AD- A283 989__________f'ELECTE

Model for Avalanches inThree Spatial DimensionsComparison of Theory to ExperimentsRenee M. Lang and Brian R. Leo April 1994

T~YPT(r nTT ATr"T1V TPP'17D I

A three-dimensional theoly Is derived to describe the lemporal behavior ofgravy cirre of cohesloniess granular media, In an attempt to model themotoon of dense, flow-type snow avalanches, Ice and rock slides. A Mohr-Coulomb yield crllerion Is assumed to describe the consftitlve behavior of themalerial, and tie basal bed fricMion Is described similarly by a Coulomb type offfiction. A drag term Is Included In order to model the occurrence of flow regimeswhere boundary drag becomes non-negllglble. Data from laboraloy simula-tlons am compared to a series of numerical studies based on the aformen-tioned theory. A nondimenslonal, depth and wkih averaged form of the theoyis considered. A Lagranglan finite difterence scheme Is then applied to numeri-cally model some limiting cases of the governing equations. Two differentnumerical models are deloped, tested and compared o experimental values.The results Indicate that the model can account for flow transitions by Inclusionofthe drag term when the Initial Inclination angle is large enough to affect bound-ary drag. Furthermore, the temporal and spatial evolution of the granulate andfinal runout position can be predicted to values well within tie experimentalerror.

Clowe Avruftve =Ossing MIoxd Hlghw, New ZeolOn4 August 1988.(PthoA by Am DennW.)

Forcmvsi of Si meic unl to U.SJBrlsh cusomayunis of mwsrconsull Shinr dProcflkforUsmof lnt tonl W eSys•nof Un#ls (SO., ASTMSMandard E380-89a, published by the American Socit for Testing and Mater-lats, 1916 Race S., Philadelphla, Pa. 19103.

CRREL Report 94-5

US Army Corpsof EngineersCold Regions Research &Engineering Laboratory

Model for Avalanches inThree Spatial DimensionsComparison of Theory to ExperimentsRenee M. Lang and Brian R. Leo April 1994

94-28485

Approved for public release; distriutton is unllmited. '94 9 01 00 %,

PREFACE

This report was prepared by Dr. Renee M. Lang, Research Physical Scientist, of theSnow and Ice Branch, Research Division, U.S. Army Cold Regions Research and Engi-neering Laboratory, and by Brian R. Leo, Senior Staff "imber, Sigma Technologies,Inc., Gig Harbor, Washington.

This work was completed at the Technische HochsCA,.t: D - .1stadt, Federal Republicof Germany. Dr. Stuart B. Savage provided guidance in tQe, vnetion of the work. Theauthors thank Heniz Wall and all the employees in the laboratory mnd machine shop atthe Technische Hochschule Darmstadt, whose cooperation was exeiplary.

ii

CONTENTS

P reface ........................................................................................................................ iiN om enclature ............................................................................................................. ivIntrod uction ................................................................................................................ 1M athem atical form ulation ......................................................................................... 3

Limiting cases of the governing equations ........................................................... 7Nondimensionalization scheme ........................................................................... 8

Numerical solutions and comparison to experiments ............................................ 14Unconstrained Coulomb flow model with constant bed friction ....................... 14Unconstrained Coulomb flow with boundary drag term .................................... 17

Conclusions and remarks ........................................................................................ 18Literature cited ........................................................................................................ 21A bstract ...................................................................................................................... 23

ILLUSTRATIONS

Figure

1. Approximation of the pile shape in the V 'plane ............................................ 112. Lateral pile velocity ......................................................................................... 123. Theoretical predictions and experimental values of nondimensional posi-

tion and velocity of the leading and trailing edge as a function of non-dimensional time for experiment DA029 ................................................... 15

4. Sequence of nondimensional computed and experimental pile width andcomputed pile height at frame time increments for DA029 ..................... 16

5. Theoretical predictions and experimental values of nondimensional posi-tion and velocity of the leading and trailing edge as a function of non-dimensional time for experiment DA025 .................................................. 18

6. Sequence of nondimensional computed and experimental pile width andcomputed pile height at frame time increments for DA025 ..................... 19

UflrlS GR~A&I

WlIc TAB oUnannounced 0Julst if icatiton

By•Distribution/

Availability ii-esSAvail and/or

Dilst. Speci1al

NOMENCLATURE

zi Cartesian coordinatesxi curvilinear coordinatesgi natural basis for curvilinear systemgi j metric of the space_e<i> orthonormal basis for curvilinear systemv underline denotes first order tensor (vector)t double underline denotes second order tensor

angle defining longitudinal variations from the horizontalV<i> physical component of vp mass density

V del operator, operating on the argument indicated by arrowt<i j> physical component of t, for the stress tensor with the subscripts implying on the i surface

and in the j directionf function defining free surfaceb function defining bed surfacen unit normal vectorv velocity vectorb body force vectora acceleration vectort Cauchy stress tensor6 bed friction angle4 internal friction angleA boundary drag coefficient£i ith characteristic length scale9 acceleration due to gravityT characteristic time scale± overbar denotes nondimensional quantityx double overbar denotes nondimensional, depth-averaged quantityi tilde denotes nondimensional, depth- and width-averaged quantityEij ratio of 1' (ith characteristic length scale) to 1V (jth characteristic

length scale)w Voellmy coefficient

iv

Model for Avalanches in Three Spatial DimensionsComparison of Theory to Experiments

RENEE M. LANG AND BRIAN R. LEO

INTRODUCTION

Increases in land use and development in mountainous areas, anticipated warming of theearth's atmosphere, and the deforestation in alpine regions from air pollution, acid rain,previously poor forestry practices, and land misuse have renewed attention to the impor-tance of reliably recognizing natural avalanche paths and predicting deposition zones. Theimpact pressures induced by an avalanche in the runout area motivate zoning restrictionsand building code requirements. Existing avalanche models are generally inadequate. There-fore, the current trend in the research of flowing snow and other materials is still directed atdetermining the predominant mechanisms governing the motion.

The phenomena collectively referred to as avalanches can be physically characterized as

multiphase gravity flows, which consist of randomly dispersed, interacting phases, whoseproperties change with respect to both time and space. Lang and Dent (1982) describe anavalanche as "the transient, three-dimensional motion of a variable mass system made upof nonrigid, nonrotund, nonuniform assemblage of granular (snow) fragments flowing downa nonuniform slope of varying surface resistance." In this sense, an exact analysis of themotion of an avalanche is perhaps an unattainable goal.

Currently, the most widely used and applied avalanche models utilize a center of massapproach (i.e., the mass is accretized at one point) and are based on ideas first suggestedby Voellmy (1955). By using the Prandtl mixing length approach (see for example Pao1961), Voellmy related the stress at the base of the flow to the square of the velocity, andthen postulated an additional Coulomb friction force. However, he assumed uniform, steadyconditions and derived a method that is difficult to apply, in the sense that a number ofsubjective parameters must be predetermined in order to obtain results that match observa-tional data. Although Voellmy's model has been successively improved (Salm 1966, Mellor1968, Perla et al. 1980), these models have not advanced beyond the aforementioned centerof mass approach. The height of the flow may be included as a parametric value, but is notcalculated as a function of space and time.

