{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:25:09Z","timestamp":1760239509574,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N5059681\/1"],"award-info":[{"award-number":["EP\/N5059681\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>A viscoelastic turbulence model in a fully-developed drag reducing channel flow is improved, with turbulent eddies modelled under a k\u2013\u03b5 representation, along with polymeric solutions described by the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model. The model performance is evaluated against a wide variety of direct numerical simulation data, described by different combinations of rheological parameters, which is able to predict all drag reduction (low, intermediate and high) regimes with good accuracy. Three main contributions are proposed: one with a simplified viscoelastic closure for the NLTij term (which accounts for the interactions between the fluctuating components of the conformation tensor and the velocity gradient tensor), by removing additional damping functions and reducing complexity compared with previous models; second through a reformulation for the closure of the viscoelastic destruction term, E\u03c4p, which removes all friction velocity dependence; lastly by an improved modified damping function capable of predicting the reduction in the eddy viscosity and thus accurately capturing the turbulent kinetic energy throughout the channel. The main advantage is the capacity to predict all flow fields for low, intermediate and high friction Reynolds numbers, up to high drag reduction without friction velocity dependence.<\/jats:p>","DOI":"10.3390\/app10228140","type":"journal-article","created":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T11:02:12Z","timestamp":1605610932000},"page":"8140","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A FENE-P k\u2013\u03b5 Viscoelastic Turbulence Model Valid up to High Drag Reduction without Friction Velocity Dependence"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5091-7170","authenticated-orcid":false,"given":"Michael","family":"McDermott","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, University of Leeds, Woodhouse, Leeds LS2 9JT, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3410-6482","authenticated-orcid":false,"given":"Pedro","family":"Resende","sequence":"additional","affiliation":[{"name":"ProMetheus, Escola Superior de Tecnologia e Gest\u00e3o, Instituto Polit\u00e9cnico de Viana do Castelo, 4900-347 Viana do Castelo, Portugal"}]},{"given":"Thibaut","family":"Charpentier","sequence":"additional","affiliation":[{"name":"Baker Hughes, Kirkby Bank Road, Knowsley Industrial Park, Liverpool L33 7EU, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1058-2003","authenticated-orcid":false,"given":"Mark","family":"Wilson","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, University of Leeds, Woodhouse, Leeds LS2 9JT, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2825-0709","authenticated-orcid":false,"given":"Alexandre","family":"Afonso","sequence":"additional","affiliation":[{"name":"Transport Phenomena Research Center, Faculty of Engineering, University of Porto, Rua Roberto Frais s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0169-517X","authenticated-orcid":false,"given":"David","family":"Harbottle","sequence":"additional","affiliation":[{"name":"School of Chemical and Process Engineering, University of Leeds, Woodhouse, Leeds LS2 9JT, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5647-1771","authenticated-orcid":false,"given":"Gregory","family":"de Boer","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, University of Leeds, Woodhouse, Leeds LS2 9JT, UK"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,17]]},"reference":[{"key":"ref_1","first-page":"135","article-title":"Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers","volume":"2","author":"Toms","year":"1948","journal-title":"Proc. Int. Cong. Rheol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1002\/aic.690210402","article-title":"Drag reduction fundamentals","volume":"21","author":"Virk","year":"1975","journal-title":"AIChE J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1146\/annurev.fl.01.010169.002055","article-title":"Drag reduction by additives","volume":"1","author":"Lumley","year":"1969","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1002\/pol.1973.230070104","article-title":"Drag reduction in turbulent flow by polymer additives","volume":"7","author":"Lumley","year":"1973","journal-title":"J. Polym. Sci. Macromol. Rev."