On the average velocity of flow over a movable bed

LA HOUILLE BLANCHE/N° 1-1964
ON THE
AVERAGE VELOelTY
OF FLOW
OVER A MOVABLE BED
BY
SELIM YALIN
*
Table of symbols.
g acceleration due to gravity;
u* --.:. VgSil shear velocity;
v kinematic viscosity;
c =u/u* friction factor;
p
density of fluid;
q = u. il discharge pel' unit width;
P8 density of bed material;
k8
Ys specific weight of bed material in
k roughness of movable bed;
fluid;
D a typical diameter of bed material
Ce.g. Dmax , Dao, etc.);
,...,
D "sand-roughness";
À
height of a sand-wave;
À
length of a sand-wave;
il depth of uniform flow;
\jJ angle of repose ofbed material;
S slope;
c friction factor of a rough rigid bed;
u average velocity of flow;
lU
Introduction.
Observations show that the movement of bed
material, due to dynamic action of flow, is conllUonly accompanied by ripples, a wave-like defonuation of the surface of the mobile bed. Obviously
the properties of the motion of a fluid on an
undulating surface are not identical tothose on
a plane surface, and consequently the friction factor c = (average velocity) /(shear velocity) cannot
remain the same when an initially plane surface
of the bed turns into an undulating one. The
question how the quantity c depends on the shape
and size of the sand waves and thus on the characteristic parameters defining a flow-phenomenon on
terminal velocity of a grain,
a movable bed is one of the most popular subjects
of investigation of contemporary fluvial hydraulics.
The pm'pose of the present paper is to suggest how,
according to the requirements of the theory of
dimensions, the experimental data has ta be used
in order to determine the friction factor c.
1. Dimensionless vari'ables
of the phenomenon.
Consider the case of a steady and uniform twodimensional flow of a real fluid with a free surface
over a cohesionless movable bed. It is assumed
that the following geometrical properties of the
bed material are specified:
(i) The shape of grains;
* Hydraulics Rcscarch Station, Wallil1gford.
(iO The shape of the grain-size distribution curve.
45
Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1964004
SELIM YALIN
In this case the phenomenon is completely defined by the characteristic parameters:
v, P, Ps ' D, h, S, g
defined by the angle \jJ and the ratio l:::'.j'A.
(10) can be expressed a fol1ows:
(1)
w\\
(11)
')
It is possible to replace any of the parameters (1)
by any independent combinations in which they
occU!' (sec e.g. Hel'. [1]). The substitution of g and
S by:
Thus
(3)
whcre, it can be assumed that, the form of F 2 does
not depend any more on the geometry of the
ripples Bor on the roughness of their surfaces.
In spite of the difl'erent appearance (11) does not
confliet 'vith (6). For since  and À are the quantities related to the phenomenon:
(4)
.?-- = /L::" (X, Y, Z, W)
(12)
..D~= l',,.. . (X"
Y Z
' W)
,
(13)
(2)
and:
yields:
v, P, Ps, D, h, Ys, u*
Rence any mechanical quantity related to the
flow-phenomenon under consideration, must, in
general, be a certain function of ail seven characteristic parameters (4), therefore the expression for
the average velocity of the flow u = q Ih, must be :
li
= tu (v, p, PB' D, h, Ys, u*)
= .!!.. = FI (X, Y, Z, W)
u*
with:
X
=
D.u* ;
v
y
= J.lI*2
ys·D
whereas:
l.!.._ .
ks
(5)
Since the functional relationship (5) represents
a physical fact which does not depend on the choice
of the units of measurement, then according to the
TC-theorem of the them'y of dimensions this relation
can be expressed in a dimensionless fonu as
fol1ows:
c
D
(6)
Z
and \jJ
=
const. (angle of repose).
From substitution of these values in (11) it
follows that F 2 is in fact a function of X, Y, Z and
\V only; and thus (6) and (11), on principle, are
identica1.
In the fol1owing paragraph an attempt to derive
theoretically an approximate form for the funetion
F 2 is demonstrated.
2. Form of the function F 20
Z=~'
D'
Observe that a rough rigid bed can be considered
as a special ease of a mobile bed having Ps"""" co
(i.e. extremely heavy grains). In this case D--- k s
and:
,X _ les·u*.
" ---,
y = W = 0 = const (8)
v
thus:
It is known in hydraulies that the head loss duc
to a contraction is very smaIl compared with that
of an expansion (sec e.g. ReL [2]). For example,
the greatest part of the head loss caused by the
sudden contraction of the rigid boundaries (Fig. 2 a)
Q
!
