Sediment Transport in Fluvial Hydraulics
Fluvial Hydraulics
CH-6
Bed-Load Transport
Sediment Transport Equations
•
Sediment transport equations used to determine CAPACITY for set
of flow conditions
–
Capacity needed for many different analyses such as
aggradation/degradation, general scour/deposition, and lateral migration
–
First step is to select appropriate equation:
•
Predicated on an understanding of the system being studied
•
Some formulas developed for sand-bed streams with high suspended load
transport
•
Other equations pertain to conditions where bed-load transport dominates
•
Study objectives determine the portion of the sediment transport that needs
to be estimated and the level of accuracy
•
Some formulas independently predict bed load and suspended load
•
Other formulas estimated bed-material load (i.e., bed load + suspended
load) – usually these equations do not include wash load
•
Procedures exist that incorporate sediment sampling data, such as the
modified Einstein procedure, that can estimate total sediment transport rate
(including wash load)
Sediment Transport Equations
Bed Load Transport Redefined…
•
Transport of sediments where the solid particles glide, roll or briefly
jump but stay close to the bed
–
Erosion of the bed (bed load transport) commences upon exceedance
of the critical shear stress,
o,cr
–
Exists a number of formulas for predicting bed-load transport:
•
Many are empirical but have incorporated dimensionless parameters
•
More common equations for bed-load transport include:
–
Duboys-Type Equations (Kalinske)
–
Schoklitsch-Type Equations
–
Meyer-Peter et al. (1948) – sometimes called Meyer-Peter Muller (MPM)
–
Einstein’s Bed Load Equation
Theoretical Considerations
•
Consider mobile bed of uniform and
noncohesive particles
•
Forces which lead to uniform and steady motion
of a particle…
–
Hydrodynamic Force:
–
Submerged Weight of Particle:
–
7 Parameters: Fluid (density, viscosity), Solid
(density, diameter), Flow (Depth or Hydraulic Radius,
Slope and Gravity – Friction Velocity)
Theoretical Considerations
•
7 parameters can be combined into 4
dimensionless
groups:
Theoretical Considerations
•
Transport of sediment can be expressed
as a function of these 4 dimensionless
variables:
Theoretical Considerations
•
Since the terms R
h
/d and
s
/
are included in
*
and with
*
=
f
(Re
*
):
–
Expression links solid transport,
q
sb
, to shear stress
–
Increase in shear stress past critical is responsible for
an increase in
q
sb
Theoretical Considerations
•
We often assume the relation below can
be expressed in terms of a power law:
Bed-Load Transport Equations
•
Be Careful!!!
–
Formulas give “reasonably satisfying results”
with the parameter domain for which they
were derived
–
Application of formulas should be done with
great care!
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
DuBoys (1879) proposed model for bed-load
transport assuming that sediment moves in layers,
each of which has a thickness
–
1
st
layer is where tractive force balances resistance
force between layers:
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
If layer between 1
st
and nth moves according to a
linear distribution, then…
•
n
is the thickness of sediment material moving
•
Thickness moves with an average velocity of
v
s
(
n
-1)/2
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Critical condition at which sediment motion is about
to begin…(n = 1)
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Assumptions of this equation have been shown to disagree
totally with observations of bed-load transport
•
Bed load does not move as sliding layers (Schoklitsch, 1914)
•
However, there is generally good agreement between this
equation and field/laboratory data
•
Proper use of the equation depends on correct evaluation of the
characteristic sediment coefficient,
–
Sidenote: generally all bed load equations based on excess
shear stress are classified as duBoys-type equations
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Experiments of Schoklitsch (1914) proved duBoys’
model of sliding layers to be wrong, but did show
that the equation fit the data well
•
Uniform grains of various kinds of sand (limited data) and
experimental flume experiment (small dimensions):
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Straub (1935) - based on work of several
researchers determined average values of several
sand sizes
•
Criticized because all data obtained from small flumes over
a small range of particle sizes
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Zeller (1963) gives graph for metric units:
Bed-Load Transport Equations
•
duBoys-Type Equations:
–
Shields (1936) proposed
critical shear stress relation
•
Intent was to present an
abbreviated form of the
factors influencing bed
load transport
•
Developed a semi-
empirical tractive-force
equation based on
1.06<
s
s
<4.25 and
1.56<
d
<2.47 mm
Bed-Load Transport Equations
•
duBoys-Type
Equations:
–
Kalinske (1947)
emphasized
turbulence
mechanism in
flow above the
bed:
Example – Graf 6.D
An artificial channel has been constructed to
divert a certain discharge from a river. This
channel has an approximately rectangular cross-
section with a width of B = 46.5 m and a bed
slope of S
f
=6.5 x 10
-4
. Uniform flow is
established when the flow depth is 5.6 m.
