Stability Analysis of Boundary Layer in Poiseuille Flow through a Modified Orr-Sommerfeld Equation

A. V. Monwanou, C. H. Miwadinou, J. B. Chabi Orou


For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive the so-called modified Orr-Sommerfeld equation that can be applied in the layer. Contrary to what one might think of, we find that Squire’s theorem is not applicable for the boundary layer. We find also that normalization by inertial or viscous effects leads to the same order of stability or instability. For the 2-D disturbances flow, we find the same critical Reynolds number for our two normalizations. This value coincides with the one we know for neutral stability of the known Orr-Sommerfeld equation. We notice also that for all over values of k in the case , correspond the same values of  at  whatever the normalization. We therefore conclude that in the boundary layer with 2-D disturbances, we have the same neutral stability curve whatever the normalization. We find also that for a flow with high hydrodynamic Reynolds number, the neutral disturbances in the boundary layer are two dimensional. At last, we find that transition from stability to instability or the opposite can occur according to the Reynolds number and the wave number.

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Applied Physics Research   ISSN 1916-9639 (Print)   ISSN 1916-9647 (Online)

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