ON NONUNIFORM, STEADY FLOW OF AVALANCHING SNOW

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1 ON NONUNIFORM, STEADY FLOW OF AVALANCHING SNOW Bruno SALM Federal Institute for Snow and Avalanche Research Weissfluhjoch, Davos, Switzerland ABSTRACT Methods known in hydraulics are used to calculate the steady and nonuniform movement of flowing avalanches. The forces developed by the deformation of the snow in motion are calculated assuming that it behaves like an ideal dry sand. The forces transmitted to the sliding surface are proportional to the normal force and to the square of the flow velocity. The theory is compared with experiments carried out on the test slide at Weissfluhjoch. RÉSUMÉ Le calcul des écoulements stationnaircs et non uniformes, connus en hydraulique, est appliqué ici au mouvement des avalanches coulantes et glissant le long d'une pente. Les forces provenant de la déformation de la neige en mouvement sont calculées selon l'hypothèse que la neige se comporte comme du sable sec idéal. On admet que les forces agissant sur la surface de glissement sont proportionnelles d'une part à la force normale et d'autre part au carré de la vitesse d'écoulement. La théorie établie est comparée aux résultats d'essais exécutés sur un plan incliné (glissoire à avalanche) au Weissfluhjoch. 1. GENERAL COMMENTS AND ASSUMPTIONS Methods have been developed in hydraulics which are believed to be suitable also for the investigation of avalanche movements. As these methods are only suitable for /lowing avalanches ('), any other forms of movement are excluded from the study. It must be realized that the properties of snow in motion are to a certain degree different from those of water. The purpose of this paper is therefore to modify the known methods in hydraulics and to determine the factors characterizing the flow. With technical hydraulics as a base, it is self-evident that the result can only be a contribution to technical avalanche dynamics. A more complete treatment of the problem would be possible with an enlarged knowledge of the mechanical properties of moving snow. The following treatment is restricted by the subsequent assumptions : flowing avalanches consist of clods of snow with a mean diameter which is much smaller than the flow depth; the nonuniformity of the flow is created by a change in the slope angle. The flow is two-dimensional but not in a general sense, as it will be shown later; finally the processes are assumed to be steady. There is an important difference between the nonuniform flow of water and that of snow. In the latter case the change inflowdepth causes a sort of energy dissipation which does not occur in a corresponding water movement. The relative motion among the snow particles which always takes place in such a situation gives rise to frictional forces. Present knowledge about these forces is very poor. Some experimental data ( 2 ) show a dependence on the normal force as well as on the rubbing speed for a certain type of snow and temperature. 19

2 2. BASIS FOR THE CALCULATION OF NONUNIFORM MOVEMENT 2.1. Deformation of avatanching snow Observations on avalanche deposits suggest in many cases that avalanching snow has the properties of an ideal dry sand. This simple incompressible material is characterized by internal friction forces independent of the speed and given by the angle of internal friction p, a constant of the material. As long as T < a tan p 0) no deformation is possible, where z stands for the shear stress and a for the normal stress on a surface element. The stress-strain relation under this condition is given by a different constitutive equation, which describes an additional feature of our material. Experiments ( 3 ) suggest that in such a case snow may be considered as an elastic material. As soon as at a certain surface element the equation ic = a tan p (2) is fulfilled, sliding takes place along certain slip lines. The material constant p does not depend from the strain magnitude nor on any other factor. In a a, r-plane the relation can be explained as follows. The Mohr circle diagram for stress at a certain point in the medium is situated in an angular space between two straight lines symmetrical to the a-abscissa and open in the positive direction of the abscissa. Deformation takes place as soon as the Mohr circle touches the straight lines. Let us assume that in a half space the stresses on a surface element parallel to the plane z = constant are given by n z and txz (fig. 1). The two possible states of deformation arc then given by the two possible Mohr circles, equivalent to the active and passive Rankine states of stress. The principal-stress ratio is generally given by : Major principal stress Minor principal stress = tan2 1 (3) Passive RANKINE s /state of stress.xactive RANKINE s7 state of stress, \ Fig. 1 Rankine's state of stress in a material without cohesion. 20

