Hydrodynamic and hydromagnetic stability

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Андрей Геннадьевич Куликовский (к восьмидесятилетию со дня рождения)@@@A. G. Kulikovskii (on his eightieth birthday)

Kelvin-Helmholtz stability waves are of great importance in the field of fluid mechanics and for quite a while now special attention has been focused on studying them in detail. Indeed two layered fluid flows occur in many physical situations, and even now, numerous works continue to approach the matter (Bontozoglou & al [2] [3], Chimonas [4], Horton & al [12], Kulikoskii & al [13], Courtis & al [6]).

Kelvin — Helmholtz Two-Phase Flow Instability

Andrei Gennad'evich Kulikovskii (on his 80th birthday)

In this work, we explore the transition of absolute instability and convective instability in a compressible inviscid shear layer, through a linear spatial-temporal instability analysis. From linearized governing equations of the shear layer and the ideal-gas equation of state, the dispersion relation for the pressure perturbation was obtained. The eigenvalue problem for the evolution of two-dimensional perturbation was solved by means of shooting method. The zero group velocity is obtained by a saddle point method. The absolute/convective instability characteristics of the flow are determined by the temporal growth rate at the saddle point. The absolute/convective nature of the flow instability has strong dependence on the values of the temperature ratio, the velocity ratio, the oblique angle, and number. A parametric study indicates that, for a great value of velocity ratio, the inviscid shear layer can transit to absolute instability. The increase of temperature ratio decreases the absolute growth rate when the temperature ratio is large; the effect of temperature ratio is opposite when the temperature ratio is relatively small. The obliquity of the perturbations would cause the increase of the absolute growth rate. The effect of number is different when the oblique angle is great and small. Besides, the absolute instability boundary is found in the velocity ratio, temperature ratio, and number space.

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Spatial-Temporal Instability of an Inviscid Shear Layer

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 [1] 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.

I. S. Shikina

Papers

On the asymptotic behavior of localized perturbations in the presence of Kelvin-Helmholtz instability