Конференции ИВТ СО РАН

Abstracts

The stability of the two layered Poiseuille flow in a plane horizontal channel is considered. The problem in the linear order is reduced to a pair of Orr – Sommerfeld equations with homogeneous boundary conditions on the interface, lid and bottom of the channel. As is well known there are many modes of disturbances with different phase velocity. Although the linear stability problem of two superposed immiscible viscous liquids was detailed researched [1] no special attention was paid to gravity mode. However interaction of such waves with the flow is interesting.

Due to numerical stiffness of Orr – Sommerfeld equation we used special algorithms such as compound matrix method [2] and Abramov’s method [3].

As a result we received the dependences of phase velocity and amplification factor of disturbances on the wave number for different flow speed. It is shown that long gravity waves can be unstable. Moreover under some circumstances the gravity mode is only one long unstable wave. The reason of this instability is the flow velocity joint on the interface. In the case when the viscous effects are strong (α Re << 1, where α is the wave number and Re is the Reynolds number) it corresponds to well known interface instability [4]. Also profiles of velocity disturbances were calculated.

Based on the velocity profiles received from linear analysis the nonlinear evolution equation for the interface disturbance of two layered flow was obtained like it was done in the paper [5]. Coefficients in the equation are presented as integrals on layers from function depended from flow and disturbance profiles. Notable that coefficient at dissipative member changes the sign in the region of linear instability. This fact corresponds to energy pumping from basic flow to the flow disturbances.

The work was supported by the president of the Russian Federation (Grants NSh-4366.2008.8)

References:

[1] Yiantsios S.G. Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids. 1988. V.31. No 11. P. 3225-3238.

[2] Zheltuhin N.A. Compound matrix method for solving Orr – Sommerfeld equation. Aerogazodinamika: Collected articles of 1 Siberian aerodynamic conference. 1973. P.70-73

[3] Goldshtik M.A., Shtern V.N. Hydrodynamic stability and turbulence. Novosibirsk: Nauka, 1977. P.33-39.

[4] Yih C.S. Instability due to viscous stratification. J. Fluid Mech. 1967. V.27 P.337-353.

[5] Arkhipov D.G., Khabakhpashev G.A. Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel, Journal of Applied Mechanics and Technical Physics, 2007. V.48 No 4. P.508-518.

Note. Abstracts are published in author's edition

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