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Hans Albert Einstein: Innovation and Compromise in Formulating Sediment Transport by Rivers

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This paper is written to mark the hundredth anniversary of the birth of Hans Albert Einstein (1904-1973). It casts his career as that of the archetypal researcher protagonist determined to master intellectually the way water flows and conveys alluvial sediment in rivers. In that effort, Einstein personified the mix of success and frustration experienced by many researchers who have attempted to formulate the complicated behavior of alluvial rivers in terms of mechanically based equations. His formulation of the relationship between rates of bed-sediment transport (especially bedload transport) and water flow comprised an innovative departure from the largely empirical approach that prevailed at the time. He introduced into that relationship the emerging fluid-mechanic concepts of turbulence and boundary layers, and concepts of probability theory. Inevitably the numerous complexities attending sediment transport mire formulation and prompt his use of several approximating compromises in order to make estimating bed-sediment transport practicable. His formulation nonetheless is a milestone in river engineering.
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Hans Albert Einstein: Innovation and Compromise
in Formulating Sediment Transport by Rivers
Robert Ettema
1
and Cornelia F. Mutel
2
Abstract: This paper is written to mark the hundredth anniversary of the birth of Hans Albert Einstein 19041973. It casts his career
as that of the archetypal researcher protagonist determined to master intellectually the way water flows and conveys alluvial sediment in
rivers. In that effort, Einstein personified the mix of success and frustration experienced by many researchers who have attempted to
formulate the complicated behavior of alluvial rivers in terms of mechanically based equations. His formulation of the relationship
between rates of bed-sediment transport especially bedload transportand water flow comprised an innovative departure from the largely
empirical approach that prevailed at the time. He introduced into that relationship the emerging fluid-mechanic concepts of turbulence and
boundary layers, and concepts of probability theory. Inevitably the numerous complexities attending sediment transport mire formulation
and prompt his use of several approximating compromises in order to make estimating bed-sediment transport practicable. His formula-
tion nonetheless is a milestone in river engineering.
DOI: 10.1061/ASCE0733-94292004130:6477
CE Database subject headings: Sediment transport; Rivers; Alluvial streams; Fluid mechanics; Turbulence; Boundary layer.
Introduction
Hans Albert Einstein, born in May 1904, might have remained
one of countless civil engineers whose work, although locally
important, had little impact on the world as a whole. However, his
trenchant independence of spirit and famous father, Albert Ein-
stein, launched him into a productive career as a researcher and
educator fascinated with the mechanics of bed-sediment transport
and water flow in alluvial rivers. By virtue of the times in which
he lived 19041973, the trans-Atlantic span of his life, and his
name, Hans Albert Einstein’s hereinafter called Einstein career
forms a convenient course along which to view the advance of
alluvial-river mechanics as an engineering science. This paper
follows part of his career, viewing his efforts to understand and
formulate two central issues in alluvial-river behavior: the rela-
tionship between bed-sediment transport and water flow, and that
between flow depth and flow rate.
Although Einstein lived most of his youth with his mother
Mileva, who had separated from Albert when Einstein was 10
years old, his career was strongly marked by his fathers influ-
ence. Family correspondence reveals that, though Albert first dis-
suaded his son from entering civil engineering, he later fostered
and partly directed that career. Until 1927, Einstein and his
mother resided in Zurich, where he attended school and eventu-
ally earned an undergraduate degree in civil engineering from the
Swiss Federal Institute of Technology ETH. Albert then encour-
aged his son to come to Germany where Albert was a professor
at the University of Berlin. Albert facilitated this move by help-
ing him locate a job at the steel construction firm of August
Klonne, in Dortmund, where Einstein worked as a structural en-
gineer focusing on bridge construction. However, by 1931 Albert
was becoming increasingly apprehensive about the growing Nazi
power in Germany. Understanding well the threat posed to Jews,
and concerned about his son’s safety, Albert encouraged a return
to Switzerland. Seven years later, Albert would again feel the
pressure to ensure his son’s safety, and would facilitate a second
move this time to the United States and job change. Thus Ein-
stein’s career was also marked by historic movements; each shift
in its course was induced by a change in the political climate
linked with such movements.
This paper discusses how despite the politically encouraged
moves, or perhaps because of them, Einstein emerged as a leading
expert in alluvial-river mechanics, his expertise being sought
around the world. The paper does so with scant inclusion of equa-
tions. Practically every major textbook on alluvial-river mechan-
ics and sediment transport e.g., the books by Einstein’s doctoral
students Graf and Chien Graf 1971; Chien and Wan 1999兲兴
present the main equations comprising Einstein’s formulations.
For a broad technical assessment of Einstein’s contributions to
alluvial-river mechanics, the writers defer to the useful synopsis
by Shen 1975, another of his doctoral students. The Proceedings
of a symposium, to honor Einstein on the occasion of his retire-
ment, lists his publications and the graduate students with whom
he worked Shen 1972.
A theme running through this paper is innovation and compro-
mise. Though springing innovatively from emerging concepts of
turbulent flow and probability theory, concepts that were becom-
ing well established in engineering only during the early decades
of the twentieth century, Einstein’s formulation of sediment trans-
port becomes beleaguered by confounding physical details and
the natural variability of sediment and flow conditions in rivers.
1
IIHR-Hydroscience and Engineering, Dept. of Civil and
Environmental Engineering, College of Engineering, The Univ. of Iowa,
Iowa City, IA. E-mail: robert-ettema@uiowa.edu
2
IIHR-Hydroscience and Engineering, Dept. of Civil and
Environmental Engineering, College of Engineering, The Univ. of Iowa,
Iowa City, IA. E-mail: connie-mutel@uiowa.edu
Note. Discussion open until November 1, 2004. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and pos-
sible publication on January 8, 2004; approved on February 6, 2004. This
paper is part of the Journal of Hydraulic Engineering, Vol. 130, No. 6,
June 1, 2004. ©ASCE, ISSN 0733-9429/2004/6-477487/$18.00.
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 477
Inevitably, simplifying assumptions, empiricism, and other judi-
cious compromises are needed to prop formulation so that it is of
practical engineering use. It is a theme common to many efforts in
formulating sediment and water movement in alluvial rivers.
