On the average velocity of flow over a movable bed

ON THE

AVERAGE VELOelTY OF FLOW

OVER A MOVABLE BED

BY

SELIM YALIN

^*

Table of symbols.

g acceleration due to gravity;

v kinematic viscosity;

p density of fluid;

P8 density of bed material;

Ys specific weight of bed material in fluid;

D a typical diameter of bed material Ce.g.Dmax, Dao, etc.);

il depth of uniform flow;

S slope;

u average velocity of flow;

Introduction.

Observations show that the movement of bed material, due to dynamic action of flow, is conllU- only accompanied by ripples, a wave-like defonu- ation 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 fac- tor 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 charac- teristic parameters defining a flow-phenomenon on

* Hydraulics Rcscarch Station, Wallil1gford.

u* --.:.VgSil shear velocity;

c=u/u* friction factor;

q

=

u.il discharge pel' unit width;

k₈,...,D "sand-roughness";

k roughness of movable bed;

À height of a sand-wave;

À length of a sand-wave;

\jJ angle of repose ofbed material;

c friction factor of a rough rigid bed;

lU 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 two- dimensional 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;

(iO The shape of the grain-size distribution curve.

45

SELIM YALIN

In this case the phenomenon is completely defin- ed by the characteristic parameters:

defined by the angle \jJ and the ratio l:::'.j'A. Thus (10) can be expressed a fol1ows:

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:

(12)

(13)

\ (11)

w\

' )

.?-- =

/L::"(X, Y, Z, W) D

..

D_~=

l',

,....(X Y Z W)" ' , whereas:

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. 2a)

l.!.._ .

Z and \jJ

=

const. (angle of repose).

ks

From substitution of these values in (11) it follows that F2 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 F₂ is demonstrated.

whcre, it can be assumed that, the form of F₂ 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 quan- tities related to the phenomenon:

2. Form of the function F20

(1)

(6) (4) (3) (2)

(5)

Z=~'

_D'

v, P, P_{s '} D, h, S,g

v, P, Ps,D,h, Ys, u*

y

=

J.lI*2

ys·D

c

= .!!.. =

FI(X, Y, Z, W) u*

li= tu (v, p, PB'D,h, Ys, u*) yields:

Rence any mechanical quantity related to the flow-phenomenon under consideration, must, in general, be a certain function of ail seven charac- teristic parameters (4), therefore the expression for the average velocity of the flow u= q

Ih,

must be : Itis possible to replace any of the parameters(1) by any independent combinations in which they occU!' (sec e.g. Hel'. [1]). The substitution of gand S by:

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:

and:

with:

X

=

D.u* ; v

.---..- - . - ..---t ' , - - - r - - - . J

- - ~ !

- - - - /' û ·~~--t

1 /. v~~ ~.

If·

; \jt • ,._:L_

i ... ···f···~····-_, ^..J₁

!···---···-·-'-··....-·-·----+--:~--i

1 . l

Q

!

'... . ... - ....- . - - - X - ..- -.. _..._ .._.... --1

Q Q

b

/2

consists mainly of the head loss due to the expan- sion 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. 2b) 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:

(9) y

=

W

=

0

=

const (8) ,X _ les·u*.

" - - - , v thus:

/1 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 varia- tion 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:

- - - _^{. .}_ . _ - - - _ . _ - - - . _^.._ - -

'...- - - . - - - À -

c=

~~- =j'(~)

⁽¹⁰⁾

Rere the form of / depends itself on the geometry of ripples and on the relative roughness of their

-_

^{...._ - - _..}_ - - - - _ . _ - - - - -

B

surfaces hiles --- Z. The shape of the triangle ABC, approximating a ripple in Figure 1, is completely 46

(14)

whereas the condition of continuily yields:

Vii ( h -

~) ⁼

^ve(h

+ _~) =

^vh

thus:

(15)

(H seems thatilis not possible to ob tain c froin <;

by a simple multiplier, as has been assumed by many authors.)

Figure 3illustrates the result of the experimental

0.1 0.2 0.3 04 0.5

y----

, ;

: : ^:

;

'00'>';; oJ' ⁰ :

'01<0

0 ~(\0ef,:^~~

..

