SAND TRANSPORTING
STUDY OF T
AN EXPERENTAL
CAPACITY OF FLOWING WATER ON SANDY BED AND
L,fSect A
i - COMPOSITION OF TEe SAND
by
Shiou-dean Chyn
S.B., National Tsinghua University, China
1932
Submitted in Partial Fulfillment of the Requirements
for the degree of
MASTER OF SCIENCE
from the
Massaohusetts Institute of Teehnology
1935
Signature
of
... .
Author....
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Department of Civil Engineering, September 1, 1935
Signature of Professor in Charge of Research.............r
Signature of Chairman of Departmental
Committee on Graduate Students......
j
........
Cambridge, Lassachusetts
Sept. 1, l935
Prof. G. ,. Swett,
Secretary of the Faculty,
-Massachusetts Institute of Technology,
Cambridge, M assachusetts.
Dear Sir:
I submit herewith my thesis entitled "An Experimental Study of the Sand Transporting Capacity of Flowing
water on Sandy Bed and the Composition of the Sand" in
partial fulfilment of the requirement for the degree of
Master of Science in Civil Engineering.
Respectfully yours
Shiou-dean Chyn
20880O
Aknowledgement
The author wantsto express the obligation he owed
to Prof. K.C. Reynolds for his valuable hints and directions,
which made possible the author's work on this subject; also
to Prof. G.Gilboy and Mr. D.W. Taylor for their help and
instructions.
i
Contents
Synopsis
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Nomenclature
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Work
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Considerations
Procedure
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Apparatus
Used
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Procedure of Making the Tests
ii
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Previous Works on Theory of Transportation
Theoretical
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Page
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Method Used for Computation and Plotting
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Observation, Source of Error, and Probable accuracy.
20 -23
Tabulation
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of Results
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bibliography
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Discussion of Results
Conclusions
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33
Table of Plottings
Serial
Analysis curve of the mixtures,
o.
A-2
Profiles,
A-3 to A-6
Capacity curves,
A-? to A-9
Sand
A-10
onstants curves,
Analysis curves of transported material,
Sand
omposition after
un,
A-17
B-1 to B-7
Appending curves,
Total
A-11 to i-16
o. 26
Sketch of the Apparatus, plate,
2acingc p. 12
ii
Nomenclature and Units Used
b,--channel
width
D --Water deoth
d --grain diameter
dg--mean grain diameter
F --modified sand uniformity modulus
G --total bed load transported
g --bed load per unit width of channel
h --gage reading of head
Tw--water temperature
q --rate of discharge per unit width of channel
s --surface slope
Subscripts: m --mean value,
c --critical value,
b --measured at bottom.
Units for g,
1 lb./ft./hr. =
0.248 gr./cm./e.
1
Synopsis
Both in laboratory work and in field work regarding river
hydraulics, the silting and erosion of a movable channel-bed is
a serious and complicate'problem.
A movable bed becomes
stable only when an equilibrium between three factors, namely,
the rate of flow, the surface grade, and the bed material, is
reached.
condition,
This is the so-called non-silting and non-eroding
Much work has been done, both in laboratory experi-
ments and field observation, on the theory of sand transportation,
to seek for the natural law governing this equilibrium conditimn,
but the solving of the whole problem is still waiting for accumulative knowledge.
It was found that for a given sand or other bed material,
it is not only its
average fineness, but also its mechanical
composition, which governs its behavior when it is transported
as "bed load".
By mechanical composition, it means the varia-
tion of the grain diameter.
The main object of the present thesis is to determine to
what extent, and in which direction the sand composition will
affect its total amount transported along the bed by flowing
water.
The problem of sorting, or the change of relative per-
centage of the individual elements during transporting, is also
analyzed.
Three synthetic sand mixtures were made first,
hey were
the same average grain size but different composition.
C
With
each mixture as bed material, about eleven test runs were made
to determine the relationship between rate of discharge, surface
2
grade, and the capacity of sand transportation.
The sand
caught at the lower end of the test flume during each run was
analyzed by sieve analysis.
The result;of first part was plotted and the equations fitting the curves were then formulated.
The conclusions thus
reached are: -1) The sand composition enters the capacity formula in
a complex way; the coefficient, the exponent of slope factor, and
the competent constant are all function-of it.
2) The critical tractive power, or the competent constant, is higher for mixtures having more uniformly sized grains.
The result of the second part is that the mean grain diameter and the sand modulus of the material transported are higher
than those of the original material.
They decrease as the
tractive power increases until limiting values are approached.
Previous Work on the Theory of Sand Transportation
In
acDougall's research report "An Experimental Determina-
tion of Bed Sediment Transportation", he has compiled the results
by both European and American authorities on this subject.
Also
a complete bibliography of the subject was published this year
by the Bureau of Reclamation (2).
In this paper then will
be only mentioned several previous works which were studied by
the author and not included in
acDougall-'s list.
A. Regarding transported capacity: -1) Lechalas. (3)
He derived in 1871 a formula making
use of Dubuat's experiments and data observed on French rivers,
G-
m(Vb- 0.06)
3
where G is the bed load transported, m is an empirical constant,
Vb is the bottom velocity of the water, and 0.06 is a competent
Ahen water depth is constant, this
constant in metric units.
formula can be approximately represented by --
G = K(V2 - k)
where Vm is the mean velocity of the flowing water.
Gilbert in his re-
2) Gilbert's complete formula.(3)
port of experiments made during 1907-1909, after considering
three factors separately, gave a complete formula including all
the factors,
G = b (s -
-(f)p
)n (Q
where s is the water surface slope, Q is the rate of discharge,
and f is the mean fineness, or the reciprocal of the mean grain
diameter of the bed material.
competent constants.
