Linear Stability Analysis (LSA) of parallel shear flows has been a major area of researchduring the last century and continues to be so in the 21st century. The famous Orr-Sommerfeld (OS) equation, which is used for local LSA of parallel shear flows, does notyield useful results for several situations. This is due to mismatch of critical Reynoldsnumber for the onset of instability, Recro, obtained from the OS analysis and criticalReynolds number for the transition to turbulence, Recrt, obtained from experimentalanalysis. In this work, the Cross-Stream Orr-Sommerfeld (CSOS) equation is derivedfor local LSA of parallel shear flows to include disturbances moving with nonzero cross-streamvelocity component. The CSOS equation is applied to Plane Couette Flow (PCF)and Plane Poiseuille Flow (PPF). A spectral collocation method based on Chebyshevpolynomials of the first kind has been used to solve the relevant equations. This givesRecro equal to 360.63 and 916.85 for PCF and PPF respectively, which are in closeagreement with Recrt obtained from experimental analysis for the corresponding flows.The aspects of global LSA of parallel shear flows for periodic boundary conditionsin streamwise and spanwise directions are also analysed. A stabilized finite elementmethod proposed by Mittal and Kumar (2007) has been utilized to do so. It is shownthat the local and global LSA give same result. Global LSA of PCF and PPF are donefor many cases (including near the onset of instability predicted by local LSA). DirectNumerical Simulation (DNS) of the respective cases has been performed to crosscheckthe analysis and study the time evolution of the disturbances.
