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Modeling dimensionless longitudinal dispersion coefficient in natural streams using artificial intelligence methods

  • Water Engineering
  • Published:
KSCE Journal of Civil Engineering Aims and scope

Abstract

Learning the Longitudinal Dispersion (LD) mechanism in natural channel is vitally important to be able to control water pollution and to prevent different stratification in flow that crucial for water resources conservation for both human and aquatic life. Many related studies can be found in the existing literature. However, almost all studies aim to investigate the mechanism of the LD in natural channels or to model dimensional LD coefficient. The main goal of this work is to develop three models based on different artificial neural network techniques to predict dimensionless (not dimensional) LD coefficients in natural channels. Another goal of this study is to present a large and critical assessment on the existing studies made on both dimensional and dimensionless LD. The data sets obtained from the literature concerning more than 30 rivers at different times in the United States of America, include the depth, the width, and the mean cross-sectional velocity of the flow, shear velocity, and dimensionless longitudinal dispersion coefficient. The results have been compared with the data at hand, a fuzzy based model, and seven conventional equations proposed in literature by using statistical magnitudes, error modes, and contour map method. It is observed that the feed forward neural network yields the best reliable results.

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References

  • Abd El-Hadi, N. A. and Davar, K. S. (1976). “Longitudinal dispersion for flow over rough beds.” Journal of Hydraulics Division, ASCE, Vol. 102, No. 4, pp. 483–498.

    Google Scholar 

  • Adarsh, S. (2010). “Prediction of longitudinal dispersion coefficient in natural channels using soft computing techniques.” Scientia Iranica Transaction A-Civil Engineering, Vol. 17, No. 5, pp. 363–371.

    Google Scholar 

  • Afshar, A. and Kazemi, H. (2012). “Multi objective calibration of large scaled water quality model using a hybrid particle swarm optimization and neural network algorithm.” KSCE Journal of Civil Engineering, KSCE, Vol. 16 No. 6, pp. 913–918.

    Google Scholar 

  • Al Naib, S. and Sanders, J. (1997). “Oblique and vertical jet dispersion in channels.” Journal of Hydraulic Engineering, ASCE, Vol. 123, No. 5, pp. 456–462.

    Google Scholar 

  • Alp, M. and Cigizoglu, H. K. (2007). “Suspended sediment load simulation by two articial neural network methods using hydrometeorological data.” Environmental Modelling & Software, Vol. 22, No. 1, pp. 2–13.

    Google Scholar 

  • Atesman, K. M. (1970). The dispersion of matter in turbulent shear flows, PhD Thesis, Colorado State University Fort Collins, Colorado, USA.

    Google Scholar 

  • Baek, K. O. and Seo, I. W. (2008). “Prediction of transverse dispersion coefficient using vertical profile of secondary flow in meandering channels.” KSCE Journal of Civil Engineering, KSCE, Vol. 12, No. 6, pp. 417–426.

    Google Scholar 

  • Baeumer, B., Benson, D. A., Meerschaert, M. M., and Wheatcraft, S. W. (2001). “Subordinated advection-dispersion equation for contaminant transport.” Water Resources Research, Vol. 37, No. 6, pp. 1543–1550.

    Google Scholar 

  • Bansal, M. K. (1970). Dispersion and reaeration in natural streams, PhD Thesis, Kansas University Lawrence, Kansas, USA.

    Google Scholar 

  • Bansal, M. K. (1971). “Dispersion in natural streams.” Journal of Hydraulics Division, ASCE, Vol. 97, No. 11, pp. 1867–1886.

    Google Scholar 

  • Basha, H. A. (1997). “Analytical model of two-dimensional dispersion in laterally non-uniform axial velocity distributions.” Journal of Hydraulic Engineering, ASCE, Vol. 123, No. 10, pp. 853–862.

    Google Scholar 

  • Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000). “Application of a fractional advection-dispersion equation.” Water Resources Research, Vol. 36,Issue 6, pp. 1403–1412.

    Google Scholar 

  • Bogle, G. V. (1997). “Stream velocity profiles and longitudinal dispersion.” Journal of Hydraulic Engineering, ASCE, Vol. 123,Issue 9, pp. 816–820.

    Google Scholar 

  • Brewin, D. J. and Martin, J. R. (2006). “Water quality management: A regional perspective — The severn-trent area.” River Research and Applications (Regulated Rivers: Research & Management), Vol. 2, No. 3, pp. 257–275.

