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On the asymptotic behavior of localized perturbations in the presence of Kelvin-Helmholtz instability. (English. Russian original) Zbl 0601.76037
Fluid Dyn. 20, 186-193 (1985); translation from Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. 1985, No. 2, 23-30 (1985).
The asymptotic behavior of localized two-dimensional perturbations of the surface of a shear discontinuity separating two homogeneous steady flows of ideal incompressible fluid is studied in the linear approximation. The effect of surface tension and gravity forces is taken into account. Mathematically the problem reduces to the investigation by the method of steepest descent of the asymptotic behavior of a double integral for various values of parameters which are the components of the group velocity vector. In this problem the principal difficulty is to find the two-dimensional steepest descent contour in the space of two complex variables that determines which of the various saddle points gives the asymptotic form.
First, for the Fourier component with respect to one of the variables with allowance for all the saddle points we find an asymptotic form which parametrically depends on the second variable. The choice of the second variable makes it possible to prove analytically that in the absence of gravity the asymptotic behavior of the growing perturbations is determined by a single saddle point in the plane of that variable. In this way it is possible to justify the authors’ previous conclusions concerning the shape of the boundary L of the region D in the group velocity plane occupied by growing perturbations. In the presence of gravity the growth rates of perturbations corresponding to different group velocities are found numerically and the region D occupied by the growing perturbations is indicated.
MSC:
76E05 Parallel shear flows in hydrodynamic stability
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[1] A. G. Kulikovskii and I. S. Shikina, ?On the development of two-dimensional perturbations at a shear discontinuity,? Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 12 (1979).
[2] M. V. Fedoryuk, The Method of Steepest Descent [in Russian], Nauka, Moscow (1977). · Zbl 0463.41020
[3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961).
[4] A. G. Kulikovskii and I. S. Shikina, ?On the development of perturbations at the interface of two fluids,? Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 46 (1977).
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