Velocity distribution in alluvial charnels

(1)

VELOCITY DISTRIBUTION IN ALLUVIAL CHARNELS

BY R. J. GARDE *

AND A. S. PAINTAL **

Introducti'on

In the study of hydraulics of alluvial channels the engineer is often interested in finding the quan- tity of ,vatel' and the sediment load carried bv the stI:eanl under given conditions of flow. The inean velocity of flo,v.is obtained by integrating the velocity distribution over the depth and dividing by the depth of How. In order to find the total sediment load, the curve of (sediment concentration X velocity) must be integrated over the depth. In studies of scour, design of channels, silt excluders and extraetors etc., the knowledge of velocity distribution is also required.

Review and theoretical considerations

Two types of formulae have been suggested to describe the velocitv distribution for turhulent flow over rigid boundar:ies; they are exponential type and logarithmic type. The logarithmic law rather than the exponential law is more frequently used hy hydraulic engineers. The logarithmic law can he ohtained from Prandtl's hypothesis of mixing length [1:3]* hy assuming that in the vieinity of ,vaIl, the mixing length is linearly proportional to the distance from the wall, and the shear stress is assumed to be constant. lt can also be obtained from Kàrmàn's similarity law [13] by supposing that mixing length is only a funetion of velocity distribution and shear stress is constant at the houndary. The hasic 10garithmÎC hw can he stated as :

li ] y

--- - -~ lou

11* -:JC <Je y'

in which

:JC is the Kàrmàn constant;

lI. is the velocity at distance y from hed;

1I* is the shear velocity

Î ^{= ,/}

^2ll.

⁼ vnrRsl) ;

\ ' P t

(l)

"0

is the average shear stress at the hed

(=YtRS );

Yt is the unit \veight of fluid;

Pt is the mass density of fluid;

B is the mean hydraulic radius;

S is the slope; and,

y' is some length at which Il = O.

Nikllradse [13] condueted experiments with pipes which were artificially roughened by coating their inside surface with sand grains of llniform size. He found that if ll*kjv is less than 3.5, the boundary aets as hydrodynamically smooth, and equation Cl) takes the form:

(2)

* Beader in Civil Engineering, University of Boorl,ee (India) .

** Lecturer in Civil Engineering, University of Boorl,ee (India).

in ,vhich v is the kinematic viseositv of :t1uid and k₈ is the size of sand grains. .

* ,Numbers in [ ] the billli(lgrapJllY.

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1964041

(2)

If Il*kjv is greater than 67, the boundary aets as hydrodynamically rough boundary and equation (1) takes the form:

shear on the bed 1:"*. Hence the value of :J( should depend upon 1:"* and :Ji, if the sediment concentration and bed configuration govern the value of :Je.

(8) (6)

k_s= (29.3n)O There

with Manning coefficient in the following form:

seems to be more important. Hence the inclusion of Froude number in equation (6) is justified. Pre- liminary analysis indicated that for velocity distri-

bution studies .

{Œy~/PU

r

)-:dfi/2-

is preferable to U/(gR)1/2.

The parameter 1:"* is dimensionless shear which can be called as index of movabilitv of sediment.

Hence in case where ripples have grown into dunes (as it happens in most of alluvial channels), it seems that 1:"* and

U

----~-,-"._-_.__.

[(LlY

s/

Pr) .d]1/2

is important in several aspects of flow over alluvial beds. The scale of undulations is proportional to U/(gR)^1/2. On the other hand in problems of resistance and sediment transport,

Variation ofk's with manning's coefficientIl

The variable length parameter k'8 and Manning's coefficient n are dependent on resistance to flow in alluvial channels, Einstein and Chien [7J 1955, correlated the length parameter k_s of following equation:

li ') 3

'u* -17.4

+ ';e-'

^10glO

^·35.~i--k~

⁽⁷⁾

U

--[(Lly,,/Pt). dJ1;2-

should he adequate to study the variation of k',Jd.

