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Critical threshold in pipe flow transition. (English) Zbl 1221.76096
Summary: This study provides a numerical characterization of the basin of attraction of the laminar Hagen-Poiseuille flow by measuring the minimal amplitude of a perturbation required to trigger transition. For pressure-driven pipe flow, the analysis presented here covers autonomous and impulsive scenarios where either the flow is perturbed with an initial disturbance with a well-defined norm or perturbed by means of local impulsive forcing that mimics injections through the pipe wall. In both the cases, the exploration is carried out for a wide range of Reynolds numbers by means of a computational method that numerically resolves the transitional dynamics. For $$Re ~ O(10^{4})$$, the present work provides critical amplitudes that decay as $$Re^{ - 3/2}$$ and $$Re^{ - 1}$$ for the autonomous and impulsive scenarios, respectively. For $$R$$e=2875, accurate threshold amplitudes are found for constant mass-flux pipe by means of a shooting method that provides critical trajectories that never relaminarize or trigger transition. These transient states are used as initial guesses in a damped Newton-Krylov method formulated to find periodic travelling wave solutions that either travel downstream or exhibit a helicoidal advection.

MSC:
 76F06 Transition to turbulence
Full Text:
References:
 [1] J FLUID MECHANICS 451 pp 35– (2002) [2] Annual Review of Fluid Mechanics 39 pp 447– (2007) · Zbl 1106.76006 [3] Physical Review Letters 91 pp 224 502– (2003) [4] Physical Review Letters 91 pp 244 502– (2003) [5] Hof, Science 305 (5690) pp 1594– (2004) [6] Hof, Nature; Physical Science (London) 443 (7107) pp 59– (2006) [7] 18 pp 17R– (2005) [8] PHYS FLUIDS 18 pp 074 104– (2006) [9] PHYS FLUIDS 19 pp 044 102– (2007) · Zbl 1146.76485 [10] PHYS FLUIDS 18 pp 1203– (2003) [11] APPL NUMER MATH 57 pp 920– (2007) [12] J COMPUT PHYS 186 pp 178– (2003) [13] EUR PHYS J SPEC TOP 146 pp 249– (2007) [14] J FLUID MECHANICS 217 pp 519– (1990) [15] Physica. D 36 pp 137– (1989) [16] Physical Review Letters 96 pp 094 501– (2006) [17] J FLUID MECHANICS 582 pp 169– (2007) [18] Physical Review Letters 99 pp 074 502– (2007) [19] J COMPUT PHYS 142 pp 370– (1998) [20] PHIL TRANS R SOC A 174 pp 935– (1883) [21] PHYS FLUIDS 6 pp 1396– (1994) [22] J FLUID MECHANICS 277 pp 197– (1994) [23] Chaos (Woodbury, N.Y.) 16 pp 041 103– (2006) · Zbl 05359818 [24] Physical Review Letters 96 pp 174 101– (2006) [25] PHYS FLUIDS 9 pp 883– (1997) [26] PHYS FLUIDS 15 pp 1517– (2003) [27] J FLUID MECHANICS 508 pp 333– (2004) [28] PHYS FLUIDS 8 pp 2923– (1996)
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