    Google Scholar 

  • Broomhead, D. and Lowe, D. (1988). “Multivariable functional interpolation and adaptive networks.” Complex Systems, Vol. 2, No. 3, pp. 321–355.

    MATH  MathSciNet  Google Scholar 

  • Chang, Y. C. (1971). Lateral mixing in meandering channels, PhD Thesis, The University of Iowa, USA.

    Google Scholar 

  • Chatwin, P. C. (1980). “Presentation of longitudinal dispersion data.” Journal of Hydraulics Division, ASCE, Vol. 106, No. 1, pp. 71–83.

    Google Scholar 

  • Chau, K.-W., Wu, C., and Li, Y. (2005). “Comparison of several flood forecasting models in Yangtze River.” Journal of Hydrologic Engineering, ASCE, Vol. 10, No. 6, pp. 485–491.

    Google Scholar 

  • Chen, W. and Chau, K-W. (2006). “Intelligent manipulation and calibration of parameters for hydrological models.” International Journal of Environment and Pollution, Vol. 28, Nos. 3–4, pp. 432–447.

    Google Scholar 

  • Chen, S., Cowan, C. F. N., and Grant, P. M. (1991). “Orthogonal least squares learning algorithm for radial basis function networks.” IEEE Transactions on Neural Networks, Vol. 2,Issue 2, pp. 302–309.

    Google Scholar 

  • Cheng, C. and Chau, K.-W. (2001). “Fuzzy iteration methodology for reservoir flood control operation.” Journal of the American Water Resources Association, Vol. 37, No. 5, pp. 1381–1388.

    Google Scholar 

  • Cigizoglu, H. K. (2005). “Application of generalized regression neural networks to intermittent flow forecasting and estimation.” Journal of Hydrologic Engineering, Vol. 10, No. 4, pp. 336–341.

    Google Scholar 

  • Day, T. J. (1975). “Longitudinal dispersion in natural channels.” Water Resources Research, Vol. 11, No. 6, pp. 909–918.

    Google Scholar 

  • Deng, Z. Q., Bengtsson, L., Singh, V. P., and Adrian, D. D. (2002). “Longitudinal dispersion coefficient in single-channel streams.” Journal of Hydraulic Engineering, ASCE, Vol. 128, No. 10, pp. 901–916.

    Google Scholar 

  • Deng, Z. Q., Singh, V. P., and Bengtsson, L. (2001). “Longitudinal dispersion coefficient in straight rivers.” Journal of Hydraulic Engineering, ASCE, Vol. 127, No. 11, pp. 919–927.

    Google Scholar 

  • Elder, J. W. (1959). “The dispersion of marked fluid in turbulent shear flow.” Journal of Fluid Mechanics, Vol. 5,Issue 4, pp. 544–560.

    MATH  MathSciNet  Google Scholar 

  • Ferreira, R. M. L., Ferreira, L. M., Ricardo, A. M., and Franca, M. J. (2010). “Impacts of sand transport on flow variables and dissolved oxygen in gravel-bed streams suitable for salmonid spawning.” River Research and Applications, Vol 26, No. 4, pp. 414–438.

    Google Scholar 

  • Firat, M., Yurdusev, M. A., Turan, M. E. (2009). “Evaluation of articial neural network techniques for municipal water consumption modeling.” Water Resources Management, Vol. 23, pp. 617–632.

    Google Scholar 

  • Fischer, H. B. (1967). “The mechanics of dispersion in natural streams.” Journal of Hydraulics Division, ASCE, Vol. 93, Vol. 6, pp. 187–216.

    Google Scholar 

  • Fischer, H. B. (1968a). “Dispersion predictions in natural streams.” Journal of Sanitary Engineering Division, ASCE, Vol. 94, pp. 927–944.

    Google Scholar 

  • Fischer, H. B. (1968b). Methods for predicting dispersion coefficients in natural streams with applications to lower reaches of the Green and Duwanish rivers, US Geological Survey Professional Paper 582-A, Washington, D.C., USA.

    Google Scholar 

  • Fischer, H. B. (1969). “The effects of bends on dispersion in streams.” Water Resources Research, Vol. 5,Issue 2, pp. 496–506.

    Google Scholar 

  • Fischer, H. B. (1975). “Discussion of simple method for prediction dispersion in streams by R.S. McQuivey and T.N. Keefer.” Journal of the Environmental Engineering Division, ASCE, Vol. 101, No. 3, pp. 453–455.