U TellY~/Pr)-:dfi/2-

Variation of k's Ii was sho,vn [4] that following funetional relationship can be written for variation of k's in alluvial channels:

The tenu Il*d/v can be interpreted as a para- meter proportional to the ratio of sediment size to thiC,kness of laminaI' sublayer when the bed is plane, the resistance to flow is governed by this ratio. However, when ripples and dunes are formed on the bed, the height of irregularities and their spacing governs the resistance to flow in part.

Therefore the ratio of thickness of laminaI' sublayer to bed material size loses its full significance.

Hence Il*d/u can be omitted from equation (6).

Recent investigations show that Froude number defined either as U/(gR)1/2 or

(5) (4) 23

J(

^loglo ^k

Y

_s

+

^8.5 (3)

Il _ 2.3 l ' Y

- - ; - - t i r Og10 1:1

*

^t.J\,., ^l\.⁸

Il

in which :Je is dimensionless number called Kar- man constant and k's is variable length parameter.

The advantage of equation (4) over use of equation (3) is that the numerical constant is grouped with length parameter in equation (4). Hence the analysis becomes rather simpler.

Variation of Karman constant In connection with flow over artificial roughnesses in open channels, it ,vas found by Rand [11]

1952, that :Je is a function of roughness size, geo- metry and relative roughness 11/D (where 11 is the height of roughness and D is the depth of flo,v).

From this it can be stated that variation of :Je in alluvial channel can be attributed at least partly, to the undulations of the bed.

Vanoni (7) 1H46, Einstein and Chien [3J 1955 and Tusbalü [14J 1956 showed that presence of suspended sediment affects the value of:Je. The turbulence is dampedhythe presence ofsuspended sediment and Karman constant is dependent on it being an index of turbulence.

From the point of view if dimensional analysis it has beenshown [4J that for duned bed, the relation:

Keulegan [8] 1H38, applied Nikuradse's results to open channel flow. He found that when hydraulic radius is used as the charaeteristic length, Nikuradse's formula for pipe flow can be applied

Lo open channel flow.

In the past, atlempts have been made to forn1U- late the laws of velocity distribution for alluvial channels and it has been found that the velocitv distribution in alluvial channels follows the logarithmic law.

Investigations carried out by Iwagaki [7] 1954, Einstein and Chien [3] 1955, and Tsubaki [15]

1955, have shown that the length parameter k8 for alluvial channels should depend on flow, fluid and sediment charaeteristics. When the bed is plane, the length parameter k" may be correlated with bed material size d. But, when the ripples and dunes are formed on the bed, the length parameter may be several times greater than the bed material size. Therefore, this length parameter should in general depend on flow, fluid and sediment characteristics. With this idea the following equation of velocity distribution was proposed (4):

holds good. Here 17 is the average height and 1is the average spacing of bed undulations, 1:"* is dimensionless shear

1

⁼ ^{(Ps -} ^Pr) ^d

r

and :Ji

Froude number ( - it found

(3)

mean velocity of flow D, the value of k's can he computed from the following equation :

Manning coefficient n and variable length parame-

ter

k'".

As n has got the dimensions of LI/o, the

parameter

n/

dl/^O is dimensionless. It may be correlated to k',,/ d another dimensionless parame- ter by the equation:

D 2.3 R

- nc

IOglO ~J,"

ll* •.J' e ,s

(10)

Collection of data Since reasonably sufficient field and flume data were already available to the writers, no additional data were collected. A summary of field and flume data used in present analysis, is given in table 1and 2.

Presentation and analysis of data

Complltation of parameters:

The velocity profiles ,vere first dra,vn. A plot hetween ll/ll* and 10glOY was made for each set of observations (see Fig.l). The points near the surface and near the movable houndary showed some deviation from the logarithmic law but l'est of the points appeared to lie on a line. An average li ne was drawn representing the trend. It showed that velocitv distribution in alluvial channels could be describe~lby a logarithmic law.

1. From the velocity profile the value of Karman constant and variable length parameter

k'"

can be computed as follows:

(i) [ùirmém Constants: The variation of ll/ll* in one cycle interval gives the value of 2.3/j(, fron1 which jC can be computed.