Beginnings: Meyer-Peters Flume
Professor Eugene Meyer-Peter of the Swiss Federal Institute of
Technology ETHin Zurich needed to know how much sediment
moved with water flowing along the Alpine Rhine, especially the
amount of coarser sediment, gravels and sands, that moved along
the rivers bed. This need was to bring the young Einstein from
Germany, where he then worked, back to Switzerland, the country
of his birth.
In the late 1920s, the Swiss federal government and the local
cantonal government of St. Gallen, responding to concerns about
an alarming increase in the frequency with which the river
flooded, had contracted Meyer-Peter to recommend an effective
modification to the Alpine Rhine over a 20-km reach extending
from the Alps to the head of Lake Constance. The river, which
wends through the Swiss Alps to Lake Constance Fig. 1, was
aggrading, and likely would break out of its leveed banks and
disastrously flood its valley. Political considerations gave Meyer-
Peters work urgency, as the Alpine Rhine above Lake Constance
formed an international border between Switzerland, Austria, and
Liechtenstein. The key question weighing on Meyer-Peters mind
was by how much to narrow the channel so that it deepened
sufficiently to convey its loads of water and sediment.
Though many efforts at channel modification had been at-
tempted elsewhere in Europe prior to 1930, they had relied on
little more than rules of thumb aided by cut-and-fill adjustments
to arrive eventually at suitably sized, nominally stable channels.
The few formulas purporting to relate flow rate and depth for
water in alluvial channels were so empirically tied to local chan-
nel conditions that they could not reliably help Meyer-Peter.
Also, the few formulas for estimating bed-sediment transport
were sketchy and, at best, loosely related bed-sediment transport
to water discharge or depth through a particular reach of river.
Schoklitsch 1930, a leading European authority on river-
engineering at the time, wrote in his highly regarded textbook Der
Wasserbau that ‘at the present stage of research, a ‘calculation’
of sediment load was out of the question.’ Meyer-Peter realized
he faced a complicated task, and in 1927 had set about designing
and supervising the construction of ETH’s impressive hydraulics
laboratory with which to undertake it.
A French engineer, Du Boys 1879, had done some simple
flume experiments and proposed the first mechanistic formula for
estimating bed sediment transport as bedload, the portion of bed
sediment transport whereby bed particles move on or near the
bed. A difficulty with his formula, though, was its basis on a
misconceived notion of bed-particle movement. Du Boys had as-
sumed that bed sediment moves as a series of superimposed
shearing layers, and had arrived at a formula relating rate of bed-
load transport as per unit width of channel to a critical flow con-
dition beyond which flow mobilized bed sediment, and an excess
of average shear stress exerted on the bed. Subsequent flume ex-
periments showed the sliding layer view of bed-sediment move-
ment to be fallacious e.g., Schoklitsch 1914; Gilbert 1914. Nev-
ertheless, the notion of a critical shear stress or flow rate, flow
depth associated with bed-sediment transport was conceptually
appealing. Consequently formulas similar to Du Boys’ were con-
sidered best suited for estimating not only bedload transport but
also the total rate of bed-sediment transport; prior to the 1930s, it
was moot whether engineers actually distinguished between the
two transport terms.
Perhaps the most advanced at sizing alluvial channels were the
British, who had sought an improved design method for irrigation
canals dug through sandy terrain in parts of the Indian subconti-
nent and Egypt. The method, termed the Regime Method e.g.,
Lacey 1929, relied almost entirely on empirical relationships to
characterize channels under long-term equilibrium or ‘regime.’
The Regime Method was still in development, and its applicabil-
ity to the Alpine Rhine with its gravel bed was uncertain.
The problems with the Alpine Rhine clearly showed that the
few existing formulas were far from being dependable or useful.
More understanding of fundamental processes was needed. Ac-
cordingly, Meyer-Peter implemented a comprehensive plan entail-
ing field measurements in the Alpine Rhine, as well as hydraulic
modeling and flume experiments to be conducted in ETH’s new
hydraulics lab.
To recruit research assistants, Meyer-Peter placed an advertise-
ment in a Zurich newspaper. The ad caught the attention of Mil-
eva Einstein, and she contacted Albert. He had briefly thought and
written about aspects of river mechanics 共关AlbertEinstein 1926,
appreciated the importance of Meyer-Peters work, and saw a
promising, safer career opportunity for his and Mileva’s elder son.
In 1931 Einstein joined Meyer-Peter’s research effort and started
working toward the doctoral degree.
At first a rather lackadaisical and playful research assistant
Fig. 2, not that well regarded by Meyer-Peter, Einstein eventu-
ally became intrigued by gravel-particle movement along Meyer-
Peters flume. After a few years, Einstein and colleagues wrote a
series of papers presenting research findings stemming from
Meyer-Peter’s plan. The papers were on bedload transport, hy-
draulic radius and flow resistance, measurement of sediment
transport, and hydraulic modeling Einstein 1934; Meyer-Peter
et al. 1934; Einstein 1935; Einstein and Mu
¨
ller 1939.
New Insight: Railway Schedules and Galton’s Board
While observing particle movement along the flume Fig. 3, Ein-
stein realized that a distribution describing the rates of travel of
identical particles could be used to determine an ‘average travel
velocity’’ for a group of particles. He borrowed this notion from
railway-schedule terminology, implying the total distance traveled
Fig. 1. Alpine Rhine constrained to a single, straightened channel
just upstream of Lake Constance, Switzerland
478 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004
divided by the total time of travel including stops. With such a
velocity determined, it might be possible to formulate the compo-
nent of bed-sediment transport called bedload, the transport of
bed particles in successive contact with the bed of a channel.
Einstein also realized he needed a course in probability theory
if he were to define distributions and average values of particle
velocity and travel distance. Fortunately, an excellent mathemati-
cian at ETH, Professor George Polya, offered the course Einstein
needed. Polya took keen interest in the problem Einstein wished
to formulate, though he initially was daunted by its complexity.
He guided Einstein through some of the probabilistic aspects of
bed-particle movement. The two men drew closer as Einstein
sought Polya’s advice on the development of the theoretical as-
pects of his doctoral thesis George Polya, letter to Albert Ein-
stein, 1935.