3 2.2. Forces transmitted to the sliding surface of the avalanche track As was suggested in ( 4 ) the force acting on the interface of the moving snow and the motionless underground has two essential components : A frictional force proportional to the normal force on the sliding surface, but independent of velocity. The factor of proportionality /< is the coefficient of friction. If the sliding track is curved with the radius of curvature 0, this frictional force increases with decreasing positive radius 0 (reduction of the slope angle) and vice versa with negative radius. It may be noticed that generally // ^= tan p. Because of the increased production of heat on the sliding surface it is supposed that /< <ê. tan p. A frictional term applied in hydraulics in the calculation of turbulent water flow in open channels (CHEZY'S equation). This resistance increases proportional to the square of velocity and is given by the coefficient of roughness k~. When the slip surface with a mean slope angle y> extends to infinity along any level and is curved according to a radius of curvature 0,the following formula for the mean velocity v is obtained 2 sin ^ - g cos 4, (4) 2+ dk g<p d stands for the flow depth measured perpendicular to the slip surface and g for the acceleration due to gravity. The velocity distribution is unknown but is supposed to be approximately uniform except in the vicinity of the slip surface. 3. STBP-BY-STEP CALCULATION OF THE NONUNIFORM, STEADY FLOW The starting point is BERNOULLI'S equation adapted to the case of flow which is subject to frictional forces. The total head therefore decreases in the direction of movement. This adaption of an equation originally derived for frictionless flow, proved good in technical hydraulics. The maximum pressure p on the sliding surface is then: p = a x = ).a z = kyd I cos i/h. (5) V g<pj A fluctuates between the limiting values?. a ^ ). ^ ).,,. y is the mean specific gravity of the flowing snow. a x and a z denote the magnitude of the stress vector in direction of the x and the z axes respectively (fig. 1). The second term within the brackets on the right-hand side of (5) takes into account the change of a z due to the curvature of the sliding surface. Within certain restrictions it is correct to equate the scalar quantity p to a value proportional to the magnitude of the stress vector a z. Equation (5) remains valid as long as the EULER equation formulated in the direction of the tangent to the path line is accurate enough to describe the motion. In this case only the component of p in direction of the tangent is used. If the curvature exceeded certain limits, it would be necessary to consider also EULER'S equations in the direction of the normal and/or binormal to the path line. Then equation (5) would be invalidated, the pressure distribution being no more triangular. But the steady movement of a flowing avalanche along a sharply curved path seems to be rather improbable. Observations show that 21

4 avalanche tracks are usually smoothed out by "dead snow" and therefore a lower limit of the radius of curvature, depending on the properties of the moving snow, seems to exist. The BERNOULLI equation, modified for flowing avalanches becomes +hl[ cos \ji + )+// = H, (6) 2g V g<p/ where A denotes the geometrical head and H the total head, decreasing in the direction of the movement. :' Equation (4) yields the mean gradient of the total head 1 a if the inclination angle of the total head K is inserted instead of y> and furthermore (7) ; 2 g<p Now the task remains to analyze the dependence of A (eq. 6) on the angle of internal friction and on the acting stresses. So far we know only the limiting values X a and ). p. With a nonuniform flow a z and T X Z are functions of the coordinates and therefore this is true also for A (fig. 1).. The relation yields after a little manipulation 2 = ± '! - = arid J J + 4 L À p cos 2 p\<t : The relation given by equation (11) is represented in figure 2. The long axis of the ellipse is situated in the A-ordinate as long as pis smaller than about 21 (X p '4.6). If p is larger, the long axis is parallel to the r^/ctj-abscissa. For the shear stresses transmitted on the sliding surface, we obtain x xz = yd \ H cos'tjj + v A r- H = yj (n cos i/'-f-a) (12) and on the other hand for the normal stresses : 22 a, = yd[cosil/+ ) (13) \ g<pj '

5 The ratio to be inserted into equation (11) is therefore a z cos y -\ v g<p 4 a_ (14) r _ i _^J<Z Fig. 2 A as a function of the ratio Applying equation (6) to adjacent cross sections 1 and 2, between which the interval should not be too large, and furthermore making use of the equations (7), (11) and (14) it is possible to write down the general equation for the nonuniform, steady movement of a flowing avalanche as follows: = as tg + l+2/ 2 2g (15) The quantities belonging to the cross section 1 are labeled with a suffix 1 and analogously the quantities of cross section 2 with a suffix 2. Ah is the difference in the geometrical head and As denotes the horizontal distance between the adjacent cross sections, (fig. 3). 23

