{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,15]],"date-time":"2026-02-15T09:15:00Z","timestamp":1771146900318,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,8,30]],"date-time":"2017-08-30T00:00:00Z","timestamp":1504051200000},"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":["51536008"],"award-info":[{"award-number":["51536008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91441104"],"award-info":[{"award-number":["91441104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11502237"],"award-info":[{"award-number":["11502237"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Entropy generation in two-dimensional Rayleigh-B\u00e9nard convection at different Prandtl number (Pr) are investigated in the present paper by using the lattice Boltzmann Method. The major concern of the present paper is to explore the effects of Pr on the detailed information of local distributions of entropy generation in virtue of frictional and heat transfer irreversibility and the overall entropy generation in the whole flow field. The results of this work indicate that the significant viscous entropy generation rates (Su) gradually expand to bulk contributions of cavity with the increase of Pr, thermal entropy generation rates (S\u03b8) and total entropy generation rates (S) mainly concentrate in the steepest temperature gradient, the entropy generation in the flow is dominated by heat transfer irreversibility and for the same Rayleigh number, the amplitudes of Su, S\u03b8 and S decrease with increasing Pr. It is found that that the amplitudes of the horizontally averaged viscous entropy generation rates, thermal entropy generation rates and total entropy generation rates decrease with increasing Pr. The probability density functions of Su, S\u03b8 and S also indicate that a much thinner tail while the tails for large entropy generation values seem to fit the log-normal curve well with increasing Pr. The distribution and the departure from log-normality become robust with decreasing Pr.<\/jats:p>","DOI":"10.3390\/e19090443","type":"journal-article","created":{"date-parts":[[2017,8,30]],"date-time":"2017-08-30T10:40:00Z","timestamp":1504089600000},"page":"443","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["A Numerical Study on Entropy Generation in Two-Dimensional Rayleigh-B\u00e9nard Convection at Different Prandtl Number"],"prefix":"10.3390","volume":"19","author":[{"given":"Yikun","family":"Wei","sequence":"first","affiliation":[{"name":"Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China"}]},{"given":"Zhengdao","family":"Wang","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China"}]},{"given":"Yuehong","family":"Qian","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1115\/1.2911298","article-title":"Natural convection with radiation in a cavity with open top end","volume":"114","author":"Lage","year":"1992","journal-title":"J. Heat Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.compfluid.2014.06.024","article-title":"Natural convection coupled with radiation heat transfer in an inclined porous cavity with corner heater","volume":"102","author":"Khaled","year":"2014","journal-title":"Comput. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/S0140-7007(98)00055-3","article-title":"Experiments on stratified chilled-water tanks","volume":"22","author":"Nelson","year":"1999","journal-title":"Int. J. Refrig."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3551","DOI":"10.1016\/S0017-9310(03)00147-9","article-title":"Experimental benchmark data for turbulent natural convection in an air filled square cavity","volume":"46","author":"Ampofo","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_5","first-page":"1450","article-title":"Experimental uncertainty of measured entropyproduction with pulsed laser PIV and planar laser induced fluorescence","volume":"48","author":"Adeyinka","year":"2005","journal-title":"Appl. Therm. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1146\/annurev.fluid.010908.165152","article-title":"Small-scale properties of turbulent Rayleigh-B\u00e9nard convection","volume":"42","author":"Lohse","year":"2010","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1140\/epje\/i2012-12058-1","article-title":"New perspectives in turbulent Rayleigh-B\u00e9nard convection","volume":"35","author":"Chilla","year":"2012","journal-title":"Eur. Phys. J. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1103\/RevModPhys.81.503","article-title":"Heat transfer and large scale dynamics in turbulent Rayleigh-B\u00e9nard convection","volume":"81","author":"Ahlers","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"144504","DOI":"10.1103\/PhysRevLett.97.144504","article-title":"Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence","volume":"97","author":"Sun","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1017\/jfm.2012.363","article-title":"Aspect ratio dependence of heat transport by turbulent Rayleigh-B\u00e9nard convectin in rectangular cells","volume":"710","author":"Zhou","year":"2011","journal-title":"J. Fluid Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"075006","DOI":"10.1088\/1367-2630\/12\/7\/075006","article-title":"Physical and geometrical properties of thermal plumes in turbulent Rayleigh-B\u00e9nard convection","volume":"12","author":"Zhou","year":"2012","journal-title":"New J. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0735-1933(00)00089-0","article-title":"Natural convection of a high Prandtl number fluid in cavity","volume":"27","author":"Poujol","year":"2000","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.1016\/S0017-9310(02)00508-2","article-title":"Cooling of cylindrical vertical tank submitted to natural internal convection","volume":"46","author":"De","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.ijthermalsci.2007.02.008","article-title":"On the importance of thermal boundary conditions in heat transfer and entropy generation for natural convection inside a porous enclosure","volume":"47","author":"Zahmatkesh","year":"2008","journal-title":"Int. J. Therm. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3221","DOI":"10.1016\/j.ijheatmasstransfer.2006.01.032","article-title":"Entropy generation in natural convection in a symmetrically and uniformly heated vertical channel","volume":"49","author":"Andreozzi","year":"2006","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/j.icheatmasstransfer.2007.01.003","article-title":"Entropy generation for natural convection in C-shaped enclosures","volume":"34","author":"Dagtekin","year":"2007","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1017\/S002211200800013X","article-title":"Analysis of sheet like thermal plumes in turbulent Rayleigh-B\u00e9nard convection","volume":"599","author":"Shishkina","year":"2012","journal-title":"J. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1017\/S0022112008003947","article-title":"Analysis of the thermal plumes in turbulent Rayleigh-B\u00e9nard convection based on well-resolved numerical simulations","volume":"618","author":"Kaczorowski","year":"2011","journal-title":"J. Fluid Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.116","article-title":"Numerical study on mixed convection and entropy generation of Cu-water nanofluid in a differentially heated skewed enclosure","volume":"85","author":"Nayak","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.jtice.2015.04.018","article-title":"Natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation","volume":"56","author":"Selimefendigil","year":"2015","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.010","article-title":"A review on entropy generation in nanofluid flow","volume":"65","author":"Mahian","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1016\/j.rser.2014.11.104","article-title":"Entropy generation analysis as a design tool\u2014A review","volume":"43","author":"Sciacovelli","year":"2015","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1209\/0295-5075\/17\/6\/001","article-title":"Lattice BGK Models for Navier-Stokes Equation","volume":"17","author":"Qian","year":"1992","journal-title":"Europhys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1146\/annurev.fluid.30.1.329","article-title":"Lattice Boltzmann method for fluid flows","volume":"30","author":"Chen","year":"1998","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1146\/annurev-fluid-121108-145519","article-title":"Lattice-Boltzmann Method for Complex Flows","volume":"42","author":"Aidun","year":"2010","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2780","DOI":"10.1103\/PhysRevE.55.2780","article-title":"Simulation of Rayleigh-B\u00e9nard convection using a lattice Boltzmann method","volume":"55","author":"Shan","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1006\/jcph.1998.6057","article-title":"A novel thermal model for the lattice Boltzmann method in incompressible limit","volume":"146","author":"He","year":"1998","journal-title":"J. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.compfluid.2015.09.004","article-title":"Simulation of natural convection heat transfer in an enclosure using lattice Boltzmann method","volume":"124","author":"Wei","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1650364","DOI":"10.1142\/S0217984916503541","article-title":"Characteristics of heat transfer with different dimensionless distance in an enclosure","volume":"30","author":"Wei","year":"2016","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_30","unstructured":"Bejan, A. (1982). Entropy Generation through Heat and Fluid Flow, John Wiley & Sons."},{"key":"ref_31","unstructured":"Iandoli, C.L. (2000). Analysis of the Entropy Generation Fields of a Low ns Centrifugal Compressor. [Master\u2019s Thesis, University of Rome]. (In Italian)."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3441","DOI":"10.1016\/S0017-9310(03)00133-9","article-title":"Entropy generation at the onset of natural convection","volume":"46","author":"Magherbi","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1016\/j.applthermaleng.2008.07.012","article-title":"Entropy generation and natural convection in rectangular cavities","volume":"29","author":"Rejane","year":"2009","journal-title":"Appl. Therm. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Sheremet, M.A., Oztop, H.F., Pop, I., and Hamdeh, N.A. (2016). Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das\u2019Model. Entropy, 18.","DOI":"10.3390\/e18010009"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Bhatti, M.M., Abbas, T., Rashidi, M.M., Ali, M.E., and Yang, Z.G. (2016). Entropy Generation on MHD Eyring-Powell Nanofluid through a Permeable Stretching Surface. Entropy, 18.","DOI":"10.3390\/e18060224"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1166\/jon.2016.1248","article-title":"Entropy generation with nonlinear thermal radiation in MHD boundary layer flow over a permeable shrinking\/stretching sheet: Numerical solution","volume":"5","author":"Bhatti","year":"2016","journal-title":"J. Nanofluids"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Abbas, M.A., Bai, Y., Rashidi, M.M., and Bhatti, M.M. (2016). Analysis of Entropy Generation in the Flow of Peristaltic Nanofluids in Channels with Compliant Walls. Entropy, 18.","DOI":"10.3390\/e18030090"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Qing, J., Bhatti, M.M., Abbas, M.A., Rashidi, M.M., and Ali, M.S. (2016). Entropy Generation on MHD Casson Nanofluid Flow over a Porous Stretching\/Shrinking Surface. Entropy, 18.","DOI":"10.3390\/e18040123"},{"key":"ref_39","first-page":"21","article-title":"Calculation of exergetic losses in compact heat exchanger passages","volume":"10","author":"Paoletti","year":"1989","journal-title":"ASME AES"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"125107","DOI":"10.1063\/1.2140024","article-title":"Statistics and geometry of passive scalars in turbulence","volume":"17","author":"Schumacher","year":"2005","journal-title":"Phys. Fluids"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/9\/443\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:43:42Z","timestamp":1760208222000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/9\/443"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,30]]},"references-count":40,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["e19090443"],"URL":"https:\/\/doi.org\/10.3390\/e19090443","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,30]]}}}