{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T22:28:35Z","timestamp":1783117715508,"version":"3.54.6"},"reference-count":48,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T00:00:00Z","timestamp":1559347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The knowledge of the fluid discharge in free surface flows requires a great number of velocity measurements along the whole cross-section, taking up a large amount of time, using expensive equipment, and employing specialized labor. To overcome these obstacles, various models have been developed thus far that show how to estimate the mean velocity through the maximum velocity. In three-dimensional open channels, the maximum velocity can be located below the free surface because of the presence of secondary flows mainly originating by the sidewalls, an occurrence known as dip-phenomenon. In this condition, predicting the maximum velocity position is quite difficult and has always represented a challenge to most hydraulic engineers and researchers. In the present study, a mathematical model derived from the information entropy theory is proposed to evaluate the velocity-dip-position over the entire cross-section of both wide and narrow open channels, thus overcoming the limitations of the existing methods. Large literature measurement sets, collected in uniform and non-uniform flows, were used to test the validity of the model, showing good agreement with the experimental data and providing an accurate estimation of the dip-position.<\/jats:p>","DOI":"10.3390\/e21060554","type":"journal-article","created":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T02:08:40Z","timestamp":1559527720000},"page":"554","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Information Entropy Theory Applied to the Dip-Phenomenon Analysis in Open Channel Flows"],"prefix":"10.3390","volume":"21","author":[{"given":"Domenica","family":"Mirauda","sequence":"first","affiliation":[{"name":"School of Engineering, Basilicata University, Viale dell\u2019Ateneo Lucano 10, 85100 Potenza, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maria Grazia","family":"Russo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Computer Science and Economics, Basilicata University, Viale dell\u2019Ateneo Lucano 10, 85100 Potenza, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1061\/(ASCE)0733-9429(1987)113:5(583)","article-title":"Entropy and probability concepts in hydraulics","volume":"113","author":"Chiu","year":"1987","journal-title":"J. Hydraul. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1061\/(ASCE)0733-9429(1989)115:5(576)","article-title":"Velocity distribution in open channel flow","volume":"115","author":"Chiu","year":"1989","journal-title":"J. Hydraul. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1061\/(ASCE)0733-9429(1995)121:1(26)","article-title":"Maximum and mean velocities and entropy in open-channel flow","volume":"121","author":"Chiu","year":"1995","journal-title":"J. Hydraul. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1061\/TACEAT.0000315","article-title":"On the cause of the maximum velocity of water flowing in open channels being below the surface","volume":"7","author":"Francis","year":"1878","journal-title":"Trans. Am. Soc. Civ. Eng."},{"key":"ref_5","first-page":"331","article-title":"A reason why the maximum velocity of water flowing in open channels is below the surface","volume":"7","author":"Stearns","year":"1883","journal-title":"Trans. Am. Soc. Civ. Eng."},{"key":"ref_6","first-page":"111","article-title":"Accuracy of stream measurements","volume":"95","author":"Murphy","year":"1904","journal-title":"Water Supply Irrig."},{"key":"ref_7","first-page":"149","article-title":"On the depression of the filament of maximum velocity in a stream flowing through an open channel","volume":"82","author":"Gibson","year":"1909","journal-title":"Proc. R. Soc. A Math. Phys. Sci."},{"key":"ref_8","first-page":"67","article-title":"Transportation of suspended sediment by running water","volume":"111","author":"Vanoni","year":"1946","journal-title":"Trans. ASCE"},{"key":"ref_9","unstructured":"Nezu, I., and Rodi, W. (1985, January 13\u201318). Experimental study on secondary currents in open channel flows. Proceedings of the 21th IAHR Congress, IAHR, Melbourne, Australia."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.advwatres.2004.12.008","article-title":"Secondary flows over artificial bed strips","volume":"28","author":"Wang","year":"2005","journal-title":"Adv. Water Resour."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1061\/(ASCE)0733-9429(2000)126:3(210)","article-title":"Detailed study of binary law for open channels","volume":"126","author":"Sarma","year":"2000","journal-title":"J. Hydraul. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1080\/00221680109499822","article-title":"Velocity profile of sediment suspensions and comparison of log-law and wake-law","volume":"39","author":"Wang","year":"2001","journal-title":"J. Hydraul. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1061\/(ASCE)0733-9429(2002)128:4(390)","article-title":"Maximum velocity and regularities in open channel flow","volume":"128","author":"Chiu","year":"2002","journal-title":"J. Hydraul. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1061\/(ASCE)0733-9429(2004)130:12(1179)","article-title":"Velocity distribution and dip-phenomenon in smooth uniform channel flows","volume":"130","author":"Yang","year":"2004","journal-title":"J. Hydraul. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1634","DOI":"10.1016\/j.advwatres.2005.12.002","article-title":"Time-mean structure of secondary flows in open channel with longitudinal bed forms","volume":"29","author":"Wang","year":"2006","journal-title":"Adv. Water Resour."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10652-007-9051-6","article-title":"Turbulent velocity profile in fully-developed open channel flows","volume":"8","author":"Bonakdari","year":"2008","journal-title":"Environ. Fluid Mech."},{"key":"ref_17","first-page":"17","article-title":"Application of the modified log-wake law in open-channels","volume":"1","author":"Guo","year":"2008","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Absi, R. (2009, January 29\u201330). Analytical methods for velocity distribution and dip-phenomenon in narrow open-channel flows. Proceedings of the IWEH: International Workshop on Environmental Hydraulics Theoretical, Experimental and Computational Solutions, Valencia, Spain.","DOI":"10.1201\/b10999-33"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1080\/00221686.2010.535700","article-title":"An ordinary differential equation for velocity distribution and dip-phenomenon in open channel flows","volume":"49","author":"Absi","year":"2011","journal-title":"J. Hydraul. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1080\/00221686.2013.818584","article-title":"Modified log-wake-law for smooth rectangular open channel flow","volume":"52","author":"Guo","year":"2013","journal-title":"J. Hydraul. Res."},{"key":"ref_21","first-page":"413","article-title":"Universal velocity distribution for smooth and rough open channel flows","volume":"6","author":"Pu","year":"2013","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"221","DOI":"10.18869\/acadpub.jafm.73.238.26403","article-title":"Prediction of velocity-dip-position at the central section of open channels using entropy theory","volume":"10","author":"Kundu","year":"2017","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mirauda, D., Pannone, M., and de Vincenzo, A. (2018). An entropic model for the assessment of stream-wise velocity dip in wide open channels. Entropy, 20.","DOI":"10.3390\/e20010069"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"720","DOI":"10.1061\/(ASCE)0733-9429(1997)123:8(720)","article-title":"Relation between mean and maximum velocities in a natural river","volume":"123","author":"Xia","year":"1997","journal-title":"J. Hydraul. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1016\/j.proeng.2014.02.084","article-title":"Manning\u2019s roughness through the entropy parameter for steady open channel flows in low submergence","volume":"70","author":"Greco","year":"2014","journal-title":"Procedia Eng."},{"key":"ref_26","first-page":"641","article-title":"An Entropy Based Velocity Profile for Steady Flows with Large-Scale Roughness","volume":"Volume 3","author":"Lollino","year":"2015","journal-title":"Engineering Geology for Society and Territory, River Basins, Reservoir Sedimentation and Water Resources"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"04014047-1","DOI":"10.1061\/(ASCE)HE.1943-5584.0001009","article-title":"Entropy parameter estimation in large-scale roughness open channel","volume":"20","author":"Greco","year":"2015","journal-title":"J. Hydrol. Eng."},{"key":"ref_28","unstructured":"Moramarco, T., Ammari, A., Burnelli, A., Mirauda, D., and Pascale, V. (2008, January 6\u201310). Entropy Theory Application for Flow Monitoring in Natural Channels. Proceedings of the iEMSS 4th Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008), Barcelona, Spain."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1007\/s10652-015-9403-6","article-title":"Entropic approach to estimate the mean flow velocity: Experimental investigation in laboratory flumes","volume":"15","author":"Moramarco","year":"2015","journal-title":"Environ. Fluid Mech."