Thierry Faug
email: thierry.faug@irstea.fr

On depth-averaged analytic solutions for free-surface granular flows impacting rigid walls down inclines

In the present paper, flows of granular materials impacting wall-like obstacles down inclines are described by depth-averaged analytic solutions. Particular attention is paid to extending the existing depth-averaged equations initially developed for frictionless and incompressible fluids down a horizontal plane. The effects of the gravitational acceleration along the slope, and of the retarding acceleration caused by friction as well, are systematically taken into account. The analytic solutions are then used to revisit existing data on rigid walls impacted by granular flows. This approach allows establishing a complete phase-diagram for granular flow-wall interaction.

I. INTRODUCTION

The influence of obstacles on the propagation of granular flows is today a booming research area. Flow-obstacle interactions in granular flows have recently drawn attractions from different disciplines in science and technology because the flows past obstacles are locally complex and highly non-linear. Flow regimes can change from fast to slow states, as well as from diverging to converging states, and vice versa.

There are for instance several studies in geophysics which used laboratory tests on granular flows to mimic full-scale mitigation structures against avalanches, in order to analyse some complex problems yet to be solved: the runup on avalanche catching dams [1][2][3][4][5], the runout shortening caused by catching dams or dissipative structures such as retarding mounds [START_REF] Hákonardóttir | A laboratory study of the retarding effect of breaking mounds in 3, 6 and 9 m long chutes[END_REF][START_REF] Hákonardóttir | [END_REF][8][9][10][11], the deflection induced by a snow shed [START_REF] Platzer | Snow engineering[END_REF], a catching dam [START_REF] Hákonardóttir | [END_REF] or a deflecting dam [14], and the effect of a complex protection system including mounds and dams [15]. Other small-scale studies used numerical simulations based on depth-averaged models [16], discrete element method [17] or full dimensional models coupled with depth-averaged models [18] to tackle the influence of retarding mounds, wedges or walls on the flow geometry. Many authors have recently combined small-scale numerical simulations and laboratory tests [19][20][21][22][23][24]. More rarely, some controlled experiments on flows of geomaterials-such as snow-interacting with obstacles were carried out on large-scale flumes to investigate the effect of snow sheds [25], retarding mounds [26] or deflecting dams [27] but those experiments remain costly.

The overarching aim of the present paper is to pay attention to rigid walls spanning the whole width of the incoming granular flow. The numerical simulations based on discrete element method [17,23,[28][29][30][31] or the full dimensional continuum models, as recently developed by Domnik and Pudasaini [32], are important and necessary contributions to the topic. In the present paper, various analytic solutions are proposed for describing the main kinematics of the granular patterns formed when flows of granular materials hit walls down inclines. A depth-averaged approach, which ignores the details of the discrete nature of the flowing material, is proposed. The particular behaviour of flowing grains is considered through choosing relevant values of the effective parameters that characterize the assembly of grains.

The depth-averaged analytic solutions proposed in the present study systematically take into account the effects of both the potential energy caused by gravity and the energy dissipated by friction, as recently done for the kinematics of granular flows without any obstacle by Pudasaini and Domnik [33]. Previous studies on small-scale granular flows past rigid walls are revisited and cross-compared in order to draw a complete phase-diagram. The latter is helpful for evidencing the conditions needed to form each of the various granular patterns induced by flow-wall interaction.

The present paper is organized as follows. Section II describes the general depth-averaged analytic solutions for bulky frictional granular flows impacting rigid walls down inclines. In the first place, the depth-averaged equations for flows without any obstacle are shortly presented (Sec. II A). Second, a general equation for the stationary granular jump is proposed (Sec. II B). Third, general equations for propagating granular jumps, namely granular bores, are given (Sec. II C). Fourth, the critical wall height below which the granular bore does no longer propagate upstream of an obstruction, and a large airborne jet is formed downstream of the wall, is tackled (Sec. II D). Section III revisites a great number of studies on various granular patterns resulting from flow-wall interaction, such as airborne granular jets, diffuse and strong granular jumps, and granular dead zones. Finally, a complete phasediagram governing the granular flow-wall interaction is prososed in Sec. IV, in line of the depth-averaged analytic solutions described in Sec. II.

II. ANALYTIC SOLUTIONS FOR GRANULAR FLOWS IMPACTING WALLS

A. Depth-averaged equations for granular flows down inclines Gravity-driven free-surface flows of granular materials are well described by depthaveraged mass and momentum conservation equations. These equations were introduced in 1871 in order to describe the non-permanent water flows in application to river floods [34].

Then, the classical assumption of shallow water flows was applied to granular flows [35,36] and motivated the mathematical modeling of natural hazards such as landslides [37], debris flows or mudflows [38,39], rock avalanches [START_REF] Naaim | Multiple Scale Analyses Coupled Phys. Syst.: Saint-Venant Symp[END_REF] and snow avalanches [22,[START_REF] Naaim | [END_REF]. The key idea is to consider that the vertical length scale (the flow thickness h) is small with respect to the horizontal length scale L along the slope. The assumption = h/L 1 is reasonable for many geophysical flows. It is then possible to depth-average the three-dimensional local conservation equations. The full derivation for granular avalanches can be found in [35,36].

Mass [Eq.( 1) below] and momentum [Eq. (2)] balances applied to an infinitesimal portion of fluid can quickly give the depth-averaged equations for a two-dimensional flow down a slope making an angle θ with the horizontal [42]:

∂h ∂t + ∂hū ∂x = 0, (1)
ρ P φ ∂hū ∂t + β ∂hū 2 ∂x = ρ P φgh cos θ tan θ -µ b -k ∂h ∂x . (2)
h is the flow thickness and ū = 1 h h 0 u(z)dz is the depth-averaged velocity. ρ P is the particle density and φ is the volume fraction of the flowing granular material, which is constant (the fluid density is ρ = ρ P φ) as a direct result of 1. The x-coordinate corresponds to the direction of the main flow and t is the time. β is defined as the ratio between the depth-averaged value of the velocity square and the square of the depth-averaged

velocity: β = ū2 /ū 2 .
The forces at stake considered in Eq.( 2) and balancing the inertial forces are: the flow weight (ρ P φgh sin θ), the friction force (τ b = µ b φρ P gh cos θ, where µ b is the basal effective friction coefficient) and the force related to the thickness gradient (kρ P φgh cos θ ∂h ∂x , where k is the earth pressure coefficient relating the normal stresses σ xx and σ zz through σ xx = kσ zz ).

The physical processes associated with the discrete nature of the granular medium-which forms the flowing layer-are not considered in detail. Relevant values should be given to the granular constitutive parameters k, β and µ b , as it will be shortly discussed thereinafter.

The value of k can be derived from soils mechanics concepts, as early suggested by Savage and Hutter [35]:

k pass/act = 2 1 ± 1 -(1 + tan 2 δ) cos 2 ϕ cos 2 ϕ -1, ( 3
)
where δ is the bed friction angle, ϕ is the internal friction angle and k pass/act = k pass for a passive state (∂u/∂x < 0) or k act for an active state (∂u/∂x > 0). Eq.( 3) with δ = ϕ yields k pass/act = (1 + sin 2 ϕ)/(1 -sin 2 ϕ). Past studies have shown that k = 1 would be a reasonable assumption for sheared granular flows in the steady and uniform regime [43].

More recent studies have highlighted a slight anisotropy in normal stresses for the steady and uniform regime: k = σ xx /σ zz = 1.05 [44]. It is likely that k is influenced by the divergence of the flow [45], which may lead to values different from 1 when the free-surface is strongly non stationary and/or non uniform.

β depends on the shape of the velocity profile. It is generally taken to be equal to 1, which corresponds to the exact value for plug flows. It can be calculated provided an assumption on the shape of the velocity profile: β = 4/3 for linear velocity profiles and β = 5/4 for Bagnold velocity profiles.

µ b can be taken as a constant [36] or more complex velocity-dependent friction laws can be considered. The granular constitutive friction laws are expressed through either frictionalcollisional concepts [46][47][48][49] or the dependence of µ b on the inertial number, namely the µ(I)-rheology described by MiDi [50] and Jop et al. [51].

The This paper presents general depth-averaged equations in order to describe a steady granular flow down an incline and interacting with a rigid wall. The wall spans the whole width of the incident flow. Note that all the equations given in the following sections should apply to incoming flows (before impacting walls) in steady-state regime.

B. Stationary granular jumps

Granular jumps, namely large discontinuities in depth and velocity, are typically observed when a rapid granular flow hits a rigid wall. When overflow is possible downstream of the wall, a stationary granular jump (see Fig. 1 in the current section) can be produced. When the obstruction prevents overflow downstream of the wall, the granular mass is entirely blocked and a granular bore propagates upstream from the obstruction (see Fig. 3a in the next section).

Full equation for steady jumps

By analogy to hydraulic jumps [START_REF] Chow | Open-Channels Hydraulics[END_REF], a theory can predict the change in depth in the granular jump [START_REF] Morrison | [END_REF][55][56]. A sketch of a stationary granular jump is given in Fig. 1. h i , ūi and ρi are the thickness, the depth-averaged velocity and the depth-averaged density at the locations i, where the subscripts i = 1 and 2 refer respectively to locations upstream and downstream of the jump region. By making use of the continuity equation (ρ

1 ū1 h 1 = ρ2 ū2 h 2 ),
the momentum equation applied to a control volume surrounding the jump yields:

ρ1 ū1 h 1 (β 2 ū2 -β 1 ū1 ) = P 1 -P 2 + W sin θ -F f . (4)
W is the weight of the granular material in the jump:

W = 1 2 KgL * (ρ 1 h 1 + ρ2 h 2 )
, where L * is the length of the granular jump region (see Fig. 1) and K is a coefficient accounting for the shape of the jump. P 1 and P 2 are the lateral pressure forces acting on sections S 1 and S 2 (see Fig. 1):

P i = 1 2 k i ρi gh 2 i cos θ,
where k i is the earth pressure coefficient (see discussion in Sec. II A). F f is the frictional force between locations 1 and 2 which can be expressed as F f = µ e W cos θ, where µ e is the effective coefficient of Coulomb friction within the jump volume, as early proposed by Savage [55] and recently revisited by Faug [START_REF] Faug | AIP Conference Proceedings[END_REF]. Some calculation allows deriving the relation between the Froude number F r 1 = ū1 / √ gh 1 cos θ of the incoming flow (defined at section S 1 in Fig. 1), the jump depth ratio h 2 /h 1 and the density variation ρ2 /ρ 1 across the jump:

F r 2 1 = 1 2 k 2 ρ2 ρ1 h 2 h 1 2 -k 1 - KL * h 1 1 + ρ2 ρ1 h 2 h 1 (tan θ -µ e )     1 β 1 -β 2 ρ2 ρ1 h 2 h 1     , (5)
where various parameters, such as the earth pressure coefficients k i and the Boussinesq momentum coefficients β i , can be estimated for granular flows (see Sec. II A). Apart from a recent study by Faug et al. [56] (see extended discussion in Sec.III A), past studies paid little attention to the other parameters appearing in Eq.( 5): the jump shape parameter K, the length of the jump relative to the thickness of the incoming flow L * /h 1 , and the difference between the tangent of the slope angle and the effective friction within the jump (tan θ -µ e ). Froude number but the increase rate is decreased when the density variation across the jump increases. Recent experiments by Faug et al. [56] showed that, at a given slope angle (which generally controls the Froude number), dilute flows produce jumps that are thinner than those produced by dense flows. This result is consistent with predictions shown in Fig. 2a.