    Google Scholar 

  • Fischer, H. B. (1979). Mixing in inland and coastal waters, Hardbound, Academic Press, No. 302, New York, N.Y., USA.

    Google Scholar 

  • Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H. (1979). Mixing in inland and coastal waters, Academic Press, No. 302, New York, N.Y., USA.

    Google Scholar 

  • Fukooka, S. and Sayre, W.W. (1973). “Longitudinal dispersion in sinuous channels.” Journal of Hydraulics Division, ASCE, Vol. 99, No. 1, pp. 195–217.

    Google Scholar 

  • Gandolfi, C., Facchi, A., and Whelan M. J. (2001). “On the relative role of hydrodynamic dispersion for river water quality.” Water Resources Research, Vol. 37,Issue 9, pp. 2365–2375.

    Google Scholar 

  • Garay, S. B., Sauvage, S., and Vervier, P. (2001). “Hydromorphological control of phosphorus in a large free-flowing gravel bed river: The Garonne River (France).” River Research and Applications, Vol. 17, Nos. 4–5, pp. 461–472.

    Google Scholar 

  • Goodfrey, R. G. and Frederick, B. J. (1970). Stream dispersion at selected sites, US Geological Survey Professional Paper 433-K, Washington, D.C., USA.

    Google Scholar 

  • Graf, J. B. (1986). “Travel time and longitudinal dispersion in Illinois streams.” US Geological Survey Water-Supply, Denver, Vol. 71, pp. 2269.

    Google Scholar 

  • Groves, J. H., Williams, D. G., Caley, P., Norris, R. H., and Caitcheon, G. (2009). “Modelling of floating seed dispersal in a fluvial environment.” River Research and Applications, Vol. 25, No. 5, pp. 582–592.

    Google Scholar 

  • Guymer, I. (1998). “Longitudinal dispersion in sinuous channel with changes in shape.” Journal of Hydraulic Engineering, ASCE, Vol. 124, No. 1, pp. 33–40.

    Google Scholar 

  • Guymer, I. and West, J. R. (1992). “Longitudinal dispersion coefficients in estuary.” Journal of Hydraulic Engineering, ASCE, Vol. 118, No. 5, pp. 718–734.

    Google Scholar 

  • Hagan, M. T. and Menhaj, M. B. (1994). “Training feed forward networks with the Marquaradt algorithm.” IEEE Transactions on Neural Networks, Vol. 5,Issue 6, pp. 989–993.

    Google Scholar 

  • Haykin, S. (1998). Neural networks — A comprehensive foundation, Second Edition, Prentice-Hall, Upper Saddle River, NJ, pp. 26–32.

    Google Scholar 

  • Ho, D. T., Schlosser, P., and Caplow, T. (2002). “Determination of longitudinal dispersion coefficient and net advection in the tidal Hudson River with a large-scale, high resolution SF6 tracer release experiment.” Environmental Scence & Technology, Vol. 36, No. 15, pp. 3234–3241.

    Google Scholar 

  • Hossien, R.-M., Ayyoubzadeh, S. A., Khadangi, E., and Ebadzadeh, M. M. (2009). “An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS.” Expert Systems with Applications, Vol. 36, No. 4, pp. 8589–8596.

    Google Scholar 

  • Huang, K. Z. and Yu, H. (2000). “A new empirical equation of longitudinal dispersion coefficient.” 8th International Symposium of Stochastic Hydraulics, Wang, Z. Y., Hu, S. X. (Eds). Stochastic Hydraulics 2000: Beijing, China, pp. 845–851.

    Google Scholar 

  • Jackel, U. and Vereecken, H. (1997). “Renormalization group analysis of macrodispersion in a directed random flow.” Water Resources Research, Vol. 33, No. 10, pp. 2287–2299.

    Google Scholar 

  • Jha, R. and Singh, V. P. (2008). “Analytical water quality model for biochemical oxygen demand simulation in river Gomti of Ganga Basin, India.” KSCE Journal of Civil Engineering, KSCE, Vol. 12, No.2, pp. 141–147.

    Google Scholar 

  • Jobson, H. E. (1997). “Predicting travel time and dispersion in rivers and streams.” Journal of Hydraulic Engineering, ASCE, Vol. 123, No. 11, pp. 971–978.

