Flow regime prediction in stepped channels using ... - Semantic Scholar

International Journal of Science & Technology Volume 3, No 1, 109-121, 2008

Flow Regime Prediction in Stepped Channels Using Neural Computing Technique 1

Ozgur KISI1, M. Emin EMIROGLU2 and Ahmet BAYLAR2 Erciyes University, Civil Engineering Department, Kayseri, TURKIYE 2 Firat University, Civil Engineering Department, Elazig, TURKIYE k is i @er c i yes .edu . tr (Received: 07.05.2007 ; Accepted: 10.01.2008)

Abstract: A chute is characterized by a steep bed slope associated with torrential flow. This chute flow may be either smooth or stepped. The flow conditions in stepped channels are classified as nappe flow, transition flow and skimming flow. In this paper, hydraulic characteristics of flow regimes on the stepped channels are presented systematically under a wide range of discharge, channel slope and step height. The artificial neural network (ANN) was used for predicting flow regimes in stepped channels using discharge, channel slope and step height parameters. The test results indicated that the ANN could be successfully used in flow regime prediction in stepped channels. Keywords: Stepped channel, Skimming flow, Transition flow, Nappe flow, Neural networks.

Yapay Zekâ Hesaplama Teknikleri Kullanılarak Basamaklı Kanallar Üzerindeki Akım Rejiminin Belirlenmesi Özet: Bir şüt akımı, büyük bir yatak şev eğimine ve üzerinde sel rejimli bir akıma sahip olan akımlar olarak karakterize edilmektedirler. Şüt akımları ya düz bir yüzey üzerinden veya basamaklı bir yüzey üzerinden akıtılırlar. Basamaklı kanallar üzerindeki akım, nap akımı, geçiş akımı veya sıçramalı akım olarak sınıflandırılmaktadır. Bu makalede, basamaklı kanallar üzerindeki akım rejimlerinin hidrolik karakteristikleri, debi, şüt açısı ve basamak yüksekliklerinin geniş bir dizisi kullanılarak sistematik olarak sunulmuştur. Yapay sinir ağı akım rejimlerini tahmin etmek için kullanılmıştır. Yapay sinir ağı oluşturulurken debi, şüt açısı ve basamak yüksekliği parametreleri kullanılmıştır. Analiz sonuçları, basamaklı kanallar üzerindeki akım rejimini belirlemede yapay sinir ağlarının kullanımının çok başarılı sonuçlar verdiğini göstermiştir. Anahtar Kelimeler: Basamaklı kanal, Sıçramalı akım, Geçiş akımı, Nap akımı, Yapay Sinir Ağları.

1. Introduction Stepped channels have become popular in recent years mainly due to the intrinsic low-cost and the speed of construction. In a stepped channel, 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 channels are commonly used for gabion weirs, river training, irrigation channels, and storm waterways. Stepped channels are used also for in-stream re-aeration and in water treatment plants to enhance the airwater transfer of atmospheric gases (e.g. oxygen, nitrogen) and of volatile organic components (VOC). The flow on stepped channels can be classified into nappe flow, transition flow, and skimming flow (Fig. 1). The hydraulic design of

stepped channel, and in particular, characteristics of nappe flow, transition flow, and skimming The flow on stepped channels can be classified into nappe flow, transition flow, and skimming flow (Fig. 1). The hydraulic design of stepped channel, and in particular, characteristics of nappe flow, transition flow, and skimming flow over stepped channels has been studied experimentally by a number of investigators. Recently, Baylar and Emiroglu [1], Emiroglu and Baylar [2] and Baylar et al. [3, 4 and 5] did some detailed experiments on the aeration efficiency of stepped channels. Knowing the flow regime is very important for hydraulic design of stepped channels. In this study, the flow regimes in stepped channels were predicted using a neural

O. Kisi, M. E. Emiroglu , A. Baylar

computing technique. Among machine learning techniques, ANN is the one that is widely used in various areas of water-related research [6, 7, 8, 9, 10 and 11]. The Levenberg-Marquardt optimization algorithm that is more powerful than the standard back propagation is used for the training of ANN models. The ANN models were tested and the results were evaluated using the absolute relative error (ARE) and determination coefficient (R2) statistics. 2. Characteristics of Skimming, Transition and Nappe Flow over Stepped Channels 2.1. Skimming Flow For large discharges, the waters flow down a stepped-channel 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 free-surface aeration, the flow becomes rapidly aerated and the free-surface appears white.

