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Standing hydraulic jumps
The jumps have been widely studied in hydraulics, because this phenomena can be ob-served naturally in many situations, and has also been used in structures built to dissipate energy e ciently. The current knowledge on hydraulic jumps has been reviewed in [14].
The hydraulic jumps have been mostly observed and studied in standing conditions on a at smooth bottom [15, 30]. Under such conditions, mass and momentum conservation equations can be simpli ed to obtain the famous Belanger’s equation [4]: h 1 p 1 + 8F r2 1 = (1.4).
The jump height ratio h =h is directly controlled by the incoming Froude number F r. A hydraulic jump is characterized by a sudden rise in the free-surface between a supercrit-ical incoming ow and a subcritical outgoing ow. During the transition, a very turbulent zone with vortices, recirculation, and a high dissipation rate occurs, which is called the roller.
Several types of standing hydraulic jumps on a smooth at bottom have been identi-ed. Jumps with low incoming Froude numbers, just above 1, will create undular jumps, with a small height ratio, where the free-surface does not reach a constant value but os-cillates around an average value. On the other hand, jumps with high incoming Froude numbers will give a big di erence between the thickness before and after the jump, with a marked and easily identi able roller.
The hydraulic jumps have also been studied under other conditions than a at smooth bottom. In hydraulic works, the jumps are often used in order to dissipate energy. Most of the time, they are created in a sloping channel which suddenly becomes at. Figure 1.5 shows that depending on the position of the jump compared to the limit between the sloping part and the at part of the channel, the jump can form on what is called A-jump (foot of the jumps at the limit), B-jump (limit inside the jump), C-jump (end of the jump at the limit, or D-jump (all the jump in the sloping part of the channel). In particular, D-jumps, have been studied in [50, 48, 35]. The slope angle of the channel has a strong in uence on the jump height ratio. Belanger’s equation (Eq. 1.4) is not su cient anymore to obtain the height ratio directly as a function of the incoming Froude number. The height ratio of D-jumps is above the height ratio of a jump on a at channel for the same Froude number. Hydraulic jumps have also been studied on rough bases which bring dissipation by friction [13, 34]. As for sloping channels, the Belanger’s equation is not able to predict the jump height ratio. For a same Froude number, it will be lower than the one in Belanger’s equation conditions.
Other types of jumps can also be formed. A circular jump appears when a vertical jet impacts a plane surface, like the water on a sink (see [10]). The symmetry of the jet creates a jump all around the impact point of the jet. Opening a gate under the water between two basins at di erent water height will create a submerged hydraulic jump [49]. The jump exists, but takes place under the water.
In the design of avalanche protection dams, the equations for hydraulic jumps in a smooth at base are still used, including in the 2009 handbook by the European com-mission [3], in order to obtain the height ratio h =h. However, the behaviour of the snow inside a dense avalanche is much more similar to the one of a dry granular material: it occurs only on slopes, it is compressible, and frictional-collisional processes come into play. Because of some similarities with the granular jumps, hydraulic jumps in a sloping channel (D-jumps) or on a rough base (which brings friction to the water, comparable to the friction between grains) will be studied in more details in Part I. In order to take into account the slope angle, or some frictional processes, the jump cannot be considered as a shock with no length. It is necessary to take into account the nite length, nite volume, and nite weight of the jump.
General relation for standing jumps
We consider a free-surface ow|of either water or dry grains|down an incline, as shown in gure 2.1. Figure 2.1c (bottom panel) displays two pictures of granular jumps observed in laboratory tests by [19]: a di use jump at low slope angle and a steep jump, with the presence of recirculation, at high slope angle. The slope of the incline is called . The depth-averaged velocity, the density, and the height (perpendicular to the bottom) of the ow are respectively u, and h before the jump, and u , and h after the jump. The jump length L is taken parallel to the bottom (see its exact de nition in x2.3.1). When the bottom is rough, we introduce the parameter r which is the mean height of the roughness (see gure 2.1a).
