Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-10T09:27:51.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct numerical simulation of power-law fluids over smooth and rough surfaces

Published online by Cambridge University Press:  29 July 2025

Hamidreza Anbarlooei Open the ORCID record for Hamidreza Anbarlooei [Opens in a new window] ,
Daniel Onofre de Almeida Cruz Open the ORCID record for Daniel Onofre de Almeida Cruz [Opens in a new window] ,
Gustavo Eduardo Oviedo Celis Open the ORCID record for Gustavo Eduardo Oviedo Celis [Opens in a new window] ,
Matheus de Souza Santos Macedo Open the ORCID record for Matheus de Souza Santos Macedo [Opens in a new window]  and
Roney Leon Thompson Open the ORCID record for Roney Leon Thompson [Opens in a new window]
Hamidreza Anbarlooei*
Affiliation:
Department of Applied Mathematics, Institute of Mathematics, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro 21941-909, Brazil
Daniel Onofre de Almeida Cruz
Affiliation:
COPPE, Mechanical Engineering Program, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro 21941-909, Brazil
Gustavo Eduardo Oviedo Celis
Affiliation:
COPPE, Mechanical Engineering Program, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro 21941-909, Brazil
Matheus de Souza Santos Macedo
Affiliation:
COPPE, Mechanical Engineering Program, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro 21941-909, Brazil
Roney Leon Thompson*
Affiliation:
COPPE, Mechanical Engineering Program, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro 21941-909, Brazil
*
Corresponding authors: Roney Leon Thompson, rthompson@mecanica.coppe.ufrj.br; Hamidreza Anbarlooei, hamidreza@im.ufrj.br
Corresponding authors: Roney Leon Thompson, rthompson@mecanica.coppe.ufrj.br; Hamidreza Anbarlooei, hamidreza@im.ufrj.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Direct numerical simulation (DNS) studies of power-law (PL) fluids are performed for purely viscous-shear-thinning ($n\in [0.5,0.75]$), Newtonian ($n=1$) and purely viscous-shear-thickening ($n=2.0$) fluids, considering two Reynolds numbers ($Re_{\tau }\in [395,590]$), and both smooth and rough surfaces. We carefully designed a numerical experiment to isolate key effects and simplify the complex problem of turbulent flow of non-Newtonian fluids over rough surfaces, enabling the development of a theoretical model to explain the observed phenomena and provide predictions. The DNS results of the present work were validated against literature data for smooth and rough Newtonian turbulent flows, as well as smooth shear-thinning cases. A new analytical expression for the mean velocity profile – extending the classical Blasius $1/7$ profile to power-law fluids – was proposed and validated. In contrast to common belief, the decrease in $n$ leads to smaller Kolmogorov length scales and the formation of larger structures, requiring finer grids and longer computational domains for accurate simulations. Our results confirm that purely viscous shear-thinning fluids exhibit drag reduction, while shear-thickening fluids display an opposite trend. Interestingly, we found that viscous-thinning turbulence shares similarities with Newtonian transitional flows, resembling the behaviour of shear-thinning, extensional-thickening viscoelastic fluids. This observation suggests that the extensional and elastic effects in turbulent flows within constant cross-section geometries may not be significant. However, the shear-thickening case exhibits characteristics similar to high-Reynolds-number Newtonian turbulence, suggesting that phenomena observed in such flows could be studied at significantly lower Reynolds numbers, reducing computational costs. In the analysis of rough channels, we found that the recirculation bubble between two roughness elements is mildly influenced by the thinning nature of the fluid. Moreover, we observed that shear-thinning alters the flow in the fully rough regime, where the friction factor typically reaches a plateau. Our results indicate the possibility that, at sufficiently high Reynolds numbers, this plateau may not exist for shear-thinning fluids. Finally, we provide detailed turbulence statistics for different rheologies, allowing, for the first time, an in-depth study of the effects of rheology on turbulent flow over rough surfaces.