Velocity-profile measurements suggest an
average velocity of 1.8 m/s and n’ = 0.0212 (due
to bed roughness). Estimate the bed-load
transport using the Kalinske equation. Express
the solid discharge as a concentration.
Example – Graf 6.D
•
Converting q
sb
to g
sb
:
•
Sediment concentration can be expressed
in a number of different ways:
–
Concentration by volume:
–
Concentration by mass:
–
Concentration by unit mass:
Example – Graf 6.D
•
Relationship between these concentrations:
Bed-Load Transport Equations
•
Schoklitsch-Type Equations:
–
Replaced excess shear stress criterion with critical
water discharge (or depth):
–
Gilbert (1914) – expansive set of data for varying
water discharge, energy grade line, and sediment
properties
•
3 flumes of lengths 14, 31.5, and 150 ft
•
Flume width ranged from 0.23 to 1.96 ft
•
Discharge varied from 0.019 to 1.19 cfs
•
Eight different kinds of sand (0.305 < d < 7.01 mm)
Bed-Load Transport Equations
•
Schoklitsch-Type Equations:
–
Schoklitsch first proposed equations in 1935 but modified them
in 1950:
•
Critical flow rate:
Bed-Load Transport Equations
•
Schoklitsch-Type Equations:
–
Schoklitsch first proposed equations in 1935
but modified them in 1950:
•
Sediment discharge:
Example – Graf 6.D
An artificial channel has been constructed to
divert a certain discharge from a river. This
channel has an approximately rectangular cross-
section with a width of B = 46.5 m and a bed
slope of S
f
=6.5 x 10
-4
. Uniform flow is
established when the flow depth is 5.6 m.
Velocity-profile measurements suggest an
average velocity of 1.8 m/s and n’ = 0.0212 (due
to bed roughness). Estimate the bed-load
transport using the Schoklitsch equation.
Express the solid discharge as a concentration.
Bed-Load Transport Equations
•
Meyer-Peter Muller
(MPM):
–
Switzerland research
laboratory (ETH) – Meyer-
Peter et al. (1934):
•
Laboratory flume with
cross-section of 2 x 2 m
and total length of 50 m
(max discharge of 5 m
3
/s)
and sediment discharge
up to 4.3 kg/(s-m)
•
Two grain sizes: 5.05 mm
and 28.6 mm
•
For sand, resulting bed-
load equation
Bed-Load Transport Equations
•
Meyer-Peter Muller (MPM):
–
ETH experiments were extended to include data
with particle mixtures
•
First attempt to include representative grain diameter
with previous formulas failed
•
Meyer-Peter et al. (1948) proposed following equation
which fit all the data:
Bed-Load Transport Equations
•
Meyer-Peter Muller (MPM):
•
Meyer-Peter et al. (1948) :
–
R
hb
= hydraulic radius of bed
–
Equivalent diameter = d
50
–
M
is a roughness parameter:
–
Derived with d = 3.1-28.6 mm (applicable to d > 2.0 mm)
–
Derived with S
f
= 0.0004-0.020
Example – Graf 6.D
An artificial channel has been constructed to
divert a certain discharge from a river. This
channel has an approximately rectangular cross-
section with a width of B = 46.5 m and a bed
slope of
S
f
=6.5 x 10
-4
. Uniform flow is
established when the flow depth is 5.6 m.
