Solid Flux of Snow-Bearing Air Flow

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Solid Flux of Snow-Bearing Air Flow

Dyunin, A. K.; National Research Council of Canada, Division of Building

Research

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PREFACE

Snowdrifts can be a maJor source of disruption in the operation of transportation services and a general nuisance in the normal wintertime activity of a community. Such drifts are formed whenever a wind, strong enough to transport horizontally a significant amount of snow, encounters an obstacle which forces it to deposit some of this snow. The usual approach taken in defending an area or struc- ture against snowdrifting has been to locate the structure properly so that the drift problem will be a minimum and to erect obstacles, such as snow fences, to control where the snow will be deposited. The approach taken in the development of these defences has been largely empirical. Attention has been directed primarily to the character of the air flow with little attention being given to the material transported. In some circumstances, it would be an advantage to have a more complete defence against snowdrifting than is now avail- able. In their attempts to develop this defence, engineers are giving more consideration to the theoretical aspects of the problem and in particular to the relationships between the air flow and the snow being transported

.

It is one of the responsibilities of the Snow and Ice Section of the Division of Building Research to collect and make available infor- mation required for the solution of snow and ice problems. The present paper, translated from the Russian, is a contribution to the theory of snowdrifting. This paper will give to the reader an appreciation of some of the factors to be considered in the theoretical description of blowing snow and its deposition as snowdrifts.

The paper was translated by Mr. 0. Belkov of the Translatione

Section of the National Research Council Library, to whom the Division of Building Research wishes to record its thanks.

Ottawa

November 1963

R.F. Legget Director

NATIONAL RESEARCH COUNC

IL

OF CANADA Technical Translation 1102

Title: Solid flux of snow-bearing air flow

(~verdyi raskhod snegovetrovogo potoka)

Author : A.K. Dyunin

Reference: Trudy Transportno-Energicheskogo Instituta,

(4):

71-88, 1954

SOLID FLUX OF SNOW-BEARING A I R FLOW

Summary

I n t h i s paper a g e n e r a l formula i s d e r i v e d , u s i n g dimensional a n a l y s i s , f o r t h e s o l i d f l u x of a " f l u i d

+

g r a n u l a r m a t e r i a l " mix- t u r e and i s a p p l i e d t o t h e p a r t i c u l a r c a s e of snow-bearing a i r flow. For p r a c t i c a l purposes of combatting snow and d e s i g n i n g new r o a d s i n r e g i o n s s u b J e c t t o s n o w d r i f t i n g one r e q u i r e s p r e c i s e knowledge of t h e laws governing snow t r a n s f e r . It i s p a r t i c u l a r l y important t o know t h e f a c t o r s governing t h e s o l i d f l u x of a snow-bearing a i r flow.

The s o l i d f l u x ( t v e r d y i raskhod) ( q ) o r simply "snow f l u x " (raskhod snega) i s defined a s t h e weight of t h e snow c a r r i e d through a u n i t of a r e a of a t r a n s v e r s e c r o s s - s e c t i o n of t h e flow i n a u n i t of t i m e . I n s p e c i a l i z e d l i t e r a t u r e f o r d e n o t i n g t h i s concept t h e terms "magnitude of t h e snow-bearing

a i r f l u x " (moshchnost snegovetrovogo potoka ) and " r a t e of snow t r a n s f e r " ( i n t e n s i v n o s t l perenosa sneya) a r e f r e q u e n t l y used. These terms do n o t correspond t o t h e mechanics of t h e m a t t e r . The term " s o l i d f l u x " ( t v e r d y i raskhod) i n t h i s c a s e i s more p r e c i s e .

The f l u x of snow i s most c o n v e n i e n t l y measured i n grams through

a

s q u a r e metre p e r second (g/m2 s e c ) . Let u s i n t r o d u c e t h e n o t a t i o n q f o r t h e snow

f l u x .

Let u s c u t t h e flow by a p l a n e p e r p e n d i c u l a r t o i t s d i r e c t i o n and c a l l t h e l i n e of i n t e r s e c t i o n of t h i s p l a n e w i t h t h e s u r f a c e of t h e e a r t h t h e flow f r o n t .

The weight of a l l t h e snow t r a n s f e r r e d by t h e wind i n

a

u n i t of time through a u n i t of f r o n t l e n g t h i n a l a y e r of a r b i t r a r y h e i g h t h i s c a l l e d t h e t o t a l s o l i d f l u x of t h e snow-bearing a i r flow o r t h e t o t a l snow f l u x and i s measured i n grams p e r metre p e r second. The t o t a l snow f l u x w i l l be denoted by t h e l e t t e r Q o r Qh i f t h e h e i g h t h of t h e l a y e r i s d e f i n e d .

The weight of t h e snow t r a n s f e r r e d through a u n i t of f r o n t l e n g t h i n a n i n t e r n a l of time T w i l l be c a l l e d snow t r a n s f e r and denoted by t h e l e t t e r G .

The dimensions of t h e snow t r a n s f e r i s d m .