··············f······~····-. .J
..........
,
1
!······-----·······-·-'-··.... -·-·----+--:~--i
1.
l
(9)
Q
as one would expeet.
If the amount of the bed mate rial in suspension
is negligible, then from the point of view of the
flow, the mobility of the bed implies, only a variation in the geometry of the bed surfaces. In other
words the expression for c must not difIer from
that for a rigid bed if le represents the roughness
caused by ripples. Therefore assuming le,...., Â, for
the case of fu lIy developed turbulent motion, the
fol1owing relation must also be valid:
c=
Q
b
~~- =j'(~)
û
1
/.
; \jt
(10)
---_. ._ . _ - - - - - _ . _ - - - - - - - - . _.._ - B
'......- - - - - . - - - - - - - À -
_--_.._ - - - - _ . _ - - - - -
/1
surfaces hiles --- Z. The shape of the triangle ABC,
approximating a ripple in Figure 1, is completely
46
/'
!
·~~--t
v
~~
•
~.
If·
,._:L_
i
Rere the form of / depends itself on the geometry
of ripples and on the relative roughness of their
....
--~
----------
/2
-_
.-------..--.-..-------t
'
, --------------r-------.J
'.....
. ... - ....- . - - - X - ..- -.. _..._ .._....
--1
consists mainly of the head loss due to the expansion b......,. c; the cfTect of the eontraction a"""" b
heing negligihle (Hef. [2]). Accordingly the head
loss due to the presence of a ripple (Fig. 2 b) can
he assumed to he equal to that due to the expansion
b ......,. c, plus the head loss duc to the skin-friction
along a"""" b.
The head loss due to the expansion b......,. c can he
expressed as follows:
(14)
LA HOU 1LLE BLANCHE/N° 1-1964
whereas the condition of continuily yields:
~) =
h-
Vii (
+ ~) =
v e (h
(15)
vh
(H seems that il is not possible to ob tain c froin <;
by a simple multiplier, as has been assumed by
many authors.)
Figure 3 illustrates the result of the experimental
thus:
(16)
If, say,
(Â/h) :::;; 1110 then 0.997:::;; 1 - (Âlh)2 :::;; 1,
Le. the denominator of the multiplier in brackets is
practically = 1; whereas for the usual values of the
angle of repose \)J, Cf. = 1 (ReL [3]). Thus (16) can
weIl be approximated by:
*
y
':>lie
=
('
hÂ)2
2g
T'
S2 =
(With
[ln (
~l
Cl
::~)J
= 1-
+s
s,
. =. l
1-
= [~
2
~
V2
(19)
gh
2
.ctg\)J )
(Â/"Je)
ln (
tr
(J"
fc:
Cl
gh
(20)
and therefore:
V
c=-=
(TIl')
ln
[Cl
VI
v*
~~
..
~(\0ef,: ~
0
C 10
9
.1<
:
0
il;
8
,
~i~: ;,;~o2~))} :r~~~~:u~~_t~~suré)
(hlk s ) ]
(Â/"Je)
0.02
.... (5~ 1/100001) (From
1(21)- uc
0(5'1/50001 (),and-etA
:
i
0.03 0.04 005
"
:
0.1
0.2
0.3
04
:
0.5
y----
(18)
Thus:
S
J'
/3
whereas the part eorresponding to the skin-friction
along the distances aB = "Je I , is:
x2
o
'01<0
;
(-~). ~
"Je
>';;
(17)
( ). ;;1
-te = ~J-.
=
'00'
3
:
;
20
1·
V2
;
:
:
7
Consequently the part of the energy gradient corresponding to the expansions b -7 cafter each
ripple length "Je, is:
SI
,
30
(21)
(J"
check of the theoretically derived form (21). The
measurements were carried out in a two-foot flume
of the H. R. S.-Wallingford; the mobile bed consistcd of polystyrene grains of equal shape and size
(Ys = 0.03; D = 1.35 mm).
The ripple characteris tics  and "Je were measured for various values of
h (h < ,.., 20 cm); the slope of the uniform flow S
being 1/5 000 and 1110 000. Assuming k s = D and
ctg \)J = 1.60 and using measured values for  and
"Je, the quantity c was computed from (21) for various
values of h. As seen from Figure 3 the values of
c given by (21) show a satisfactory agreement with
those obtained directly from the measurements of
the flow-discharges q, Le. from c = q/(h Vg1iS).