Velocity-profile measurements suggest an
average velocity of 1.8 m/s and n’ = 0.0212 (due
to bed roughness). Estimate the bed-load
transport using the Meyer-Peter et al. formulas.
Express the solid discharge as a concentration.
Sediment Transport Equations
Einstein’s Bed-Load Transport
Equation
•
Developed in 1942 (empirical) and 1950 (analytical)
•
Probabilistic model for transport of sediment as bed load
–
Bed load transport is related to the fluctuations in velocity rather
than the average velocity
–
Beginning and end of motion are expressed in probabilistic terms
with considerations for the lift and particle’s weight
–
Built on observations/experimental evidence:
•
Bed load moves slowly downstream – motion of an individual
particle is quick step with long intermediate stops
•
Average step made by any bed load particle is independent of flow
condition, transport rate, and bed composition (always the same)
•
Different transport rates due to changes in the average time
between steps and the thickness of the moving layer
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Equilibrium condition of exchange of bed particles
between the bed layer and the bed
–
# particles deposited per time per unit bed area = #
particles eroded per unit time per unit bed area
–
Deposition:
•
Each particle (d) has steps of length A
L
d and will be
deposited over an area A
L
d long with unit width
•
Let
g
s
= bed load rate in weight per time per width, N/(m-s)
•
Let
i
s
= fraction of bed load in given grain size
•
Then
g
s
i
s
=
rate at which given size moves through the unit
width per unit time
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
•
Weight of a single particle =
s
k
2
d
3
(k
2
= constant of grain
volume)
•
Number of particles of a certain fraction deposited per unit
time and bed area:
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Erosion:
•
Whether a particle will be eroded depends on the availability
of the particle and on flow conditions (turbulence level)
•
Let
i
b
= fraction of bed material in a given grain size
•
# particles of size d in a unit area of bed surface =
i
b
/k
1
d
2
,
where k
1
= constant of grain area
•
Let p
e
/t
e
= probablility of removal, where t
e
= time consumed
by each exchange
•
# particles eroded per unit time and unit area:
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Erosion:
•
No direct method is available for the determination of t
e
–
Einstein (1942) suggests a function of
v
ss
:
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Bed Load Equation (Equilibrium):
•
Rate of deposition balances rate of erosion:
–
Exchange probability:
•
p
e
= fraction of time during which the lift exceeds the weight
of the particle at any one location
•
Non-intensive sediment transport
p
e
is small and
deposition is everywhere possible
•
Strong sediment transport
p
e
becomes larger and
deposition is not everywhere possible
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Exchange probability:
•
Einstein (1950) suggests that p
e
can be used to evaluate A
L
d :
–
If p
e
is small, distance traveled is constant: A
L
d =
b
d where
b
is
approximately 100
–
For larger p
e
:
»
Only (1-p
e
) particles can deposit after traveling
b
d
»
And p
e
particles stay in motion – would like to deposit but
can’t because after
b
d lift forces exceed weight
»
Of these p
e
particles, p
e
(1-p
e
) are deposited after traveling
2
b
d while p
e
2
particles are still not deposited
»
We can express the total travel distance as a series:
Einstein’s Bed-Load Transport
Equation
•
Physical Model:
–
Substituting the series into the bed load equation:
Einstein’s Bed-Load Transport
Equation
•
Empirical Relation (Einstein, 1942):
Einstein’s Bed-Load Transport
Equation
•
Empirical Relation (Einstein, 1942):
–
Probability determination:
Einstein’s Bed-Load Transport
Equation
•
Empirical Relation (Einstein, 1942):
–
Probability determination:
Einstein’s Bed-Load Transport
Equation
•
Empirical Relation (Einstein, 1942):
–
Weak Sediment Transport:
•
Einstein determined these constants using the
data of Gilbert (1914) and Meyer-Peter et al.