Other models attempt to idealize these complicated materials as linear Newtonian fluids(Brugnot 1979, Dent and Lang 1980), leading to the Navier-Stokes equations that may besolved numerically. Although this type of constitutive relation perhaps should not be assumedto adequately describes these media, some success has been achieved in modeling certainaspects of avalanche behavior with this approach. Apparent viscosities may be measured forfluidized solids (e.g. Siemes and Hellmer, 1962). The easc. with that particles fluidize and therange of conditions that sustain fluidization vary greatly among solid-fluid gravity currentsystems.

When considering snow avalanches in particular, due to the nonlinear behavior, andstrong stress and strain history dependence of snow, constitutive equations tend to be quitecomplicated. Both continuum theories (Salm 1971, 1975, 1977, Brown et al. 1973, Brownand Lang 1973, Brown 1976, 1977, 1979, 1980a, 1980b) and microstructural theories (St.Lawrence 1977, St. Lawrence and Lang 1981, Hansen and Brown 1986, Hansen 1985) have

been proposed and applied successfully to various, specific problems in snow mechanics. Butto some extent all require detailed information on the particular snow type in question, thatentails a wide range of possible parametric values.

Another hindering factor has been the availability of experimental data on avalanchemotion. Field measurements are limited and difficult to obtain (see, for example, Norem etal. 1985, Gubler 1987), and similitude is difficult to achieve when attempting to scale-downand model such flows in the laboratory. For example, Lang and Dent (1980) utilized a 1/100scale model of snow, and successfully matched the Froude number, but not the Reynoldsnumber. This is not an unusual problem in scale modelling when the same "fluid" is used.For a scale model with length 1/n of the prototype, equality of the Reynolds number requiresa model characteristic speed of v,,, = nvp, where vp is the prototype characteristic speed.Equality of the Froude number requires vm = vp / Vfn/. Both requirements cannot be satisfiedby using the same "fluid" in both model and prototype. Other recent attempts to scale modelavalanche flow include Nakamura et al. (1987), Pliiss (1987), Hutter et al. (1988), and Langet al. (1989).

Savage and Hutter (1989) demonstrated that one may adequately describe a flowingmass of cohesionless, granular medium as an incompressible continuum that obeys a Mohr-Coulomb yield criterion. If the moving mass is assumed to be long and thin, a nondi-mensionalization scheme (resembling shallow-water theory) may be employed, which rendersmore tractable governing equations. By neglecting the boundary layer, a classical approx-imation method (first introduced by von Kirm~n (1921) and later refined by Pohlhausen(1921)) is employed, in that the equations are averaged over depth and a velocity profile isassumed. By defining a depth-averaged streamwise velocity and assuming incompressibility,these equations may be reliably solved with a Lagrangian finite difference scheme, such thatthe computational grid is advected with the material.

The purpose of this investigation was to extend this theory to three spatial dimensions andallow for the inclusion of a boundary drag term, and test its applicability against experimentalresults. The governing equations are derived in a curvilinear coordinate system that allowsfor topography to vary as in a natural avalanche path. The only assumptions included in thedevelopment are incompressibility and the application of a Mohr-Coulomb type material law,while also ailowing the inclusion of a boundary drag term. A nondimensionalization schemeis employed, choosing three characteristic length scales. By depth and width averaging thebalance equations for the theory in three spatial dimensions, sidewise spreading of the massmay be included. A Lagrang~an finite difference scheme is applied to numerically integrate

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the sets of equations that are derived. The numerical simulations are then compared to thelaboratory findings. The experimental results have been presented in Lang et al. (1989).

The analytical results indicate that a flow transition regime exists that is analogous to alaminar to turbulent transition in a fluid, and determines the applicability of the boundarydrag term. Results are reviewed and suggestions are made for appropriate investigationsimplied by these results.

MATHEMATICAL FORMULATION

The governing equations and closure relations are proposed to represent the flow of acohesionless granular medium on any given topography. By describing granular media asmultiphase materials, intraphase balance equations for mass, momentum, and energy and theassociated boundary and initial conditions may. in principle, be written. But the resulting setof equations is usually too complicated to allow for a solution. For example, the application ofkinetic theories for granular flow would in general require the solution of the energy equation,determination of granular temperatures, and velocity and density gradients. Szidarovsky etal. (1987) discuss the inherent difficulties in attempting to apply these theories to chute flowsof granulate. Furthermore, such information is inappropriate for an engineering application.

The goal is therefore to develop a simple model to predict the flow characteristics of themass of granulate. A purely mechanical theory is proposed and the granular mass is treatedas a continuum. The concept of representing the granulate as a continuum should poseno conceptual problem, as a mixture theory approach would require the sum of the phaseinteraction terms to vanish in the overall balance equations, in any case.

The requisite is a suitable Constitutive assumption. In order to justify the followingapproach, it is necessary to consider the different flow regimes in question, and also theexperimental results from other investigations.

First, two extreme flow regimes for granular media are commonly recognized (see, forexample, Savage 1984). At very low shear rates, a granulate may be characterized as aMohr-Coulomb material. This implies quasistatic deformation, to some degree. The Mohr-Coulomb law is the basis of Critical State Soil Mechanics (CSSM), and is commonly referredto as the rupture condition; i.e., it implies impending plastic flow for a static soil mass, asit predicts the limit equilibrium when the shear stress is equal to the shear strength of thenoncohesive soil. Simply stated, the maximum ratio of the shear to normal stress is equatedto the tangent of the internal friction angle po of the granulate. For slowly flowing masses ofgranulate, the bed shear stress results from the dry friction forces at the particle contacts,and the Coulomb failure criterion adequately describes these types of flows.

When shear rates are very high and particles are widely dispersed, instantaneous collisionsdominate the behavior of the granulate. Interparticle and particle-bed contacts are brief,and the fluctuating part of the particle motion cannot be neglected. Kinetic theories predictquadratic dependence of stress on shear-rate, for rapidly shearing highly dispersed particulate,and this seems to provide good agreement with the available laboratory data. This extremecase may be used to describe the behavior of the dust cloud, which only serves to obscurethe true nature of the avalanche, but does not represent any type of snow avalanche activitythat could be construed as destructive, and need not be considered.

The real avalanche flow situation in most cases may develop from a low shear rate flow intoan intermediate regime where both long-term particle contact and instantaneous interparticlecollisions contribute to the generation of stresses. This was shown to be the case in the

3

previously discussed experiments in Lang et al. (1989), for the artificial avalanches released atangles exceeding 350. Observations (Dent and Lang 1981) and Doppler radar measurements(Guler 1987) on natural snow avalanches confirm that a dense slowly deforming mass rideson a rapidly shearing layer, and may or may not be encompassed by a highly suspended dustcloud. This type of flow is more difficult to handle theoretically, as the exact mechanisms havenot yet been clearly defined. Savage (1983) suggested representing the total stresses as a linearsum of the rate-independent dry friction and a rate-dependent "viscous" part. The questionremains as to how the dynamic friction angle actually changes at lower volume fractions andhigher shear rates. Following the work of Savage and Hutter (1989), the governing equationsof mass and momentum balance are derived in three spatial dimensions in a coordinate systemthat allows for any given topography. An angle ( is defined such that ( is a function of x1 ,the longitudinal direction, and defines the local inclination angle of the generatrix,

( = ( (xl) (1)

It is then possible to define Cartesian components z' in terms of the curvilinear coordinates xi

and the natural basis { g, }. An orthonormal basis { Ej } in terms of the covariant componentsof the metric or fundamental tensor of the space gm , = g " g is defined and applied.