A s y s t e m a t i c measurement of t h e snow f l u x was s t a r t e d i n 1927 a t t h e Vodenyapino Experimental S t a t i o n * of t h e C e n t r a l Research I n s t i t u t e of t h e M i n i s t r y of Communications (TSNII MPS) by means of s p e c i a l d e v i c e s ( b l i z z a r d

m e t e r s ) ( 7 , 3 ) , A t f i r s t t h e s c i e n t i s t s a t t h e s t a t i o n used v e r y l o n g exposure t i m e s f o r t h e i r i n d i v i d u a l measurements. It was t h e r e f o r e v e r y d i f f i c u l t t o e s t a b l i s h any r e l a t i o n s h i p between snow f l u x and any p a r t i c u l a r f a c t o r . A r e d u c t i o n i n exposure time made i t p o s s i b l e t o e l u c i d a t e t h e p r o g r e s s i v e I n c r e a s e i n snow f l u x a s t h e wind v e l o c i t y i n c r e a s e d . A number of a t t e m p t s were made t o f i n d e m p i r i c a l formulae f o r c a l c u l a t i n g t h e t o t a l f l u x Qh

a t a

g i v e n wind v e l o c i t y ( v y ) a t a g i v e n h e i g h t y . U s u a l l y t h e h e i g h t i s l i m i t e d t o h = 2

m

s i n c e t r a n s t e r of snow a t g r e a t e r h e i g h t s i s n e g l i g i b l e .

The f i r s t of t h e known formulae was d e r i v e d by A.Kh. Khrgian (12

Q

=

-

3,47

+

O,15v2+0,O652v,2

A.

CM LIJB

B.V. Ivanov, a s c i e n t i s t of t h e West S i b e r i a n branch of t h e Academy o f S c i e n c e s , u s i n g t h e e x p e r i m e n t a l d a t a of t h e Vodenyapino S t a t i o n and h i s own d a t a , suggested i n 1951 t h e formula

q,,,= 0,003~,3 A-

c ~ l ~ i n . ( 2 )

( 9 )

A formula of s i m i l a r s t r u c t u r e

was

p u b l i s h e d by

D.M.

M e l l n i k i n 1952

,

The a u t h o r of t h i s paper a t v a r i o u s t i m e s h a s suggested t h e f o l l o w i n g e x p r e s s i o n s

A . A . Komarov*, a s c i e n t i s t of t h e West S i b e r i a n branch o f t h e Academy of S c i e n c e s , i n g e n e r a l i z i n g b l i z z a r d measuring o b s e r v a t i o n s of t h e t r a n s p o r t a - t i o n and power i n s t i t u t e of t h e West S i b e r i a n branch of t h e Academy of

S c i e n c e s , ( ~ ~ ~ ZSPAN), d e r i v e d t h e f o l l o w i n g e m p i r i c a l r e l a t i o n s h i p

To compare t h e s e formulae one must reduce them

a l l

t o t h e same u n i t s and I n terms of t h e wind v e l o c i t y a t a h e i g h t of 1 m. To c o n v e r t t o t h e l e v e l of 1 m a wind v e l o c i t y measured a t a h e i g h t of 11 m ( v , , ) , we u s e t h e formula of P r o f e s s o r V . N . O b o l e n s k i i ( r e f . 1 0 , p . 529):

*

See h l s p a p e r "Some r u l e s on t h e m l g r a t i o n and d e ~ o s i t i o n of snow i n w e s t e r n S i b e r i a and t h e i r a p p l l c a t i o n t o c o n t r o l measures

.

where according t o t h e d a t a of Peshke f o r snow cover n =

5.

Consequently v , , = ~ , ~ I 1 ~ ~ ~ = 1 , 6 2 ~ ~ . For t h e l a y e r a d j a c e n t t o t h e e a r t h we u s e t h e formula of S . A . Sapozhnikova ( r e f . 11, p . 72) where l o g 6 =

-4.85

( s e e our r e p o r t " V e r t i c a l d i s t r i b u t i o n of s o l i d f l u x i n

a

snow-wind flow", ( ~ i g

.

6 )

-

NRC TT-999)

.

Consequently

I n reducing t h e formula of B.V. Ivanov f o r t h e t o t a l f l u x one should keep i n mind t h a t

it

i s c o n s t r u c t e d from t h e d a t a of

N.N.

~ z ~ u m o v ( ~ ) a c c o r d i n g t o which

Q

e

4,9q,n.x

A f t e r c a r r y i n g o u t t h e n e c e s s a r y c a l c u l a t i o n s we p r e s e n t

a

summary of t h e r e s u l t s i n Table I .

I n F i g . 2 t h e c u r v e s f o r Q = f ( v , ) were c o n s t r u c t e d from t h e formula i n Table I and a p l o t t i n g of t h e mean d a t a obtained from f i e l d measurements of b l i z z a r d s i n c l u d i n g 155 measurements from t h e Vodenyapino Snowdrift S t a t i o n of TsNII and

31

measurements of TEI ZSFAN, a d j u s t e d t o g i v e t h e t o t a l snow f l u x f o r a snow-bearing

a i r

flow of u n l i m i t e d h e i g h t . D .M

.

Me1 n i k checked t h e r e s u l t s obtained by t h e s t a n d a r d b l i z z a r d m e t e r s of t h e t y p e VO and found t h a t t h e a c t u a l weight of t h e snow t r a n s f e r r e d

was

t w i c e t h a t c a l c u l a t e d from t h e r e a d i n g s of t h e b l i z z a r d m e t e r s . He took t h i s i n t o account i n determin-ing t h e c o e f f i c i e n t i n formula

( 3 ) .

Therefore t h e o r d i n a t e s of t h e curve o b t a i n e d

by D . M . M e l l n i k a r e approximately two times g r e a t e r t h a n t h o s e of t h e e x p e r i - mental p o i n t s .

I n t h e l i g h t of experimental d a t a obtained r e c e n t l y t h e formulae of

A.m.

Khrgian, B.V. Ivanov and t h e f i r s t formula of t h e a u t h o r g i v e v e r y low r e s u l t s even i f t h e e r r o r of t h e b l i z z a r d meter

i s

t a k e n i n t o a c c o u n t .

