Abstract

Because of climate change, flood-prone areas are more and more frequently exposed to potential casualties and damage. The capability of the flow to carry relevant quantities of sediments during these critical events adds to the further complexity of the resulting scenarios. The interaction between the flow and the obstacles in flood-inundated areas contributes to an increase in the hazard level and constitutes a relevant concern in the framework of risk analysis. Despite this relevance, the existing literature on the topic is relatively scarce, especially for the estimation of the forces acting on rigid obstacles in the presence of a mobile bed. In the present paper, a recent two-phase shallow-water morphodynamical model, particularly suited for the analysis of fast geomorphic transients, is applied for the numerical simulation of the propagation of a dam-break wave over an erodible floodplain in the presence of a rigid obstacle. The geometry of the test-case is inspired by a recent fixed-bed study reported in the literature, for which extensive experimental and numerical data concerning the flow field and the dynamic loading against the obstacle are available. Results of the numerical simulations contribute to highlight the effects of the obstacle on the changes in the bottom topography.

INTRODUCTION

The presence of obstacles, such as bridge piers, is common in rivers and channels, but they may also be encountered in potentially flooded areas. The magnitude of floods caused by intense rainfall is becoming more and more severe owing to climate change (Hoerling et al. 2001). In turn, the exposition of initially dry areas to flood waves arising from the failure of embankments or dikes has increased. In these cases, the wave impacts on a variety of obstacles such as buildings, piles, industrial and commercial structures, with damaging consequences. The presence of structures in flooding areas has many aspects that require adequate investigation, especially in case of a severe flood like that following a dam break. First, an appropriate risk control strategy, essential to prevent damage and to reduce the losses, requires the individuation of risk areas and the selection of adequate measures for its mitigation, such as structural countermeasures including longitudinal walls, deflection and redirecting structures (Hung et al. 1984). Second, obstacles may considerably affect the flood wave propagation, especially when they are close to the dam and in the first moments after the break (Soares-Frazão & Zech 2007). Third, the impacting force on the obstacles may produce structural damage even compromising their stability.

Finally, the capability of the flood to carry relevant quantities of sediment in the flow is another important aspect. In this case, the sediment concentration influences the characteristic of the mixture flow, moving from the bed and suspended load to debris transport. The interaction between the liquid–solid mixture and the presence of obstacles in the inundated areas both contribute to affect the flood wave propagation. On the one hand, the obstacles deviate the flow pattern, and, on the other hand, the morphological change induced by the flow may expose the obstacle to unexpected loading condition, therefore contributing to its structural damage. For these reasons, the propagation of flood waves over loose sediment bed and its interaction with obstacles constitute a relevant concern in the framework of risk analysis.

In conclusion, the study of the complex interaction between the flood wave and the obstacles is extremely important to evaluate the hazard levels and to support the design of structural countermeasures. This requires the knowledge of both the hydro- and sediment dynamics and the force acting on the structure by the flow (Thieken et al. 2005).

The dynamic impact of a flood wave on a structure and the consequent damage is, in some cases, evaluated by empirical formulas or adopting simplified analysis (e.g. Kelman & Spence 2004). However, these approaches may produce incorrect indications since they are usually derived through site-specific studies. Alternatively, the principal kinematics characteristics of the dynamic impact and the acting force may be numerically reproduced.

The study of a wave impacting on an obstacle has been widely investigated both experimentally and numerically considering clear water flowing on non-erodible beds (e.g. Bukreev 2009; Chen et al. 2014); debris flows (e.g. Canelli et al. 2012; Scheidl et al. 2013), and snow and dry avalanches (e.g. Tai et al. 2001; Chiou et al. 2005; Teufelsbauer et al. 2009; Cui & Gray 2013; Faug 2015).