The jump is de ned as the part of the ow between the upstream supercritical ow and the downstream subcritical ow, both at equilibrium. In this de nition, the jump cannot be considered as a shock, as done for instance in [32], [27], or [69]. In contrast to a shock, the jump has a nite length L and then a volume on which forces come into play. We apply mass and momentum conservation equations in their depth-averaged forms under steady state conditions to this jump volume, in a similar fashion as initially proposed by [61] for granular ows, and earlier by [16] for water ows. In addition, we take into account the density change across the jump, as recently considered by [19]. The forces that apply on the jump volume are the weight of the jump w (see its expression in x2.2.1), the e ective frictional force acting over the jump length b (see its expressions in x2.2.1 and x2.2.2 for dry grains and water, respectively), and the pressure forces acting on each side of the jump, which are assumed to be hydrostatic. The depth-averaged equations of mass and momentum conservation projected on the x axis (along the slope of the incline) read as follows:
1 1 uh = u h ; (2.2).
u2h u2h = k gh2 cos k gh 2 cos + w sin L; (2.3).
where k (respectively k ) and (respectively ) hold for the earth pressure and Boussi-nesq momentum coe cients before (respectively after) the jump, as will be discussed in more detail in x2.3.6.
Hydraulic and granular jumps data revisited and compared
In the present section, we compare the parameters discussed at the end of the previ-ous section, which were measured in a number of experiments under di erent boundary conditions with either water or dry granular materials.
Types and de nitions of the jumps
The data revisited in our study include (i) hydraulic jumps formed down smooth inclines measured by [35], [48], and [50], (ii) hydraulic jumps on a rough bottom by [13] and [34], and (iii) granular jumps down smooth inclines recently studied by [19]. Both granular and hydraulic jumps considered here were formed on a portion of constant slope, with a gate at the outlet. In hydraulics, they are called D-jumps (only the data from D-jumps from the above references about water jumps are studied here). It is also possible to form jumps in the middle of a slope break (the so-called A,B, or C-jumps in hydraulics).In the present paper, the jump we consider separates two ow conditions: the incoming supercritical ow where the free-surface is parallel to the bottom (i.e. tan dZ=dx = tan ), and the downstream subcritical ow where, the angle of the free surface becomes constant (tan dZ=dx = 0 for hydraulic ows, or tan dZ=dx = tan 0 for dry grains). The jump is de ned as the part of the ow between those two zones, thus corresponding to the ow region 0 x L in gure 2.1. This allows to de ne the length L, and the heights h Z(x = 0) and h Z(x = L). Note that the ows of dry granular materials are compressible, thus producing compressible jumps. In particular, compressible jumps were recently addressed by [19] who carefully estimated the density change across the jump = in their experiments. Based on this de nition of L, gure 2.2a shows examples of the shape of granular jump pro les evolving with the Froude number, as measured in the laboratory by [19].
Variation of the shape factor K0
The shape factor K0, de ned by (2.6), is useful to estimate the averaged value over the jump length of physical quantities (see for instance the estimation of uJ ) that depend on the position within the jump. The shape factor was only measured by [35] and [48] for hydraulic jumps, and by [19] for granular jumps. Figure 2.2b shows how the shape factor K0 measured in the aforementioned studies varied with the Froude number F r of the incoming ow. Data by [35] for water, and by [19] for granular uids, are remarkably of the same order of magnitude. Both data show that K0 increases with F r. This trend was however not observed by [48] who found a nearly constant K0 (equal to 1) in their experiments with water, though the range of F r which they investigated was similar. Note that a slight decrease of K0 with slope was observed by [48]. However this decrease is not observed by [35]. In the present paper (see x2.4), we will consider K0 = 1 for other experimental studies for which K0 was not directly measured or estimated: this concerns the data by [50] for water jumps down a smooth incline, and by [13] and [34] for water jumps on a rough bed.