D.M.

M e l l n i k t r i e d t o g i v e a t h e o r e t i c a l b a s i s f o r h i s f o r m u l a ( 9 ) . He assumed t h a t t h e snow f l u x was p r o p o r t i o n a l t o wind energy, 1 . e .

where H

-

a

c o e f f i c i e n t having t h e dimension of cmi;

m

-

a i r mass p a s s i n g over

a

s p e c i f i c c r o s s - s e c t i o n F p e r second, 1 . e . t h e mass f l u x of t h e a i r flow.

Because m = pFV, where p is the mass density of the air,

Q

is propor- tional to the cube of wind velocity. It should be noted that the hypothesis

formulated by

D.M.

Meltnik is not precise, i.e. he in fact assumes that snow

flux is proportional to the force of the wind and not to energy.

Such are the results of investigation of the solid flux of a snow-bearing air flow available at the present time.

Up to the present time no use has been made of the large amount of experimental data on the transfer of sand particles by water and air. The transfer of snow by air is a particular case of the transfer of a granular material by a fluid whose specific gravity is less than that of the particle being transferred. In the first stqge of investigation one should find the general criteria of this phenomenon.

Let qo be the maximum solid flux at the height yo+ . ,

where

%

-

total solid flux in a layer of height y,

c l

-

a linear magnitude depending on yo, the linear characteristic of

the roughness of the surface 6 and the linear characteristic of

saturation of the flow by particles c.

The values of c, and yo are determined by the formulae

The magnitude c depends not only on the saturation of the flow by particles but also on the variation in the vertical component of the velocity of turbu- lent flow with respect to its height(').

q o [ ~ ~ - 2 ~ - 1

1

depends on the following factors (in the square brackets are given the dimension formulae):

~ [ F L - ~ T ~ ]

-

mass density of the particles being transferred; d

[L]

-

size of the particles;

~ [ L T - ~

1

-

acceleration due to gravity;

*

See our paper "Vertical distribution of solid flux in a snow-wind flow"

where ~r

-

r e l a t i v e volume c~f t h e p a r t i c l e i n t h e flow,

q,

-

components of t h e s o l i d f l u x I n t h e d i r e c t i o n of t h e c o o r d i n a t e a x i s

%

w ~ t h valueti of t h e Index a being 1 , 2 , 3 .

Let us c o n s i d e r two 3:ysl;ems of c o o r d i n a t e s . Let e q u a t i o n ( 1 2 ' ) i n t h e f i r s t system have t h e form

I n t h e second system

A f t e r d i v i d i n g and m u l t i p l y i n g a l l t h e v a l u e s of t h e f i r s t e q u a t i o n b y t h e s i m i l a r i t y c o n s t a n t s g , A ,

t ,

q, 1, we g e t

P r e c i s e l y t h e same r e s u l t w i l l o b t a i n f o r t h e right-hand p a r t i f t h i s o p e r a t i o n

i s

c a r r i e d o u t with t h e e q u a t i o n of t h e second system of coordinates. Consequently i n t h e right-hand s i d e we have a dimensionless s i m i l a r i t y

1

c r i t e r i o n . Let u s w r i t e i n t h e form

1

where

,

i s a c c e l e r a t i o n .

t

Assuming t h a t

-

-

-

g and 1 = d we g e t c r i t e r i o n

a,.

t 2

Let u s use t h e d a t a of V.N. Goncharov and G.N. Lapshin on t h e t r a n s f e r of ( 3

1

sand i n water t o e l u c i d a t e t h e r e l a t i o n s h i p between t h e c r i t e r i a @, and Q,

.

The l i n e a r roughness c h a r a c t e r i s t i c 6 i s determined a c c o r d i n g t o

A .P. Zegzhda ( r e f .

6,

p . 102, 10'1).

The v a l u e s of k , found by A.P. Zegzhda e x p e r i m e n t a l l y , a r e shown i n F i g . 2 .

The v a l u e of c Is determined by t h e e m p i r i c a l formula o f

V.N.

Goncharov ( r e f .

3 ,

p . 41, 2 0 2 ) .

c = 0,14. ~"(0,0005

+

0 , 3 5 d ) ~ " l ~ _{( 1 5 )}

where y

-

d e p t h of t h e flow i n m e t r e s .

The r e s u l t s of c a l c u l a t i o n a r e shown i n Table 11.

I n t h e c a s e of " a i r

+

sand" we w i l l u s e t h e d a t a of R . Bagnold (13,141 who measured t h e f l u x of sand i n a n a i r flow i n a s p e c i a l wind t u n n e l and under f i e l d conditions d u r i n g sand storms i n t h e Egyptian d e s e r t .

According t o t h e e x p e r i m e n t a l d a t a of Bagnold

I O O Q q O = - a

5

The predominant d i a m e t e r of t h e sand g r a i n s was

Consequently

s e c 2

L e t u s assume t h a t p = 123

,+.

The c r i t i c a l v e l o c i t y v,

,

i s

d e f i n e d

m

by formula

( 9 )

keeping i n mind t h a t f o r d r y sand ( s e e f o o t n o t e on p.

6 ) :

where p

i s

t h e v i s c o s i t y of t h e a i r , b e i n g e q u a l t o 1 . 8 5

los-

a t

m2

t o = 20° and a p r e s s u r e of 760 mm & , v = 1 5 - l o 4

-

P s e c m 2 ( k i n e m a t i c v i s c o s i t y ) . Then

T a b l e I11 shows t h e r e s u l t s o f c a l c u l a t i n g

a,

and Q2 f o r t h e t r a n s f e r of

sand i n a i r . Wind v e l o c i t y i s converted t o t h e l e v e l of y = 1 m e t r e .