Among the numerical studies concerning debris flows, Kattel et al. (2018) modelled debris impacting tetrahedral obstacles of different dimensions, number and orientation, reproducing the behaviour of the liquid–solid mixture with the quasi-three-dimensional two-phase model of Pudasaini (2012). The presence of obstacles may increase the solid- and fluid-phase separation, and it strongly influences the wave propagation. Debris flow propagation in urban areas has been recently simulated by Gao et al. (2017) with a depth-integrated continuum model, demonstrating that the obstacles increase the depth and the velocity of the flow and the impact pressure. More recently, Di Cristo et al. (2019) reproduced the impact force of a mud flow on a rigid obstacle with both a single-phase, shear-thinning power-law model and a two-phase model, which separately considers the liquid and solid phases. The comparison suggests that in the situations where the liquid and solid phases remain mixed, the essential features of the interaction with the obstacles and the maximum impact force are similarly predicted by the two models, while some differences are registered in the cases characterized by phase separation.

Concerning clear water flows over non-erodible beds, among the different experimental studies, only few evaluated the impact force. Lobovsky et al. (2014) performed laboratory tests measuring the load produced from a dam break on a vertical wall using miniaturized pressure sensors, while Aureli et al. (2015) evaluated the impact force on a single rigid obstacle.

With regard to numerical studies, Wang et al. (2000) used Saint Venant Equations with a second-order total variation diminishing finite-difference method for solving 1D and 2D dam-break in the presence of obstacles, also discussed the effect of bed slope, bottom friction and the tailwater/reservoir depth ratio. Successively, many other works considered the shallow-water formulation as a reasonable approach for the simulation of these complex processes. For example, Shige-eda & Akiyama (2003) investigated two-dimensional (2D) flood flows and the hydrodynamic force acting on structures, showing that the prediction accuracy of the shallow-water model is reasonable for both hydrodynamics and force on structures. Bukreev (2009) successfully applied a shallow-water model to predict the dynamic action exerted by a dam-break wave on a vertical wall. Soares-Frazao & Zech (2008) and El Kadi Abderrezzak et al. (2009) investigated the potentiality of a depth-averaged shallow-water model in reproducing the propagation in urban areas of both flash floods and dam-break waves. Numerical results, in terms of both flow depths and velocities, fairly agree with experimental ones, although some discrepancies were observed around buildings, where the flow is strongly 3D. More recently, Aureli et al. (2015) compared the predictions of a 2D depth-averaged model, a 3D Eulerian two-phase model and a 3D smoothed particle hydrodynamics model, in the simulation of the forces exerted by a dam-break wave on a rigid squat structure. Comparing the numerical and the experimental results, the authors found that the error in the peak load for the 2D model based on the shallow-water approach was in the order of 10%.

As far as the simultaneous presence of obstacles and of an erodible floodplain are concerned, from the experimental point of view, the tests by Palumbo et al. (2008) and the test-cases studied under the NSF-Pire project (Soares-Frazão et al. 2012), regarding a dam-break in an erodible channel with a sudden enlargement, contributed to shed some light on the basic elements of the interaction of the flood wave with rigid walls. However, they neither considered any obstacle in the floodplain nor evaluated the impact forces.

The numerical analyses which were subsequently developed, support the idea that for these kinds of processes, the shallow water approximation represents a suitable framework, provided that adequate modelling of the non-equilibrium sediment transport is accounted for. To match this latter requirement, several non-equilibrium models have been proposed, which can be grouped into mixture single layer (e.g. Wu & Wang 2007), multi-layer (e.g. Capart & Young 2002; Savary & Zech 2007; Li et al. 2013; Swartenbroekx et al. 2013) or multi-phase (e.g. Dewals et al. 2011; Greco et al. 2012; Rosatti & Begnudelli 2013; Di Cristo et al. 2016).

The objective of this paper is to numerically analyse the propagation of a dam-break wave over an erodible floodplain in the presence of a rigid obstacle. The results of both fixed and erodible bed conditions are compared for understanding the role of the sediment mobility on the hydrodynamics and the impact force. The shallow-water two-phase model of Di Cristo et al. (2016) is adopted for reproducing the impact of a dam break considering the same geometry of the fixed-bed test case by Aureli et al. (2015). In this study, both fixed bed and mobile bed conditions are reproduced, and in the latter case, two different tests characterized by two different sediments constituting the loose bed are considered. The preliminary study of Di Cristo et al. (2018a) shows a comparison between the fixed and mobile bed tests. In the present paper, the comparison is completed with the results presented in terms of the temporal evolution of the flow field, the bottom deformation and the impact force on the obstacle.