    Google Scholar 

  • Kashefipour, S. M. and Falconer, R. A. (2002). “Longitudinal dispersion coefficients in natural streams.” Water Research, Vol. 36, No. 6, pp. 1596–1608.

    Google Scholar 

  • Kim, T. and Sheng Y. P. (2010). “Estimation of water quality model parameters.” KSCE Journal of Civil Engineering, KSCE, Vol. 14, No. 3, pp. 421–437.

    Google Scholar 

  • Kim, S., Shiri, J., and Kisi, O. (2012). “Pan evaporation modeling using neural computing approach for different climatic zones.” Water Resources Management, Vol. 26, No. 11, pp. 3231–3249.

    Google Scholar 

  • Kisi, O. (2004). “Multi-layer perceptrons with Levenberg-Marquardt training algorithm for suspended sediment concentration prediction and estimation.” Hydrological Sc ences Journal, Vol. 49, No. 6, pp. 1025–1040.

    Google Scholar 

  • Kisi, O. (2005). “Suspended sediment estimation using neuro-fuzzy and neural network approaches.” Hydrological Sc ences Journal-Journal Des Scences Hydrolog ques, Vol. 50,Issue 4, pp. 683–696.

    Google Scholar 

  • Kisi, O. (2007). “Streamflow forecasting using different artificial neural network algorithms.” Journal of Hydrologic Engineering, ASCE, Vol. 12. Issue 5, pp. 532–539.

    Google Scholar 

  • Koussis, A. D. and Rodriguez-Mirasol, J. (1998). “Hydraulic estimation of dispersion coefficient for streams.” Journal of Hydraulic Engineering, ASCE, Vol. 124, No. 3, pp. 317–320.

    Google Scholar 

  • Lee, G. C. and Chang, S. H. (2003). “Radial basis function networks applied to DNBR calculation in digital core protection systems.” Annals of Nuclear Energy, Vol. 30,Issue 15, pp. 1561–1572.

    Google Scholar 

  • Li, S. G. and Zhou, X. (1997). “Stochastic theory for Irregular stream modeling II: Solute transport.” Journal of Hydraulic Engineering, ASCE, Vol. 123, No. 7, pp. 610–616.

    Google Scholar 

  • Lin, G.-F. and Wu, M.-C. (2011). “An RBF network with a two-step learning algorithm for developing a reservoir inow forecasting model.” Journal of Hydrology, Vol. 405, Nos. 3–4, pp. 439–450.

    Google Scholar 

  • Liu, H. (1977). “Predicting dispersion coefficient of stream.” Journal of Hydraulic Engineering, ASCE, Vol. 103, No. 1, pp. 59–69.

    Google Scholar 

  • Liu, X., Hua, Z., Xue, H., Gu, L., and Xie, Z. (2011). “Inverse optimization method to determine longitudinal dispersion coefficient and selection of sampling time in tracer tests.” Fresenius Environmental Bulletin, Vol. 20, No. 5, pp.1149–1154.

    Google Scholar 

  • Makinia, J. and Wells, S. A. (2005). “Evaluation of empirical formulae for estimation of the longitudinal dispersion in activated sludge reactors.” Water Research, Vol. 39, No. 8, pp. 1533–1542.

    Google Scholar 

  • McQuivey, R. S. and Keefer, T. N. (1974). “Simple method for prediction dispersion in streams.” Journal of the Environmental Engineering Division, ASCE, Vol. 100, No. 4, pp. 997–1011.

    Google Scholar 

  • Mutlu, E., Chaubey, I., Hexmoor, H. and Bajwa, S.G. (2008). “Comparison of articial neural network models for hydrologic predictions at multiple gauging stations in an agricultural watershed.” Hydrological Processes, Vol. 22, No. 26, pp. 5097–5106.

    Google Scholar 

  • Muttil, N. and Chau, K.-W. (2006). “Neural network and genetic programming for modelling coastal algal blooms.” International Journal of Environment and Pollution, Vol. 28, Nos. 3–4, pp. 223–238.

    Google Scholar 

  • Noori, R., Karbassi, A. R., Farokhnia, A., and Dehghani, M. (2009). “Predicting the longitudinal dispersion coefficient using support vector machine and adaptive neuro, fuzzy inference system techniques.” Environmental Engineering Science, Vol. 26, No. 10, pp. 1503–1510.