JFM classification

Non-Newtonian Flows: Rheology Non-Newtonian Flows: Polymers Turbulent Flows: Turbulence simulation

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1016 , 10 August 2025 , A13
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Allen, J.J., Shockling, M.A., Kunkel, G.J. & Smits, A.J. 2007 Turbulent flow in smooth and rough pipes. Phil. Trans. R. Soc. A: Math. Phys. Engng Sci. 365 (1852), 699–714.10.1098/rsta.2006.1939CrossRefGoogle ScholarPubMed
Anbarlooei, H.R., Cruz, D.O.A. & Freire, A.P.S. 2014 The logarithmic solution of purely viscous fluid. In 10th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements. Springer.Google Scholar
Anbarlooei, H.R., Cruz, D.O.A. & Freire, A.P.S. 2015 a Fully turbulent mean velocity profile for purely viscous non-Newtonian fluids. In 15th European Turbulence Conference. European Mechanics Society.Google Scholar
Anbarlooei, H.R., Cruz, D.O.A. & Ramos, F. 2020 New power–law scaling for friction factor of extreme Reynolds number pipe flows. Phys. Fluids 32 (9), 095121.10.1063/5.0020665CrossRefGoogle Scholar
Anbarlooei, H.R., Cruz, D.O.A., Ramos, F. & Freire, A.P.S. 2015 b Phenomenological Blasius-type friction equation for turbulent power–law fluid flows. Phys. Rev. E 92 (6), 063006.10.1103/PhysRevE.92.063006CrossRefGoogle ScholarPubMed
Anbarlooei, H.R., Cruz, D.O.A., Ramos, F., Santos, C.M.M. & Freire, A.P.S. 2017 Phenomenological friction equation for turbulent flow of Bingham fluids. Phys. Rev. E 96 (2), 023107.10.1103/PhysRevE.96.023107CrossRefGoogle ScholarPubMed
Anbarlooei, H.R., Cruz, D.O.A., Ramos, F., Santos, C.M.M. & Freire, A.P.S. 2018 On the connection between Kolmogorov microscales and friction in pipe flows of viscoplastic fluids. Physica D: Nonlinear Phenom. 376–377, 69–77.10.1016/j.physd.2017.11.005CrossRefGoogle Scholar
Anbarlooei, H.R., Ramos, F. & Cruz, D.O.A. 2022 Connection between attached eddies, friction factor, and mean-velocity profile. Phys. Rev. Fluids 7 (2), 024602.10.1103/PhysRevFluids.7.024602CrossRefGoogle Scholar
Arosemena, A.A., Andersson, H.I. & Solsvik, J. 2020 Turbulent channel flow of generalized Newtonian fluids at a low Reynolds number. J. Fluid Mech. 908, A43.10.1017/jfm.2020.903CrossRefGoogle Scholar
Ashrafian, A., Andersson, H.I. & Manhart, M. 2004 DNS of turbulent flow in a rod-roughened channel. Intl J. Heat Fluid Flow 25 (3), 373–383.10.1016/j.ijheatfluidflow.2004.02.004CrossRefGoogle Scholar
Beris, A.N. & Dimitropoulos, C.D. 1999 Pseudospectral simulation of turbulent viscoelastic channel flow. Comput. Meth. Appl. Mech. 180 (3), 365–392.10.1016/S0045-7825(99)00174-7CrossRefGoogle Scholar
Burattini, P., Leonardi, S., Orlandi, P. & Antonia, R.A. 2008 Comparison between experiments and direct numerical simulations in a channel flow with roughness on one wall. J. Fluid Mech. 600, 403–426.10.1017/S0022112008000657CrossRefGoogle Scholar
Busse, A., Lützner, M. & Sandham, N.D. 2015 Direct numerical simulation of turbulent flow over a rough surface based on a surface scan. Comput. Fluids 116, 129–147.10.1016/j.compfluid.2015.04.008CrossRefGoogle Scholar
Clapp, R.M. 1961 Turbulent heat transfer in pseudoplastic non-Newtonian fluids. In Conference of International Developments in Heat Transfer, pp. 652–662. ASME.Google Scholar
Coleman, G.N. & Sandberg, R.D. 2010 A primer on direct numerical simulation of turbulence – methods, procedures and guidelines. Tech. Rep, Aerodynamics & Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, UK.Google Scholar
Cruz, D.O.A. & Pinho, F.T. 2003 Turbulent pipe flow predictions with a low Reynolds number k − ɛ model for drag reducing fluids. J. Non-Newtonian Fluid Mech. 114, 109–148.10.1016/S0377-0257(03)00119-8CrossRefGoogle Scholar