The Eulerian form for mass balance is

d p( x,t +pt) ) = ~ ~)0(2)

d t . . . .In terms of the physical components of v and assuming that the granulate-air mixture isincompressible, mass balance is written as

OV<I> + V<2> V<3> (3)Ox' + Ox 2 + X3

+ X3 ( 2 + 8-X3 ) + a-xlV<3> = 0

The principle of balance of linear momentum can be written as

S.t~pb~p(4)

where t is the Cauchy stress tensor, b is the body force vector and a is the accelerationvector. The equations of motion can be expressed in terms of the physical components asfollows:in the x1 length direction as

49t<1¢ 1> +( 2-(-< 1>+t<2 1> + &<3 1> + p~sin (Ot~ 1 +2•i~lt3 > 1 + X3 ax

Ox1 <3iL -x2 + Ox3

P [V< I___l> + V<L>V<3> a(1( V I>OX1 O~>Ov, v~

+f.1+ _L- t) Ot +V< Ox2 +V<3> Ox3 (5)

in the x 2 width direction as

4

&t<I 2> +o (I + ± 3 ) at<22> &<3 2>1ax1 + iax <3>+Va~x ) a2 + aX3j

f aV<2>1

+V1+ 3 ) V<2> + V<2> + V<3>, (6)

and in the x 3 depth direction as

&<I 3> (t<3 3> -- t<1i1>) + 1l+ ±-X3 L-§23 + t<33> pcosSOx x xx1 Ox2 +- 1

P v<1> OV<3> v2 a(

axi [a<3 +V<>9<3._> + V r<3> (7

The above forms for the balance of momentum equations are shown in order to emphasizethe coupling of many terms with the degree ci curvature OC/Ox1 and the depth of the flowX3 .

The boundary conditions need to be defined at the solid surface, or base of the flow, andat the free surface.

By defining the free surface as

f = f(x 1 , x 2 , t) = x 3 , (8)and requiring the local and convected rates of change of f to balance, the well-known kine-matic condition is obtained,

Of -a f•+ . v=O , (9)

which simply states that the free surface is material, and applies when f is generated bystreamlines.

In terms of the physical components, the kinematic condition reads

Of 1( ý X3 \)r[f O faxiv<l> + l axi L• at aX2 V<2> -- V<3>1 --- 0 (10)

A balance of forces at the free surface requires the total traction to vanish there, i.e.

nt_= 0 ,at x3 =f(x',x 2 ,t) 0(11)

The above condition also applies to a continuous material. Reportedly (see Johnson andJackson 1987) when this boundary condition is applied to problems concerning granularmedia, it tends to introduce a singular behavior of the traction in the neighborhood ofthe interface that is physically unrealistic, as the stress tensor vanishes rapidly through athickness that is typically less than the thickness of one particle diameter. However, obtaining"depth-averaged" solutions is intended ,and this difficulty should not be a problem, exceptat the flow boundaries.

5

Furthermore, it is important to discuss tile meaning of a "free surface" in a physicalsense. As discussed, it is generally accepted that an av:'lanche may be considered to be aflow consisting of thiee distinct regimes. The dense core of the flow is normally obscuredby a "dust cloud," which is a layer of highly ,uspended particulate. From observation, this

layer is dominated by the interstitial fluid behavior (Lang et al. 1989). Intuitively, if onedefines the "free surface" as the interface of the core of the flow Nxith the turbulent dustcloud, it would be more appropriate to consider the drag at this rough interface rat'ier thana vanishing value of traction. However, it can be demonstrated (McClung and Schaerer 1983)that inclusion of such a term has a negligible effect on the overall resistance to motion of themass.

With these considerations in mind, the stress-free condition is assumed, and the resultingboundary conditions at the free surface are

fxt<lI>+(l+ •( ) [1 t<2 -I > , (12)

f t<2>+ 1 +E f) [ -- 2 t<2 >t<3 2> 0

X, t< 1 3> + (i+t< 2 3> 0=-~<13+ (i~~ [~< 2 3 >- t<3 3>j

written in the x1 , x2 and x3 directions respectively.

At the base of the flow, the surface may be described by a function b = b(xI , x 2 ), suchthat the function b may be arbitrarily defineo lor a given avalanche bed.

The tangency of flow at the base is expressed by the kinematic condition

v" n=O ,at x3 = b(x 1 , x2 ) (13)

Expanding the above in three spatial dimensions yields

(xlV<l>1+ [+ ax 1) xV<2> - V<3> =0 ,at x3 = b(x 1,x 2 ) (14)

Applying the assumption that the basal boundary layer is of a thickness proportionalto only a few particle diameters, this layer is neglected. This assumption greatly simplifiesthe mathematical formulation, as in this layer collisional and frictional stresses must beconsidered. An appropriate boundary condition has been derived by Hui et al. (1984) forsuch a case, based on kinetic theory. However, some knowledge of the specularity of particle-wall collisions is required.

By considering only cohesionless, dry granulate, the basal shear traction may be expressedas

nt- n(nt- n)-nt=n- R -tan6 -DvI v at x 3 =b(x, x2 ) (15)

where the first term on the right hand side is the Coulomb sliding law, and tan 6 is the bedfriction angle whose value(s) depend on the nature of the particles, degree of packing, andthe contact surface. The remaining term on the right-hand side is the bounda:y drag term.It should be noted that the above is compatible with the kinematic condition at the base (eq

6

13). When the basal surface agrees with the coordinate generating surface, n = F<3> andthe kiriematic ctndition at the base eq 13 reduces to

V<3> = 0 (16)

Equation 15 at the base reduces to

S= - ta, 6 + D° I (17)

rtan6 1

t<32>= -v< 2 > [ 'tan t<3 3 >+DI 1il.

In view of the experimental work (Lang et al. 1989), which encompassed only the caseof the basal surface replicating the coordinate surface, the following discussion will focus onthis limiting case. Also, the scaling arguments will demonstrate the inconsequence of weakcurvature. For strong curvature in the lateral direction, the theory may be easily reduced tochannel flow equations.

Limiting cases of the governing equations

The governing equations that were presented are now nondimensionalized and averagedover the depth and width, in order to provide a set of tractable equations. The nondi-mensionalization scheme allows for any number of characteristic length scales, that differsfrom more conventional approaches. Becker (1976) presents a thorough treatment of thismethod, commonly termed a "distorted model." However he prefers the more distinctivenomenclature of "Pythagorean method," as the Pythagoreans were motivated by the idea ofgeometrical proportion being the basis of all things. By applying this method, inspectionalanalysis indicates limiting forms of the equations that may be more easily treated. The ap-plication of this method has been accepted as the appropriate procedure when attempting tomodel large-scale, geophysical phenomena. By adopting the concept of a "distorted model,"consideration of more than one characteristic length scale is allowable. Such an approach iscommonly used and applied to large-scale, free-surface flow phenomena such as atmosphericand oceanic motions. A specific example is the well-known "shallow water model." Scalingprovides the means to neglect small variations in topography that do not significantly affectthe lateral or longitudinal spread of the flowing material. Curvature in the lateral directionwill be neglected, as discussed in a later section. For strong curvature in the lateral direction,the equations are easily reduced to a channel flow model.

