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1. SOFT COMPUTING
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Soft Comput (2009) 13:701–707
DOI 10.1007/s00500-008-0343-7
ORIGINAL PAPER
Predicting flow conditions over stepped chutes based on ANFIS
Davut Hanbay · Ahmet Baylar · Emrah Ozpolat
Published online: 15 July 2008
© Springer-Verlag 2008
Abstract Chute flow may be either smooth or stepped. The
flow conditions in stepped chutes have been classified into
nappe, transition and skimming flows. In this paper, characteristics
of flow conditions are presented systematically
under a wide range of critical flow depth, step height and
chute slope. The Adaptive Network Based Fuzzy Inference
System (ANFIS) is used to predict flow conditions in stepped
chutes using critical flow depth, step height and chute slope
information. The proposed model performance is determined
by threefold cross validation method. The evaluated classification
accuracy of ANFIS model is 99.01%. The test results
showed that the proposed ANFIS model can be used successfully
for complex process control in hydraulic systems.
Keywords ANFIS · Fuzzy membership function · Stepped
chute · Flow conditions · Cross validation
1 Introduction
In a stepped chute, the chute face is provided with a series of
steps, from near the crest to the toe. The provision of steps
can produce significant energy dissipation. Stepped chutes
are commonly used for gabion weirs, river training, irrigation
channels, and storm waterways. Stepped chutes are used
also for in-stream re-aeration and in water treatment plants
to enhance the air–water transfer of atmospheric gases (e.g.
oxygen, nitrogen) and of volatile organic components.
D. Hanbay
Electronic and Computer Science Department, Firat University,
Elazig, Turkey
A. Baylar (B) · E. Ozpolat
Civil Engineering Department, Firat University, Elazig, Turkey
e-mail: abaylar@firat.edu.tr
Three distinct flow conditions are found on stepped chutes,
so-called nappe flow, transition flow, and skimming flow
(Fig. 1). Rajaratnam (1990), Chamani and Rajaratnam (1994),
Chamani and Rajaratnam (1999a), Chamani and Rajaratnam
(1999b), Chanson (1994a,b, 1996) and Hewlett et al. (1997)
have focused mainly on the hydraulic design of stepped chute,
and in particular, characteristics of nappe flow, transition
flow, and skimming flow over stepped chutes. Recently,
Baylar and Emiroglu (2003), Emiroglu and Baylar (2003,
2006) and Baylar et al. (2006, 2007a,b,c) did some detailed
experiments on the aeration efficiency of stepped chutes.
In recent years the developments in intelligent methods
make them possible to use in complex systems modeling.
One of these intelligent methods is artificial neural network.
Artificial neural network performance depends on the size
and quality of training samples. When the number of training
data is small, not representative of the possibility space,
standard neural network results are poor. The other intelligent
method is fuzzy theory, which has been used successfully in
many applications (Frayman and Wang 1998; Kecman 2001;
Held et al. 2006; Wang and Fu 2005; Hanbay et al. 2006a,b).
Fuzzy methods provide linguistic labels for complex system
modeling. There are many advantages of fuzziness, one of
which is the ability to handle blur data. Both artificial neural
network and fuzzy logic advantages are used in ANFIS’s
architecture. ANFIS uses a hybrid learning algorithm to identify
parameters of Sugeno-type fuzzy inference systems. It
applies a combination of the least-squares method and the
backpropagation gradient descent method for training membership
function parameters to emulate a given training data
set.
ANFIS is widely used in complex system studies for
modeling, control or parameter estimating. However, its
application to the hydraulic systems related studies are very
limited (Hanbay et al. 2007). In this study, the flow conditions
123

702 D. Hanbay et al.
Fig. 1 Flow conditions on a stepped chute. a Skimming flow, b transition
flow, c nappe flow
in stepped chutes are predicted based on the ANFIS. The
paper is organized as follows. In Sect. 2, wereviewsome
basic properties of stepped chute flow and the ANFIS. In
Sect. 3, experimental data are given and the effectiveness
of the proposed structure is demonstrated. Finally, Sect. 4
presents conclusion.
2 Preliminaries
The theoretical foundations of the presented study are given
in the following subsections.
2.1 Flow conditions over stepped chutes
2.1.1 Skimming flow condition
For large discharges, the waters flow down a stepped chute
as a coherent stream “skimming” over the steps. The external
edges of the steps form a pseudo-bottom over which
the flow skims. Beneath the pseudo-bottom, recirculating
vortices develop and recirculation is maintained through the
transmission of shear stress from the main stream (Fig. 1a).