The article is organized as follows. In the ‘Material and Methods’ section, both the test-case and the morphodynamical model adopted for the numerical simulations are described. Some details concerning the numerical solution method are also given. In the section ‘The Considered Test-Case’, results of the numerical simulations are presented and discussed. Finally, the main conclusions are drawn.

MATERIAL AND METHODS

The geometry of the test-case considered for the numerical simulations has been selected from the existing literature on clear water dam-break waves impacting against rigid obstacles over a fixed bed. Namely, the study carried out by Aureli et al. (2015) has been considered. In what follows, both the test-case and the morphodynamical model adopted for the numerical simulations are described.

THE CONSIDERED TEST-CASE

Figure 1 reproduces the plan view of the test-case by Aureli et al. (2015). The flow is caused by the sudden removal of the gate, which triggers the propagation of a dam-break wave over the downstream floodable area.

Figure 1

Sketch of the considered test-case (dimensions are in metres).

Figure 1

Sketch of the considered test-case (dimensions are in metres).

In the numerical simulations, the floodplain can be assumed as non-erodible or constituted by loose sediment, namely uniformly graded sand. In the former case, the resulting layout exactly coincides with the scheme investigated by Aureli et al. (2015), while in the latter case two scenarios have been considered by changing the sediment diameter.

THE MATHEMATICAL MODEL

A brief description of the model presented in Di Cristo et al. (2016) is provided below, whereas the reader is referred to the original reference for further details about the underlying assumptions and the subsequent limitations, along with a more exhaustive description of the numerical method employed for the simulation. Since in the present application the suspended load has been neglected, the mathematical model is constituted of the equations expressing the mass and momentum conservation for the liquid (Equations (1) and (3)) and the solid phase transported as bedload (Equations (2) and (4)). Finally, Equation (5) describes the evolution of the loose bed. They read as follows: 
formula
(1)
 
formula
(2)
 
formula
(3)
 
formula
(4)
 
formula
(5)
in which t is the time, g denotes the gravity acceleration, r = (ρsρl)/ρl with ρl and ρs are constant liquid and solid densities, respectively. As far as the flow variables are concerned, δl denotes the liquid-phase volume for the unit bottom surface and δs is the solid-phase volume transported as bedload for the unit bottom surface. Hence, the free surface and bottom elevation are denoted by zw and zB, respectively, h = zwzB = δl + δs. U is the phase-averaged velocity vector, with the subscript l and s used to denote the water and the solid phase, respectively. For both phases, the second-order tensor UU denotes the diadic product of the phase-averaged velocity with itself. Cs is the bedload volume concentration and eB is the bottom erosion/deposition rate.
The source terms of momentum equations Sl and Ss are given by the following formulas: 
formula
(6)
 
formula
(7)
where μd is the dynamic friction coefficient, CCh denotes the dimensionless Chezy coefficient, sB is the bottom slope, D is the drag force of the liquid on the solid particle and α is the coefficient needed to evaluate the collisional stresses. In Equation (7), the first two terms represent the ratio τB,s/ρs, where the bottom shear stress, τB,s accounts for both frictional and interparticle collisional stresses. In Equation (6), the first four terms represent the ratio τB,l/ρl, where the bottom shear stress on the liquid phase, τB,l, is expressed as the difference between the shear stress which would act upon the bottom in the absence of sediment transport and the momentum transferred to the solid phase τB,s. The drag force D is evaluated as follows: 
formula
(8)
where CD is a bulk drag coefficient and d is the sediment particle diameter. The bottom entrainment/deposition is evaluated by means of the following formula (Pontillo et al. 2010): 
formula
(9)
in which ws denotes the sediment settling velocity, corrected to account for its dependence on Cs through the well-known semi-empirical formula by Richardson & Zaki (1954). The dimensionless mobility parameter T, which accounts for the excess of the mobilizing stresses onto the bottom surface with respect to the resisting ones (Van Rijn 1984), is evaluated through the following formula: 
formula
(10)
where τc is the threshold shear stress for particle motion and τB is the magnitude of Mohr–Coulomb stress at the bottom, depending on the static friction coefficient μs. Even if the coefficients α and CD could be estimated from the existing empirical formulas, they are evaluated in the model by means of the following relations, deduced to comply with the equilibrium empirical formulas for transport at capacity (Di Cristo et al. 2016): 
formula
(11)
where k1 and k2 are the following two dimensionless coefficients: 
formula
(12)
 