Also, assumptions are made to simplify the balance equations and boundary conditions.Many of these result directly from the scaling arguments, and others are based on experiencewith the physical problem at hand The nondimensionalized equations are then averagedover the depth and width of the flow. This is a classical approximation method, commonlyapplied in evaluating boundary layer flows. The result is a set of equations that represent abalance between the average inertial and resistant forces for each position in the longitudinaldirection. Ordering of arguments allows further simplification. The need to consider thedetailed character of the flow is thus avoided, and sufficient information for an engineeringapplication is obtained.

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Nondimensionalization scheme

At this point, a scheme for nondimensionalization of the governing equations is presented.This step of modeling geophysical phenomena is always to some extent a "leap of faith."That is to say, choosing an appropriate model depends on experience, intuition and explicitdynamic reasoning. The essential dynamic character of the geophysical system must beretained but the validity of the model is examined after the fact.

Although the depth, length and width of an avalanche varies spatially and temporally, itis assumed that characteristic values C' may be sensibly chosen such that

xi = £it, (18)

where the t' are the nondimensional lengths. The concept of an avalanche as a gravitycurrent suggests the following representation,

> -: [gec1 I 1/2 <2 > =

= < = D<i> (19)

The above may be interpreted with respect to the fluid dynamic concept of "head" or thefree fall of a mass point in a gravity field through a given distance, although the length scalemay differ. This leads to an expression for time as

t = T f = [ T I (20)

As the Froude number is the governing parameter for the problem under consideration, thestresses are scaled according to a hydrostatic relationship,

t<ii> = PogC3 Coso t<i i> (21)

t<,j> = pog£sin(ot<ij>, i:j (22)

where the null subscript denotes a characteristic value for the quantity under consideration,while

sin c=cos(,, tan6 (23)

The expression for sin c, is a logical choice, as both the driving force and the resistance arerepresented in eq 23, which both influence the order of magnitude of the resultant shearstresses. The angle ( may also be characterized as

(24)

Also, the following definitions are applied:

ij= (25)

X 1C_W = (26)

In order to evaluate the basal boundary conditions the drag coefficient VD0 is defined in

8

terms of the Voellmy coefficient w , which is traditionally applied to flowing snow, as follows;

9(27)

tu has the units of m/s 2 such that it is a natural choice to scale it with the gravitationalconstant such that w is related to the dimensionless form f by

ta = Gtu (28)

Depth and width averaging

When a functional form of solution to a set of governing equations cannot be found (i.e.a solution that satisfies the equations identically at each point in space and time and thatapproaches the appropriate boundary values) it may be advantageous to try and satisfy theequations on the average, if the geometry of the problem allows for this approach to result ina meaningful solution. If the equations may be integrated with respect to t3 across the depthof flow and with respect to t2 across the width of flow, the resulting equations will representa balance between the average inertial forces and resistant forces for each X1 location. Inthis manner, a velocity distribution may be obtained that satisfies this averaged balance offorces but that does not satisfy the balance at each point across the height and width of theflowing mass.

This method is a classical approximation method that is widely used in examining bound-ary layer behavior, and was first introduced by von Kirmtn (1921) and was further improvedby Pohlhausen (1921). The results that are obtained from such an approximation are foundto be reasonably accurate in many instances. By acknowledging that fj<1> and f)<2> are con-tinuous on the interval [0, 1], the first mean value theorem of calculus ascertains the existenceof a point p contained in the open interval (0, 1) such that

= '<i>>(p) Io

= hV<i>, (29)

and furthermore, it may be assumed that

fJI j<,> j3 a3 _0 t<i>(P) f2 3 da3

h2== -'v<i>, (30)

where the notation V<j> implies the "depth-averaged" value of V<i> and 1-0 = h, thenondimensional flow height. The latter equation is exact if the velocity profile over the depthof the flow is uniform.

9

Let

E3 1E3 2 A2 3 = 0(E2 1 ) (31)

and retain only terms of O(E3 i) . The depth-averaged velocity squared is

<> = V 2<>d3 (32)

where the Boussinesq momentum coefficient will be taken to be near unity such that

<<> 2(33)

To be realistic, the Boussinesq coefficient is greater than one, but this approximation issufficient when the velocity profile is blunt. For a power law velocity distribution (see,for example, Burgers 1924) that typifies a pure Newtonian turbulent fluid, the Boussinesqmomentum coefficient is 1.05 (Savage 1989b). The normal stress at the bed is then

1< OS> "• Cos C +<I>1 = (34)I cos O

23_--0

The use of an "earth pressure coefficient" is employed, such that the normal stressesparallel and perpendicular to the path may be equated by employing an earth pressurecoefficient kactpass

f<1 1> = kactpasat<3 3> (35)

katpaas has a value of kat or kp,,s, determined by the spatial evolution of the pile as eitherelongation (active case) or contraction (passive case) with respect to the flow direction x1 .From a standard Mohr's circle interpretation, it may be shown that

kpa:t }=2 T1 : ý _ (I + tan2 6) cos2 ]/cos2 _1 (36)

for ab<i>/&, x=- > 0<0.

It is also assumed that

V<=>V<2> - • v<>v<2>dx3 (37)

and

V<1>V<2> <1> V<2> (38)

Also

dP3 V2<1>d23 ce 02 1> f' a3 d23 V<1>

(39)

10

The integrand i<2 1> is neglected by ordering A2 I E3 2 < 0 (E1 2), as this shear stresswould only be locally effective in the case of confined narrow channel flows. It has been shown(Savage 1989 and Lang 1988) that sidewall shearing may be incorporated, if necessary, as aneffective friction.

Also

V<l>V<2> = h f<1>T<2>d3 (40)

and assume that

T<I>fv<2> - V<1>V<2>. (41)By neglecting the t<1 2> integrand, the lateral pressure gradient may be integrated as

a± [ kactpassh2 (c( =2 (42)a•--- 2 t <2 2 >d2r3 • X-=2 jcos (Co cs(•+ Ka•< 1> )]2

based on the value of the t< 3 3> stress component. An approximation for the t<3 2> stressmay be made from eq 42 from relating the shear to normal stress such that

i< 3 2>d±3 {2 <2-->OcosCtan [cos(+K: ) <,>) tan6 + E32 (43)

The nondimensionalized depth-averaged equations now may be further simplified bywidth averaging. By defining the (nondimensional) half-width of the pile as

bm = bm(•', i) (44)

and assuming that the pile of granulate retains a parabolic shape in the 21 plane, the heightmay be expressed as

t ) -- ho(± t) 1- (b 2( ) 2 ) (45)

where ho( 21, t) is the centerline depth (refer to Figure 1). This assumption is fully supportedby the experimental results discussed in Lang et al., 1989. Although natural avalanches donot retain such perfect symmetry due to uneven topography and particle size variations,some degree of symmetry is usually observed in natural avalanche deposition and a parabolicdistribution is reasonably replicated.

ho bh= ho _..)2)

. .............

x 2

-bm(±1, g) 0 bm(±', g)

