Direct Numerical Simulation of Laminar-Turbulent Transition in a Non-Axisymmetric Stenosis Model for Newtonian vs. Shear-Thinning Non-Newtonian Rheologies


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. 1.

    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)

    Article  Google Scholar 

  2. 2.

    Wootton, D.M., Ku, D.N.: Fluid mechanics of vascular systems, diseases, and thrombosis. Annu. Rev. Biomed. Eng. 1, 299–329 (1999)

    Article  Google Scholar 

  3. 3.

    Berger, S.A., Jou, L.D.: Flows in stenotic vessels. Annu. Rev. Fluid Mech. 32, 347–382 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Ku, D.N.: Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434 (2000)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Young, D.F.: Fluid mechanics of arterial stenoses. J. Biomech. Eng. 101, 157–175 (1979)

    Article  Google Scholar 

  6. 6.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    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)

  8. 8.

    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)

    Article  Google Scholar 

  9. 9.

    Young, D.F., Tsai, F.Y.: Flow characteristics in models of arterial stenoses - I. Steady flow. J. Biomech. 6, 395–410 (1973)

    Article  Google Scholar 

  10. 10.

    Kim, B.M., Corcoran, W.H.: Experimental measurements of turbulence spectra distal to stenoses. J. Biomech. 7, 335–342 (1974)

    Article  Google Scholar 

  11. 11.

    Cassanova, R.A., Giddens, D.P.: Disorder distal to modeled stenoses in steady and pulsatile flow. J. Biomech. 11, 441–453 (1978)

    Article  Google Scholar 

  12. 12.

    Ahmed, S.A., Giddens, D.P.: Velocity measurements in steady flow through axisymmetric stenoses at moderate reynolds numbers. J. Biomech. 16, 505–516 (1983)

    Article  Google Scholar 

  13. 13.

    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)

    MATH  Article  Google Scholar 

  14. 14.

    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)

    MATH  Article  Google Scholar 

  15. 15.

    Clark, C: The fluid mechanics of aortic stenosis - I. theory and steady flow experiments. J. Biomech. 9, 521–528 (1976)

    Article  Google Scholar 

  16. 16.

    Ahmed, S.A., Giddens, D.P.: Flow disturbance measurements through a constricted tube at moderate reynolds numbers. J. Biomech. 16, 955–963 (1983)

    Article  Google Scholar 

  17. 17.

    Deshpande, M.D., Giddens, D.P.: Turbulence measurements in a constricted tube. J. Fluid. Mech. 97, 65–89 (1980)

    Article  Google Scholar 

  18. 18.

    Khalifa, A.M.A., Giddens, D.P.: Characterization and evolution of poststenotic flow disturbances. J. Biomech. 14, 279–296 (1981)

    Article  Google Scholar 

  19. 19.

    Ahmed, S.A., Giddens, D.P.: Pulsatile poststenotic flow studies with laser doppler anemometry. J. Biomech. 17, 695–705 (1984)

    Article  Google Scholar 

  20. 20.

    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)

    Article  Google Scholar 

  21. 21.

    Deshpande, M.D., Giddens, D.P., Mabon, R.F.: Steady laminar flow through modelled vascular stenoses. J. Biomech. 9, 165–174 (1976)

    Article  Google Scholar 

  22. 22.

    Lee, J.S., Fung, Y.C.: Flow in locally constricted tubes at low Reynolds numbers. J. App. Mech. 37, 9–16 (1970)

    MATH  Article  Google Scholar 

  23. 23.

    Varghese, S.S., Frankel, S.H., Fischer, P.F.: Modeling transition to turbulence in eccentric stenotic flows. J. Biomech. Eng. 130, 014,503 (2008)

    Article  Google Scholar 

  24. 24.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    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)

    Article  Google Scholar 

  26. 26.