Dodge, D.W. & Metzner, A.B. 1959 Turbulent flow of non-Newtonian systems. AIChE J. 5 (2), 189–204.10.1002/aic.690050214CrossRefGoogle Scholar
Dubief, Y., White, C.M., Terrapon, V.E., Shaqfeh, E.S.G., Moin, P. & Lele, S.K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280.10.1017/S0022112004000291CrossRefGoogle Scholar
Ferziger, J.H. & Peric, M. 1999 Computational Methods for Fluid Dynamics. Springer.10.1007/978-3-642-98037-4CrossRefGoogle Scholar
Gavrilov, A.A. & Rudyak, V.Y. 2016 Direct numerical simulation of the turbulent flows of power–law fluids in a circular pipe. Thermophys. Aeromech. 23 (4), 473–486.10.1134/S0869864316040016CrossRefGoogle Scholar
Gioia, G. & Chakraborty, P. 2006 Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory. Phys. Rev. Lett. 96 (4), 044502.10.1103/PhysRevLett.96.044502CrossRefGoogle ScholarPubMed
Gioia, G., Guttenberg, N., Goldenfeld, N. & Chakraborty, P. 2010 Spectral theory of the turbulent mean-velocity profile. Phys. Rev. Lett. 105 (18), 184501.10.1103/PhysRevLett.105.184501CrossRefGoogle ScholarPubMed
Gnambode, P.S., Orlandi, P., Ould-Rouiss, M. & Nicolas, X. 2015 Large-Eddy simulation of turbulent pipe flow of power–law fluids. Intl J. Heat Fluid Flow 54, 196–210.10.1016/j.ijheatfluidflow.2015.05.004CrossRefGoogle Scholar
Greenshields, C.J. 2015 OpenFOAM User Guide. The OpenFOAM Foundation Ltd.Google Scholar
Guang, R., Rudman, M., Chryss, A., Slatter, P. & Bhattacharya, S. 2011 Direct numerical simulation (DNS) investigation of turbulent open channel flow of a Herschel-Bulkley fluid. In Paste 2011: Proceedings of the 14th International Seminar on Paste and Thickened Tailings, pp. 439–452. Australian Centre for Geomechanics.10.36487/ACG_rep/1104_38_GuangCrossRefGoogle Scholar
Ikeda, T. & Durbin, P.A. 2007 Direct simulations of a rough-wall channel flow. J. Fluid Mech. 571, 235–263.10.1017/S002211200600334XCrossRefGoogle Scholar
Inagaki, M. 2021 Large eddy simulation of non-Newtonian viscous fluids with low grid dependency using an anisotropy-resolving subgrid-scale model. J. Non-Newtonian Fluid Mech. 295, 104603.10.1016/j.jnnfm.2021.104603CrossRefGoogle Scholar
Ismail, U., Zaki, T.A. & Durbin, P.A. 2018 Simulations of rib-roughened rough-to-smooth turbulent channel flows. J. Fluid Mech. 843, 419–449.10.1017/jfm.2018.119CrossRefGoogle Scholar
Issa, R.I. 1986 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 40–65.10.1016/0021-9991(86)90099-9CrossRefGoogle Scholar
Jagadeesan, K. & Narasimhamurthy, V.D. 2019 Reynolds number effects in rib-roughened turbulent channel flow. Intl J. Adv. Engng Sci. Appl. Maths 1 (4), 254–262.10.1007/s12572-020-00258-6CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.10.1017/S0022112095000462CrossRefGoogle Scholar
Karahan, D.T., Ranjan, D. & Aidun, C.K. 2023 Turbulent channel flow of generalized Newtonian fluids at Re τ = 180. J. Non-Newtonian Fluid Mech. 314, 105015.10.1016/j.jnnfm.2023.105015CrossRefGoogle Scholar
Komen, E., Shams, A., Camilo, L. & Koren, B. 2014 Quasi-DNS capabilities of OpenFOAM for different mesh types. Comput. Fluids 96, 87–104.10.1016/j.compfluid.2014.02.013CrossRefGoogle Scholar
Krogstad, P.-A., Andersson, H.I., Bakken, O.M. & Ashrafian, A. 2005 An experimental and numerical study of channel flow with rough walls. J. Fluid Mech. 530, 327–352.10.1017/S0022112005003824CrossRefGoogle Scholar
Kuwata, Y. & Kawaguchi, Y. 2019 Direct numerical simulation of turbulence over systematically varied irregular rough surfaces. J. Fluid Mech. 862, 781–815.10.1017/jfm.2018.953CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to Re τ = 5200. J. Fluid Mech. 774, 395–415.10.1017/jfm.2015.268CrossRefGoogle Scholar