Figure 1: Approximation of pile shape in the ±1 plane

11

V<2> •Z

_bn ii bm

Figure 2: Lateral pile velocity

Assuming parabolic symmetry of the pihe implies that i,<2>, the nondimensional depthaveraged lateral pile velocity, varies linearly with respect to t2 as in Figure 2 such that

dbm t2V<2> =-- (46)dt b,,

Integrating the continuity eq 3 over the depth and half-width, applying Leibniz' rule, andinvoking the symmetry conditions in (45 and 46) gives

a -- M- h< j1> a2 + hd 2 + ,C E31 a( a b- h2- d t2 = 0 (47)

Define a "width"-averaged depth h

h= m h id 2 (48)

By assuming that the depth has a parabolic distribution in the X1 = constant planes

-2-

h 2- ho (49)3

Furthermore, a depth- and width-averaged velocity may be defined again by use of the meanvalue theorem of calculus such that

1 [b • h•

<>= g <1>hdt2 (50)

With these, continuity becomes

a (hobm•i<l>) + (hobm) 2 X E3(h1bm) =0 (51)+K 1 R 3 -0,i (5t

Width-averaging of the momentum balance equations requires an approximation for the

norm of the velocity vector. By acknowledging that sidewise spreading contributes in a

neglible and fairly constant manner to the value of the depth-averaged velocity, one approx-

imation to De considered is

Z <1> (52)

Furthermore, let

12

A = E3 2 kactpass ,

C I KE 3 17 I

Fi 1E3 1 1V

F2 I

E3 2 f '

hj <1>b <2> d> h<2>d (53)

where the last approximation is physically realistic, as the longitudinal velocity has beenassumed to have a blunt profile over depth and does not appear to vary over the width atany one transverse section. Applying the aforementioned definitions and approximations tothe momentum equation in the i<2> direction eq 6, and by application of Leibniz' rule,termwise integration from t3 = 0 to t3 = h and 22 - 0 to t2 = bm gives

0~ + -~) (cos( + Bi <1 -bmh (V<2> Ch(o) tan6 (cos( +h f1B1ib f2 1 >)

bm

F2 b..ChV<1>(=<2> 2 2(54)

-2 2 (- )bm

- ~!bmho [i<1>j-=(<2> +) + (<2> )+ Cho- )] -=<2(bh')}b.• b,• bm

The above equation is easily obtained by simple integration by noting that

=0

V<2> = 0

0

Similarly, letG =E3 I kactPass (55)

By the same method the width-averaged momentum balance in the _<l> direction is

13

3G [2bh 2 COS 8bM Bf021 l (1- )t.Co D2 0l4 08b; [ hho C5OSl 3 + 15 ] ) [cosJtanb+BI>tanb]

- Flf)21> (3h 1 ) + 2h.C sn

af,<1> ab<i> 2Cho 0 &)<1>- f + - o±1 + 5 at

< (56)

Equations 51, 56, and 54 comprise a complete boundary value problem for the free surfaceflow of a cohesionless granulate down an open terrain of variable curvature in the longitudinaldirection. The requirement of pile symmetry has been imposed, which is satisfied by theexperiments in Lang et al. (1989) that meet the criteria of being friction-governed flows.

NUMERICAL SOLUTIONS AND COMPARISON TO EXPERIMENTS

Because a closed form solution is not available for the system of equations which wasoutlined, a numerical solution must be sought. Savage and Hutter (1989) show that a La-grangian formulation is the natural choice for this system of equations in order to determinethe position of the granulate surface through time. With this method, the boundaries ofthe finite difference grid are convected with the depth- and width-averaged velocities of thegranular material.

Two different models are developed. First, the boundary drag term is neglected andpredictions are based solely on modelling the material with the Coulomb friction law. Then,it is demonstrated that a flow transition regime exists where the boundary drag term mustbe included in the model in order to predict the position of the pile through time, and morespecifically, the final runout zone position. An artificial viscosity term /u 2 u/Ox2 is added tothe equation of motion in order to smooth the solution.

Unconstrained Coulomb flow model with constant bed friction

Now the depth- and width-averaged balance equations are solved. In addition to pre-dicting the length, height and longitudinal velocity through time, the sidewise spreading andrate of sidewise spreading may be determined. The results from channel flow models (Lang1988) are used as a guideline. A first approximation is applied, where it is assumed thatthe boundary drag term may be neglected, and that b remains constant through the lengthof the pile. This provides the simplest set of width-averaged equations. For this system ofequations, the condition is applied that C < < 1. The depth- and width-averaged equationsas presented are then simplified. For comparison with simple laboratory findings a constantbed friction angle provides good results. This model was tested against the experimentalresults presented in Lang et al 1989. The numerical model and laboratory results show rea-sonable agreement. For example, experiment DA029 was released at an initiation zone angleof 350. DA029 consisted of 20.0 kg of 2- to 3- mm quartz particles, reaching a maximumvelocity of 3.6 m/s at 0.9 s. Figure 3 shows the experimental values of velocity and positionfor the leading and trailing edges through time as compared with the theoretical predictions.

14

DA029 DA029

10.0 4.0-

8.0'a 6.0 " ."" "2 0

• • > 2.0-,I 4.0 ••"J.--"'+ • • 1.o0.

2.0 0

0. 00

6.0

1.0 -1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 2.0 4.0 6.0 8.0 10.C 12.0

dimensionless time dimensionless time

- theory rear 0 experimental rear - theory rear 0 experimental rear- theory toe A experimental toe - theory toe & experimental toe

Figure 3: Theoretical predictions and experimental values of nondimensional position and

velocity of the leading and trailing edge as a function of nondimensional time for experiment

DA029

The leading edge is the point on the t2 = 0 axis that travels the farthest, and the trailing

edge is the point on the t2 = 0 axis that travels the least amount of distance through time.

Due to the method of photography (see Lang et al 1989), a nondimensional time error of

0.52 may be incurred in determining the initial frame, which may result in some of the offset.

Nevertheless, the theory predicts the general characteristics of the longitudinal progress of

the granular mass, and the final position of the forward edge in the runout zone is accurately

predicted. The prediction of the position and velocity of the leading edge of an avalanche

is the critical information required to calculate the impact pressure on a structure and to

motivate zoning restrictions.

In Figure 4 the progression of the width and height at selected time increments is pre-

sented. The time increments were chosen to correspond with frame times. The left hand side

of the figure represents the progression of the half width of the model and experiment (see

Lang et al 1989) at selected time increments, and the right hand side is the predicted depth

profile through time. Due to limited space in the laboratory, it was not possible to record

the height of the pile through time, but the general features are reasonably reproduced by

the theory; that is, the pile collapses rapidly, elongates and flattens to a maximum prior to

encountering the runout zone. Then as the forward portion of the pile stops, the trailing

edge retains momentum and piles up over the static portion. When the angle of repose is

exceeded, the pile relaxes back to the internal angle of friction, and the tail comes to rest

at some angle less than the angle of repose (Fig. 4). The maximum final (nondimensional)

pile height was measured to be 0.21, and the model predicts 0.23. The model overestimates

the (dimensional) lateral spread of the pile, by approximately 9%, and the overestimation of

the lateral spread causes the underestimation of the pile length through time, particularly in

the runout zone. The pile is predicted to spread more rapidly in the lateral direction in the

initiation zone.For experiments that were conducted at an initial bed angle (or starting zone angle) of

35', the constant bed friction angle model is a good predictor. For laboratory simulations

15

DA029

2.0 -

t = 0.01.0-

t = 2.583.0-

&A2.0- A

DA029t =5.464.0 1.0-

t =0.0

. 3.0- 0.8

* 2.0 0.6

. 1.0 0ES

0.2

t 8.254.0-

A t =2.58

3.0 . 0.2-.02.0-• ••2.0,- t =5.46

01.0 - " 0.

