Abstract
Steady inflow through a non-axisymmetric stenotic model at Re = 500-1000 for Newtonian and shear-thinning non-Newtonian rheologies was studied numerically to investigate the experimental evidence of stabilizing effect of shear-thinning fluids. A minimally-dissipative and energy-preserving finite-element based code was used, and results were verified against a higher-order spectral element code. Below a critical Reynolds number (Recrit), both rheology models showed non-stationary and intermittent flow in time, with successive phases of laminar and turbulent regions that were quasi-periodic with long observation times. Using the conventional definition of Reynolds number based on high-shear viscosity, transition was delayed for the shear-thinning model, with Recrit of 760 vs. 700 for the Newtonian rheology, a delay broadly consistent with previous reports. However, using domain-averaged viscosity aposteriori, the Recrit for the shear-thinning model dropped to 710, closer to the Newtonian value. The transition process and the vortical structures for both rheologies were similar, albeit with some differences in the turbulent kinetic energy and evolution of non-stationary perturbations near the transition point. This suggests that previously-reported delays in transition to turbulence for blood vs. Newtonian fluids may be due to rheological factors other than shear-thinning, such as viscoelasticity. Our study also further highlights the challenges of defining non-Newtonian Reynolds numbers for flows in non-trivial geometries.
Similar content being viewed by others
References
Caro, C.G., Fitz-Gerald, J.M., Schroter, R.C.: Atheroma and arterial wall shear observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. R. Soc. Lond. B Biol. Sci. 177, 109–133 (1971)
Wootton, D.M., Ku, D.N.: Fluid mechanics of vascular systems, diseases, and thrombosis. Annu. Rev. Biomed. Eng. 1, 299–329 (1999)
Berger, S.A., Jou, L.D.: Flows in stenotic vessels. Annu. Rev. Fluid Mech. 32, 347–382 (2000)
Ku, D.N.: Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434 (2000)
Young, D.F.: Fluid mechanics of arterial stenoses. J. Biomech. Eng. 101, 157–175 (1979)
Varghese, S.S., Frankel, S.H., Fischer, P.F.: Direct numerical simulation of stenotic flows. Part 1. Steady flow. J. Fluid. Mech. 582, 253–280 (2007)
Robertson, A.M., Sequeira, A., Owens, R.G.: Rheological models for blood. In: Formaggia L, Quarteroni, A, Veneziani, A (eds.) Cardiovascular Mathematics, pp 211–241. Springer (2009)
Forrester, J.H., Young, D.F.: Flow through a converging-diverging tube and its implications in occulsive vascular disease - II. J. Biomech. 3, 307–316 (1970)
Young, D.F., Tsai, F.Y.: Flow characteristics in models of arterial stenoses - I. Steady flow. J. Biomech. 6, 395–410 (1973)
Kim, B.M., Corcoran, W.H.: Experimental measurements of turbulence spectra distal to stenoses. J. Biomech. 7, 335–342 (1974)
Cassanova, R.A., Giddens, D.P.: Disorder distal to modeled stenoses in steady and pulsatile flow. J. Biomech. 11, 441–453 (1978)
Ahmed, S.A., Giddens, D.P.: Velocity measurements in steady flow through axisymmetric stenoses at moderate reynolds numbers. J. Biomech. 16, 505–516 (1983)
Vetel, J., Garon, A., Pelletier, D., Farinas, M.I.: Asymmetrey and transition to turbulence in smooth axisymmetric constriction. J. Fluid Mech. 607, 351–386 (2008)
Griffith, M.D., Leweke, T., Thompson, M.C., Hourigan, K.: Steady inlet flow in stenotic geometric: Convective and absolute instabilities. J. Fluid. Mech. 616, 111–113 (2008)
Clark, C: The fluid mechanics of aortic stenosis - I. theory and steady flow experiments. J. Biomech. 9, 521–528 (1976)
Ahmed, S.A., Giddens, D.P.: Flow disturbance measurements through a constricted tube at moderate reynolds numbers. J. Biomech. 16, 955–963 (1983)
Deshpande, M.D., Giddens, D.P.: Turbulence measurements in a constricted tube. J. Fluid. Mech. 97, 65–89 (1980)
Khalifa, A.M.A., Giddens, D.P.: Characterization and evolution of poststenotic flow disturbances. J. Biomech. 14, 279–296 (1981)
Ahmed, S.A., Giddens, D.P.: Pulsatile poststenotic flow studies with laser doppler anemometry. J. Biomech. 17, 695–705 (1984)
Ojha, M., Cobbold, R.S.C., Johnston, K.W., Hummel, R.L.: Turbulence measurements in a constricted tube. J. Fluid Mech. 203, 173–197 (1989)
