SEDIMENT DISCHARGE 

AND STREAM POWERA PRELIMINARY ANI'~OUNCEMENT 

CIRCULAR 421 

Sediment Discharge and Stream Power 

A Preliminary Announcement 

By Ralph A. Bagnold 

Geological Survey Circular 421 

Washington, 1960 
United States Department of the Interior 

FRED A. SEATON, SECRETARY 

Geological Survey 

THOMAS B. NOLAN, DIRECFOR 

Free on application to the U.S. Geological Survey, Washington 25, D. C. 
Sediment Discharge and Stream Power 

A Preliminary Announcement 

By Ralph A. Bagnold 

In a previous paper (1956) I attempted to treat the problem of the movement of granular solids in or by fluids as broadly as possible for the sake 

of general scientific interest and incidentally for the sake of embracing as 

wide a range of factual evidence as possible. The section on sedtment transport by water streams was therefore treated merely as a special case. So no 

attempt was made to adapt the reasoning in this special case to the needs of 

practice. 

The treatment was deliberately confined to the simplest conditions of 

uniform grain size and steady flow. As regards uniformity of grain size it 

appeared, however, on the basis of the evidence given by Paper No. 17 of the 

u. s. Waterways Experiment Station (1935) that the ~esults might be extended 

tentatively to natural materials. The width of the size distribution in the 

Waterways Station experiments was narrow and happened not to include any 

appreciable proportion of grains small enough to be within the Stokes' law 

range of size. 

As regards steadiness of the flow, no attempt was made to extend the argument to unsteady flow such as occurs when the Froude number approaches 

unity and the water surface be canes very much disturbed. On the evidence 

of data from the Gilbert (1914) experiments unsteadiness of the flow seemed 

1 
to have no effect ~n the transport rate. But again, Gilbert's sands being 

unifor.m, they did not include any proportion of Stokes' law sizes. 

The general case included windblown sand which provided a good deal of 

useful evidence. This meant that the only relevant flow quantity available 

for use in the general treatment as an independent variable was the boundary stress. In the stream case, where the gravity slopes are very small, 

the boundary stress can be written as the fluid stress 1: • Further, I was 

led naturally to think in ter.ms of boundary stress ( because the threshold of grain movement, which must clearly be introduced, has for a long 

time been assumed to be definable as a critical value of the boundary stress. 

This definition seemed axiomatic. 

The key idea underlying the theory was as follows: 

When any real substance (water) impels any other real substance ( sed:iment) 

to move, all experience shows that energy must be expended by the first substance in maintaining the motion of the second against same kind of dynamic 

opposition. And power--that is, a time rate of energy expenditure--is neeessary to maintain the motion at a given time rate. Thus a stream can be 

regarded as a transporting machine; and we have the dynamic relation 

rate of work done = efficiency x available power (1) 

The work rate is a certain proportion of the available power, and both 

work rate and power are the same thing and measurable in the same units. 

Both have the nature of force times velocity with which something is pushed 

along in the direction of the force. 

Since the sediment is ~ersed, the work rate is to be measured in ter.ms 

of ~~(0 gMU where M is the dry mass of the sediment, U its velocity of 

·<T 

2 
transport, and MU is its transport rate G. 

Hence, the sediment transport rate is essentially a measure of the rate 

of work done in transporting the sediment. 

The available power of an open turbulent stream is the time rate of conversion of potential gravity energy into kinetic turbulent eddy energy (and 

thence into heat energy}. Hence the available power of an open stream flowing steadily down a constant slope s is t'gQs, or ;ogRS x u = t u per unit 

boundary area where Q is water d~scharge, R is hydraulic mean depth, a is 

energy slope' u is mean velocity' and r is mean distributed stress. 

But in the general approach neither Q nor u could be used, because in 

the case of atmospheric wind neither had any definable meaning. So I expressed the available power in tenns of f times the drag-velocity v..,..=~ 

times an experimental constant 1 A 1 • Thus 

rr-~ J Q. = e f~·~ tr p r' (2) 

-

- 'A'e r1tp 

where G is the whole transport rate over the perimeter P, and e is an efficiency factor. 

When only the bedload was considered, the efficiency e appeared as 

whence work rate ib = (3) 

where Bb depended on grain size via the drag coefficient. 