The constants
d
/t, and
are
Each competent constant is related to the
other two factors, yet the definite function of such relation is
still unknown.
The mean value of the three exponents resulted
from his experiments are
n = 1.59,
o = 1.02,
p = 0.58
respectively.
Gilbert also have made experiments on mixed grades and
found that,
i) addition of finer debris increases the capacity
for coarser bed load; and
ii) capacity of transportation varies as slope in the
same power for uniform material but with lower competent value.
3) C.H. MacDougall, M.I.T. (1)
During 1932-1933,
4
MacDougall made experiments on three sand mixtures, the results
were, -i) a capacity formula similar to Schoklitsch's
g = kl
in which s
and
n- (sq - Soqo)
o are the critical slope and rate of discharge
respectively;
ii) the three constants in the above formula, viz., kl,
m, and soqo, can be expressed in terms of the mean grain diameter
and the sand modulus of the material transported.
All the
constants vary approximately with the reciprocal of the sand
modulus.
4) Straub's rational formula. (4)
For the bed sedi-
ment transportation, Prof. Straub of the University of Minesota
has suggested from theoretical consideration the following
formula,
g = 8'4 n'l.; e((e
_qS)
when slope is constant, and
g = q!n.s07(s7
S'
7
The constant "n" is the same
when the discharge is constant.
as the roughness factor in Manning's formula for mean velocity.
He also showed by comparison that these rational formulas are
consistent with Gilbert's experiments and
ennedy's erosion
formula.
5) Vicksburg U.S. Waterway Experiment Station.(8 )
Based
on an elaborated series of experiments made with river sands,
the Vicksburg Experiment Station has derived a formula for the
bed load capacity -g =
D - s0Do
>
where n is the roughness factor, m is a constant
hich varies only
with the type of sand within narrow limits, namely 1.5 to 1.3.
kl is a sand property factor.
B. Regarding critical conditions: -1) Airy's sixth power theory, (5)
W=
V b
where W is the weight of a single sand particle transported, Vcb
is the critical bottom velocity.
2) Ho's empirical formula for critical bottom velocity,(6)
Vcb = 0.204 d0.35
where Vcb is the critical velocity measured right at the bottom
in meters per second and
is the grain diameter in millimeters.
3) Vicksburg Station's tractive force formula, (8)
T = 29
where T
(, )
is the critical tractive force in grams per square meter,
dg is the mean grain diameter in millimeters, M the sand modulus,
?rand ?p are densities of sand and water respectively.
This formula is a revision of Kramer's with the data from
the Station added.
4) Leighly.(7)
Prof. Leighly has developed a theory
stating that the imagined water prism acting on a unit area of
bed is not "1lsquare foot x depth" but a distorted prism with
bed as the lower base, the line of maximum -velocityas the upper
base, and the transversal and longitudinal planes perpendicular
to velocity contours as boundary surfaces.
Thus the tractive
force is not simply a function of slope and depth, but also a
function of form ratio (the ratio between depth and width) and
6
the location of maximum velocity.
Theoretical Consideration
The tractive power
Quite comparable to the idea of tractive
force, MacDougall has proposed the term 'tractive power' of a
channel.
From a theoretical point of view, he has
determined
that the tractive power of a stream flow is a product of slope and
rate of discharge.
In other
words, with a given channel bed,
the sediment transported per unit width of channel bed is proportional to the product of slope and discharge, with a competent
value to be deducted.
g
Expressed in an equation, it will be
=
k (sq - soqo)
where the value of k and soq
o
depend upon the bed material.
The competent constant soq
o
may not be exactly the critical
condition in case of a mixture, because the sorting effect occurs
s or q converges
to small value.
Bed material composed of a mixture
It is a definite known
fact that a certain size grain needs a certain tractive force to
set it in motion, the magnitude of the critical tractive force
being proportional to the 1.5 power of the grain diameter.
But
in a mixture, the different grains do not act according to the
above law.
In the presence of fine particles, the coarser
grains may be set in motion at a much lower stage, because the
finer particles pave the bed to reduce the friction, also because
the coaser particles, when the finer particles around them become
eroded away, project higher and produce more turbulence, thus
increasing the tractive force locally.
It is therefore nece-
ssary to introduce another factor beside the mean grain diameter
7
to evaluate the mutual action between finer and coaser grains.
This factor would make it possible to reduce any mixture to an
equivalent sigle grained bed material so far as the transported
capacity is concerned.
Void ratio is a property of bed material, which is closely
It defined the roughness and
related to the bed load motion.
The former factor tends to an dor
the percentage of interstices.
the particles but the later tends to help to dislodge them.
The two factors counteract each other to make the effect of void
ratio on sand movement uncertain.
This is why Gilbert did not :-,"
success on this feature.
D, and
, these two constants were introduced by Krey and
Kramer and later used by M.I.T. and Vicksburg Experiment Station.
With the factor M, it is thought possible to reduce any mixture
with known mechanical composition to a single grained material
with mean diameter Dg.
Expressed mathmatically, the two quantities defined by
Kramer are as follows;
D4P
p< D~
g
where D is the grain diameter as measured by the size of the sieve
opennings, p is the percentage by weight of the grains larger
than that particular D.
M
=
pD50%
S
Pro%.
· In
Dap
its physical meaning,
is the ratio of Dg of the finer
8
The value D
half over that of the coaser half.
measures the
average fineness of the material, while M measures the relative
diversity of fineness relative to D g .
We cannot make any substantial change in the mechanical
composition of a bed material without altering the values Dg and
M of it.
Therefore these quantities are a unique function of
the material.