.1< ~

il;

,

"

1· ~i~: ;,;~o2~))} :r~~~~:u~~_t~~suré)

; ....(5~1/100001)(From 1(21)-uc 0(5'1/50001 (),and-etA

i ^: _:

:

30.02 0.03 0.04 005 20

30

C 10 9 8 7

(16)

y ('

Â)2

^V2

':>lie

= h

2g (17)

Consequently the part of the energy gradient cor- responding to the expansions b-7cafter each ripple length "Je, is:

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:

SI

= -te ⁼ ~J-. (-~). ( ~ ). ;;1

⁽¹⁸⁾

whereas the part eorresponding to the skin-friction along the distances aB

=

"Je_{I ,} is:

Thus:

S

s, + s

1 - (Â/"Je) (J"

. =.

^{l 2}

= [~

^{ln ( Cl}

fc: tr

and therefore:

"Je x² V2

S2 =

T'

^{[ln ( Cl}

_::~)J

^{2 gh}

(With

~l

^{= 1 -}

~

^{.ctg\)J )}

/3

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 consist- cd of polystyrene grains of equal shape and size (Ys

=

0.03; D

=

1.35 mm). The ripple characte- ris tics  and "Je were measured for various values of

h (h

< ,..,

20cm); 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 quantitycwas 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.)

(21) (20) (19)

gh

(TIl') ln [Cl (hlk_{s) ]}

VI

(Â/"Je) ^(J"

c = - =V

v*

where:

• c and ëcan also be interpreted as friction factors of a mobile bcd "with" and "without" ripples respectively.

~ ⁼ V

^{1 -}

~

. ( ctg \)J -

~

^{(2) (24)}

s ⁼

^{X2/Y; Of)= YIZ; Y; W} (25) and thus (6) can be replaced by:

3. Experimental values of c for (water+sand).

s

^{= Ys Dil/pv2 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 cons- tant. The hydraulic state of the flow-phenomenon 47 Although (6) and (26) are, mathematically CO!l-

sidered, equivalent; from the practical poin t of view, (26) is more convenient. For, both:

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:

(22)

(23)

(J"= ct^0' \)J --

~ ~

^.

[-.!.

^III(Cl

J!....)\ -.1

²

n 2 h _x k_s __

Berc x and Cl are constants

_ V 1 (

h)

c = - = - l n C l -

V* Je ks

as one vvould expect. From ('21) and (23) il follows that the relation between the friction factors for a mobile bed cand for a rigid bed (having the same roughness ks)*

c

^is:

(x

=

0.4, Cl

=

e2 .4

=

11.00) (Ref. [4]).

Thus c is a function of hlÂ, hlk_s' Â/"Je and \)J only as predicted by ('11). If there are no ripples,

Le. = 0, then (21) reduces to:

SELIM YALIN

and c reduces into a function of three variables only:

is reflected by 1)

=

(Y/Ys).S and Y

=

pu*2/YsD only.

Suppose now that itis intented to de termine the form of the function Fa experimentally for (water

+

sand), i.e. for:

p

=

101.70 kg.s².m-^1, v

=

1.01.10-^Hm²/s and Ys= 1.65.

In this case:

/4

W

=

1/2.65 - const.

S =

16180 Da [D in (mm) ]

1)= 0.606 S y

=

0.606 Sh/D

(27)

(28)

The corresponding

S =

const values are:

(3l) (2H)

1 6H1.1;

4401.0.

1478.5;

3463.4;

t'

=

!.!.!l!,!~^----2.50 u*

654.5;

2003.5 ;

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 = D50 •

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 D₅₀ 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

= 1..-

ln Z

+

^B

= _~

^ln

_~ +

^{B (aO)}

u* x x 1)

and by relation:

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].

and can be represented by a family of curves, hav- ing, say, Y as abscissa, c as ordinate and '1)as para- meter. 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 repre- sents àn experimental solution of the problem, Le.

the function F,! is experimentally determined.