The relation was expressed in non-dimensional form by dividing both sides 

of the equation by a standard power, (o--;o)gD~-e,o. The purpose was 

to reduce the terms to the same scale in order that different sets of experimental results could be canpared. This procedure also avoids the continual 

3 
repetition of the term (o--;o)~ . The idea had previously been used by 

Shields (1936) and was further developed by Einstein (1950 ). 

Owing to the fact that the threshold of bed movement was assumed definable as a critical value of the stress ~ , it was necessary to plot experimental transport rate values against f' . The natural method of plotting 

would have been to plot transport rate against available power direct, according to equation (1), in order to plot like quantities against each 

other. 

Now several considerations, including the inconsistency of experimental 

measurements of the threshold in terms of stress, point to the probability 

that the threshold of bed movement should be defined as a threshold power 

and not as a threshold stress. If we assume a power threshold we have 

L = b = e~ (w- wt:) (4) 

.(l ' in which w is power per unit boundary area, equal to total power -p· e b 

should have a value of the same order as AbBb in the earlier notation of 

Bagnold ( 1956); e 1 b would be equal to AbBb were it not for the change in 

definition of the threshold. 

We can now plot like against like. This has the great advantage that if 

the parameter e' is constant over any part of the experimental range, its 

constancy will appear at once as a straight line if we plot on ordinary 

graph paper instead of on log-log paper. 

Further, confining attention to water streams, we can evaluate the available power W directly as ~gQs/P--both Q and s being directly measured 

quantities, instead o:f t' 1:; times some in:ferred numerical :factor. Incidentally, this avoids altogether the often difficult and unreliable meas4 
urement o:f :flow depth in order to evaluate 1.' • 

Transport o:f the bedload uses up a portion o:f the available kinetic 

power ~ • This portion is converted directly into heat in the process or 

intergranular :friction. 

The remainder W - ib being still in kinetic eddy-:form is available to 

maintain a suspended load. And i:f every turbulent eddy could play its part 

we should have 

i = ib + is = CO = fgQs/P (5) 

That is, regardless of grain size the transport rate expressed in the dynamic fonn [ o:ro fj " measured rate per unit width] would beccme equal. to 

the directly measured stream power. The efficiency of the machine would 

then be unity. 

Under what conditions might this ideal state be possible? 

The eddies may be assumed to have a random set of eddy-velocities, ranging from zero to some upper limit determined by the size of the bed grains 

and by the velocity of flow. In order :for any grain suspension to occur, 

some eddies at least must have an eddy velocity equal to or exceeding the 

fall velocity of some grains. With sand of uniform size, we therefore find 

a definite threshold state at which suspension begins. But with uni:form 

grains of finite size it appears that the ideal "equality" state of perfect 

efficiency could never be reached however great we make the power, because 

there would always be some eddies having insufficient eddy velocity. 

Plotting Gilbert 1 s data for uniform sands as i = ~ g x sediment disqcharge per unit flume width, against W = f gQ s/width, on log-log scales for 

comparison with figures 13-17 of my 1956 paper, one gets a pattern or which 
figure lA is representative. 

The plot appears to be asymptotic to the equality line. i fails to 

reach equality with ~ • On the basis of the previous method of plotting, 

the work rate ib for bedload alone would be as sketched by the broken curve. 

And the threshold of suspension would be indicated by the point at which the 

plot breaks away from this curve. (The suspension threshold is better indicated in figure 5A. ) 

But if the sedtment contains a range of smaller grain sizes, the plot 

would approach the equality line much more steeply, and the threshold of 

suspension would occur earlier and might coincide with the threshold of bed 

movement. 

It would however not be expected to reach equality with w unless the 

sediment contains a sufficient proportion of grains so small as to lie within the true Stokes' law range of size; that is, of grains whose fall through 

the water dissipates energy directly into heat without creating "wasted" 

wake turbulence. The required proportion appears to be such that the lowgrade wake turbulence created by larger grains is all utilized in suspending smaller ones. 