Incompressible jumps (ρ 2 /ρ 1 = 1) are considered in the following of the present section.

How the relation between h 2 /h 1 and F r 1 is influenced by k i , β i , L * /h 1 , K and tan θ -µ e can be analysed then. then a reasonable assumption. The influence of K and tan θ -µ e remain generally weak.

A significant influence appears if the length of the jump relative to the thickness of the incoming flow, L * /h 1 , is large.

It can be concluded that the relation between h 2 /h 1 , F r 1 and ρ2 /ρ 1 is strongly influenced by the length of the jump, and, to a lesser extent, by a possible change in the earth pressure coefficient across the jump.

Simplified solution for incompressible steady jumps

This section considers incompressible, or slightly compressible jumps, for which ρ2 = ρ1 .

Such jumps are obtained if the incoming flow are not too dilute. Dilute flows produce compressible jumps for which the change in density across the jump cannot be ignored, as recently evidenced by [56]. Under the additional assumptions k i = k (no change in earth coefficient across the jump) and β i = β (no variation of velocity profiles across the jump), it is worth establishing that Eq.( 5) can be written in the following form:

h 2 h 1 2 + h 2 h 1 - 2βF r 2 1 k -K L * h 2 -h 1 (tan θ -µ e ) = 0. (6)
In hydraulics [START_REF] Chow | Open-Channels Hydraulics[END_REF], it is generally assumed that the product K L * (h 2 -h 1 ) , which accounts for the geometry of the jump, can be expressed as a simple function of F r 1 . Recent experiments on the shape of standing granular jumps [56] suggested that it is a reasonable assumption for granular flows too, provided that the jumps are incompressible or slightly compressiblemeaning that the density variation across the jump is nil or weak. The assumption that the Froude number of the incoming flow strongly controls the shape of the jump will be further discussed at section III A 1. Eq.( 6) has the following solution:

h 2 h 1 = 1 2   1 + 8βF r 2 1 k -K L * h 2 -h 1 (tan θ -µ e ) 1/2 -1   . (7)
For frictionless flows (µ e = 0) with no deviation from a purely hydrostatic pressure (k = 1) and uniform velocity profiles (β = 1), Eq.( 7) leads to the traditional equation proposed for hydraulic jumps on steep slopes by Chow [START_REF] Chow | Open-Channels Hydraulics[END_REF]:

h 2 h 1 = 1 2   1 + 8F r 2 1 1 -K L * h 2 -h 1 tan θ 1/2 -1   . ( 8
)
3. Asymptotic solution for a granular flow over a small bump Eq.( 7) can be used to describe a granular flow passing slowly over a bump of height H 0 .

The latter situation is similar to a very diffuse granular jump for which h 2 = H 0 + h 3 (where h 3 is the critical depth at the top of the bump), h 3 -h 1 = 0 (where 0 ∼ 0) and K → 1.

Eq.( 7) yields:

H 0 h 1 + 0 h 1 + 1 = 1 2   1 + 8βF r 2 1 k -L * H 0 + 0 (tan θ -µ e ) 1/2 -1   . (9)
By considering 0 → 0 and H 0 /h 1 → 0 (small bump), one can derive a critical value F r c

for the Froude number of the incoming flow:

F r c = 1 β k - L * H 0 (tan θ -µ e ) . (10)
The above equation suggests that the critical value of the Froude number-classically equal to 1 for horizontal water or air flows over a bump-would depend on β, k, (tan θ -µ e ), and L * H 0 in the case of frictional granular flows down inclines.

C. Propagating jumps upstream of an obstruction

The granular jump propagating upstream of an obstacle at speed U (see sketch in Fig. 3a), namely the granular bore, is for instance observed when a rapid granular flow hits a wall spanning the entire width of the flow [3,[START_REF] Gray | [END_REF]. Detailed features of that type of granular bore have been investigated with the help of computations by Pudasaini and Kröner [4]. The granular bore can be also observed when the material flows past a change in slope from a high slope to a lower slope [3,59]. In geophysics, this situation is typical of some avalanches paths with a sharp change in slope which occurs at the transition from the flowing zone to the runout zone.

Full equation for bores

By choosing a reference frame travelling at the speed U in the direction opposite to the incoming flow, mass and momentum conservation equations across the jump give (Fig. 3a):

ρ1 h 1 (ū 1 + U ) = ρ2 h 2 U, (11)
ρ2 h 2 U 2 -ρ1 h 1 (ū 1 + U ) 2 = 1 2 k 1 ρ1 gh 2 1 cos θ - 1 2 k 2 ρ2 gh 2 2 cos θ + W sin θ -F f . (12)
Vertical velocity profiles are assumed uniform for a sake of simplicity (β = 1). The weight of the jump W and its effective friction F f with the bottom (and side-walls for confined flows) are determined by the equations proposed by Savage [55] and detailed in the previous section. A relation between the Froude number F r 1 = ū1 / √ gh 1 cos θ, the depth ratio h 2 /h 1 and the density ratio ρ2 /ρ 1 can be derived from Eqs.( 11) and ( 12):

F r 2 1 = 1 2 k 2 ρ2 ρ1 h 2 h 1 2 -k 1 - KL * h 1 1 + ρ2 ρ1 h 2 h 1 (tan θ -µ e ) 1 - 1 ρ2 ρ1 h 2 h 1 . ( 13
)
Similarly to the stationary granular jump, one can analyse how the depth ratio of the propagating bore is influenced by the various parameters: k i , tan θ -µ e , K and L * /h 1 . Such an analysis, which gives similar results to those obtained for the stationary granular jump, is not detailed here.

Eq.( 13) is slighly different from Eq.( 5). Figure 3b shows the depth ratio h 2 /h 1 versus F r 1 for a propagating jump [Eq.( 13)] compared to a steady granular jump [Eq.( 5)]. It is considered that the density variation across the jump is nil (ρ 2 /ρ 1 = 1) and the following values for the various parameters are used:

k i = 1, K=1, L * /h 1 = 10 and tan θ -µ e = 0.11.
At a given value of the Froude number of the incoming flow, the propagating bore is thicker than the stationary jump. This result is not general but depends on the combination of parameters, as detailed in [START_REF] Faug | AIP Conference Proceedings[END_REF].

The equations proposed above are a slight extension of the equations which were initially proposed by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] but did not consider the source term related to friction acting within the jump volume. Through accounting only for the source term related to the component of the gravitational acceleration along the slope, Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] distinguished two cases: the angle of repose of the granular material ϕ is either smaller or greater than the slope θ. The analysis by [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] leads to two schematic diagrams of the granular bore. When ϕ ≥ θ, the depth h 2 of the bore is constant, and the free-surface downstream of the jump is parallel to the bottom, as drawn in Fig. 3a. In contrast, when ϕ ≤ θ, the depth h 2 is increased while the flow approaches the wall [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF], and the slope of the free-surface is equal to ϕ. The latter situation was studied in detail by Pudasaini et al. [3] through small-scale laboratory tests. The shape of the granular bore evolving over time could be reproduced well by depth-averaged numerical simulations-including shock capturing technique-once the measured model parameters were carefully implemented, as demonstrated by Pudasaini

and Kröner [4]. In their study, Pudasaini and Kröner [4] evidenced a strong influence of the internal friction angle of the granular material. 13) and ( 5).

Simplified solution for bores

For frictionless materials (µ e = 0) flowing down an horizontal plane (θ = 0), and under the assumption k i = 1, an estimate of the depth ratio can be obtained:

F r 1 = h 2 h 1 1 2 ρ2 /ρ 1 - 1 h 2 /h 1 - 1 h 2 /h 1 2 + 1 h 2 /h 1 3 1 ρ2 /ρ 1 . (14)
Eq.( 14) with ρ2 /ρ 1 = 1 yields a cubic function of h 2 /h 1 :

h 2 h 1 3 - h 2 h 1 2 -(1 + 2F r 2 1 ) h 2 h 1 + 1 = 0, (15)
which can be solved with the Cardano's method. Eq.( 15) has only one physically meaningful solution, as earlier discussed by Jóhannesson et al. [START_REF] Jóhannesson | Background for the determination of dam height in the SATSIE dam design guidelines[END_REF]:

h 2 h 1 = 1 3 2 cos δ 0 6F r 2 1 + 4 + 1 , (16)
where δ 0 is given by:

δ 0 = 1 3 arctan F r 1 27(16 + 13F r 2 1 + 8F r 4 1 ) 9F r 2 1 -8 . (17)
Note that Eqs.( 16) and ( 17) are currently considered as important criteria in the European guidelines for the design of protection dams against avalanches [START_REF] Jóhannesson | The design of avalanche protection dams. Recent practical and theoretical developments[END_REF]. However, it is worthy to stress that those equations do not take into account some important aspects of granular flows, such as possible changes in density and velocity profiles across the jump, and the effect of friction as well. The equation proposed to derive the height of a granular bore [Eq.( 13)]

needs to be verified on new relevant laboratory tests and/or discrete numerical simulations.

Attention should be paid to low and moderate values of F r 1 , and to all parameters involved in Eq.( 13): k i , K, L * /h 1 and µ e .

D. From a granular bore (upstream of the wall) to a large airborne jet (downstream of the wall)

The previous section addressed the granular bore upstream of a wall while overflow downstream of the wall was prevented. One can wonder what might occur when the wall height H relative to the incident flow h 1 is decreased, which causes overflow downstream of the wall. The latter situation is depicted in Fig. 4a. As H/h 1 decreases, an increasing proportion of the flow can escape. Finally the bore no longer propagates upstream of the wall, and a large jet forms downstream of the obstacle, as displayed in Fig. 4b. Hence, as suggested by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF], there exists a minimum height of the wall for which a bore is present upstream of the wall. A mathematical description of the bore upstream of the wall when some flow overtops the wall can be formulated indeed. A description, which accounts for the source terms (gravitational acceleration along the slope and retarding acceleration caused by friction), is proposed in the following. This is again a slight extension of the earlier analytical solution proposed by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] for frictionless flows down a slope.

The conservations of mass and momentum fluxes across the jump in the reference frame travelling upstream with the bore at speed U give (Fig. 4a): where notation similar to that used for the stationary granular jump equation (see Eq.( 4)

ρ1 h 1 (ū 1 + U ) = ρ2 h 2 (ū 2 + U ), ( 18
) ρ2 h 2 (ū 2 + U ) 2 -ρ1 h 1 (ū 1 + U ) 2 = 1 2 ρ1 gh 2 1 cos θ - 1 2 ρ2 gh 2 2 cos θ + W sin θ -F f , (19)
in section II B 1) is considered.

