INTRODUCTION
A longitudinal profile of a fall has shown in the Fig. 1.
In this figure y_{c} is the critical depth, h is the height of fall,
y_{1} and y_{2 }are the conjugate depths, L_{p} is the
length between fall and the incident point of the jet in the stilling basin,
Y_{p} is the depth of stilling basin, H_{t} is the total energy
of the flow upstream of the fall and H_{1} and H_{2 }are_{
}the flow energy before and after the hydraulic jump, respectively. The
flow, after falling, enters into the pool downstream of the fall. Due to mixing
with the water in the pool, a part of total energy H_{t} is dissipated.
Then, a supercritical flow is established and due to the downstream sub critical
flow, the hydraulic jump is formed and as a result, a part of the flow energy
is dissipated. A stepped spillway includes several falls (vertical drops) which
are connected to each other on the steep slope.
Two forms of the flow including; Skimming and Nappe flows are established over
stepped spillways. In the Nappe flow; the flow hits the horizontal part of each
step of the spillway consequently from upstream to the downstream. Due to the
slope of the drop, the complete hydraulic jump may form or may not. The Nappe
flow can be developed in the low flow discharge and large depth of the step.
The energy dissipation in the steps is due to the kind of hydraulic jump and
mixing of air with the flow water (Chamani and Rajaratnam, 1994). In the Skimming
flow, steps act as a large roughness against the flow over steps.Ohtsu et
al. (2000) have described more about the flow characteristics in the skimming
flow over stepped spillways.