These ideas seem to be confirmed both by existing laboratory data and by 

a small amount of river data so far examined. Provided sufficient proportion of Stokes' law grains are present in the sediment, the sediment discharge in terms of the work rate rises abruptly until it does, in fact, 

reach equality with the stream power W • At that point there is a sharp 

discontinuity, so that for all higher values of the stream power, the sediment discharge remains equal to the stream power. Comparative plots are 

6 
100 r-----------.---------, 

FLOW DEPTHS FLOW DEPTHS ADJUSTED TO 

0.4 FT / 0 

00 

gDeo 

~00 

oo/ 

"'e "o't oO 0 0 ..<,.-\. 

;;. 0 ° 0 / f C:,oo l/ 0 

0 // 0 0 

0 0 / 00 / o Oo _/ 

'l.oo / 

oo/ 

0 0)/ 

0 v: ~ / 

/ 0 00 

/ 

0 ~ 

0/ 

i' 

I 

I 

I 

fa 

I 

5 10 50 I 5 10 

POWER INTENSITY, w :;'gQsjwidth, IN LBS/SEC3 

Figure l. --PLOTS USING DATA FOR GILBERT 1 S EXPERIMERrS, SAND D, 

O. 786 MM (UNIFORM) 

B 

50 100 

A. Sediment discharge G lbs/sec expressed as the work rate per unit 

width 1 = ~-;,<0. gG/width, lbsfsec-3, plotted against stream 

power w = ;og~/width, lbs/sec-3 showing gradual approach to 

equality; and showing correlation of scatter vi th flow depth. 

B. Same data reduced to constant flow depth of 0.4 ft by multiplying each value of i and w by 0. 4 / experimenial. flow depth. 

No corrections have been made for wall effects, which appear f'rcm 

figure lB to be negligible, since this plot shows i1o significant 

correlation between sediment discharge and either f'low depth or 

depth-to-width ratio. Residual scatter attributable to errors 

in the estimation of the true energy slope. 

7 
shown in figure 2. 

The new modified approach seems entirely consistent with the reasoning 

in my earlier paper (1956) but is more direct and practical. Moreover, the 

previous restrictions to uniform grain size are removed, and attention can 

be directed to the effects of varying the degree of heterogeneity, especially as regards the proportion of very fine sizes present. The only real innovation is the assumption that the threshold of bed movement should be regarded as a threshold of power instead of a threshold of bed stress. 

In figure 2 the Brooks-Nomicos data (tabulated in Brooks, 1957) plotted 

as work rate against stream power show but little scatter. Whereas Brooks 

(1955 and 1957) pointed out that a very large and inexplicable scatter resulted when the same data were plotted against bed stress. He rightly insisted that velocity of flow must play an important part, but was not able 

to explain why. Since the power is the product of stress and velocity, the 

part played by the velocity is now clear. 

At the other end of the scale, not far above the threshold of bed movement, interesting ideas emerge from plotting sediment discharge against power on an ordinary linear scale. A plot of data from Waterways Station (1935) 

Sand No. 1, shows that equation (4) is indeed a linear relation (see figure 

3A) except very near the threshold where one finds a "tail" which results 

from the heterogeneity of the sand. The bed movement in the tail was selective (reported as not "general movement 11 ), whereas all the points on the 

linear part of the plot correspond to movement reported as "general." 

The intercept of the straight line plot with the axis gives the effective 

8 
......--.... 

~ 

z 

w 

~ 

0 

w 

(f) 

...__. 

..., 

u 

w 

~ <I) 

co 

_J 

z 

....c 

-+-

~ 

~ <.!J 

en 

~~ II 

....... 

w 

I-

<( 

0::: 

~ 

0::: 

0 

5 

I~ A/ I~ IY 

100 I-- ------------L-------------~~----~20=~~--~A_JY'~------------~ 

10 I--

0.1 

LABORATORY DATA 

BROOKS (1955) 

Sand No.4, 13~ 

Depth 0.25 Ft 

Depth 0.53 Ft 

Sand No.2, 10~ 

NOMICOS (1956) 

Sand No.5, 152/' 

NIOBRARA RIVER DATA 

X 

+ 

• 

0 

A 

A 

A 

A 

..--1-- DEPTH 0.9- 3.0 FT..-+----+-

0.01~------------~L-------------~--------------~--------------~ 

0.1 I 10 100 1000 

POWER INTENSITY, w = _,ogQsjwidth, IN LBS /5EC3 (WATER) 

Figure 2.--Com.parison of discharge of fine sediment in laboratory 

experiments and in rivers. Sediment work rate reaches equal! ty 

w1 th stream power. 