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Air and water are fully mixed forming a homogeneous two-phase flow [12]. 2.2. Transition Flow Ohtsu and Yasuda [13] were probably the first to introduce the concept of a “transition flow” regime, although they did not elaborate on its flow properties. For a given stepped-channel chute geometry, a range of flow rates gives an intermediary flow regime between nappe flows at low discharges and skimming flows at large flow rates. In the transition flow regime, 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 [12]. 2.3. Nappe Flow 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 regime. 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 [12].

Flow Regime Prediction in Stepped Channels Using Neural Computing Technique

Figure 1. Air entrainment mechanisms above stepped-channel chute a) Skimming flow regime, b) Transition flow regime, c) Nappe flow regime

3. Experiments The data used in this study were taken from studies conducted by Baylar and Emiroglu [1] and Baylar et al. [3] on a large model of a stepped channel. A schematic representation of the experimental set-up is shown on Fig. 2 that shows a prismatic rectangular chute channel, 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 regime. Water was pumped from the storage tank to stilling tank, from which water entered the chute

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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 x 10-3 m2/s and 166.67 x 10-3 m2/s. The discharge was measured by means of a flow meter installed in the supply line. Channel 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.


Grid Upstream channel

ep St

Stilling tank

-ch

Water feed line

d pe an lc ne hu te

Water flowmeter Storage tank Water pump

α

Flow control valve

Downstream channel

Figure 2. Laboratory stepped channel apparatus

4. Results and Analysis Fig. 3 shows classification of flow conditions on stepped-channel chutes. It is observed from Fig. 3 that the type of flow regime is a function of the step height, channel slope and flow rate. Three different flow regimes, namely the nappe, transition and skimming flow regimes occur on stepped-channel chutes. A tendency towards the nappe flow regime is observed with increasing step height and decreasing unit discharge and channel slope. However, the results show a tendency towards the transition and skimming

flow regimes as unit discharge and channel slope increase and as step height decreases. Ohtsu et al. [14] showed that the upper limit of the step height for the formation of the nappe flow and the lower limit of the step height for the formation of the skimming flow can be predicted by Eqs. 1 and 2. A perfect agreement is observed between the experimental results and the results of Eqs. 1 and 2, as shown in Fig. 3.

1 ⎛ hc ⎞ = ⎜ ⎟ ⎝ h ⎠ Nappe 0.57 (tan α)3 + 1.30

for 5.7° ≤ α ≤ 55°

(1)

1 ⎛ hc ⎞ = ⎜ ⎟ ⎝ h ⎠Skimming 1.16 (tan α) 0.165

for 5.7° ≤ α ≤ 55°

(2)

where the critical flow depth h c = 3 q 2 / g is in meters, water discharge per unit width q is in m2/s, the acceleration of gravity g is in m/s2, step

height h is in meters and channel slope α is in degrees.

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3.0

Skimming flow

2.5

Transition flow

1.5

1.0

0.5

Eq. 2

Nappe flow

hc / h

2.0

Eq. 1

0.0 0

10

20

30

40

50

60

α (degrees) Figure 3. Flow conditions on stepped-channel chutes [3] (Solid lines indicate experimental data and dotted line data of Ohtsu et al. [14]