Fig. 1: 
Longitudinal profile of a stepped spillway 
In the Skimming flow over steps a pseudobottom which connects the end of
steps to each other is established. Researchers believed that the secondary
flow under this pseudobottom is the cause of the most of the energy dissipation
in this flow regime over the stepped spillway (Chanson, 1994a, c). The amount
of the dissipated energy due to the hydraulic jump is not reported in the literature
(Chamani and Rajaratnam, 1994; Chanson, 1994a). This research intends to develop
a model for estimation of the energy dissipation in the stepped spillways. The
energy dissipation due to hydraulic jump in the low longitudinal slope stepped
spillways is more important than the steep spillways. Due to this reason, several
tests over seven stepped spillways with low slopes including 25° (5, 10
and 15 steps) and 15° (5, 10, 15 and 30 steps) have been undertaken. The
results of this study have been compared with the other researchers’ findings.
For instance, Rand (1955) has undertaken several experiments over vertical drops
and reported several equations for computation of the flow characteristics.
Chanson (1994a) has presented the following equation for the energy dissipation
in the vertical drops.
In Eq. 1 H_{Dam} = the total head of the weir which
is equal to N multiplied by h (the height of each step). Equation 1 can be rewritten
as function of (y_{c}/h) as follows:
Equation 2 can be applied to calculate the dissipated energy in the Nappe flow
(Chanson, 1994a). Chanson has compared Eq. 2 with findings
of Stephenson (1979). It has been shown that there is good agreement between
results of Eq. 2 and the experimental results. The relative
head loss equation has been expressed as follows (Chamani and Rajaratnam, 1994):
Chamani and Rajaratnam (1994), using experimental data of Horner for h/l =
0.421 with 8, 10, 20 and 30 steps; h/l = 0.526 with 10 and 30 steps; h/l = 0.736
with 10 and 30 steps and h/l = 0.842 with 10 and 30 steps have calculated α.
They have shown that α decrease with increasing y_{c}/l. Chanson
(1994b) has discussed that for calculating the relative energy loss, Eq.
2 is more reliable than Eq. 3. It does not need the energy
loss to be calculated in each step. Several researchers including Ellis (1989),
Peyras et al. (1991) and Chamani and Rajaratnam (1994) believe that the
energy loss in the Nappe flow is more than the Skimming flow. Chanson (1994b,
c) believes that in the long stepped spillways with uniform flow, the energy
loss in the Skimming flow is more than the Nappe flow. Chanson and Gonzalez
(2004) have reviewed recent advances in stepped spillway design. There is a
lack of mathematical modelling of the stepped spillways in the prediction of
the stepped spillway head loss in the literature. In this research an attempt
has been paid to simulation of the stepped spillway hydraulics using ANSYS 10.0
(2005).
MATERIALS AND METHODS
This research program has been undertaken in the hydraulic laboratory of Shahid Chamran University, Iran. Laboratory flume was made of glass with 10 mm thickness and its properties are as follows: length = 10 m; width = 25 cm; height = 48 cm. Seven models of stepped spillways with 25° slope (5, 10, 15 steps) and 15° slope (5, 10, 15, 30 steps) were built of galvanized sheet. They were installed 4 m downstream of the upstream wall of the laboratory flume, individually. The water flow was supplied into the flume by a tank with a control valve. The height of the tank is 4.5 m. To control the hydraulic jump, a slide gate was used in the end of the laboratory flume. The location of the hydraulic jump was fixed at 0.5 m downstream of the toe of the stepped spillway. The depth of flow before and after the hydraulic jump was measured with a point gauge (0.1 mm accuracy) in the center of the flume. The discharge of the flow was measured with 53° triangle weir. A large amount of air was entered in the flow before hydraulic jump. Due to two phase flow, there was fluctuation in the water surface and the measurement of the water surface level was involved with 3 mm error. The water surface level immediately after hydraulic jump was measured somewhere downstream which air bubbles were observed.
Head loss due to hydraulic jump over each step is summation of two parts; the head loss between upstream of fall and the section before the hydraulic jump. It is due to impact of the flow jet to the water of pool. Another part of head loss is due to the hydraulic jump which may be a complete hydraulic jump or incomplete hydraulic jump. Several methods have been proposed for the head loss in step spillways have not taken into account the head loss of the hydraulic jump in the Nappe flow. In the present study the head loss due to the hydraulic jump is also taken into account. Barani (2005) tried to optimize the geometry of the stepped spillway using physical modeling technique.
In addition, ANSYS software has been used to simulate flow over stepped spillways.
ANSYS is a finite element model which solves NavierStokes and continuity equations
using finite element method. It can simulate turbulence flow with several turbulent
sub modules. Sub modules of ANSYS were examined for the best simulation process
of the phenomenon. Finally, the NKE turbulent model of ANSYS was selected to
simulate the same flow conditions of the physical modeling in this research
program.
RESULTS AND DISCUSSION
Physical modelling: Energy loss between upstream and downstream of the
fall before the hydraulic jump can be calculated by the following equation:
derivation of Eq. 4 over dy_{1} gives y_{1}
= y_{c} which is equal to h, the height of fall. Dimensionless equation
of Eq. 4 over h can be expressed as follows:
In Eq. 5 and 6 ΔH_{1}/h is
a function of y_{1}/h and y_{c}/h. For a y_{1}/h with
increasing of y_{c}/h, ΔH_{1}/h is increased up to one
which gives the ΔH_{1}= height of the pool is increased. Then ΔH_{1}/h
is decreased down to zero. The maximum of these curves are observed when y_{c}/h
= 0.1, 0.3 and 0.5. Figure 2 demonstrates results of
Eq. 5.
Another form of Eq. 5 is as follow:
Figure 3 illustrates the relative energy loss against y_{1}/h
and y_{c}/h of Eq. 7. Figure 3 shows
that there is a minimum ΔH_{1}/h when y_{c}/h = 0.85. y_{c}/h
= 0.85 is the border line for Nappe and Skimming flow. It denotes that with
increasing in the flow discharge next to the border line the flow regime will
change from one to the other case. Therefore, in a Nappe flow, increase in the
flow discharge in the border line when it is near to the Skimming flow, minimum
relative energy loss equal to 24.23% is exist.
In the complete hydraulic jump, energy loss can be estimated by Eq.
8 as below:

Fig. 2: 
Illustration of the relative energy loss against y_{1}/h
and y_{c}/h 

Fig. 3: 
Demonstration of the relative energy loss against y_{c}/h 
Dimensionless equation of above equation can be as the following equation:
Equation 9 shows that this relative energy is also function of y_{c}/h.
Figure 4 shows the correlation between these two parameters.
It is observed that this relative energy in maximum (0.485) when y_{c}/h
= 0.41. The energy loss due to the hydraulic jump starts from 0.2 to its maximum
with increase in y_{c}/h. Then, it will decrease with increase in y_{c}/h.
The value of y_{c}/h = 1 is the limit for transformation of Nappe flow
regime to Skimming flow regime. The rest of the curve from y_{c}/h =
1 to y_{c}/h = 4.5 due to lack of formation of the hydraulic jump is
not applicable.
Summation of ΔH_{1} and ΔH_{2} is shown in the Fig.
5. This Fig. 5 gives very good criteria for analysis of
the energy loss in the Nappe and Skimming flows. The variation of y_{c}/h
can be classified to three zones as follows:

Fig. 4: 
Variation of the relative energy loss and y_{c}/h
Eq. 9 

Fig. 5: 
Summation of relative energy loss against y_{c}/h 
A: Zone one for y_{c}/h = 0.0 to 1.0 which in this range, the
Nappe flow regime is taken place. In this zone the energy loss is decreased.
Comparing with the two other zones in the first zone the energy loss is maximum.
B: Zone two for y_{c}/h = 1.0 to 1.79 which in this range the energy loss decrease with increasing in y_{c}/h. In y_{c}/h = 1.79 the relative energy loss equals 0.618. The left limit shows that the nappe flow starts to be established and the right limit of this range denotes the minimum of the energy loss in the Nappe flow regime. Comparison of the first and second zones shows in the Skimming flow regime the head loss is less than the Nappe flow (y_{c}/h = 1 up to 1.79). This result is in agreement with findings of Chamani and Rajaratnam (1994) and Matos and Quintela (1994).
C: Zone three which y_{c}/h > 1.79. In this zone the head loss increases and for the Skimming flow is more than the Nappe flow. There is a good agreement between result of this research program and Chanson results.
Actually, in the long stepped spillways with uniform flow, the head loss in
the Skimming flow is more than the Nappe flow. Figure 6 compares
findings of some research programs.

Fig. 6: 
Comparison of the relative energy loss in different studies 
Mathematical modelling: ANSYS software has been selected to simulate
relative energy loss in the Nappe flow on the stepped spillways. ANSYS is a
finite element model which solves NavierStokes and continuity equations using
finite element method. Flow conditions in the upstream and downstream of the
physical models were given to ANSYS as boundary and initial conditions. Figure
7 demonstrates simulated flow velocity vectors over one of physical model
established in this study.
As it is shown in Fig. 7, ANSYS software is able to simulate
the flow hydraulics over stepped spillways very successfully. Several mathematical
models based upon established physical models were developed in ANSYS. Using
data of the physical models as boundary conditions, relative head losses due
to different number of steps in the spillway are compared with the observed
data and the same results of Chanson's proposed experimental equation. Figure
8 shows this comparison for two very important dimensionless parameters
in the dimensional analysis of the phenomenon.
As can be seen in the Fig. 8, a very good agreement is seen
between observed data and ANSYS outputs. As it is shown, Chanson's experimental
equation results are less agreed with the observed data. In addition, the same
comparison is undertaken for relative energy loss in the Nappe flow due to number
of steps in the stepped spillway. Figure 9 demonstrates the
results of this part of the study.
In Fig. 9, Chanson's experimental equation has shown less
accuracy than ANSYS. However, there is very good agreement between ANSYS outputs
and experimental observation data.

Fig. 7: 
Predicted velocity vectors for the collected experimental
data 

Fig. 8: 
Comparison of ANSYS outputs and Chanson's experimental equation
with observed data for the Nappe flow 

Fig. 9: 
Comparison of ANSYS outputs and Chanson's experimental equation
with observed data for different number of steps 
CONCLUSIONS
In the present study, the Nappe flow hydraulics has been studied at, Shahid Chamran University, Iran using physical and mathematical models. The energy loss in the Nappe flow is classified into two parts and dimensionless equations have been derived for both of them. In the stepped spillways with mild slope, complete hydraulic jump or incomplete hydraulic jump can be formed. Therefore, the energy loss of hydraulic jump has been entered in the equations. Then, for assessment of the accuracy of the equations, several experiments have been set up in the hydraulic laboratory. The experiments have been undertaken in two models with slope angle of 15 and 25°. In the Nappe flow with high discharge (Nh/y_{c}<15), the equations presented by Chamani and Rajaratnam is in very good agreement with results of this research program.
The hydraulic jump effect is not considered in the presented equations by the researchers in their previous. It denotes that in the Nappe flow with high flow discharge, the hydraulic jump does not form. Therefore, the head loss due to the hydraulic jump is not considered in the previous studies. However increasing of Nh/y_{c} in the lower flow discharge, the hydraulic jump is formed over each individual step. It is considered in the present study. Finally, ANSYS is employed to simulate hydraulics over stepped spillways. It is concluded that ANSYS has capability to cope with the phenomenon in the stepped spillway with less than 6% error.