6 Fig. 3 Step-by-step calculation of the nonuniform, steady flow. The easiest way of solving equation (15) is by trial and error using a computer. If the calculation proceeds from cross section 1 to cross section 2, a flow depth d% has to be estimated. From this all unknown quantities of cross section 2 can be calculated. First of all where Q [m 3 sec" 1 ] is the (known) rate of flow and B the width of the stream. This yields on one hand the new velocity head v\l2g and on the other hand from equation ( 14) txzloz- fa from this follows by using equation (11). Equation (7) yields the mean gradient of the total head, when in equation (8) the mean velocity is : and the mean flow depth is : 24

7 The calculation must be continued until for an estimated value of d-i the right side of (15) equals its left side. To obtain unique solutions of (15) a boundary value must be given. This means that the starting point of the calculation is a cross section with known quantities. But does such a cross section always exist? As long as in a surface element equation (2) is fulfilled, a local disturbance can propagate at the critical speed (*) v cr. = -Jd). cosi^ g.. (16) «or. defines the wave velocity for our idealised avalanching snow. We can therefore also distinguish between two types of flow, as is usually done in hydraulics : the "streaming flow" (v < Der.) and the "shooting flow" (u > Der.). In analogy with hydraulics the calculation has to proceed upslope in the case of the " streaming flow" and downslope in the other case; starting in both cases from a known flow depth. Further analogies with hydraulics arc as follows. The transition from the shooting flow to the streamingflowhappens always in a discontinuous way, which is called the hydraulic jump. The flow depth after this jump d u can be calculated from the following equation, obtained from the theorem of momentum:,2 K cos \j/ u... 1 rfo^o/, i.a d u - -^Hsmft.-pcos i/o+ --^- (l+/i - JiH 2 d u g \ 180 / 1 i^o "o,,, {' <> d u cos t/'o, y o\ n ni\ 4 ; «o cos zli// 1 I = U (17) ^ /c V 2 ff/ The quantities with suffixes "0" and "w" refer to the upper, lower cross sections respectively. L is the distance between these sections and/iy the difference between the slope angles yo and yv The change from streaming flow to shooting flow is continuous. At the point of deflection the flow depth attains the critical value gb 2 ;.cos (18) corresponding to the critical velocity o C r.- The calculation according to (15) starts therefore upstream and downstream from the point of deflection. Finally the conclusion can be drawn, that in every case a known boundary value exists as a starting point for the stcp-by-step calculation with equation (15). 4. EXPERIMENTS WITH THE TEST SLIDE The test slide, situated on the Weissfluhjoch, allows the artificial release of flowing snow masses. The upper part of this plant, the acceleration section, is 20 m long and 2,5 m wide. The inclination of this section can be varied between 30 and 45. In the topmost part there is a snow collector where the test snow is filled in (Maximum volume 12 m 3 ). During the short time while the snow accumulates it developes a small (*) The conclusions drawn in ( 4 ) concerning the wave velocity are only valid in special cases. 25

8 amount of cohesion, so that, as the sliding surface consists of smooth aluminium profiles, the snow moves on the whole acceleration section in a translators way. Two light barriers arranged at the end of this section serve to detect the frontal velocity just before the snow reaches the test section. The surface of the latter ascends slightly in the direction of movement with a slope angle of 5 30'. For constructional reasons this inclination can not be changed. At the deflection point the flow pattern changes to one of a flowing avalanche. i n m / *:-..]. Hu-i- 38m. Normal stress.shearing stress Fig. 4 Experiment 8/67 (Test slide Weissfluhjoch). 26

