and without the debris flow breaker. In this case the impact force in the presence of the breaker is higher again, but the difference between the two cases is smaller. Finally, in Fig. 14 we compare the impact force acting on the whole structure (check dam plus debris flow breaker, black circles) with that corresponding to the only check dam with the same net width of the opening (green triangles). For the only check dam, the impact force is significantly lower, so that the solution with the flow breaker implies a more burdensome condition for the structure. The material impacting the debris flow breaker is deviated to the check dam wings, so that the total impact force increases, and less material flows downstream the check dam with respect to the case without breaker. So, in conclusion, regarding the dynamic impact estimation, the presence of the flow breaker implies an impact force on the structures between the plain wall (R = 0) and the presence of the slit

Fig. 13. Scheme of the streamlines path, in presence of a debris flow breaker within the opening check dam, upstream front view and upper view.

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instead clear that the grain size distribution directly influences the value of flow depth ho (Armanini, 2015), which in our relations is a variable that must be known. Regarding the presence of boulders in a debris flow surge, according to our observations, if the boulders are at least three times smaller than the slit width, they do not influence the check dam functioning. Otherwise, for bigger boulders, the opening could be clogged and its efficiency may be compromised. In this case a possible solution could be to intercept the big boulders through other types of structures in order to avoid the check dam clogging.

6. Conclusions Fig. 14. Comparison of the dynamic impact acting on the whole structure, with and without the debris flow breaker.

(R = 0). This means that, from a structural point of view, the flow breaker does not relieve the check dam, instead it could lead to higher loads. 5. Considerations on the grain size influence In this paper we treated the debris flows as a mixture of grains and water (Armanini, 2013; Takahashi, 2014) composed by a Newtonian interstitial fluid (water) and a granular fluid (sediments). This approach is different from the geotechnical one, which considers the debris flow as a soil, that is an elasto-plastic medium. Moreover, in order to take into account the situations in which it is necessary to distinguish between the pressure of the solid phase and that of the liquid phase, the geotechnical scheme adopts the concept of pore pressure, which in the two-phase approach coincides with the fluid phase pressure (Pudasaini, 2012). In the granular fluid mechanics, the granular medium is treated as a continuum fluid, with the same equations of mass, momentum and energy conservation used for clear water, except for the constitutive relations, which are different from those of a Newtonian fluid. In the literature there are different constitutive models for granular flows, which consider the normal and tangential stresses dependent on the grain size (Bagnold, 1954; Jenkins and Savage, 1983). With reference to Eqs. (1), (2) and Fig. 4 and according to the hypotheses done, the grain size could affect mainly the normal stresses p and the tangential stresses to . However, one of the hypothesis is that the length of the control volume is very small, so that the tangential stresses, acting on a very small distance, are negligible. Regarding to normal stresses, they acts on the inflow section, the outflow section and the wall of the check dam. In the inflow section, by doing the balance of the vertical forces in the neighborhood of the generic point, we obtain that the pressure on the point is equal to the weight of the volume of total fluid above that point: p = qm g(ho − y), with y the elevation of the point. Therefore, according to this relation, the value of the pressure is independent of the size distribution, but it is a function of the flow depth of the incoming flow, the elevation of the generic point considered and the density of the mixture, which depends on the solid concentration. This seems to be a paradox. We may explain it by observing that the constitutive relations determine the flow depth in the inflow section, while if we consider the impact force as a function of the flow depth (hydrostatic force), this dependence disappears. The same reasoning may be applied to pressures on the outflow section. Regarding the pressures on the wall of the check dam, we did not make any hypothesis on them; we only apply the momentum conservation equation, according to which, even if the grain size could affect the final value, it does not affect the balance of the forces. It is

We analyzed the dynamic impact of a debris flow surge against a slit check dam, starting from the rational approach developed by Armanini et al. (2019), by considering that the streamlines of the surge continue undisturbed until the moment of impact, i.e. that the momentum exiting the slit is equal to that of the incoming flow portion of a width equal to the slit. However, we observed that near the opening, the streamlines assume a curvature that allows a greater portion of flow to pass through the slit. We introduced a coefficient c that accounts for this effect, and we calibrated it with experimental data for different hydraulic conditions (different Froude numbers) and different ratios R between the slit width and the channel width. The coefficient c resulted to be a function of the ratio R only. Its meaning is similar to the contraction coefficient: due to the opening of the check dam, there is a contraction of the streamline in it, so that the quantity of the incoming material that pass through the check dam is larger than those located just along the width corresponding to the check dam opening. In the second part of the paper, we analyzed the influence of a debris flow breaker within the opening of the check dam. The impact force is registered both on the check dam and on the breaker. By comparing the force acting in the presence of the breaker or without it, we observed that the impact is more onerous in the first case. Indeed, the incoming flow is deviated from the flow breaker to the check dam wings, so that more material impact on the structure than when flow breaker is not present. This effect is maybe responsible for a major local energy dissipation. Furthermore, the diversion of the flow towards the wings could allow to trap a major quantity of material and further reducing the peak of discharge. Anyway, the debris flow breaker induces major stresses acting on the structure (up to 30% greater), and this should be taken into account in the design phase of the structure.

Acknowledgments The authors are grateful to Giulia Grazioso, who made a great contribution to the work during the course of her Master thesis. The authors would thank the Technicians of the Hydraulic Laboratory of the University of Trento, Andrea Bampi, Lorenzo Forti, Fabio Sartori and Paolo Scarfiello, for their valuable support during the experimental work. References Armanini, A., 2013. Granular flows driven by gravity. J. Hydraul. Res. 51 (2), 111–120. Armanini, A., 2015. Closure relations for mobile bed debris flows in a wide range of slopes and concentrations. Adv. Water Resour. 81, 75–83. Armanini, A., Fraccarollo, L., Rosatti, G., 2009. Two-dimensional simulation of debris flows in erodible channels. Comput. Geosci. 35 (5), 993–1006. Armanini, A., Larcher, M., 2001. Rational criterion for designing opening of slit-check dam. J. Hydraul. Eng. 127 (2), 94–104. Armanini, A., Rossi, G., Larcher, M., 2019. Dynamic impact of a water and sediments surge against a rigid wall. J. Hydraul. Res. 1–12.

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