For t h e c a s e " a i r + snow" we w i l l u s e t h e b l i z z a r d m e t e r i n g measurements of TEI ZSFAN, t h e Vodenyapino S t a t i o n of TsNII MPS and t h o s e of L.M. Danovskii of t h e NIIZhT ( n a i l T r a n s p o r t Research I n s t i t u t e ) ( 4 , 5 )

According t o formula ( 7 ) when y-m, i . e . f o r a s t r a t u m o f u n l i m i t e d h e i g h t

\$hen l o g 6 = -4.35 and c = 0 . 2 8 cm ( s e e o u r p a p e r " V e r t i c a l d i s t r i b u t i o n of s o l i d f l u x i n a snow-wind f l o w " ) a c c o r d i n g t o formulae

( 3 )

and ( 3 1 ) we g e t c, Z

4.7

cm, yo = 0.28 cm. The predominant s i z e of snow p a r t i c l e s n e a r t h e s u r f a c e i s d = 0 . 5 mm. T h e i r average mass d e n s i t y a c c o r d i n g t o Nakaya i s

h = 4 0 . 103 SCC'

m4

Consequently

I n t h e c a s e of l o o s e snow

( s e e o u r paper

on he

mechanical p r o p e r t i e s of snow e r o s i o n " ) . When t o =

-lo0

and t h e p r e s s u r e L s

,760

mm p = 177 s = 1 2 . 1 m2/sec.

m

L e t u s determine v t l a c c o r d i n g t o formula

( 9 ) :

The wind v e l o c i t y was measured a t t h e Vodenyapino S t a t i o n a t a h e i g h t of 2 m e t r e s . We have converted them t o a h e i g h t of 1 m e t r e u s i n g t h e p r e v i o u s l y d e r i v e d r e l a t i o n

The v a l u e s of Q 1 and @, d e f i n e d from b l i z z a r d m e t e r i n g measurements a r e given i n Table I V .

F i g u r e

3

shows i n l o g a r i t h m i c c o o r d i n a t e s t h e e x p e r i m e n t a l r e l a t i o n s h i p s of 0 , and 0 , a c c o r d i n g t o t h e d a t a of T a b l e s 11, I11 and I V . The v a l u e of

( 0

-

1) becoming 0 a t v = v1 i s p l o t t e d a l o n g t h e a b s c i s s a . The e x p e r i m e n t a l p o i n t s f o r a water-sand and wind-sand flow a r e a r r a n g e d i n a s t r a i g h t l i n e . On t h e average t h e f u n c t i o n of Q2 =

= ( a l )

can be p r e s e n t e d i n t h e form of a

I n a l l p r o b a b i l i t y t h e s o l i d f l u x i n a flow w i t h heavy p a r t i c l e s depends l i t t l e on t h e F o u r i e r number Fr, which e n t e r s i n t h e formula ( 1 0 ) . I n f a c t f o r " w a t e r

+

sand" ( w i t h d = 0.002 m ) t h i s number i s e q u a l t o

F o r " a i r

+

sand"

i . e . g r e a t e r by t h e f a c t o r of 45, and t h e c o r r e s p o n d i n g e x p e r i m e n t a l p o i n t s on t h e graph of

a,

= f ( Q l ) f i t s a t i s f a c t o r i l y a l t h o u g h t h e graph was c o n s t r u c - t e d w i t h o u t t a k i n g i n t o account t h e P r , number.

Experimental p o i n t s c o r r e s p o n d i n g t o b l i z z a r d m e t e r i n g d a t a w i t h v a l u e s of 0 , 4 2 . 6 do n o t go o u t s i d e t h e l i m i t s of t h e r e g i o n occupied by e x p e r i - mental p o i n t s of t h e water-sand and wlnd-sand f l o w s . When Ql >

2.6

t h e y

d e v i a t e g r e a t l y from t h e s t r a i g h t l i n e A A ' b u t r e t a i n i t s s e n s e .

T h i s d e v i a t i o n i s explained by t h e l a c k of p r e c i s i o n i n b l i z z a r d m e t e r i n g measurements. A s a l r e a d y i n d i c a t e d , a c c o r d i n g t o t h e d a t a of

D.M.

M e l l n i k , t h e a c t u a l f l u x of snow i s t w i c e t h a t o b t a i n e d from t h e m e t e r r e a d i n g s . The v a l u e of Q , a l o n g t h e l i n e A A ' when 0, > 2 . 6 i s g r e a t e r t h a n t h e o r d i n a t e s of t h e b l i z z a r d m e t e r i n g p o i n t s by a f a c t o r o f 2 . 5 and more, i . e . t h e t h e o - r e t i c a l l y determined e r r o r of t h e b l i z z a r d m e t e r i s g r e a t e r t h a n t h a t o b t a i n e d by D . M . M e l l n i k .

IIowever, a t low v e l o c i t i e s , below 2 . 6 - 2 . 7 1

7

m/sec, b l i z z a r d m e t e r s , a s can be s e e n from F i g .

3 ,

a r e more o r l e s s a c c u r a t e . The i n a c c u r a c y i n - c r e a s e s n l t h v e l o c i t y . T h i s i s confirmed by e x p e r i m e n t s of G . A . Nizovkin working a t T s N I I MPS who e s t a b l i s l ~ e d th a t t h e g r e a t e r t h e v e l o c i t y o f t h e wind t h e more snow i s blovm o u t of t h e box of' t h e b l i z z a r d m e t e r .