Einstein viewed gravel particle movement as a succession of
alternating forward leaps and rest pauses. Einstein assumed the
forward leaps to be relatively brief compared to the rest pauses. In
analogy with the movement of a commuter train, the particle is
taken to move over a distance that is long compared to the indi-
vidual distances between stops, and that travel periods were neg-
ligibly brief compared to stop periods. To simplify the formula-
tion he assumed that the forward leaps take no time. Certainly this
would be the case for low rates of gravel transport, for which
water flow dislodges particles from the bed and moves them
downstream until lodging in some momentarily secure seating. At
intense transport rates, however, a blizzard of particles would
bounce along the bed, each particle pausing for the barest of
moments, if pausing at all. The assumption simplified the proba-
bilistic analysis and lent it symmetry.
Polya suggested that Galton’s board would be a convenient
statistical device for describing and tracking the movement of a
bed particle. Galton, a cousin of Charles Darwin, was a statisti-
cian who developed a board comprising two perpendicular axes,
of which one represents distance traveled, and the other repre-
sents duration of pause Fig. 4.
To play bed-particle movement on Galton’s board, Einstein
first had to define the board’s properties in terms of movement on
the bed. Since water flow conditions along the flume were con-
stant, he assumed that the likelihood of particle motion was the
same at any point and at any time on the bed. Using Galton’s
board Einstein arrived at a formula giving the travel distribution
of a set of particles along the bed or board. The formula is
expressed in terms of a probability distribution that describes the
number of particles located a distance increment downstream
from the particle source the origin of the board since a given
period had elapsed. From the known characteristic distribution
and the distribution moments, Einstein could determine the av-
erage distance traveled and average resting period of bed particles
moving across the board, and thereby potentially estimate the rate
of sediment transport.
In correspondence with his father, Einstein described his re-
search, explaining its objective, the difficulties he faced, and the
approach he was taking. Albert responded with interest and en-
couragement, offering suggestions intended to clarify the process
Einstein was attempting to formulate. Albert gave considerable
thought to his son’s research subject, and took pleasure in sug-
gesting ways to formulate particle motion. For example in one
letter from Princeton in 1936, Albert proposed a way to eliminate
the approximating assumption whereby the periods of particle
motion were taken as negligibly short compared to the periods
that the particles were at rest on the bed. That assumption be-
comes weak at high intensities of bedload transport for which
almost the entire bed is mobilized. Usually Albert’s suggestions,
though probing Einstein’s formulation, were not fruitful. They led
to complicated mathematical equations whose solution then en-
tailed dubious simplifications.
Einstein approached the probabilistic formulation of the bed-
load transport of bed particles from two standpoints. One stand-
point aimed at determining the distribution of particle travel dis-
Fig. 2. A playful Einstein amidst a pile of fellow Swiss Federal
Institute of Technology students
Fig. 3. Observation, by Einstein, of 22-mm-diameter gravel from the
Alpine Rhine moving in Meyer-Peters new flume at Swiss Federal
Institute of Technology
Fig. 4. Galton’s board
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 479
tances in the flume, whence Einstein could attempt to relate
average particle velocity to the water flow parameters. The second
one sought to simulate the capture of bedload particles by a bed-
load basket, such as he had tested in the ETH flume. His experi-
ments with the flume led to data curves, like those in Fig. 5,
relating bedload transport rate per unit width of channel versus an
average speed of particle travel, with particle shape as a third
variable.
A major step in his formulation required relating the average
characteristics of particle travel to turbulent flow behavior. This
step also required that Einstein deal with the experimental diffi-
culty of particles departing the end of the flume during an experi-
ment. As big as it was, the ETH flume was too short. Without
information on the distances traveled by those lost particles it was
difficult to define the average characteristics of particle move-
ment. Einstein earned Polya’s commendation by working around
this conundrum statistically.
Einstein attained the doctorate degree from ETH in 1937 Ein-
stein 1937, submitting a thesis in which he novelly applied prob-
ability theory to describe bed-particle movement in turbulent
flow. Though the scope of his thesis research did not include
formulation of a method for estimating rates of bed sediment
transport under given water flow and channel conditions, Ein-
stein’s insights into individual motion of bed particles formed the
basis for his new approach to formulating bedload transport. In a
letter to Einstein, Meyer-Peter described Einstein’s doctoral study
as producing ‘some intriguing ideas, but not exactly useful for
my Alpine Rhine study.’
Formulation: Enoree-River Flume
Albert, who had moved to the United States in 1934 because of
his concern about political movements in Germany, persuaded
Einstein to come to the United States in 1938. His arrival coin-
cided with the recent establishment of the U.S. Soil Conservation
Service SCS, reflecting a great national concern about soil ero-
sion and the condition of many rivers. Albert assisted his son in
securing a position as a cooperative agent with SCS’s newly es-
tablished field laboratory on the Enoree River Fig. 6, near
Greenville, S.C. The lab was established for measuring sediment
loads in the Enoree River in order to better understand the rela-
tionships between sediment transport and water flow. It was lo-
cated in a region of South Carolina that had experienced severe
soil erosion problems incurred with intensive farming. Einstein
worked with colleagues Joe Johnson and Alvin Anderson on ways
to measure sediment transport. They quickly realized the need to
distinguish two distinct populations of sediment conveyed by
water in the river. In one paper Einstein et al. 1940 they coined
the term ‘washload’ to describe the rivers load of suspended
fine silt and clay-size particles derived from soil erosion and usu-
ally not comprising the rivers bed, the source of bed-sediment
load.
Einstein continued trying to translate the findings from his
thesis research into a practical method for describing and predict-
ing bedload transport of sediment in rivers and streams. He was
unconvinced by the critical-shear-stress approach used by several
prior formulations e.g., Du Boys 1879; Shields 1936.Inhis
opinion, bedload movement was better related to flow turbulence
near the bed. Accordingly he took the principal conclusions from
his thesis and used them as a basis for a new approach that
equated the volumetric rate of bedload transport to the total num-
ber and volume of particles likely to be in motion. In turn, the
number and volume of moving particles depended on the prob-
ability that water flow would lift or eject an individual particle
from its seating on the bed and move it downstream in a given
period. Einstein viewed that probability as reflecting the stochas-
tic nature of water velocities close to the bed.