(ii) Variable Zength parame ter k',,:

a) From the plot of ll/ll* versus loglo y, the value of ll/ll~,is read at y

=

1. The value of jC is already lmown. Substituting the value of y, jC and lllll*in equation (4), k's may be computed.

b) Knowing the hydraulic mean radius Rand

(11)

Variation of Karman constant jC The coefficient jC initially appeared in the lite- ralure as a universal constant of turbulence exchange in Von-Karman's hypothesis of turbulence similitude. From Nikuradse's experiments of artificially roughened pipes, the value of jC has heen found to be 0.4, although Vanoni (quoted in 1) while reanalysing the data of Nikuradse, found jC

[0 vary from 0.32 to 0.42. The review of literature indicates that the acceptcd value of je is in the vicinity of 0.4 in case of pipes.

There is evidence to show that the value of je in rigicl houndary open channels with artificial roughnesses is not constant but varies appreciably.

It has been found by Rand [11

J

1952, and Albert- son and Sayre [1] 1961, that the value of .'iC is difl'erent for difl'erent roughness patterns and it is constant for one type of roughness pattern.

On analysing the data for alluvial channels it has heen found that the value of :JC varied appreciably and sometimes the values are even greater than unity. Figure 2 shows the variation of jC with h/Z for the data collected hy D.S.G.S., Barton- Lin, and Laursen. Furthermore in majorityof l'uns in which value of jC are greater than 0.4, the value of h/D is of the order of 0.1 or so. Fi- gure 3 shows that jC increases wifh the increase of in hlD. On the other hand Rand [l1J 1952, has found that .lfC decreases with increase in blD for artificial roughnesses in open channels. This apparent discrepancy may he explainedon the hasis of combined efl'ect of relative roughness and sediment on the Karman's constant.

The collected field and flume data were analys- cd to study the variation of :JC and k's with change in flow, fluid and sediInent characteristics.

Y/RS

1:"*= -,---' -"'-, -

. (Ys-YI) d

Since the wall effects could be easily computed for flume data, hydraulic radius with respect to hed was used instead of R in the ahove expression.

In case of canals and river s, side or bank effects were not known; hence R was used. 'Vhere the width of canal or river was large; D, the deplh of 11ow, has been used instead of R.

iL Froude number: It is defined as follows:

This equation is obtained hy Integration of equa- lion 4, over the depth of flow.

The value of k's was computed from hoth the methoc!s and the mean value of it has been used in this analysis.

2. Dimensionless shear parameter:

It is defined as follows:

4 6

20

u

U, 22 24 26 li U,

(9)

101 y

k's ( n )

(f- =

f

,dI/O

1 0"

Variations de la loi logarithmique /1 Variation of logarithmiclaw

0.1

0Canal 2 (SimonaBender)

lt

Canal 19 (Simon a¹ lo~1 ¹ 1

Bender)

Profile - ProflÏ - n° 3 ¹ Profile-Profil-n°1

8 1

1 1

/

¹

^,

u

/

1/1{

Il"

¹

6 ¹

V

I!/f

1 ¹

1

/ i

/1

ⁱ ¹

4 1

V 1

1 1 1

OJ 1 10

0" Y 1 10

8

U.E Canal(E.Y.Cl

1 ! Middle Loup River

Profile-Profi/-nO2

/ J

^{x Profile}_Profl/_noS

6 i

V

¹

^xj

¹ ¹

1

/i

¹ ^1/ ¹ ¹

4 i

i 1/

¹ ^1/

Ji

^/

2

u

uu

(4)

x

~ ⁾⁽

X

li:

•

Xx ^x" _~ x

- ^•

x _x ^x

- _-4 ^• .- - _-

^g g

_- -

^g

•

^g ~ g

X "g

"

g

"

x ~

lx ^v x

x x x

xx ^{IX ...} x x x

1& X •• -.• •

•

x x

• • ^;" _•

^li

. " " . " " _• ^•• _•

^x

Varia,tions de K en fonction dehl1 Variation of K with h/l

.06 .08 .1

tion in em'lier stages was exclusively hased on either smooth pipes or pipes roughened with uni- form sand grains. For sueh houndary charaeteristics, the fio,v was of wake interference type and the values were in the vicinitv of 0.4. If the roughness heights as comparerl to the depth of 110w can he varied (as in case of artificial roughnesses) it is expeeted that ;'je values will be dilIerent than 0.4. The reason for laI'ger values of k under these eircumstances is that the local vortices are set up by the roughnesses and these increase the level of turbulence. In such cases there is not only the turbulent exchange in its customary sense hut also an ex change of momenturn due to "Conveeiive currents" induced by these roughnesses.