The f a c t t h a t t h e a c c u r a c y of t h e b l i z z a r d m e t e r d e c r e a s e s r a p i d l y when

a ,

> 2 . 6 can be e x p l a i n e d by changes i n t h e r e s i s t a n c e t o t h e motion of p a r t i c l e s f a l l i n g i n t o t h e n o z z l e of t h e b l i z z a r d m e t e r . The c o r r e s p o n d i n g Reynolds number i s

T h i s Reynolds number i s c h a r a c t e r i s t i c o f t h e t r a n s i t i o n boundary between t h e i n t e r m e d i a t e r e g i o n and t h e r e g i o n where t h e r e s i s t a n c e i s p r o p o r t i o n a l t o t h e s q u a r e of t h e v e l o c i t y ( r e f . 2 , p . 3 0 1 ) . The same Reynolds number f o r a

wind-sand flow i s reached (when d = 0.00025 m) only when

a,

=

3.5,

1 . e . f a r from t h e limits obtained by R . Bagnold.

I t

i s

h e r e t h a t one should seek t h e reason f o r t h e g r e a t e r accuracy of t h e "sand metering" f i e l d measurements of R. Bagnold, which correspond w e l l w i t h t h e p r e c i s e d a t a he obtained i n a wind t u n n e l .

Expanding formula (16) we g e t , i n t h e f i r s t approximation, t h e f o l l o w i n g g e n e r a l formula f o r t h e t r a n s f e r of heavy p a r t i c l e s i n a f l u i d medium

where 61

= f

(YO, c, ;)andyo

=

f (c, 6)

a r e determined from formulae

(8)

and (81).

Consequently t h e f l u x of heavy p a r t i c l e s , i n c l u d i n g snow, c a r r i e d by a c u r r e n t depends on: ( 1 ) d e n s l t y of t h e p a r t i c l e A, ( 2 ) dimension of t h e p a r t i c l e d ,

( 3 )

t h e l i n e a r c h a r a c t e r i s t i c of s u r f a c e roughness 8, ( 4 ) l i n e a r c h a r a c t e r i s t i c of t u r b u l e n c e c ,

( 5 )

d e n s i t y of t h e f l u i d P, ( 6 ) t h e t r a n s l a - t i o n a l v e l o c i t y of t h e flow v , , ( 7 ) adhesion between t h e s u r f a c e p a r t i c l e s T.

S u b s t i t u t i n g i n formula ( 1 7 ) t h e average v a l u e s f o r a snow-bearing a i r 2

flow we obtained p r e v i o u s l y A = 40

l o 3

* g d = 0.0005

m;

c , = 0.047

m,

m

yo = 0.0028

m,

v t , = 2.71 m/sec and assuming t h a t y = 2

m

we o b t a i n f o r average c o n d i t i o n s and l o o s e snow a formula t h a t depends only on v , :

Q, = 0,255 ( v ,

-

2,71)' e / ~ s e c

.

( 1 8 ) F i g u r e 4 shows a comparative curve of ( 1 8 ) and a curve f o r t h e M e l t n i k e q u a t i o n

( 3 )

.

Let u s check formula (18) w i t h f i e l d measurements.

Example No. 1. I n February 1952 D .M. M e l l n i k , i n checklng b l i z z a r d m e t e r s , c a r r l e d o u t c a r e f u l measurements of t h e weight of snow t r a n s f e r r e d by a 22 hour b l i z z a r d . The t o t a l t r a n s f e r o v e r a 1 metre c o n t r o l c r o s s - s e c t i o n was 5900 kg/m, and on t h e second c r o s s - s e c t i o n i t was 5700 kg/m. The average was 5800 kg/m.

A graph showing h o u r l y changes i n wind v e l o c i t y a r e shown i n F i g .

5.

Formula (18) f o r v e l o c i t i e s , measured w i t h an anemometer 11 metres above

V l l v ,

t h e s u r f a c e , a f t e r s u b s t i t u t i o n of v , = and = 2.71 1.62 = 4.44 m/sec, becomes

Let t h e v e l o c i t y of t h e trind d u r i n g a time i n t e r v a l of T change l i n e a r l y ( s e e F i g .

6 )

from v , t o v,

+

Av.

Then t h e i n t e r m e d i a t e v e l o c i t y v , corresponding t o a p e r i o d of time t ,

i s

e q u a l t o

/ V

=

7'1

-+

-

, A v .

T

The t o t a l t r a n s f e r of snow G

%

f o r

a

tlme T

i s

equal t o

Here T i s expressed i n h o u r s .

S u b s t i t u t i n g t h e d a t a of F i g .

5

we g e t

According t o t h e formula of D

.M

.

Mel'nik

Example No. 2 . I n t h e w i n t e r 1946-47 s c i e n t i s t s of TsNII, u s i n g a double snow f e n c e , made c a r e f u l measurements of t h e t o t a l snow t r a n s f e r throughout t h e w i n t e r and found t h e v a l u e t o be 91,000 kg/m.

With t h e method used i n Example No. 1, u s i n g d a t a of t h e m e t e o r o l o g i c a l s t a t i o n s , t h e t h e o r e t i c a l l y expected snow d e p o s i t was c a l c u l a t e d .

They were:

according t o formula ( 1 9 )

,

87,932 k@;/m

according t o formula D.M. Mel'nik, 93,051 k g / m .

Example No.

3.

