Contribution of theoretical and experimental results to powder-snow avalanche dynamics

Cold Regions Science and Technology, 8 (1983) 67 - 7 3

67

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

C O N T R I B U T I O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L R E S U L T S TO POWDER-SNOW AVALANCHE D Y N A M I C S P. Beghin and G. Brugnot Centre du Machinisme Agricole, du G~nie Rural, des Eaux et des For~ts, Grenoble (France)

(Received March 31,1982; accepted in revised form December 22, 1982)

ABSTRACT A powder-snow avalanche can be considered as the flow o f a turbulent buoyant volume o f heavy fluid (air-snow suspension) in an ambient fluid, the air. In the dynamics o f such a flow, two mechanisms must be taken into account: the air entrainment and the snow entrainment inside the avalanche. From fluid mechanics equations (mass conservation and momentum equations)formulae were obtained giving velocity and density o f the avalanche as a function o f the slope path, the growth rate o f the avalanche and fresh snow-cover characteristics. On the other hand, laboratory simulations gave (among others) experimental results about the growth rates o f buoyant clouds. From these theoretical and experimental studies, practical examples are proposed with given path profiles and snow-cover characteristics. Such examples can be generalised to any other cases.

1. A QUALITATIVE ANALYSIS OF THE P.S.A. PHENOMENON

INTRODUCTION Among all avalanche types, the powder-snow avalanche (P.S.A.) is, as a matter of fact, one of the most frightening and the least known. Field measurements are not easy and remain scarce. Observations and a posteriori witnesses often lack accuracy and are difficult to use. Therefore C.E.M.A.G.R.E.F.*

*C.E.M.A.G.R.E.F.: Centre du Machinisme Agricole, du G6nie Rural, des Eaux et des For6ts. 0165-232X/83/$03.00

and I.M.G.** have developed, since 1974, a physical model of the phenomenon. We have considered a powder-snow avalanche as the down slope motion of a turbulent buoyant cloud denser than the surrounding air which is entrained; it consists of snow particles suspended in air. The model study was carried out in a closed inclinable clean water tank in which a denser fluid was injected, first salt solution, then sand and barium sulphate. The purpose of the present study was to determine the laws of the dynamics of the powder-snow avalanche in two different ways: from theoretical considerations of fluid mechanics on one hand, from model results on the other hand. Several formulae were established for the avalanche velocity, density and dimensions, calculated from the slope configuration and snowpack characteristics (thickness and density of recent snow).

© 1983 Elsevier Science Publishers B.V.

Description of a powder-snow avalanche A P.S.A. may be considered as a finite volume of very turbulent buoyant flowing fluid down a slope and may be referred to as an inclined thermal. The snow particles remain suspended due to large turbulent eddies which have a characteristic velocity higher than the fall velocity of snow particles in air. The flow dynamics are ruled by the gravity **I.M.G.: Institut de M~canique de Grenoble.

68

~~.

air entrainment

oese~[arge

,,-:~-:: "

turbuleeddi nt

Large turbulent eddies

Fig. 1. Sketch showing the mechanism of air entrainment within a P.S.A. forces and the entrainment mechanisms which divide into: • Ambient air entrainment: at the avalanche-air interface, surrounding air is entrained within the cloud by the large turbulent eddies (Fig. 1). • Snowpack entrainment: when the intergranular cohesion of the upper layer is rather weak, it can be entrained within the avalanche. Hopflnger and Tochon-Danguy (1977) consider that a snowpack fraction is lifted above the avalanche front, then incorporated due to gravitational instability. It is interesting to remark that snow entrainment is both the driving and retarding mechanism: driving because it provides an increasing flux of snow to accelerate the avalanche and retarding because the entrained snow is motionless.

the forces we have to consider are gravity forces, turbulent ground friction and fresh snow layer entrainment forces. The density of the avalanche is probably nearly that of the fresh snow layer. The entrainment of ambient air is negligible. We suppose a P.S.A. to develop from this turbulent flow but we must acknowledge the lack of experimental evidence of this transition. Some authors, for instance Mellor (1978), consider that the turbulent boundary layer appears above the sliding mass of snow as soon as the average velocity reaches 10 m/s; in that layer some particles are set into suspension. This process of suspension mechanism was also discussed by Hampton (1972), related to turbidity currents. They are of the opinion that the above explanation is not entirely satisfying, since the described process is not efficient enough to explain how a large mass of particles is set into suspension. They show how the mixing with the ambient fluid occurs in the flow head (Fig. 3).

turbulent boundary l

a

~

=i:i"~":

"

air entrainment ~fresh snow cover Fig. 3. Sketch showing the mechanism of P.S.A. formation from sliding snow, snow lifting lifting

,~;::::., ~ ~ n' t r ~snow n ~ oentrainment t 2. THEORETICAL AVALANCHE

STUDY

OF POWDER-SNOW

Fig. 2. Sketch showing the mechanism of snow and air entrainment within a P.S.A.