The mass flux conservation between the granular jump and the top of the wall gives

h 3 ū3 = h 2 ū2
. Following the assumption of a critical flow at the top of the wall [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF], with F r 3 = F r c (it is assumed a generalized critical value of the Froude number likely to be different from 1), it yields:

ū3 = F r c gh 3 cos θ. (20)
As described by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF], the bore propagates upstream until the mass flux over the jump equals the mass flux over the obstacle (ū 1 h 1 = ū2 h 2 = ū3 h crit 3 ) and the critical depth of the flow over the obstacle is then given by:

h crit 3 = ū1 h 1 √ g cos θ 2/3 F r -2/3 c , (21)
by considering the critical value F r c of the Froude number, which can be different from 1

for frictional granular flows down inclines. The lower the obstacle height is, the more mass is transported over the obstacle, and when:

H h 1 h 2 h 1 - h crit 3 h 1 , (22)
all the approaching mass flux is transported directly over the obstacle and the bore does not need to propagate upstream. In this context, ones moves from a granular bore to a standing granular jump [U = 0 in Eqs.( 18) and ( 19)]. The critical wall height below which the bore does not exist is then given by the following relation:

H jet c h 1 = h 2 h 1 - F r 1 F r c 2/3 , (23)
where the ratio h 2 /h 1 is given by Eq.( 5) (full equation) or Eq.( 7) (approximate solution for incompressible granular fluids). Eq.( 23) includes the effect of the slope and the effective friction, and will be used to interpret some of the existing experimental data described in section III B.

The propagation of granular bores at high slope over a smooth bottom (meaning that the typical size of roughness is much smaller than the grain size) has been studied by laboratory tests [3] and successfully captured by depth-averaged numerical simulations [4], provided that the constant friction parameters, measured independently, were carefully implemented to more recent studies [3-5, 10, 20, 56, 60, 65]. The following sub-sections revisits some of the existing data on granular jumps, in line with the analytic solutions proposed in section II.

The stationary granular jump

Brennen et al. [START_REF] Brennen | [END_REF] distinguished two types of stationary granular jumps and identified three types of flows between the jump and the wall (flat plate weir in their tests). At low F r 1 , the flow in the jump consisted of a smooth expansion whereas, at high F r 1 , the jump contained a recirculating eddy necessarily accompanied by a stagnation line. The smooth expansion, namely the very diffuse jump, characterized by a low value of the ratio h 2 /h 1 would correspond to the so-called granular dead zone process described by Faug et al. [66],

as it will be discussed in section III C. At the lowest slopes, the flow between the jump and the weir was shearing over its entire depth. At highest slopes, the jump was followed immediately by the formation of a base wedge. An intermediate situation corresponded to a more complex jump structure, as detailed by Brennen et al. [START_REF] Brennen | [END_REF]. However, Brennen et al.

[63] found that the comparison between theory and their experimental data was not affected by the type of jump (smooth expansion or jump with recirculating eddy) or of downstream flow. Their experimental data were compatible with data reported earlier by Savage [55].

Figure 5 shows the depth ratio h 2 /h 1 measured by Savage [55] and Brennen et al. [START_REF] Brennen | [END_REF] as a function of F r 1 . Faug et al. [56] have recently designed a laboratory chute equipped with a discharge gate at the exist of the chute, able to produce standing jumps over a wide range of slope angle and mass discharge. They identified compressible jumps for which the density variation across the jump cannot be neglected. Above a critical mass discharge for which the incoming flow are not too dilute, the jumps were incompressible. Their data on incompressible jumps is also included in Fig. 5.

The collection of data available in literature [55,56,[START_REF] Brennen | [END_REF] show that all data merge into one single group. The prediction from the hydraulic jump equation, namely h 2 /h 1 =

( 1 + 8F r 2 1 -1)/2, which is strictly valid for frictionless flows down a horizontal plane, is also shown by the line drawn in Fig. 5. The line is systematically below the laboratory data. This result clearly highlights the need of an equation suitable for granular media.

As a first step, and under the assumption of incompressible jumps (ρ 2 /ρ 1 = 1), Eq.( 7) is fitted on all laboratory data reported in Fig. 14. The following parametrization is considered: where χ = β/ k -KL * (tan θ-µe)

h 2 h 1 = 1 2 1 + 8χF r 2 1 -1 , (24)
(h 2 -h 1 )
is used as a tuning parameter. Figure 6 depicts the exact values for χ as a function of the Froude number for each couple (F r 1 ,h 2 /h 1 ) measured by Savage [55], Faug et al. [56], Brennen et al. [START_REF] Brennen | [END_REF]. Though the data remain scattered, one can note the following trend: χ is close to 1 at high F r 1 while it starts diverging when F r 1 is decreased. This is the proof of a transition-which occurs around F r 1 ∼ 3 -4, as clearly demonstrated by the recent experiments by Faug et al. [56]. By decreasing F r 1 , there is a transition from granular jumps (in rapid flows) relatively well predicted by the traditional hydraulic equation toward granular jumps (in slower flows) whose height is much higher than the height predicted by the traditional hydraulic equation. Fig. 6 shows that this finding could be extracted from the existing data provided in earlier studies by Savage [55] and Brennen et al. [START_REF] Brennen | [END_REF].

In contrast to previous studies by Savage [55] and Brennen et al. [START_REF] Brennen | [END_REF], Faug et al. [56] paid a detailed attention to the shape of granular jumps and found that the jumps were steep at high F r 1 , while they became much more diffuse at lower F r 1 , as depicted in the pictures provided in Fig. 7. while L * /(h 2 -h 1 ) depicts a sharp increase when the Froude number decreases. By fitting Eq.( 7) (with k = 1 and β = 1) to the experimental data, the friction µ e could be derived (see detail in [56]) and compared to tan θ, as shown in Fig. 8c. The difference tan θ -µ e remains tiny and weakly influenced by F r 1 (a small increase with F r 1 may be detectable but remains within the uncertainty of measurements), as drawn in Fig. 8d. From the curves drawn in Fig. 8, it can be concluded that the dramatic increase of L * /(h 2 -h 1 ) when F r 1 is below 3 -4 prevails. This produces an increase of the product K L * (h 2 -h 1 (tan θ -µ e ), which makes χ higher than 1 in Eq. (24). The height of the diffuse jumps is then higher than the height predicted by the traditional hydraulic equation. This result is interpreted as a direct consequence of the forces caused by the gravity acceleration balanced by friction, and acting within the jump volume. Not considering those forces would lead to an underestimate of the height of the diffuse granular jumps observed at low F r 1 , typically smaller than 3 -4.

The traditional hydraulic equation is only suitable for (incompressible) granular jumps at high F r 1 . Predicting the height of granular jumps at low F r 1 is yet a problem to be fully solved for two reasons: (i) Eq.( 7) remains implicit (h 2 /h 1 being a function of h 2 -h 1 ) and (ii) it does not exist any theory to predict the length of the jump L * .

In want of a better knowledge of the geometry of jumps, one can propose an empirical approach by fitting the following relations on the measured values of K and L * /(h 2 -h 1 ), which are assumed to depend primarily on F r 1 (as usually stated in hydraulics [START_REF] Chow | Open-Channels Hydraulics[END_REF]):

K = 1 + a(F r 1 -F r c ), ( 25
) L * h 2 -h 1 = b (F r 1 -F r c ) c , ( 26
)
where a, b, and c are constant parameters. The relation for L * /(h 2 -h 1 ) is a power law able to capture the increase of L * /(h 2 -h 1 ) when F r 1 approaches the critical value F r c .

Note that F r → F r c , yielding h 2 → h 1 , is a singular point. Describing what happens at tiny values of h 2 -h 1 is challenging because the processes are strongly driven by the complex interplay between hydrodynamic properties and the frictional nature of the granular fluid.

The relation for K is an affine function which satisfies the following property: K → 1 when F r → F r c (very diffuse jumps). The fitted lines for K and L * /(h 2 -h 1 ) are reported in Fig. 8a andb, respectively, and the values of parameters a, b and c are indicated as well.

F r c = 0.6 provided the best fits. Using F r c = 1 does little to change the fitted curves but it is not consistent with the laboratory results by Faug et al. [56] who could observe very diffuse jumps at values of F r 1 lower than 1. In want of any further knowledge at the present time, the empirical relations for K and L * /(h 2 -h 1 ) will be used in the following to estimate the bore height and the critical wall height below which jumps no longer exist.

Bores and critical wall height below which jumps no longer exist

In particular studies conducted in geophysics to understand the interaction between avalanches of granular materials and walls, the key role played by the occurrence-or the non-occurrence-of granular bores has been pointed out. Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] and Faug et al.

[10] investigated the impact of granular avalanches with walls that mimick avalanche barriers, and observed different granular patterns depending on the wall height relative to the incoming flow depth.

Faug et al. [10] designed laboratory tests on a granular avalanche (F r 1 ∼ 3.75), which overflowed dams of various heights from H/h 1 = 0.7 to H/h 1 = 7.4. Those tests were initially performed to study avalanche run-out shortening caused by dams, which is crucial for avalanche protection in geophysics [START_REF] Hákonardóttir | A laboratory study of the retarding effect of breaking mounds in 3, 6 and 9 m long chutes[END_REF][8][9][10][START_REF] Hákonardóttir | Retarding effects of braking mounds: Granular flows[END_REF]. In their experiments, Faug et al. [10] observed a distinct transition, around H/h 1 = 2 -3, from a gentle overflow of the granular material to a bore regime (see figure 4 in [10]). A sharp jump propagated far upstream of the wall in the latter situation. In the former situation, a quasi-steady dead zone formed against the wall and other incoming grains were able to pass smoothly on the dead zone to overflow the wall. This situation is discussed in more detail in section III C. Eq.( 23) that predicts the critical wall height H jet c below which bores no longer exist can be checked against the experimental observations by [10]. Each couple (F r 1 ,H/h 1 ) tested by [10] is reported in Fig. 10a, by distinguishing between the experiments with a bore propagating upstream (crosses) and the ones without any bore (circles). The prediction from Eq.( 23) is reported as well, where h 2 /h 1 is calculated with the approximate solution given by Eq. [START_REF] Hákonardóttir | [END_REF]. The following values were considered for the parameters needed: k = 1, β = 1, and L * /(h 2 -h 1 ) and K were derived from the empirical relations proposed in Sec III A 1. Two values of the difference tan θ -µ e were tested: 0.05 and 0.10, which are typical values recently measured by [56] (see Fig. 8d). Eq.( 23) is found to be quite efficient at demarcating the experiments with the sharp jump (crosses in Fig. 10a) from the experiments with the more diffuse overflow (circles in Fig. 10a).

Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] designed laboratory tests on much more rapid granular avalanche (F r 1 ∼ 12) impacting dam-like obstacles of four different heights:

(i) H/h 1 = H ∞ /h 1 (H = H ∞ prevented overflow), (ii) H/h 1 = 18, (iii) H/h 1 = 15,
and (iv) H/h 1 = 6.7. As shown in Fig. 9b, a granular jump propagating upstream was observed for test (i) for which the incoming flow was entirely blocked upstream of the dam (H = H ∞ ). For tests (ii) and (iii), the granular bore was still present, while some material was able to overflow the smaller dam. In contrast, a long airborne granular jet formed downstream of the dam in test (iv), when H/h 1 was reduced to 6.7. Eq.( 23) that predicts the critical wall height H jet c below which bores no longer exist can be checked against the experimental observations by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF]. Fig. 10a shows each point (F r 1 ,H/h) corresponding to tests (ii)-(iv) conducted by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF]. The prediction from Eq.( 23) is reported as well.

Again, the following values were considered for the parameters needed: k = 1, β = 1 and L * /(h 2 -h 1 ) and K were derived from the empirical relations proposed in Sec III A 1.

Two values of tan θ -µ e were tested: 0.05 and 0.10. Again, Eq.( 23 It is worthy to note that the predictions from traditional hydraulics (F r c = 1 and tan θµ e = 0), reported in dashed lines in Fig. 10, are very close to predictions from the approximate solution stemming from Eq.( 7) implemented with the empirical laws for K and L * /(h 2 -h 1 ).

They are thus efficient at capturing the transitions observed by [10,[START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF] for experiments at high Froude numbers. However, it is likely that hydraulic theory would fail for lower values of the Froude number for which the term K L * h 2 -h 1 starts playing a role in Eq.( 7).

B. The granular airborne jet

Granular airborne jets are observed in very rapid granular flows (F r 1 1) interacting with a relatively small wall (H/h 1 1), as shown in Fig. 11a. The length of the influence zone upstream of the obstacle is small and reduced to a small wedge of stagnant grains (the size of the wedge is typically close to the wall height). In geophysics, the length of granular airborne jet is for instance relevant for the design of the minimum distance between two rows of dissipative structures against snow avalanches [26,[START_REF] Naaim-Bouvet | [END_REF]. Predicting accurately the jet length avoids designing dissipating structures that are ineffective because they would be overtopped by the jet. The trajectory of these airborne jets was studied in detail by Hákonardóttir et al. [START_REF] Hákonardóttir | [END_REF] (note that airborne snow jets have been also studied [26]). The experiments showed that the rapid flow detached from the top of the obstacle as a coherent granular jet, the motion of which being well described by theory for a two-dimensional irrotational flow of an inviscid fluid over a dam [START_REF] Yih | Fluid mechanics[END_REF]. As suggested by Hákonardóttir [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF], the theory from Yih [START_REF] Yih | Fluid mechanics[END_REF] gives a relation between the launch angle β jet , the angle of the uptream face of the wall α and the obstacle height relative to the incoming flow H/h 1 . Notation used is shown in Fig. 11a, and further details for the theory of Yih can be found in [START_REF] Hákonardóttir | The interaction between snow avalanches and dams[END_REF]. Hákonardóttir et al. [START_REF] Hákonardóttir | [END_REF] showed that on colliding with a barrier, a shallow granular flow of high Froude number becomes airborne and follows a coherent ballistic trajectory, with negligible effect of resistive forces within the deflection region. Thus, for a given flow speed and depth of the avalanche relative to that of the obstacle, the length of the airborne jet can be estimated by combining the prediction of the angle of deflection β jet with the parabolic trajectory of the jet, on the assumption that air resistance plays only a negligible role for these trajectories (note that similar conclusion was obtained for rapid snow-chute flows [26]). Hákonardóttir et al. [START_REF] Hákonardóttir | [END_REF] (note that tests from snow-chute flows are also included [26]).

Those values correspond to tests for which large airborne granular jets were observed downstream of the wall, without any granular jump formed upstream of the wall, thus offering a test for Eq.( 23) described in Sec. II D. Eq.( 23) gives the critical wall height, H jet c , which demarcates the granular bores propagating upstream of the wall from the large airborne jets formed downstream without any granular jump formed upstream. The prediction of that equation should be above all the points corresponding to couples (F r 1 , H/h 1 ) tested by Hákonardóttir et al. [START_REF] Hákonardóttir | [END_REF]26]. The values of each parameter needed for Eq.( 23) implemented with Eq.( 7) are given in caption of Fig. 11. It is verified that all the points corresponding to couples (F r 1 , H/h 1 ) tested by Hákonardóttir et al. [START_REF] Hákonardóttir | [END_REF], and by Hákonardóttir et al. [26] for snow-chute flows, are below the prediction of H jet c shown by the group of lines drawn in Fig. 11b.

C. The granular dead zone regime

Relatively slow flows (F r 1 ∼ 1) impacting relatively small walls (H/h 1 ∼ 1) produce the formation of a quasi-static stagnant zone upstream of the wall, which can reach a stationary length, while the incoming grains start to flow gently over the static pile and form a stationary jet whose energy is much smaller than the airborne jet discussed in section III B.

This granular pattern is illustrated in Fig. 12a (see also Fig. 9b).

The co-existence between this stagnant zone and the flowing grains above the dead zone has been observed and studied in detail by Faug et al. [66]. This so-called dead zone process [28,66,[START_REF] Lachamp | [END_REF] F r H is defined by F r 2 1 = F r 2 H (H/h 1 ). Fig. 13 shows that L dz /h 1 is strongly correlated to F r H , which demonstrates the importance of the dimensionless number F r H . For F r H ∼ 1, the dead zone length starts diverging. For the largest values of F r H , L dz /h 1 reaches a constant value around 10. Most of the studies about flow-wall interactions investigated rapid granular flows down steep slopes and/or over smooth bottoms (typical size of roughness much smaller than the grain size). That rapid dynamics remains of course important but attention should be paid to the slower dynamics too. New tests should be designed to investigate slow granular flows on lower slopes and/or over a rough bottom. The dead zone process stems from the ability of granular materials to form stagnant zones (solid-like behaviour) which can co-exist with inertial zones (fluid-like behaviour). It is worthy to note that recent developments have been conducted to predict the force experienced by the wall when a dead zone is present, as discussed in [71] and references therein. flowing grains above can be formed) and a need for further investigations. In particular, the interaction between slow granular flows and walls should be investigated in more detail because it is entirely driven by the complex interplay between the hydrodynamic properties and frictional nature of the granular fluid. This slow dynamics is for instance crucial in geophysics and natural hazards mitigation, because it is relevant to situations where gravity mass movements interact with protection structures in the runout zones while the Froude number becomes relatively low.

L + d = f F r 1 , H h 1 , µ e , ϕ, k , (27)
L - d = g F r 1 , H h 1 (28

  flow disturbed by the presence of an obstacle may have spatial variations L whose scale is close to the typical flow thickness h. The shallow water flows assumption ( = h/L 1) is then no longer valid, as clearly evidenced by Pudasaini et al. [3], Pudasaini and Kröner [4], Faug et al. [10], Gray et al. [20], Johnson and Gray [52], etc.

FIG. 1 .

 1 FIG.1. Sketch of a stationary granular jump down an incline. The subscripts i = 1 and 2 refer respectively to locations upstream (section S 1 ) and downstream (section S 2 ) of the jump.

Fig.2a shows

  Fig.2a shows the jump depth ratio as a function of the Froude number of the incoming flow for different values of the density variation across the jump. The following values are used for the parameters in Eq.(5): k i = 1, β i = 1, L * /h 1 = 10 (typical value measured in the experiments by Faug et al. [56]), K = 1 and tan θ -µ e = 0.11 (which is for instance obtained from θ = 30 • and µ e = tan δ where δ = 25 • ). The jump height increases with the

Fig.2b shows

  Fig.2bshows the jump depth ratio as a function of the Froude number for different values of L * /h 1 . Other parameters have the same values as in Fig.2a. The results show that the influence of L * /h 1 is significant, in particular for low values of the Froude number of the incoming flow. Increasing L * /h 1 generally produces an increase of the jump height relative to the incoming flow thickness, and the effect is enhanced at low Froude number.

Fig.2c depicts h 2 /h 1

 21 Fig.2c depicts h 2 /h 1 versus F r 1 for two values of k 2 . k 1 is kept equal to 1 (isotropic flow conditions regarding normal stresses), which appears to be a reasonable assumption for the incoming flows in steady regime, as previously discussed in Sec. II A. Moreover, it can be verified that the influence of k 1 remains weak with respect to the influence of k 2 . The jump height decreases while k 2 is increased from 1 up to 1.43, as shown in Fig.2c. The latter value for k 2 would correspond to an anisotropy in normal stresses under the following assumptions: (i) ϕ = δ = 25 • and (ii) passive state downstream of the jump where ∂u/∂x < 0, which yields a sign + in Eq.(3). Other parameters have the same values as in Fig.2a with ρ2 /ρ 1 = 1. Changes in β i from 1 to 5/4 or 4/3 (see discussion about those values in Sec.II) have very weak influence on the relation between h 2 /h 1 and F r 1 (not shown here). Keeping β i = 1 is

FIG. 3

 3 FIG. 3. a) (Color online) Sketch of a granular jump propagating upstream of a rigid wall (namely a granular bore). b) Depth ratio h 2 /h 1 versus the Froude number F r 1 for a propagating granular jump [Eq.(13)] and for a steady granular jump [Eq.(5)]. Incompressible jumps and bores are considered (ρ 2 /ρ 1 = 1). Identical values of the parameters (β i = 1, k i = 1, K=1, L * /h 1 = 10 and tan θ -µ e = 0.11) are considered for Eqs.(13) and (5).

FIG. 4 .

 4 FIG. 4. (Color online) a) Sketch of a propagating granular jump (or bore) upstream of a wall with an overflow downstream of the wall; b) Large airborne jet formed downstream of the wall when the wall height is decreased and the bore no longer exists.

  in the numerical model. It would be of interest to design new tests on the propagation of bores on lower slopes, and/or over rough bottoms. The latter conditions would reduce the Froude number of the flows and produce more elongated and diffuse bores. It is likely that depth-averaged models implemented with constant friction parameters would not describe those diffuse bores as well as they can do for bores in very fast granular flows.III. EXPERIMENTS ON GRANULAR PATTERNS NEXT TO WALLS REVIS-ITED A. The granular jumpThe stationary jumps and the propagating jumps (or bores) generated by an obstruction normal to the incident flow have been investigated by several authors, from the pioneering works of Morrison and Richmond[START_REF] Morrison | [END_REF], Savage[55], Brennen et al.[START_REF] Brennen | [END_REF], Campbell et al.[64]

FIG. 5 .

 5 FIG. 5. h 2 /h 1 versus F r 1 measured experimentally by Savage [55], Faug et al. [56], Brennen et al. [63]. The line shows the prediction from the traditional hydraulic equation h 2 /h 1 = ( 1 + 8F r 2 1 -1)/2.

Fig.8a andFIG. 7 .