From t h e d a t a of TEI ZSFAN, d u r i n g t h e w i n t e r 1950-51 snow f e n c e s , i n two r e g i o n s c l o s e t o each o t h e r , placed p e r p e n d i c u l a r t o t h e d i r e c t i o n of t h e wind, c o l l e c t e d an average amount of snow of 211,000 k d m .

The snow d e p o s i t s determined t h e o r e t i c a l l y were: according t o formula ( l g ) , 213,294 kg/m

according t o formula D .M. M e l 1 n i k , 193,077 k d m .

R e s u l t s o b t a i n e d with formula ( 1 9 ) and w i t h t h e M e l l n i k formula

( 3 )

a r e s i m i l a r . It should be noted t h a t i f wind v e l o c i t y i s determined by a meteor- o l o g i c a l anernometcr, t h e M e l l n i k formula g i v e s b e t t e r r e s u l t s t h a n f o r m u l a ( 1 9 ) .

v 1 I , assumed by u s t o b e

T h i s i s e x p l a i n e d by t h e f a c t t h a t t h e r e l a t i o n

-

v1

c o n s t a n t and e q u a l t o 1 . 6 2 , d o e s i n f a c t i n c r e a s e w i t h v , , and a c c o r d i n g t o t h e d a t a of TsIJII MPS a t h i g h wind v e l o c i t i e s can r e a c h a v a l u e c l o s e t o two. I f t h i s r e l a t i o n i s

t h e n

S u b s t i t u t i n g t h i s v a l u e of v, i n formula

(18)

we g e t t h e M e l l n i k formula e x p r e s s e d i n grams/* s e c . F i g u r e 7 shows e x p e r i m e n t a l v a l u e s of

5

d e t e r -

v1

mined a t v a r i o u s wind v e l o c i t i e s by D.M. M e l l n i k and TEI ZSFAN, and a c u r v e i s

"1 1

c o n s t r u c t e d f o r

-

= f ( v l l ) a c c o r d i n g t o e q u a t i o n ( 2 0 ) . T h i s c u r v e c o r r e s - v1

ponds f a i r l y w e l l t o e x p e r i m e n t a l d a t a .

Thus i n u s i n g d a t a from m e t e o r o l o g i c a l s t a t i o n s one should u s e t h e Melt nilc f o r m u l a . I f , however, wind v e l o c i t i e s a r e measured i n t h e l a y e r a d J a c e n t t o t h e e a r t h , one should u s e formula ( 1 8 ) .

From t h e above one c a n d e r i v e t h e f o l l o w i n g c o n c l u s i o n s :

(1) The t r a n s f e r of snow i n a n a l r f l o w i s s u b J e c t t o t h e g e n e r a l laws governing t h e t r a n s p o r t of heavy g r a n u l a r m a t e r i a l i n a f l u i d medium and depends on u n i v e r s a l c o n s t a n t s a p p r o p r i a t e t o t h e s e mechanisms.

( 2 ) Formula ( 1 7 ) c o r r e s p o n d s w e l l t o e x p e r i m e n t a l d a t a . Fonnula ( 1 8 ) and formula

( 3 )

of

D.M.

M e l ' n i k a r e t h e most a p p l i c a b l e f o r rough c a l c u l a t i o n s of snow t r a n s f e r o r t h e t o t a l d e p o s i t of a number o f b l i z z a r d s o r f o r t h e e n t i r e w i n t e r .

( 3 )

I n t h e g e n e r a l c a s e t h e snow f l u x o v e r

a

s p e c i f i c a r e a o r f o r e a c h b l i z z a r d depends n o t o n l y on t h e t r a n s l a t i o n a l v e l o c i t y of t h e wind b u t a l s o on t h e s t a t e of t h e snow s u r f a c e ( e x t e n t of compactness, t h e e x t e n t t o which t h e p a r t i c l e s a r e f r o z e n t o each o t h e r and roughness of t h e s u r f a c e ) , on t h e t u r b u l e n c e of t h e f l o w , s i z e and d e n s i t y of t h e p a r t i c l e s and on a i r tempera- t u r e . A number of t h e s e f a c t o r s c a n be c o n t r o l l e d w i t h t h e aim of d e c r e a s i n g o r e l i m i n a t i n g snow t r a n s f e r o r i n c r e a s i n g i t .

I n c o n c l u s i o n t h e a u t h o r would l i k e t o extend h i s s i n c e r e t h a n k s t o f e l l o w workers of t h e TsNII MPS and t o

D.M.

M e l ' n i k , G.A. Nizovkln and F . I . Antonov f o r k i n d l y s u p p l y i n g e x t e n s i v e primary d a t a c o l l e c t e d by them a s w e l l a s f o r a d v i c e and c o n s u l t a t i o n s , w i t h o u t which t h l s work c o u l d n o t have been done.

References

1. B a r e n b l a t t , G . P . The motion of suspended p a r t i c l e s i n

a

t u r b u l e n t f l o w . Prikladnaya Matematika 1 Mekhanika, 17: 1953.

2 . Vclikanov, M . A . Dynamics of r i v e r - b e d , flow (Dinamika ruslovykh potokov)

,

1946.

3.

Goncharov, V . N . Motion of s i l t d e p o s i t s (Dvizhenie nanosov), 1933.

4 .

Danovskii, L.M. Operation of snow f e n c e s on r o a d s of Western S i b e r i a .

Trudy NIIZHT, ( g ) ,

1353.

5 .

Danovslcii, L.M. Snow f e n c e s on r o a d s of 'vlcstern S i b e r i a @negozashchitnye

zabory na dorogakh Zapadnoi ~ i b i r i ) , 1950.