formula
(13)
In Equation (13), KMPM is the Meyer–Peter and Müller formula coefficient and c1 is a dimensionless model parameter. With the above closures, the model application requires the definition of only three independent dimensionless parameters. Two of them, CCh and KMPM, may be evaluated from the standard literature. The remaining free model parameter c1 is bounded between the lower and upper limits theoretically deduced, with very little sensitivity of the results to its variation (Di Cristo et al. 2016).

The system of Equations (1)–(5) is solved with a numerical code developed by the authors employed also in Di Cristo et al. (2016) and Di Cristo et al. (2018b). The numerical method, using 2D unstructured quadrilateral meshes, relies on a mixed cell-centred (CCFV) and node-centred finite-volume discretization. The former is adopted for the variables δl, δs, Ul and Us, defined at the grid cell centres, the latter for the bed elevation zB, with the control volumes constructed around the mesh nodes by the median-dual partition (Barth & Jespersen 1989; Delis et al. 2011). The numerical fluxes in the CCFV discretization of Equations (1)–(4) are calculated through the first-order Harten–Lax–Van Leer (HLL) scheme (Harten et al. 1983), with second-order reconstruction of the free-surface elevation for subcritical flows, and an appropriate treatment of the bed slope source term is included (Greco et al. 2008).

RESULTS AND DISCUSSION

The model discussed in the previous section has been applied to the numerical simulation of the problem sketched in Figure 1. In the first simulation, the floodplain is assumed to be non-erodible, while in the second series of simulations it is assumed constituted by loose sediments. Two scenarios have been considered by changing the sediment diameter from 5 × 10−4 m (denoted as Test 1), representative of fine sand, to 5 × 10−3 m (denoted as Test 2), which is typical of very coarse sand. In the simulations, the following values of the parameters are assumed: ρs= 2,650 kg/m3, μd = 25°, μs= 38°, KKMPM = 8 and c1 = 0.25. To isolate the effect of bed mobility, in both erodible and non-erodible tests, the same value of the dimensionless Chezy coefficient, CCh = 12, is adopted. A computational mesh with Δx = Δy = 5 × 10−3 m used, along with Δt = 1/1024 s. Based on the simulated flow fields, the impact force is evaluated by numerically computing the following integral (Di Cristo et al. 2018b): 
formula
(14)
where σ denotes the boundary of the obstacle and is the corresponding normal unit vector. The values of the flow variables in the cells adjacent to the obstacle are used.

Figure 2 shows the time history of the impact force F on the obstacle for the fixed and mobile bed cases, along with the simulation results from the shallow-water model of Aureli et al. (2015). As far as the non-erodible case is concerned, the results fairly agree with those of Aureli et al. (2015), even though the force values predicted herein are slightly smaller. This is attributed to the different resistance formula employed in Aureli et al. (2015), i.e., Manning instead of Chezy, and to the value of the dimensionless Chezy coefficient used in the present calculations. Indeed, as shown in Di Cristo et al. (2019), increasing the CCh value up to 25, the agreement between the results of two simulations is strongly improved. After the impact of the wave on the obstacle (t ∼ 0.5 s), an abrupt increase of the force is observed, which reaches its maximum value (F ∼ 6 N) at t ∼ 1.25 s, and then it quasi-monotonically decreases, reaching half of the peak value at t ∼ 3.0 s, when the flood wave completely surrounds the obstacle. Bottom mobility only marginally influences the load dynamics on the obstacle, especially in the first instants. The time at which the impact and the peak force occur, along with the peak force, only slightly depends on the sediment composing the mobile bottom.