    Google Scholar 

  • Noori, R., Karbassi, A. R., Mehdizadeh, H., Vesali-Naseh, M., and Sabahi, M. S. (2011). “A framework development for predicting the longitudinal dispersion coefficient in natural streams using an artificial neural network.” Environmental Progress & Sustainable Energy, Vol. 30, No. 3, pp. 439–449.

    Google Scholar 

  • Nordin, C. F. and Sabol, G. V. (1974). “Empirical data on longitudinal dispersion in rivers.” US Geological Survey Water Resources Investigations, Washington, D.C., pp. 20–74.

    Google Scholar 

  • Parker, F. L. (1961). “Eddy diffusion in reservoirs and pipe lines.” Journal of Hydraulics Division, ASCE, Vol. 87, No. 3, pp. 151–171.

    Google Scholar 

  • Pasquero, C., Provenzale, A., and Babiano, A. (2001). “Parameterisation of dispersion in two-dimensional turbulence.” Journal of Fluid Mechanics, Vol. 439, No. 1, pp. 279–303.

    MATH  Google Scholar 

  • Piotrowski, A., Napiorkowski, J. J., and Rowinski, P. M. (2006). “Flashflood forecasting by means of neural networks and nearest neighbour approach-a comparative study.” Nonlinear Processes in Geophysics, Vol. 13, Vol. 4, pp 443–448.

    Google Scholar 

  • Pujol, L. and Sanchez-Cabeza, J. A. (1999). “Determination of longitudinal dispersion coefficient and velocity of the Ebro river waters (Northeast Spain) using tritium as a radiotracer.” Journal of Env ronmental Radioactivity, Vol. 45, No. 1, pp. 39–57.

    Google Scholar 

  • Ranjan, S. R. (2011). “Prediction of longitudinal dispersion coefficients in natural rivers using artificial neural network.” Environmental Fluid Mechanics, Vol. 11. No. 3, pp 247–261.

    MathSciNet  Google Scholar 

  • Riahi-Madvar, H., Ayyoubzadeh, S. A., Khadangi, E., and Ebadzadeh, M. M. (2009). “An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS.” Expert Systems with Appl cations, Vol. 36, No. 4, pp. 8589–8596.

    Google Scholar 

  • Ribeiro, C. B. D. M., Silva, D. D. D., Soares, J. H. P., and Guedes, H. A. S. (2010). “Development and validation of an equation to estimate longitudinal dispersion coefficient in medium-sized rivers.” Engenharia Sanitaria E Ambiental, Vol. 15, No. 4, pp. 393–400.

    Google Scholar 

  • Rowinski, P. M., Guymer, I., and Kwiatkowski, K. (2008). “Response to the slug injection of a tracer-a large-scale experiment in a natural river.” Hydrolog cal Scences Journal-Journal Des Scýences Hydrologques, Vol. 53, No. 6, pp. 1300–1309.

    Google Scholar 

  • Rowinski, P.M., Piotrowski, A., and Napiorkowski, J.J. (2005). “Are artificial neural network techniques relevant for the estimation of longitudinal dispersion coefficient in rivers?.” Hydrological Sciences Journal-Journal Des Sciences Hydrologiques,Vol. 50, No. 1, pp. 175–187.

    Google Scholar 

  • Savci, M. E. (1972). Analysis Submerged Jumps and Longitudinal Dispersion of Soluble Matter in Submerged Jumps, PhD Thesis, Istanbul Technical University, Istanbul.

    Google Scholar 

  • Savci, M. E. (1973). “Longitudinal dispersion of soluble matter in submerged jumps.” Proceeding of 15 th Congress of the International Association for Hydraulic Research (IAHR, AIRH), Istanbul, Turkey, Vol. 2, pp. 71–78.

    Google Scholar 

  • Sayre, W. W. (1968). Dispersion of mass in open channel flow, Colorado State University Hydraulics (Fort Collins), Colorado, Vol. 3, No. 4, p. 73.

    Google Scholar 

  • Sayre, W. W. and Chang, F. M. (1968). “A laboratory investigation of open-channel dispersion processes for dissolved, suspended and floating dispersants.” US Geological Survey Professional Paper 433-E, pp. E1–E71.

    Google Scholar 

  • Scheidegger, A. E. (1954). “Statistical hydrodynamics in porous media.” Journal of Applying Physics, Vol. 25, No. 8, pp. 994–1001.