E

t =9.79 t =8.254.0 0.2

3.0 -- -t =9.79

2.0 0.4

1.0i 0.2-

0.0 -0.0-

0.0 2.0 4.0 6.0 8 0 10.0 0.0 2.0 4.0 6.0 8.0 10.0

dimensionless length dimensionless length

- theory A experimental - theory

Figure 4: Sequence of nondimensional computed and experimental pile width and computedpile height at frame time increments for DA029

16

conducted with greater starting zone angles, the model greatly overpredicts the final positionof the granulate mass in the runout zone. For example, experiments DA025 and DA033 werealso modeled with the constant bed friction model. Both of these experiments were releasedat an initiation angle of 45°. For DA025, 20.0 kg of 2- to 3- mm quartz particles were used.The resulting predictions of position and velocity through time show that the final positionsof the leading edge and the tail are greatly overpredicted. Position and velocity profiles forDA033 18.4 kg of 2-3 mm marmor show also that the toe and tail positions are overpredictedwith this model (for details, see Lang 1992).

This result agrees with problems encountered when attempting to model natural snowavalanches. Gubler (1987) discusses the problem of modeling higher speed avalanches withtraditional avalanche models. The friction coefficients must be increased by a factor of 2, ingeneral, in order to force the avalanche to come to rest in the appropriate position in therunout zone.

Because this numerical model accurately predicts the experiments conducted at 350 whichresults in lower velocities and accelerations than for larger angles, it is possible that a tran-sition in flow regime occurs in the boundary layer where the boundary drag force becomesnon-negligible at higher velocities. Its inclusion in the numerical model is presented in thefollowing section. Gubler (1987) discusses a similar flow regime transition for natural snowavalanches from flexible sliding body to partial fluidization. Furthermore, Buser and Frutiger(1980) found that the inclusion of both sliding bed friction and boundary drag underpredictvelocities for very large natural snow avalanches. This result may also be an indication offlow transition behavior, in the sense that an avalanche may initiate in the flexible slidingbody type flow and reach a transitive velocity where partial fluidization occurs. If the ad-ditional resistance due to boundary drag is applied throughout the flow time, the theorywould underpredict velocities. However, if a critical velocity could be determined that wouldpredict the transition, and hence the appropriate application of an additional boundary dragterm, perhaps this problem could be circumvented.

Unconstrained Coulomb flow with boundary drag term

It was mentioned in the previous discussion that the constant bed friction angle modelbecomes a poor predictor for initiation zone angles greater that 35'. At this point it seemssensible to attempt to model the additional resistance to flow by inclusion of the boundarydrag term. Again, average values are used to avoid the possibility of the pile height approach-ing zero. Also, the depth- and width-averaged drag term includes the width of the pile in thedenominator, that is zero at the midpile. Therefore, the following average of the drag termis included:

3 2 ~3 Qua (57)

2 E31 h,, ba57

As an example, experiment DA025 was modeled with the inclusion of the drag term, andresults are presented in Figures 5 and 6. DA025 was conducted at an initiation angle of 45°and 20.0 kg of 2-3 mm quartz particles were used. Figure 5 depicts the resulting predictionsof position and velocity through time. The final position of the leading edge is somewhatunderpredicted, but the general motion of the pile through time is well mirrored. The modeldoes not allow the maximum leading edge velocity to be attained. It may be restated thatideally, the boundary drag term should perhaps be applied only when a critical velocity (orcritical Froude number) is attained.

17

DA025 DA025

10.0 4.0-1o*o 4

8.0 t> 3.0-Z

6.0 2.0-J 4.0 A.2 1.0-

2.0 .0.00

0.0-1.0 i i1.0 i , i

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0

dimensionless time dimensionless time

- theory rear 0 experimental rear - theory rear 0 experimental rear

- theory toe &I experimental toe - theory toe &I experimental toe

Figure 5: Theoretical predictions and experimental values of nondimensional position andvelocity of the leading and trailing edge as a function of nondimensional time for experiment

DA025

As shown in Figure 5, the initial spread of the pile is overpredicted, but in the runout zone,

the shape is quite accurately replicated. The pile is predicted to be shorter in the longitudinal

direction and slightly wider than the actual pile shape. The final maximum (nondimensional)

pile height is extremely accurate: the theory predicted 0.205 and the experimental result was

0.206.

CONCLUSIONS AND REMARKS

Governing equations and closure relations were proposed to represent the flow of a co-hesionless granulat medium on any given topography. The goal was to develop an easilyapplied model to predict the bulk flow characteristics of the mass of granulate. Indeed, theanticipated result of every avalanche model is to predict the final runout zone position ofthe mass and the possible impact pressures on structures in the path of an avalanche. Apurely mechanical theory was proposed and the granular mass was treated as a continuum.The balance equations in terms of physical components in an adaptable and appropriatecoordinate system were derived. A constitutive assumption was justified by considering thedifferent flow regimes in question, and the experimental results from Lang et al. (1989), andother investigations.

The governing equations were nondimensionalized and averaged over the depth and width,in order to provide a set of tractable equations. The nondimensionalization scheme allows forany number of characteristic length scales, which differs from more conventional approaches.By applying this method, inspectional analysis indicates limiting forms of the equationsthat may be more easily treated. The application of this method has been accepted asthe appropriate procedure when attempting to model large scale, geophysical phenomena.Assumptions were made to simplify the balance equations and boundary conditions, whichresulted directly from the scaling arguments or were based on experience with the physical

18

DA025

2.0-7-

t = 0.01.0

t = 2.063.0-

2.0-

1.0-

DA025t = 4.79

4.0- 1.0-

t =0.0S 3.0- 0.8-

2.0- 0.6-U

1.0C 1.0 " 0.4 -

- 0.2-

t = 7.634.0-

t ='2.06

3.0- j 0.2-2.0- A

t =4.79

1.0- 02

t=' t = 7.63t =9.28

4.0- 0.2

3.0-t =9.28

2.0 0.4

1.0 0.2

0.0 - I I I I I 0.0 I I I I I0.0 2.0 4.0 6.0 8.0 10 0 0.0 2.0 4.0 6.0 8.0 10.0

dimensionless length dimensionless length

- theory A experimental - theory

Figure 6: Sequence of nondimensional computed and experimental pile width and computed pile height at frametime increments for DA025

19

problem at hand. Changes in the volume fraction of the granulate and variations in topog-raphy in the lateral direction wer-, neglected. The nondimensionalized equations were thenaveraged over the depth and width of the flow, a classical approximation method, commonlyapplied in evaluating boundary layer flows resulting in a set of equations which represent abalance between the average inertial and resistant forces for each position in the longitudi-nal direction. With this method, no information can be provided to determine the detailedcharacter of the flow, but sufficient information for an engineering application is obtained.

A Lagrangian discretization scheme was applied in order to numerically evaluate themagnitude of velocity, and predict the spatial evolution of the granular pile through time.This type of analysis allows for the particulate to be advected with the finite difference grid,and was shown by Savage and Hutter (1989) to be the appropriate choice for such a problem.

