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Transition to turbulence in pulsating pipe flow

  • Duo Xu (a1) (a2) (a3), Sascha Warnecke (a1) (a2), Baofang Song (a1), Xingyu Ma (a1) and Björn Hof (a1)...


Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $\unicode[STIX]{x1D6FC}$ ), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $\unicode[STIX]{x1D6FC}$ . In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence (‘puffs’) analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero $\unicode[STIX]{x1D6FC}$ limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing $\unicode[STIX]{x1D6FC}$ ) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.


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Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192196.
Avila, M., Willis, A. P. & Hof, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127136.
Barkley, D., Song, B. F., Mukund, V., Lemoult, G., Avila, M. & Hof, B. 2015 The rise of fully turbulent flow. Nature 526 (7574), 550553.
Bogren, H. G. & Buonocore, M. H. 1994 Blood flow measurements in the aorta and major arteries with MR velocity mapping. J. Magn. Reson. Imag. 4 (2), 119130.
Brown, L. D., Cai, T. T. & Dasgupta, A. 2001 Interval estimation for a binomial proportion. Stat. Sci. 16 (2), 101133.
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
Falsetti, H. L., Carroll, R. J., Swope, R. D. & Chen, C. J. 1983 Turbulent blood flow in the ascending aorta of dogs. Cardiovasc. Res. 17 (7), 427436.
Flaherty, J. T., Pierce, J. E., Ferrans, V., Patel, D. J., Tucker, W. K. & Fry, D. L. 1972 Endothelial nuclear patterns in the canine arterial tree with particular reference to hemodynamic events. Circulat. Res. 30 (1), 2333.
Florio, P. J. & Mueller, W. K. 1968 Development of a periodic flow in a rigid tube. Trans. ASME J. Basic Engng 90 (3), 395399.
Freis, E. D. & Heath, W. C. 1964 Hydrodynamics of aortic blood flow. Circulat. Res. 14 (2), 105116.
Gerrard, J. H. & Hughes, M. D. 1971 The flow due to an oscillating piston in a cylindrical tube: a comparison between experiment and a simple entrance flow theory. J. Fluid Mech. 50 (01), 97106.
Gilbrech, D. A. & Combs, G. D. 1963 Critical Reynolds numbers for incompressible pulsating flows in tubes. Dev. Theor. 1, 292304.
Hershey, D. & Im, C. S. 1968 Critical Reynolds number for sinusoidal flow of water in rigid tubes. AIChE J. 14 (5), 807809.
Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), 244502.
Hof, B., de Lozar, A., Kuik, D. J. & Westerweel, J. 2008 Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101, 214501.
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443, 5962.
von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory stokes layers. J. Fluid Mech. 62, 753773.
Kuik, D. J., Poelma, C. & Westerweel, J. 2010 Quantitative measurement of the lifetime of localized turbulence in pipe flow. J. Fluid Mech. 645, 529539.
de Lozar, A. & Hof, B. 2009 An experimental study of the decay of turbulent puffs in pipe flow. Phil. Trans. R. Soc. Lond. A 367, 589599.
Nerem, R. M. & Seed, W. A. 1972 An in vivo study of aortic flow disturbances. Cardiovasc. Res. 6, 114.
Nerem, R. M., Seed, W. A. & Wood, N. B. 1972 An experimental study of velocity distribution and transition to turbulence in the aorta. J. Fluid Mech. 52, 137160.
Peacock, J., Jones, T., Tock, C. & Lutz, R. 1998 The onset of turbulence in physiological pulsatile flow in a straight tube. Exp. Fluids 24, 19.
Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110 (26), 1055710562.
Samanta, D., de Lozar, A. & Hof, B. 2011 Experimental investigation of laminar turbulent intermittency in pipe flow. J. Fluid Mech. 681, 193204.
Sarpkaya, T. 1966 Experimental determination of the critical Reynolds number for pulsating poiseuille flow. Trans. ASME J. Basic Engng 88, 589598.
Silver, M. D. 1978 Late complications of prosthetic heart valves. Arch. Pathol. Lab. Med. 102, 281284.
Stein, P. D. & Sabbah, H. N. 1976 Turbulent blood flow in the ascending aorta of humans with normal and diseased aortic valves. Circulat. Res. 39, 5865.
Stettler, J. C. & Hussain, A. K. M. F. 1986 On transition of the pulsatile pipe flow. J. Fluid Mech. 170, 169197.
Thomas, C., Bassom, A. P., Blennerhassett, P. J. & Davies, C. 2011 The linear stability of oscillatory poiseuille flow in channels and pipes. Proc. R. Soc. Lond. A 467, 26432662.
Trip, R., Kuik, D. J., Westerweel, J. & Poelma, C. 2012 An experimental study of transitional pulsatile pipe flow. Phys. Fluids 24, 014103.
Willis, A. P. & Kerswell, R. R. 2009 Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised ‘edge’ states. J. Fluid Mech. 619, 213233.
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