6 . Zegzhda, A.P. S i m i l a r i t y t h e o r y and method of d e s i g n i n g h y d r o e n g i n e e r i n g models ( ~ e o r i y a podobiya i metodika r a s c h c t a g i d r o t e k h n i c h e s k i k h m o d e l e i ) , 1938.

-(. Izyumov, N . N . Measurement of snow t r a n s f e r . Trudy NIS TsPTEU NKPS

( ~ e s e a r c h S t a t i o n , C e n t r a l E n g i n e e r i n g and Economic Planning A u t h o r i t y , Commissariat of ailr roads), ( 1 3 9 ) , 1931.

3 .

Izyumov, N . N . Program of g e o p h y s i c a l o b s e r v a t i o n s a t a s p e c i a l s n o w d r i f t s t a t i o n . Trudy NIS TsPTEU.

9 . M e l ' n i k , D.M. Laws governing snow t r a n s f e r and t h e i r u t i l i z a t i o n i n combatting s n o w d r i f t i n g . Tekbnika Zheleznykh Dorog, ( l l ) , 1952.

1 0 . Obolensltii, V .N

.

Meteorologiya, Ch. 1, 1938.

11. Sapozhnikova S.A. Mikroklimat i mestnyi k l i m a t ( ~ i c r o c l i m a t e and l o c a l c l i m a t e

j ,

1950.

1 2 . Khrgian, A.Kh. The e f f e c t of s m a l l r a i l r o a d p r o f i l e s on snow d e p o s i t i o n .

Trudy NIIPS,

(33),

1934.

13.

Bagnold, R . A . The measurement of sand s t o r m s . Proc

.

Royal Soc. London,

(9291, 1938.

Table I

Summary of c a l c u l a t e d snow f l u x using v a r i o u s formulae

No. 1 2

3

4

5

6

Author of formula A .Kh. Khrglan D .M. M e l l n l k B .V

.

Ivanov The a u t h o r The a u t h o r A . A . Komarov

The formulae have been converted t o v, and expressed I n g/m s e c Q =

-5.8

+

0 . 2 6 7 ~ ~

+

0 .123v, 2 Q = 0 .092v, 3 Q = 0 .O2g5v1 Q = 0.0234(1.062v1

-

4 ) '

Q = O.O3J4(1

-

~ ) v ,

3 "1 Q = 0 . O l l v l 3.5

-

0

.68

T a b l e I1

Values of @, and Qi, c a l c u l a t e d from t h e a v e r a g e d a t a o f

530

measurements of t h e f l u x of sand i n w a t e r c a r r l e d o u t by V . N . Goncharov and G . M . Lapshin

Continued I 1 Mean v e l o c i t y

,,

m/sec m 1 0.05 0.65 0.86 1,23 0.64 0.84 1.15 1 39 1.37 0.66 0.73 0.83 0.97 1.12 1.14 1.37 0.67 0,79 0.90 1 . O I 1.14 1.18 1.33 0 62 0.78 1.06 1.30 1.34 0 68 0.75 0.90 0.92 0.96 1.17 1.12 1.23 0.74 0.78 0.92 1.08 1.18 1.25 1.33 Flux of Q g/* ss% 4 84 66,6 292 1390 69 24 1 701 1323 2318 64.4 111.1 23 1 354 624 688 1221 44.1 118.8 182 354 572 696 930 54.1 1 65 547 1148 1522 59 98.2 229.5 297 310 542 700 968.4 59 104.6 215.3 416 618 902 946 Of fl0W Cm 2 9.0 9.9 10.1 1 1 , l 9.9 9 . 6 10.6 10.9 12.7 9.9 9 . 2 10.0 10.1 11.4 10.3 10.6 9.5 10.1 9 9 11.0 11.5 10.1 10.3 10.0 9.9 10.3 10.1 9.8 9.9 9.7 10.8 9.5 11.8 11.0 11.7 12.0 10.6 9.8 9.8 10.9 10.8 10.4 9,4 Observed c r i t i c a l velocity v,,, rn/sec 5 0.25 0.24 0.28 0.285 0.286 0,30 0.30 l a m . of ~ : ~ n d p a r t i c l e s ( j mm 3 0.75 1.33 1.50 1.87 1.91 2.16 2.33 Q , 6 2.6 2.6 3.44 4.92 2.66 3 5 4.8 5.8 5.7 2.36 2 60 2.90 3.46 4.0 4.07 4.9 2.35 2.78 3.16 3.55 4.0 4.15 4.67 2.18 2 74 3.72 . 4.56 4.70 2.27 2.50 3.0 3.06 3.2 3.9 3.74 4.1 2.47 2.6 3.06 3.6 3.94 4.16 4.44 9, 10-3 7 24.6 19.4 83 380 14.3 50.5 131 257 417 12.3 22.2 44.1 65.8 112 127 2 23 7.63 19.5 30.5 33 76.8 115 154 0.86 26.7 85.5 189 238 8.42 16.4 35.8 44.2 47 79,2 106 146 8.15 14.8 29.1 56 84 125 139

-15-

Table I1

-

continued C o n t i n u e d k i n v e l o c i t y vm rn/scc 1 0.48 0.64 0.69 0.75 0.97 0.98 1 .07 1.26 0 66 0.87 1.10 1.31 1.46 0.69 l .Ol 1.17 1.27 1.35 O.G2 0.81 1.09 1.28 1.41 0.64 0.74 1.12 1.30 1.24 0.62 0.48 1.04 1.31 1.45 0.65 0.77 0.87 0.95 1.02