The difficulty lay in determining the probability that the hy-
drodynamic lift on any particle on the bed is about to exceed the
particle’s weight within a given period of time. From a different
perspective, the probability could be viewed as the part of the bed
for which hydrodynamic lift force exceeds particle weight. The
probability problem is comprised of two parts. One part con-
cerned the need for an equation for hydrodynamic lift; particle
weight is relatively easy to formulate. The other part concerned
finding a meaningful expression of time; transport rate implies
movement per unit time. Einstein adapted a well-known and stan-
dard formula for hydrodynamic lift, writing it in terms of a local
velocity of water flow at a level near the bed. Here, though, as-
sumptions were needed regarding estimation of the velocity and
lift coefficient.
The trickier problem concerned the inclusion of a time period.
The most reasonable period to use was the average time required
for the water to remove one particle from the bed. Unfortunately
there is no way to express the time required for hydrodynamic lift
to pick up a particle. Einstein assumed that lift involves some
characteristic dynamic feature of the flow field around a particle
Fig. 5. Sample of data from Einstein’s work with Meyer-Peter’s
flume at Swiss Federal Institute of Technology
Fig. 6. U.S. Soil Conservation Service’s Enoree-River Flume, South
Carolina, 1939. The flume was designed for measuring sediment
loads and flow in an actual river
480 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004
falling in still water. Particle diameter divided by particle fall
velocity expresses a characteristic time. Up to this stage, his for-
mulation was reasonably rigorous, once the under-girding as-
sumptions about average particle step length were accepted. But
the subjective use of fall velocity for particles in a description of
particles rolling and bouncing along the bed was unsettling.
By combining formulas for the weight rate of bed particles
moving as bedload, hydrodynamic lift on a bed particle, bed par-
ticle weight, and characteristic time based on bed particle fall
velocity, Einstein arrived at a seemingly simple relationship be-
tween intensity of sediment discharge and the probability of par-
ticle entrainment from the channel bed. In terms of a more de-
tailed formulation repeated in most textbooks, Einstein
expressed this relationship as
A f
B
(1)
The two parameters, and , were central to Einstein’s charac-
terization of bedload transport; dimensionless expression for
intensity of sediment transport; and dimensionless expression
for flow intensity or gross shear force exerted on the bed. A and
B constants incorporating awkward details about particle shape
and step length, as well as water velocity distribution.
Einstein could not derive the exact form of the relationship
between and . Too many variables were unknown. He instead
had to find the relationship from plots of bedload data interpreted
as versus . If his formulations were correct conceptually, the
data would lie systematically along a single curve signifying a
single general relationship, or ‘law,’’ for bedload transport.
Validation Test: Gilbert’s Data
Obtaining reliable data from which to determine the relationship,
however, was not straightforward. Einstein used the only two
comprehensive sets of lab flume data readily available to him at
the time: his from ETH Meyer-Peter et al. 1934, and those pub-
lished by Karl Grove Gilbert about 20 years earlier Gilbert
1914. Gilbert, a prote
´
ge
´
of John Wesley Powell, had conducted
novel and comprehensive flume experiments at the University of
California-Berkeley. The great river surveys of the 1800s notably
Humphreys and Abbot 1861, Powell 1875兲兴 were accompanied
by engineering and scientific desire to know more about the me-
chanics of rivers in the United States Gilbert’s data encompassed
a greater range of sediment and flow conditions than did Ein-
stein’s ETH data. Incidentally, Gilbert too had used a railway
analogy to characterize water and flow and sediment transport in
rivers; the term ‘grade,’ meaning channel slope, was borrowed
from the grade of railway tracks, which commonly were laid
along the relatively level ground of flood plains alongside rivers
and streams Pyne 1980.
By and large the two sets of data fell along a curve in accor-
dance with his formulation, except for a range of conditions re-
flecting high intensities of sand transport. Those data veered sub-
stantially away from Einstein’s postulated curve, and clustered
along their own curve Fig. 7. The deviant data suggested that
bedload transport, or rather bed-sediment transport, could not be
fully described using his method. What disconcerted him was the
realization that the deviant data were not merely a batch of results
from a set of extreme hydraulic conditions, but in fact were rep-
resentative of flow and sediment transport in the sand-bed chan-
nels representative of most rivers in the United States.
The deviation caused Einstein to review the formulation of his
method, and to question the accuracy of Gilbert’s data for sand
bed channels. He queried his own assumption that all bedload
particles moved in steps of constant length proportional to particle
diameter, unaffected by flow conditions. His work at ETH had
suggested this to be the case for the gravel beds at fairly low
intensities of transport for which the probability of particle en-
trainment was moderate or low. He conjectured that, with increas-
ing intensity of transport, the probability of entrainment is high
and the step lengths increase from the constant length at low
intensities. As step length increases, the area and number of par-
ticles starting movement together increases, and consequently so
does the rate of bedload transport. This refinement of his theory
modified the relationship between and , and led to a second
curve with a common stem as the original curve, but which
veered away in almost the same manner as the cluster of Gilbert’s
sand-bed data. The new curve, though, still did not run through
those data. Einstein wondered if Gilbert’s data were tainted with
measurement error.
By 1941, Einstein had sufficiently ordered his thoughts on a
method for describing and predicting bedload transport that he
was able to get them published as an ASCE Proceedings paper
Einstein 1942. As was the practice of the ASCE Transactions
Journal, which subsequently published his paper, his paper was
accompanied by discussions by researchers interested in alluvial
sediment transport. It drew praise for its attempt to relate sedi-
ment movement and flow mechanics, but it raised questions about
the main assumptions spanning the gap between formulation con-
cepts and presentation of a practical predictive method. In par-
ticular, it was criticized for purporting to be on greater rational
basis than were the current formulations based on the concept that
a critical value of bed shear stress or rate of water flow be ex-
ceeded before bed particles could move. Difficult questions in-
cluded: Why base the formulation on lift force alone? Why should
settling velocity be included in a formulation of bedload trans-
port? One discusser, Anton Kalinske, remarked that Einstein evi-
dently had ‘stepped over into the realm of abstract dimensional
analysis’’ when he used particle settling velocity as a convenient
parameter to put the probability of particle motion in a time con-
text. Kalinske had attempted to include turbulence in formulating
the movement of bed particles Kalinske 1942, and provided
insightful comments that Einstein eventually would have to con-
sider in advancing his formulation.