(The actual water depUIs h, which were always less
than 1/3 of the width of the flume (60 cm), wel'e
reduced into those of the two-dimensionaI fiow
according to the method given by H. A. Einstein
in Ref. [5], Appendix II.)
where:
(J"
=
[-.!.
ct 0' \)J -- ~ ~ .
III (Cl J!....)\ -.1
n
2 h
_x
k s __
Berc x and
(x
=
Cl
0.4,
2
(22)
are constants
Cl
= e =
11.00) (Ref. [4]).
2 .4
Thus c is a function of hl Â, hlk s' Â/"Je and \)J only
as predicted by ('11). If there are no ripples,
Le.
Â
= 0,
then (21) reduces to:
c = - = -1l n (
_
V
V*
Je
V
1-
~
(23)
Clks
. ( ctg \)J -
~
The form (21) shows how c = vlv* depends upon
the properties of the ripples which are themselves
the functions of X, Y, Z and \V. Thus c can be
considered directly as a certain function of X, Y,
Z and 'Xl. This fact is expressed by (6).
Observe that the consideration of ,X, Y, Z and VV
is equivalent to that of:
s=
h)
as one vvould expect. From ('21) and (23) il follows
that the relation between the friction factors for
a mobile bed c and for a rigid bed (having the same
roughness ks) * c is:
~ =
3. Experimental values
of c for (water+sand).
(2
)
(24)
• c and ë can also be interpreted as friction factors of a
mobile bcd "with" and "without" ripples respectively.
X2 /Y;
Of)
= Y IZ;
Y;
W
(25)
and thus (6) can be replaced by:
Although (6) and (26) are, mathematically CO!lsidered, equivalent; from the practical poin t of
view, (26) is more convenient. For, both:
s=
Ys Dil/ pv 2
and \V = p/Ps
do not depend on the hydraulical properties (they
do not involve h, S or v*) and therefore if the fiuid
and bed material are specified they remain constant. The hydraulic state of the flow-phenomenon
47
SELIM YALIN
is reflected by 1) = (Y/Ys).S and Y = pu*2/YsD only.
Suppose now that it is intented to determine the
form of the function Fa experimentally for (water
sand), i.e. for:
+
p
=
101.70 kg.s 2 .m- 1, v
=
1.01.10- H m 2 /s
and Ys = 1.65.
In this case:
W
=
1/2.65 -
S = 16180 Da
const.
[D in (mm) ]
(27)
= 0.606 S
y = 0.606 Sh/D
1)
/4
and c reduces into a function of three variables
only:
(28)
Obviously the form of F 4 depends on the constant value of VV and, theoretically, must change if
W changes. If D = const., Le. a certain sand is
selected, then S = const also and c varies as a function of 1) and Y only:
(2H)
/5
and can be represented by a family of curves, having, say, Y as abscissa, c as ordinate and '1) as parameter. In this case a certain CLIrVe corresponds to
a certain value of 1) (Le. certain value of slope S).
'1'0 another grain size D, i.e. to another constant
values of S correspondsanother family of curves.
Hence, if from measurements carried out for
various grain sizes such families of CLIrVeS are
obtained then the totality of these families represents àn experimental solution of the problem, Le.
the function F,! is experimentally determined.
The experimental points on Figures 4-H represent
the curve-families (2H) for the following values of
D (computed from the data in Ref. [6]) :
0.31;
0.48;
0.51;
The corresponding
0.52;
0.54;
0.59; (mm)
S = const values are:
654.5; 1478.5; 1 6H1.1;
2003.5 ; 3463.4; 4401.0.
/6
The grain-size distribution CLIrVeS of the bed
materials above have the same S-like shape (see
Ref. [6]): the grain size D being chosen as D = D 50 •
If the shape of the grain-size distribution CLll'Ve
varies; then theoretically, the corresponding family
of curves (2H) must also vary even if the value of
D 50 remains the same.
C- CLlrveS on Figures 4-9 represent the values
of the friction factor fOL' the case of a rigicl bed
(see footnote *) as given by the logarithmic fOrIn
lIma x
u*
= 1..- ln
x
Z
+B=
~
x
ln
~
+B
(aO)
1)
and by relation:
t'
=
!.!.!l!,!~
----
2.50
(3l)
u*
y, ....•
Il
48
where B is a certain function of X = ylsY given
hy Figure 20-20 in Ref. [4].