Figure 2

Time history of the impact forces over the downstream obstacle for fixed and mobile bed cases.

Figure 2

Time history of the impact forces over the downstream obstacle for fixed and mobile bed cases.

A preliminary discussion of the local characteristics of the flow field has been presented in Di Cristo et al. (2018a). That discussion focused mainly on the results at t = 3 s, at which the flood wave reattaches behind the obstacle. This time is representative of the final part of the investigated process, as the bottom geometry changes slightly going on. In the previous paper, it has been noted that the fine sediment promotes the reattachment of the flow and a reduced flow height in front of the obstacle is observed, which is consistent with the lower values of the impact force (Figure 2). As far as the bottom topographies are concerned, the case with the fine sediment is characterized by a more pronounced scour hole just downstream the original dam position and at the upstream corners of the obstacle (with depths of about 0.07 m). Finally, it is observed that the interaction with the coarser sediment produces a reduction of the momentum in the region close to the original dam.

In what follows a more detailed analysis is provided, which is mainly focused on the near- and mid-field phase of the evolution of the process (t ≤ 2.0 s). Namely, the free- and bed-surface elevation of the solid-phase volume (for the unit bottom surface) transported as bed load (δs) for the two cases d = 5 × 10−4 m (denoted as Test 1) and d = 5 × 10−3 m (denoted as Test 2) are discussed in Figures 37 for different instants. Moreover, for the sake of comparison, the free-surface elevation of the fixed bed (denoted as Test 0) is also shown.

Figure 3

Results at t= 0.25 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 3

Results at t= 0.25 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 4

Results at t= 0.5 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 4

Results at t= 0.5 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 5

Results at t= 1.0 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 5

Results at t= 1.0 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 6

Results at t= 1.5 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g). Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2019.014.

Figure 6

Results at t= 1.5 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g). Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2019.014.

Figure 7

Results at t= 2.0 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

Figure 7

Results at t= 2.0 s. Free-surface elevation for d = 5 × 10−4 m (a), d = 5 × 10−3 m (b) and fixed bed (c). Bedload solid-phase volume (for the unit bottom surface) for d = 5 × 10−4 m (d) and d = 5 × 10−3 m (e). Bottom erosion/deposition for d = 5 × 10−4 m (f) and d = 5 × 10−3 m (g).

At t= 0.25 s, Figure 3(b) and 3(c) indicate that the free-surface elevation for Test 2 and Test 0 shows only marginal differences. On the other hand, in the d = 5 × 10−4 m case (Figure 3(a)), the shape of the wave is characterized by a more pronounced diffusion in the transversal direction. Namely, the longitudinal distance reached by the wave front does not substantially differ from the other two tests, whereas the width of the wave body in the transversal direction is sensibly larger. Indeed, while in Test 1 the bed load is already appreciable, in the d = 5 × 10−3 m case, it is only perceivable (Figure 3(d) and 3(e)). At the same instant, the erosion process is taking place in the d = 5 × 10−4 m case while it is negligible in Test 2 (Figure 3(f) and 3(g)).

