Wall-transpiration-induced instabilities in plane Couette flow

Abstract

Linear stability of Couette flow modified by transpiration applied at the lower wall is considered. It is shown that transpiration can induce flow instability resulting in the appearance of streamwise-vortex-like structures. It is argued that the instability is driven by centrifugal forces associated with streamline curvature. The conditions leading to the onset of the instability depend on the amplitude and wavelength of the transpiration and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required for the onset of the instability for the specified amplitude of the transpiration regardless of its wavelength are also given. The threshold amplitude required for the onset varies approximately as ${\sim}\hbox{\it Re}^{-1.15}$ for large $Re$, where the Reynolds number used is based on the velocity difference between the walls and the channel half-width. The existence of a global threshold, below which the instability cannot occur regardless of the amplitude of the transpiration, has been demonstrated. This threshold corresponds approximately to $\hbox{\it Re}=84$.