The only defect is t hat if we make the change
carefully without changing the 'Dg's of both the upper 50
the lower 50%, they will not be affected.
and
This means, we still
have the freedom to have any kind of sand distribution within
within the finer 50% and coaser 50% when Dg and M are both fixed.
In order to have a more representative index to the size variation,
a modified modulus is suggested:
TF
koth
Fs
Thus the F value takes care of extreme diameters and limits the
freedom of change of distribution to within each 25%.
.it is reasonable to use the
and 4 power.
In addition,
A slight change of
Dg in any 50% changes the value of F approximately one half as
much, and slight change of Dg in any 255% dhanges F one fourth as
much.
It is also not laborious to determine F from a mechanical
analysis curve.
Complete
movement of a mixture.
Schaffernak has deduced from
his experiments the following conclusion: "According to size of
grain, each mixture corresponds to a certain type of curve of
material movement which consists of a flat portion and a steeper
portion of curve common to all types of mixtures.
The point of
flexure corresponds to the critical bottom velocity which is just
9
able to set in motion the entire mixture."
Obviously there are two limiting "critical conditions" for a
given mixture.
The upper limiting condition is the critical for
the coaser grains, or rather the coarsest, above which all particles can be moved with ease.
The lower limiting condition is the
critical for the finest grains, below which no particle can be moved.
Between these limits, there must be a critical bottom ve-
locity which is
ust able to move the coarsest grains in the pre-
sence of the finer elements, and therefore is able to move the
mixture as a whole.
Procedure of Experimental Work
The problem
To study how the uniformity of a sand mixture
affects the capacity of bed-load transportation.
By limiting the problem to bed-load, that part of sand transported in the mode of suspension is entirely neglected.
This
assumption is permissible when the mixture used does not contain
very fine grains.
iethod of attack.
Because the present objectris to relate th e
sand transporting capacity to those properties of sand apart from
its fineness, it will be advantageous to eliminate Dg first.
qb
accomplish this, three artificial mixtures were made from three
sieved sands, each sand having a different mean grain diameter
from the others.
natural sand, --
The three kinds of sand were obtained from
the Plum Island sand --
by successive sieving.
By mixing them in proper proportions, the three mixtures were
made with approximately equal mean grain diameters but wide varying uniformity, or sand modulus.
10
With each one of the artificial mixtures as bed material, 2
sets of test runs were made to determine the sand transporting capacity under different hydraulic condition.
The hydraulic quan-
tities directly measured were the slope and the rate of discharge.
Using these as variables, the two sets were so controlled that the
slope was held constant and the discharge varied in the first set
and vice versa in the second set.
The reason for using q instead of depth as an independent
variable is that it can be precisely measured by a sharp-crested
trapazoidal weir.
On the other hand the quantity d is very
hard to measure when the xottom becomes rough and the water surface is undulating.
Throughout the experiment, all test runs are made with a
given slope and discharge and then adjust the sand feeding to kee p
the bottom at the upper end at a definite elevation.
This me-
thod was claimed by MacDougall to be preferable to Gilbert's
method which is to fix the discharge and rate of sand feeding
first and then measure the surface slope after both the bottom
and surface of water reached an equilibrium condition.
However,
one of the test runs, No. 12, was made with the second method
with comparable result, which indicates the possibility of the
equilibrium.
It was thought that this equilibrium takes a long
time and waste more sand, therefore in all runs except 12, the
procedure is to start-with a uniform bed and parallel surface.
The two sets of readings, if properly taken, serve to determine the capacity formula for each mixture.
The initial or
critical condition can be computed from each set of data and
then checked with each other.
Some visual readings of the con-
11
dition for start of motion were also taken, but due to the difficulty of the criterion, and the diversity between upper and lower
limits of initial condition, the visual readings are only used as
a check.
In order to get detailed information of the sand composition,
each mixture was analyzed with closely spaced series of standard
sieves before the run and the results from three samples were
averaged.
After each run, the sand caught at the lower end of
the channel was dried in air and analyzed with mechanicalshaked
sieves.
Use of other methods than the sieve analysis may get more
accurate result as to the mechanical composition, but due to the
character of the sands used, which are comparatively coarse, the
sieve method is more convienent.
Range of the experiments.
1:300 to 1:900.
The slope varies in the range from
The upper limit was fixed by the channel outlet
construction, because the tilting gate, when swung to its lowest
position, can only allow a slope slightly flatter than 1:300.
The lower limit was fixed by the accuracy of tubes and gages.
The lower limit of rate of discharge q was determined by the
minimum discharge required to start the general motion of the bed
load.
The upper limit for q, as well as that for s, was fixed by
the possibility of control.
Above the upper limits, the excessive
high dunes occuring on the bed would make the slope readings not
representative.
The rapid -change in the roughness of the bed
also would make the hydraulic condition deviate further from a
uniform flow.
liz
0
Q
K
Ut
"I
b
t',
ta
.3
12
Those grains coarser than 4.699 mm. in diameter were sieved
out because they would require higher tractive force to start
general motion.
As shown by Schoklitsch's formula and data com-
piled by Kramer, the sand particle of such a diameter requires a
slope of 0.00216 at depth of 5 cm. to set.in motion.
This con-
dition is very near to the upper physical limit of this equipmert.
The finer elements were not sieved out because they formed a rela tively small percent or the total volume.
The water temperature was also taken during each run.
enables one to compute the Reynolds number of each run.
This
It was
intended that all the runs are made with turbulent but not shooting flow, to ensure the type of movement of bed material corresponding to the second and third types described below, which is
the condition prevailing in nature.
Types of sand movement
The sand movement on bed can be divided
into the following 5 types:
1) Only isolated grains moved occasionally;
2) General motion occurs, but no riffles developed;
3) Movement with riffle formation;
4) Riffles flattened by strong current; and
5) Unti-dunes formed, the riffles travel in upstream
direction.