(1934) –
See Figure on Next Slide
•
All data with
<0.4 plots on a single curve
(Curve 1) with:
Einstein’s Bed-Load Transport
Equation
Einstein’s Bed-Load Transport
Equation
•
Empirical Relation (Einstein, 1942):
–
Strong Sediment Transport:
•
For
>0.4: Curve 2
–
For sand mixtures, Einstein suggested an
effective diameter of 35-45% finer (d
35
commonly used)
Einstein’s Bed-Load Transport
Equation
•
Einstein (1950) replaced empirical solution with an
analytical solution:
–
Based on same concepts of probability of motion:
Einstein’s Bed-Load Transport
Equation
•
Einstein (1950) replaced empirical solution with an
analytical solution:
–
Based on same concepts of probability of motion:
Einstein’s Bed-Load Transport
Equation
Einstein’s Bed-Load Transport
Equation
•
Einstein (1950) replaced empirical solution with an
analytical solution:
–
The earlier proposed functional relationship is valid for uniform
grains:
–
Einstein extended this function for non-uniform sediment:
Einstein’s Bed-Load Transport
Equation
Einstein’s Bed-Load Transport
Equation
•
Einstein (1950) replaced empirical solution
with an analytical solution:
–
We can even rewrite the function in simpler
format:
Einstein’s Bed-Load Transport
Equation
–
Proposed the following solution for p
e
that
resembled normal distribution:
Einstein’s Bed-Load Transport
Equation
–
But we don’t want to have to evaluate this
function:
Einstein’s Bed-Load Transport
Equation
Einstein’s Bed-Load Transport
Equation
Einstein’s Bed-Load Transport
Equation
•
Note that Einstein never uses a “critical value for
erosion”:
–
However, the critical shear stress from Shields
matches with good agreement the value of
with
small
–
Einstein equation valid for d > 0.7 mm (0.8-28.6 mm)
across large range of bed slopes
–
Applied world-wide with “great success”
Example – Graf 6.D
An artificial channel has been constructed to
divert a certain discharge from a river. This
channel has an approximately rectangular cross-
section with a width of B = 46.5 m and a bed
slope of
S
f
=6.5 x 10
-4
. Uniform flow is
established when the flow depth is 5.6 m.
Velocity-profile measurements suggest an
average velocity of 1.8 m/s and
n
’ = 0.0212 (due
to bed roughness). Estimate the bed-load
transport using Einstein’s formula. Express the
solid discharge as a concentration.
Armoring
•
Non-cohesive beds consist
of a number of different
particle sizes
PSD:
–
Curve can be divided into
fractions
–
Usually 4 or 5 unequal
fractions after elimination of
smallest 5% of finest
particles (wash load) and 5%
of coarsest particles
–
For each fraction, you can
then determine:
•
d
i
= average diameter
•
i
sb
i
q
sb
i
= bed-load transport
Armoring
–
PSD for bed material is different than that moving
as bed load or suspended load:
•
Given fraction of the PSD of bed material (i
b
i
) different
from corresponding fraction of solid discharge curve
(i
sb
i
)
Armoring
–
Smaller particles are more easily eroded than larger ones:
•
Grain-size sorting process
–
A
c
c
u
m
u
l
a
t
i
o
n
o
f
t
h
e
r
e
m
a
i
n
i
n
g
l
a
r
g
e
r
p
a
r
t
i
c
l
e
s
c
a
l
l
e
d
a
r
m
o
r
i
n
g
(
a
r
m
o
u
r
i
n
g
)
–
p
r
o
t
e
c
t
s
t
h
e
u
n
d
e
r
l
y
i
n
g
s
e
d
i
m
e
n
t
•
Prevents erosion during subsequent flood events due to larger
particles at bed surface
•
Capacity for sediment transport not satisfied until armor layer is
destroyed (original sediment reappears and begins to form a new
armor layer)
•
Samples taken when armoring has occurred must be interpreted with
caution!
–
Less than a single complete covering layer suffices for total
armoring effect
–
Field observations suggest that a relatively stable armor layer
requires a minimum of two layers of armoring particles
Question: At what location in the channel cross-section would
armoring begin?