Deshpande, M.D., Giddens, D.P., Mabon, R.F.: Steady laminar flow through modelled vascular stenoses. J. Biomech. 9, 165–174 (1976)
Lee, J.S., Fung, Y.C.: Flow in locally constricted tubes at low Reynolds numbers. J. App. Mech. 37, 9–16 (1970)
Varghese, S.S., Frankel, S.H., Fischer, P.F.: Modeling transition to turbulence in eccentric stenotic flows. J. Biomech. Eng. 130, 014,503 (2008)
Sherwin, S.J., Blackburn, H.M.: Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows. J. Fluid Mech. 533, 297–327 (2005)
Samuelsson, J., Tammisola, O., Juniper, M.P.: Breaking axi-symmetry in stenotic flows lowers the critical transition Reynolds number. Phys. Fluids 27, 104,103 (2015)
Mittal, R., Simmons, S.P., Najjar, F.: Numerical study of pulsatile flow in a constricted channel. J. Fluid Mech. 485, 337–378 (2003)
Griffith, M.D., Leweke, T., Thompson, M.C., Hourigan, K.: Effect of small asymmetries on axisymmetric stenotic flows. J. Fluid. Mech. 721, R1 (2013)
Castro, W., Squire, W.: The effect of polymer additives on transition in pipe flows. Appl. Sci. Res. 18, 81–96 (1967)
Chung, J.S., Graebel,W.P.: Laser anemometer measurements of turbulence in non-newtonian pipe flows. Phys. Fluids 15, 546–554 (1972)
Pereira, A.S., Pinho, F.T.: Turbulent pipe flow characteristics of low molecular weight polymer solutions. J. Non-Newtonian Fluid Mech. 55, 321–344 (1994)
Pinho, F.T., Whitelaw, J.H.: Flow of non-newtonian fluids in a pipe. J. Non-Newtonian Fluid Mech. 34, 129–144 (1990)
Vaseleski, R.C., Metzner, A.B.: Drag reduction in the turbulent flow of fiber suspensions. AIChE J. 20, 301–306 (1974)
Draad, A.A., Kuiken, G.D.C., Nieuwstadt, F.T.M.: Laminar-turbulent transition in pipe flow for newtonian and non-newtonian fluids. J. Fluid. Mech. 377, 267–312 (1998)
Virk, P.S., Wagger, D.L.: Aspects of mechanisms in type b drag reduction. In: Structure of Turbulence and Drag Reduction, pp. 201–213 (1990)
Esmael, A., Nouar, C., Lefevre, A., Kabouya, N.: Transitional flow of a non-newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shear-thinning behavior. Phys. Fluids. 22(10), 101,701 (2010)
Güzel, B, Burghelea, T, Frigaard, I A, Martinez, D M: Observation of laminar–turbulent transition of a yield stress fluid in Hagen-Poiseuille flow. J. Fluid. Mech. 627, 97–128 (2009)
Walker, A M, Johnston, C R, Rival, D E: On the characterization of a non-newtonian blood analog and its response to pulsatile flow downstream of a simplified stenosis. Ann. Biomed. Eng. 42(1), 97–109 (2014)
Biswas, D, Casey, D M, Crowder, D C, Steinman, D A, Yun, Y H, Loth, F: Characterization of transition to turbulence for blood in a straight pipe under steady flow conditions. J. Biomech. Eng. 138(7), 071,001 (2016)
Nouar, C, Bottaro, A, Brancher, J P: Delaying transition to turbulence in channel flow: Revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177–194 (2007)
Roland, N, Plaut, E, Nouar, C: Petrov–Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: First theoretical evidences for a delayed transition. Comput. Fluids 39(9), 1733–1743 (2010)
Rudman, M, Blackburn, H M: Direct numerical simulation of turbulent non-newtonian flow using a spectral element method. Appl. Math. Model 30(11), 1229–1248 (2006)
Rudman, M, Blackburn, H M, Graham, L J W, Pullum, L: Turbulent pipe flow of shear-thinning fluids. J. Non-Newtonian Fluid Mech. 118(1), 33–48 (2004)
Chikkadi, V, Sameen, A, Govindarajan, R: Preventing transition to turbulence: a viscosity stratification does not always help. Phys. Rev. E 95(26), 264,504 (2005)
Singh, J, Rudman, M, Blackburn, H, Chryss, A, Pullum, L, Graham, L: The importance of rheology characterization in predicting turbulent pipe flow of generalized newtonian fluids. J. Non-Newtonian Fluid Mech. 232, 11–21 (2016)
Wall, D P, Wilson, S K: The linear stability of channel flow of fluid with temperature-dependent viscosity. J. Fluid Mech. 323, 107–132 (1996)
Gijsen, F J H, van de Vosse, F N, Janssen, J D: The influence of the non-newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. J. Biomech. 32(6), 601–608 (1999)
Lee, S W, Steinman, D A: On the relative importance of rheology for image-based CFD models of the carotid bifurcation. J. Biomech. Eng. 129(2), 273–278 (2007)
Han, S I, Marseille, O, Gehlen, C, Blümich, B: Rheology of blood by NMR. J. Magn. Reson. 152(1), 87–94 (2001)