Another discusser, Samuel Shulits, who in the early 1930s had
prepared an English translation of Schoklitsch’s book Der Wass-
erbau, wrote that Einstein’s ‘scholarly probe into the universal
Fig. 7. Relationship between and ; modified from Einstein
1942
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 481
law for the transportation of bed load is inspiring.’ Shulits then
quickly tempered his praise by comparing Einstein’s formula, Eq.
1, with bedload formulas based on the notion of a critical flow
condition beyond which bed particles of a characteristic size
began moving. He thought the latter formulas more directly ex-
pressed the relationship between flow and sediment transport,
whereas Eq. 1, notwithstanding all the interesting background to
its formulation, essentially devises a functional relationship be-
tween two dimensionless parameters. In his closure to the discus-
sions, Einstein 1942 stoutly defended his approach and dispar-
aged the notion of a critical flow condition, calling it ‘‘a condition
that does not exist in nature.’The significance of his approach, he
argued, was not so much its immediate outcome, Eq. 1and Fig.
7, but rather its grappling with the problem of formulating bed-
sediment transport in terms of actual turbulent-flow behavior. He
acknowledged that ‘the problem is far from solved.’
Mountain Creek: A Little River
In contrast with the Alpine Rhine and the Enoree, Mountain
Creek in South Carolina was a mere ditch Fig. 8. Yet, to Ein-
stein, Mountain Creek was an ideal little river. The creek pos-
sessed most of the characteristics of alluvial rivers that Einstein
sought to understand and formulate. Moreover, it was conve-
niently small so that Einstein could measure its water and sedi-
ment loads. The Enoree River field station had proven disappoint-
ing for obtaining field data on bedload because of insufficiently
frequent large flows.
Mountain Creek could help in calibrating or linking his
laboratory-flume insights and equations to the behavior of a sand-
bed river. He had learned from his Alpine-Rhine work at ETH
that laboratory results, and formulations based only on the results
of laboratory idealizations of rivers, usually are regarded skepti-
cally by practical engineers dealing with real rivers. If he could
show that his ideas worked for sediment movement in Mountain
Creek as well as in his ETH flume, then showing that they worked
for a river would be a matter of simple geometry.
As perverse luck would have it, the summer and autumn of
1941 were relatively dry in South Carolina. Flows in the creek
barely moved any sediment. Only a few inches of rain fell, though
a single storm did drop an inch-and-a-half of rain during the
evening of almost the last day Einstein intended to monitor the
creek. He and technicians were out at the enlivened creek imme-
diately the next morning, recording its discharges of water and
sediment. The equipment worked well and the measurements
proved, at least to Einstein’s satisfaction, that his concepts were
valid for a little river like Mountain Creek Einstein 1944.He
subsequently obtained further data from another little river, West
Goose Creek in Mississippi.
Method Extended: Caltech Flume
With the entry of the United States into World War II, and after
the modest yield of results from the Enoree River, the SCS wound
down its work at Enoree Field Station and reassigned the station’s
personnel. In 1943 Einstein was transferred to SCS’s laboratory at
the California Institute of Technology, Pasadena. Albert was en-
thused about the move and encouraged his son to contact The-
odore von Ka
´
rman, a renowned Caltech fluid mechanician. Von
Ka
´
rman, however, was busy with war-related matters, and he
never developed Albert’s hoped-for relationship with Einstein.
As his part of the war effort, Einstein was seconded to
Caltech’s Hydrodynamics Laboratory to work on shock waves
produced by explosives and projectiles breaking the sound barrier.
However, he still had opportunities to continue developing his
bedload method and to investigate several pressing problems
emerging in the wake of dam building and other engineering ac-
tivities along rivers, in particular along the Rio Grande River. As
Einstein saw things, accurate estimation of bed-sediment load, not
just bedload, was the most important problem in alluvial-bed river
engineering. The ability to predict bed-sediment load in a river
would enable engineers to predict the rivers response to changes
in its water and sediment loads, thereby reducing the uncertainty
associated with utilizing the river as a resource for water and
hydropower.
Convinced of the essential correctness of his bedload method,
Einstein set about extending it by addressing several complicated
aspects of bed-sediment transport: bedform development, trans-
port of nonuniform bed sediment, and combined bedload and
suspended-load transport of bed sediment i.e., total bed-sediment
load.
SCS researchers at Caltech, Arthur Ippen and notably Hunter
Rouse, had formulated an equation for the vertical distribution of
suspended bed sediment over the depth of flow. The equation, and
lab data supporting it, were written up by Rouse 1939, and
subsequently elaborated by another SCS researcher, Vito Vanoni
1946. Commonly called the Rouse equation, it is one of the
more successful formulations of sediment transport. However, it
gives only the distribution of suspended-sediment concentration
relative to some reference elevation near the bed, showing that the
concentration decreases rapidly with higher elevation in a flow. A
practical difficulty was that the equation does not give the abso-
lute suspended-sediment load. To get that, the relative distribution
has to be tied to a known, or estimated, sediment load concentra-
tion at some level near the bed. Here, Einstein saw an opportunity
to link the Rouse equation with his formulation of bedload and, in
his words, to produce ‘a unified method for calculating the part
of the sediment load in an alluvial river that is responsible for
maintaining the channel in equilibrium’Einstein 1950.
Einstein surmised that the suspended-load distribution as de-
scribed by Rouse’s equation could be spliced to the top of the
bedload layer as described by the bedload formulation. The con-
centration of particles moving at the top of the bedload layer
would serve as a convenient and reasonable reference concentra-
Fig. 8. Mountain Creek, Miss., 1941. The creek was fitted with a
size-reduced version of the sediment-measurement apparatus used for
the Enoree River Fig. 6.
482 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004
tion with which to set the maximum concentration at the bottom
of the suspended load distribution. However, the splicing of bed-
load and suspended load is easier said than accurately done. It
meant also that Einstein needed to strengthen the rigor of his
bedload formulation. His statistical observations on particle mo-
tion would have to be modified, including his earlier assumption;
the average step of a certain particle is the same even if the
hydraulic conditions or the composition of the bed changes. Gil-
bert’s data and the data from Mountain Creek showed that this
assumption did not hold at the intense rates of sediment transport
occurring for sandy rivers.
To see how particles move under conditions of intense rates of
sediment transport, and to get adequately detailed data on bed-
sediment transport at high intensity rates for which bed sediment
would be transported in suspension as well as along the bed,
Einstein needed more flume experiments. During the period
19441946, while others at Caltech were largely occupied by
defense-related research, Einstein used his spare time to churn
water and sediment through a small recirculating flume in the
SCS lab.
