## Absolute and convective instabilities in free shear layers.(English)Zbl 0588.76067

The authors introduce the important concept of absolute and convective instability to hydrodynamic problems where up to now the general procedure was to choose between a temporary (real wave number) or spatial (real frequency) theory. According to a criterion developed for plasma instabilities by A. Bers [in: C. DeWitt and J. Peyrand (eds.), Physique des plasmas, Gordon and Breach, New York (1975) on pp. 117-213] and R. J. Briggs [Electron-stream interaction with plasmas, Research Monograph No.29, MIT Press, Cambridge Mass. (1964)] the character of any linear instability can be determined by the singularities of the dispersion relation for complex frequencies and complex wave numbers: if the branch point of the unstable mode in the complex wave number plane corresponds to a temporally damped mode (i.e. lies in the lower half of the complex frequency plane) the flow is convectively unstable, otherwise it is absolutely unstable. The physical importance of this distinction comes from the fact that (within linear theory) the impulse response of an absolutely unstable flow becomes unbounded for large time at all points. (Nonlinear effects naturally limit the response resulting in a nonlinear oscillator). For a convectively unstable flow the impulse response decays to zero for large time at all points of the flow, and if such a flow is forced periodically it acts as amplifier. As a particular application these concepts are applied to a family of free shear layers with varying velocity ratio. It is demonstrated that a shear layer is convectively unstable for coflowing streams but can become absolutely unstable if the two streams flow in apposite directions and the velocity ratio $$\Delta U/2U_{mean}$$ is larger than 1.315. Finally the implications for wakes and jets are discussed briefly.
Reviewer: W.Koch

### MSC:

 76E05 Parallel shear flows in hydrodynamic stability 76F10 Shear flows and turbulence 76E15 Absolute and convective instability and stability in hydrodynamic stability
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### References:

 [1] Merkine, Geophys. Astrophys. Fluid Dyn. 9 pp 129– (1977) [2] DOI: 10.1017/S0022112072001193 · Zbl 0251.76029 [3] DOI: 10.1017/S0022112081001560__S0022112081001560 [4] DOI: 10.1017/S0022112079002056 · Zbl 0396.76078 [5] DOI: 10.1063/1.1706010 [6] DOI: 10.1017/S0022112080001425 · Zbl 0447.76041 [7] DOI: 10.1017/S0022112066001241 [8] DOI: 10.1063/1.1761679 [9] Weissman, Phil. Trans. R. Soc. Lond. 290 pp 639– (1979) [10] DOI: 10.1017/S0022112075003394 [11] DOI: 10.1017/S0022112071000557 [12] Tatsumi, J. Phys. Soc. Japan 19 pp 1966– (1964) [13] Koch, J. Sound and Vib. 99 pp 443– (1985) [14] Ho, J. Fluid Mech. 119 pp 443– (1982) [15] Gaster, Proc. R. Soc. Lond. 347 pp 271– (1975) [16] DOI: 10.1063/1.1691990 [17] DOI: 10.1017/S0022112065000873 · Zbl 0129.20205 [18] DOI: 10.1017/S002211206600034X [19] Drazin, Adv. Appl. Mech. 9 pp 1– (1966) [20] DOI: 10.1017/S0022112078002645 · Zbl 0399.76057 [21] DOI: 10.1063/1.863880 [22] DOI: 10.1017/S0022112072002617 [23] Michalke, Z. Flugwiss. 19 pp 319– (1971) [24] DOI: 10.1017/S0022112065001520 [25] DOI: 10.1017/S0022112064000908 · Zbl 0129.20302
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