Leonardi, S., Orlandi, P. & Antonia, R.A. 2007 Properties of d- and k-type roughness in a turbulent channel flow. Phys. Fluids 19 (12), 125101.10.1063/1.2821908CrossRefGoogle Scholar
McKeon, B.J., Li, J., Jiang, W., Morrison, J.F. & Smits, A.J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.10.1017/S0022112003007304CrossRefGoogle Scholar
Metzner, A.B. & Reed, J.C. 1955 Flow of non-Newtonian fluids – correlation of the laminar, transition, and turbulent-flow regions. AIChE J. 1 (4), 434–440.10.1002/aic.690010409CrossRefGoogle Scholar
Moser, R.D., Kim, J. & Mansour, N.N. 1999 Direct numerical simulation of turbulent channel flow up to Re τ = 590. Phys. Fluids 11 (4), 943–945.10.1063/1.869966CrossRefGoogle Scholar
Narayanan, C., Singh, J.S., Nauer, S., Belt, R.J., Palermo, T. & Lakehal, D. 2024 Turbulent flow of non-Newtonian fluid in rough channels. J. Fluid Mech. 1000, A55.10.1017/jfm.2024.891CrossRefGoogle Scholar
Nikuradse, J. 1933 Stromungsgesetze in rauhen rohren. Forsch. Arb. Ing. Wes. 356, 1.Google Scholar
Ohta, T. & Miyashita, M. 2014 DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids. J. Non-Newtonian Fluid Mech. 206, 29–39.10.1016/j.jnnfm.2014.02.003CrossRefGoogle Scholar
Orlandi, P. & Leonardi, S. 2006 DNS of turbulent channel flows with two- and three-dimensional roughness. J. Turbul. 7, N73.10.1080/14685240600827526CrossRefGoogle Scholar
Peixinho, J., Nouar, C., Desaubry, C. & Théron, B. 2005 Laminar transitional and turbulent flow of yield stress fluid in a pipe. J. Non-Newtonian Fluid Mech. 128 (2), 172–184.10.1016/j.jnnfm.2005.03.008CrossRefGoogle Scholar
Pereira, A.S., Mompean, G., Thais, L. & Thompson, R.L. 2017 Statistics and tensor analysis of polymer coil–stretch mechanism in turbulent drag reducing channel flow. J. Fluid Mech. 824, 135–173.10.1017/jfm.2017.332CrossRefGoogle Scholar
Pereira, A.S., Thompson, R.L. & Mompean, G. 2019 Common features between the Newtonian laminar–turbulent transition and the viscoelastic drag-reducing turbulence. J. Fluid Mech. 877, 405–428.10.1017/jfm.2019.567CrossRefGoogle Scholar
Pinho, F.T. & Whitelaw, J.H. 1990 Flow of non-Newtonian fluids in a pipe. J. Non-Newtonian Fluid Mech. 34 (2), 129–144.10.1016/0377-0257(90)80015-RCrossRefGoogle Scholar
Pinho, F.T. 2003 A GNF framework for turbulent flow models of drag reducing fluids and proposal for a k − ɛ type closure. J. Non-Newtonian Fluid Mech. 114 (2), 149–184.10.1016/S0377-0257(03)00120-4CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Rahmani, H. & Taghavi, S.M. 2022 a A comprehensive model for viscoplastic flows in channels with a patterned wall: longitudinal, transverse and oblique flows. J. Fluid Mech. 984, A32.10.1017/jfm.2024.197CrossRefGoogle Scholar
Rahmani, H. & Taghavi, S.M. 2022 b Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall. J. Fluid Mech. 948, A34.10.1017/jfm.2022.700CrossRefGoogle Scholar
Rudman, M. & Blackburn, H.M. 2006 Direct numerical simulation of turbulent non-Newtonian flow using a spectral element method. Appl. Math. Model. 30 (11), 1229–1248.10.1016/j.apm.2006.03.005CrossRefGoogle Scholar
Rudman, M. & Blackburn, H.M. 2012 Turbulence modification in shear thinning fluids: Preliminary results for power–law rheology. In 18th Australasian Fluid Mechanics Conference. Australasian Fluid Mechanics Society.Google Scholar
Rudman, M., Blackburn, H.M., Graham, L.J.W. & Pullum, L. 2004 Turbulent pipe flow of shear-thinning fluids. J. Non-Newtonian Fluid Mech. 118 (1), 33–48.10.1016/j.jnnfm.2004.02.006CrossRefGoogle Scholar