Armoring
•
Graf describes armoring as an asymptotic process:
–
As u
*
increases, smaller particles are eroded and larges
ones stay in place
–
Corresponding friction velocity once larger ones stay in
place = critical friction velocity for armoring (u
*a,cr
)
–
Maximum possible armored bed formed by largest
particles: d
90
or larger
–
For discharges when u>u
*a,cr
, armored bed becomes
unstable and will be destroyed
–
Original sediment sizes will be exposed at surface and
active erosion begins again
–
Armor can then be restored under moderate flows
Armoring
•
SCS (1977) suggests d
95
as representative
size “paving the channels”
•
From Shields criterion (verify this using
Shields-Yalin diagram):
–
SCS (1977) says that armoring is probable when
d
c
is equal to or smaller than the d
95
size
•
Percentage of bed material equal to or larger
than the armor particle size (d
a
)
Armoring
•
USBR (1984) provides equation for depth of
scour necessary to establish an armor layer
(
Z
a
):
Example
Consider a case where the critical particle
size is 1.5 in and a representative bed-
material gradation curve shows that this is
the d
90
size. What is the depth to formation
of an armor layer?
Armoring
•
Correia and Graf (1988) proposed
relationship between original PSD and armor
layer PSD:
•
Empirical relationship for prediction of the
stability of the armor layer (Raudkivi, 1990):
Example – Graf 6.E
A mountain river with a bottom slope of
0.0062 has an approximately rectangular
cross-section, being B = 23.5 m wide.
Analysis of the sediment samples taken
from well below the armor layer show that
d
50
= 60 mm and d
90
= 200 mm and the
density of sediment is 2.65. Determine the
diameter of maximum possible armor. At
what flow depth does the armor layer
become unstable?
Sediment transport equations play a crucial role in predicting sediment capacity under different flow conditions, aiding in various analyses like aggradation, degradation, scour, deposition, and migration. Different formulas cater to varied scenarios, distinguishing between bed load and suspended load, with some considering total sediment transport rates. The redefined concept of bed load transport involves particles gliding and rolling close to the bed, driven by critical shear stress. Theoretical considerations delve into forces affecting particle motion, emphasizing key parameters and dimensionless groups for analysis.
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Presentation Transcript
Fluvial Hydraulics CH-6 Bed-Load Transport
Sediment Transport Equations Sediment transport equations used to determine CAPACITY for set of flow conditions Capacity needed for many different analyses such as aggradation/degradation, general scour/deposition, and lateral migration First step is to select appropriate equation: Predicated on an understanding of the system being studied Some formulas developed for sand-bed streams with high suspended load transport Other equations pertain to conditions where bed-load transport dominates Study objectives determine the portion of the sediment transport that needs to be estimated and the level of accuracy Some formulas independently predict bed load and suspended load Other formulas estimated bed-material load (i.e., bed load + suspended load) usually these equations do not include wash load Procedures exist that incorporate sediment sampling data, such as the modified Einstein procedure, that can estimate total sediment transport rate (including wash load)
Bed Load Transport Redefined Transport of sediments where the solid particles glide, roll or briefly jump but stay close to the bed Erosion of the bed (bed load transport) commences upon exceedance of the critical shear stress, o,cr Exists a number of formulas for predicting bed-load transport: Many are empirical but have incorporated dimensionless parameters More common equations for bed-load transport include: Duboys-Type Equations (Kalinske) Schoklitsch-Type Equations Meyer-Peter et al. (1948) sometimes called Meyer-Peter Muller (MPM) Einstein s Bed Load Equation
Theoretical Considerations Consider mobile bed of uniform and noncohesive particles Forces which lead to uniform and steady motion of a particle Hydrodynamic Force: u d 2 ( g 2 * FH f d u * ) Submerged Weight of Particle: 3 W d p s 7 Parameters: Fluid (density, viscosity), Solid (density, diameter), Flow (Depth or Hydraulic Radius, Slope and Gravity Friction Velocity)