9 
threshold power wt . 

Plotting the same sediment discharge data against the power expressed as 

"t t; instead o:f as t'gQS /width (figure 3B) one obtains another linear 

plot on a different scale because the constant C = u/~ is omitted, but 

{" 

the detailed pattern of the scatter is nearly identical. This shows that 

such scatter cannot be due to errors in either the measurement of discharge 

or the measurement of flow depth. The slope being fixed and unadjusted, the 

scatter seems to be due to the assumption that the energy slope was equivalent to the slope of the flume~. The same data are plotted in figure 4 on 

log-log scale for comparison of the relative scatter. It should be noticed 

that threshold conditions are better seen from the linear plottings. 

When one makes the same comparison in the case of the finer sand No. 8 

( 0. 20 llDil), the scatter in a plot o:f 't' ~ is hopelessly large, but i:f those 

same data are plotted on the basis of /' gQs/width, it is reduced in a quite 

startling way. (See figures 5A and B~) This finding suggests that large 

dune features on the bed and great difficulty in measuring the mean flow 

depth prevent a good measurement of 1: . 

By comparing these linear plots for sand No. 1 (diameter, 0.586 mm) and 

for sand No. 9 (4.0 mm), both sizes being such that there was no appreciable 

form drag, we see that the threshold powers are approxtmately in the ratio 

n3/2, which is consistent with the theory; that is, that threshold stress 

is proportional to D, and threshold power varies as n3/2. Hence, it appears correct to compare experimental results relating to different scales 

~ The bed surface having been initially made parallel to the flume floor, 

appreciable changes in the bed slope would not occur within the limited experimental flow period until general grain movement took place. This accounts for the relative lack of scatter in the tail. 

10 
l'tl 

u 

w 

.20 

.15 

~ lf) 

co 

_J 

.10 

2 

,. 

....c 

-+-

:-2 

z 

(.!} 

en 

~ .05 

\lb 

II 

·r-' 

0 

A I c /0 

0/ 

joo 

0 I/ 

o I 0 

0/ 

0 

0 I 

~g 8 I 

00 I 0 

0 0 

oo I oo 

oo /oo 

0 00 

0 

I oo8o 0 

0 0 

cooo 

"o0f "-Og.fy 

0 .02 .04 .06 .08 .10 

't'\[f_ = wjc, IN LBS 

to 

I 

B 0 

II 

0 I 

v 

/' 0 0 

0 I Vo ol 

I 

0 

0 )t 

Ia 0 0 

I 'b /0 0 

0 0 ,o 0 

a} 0 

0 

I o o o I 0 

0 

2o! 0 0 

0 0 %01 

t cf\f' oo 

'------0~0 :0 I 

0 .2 .4 .6 .5 1.0 1.2 1.4 1.6 

w =;:>g Qsjwidth, IN LBS/SEC 3 

Figure 3. --PLOTS OF U. S. WATERWAYS EXPERIMENT STATION DATA FOR SAND NO. 1, 

0.586 MM 

A. Experimental bedload work rate i plotted against stream power w = 

;ogQs/width on a linear scale to show the linear relation i oC w-wt 

(Threshold power u>t substituted for conventional threshold tractive 

stress Tt .) 

0 

B. Same data as A above, plotted as i against the power function r:r'fo = ~~ the tractive stress 'l' being obtained fran measurement of slope and flow 

depth, whereas the power in A was obtained fran measurement of slope and 

discharge. The velocity coefficient C is given by the ratio of the 

gradients in the tvo plots, c. " ~ 

11 
A B 

oo c;(J 

0 0 

0.1 

....c 

-+-

-o 

00 

go 

0 

o/oo 

ooOo 

~oooo 

0 oO 

0 0 

0 y, 

dloo 

'iJPaoo 

o Do 

{ 

\!)-

Qlo 0 

ooo 

0 

~ 00 

00 

0 

~ 

~~ 

0 ~ 

oo 

Oo 

00 

0 

ooo8o 

0 0 

Oo oo 

0 

0.01 OQ:) 

00 0(\:1 

_oo 

....... 0 0 

0 0 

0 0 

0 0 0 0 

0 

0.001 0 0 

0.1 0.5 I 5 0.01 

W = f <] Q s j width 

Figure· 4. --Values in figure 3A (lett) and 3B (right) replotted on loglog scales to compare relative scatter. 

