Elsevier

Journal of Hydrology

Volumes 434–435, 20 April 2012, Pages 142-148
Journal of Hydrology

Technical Note
Gene-expression programming for transverse mixing coefficient

https://doi.org/10.1016/j.jhydrol.2012.02.018Get rights and content

Summary

This study presents gene-expression programming (GEP), which is an extension of genetic programming (GP), as an alternative approach to predict the transverse mixing coefficient in open channel flows. Laboratory data were collected in the present study and also from the literature for the transverse mixing coefficient covering wide range of flow conditions. These data were used for the development and testing of the proposed method. A functional relation for the estimation of transverse mixing coefficient has been developed using GEP. The proposed GEP approach produced satisfactory results compared to the existing predictors for the transverse mixing coefficient.

Highlights

► This study presents gene-expression programming (GEP) to predict the transverse mixing coefficient in open channel flows. ► A functional relation for transverse mixing coefficient has been developed using GEP. ► The proposed GEP approach produces satisfactory results compared to the existing predictors for the mixing coefficient.

Introduction

Quantitative understanding of mixing of pollutants in streams is a matter of concern in recent years for effective control of pollution in the streams. Transverse mixing is arguably more important in water quality management than either vertical or longitudinal mixing, especially when dealing with the discharge of pollutants from point sources or the mixing of tributary inflows. Beltaos (1980) showed that, except for small streams, complete cross-sectional mixing was not achieved until the pollutant had traveled a long distance, which was generally not within the length of practical interest. The length required for complete cross-sectional mixing of pollutants is about 20 times the top width for a rough stream and about 200 times the top width for a smooth stream (Fischer, 1967). Therefore, modeling of transverse mixing is necessary to compute the distribution of pollutant concentration in streams.

The process of transverse mixing of a conservative and neutrally buoyant substance in steady flow through a straight channel is modeled by the principle of conservation of mass and written as (Ahmad, 2008, Ahmad, 2009):x(UDC)=zDEZCzwhere C = depth-averaged concentration; D = depth of flow; U = depth-averaged velocity in longitudinal direction; x and z = longitudinal and transverse distances, respectively; Ez = depth-averaged mixing coefficients in the transverse directions.

Solution of the above equation yields concentration of the pollutants at a location downstream of the injection site. The computed pollutants concentration is requisite for the management of water quality in the river system. Several investigators (Ahmad et al., 2011, Ahmad and Kothyari, 2001, Guan et al., 2002, Stefanovic and Stefan, 2001; and Sullivan, 1968) solved Eq. (1) numerically for different boundary conditions. However, solution of Eq. (1) for pollutant concentration needs the value of transverse mixing coefficient in advance, in addition to the channel and flow properties. Therefore, it is essential to estimate the transverse mixing coefficient for known flow conditions in a channel for getting the pollutant concentration at any location downstream of the injection site of the pollutant. The present paper deals with the estimation of transverse mixing coefficient for its use in numerical or analytical models.

Based on the experimental and field data, several investigators attempted to establish the relationship between the transverse mixing coefficient and bulk channel parameters such as width, depth, shear velocity U, friction factor, curvature and sinuosity (Beltaos, 1980, Boxall and Guymer, 2003, Fischer, 1967, Lau, 1981, Stefanovic and Stefan, 2001). The transverse mixing coefficient is high in meandering and curved channels due to the presence of secondary currents (Albers and Steffler, 2007; and Fischer, 1967). Fischer et al. (1979) reported Ez = 0.23DU in a straight irrigation canal.

Sayre (1969) measured an average value of Ez/DU = 0.17 in a laboratory channel, which was identical to the value given by Orlob (1983). Fischer et al. (1979) reviewed several studies and suggested that Ez/DU = 0.15. Lau and Krishnappan (1977) found that the non-dimensional diffusivity Ez/HU did not vary with the width of the channel. Based on the experimental study in a rectangular channel, Chau (2000) proposed Ez/DU = 0.18 and Ahmad (2007) proposed Ez/DU = 0.15.

Alternative fitting approaches such as artificial neural networks (ANNs) (Lee et al., 2007) and adaptive neuro fuzzy inference system (Bateni et al. (2007) have recently been shown to yield effective estimates of scour depth at bridge piers. ANNs have been reported to provide reasonably good solutions for hydraulic engineering problems where highly nonlinear and complex relationship existed among the input–output pairs in the corresponding data (Azmathullah et al., 2005, Azmathullah et al., 2006). While ANN-based models are powerful in that they provide a good fit to the data used in model training and validation, such models often do not result in compact and explicit equations for use by designers. The resulting ANN-based model structure is often a long expression consisting of activation functions with variable complexity depending on the number of hidden layers used in the model structure.