(See for (30) and (:il) H.ef. [4] chapt. XX.)
c- curves for laminar flow were obtained from eqn
01.22) in Ret'. [7].
LA HOUILLE BLANCHE/N° 1-1964
polystyrene used in the experiments at H. R. S.
The 1; - value of this polystyrene was 1; = 5H6.55
whereas the 1; - value of the sand D 5Ü = 0.347 mm
(Sand No. 6 in ReL [6]) is = 676.30). Since the 1;values for both materials are of the same order, the
e - values of the sand D 5Ü = 0.347 mm can be plolted versus Y together with those of the polystyrene
on Figure 10. As seen from Figure 10 the e - values
30
!
a
,
!
i
-ool >~
i
!
(0.1-<:
0
1
C la 1-.......
/8
1
-,
8
:W,'
D·0.59mm l!:'440rOI
-<:> 1 5°1/100001
05'0.00101')'6.06.10- 41
Oi!
S : 0.0015 (7'J: 9.09,10- 4 )
i_
50 . _30 1--..
,~l_u_n::;_~a!o
4
• S '0,0020(7]' 12.12-\0-4)
iT
T
rSlo9/
,
0/
01-
/0
CI0
-
1
00
. 2
:~ ~i~~ I-~
curves
1
0°
~
!
:
courbes
~
i
,Ii
001
l
)
T
i
0.04 0.05
x ($= 1/10COO) } fram - de C = v/v-"
,
. (0 meosured - mesuré)
0.1
0.2
0.3
) Sand -5ab,
O=O,347mm
OA
0.5
110
-~---_._----_._------
1
/
0.001
a.03
---
;7
T
i;
C 0""/,,,
o 1 $0 1/5000 1 ( q meosured - mesure)
P
I ' yrene
o ys
. . 1 5 ° 1/10000) \ (tram eqn 1211) - (de l'équation 12/1) 1Do 1.35 mm
015 01/50001 J (Àond-etLlmeosured-me.surés) .
1
1
40
1from·de
i
,
il :
7
00
0
0
F~"
~~~~
, 8"
... ' ,
9
10
) 00
-
1
i
.
•. U
;
Y
.
1,1
for both materials are of the same order (if 1; and
Y, i.e. if X and Y are the same) and thus the
influence of VV can be negleeted.
4. General remarks.
,
i
0
/9
As seen from Figures 4-H. e - values given hy
experiment
for a movable hed ao'ree
weIl with the
.
b
tllCoretIcally obtained t' - curves whell the values
of Y are smaIl (Le. before the ripples are fOTmed).
As Y increases, ripples form, and the e - values of
the movablehed deviate cOllsiderably from c - curves.
Figures 4-H show that the experimental points
c.orresponding to the various values of the slope S
(Le. of 1]) are very close to each other, so close that
the difIerence between the experimental CIll'VeS
S = const. is almost of the same order as the scaUer
due to the errors of measurement, i.e. the variation
of e with '!) seems to be considerahly weaker than
the variation with Y. On the other hand the paUls
~ormed by ~xperimental points change considerably
from one FIgure to another, Le. if 1; varies. Hence
it is pos~ihle to conclude that 1; and Y more important varIable than 1] {at least within the scope of
the experiments ploUed on Figures 4-H).
As seen from (25) Z is involved in 1] onlv. Thus
to state that "'!) is unimportant" means a~Itomati­
cally that "Z is unimportant" [see (7)] * *.
Consider now the comhination VV. Suppose that
the influence of VV isalso negligible in comparison
with 1; and Y. If this assumption is correct, then,
two flows flowing on ditIerent hed materials, i.e.
having difTerent values of \V, must have the same,
or nearly the same, values of e if their values of
1; and Y are equal and their lessimportant comhinations 1] are of the same order. VV = 1/2.65 for
sand differs considerably from W = 1/1.03 for
. Since the flow-phenomenon studied in this paper
IS defined by seven independent mechanical quantities, according to the theory of dimensions the
dimensionless quantity e must be a function of four
dimensionless variables X, Y, Z, VV. However the
theOl'y of dimensions is not able to provide the
form of such a funetion. On the other hand
depending on the fonu of this funetion, e, may vary
to a greater extent with some of the variables and
to a lesser extent with others. Therefore if the consideration of SOlue variables can be omiUed, this is
a purely practical decision and does not mean that
the theOl'y of dimensions provides the wrono' number of variables. ConsequenUy the stateme~t that
"Z and VV can be neglected" (within the scope of
the experimental data used) does not imply that
"e does not depend on Z and VV".