Apparatus Used
The main equipment is a wooden flume 32 ft. long, 24" wide
and 15" deep, with an inlet tank and an outlet tank.
Of the 24"
width, only the central portion, about 16" wide, leads to the
measuring weir.
Therefore only the discharge and bed load
flowing down that part is to be measured.
13
The flume is supported by a jack screw at center,and by a roller at
its upper end under the inlet tank, so that it is possible to
tilt the flume at any slope.
The inlet flow is controlled by a valve which discharges
water for a given valve opening at a constant rate from an elevated
constant level supply tank.
capacity
The pump supplying water has a
of 2 c.f.s.
The outlet flow is measured by a trapazoidal weir in the
tank under the lower end of the flume.
The head over weir is
read by a point gage.
The sand bed is molded by a board which is suspended from
a rolling frame resting on rails mounted on top edges of the flume.
The rails are fixed at 2 ft. intervals by adjustable screws so
that the slope of the top of the rails can be accurately adjusted.
A point gage mounted on rollers is used for measuring profiles.
At the lower end of the flume, an overflow tilting gate
regulated by worm and gears is used as a controlling section.
The surface slope is measured by six piezometer tubes at 1.5meters intervals along the center line of the channel.
The sand feeding is done by an automatic drum feeding
apparatus suggested by Gilbert.
The sand transported down
the lower end is caught in an enlarged section by a basket
15" by 24", which is suspended from a balance arm.
of the arm, a spring balance and a wireleading
At the end
to automatic
recording machine are connected to it.
The sand recorder consists of a metal plate moving by
synchronous motor at the rate of 1/5" per minute, and a metal
loop wire with sparking point which gives a continuous record
of the sand weight.
It was thought that the total energy lost in the channel
may be measured by the difference between end levels, so that
two automatic floats were added to the piezometer tubes No.1
and No.6 and two adjustable screws were attached on the bottom
to measure the surface and bed elevations at the ends.
The rate of sand feeding can be adjusted both by the rheostat on shunt motor and the feeding slit under the sand box.
In the first ten runs the feeding was too high, even using the
lowest rate, when the tractive power is low.
to stop the feeding at times.
It was necessary
This was improved later by add-
ing resistence to the motor.
In making the sand mixtures, the sands were sieved by
screens made with commercial screen cloth, but all the following analyses were made with Tyler's standard sieves.
machine used is a Rotap made by Tyler
The shaking
o.
The specific gravity of the mixtures was determined by a
pychnometer.
Procedure of Making the Tests
The channel bottom was tilted first to a medium slope of
1:600 and kept thus through all the runs.
After the flume Was
properly tilted, wedges are put under the flume to make it supported on several wooden frames and relieted from strain.
Zero point of measurin gage.
The ero point
of the measuring
trapezoidal weir was determined by means of a hook gage and
level bubble.
Three sets of readings at three points on the
weir edge were taken and averaged.
15
The flume was then filled at the bottom
Making the sand bed.
vwithabout a 2" thickness of the sand mixture to be used.
The
sand was loosely dumped but later compacted slightly and covered
overnight with water 10" deep.
Leveling the top rails.
For each particular slope used, the
top rails were adjusted as follows:
Raise the tailgate to its
Is
maximum height, fill the leaks.
flow occurs.
Open inlet valve until over-
Wait for half an hour till the surface is quiet.
Keep the adjusting screws to an amount required by the computed
grade.
After that, check the elevation at all points again.
All the elevations were read by the travelling point gage.
The maximum erreo allowed in adjustingrails
Preparing the bed.
was 0.04 cm.
The bed was then scraped to the desired
height and slope by a board, the bottom of which was adjusted
by means of,level bubble.
After perfectly smoothed, two read-
ings of the bed elevation at the ends were taken.
After each
run, the bed material on top layer was turned over and mixed with
the lower layer; after each three runs, the sand in each of the
-one third stratch of the channel was thoroughly mixed again.
The material at the upper end was replaced each run with new
sand, because the sorting effect was prominent there.
Starting the run.
\
The tailgate was then raised to maximum
top level, the inlet valve was opened gradually until the desired
discharge was indicated by the measuring gage.
lowered until the pezometer
ed slope.
hour.
The gate was
readings conformed with the desir-
This condition was maintained at least for half an
During the riffle formation, the slope and depth varied
16
quite an amount, it was therefore necessary to wait until this
period was over.
Nevertheless, in some of the runs, the change
took place very slowly so that it was completed in the middle of
the runs.
As soon as the equilibrium condition was reached, which was
shown by a uniform flow condition and a permanent slope under the
fixed rate of discharge and sand feeding, the sand recording machine
was switched on, and piezometer readings and gage readings were
read.
All the readings were repeated at about 5-minutes inter-
vals.
Fixing the sand feeding rate.
The sand feeding rate was fixed
at first by comparison with that of previous runs.
Because the
discharge or slope varied gradually from the previous one, it
was possible to adjudge the feeding rate fairly closely.
Then
by watching continuously the bed elevation , the sand feeding was
adjusted to keep a constant bed at the upper end of the flume.
Profile reading.
About 30 minutes after the starting, when
the sand curve obtained was enough to show a straight line
or
steady tendency, the tailgate was then swung up and the run ended.
The bed elevation after the run was taken by a round plate gage
of 1 inch diameter at 13 section, 3 points were measured at each
section.
Getting sand sample.
The sand caught in the weighing basket
was then thoroughly mixed and representative sample, about 600 gr.
when dried, was taken.
17
Method Used for Computing and Plotting
Averaging the slope.