Rudman, M., Graham, L.J.W., Blackburn, H.M. & Pullum, L. 2002 Non-Newtonian turbulent and transitional pipe flow. In 15th International Conference on Hydrotransport, pp. 271–286. BHR Group.Google Scholar
Santos, C.M.M. 2018 Experimental analysis and friction modeling in fully developed turbulent flow of non-Newtonian fluid in smooth and rough surface pipe. PhD thesis, Universidade Federal do Rio de Janeiro, Brazil.Google Scholar
Santos, C.M.M., Anbarlooei, H.R., Cruz, D.O.A. & Ramos, F. 2018 Post roughness dominant regime in the turbulent flow through rough pipes. In Proceedings of the Ninth International Symposium on Turbulence Heat and Mass Transfer. Begell House Inc.Google Scholar
Singh, J., Rudman, M. & Blackburn, H.M. 2017 a The effect of yield stress on pipe flow turbulence for generalised newtonian fluids. J. Non-Newtonian Fluid Mech. 249, 53–62.10.1016/j.jnnfm.2017.09.007CrossRefGoogle Scholar
Singh, J., Rudman, M. & Blackburn, H.M. 2017 b The influence of shear-dependent rheology on turbulent pipe flow. J. Fluid Mech. 822, 848–879.10.1017/jfm.2017.296CrossRefGoogle Scholar
Singh, J., Rudman, M. & Blackburn, H.M. 2018 Reynolds number effects in pipe flow turbulence of generalized Newtonian fluids. Phys. Rev. Fluids 3 (9), 094607.10.1103/PhysRevFluids.3.094607CrossRefGoogle Scholar
Soares, E.J. 2020 Review of mechanical degradation and de-aggregation of drag reducing polymers in turbulent flows. J. Non-Newtonian Fluid Mech. 276, 104225.10.1016/j.jnnfm.2019.104225CrossRefGoogle Scholar
Sprague, M.A. 2010 A comparison of Nek5000 and OpenFOAM for DNS of turbulent channel flow. In Nek5000 Users Meeting, Argonne National Lab.Google Scholar
Thakkar, M., Busse, A. & Sandham, N.D. 2018 Direct numerical simulation of turbulent channel flow over a surrogate for Nikuradse-type roughness. J. Fluid Mech. 837, R1.10.1017/jfm.2017.873CrossRefGoogle Scholar
Thompson, R.L. & Oishi, C.M. 2021 Reynolds and Weissenberg numbers in viscoelastic flows. J. Non-Newtonian Fluid Mech. 292, 104–550.10.1016/j.jnnfm.2021.104550CrossRefGoogle Scholar
Varma, H., Jagadeesan, K., Narasimhamurthy, V.D., Kesarkar, A.P. & Andersson, H.I. 2021 LES and DNS of symmetrically roughened turbulent channel flows. Acta Mechanica 232 (12), 4951–4968.10.1007/s00707-021-03082-6CrossRefGoogle Scholar
Vilquin, A., Jagielka, J., Djambov, S., Herouard, H., Fisher, P., Bruneau, C.-H., Chakraborty, P., Gioia, G. & Kellay, H. 2021 Asymptotic turbulent friction in 2D rough-walled flows. Sci. Adv. 7 (5), eabc6234.10.1126/sciadv.abc6234CrossRefGoogle ScholarPubMed
Vo, S., Kronenburg, A., Stein, O.T. & Hawkes, E.R. 2016 Direct numerical simulation of non-premixed syngas combustion using OpenFOAM. In High Performance Computing in Science and Engineering, 16 (ed. W.E Nagel, D.H. Kröner & M.M. Resch) pp. 245–257. Springer International Publishing.Google Scholar
Warholic, M.D., Massah, H. & Hanratty, T.J. 1999 Influence of drag-reducing polymers on turbulence: effects of Reynolds number, concentration and mixing. Exp. Fluids 27 (5), 461–472.10.1007/s003480050371CrossRefGoogle Scholar
Wei, T., Fife, P., Klewicki, J. & Mcmurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.10.1017/S0022112004001958CrossRefGoogle Scholar
White, F.M. 1991 Viscous Fluid Flow. 3rd edn. McGraw-Hill.Google Scholar
Yimprasert, S., Kvick, M., Alfredsson, P.H. & Matsubara, M. 2021 Flow visualization and skin friction determination in transitional channel flow. Exp. Fluids 62 (2), 31.10.1007/s00348-020-03102-6CrossRefGoogle Scholar
Zheng, E.Z., Rudman, M., Singh, J. & Kuang, S.B. 2019 Direct numerical simulation of turbulent non-Newtonian flow using OpenFOAM. Appl. Math. Model. 72, 50–67.10.1016/j.apm.2019.03.003CrossRefGoogle Scholar