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A frequency selection criterion in spatially developing flows. (English) Zbl 0716.76041
Summary: The possible existence of global modes or self-excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency $$\omega_ 0(X)$$ with distance X. The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an 0(1) growth rate. (2) If the singularity X, closest to the real X-axis of the complex function $$\omega_ 0(X)$$ is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency $$\omega_ 0(X_ s)$$. Thus, it will be temporally growing only if Im $$\omega$$ $${}_ 0(X_ s)$$ is positive.

##### MSC:
 7.6e+31 Nonlinear effects in hydrodynamic stability 7.6e+06 Parallel shear flows in hydrodynamic stability
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##### References:
 [1] Chomaz, Wave selection mechanisms in open flows, Bull. Amer. Phys. Soc. 31 pp 1696– (1987) [2] Chomaz, Proceedings of the Symposium on Turbulent Shear Flows pp 3.2-1– (1987) [3] Chomaz, Bifurcation to local and global modes in spatially-developing flows, Phys. Rev. Lett. 60 pp 25– (1988) · doi:10.1103/PhysRevLett.60.25 [4] Monkewitz, Preferred modes in jets and global instabilities, Bull. Amer. Phys. Soc. 33 pp 2273– (1988) [5] Chomaz, NATO ASI Series B: Physics 237, in: Proceedings of the Conference on New Trends in Nonlinear Dynamical and Pattern-forming Phenomena (1990) [6] Koch, Local instability characteristics and frequency determination of self-excited wake flows, J. Sound Vibration 99 pp 53– (1985) · doi:10.1016/0022-460X(85)90445-6 [7] Pierrehumbert, Local and global baroclinic instability of zonally varying flow, J. Atmospheric Sci. 41 pp 2141– (1984) · doi:10.1175/1520-0469(1984)041<2141:LAGBIO>2.0.CO;2 [8] Bar-Sever, Local instabilities of weakly non-parallel large scale flows: WKB analysis, Geophys. Astrophys. Fluid Dynamics 41 pp 233– (1988) · Zbl 0708.76063 · doi:10.1080/03091928808208852 [9] Plasma Physics pp 124– (1961) [10] Briggs, Electron-Stream Interaction with Plasmas (1964) [11] Lifshitz, Physical Kinetics (1981) [12] Bers, Handbook of Plasma Physics pp 1:451– (1983) [14] Rockwell, Self-sustained oscillations of impinging free-shear layers, Annual Rev. Fluid Mech. 11 pp 67– (1979) · doi:10.1146/annurev.fl.11.010179.000435 [15] Huerre, Absolute and convective instabilities in free shear layers, J. Fluid Mech. 159 pp 151– (1985) · Zbl 0588.76067 · doi:10.1017/S0022112085003147 [16] Balsa, Three-dimensional wavepackets and instability waves in free shear layers, J. Fluid Mech. 201 pp 77– (1989) · Zbl 0667.76061 · doi:10.1017/S0022112089000844 [17] Gaster, A theoretical model of a wavepacket in the boundary layer on a flat plate, Proc. Roy. Soc. London Ser. A 347 pp 271– (1975) · doi:10.1098/rspa.1975.0209 [18] Ho, Perturbed free shear layers, Annual Rev. Fluid Mech. 16 pp 365– (1984) · doi:10.1146/annurev.fl.16.010184.002053 [19] Monkewitz, The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers, Phys. Fluids 31 pp 999– (1988) · doi:10.1063/1.866720 [20] Triantafyllou, On the formation of vortex street behind stationary cylinders, J. Fluid Mech. 170 pp 461– (1986) · doi:10.1017/S0022112086000976 [21] Hannemann, Numerical simulation of the absolutely and convectively unstable wake, J. Fluid Mech. 199 pp 55– (1989) · Zbl 0659.76052 · doi:10.1017/S0022112089000297 [22] Yang, Absolute and convective instability of a cylinder wake, Phys. Fluids A 1 pp 689– (1989) · doi:10.1063/1.857362 [23] Provansal, Bénard-von Kármán instability: Transient and forced regimes, J. Fluid Mech. 182 pp 1– (1987) · Zbl 0641.76046 · doi:10.1017/S0022112087002222 [24] Sreenivasan, Proceedings of the Forum on Unsteady Flow Separation 52 (1987) [25] Monkewitz, Absolute instability in hot jets, AIAA J. 26 pp 911– (1988) · doi:10.2514/3.9990 [26] Monkewitz, Self-excited oscillations and mixing in a heated round jet, J. Fluid Mech. 213 pp 611– (1990) · doi:10.1017/S0022112090002476 [27] Sreenivasan, Absolute instability in variable density round jets, Exp. in Fluids 7 pp 309– (1989) · doi:10.1007/BF00198449 [29] Huerre, Instabilities and Nonequilibrium Structures pp 141– (1987) · doi:10.1007/978-94-009-3783-3_7 [30] Monkewitz, The role of absolute and convective instability in predicting the behavior of fluid systems, European J. Mech. B / Fluids 9 pp 395– (1990) · Zbl 0705.76038 [31] Huerre, Local and global instabilities in spatially-developing flows, Annual Rev. Fluid Mech. 22 pp 473– (1990) · doi:10.1146/annurev.fl.22.010190.002353 [32] E. C. Titshmarsh Eigenfunction Expansions Associated with Second Order Differential Equations [33] Pokrovskii, On the problem of above-barrier reflection of high energy particles, Soviet Phys. JETP 13 pp 1207– (1961) [34] Wasow, Asymptotic Expansions for Ordinary Differential Equations (1965) · Zbl 0133.35301 [35] Hille, Lectures on Ordinary Differential Equations (1969) [36] Hille, Ordinary Differential Equations in the Complex Domain (1976) [37] Bender, Advanced Mathematical Methods for Scientists and Engineers (1978) [38] Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 [39] Soward, The linear stability of the flow in the narrow gap between two concentric rotating spheres, Quart. J. Mech. Appl. Math. 36 pp 19– (1983) · Zbl 0544.76039 · doi:10.1093/qjmam/36.1.19 [40] Papageorghiou, Stability of the unsteady viscous flow in a curved pipe, J. Fluid Mech. 182 pp 209– (1987) · Zbl 0651.76016 · doi:10.1017/S0022112087002313 [41] Bensimon, Dynamics of curved fronts and pattern reflection, J. Physique 48 pp 2081– (1987) · Zbl 0634.46004 · doi:10.1051/jphys:0198700480120208100 [42] Fuchs, Theory of mode conversion in weakly inhomogeneous plasma, Phys. Fluids 24 pp 1251– (1981) · Zbl 0459.76096 · doi:10.1063/1.863528
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