The experimental points on Figures4-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; 0.52; 0.54; 0.59;(mm) Obviously the form of F4 depends on the cons- tant 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 func- tion of 1) and Y only:

Il /6 /5

y, ....•

48

0.3 OA 0.5 0.2

. 2 .03 0.04 0.05 0.1

!

a ^, ! ⁱ

-ool>~ i _! ) 00¹ 0

0 i

(0.1-<: ⁰

1

10

~~~~F~"

1-...

-,

:W,' , 8" ^,

...

^',

i l : ⁾

-<:>1 5°1/1000011from·deC0""/,,,

l

o 1$0^{1/5000 1 (}q meosured -mesure) P I '

T

. . 1 5 ° 1/10000) \(tram eqn1211) -(de l'équation12/1)1Do^{o ys yrene}1.35 mm

i

01501/50001 J(Àond-etLlmeosured-me.surés) . x($=1/10COO) }fram -deC=v/v-" ) Sand-5ab,

, . (0meosured -mesuré) O=O,347mm

00 a

4

110

4. General remarks.

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.

30

* * Attention is drawn to the fact that the points, ^CO!TCS- ponding 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 river- depths (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).

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 plolt- ed versus Y together with those of the polystyrene on Figure 10. As seen from Figure 10 the e -values

. Since the flow-phenomenon studied in this paper IS defined by seven independent mechanical quan- tities, 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. Thereforeifthe consi- deration 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' num- ber 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 el se but the reciprocal of Y (Le. '1'= l/Y) the method given in ReL [8] assu- mes, 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

- - --~---_._----_._---

Cla 9 8 7

00 D·0.59mm l!:'440rOI

05'0.00101')'6.06.10- 411

i_

Oi!S : 0.0015(7'J:9.09,10- 4 ),~l_u_n::;_~a!o

._- • S '0,0020(7]' 12.12-\0-4)

0 ¹

iT

^T:

1--.. _, rSlo9/ ! - 1

0/ ₁

:~ ~i~~ I-~

curves

01- courbes

/0

_~0° ^~

0 ¹

;7

/ i ₁ - i

•. U ^.

T

, I i i; ; 1,1 , i

0.001 001 Y

.

⁰

50 4 30

CI

/8

/9

As seen from Figures 4-H.e -values given hy experiment for a movable hed ao'ree weIl with the

tl . b

lCoretIcally 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 -

cur- ves.

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

~ormedby ~xperimentalpoints change considerably from one FIgure to another, Le. if 1; varies. Hence it is pos~ihleto conclude that 1; and Y more impor- tant 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 comhi- nations 1] are of the same order. VV= 1/2.65 for sand differs considerably from W

=

1/1.03 for

SELIM YALIN

the theoretical curves given by the method of Ref. [8] disagree with the results of the meas ure- ment.

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:

Therefore c=

f

[X, Y, Z,vV,cp(X, Y)]; i.e. accord- ing 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.

c

(w )1.85

(W- _1)08". 0

=

\ ^{al -}v* ^X

+

^a2c2 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 indepen- dent variable ifX and Y are given. For:

w/v*

=

cp(X, Y).

[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}IcGraw- Hill 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.

[6] D.S.W.E.S. expts. - Paper No. 17, Studies of River Bed. Materials under Movemcnt, with special refercnce ta the lower Mississipi River, 1935.

[7] N.J. KOTSCHIN, I.A.KlBEL and N. \V. ROSE. - Theore- tische Hydromechanik. Band 2, Akademie-Verlag, Ber- lin, 1955.

, - - - -

References.

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.

[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. - Lec- ture 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 Transpor- tation. Proc. A..S.C.E., vol. 8D, HY3, lD(i3.

[12J L.B. LEOPOLD and T. MADDOCK. - The Hydraulie Geo- metry of Stream Channels and some Physiographie Implications. lJ.S.G.S. profession al paper 252, \Vash- ington, 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 considé- rations 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 ciné- matique vet 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 F2 des paramètres Il//::", Il/les' /::"/1. et tJ; (équ. 11).

50

---_._-_.

__

._---

Forme de la fonction F_{2 :}

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 /?V2est 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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