The following least squares formula was
used for computing surface slope from a set of piezometer reading,
s
=
1
(h
-h,
+ (ha- h)3
52+ 32
1
4
(h
- h)
520 (h6 - h,) 5 + (h - h) 3 -+(h,- h)]
where the 'h' s are piezometer readings in centimeter, with
the subscripts indicating the number of tubes.
The constant
150 is the length of each interval between successive tubes.
Afterwards the average slope of the run was computed by taking I<>
arithmetical mean of the several slope readings.
Averaging the depth.
The bed elevations at the 13 sections
were plotted as a profile.
straight ,
By adjusting the position of a
the average depth of water at the end of each run
could be read directily from the profile.
The mean depths
at the beginning and the end of each run were then averaged to
obtain the mean depth of the whole run.
Averaging the discharge.
The several measuring gage read-
ings were first averaged with weighting factors according to
the length of interval in which they were taken.
Using the
average head over,weir, the discharge could be directly read
from a calibrating chart.
Rate of sand discharge.
To get the real weight of sand trans-
ported, the apparent weight recorded was divided by the specific
l
gravity of the particu
ar sand mixture used.
18
Because the measured discharge is only that
Roughness "n".
part from the central portion of 16i'"wide, the mean hydraulic
radius R is neither the mean depth of water nor the wet area
divided by 2d- b.
It lies somewhere between these values.
Accordingly, the computed n varies between two extremities, but
it would be closer to that one computed from the mean depth.
Plotting the capacity curves.
The data was first plotted
with the straight line formula,* (Curves B3, B4) but it was
found later that the straight line formula does not apply to
this particular set of data.
$
The formula referredon
..
namely,
g-
.
p.6 has two special forms,
.
kls(q - k 2 /s) = kl(q - qo)s,when s is constant, r~s
,.s, '
g = klq( - k2/q) = kl(s - so)q,when q is constant.-lk-
Both te
forms represent a straight line.
applies, the value of qos mustidentical
If this formula
with soq, but the
values obtained from the plottings are:
MixtureNo. 1,
sqo= 0.18, Soq
Mixture No. 2,
sqo = 0.213,
soq = o.156;
Mixture No. 3,
sqo = 0.170,
soq = 0.237.
0.226;
The final formula adopted for plotting is one similar to
MacDougall's.
Its general form is
g = k
where m, kl
( s q
k2 )
(I)
and k 2 are constants depending on the nature of I
sand, or more exactly, on the uniformity and density of sand.
Because in each constant slope group, only five test runs were
made, it was intended to corelate the data with MacDougall's
19
and his result of using unit power for q was assumed first.
The six groups of data were plotted separately on a sheet
of logarithmic paper, using the two special forms of formula (I),
namely,
k1 sm(q - qo ) ,
gs = klsm( - k2 /sm)
gs
or,
gg
=I
klk2
=
(ksm)
q
(II)
; and
k 1 c (sm- k 2 /q) = k1q (m
-
o),
g + klk 2 = (klq)sm .
or,
(III)
Formula (IID was plotted with q varying, and (III) plotted with
s varying.
The conditions which fix the three constants in
(I) are therefore:
i) the slope of q curve must be unity;
ii) klk 2 in the two.cases must be equal; and
iii) g
+ klk 2
gq + klk2 when both s and q are equal.
This method of plotting is shown on plate A-7a that the two vertical lines meet the two curves on the same horizontal.
After the two logarithmic curves had been determined, the
constants in formula (I) were determined at once.
Thus,
k 1 = (kls )/ m = (klq)/q,and
k2 = (klk2) /k ·
The resulted formulawas
then plotted on linear plotting paper.
(Plates A-7 to A-9)
In making the constant slope or constant discharge plotting
an adjustment was made when the actual slope or discharge deviates
too much from the normal slope or the normal discharge of the
group.
that
The correction was computed approximately by assuming
g -- kl ( sq - soqo') was valid.
The last step was to plot k,, k2, and m against
F' of
20
the three mixtures.
Plotting the sorting curves.
The result from the mechanical
analysis of sand transported was plotted in constant slope and
The fine
constant discharge groups. (Plates A-11 to A-16)
curve represents the original composition before the run, to be
used as a comparison basis.
From careful study of the result,
a modification was made on composition curves of mixtures No. 1
Of the origin-
and No. 3 to eliminate the effect of suspension.
al percentages,
of those passing through 48-mesh sieve, and
1/8 of those passing through 100-mesh sieve were assumed to be
brought into suspension in every run, hence were not caught in
the weighing basket.
This assumption only made the upper part
of the curves round-ended, but did not alter much the general
shape.
From the curves, the values dg and M were determined and
plotted against slope and discharge respectively.
Observation, Source of error, and Probable Accuracy
Controlling factors; their steadiness.
Of the three factor ,
the slope is the most difficult one to handle.
iot only it is
very sensitive to discharge change or bed change, but also it is
hard to decide
hether to raise or to lower the tailgate when
maintaining the slope.
0.5 mm. which means 4
The accuracy of piezometers reads to
of 1:600 slope and higher percentage for
gentle slopes, but the dune height at tha bottom may bary sometimes from -1.5 cm. to +1.5 cm.
This of course causes a local
change of depth and hence the grade line.
may occur in a short stretch.
Even a reverse slope
21
In most of the test runs, the surface slope consisted of a
steeper upper half and a gentler lower half.
from the profile plottings.
This may be seen
The causes of such deviation are:
i) The channel flow was not strictly a uniform one, its
upper part was somewhat like an approach, hence excessive slope
was required to accelerate the flow.
ii) The enlerged
ection had a backwater effect.
Second to the slope was the bed material factor.
