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Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. (English) Zbl 1191.76053
Summary: The stability behaviour of a low-Reynolds-number recirculation flow developing in a curved channel is investigated using a global formulation of hydrodynamic stability theory. Both the resonator and amplifier dynamics are investigated. The resonator dynamics, which results from the ability of the flow to self-sustain perturbations, is studied through a modal stability analysis. In agreement with the literature, the flow becomes globally unstable via a three-dimensional stationary mode. The amplifier dynamics, which is characterized by the ability of the flow to exhibit large transient amplifications of initial perturbations, is studied by looking for the two- and three-dimensional initial perturbations that maximize the energy gain over a given time horizon. The optimal initial two-dimensional perturbations have the form of wave packets localized in the upstream part of the recirculation bubble. It is shown that they are first amplified while travelling downstream along the shear layer of the recirculation bubble and then decay when leaving the recirculation bubble. Maximal energy gain is thus achieved for a time horizon approximately corresponding to the propagation of the wave packet along the whole recirculation bubble. The resonator and amplifier dynamics are associated with different types of structures in the flow: three-dimensional steady structures for the resonator dynamics and nearly two-dimensional unsteady structures for the amplifier dynamics. A comparison of the strength of the two dynamics is proposed. The transient energetic growth of the two-dimensional unsteady perturbations is large at moderate time, compared to the very weak exponential growth of the three-dimensional stationary mode. This suggests that, as soon as there is noise in the system, the amplifier dynamics dominates the resonator dynamics, thus explaining the appearance of unsteadiness rather than the emergence of stationary structures in similar experimental flows.
Reviewer: Reviewer (Berlin)

76E09 Stability and instability of nonparallel flows in hydrodynamic stability
Full Text: DOI
[1] DOI: 10.1007/BF00127673 · doi:10.1007/BF00127673
[2] DOI: 10.1017/S0022112096007689 · Zbl 0875.76111 · doi:10.1017/S0022112096007689
[3] DOI: 10.1017/S0022112091003488 · Zbl 0728.76057 · doi:10.1017/S0022112091003488
[4] DOI: 10.1017/S0022112087002234 · Zbl 0639.76041 · doi:10.1017/S0022112087002234
[5] DOI: 10.1098/rsta.2000.0704 · Zbl 0981.76506 · doi:10.1098/rsta.2000.0704
[6] DOI: 10.1002/fld.1650090809 · Zbl 0683.76027 · doi:10.1002/fld.1650090809
[7] DOI: 10.1017/S0022112001003627 · Zbl 0987.76021 · doi:10.1017/S0022112001003627
[8] DOI: 10.1017/S0022112006002898 · Zbl 1105.76028 · doi:10.1017/S0022112006002898
[9] DOI: 10.1146/annurev.fluid.37.061903.175810 · Zbl 1117.76027 · doi:10.1146/annurev.fluid.37.061903.175810
[10] DOI: 10.1017/S0022112005005112 · Zbl 1073.76027 · doi:10.1017/S0022112005005112
[11] DOI: 10.1063/1.858386 · doi:10.1063/1.858386
[12] DOI: 10.1023/B:JAMT.0000030328.95959.1b · doi:10.1023/B:JAMT.0000030328.95959.1b
[13] DOI: 10.1002/(SICI)1097-0363(19971230)25:123.0.CO;2-A · doi:10.1002/(SICI)1097-0363(19971230)25:123.0.CO;2-A
[14] Denham, Trans. Inst. Chem. Engrs 52 pp 361– (1974)
[15] DOI: 10.1016/j.euromechflu.2003.09.010 · Zbl 1045.76501 · doi:10.1016/j.euromechflu.2003.09.010
[16] DOI: 10.1103/PhysRevLett.78.4387 · doi:10.1103/PhysRevLett.78.4387
[17] DOI: 10.1017/S002211200200232X · Zbl 1026.76019 · doi:10.1017/S002211200200232X
[18] DOI: 10.1017/S0022112083002839 · doi:10.1017/S0022112083002839
[19] DOI: 10.1017/S0022112096002807 · doi:10.1017/S0022112096002807
[20] DOI: 10.1017/S0022112007005496 · Zbl 1175.76049 · doi:10.1017/S0022112007005496
[21] DOI: 10.1126/science.261.5121.578 · Zbl 1226.76013 · doi:10.1126/science.261.5121.578
[22] DOI: 10.1098/rsta.2000.0706 · Zbl 1106.76363 · doi:10.1098/rsta.2000.0706
[23] DOI: 10.1017/S002211200200873X · Zbl 1015.76027 · doi:10.1017/S002211200200873X
[24] Schmid, Stability and Transition in Shear Flows. (2001) · Zbl 0966.76003 · doi:10.1007/978-1-4613-0185-1
[25] DOI: 10.1146/annurev.fluid.38.050304.092139 · doi:10.1146/annurev.fluid.38.050304.092139
[26] Pauley, J. Fluid Mech. 251 pp 1– (1990)
[27] Orr, Proc. R. Irish Acad. 27 pp 9– (1907)
[28] DOI: 10.1017/S0022112003005287 · Zbl 1063.76537 · doi:10.1017/S0022112003005287
[29] DOI: 10.1002/(SICI)1097-0363(19970615)24:113.0.CO;2-R · doi:10.1002/(SICI)1097-0363(19970615)24:113.0.CO;2-R
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