9 strain gauges. A mechanical method of measuring forces using a steel cone penetrating an aluminium plate is provided in box C. This method gives only the maximum value of both force components, it is used to check its reliability by comparing with the electronic measurements. Fig. 5 Test section with measuring boxes and a light barrier (Photo SLF). The forces transmitted on the sliding surface of the test section are measured with 4 boxes arranged in a row (fig. 5). In the boxes A, B, and D (fig. 4) the normal forces at two points and the shear forces at one point are measured in each box by means of Only few comments will be given here about the 17 tests carried out during winter 1966/67. One typical test will be selected for more detailed discussion. The final velocity before the point of deflection in every case exceeded (; cr.. As the inclination of test section remained slightly ascending, we had in each test the conditions for a hydraulic jump. So the flow depth on the test section can be calculated with equation (17). Since the snow after the end of the section drops in a free fall to the ground, the active RANKINE state of stress with X u = X a is therefore reached at that point. Equation (17) yields for experiment 8/67, d u = 1.38 m. Under the existing test conditions the snow must move with a constant flow depth d u over the test section, except in the vicinity of the deflection point. a z can not decrease in the case of 0 = oo and a negative tp, because a reduction in theflowdepth would give rise to an accelerated movement. On the other hand an increase of o z is just impossible, because the normal stress n x must reach its minimum magnitude at the end of the section, and an increase of the flow depth can not take place with decreasing a x. It follows that with a constant flow speed the gradient of the total head e remains constant within the range considered. Equation (15) is applicable only in the vicinity of the deflection point. Forexp. 8/67 the nonuniform movement was calculated at the cross sections I, II and III (fig. 4). The radius of curvature 0 was obtained from equation (13) by inserting the corresponding measured magnitude of a z. The maximum normal stress for the stationary case, 27

10 Fig. 6 Flow of the test snow over the test section in experiment 8/67. The effect of the deflection point on the snow can be seen. (Photo SLF). Fig. 7 Oscillograph curve of.experiment 8/67. The numbers refer to the measuring points indicated in figure 4. 28

11 6030 kp/m 2, yields a minimum radius of 0.57 m. Here we take note of the fact that equation (14) gives generally no limitation on <P because of /u <g tan />. Only a restriction on 80, : ds(where s denotes the length of the path line) is given by equation (15). The result of the calculation shows a large rise in the flow depth between cross sections I and II. At the latter section the flow depth d u is already reached. This leads one to suppose that the forces acting on the slip surface must be concentrated near the deflection point, and that on the major part of the test section <r 2 is not larger than yd u = 500 kp/m 2. As regards the distribution of the normal stresses we notice that apparently a flow parallel to the surface of the test section is reached not earlier than in the middle of box B. Closer to the deflection point the theoretical flow pattern does not seem in reality to have been reached. Figure 7 shows the oscillograph curve of test 8/67. The time axis is horizontal and the time sweep is 0.1 sec/cm. On the vertical force axis one division is equal to about 2200 kp. The distance between two adjacent lines of the coordinate scheme is one cm or one division respectively. The distinct nonsteady impact pressure may be interpreted as a consequence of the elasticity of the snow. At the moment when the snow front reaches the deflection point the radius of curvature is very close to zero because of the discontinuous change in the slope angle. With a rigid body the magnitude of the normal stress would increase locally to infinity. Of course this would never happen. If it is presumed that snow is deformed entirely elastically at the first moment, similar pressure curves to those in figure 7 are obtained ( 3 ). A very short time after the impact, a finite radius of curvature is formed with "dead snow" and so the steady phase is attained. REFERENCES (!) DE QUERVAIN, M.R., (1966): On Avalanche classification, a further contribution. Scientific aspects of Snow and Ice Avalanches, Davos, IASH Publ-, 69. ( 2 ) BÛCHER, E., and ROCH, A., (1946): Reibungs- und Packungswiderstande bei raschen Schneebewegungen, Interner Bericht SLF, Nr. 31. ( 3 ) SALM, B.,(1964): Anlage zur Untersuchung dynamischer Wirkungen von bewegtem Schnee, ZAMP, 15, ( 4 ) SALM, B., (1966): Contribution to Avalanche dynamics. Scientific Aspects of Snow and Ice Avalanches, Davos, [ASH Publ., 69. DISCUSSION A.K. DIUNIN This report is a very interesting new step in the understanding of avalanche mechanics. The author's works are well known in our country. I can advise the author to investigate the influence of turbulence because it can lead to a more correct physical picture. REPLY BY THE AUTHOR Turbulence in flowing avalanches (airborne powder avalanches and slab avalanches were excluded from the study) has been taken into account by using CHEZY'S equation for turbulent water flow in open channels. This simple assumption seemed to be the most reasonable one but it is conceivable to replace it by another correlation. A discussion about this problem seems to be fruitless as long as we do not know more about the mechanical properties of moving snow and about the phenomena within a snow stream. 29

Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 51 Earth Pressure Theories II

Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 51 Earth Pressure Theories II Welcome to lecture number two on earth pressure theories.