Making
series of tests of constant slope without disturbing the bed
would be a quicker way, but much care would have to be taken to
keep the composition of bed material as constant as possible.
Therefore , the bed was remolded each time and allowed to develop
riffles again.
large extent.
In this way the sorting effect was avoided to a
At the upper end, or in case of very weak flow,
the bed seems to be paved with coarse particles which remain
untouched.
At very high stage, the discharge measuring gage does not
read accurately, because the level in the measuring tank vebrates rhythmically, partly due to the lack of approach, partly due
to the unsteady flow over the weir crest.
Transversal condition.
The bed velocity along center line
was always higher than that along side walls, therefore the
sediments carries along the middle third was heavier and moved
quicker.
When snd dunes were formed, they usually assumes a
.?,
:-,
tongue shape with its point directed downstream.
Another transverse difference was
that the finer grains along side wall
A were usually eroded off and brought
22
to a stripe B, about 4" apart from A.
The accumulation of fine
particles at B sometimes produced sub-dunes of small height and
closer spacing.
Problem of suspension.
the sand
sieve.
In making two of the three mixtures,
used contained 5.5% of grains passing through 100-mesh
This caused part of the bed material to go into suspension,
which could not be caught in the sand basket and therefore formed
6 lost.
It did not, however, interfered with the bed load capa-
city assuming that the small percentage of suspension only changed
the flow velocity in smaller order than the bed load did.
Sorting effect
occured even at highest stage.
There was no
such chance that the bed material moved as a whole unit.
A se-
tion from one dune to next dube formed a complete cycle of sorting.
The grains paved layer over layer according to their sizes with
the coarsest settling first.
On the top of dunes, there was a
layer of finest particles, this made a smooth bed for coarser
ones to roll upon.
The coarser particles moved with higher veb -
city but severally, and the finer sand grains moved slower but
collectively.
The more the uniformly sized fine elements are
contained in the mixture, the easier the formation of sand dunes.
Because most of the graines would move and settle simultaneously
at a small change of bed velocity.
The paving action of coarser
grains seemed to counteract the dune formation.
Accuracy of the measuring apparatus.
The measuring gage for
outflow can measures to 0.05 mm. of head except at highest stage,
which approximately indicates 0.012 liter per second of discharp .
The piezometers can be read to 0.5 mm., but sometimes the paraliax
effect comes in, which may cause an error of
0.5mm.
In most of
23
runs, the level in the piezometer tubes represented a fair
average of the oscillating water surface in flune.
It usually
fluctuated between limits 2mm. apart periodically.
The sand weighing devices worked pretty dependable except
that the basket often touched the tank wall due to eccentric
loading, it was corrected as-soon as perceived.
The accuracy of the spring balance canbe kept within t
of a
pound of sand weight in water, or 0.4 pound-weight of sand in air,
which is about 7% of the smallest reading.
When water is clear, the method.of using bottom screws for
measuring depth directly gives elevation to nearest millimeter,
but it seemsimpossible
to get the whole profile reading during
the run, the isolated readings can only serve as a check.
rWI2
24
Tabulation
of Results
I. Mechanical Analysis of the sand mixtures.
Mixture No. 1
Sieve No.
Size of
opening
4
4.699
2.362
1.168
0.833
0.701
0.589
8
14
20
24
28
32
48
100
pan
0.495
0.295
0.147
Accumulative percentages
Average
2nd S.
1st S.
0.1
1.e
14.15
42.8
63.8
76.0
87.0
97.5
99.8
100.0
0.0
0.9
13.75
43.1
63.4
73.1
81.7
92.0
98.1
100.0
0.1
1.4
13.95
42.95
63.1
74.5
84.35
94.75
98.95
100.0
Mixture No. 2
Sieve No.
Size of
opening
14
20
24
28
48
pan
1.168
0.833
0.701
0.589
0.295
_..._
Ac cumulative percentage
Average
2nd S.
1st S.
4.46
51.72
86.6
95.62
99.89
100.0
3.62
47.89
82.15
93.45
99.35
100.0
4.04
49.81
84.35
94.54
99.62
100.0
Mixture No. 3
Sieve No.
4
8
Size of
openting
Accumulative percentage
Average
2nd S.
1st S.
0.1
3.73
14
24.55
20
24
26.08
29.05
28
32
45.75
50.99
48
82.80
100
pan
96.20
100.0
0.04
4.0
29.7
31.9
34.9
51.25
67.40
87.40
97.10
100.0
0.07
3.88
27.13
28.99
31.98
48.50
63.70
85.10
96.65
100.0
r..
25
II.Test Runs.
Table
{
Ii
I
c.f.sf.
Dm
cm.
Run No.
Assum.S
Actu. S.
cm3s/ cm
1
2
3
4
5
1:600
1:600
1:600
1:600
1:600
0.00152
0.00157
0.00161
0.00157
0.00200
202
224
252
308
590
0.217
0.241
0.275
0.332
0.636
5.38
5.64
6.09
6.98
10.07
6
7
9
10
11
12
13
1:500
1:400
1:300
1:700
1:800
1:500
1:550
0.00185
0.00247
0.00300
0.00146
0.00120
0.00200
0.00181
246
247
243
258
255
252
251
0.265
0.267
0.261
0.277
0.274
0.271
0.270
5.67
5.21
4.68
5.90
6.32
5.49
5.51
14
15
16
17
18
1:400
1:500
1:700
1:800
1:900
0.00238
0.00206
0.00150
0.00133
0.00110
317
307
309
312
310
0.341
0.330
0.333
0.336
0.334
6.29
6.34
7.00
7.27
7.35
19
20
21
22
23
1:600
1:600
1:600
1:600
1:600
0.00159
0.00163
0.00169
0.00168
0.00178
206
255
310
367
490
0.221
0.274
0.335
0.395
0.528
5.06
5,83
6.80
7.80
8.91
24
25
26
27
28
1:400
1:500
1:700
1:800
1:900
0.00247
0.00191
0.00145
0.00129
0.00111
313
314
312
309
307
0.3370.338
0.335
0.332
0.331
6.08
6.51
6.90
7.13
7.24
29
30
32
33
34
1:600
1:600
1:600
1:600
1:600
0.00160
0.00163
0.00161
0.00163
0.00166
255
208
313
367
503
0.274
0.224
0.337
0.394
0.542
5.78
5.04
6.65
7.52
9.09
-1
ir,
26
I
Table II
.- Run No.