In order to study the variation of ;'j(, Figure 4 is drawn which shows the variation of ;'je with

"",. Preliminarv analvsis failed to systematise the scatler on Figu;'e 4 hy~inclusion of Fi-rHlde numher . Bence Figure 4 shows the plot of ;'je against "t"

only. Il can he seen from this plot thnt with increase in """ ;'je decreases. \Vith the increase in

""" the sediment load increases whieh damps the turhulence and causes the decrease in the value of;'j(. The size of ripples and dunes is also the function of "", and thus with the increase in "or. the value of ;'je ,viII also change. These efl'eets are accounted by single parameter "",.

Tentatively a li ne is drawn on this figure to show that up to T,= 0.05 the value of k is 0.4. For "Oi<

values

<

0.05 the sediment ,viII not rnove and therefore, the values of ;'je in such case, would he ident- ical to those in case of rigicl boundary open channel.

Contrary to common belief, it can he seen that ;'je in alluvial channels can assume any value from 1 to as low as 0.15. In spite of appreciable seatIer there is a definite tendency for ;'je to deerease with

"Oi<' This figure can he used for predicting the value of ;'j(.

12 .4

8 .6 .4

.2

..b.. .2 D .06 .08 .1 h

l .04

.04 2

2

.2 .02 .6 .8

.4

.8

.2 .02 .6

'X.

.4

Variation of lenght parameters k'.<

As discussed em'lier, the length parameter varied with change in 11ow, 11uid and sediment charael- eristics. The length parame ter k'" is dependent on resistance to 1low. The size of ripples and dunes govern the resistance to 11ow. Figure 5 shows that with the increase in h/l, the values of k'jh also increase. Figure (j indieates that k'_{s /}d increases with the increase of h/l. These figures clearly show that values of k', are dependent on sizes of ripples and dunes.

a) VAlUATION OF k's/d, \VITH "Oi< & §l.

The length parameter k', is governed by size of ripples and dunes. The size of ripples and dunes depends upon "Oi< and §l. Therefore k'3/d is a func- tion of "", and §i, as discussed earlier.

The figure 7 sho,vs the plot k'8/ d versus T.~, with Froude number as thinl variable. In spite of scaU- cr, there is a definite tendency for J'-'.jd to vary with §i also. There is a deviation in case of Laur- sen's data for 0.04 mm size., it does not follow the trend. In case of fine sediment 10ad there is a zone of heavy sediment concentration near the bed.

The mixture of fine sediment and water behaves 1ike a 1luid and is no clear n(>11"''''_

cationhehveen the /5

/4 /3

6 8 10 4 .2 T

.

A .6.8 1

Variations de K en fonction de "t*

Variation of K with"t*

..::I- d

Variations deII, en fonction de h/l 1'ariation of Jvithhll Variations de K en fonction de hlD

Variation of K witl! hlD

.04 .06 .08 .1 .02

1

..-HI:

----

...

i•

--1---'-- - - -

. .. _.

^-

00.:. o'. ° --1

x x

1

. -

x ⁰

x x xX x !

x

T

¹

.1 0.2 004 0.60.81.0 2 k 4 6

-e

¹⁰ ²⁰ ⁴⁰ ⁶ ro10

8 1 x

.

x ,;Fond mobiïeMovable bed1

=t=1--

6 x Xx

01 x x x o x

8:,j:x~ ¹

0

.

^{x x x}^p<j?Q. 1'.° _{.'" +}

Rigid

~~~

⁰ ⁰ ⁰⁰ ^·~4li~~^~,"i. ^.+

Fond

~~~i

⁰⁰⁰ ^+: ^0+°

2 1 ^• ⁰i:

!

¹ 1

1.[

^{0 100}^{! •}^,^!