    MATH  Google Scholar 

  • Seo II, W. and Baek, K. O. (2004). “Estimation of the longitudinal dispersion coefficient using the velocity profile in natural streams.” Journal of Hydraulic Engineering, ASCE, Vol. 130, No. 3, pp. 227–236.

    Google Scholar 

  • Seo II, W. and Cheong, S. (1998). “Predicting longitudinal dispersion coefficient in natural streams.” Journal of Hydraulic Engineering, ASCE, Vol. 124, No 1, pp. 25–32.

    Google Scholar 

  • Shen, C., Niu, J., Anderson, E. J., and Phanikumar, M. S. (2010). “Estimating longitudinal dispersion in rivers using acoustic doppler current profilers.” Advances in Water Resources, Vol. 33, No. 6, pp. 615–623.

    Google Scholar 

  • Shieh, H. Y., Chen, J. S., Lin, C. N., Wang, W. K., and Liu, C. W. (2010). “Development of an artificial neural network model for determination of longitudinal and transverse dispersivities in a convergent flow tracer test.” Journal of Hydrology, Vol. 391, Issues 3–4, pp. 367–376.

    Google Scholar 

  • Smith, P., Beven, K., Tawn, J., Blazkova, S., and Merta, L. (2006). “Discharge dependent pollutant dispersion in rivers: Estimation of aggregated dead, zone parameters with surrogate data.” Water Resources Research, Vol. 42, No. 4, p. W04412.

    Google Scholar 

  • Sooky, A. A. (1969). Longitudinal dispersion in open channels.” Journal of Hydraulics Division, ASCE, Vol. 95, No. 4, pp. 1327–1346.

    Google Scholar 

  • Specht, D. F. (1991). “A general regression neural network.” IEEE Transactions on Neural Networks, Vol. 2, No. 6, pp. 568–576.

    Google Scholar 

  • Strack, O. D. L. and Fairbrother, M. D. (1997). “Numerical solution of the differential equation for moving front dispersion.” Journal of Hydrology, Vol. 194, Issues 1–4, pp. 164–179.

    Google Scholar 

  • Swamee, P. K., Pathak, S. K., and Sohrab, M. (2000). “Empirical relations for longitudinal dispersion in streams.” Journal of Environmental Engineering, ASCE, Vol. 126, No. 11, pp. 1056–1062.

    Google Scholar 

  • Sümer, M. (1970). Longitudinal dispersion of weight or weightless matters, PhD Thesis, Istanbul Technical University, Civil Engineering Faculty, Istanbul.

    Google Scholar 

  • Taormina, R., Chau, K-W., and Sethi, R. (2012). “Artificial Neural Network simulation of hourly groundwater levels in a coastal aquifer system of the Venice lagoon.” Engineering Applications of Artificial Intelligence, Vol. 25, No. 8, pp. 1670–1676.

    Google Scholar 

  • Tayfur, G. (2006). “Fuzzy, ANN, and regression models to predict longitudinal dispersion coefficient in natural streams.” Nordic Hydrology, Vol. 37, No. 2, pp. 143–164.

    Google Scholar 

  • Taylor, G. I. (1953). “The dispersion of soluble matter in solvent flowing through a tube.” Proceedings of the Royal Society of London A 1953, Vol. 219(A), pp. 186–203.

    Google Scholar 

  • Taylor, G. I. (1954). “The dispersion of matter in turbulent flow through a pipe.” Proceedings of the Royal Society of London A 1954, Vol. 223(A), pp. 446–468.

    Google Scholar 

  • Thackston, E. L. and Krenkel, P. A. (1967). “Longitudinal mixing in natural streams.” ASCE Proceedings Journal of the Sanitary Engineering Division, Vol. 93(SA5), pp. 67–90.

    Google Scholar 

  • Toprak, Z. F. (2004). Determination of longitudinal dispersion coefficients in natural channel using fuzzy logic method, PhD Thesis, Istanbul Technical University, Institute of Science and Technology, Istanbul

    Google Scholar 

  • Toprak, Z. F. and Cigizoglu, H. K. (2008). “Predicting longitudinal dispersion coefficient in natural streams by artificial intelligence methods.” Hydrological Processes, Vol. 22, No. 20, pp. 4106–4129.