vm
VM
vc
Tw
m/s
tt/s
m/s
1
0.375
1.23
0.725
22.5
3
4
0.413
0.443
1.36
1.46
0.777
0.827
20.2
18.9
2
5
6
7
0.398
0.586
1.31
1.93
0.742
0.992
21.2
18.5
0.434
0.476
1.43
1.56
0.744
0.714
18.4
21.0
10
0.437
1.44
0.760
22.2
13
0.456
9
11
12
0.518
0.404
0.459
1.70
0.677
22.2
R
C
21200
22800
25000
29800
56250
23400
25130
25320
26950
1.33
1.51
0.787
0.733
21.8
21.8
26330
26100
1.50
0.734
22.4
26400
14
15
0.504
0.484
1.68
1.59
0.784
0.788
22.4
24.8
33900
34370
18
0.422
1.38
0.848
23.0
32970
19
20
21
22
23
0.407
0.436
0.456
0.470
0.550
1.34
1.43
1.48
1.54
1.81
0.703
0.755
0.815
0.873
0.933
24.7
27.0
26.0
26.6
26.5
22780
29660
36150
38750
56500
24
0.515
1.69
0.772
23.6
33760
26
0.452
1.48
0.822
27.0
36400
28
0.424
0.841
25 5
34590
16
17
25
27
29
30
32
33
34
0.441
0.429
0.482
0.433
1.44
1.41
1.58
1.42
1.39
0.832
0.843
0.799
0.835
23.9
25.0
23.6
26.9
0.441
1.45
0.752
24.5
0.553
1.82
0.942
24.8
0.412
0.470
0.488
1.35
1.54
1.60
0.702
0.807
0.857
25.0
24.0
24.6
33560
34760
33900
35900
28080
23200
34070
40500
56140
27
Table III
mi~~~o
Run
g
'o.
g
I
gr/cm/*;' adjusted
#/ft/hr
mq
1
2
3
4
5
6.76
9.18
11.72
15.82
49.20v
1.68
2.31
2.91
3.93
12.20
2.05
2.59
3.09
4.31
9.82
0.0000173
182
190
182
275
0.00349
408
480
562
1620
6
7
9
10
11
17.40
34.50,
44.00
15.15
10.78
4.31
8.55
10.90
3.76
2.68
4.45
8.74
11.30
3.58
2.34
0.0000241
392
545
163
116
0.00592
970
1324
421
295
12
29.00
7.20
275
692
13
22.60
5.61
232
582
14
15
16
17
18
26.80
24.30
13.75.
12.52
9.23
6.64
6.03
3.41
3.11
2.29
19
20
21
6.7613.33
18.70^
1.68
3.31
4.63
22
8
0.00475
421
317
287
241
1.505
1.293
0. 980
0.895
0.745
1.84
3.43
4.54
335
321
353
0.688
0.819
1.093
6.97
6.88
351
1.288
23
36.00.
8.92
8. 2
369
1.810
24
25
26
27
28
54.40
39.40
18.8013.95
7.37
13.50
9.76
4.67
3.46
1.83
0.000224
156
106
90
73
0.0701
490
331
278
224
29
17.10
4.24
122
311
2.10
125
260
5.46
7.19
9.30
123
125
128
385
459
645
30
U
s
s
32
33
34
Z8./o
..-I
22.00
29.00
37.50
* The exponent m
.47
1.69
0.807
1.40
for
No. 1-13
14-23
24-34
28
Table
Run No.
ixture
No.
3
3
3
3
3
Moderate
5
3
6
7
9
i.
*Description
of dunes
1
2
4
IV
Not formed
Neglegible
n
use Dm
Material transp.
dg
0.0149
0.0146
0.0154
0.0948
0.0948
0.366
0.336
0.0152
0.0800
0 .908
0.335
Excessively
high
0.0165
0.0820
0.323
3
Neglegible
0.0146
0.948
0.339
3
i"
0.0146
0.898
0.340
t
0.307
3
Moderate
0.0137
0.864
0.301
10
11
12
13
3
3
3
3_
Neglegible
Not formed
Moderate
0.0134
0.0136
0.0141
0.0135
0.918
0.950
-0.934
0.310
0.354
-0.346
14
15
16
17
18
2
2
2
2
2
High dunes
Moderate
High
High
Moderate
0.0151
0.0149
0.0149
0.0148
0.0138
0.858
0.842
-0.890
0.850
0.675
0.644
-0.698
0.743
19
20
21
22
23
2
2
2
2
2
Moderate
High
Exoessive
Excessive
"
0.0134
0.0139
0.0151
0.0159
0.0153
0.882
0.846
0.846
0.838
0.856
0.777
0.740
0.733
0.738
0.712
24
1
High
0.0149
0.850
0.512
25
26
27
28
1
1
1
1
Moderate
"
"
"
0.0147
0.0141
0.0136
0.878
0.878
0.910
0.963
0.514
0.557
0.521
0.541
0.0142
29
1
ivModerate
0.0136
0.884
0.542
30
1
0.0134
0.900
0.630
32
1
"
0.0L40
0.818
0.542
33
34
1
1
0.0147
0.0149
0.850
0.806
0.557
0.580
High
t
Excessive
i.Voderate,
High,
'
maximum dune height lessthan
* Neglegible,
"
t
"
TV
TV
,T
tt
tf
over
1
'
Excessi'lely high,
O.Scm.