Two models were developed for the free surface flows. The first assumed a constant bedfriction angle, and neglected any boundary drag effects. This model proved to be adequatein predicting the experiments discussed in Lang et al. (1989), which were conducted at bedsurface angles of 350, but failed for larger values of initiation zone angles. The method ofphotography incurred a nondimensional time error of 0.52 in determining the initial frame,which may have resulted in some of the inaccuracy. Nevertheless, the theory accuratelypredicts the general characteristics of the longitudinal progress of the granular mass, thevelocity with time, and the final position of the forward edge in the runout zone for the350 experiments (e.g., DA029). The progression of the height and width at selected timeincrements, corresponding with frame times, was also well reproduced by the theory. Thepile collapses rapidly, elongates and flattens to a maximum prior to encountering the runoutzone. Then as the forward portion of the pile steps, the trailing edge retains momentumand piles up and over the static portion. When the angle of repose is ex-ceded (Fig. 4)the tail reverts back to an angle equivalent to or less than the internal angle of friction.The final angle (actual or predicted) depends on the amount of backwashing that occurs.The maximum final (nondimensional) pile height was predicted to within 8%. The modeloverestimates the lateral spread of the pile and this causes the underestimation of the pilelength through time, particularly in the runout zone. The pile is predicted to spread morerapidly in the initiation zone. But the average flow procession of the pile is well predicted.

For laboratory simulations conducted with greater starting zone angles, the model tLatneglects any flow resistance due to drag overpredicts the final position of the granulate massin the runout zone. This result agrees with problems encountered when attempting to modelnatural snow avalanches. Attempting to model higher speed avalanches with a basic, or all-encompassing, avalanche model, and producing erroneous results is not a new or surprisingproblem. Gubler (1987) discusses the problem of incre:• ;ing friction coefficients by a factor of2, in general, in order to force avalanche models to come to rest in the appropriate positionin thc run-out zone, when attempting to model snow avalanches that attain higher velocities.

Since the first model breaks down for initiation zone angles greater that 350, an attemptwas made to model the additional resistance to flow by inclusion of the boundary drag term.Experiment DA025 was modeled with the inclusion of the boundary drag term, and resultswere preiented in Figures 5 and 6. This experiment was released at an initiation angle of450.

For DA025, the final position of the leading edge was underpredicted, but the generalmotion of the pile through time is well mirrored. The model does not allow the maximumleading edge velocity to be attained. It is not an unreasonable idea that the boundary dragterm should perhaps be applied only when a critical velocity (or critical Froude number) isattained. Also, the initial spread of the pile was overpredicted, but in the runout zone, the

20

shape is quite accurately replicated. The pile is predicted to be shorter in the longitudinaldirection and slightly wider than the actual pile shape. The final maximum (nondimen-sional) pile height was accurately predicted, with a theoretical value of 0.205 compared tothe experimental result of 0.206.

In general, the numerical models and experiments show good agreement and providevaluable information. These results are justification for the characterization of the flow, and

persuance of averaged quantities. The resulting predictions of geometry and velocity throughtime are encouraging. However, some subjectivity is still required to determine appropriateconstitutive parameters and it is unknown if the models can represent naturally occurringevents.

Future work should include testing these models against naturally occurring events, inparticular, snow avalanches, since experimental data are available (Gubler, 1987). This wouldbe no minor task, considering one would have to attempt to determine in-situ properties ofthe snowpack. Also, the model presented was developed to include curvature variations of theslide path in the lateral direction. If one were to completely test the model against a naturallyoccurring event, it would have to be dctermined how important the topographical variationsin the lateral direction are in affecting the final runout position. Certainly, constraining theflow in a channel prohibits dilatation to some degree. This would entail more experimentalwork on simple materials. It may be sufficient to characterize a given natural topographyby simply combining the channel flow and unconstrained flow models, and specifying thechannel width as a function of longitudinal direction, and the position(s) where chanitel flowis no longer appropriate. Asymmetry in the spreading direction may or may not be a decisivefactor in determining the final runout position of the flow.

Also, in order to compare the model to snow avalanches one must remain cognizant ofthe fact that the cohesive nature of snow has been neglected. This omission is probably mostimpractical for modeling the initiation of the avalanche and the final stages of compaction inthe runout zone, but it is unknown if neglecting cohesion plays a significant role in determiningthe salient features of the flow. The same statement may be made concerning particle sizedistribution. It is sufficient to say that some progress has been made in modeling granularflows, but efforts should continue to resolve these and many other issues.

LITERATURE CITED

Becker, H.A. (1976) Dimensionless parameters: Theory and methodology. Applied

Science Publishers, Ltd., London 128 pp. 24-29.Brown, R.L. (1976) A thermodynamic study of materials representable by integral

expansions. International Journal of Engineering Science, 14(11) 1033-1046.

Brown, R.L. (1976) A fracture criterion for snow. Journal of Glaciology, 19(81) 111-121.

Brown, R.L. (1979) A volumetric constitutive law for snow subjected to large strains

and strain rates. U.S. Army Cold Regions Research and Engineering Laboratory Report 79-20

Brown, R.L. (1980a) Pressure waves in snow. Journal of Glaciology, 25(91) 99-107.

Brown, R.L. (1980b) An analysis of non-steady plastic shock waves in snow. Journal

of Glaciology, 25(92) 279-287.Brown, R.L. and T.E. Lang (1973) On the mechanical properties of snow and its

relation to the snow avalanche. Proceedings, Annual Engineering Geology Soils Engineering

Symposium 1 1 th. 19-36.

Brown, R.L., T.E. Lang, W.F. St. Lawrence and C.C.Bradley (1973) A failure

criterion for snow. Journal of Geophysical Research, 78(23). 4950-4958.

21

Brugnot, G. (1979) Recent progress and new applications of the dynamics of avalanches.Proceedings Snow in Motion Symposium, International Glaciological Society

Buser, 0. and-H. Frutiger (1980) Observed maximum runout distance of snow avalanchesand the determination of the friction coefficients tz and ý. Journal of Glaciology, 26(94) 121-130.

Dent, J.D. and T.E. Lang (1980) Modelling of snow flow. Journal of Glaciology,26(94). 131-140.

Dent, J.D. and T.E. Lang (1981) Experiments on the mechanics of flowing snow. ColdRegions Science and Technology

Gubler, H.-U. (1987) Measurements and modelling of snow avalanche speeds. InAvalanche Formation, Movement and Effects, International Association for the Hydrolog-ical Sciences Publ. No. 162 (ed. B. Salm and H.-U. Gubler), 405-420.

Hansen, A.C. (1985) A constitutive theory for high rate multiaxial deformation of snow.Ph.D. Dissertation, Montana State University

Hansen, A.C. and R.L. Brown (1986) The granular structure of snow: an internalstate variable approach. Journal of Glaciology,32(112). 434-439.

Hui, K., P.K. Haft, J.E. Ungar and R. Jackson (1984) Boundary conditions forhigh shear grain flows. Journal of Fluid Mechanics 145 22"-233.

Hutter, K., Ch. Pliiss and N. Maeno (1988) Some implications deduced fromlaboratory experiments on granular avalanches. No. 94 der Versuchsanstalt fdr Wasserbau,Hydrologie und Glaziologie an der ETH, 323-344.