 7 Fig.8a and bsummarize the key experimental results by Faug et al.[56] through depicting how the shape coefficient K and the ratio L * /(h 2 -h 1 ) evolve with F r 1 (see details in[56] on how the jump length L * was defined experimentally). K increases roughly linearly with F r 1 ,

FIG. 8 .

 8 FIG. 8. (a) K versus F r 1 ; the continuous line shows the fit with 1 + a(F r -F r c ), where a = 0.07 and F r c = 0.6. (b) L * /(h 2 -h 1 ) versus F r 1 ; the continuous line shows the fit with

  ) allows to demarcate roughly the experiments with bores [tests (ii) and (iii) represented by cross symbols] from the experiment without a bore and a large jet formed downstream of the wall [test (iv) shown in a circle symbol].

FIG. 9 .

 9 FIG. 9. (Color online) Hákonardóttir [60] studied granular flows at high Froude number (F r 1 ∼ 12) impacting a rigid wall. At large wall height preventing overflow, a granular jump propagate upstream [test(i)]. While decreasing the wall height, a clear transition from a granular bore propagating upstream of the wall with some overflow downstream of the wall [tests (ii) and (iii)], to a large airborne jet formed downstream of the dam [test (iv)] was observed. The photographs by K.M. Hákonardóttir were adapted from Hákonardóttir [60].

FIG. 10 .

 10 FIG. 10. Points (F r 1 , H/h 1 ) tested by Faug et al. [10] (top graph) and Hákonardóttir [60] (bottom graph). Cross symbols (+) correspond to tests for which a bore propagated upstream and circle symbols (•) represent tests for which the bore was not present. The continuous lines show predictions from Eq.(23) implemented with Eq.(7) for various values of tan θ -µ e (see text). The dashed lines show the prediction from traditional hydraulics for horizontal flows.

FIG. 11

 11 FIG. 11. a) (Color online) Photograph of the granular jet as it detaches from the top of the obstacle (photograph by K.M. Hákonardóttir); b) Values of the couple (F r 1 , H/h 1 ) tested in the laboratory experiments on airborne jets with granular materials [13] and snow [26]. The predictions from Eq.(23) for the critical height H jet c , below which the granular bore no longer exists, are drawn for k = 1 and β = 1. L * /(h 2 -h 1 ) and K are derived from the empirical functions of F r 1 . Three different values are considered for tan θ -µ e : 0.05, 0.10 and 0.15. The dashed line shows the prediction from traditional hydraulics.

Figure 11b displays the

  Figure 11b displays the experimental values of the couple (F r 1 , H/h 1 ) tested by

FIG. 12

 12 FIG. 12. a) (Color online) Photograph of the granular dead zone created upstream of a wall: creation of the dead zone (Inset: stationary dead zone [66]); b) Values of couples (F r 1 , H/h 1 ) which correspond to the experiments on dead zones by Faug et al. [66]. The predictions from Eq.(23) for the critical height H jet c , below which the granular bore no longer exists, are drawn for k = 1, β = 1 and tan θ -µ e = 0.15. L * /(h 2 -h 1 ) and K are derived from the empirical functions of F r 1 . Two values of the b-constant are considered: b = 8 and b = 12. The prediction from traditional hydraulics is reported too.

  differs from the granular jump because it is not accompanied by a sharp discontinuity in depth (strong jump). It consists more of a dense granular stream flowing down a pile made of quasi-static grains. The thickness and the depth-averaged velocity, and the density as well, of the flowing zone vary smoothly. The values of couples (F r 1 ,H/h 1 ) tested by Faug et al.[66] are drawn in Fig.12a. The reasons why such a dead zone (without any sharp jump formed) was observed even for relatively high F r 1 and H/h 1 in the experiments by Faug et al.[66] still remains unclear. However, it is worthy to note that increasing b, which corresponds to an increase of the length of the jump, produces an increase of H jet c . Increasing H jet c then leads to an increase of the area for which dead zones (in absence of a bore) can be observed. The predictions from Eq.(23) for the critical height H jet c , below which the granular bore no longer exists, are drawn in Fig.12b for k = 1, β = 1 and tan θ -µ e = 0.15. L * /(h 2 -h 1 ) and K are derived from the empirical functions of F r 1 . Two values of the b-constant are considered: b = 8 and b = 12.

Fig. 13

 13 Fig.13displays the measured length of the dead zone[66] scaled by the thickness of the incident flow, L dz /h 1 , as a function of the Froude number of the obstacle, F r H = ū/ √ gH cos θ.

  FIG. 13. Dead zone length scaled by the thickness of the incident flow, L dz /h 1 , versus the Froude number of the obstacle, F r H , as measured by Faug et al. [66].

)

  Including any influence of µ e , ϕ and k on L -/d is considered as negligible-or even irrelevant, because granular flows downstream of the wall essentially consist of detached jets formed of free falling particles in a relatively dilute regime. The three important steady regimes are defined below depending on the magnitude of the scaled lengths L + /d and L -/d, and the value of F r H as well. This allows establishing a complete phase-diagram of the granular flow-wall interaction.

A. 1 )

 1 The granular airborne jet regime: L -/d L + /d (F r H At low H/h 1 and high F r 1 , the incoming flow is able to flow easily over the wall forming a large granular airborne jet downstream of the wall, as depicted in Fig.4b. A steady state can be achieved. L -/d is high and approximately determined by the length of the stationary airborne jet formed downstream of the wall, while L + /d is small (typically close to the wall height from laboratory tests: L + ∼ H) and roughly reduced to the length of the stationary wedge formed upstream of the wall. The airborne jet regime, mainly controlled by the inertia of the incoming flow, is defined byL -/d L + /d and F r H 1.B. The granular dead zone regime:L -/d ∼ L + /d (F r H ∼ 1)At intermediate values ofH/h 1 and F r 1 , all typical energies involved in the flow-wall interaction (both potential and kinetic energies of the incident flow, and the potential energy based on the wall height H as well) are of the same magnitude. A steady-state pattern can be formed: it consists of a quasi-static stagnant zone which co-exists with an inertial flowing zone above able to flow over the obstacle, as displayed in Fig.12a. L + /d is determined by the length of the dead zone formed upstream of the wall, while L -/d corresponds to a jet formed upstream of the wall. The jet formed is shorter, denser and much less energetic than the granular airborne jet discussed above. The dead zone regime is defined by L + /d ∼ L -/d and F r H ∼ 1.C. The granular jump regime:L -/d L + /d (F r H undetermined)At high F r 1 and H/h 1 , the incoming flow has a strong kinetic energy but the potential energy associated with the obstacle height (∼ gH) is simultaneously high, which makes F r H undetermined. A thick layer of grains is able to propagate upstream of the wall (subcritical flow), and encounters the grains from the incoming rapid flow (supercritical flow). A granular bore, at the sharp transition between the sub-critical flow (propagating upstream) and the supercritical (incoming) flow, is formed, as depicted in Fig.4a. The distance at which the bore is able to propagate upstream defines L + /d. Under certain conditions (H < H jet c ), a stationary granular jump can be formed leading to a steady value of L + /d while a part of the incoming flow is able to overtop the wall, forming a jet of very low energy (L -/d much

FIG. 14

 14 FIG. 14. (Color online) Phase-diagram for granular flow-wall interaction. The zones defining the conditions under which each of the granular patterns discussed in the present paper are drawn in the plane (F r 1 , H/h 1 ). Predictions from Eq.(7) for the height of steady jumps (thick lines) and from Eq.(23) for the critical height below which the granular bore no longer exists (thin lines) are reported (k = 1, β = 1 and tan θ -µ e = 0.10). K and L * /(h 2 -h 1 ) are derived from the empirical functions of F r 1 . Two values of b are considered for H jet c and h 2 /h 1 : b = 8 (dashed line) and b = 12(dotted line). Also, the prediction from traditional hydraulics for horizontal flows is reported in continuous line for H jet c .

ACKNOWLEDGMENTS

The author is grateful to the financial support by French National Research Agency through the project ANR-MOPERA and by the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme under REA Grant Agreement No. 622899 (FP7-PEOPLE-2013-IOF, GRAINPACT).

smaller than L + /d).

D. Phase-diagram for granular flow-wall interaction

In line of the above qualitative considerations and the analytic solutions discussed in the present study, a phase-diagram for granular flow-wall interaction can be proposed, as drawn in Fig. 14.

Eq.( 7) gives the scaled height h 2 /h 1 of the steady granular jumps. Eq.( 23) makes it possible to predict the critical wall height H jet c , which defines the frontier between large airborne jets (F r 1 > F r c and H < H jet c ) and steady granular jumps (F r 1 > F r c and H > H jet c ). There exists a line above which steady-state conditions cannot be achieved and a granular bore will propagate upstream of the wall provided that the Froude number is larger than F r c . This line corresponds to the jump height defined by Eq.( 7). Once H/h 1 is larger than h 2 /h 1 , a bore propagates upstream and its height is given by Eq.( 13). Below F r c , the granular bore does not form. Still below F r c but at sufficiently low values of H/h 1 (H < H jet c ), a steady-state can be reached through the formation of a granular dead zone co-existing with the flowing grains above able to overtop the wall.

The following values of parameters are used for the predictions given in Fig. 14 [Eq.( 7) for Empirical closure relations were used to estimate the geometry of the granular jump, i.e. the relations for K and L * /(h 2 -h 1 ), and the difference tan θ -µ e was considered as a constant in the range 0.05 -0.15, in accordance to recent laboratory results by [56]. At high Froude numbers, the semi-empirical solutions obtained provide results which are close to the


Benoit Camenen
email: benoit.camenen@irstea.fr

Albert Herrero
email: albert.herrero@irstea.fr

Guillaume Dramais
email: guillaume.dramais@irstea.fr

Fabien Thollet
email: fabien.thollet@irstea.fr

Chloe Le Bescond
email: chloe.le-bescond@irstea.fr

Emeline Perret
email: emeline.perret@irstea.fr

Céline Berni
email: celine.berni@irstea.fr

FIELD EXPERIMENT ON THE DYNAMICS OF FINE AND COARSE SEDIMENTS OVER A GRAVEL BAR IN AN ALPINE RIVER

Keywords: sediment mixture, gravel bar, infiltration, incipient motion 1

Alpine gravel-bed rivers are characterized by very poorly sorted sediments and significant grain sorting is generally observed on gravel bars. The grain size variability may significantly influence the inception of movement for each of the classes represented over a gravel bar. The purpose of this paper is to present some field measurements achieved on the Arc en Maurienne River, France. The main objective is to characterize the dynamics of fine and coarse sediments for conditions close to the incipient motion of gravels and for different degrees of clogging of the river bed. The field campaign was carried out on a gravel bar located 10 km downstream of St Jean de Maurienne during the dam flushing event on June 17th, 2014. Six patches including tagged pebbles (using PIT-tag) were built up with three different characteristics: clean patches formed with gravels and pebbles only, patches with a natural mixture where tagged pebbles were substituted to naturally arranged ones, and patches artificially clogged with fine sediments. These patches were located on the side of a secondary channel on the gravel bar. Surface grain size analyses of the gravel (using the Woolman method and photo analyses using BaseGrain) as well as topographic measurements were achieved the day before the event. During the flushing event, surface velocities over the gravel bar were measured thanks to video analysis (LSPIV). Water levels and mean slope in the secondary channel were measured thanks to pressure gauges positioned on the side of the secondary channel. An estimation of the bed shear stresses over the patches was thus possible all along the event. Intense water sampling was also carried out both upstream and downstream the secondary channel. During the postevent survey the day after, the PIT-tag search showed that the coarse particles did not move. On the other hand, a significant amount of fine sediments were deposited on the patches and infiltrated on the clean patches. Grain size analysis showed that deposited sediments were much coarser than sediments in suspension. An estimation of the fine sediment deposition rate and a discussion on the sediment dynamics is proposed from these measurements.