1.0cm.
it
2.0cm.
2.Ocm.
29
Discussion of
esults
Under fixed hydraulic conditions, using sand mixture of
same mean grain diameter, the sand transporting capacity may
be expected to be simply a function of the sand composition,
thus,
()
=
f (s q, dg)
(F)
If the experiments were so arranged that the factor F was chosed
as the only variable, the result came out would determine
directly.
(')
The nature of the experiment, however, prohibits
such doing, because it would require a huge amount of sand to
pave the channel bed each time.
Also since the result of the
present experiments shows clearly that the sand transpotation
cannot be expressed in any simple form of explicit function of
'F',* the method adopted seems to be superior to the method of
varying F.
*(
This relation may be seen from the plottings B3 and
,
that for same s and q, the capacity g for the mixtures is in the
following order according to the magnitude,
Mixture No. 1, No. 2, No.3
in one case, and in the order
Mixture No. 1, No. 3, No. 2
in the other, also that below a certain point, where the straight
lines cross each other, the above orders are reversed.
The plotting of the capacity curve for mixture No. 3 has
been done with great difficulty, because the data from the two
separate groups did not consistent in any way.
error was not yet found definitely.
The systematic
One of the reasons may be
30
mentioned is the change made on the lower end sill or the channel bed.
Its top was flushed with the bed during the first 7
runs, but was depressed later to about one centimeter lower
than the bed elevation.
The final curve was plotted by consi-
dering the consistent points only.
The result compares not favorably with that of Gilbert's
experiments in one phase, i.e., the exponent "o"tin the formula
g = b 2 (q -)
is in general too low.
(Gilbert's values vary from 0.79 to 1.24,
while the results from this experiment are 0.55, 0.66, and 0.95.)
On the other hand, the three exponents for s are nearly equal
in spite of the variation of the sand uniformity. (Plate B2)
This agree s with Gilbert's result on mixtures.
It is stated in Vicksburg Station's report that when the
tracive force is low, the mean grain diameter of the transpoted
material is higher than that of the original, though the usual
J33
idea believes that it should be lower.
( p. 32, ref. 8)
Another matter of interesting is that when capacity g is
plotted against the mean velocity Vm, the curve resulted consists of two parts: the first part assymptotes a straight line
but with increasing slope; the second part starts with a very
steep slope and then bends downward.
Quite accidentally, the
velocity at the point of intersection of these two parts is
just the velocity at which the reffles changes from moderate
or neglegible height to high dunes.
The same curve for mix-
ture No.2 however, does not show the diffirence before and
after that point.
It may be interpreted as this:
The sorting
effect becomes prominent after the riffles have been completely
developed.
31
When the completely sorted material advances in the form
dunes, the capacity g is materially changed and does
ofand
not follow the same law as before.
The higher the uniformity
of the sand, the less would be the effect of sorting on capacity.
From table II, it may be concluded that all the runs were
made with streaming turbulent flow, as all the'Vm's were less
than V critical, and all the Reynolds numbers were higher than
2000.
Conclusions
First part, --
capacity of sand transportation.
The
capacity of transportation of a channel flow can be expressed
1sa2 JU1L
A
-4.1
_
I u m.LU
g =
k(smq
- k2)
With the mean grain diameter of the sand held fast, the constants k1 , k 2 , and m are functions of the uniformity modulus
F (or M) of the sand.
i) k
decreases with increasing F, in an order higher
hnn the sr i he
ii) k
lin? redsti n
increases with increasing F, in an order higher
than the straight line relation.
iii) m decreases with increasing F, in an order higher
than the straight line relation.
When the curve is extrapolated
to uniform material, that is, F - 1, the exponent m will be less
than unity-.
Second part, hydraulic condition and sortirgeffect.
i) Both dg and F of the transported material, when
the tractive force is low, are higher than those of the original
1
32
material.
As the tractive force
or tractive power is increased,
dg and F decrease gradually until a point is reached, after which
they remain almost constant.
This point corresponds to the
state of excessive dune formation.
ii) As the tractive power increases, the percentage of
coarser elements decreases and that of the finer elements increases.
iii) The more the uniformly sized fine elements the
mixture contains, the easier the developement of sand dunes.
word'fine' has a relative meaning to the tractive power.)
r
I
I
I
I
I
I
The
Bibliography
1.
An Experimental Determination of ed Sediment Transportation, by C.H. MacDougall.
Unpublished Research Report, M.I.T., 1933.
2.
Bibliography on Transportation of Solids by
U.S. Bureau of Reclamation, Denver. 1935.
3.
The Transportation of Debris by Running Water, Gilbert.
U.S.G.S. Professional Paper 86. 1914.
4.
Transaction of Geophysical Union,
National Research Council, 1933.
5.
Formula for the power of a stream to move substances, by
Airy.
Min. I.C.E.,.
lowing Water.
Vol. 82, 1882.
6.
Bed Velocity and erosion, by C.T. Ho,
Unpublished thesis, Univ. of Iowa.
7.
Toward a Theory of the Morphologic Significance of Turbulence, by J.B. Leighly.
Univ. of California, 1932.
8.
Studies of River Bed Materials and 'Their Movement.
Paper 17, U.S. Waterway
-xperimentStation, 1935.
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