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Table 1. Experimental data of stepped channel for h = 0.05 m h hc hc/h q α (-) (m) (m) (m2/s x 10-3) (deg.) 16.67 0.05 0.030 0.600 14.48 33.33 0.05 0.048 0.960 14.48 50.00 0.05 0.063 1.260 14.48 66.67 0.05 0.077 1.540 14.48 100.00 0.05 0.101 2.020 14.48 133.33 0.05 0.122 2.440 14.48 166.67 0.05 0.141 2.820 14.48 16.67 0.05 0.030 0.600 18.74 33.33 0.05 0.048 0.960 18.74 50.00 0.05 0.063 1.260 18.74 66.67 0.05 0.077 1.540 18.74 100.00 0.05 0.101 2.020 18.74 133.33 0.05 0.122 2.440 18.74 166.67 0.05 0.141 2.820 18.74 16.67 0.05 0.030 0.600 22.55 33.33 0.05 0.048 0.960 22.55 50.00 0.05 0.063 1.260 22.55 66.67 0.05 0.077 1.540 22.55 100.00 0.05 0.101 2.020 22.55 133.33 0.05 0.122 2.440 22.55 166.67 0.05 0.141 2.820 22.55 16.67 0.05 0.030 0.600 30.00 33.33 0.05 0.048 0.960 30.00 50.00 0.05 0.063 1.260 30.00 66.67 0.05 0.077 1.540 30.00 100.00 0.05 0.101 2.020 30.00 133.33 0.05 0.122 2.440 30.00 166.67 0.05 0.141 2.820 30.00 16.67 0.05 0.030 0.600 40.00 33.33 0.05 0.048 0.960 40.00 50.00 0.05 0.063 1.260 40.00 66.67 0.05 0.077 1.540 40.00 100.00 0.05 0.101 2.020 40.00 133.33 0.05 0.122 2.440 40.00 166.67 0.05 0.141 2.820 40.00 16.67 0.05 0.030 0.600 50.00 33.33 0.05 0.048 0.960 50.00 50.00 0.05 0.063 1.260 50.00 66.67 0.05 0.077 1.540 50.00 100.00 0.05 0.101 2.020 50.00 133.33 0.05 0.122 2.440 50.00 166.67 0.05 0.141 2.820 50.00

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Flow Regime Nappe Transition Skimming Skimming Skimming Skimming Skimming Nappe Transition Skimming Skimming Skimming Skimming Skimming Nappe Transition Skimming Skimming Skimming Skimming Skimming Nappe Skimming Skimming Skimming Skimming Skimming Skimming Transition Skimming Skimming Skimming Skimming Skimming Skimming Transition Skimming Skimming Skimming Skimming Skimming Skimming



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Flow Regime Nappe Nappe Nappe Transition Transition Skimming Skimming Nappe Nappe Nappe Transition Transition Skimming Skimming Nappe Nappe Nappe Transition Skimming Skimming Skimming Nappe Nappe Nappe Transition Skimming Skimming Skimming Nappe Nappe Transition Transition Skimming Skimming Skimming Nappe Transition Transition Transition Skimming Skimming Skimming



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Flow Regime Nappe Nappe Nappe Nappe Nappe Transition Transition Nappe Nappe Nappe Nappe Nappe Transition Transition Nappe Nappe Nappe Nappe Nappe Transition Transition Nappe Nappe Nappe Nappe Nappe Transition Skimming Nappe Nappe Nappe Nappe Transition Transition Skimming Nappe Nappe Nappe Transition Transition Skimming Skimming


5. Neural Networks weights Wij and Wjk between layers of neurons. Initial estimated weight values are progressively corrected during a training process that compares predicted outputs to known outputs, and back propagates any errors (from right to left in Fig. 4) to determine the appropriate weight adjustments necessary to minimize the errors. The Levenberg-Marquardt (LM) training algorithm was used here for adjusting the weights. The adaptive learning rates were used for the purpose of faster training speed and solving local minima problem. For each epoch, if performance decreases toward the goal, then the learning rate is increased by the factor learning increment. If performance increases, the learning rate is adjusted by the factor learning decrement. The numbers of hidden layer neurons were found using simple trial and error method.

