{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T22:08:40Z","timestamp":1761948520125,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T00:00:00Z","timestamp":1558569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51509004"],"award-info":[{"award-number":["51509004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Open Research Foundation of Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation, Ministry of Water Resources, China","award":["2018KJ01"],"award-info":[{"award-number":["2018KJ01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the context of river dynamics, some experimental results have shown that particle velocity is different from fluid velocity along the stream-wise direction for uniform sediment-laden open-channel flows; this velocity difference has been termed velocity lag in the literature. In this study, an analytical expression for estimating the velocity lag in open-channel flows was derived based on the Tsallis entropy theory together with the principle of maximum entropy. The derived expression represents the velocity lag as a function of a non-dimensional entropy parameter depending on the average and maximum values of velocity lag from experimental measurements. The derived expression was tested against twenty-two experimental datasets collected from the literature with three deterministic models and the developed Shannon entropy-based model. The Tsallis entropy-based model agreed better with the experimental datasets than the deterministic models for eighteen out of the twenty-two total real cases, and the prediction accuracy for the eighteen experimental datasets was comparable to that of the developed Shannon entropy-based model (the Tsallis entropy-based expression agreed slightly better than the Shannon entropy-based model for twelve out of eighteen test cases, whereas for the other six test cases, the Shannon entropy-based model had a slightly higher prediction accuracy). Finally, the effects of the friction velocity of the flow, the particle diameter, and the particles\u2019 specific gravity on the velocity lag were analyzed based on the Tsallis entropy-based model. This study shows the potential of the Tsallis entropy theory together with the principle of maximum entropy to predict the stream-wise velocity lag between a particle and the surrounding fluid in sediment-laden open-channel flows.<\/jats:p>","DOI":"10.3390\/e21050522","type":"journal-article","created":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T02:22:00Z","timestamp":1558664520000},"page":"522","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0579-608X","authenticated-orcid":false,"given":"Zhongfan","family":"Zhu","sequence":"first","affiliation":[{"name":"Beijing Key Laboratory of Urban Hydrological Cycle and Sponge City Technology, College of Water Sciences, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingshan","family":"Yu","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Urban Hydrological Cycle and Sponge City Technology, College of Water Sciences, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5930-199X","authenticated-orcid":false,"given":"Jie","family":"Dou","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Nagaoka University of Technology 1603-1, Kami-Tomioka, Nagaoka 940-2188, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2118-5174","authenticated-orcid":false,"given":"Dingzhi","family":"Peng","sequence":"additional","affiliation":[{"name":"Beijing Key Laboratory of Urban Hydrological Cycle and Sponge City Technology, College of Water Sciences, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1061\/(ASCE)0733-9429(2004)130:7(657)","article-title":"Analysis of velocity lag in sediment-laden open channel flows","volume":"130","author":"Cheng","year":"2004","journal-title":"J. Hydraul. Eng."},{"key":"ref_2","unstructured":"Chien, N., and Wan, Z. (1983). Sediment transport mechanics, Science Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1098\/rspa.1973.0038","article-title":"The nature of saltation and bedload transport in water","volume":"332","author":"Bagnold","year":"1973","journal-title":"Proc. Royal Soc. Lond. Ser. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1061\/(ASCE)0733-9429(1996)122:5(282)","article-title":"Error estimate in Einstein\u2019s suspended sediment load method","volume":"122","author":"Aziz","year":"1996","journal-title":"J. Hydraul. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1061\/(ASCE)0733-9429(1997)123:9(742)","article-title":"Velocity profiles for particles and liquid in open-channel flow with suspended sediment","volume":"123","author":"Muste","year":"1997","journal-title":"J. Hydraul. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1061\/(ASCE)0733-9429(1997)123:12(1118)","article-title":"Turbulence modulation and particle velocities over flat sand beds at low transport rates","volume":"123","author":"Best","year":"1997","journal-title":"J. Hydraul. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1016\/0301-9322(90)90099-5","article-title":"Particle-turbulence interaction in a boundary layer","volume":"16","author":"Rashidi","year":"1990","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s003480050136","article-title":"On the behavior of solid particles in a horizontal boundary layer with turbulence and saltation effects","volume":"23","author":"Taniere","year":"1997","journal-title":"Exp. Fluids"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kiger, K.T., and Pan, C. (2002). Suspension and turbulence modification effects of solid particulates on a horizontal turbulent channel flow. J. Turbul.","DOI":"10.1615\/TSFP2.480"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chauchat, J., and Guillou, S. (2008). On turbulence closures for two-phase sediment-laden flow models. J. Geophys. Res. Oceans.","DOI":"10.1029\/2007JC004708"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s10652-008-9091-6","article-title":"Hierarchical modeling of the dilute transport of suspended sediment in open channels","volume":"9","author":"Bombardelli","year":"2009","journal-title":"Environ. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1080\/00221686.1999.9628264","article-title":"Two-phase formulation of suspended sediment transport","volume":"37","author":"Greimann","year":"1999","journal-title":"J. Hydraul. Res."},{"key":"ref_13","first-page":"273","article-title":"Two-phase modeling of suspended sediment distribution in open channel flows","volume":"42","author":"Jiang","year":"2004","journal-title":"J. Hydraul. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.euromechflu.2015.11.003","article-title":"Velocity lag between particle and liquid in sediment-laden open channel turbulent flow. Eur","volume":"56","author":"Pal","year":"2016","journal-title":"J. Mech. B Fluids"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kumbhakar, M., Kundu, S., Ghoshal, K., and Singh, V.P. (2016). Entropy-based modeling of velocity lag in sediment-laden open channel turbulent flow. Entropy, 18.","DOI":"10.3390\/e18090318"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Singh, V.P., Sivakumar, B., and Cui, H.J. (2017). Tsallis entropy theory for modelling in water engineering: A review. Entropy, 19.","DOI":"10.3390\/e19120641"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1061\/(ASCE)HE.1943-5584.0000793","article-title":"One dimensional velocity distribution in open channels using Tsallis entropy","volume":"19","author":"Cui","year":"2014","journal-title":"J. Hydrol. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/j.jhydrol.2018.06.010","article-title":"Comparative study of 1D entropy-based and conventional deterministic velocity distribution equations for open channel flows","volume":"563","author":"Luo","year":"2018","journal-title":"J. Hydrol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1061\/(ASCE)HE.1943-5584.0000319","article-title":"Entropy theory for two-dimensional velocity distribution","volume":"16","author":"Luo","year":"2011","journal-title":"J. Hydrol. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1061\/(ASCE)HE.1943-5584.0000610","article-title":"Two-dimensional velocity distribution in open channels using the Tsallis entropy","volume":"18","author":"Cui","year":"2013","journal-title":"J. Hydrol. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"447","DOI":"10.13031\/2013.29585","article-title":"Tsallis entropy theory for derivation of infiltration equations","volume":"53","author":"Singh","year":"2010","journal-title":"Trans. ASABE"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (2010). Entropy theory for derivation of infiltration equations. Water Resour. Res., 46.","DOI":"10.1029\/2009WR008193"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (2010). Entropy theory for movement of moisture in soils. Water Resour. Res., 46.","DOI":"10.1029\/2009WR008288"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1061\/(ASCE)0733-9429(2000)126:1(16)","article-title":"Mathematical models of distribution of sediment concentration","volume":"1","author":"Chiu","year":"2000","journal-title":"J. Hydraul. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1061\/(ASCE)HE.1943-5584.0000865","article-title":"Suspended sediment concentration in open channels using Tsallis entropy","volume":"19","author":"Cui","year":"2013","journal-title":"J. Hydrol. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1061\/(ASCE)HE.1943-5584.0000930","article-title":"Flow duration curve using entropy theory","volume":"19","author":"Singh","year":"2014","journal-title":"J. Hydrol. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.physa.2014.10.075","article-title":"Modelling sediment concentration in debris flow by Tsallis entropy","volume":"420","author":"Singh","year":"2015","journal-title":"Phys. A"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Zhu, Z., Wang, H., Pang, B., Dou, J., and Peng, D. (2019). Comparison of conventional deterministic and entropy-based methods for predicting sediment concentration in debris flow. Water, 11.","DOI":"10.3390\/w11030439"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhu, Z., and Yu, J. (2019). Estimating the bed-load layer thickness in open channels by Tsallis entropy. Entropy, 21.","DOI":"10.3390\/e21020123"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1017\/S0022112004008304","article-title":"Particle-fluid interactions in a plane near-wall turbulent flow","volume":"505","author":"Righetti","year":"2004","journal-title":"J. Fluid Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"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","first-page":"959","article-title":"Using Shannon entropy to model turbulence-induced flocculation of cohesive sediment in water","volume":"11","author":"Zhu","year":"2018","journal-title":"Environ. Sci. Pollut. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1063\/1.868552","article-title":"Particle behavior in the turbulent boundary layer velocity and distribution profiles","volume":"7","author":"Kaftori","year":"1995","journal-title":"Phys. Fluids"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"W10402","DOI":"10.1029\/2004WR003595","article-title":"Two-phase versus mixed-flow perspective on suspended sediment transport in turbulent channel flows","volume":"41","author":"Muste","year":"2005","journal-title":"Water Resour. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1061\/(ASCE)0733-9429(1997)123:2(149)","article-title":"Simplified settling velocity formula for sediment particle","volume":"123","author":"Cheng","year":"1997","journal-title":"J. Hydraul. Eng."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/522\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:54:40Z","timestamp":1760187280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/522"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,23]]},"references-count":38,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["e21050522"],"URL":"https:\/\/doi.org\/10.3390\/e21050522","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,5,23]]}}}