12
2 Similarity of Model Experiments
experiment, the gravity effect is greater than viscosity for turbulent flows caused
by the rainfall. Consequently, the time scale number of the rainfall duration, λ t , is
related to Fr similarity; that is,
λ t = t p /t m = λ
0.5
L
(2.1)
where t is rainfall duration (s); subscripts p and m stand for prototype parameter and
model parameter respectively.
The emphasis of the proposed method is that the ratio of geomorphologic evolvement ratio of the model corresponds to the prototype ratio, which remains constant
after rainfall events, such that the prototype soil erosion processes can be measured
based on the model experimental results:
¯
Y m(i) /L m
¯
Y p(i) /L p
=
λ L
λ Y
= R
(2.2)
where i is the sequence of rainfall events; ¯
Y (i) is the mean erosion/deposition depth
of the landform during rainfall events (m); L is the length of the small watershed
(m); Y /L is a dimensionless term, which stands for the geomorphologic evolvement
ratio of a single rainfall event; and R is the ratio of erosion extent between the
model and prototype. When R = 1, the erosion/deposition depth scale number, λ Y,
equals the length scale number, λ L , illustrating that the erosion/deposition extent
¯
Y p /L p for a single simulated rainfall event in the prototype is equal to the product
of the scaled model ¯
Y m /L m . However, when R = 1, a model is distorted because
the erosion/deposition scale differs from the length scale of the physical soil erosion
model. Thus, the length scale number, λ L , in the scaled model can be obtained by
multiplying the erosion/deposition depth scale number, λ Y , by ratio R.
Since the soil density ρ s (kg/m
3 ) in the prototype is close to the model one, the
soil loss scale number λ S could be calculated as follows:
λ S =
S p
S m
=
A p × ¯
Y p × ρ S p
A m × ¯
Y m × ρ S m
=
λ
3
L
R
(2.3)
where S is the soil loss, and A is the drainage area. When R = 1, the scale number of
soil loss, λ S , equals the volume-scale number, λ
3
L , which could estimate the volume
of the prototype soil loss based on the volume-scale number in the model experiment.
When R > 1, the model erosion volume must be multiplied by the coefficient λ
3
L /R to
estimate the amount of soil loss in the prototype, and the model landform varies more
serious than the prototype landform under the same rainfall scale. When R < 1, the
model erosion volume must be multiplied by a large coefficient to estimate the amount
of soil erosion in the prototype. A comparison of the sediment transport capacity and
sedimentation similarity in the river engineering indicates that the conditions of
kinematic similarity and dynamic similarity are embodied in the scale of cumulative
erosion volume for each rainfall event (Eq. 2.3). Four measures are required to ensure
2 Similarity of Model Experiments
experiment, the gravity effect is greater than viscosity for turbulent flows caused
by the rainfall. Consequently, the time scale number of the rainfall duration, λ t , is
related to Fr similarity; that is,
λ t = t p /t m = λ
0.5
L
(2.1)
where t is rainfall duration (s); subscripts p and m stand for prototype parameter and
model parameter respectively.
The emphasis of the proposed method is that the ratio of geomorphologic evolvement ratio of the model corresponds to the prototype ratio, which remains constant
after rainfall events, such that the prototype soil erosion processes can be measured
based on the model experimental results:
¯
Y m(i) /L m
¯
Y p(i) /L p
=
λ L
λ Y
= R
(2.2)
where i is the sequence of rainfall events; ¯
Y (i) is the mean erosion/deposition depth
of the landform during rainfall events (m); L is the length of the small watershed
(m); Y /L is a dimensionless term, which stands for the geomorphologic evolvement
ratio of a single rainfall event; and R is the ratio of erosion extent between the
model and prototype. When R = 1, the erosion/deposition depth scale number, λ Y,
equals the length scale number, λ L , illustrating that the erosion/deposition extent
¯
Y p /L p for a single simulated rainfall event in the prototype is equal to the product
of the scaled model ¯
Y m /L m . However, when R = 1, a model is distorted because
the erosion/deposition scale differs from the length scale of the physical soil erosion
model. Thus, the length scale number, λ L , in the scaled model can be obtained by
multiplying the erosion/deposition depth scale number, λ Y , by ratio R.
Since the soil density ρ s (kg/m
3 ) in the prototype is close to the model one, the
soil loss scale number λ S could be calculated as follows:
λ S =
S p
S m
=
A p × ¯
Y p × ρ S p
A m × ¯
Y m × ρ S m
=
λ
3
L
R
(2.3)
where S is the soil loss, and A is the drainage area. When R = 1, the scale number of
soil loss, λ S , equals the volume-scale number, λ
3
L , which could estimate the volume
of the prototype soil loss based on the volume-scale number in the model experiment.
When R > 1, the model erosion volume must be multiplied by the coefficient λ
3
L /R to
estimate the amount of soil loss in the prototype, and the model landform varies more
serious than the prototype landform under the same rainfall scale. When R < 1, the
model erosion volume must be multiplied by a large coefficient to estimate the amount
of soil erosion in the prototype. A comparison of the sediment transport capacity and
sedimentation similarity in the river engineering indicates that the conditions of
kinematic similarity and dynamic similarity are embodied in the scale of cumulative
erosion volume for each rainfall event (Eq. 2.3). Four measures are required to ensure