A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

Abstract

The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberkc>0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk>kc—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3].