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Corresponding author: T. Faug, Irstea, UR ETGR, 2 rue de la Papeterie, BP 76, FR-38402 Saint-Martin-d'Hères CEDEX, France. (email@example.com)
 A scaling law is proposed to describe the force experienced by a wall-like obstacle overflowed by a granular avalanche-flow when a stagnant zone is formed upstream from the wall and co-exists with an inertial zone above without granular jump. It relates the force on the wall relative to the force due to the kinetic energy of the undisturbed incident flow to (i) the Froude number, (ii) the wall height relative to the flow thickness, (iii) the slope angle and (iv) the various parameters associated with the properties of the flowing granular material. The scaling law is compared to small-scale discrete numerical simulations in two dimensions and data from granular laboratory tests. Finally, we discuss on the applicability of the new model to the full-scale granular flows.
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 Many geophysical flows occur as dense granular avalanches that are driven downslope under balance between gravity and friction, such as debris flows, snow avalanches, lahars and dense pyroclastic flows [Savage and Hutter, 1989; Iverson, 1997; Pudasaini and Hutter, 2007]. Finite-sized granular avalanches are time-varying processes. They involve transitions from a rapid dilute avalanche front to a long-lasting avalanche tail, as investigated in detail byBartelt et al. , with a possible quasi-steady avalanche body. When such a flow interacts with a wall-like obstacle, grains are trapped upstream of the wall and form a stagnant zone that can co-exist with an inertial zone above with a smooth variation of the free-surface in the vicinity of the wall (in contrast to the granular bore that implies a sharp variation of the free-surface, as is reported inGray et al. , Pudasaini et al.  or Pudasaini and Kroener ). The transition from the dead zone process to the granular bore, and reciprocally, is mainly controlled by the Froude number of the incident flow, , and the wall height H relative to the thickness h of the incoming undisturbed flow, where is the depth-averaged velocity,θ the slope angle and g the gravity. For extended and generalized Fr that include the depth variations of flow properties, we refer to Pudasaini and Domnik  and Domnik and Pudasaini .
 The critical height, Hc, below which the granular bore does no longer exist can be expressed as Hc/h = h2/h1 − Fr2/3 [Hákonardóttir, 2004], where h2/h1 is the ratio of the heights before and after the jump: . Based on the pioneering work of Savage  on granular jumps, a general expression of χ can be derived for frictional fluids down an inclined slope θ: . Here, μe is the effective friction coefficient, the length of the granular jump (see inset in Figure 1) and a shape-coefficient of the jump. This expression is derived from a balance between (i) the momentum flux across the jump, (ii) the pressure force gradient across the jump, (iii) the weightW of the jump ( with the assumption that the density is unchanged before and after the jump: ) and (iv) the friction force (μeWcosθ) with the bottom and side-walls, as detailed inSavage . As an example, Figure 1 displays the ratio H/h over time measured by Caccamo et al.  and the corresponding ratio Hc/h with χ = 1 or χ = 1.27. Here, χ = 1 refers to horizontal channels and frictionless fluids or is likely to stand for flow conditions for which μe would exactly balance tanθ. Also, note that, χ = 1.27 is calculated with θ = 30° (slope corresponding to tests of Caccamo et al.  drawn in Figure 1) and the following typical values: μe = 0.36 (μe = tanδ where δ = 20° is the typical friction angle between walls and grains) and , as prescribed by Savage in lack of the precise geometry of the jump. The cross-comparison ofH/h and Hc/h in Figure 1 shows that granular bores are expected to be formed at very short times (H > Hc for t < t1: first impact of the dilute avalanche front) and at final times just before the avalanche comes to rest (t > t2) while a dead zone process is likely to occur during most of the time of the flow-wall interaction (H < Hc for t1 < t < t2). A detailed experimental analysis of the transition at time t2 was provided by Caccamo et al.  and demonstrated that this analysis was relevant and promising. Investigating the transition at time t2 from a dead zone process(solid-like, stagnant zone that co-exists with a fluid-like, inertial zone above) towards agranular jump remains a great challenge. Detecting the transition from a granular jump to a dead zone process and measuring the time t1 is also a challenge which is made more difficult by the nature of the dilute avalanche front. This letter aims to lighten a recent analytical model that was developed by Chanut et al.  to predict the force on the wall between times t1 and t2. This model has important applications for geophysical flows which are likely to hit structures in relatively low slope runout zones while granular jumps are not systematically formed.
Section 2 discusses how a scaling law can be derived from the theory described in Chanut et al. . The robustness of the scaling law is demonstrated by revisiting small-scale numerical simulations in two dimensions (section 3) and laboratory data (section 4) on granular flows past walls. Finally, we conclude on how the present study opens the path to future theoretical studies and could be applied to full-scale avalanche-flows.