For all the tests, the time t= 0.5 s is very close to the moment in which the wave impacts against the obstacle (see Figure 2). Soon after, the wave is deflected and splits into two symmetrical bodies flowing laterally to the obstacle (Figure 4(a)–4(c)). The comparison among Figure 4(a)–4(c) indicates that the fixed-bed case is characterized by the highest value of the celerity propagation, as suggested by the time evolution of the impact force shown in Figure 2. Interestingly, the presence of erodible bottom appears to slow down the celerity of the wave, even though this effect is marginally influenced by the sand diameter value. Globally, independently of the sand diameter, the presence of erodible bed only marginally influences the hydrodynamics of the wave at this instant (see Figure 4(a)–4(c)). However, the main peculiarity of the erodible bed cases at this instant is the presence of a recirculation zone in the region very close to the gate. This flow feature is particularly evident in Test 1 (Figure 4(a)), whereas it is not found in Test 0. The δs maps (Figure 4(d) and 4(e)) put in evidence that, owing to the large diameter of the sand, in Test 2, the bedload transport occurs mainly in the centre of the wave, where the high velocity values are able to entrain sediments from the bottom (Figure 4(e)). In contrast, in the d = 5 × 10−4 m case (Figure 4(a)), the δs footprint strongly resembles one of the liquid phases and therefore almost everywhere the fluid flow is able to entrain sediment from the erodible bottom inducing the bedload movement. The differences in the erosion/deposition process between Tests 1 and 2 are clearly shown in Figure 4(f) and 4(g). The high velocity values very close to the gate produce an intense erosion process, which is more evident for the simulation with fine sand (Figure 4(f)) rather than in the d = 5 × 10−3 m case (Figure 4(g)).

At t = 1.0 s (Figure 5), in all tests, the wave surrounds the obstacle, laterally deflecting and overcoming it. However, the deflected wave has not reached the lateral walls yet. Moreover, created by the reflection of the impacting wave against the obstacle, a discontinuous front just upstream the obstacle is visible in all cases. While in Test 2 the shape of the wave still strongly resembles one of the fixed-bed simulation (Figure 5(b) and 5(c)), the small value of sand diameter in Test 1 causes a transversal diffusion stronger than in Test 0. In the d = 5 × 10−4 m case, the intense bedload, witnessed by the high δs values just upstream of the obstacle, suggests that a strong momentum transfer from the liquid phase towards the solid one is occurring (Figure 5(d)). The erosion process, which is particularly violent in the zone just downstream of the gate, induces a consistent reduction of the surface bed elevation (Figure 5(f)). Moreover, starting from the upstream edges of the obstacle, two thin eroded limbs are clearly appreciable. In Test 2, the bed load activity is weaker than in Test 1, and it takes place in a smaller portion of the channel (Figure 5(e) and 5(g)).

At t= 1.5 s in all tests (Figure 6(a)–6(c)), the impact of the wave with the lateral walls has taken place because of the lateral deflection, and moreover, two additional discontinuous wave fronts are generated symmetrically with respect to the obstacle. The wave reaches the flume outlet and leaves a large dry region behind the obstacle. The comparison between Figure 6(a) and 6(c) demonstrates that the bed erodibility promotes the lateral wave distortion. The movement of the solid phase is seen to strictly follow the water movement in the d = 5 × 10−4 m case, although, due to the interaction of the two-phase current with the solid walls (obstacle and lateral walls), three localized islands with high values of δs have been created (Figure 6(d)). In Test 2, the solid transport is less vigorous than in Test 1, and it occurs only dowstream of the obstacle (Figure 6(e)). The intense bedload leads in Test 1 contributes to a solid deposition in a very concentrated zone just upstream of the obstacle (Figure 6(f), red colour). In contrast, in the d = 5 × 10−3 m case, the deposition process occurs, still upstream of the obstacle, but in a larger part of the channel than in Test 1 (Figure 6(g)).

Figure 7 shows that, independently of the bed erodibility, at t = 2.0 s the discontinuous fronts created by the interaction of the two currents with the lateral walls propagate symmetrically towards the centre of the channel, significantly reducing the extension of the dry zone present in the back of the obstacle. Such a modification is particularly evident in the d = 5 × 10−4 m case (Figure 7(a)). In Test 1, the erodibility of the bed determines the presence of superficial waves characterized by small values of wavelength which are seen to be associated with bed forms (see Figure 7(a), 7(d) and 7(f)). Similar bed forms have been frequently found in analogous geomorphic transients (e.g. Spinewine & Zech 2010) and, as it has been theoretically shown, their formation may be described even by a shallow-water approach, provided that a dynamic description for the sediment transport is considered (Di Cristo et al. 2006; Vesipa et al. 2012; Greco et al. 2018). Finally, the zones close to the upstream edge of the obstacle in which a deposition started at t = 1.0 s increase their extension and, moreover, a deposition process occurs symmetrically close to the lateral walls. In Test 2, the presence of bed forms is not detected, and only a deposition process is found upstream of the obstacle but in a smoother way than in Test 1. The geometry of this depositional pattern is elongated in the transversal for an extension approximately three times the obstacle width. No appreciable deposition is detected alongside the lateral walls.