1

1

.o

Fpth

of low cn 2 6.1 7.5 5.9 7.3 6.9 7.8 7.5 7.1 10.1 10.2 10.6 1 1 ,O 10.4 19 3 17.8 20.1 20 6 19.8 10.4 9 - 5 9.6 10.5 9.9 10 5 9.5 10.6 10.6 8.9 10.1 9.5 9.72 10 7 10,3 8.3 11.2 10.1 12 8 11.2 9.9 I dc:i n di::~dof p ; i r t i c t e r d rnv 3 2.5 2.5 2.5 2.5 2.5 2.94 2.99 0.96 1.02 1.06 1.16 1.u 1.27 0.60 0.77 1.04 1.27 1.39 0.70 0.76 0.85 0.95 1.11 1.21 1.33 1.39 1.82 2.99 3.08 3.16 10.5 9.5 10 8 11.2 8.2 10.3 9.6 9.8 9.3 10 4 10.5 10.2 9.6 10.2 10.1 12.1 11.4 9.9 10.0 10.3 Flux of s a n d Q g/m

seg

4 0.56 25.5 91.3 74.3 359 272 533 981 29.1 175 465.7 1033.5 1453 4 . 8 255 555.6 810 943 5 2 , 4 253 7 25 1420 2250 54.9 158.9 691 1383 1357 4 0 169 .50 7 1129 1619 16.7 35 9 129 8 199 6 245.5 350 395 274.3 472 GO 1 740 643 22.8 1 87 653 1020 1673 49.3 85.9 179 322 529.3 7 16 916 4 70 1169 Observed c r i t i c a l v e l o c i t y

?lSec

m 5 0*29 0,312 0.36 0.263 0.263 0.29 0,32 0.32 0.296 0.318

*,

6

1

1.65 2.20 2.38 2.58 3.34 3.38 3.69 4.35 2.11 2.79 3.53 4.2 4.7 1.02 2 81 3 , 2 5 3.53 3-76 2.35 3.08 4.15 4.88 5.37 2.43 2.81 4 25 4 95 4.72 2.13 1 .65 3 GO 4.53 5.0 2.4 2'03 2.72 2.97 3.18 3. I4 <P, lo-= 7 0.1 4.1 17 12 60.5 42 86 164 4.0 23.4 62 134 195 1 4 25.5 61 73 87.5 7 0 35.8 102 189 300 7.3 22 4 91.5 183 196 4.9 21 64 132 191 2.26 4.29 15.7 21.6 29.4 43 4 3 .o 3.18 3.32 3.62 3.88 3.96 2.02 2.6 3.52 4*3 4.7 2.2 2.39 2.67 2.99 3.46 3.8 4.19 4.37 4.46 47.9 35.8 58 70.5 101 77.5 2.76 226 86 121 197 5.5 9,Q 36 193 54.5 76 103 165 131

Table I1

-

continued

Table

I11

Values of

a,

and

@,

c a l c u l a t e d from d a t a of

measuring a i r - s a n d flow

Location of

measurements

1

Wlnd t u n n e l

S u r f a c e of sand

dunes i n a d e s e r t

11

ind

v e l o c i t y

v1 m/sec

2

6.22

6 . 3 3

8.65

g .81

11.15

1 4 . 1 3

'7.08

7 . 6 3

11.54

11

.80

13.28

C r i t i c a l

v e l o c i t y

v f l m/sec

3

4 . 3 1

@,

lo-'

6

0.063

0 . 4 3

1 . 7 5

3

.72

6 . 6

1 8 . 2

1.03

1.34

9 - 7

9 - 4

18.2

,

4

1.2':

1.40

1 . 7 6

2

.oo

2 . 2 7

2

.a8

1.44

1 . 5 6

2.35

2 . 4 1

2 . 7 1

S o l i d f l u x

of t h e

wind-sand

flow

Q

g/m

s e c

5

0.42

2 . 3

11.8

52

44

122

7 - 5 6

8.96

64.85

62.84

122.2

T a b l e I V

Values of

@,

a n d 0, c a l c u l a t e d from t h e d a t a of b l i z z a r d m e t e r i n g measurements

Q;/m sec Graph 9. = f ( v , )

,..-,Formula d A.A. Komarov(6)

,--.-

Formu fa d B.V. Iranov ( 2 )

F o r m u l a of the a o h a r ( 5 )

----

T'ovmutr O F the authov ((C)

-

-

Grmvlr dA.Kh. ~ h r ~ i a n ( ~ ) - A r r n u ~ a 4 D.u.T?el'nik 0)

x blizzard metering data af T E I

o blizzard r n e t e r i ~ daf a 04 &he

~ o d e n ~ a p i n o Station lq+7-48 F i g . 1 Comparison of v a r i o u s f o r m u l a e g i v i n g t h e dependence o f v e l o c i t y o f t h e s o l i d f l u x 02 a snovr-bearing a i r f l o w Fig. 2 Graph o f t h e f u n c t i o n

K =

f ( d ) a c c o r d i n g t o A . P . Zegzhda

Experimental r e l a t i o n between c r i t e r i a

Fig.

4

Curves

Q,

= f(v,) according to formula (13) and D

.M.

Mel'nik hours I8

-

16.- 1 4 - Fie.

5

4 -- 2

-

Variations in wind velocity according to anemometer during the period of the blizzard

( ~ x a m p l e No. 1 )

H I O N N U 1 4 1 z J t 3 6 7 d 9 / o ~ n n u ~ l c 1 7

TwJ2hr. T-6 hr. T.9 hr.

Fig.

6

Schematic diagram for the derivation of formula (20)

Fig. 7

V l l

Variations in the ratio

-

in relation

V l