12 
1.0 0 

USWES Sand No. 8 USWES Sand No.8 

0.205mm. 

1.0 

! 

.9 I .9 

/ 

.8 

I 

I .a 

l .7 / .1 

I 

:·o: 0 .r.;·. ! .6 .6 

II 

I /v 0 

.5 .5 

.4 .4 (/ 

/'T 

o:::::f-- Suspens•on 

Thre~hold ~ t--

./ 

.3 .3 

/o 0 

.2 ,...v .Z. 

0 

0 

I 

--"-

I h~ 

/ 0 0 

y 0/ 

0 ooc o"' yf i '-- oo OoLo 

~2. A .6 .8 1.0 1.2. 1.4 1.6 1.8 2..0 2..2. 2..4 2..6 2..6 3.0 3.2 3A 3.6 3.8 ?oz. .04 .06 .08 .10 .IZ. .14 .If. .18 .z.p .z.z. .24 .26 .U .30 .32. .34 .36 -~ 

w =_,.o9Qsjwidth T{J 

Figure 5.--PIDrS OF U. S. WATERWAYS EXPERIMENT STATION DATA FOR 

SAND NO. 8, FINE GRAINED, O. 205 MM 

A. Experimental discharges ot finer grains, plotted as 1n figure 

3A, 1 against w = jCJ8Qs/width, showing the departure fran 

the linear bedload relation at a definite suspension threshold. Note that this threshold had not been reached in the 

case of the coarser sediment of figure 3· 

B. Same data as figure 5A plotted against the power function 

showing the gross scatter introduced by difficulties in measuring the true flow depth. 

13 
of mean grain size and mean density by dividing both i and~ by the appropriate unit power function.fl( 

q ~ : ( <r -r 1 'J v i <r; r- ':1 o f 

So that i = v-f.? '3 ~ becomes 

q- p 

o--(! , GP {lr-~}J 0 /<rf" , 0 = J;" 'of Bagnold (1956) (T I (l I ' I' . % 

and w becomes = c e 3

/:2. whert 

A number of queries arise about the effects of channel shape and scale. 

The most direct relation is that between the whole sedtment discharge in 

terms of I = i x width, and the whole stream power .1J. = W x width. But 

the absolute values of I and .1:2 will then vary with the size of the stream, 

those for rivers being enormously greater than those for flumes. And if one 

compares values for the same river at different stages, one is really comparing conditions in effectively different rivers, the cross sections being 

different. 

The scale effect is reduced by dealing in quanti ties per unit width, but 

it is not removed, because the unit cross sections still vary in flow depth. 

If at low stage the sediment were to move as bedload only, as in the case 

of Waterways Station sand No. 1, there appears from these experimental data 

to be no independent correlation between transport rate and flow depth. Flow 

depth seems to be involved only as determining the discharge Q. In this 

"low-stage," or bedload, region no evidence seems to exist that scatter is 

improved by making a systematic allowance for wall effect as some function 

of the width-depth ratio of a flume. Width-to-depth ratios have generally 

been too small. 

14 
It is very clear, however, when plotting Gilbert's data, most of which 

cover the "high-stage" or suspension region, that the sediment transport 

rate figures have a pronounced correlation with flow depth. Gilbert was 

the only investigator who extended his experimental conditions to include 

large width-to-depth ratios by varying width as well as depth. Unfortunately the available water discharge did not permit him to combine large 

flow depth with large width. So large depth had to be associated with narrow width. 

The correlation between sediment discharge and flow depth has been known 

for a long time, but it has always been attributed to wall effect. The carrelation is so large, however, that in correcting for it as a wall effect 

one has to assume the smooth walls to exert the same drag as the grain bed; 

one has to make the full "hydraulic mean depth allowance b/ (b + 2d). " This 

has always seemed to me to be irrational, especially when the bed is rippled and must have a very large form drag. 

It seems more reasonable to assume the drag effect of smooth walls to be 

negligible compared with the drag of a very rough bed, and to assume Gilbert's wide scatter of transport rate values to be due to the variation in 

cross-sectional shape as the depth was increased. 