1

.04o

'.01 .4

.2

h

l .1 .08 .06 A

(5)

f-X---+--~-+--+--I---+----+- -!----I-+---.---/---.--,---I--- I---+---t----j--HI-···---t-··-.--+----t---- -- g--

- --- -I-II--j··-···---+----j----j--

/8

- t----l--

'---,..'

1

\

010"-.,:---L-..L....l...l.,l;;.0-~--f,k.-L.~'on-..L..---'--.L-~1!}1

I f

Variations de k' ,1d en fonction de hld Variation of [k'.1dwith hld 10~--.----r---'--'-''--'---'-'-T1---I-,-,n

/6

o·

, , x

2

1 1

i ⁰

.

¹ ⁰

<1

1

00 000

. .

^.~oo "

8 '1 ^l ^(l00 , _:~~".-

6' ^oc f -

I ^,

.

^: ^'1 ^'i^~:l ^{' 0 0}⁰8 ⁰⁰ ¹

l

1 1

1

,.

Il

¹ ¹

1 1 1

2.-' " ^,

h

TI---_-j--_~-+--+-+

x 1

0.1

f---+---+---,--+-+---.--+1 ---1----1-+-+ -.-+---+.---1-+-

1 1

.01.~0~01--.L----'-..L.-'-.:.0!;-1---'-;k"--'--...L--'-;o:;-;.,:----'---'--'--L..;

T

Variations de k'./11 en fonction de ]1/1 Variation of k'./h with hl1

.0 10 .02 .04 .06.0810 ,2 .4.6.810k, 2 4 6 810 20 40 60 80102

d

Variations de k'.ld en fonction de

'*

^/1

et de F

=

U/Y(/:'y./fI').d- Variation of [k'.1d with

'*

and F

=

U /Y(/:'y.1fi') .d

.0 n_0

d'-"_.04

10 200 10 1

d (mm)

10

Plane bed

-

^t-t-

_r--

V

^VI--'

Fond ridé V

Dune bed

--- ^vI--'

Variation of x, y andc with d mm (inF.P.S. system) ,5

.110-² .4

Fond plan .6

.3 .2 y

10 10-2 1

d(mm)

10 1

dlmm) Variations de x, II et c /9/

en fonction de d en mm (en dimensions F.P.S.) Fond plan

Plane bed 1

...

V

li

Il

Fond ridé _{/ '}

V

Dune _b~i'"

v--

^:---

- ^r-- -- ^r----..

...-

/

^...

Résultats "nature" _{, . /}V

20 _For _field _data

-

-2 ^-1 ,1 2

.4 .6

.3 .210-²

o

10 80

,8 ,7

X .5

C 40

of constant 51 can be drawn. Thus k'sld is related

with

"*

^and 51. For known values of

"*

^and 51,

k'sld can be predicted froll1 Figure 7. Usually R, S, d and J::,.Ys/Pr are known in alluvial channels;

later a ll1ethod has been suggested to predict the approxill1ate mean velocity of flow.

b) VARIATION OFk'sldWITH MANNING COEFFICIENTn:

The value of Manning's n has been cOll1puted from the following equation:

n = 1.486 R2/3 Sl/2

U

723

(6)

Figure 8 shows the plot of k'./d versus n/d^1!G.

There is a trend for increase in k',,/ d with increase inn/d1!G but the scaUer seems to be very great and no definite law can be formulated.

Prediction of mean velocity of flow U In order to known k'" one must know the mean velocity of flow U. For computing U, the slightly modified version of Liu-Hwang [lH] 1H61, formula is used. It is as follows:

where C, x and y

= f

(d). The value of x and 11 can be obtained l'rom author's graph (see Fig. H).

The c-d graph of author was 1110dified in 1H61 [5].

Knowing the values of R, S and d, mean velocity of flow can be predicted.

As a result of the present investigation, the significance of various factors influencing the flow in alluvial channels has been brought to lighL The parameters of primary importance are

"*

^and

Froude number. The Froude number of this form U

TABLE 1

Velocity distribution in vertical After studying the variation of :JC and k'" the following procedure is recommended for computing the velo city distribution of D, S, d, PB and Pt are known.

Steps:

1. Knowing the bed material size, find the values of C, x and y (Fig. H) and compute U.

2. Compute

"*

^and ^§I.

