Central Aero-Hydrodynamic Institute (TsAGI)
Zhukovsky, Russia
The particles slightly affect on the fluid mean flow when their mass content is low. However their influence on the instability wave may be significant because of the unsteady momentum transfer between phases. In the unsteady perturbed velocity field the particles cannot accommodate rapid acceleration but tend to slip through the gas since the density of their material is much greater than that of the gas. They experience a force and equal but opposite volume forces act on a gas. Linear stability of 2D flow of a dusty gas in a boundary layer over a flat plate was investigated recently by Asmolov \& Manuilovich (1998). Particles were assumed homogeneously distributed, and to be under the action of the Stokes drag only. Following to Saffman (1962), the stability problem was reduced to a single ordinary differential equation. From the numerical solution of this equation it was shown that particles suppress Tollmien-Schlichting (TS) wave.
In the present work the theory is extended to the flows with non-uniform distribution of particle density. Stability of dusty-gas flow in a channel is considered. Momentum transfer due to both the drag and the lift on the particles is taken into account. The eigenvalue problem for modified Orr-Sommerfeld equation is solved using two approaches: the perturbation theory and direct integration. The dust influence is considerably more than for homogeneously distributed particles. However particles can either stabilize or destabilize TS wave depending on the position of maximum of density distribution. The most stabilizing effect takes place for sufficiently sharp distribution with the maximum close to critical layer.
The discontinuity of eigenvalue dependence arises because of the resonant acceleration of the particles in the critical layer. It may take place for both stable and unstable TS wave. The kinetic energy balance between mean and disturbance flow is studied. It is shown that strong effect of the dust on the stability characteristics is owing to the change of the shift in the phases of the two components of the disturbance velocity.
The work was supported by Russian Foundation for Fundamental Research (Grant No. 96-01-01245).
References:
[1] {Asmolov, E.S. and Manuilovich, S.V.} 1998 Stability of dusty-gas laminar boundary layer on a flat plate, {\it J. Fluid Mech.} Accepted for publication.
[2] {Saffman, P.G.} 1962 On the stability of laminar flow of a dusty gas. {\it J. Fluid Mech.} {\bf 13}, pp. 120-128.