Taking grain suspension to be a volume effect, as distinct from bedload 

movement which is a boundary effect, one may reduce Gilbert's data--both 

transport rate and power--from a "per unit width" basis to a nper unit 

volume 11 or unit depth bas is by dividing both i and W by the flow depth of 

the run concerned. The correlation with depth then disappears without any 

recourse to an arbitrary correction for wall effect. In figure lB the data 

15 
of figure lA have been reduced to the common basis of a 0.4-ft depth. The 

residual scatter is no greater than can be attributed to errors in slope 

estimation. 

It seems significant that the Brooks-Nomicos results, which cover the 

same suspension region up to the approach to "equality," have no appreciable scatter when plotted as i against w = ~ gQ;·s/width, whereas Gilbert's 

unreduced data have a wide scatter. The apparent reason is that Nomicos 

ran his experiments at constant depth--that is, constant cross section--

and Brooks' variation of depth was comparatively small. 

This suggests that consistent results from sediment transport experiments 

require constant cross section, just as do consistent results from water flow 

experiments without sediment. This aspect seems to have been overlooked 

previously, presumably because it was difficult to adjust the energy slopes 

of old-fashioned fixed flumes, experiments could not be made at constant 

depth. 

16 
Sediment Discharge as a Function of Stream Power Measured by Qs Compared 

With That Measured by Tractive Stress 

The stream power w fgQs/width can be expressed as 

= fgRs X U = 't' Ll 

And by introducing a conduction coefficient C of the Chezy type so that 

= u/~ ;o 

c't'{f ;C 

c 

we have w 

For clear water flowing in the same channel, provided the depth remains 

reasonably constant, C should remain constant as u is varied ~Aquve law). 

When figures 3A and 3B are compared for flow with bedload sediment transport only, it is evident that C does remain constant, for both plots give 

straight lines, and C is given by the ratio of the two constant gradients. 

The transport of bedload, therefore, does not appear to invalidate the 

square law for turbulent flow. 

If, however, we plot Nom.icos 1 sediment discharge data (sand No. 5, Brooks 

1957), obtained at constant flow depth and therefore most likely to be consistent, against 1:fj., as in :figure 3B, the results are highly anomalous. 

(See figure 6.) Assuming the effective values of a in 'l' = f>gRs were 

correctly estimated; then, as Brooks has pointed out, the sediment discharge 

may for a given value of 1: have two values, the second being as much as 

10 times the first. 

The reason for this apparent anomaly is now cl.ear. It fits in with much 

other evidence. Water flow containing varying concentrations of fine solids 

17 
to 

/ 

// 

c:> I/ 

// 

v 0 / Ci) ( Ci) 

Ci) C!) 

1.0 / 

.., I 

1 ~~ .s 

I 

I 

Ci) 

I 

j 

I 

a/ 

I 0.1 I 

I 

• I 

.OS" 

I/ 

I 

0.1 o.z 0.4 0.5" o. 7 1.0 z.o 

1:.j T 

/) 

Figure 6. --Data :for Nomicos 1 Sand No. 5, plotted as 

work rate, 1, as a tunction o:f tangential stress, 

t . These data were obtained at a nearly constant depth o:f :flow. 

18 
in suspension does not obey the square law. The velocity coefficient C is 

not at all constant. 

This may be seen as follows : If C were constant, plots of' sediment discharges on l.og-l.og acal.ea against w = f gQa/width and again:;t w jc = 1:fj, 

should have identical gradients, as in figure 4. The gradients of' the 

Namicos plot in figure 2 have therefore been transferred to figure 6 (the 

two broken lines). They have been positioned with reference to the four 

lowest of the plotted points; and the agreement here is very good. But as 

the real. power ia increased, the apparent power aa given by 'r fJ becanes 

relatively smaller, because a real velocity increase has not been taken into account. 

u. The values of' the velocity coefficient C = ~ may be calculated directly 

fi from the velocity and tractive force data. These values are shown in figure 

7 plotted first against the ratio i / w and also against the Froude number 

u/ {id. The similarity between the two curves is remarkable. It seems important to discover how far the variation in C depends on Froude number, 

sediment size and canposition, and other possible factors. It may be significant that the apparent peak in figure 7B occurs at F = 0.8 which would 

be very close to unity f'or a Froude number basis or water surface velocity 

instead of mean velocity. 