    Google Scholar 

  • Toprak, Z. F. and Savci, M E. (2004). “Predicting dimensionless longitudinal dispersion coefficient in natural streams by fuzzy-logic approach.” Proc. of International Conference on Water Observation and Information System for Decision Support, Ohrid, Republic of Macedonia, (ffp-80-049).

    Google Scholar 

  • Toprak, Z. F. and Savci, M. E. (2007). “Longitudinal dispersion modeling in natural channels by fuzzy logic.” CLEAN Soil Air Water, Vol. 35,Issue 6, pp. 626–637.

    Google Scholar 

  • Toprak, Z. F., Savci, M. E., and Avci, C. (2004b). “Comparison of the dispersion model results using contour map method.” Proceeding of 6 th International Congress on Advances in Civil Engineering (ACE2004), Bogazici University, Istanbul, Turkey, pp. 1407–1417.

    Google Scholar 

  • Toprak, Z. F., Şen, Z., and Savci, M. E. (2004a). “Comment on longitudinal dispersion coefficients in natural channel.” Water Research, Vol. 38,Issue 13, pp. 3139–3143.

    Google Scholar 

  • Tsoukalas, L. H. and Uhrig, R. E. (1997). Fuzzy and neural approaches in engineering, John Wiley and Sons, New York, N.Y.

    Google Scholar 

  • Willis, I., Lawson, W., Owens, I., Jacobel, B., and Autridge, J. (2009). “Subglacial drainage system structure and morphology of Brewster Glacier, New Zealand.” Hydrological Processes, Vol. 23, No. 3. pp. 384–396.

    Google Scholar 

  • Wu, C. L., Chau, K.-W., and Li, Y. S., (2008). “River stage prediction based on a distributed support vector regression.” Journal of Hydrology, Vol. 358, Nos. 1–2, pp. 96–111.

    Google Scholar 

  • Xiaodong, L., Zulin, H., Hongqin, X., Li, G., and Zengfang, X. (2011). “Inverse optimization method to determine longitudinal dispersion coefficient and selection of sampling time in tracer tests.” Fresenius Environmental Bulletin, Vol. 20, No. 5, pp. 1149–1154.

    Google Scholar 

  • Yearsley, J. R. (2009). “A semi-Lagrangian water temperature model for advection-dominated river systems.” Water Resources Research, Vol. 45,Issue 12, W12405, p. 19.

    Google Scholar 

  • Yotsukura, N., Fischer, H. B., and Sayre, W. W. (1970). Measurement of mixing characteristics of the Missouri River between Sioux City, Iowa and Plattsmouth, Nebraska. US Geological Survey Water-Supply, Paper 1899-G, Washington, D.C., USA.

    Google Scholar 

  • Yu, Y. S. and Wenzhi, L. (1989). “Longitudinal dispersion in rwers: A dead-zone model solution.” JAWRA Journal of the American Water Resources Association, Vol. 25,Issue 2, pp. 319–325.

    Google Scholar 

  • Zhang, Q. (1995). “Transient behaviour of mixing induced by a random velocity field.” Water Resources Research, Vol. 31,Issue 3, pp. 577–591.

    Google Scholar 

  • Zhang, W. (2011). “A 2-D numerical simulation study on longitudinal solute transport and longitudinal dispersion coefficient.” Water Resources Research, Vol. 47,Issue 7, W07533, p. 13.

    Google Scholar 

  • Zhang, Y. J, Jha, M, Gu, R., Wensheng, L., and Alin, L. (2012). “A DEM-based parallel computing hydrodynamic and transport model.” River Research and Applications, Vol. 28,Issue 5, pp. 647–658.

    Google Scholar 

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Authors and Affiliations

  1. Civil Engineering Dept., Dicle University, Diyarbakir, 21280, Turkey

    Z. Fuat Toprak & Nizamettin Hamidi

  2. Canik Başarı University, Samsun, 55020, Turkey

    Ozgur Kisi

  3. Civil Engineering Dept., Harran University, Sanliurfa, 63300, Turkey

    Reşit Gerger

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  1. Z. Fuat Toprak
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  3. Ozgur Kisi
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Toprak, Z.F., Hamidi, N., Kisi, O. et al. Modeling dimensionless longitudinal dispersion coefficient in natural streams using artificial intelligence methods. KSCE J Civ Eng 18, 718–730 (2014). https://doi.org/10.1007/s12205-014-0089-y

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  • DOI: https://doi.org/10.1007/s12205-014-0089-y

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