Small-scale vorticity is also generated at the corner of the
steps. The aerated flow region follows a region where the
free-surface is smooth and glassy. Next to the boundary however,
turbulence is generated and the boundary layer grows
until the outer edge of the boundary layer reaches the surface.
When the outer edge of the boundary layer reaches the
free surface, the turbulence can initiate natural free surface
aeration. The location of the start of air entrainment is called
the point of inception. Downstream of the inception point of
123
free-surface aeration, the flow becomes rapidly aerated and
the free-surface appears white. Air and water are fully mixed
forming a homogeneous two-phase flow (Chanson 2002).
2.1.2 Transition flow condition
Ohtsu and Yasuda (1997) were probably the first to introduce
the concept of a “transition flow” condition, although they
did not elaborate on its flow properties. For a given stepped
chute geometry, a range of flow rates gives an intermediary
flow condition between nappe flows at low discharges and
skimming flows at large flow rates. In the transition flow
condition, air bubble entrainment takes place along the jet
upper nappe and in the spray region downstream of the stagnation
point. The flow highly turbulent, air and water are
continuously mixed (Fig. 1b). The air entrainment process in
transition flow is not yet fully understood (Chanson 2002).
2.1.3 Nappe flow condition
For a given flat step geometry, low flows behave as a series
of free-falling jets with nappe impact onto the downstream
step: i.e. nappe flow condition. At the upstream end of each
step, the flow is characterized by a free-falling nappe, an air
cavity and a pool of recirculating fluid (Fig. 1c). In a nappe
flow, air is entrained at the jet interfaces and by a plunging
jet mechanism at the intersection of the lower nappe with
the recirculating pool, while de-aeration is often observed
downstream. In the free-falling nappe, interfacial aeration
takes place at both the upper and lower nappes. At the lower
nappe, the developing shear layer is characterized by a high
level of turbulence and significant interfacial air entrainment
is observed (Chanson 2002).
2.2 Adaptive network based fuzzy inference system
(ANFIS)
The architecture and learning rule of ANFIS have been
described in detail in (Jang 1993). ANFIS is a multilayer
feed forward network where each node performs a particular
function on incoming signals. Both square and circle
node symbols are used to represents different properties of
adaptive learning. To perform desired input-output characteristics,
adaptive learning parameters are updated based on
gradient learning rules (Jang 1993). ANFIS model is one of
the implementation of a first order Sugeno fuzzy inference
system (Kulkarni 2001). The rules are of the form in Eq. (1).
In this system
If x1 is A1, x2 is A2, then y = px1 + qx2 + r (1)
where x1 and x2 inputs A1 and A2 corresponding term set,
y is output, p, q, r are constant. An ANFIS model is shown
in Fig. 2. It is a multi-input, one-output model; a multi-output

Predicting flow conditions over stepped chutes based on ANFIS 703
X1
X 2
L1
A1
A2
A3
B1
B2
B3
L2
π
π
π
W1
W n
Fig. 2 ANFIS model structure
L3
N
N
N
W11
W 1n
model can be designed by connecting few single output
models. The node functions in the same layer are similar
and described as below
Layer-1 Every node i in this layer is a square node with
a node function. Nodes in layer 1 implement fuzzy membership
functions, mapping input variables to corresponding
fuzzy membership values. Outputs of this layer can be
described as Eq. (2)
O 1
i
= µAi (x) (2)
where x is input to node i, and Aiis linguistic label associated
with this node function. O1 i is the membership function of
Ai, the fuzzy membership functions can take any form, such
as triangular, Gaussian but usually µAi (x) is chosen bellshaped
with maximum equal to 1 and minimum equal to 0.
Detail information on the types of membership functions was
described by Bhattacharya and Vasant (2007), Vasant and
Bhattacharya (2007), Vasant et al. (2007) and Bhattacharya
et al. (2008).
Layer-2 Every node in this layer is a circle node labeled
which multiplies the incoming signals and sends the product
out. For instance,
wi = µA1 (x)µA2 (y)... i = 1, 2, 3,...,N (3)
Each node output represents the firing strength of a rule.
Layer-3 Every node in this layer is a circle node labeled
N. Theith node calculates the ratio of the ith rules firing
strength to the sum of all rule’s firing strengths. Where ¯w is
the normalized firing strength of rules.
¯w =
wi
w1 + w2 ···wN
L4
L5
i = 1, 2, 3,...,N (4)
Layer-4 Every nodei in this layer is a square node with a
node function,
O 4 i = ¯wi fi = ¯wi(px + qy + ...+ r)
i = 1, 2, 3,...,N (5)
Σ
Y
where ¯wi is the output of layer 3 and {p, q, r} is the parameter
set. Parameters in this layer will be referred to as consequent
parameters.