Theoretical Considerations 7 parameters can be combined into 4 dimensionless groups: d u = Re * * 3 / 1 g ( ) = 1 d d s 2 * R S s 2 u h f = = = * o ( ) ( ) ( ) * or d d d s s s u = = = o * Fr ( ) ( R ) * * D 1 s gd d s s h = Relative Depth h or d d = = Relative Density s s s
Theoretical Considerations Transport of sediment can be expressed as a function of these 4 dimensionless variables: R d f = , , , h s d * * 2 q L = = = sb 1 q ( ) * sb T 3 s gd s q q = = sb sb u d Ud *
Theoretical Considerations Since the terms Rh/d and s/ are included in * and with *= f(Re*): ( ) ( ) = = f f * q ) d = sb 1 o f ( ( ) 3 s gd s s Expression links solid transport, qsb, to shear stress Increase in shear stress past critical is responsible for an increase in qsb
Theoretical Considerations We often assume the relation below can be expressed in terms of a power law: ( ) * = U 8 f = / o 2 U * o ( ) U b = = 2 q a b s sb s s
Bed-Load Transport Equations Be Careful!!! Formulas give reasonably satisfying results with the parameter domain for which they were derived Application of formulas should be done with great care!
Bed-Load Transport Equations duBoys-Type Equations: DuBoys (1879) proposed model for bed-load transport assuming that sediment moves in layers, each of which has a thickness 1stlayer is where tractive force balances resistance force between layers: ( ) = = DS c n o f f s
Bed-Load Transport Equations duBoys-Type Equations: If layer between 1stand nth moves according to a linear distribution, then n is the thickness of sediment material moving Thickness moves with an average velocity of vs(n-1)/2 3 m ) 1 ( n n s = = q v sb s 2 m
Bed-Load Transport Equations duBoys-Type Equations: Critical condition at which sediment motion is about to begin (n = 1) ( ) ( ) ( )cr o o n = v ( ) o = c = s q ( ) 2 sb o o o f s cr 2 cr o cr v = = characteri stic sediment coefficien t s ( ) 2 2 o cr
Bed-Load Transport Equations duBoys-Type Equations: Assumptions of this equation have been shown to disagree totally with observations of bed-load transport Bed load does not move as sliding layers (Schoklitsch, 1914) However, there is generally good agreement between this equation and field/laboratory data Proper use of the equation depends on correct evaluation of the characteristic sediment coefficient, Sidenote: generally all bed load equations based on excess shear stress are classified as duBoys-type equations
Bed-Load Transport Equations duBoys-Type Equations: Experiments of Schoklitsch (1914) proved duBoys model of sliding layers to be wrong, but did show that the equation fit the data well Uniform grains of various kinds of sand (limited data) and experimental flume experiment (small dimensions): 1 54 . 0 s = ( ) metric
Bed-Load Transport Equations duBoys-Type Equations: Straub (1935) - based on work of several researchers determined average values of several sand sizes Criticized because all data obtained from small flumes over a small range of particle sizes 173 . 0 4 / 3 English d = ( )
Bed-Load Transport Equations duBoys-Type Equations: Zeller (1963) gives graph for metric units:
Bed-Load Transport Equations duBoys-Type Equations: Shields (1936) proposed critical shear stress relation Intent was to present an abbreviated form of the factors influencing bed load transport Developed a semi- empirical tractive-force equation based on 1.06<ss<4.25 and 1.56<d<2.47 mm ( ) q = o o 10 sb s cr ( ) qS d e s
Bed-Load Transport Equations duBoys-Type Equations: Kalinske (1947) emphasized turbulence mechanism in flow above the bed: = f d u * ( ) o q sb cr o
Example Graf 6.D An artificial channel has been constructed to divert a certain discharge from a river. This channel has an approximately rectangular cross- section with a width of B = 46.5 m and a bed slope of Sf=6.5 x 10-4. Uniform flow is established when the flow depth is 5.6 m. Velocity-profile measurements suggest an average velocity of 1.8 m/s and n = 0.0212 (due to bed roughness). Estimate the bed-load transport using the Kalinske equation. Express the solid discharge as a concentration.
