# zbMATH — the first resource for mathematics

The critical point of the transition to turbulence in pipe flow. (English) Zbl 1419.76270
Summary: In pipes, turbulence sets in despite the linear stability of the laminar Hagen-Poiseuille flow. The Reynolds number ($$Re$$) for which turbulence first appears in a given experiment – the ‘natural transition point’ – depends on imperfections of the set-up, or, more precisely, on the magnitude of finite amplitude perturbations. At onset, turbulence typically only occupies a certain fraction of the flow, and this fraction equally is found to differ from experiment to experiment. Despite these findings, Reynolds proposed that after sufficiently long times, flows may settle to steady conditions: below a critical velocity, flows should (regardless of initial conditions) always return to laminar, while above this velocity, eddying motion should persist. As will be shown, even in pipes several thousand diameters long, the spatio-temporal intermittent flow patterns observed at the end of the pipe strongly depend on the initial conditions, and there is no indication that different flow patterns would eventually settle to a (statistical) steady state. Exploiting the fact that turbulent puffs do not age (i.e. they are memoryless), we continuously recreate the puff sequence exiting the pipe at the pipe entrance, and in doing so introduce periodic boundary conditions for the puff pattern. This procedure allows us to study the evolution of the flow patterns for arbitrary long times, and we find that after times in excess of $$10^7$$ advective time units, indeed a statistical steady state is reached. Although the resulting flows remain spatio-temporally intermittent, puff splitting and decay rates eventually reach a balance, so that the turbulent fraction fluctuates around a well-defined level which only depends on $$Re$$. In accordance with Reynolds’ proposition, we find that at lower $$Re$$ (here 2020), flows eventually always resume to laminar, while for higher $$Re\, (\geqslant 2060$$), turbulence persists. The critical point for pipe flow hence falls in the interval of $$2020<Re<2060$$, which is in very good agreement with the recently proposed value of $$Re_c=2040$$. The latter estimate was based on single-puff statistics and entirely neglected puff interactions. Unlike in typical contact processes where such interactions strongly affect the percolation threshold, in pipe flow, the critical point is only marginally influenced. Interactions, on the other hand, are responsible for the approach to the statistical steady state. As shown, they strongly affect the resulting flow patterns, where they cause ‘puff clustering’, and these regions of large puff densities are observed to travel across the puff pattern in a wave-like fashion.

##### MSC:
 76F06 Transition to turbulence 76F10 Shear flows and turbulence
##### Keywords:
turbulent flows; turbulent transition
Full Text:
##### References:
 [1] Avila, K.; Moxey, D.; De Lozar, A.; Avila, M.; Barkley, D.; Hof, B., The onset of turbulence in pipe flow, Science, 333, 6039, 192-196, (2011) · Zbl 1411.76035 [2] Avila, M.; Willis, A. P.; Hof, B., On the transient nature of localized pipe flow turbulence, J. Fluid Mech., 646, 127-136, (2010) · Zbl 1189.76262 [3] Barkley, D., Simplifying the complexity of pipe flow, Phys. Rev. E, 84, 1, (2011) [4] Barkley, D.2016Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech.803, P1. · Zbl 1454.76047 [5] Barkley, D.; Song, B.; Mukund, V.; Lemoult, G.; Avila, M.; Hof, B., The rise of fully turbulent flow, Nature, 526, 7574, 550-553, (2015) [6] Binnie, A. M., A double-refraction method of detecting turbulence in liquids, Proc. Phys. Soc., 57, 5, 390-402, (1945) [7] Binnie, A. M.; Fowler, J. S., A study by a double-refraction method of the development of turbulence in a long circular tube, Proc. R. Soc. Lond. A, 192, 32-44, (1947) [8] Bottin, S.; Chaté, H., Statistical analysis of the transition to turbulence in plane Couette flow, Eur. Phys. J. B, 6, 1, 143-155, (1998) [9] Bottin, S.; Daviaud, F.; Manneville, P.; Dauchot, O., Discontinuous transition to spatiotemporal intermittency in plane Couette flow, Europhys. Lett., 43, 2, 171, (1998) [10] Brosa, U., Turbulence without strange attractor, J. Stat. Phys., 55, 5-6, 1303-1312, (1989) [11] Chaté, H.; Manneville, P., Spatio-temporal intermittency in coupled map