Layer-5 The single node in this layer is a circle node
labeled that computes the overall output of ANFIS as the
summation of all incoming signals.
wi fi
i
= overall output = ¯wi fi =
(6)
O 5 i
3 Application
3.1 Experiments
i
The data used in this study were taken from studies conducted
by Baylar and Emiroglu (2003) and Baylar et al. (2006)ona
large model of a stepped chute. A schematic representation
of the experimental set-up is shown on Fig. 3 that shows a
prismatic rectangular chute, 0.30 m wide and 0.50 m deep,
in which the steps were installed. The side walls were made
of transparent methacrylate to follow flow condition. Water
was pumped from the storage tank to stilling tank, from which
water entered the chute through an approach channel, with
its bed 1.25 and 2.50 m above the laboratory floor. Downstream
channel used in this study was 3.0 m long, 0.35 m
wide and 0.45 m deep. The experiments reported here were
carried out, with unit discharges ranging between 16.67 ×
10 −3 and 166.67 × 10 −3 m 2 /s. The discharge was measured
by means of a flow meter installed in the supply line. Chute
slopes were equal to 14.48 ◦ , 18.74 ◦ , 22.55 ◦ ,30 ◦ ,40 ◦ , and
50 ◦ and for all slopes tested, steps with h equal to 5, 10, and
15 cm were used.
3.2 Experimental results
The experimental results indicated that the type of flow condition
is a function of the flow rate, step height and chute slope.
Three different flow conditions, namely the nappe, transition
and skimming flow conditions occur on stepped chutes.
A tendency towards the nappe flow condition is observed
with increasing step height and decreasing unit discharge
and chute slope. However, the results show a tendency
towards the transition and skimming flow conditions as
unit discharge and chute slope increase and as step height
decreases. Figure 4 shows classification of flow conditions
on stepped chutes.
3.3 ANFIS modeling results
All program code was written in MATLAB. The parameters
considered in the study are the ratio between the critical
i
wi
123

704 D. Hanbay et al.
Fig. 3 Laboratory stepped
chute apparatus
hc/ h
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
Skimming
flow
Transition
flow
Nappe
flow
10
20
30
Tank
α (degrees)
40
Water
pump
Water
flowmeter
Fig. 4 Flow conditions on stepped chutes (Baylar et al. 2006)
flow depth and step height (hc/h), chute slope (α) and flow
condition. The parameters (hc/h) and α are used as inputs to
the ANFIS for the prediction of flow condition. Three flow
conditions, nappe, transition and skimming, are denoted as
the numbers 1, 2 and 3, respectively. Total 126 experimental
data sets were used in this study. Threefold cross validation
method was used for proposed model validation. The training
performance of ANFIS model is shown in Fig. 5. InFig.5
error is the array of root mean square training errors.
The test results are represented in Table 1 and graphically
showed in Fig. 6. In the fourth column of Table 1, the numbers
1, 2 and 3 indicate the flow conditions, nappe, transition
and skimming, respectively. In the fifth column, the ANFIS
predictions are presented. In the sixth column, the classifica-
123
50
Water feed line
Flow
control valve
60
Error
0.194
0.192
0.190
0.188
0.186
0.184
0.182
Grid
Upstream
channel
Stilling
tank
Stepped chute
α
Downstream
channel
0.180
0 20 40 60
Epochs
80 100
Fig. 5 Training performance of ANFIS model
tion result of ANFIS model is presented. It can be obviously
seen from Table 1 that the ANFIS predicts observed flow
conditions at high accuracy. The correct classification
success rate was found as 99.01%.
The ANFIS structure is described in Table 2 for gbellmf
membership function. Different types of membership functions
were used. The best performance is obtained with
gbellmf as seen from Table 3. Each input has three membership
function. The training epoch for all membership is
100.