},{"key":"ref_30","unstructured":"Termini, D., and Moramarco, T. (2016). Application of entropic approach to estimate the mean flow velocity and Manning roughness coefficient in a high-curvature flume. Hydrol. Res., 48."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Shannon, C.E. (1948). The Mathematical Theory of Communications, Bell System Technical Journal.","DOI":"10.1002\/j.1538-7305.1948.tb00917.x"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics: I","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical mechanics: II","volume":"108","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1109\/PROC.1982.12425","article-title":"On the rationale of maximum entropy methods","volume":"70","author":"Jaynes","year":"1982","journal-title":"Proc. IEEE"},{"key":"ref_35","unstructured":"Hu, C., and Hui, Y. (1995). Mechanical and Statistical Laws of Open Channel Sediment-Laden Flow, Science Press."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1680\/iwtme.1994.27234","article-title":"Turbulence characteristics in rough uniform open-channel flow","volume":"106","author":"Kironoto","year":"1994","journal-title":"Proc. Inst. Civ. Eng. Water Marit. Energy"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1051\/lhb:200504012","article-title":"Applicability of velocity distribution equations in rough-bed open-channel flow","volume":"4","author":"Ardiclioglu","year":"2005","journal-title":"La Houille Blanche"},{"key":"ref_38","unstructured":"Einstein, H.A., and Chien, N. (1955). Effects of Heavy Sediment Concentration Near the Bed on Velocity and Sediment Distribution, University of California at Berkeley and Missouri River Division, U.S. Army Corps of Engineers. [3rd ed.]."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Guy, H.P., Simons, D.B., and Richardson, E.V. (1966). Summary of Alluvial Channel Data from Flume Experiments, 1956-61, Professional Paper.","DOI":"10.3133\/pp462I"},{"key":"ref_40","unstructured":"Lyn, D.A. (1987). Turbulence and Turbulent Transport in Sediment-Laden Open-Channel Flows. [Ph.D. Thesis, California Institute of Technology]."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1029\/WR022i010p01377","article-title":"Effects of suspended sediment on the Open-Channel velocity distribution","volume":"22","author":"Coleman","year":"1986","journal-title":"Water Resour. Res."},{"key":"ref_42","unstructured":"Valiani, A. (1988). Nuove esperienze su correnti uniformi con trasporto di sedimenti in sospensione: La metodologia sperimentale ed alcuni risultati. Seminario Su Trasporto Solido Ed Evoluzione Morfologica Dei Corsi D\u2019acqua, IRIS Universit\u00e0 degli Studi di Ferrara. (In Italian)."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1080\/00221680209499886","article-title":"Suspension flows in open channels; experimental study","volume":"40","author":"Graf","year":"2002","journal-title":"J. Hydraul. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1061\/(ASCE)1084-0699(2004)9:1(42)","article-title":"Estimation of mean velocity in natural channels based on Chiu\u2019s velocity distribution equation","volume":"9","author":"Moramarco","year":"2004","journal-title":"J. Hydrol. Eng."},{"key":"ref_45","first-page":"355","article-title":"Practical method for flow velocity measurements in fluvial sections","volume":"Volume 146","author":"Brebbia","year":"2011","journal-title":"River Basin Management VI, WIT Transactions on Ecology and the Environment"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2166\/nh.2013.122","article-title":"Effect of bed roughness on 1-D entropy velocity distribution in open channel flow","volume":"46","author":"Greco","year":"2015","journal-title":"Hydrol. Res."},{"key":"ref_47","unstructured":"Mirauda, D., de Vincenzo, A., and Pannone, M. (2017). Statistical characterization of flow field structure in evolving braided gravel beds. Spat. Stat."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1061\/(ASCE)0733-9429(1991)117:5(615)","article-title":"Application of entropy concept in open-channel flow study","volume":"117","author":"Chiu","year":"1991","journal-title":"J. Hydraul. Eng."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/6\/554\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:30Z","timestamp":1760187330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/6\/554"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,1]]},"references-count":48,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["e21060554"],"URL":"https:\/\/doi.org\/10.3390\/e21060554","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,1]]}}}