Recognition: Rio Grande River
While concerns about soil conservation and sediment transport
had been set aside during World War II, these concerns returned
urgently right after the war, at which time Einstein was well po-
sitioned to play a leading role in addressing them. In May 1947
the various federal agencies concerned with rivers and their wa-
tersheds convened at the Denver headquarters of the U.S. Bureau
of Reclamation to hold the nation’s first meeting focused on the
sedimentation troubles facing engineers and soil conservationists
in the United States. All the federal agencies sent representatives.
Also in attendance were engineers and scientists from diverse
state agencies, universities, and a number of overseas organiza-
tions. The conference placed Einstein center-stage as one of the
nation’s leading authorities on sediment transport at a time when
the full implications of the nation’s sediment troubles were be-
coming pressing.
The need for the conference had arisen from a growing na-
tional recognition of the widespread, adverse consequences that
sediment troubles were posing for river-basin development and
for the conservation of land and water resources. The national
scope of the troubles had become increasingly worrisome during
the mid-1930s, shortly after the federal government had initiated
numerous programs to enhance irrigation, hydropower, naviga-
tion, flood control, and soil conservation. Severely eroded water-
sheds, river-channel aggradation or degradation, reservoir sedi-
mentation, and the adverse environmental effects of muddied
waters all indicated that much more needed to be learned about
watershed and river behavior. Prominent among the sediment
troubles discussed were those along the Rio Grande River.
SCS colleague Vito Vanoni 1948 outlined for conference par-
ticipants a history of the development of predictive relationships
for sediment-transport and water flow in alluvial rivers. He laid
out the big questions to be addressed in order to better understand
how rivers move sediment, and ended his presentation by lament-
ing the lack of scientific and engineering attention given in the
United States to sediment-transport problems, problems whose
national importance he ranked with the more popular contempo-
rary problems of atomic energy and rocket propulsion. Fewer than
10 professionals in the United States, he estimated, were devoting
the major part of their time to the study of sediment transport.
Prominent among the 10 was Einstein, whose new approach to
bedload transport estimation Vanoni described as ‘a radical de-
parture from all previous bed-load formulas.’
Einstein addressed the participants on two issues of keen in-
terest with regard to the sediment troubles along the Rio Grande:
measuring and predicting the rate at which rivers move sediment
along their bed Einstein 1948. He likened the middle Rio
Grande to Mountain Creek, asserting that it behaved essentially
like the creek. Unlike the other speakers, Einstein could draw on
and describe European as well as U.S. experience. Moreover, for
many participants the name Einstein held beguiling promise of
major breakthroughs in understanding and formulating the me-
chanical laws of sediment transport by rivers. Within several
months of the Interagency Sedimentation Conference, former
SCS colleague Joe Johnson facilitated Einstein joining the engi-
neering faculty of the University of California at Berkeley.
Over the following 2 years, Einstein completed a detailed
write-up of his bed-sediment transport method and published it as
U.S. Department of Agriculture Report 1026 Einstein 1950, now
widely recognized as a milestone in alluvial-river mechanics. His
method became widely known thereafter as the Einstein method,
and was used extensively by the Bureau, the U.S. Corps of Engi-
neers USACE, the U.S. Geological Survey, and many others.
Report 1026 elaborated and better explained Einstein’s probabil-
ity approach to bed-sediment transport. Moreover, it presented an
elegant splicing of the bedload and suspended-load components
of bed-sediment transport, and it introduced new concepts aimed
at reducing some of the empiricism in the versus relationship
introduced by Einstein 1942.
The concepts included modifying the flow intensity parameter
so that it could be used for estimating the transport rates of
particle-size fractions comprising a bed of nonuniform sediment.
Further, it involved estimation of the flow energy expended on
bed-particle roughness, not on the entire bed; introduction of two
adjustments to account for the velocity of flow locally around a
particle; and pressure distributions at a bed surface of nonuniform
sized sediment. The modified parameter is
*
Y
2
/
x
2
(2)
in which factor intended to account for the sheltering of
smaller particles amidst larger particles in a bed of nonuniform
sediment; Y pressure-correction factor, which together with
2
/
x
2
, is intended to account for the influence of particle size
nonuniformity on hydrodynamic lift; and
modified in an
effort to account for bed sediment development of bedforms on
channel beds. Though quite readily conceived, near-bed com-
plexities in flow and particle disposition meant that and Y have
to be determined empirically from flume data. Textbooks on sedi-
ment transport e.g., Chien and Wan 1999 explain the details
associated with the terms in Eq. 2, and elaborate on the subse-
quent work examining and Y. Using Gilbert’s data and his own
ETH data, Einstein arrived at the following relationship between
probability for motion, p, the parameter
*
, and the intensity of
bedload transport for particles in the particle-size fraction,
*
i
(i
B
/i
b
), where i
b
and i
B
fractions are the fractions of a
given particle size in the bed and the bedload, respectively:
p 1
1
1.2
(1/7)
*
i
2
(1/7)
*
i
2
e
t
2
43.5
*
i
1 43.5
*
i
(3)
This equation is commonly referred to as the Einstein bedload
function e.g., Chien and Wan 1999. Fig. 9 shows a data curve
relating
*
and
*
i
. Though the Einstein method in Report
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 483
1026 was comprehensive, it was somewhat cumbersome to apply,
and some of its components were found to need adjustment.
Application: Missouri River
For 2 months during the summer of 1948, about 4 months after
his first meeting as a board member for the Missouri River Divi-
sion’s Sedimentation Studies program, Einstein was in North Da-
kota and Montana, at the headquarters of the USACE’s Garrison
District. Over the remainder of his career he maintained a produc-
tive relationship with USACE, assisting it with sediment concerns
along the Missouri, the Arkansas, and other rivers, and working to
better formulate sediment transport and water flow in rivers.
USACE’s Missouri River Division was charged to oversee a
vast watershed that covered about one-fifth of the continental U.S.
In 1948, the Division’s efforts were concentrated largely on
implementing the Pick-Sloan Act, which prescribed a plan to con-
trol and regulate the rivers flow for the purposes of flood control,
hydropower generation, and navigation. However, the Division
found implementing Pick-Sloan to be fraught with more difficul-
ties than the plan had envisioned. All of the dam projects called
for were facing difficulties and setbacks attributable to the rivers
sediment Fig. 10.