Lang, R.M. (1988) A numerical study on channel flows of cohesionless granular mediawith variable basal friction and Voellmy type drag. Eidgenossiche Institut fuer Schnee undLawinen Forschung Report.

Lang, R.M., B.R. Leo and K. Hutter (1989) Flow characteristics of an unconstrainednon-cohesive granular medium down an inclined, curved surface: Preliminary experimentalresults. Annals of Glaciology 13 146-153.

Lang, R.M. (1992) An experimental and analytical study on gravity driven free surfaceflows of cohesionless granular media. Doctoral Dissertation, Technische Hochschule Darm-stadt.

Lang, T.E. and J.D. Dent (1980) Scale modelling of snow avalanche impact on struc-tures. Journal of Glaciology 26(94) 189-196.

Lang, T.E. and J.D. Dent (1982) Review of surface friction, surface resistance, andflow of snow. Reviews of Geophysics and Space Physics 20(1). 21-37.

McClung, D. and P.A. Schaerer (1983) Determination of avalanche dynamics frictioncoefficients from measured speeds. Journal of Glaciology, 26(94) 109-120.

Mellor, M. (1968) Avalanches. U.S. Army Cold Regions Research and EngineeringLaboratory Monograph III A3d.

Nakamura, H. et al. (1987) A newly designed chute for snow avalanche experiments. InAvalanche Formation, Movement and Effects, International Association for the HydrologicalSciences Publ. No. 162 (ed. B. Salm and H.-U. Gubler), 441-451.

.rem, H., T. Kvisteroy and B.D. Evensen (1985) Measurement of avalanchespeeds and forces: instrumentation and preliminary results of the Ryggfonn project. Annalsof Glaciology, 6, 19-22.

Pao, R.H.F. (1961) Fluid Mechanics. New York: John Wiley.Perla, I.P., T.T. Cheng and D.M. McClung (1980) A two-parameter model of snow

avalanche motion. Journal of Glaciology, 26 (94), 197-207.

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Pliiss, Ch. (1987) Experiments on granular avalanches. Diplomarbeit, Abteilung X,Eidgenossische Technische Hochschule, Ziirich, 113 pp.

Pohlhausen,K. (1921) Zur ndherungsweisen Integration der Differentialgleichung derlaminaren Grenzschicht. Zeitung fuer Angewandte Mechanik und Mathematik 1 252-268.

Salm, B. (1966) Contribution to avalanche dynamics. International Association for theHydrological Sciences AISH Publ., 69. 199-214.

Salm, B. (1971) On the rheological behaviour of snow under high stresses. Contrib.Inst. Low Temp. Sci. Hokkaido Univ., Ser. A, 23. 1-43.

Salm, B. (1975) A constitutive equation for creeping snow. International Associationfor the Hydrological Sciences AISH Publ., 114. 222-235.

Salm, B. (1977) Eine Stoffgleichung ffir die kriechende Verformung von Schnee. Disser-tation 5861, Eidgenossische Technische Hochschule Ziirich

Savage, S. B. (1983) Granular flows down rough inclines: Review and extension. Me-chanics of Granular Materials: New Models and Constitutive Relations (ed J.T. Jenkins andM. Satake). Amsterdam: Elsevier, 261-82.

Savage, S. B. (1984) The mechanics of rapid granular flows. Advances in AppliedMechanics 24 (ed T.Y Wu and J. Hutchinson). New York: Academic, 289-366.

Savage, S. B. and K. Hutter (1989) The motion of a finite mass of granular materialdown a rough incline. J. Fluid Mech. 199, 177-215.

Savage, S. B. (1989) Flow of granul,.x materials. Theoretical and Applied Mechanics,Elsevier Sci. Pub., IUTAM. 241-266.

Siemes, W. and L. Hellmer (1962) Chem. Eng. Sci., 17, 555.St. Lawrence, W.F. (1977) A structural theory for the deformation of snow. Ph.D.

Dissertation, Montana State UniversitySt. Lawrence, W.F. and T.E. Lang (1981) A constitutive relation for the deformation

of snow. Cold Regions Science and Technology, 4 3-14.Szidarovszky, F., K. Hutter and S. Yakowitz (1987) A numerical study of steady

plane granular chute flows using the Jenkins-Savage model and its extension. InternationalJournal for Numerical Methods in Engineering, 24 1993-2015

Voellmy, A. (1955) Uber die Zerst6rungskraft von Lawinen. Schweizerische Bauzeitung73, 159-162, 212-217, 246-249, 280-285

von K'irm.n, Th. (1921) Uber laminare und turbulente Reibung. Zeitung fuer Ange-wandte Mechanik und Mathematik 1, 233-252.

23

REPORT DOCUMENTATION PAGE I 1W 70BBm PM, No. 7408

Mei~* em afneedad, -id cp eigmdsele2i ftoti cisalndk*nndAm. smidowmmitm regdn V#ebudm emeb ora my ohws peci~x -lbc ol I, mdonmi.

VA2 0430.wmis ~lmacMa.wmugmut mid SiKLPerowik RsdxdAaonPc (0704-0100). Wad*VWo. DC 206031. AGENCY USE ONLY . REPORT DATE. REPORT TYPE AND DATES COVERED1.AGNC UE NL AS bknQ -T R April 194. TITLE AND SUBTITLE 6. FUNDING NUMBERS

Model for Avalanches in Three Spatial Dimensions:Comparison of Theory to Experiments

6. AUTHORS

Renee M. Lang and Brian R. Leo

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) S. PERFORMNG ORGANIZATIONREPORT NUMER

U.S. Army Cold Regions Research and Engineering Laboratory72 Lyme Road CRREL Report 94-5Hanover, New Hampshire 03755-1290

9. SPONSORINGIONITORING AGENCY NAME(S) AND ADORESS(ES) 10. SPONSORINGAONITORINGAGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES

12. DISTRIBUTONIAVALAUTY STATEMENT 12b. DISTRIBUTION CODE

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161.

1I& ABSTRACT (ahianm 2W was)

A three-dimensional theory is derived to describe the temporal behavior of gravity currents of cohesionless granular me-dia, in an attempt to model the motion of dense, flow-type snow avalanches, ice and rock slides. A Mohr-Coulomb yieldcriterion is assumed to describe the constitutive behavior of the material, and the basal bed friction is described similarlyby a Coulomb type of friction. A drag term is included in order to model the occurrence of flow regimes where boundarydrag becomes non-negligible. Data from laboratory simulations are compared to a series of numerical studies based on theaforementioned theory. A nondimensional, depth and width averaged form of the theory is considered. A Lagrangian fi-nite difference scheme is then applied to numerically model some limiting cases of the governing equations. Two differentnumerical models are developed, tested and compared to experimental values. The results indicate that the model can ac-count for flow transitions by inclusion of the drag term when the initial inclination angle is large enough to affect bound-ary drag. Furthermore, the temporal and spatial evolution of the granulate and final runout position can be predicted tovalues well within the experimental error.

14. SUBJECT TERM 15. NUMBER PAGES

Avalanches Mathematical modelsGranular media Snow I& PRICE CODE

17. SECURITY CLASSIFICATION 16. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LUMIATION OF ABSTRACT

OF REPORT OF ThIS PAGE OF ABSTRACT

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

NSH 7540.01-290-5500 Stand Form 296 (Rev. 2-M)PMrebed by ANSI 81d. SA*-12o-10