3. Kno""\ving

"*,

read the value of :JC l'rom Fi- gure 4.

4. Knowing

"*

^and ^§l, read the value of k'8 l'rom Figure 7.

5. Knowing :Je and k'8, velocity distribution can be predicted l'rom equation (4).

Conclusion

has been fûlll1d to be more adequate.

The equation of velocity distribution in alluvial channel has been found to be of following fonn:

2.3 l ' Il - ogIO-~

:JC . k'8

in which :JC and k'8 depend on flow, fluid and sediment charaeteristics.

Summal'Y of tlle field data usaI in tlle analysis

No. STlIEA~[ on CANAL cl RANGE OF D RANGE OF Q _SYMBOL

(in mm) (in ft) (in cfs)

1 U.P. Canals (India) [10ⁱJ . . . ... 0.189-0.365 2.96-13'.07 376-9703 0

2 American Canals [12J .. ... 0.029-0.805 2.61- 8.50 43-1031 0

3 Niobrara River [2J (Nebr. USA) .... 0.28 1.40- 3.60 215-916 e

4 Middle Loup River [6J (USA). .... 0.30 1.10- 2.40 393-479 D

TABLE 2

Summal'Y of tlze flume data used in tlle analysis

724 No.

1 2 3 4 5 6

INVESTIGATION

..l...auJ;scn..[4-J. . . .

P-ien{4] ;. . . . .; .

Barton & Lin. [4J. . . . .

Einstein&ClIien [4J .

U.S.G.S. [4J .

Vanoni & Brooks [18J .

cl (in mm)

0.04 & 0.10 0.18 O.lS 0.27 0.45 0.091

FLUME

105' X 3' & 1.5' deep 70' X 0.94' & 0.82 deep

65' X 4' & 2.0' deep.

40' X 1.0' 1.0 ft deep.

150 ft X 8ft & 2.0 ft deep.

40' X 10.5 in. & lOin deep.

SYMBOL

Q lB

•

+

X /).

(7)

Bibliography

[1] ALIlEHTSON and SAYHE. - Houghness spacing in rigi<!

bounclary open channel.Journal of Hydraulic Division.

Proc. ASCE, May, 1961.

[2] COLIlY and HEMIlHEE. -- Computation of total sediment discharge Niobrara Hiver, Near Cody, Nebraska. Geolo- yieal Survey Water Supply paper, 1357, Govt. Printing Office U.S.A. HJ55.

[3] EINSTEIN and CHIEN. - Effect of heavy sediment concentration near the bed on velocity and sediment distri- lmtion. Series No. 3:l, Issue No. 2, University of Cali- [ornia, 1955.

[4] GAHDE (H.J.). - Total sediment transport in alluvial channels. Ph. D. Dissertation submitted to Colorado State University, Fort Collins, U.S.A. ,Jan., 195!J.

[5] GAHDE (H.J.). - Discussion to : Discharge formula for straight alluvial channels; by LIU (H.K.) and HWANG (S. Y.). Trans. A.S.C.E., vol. 126, Part l, 1961.

[fi] HUIlIlEL (D. 'V.). - Basic sediment transport data l\Iid- die Loup river, near Dunning, Nebraska, Progress Heport No. 1 Investigation of some sedimentation characteristics of sand becl streams. U.S. Geoloyical Survey, W. H. Div. 1956.

[7] IWAGAIU (Y.). - On the la'ws of resistanee to turbulent flow in open channels. Proe. of 4th Japan National ConYl'ess of Applied Mec1wnics, 1954·.

[8] KEULEGAN. - Laws of turbulent flow in open channel.

JOUJ"1lal of Research, vol. 21.21, Dee., 19:18 (Nat. BUI·.

of Std. U.S.A.).

[9] LIU and HWANG. - Diseharge formula for straight alluvial chanl1els. Trans. A.S.C.E., 1961, Part l, vol. 126.

[lû] pHAKASH and DAYA. - Sediment Concentration and 'Kàrmàn constant in alluvial ehannels. Ir1'. Res. Ins!.

Hoorkec. 1962.