INTRODUCTION

Alpine gravel-bed rivers are often characterized by very poorly sorted sediments and significant grain sorting is observed on gravel bars with relatively coarse sediments on the bar head and transverse channel and fine sediments on the bar tail [START_REF] Lisle | Formation of stationary alternate bars in a steep channel with mixed-size sediment : a flume experiment[END_REF][START_REF] Lisle | Response of a channel with alternate bars to a decrease in supply of mixed-size bed load : a flume experiment[END_REF][START_REF] Diplas | Modelling of fine and coarse sediment interaction over alternate bars[END_REF][START_REF] Lanzoni | Experiments on bar formation in a straight flume: 2. Graded sediment[END_REF][START_REF] Eekhout | Field experiment on alternate bar development in a straight sand-bed stream[END_REF]. The grain size variability significantly influences the inception of movement for each of the classes represented over a gravel bar [START_REF] Konrad | Partial entrainment of gravel bars during floods[END_REF]. In particular, fine sediment infiltration in the bed and fine sediment deposits are often observed over gravel bars in alpine rivers [START_REF] Diplas | Modelling of fine and coarse sediment interaction over alternate bars[END_REF][START_REF] Lisle | Fine bed material in pools of natural gravel bed channels[END_REF]. Depending on the fine sediment content, fine sediments may limit or enhance gravel mobility [START_REF] Diplas | Deposition and removal of fines in gravel-bed streams[END_REF]. The purpose of this paper is to present some field measurements performed on the Arc en Maurienne River, France. Main objectives are to characterize the dynamics of fine and coarse sediments for conditions close to the incipient motion of gravels and for different degrees of clogging of the river bed. The field campaign was carried out on a gravel bar located 10 km downstream of St Jean de Maurienne during the dam flushing event on June 17th, 2014. Patches with different degrees of clogging were built with tag gravels. Main hydraulic parameters measured during the event were water surface levels and surface velocities. Sediment characteristics of the patches were estimated before and after the event. A discussion of the impact of this event on the bed evolution, more especially on the infiltration dynamics is provided.

2.

EXPERIMENTAL SITES AND DATA

Description of the experimental site

The fieldwork took place in the Arc River, located in the French Alps (Figure 1). The catchment area is 1957 km 2 (grey area in Fig. 1) with a nival hydrologic regime and a mean water discharge from 6-8 m 3 /s in winter to 15-20 m 3 /s in summer [START_REF] Jodeau | Morphodynamique d'un banc de galets en rivière aménagée lors de crues[END_REF][START_REF] Jaballah | Alternate bar morphodynamics in an engineered mountainous river[END_REF]. The river bed has been largely modified and artificially straightened in many places in order to allow the 1 km wide valley to contain a road, a highway and a railway. Consequently, only 5% of the river reaches remain with their natural flow patterns and morphology. In addition the flow is regulated by many hydraulic constructions (several dams and pipes) for hydroelectricity production. The present regulated flow regime has significantly altered the natural river discharges and sediment transport. Although its bed is made of gravel, the Arc en Maurienne River has unusually high fine sediment transport, supplied by tributaries. The main ones are the Arvan and Glandon streams, which supply large amounts of predominantly black Lias schist.

The area of interest is a system of alternate bars located 18 km from the downstream reservoir at Sainte-Marie-de-Cuines (see Figure 1). This system is apparently forced due to the presence of a bend and a bridge pier at the downstream boundary [START_REF] Jaballah | Alternate bar development in an alpine river following engineering works[END_REF]. Lateral limits are set by 5 m high embankments made of boulders and scattered young trees leading to a river width of 50m in average. The mean slope of this reach is approximately 0.6 %. The fieldwork was more precisely carried out on a gravel bar located on the left side of the river 2.5 km upstream the confluence with the Glandon tributary (2 km upstream the bridge).

Figure 1. Map of the study site (green circles correspond to the three river dams).

Experimental set-up

A procedure is proposed here to study incipient motion of gravel particles for different conditions of clogging. Six patches were built on a gravel bar, which separates a secondary channel from the main channel (see Fig. 3). This secondary channel diverts a low percentage of the main channel flow discharge (about 10%). The gravel bar is totally covered for high flow conditions (over 110 m 3 /s approximately). The six patches included 20 tracers (natural particle with a Pit-tag inserted) laid following a regular grid (Fig. 3, [START_REF] Camenen | An estimation of gravel mobility over an alpine river gravel bar (Arc en Maurienne, France) using PIT-tag tracers[END_REF]. Their size was one square meter approximately. The six patches were prepared in two rows parallel to the flow on the side of the secondary channel (cf. Fig; 3d). They were built in order to obtain three different characteristics:

• two clean patches made of cobbles of 4 to 10 cm in diameter (approx.), see Fig. 2a.

• two unmodified patches where natural cobbles are substituted by tagged cobbles, see Fig. 2b.

• two clogged patches for which a large amount of very fine sediments surrounds the tagged cobbles, see Fig. 2c.

Preparation of the different patches is detailed as follows. For unmodified patches, the bed structure was not modified and twenty cobbles initially present in the bed were manually substituted by tagged cobbles. The tagged cobbles were distributed in four rows of five cobbles each, with a distance of more than 20-25 cm between them. For the other type of patches, a surface of approximately one square meter was first excavated to a depth of approximately 10 cm. In case of clean patches, the resulting hole was then filled with cobbles and pebbles between 5 and 10cm in diameter including twenty tagged cobbles distributed similarly as for the unmodified patches. In case of clogged patches, it was filled with silt and clay sized material found in nearby (fine deposits on the gravel bar); and eventually, twenty tagged cobbles were introduced within this layer forming four rows of five cobbles each. The position and code of each tagged particles was identified within each patch. Coordinates (in the Lambert 93 system) of the four corners of each patch were measured using DGPS tool. A topographic survey of the gravel and secondary channel was also achieved before the flushing event.

An estimation of the grain size distribution for each of these patches was made using photo analysis thanks to the software Basegrain [START_REF] Detert | Automatic object detection to analyze the geometry of gravel grains -a free stand-alone tool[END_REF]) to be compared with the [START_REF] Wolman | A method of sampling coarse river-bed material[END_REF] pebble count procedure achieved on several places of the gravel bars. The median grain size varies from 1 to 3 cm around the patches with a standard deviation of approximately 3.5; detailed results are not presented in this paper. During the flushing event, several measurements were achieved to obtain a better idea of the hydro-sedimentary conditions over the patches. A system for video analysis was implemented using a camera fixed at the top of a telescopic mast on the side embankment (Fig. 3). The Large Scale Particle Image Velocimetry (LSPIV) method was used to measure the surface flow velocities during the event. Ground Reference Points (GRPs) were positioned in the view field using square targets (red and white) on the gravel bar and on the bank. They allow a geometrical correction of the pictures [START_REF] Jodeau | Application and evaluation of LS-PIV technique for the monitoring of river surface velocities in high flow conditions[END_REF]. The software Fudda-LSPIV (Le [START_REF] Coz | Image-based velocity and discharge measurements in field and laboratory river engineering studies using the free FUDAA-LSPIV software, River Flow[END_REF] was eventually used to process the movies in order to get the velocity field and estimate the discharge within the secondary channel. 10 movies of 30 s were eventually taken during the event, which corresponds to a movie every 40 minutes in average.

Two pressure gauges were installed upstream and downstream the secondary channel in order to obtain an estimation of the water level and slope within the secondary channel. Atmospheric pressure effects were taken into account using a third pressure gauge outside water.

Since fine sediment concentrations may be very high during these events (up to 30 g/l, [START_REF] Antoine | Assessment of the suspended sediment load due to dam flushing along the Arc and Isère rivers, France[END_REF], a possible deposition of fine particles was expected along the secondary channel during the event. Intense water sampling (every half an hour) was performed at both upstream and downstream ends of the secondary channel. Sediment concentration was measured for each sample using the filtration method and grain size analysis was made by laser diffractometry.

After the event, we intended to make a search of the tagged particles using a specific antenna to detect each particle with their own code together with a DGPS to note the location of each particle detected [START_REF] Camenen | An estimation of gravel mobility over an alpine river gravel bar (Arc en Maurienne, France) using PIT-tag tracers[END_REF]. However, as discussed later, the flushing event was interrupted unexpectedly and the discharge did not reach 130 m 3 /s as planned. As a consequence, three of the patches remained outside water during the event and velocities were too low over the three other patches to transport coarse particles. None of the tagged particles moved but a significant infiltration / deposition of fine particles was eventually observed.

EXPERIMENTAL RESULTS

Discharge and sediment concentrations

The discharge was measured at the hydrometric station of Pontamafrey located 9 km upstream. It was estimated that the time for the wave to travel from Pontamafrey to Sainte-Marie-de-Cuine is ∆t = 45mn. Flushing operations are conducted yearly in June with the exception of years for which a large flood occurred previously. Since the same procedure is followed, similar discharge time series are generally observed with a first plateau at 80-100 m 3 /s lasting approximately 4 hours and a second plateau at 120-130 m 3 /s lasting approximately 4 hours (see red line in Fig. 4a corresponding to the 2011 flushing event). It appears clearly that the 2014 flushing event was shorten with nearly no existence of the second plateau, which obviously affected our experiment. This was explained later due to an incident in one of the dam reservoirs. In Fig. 4b, the slope in the secondary channel during the event is presented, as estimated from the two pressure gauges installed. Interestingly, it appeared that the slope first decreased with the increase of discharge, maybe due to the downstream boundary influenced by the main channel level. It varied between 0.6 and 0.7% during the event. From the water level and LSPIV measurements, we also estimated the duration for which patches P1, P3 and P5 were under water:

T =60 mn.
Results for concentration measurements are presented in Fig. 5. Concentration levels clearly follow the discharge timeseries. A plateau at 4 g/l is observed during the first plateau for Q followed by a pick at 12 g/l at the pick of discharge for which supercritical flow occurs in dam reservoirs. This may explain the much larger concentrations observed. However, these concentrations remain relatively low compared to previous flushing events [START_REF] Antoine | Assessment of the suspended sediment load due to dam flushing along the Arc and Isère rivers, France[END_REF]. Grain size analyses (Fig. 5b) show that the composition of the suspension did not evolve significantly during the event (d50 = 17 μm).