Khan, M O, Chnafa, C, Gallo, D, Molinari, F, Morbiducci, U, Steinman, D A, Valen-Sendstad, K: On the quantification and visualization of transient periodic instabilities in pulsatile flows. J. Biomech. 52, 179–182 (2017)
Mortensen, M, Valen-Sendstad, K: Oasis: A high-level/high-performance open source Navier-stokes solver. Comput. Phys. Commun. 188, 177–188 (2015)
Khan, M, Valen-Sendstad, K, Steinman, D: Narrowing the expertise gap for predicting intracranial aneurysm hemodynamics: Impact of solver numerics versus mesh and time-step resolution. Am. J. Neuroradiol. 36(7), 1310–1316 (2015)
Khan, M, Steinman, D, Valen-Sendstad, K: Non-newtonian versus numerical rheology: practical impact of shear-thinning on the prediction of stable and unstable flows in intracranial aneurysms. Int. J. Numer. Methods Biomed. Eng. 33(7) (2016)
Khan, M O, Valen-Sendstad, K, Steinman, D A: Cerebral aneurysm blood flow simulations: There’s solver settings and then there’s solver settings. J. Biomech. 61, 280 (2017)
Berg, P, Roloff, C, Beuing, O, Voss, S, Sugiyama, S I, Aristokleous, N, Anayiotos, A S, Ashton, N, Revell, A, Bressloff, N W, et al: The computational fluid dynamics rupture challenge 2013—Phase II: Variability of hemodynamic simulations in two intracranial aneurysms. J. Biomech. Eng. 137(12), 121,008 (2015)
Simo, J C, Armero, F: Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-stokes and euler equations. Comput. Methods Appl. Mech. Eng. 111(1), 111–154 (1994)
Sirovich, L: Turbulence and the dynamics of coherent structures part i-iii. Quart. Appl. Math. 45(3), 561–571 (1987)
Nobach, H, Tropea, C, Cordier, L, Bonnet, J P, Delville, J, Lewalle, J, Farge, M, Schneider, K, Adrian, R: Review of some fundamentals of data processing. In: Tropea, C, Yarin, A, Foss, J F (eds.) Springer handbook of experimental fluid mechanics, pp 1337–1398. Springer (2007)
Karniadakis, G, Sherwin, S: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press (2013)
Loken, C, Gruner, D, Groer, L, Peltier, R, Bunn, N, Craig, M, Henriques, T, Dempsey, J, Yu, C H, Chen, J, Dursi, L J, Chong, J, Northrup, S, Pinto, J, Knecht, N, von Zon, R: Scinet: Lessons learned from building a power-efficient top-20 system and data centre. J. Phys. Conf. Series 256(1), 012,026 (2010)
Molla, M M, Paul, M C: Les of non-newtonian physiological blood flow in a model of arterial stenosis. Med. Eng. Phys. 34(8), 1079–1087 (2012)
Grinberg, L, Yakhot, A, Karniadakis, G E: Analyzing transient turbulence in a stenosed carotid artery by proper orthogonal decomposition. Ann. Biomed. Eng. 37 (11), 2200–2217 (2009)
Klebanoff, P S, Tidstrom, K, Sargent, L: The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12(1), 1–34 (1962)
Zhou, J., Adrian, R.J., Balachandar, S., Kendall, T.: Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999)
Antiga, L., Steinman, D.A.: Rethinking turbulence in blood. Biorheology 46 (2), 77–81 (2009)
Acknowledgements
The authors thank the reviewers for their helpful comments and suggestions, especially regarding the re-interpretation of our findings in light of Reynolds number definitions. Computations were performed on the GPC supercomputer at the SciNet [59] HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. The study was supported by The Research Council of Norway through a Centres of Excellence grant to the Center for Biomedical Computing at Simula Research Laboratory, project number 179578 and as part of the Centre for Cardiological Innovation, also funded by The Research Council of Norway. MOK acknowledges funding support from Barbara and Frank Millgan Fellowship, Weber and Mariano Scholarship, Mary Gertrude I’ Anson Scholarship, Frank Howard Fellowships, and Simula Research Laboratory Fellowship. DAS acknowledges grant (G-13-0001830) and salary (MC7455) support from Heart & Stroke Foundation.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The authors declare that they have no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
(MP4 14.7 MB)
(MP4 14.7 MB)
Rights and permissions
About this article
Cite this article
Khan, M.O., Valen-Sendstad, K. & Steinman, D.A. Direct Numerical Simulation of Laminar-Turbulent Transition in a Non-Axisymmetric Stenosis Model for Newtonian vs. Shear-Thinning Non-Newtonian Rheologies. Flow Turbulence Combust 102, 43–72 (2019). https://doi.org/10.1007/s10494-018-9905-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-018-9905-7