In ReL [8] it is assumed that the "irregularities
on the bed surface" (ripples) and consequenUv e
is a function of the Einsteinian function '1' ol;ly.
Since '1' is nothing else but the reciprocal of Y
(Le. '1' = l/Y) the method given in ReL [8] assumes, in tenus of this paper, that e is a funetion of
Y only. ApparenUy such a consideration is too
simple. Indeed this method does not cover aIl the
praetical cases it purports to do. For example the
Figures 7,2 and 7,3 in Ref. [H] show distinctly how
* * Attention is drawn to the fact that the points, CO!TCSponding to the river-data (from Ref. [12]), in Figure 4, form
a natural extension of the path formed hy the points of the
Hume-data; in spite of the mu ch lm'ger values of the riverdepths (1.28 m :( 11 :( ,u;a m = 12ft) and smaller value of the
river-slope (S
0.000284). This confirms once more, that is,
n()~ 11 and S separately, but the proclllct 1I.S (i.e. u,'
ghS)
whlCh matters. Consequently the comhinations involvin" 11
and S separately (i.e. 11 and Z) must be less important than
those involving u. (Le. X and Y).
=
=
SELIM YALIN
Therefore c = f [X, Y, Z, vV, cp (X, Y)]; i.e. according to Ref. [10] c is a function of X, Y, Z, W only,
which agrees completely with the considerations of
the present paper. However it is not intended to
discuss whether the mathematical form, connecting
X, Y, Z, W in eqn (9) of Ref. [10}, can also be
correct.
the theoretical curves given by the method of
Ref. [8] disagree with the results of the measurement.
In a more recent work-Ref. [10] a mathematical
form for c is determined (eqn (9) in Ref. [10]).
Using the notations of the present paper eqn (9)
in Ref. [10] can be written as follows:
(W
c
-_
1)08"
. 0
= (w
al - v*
\
X
+ a2 c2)1.85
Acknowledgement.
(with al> a2 consts).
Thus c is given as c = f CoX, Y, Z, W, w/v*). But
as proved in Ref. [l1J w/v* cannot be an independent variable if X and Y are given. For:
w/v*
=
The Author wishes to acknowledge the work of
Ml'. B. A. Say, Assistant Experimental Officer, in
carl'ying out aIl the numerical calculations. This
study is published with the permission of Hw
Direetor of Hydraulics Research.
cp (X, Y).
References.
[1] L. I. SEDOV. Similarity and Dimensional Methods
in Mechanics. Infosearcll Limited, London, 1959.
[2] N. I. PAVLOVSIW. - Collected \Vorks. Vol. 1, Academy
of Sciences of tlle U.S.S.R., Moscow-Leningrad, 1955.
[3] 1. E. IDELCHIK. - Hydl'aulie Losses. Gros. En. Izd., Mos~
cow, 1954.
[4] H. SCHLICHTING. - Boundary Layer Theory. i}IcGrawHill Book Company Inc., New York, Toronto, London,
1960.
[5] H. A. EINSTEIN. - Formulas for the Transportation of
Bed Load. Trans. A.S.C.E., vol. 107, pp. 561-577, 1942.
Paper No. 17, Studies of River
[6] D.S.W.E.S. expts. Bed. Materials under Movemcnt, with special refercnce
ta the lower Mississipi River, 1935.
[7] N. J. KOTSCHIN, I. A.KlBEL and N. \V. ROSE. - Theoretische Hydromechanik. Band 2, Akademie-Verlag, Berlin, 1955.
[8J H. A. EINSTEIN and N. L. BARBAHOSSA.
Hiver Channel
Houghness. l'mns. iLS.C.E., vol. 117, pp. 1121-114(;,
1952.
[DJ V. A. VANONI, N. H. 13HooKs and J. F. KENNEDY. - Lecture Notes on Sediment Transportation and Channel
Stability. California Inst. of l'echnology, Heport No.
IUI-H-l, Pasadena, Calif., lD(;1.
[10J D. B. SnwNs and E. V. HICHAHDSON. Resistance to
Flow in Alluvial Channels. Proc. A.S.C.E., vol. 8(i,
HYS, 1960.
[11 J M. S. YALIN. - An Expression for Bed Load Transportation. Proc. A..S.C.E., vol. 8D, HY3, lD(i3.