Artificial neural networks (ANNs) are based on the present understanding of biological nervous system, though much of the biological detail is neglected. ANNs are massively parallel systems composed of many processing elements connected by links of variable weights. The multi-layer backpropagation network (MLP) is by far the most popular among the many ANN paradigms [15]. The network consists of layers of parallel processing elements, called neurons, with each layer being fully connected to the proceeding layer by interconnection fully connected to the proceeding layer by interconnection strengths, or weights (W). Fig. 4 illustrates a three-layer neural network consisting of layers i, j, and k, with the interconnection

i 1

j

Wij

1

Wjk

k 1

2

. Input . . . . . .

2

. . . . . . .

2

L

M

N

. . Output . . . . .

Figure 4. A three-layer neural network structure

5.1. The Levenberg-Marquardt Algorithm While back propagation with gradient descent technique is a steepest descent algorithm, the Levenberg-Marquardt algorithm is an

approximation to Newton’s method [16]. If we have a function V(x) which we want to minimize with respect to the parameter vector x, then Newton’s method would be

−1

∆x = - ⎡∇ 2V( x )⎤ ∇V( x ) ⎢⎣ − ⎥ − ⎦

(3)

where ∇ 2 V( x ) is the Hessian matrix and ∇V ( x ) is the gradient. If we assume that V(x) is a sum of −

−

squares function

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N

V(x) =

∑ e i 2 (x)

(4)

i =1

then it can be shown that ∇V( x ) = JT(x)e(x)

(5)

−

∇ 2 V( x ) = JT(x) J(x) + S (x)

(6)

−

where J(x) is the Jacobean matrix and N

S(x) =

∑ e i ∇ 2 e i (x− )

(7)

i =1

For the Gauss-Newton method it is assumed that S(x) ≈ 0, and the update (3) becomes ∆x = [JT(x) J(x)]-1 JT(x) e(x) (8) The Marquardt-Levenberg modification to the Gauss-Newton method is ∆x = [JT(x) J(x) + µI]-1JT(x)e(x) (9) The parameter µ is multiplied by some factor (β) whenever a step would result in an increased V(x). When a step reduces V(x), µ is divided by β. When µ is large the algorithm becomes steepest descent (with step 1/µ), while for small µ the algorithm becomes GaussNewton. The Marquardt-Levenberg algorithm can be considered a trust-region modification to Gauss-Newton. The key step in this algorithm is the computation of the Jacobean matrix. For the neural network-mapping problem the terms in the Jacobean matrix can be computed by a simple modification to the back propagation algorithm [17]. 5.2. Application and Results A program code including neural networks toolbox, were written in MATLAB language for the ANN simulation. Different ANN architectures were tried using this code and the appropriate model structure was determined. A difficult task with ANN involves choosing parameters such as the number of hidden nodes, the learning rate, and the initial weights. Determining an appropriate architecture of a neural network for a particular problem is an important issue, since the network topology directly affects its computational complexity and its generalization capability. The optimum network geometry is obtained utilizing a trialand-error approach in which ANN are trained with one hidden layer. It should be noted that one hidden layer could approximate any

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continuous function, provided that sufficient connection weights are used [18]. Here, the hidden layer node number of ANN model was determined after trying various network structures since there is no theory yet to tell how many hidden units are needed to approximate any given function. In the training stage, same initial weights were used for each ANN networks. The sigmoid activation function was used for the hidden and output nodes. The parameters considered in the study are the ratio between the critical flow depth and step height (hc/h), channel slope (α) and flow regime. The parameters (hc/h) and α are used as inputs to the ANN for the estimation of flow regime. Three flow regimes, nappe, transition and skimming, are denoted as the numbers 1, 2 and 3, respectively. Of the 126 experimental data sets, the 110 data are used to train the ANN and the remaining data are used for validation. The remaining 16 data sets are randomly selected among the whole data. The model results are evaluated using absolute relative error (ARE) and determination coefficient (R2) statistics. Before applying the ANN to the data, the training input and output values were normalized using the equation

a

x i − x min +b x max − x min

(10)

where xmin and xmax denote the minimum and maximum of the stage and discharge data. Different values can be assigned for the scaling factors a and b. There are no fixed rules as to


respectively. In the fifth column, the ANN (2,6,1) denotes an ANN model comprising 2 input, 6 hidden and 1 output layer neurons. It can be obviously seen from Table 4 that the ANN estimates observed flow regimes with a quite high accuracy. The mean ARE of the ANN estimates is as low as % 0.9.