[11] HAND. Discussion Proceediny of 5th l1ydraulic Con- ference, Iowa University, 1952.

[12] SIMONS. - Theory and design of stable channels in alluvial material. Ph. D. Thcsis, State University of Colorado, Fort Coll ins.

[13] SCHLICHTING (H.). - Boundary layer theory, Chapters : Theoretical assumptions for calculations of turbulent flows; and: Turbulent flow through pipes. McGraw Hill Book Co., Inc., New York.

[14] TSUIlAlU. - On the effect of suspended sediment on flow characteristics. Japaneese Society of Civil Ellyineers;

Sept. 1956.

[15] TSUIlAKI. - On sediment transportation with sand ripples. Japaneese Society of Civil Ellyineers Journal, 1955.

[Hi] TsuBAKI and FUHUYA. - Velocity law in alluvial rivers.

Hydrodynam.ie Researeh Institute, 1(11/105110 University, vol. VII, No. IV, 1951.

[17] VANONI (V. A.). - Transportation of suspended sediment by'Vater. Trans. A.S.C.E., Vol. III, 1946.

[18] VANONI and BHOOKS. - Laboratory studies of the roughness and suspended load of alluvial streams. Cali- fOJ"1lia Institute of Teehnology Pasadena, California.

Dec., 1957.

Résumé

Distribution des vitesses dans les canaux alluvionnaires par R. J. Garde

*

et A. S. Paintal

* *

Il est nécessaire de connaître la distribution verticale des. vitesses dans les canaux alluvionnaires pour trai- ter de nombreux problèmes concernant le débit solide en suspension, la stabilité du profil...

Dans le cas d'un écoulement turbulent sur paroi rigide, les théories de longueur de mélange de Prandtl et de similitude de Karman conduisent à la loi de distribution logarithmique des vitesses (1), vérifiée par Nikuradse pour les conduites artificiellement rugueuses (équations 2 et 3). Keulegan a montré d'autre part que les lois de Nikuradse étaient transportables à des écoulements à surface libre.

Mais, pour les canaux alluvionnaires, la constante de Karman :Je et le paramètre de longueur k_s dépendent des caractéristiques de l'écoulement, du fluide et des sédiments. Les auteurs proposent l'équation (4) dérivée de l'équation

on

de Nikuradse pour les parois hydrauliquement rugueuses. Utilisant les résultats publiés par de nombreux chercheurs, ils ont déterminé les lois de variation des paramètres :J\,' et k's'

Reader in Civil Engineering, University of Hoorkee (India).

Leeturer in Civil Engineering, University of Hoorhee (Inclia).

725

(8)

Variation de la constante de Karman :Je :

Ce paramètre dépend de la rugosité des ondulations du lit et est influencé par la présence de sédiments en suspension. L'analyse dimensionnelle et l'expérience montrent que :Je ne dépend que de l'indice de mobilité du lit :

et du nombre de Fraude:

Voisine de 0,4 pour des écoulements sur fond fixe, la constante de Karman varie dans de larges proportions dans le cas d'écoulements sur fonds alluvionnaires (fig. 4). En effet, si '1". augmente, lc débit solide croit et la turbulence, caractérisée par :Je, diminue. L'influence du paramètre g n'a pas été examinée.

Variation de k's:

Le paramètre k',,1 cl ne dépend également que des paramètres '1". et g, clans le cas d'un écoulement sur fond alluvionnaire cOlnportant des rides et des dunes. Les résultats expérimentaux ont permis de définir la loi de variation de k'/cl (fig. 7).

Les auteurs ont pensé qu'il serait possible de définir une relation (équation 9) entre le paramètre k's et le coefficient de rugosité de Manning 11, à l'image de l'équation (8) correspondant au paramètre k_s d'Einstein-Chien (équation 7); mais la dispersion expérimentale ne permet pas de préciser une telle relation (voir fig. 8).

Remarque:

Les figures 4 et 7 donnent les paramètres de l'équation de distribution verticale des vitesses, connaissant 't'.

et g. Les auteurs proposent pour le calcul de la vitesse moyenne U figurant dans g la relation U= CH.xSY de Liu- Hwang (fig. 9).

726

Velocity distribution in alluvial charnels