We suppose the flow to be two-dimensional, i.e., its width is constant. (Channeled track or very broad avalanche.)

The formation of a powder-snow avalanche

Calculation hypothesis

First, due to its instability, the upper fresh snow layer sets into motion. After Mellor (1978), it grows rapidly into a turbulent flow. In this stage of flow,

Since the P.S.A. flow is considered as a turbulent buoyant fluid cloud, we use the hypothesis of Beghin, Hopfinger and Britter (1981) that the ground friction

69 is negligible if compared to the entrainment drags whenever the slope angle is greater than 5 ° . Furthermore, we suppose the sedimentation of the snow particles suspended to be negligible and the growth coefficients to depend only on slope angle and on snow entrainment. The following notation will be used: H avalanche flow height ( H = H o for x f = 0 ) L avalanche flow length (L =Lo for x f = 0 ) A avalanche flow volume per width unit (A = A o for xf = 0) k form coefficient (A = k .HL) Pa ambient fluid density avalanche average density with A~ = p - P a (P= Po f o r x f = O) x distance travelled by the avalanche mass centre (from the flow origin) xf distance travelled by the avalanche front (from the flow origin) U avalanche mass centre velocity Uf avalanche front velocity (Uf = Ufo for xf = 0) al height growth coefficient (al = dH/dxf) a2 length growth coefficient (a2 = dL/dxf) 0 slope angle ON new snow layer density h N new snow layer thickness c~ the new snow thickness fraction entrained by avalanche m avalanche mass Kv added mass coefficient (see Batchelor, 1967)

/

\

'4 Fig. 4. Sketch showing the characteristics of a P.S.A. d _-- [(KvPa + p)A U] = (~ , Pa)AgsinO td

(2)

KvPa is the added mass term which accounts for the motion induced by the avalanche on the surrounding air. Since U = Uf[1 - (t~/2)] and d

d dxf --

dt

d --

dxf dt

Vf

- -

dxf

we can transform (2) into: dUf d (Kvp a + p~A Uf dxf = A~Ag * - U~ ~ f [(KvPa +p--)A]

g* = g sinO/[1 - (a2/2)]

(3) Flow equations Mass conservation

From the origin to the frontal position of the avalanche, at xf (Fig. 4) the mass of snow incorporated in the flow is CzhNPNX f. Since the volume concentration of snow can be neglected, we can consider that the mass of entrained air is equal to Pa(A - A o). Thus: m = pa(A - A o ) +CzPNhNXf +-PoAo

(1)

Momentum conservation During the time interval dt, the momentum variation equals the slope parallel components of the gravity force, so:

The last term of eqn. (3) is actually the resisting force corresponding to snow and air entrainment. If the growth rates al and a2 are supposed to be independent of xf (Beghin, Hopfinger and Britter, 1981) it results that: H = Ho + OtlXf L = Lo+ot2x f We suppose the longitudinal cross-section of the avalanche to be an elliptic half-cylinder. Then: Kv = 2H/L Using eqn. (1), we can write: (KvPa + p~A = kPa(aX ~ + b x f + c)

(Batchelor, 1967)

70

3. EXPERIMENTALRESULTSANDDISCUSSION

where: a

= 2 o ~ + oq o~2

Following Hopfinger and Tochon-Danguy (1977), we consider that the two adimensional numbers to be considered in the simulation of the powder snow avalanches are the densimetric Froude number U/[(A~/pa)gH] I/2 and the density ratio A~/pa. If both parameters are considered, the velocity in the laboratory is too high if referred to the channel dimensions. On the other hand, it is impossible to find such fluids with A-P/pa as high as in nature. Therefore we introduced a distortion of the density ratio.

ClhN

b = 4aiHo +a2Ho + a , L o + HoLo c = -Po ~ + 2H~ Pa

Then (3) transforms into: 1

d

kPa(aX~ +bxf+c)'-~ ~ (U~) = (C,hNPNXf+ A-~k HoLo)g * - k Pa( 2ax f + b) U~

Experimental results (Hopfinger and Tochon-Danguy, 1977; Beghin, Hopfinger and Britter, 1981)

The solution of which is: 4

3

2

Ut2= 2g_____** [34"~-+[3'"x'~" +[32"~" +[3'xf+[3° (ax~ + b x f +c) 1

kPa

(4)

inflowrate Releasegate~~(heavy fluid)~

When xf = 0, Uf = f f o , [3o-

kpa c2 2g* Ui2°

Water tank _~J

,~/"

and ~' ~.

