Forces in granular flow

MECH. RES. COMM.

FORCES

Vol.

I, 3-7, 1974.

Pergamon Press.

IN GRANULAR FLOW

E. W i e g h a r d t Institut fGr Schiffbau der UniversitMt Hamburg (Received 18 September 1973; accepted as ready for print 14 December 1973)

Introduction Measurements of the forces on a rod dipped into a rotating sand bed are described. Any references to similar work would be highly welcomed.

Cylindrical rod The drag of a cylindrical rod dipped vertically into a rotating sand bed (vessel diameter 1.4 m, sand depth o.5 m) has been measured previously by J.Glinz [I]. Approximately it is

D = 5.12~h5/2d1/2(1

- 0.57 v+ + o.51 v+2)(1 + 1.2 d , ~ )

(1)

with ~ bulk density of sand in loose packing, h immersion depth (5 to 17 cm), d rod diameter (1 to 5 cm), r radial distance from axis of revolution (2o to 5o cm); v+ is a Froude number

v+ = (d/h)l/4.v/v'r'~gh

(2)

om with v velocity of sand at the rod (2o to 18o cm/s), g=981 s-, z .

Hence, main results are: a) the drag depends only little on speed (D has a flat minimum at v+=o.56), b) the drag is proportional to h 5/2 and not to h 2 as one would expect when the pressure and with it Coulomb friction forces would increase linearly with depth. Plate flow New t e s t ~ w e r e

made with a rod of rectangular cross section

a=o.3 cm by b=2 cm to resemble flat plate flow at an incidence Scientific Communication

4

K. WIEGHARDT

angle~

(with r = 5o c m , ~ = 1.61

i m m e r s i o n depths we may assume surface

in p l o u g h i n g

underneath flow.

Vol.

p/cm3).

for greater

that the work done at the sand

a furrow is small compared with the work

and that - k i n e m a t i c a l l y

- there

is mainly plane

force R is again p r o p o r t i o n a l

to h 5/2

is shown in FIG.I

Here,

by ~ h5/2bl/2

and plotted vs. v+=

(b/h)I/4v/gvf~

and three i m m e r s i o n

depths;

to

The resultant

At least

i, No.

R is made d i m e n s i o n l e s s for four angles

the dotted

(1-o.57v++o.51v+ 2) as in equ.(1).

curves

are here h i g h e r

0nly for the smallest

gives

the rod along

only mean values For dimensional the effective

bef f = b s i n ~ Now,

+ a cos~

of straight and - ~

~beff+o.2

vs.~

in FIG.2

Hence,

of the plate. cm/vZ2.2

vs.o(

at ~reat

or v+(~) become

a good ap-

0 the breadth not great

as a c o r r e c t i o n

or thick-

compared

one might

~rain diameter

of e.g.

add I mm

cm the ratio:

cross

force L

to

depth h = 12 cm; it is independent

of

(The Froude number with beff(~)

.) Only for the highest

: o.67 to 1.16 the inertia

important

with

This would ~ive the dotted curve

speed for v = o.19 to o.~:11 m/s. is here o . o 7 ~ v + ~ o°52

f

ploughshare

is, indeed,

For O ( =

an effective

The lower ~art of FIG.2 gives drag D

were evaluated.

(3)

of the rod is only a = 0.3 cm, i.e.

on both sides

lines,

.

to the test points.

to bef f after equ.(3)

at the

the sand approaches

of this r e c t a n g u l a r

with the grain diameter.

for

for + ~

depths.

angle FIG.2

one would like to find R ~ b e f

the curve V ~ e f f / b

proximation ness

(r=5o cm) instead

reasons

breadth

for R/~h5/2b1/2

for four sand speeds v

of the forces

the

ones for greater

depth of h = 12 cm. Since circles

angles,

of the force on incidence

R ( ~ ) / R ( ~ =9o °) vs. ~

same i m m e r s i o n

and the values

than the c o r r e s p o n d i n g

As for the dependence

of incidence

are proportional

depth h = 4 cm and then for greater incidence furrow work is not negligible

as

forces

and yield smaller values

speed v=1.82m/s

of the sand motion for

LID.

Plate flow in wet sand Other tests have

shown ~hat the forces

for a sand-water mixture density ~ ) w h e n there

are p r a c t i c a l l y

(with a c o r r e s p o n d i n g

is a surplus

of water.

increase

the same in

But, when at rest

I

Vol.

I, No.

1

FORCES

IN G R A N U L A R

FLOW

5

i.e. before the test, the water is just covering the sand then the forces become much greater, almost doubled. This might be an effect of Reynolds' dilatancy [2~. Because of unavoidable vibrations the rotating sand usually is in a close packing. Yet, to enable the grains to relative motions near the rod the packing there must loosen. ~hen there is not enough water nearby to fill at once the locally enlarged interstices there will be regions with wet sticky sand, mainly near the sand surface, and greater sand masses will be involved than with either no water at all or with abundant water. Discussion As a model for such granular flows seems to be missing, the tesl results were first analysed naively by simple power formulae. Yet, in the limit for vanishing speed v-~0 the results should fit Coulomb's theory for passive earth resistance or the refined theory for a curved surface of slip in front of a gliding wall. An appropriate evaluation gives for the plate at v-~0 Ro= ½ ~ h 2 b e f f ' ( 1 7

+ 2 h/bef f) instead of Ro= 6 . 2 ~ h 2 " 5 b e ~ } 5 ;

and, indeed, 8.5 + h/bef f = 6 . 2 ~ h / b e f f ± Io% for 2.7< h/bef f < 27. With angles of friction in the sand (~) and at the body (~)

between 3o and 35 o the first term 1 7 . ~ h 2 b e f f ~

corresponds

to a plane stress acting on the breadth bef f. The second term 2 . ~ h 3 might be explained by inner friction on the lateral surfaces of the sand mass starting to slide in front of the plate. Assuming there at depth y a shear stress

~y.tan¢,

a

rough calculation gives approximately this term for 3 o ~ < 3 5

°.

The simple addition of force terms seems possible since the equation for Airy's stress function is thoroughly hyperbolic.With increasing speed at first solid friction and forces decrease somewhat, and at higher speeds the momentum of flowing sand yields a quadratic term.

References [I] Wieghardt,K. ZAMM 48, 1968, T 248/9 [2] Reynolds, 0. Phil.~iag. Ser.5, vol.2o, 469, 1885

6

K. W I E G H A R D T

Vol.

i, No.

+

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30 °

x

V

T

60°

+

A

i

90 °

•~

E)

•

hS/2 32

FIG. I

o,5

1

I

I

Resultant force R on a rectangular rod in sand flow vs speed. (~" sand density; h immersion depth, b breadth of rod,~, angle ) of incidencej v - - ( b/h ) 1 / & ' v / ~ - .

181 499

v,

1.5 I

I

Vol.

1, No.

i

FORCES IN G R A N U L A R FLOW

7

i; ®

-o.8

V b ,in,~., ~'~o. , ~ / ~ -

/

X

sin d~ . a cos c~ + 0,2cm / b/b .0 :2 cm

+

X

l

beff

~

m

~a=0,3cm V

m/s O,lg

0,40 + 0,81 X 1,82

0.2

®

30 °

=d" I

FIG. 2

I

60 ° II

1

I

oC II

Force ratio R(~,)I R(~,:90 o) and ratio" cross force L to drag D vs angle of incidence =6 at four speeds ~ immersion depth h = 12 cm.

"~90 o !