4 Conclusions
Stepped chute flows are characterized by a high level of turbulence
and large amounts of entrained air. The mechanisms
by which air is entrained into water because of a stepped chute
are several and complex. Basic air entrainment mechanisms
occur in nappe, transition and skimming flow conditions. For
the hydraulic design of a stepped channel, it is important to

Predicting flow conditions over stepped chutes based on ANFIS 705
Table 1 The test data and ANFIS outputs for predicted flow condition
hc/h α Flow condition Flow condition Flow condition Flow condition Prediction
(deg.) observed observed predicted by ANFIS classified by ANFIS error (%)
0.960 14.48 Transition 2 2.2585 2 12.9250
2.020 14.48 Skimming 3 2.9928 3 0.2400
0.600 18.74 Nappe 1 1.0670 1 6.7000
1.540 18.74 Skimming 3 3.0122 3 0.4067
2.820 18.74 Skimming 3 3.1594 3 5.3133
1.260 22.55 Skimming 3 2.8854 3 3.8200
2.440 22.55 Skimming 3 2.9654 3 1.1533
0.960 30.00 Skimming 3 2.8511 3 4.9633
2.020 30.00 Skimming 3 3.0531 3 1.7700
0.600 40.00 Transition 2 1.4657 1 46.5700
1.540 40.00 Skimming 3 3.0196 3 0.6533
2.820 40.00 Skimming 3 2.9737 3 0.8767
1.260 50.00 Skimming 3 3.0539 3 1.7967
2.440 50.00 Skimming 3 2.9654 3 1.1533
0.480 14.48 Nappe 1 1.0142 1 1.4200
1.010 14.48 Transition 2 2.4281 2 21.4050
0.300 18.74 Nappe 1 0.9513 1 4.8700
0.770 18.74 Transition 2 1.4971 1 49.7100
1.410 18.74 Skimming 3 2.9742 3 0.8600
0.630 22.55 Nappe 1 1.1473 1 14.7300
1.220 22.55 Skimming 3 2.8626 3 4.5800
0.480 30.00 Nappe 1 0.9100 1 9.0000
1.010 30.00 Skimming 3 2.9849 3 0.5033
0.300 40.00 Nappe 1 1.0433 1 4.3300
0.770 40.00 Transition 2 2.1442 2 7.2100
1.410 40.00 Skimming 3 2.9828 3 0.5733
0.630 50.00 Transition 2 2.0776 2 3.8800
1.220 50.00 Skimming 3 3.0504 3 1.6800
0.320 14.48 Nappe 1 0.9874 1 1.2600
0.673 14.48 Nappe 1 1.2924 1 29.2400
0.200 18.74 Nappe 1 1.0168 1 1.6800
0.513 18.74 Nappe 1 0.9667 1 3.3300
0.940 18.74 Transition 2 2.1103 2 5.5150
0.420 22.55 Nappe 1 0.9218 1 7.8200
0.813 22.55 Transition 2 1.8099 2 9.5050
0.320 30.00 Nappe 1 0.9260 1 7.4000
0.673 30.00 Nappe 1 1.3815 1 38.1500
0.200 40.00 Nappe 1 0.9901 1 0.9900
0.513 40.00 Nappe 1 1.2656 1 26.5600
0.940 40.00 Skimming 3 2.8379 3 5.4033
0.420 50.00 Nappe 1 1.4913 1 49.1300
0.813 50.00 Skimming 3 2.5796 3 14.0133
predict flow conditions. In this study, an ANFIS model was
used in prediction of flow condition over stepped chutes. The
most important aspect of the intelligent model is the ability
of self-organization of the ANFIS without requirements of
programming and the immediate response of a trained net
during real-time applications. These features make the
123

706 D. Hanbay et al.
Flow regime
3.5
3
2.5
2
1.5
1
0.5
Measured Data
ANFIS Output
0 10 20 30 40 50
Samples
Fig. 6 Test performance of ANFIS model (The membership function
number is 3 for each input)
Table 2 The ANFIS structure for (gbellmf)
Number of nodes 35
Number of linear parameters 27
Number of nonlinear parameters 18
Total number of parameters 45
Number of training data pairs 84
Number of checking data pairs 42
Number of fuzzy rules 9
Table 3 The performance of different type membership functions
Membership Classification accuracy (%) Average
functions classification
Test dataset-1 Test dataset-2 Test dataset-3 accuracy (%)
Dsigmf 98.84 99.08 98.68 98.8667
Gauss2mf 98.99 98.82 98.87 98.8933
Gbellmf 98.97 99.16 98.91 99.0133
Pimf 98.90 98.05 98.05 98.3333
Psigmf 98.83 99.08 98.68 98.8633
Trapmf 98.81 98.94 97.75 98.5000
Trimf 98.63 99.07 99.03 98.9100
intelligent model suitable for complex systems modeling.
The modeling performance of this study shows the advantages
of intelligent modeling. It is rapid and easy to operate.
This model offers advantage in commercial application.
Although our intelligent model was carried out on the hydraulic
systems, similar results for the biomedical, biological and
industrial systems can be expected.
123
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