},{"key":"ref_5","first-page":"258","article-title":"A Freeman scholar lecture: The effect of additives on fluid friction","volume":"94","author":"Hoyt","year":"1972","journal-title":"J. Fluids Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1017\/S0022112067001442","article-title":"The Toms phenomenon: Turbulent pipe flow of dilute polymer solutions","volume":"30","author":"Virk","year":"1967","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1146\/annurev.fluid.40.111406.102156","article-title":"Mechanics and prediction of turbulent drag reduction with polymer additives","volume":"40","author":"White","year":"2008","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1103\/RevModPhys.80.225","article-title":"Colloquium: Theory of drag reduction by polymers in wall-bounded turbulence","volume":"80","author":"Procaccia","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10494-005-9002-6","article-title":"New answers on the interaction between polymers and vortices in turbulent flows","volume":"74","author":"Dubief","year":"2005","journal-title":"Flow Turbul. Combust."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1063\/1.869229","article-title":"Direct numerical simulation of the turbulent channel flow of a polymer solution","volume":"9","author":"Sureshkumar","year":"1997","journal-title":"Phys. Fluids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"244503","DOI":"10.1103\/PhysRevLett.92.244503","article-title":"Drag reduction by polymers in wall bounded turbulence","volume":"92","author":"Pomyalov","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"055301","DOI":"10.1103\/PhysRevE.70.055301","article-title":"Drag reduction by a linear viscosity profile","volume":"70","author":"Casciola","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"066303","DOI":"10.1103\/PhysRevE.82.066303","article-title":"Strong polymer-turbulence interactions in viscoelastic turbulent channel flow","volume":"82","author":"Dallas","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1017\/S0022112003004610","article-title":"Drag reduction by polymer additives in a turbulent channel flow","volume":"486","author":"Min","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0377-0257(03)00120-4","article-title":"A GNF framework for turbulent flow models of drag reducing fluids and proposal for a k\u2013\u03b5 type closure","volume":"114","author":"Pinho","year":"2003","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.jnnfm.2004.05.004","article-title":"Modelling the new stress for improved drag reduction predictions of viscoelastic pipe flow","volume":"121","author":"Cruz","year":"2004","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ijheatfluidflow.2005.08.002","article-title":"Numerical predictions and measurements of Reynolds normal stresses in turbulent pipe flow of polymers","volume":"27","author":"Resende","year":"2006","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/14685248.2013.851385","article-title":"A Reynolds stress model for turbulent flows of viscoelastic fluids","volume":"14","author":"Resende","year":"2013","journal-title":"J. Turbul."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Leighton, R., Walker, D.T., Stephens, T., and Garwood, G. (2003, January 6\u201310). Reynolds stress modeling for drag reducing viscoelastic flows. Proceedings of the ASME\/JSME 2003 4th Joint Fluids Summer Engineering Conference, Honolulu, HI, USA.","DOI":"10.1115\/FEDSM2003-45655"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.jnnfm.2008.02.008","article-title":"A low Reynolds number turbulence closure for viscoelastic fluids","volume":"154","author":"Pinho","year":"2008","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"035106","DOI":"10.1063\/1.1850920","article-title":"Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow","volume":"17","author":"Housiadas","year":"2005","journal-title":"Phys. Fluids"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.jnnfm.2006.04.012","article-title":"Turbulent channel flow of dilute polymeric solutions: Drag reduction scaling and an eddy viscosity model","volume":"139","author":"Li","year":"2006","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.jnnfm.2005.12.012","article-title":"Influence of rheological parameters on polymer induced turbulent drag reduction","volume":"140","author":"Li","year":"2006","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/j.jnnfm.2011.02.012","article-title":"A FENE-P k\u2013\u03b5 turbulence model for low and intermediate regimes of polymer-induced drag reduction","volume":"166","author":"Resende","year":"2011","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s10494-012-9424-x","article-title":"Development of a Low-Reynolds-number k-\u03c9 Model for FENE-P Fluids","volume":"90","author":"Resende","year":"2013","journal-title":"Flow Turbul. Combust."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.jnnfm.2010.01.013","article-title":"Reynolds-averaged modeling of polymer drag reduction in turbulent flows","volume":"165","author":"Iaccarino","year":"2010","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_27","unstructured":"Dubief, Y., Laccarino, G., and Lele, S. (2004). A Turbulence Model for Polymer Flows, Center for Turbulence Research."