Canparing figure 6 with figure 5B for o. 2 mm sand of narrow size distribution, it now appears that the latter is not so hopelessly inconsistent as 

it seemed at first. The points curl backwards in the same way. The Froude 

number of' the top point was 0.64. 

On the other hand, there is no definite indication of any such effect 

19 
'' ~/o-\ 

I 0 

A 

7 

7 

v 

15 

14 

1.3 

12. 

I / 0 

1/ 

/ 

[7 

8 /0 

v v 

-0 o-o 

-( 

7 

6 .I .z .3 .4 .5 .6 .7 . 8 .9 1.0 

Vw - ....... - 1/0 1"0, I 

15 

B l 

If 14 

I 

I l 

II 

13 

IZ 

01 

0 10 I 

1/ 9 

8 

I kd' _...ol 

7 

b 0 0.1 o.z 0.3 0.4 0.5_ 0., 0.7 0.8 0.9 1.0 

F=~ 

Figure 7. --Data from Nomicos 1 experiments, sand No. 5· 

Upper graph, A, shows velocity coefficient as a function of the ratio of work rate to stream power. Lower 

graph, B, shows same coefficient as a function of 

Froud.e number. 

20 
fran Gilbert 1 s data for uniform sands. And it may well be that the effect 

is in same way due to the presence of a proportion of fine grains within the 

Stokes 1 law range of size. In support of this notion it may be pointed out 

that in the Brooks-Vanoni-Namicos experiments four different sands were used, 

each having approximately the same mean size, but differing widely in the 

proportion of fine sand present. All four sands exhibited the effect of a 

progressive increase in the velocity coefficient c. The relative magnitude 

of the increase is in one for one correlation with the proportion of fine 

sand. 

21 
REFERENCES 

Bagnold, R. A., 1956, The flow of' cohesionless grains in fluids: {London) 

Philos. Trans., Royal Soc., v. 249, P• 235--297· 

Brooks, N. H., 1955, Mechanics of streams with movable·beds of' fine 

sands: Amer. Soc. Civll Engineers, Proc., v. 81, Separate 668. 

1957, Discussion of' Brooks, 1955: Amer. Soc. Civil Engi- ____ n_e_e_r_s-,~Proc., v. 83, Separate 1230. 

Einstein, H. A., 1950, The bedload function for sediment transportation 

in open channel flows: u. s. Dept. Agri. Tech. Bull. 1026, 71 p. 

Gilbert, G. K., 1914, Transportation of' debris by running water: u. s. 

Geol. Survey Prof. Paper 86. 

Namicos, G. N., 1956, Effects of' sediment load on the velocity field 

and friction factor of turbulent flow in an open channel, in 

Vanoni and Brooks, 1957. -

Shields, A., 1936, Anwendung der Aehnlichkeitsmechanik und der 

Turbulenzf'orschung auf' die Geschiebebewegung: Mitt. Prouss. 

Ver-Anstal.t., Berlin, n. 26. u. 

u. s. Waterways Experiment Station, 1935, Studies of river bed materials 

and their movement with special reference to the lower Mississippi 

River: Paper No. 17, Vicksburg, Miss., 161 p. 

Vanoni, V. A., and Brooks, N. H., 1957, laboratory studies of' the roughness and suspended load of' alluvial streams: California Inst. Technology, Sedimentation Laboratory Rept. E-68. 

22 
LIST OF SYMBOLS 

A Experimental constant R Hydraulic mean depth 

B Factor dependent on grain size s Slope of water surface 

b Refers to bedload t Subscript signif.ying threshold 

c Velocity coefficient of movement 

D Diameter of grain u Velocity of transport 

d Depth u. Mean velocity of water 

e An efficiency factor lA.-il' Drag velocity 

F Froude number for whole channel y Unit stress 

g Gravitational acceleration a- Grain density 

G Transport rate ~ Fluid density 

I Work rate for whole width r Tangential boundary stress 

{sediment) r: Mean distributed tangential 

ib Work rate per unit width boundary stress 

M Dry mass of sediment w Power intensity per unit width 

p Perimeter !l Whole stream power for full 

Q Water discharge width 

23 

GPO 1176.41