which standardization approach should be used in particular circumstances [19]. The a and b were taken as 0.6 and 0.2 herein, respectively. The ARE statistics of the ANN model in test period are given in Table 4. In the fourth column, the numbers 1, 2 and 3 indicates the flow regimes, nappe, transition and skimming,

Table 4. The ARE statistics for the estimated flow regime using ANN model - test period hc/h (-)

α (deg.)

Flow Regime Observed

Flow Regime Observed

1.540

14.48

Skimming

3

Flow Regime Estimated by ANN (2,6,1) 3.02

1.260

18.74

Skimming

3

2.99

0.45

2.020

22.55

Skimming

3

3.01

0.29

2.020

30.00

Skimming

3

3.00

0.09

1.540

40.00

Skimming

3

3.01

0.18

2.440

50.00

Skimming

3

3.01

0.18

0.480

14.48

Nappe

1

0.98

2.24

0.770

18.74

Transition

2

1.97

1.56

0.480

22.55

Nappe

1

1.00

0.40

1.220

30.00

Skimming

3

2.99

0.35

1.220

40.00

Skimming

3

3.01

0.18

0.420

14.48

Nappe

1

0.98

1.86

0.420

18.74

Nappe

1

0.99

0.50

0.513

30.00

Nappe

1

1.00

0.39

0.420

40.00

Nappe

1

1.00

0.33

0.320

50.00

Nappe

1

0.95

4.71

The ANN estimates were compared with the observed flow regime values in Fig. 5 in the form of hydrograph and scatter plots. As can be seen from these graphs, the ANN estimates catch

ARE (%) 0.54

the observed values with a high accuracy. The coefficients of the fit line equation, 1.009 and 0.025, are quite close to the 1 and 0, respectively, with a high R2 value of 0.9998.

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4

4

observed

R2 = 0.9998

3

model

Flow regime

y = 1.009x - 0.025

ANN (2,6,1)

3 2 1

2 1

0 0

2

4

6

8

10

12

14

16

18

Experiment

0 0

1

2

3

4

observed

Figure 5. The plotting of ANN estimates and observed flow regimes in test period

6. Conclusions In stepped channels, the amount of entrained air is an important design parameter. The amount of entrained air depends on flow condition on stepped channels. Three distinct flow regimes are found on stepped channels, socalled nappe flow, transition flow, and skimming flow.

A three layer neural network technique was used in estimation of flow regime on stepped channels. The results indicated that the ANN method provided flow condition estimates with a quite high accuracy. Therefore, the ANN can be used to estimate flow regimes in stepped channels.

7. References 1.

2.

3.

4.

5.

Baylar, A., & Emiroglu, M. E., (2003). Study of Aeration Efficiency at Stepped Channels. Proc., Inst. Civ. Engrs. Water and Marit. Engrg., 156(WM3), 257-263. Emiroglu, M. E., & Baylar, A., (2003). An Investigation of Effect of Stepped Chutes with End Sill on Aeration Performance. Water Quality Research Journal of Canada, 38(3), 527-539. Baylar, A., Emiroglu, M. E., & Bagatur T., (2006). An Experimental Investigation of Aeration Performance in Stepped Spillways. Water and Environment Journal, 20(1), 35-42. Baylar, A., Bagatur T., & Emiroglu, M. E., (2007a). Prediction of Oxygen Content of Nappe, Transition, and Skimming Flow Regimes in Stepped-Channel Chutes. Journal of Environmental Engineering and Science, 6(2), 201-208. Baylar, A., Bagatur T., & Emiroglu, M. E., (2007b). Aeration Efficiency with Nappe Flow over Stepped Cascades. Proceedings of the Institution of Civil Engineers-Water Management, 160(1), 43-50.

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