lattices, Phys. D, 32, 3, 409-422, (1988) · Zbl 0656.76055 [12] Faisst, H.; Eckhardt, B., Sensitive dependence on initial conditions in transition to turbulence in pipe flow, J. Fluid Mech., 504, 343-352, (2004) · Zbl 1116.76362 [13] Grassberger, P., On phase transitions in Schlögl’s second model, Z. für Phys. B, 47, 4, 365-374, (1982) [14] Hinrichsen, H., Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys., 49, 7, 815-958, (2000) [15] Hof, B.; De Lozar, A.; Avila, M.; Tu, X.; Schneider, T. M., Eliminating turbulence in spatially intermittent flows, Science, 327, 5972, 1491-1494, (2010) [16] Hof, B.; De Lozar, A.; Kuik, D. J.; Westerweel, J., Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow, Phys. Rev. Lett., 101, 21, (2008) [17] Hof, B.; Westerweel, J.; Schneider, T. M.; Eckhardt, B., Finite lifetime of turbulence in shear flows, Nature, 443, 7107, 59-62, (2006) [18] Janssen, H.-K., On the nonequilibrium phase transition in reaction – diffusion systems with an absorbing stationary state, Z. für Phys. B, 42, 2, 151-154, (1981) [19] Kaneko, K., Spatiotemporal intermittency in coupled map lattices, Prog. Theor. Phys., 74, 5, 1033-1044, (1985) · Zbl 0979.37505 [20] Kuik, D. J.; Poelma, C.; Westerweel, J., Quantitative measurement of the lifetime of localized turbulence in pipe flow, J. Fluid Mech., 645, 529, (2010) · Zbl 1189.76031 [21] Lemoult, G.; Shi, L.; Avila, K.; Jalikop, S. V.; Avila, M.; Hof, B., Directed percolation phase transition to sustained turbulence in Couette flow, Nat. Phys., 12, 3, 254-258, (2016) [22] Lindgren, E. R., Some aspects of the change between laminar and turbulent flow of liquids in cylindrical tubes, Ark. Fys., 7, 23, 293-308, (1953) [23] De Lozar, A.; Hof, B., An experimental study of the decay of turbulent puffs in pipe flow, Phil. Trans. R. Soc. Lond. A, 367, 1888, 589-599, (2009) · Zbl 1221.76009 [24] Manneville, P., On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular, Eur. J. Mech. (B/Fluids), 49, 345-362, (2015) · Zbl 1408.76274 [25] Meseguer, A.; Trefethen, L. N., Linearized pipe flow to Reynolds number 107, J. Comput. Phys., 186, 1, 178-197, (2003) · Zbl 1047.76565 [26] Moxey, D.; Barkley, D., Distinct large-scale turbulent – laminar states in transitional pipe flow, Proc. Natl Acad. Sci. USA, 107, 18, 8091-8096, (2010) [27] Pavelyev, A. A.; Reshmin, A. I.; Teplovodskii, S. Kh.; Fedoseev, S. G., On the lower critical Reynolds number for flow in a circular pipe, Fluid Dyn., 38, 4, 545-551, (2003) [28] Pavelyev, A. A.; Reshmin, A. I.; Trifonov, V. V., Effect of the pattern of initial perturbations on the steady pipe flow regime, Fluid Dyn., 41, 6, 916-922, (2006) [29] Peixinho, J.; Mullin, T., Decay of turbulence in pipe flow, Phys. Rev. Lett., 96, 9, (2006) · Zbl 1114.76304 [30] Pomeau, Y., Front motion, metastability and subcritical bifurcations in hydrodynamics, Phys. D, 23, 1, 3-11, (1986) [31] Reynolds, O., An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Phil. Trans. R. Soc. Lond. A, 174, 935-982, (1884) · JFM 16.0845.02 [32] Reynolds, O., On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. R. Soc. Lond. A, 186, 123-164, (1895) · JFM 26.0872.02 [33] Von Rotta, J., Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr, Arch. Appl. Mech., 24, 4, 258-281, (1956) [34] Samanta, D.; De Lozar, A.; Hof, B., Experimental investigation of laminar turbulent intermittency in pipe flow, J. Fluid Mech., 681, 193-204, (2011) · Zbl 1241.76047 [35] Sano, M.; Tamai, K., A universal transition to turbulence in channel flow, Nat. Phys., 12, 3, 249-253, (2016) [36] Schiller, L., Experimentelle Untersuchungen zum Turbulenzproblem, Z. Angew. Math. Mech.-J. Appl. Math. Mech., 1, 6, 436-444, (1921) [37] Sibulkin, M., Transition from turbulent to laminar pipe flow, Phys. Fluids, 5, 3, 280-284, (1962) [38] Stern, E., Beitrag zur Untersuchung der Intermittenz einer Rohrströmung, Acta Mech., 10, 1-2, 67-84, (1970) [39] Willis, A. P.; Kerswell, R. R., Critical behavior in the relaminarization of localized turbulence in pipe flow, Phys. Rev. Lett., 98, 1, (2007) [40] Wygnanski, I. J.; Champagne, F. H., On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug, J. Fluid Mech., 59, 2, 281-335, (1973) [41] Xiong, X.; Tao, J.; Chen, S.; Brandt, L., Turbulent bands in plane-Poiseuille flow at moderate Reynolds numbers, Phys. Fluids, 27, 4, (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.