Example Graf 6.D Converting qsbto gsb: g = = = sb q Q q B G g B sb sb sb sb sb s Sediment concentration can be expressed in a number of different ways: Concentration by volume: Concentration by mass: Concentration by unit mass: 3 m Q s = = sb C s 3 m Q s kg G s = = sb C s kg G s kg G s = = sb C s 3 m Q s
Example Graf 6.D Relationship between these concentrations: = C C s s s C = = s s s C C ( ( ) ) s s + C s s m = + C m s s
Bed-Load Transport Equations Schoklitsch-Type Equations: Replaced excess shear stress criterion with critical water discharge (or depth): ( sb q q = ) q cr Gilbert (1914) expansive set of data for varying water discharge, energy grade line, and sediment properties 3 flumes of lengths 14, 31.5, and 150 ft Flume width ranged from 0.23 to 1.96 ft Discharge varied from 0.019 to 1.19 cfs Eight different kinds of sand (0.305 < d < 7.01 mm)
Bed-Load Transport Equations Schoklitsch-Type Equations: Schoklitsch first proposed equations in 1935 but modified them in 1950: Critical flow rate: 1 n / 1 2 = 3 / 5 cr q D S cr f d = = . 0 076 . 0 006 s D d m d d 40 cr S e = / 1 6 . 0 0525 n d 3 / 5 / 3 2 d = . 0 26 s q cr 7 / 6 S e
Bed-Load Transport Equations Schoklitsch-Type Equations: Schoklitsch first proposed equations in 1935 but modified them in 1950: Sediment discharge: 5 . 2 s ( ) / 3 e 2 = q S q q sb cr s : Establishe d for = 0 . 7 3 . 0 = d mm . 0 0004 . 0 020 S f
Example Graf 6.D An artificial channel has been constructed to divert a certain discharge from a river. This channel has an approximately rectangular cross- section with a width of B = 46.5 m and a bed slope of Sf=6.5 x 10-4. Uniform flow is established when the flow depth is 5.6 m. Velocity-profile measurements suggest an average velocity of 1.8 m/s and n = 0.0212 (due to bed roughness). Estimate the bed-load transport using the Schoklitsch equation. Express the solid discharge as a concentration.
Bed-Load Transport Equations 3 / 2 3 / 2 g g S = 4 . 0 17 sb e Meyer-Peter Muller (MPM): Switzerland research laboratory (ETH) Meyer- Peter et al. (1934): Laboratory flume with cross-section of 2 x 2 m and total length of 50 m (max discharge of 5 m3/s) and sediment discharge up to 4.3 kg/(s-m) Two grain sizes: 5.05 mm and 28.6 mm For sand, resulting bed- load equation d d = g q sb = sb s g q
Bed-Load Transport Equations Meyer-Peter Muller (MPM): ETH experiments were extended to include data with particle mixtures First attempt to include representative grain diameter with previous formulas failed Meyer-Peter et al. (1948) proposed following equation which fit all the data: ) ( ) ( s = ( 2/3 g R S sb = 1/3 hb M e 0.25 0.047 ( ) d d s s ( ) ) = s g q sb sb g sb q ( ) sb s
Bed-Load Transport Equations Meyer-Peter Muller (MPM): Meyer-Peter et al. (1948) : Rhb= hydraulic radius of bed Equivalent diameter = d50 Mis a roughness parameter: / 3 2 K ( ) = = 1 s no bedforms M M K s ( ) 1 bedforms M 21 d 1 . 26 = = = K grain roughness s / 1 6 / 1 6 d 50 90 U = = K total roughness s 3 / 2 / 1 e 2 R S hb Derived with d = 3.1-28.6 mm (applicable to d > 2.0 mm) Derived with Sf = 0.0004-0.020
Example Graf 6.D An artificial channel has been constructed to divert a certain discharge from a river. This channel has an approximately rectangular cross- section with a width of B = 46.5 m and a bed slope of Sf=6.5 x 10-4. Uniform flow is established when the flow depth is 5.6 m. Velocity-profile measurements suggest an average velocity of 1.8 m/s and n = 0.0212 (due to bed roughness). Estimate the bed-load transport using the Meyer-Peter et al. formulas. Express the solid discharge as a concentration.