. ...'."~"'

[3t = cA'PokHoLo [32 = b A-~oktloL o + c ' C l h N P N 133 = a A ~ o k H o L o + b . c t h N P N [34 = a ' C l h N P N

on the other hand, from eqn. (1), with m = k(Pa +

,ao-)nL, Fig. 5. Sketch of the experimental set-ups (gravity current). -Pa =

ClhNPNXf + (-Po - pa)kHoLo k ( n o + OqXf) (L o + a2Xf)

(5)

The flow is completely defined if we know the following array of values: • at the origin: H0, Lo, Po, Ufo • alongthe slope: PN, hN, cl, oq, o~2 When xf is large enough, Uf has an asymptotic value proportional to (ClhNPN) 1/2 if cl g:0 (as predicted by Voellmy, 1955) and Uf decreases as 1/x/xf if ct=O.

The experiments were carried out in a perspex water tank (300 cm long, 50 cm deep, 30 cm wide). Two types of flow were studied (Figs 5 and 6): • The gravity current type, with continuous buoyant fluid injection. The head of the flow is followed by the quasi-steady flow of the body (Fig. 5). • The buoyant cloud type where a finite volume of buoyant fluid is released instantaneously. Behind the head, there is no steady layer (Fig. 6).

71

dH --

dxf

~

10 -~ (5 + o)

(8)

and also that the length growth rate is given by eqn. (7).

~

ns calculatiohypothesi n s volume Discussioabout

~~

0f heavy f l u i d

Fig. 6. Sketch of the experimental set-ups (buoyant cloud).

For a gravity current, the height growth rate of the layer is

dh -- ~ 10-3(5+0) dx The height growth rate of the head depends on the inflow rate and is always greater than that of the layer. For a buoyant cloud, it was reckoned that: dH --

~ 3.5 × 10 -3 (0 + 10)

The hypothesis of a negligible sedimentation is only valid if the characteristic velocity of the large eddies Ut is higher than the free-fall velocity of snow particles Uc. If we take Uc -~ 50 cm/s (Mellor, 1965) and Ut ~ (dH/dxf)Uf (Tekkenes and Lumley, 1972), for 0 = 10°, the hypothesis is valid when Uf is higher than about 10 m/s. In tlae second hypothesis, the growth rates at and a2 are supposed to depend only on the slope and on the snow entrainment. In fact, the surrounding fluid entrainment seems to depend also on the ratio

(~/pa) ''~.

In the laboratory it was shown that the height growth rate was a little greater when ff ~ 2pa than when ~ ~ Pa. In the first approximation the experimental results from our model study will be used for practical applications.

Uf (cmls) 1

(6)

dxf P

and dL - - ~ 4 × 10 -a (0 + 60) as soon as0 ~>50. dxf

~'

v

1

(7)

Experimental results about the front velocity Uf of buoyant clouds are compared with theoretical ones from eqn. (4), showing eqn. (4) to be relevant only when Cl = 0. On the other hand, some experiments on gravity currents (Hopfinger and Tochon-Danguy, 1977)were run with a thin layer of dense fluid distributed along the floor: they indicated that the layer was to a large extent lifted up above the head and so the height growth rate of the head became equal to that of the steady layer behind: 10 -3 (5 + 0). By extrapolation we suppose that the height growth rate of a buoyant cloud when ground entrainment occurs is also

SLOPE EXPERIMENTAL POINTS THEORETICAL CURVE 5~ I

i o + 1 5 o

i

0

20

. . . .

(i) [2} x

(3)

5'o . . . .

100' . . . .

1;,~xfl cm )

Fig. 7. Front velocity Uf as a function of the distance x f for flows on different slopes (from Beghin, Hopflnger and Britter experiments). Comparison with theoretical results from eqn. (4). Origin x f = 0 is taken where the flow is completely established (about 50 to 100 cm downstream from the release gate (Fig. 6)). Ufo is the experimental value of Uf at this origin.