One can observe a small pick at the pick discharge with (d50 = 20 μm).

Surface velocity and bed shear stresses

LS-PIV is an image-based method, which consists in calculating the most likely displacements of patterns between two successive images using statistical identification [START_REF] Fujita | Large-scale particle image velocimetry for flow analysis in hydraulic engineering applications[END_REF][START_REF] Jodeau | Application and evaluation of LS-PIV technique for the monitoring of river surface velocities in high flow conditions[END_REF]. Fig. 6 presents some typical results from the LSPIV post-processing using Fudaa-LSPIV (Le [START_REF] Coz | Image-based velocity and discharge measurements in field and laboratory river engineering studies using the free FUDAA-LSPIV software, River Flow[END_REF]. If a cross-section of the river bed is available (see yellow line in Fig. 6a), it is also possible to estimate the discharge assuming a ratio between the depth averaged velocity and the surface velocity of 0.85 (rough bed). Similarly, assuming a roughness length ks = 2d90 ≈ 8 cm, it is also possible to estimate the local bed shear stress. Focusing on the side of the secondary channel next to the patches (Fig. 6b), it is interesting to see the deflection of the surface velocities toward to side of the channel. It actually corresponds to small waves propagating toward the gravel bar with a celerity of 10 cm/s approximately. Because of mass conservation, a small undertow current should be present with a main direction toward the center of the channel. E-proceedings of the 36 th IAHR World Congress, the Netherlands 28 June -3 July, 2015, The Hague, the Netherlands Best shear stress estimates using both LSPIV and water level measurements are presented in Fig. 7. As discussed before, LSPIV allows an estimation of local bed shear stress as soon as bed topography measurements are available. Fig. 7a presents the bed shear stress distribution throughout a secondary channel section. Values over 100 Pa were observed in the middle of the secondary channel that is sufficient to mobilize the coarsest particles forming the bed. Sectionaveraged bed shear stress are presented in Fig. 7b, either using the "law of the wall" method with data from Fig. 7a or using the reach averaged method:

τ = ρC f V 2 =ρg R h I [1]
with ρ the water density, C f = f ( R h , k s ) the friction coefficient, V the section-averaged velocity, g the acceleration of gravity, Rh the hydraulic radius, and I the water slope. The "law of the wall" method was applied locally using the local depth-averaged velocity and assuming R h =h with h the local water depth. Good agreement is observed between the two methods used to estimate the section averaged bed shear stress in the secondary channel. Results from the LSPIV measurements are however very sensitive to the water level estimation, especially when water depths are shallow. This explains the very low value obtained at t = 07:23. In the zone over the patches, it was difficult to estimate the local bed shear stress. By extrapolation to what we could measure, we estimate it close to zero. However, some possible erosion of fine sediments could have been possible because of the propagation of the small waves described above.

Fine sediment infiltrated

Although no movement of the tagged particle could be observed, a significant amount of fine sediments infiltrated the clean patch P3. An estimation of the amount of fine sediments infiltrated was made using the following method (Fig. 8): a surface of approximately 0.2 m² (0.45 × 0.45 m) was selected within patch P3. All fine sediments infiltrated in cobble layer (approximately 10 cm thick, see Fig. 8c) were collected in a basket. Coarse particles were taken out and cleaned in the same basket. Eventually, a sample of 5.23 kg was collected. Since some sandy particles may have been collected at the bottom of the layer, this sample was sieved at 0.5 mm. It remained 4.56 kg. When fine sediments are much smaller than coarse sediments, fines can percolate without getting trapped as they pass through the pore throats, therefore being deposited only on the top of coarse grains or on an eventual impermeable layer located under the gravel bed. In this situation, the bed is filled from the bottom upwards and tends to a relatively uniform distribution of fines over the bed depth. This infiltration mechanism that corresponds typically to our case is commonly known as unimpeded static percolation [START_REF] Gibson | Vertical gradational variability of fines deposited in a gravel framework[END_REF][START_REF] Evans | Fine sediment infiltration dynamics in a gravel-bed river following a sediment pulse[END_REF]. Assuming an initial porosity of 0.4 for the patch P3, and a porosity of 0.4 for the fine sediments infiltrated, the maximum amount of fine sediments that could infiltrate the 0.2 m² is M =2650× 0.4² × 0.1× 0.2=8.5 kg. This means that approximately 50% of the pores were clogged with fine sediments in 60 minutes. In Fig. 9, the grain size distributions of the suspended sediments and of the sediments infiltrated in the patch 3 are plotted.

For the sediment in suspension, 6 classes predominate:

•

CONCLUSIONS

A field survey to study the dynamics of fine and coarse sediments in an alpine river during a flushing event is presented. It included flow velocity measurements and surface water sampling during the event and characteristics of the bed and reference patches before and after the event.

The proposed methodology appeared successful to better understand the local dynamics of a mixture of sediment. Estimation of bed shear stress in the secondary channel using both pressure gauges and LSPIV measurements were in good agreement. An intense sampling frequency for the fine sediment in suspension is necessary for such event since the concentration values can change rapidly.

However, the flushing event was interrupted unexpectedly and bed shear stress remained too low to transport coarse particles. None of the tagged particles moved but a significant infiltration / deposition of fine particles was eventually observed. The infiltration rate was estimated at 50% in the clean patch after a period under water of 60 minutes only. Grain size analyses of both fine sediment in suspension and infiltrated sediment gave an interesting view of the fine sediment dynamics over the gravel bar.

A similar experiment is planned for the 2015 flushing event with the objective of better estimating the inception of motion of coarse particles for different fine sediment contents.

Figure 2 .

 2 Figure 2. Photos of the different patches built: (a) cleaned patch P6 (b) natural patch P1 (c) clogged patch P5, and (d) schematic view of the experimental set-up.

Figure 3 .

 3 Figure 3. Photo of the set-up for video analysis showing GRPs (photo inserted: mast on which the camera is fixed).

Figure 4 .

 4 Figure 4. Discharge time series (TU+2) measured at the hydrometric station of Pontamafrey with ∆t = 45mn (a) and water slope on the secondary channel (red line, the blue line corresponds to the discharge) (b).

Figure 5 .

 5 Figure 5. Concentration time series measured upstream and downstream the gravel bar, 1km downstream from a bridge and 10 km upstream at the hydrometric station of Pontamafrey with ∆t = 45mn (a) and grain size analysis of samples next to the gravel bar (b).

Figure 6 .

 6 Figure 6. Estimation of the surface velocity using LSPIV in the secondary channel at 13:30 over the whole section (a) and making a zoom close to the patches (b).

Figure 7 .

 7 Figure 7. Local bed shear stress distribution throughout the secondary channel next to the patches from LSPIV measurements (a) and mean bed shear stress in the secondary channel from slope measurement and LSPIV measurement (green square) (a).

Figure 8 .

 8 Figure 8. Estimation of the fine sediment infiltration in patch P3; photos of the sampling surface after taking off pebbles (a) and cleaning all deposits (b), side picture of the sampling surface (c).

  one class of clay with d ≈ 5 μm, • one class of fine silts with d ≈ 12 μm, • four classes of coarse silts with d ≈ 25, 35, 47, and 68 μm, respectively The very same classes are observed for the sediments infiltrated in patch 3 but with different proportions. A class of poorly sorted fine sands with d ≈ 120 μm is also present in the deposits. It clearly shows the grain sorting occurring over the bed for the infiltration process. Coarser particles with larger settling velocity do settle and infiltrate easier even when bed shear stresses are very low. The presence of the fine sand class indicates that this sediment fraction travels in a graded suspension with concentration much larger close to the bottom. Surface water sampling on the side of the river do not allow to detect this class of sediments.

Figure 9 .

 9 Figure 9. Grain size distribution comparison between the sediments in suspension and the sediments infiltrated in the patch 3.


B Camenen
email: benoit.camenen@irstea.fr

A Herrero

G Dramais

F Thollet

C Le Bescond

E Perret

C Berni

Field experiment on the dynamics of fine sediments over a gravel bar in an alpine gravel-bed river

Introduction

Alpine gravel-bed rivers are often characterized by very poorly sorted sediments and significant grain sorting is observed on gravel bars with relatively coarse sediments on the bar head and transverse channel and fine sediments on the bar tail [START_REF] Lisle | Formation of stationary alternate bars in a steep channel with mixedsize sediment: a flume experiment[END_REF][START_REF] Diplas | Modelling of fine and coarse sediment interaction over alternate bars[END_REF][START_REF] Lanzoni | Experiments on bar formation in a straight flume 2. graded sediment[END_REF][START_REF] Eekhout | Field experiment on alternate bar development in a straight sand-bed stream[END_REF]. The grain size variability significantly influences the inception of movement for each of the classes represented over a gravel bar [START_REF] Konrad | Partial entrainment of gravel bars during floods[END_REF]. In particular, fine sediment infiltration in the bed and fine sediment deposits are often observed over gravel bars in alpine rivers [START_REF] Diplas | Modelling of fine and coarse sediment interaction over alternate bars[END_REF][START_REF] Lisle | Fine bed material in pools of natural gravel bed channels[END_REF]. Depending on the fine sediment content, fine sediments may limit or enhance gravel mobility [START_REF] Diplas | Deposition and removal of fines in gravel-bed streams[END_REF]. The purpose of this paper is to present some field measurements performed on the Arc en Maurienne River, France. Main objectives are to characterize the dynamics of fine sediment infiltration in the gravel matrix for conditions close to the incipient motion of gravels. The field campaign was carried out on a gravel bar located 10 km downstream of St Jean de Maurienne during the dam flushing event on June 17th, 2014, where patches with clean gravels were previously built. Water surface levels and surface velocities were measured during the event as well as sediment concentrations. Volume and characteristics of the sediment infiltrated in the patches were estimated after the event. A discussion of the impact of this event on the infiltration dynamics is provided.

Experimental site and data

Location of the experimental site

The fieldwork took place in the Arc River, located in the French Alps. The catchment area is 1957 km 2 (grey area in Fig. 1) with a nival hydrologic regime and a mean water discharge from 6-8 m 3 /s in winter to 15-20 m 3 /s in summer [START_REF] Jodeau | Morphodynamique d'un banc de galets en rivière aménagée lors de crues[END_REF][START_REF] Jaballah | Alternate bar morphodynamics in an engineered mountainous river[END_REF]. The river bed has been largely modified and artificially straightened in many places in order to allow the 1 km wide valley to contain a road, a highway and a railway. In addition the flow is regulated by many hydraulic constructions (several dams and pipes) for hydroelectricity production. The present regulated flow regime has significantly altered the natural river discharges and sediment transport. Although its bed is made of gravel, the Arc en Maurienne River is characterised by high fine sediment transport (black Lias schist mainly), supplied by tributaries (such as the Arvan and Glandon streams). The area of interest is located in a system of alternate bars 18 km from the downstream reservoir at Sainte-Marie-de-Cuines (see Fig. 1). This system is apparently forced due to the presence of a bend and a bridge pier at the downstream boundary [START_REF] Jaballah | Alternate bar development in an alpine river following engineering works[END_REF]. Lateral limits are set by 5 m high embankments made of boulders and scattered young trees lead-ing to a river width of 50 m in average. The mean slope of this reach is approximately 0.6%. More precisely, the fieldwork was carried out on a gravel bar located on the left side of the river 2.5 km upstream the confluence with the Glandon tributary (2 km upstream the bridge).

