Local and Biglobal Linear Stability Analysis of Parallel Shear Flows

Linear Stability Analysis (LSA) of parallel shear flows, v is local and global approaches, is presented. The local analysis is carried out by solving the Orr-Sommerfeld (OS) equation using a spectral-collocation method based on Chebyshev polynomials. A stabilized finite element formulation is employed to carry out the global analysis using the linearized disturbance equations in primitive variables. The local and global analysis are compared. As per the Squires theorem, the two-dimensional disturbance has the largest growth rate. Therefore, only two-dimensional disturbances are considered. By its very nature, the local analysis assumes the disturbance field to be spatially periodic in the streamwise direction. The global analysis permits a more general disturbance. However, to enable a comparison with the local analysis, periodic boundary conditions, at the inlet and exit of the domain, are imposed on the disturbance. Computations are carried out for the LSA of the Plane Poiseuille Flow (PPF). The relationship between the wavenumber, a, of the disturbance and the streamwise extent of the domain, L, in the global analysis is explored for Re = 7000. It is found that alpha and L are related by L = 2 pi n/alpha, where n is the number of cells of the instability along the streamwise direction within the domain length, L. The procedure to interpret the results from the global analysis, for comparison with local analysis, is described.