This relation between shear stress and flow velocity
can also be used for flow in channels where we have
bedload transport (Fig. 2.8). Solving the two equations
above with respect to the velocity (v) we obtain:
v ¼ C R sin α
ð
Þ
1=2
This is the Chezy equation and C is the Chezy
number.
The value of C depends on the roughness of the bed
and on the shape of the channel, particularly its sinuosity.
Often used in engineering for calculating the velocity of water in channels, is Manning’s formula:
v ¼ R
2=3
Á sin α
ð
Þ
1=2 =η
where n is the coefficient of roughness of the bed: n ¼
0.01 corresponds to a smooth metal plate and n ¼ 0.06
to a shifting bed of gravel. It is of great practical
importance to be able to calculate water velocity and
thereby the erosion potential of artificial channels.
The Froude number is a parameter which is often
used to describe water flow:
F ¼ v= g Á h
ð
Þ
1=2
where v is the average velocity, h is depth of water and
g the force of gravity. The Froude number is the ratio
between the kinetic energy of the water masses (which
is proportional to the square of the velocity) and the
force of gravity, which is proportional to the depth, h.
For low Froude numbers the water flows out of phase
with the bedforms, and current ripples or crossbedding develop. This is called the lower flow regime.
When the velocity, v, becomes high in relation to the
depth of water, h, rapid or shooting flow develops
where the waves come into phase with the boundary
irregularities (Fig. 2.9); this represents the upper flow
regime.
The transition between lower and upper flow
regimes corresponds to a Froude number of 0.6–0.8.
2.9
Sediment Transport Along the Bed
Due to Water Flow
What actually gives flowing water the capacity to
carry sediment, and how are sediment particles
transported?
We have seen that flowing water exerts shear forces
against the stream bed. Frictional forces are converted
into turbulence in the overlying water, and have the
effect of transporting particles along the stream bed.
Under moderate flow conditions the largest particles
will be transported along, or just above, the bed as
bedload (Fig. 2.8). This takes place partly through
rolling or slow creep, partly through saltation, i.e. the
grains jump along the bed.
Saltation can be partly explained through
Bernoulli’s equation:
P þ g Á h þ
v
2
2
¼ CðconstantÞ
Here P ¼ pressure, h ¼ height above the stream
bed, and v ¼ velocity. We see that water which flows
over a sediment grain on the bed will have a greater
velocity than water which flows under the grain.
Bernoulli’s equation predicts that the pressure above
the grain must be less than the pressure adjacent to the
grain (P), and when this difference becomes sufficiently great it will be possible to lift the grain from
the stream bed. This “airplane wing effect” does not
work once the grain is in the water above the stream
bed, and the grain will then drop to the bed again.
The condition for sediment grains being
transported in suspension is that their settling velocity
must be less than the upward vertical turbulence component. This means that the grain must be transported
upwards through the water at least as fast as it falls
downwards. The magnitude of the vertical turbulence
upwards will be a function of the horizontal velocity
Dissolved ions
Dissolved load
Turbulence
Suspended load
Bottom transport
Bed load
(Na
+ , Cl
− etc.)
Saltation
Fig. 2.8 Different forms of transport in water. Sediment grains
may be carried in suspension if the vertical component of the
turbulence is equal to the falling velocity of the grains. Larger
grains are carried along the bottom due to the shear stress
42
K. Bjørlykke
can also be used for flow in channels where we have
bedload transport (Fig. 2.8). Solving the two equations
above with respect to the velocity (v) we obtain:
v ¼ C R sin α
ð
Þ
1=2
This is the Chezy equation and C is the Chezy
number.
The value of C depends on the roughness of the bed
and on the shape of the channel, particularly its sinuosity.
Often used in engineering for calculating the velocity of water in channels, is Manning’s formula:
v ¼ R
2=3
Á sin α
ð
Þ
1=2 =η
where n is the coefficient of roughness of the bed: n ¼
0.01 corresponds to a smooth metal plate and n ¼ 0.06
to a shifting bed of gravel. It is of great practical
importance to be able to calculate water velocity and
thereby the erosion potential of artificial channels.
The Froude number is a parameter which is often
used to describe water flow:
F ¼ v= g Á h
ð
Þ
1=2
where v is the average velocity, h is depth of water and
g the force of gravity. The Froude number is the ratio
between the kinetic energy of the water masses (which
is proportional to the square of the velocity) and the
force of gravity, which is proportional to the depth, h.
For low Froude numbers the water flows out of phase
with the bedforms, and current ripples or crossbedding develop. This is called the lower flow regime.
When the velocity, v, becomes high in relation to the
depth of water, h, rapid or shooting flow develops
where the waves come into phase with the boundary
irregularities (Fig. 2.9); this represents the upper flow
regime.
The transition between lower and upper flow
regimes corresponds to a Froude number of 0.6–0.8.
2.9
Sediment Transport Along the Bed
Due to Water Flow
What actually gives flowing water the capacity to
carry sediment, and how are sediment particles
transported?
We have seen that flowing water exerts shear forces
against the stream bed. Frictional forces are converted
into turbulence in the overlying water, and have the
effect of transporting particles along the stream bed.
Under moderate flow conditions the largest particles
will be transported along, or just above, the bed as
bedload (Fig. 2.8). This takes place partly through
rolling or slow creep, partly through saltation, i.e. the
grains jump along the bed.
Saltation can be partly explained through
Bernoulli’s equation:
P þ g Á h þ
v
2
2
¼ CðconstantÞ
Here P ¼ pressure, h ¼ height above the stream
bed, and v ¼ velocity. We see that water which flows
over a sediment grain on the bed will have a greater
velocity than water which flows under the grain.
Bernoulli’s equation predicts that the pressure above
the grain must be less than the pressure adjacent to the
grain (P), and when this difference becomes sufficiently great it will be possible to lift the grain from
the stream bed. This “airplane wing effect” does not
work once the grain is in the water above the stream
bed, and the grain will then drop to the bed again.
The condition for sediment grains being
transported in suspension is that their settling velocity
must be less than the upward vertical turbulence component. This means that the grain must be transported
upwards through the water at least as fast as it falls
downwards. The magnitude of the vertical turbulence
upwards will be a function of the horizontal velocity
Dissolved ions
Dissolved load
Turbulence
Suspended load
Bottom transport
Bed load
(Na
+ , Cl
− etc.)
Saltation
Fig. 2.8 Different forms of transport in water. Sediment grains
may be carried in suspension if the vertical component of the
turbulence is equal to the falling velocity of the grains. Larger
grains are carried along the bottom due to the shear stress
42
K. Bjørlykke