This paper deals with estimation of transverse mixing coefficient using a new soft computing technique, i.e. Gene-expression programming, the details of which are given in the next section. The accuracy of the proposed equation was checked with unused experimental data. The estimated value of transverse mixing coefficient is required for the computation of pollutant concentration from Eq. (1) for the pollution management of the riverine system.

Section snippets

Overview of GEP

GEP, which is an extension of GP (Ferreira, 2001a) is a search technique that involves computer programs (e.g., mathematical expressions, decision trees, polynomial constructs, and logical expressions). GEP computer programs are encoded in linear chromosomes, which are then expressed or translated into expression trees (ETs). ETs are sophisticated computer programs that have usually been evolved to solve a particular problem and are selected according to their fitness in solving that problem.

Description of collected data and dimensional analysis

Several experimental studies for transverse mixing in straight rectangular laboratory channels have been conducted (Elder, 1959, Engelund, 1969, Engmann, 1974; Lau, 1981, Luk et al., 1990, Mcnulty and Wood, 1984, Nokes, 1986, Prych, 1970, Rutherford, 1994, Seo et al., 2006; and Shen, 1978). These provide estimates of the transverse mixing coefficient (Ez) provided the flow does not depart significantly from the plane shear flow. Ahmad (2007) performed experiments to measure the concentration

Gene-expression programming for transverse mixing coefficient

In this section, the transverse mixing coefficient is modeled using the GEP approach.

Initially, the “training set” was selected from the entire data set, and the rest was used as the “testing set”. Selection of training set means learning environment for GEP. After data division, different parameters for the model were decided which are demonstrated in the following six steps

  • 1.

    Like most other evolutionary algorithms, GEP starts with an initial population of individuals. The population of

Results and discussions

Results of the empirical equations for transverse mixing coefficient proposed by Prych, 1970, Fischer, 1967, and Ahmad (2007) were calculated for all the data sets and compared with measured data (Fig. 3). As can be seen from Fig. 3, it is apparent that none of these empirical equations have good results and shows considerable errors in comparison with measured data. The values of statistical index show poor performance of the existing empirical equations, i.e. Fischer et al. (1979), R2 = 0.78;

Conclusions

The gene-expression programming approach was used to derive a new expression for the prediction of transverse mixing coefficient in open channel flows. The expression made use of few geometric (river width, flow depth) and hydraulic parameters (cross-sectional average and shear velocities), commonly known to water engineers and planners. A performance evaluation of the new expression was carried out by comparing the predictions from the new formula with the existing equations, using data

Acknowledgements

The 2nd author expresses deep appreciation to Dept of Science and Technology, Government of India, for providing research Grant No. III.5(60/2000-ET). The authors wish to thanks to Robert D. Jarrett, US Geological Survey (USGS) for his suggestions in preparation of this manuscript and also review.

References (41)

  • H.Md. Azmathullah et al.

    Estimation of scour below spillways using neural networks

    IAHR, J. Hydraulic Res.

    (2006)
  • H.Md. Azmathullah et al.

    Neural networks for estimation of scour downstream of ski-jump bucket

    J. Hydraulic Eng.

    (2005)
  • S. Beltaos

    Transverse mixing tests in naturals streams

    J. Hydraulic Div. ASCE

    (1980)
  • J.B. Boxall et al.

    Analysis and prediction of transverse mixing coefficients in natural channels

    J. Hydraulic Eng. ASCE

    (2003)
  • J.W. Davidson et al.

    Method for identification of explicit polynomial formulae for the friction in turbulent pipe flow

    J. Hydroinformat.

    (1999)
  • J.W. Elder

    The dispersion of marked fluid in turbulent shear flow

    J. Fluid Mech.

    (1959)
  • F. Engelund

    Dispersion of floating particles in uniform channel flow

    Am. Soc. Civ. Eng.

    (1969)
  • Engmann, J.E.O., 1974. Transverse Mixing Characteristics of Open an Ice-Covered Channel Flows. Ph.D. Thesis, University...
  • Ferreira, C., 2001a. Gene expression programming in problem solving. In: 6th Online World Conference on Soft Computing...
  • C. Ferreira

    Gene expression programming: a new adaptive algorithm for solving problems

    Complex Syst.

    (2001)
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