72

4. PRACTICAL

APPLICATIONS

In this Section, a real avalanche is considered to present, from the starting zone, the structure of a buoyant cloud (Fig. 6) rather than that of a gravity current. The theoretical formulae (4) and (5) are used to calculate the avalanche velocity and density. With the assumption that the laboratory results are valid for the real case, the growth rates used in eqns. (4) and (5) are given by the formulae (6) and (7) and when snow entrainment occurs (cl =~ 0), by (7) and (8). From the starting zone the avalanche path is divided into successive parts where the slope and snow entrainment are constant. The flow origin (index O) is taken at the beginning of each part. Therefore, using eqns. (4) and (5), it is possible to plot the front velocity and mean density of the avalanche as a function of the distance from the flow origin. The following numerical examples illustrate these considerations. Curves were obtained from a Hewlett-Packard calculating plotting system.

Example I (Fig. 8) Uniform slope, 0 = 30°; PN = "75 kg/m 3. Fresh snow layer thickness, hN = 1 m.

Uf , A/~ (m/s](kg/m3] 80'.

f

\ /(Uf, ~ /

"w. . . .

-..

t

t

\

Fig. 9. Numerical examples o f P.S.A. flows. F r o n t velocity and m e a n density as a function o f distance.

Part 1 : Half-thickness is entrained into the avalanche (c, -- 0.5). Lengths of part 1 are 300 m, 500 m, 700 m. Part 2: No entrainment occurs (Cz = 0). Lengths of part 2are 1200 m, 1000 m, 800 m. Example 2 (Fig. 9) Uniform slope, 0 = 30 ~, PN = 75 kg/m3; h N = 1 m. Part 1: Cl =0.5; length= 500m. Part 2: c~ = 0.25; length = 500 m. Part 3: c~ = 0; length = 800 m. In Figs. 8 and 9, we see how the avalanche starts accelerating when snow entrainment stops. It can be explained by the fact that the snow entrainment has both a driving and a retarding effect on the flow (see Section 1).

- - .~.:o

-.-~-_

' c 1=0

CONCLUSION

1(] 0

uf ,~,(s (mls) (kglm 3] loo'.

sbo-- 7'o~- - - T~oo'-~oo-

- -~oo

"xf~°l

Fig. 8. Numerical examples o f P.S.A. flows. F r o n t velocity and m e a n density as a f u n c t i o n o f distance.

By considering a powder-snow avalanche as a buoyant cloud of finite volume, it was possible to quantitatively determine its flow parameters from fluid mechanics equations and laboratory results. For any slope profile and fresh snow cover property, avalanche characteristics can be determined along the whole path. From the velocity and density evalua-

73 tions it is possible to calculate the dynamic pressures on any structure. Little is known about snow entrainment conditions and it would be interesting to do further studies on the role o f parameters such as cohesion and density o f the snow cover in the entrainment mechanism. We are aware o f the importance o f performing field measurements o f P.S.A. which will be the only way for testing some controversial hypotheses in this paper. Therefore, we plan to achieve some stereophotogrammetric observations on high speed avalanches during next winter.

ACKNOWLEDGEMENT The authors gratefully acknowledge the scientific support o f Mr. E.J. Hopfinger and the financial support o f the Centre du Machinisme Agricole, du Gdnie Rural, des Eaux et des For~ts.

REFERENCES Allen, J.R.L. (1971), Mixing at turbidity current heads, J. of Sedim. Petrol., 41 (1): 97-113. Batchelor, G.K. (1967), An introduction to Fluid Dynamics, Cambridge University Press, 431 pp. Beghin, P., Hopfinger, E.J. and Britter, R.E. (1981), Gravitational convection from instantaneous sources on inclined boundaries, J. Fluid Mechanics, 107: 407-422. Hampton, M.A. (1972), The role of subaqueous debris flow in generating turbidity currents. J. of Sedim. Petrol., 42(4): 775-793. Hopfinger, E.J. and Tochon-Danguy, J.C. (1977), A model study of powder-snow avalanches, J. of Glaciology, 19: 343-356. MeUor, M. (1965), Blowing Snow, Cold Regions Science and Engineering Monograph, CRREL, Hanover, Part 3, Section A 3C. MeUor, M. (1978), Dynamics of Snow Avalanches, Ch. 23, pp. 753-792, in: B. Voight (Ed.), Rockslides and Avalanches, Developments in Geotechnical Engineering, Vol. 14A, Elsevier, Amsterdam, 833 pp. Tennekes, H. and Lumley, J.L. (1972), A First Course in Turbulence, M.I.T. Press. Voellmy, A. (1955), Uber die Zerst6rung yon Lawinen, Schweizerische Bauzeitung, Heft 73.