The analysis of the present process with a two-phase morphodynamical model provides the opportunity to discuss the effects of the grain size on the dynamics of the solid phase during the impact against the obstacle. To this aim, Figure 8 compares the velocity vector plots referring to the two Test Cases 1 and 2, representing the velocity field of both the liquid and the solid phases at t = 1.0 s. At this time, the wave overcomes the obstacle, and the fluid/sediment mixture is deflected in the lateral direction.

Figure 8

Velocity vector plots for d = 5 × 10−4 m (a) and d = 5 × 10−3 m (b). Black (red) arrows refer to liquid (solid)-phase velocities. Vector density has been reduced to improve readability. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2019.014.

Figure 8

Velocity vector plots for d = 5 × 10−4 m (a) and d = 5 × 10−3 m (b). Black (red) arrows refer to liquid (solid)-phase velocities. Vector density has been reduced to improve readability. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/hydro.2019.014.

The results obtained with the fine sand (Figure 8(a)) show that the velocity vectors of the solid phase have a smaller magnitude of the liquid-phase ones, but they present nearly the same direction. The magnitude of maximum difference between the velocities, i.e. the maximum of the module of the difference between the liquid and the solid velocities, is 0.44 m/s. In contrast, the coarse sand (Figure 8(b)) is characterized by a much more pronounced resistance to adapt to the change in the flow direction induced by the impact against the obstacle, and a sensible disalignment of the velocity vectors of the two phases is detected in the region close to the obstacle corner. In this case, the magnitude of maximum difference between the velocity of the solid and liquid phases is 0.64 m/s. This result and the comparison between Figure 8(a) and 8(b) demonstrate that the hypothesis of collinearity of the motion direction of the two phases, which is assumed in the single phase and the mixture models, may fail in the presence of coarse sand, whereas it is reasonably satisfied if the sediment is sufficiently fine.

CONCLUSIONS

In the present paper, a numerical study of the impact of a dam-break wave propagating over an erodible floodplain against a rigid obstacle has been discussed. The investigated condition has been inspired by a laboratory experiment recently reported in the literature with a non-erodible bed. To cope with the features of the complex morphodynamics characterizing the present test case, a recent shallow-water two-phase model, numerically integrated with a proper method, has been adopted. Results have been analysed in terms of impact force on the obstacle, flow field and bottom deformation. The effect of the bottom erodibility has been discussed by comparing with the corresponding fixed-bed condition. Moreover, the effect of the diameter of the sediment constituting the loose bed on the two-phase flow field and on the bottom evolution has been discussed for fine and coarse sand.

Bottom mobility has been found to affect the shape of the wave starting from the very beginning of the geomorphic transient, influencing the propagation of the wet-dry front in the transversal direction, especially for fine sand. As far as the evolution of the erodible bottom is concerned, fine sand has been found to determine more pronounced scour at the original dam location and at the upstream corners of the impacted obstacles compared to the case with coarse sand. On the other hand, the coarse sediment induces the appearance of a larger deposit area immediately upstream of the obstacle, elongated in the transversal direction with a width of about three times the obstacle. Additionally, the analysis of the flow field of solid and liquid phases has shown that under similar circumstances, the hypothesis of collinearity of the motion direction of solid and liquid phases, adopted in the single phase and mixture models, may fail in the presence of coarse sand.

ACKNOWLEDGEMENTS

The work described in the present paper was realized in the framework of the project MISALVA and financed by the Italian Minister of the Environment, Land Protection and Sea (CUP H36C18000970005).

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