CHEBYSHEV TAU-QZ ALGORITHM METHODS FOR CALCULATING SPECTRA OF HYDRODYNAMIC STABILITY PROBLEMS

The Chebyshev tau method is examined in detail for a variety of eigenv alue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourt h derivative operator, D-4, is compared with those involving the secon d and first derivative operators, D-2 and D, respectively. The latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. Ph ysical problems explored are those of Poiseuille flow, Couette flow, p ressure gradient driven circular pipe flow, and Couette and Poiseuille problems for two viscous, immiscible fluids, one overlying the other.