Variation of the relative length L=(h h)
The relative length L=(h h), which appears in the simpli ed implicit relation (2.8) derived from the assumption = = 1 in the case of granular jumps, is analysed here. With both grains and water, L=(h h) is a decreasing function of the slope, as shown in gure 2.2c. It is worth noticing that L=(h h) reaches a nite value around 6 8 for water ows at = 0, as shown in gure 2.2c. Such a range is consistent with values reported by a number of earlier studies, as for instance in [55] and [2]. For valuable reviews which summarize the great number of experimental data on the length of hydraulic jumps, one can mention the works by [29] and, more recently, by [63]. It should be noted that in the paper of [13] it is not the total length of the jump L, but the length of the roller Lr which was measured (see sketch in gure 2.1a). However, the sequent depth was not measured at the end of the roller but|like in the other papers|at the end of the jump. Strictly, (2.15) should apply to L and h or to Lr and the ow height at the end of the roller, but should not apply to Lr and h . The length of the roller is generally smaller than the length of the jump and approaches the latter when the Froude number increases, as reported by [54]. Such a di erence between L and Lr becomes clear at = 0 and r = 0. For an horizontal smooth bed, L=(h h) is around 7 (data of [35], [50], and [34] shown by yellow points in gure 2.2c), while Lr=(h h) is around 4 (data of [13] shown by turquoise points in gure 2.2c).
Steady granular ows can only occur above a slope angle 0, which was about 23 in the laboratory test by [19]. No jump can form below this limit angle, and when 7!0, the relative length of the jump L=(h h) starts diverging, as seen in gure 2.2c. However, for higher values of , the values seem to align with the data with water. The Froude number of granular ows is mainly controlled by the slope angle, and decreases with slope. In contrast, it is possible to form supercritical water ows at high F r (far above the critical Froude number) on low slope angles, which can then produce strong hydraulic jumps of nite length. While approaching the slope 0, F r tends towards the critical Froude number, meaning that simultaneously h approaches h: the jump is more and more di use and nally disappears, as observed by [19]. The behaviour of granular jumps close to 0 would need further investigation.
Variation of the relative length L=h
Figure 2.2d shows how the jump length relative to the height of the incoming ow evolves with the Froude number. The behaviour of the relative length of the jump, L=h, changes strongly between dry granular ows and hydraulic ows. For granular jumps, L=h is almost independent of F r, and relatively small (around 10 20 in the tests by [19]), by comparison to the value obtained with hydraulic jumps (from 10 to 160). Those results suggest that the friction between grains is able to dissipate e ciently the energy of the incoming ow, over a relatively small distance.
On the contrary, gure 2.2d suggests that the relative length of the jump L=h is highly correlated with the Froude number in hydraulic ows, whether the channel is inclined or horizontal, smooth or rough. At a given F r, it is clear from gure 2.2d that the relative length of the jump is smaller if the bottom is rough, and it decreases as the typical roughness size increases. This means that a rough channel bottom is more able to dissipate energy than a smooth one. This result shows the key role played by the friction forces acting within the jump volume, thus justifying the general jump relation which we are proposing here.
Interestingly, the slope has no e ect on the relative length L=h of the jumps formed on a smooth bottom: the blue points for the jumps in water ows down a smooth incline, and the yellow points for the jumps in water ows on a horizontal smooth bottom, are aligned in gure 2.2d. This means that at a given F r, the jump length is only proportional to the height of the incoming ow and does not depend on the slope angle. We will propose simple closure relations for L=h in x2.4.4.
Depth-averaged friction laws derived using minimal as-sumptions
Determining the dimensionless number e, namely the e ective friction coe cient e within the granular jump or the hydraulic friction coe cient fe within the hydraulic jump, is of course a very challenging question, linked to the rheology of the uid at stake, the system size and/or the boundary conditions (bottom roughness in particular). We adopt here a back-analysis strategy. In the rst step, we compute the exact values of e needed to match (2.15) to the experimental data. In the second step, we compare those exact values to the predictions from friction laws derived using minimal assumptions under which the dimensionless number e is a function of the incoming ow features. In the present section, we consider = = 1 and k = k = 1 in (2.15). Those assumptions will be discussed in x2.3.6.