By September 1948, the technical problems facing the division
were clear to Einstein. He summarized them in a brief report to
the division, stating ‘every attempt must be made to study the
questions in the Missouri itself and in its tributaries. This is the
only way to find the relative importance of the various influ-
ences’Einstein, unpublished report to Missouri River Division,
U.S. Army Corps of Engineers, Omaha, 1948. He realized that
the efficacy of the predictive methods would have to be checked
by the simultaneous measurement of sediment transport and the
flow variables of the river itself.
One immediate matter was a little delicate. The method se-
lected for estimating the rates of bed-sediment transport through
the river was the bedload equation proposed by Professor Lorenz
Straub, a prominent hydraulics engineer who had been a USACE
engineer in the 1930s. Straub 1935 had proposed the method
while working on House Document 238 Missouri River Report,
a detailed assessment of the flow and sediment problems posed to
engineering use of the Missouri River. As Straub was the senior,
and initially the most vocal, board member, and since his method
had been developed expressly with the Missouri River in mind,
his was the method that the division had decided to adopt. Ein-
stein was uncomfortable with the method. In a long letter report
to the division, he outlined the steps that needed to be taken to
gauge the sediment load conveyed by the river, and he went
through the shortcomings of Straub’s method. Besides being es-
sentially an extension of the shear-stress or discharge excess
approach proposed earlier by Du Boys and others e.g., Schokl-
itsch 1934, Straub’s method assumed that the river kept its cross-
sectional shape and its roughness for the full range of water flow
and while the rivers bed degraded or aggraded. These assump-
tions seemed unreasonable to Einstein, and they were not sup-
ported by measurements of flow depth and flow rate for selected
reaches of the river.
Einstein together with the division’s engineers had examined
data on the Missouri River and several of its tributaries in an
effort to better understand the relationship between flow depth
and flow rate for these rivers. An explanation ventured in terms of
changing channel shape failed because channel shapes were
found not to change appreciably as flow varied. A more promising
explanation related flow energy loss to the intensity of bed-
sediment transport and thereby Einstein’s modified flow-intensity
parameter
), and variations in bedforms and thereby bed
roughness.
Both Straub’s and Einstein’s methods were used for estimating
bed-sediment transport, though Straub’s was soon abandoned.
Einstein, though, encountered an unexpected complication: down-
stream of Fort Peck Dam, the rivers degrading bed became ar-
mored with coarser bed sediment. No sediment-transport method
had taken armoring into account.
Further Refinement: Berkeley’s Flumes
The University of California-Berkeley’s ability to attract talented
graduate students, combined with Einstein’s link to USACE, en-
abled him to undertake at Berkeley a sustained research effort
aimed at better understanding and formulating sediment-transport
processes. It was an effort largely undertaken by graduate stu-
dents and USACE engineers working under Einstein’s guidance.
They embarked on a comprehensive series of flume investigations
aimed at illuminating key aspects of sediment and flow behavior.
Moreover, Einstein’s Berkeley appointment enabled him to teach,
something he enjoyed Fig. 11.
Briefly mentioned here are two examples illustrative of that
effort. An especially important issue, and one that has challenged
formulation of sediment transport in rivers, concerns what hap-
Fig. 9. Relationship between
IL
and
*
from Einstein 1950
Fig. 10. Board members Einstein, Straub, Vanoni, Laneand Corps
engineers ponder bed degradation of the Missouri River downstream
of Fort Peck Dam, Mont., 1948
484 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004
pens if the bed sediment comprises a wide range of particle sizes.
This situation, of course, is the norm for most riverbeds. A basic
assumption underpinning the Einstein method needed further
work; i.e., that all particle sizes in a river may be equally avail-
able at the bed surface and within the bed. Over about 1 year,
1950 to 1951, graduate student Ning Chien and Einstein carried
out a series of flume experiments and were in a position to pro-
vide detailed descriptions of the processes whereby different sized
bed particles segregate in the upper layer of a river bed, how
armoring occurs, and how a riverbed acts much like a reservoir
for sediment, storing it during periods of reduced water flow, and
releasing it during periods of greater water flow.
With doctoral student Robert Banks, Einstein began investigat-
ing how several factors contribute to flow resistance in rivers.
This effort would provide more insight for his sediment-transport
method, and it would help address a crucial companion issue
concerning the relationship between water discharge and depth in
alluvial rivers. The total resistance opposing the flow consists of
the combined effect of resistance attributable to surface rough-
ness, bedforms or bar resistance as he expressed it, and vegeta-
tion. Einstein wondered if the total resistance could be expressed
as the sum of these components. This thought was not new. It had
been used successfully in determining flow resistance in flow
around bodies.
The notion of dividing flow resistance into two parts, particle
roughness drag and bedform drag, was new for alluvial-river me-
chanics. Its first practical implementation is the Einstein-
Barbarossa method for estimating the relationship between flow
depth and flow rate in alluvial channels. Einstein and Nicholas
Barbarossa, an engineer with the USACE’s Omaha District, used
data from the Missouri, several of its tributaries, and two Califor-
nia rivers to find a relationship between Einstein’s parameter
*
and that part of flow-energy loss attributable to bedforms. Addi-
tionally, they used the Manning-Strickler equation to estimate en-
ergy loss attributable to surface roughness. Publication of their
method Einstein and Barbarossa 1952was a further milestone in
formulating alluvial-river mechanics.
Confronting Complexity
The complex mix of processes at play in natural alluvial rivers
has defied so far at least reliable prediction of bed-sediment
transport and flow depth; uncertainties of 100% or more are com-
mon for predicting rates of bed-sediment transport. Engineers and
scientists have long recognized that rivers are complex, and ac-
cordingly have used largely empirical as well as analytical ap-
proaches to characterize alluvial-river behavior. Commonly, the
practical design engineer and the scientist in the field have found
the empirical approach more practicable and have been skeptical
of sophisticated, predictive methods based on advanced fluid me-
chanics and data from laboratory flumes. Proponents of the more
empirical approach and those of the largely mechanistic approach
are quick to debate each others methods, especially when one
claims to be the superior. The following exchange is an example
of the debate, and illustrates Einstein’s conviction about the ulti-
mate truth of the concepts supporting his method for estimating
bed-sediment load. The exchange follows a paper published by
Ning Chien, Einstein’s student.
