Stability analysis of Boundary Layer in Poiseuille Flow Through A Modified Orr-Sommerfeld Equation

摘要

For applications regarding transition prediction, wing design and control ofboundary layers, the fundamental understanding of disturbance growth in theflat-plate boundary layer is an important issue. In the present work weinvestigate the stability of boundary layer in Poiseuille flow. We normalizepressure and time by inertial and viscous effects. The disturbances are takento be periodic in the spanwise direction and time. We present a set of lineargoverning equations for the parabolic evolution of wavelike disturbances. Then,we derive modified Orr-Sommerfeld equations that can be applied in the layer.Contrary to what one might think, we find that Squire's theorem is notapplicable for the boundary layer. We find also that normalization by inertialor viscous effects leads to the same order of stability or instability. For the2D disturbances flow ($\theta=0$), we found the same critical Reynolds numberfor our two normalizations. This value coincides with the one we know forneutral stability of the known Orr-Sommerfeld equation. We noticed also thatfor all overs values of $k$ in the case $\theta=0$ correspond the same valuesof $Re_\delta$ at $c_i=0$ whatever the normalization. We therefore concludethat in the boundary layer with a 2D-disturbance, we have the same neutralstability curve whatever the normalization. We find also that for a flow withhight hydrodynamic Reynolds number, the neu- tral disturbances in the boundarylayer are two-dimensional. At last, we find that transition from stability toinstability or the opposite can occur according to the Reynolds number and thewave number.