Table of contents :
I Theoretical relations for jumps in both hydraulic and dry granular ows
2 A general relation for standing normal jumps in both hydraulic and dry granular ows
2.1 Introduction
2.2 General relation for standing jumps
2.2.1 Standing granular jumps
2.2.2 Standing hydraulic jumps
2.2.3 Cubic relation for both granular and hydraulic jumps
2.3 Hydraulic and granular jumps data revisited and compared
2.3.1 Types and denitions of the jumps
2.3.2 Variation of the shape factor K0
2.3.3 Variation of the relative length L=(h h)
2.3.4 Variation of the relative length L=h
2.3.5 Depth-averaged friction laws derived using minimal assumptions
2.3.6 Discussion on other parameters in general jump relation: =, k, k, and
2.4 General jump relation versus laboratory experiments
2.4.1 Hydraulic, incompressible, jump down a smooth incline: e = 0, = = 1
2.4.2 Hydraulic, incompressible, jump on rough horizontal channel: e 6= 0, = = 1
2.4.3 Granular, compressible, jump on a smooth incline: e 6= 0, = 6= 1 32
2.4.4 Analysing both water and granular uids
2.5 Discussion and conclusion
3 Discussion on resistive forces across water jumps
3.1 Correction of the equation for water jumps
3.1.1 An error in the resistive force term
3.1.2 Possible eect on the general relation
3.2 Resistive force across rough water jumps
3.2.1 New expression for velocities inside water jumps
3.2.2 Other options for resistive forces across water jumps
3.2.3 New empirical expression that t the data
II Two Dimensional numerical simulations
4 Discrete Element Method simulations of standing jumps in granularows down inclines
4.1 Introduction
4.2 DEM simulations of standing jumps
4.2.1 Microscopic contact laws
4.2.2 DEM set-up to produce the jumps
4.2.3 Measurement techniques
4.3 A rich variety of granular jumps
4.3.1 Jump steepness
4.3.2 Jump compressibility
4.4 Conclusion
5 Length of standing jumps along granularows down inclines
5.1 Introduction
5.2 Methods
5.2.1 Numerical set-up to produce standing jumps
5.2.2 From micro to macro: coarse-graining
5.2.3 Dening the length(s) of the jump
5.3 The dierent types of jump
5.3.1 Laminar granular jumps
5.3.2 Steep colliding granular jumps
5.3.3 Hydraulic-like granular jumps with an internal roller
5.4 Jump length variation with input parameters
5.4.1 Slope angle
5.4.2 Mass discharge
5.4.3 Grain diameter
5.4.4 Interparticle friction
5.5 Discussion and conclusions
5.5.1 Jump height ratio
5.5.2 Length of the jumps
5.5.3 Eective friction depending on the jump type
6 Energy dissipation in numerical granular jumps
6.1 Introduction
6.2 Variation of energy along the incline
6.3 Energy loss within the jump
6.4 Comparison between coarse-grained exact energy and hydrodynamic energy from depth-averagedow properties
6.5 Discussion
6.6 Conclusion
III Experimental study
7 X-ray radiography of standing jumps down inclines
7.1 Experimental set-up and procedure
7.2 Granular jumps with spherical grains
7.2.1 Density elds
7.2.2 Macroscopic properties of jumps
7.3 Jumps with elongated grains
7.3.1 Similarities and dierences between spherical and elongated grains .
7.3.2 The role of the grain alignment
7.3.3 A new type of granular jump
7.4 Discussion and conclusion
8 Conclusions and perspectives
8.1 Snow avalanches and design of protection dams
8.2 Innovative tools
8.3 Dierent types of granular jumps
8.4 Towards a general theoretical framework