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.jnnfm.2013.09.007","article-title":"A viscoelastic k\u2013\u03b5\u2013v2\u2013f turbulent flow model valid up to the maximum drag reduction limit","volume":"202","author":"Masoudian","year":"2013","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.ijheatfluidflow.2015.05.017","article-title":"A Reynolds stress model for turbulent flow of homogeneous polymer solutions","volume":"54","author":"Masoudian","year":"2015","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.ijheatmasstransfer.2016.04.053","article-title":"A RANS model for heat transfer reduction in viscoelastic turbulent flow","volume":"100","author":"Masoudian","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_31","unstructured":"Tsukahara, T., and Kawaguchi, Y. (2013). Proposal of damping function for low-Reynolds-number-model applicable in prediction of turbulent viscoelastic-fluid flow. J. Appl. Math., 2013."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.ijheatfluidflow.2018.07.004","article-title":"An improved k-\u03b5 turbulence model for FENE-P fluids capable to reach high drag reduction regime","volume":"73","author":"Resende","year":"2018","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_33","unstructured":"Bird, R.B., Armstrong, R.C., and Hassager, O. (1987). Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, Wiley."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"040802","DOI":"10.1115\/1.3124648","article-title":"Reynolds-averaged Navier\u2013Stokes equations for turbulence modeling","volume":"62","author":"Alfonsi","year":"2009","journal-title":"Appl. Mech. Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1115\/1.3242636","article-title":"Improved form of the k-\u03b5 model for wall turbulent shear flows","volume":"109","author":"Nagano","year":"1987","journal-title":"J. Fluids Eng."},{"key":"ref_36","unstructured":"Nagano, Y. (1993, January 16\u201318). Modeling the dissipation-rate equation for two-equation turbulence model. Proceedings of the 9th Symposium on Turbulent Shear Flows, Kyoto, Japan."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"N19","DOI":"10.1080\/14685248.2012.685522","article-title":"Some dynamical features of the turbulent flow of a viscoelastic fluid for reduced drag","volume":"13","author":"Thais","year":"2012","journal-title":"J. Turbul."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.ijheatfluidflow.2013.05.016","article-title":"Analysis of polymer drag reduction mechanisms from energy budgets","volume":"43","author":"Thais","year":"2013","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1080\/14685248.2015.1125492","article-title":"Grid and subgrid-scale interactions in viscoelastic turbulent flow and implications for modelling","volume":"17","author":"Masoudian","year":"2016","journal-title":"J. Turbul."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"013103","DOI":"10.1063\/1.3294574","article-title":"Temporal large eddy simulations of turbulent viscoelastic drag reduction flows","volume":"22","author":"Thais","year":"2010","journal-title":"Phys. Fluids"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1016\/j.physd.2009.07.013","article-title":"A short review on drag reduction by polymers in wall bounded turbulence","volume":"239","author":"Benzi","year":"2010","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"025508","DOI":"10.1088\/1873-7005\/aa9e37","article-title":"New developments in isotropic turbulent models for FENE-P fluids","volume":"50","author":"Resende","year":"2018","journal-title":"Fluid Dyn. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2657","DOI":"10.1016\/0017-9310(95)00009-X","article-title":"A nonlinear low-Reynolds-number \u03ba-\u03b5 model for turbulent separated and reattaching flows\u2014I. Flow field computations","volume":"38","author":"Park","year":"1995","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1017\/S0022112099007004","article-title":"An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows","volume":"403","author":"Wallin","year":"2000","journal-title":"J. Fluid Mech."},{"key":"ref_45","first-page":"202","article-title":"The incomplete swak4Foam reference","volume":"131","author":"Gschaider","year":"2013","journal-title":"Tech. Rep."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/10\/22\/8140\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:34:38Z","timestamp":1760178878000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/10\/22\/8140"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,17]]},"references-count":45,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["app10228140"],"URL":"https:\/\/doi.org\/10.3390\/app10228140","relation":{},"ISSN":["2076-3417"],"issn-type":[{"type":"electronic","value":"2076-3417"}],"subject":[],"published":{"date-parts":[[2020,11,17]]}}}