[12J L. B. LEOPOLD and T. MADDOCK. - The Hydraulie Geometry of Stream Channels and some Physiographie
Implications. lJ.S.G.S. profession al paper 252, \Vashington, 1D53.
,--------------------
Résumé
Vitesse moyenne d'un écoulement sur fond mobile
par M. Selim YaUn
INTRODUCTION
L'un des problèmes d'actualité de l'hydraulique fluviale est celui de l'influence des ondulations d'lm lit
alluvionnaire sur les pertes de charge de l'écoulement. L'auteur se propose de déterminer, à partir de considérations d'analyse dimensionnelle, les paramètres à adopter pour l'étude expérimentale de ce problème.
Variables sans dimensions du phénomène:
La vitesse moyenne v de l'écoulement est fonction des sept paramètres définissant le fluide (viscosité cinématique v et masse volumique ~), le matériau de fond (masse volumique ~s et diamètre D), et l'écoulement (tirant
d'eau Il, pente S et accélération de la pesanteur g), ou encore des paramètres dérivés v, ~, ~s' D, Il,
Selon le théorème des II, le coefficient de frottement c = v/v* est alors fonction des quatre variables sans
dimensions X, Y, Z et W (équ. 7).
Dans le cas d'un lit fixe plat et rugueux, c est seulement fonction de X et Z, puisqu'on peut considérer que
ce cas correspond à ~s et Ys ---?> 00. La hauteur de rugosité les est le diamètre caractéristique des grains de matériau.
On obtient alors la relation (9),
Quand le fond est ondulé, la hauteur de rugosité à prendre cn compte est la hauteur des rides ou dunes;
la forme de l'ondulation est définie d'autre part par l'angle de repos sous l'eau du matériau tJ; et la cambrure /::"/1.
(fig. 1). Le coefficient de frottement cest donc une fonction F 2 des paramètres Il//::", Il/les' /::"/1. et tJ; (équ. 11).
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LA HOUILLE BLANCHE/N° 1-196<
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Forme de la fonction F 2
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:
L'auteur détermine le coefIicient de frottement c sur le fond à partir des pertes de charge de l'écoulement,
lesquelles se subdivisent en :
-
pertes de charge à la Borda au passage de la crête des dunes (fig. 2, équ. 17);
pertes de charge par frottement sur la face amont des dunes (équ. 19).
L'expression finale de c est donnée par la relation (21).
Les valeurs de c calculées par cette formule sont en han accord avec celles déterminées en partant des valeurs
expérimentales de v/v* (fig. 3), pour les essais en canal effectués au Laboratoire de vVallingford.
Valeurs expérimentales de c (eau
+
sable) :
Le coefficient c est fonction des paramètres X, Y, Z et vV, c'est-à-dire encore de : ç = X2/Y, 'fJ = Y/Z, y et W.
Le paramètre ç = Y8IP / ?V2 est pratique, car il est indépendant de l'écoulement.
Les figures 5 à 9 indiquent les résultats obtenus avec du sable (VV = Cte), pour diverses granulométries (ç
variable). Les valeurs expérimentales de c sont très voisines des valeurs de c calculées à partir de la loi de vitesse
il la paroi de Prandtl, quand il n'y a pas de dunes. L'influence de la pente S (c'est-à-dire de .~) est négligeable,"
compte tenu des erreurs de mesure; par contre, l'allure des courbes dépend beaucoup de ç.
Pour deux matériaux avec .~ = S (y/ Y,) voisins, les valeurs expérimentales de c correspondant il des ç et Y
égaux sont très proches (voir fig. 10); le paramètre vV = ?/?8 n'a donc pas d'influence très marquée.
REMARQUES GÉNÉRALES:
Les paramètres Z (ou -~) et VV peuvent être négligés, du moins dans le domaine expérimental envisagé; ce
qui ne signifie pas néanmoins que le coefficient c soit indépendant de ces deux variables.
Einstein (réf. 8) supposait en fait que c ne dépendait que de Y; la divergence entre ses résultats théoriques
et expérimentaux tient il cette hypothèse.
L'expression établie par Simons et Richardson (réf. 10) peut se mettre sous la forme c = f (X, Y, Z, vV, w/v*),
w étant la vitesse de chute du grain. Comme w/v* ne dépend que de X et Y, on est donc ramené à une fonction des
quatre paramètres proposés par l'auteur.
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