{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T03:10:43Z","timestamp":1780629043967,"version":"3.54.1"},"reference-count":51,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T00:00:00Z","timestamp":1580947200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11672124"],"award-info":[{"award-number":["11672124"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shenzhen Key Laboratory of Complex Aerospace Flows","award":["ZDSYS201802081843517"],"award-info":[{"award-number":["ZDSYS201802081843517"]}]},{"name":"the Shenzhen Peacock Plan","award":["KQTD2016022620054656"],"award-info":[{"award-number":["KQTD2016022620054656"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The effect of the Prandtl number (Pr) on the flow and heat transfer from a porous circular cylinder with internal heat generation in the mixed convection regime is numerically investigated. The steady flow regime is considered over the ranges of the Reynolds number (Re), Darcy number (Da), and Richardson number (Ri), varying from 5 to 40, 10\u22126 to 10\u22122, and 0 to 2, respectively. The wake structure, the temperature distribution, and the heat transfer rate are discussed. Besides precipitating the growth of the recirculating wake, the Prandtl number is found to have a significant impact on the thermal characteristics. The concave isotherms, resembling a saddle-shaped structure, occur behind the cylinder at larger Pr, resulting in swells of the isotherms pairing off at the lateral sides. These swells are found to have a negative effect on heat transfer owing to a relatively smaller temperature gradient there. Then, the heat transfer rate in terms of the local Nusselt number (Nu) and enhancement ratio (Er) is calculated, which is closely related to Pr, Re, Da, and Ri. The local minimum heat transfer rate along the cylinder surface is found at the position where the swells of the isotherms form.<\/jats:p>","DOI":"10.3390\/e22020184","type":"journal-article","created":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T03:13:27Z","timestamp":1581045207000},"page":"184","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Effect of Prandtl Number on Mixed Convective Heat Transfer from a Porous Cylinder in the Steady Flow Regime"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7034-8681","authenticated-orcid":false,"given":"Shimin","family":"Yu","sequence":"first","affiliation":[{"name":"Harbin Institute of Technology, Harbin 150001, China"},{"name":"Shenzhen Key Laboratory of Complex Aerospace Flows, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tingting","family":"Tang","sequence":"additional","affiliation":[{"name":"Shenzhen Key Laboratory of Complex Aerospace Flows, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China"},{"name":"Center for Complex Flows and Soft Matter Research, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianhui","family":"Li","sequence":"additional","affiliation":[{"name":"Shenzhen Key Laboratory of Complex Aerospace Flows, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2073-7512","authenticated-orcid":false,"given":"Peng","family":"Yu","sequence":"additional","affiliation":[{"name":"Shenzhen Key Laboratory of Complex Aerospace Flows, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China"},{"name":"Center for Complex Flows and Soft Matter Research, Southern University of Science and Technology, Shenzhen 518055, China"},{"name":"Guangdong Provincial Key Laboratory of Turbulence Research and Applications, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,6]]},"reference":[{"key":"ref_1","unstructured":"Nield, D.A., and Bejan, A. (2006). Convection in Porous Media, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Vad\u00e1sz, P. (2008). Emerging Topics in Heat and Mass Transfer in Porous Media: From Bioengineering and Microelectronics to Nanotechnology, Springer Science & Business Media.","DOI":"10.1007\/978-1-4020-8178-1"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3438","DOI":"10.3390\/e17053438","article-title":"Heat transfer and pressure drop characteristics in straight microchannel of printed circuit heat exchangers","volume":"17","author":"Seo","year":"2015","journal-title":"Entropy"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ma, H., Duan, Z., Su, L., Ning, X., Bai, J., and Lv, X. (2019). Fluid flow and entropy generation analysis of Al2O3\u2014Water nanofluid in microchannel plate fin heat sinks. Entropy, 21.","DOI":"10.3390\/e21080739"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.rser.2014.08.040","article-title":"Review of convection heat transfer and fluid flow in porous media with nanofluid","volume":"41","author":"Mahdi","year":"2015","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1016\/j.applthermaleng.2016.05.096","article-title":"A review of the capabilities of high heat flux removal by porous materials, microchannels and spray cooling techniques","volume":"104","author":"Smakulski","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Chen, H., Zheng, Z., Chen, Z., and Bi, X. (2016). A lattice gas automata model for the coupled heat transfer and chemical reaction of gas flow around and through a porous circular cylinder. Entropy, 18.","DOI":"10.3390\/e18010002"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2170","DOI":"10.1016\/j.enconman.2010.12.027","article-title":"Heat transfer from a permeable square cylinder to a flowing fluid","volume":"52","author":"Dhinakaran","year":"2011","journal-title":"Energy Convers. Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.oceaneng.2016.08.031","article-title":"Vortex shedding suppression and wake control: A review","volume":"126","author":"Rashidi","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.ijheatmasstransfer.2013.01.067","article-title":"Forced convection from a circular cylinder in pulsating flow with and without the presence of porous media","volume":"61","author":"Thompson","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1017\/jfm.2011.313","article-title":"Thermal effects on the wake of a heated circular cylinder operating in mixed convection regime","volume":"685","author":"Hu","year":"2011","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"277","DOI":"10.2514\/3.356","article-title":"Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow","volume":"6","author":"Ahmad","year":"1992","journal-title":"J. Thermophys. Heat Transf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1522","DOI":"10.1016\/j.applthermaleng.2006.09.023","article-title":"Experimental investigation of mixed convection heat transfer for thermally developing flow in a horizontal circular cylinder","volume":"27","author":"Mohammed","year":"2007","journal-title":"Appl. Therm. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9735","DOI":"10.1021\/ie801892m","article-title":"Mixed convection heat transfer from a cylinder in power-law fluids: Effect of aiding buoyancy","volume":"48","author":"Srinivas","year":"2009","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/0017-9310(83)90014-5","article-title":"A theoretical study of laminar mixed convection from a horizontal cylinder in a cross stream","volume":"26","author":"Badr","year":"1983","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1016\/j.ijheatmasstransfer.2011.12.034","article-title":"Mixed convection flow and heat transfer across a square cylinder under the influence of aiding buoyancy at low Reynolds numbers","volume":"55","author":"Sharma","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wu, H.-W., and Wang, R.-H. (2011). Mixed convective heat transfer past a heated square porous cylinder in a horizontal channel with varying channel height. J. Heat Transf., 133.","DOI":"10.1115\/1.4002632"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1016\/j.ijheatmasstransfer.2018.06.056","article-title":"Effect of thermal buoyancy on flow and heat transfer around a permeable circular cylinder with internal heat generation","volume":"126","author":"Yu","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1080\/10407780500283325","article-title":"Effects of Reynolds and Prandtl numbers on heat transfer across a square cylinder in the steady flow regime","volume":"49","author":"Dhiman","year":"2006","journal-title":"Numer. Heat Transf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"113603","DOI":"10.1063\/1.4966937","article-title":"Effects of Prandtl number on the laminar cross flow past a heated cylinder","volume":"28","author":"Mathur","year":"2016","journal-title":"Phys. Fluids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1063\/1.1692061","article-title":"Steady laminar forced convection from a circular cylinder at low Reynolds numbers","volume":"11","author":"Dennis","year":"1968","journal-title":"Phys. Fluids"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/0017-9310(90)90180-3","article-title":"Heat transfer in Newtonian liquids around a circular cylinder","volume":"33","author":"Vilimpoq","year":"1990","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s00231-006-0155-1","article-title":"A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder","volume":"43","author":"Bharti","year":"2007","journal-title":"Heat Mass Transf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4598","DOI":"10.1016\/j.ijheatmasstransfer.2005.04.033","article-title":"Flow and heat transfer across a confined square cylinder in the steady flow regime: Effect of Peclet number","volume":"48","author":"Dhiman","year":"2005","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.ijheatmasstransfer.2008.07.032","article-title":"Effects of Reynolds and Prandtl numbers on heat transfer from a square cylinder in the unsteady flow regime","volume":"52","author":"Sahu","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"79","DOI":"10.2298\/TSCI100914057S","article-title":"Heat transfer from a rotating circular cylinder in the steady regime: Effects of Prandtl number","volume":"16","author":"Sharma","year":"2012","journal-title":"Therm. Sci"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"937","DOI":"10.4208\/cicp.2014.m314","article-title":"Effects of Reynolds and Prandtl numbers on heat transfer around a circular cylinder by the simplified thermal lattice Boltzmann model","volume":"17","author":"Chen","year":"2015","journal-title":"Commun. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.1007\/s12206-015-0351-3","article-title":"Free stream flow and forced convection heat transfer across rotating circular cylinder in steady regime: Effects of rotation, Prandtl number and thermal boundary condition","volume":"29","author":"Sufyan","year":"2015","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3885","DOI":"10.1016\/S0017-9310(02)00109-6","article-title":"Effects of the Darcy number, the Prandtl number, and the Reynolds number on local thermal non-equilibrium","volume":"45","author":"Kim","year":"2002","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1358","DOI":"10.1016\/j.ijheatmasstransfer.2018.06.003","article-title":"Effects of Prandtl number on the forced convection heat transfer from a porous square cylinder","volume":"126","author":"Anirudh","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Khan, A., Khan, D., Khan, I., Taj, M., Ullah, I., Aldawsari, A.M., Thounthong, P., and Sooppy Nisar, K. (2019). MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana\u2013Baleanu Derivative of Non-Local and Non-Singular Kernel. Symmetry, 11.","DOI":"10.3390\/sym11101295"},{"key":"ref_32","unstructured":"Steinberg, D.S. (1980). Cooling Techniques for Electronic Equipment, Wiley-Interscience."