Figure 1: Map of the study site (green circles correspond to the three river dams).

Experimental set-up

A procedure is proposed here to study infiltration of fine sediments in a matrix made of gravel particles. Two patches were built on a gravel bar, which separates a secondary channel from the main channel, using gravels of 4 to 10 cm in diameter (see Fig. 2a). This secondary channel diverts a low percentage of the main channel flow discharge (about 10%). The gravel bar is totally covered for high flow conditions (over 110 m 3 /s approximately). Actually, six patches were built with an initial objective to study inception motion of coarse particles in different initial stage of clogging [START_REF] Camenen | Field experiment on the dynamics of fine and coarse sediments over a gravel bar in an alpine river[END_REF]. Because the peak discharge was lower than expected, no movement of the gravel was observed [START_REF] Camenen | Field experiment on the dynamics of fine and coarse sediments over a gravel bar in an alpine river[END_REF]. The patch size was one square meter approximately; and the patches were located on the side of the secondary channel (cf. Fig. 2b). Preparation of the clean patches is detailed as follows: A surface of approximately one square meter was first excavated to a depth of approximately 10 cm. The resulting hole was then filled with gravels and pebbles between 4 and 10 cm in diameter. Coordinates (in the Lambert 93 system) of the four corners of each patch were measured using DGPS tool. A topographic survey of the gravel and secondary channel was also achieved before the flushing event.

During the flushing event, several measurements were performed to estimate hydro-sedimentary conditions over the patches. A system for video analysis was implemented using a camera fixed at the top of a telescopic mast on the side embankment. The Large Scale Particle Image Velocimetry (LSPIV) method was used to measure the surface flow velocities during the event. Ground Reference Points (GRPs) were positioned in the view field using square targets (red and white) on the gravel bar and on the bank in order to make geometrical correction of the pictures (see Fig. 2b). The software Fudda-LSPIV [START_REF] Le Coz | Image-based velocity and discharge measurements in field and laboratory river engineering studies using the free fudaa-lspiv software[END_REF] was eventually used to process the movies in order to get the velocity field and estimate the discharge within the secondary channel. 10 movies of 30 s were taken during the event, which corresponds to a movie every 40 mn in average. Two pressure gauges were installed upstream and downstream the secondary channel in order to obtain an estimation of the water level and slope within the secondary channel. Intense water sampling (every half an hour) was performed at both upstream and downstream ends of the secondary channel. Sediment concentration was measured for each sample using the filtration method and grain size analysis was made by laser diffractometry.

Experimental results

Discharge and sediment concentrations

The discharge was measured at the hydrometric station of Pontamafrey located 9 km upstream (see Fig. 3). It was estimated that the time for the wave to travel from Pontamafrey to Sainte-Marie-de-Cuine is ∆t p = 45 mn. Flushing operations are conducted yearly in June with the exception of years for which a large flood occurred previously. Since the same procedure is followed, similar discharge time series are generally observed with a first plateau at 80-100 m 3 /s lasting approximately 4 hours and a second plateau at 120-130 m 3 /s lasting approximately 4 hours. The 2014 flushing event was actually shorten with nearly no existence of the second plateau due to an incident in one of the dam reservoirs. Concentration levels clearly follow the discharge time series (see Fig. 3). A plateau at 4 g/l is observed during the first plateau for Q followed by a pick at 12 g/l at the pick of discharge for which supercritical flow generally occurs in dam reservoirs. This may explain the much larger concentrations observed. However, these concentrations remain relatively low compared to previous flushing events [START_REF] Antoine | Assessment of the suspended sediment load due to dam flushing along the Arc and Isère rivers, France[END_REF]. Grain size analyses show that the composition of the suspension did not evolve significantly during the event (d 10 = 3 µm, d 50 = 17 µm, d 90 = 50 µm).

Velocities and bed shear stresses

Surface velocity measurements were obtained using LS-PIV method [START_REF] Le Coz | Image-based velocity and discharge measurements in field and laboratory river engineering studies using the free fudaa-lspiv software[END_REF][START_REF] Camenen | Field experiment on the dynamics of fine and coarse sediments over a gravel bar in an alpine river[END_REF]. If a cross-section of the river bed is available, it is also possible to estimate the discharge assuming a ratio between the depth averaged velocity and the surface velocity of 0.85 (rough bed). Similarly, assuming a roughness length k s = 2d 90 ≈ 8 cm, it is also possible to estimate the local and section averaged bed shear stress using the "law of the wall" method :

τ m = ρC f V 2 (1)
with C f = f (R h , k s ) the friction coefficient, R h ≈ H the hydraulic radius, H the mean water depth in the section, V the section-averaged velocity. Based on slope measurements, one can also compute section-averaged bed shear stress :

τ m = ρgR h S (2)
with ρ the water density, g the acceleration of gravity, and S the water slope. Section-averaged bed shear stress in the secondary channel is presented in Fig. 4 using both methods. Good agreement is observed between the two methods. Results from the LSPIV measurements are however very sensitive to the water level estimation, especially when water depths are shallow. This explains the very low value obtained at t = 07:23. In the zone over the patches, it was difficult to estimate the local bed shear stress. By extrapolation to what we could measure, we estimate it close to zero. However, some possible erosion of fine sediments could have been possible because of small waves propagating toward the gravel bar with a celerity of 10 cm/s approximately.

Figure 4: Mean bed shear stress in the secondary channel based on slope measurement and LSPIV measurement.

Fine sediment infiltration

A significant amount of fine sediments infiltrated the clean patch located close to the secondary channel (Figs. 2a and5a). An estimation of the amount of fine sediments infiltrated was made using the following method (Fig. 5b andc): a surface S ≈ 0.2 m 2 (0.45 × 0.45 m) was selected within the patch. All fine sediments infiltrated in the cobble layer (δ ≈ 10 cm) were collected in a basket. Coarse particles were taken out and cleaned in the same basket. Eventually, a sample of 5.23 kg was collected. Since some sandy particles may have been collected at the bottom of the layer, this sample was sieved at 0.5 mm. It remained 4.56 kg. When fine sediments are much smaller than coarse sediments, fines can percolate without getting trapped as they pass through the pore throats, therefore being deposited only on the top of coarse grains or on an eventual impermeable layer located under the gravel bed. In this situation, the bed is filled from the bottom upwards and tends to a relatively uniform distribution of fines over the bed depth. This infiltration mechanism that corresponds typically to our case is commonly known as unimpeded static percolation [START_REF] Gibson | Vertical gradational variability of fines deposited in a gravel framework[END_REF][START_REF] Evans | Fine sediment infiltration dynamics in a gravel-bed river following a sediment pulse[END_REF]. Assuming an initial porosity p patch = 0.4 for the patch, and a porosity of p f ines = 0.4 for the fine sediments infiltrated, the maximum amount of fine sediments that could infiltrate the selected surface surface is :

M = ρ s p patch (1 -p f ines )δ S = 12.7 kg (3)
This means that approximately 30% of the pores were clogged with fine sediments in ∆t s ≈ 60 mn (time period for which the patch was underwater). In Fig. 6, the grain size distributions of the suspended sediments and of the sediments infiltrated in the patch 3 are plotted. For the sediment in suspension, 5 classes of sediments (clay and silts) predominate (see Tab.

M dep = S∆t s ∑ W si C i (4)
where W si and C i are the settling velocity (estimated using [START_REF] Camenen | A simple and general formula for the settling velocity of particules in suspension[END_REF] and concentration for each class of sediment, respectively. We assumed here a constant total concentration C = 10 g/l during the period when the patch was submerged (see Fig. 3). As shown in Tab. 1, Eq. 4 yields overall good results for the estimation of the mass deposited. However, because of the high sensitivity of settling velocity to the grain size, such a computation leads to an overestimated sediment sorting. This may be explained by some possible flocculation effects for fine particles as well as some hindered effects for the coarser particles (see Tab. 1).

Conclusion

A field survey to study the dynamics of fine sediment dynamics in an alpine river during a flushing event is presented. It included flow velocity measurements and surface water sampling during the event and characteristics of the bed and reference patches before and after the event. Estimation of bed shear stress in the secondary channel using both pressure gauges and LSPIV measurements were in good agreement. An intense sampling frequency for the fine sediment in suspension is necessary for such event since the concentration values can change rapidly. A significant infiltration / deposition of fine particles was eventually observed in the patches. The infiltration rate was estimated at 0.1 g/s for C ≈ 10 g/l resulting in a clogging of approximately 30% of the pores in one hour. Grain size analyses of both fine sediments in suspension and infiltrated sediments gave an interesting view of the fine sediment dynamics over the gravel bar, especially for the grain size interaction (flocculation, hindered effects). A similar experiment is planned for the 2015 flushing event with the objective of better estimating the inception of motion of coarse particles for different fine sediment contents.

Figure 2 :

 2 Figure 2: Photos of a clean patch (a) and of the disposition of all patches prepared for the experiment (b).

Figure 3 :

 3 Figure 3: Water discharge and concentration time series measured upstream and downstream the gravel bar, 1 km downstream from a bridge and 10 km upstream at the hydrometric station of Pontamafrey with ∆t p = 45 mn.

Figure 5 :

 5 Figure 5: Photos of a clean patch after the flushing event (a), photos of the sampling surface after taking off pebbles (b) and cleaning all deposits (c).

Figure 6 :

 6 Figure 6: Comparison of the grain size distributions of the sediments in suspension (SSM 11:30) and the sediments infiltrated in the patch (deposits).

Table 1 :

 1 Simplified grain size distribution of the sediments in suspension and estimation of the infiltrated flux (W s : settling velocity; M dep : mass infiltrated).

	Class	d	% mass	W s	M dep,mod M dep,exp
		[µm]		[m/s]	[kg]	[kg]
	1	5	16	1.4 × 10 -5	0.02	0.45
	2	12	35	8.0 × 10 -5	0.20	0.89
	3	30	24	5.0 × 10 -4	0.85	0.79
	4	45	15	1.1 × 10 -3	1.16	0.68
	5	70	10	2.7 × 10 -3	1.99	1.14
	6	120	0.1	7.0 × 10 -3	0.08	0.46
	total		100		4.22	4.42
	els in a graded suspension with concentration much larger
	close to the bottom. Surface water sampling on the side
	of the river do not allow to detect this class of sediments.
	It is possible to estimate the mass deposited (or infiltrated)
	from the theoretical settling flux :

9th symposium on river, coastal and estuarine morphodynamics, RCEM, Iquitos City, Peru, 30August -03 September, 2015