2. A Scaling Law
 For a relatively low channel slope, when a granular flow hits a wall spanning the whole width of the incident stream, a stagnant zone is formed upstream of the obstacle (grains trapped upstream from the wall) and co-exists with an inertial zone flowing above (grains overflowing the wall), in absence of a granular jump (seeFigure 2). For higher slope inclinations and strong granular jumps, we refer to Pudasaini et al.  and Pudasaini and Kroener . The stagnant zone process and the induced force on the wall can be described by mass and momentum conservation equations applied to the control volume delimited by sections S and S* (hatched zone drawn in Figure 2), as described by Chanut et al. (under time-varying flow conditions: = (t), h = h(t), , etc.):
where is the flow width and ρP is the particle density. Exponents N and T denote the normal and tangential components of force with respect to the wall (corresponding to x- and y-axis directions in Figure 2). h, and are the time-varying flow thickness, depth-averaged velocity and depth-averaged volume fraction of the undisturbed incoming flow at sectionS.
Equation (2) defines which is the x-component (normal to the wall) of the force due to momentum flux between sections S and S*. In order to derive , Chanut et al.  defined the Boussinesq coefficient β related to the velocity profile in depth of the incident flow, the average deflection angle α (associated with the jet formed at the top of the wall) with respect to the chute bottom and the coefficient κ in linkwith energy dissipation. κ can be related to the restitution coefficient of particles (e): .
Equation (4) defines that is the normal component of the weight of the control volume V0. The control volume V0 is derived from the geometry (hatched zone in Figure 2): , where . The dead zone length is simply derived from the triangular shape of the stagnant zone: , where αdz is the dead zone angle (see Figure 2). Analytical expressions are provided by Chanut et al. for time-dependent anglesα and αdz. , in equation (4), is defined as the mean mass in the control volume: , accounting for the possible compaction of the stagnant zone with respect to the flowing zone above. ϕmax is the mean volume fraction of solid grains inside the dead zone. ϕmax is greater than and can be derived from the random close packing, as is defined by Cumberland and Crawford .
 is the coefficient of effective friction between the stagnant zone and the rough bottom of the channel, which makes it possible to define the basal friction force . In addition to the tity-component (tangential to the wall) of the weight of the control volume (equation (5) for ), the basal friction force also includes the y-component of the force due to momentum flux between sections S and S∗ (equation (6) for ). is estimated by whatever the slope, as detailed in Chanut et al. .
Fmv is the force due to momentum variation over time inside the control volume (Fmv = 0 in steady regime). Detailed calculation to obtain equation (7) is given by Chanut et al. . The mass flux from the inertial zone to the quasi-static zone is neglected: for any time t, where , * and h* are depth-averaged volume fraction, depth-averaged velocity and thickness of the outgoing flow at sectionS*. This assumption is reasonable except for times shorter than the characteristic time τ of the dead zone formation.
 1. We do not consider the avalanche front but only the avalanche body and tail when the stagnant zone is formed (constant length L over time while and h can change over time, as reported in Caccamo et al. ): the dead zone angle αdz is equal to the steady value defined by αdz = θ − θmin, as described by Chanut et al. . θminis the angle associated with quasi-static deformations, i.e., the minimum angle above which steady flows can occur (as is defined byPouliquen ). This assumption may fail at low slope angles for which the characteristic time τ of dead zone formation is large (see discussion in section 4).
 2. The free-surface angleαfs (see Figure 2) is assumed to be equal to the dead zone angle αdz: according to assumption 1, the deflection angle α (mean value of the both aforementioned angles) can be expressed as α ≃ θ − θmin.
 3. For typical values from numerical and laboratory data (described in sections 3 and 4), the terms (≃ α) and ( ) are generally small compared to large values of L/h in expression of V0; this assumption, combined with assumptions 2 and 3, gives .
 4. The term Fmvis assumed to be negligible: this assumption is reasonable for both body and tail of the finite-sized avalanche-flow, but should fail for the avalanche front.
 These assumptions lead to a scaling law giving the force F relative to the force due to the incoming kinetic energy, , as a function of the Froude number:
can be interpreted as an equivalent drag coefficient that predicts an increase of when Fr is decreased. depends on the slope θ but also on parameters associated with the mechanical properties of the flowing grains: the shape of the velocity profile β, the restitution coefficient e (via κ) and the minimum friction angle θmin. is the sum of the classical earth pressure coefficient (k) and a term (K*) that accounts for the presence of the stagnant zone behind the wall. K* depends on the slope θ, the minimum friction angle θmin and the ratio of the wall height to the thickness of the incident flow, H/h. The equation for K* show that high values of H/h and θmin contribute to enhance the equivalent drag coefficient.