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"119","DOI":"10.13182\/NSE77-A27015","article-title":"Natural convection in porous media with heat generation","volume":"63","author":"Hardee","year":"1977","journal-title":"Nucl. Sci. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/S1385-8947(00)00349-1","article-title":"A catalytic heat-exchanging tubular reactor for combining of high temperature exothermic and endothermic reactions","volume":"82","author":"Ismagilov","year":"2001","journal-title":"Chem. Eng. J."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Khan, U., Zaib, A., Khan, I., and Nisar, K.S. (2019). Activation energy on MHD flow of titanium alloy (Ti6Al4V) nanoparticle along with a cross flow and streamwise direction with binary chemical reaction and non-linear radiation: Dual Solutions. J. Mater. Res. Technol.","DOI":"10.1016\/j.jmrt.2019.10.044"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2635","DOI":"10.1016\/0017-9310(94)00346-W","article-title":"Momentum transfer at the boundary between a porous medium and a homogeneous fluid\u2014I. Theoretical development","volume":"38","author":"Whitaker","year":"1995","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2647","DOI":"10.1016\/0017-9310(94)00347-X","article-title":"Momentum transfer at the boundary between a porous medium and a homogeneous fluid\u2014II. Comparison with experiment","volume":"38","author":"Whitaker","year":"1995","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_38","first-page":"201","article-title":"Momentum jump condition at the boundary between a porous medium and a homogeneous fluid: Inertial effects","volume":"1","author":"Alberto","year":"1998","journal-title":"J. Porous Media"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1002\/fld.1383","article-title":"A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump","volume":"53","author":"Yu","year":"2007","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1002\/fld.1575","article-title":"A numerical method for forced convection in porous and homogeneous fluid domains coupled at interface by stress jump","volume":"56","author":"Chen","year":"2008","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1115\/1.2910291","article-title":"Perspective: A method for uniform reporting of grid refinement studies","volume":"116","author":"Roache","year":"1994","journal-title":"J. Fluids Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1146\/annurev.fluid.29.1.123","article-title":"Quantification of uncertainty in computational fluid dynamics","volume":"29","author":"Roache","year":"1997","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"696","DOI":"10.2514\/2.457","article-title":"Verification of codes and calculations","volume":"36","author":"Roache","year":"1998","journal-title":"AIAA J."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.ijheatfluidflow.2009.12.009","article-title":"Wake structure for flow past and through a porous square cylinder","volume":"31","author":"Yu","year":"2010","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compfluid.2010.09.040","article-title":"Steady flow around and through a permeable circular cylinder","volume":"42","author":"Yu","year":"2011","journal-title":"Comput. Fluids"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.ijheatfluidflow.2012.03.002","article-title":"Numerical simulation on steady flow around and through a porous sphere","volume":"36","author":"Yu","year":"2012","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1108\/09615530910930982","article-title":"Numerical analysis for the flow past a porous trapezoidal-cylinder based on the stress-jump interfacial-conditions","volume":"19","author":"Chen","year":"2009","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3799","DOI":"10.1016\/j.ijheatmasstransfer.2007.02.021","article-title":"A numerical study of momentum and forced convection heat transfer around two tandem circular cylinders at low Reynolds numbers. Part II: Forced convection heat transfer","volume":"50","author":"Juncu","year":"2007","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1017\/S0022112069002448","article-title":"The effect of base bleed on the steady separated flow past bluff objects","volume":"39","author":"Leal","year":"1969","journal-title":"J. Fluid Mech."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s00231-001-0279-2","article-title":"Entropy generation due to cross-flow heat transfer from a cylinder covered with an orthotropic porous layer","volume":"39","year":"2002","journal-title":"Heat Mass Transf."},{"key":"ref_51","unstructured":"Ya, C., Ghajar, A., and Ma, H. (2015). Heat and Mass Transfer: Fundamentals & Applications, McGraw-Hill."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/184\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:55:24Z","timestamp":1760172924000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/184"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,6]]},"references-count":51,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020184"],"URL":"https:\/\/doi.org\/10.3390\/e22020184","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,6]]}}}