    Mittal, R., Simmons, S.P., Najjar, F.: Numerical study of pulsatile flow in a constricted channel. J. Fluid Mech. 485, 337–378 (2003)

    MATH  Article  Google Scholar 

  27. 27.

    Griffith, M.D., Leweke, T., Thompson, M.C., Hourigan, K.: Effect of small asymmetries on axisymmetric stenotic flows. J. Fluid. Mech. 721, R1 (2013)

    MATH  Article  Google Scholar 

  28. 28.

    Castro, W., Squire, W.: The effect of polymer additives on transition in pipe flows. Appl. Sci. Res. 18, 81–96 (1967)

    Article  Google Scholar 

  29. 29.

    Chung, J.S., Graebel,W.P.: Laser anemometer measurements of turbulence in non-newtonian pipe flows. Phys. Fluids 15, 546–554 (1972)

    Article  Google Scholar 

  30. 30.

    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)

    Article  Google Scholar 

  31. 31.

    Pinho, F.T., Whitelaw, J.H.: Flow of non-newtonian fluids in a pipe. J. Non-Newtonian Fluid Mech. 34, 129–144 (1990)

    Article  Google Scholar 

  32. 32.

    Vaseleski, R.C., Metzner, A.B.: Drag reduction in the turbulent flow of fiber suspensions. AIChE J. 20, 301–306 (1974)

    Article  Google Scholar 

  33. 33.

    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)

    MATH  Article  Google Scholar 

  34. 34.

    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)

  35. 35.

    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)

    Article  Google Scholar 

  36. 36.

    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)

    MATH  Article  Google Scholar 

  37. 37.

    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)

    Article  Google Scholar 

  38. 38.

    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)

    Article  Google Scholar 

  39. 39.

    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)

    MATH  Article  Google Scholar 

  40. 40.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    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)

    MATH  Article  Google Scholar 

  42. 42.

    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)

    MATH  Article  Google Scholar 

  43. 43.

    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)

    Google Scholar 

  44. 44.

    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)

    MathSciNet  Article  Google Scholar 

  45. 45.

    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)

    MATH  Article  Google Scholar 

  46. 46.

    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)

    Article  Google Scholar 

  47. 47.

    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)

    Article  Google Scholar 

  48. 48.

    Han, S I, Marseille, O, Gehlen, C, Blümich, B: Rheology of blood by NMR. J. Magn. Reson. 152(1), 87–94 (2001)

    Article  Google Scholar 

  49. 49.

    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)

    Article  Google Scholar 

  50. 50.

    Mortensen, M, Valen-Sendstad, K: Oasis: A high-level/high-performance open source Navier-stokes solver. Comput. Phys. Commun. 188, 177–188 (2015)

    MATH  Article  Google Scholar 

  51. 51.

    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)

    Article  Google Scholar 

  52. 52.

    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)

  53. 53.

    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)

    Article  Google Scholar 

  54. 54.

    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)

    Article  Google Scholar 

  55. 55.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Sirovich, L: Turbulence and the dynamics of coherent structures part i-iii. Quart. Appl. Math. 45(3), 561–571 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    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)

  58. 58.

    Karniadakis, G, Sherwin, S: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press (2013)

  59. 59.

    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)

    Article  Google Scholar 

  60. 60.

    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)

    Article  Google Scholar 

  61. 61.

    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)

    Article  Google Scholar 

  62. 62.

    Klebanoff, P S, Tidstrom, K, Sargent, L: The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12(1), 1–34 (1962)

    MATH  Article  Google Scholar 

  63. 63.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

    Antiga, L., Steinman, D.A.: Rethinking turbulence in blood. Biorheology 46 (2), 77–81 (2009)

    Google Scholar 

Download references


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



Corresponding author

Correspondence to M. O. Khan.

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)

(DOCX 30.0 MB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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).

Download citation


  • Stenosis
  • Shear-thinning
  • CFD
  • DNS
  • Transition
  • Turbulence
  • Hemodynamics