3. Scaling Law Versus Numerical Data
Figure 3 shows the force derived from equations (8), (9), and (10) as a function of the force calculated from discrete numerical simulation(DNS) in two dimensions (2D) by Chanut et al. . Summarized information on DNS is provided in Table 1 (see also inset of Figure 2 that displays a sketch of the flow geometry). We consider the force per unit width relative to the volume fraction of the incoming flow in order to compare the numerical tests to the laboratory tests for which the flow density was not measured. All the data tend to merge in one single group, as shown in Figure 3. We applied a linear fit (y = ax) to the data, which makes it possible to determine the average volume fraction of the numerical granular flows: (see line in Figure 3). The value 0.66 is fully compatible with typical volume fraction from 2D dense granular flows. In inset of Figure 3, results are compared with a prediction based on a purely inertial force: . The latter analysis leads to a large spreading of the data compared to the data grouping from equations (8), (9), and (10), which shows the robustness of the scaling applied to 2D DNS.
Table 1. Summarized Information on Discrete Numerical Simulations (DNS) and Laboratory Tests (LT)a
d is the particle diameter, H0 is the height of the release gate (reservoir exit) and xwall/d is the position (from the release gate) at which measurements (h and for control flows without any obstacle, and F in presence of the wall) were made, as depicted in inset of Figure 2. Other symbols as defined in section 2.
Figure 4 shows the force derived from equations (8), (9), and (10) as a function of the force measured in laboratory tests (LT) by Caccamo et al. . Summarized information on LT is provided in Table 1 (see also inset of Figure 2). For slopes greater than 24°, all the laboratory data collapse into one single group, as shown in Figure 4, which shows the relevance of the scaling. For slopes in the range [21°–24°], the data deviate from the single master group. The force predicted by equation (8)is systematically larger than the measured force. This result is caused by side-wall effects that decreaseK* in equation (10) and that are not considered in the theoretical scaling law proposed here. Faug et al.  showed that the model's prediction could be improved at low slope angles by increasing the characteristic time τ of dead zone formation in the expression that gives the dead zone angle over time. As a result, the length of the stagnant zone increased slower and did not reach its steady value, as it is assumed here to derive the scaling law (assumption 1 in section 2). Again, we applied a linear fit (y = ax) to the data (for θ > 24°), which makes it possible to determine the average volume fraction of the laboratory granular flows: (see line in Figure 4). The value 0.44 is fully compatible with typical volume fraction of 3D dense granular flows [see Forterre and Pouliquen, 2008]. It is notable that the fitted ratio is exactly equal to typical ratios of volume fractions stemming from 2D and 3D granular systems, as described in Cumberland and Crawford . Inset of Figure 4shows a cross-comparison with a prediction based on . Again, the analysis based on a purely inertial force leads to a large spreading of the data compared to prediction from equations (8), (9), and (10), which shows the robustness of the scaling applied to LT (see in the inset).
 This study opens a path to future theoretical developments for narrower obstacles when lateral fluxes occur, and comparison with well-documented field data. Work is under progress to extend the scaling to more complex shapes of obstacles by generalizing theH/h − dependent term that describes the important contribution of the dead zone process in the force. The network of chain forces formed inside the dead zone might explain the enhancement of drag coefficients for narrower obstacles such as masts in low-velocity regimes, as observed bySovilla et al. [2008, 2010]for snow avalanches. Accounting for more complex shapes of dead zone and the associated network of force chains via a mean correlation length will be a major step towards next models of avalanche force. Data from snow avalanches that interacted with plate-like obstacles byThibert et al. or pylon-like obstacles bySovilla et al.  will be useful.
 This letter described a scaling law giving the force experienced by a wall overflowed by a dense granular avalanche. The force on the wall relative to the force due to kinetic energy is derived as a function of the Froude number and the obstacle height relative to the flow-depth. This scaling is suitable when a stagnant zone is formed upstream of the wall and co-exists with an inertial zone above. The theoretical scaling law was successfully compared to small-scale data from both discrete numerical simulations (2D) and laboratory tests (channelized flows), in a wide range of slopeθ (14°–32° for DNS and 25°–33° for LT), of Froude number (0.06–3.8 for DNS and 0.4–8.8 for LT) and of wall height relative to flow thickness, H/h (1–7 for DNS and 1.7–9 for LT).
 The authors acknowledge financial support from the French National Research Agency (ANR-MOPERA) and from Interreg Alcotra project (MAP3).
 The Editor thanks Betty Sovilla and Shiva Pudasaini for their assistance in evaluating this paper.