Einsteins Bed-Load Transport Equation Developed in 1942 (empirical) and 1950 (analytical) Probabilistic model for transport of sediment as bed load Bed load transport is related to the fluctuations in velocity rather than the average velocity Beginning and end of motion are expressed in probabilistic terms with considerations for the lift and particle s weight Built on observations/experimental evidence: Bed load moves slowly downstream motion of an individual particle is quick step with long intermediate stops Average step made by any bed load particle is independent of flow condition, transport rate, and bed composition (always the same) Different transport rates due to changes in the average time between steps and the thickness of the moving layer
Einsteins Bed-Load Transport Equation Physical Model: Equilibrium condition of exchange of bed particles between the bed layer and the bed # particles deposited per time per unit bed area = # particles eroded per unit time per unit bed area Deposition: Each particle (d) has steps of length ALd and will be deposited over an area ALd long with unit width Let gs= bed load rate in weight per time per width, N/(m-s) Let is= fraction of bed load in given grain size Then gs is = rate at which given size moves through the unit width per unit time
Einsteins Bed-Load Transport Equation Physical Model: Weight of a single particle = sk2d3(k2= constant of grain volume) Number of particles of a certain fraction deposited per unit time and bed area: i g s s g i = s s )( ) ( 3 4 A d k d A k d 2 2 L s L s
Einsteins Bed-Load Transport Equation Physical Model: Erosion: Whether a particle will be eroded depends on the availability of the particle and on flow conditions (turbulence level) Let ib= fraction of bed material in a given grain size # particles of size d in a unit area of bed surface = ib/k1d2, where k1= constant of grain area Let pe/te= probablility of removal, where te= time consumed by each exchange # particles eroded per unit time and unit area: p i b d 1 e t 2 k e
Einsteins Bed-Load Transport Equation Physical Model: Erosion: No direct method is available for the determination of te Einstein (1942) suggests a function of vss: d t = k ( ) 3 e v g ss s
Einsteins Bed-Load Transport Equation Physical Model: Bed Load Equation (Equilibrium): Rate of deposition balances rate of erosion: ( ) g i i p g = s s b k e s 2 4 2 A k d k d 1 3 L s Exchange probability: pe= fraction of time during which the lift exceeds the weight of the particle at any one location Non-intensive sediment transport peis small and deposition is everywhere possible Strong sediment transport pebecomes larger and deposition is not everywhere possible
Einsteins Bed-Load Transport Equation Physical Model: Exchange probability: Einstein (1950) suggests that pecan be used to evaluate ALd : If peis small, distance traveled is constant: ALd = bd where bis approximately 100 For larger pe: Only (1-pe) particles can deposit after traveling bd And peparticles stay in motion would like to deposit but can t because after bd lift forces exceed weight Of these peparticles, pe(1-pe) are deposited after traveling 2 bd while pe2particles are still not deposited We can express the total travel distance as a series: d ( ) ( ) = n = + = n 1 1 b A d p p n d L b 1 p 0
Einsteins Bed-Load Transport Equation Physical Model: Substituting the series into the bed load equation: 1 p k k i g = 1 3 e s s ( ) 3 1 p k i gd 2 e b b s s 1 g = = Intensity of Bed Load Transport s ( ) 3 gd s s p i = = e s A A * * * 1 p i e b = Intensity of Bed Load Transport for individual an grain size *
Einsteins Bed-Load Transport Equation Empirical Relation (Einstein, 1942): p 42 42 = e A * * 1 p e 42 1 1 k k g 42 42 = 1 3 s A ( ) * * 3 k F gd 2 b s 1 s ( ) = . 0 816 F sand with d mm