Shortly before he returned to China where he was to play a
leading role addressing that country’s river problems, Chien in
1954 published two ASCE Proceedings papers that drew a salvo
of criticism from a leading exponent of the empirical Regime
method approach to river behavior. One paper ‘The present sta-
tus of research on sediment transport’ see Chien 1956兲兴 ad-
dressed the relationship between water discharge and bed-
sediment load. In it, Chien described the reliance of sediment-
transport formulation on accurate formulation of water flow.
Appended to Chien’s paper was a stern discussion criticizing
Chien’s neglect of the body of understanding collectively termed
the Regime Method. The discusser, Thomas Blench, a very ca-
pable hydraulic engineer and a leading proponent of that method,
argued that Chien’s paper presented knowledge limited only to
findings from ‘laboratory flumes with trifling flows.’ He further
argued that Chien had neglected ‘the vast amount of observations
on canals in the field, the dynamical aspect of the formulas
evolved there-from, and the fact that these formulas provide a
simple and adequate means of practical design that has been used
widely for many years.’’ The Regime Method’s formulas, Blench
claimed, ‘represent what real channels actually do.’
Einstein, though not a coauthor of Chien’s paper, wrote an
additional closure discussion to that by Chien. He took issue with
the Blench’s claim about the sufficiency of the ‘superiority of the
‘simple and adequate’ Regime formulas. Those formulas, he
pointed out, were developed by curve-fitting of data from ‘a very
narrow range of bedload conditions.’ He went on to express,
among other things, his doubt that the Regime formulas would
work for rivers in the United States. In his closure following
Blench’s cutting discussion of another ASCE Proceedings paper
see Einstein and Chien 1956, Einstein presented figures showing
the inadequate performance of the Regime formulas.
Since Einstein 1950numerous methods have been developed
for estimating the relationships between water discharge and bed-
sediment transport in alluvial rivers. Some methods have built on
the Einstein method laid out in Report 1026, or modified the
method for better accuracy and more convenient use e.g., Colby
and Hembree 1955; Bishop et al. 1965. Others have developed
from improved insights, and still others have remained resignedly
empirical e.g., Brownlie 1981. Meyer-Peter’s research plan for
the Alpine Rhine led to another quite widely used method for
estimating bedload transport Meyer-Peter and Mu
¨
ller 1948.
Ironically, Einstein through his early work at ETH Meyer-Peter
et al. 1934 had a significant role in developing that method.
Fig. 11. Einstein at a Berkeley flume explaining flow processes to
students
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 485
Latter Years
Much of Einstein’s career can be cast as the archetypal story of
the researcher protagonist determined to master intellectually the
way water flows and conveys alluvial sediment in a river. In that
effort, he personifies the mixed success and frustrations experi-
enced by many researchers who have attempted to describe the
complicated behavior of alluvial rivers in terms of rationally
based equations. The effort begins keenly with apparent good
promise of success, based on innovative new insights into com-
ponent processes. Formulation seems within reach and progress is
made, but then judicious assumptions and curve-fitting empiri-
cism have to be invoked as approximating compromises to ac-
commodate the many confounding complexities inevitably faced.
Einstein retained his long fascination with alluvial rivers Fig. 12
and continued his efforts to understand and formulate how they
convey sediment. During his latter years, his fascination broad-
ened to include sediment transport in coastal waters.
Einstein retired from his active professorship at the University
of California on July 1971, at the age of 67. His retirement earned
him the Berkeley Citation, an award ‘for distinguished achieve-
ment and notable service to the university,’ and a Certificate of
Merit from the U.S. Department of Agriculture, ‘for pioneering
research in developing the bed-load function of sediment trans-
port by streams, and leadership in developing application of fluid
dynamics theory in solving engineering problems in the field of
soil and water conservation.’ Eight months later the American
Society of Mechanical Engineers presented him a certificate of
recognition for his 20 years of ‘devoted and distinguished ser-
vices to applied mechanics reviews.’
However, none of these accolades seem to have meant as
much to him as the sedimentation symposium that had been held
in his honor a few weeks earlier in June 1971. About 80 profes-
sors, researchers, and former students of Einstein’s had attended,
many of the students flying in with their spouses from distant
locations to honor their former professor. The symposium report-
edly was very moving for Einstein, who greatly enjoyed the oc-
casion. Perhaps foremost among his contributions were the 20
doctoral graduates he guided while at Berkeley. Many of them
became leading figures in the study of alluvial rivers and hydrau-
lic engineering.
Late June 1973, while a Visiting Scholar at Woods Hole
Oceanographic Institute in Massachusetts, Einstein suffered a
heart attack and shortly thereafter died. At the time of his death,
he and friend Don Bondurant, a retired USACE engineer, had
outlined a book on alluvial rivers. It was not to be the usual
format of textbook, but rather an approach that introduces typical
engineering problems arising between people and alluvial rivers,
then explains the knowledge and methods needed to solve the
problems. The book, like Einstein’s work to formulate sediment
transport, was a task that unfortunately remained unfinished.
In 1988, the American Society of Civil Engineers established
the Hans Albert Einstein Award ‘to honor Hans Albert Einstein
for his outstanding contributions to the engineering profession
and his advancement in the areas of erosion control, sedimenta-
tion and alluvial waterways.’
Acknowledgments
The writers thank Professor Daniel Vischer of ETH-Zurich for his
assistance with background material used in preparing this paper.
They also thank the papers reviewers and Pierre Julien, JHE
Editor.
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JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 487
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... Firstly, the bed load transport rate was found according to the researchers' equations of Ettema and Mutel [11], Lajeuness et al. [12], Nielsen [13]. Secondly, the bed load transport rate was calculated according to Eq. (14). ...
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... Sediment transport under the influence of gravity alone can usually occur on steep surfaces, including hills, cliffs, and continental slopes. [1][2][3][4][5]. In particular, many scientists around the world Graf 1971, Vanoni 1975, and Yalin 1977, Soulsby 1997 (Rijn, Engelund and Hansen, Ackers and White (1973), Sleath 1984, Nielsen 1992, Fredse and Deigaard 1992, Brownlie, K.Sh. ...
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