© 1980 by Oxford University Press
ON THE STABILITY OF THE DEVELOPING FLOW IN A CHANNEL OR CIRCULAR PIPE
(
1Applied Mathematics Dept., University of Western Ontario London, Ontario, Canada
2Dept. of Mathematical Sciences, Indiana University, Purdue University at Indianapolis Indianapolis, Indiana, U.S.A.
)
Permanent Address: Mathematics Dept., Imperial College, London SW7 2BZ, U.K.
The stability of the tube flow developing from uniform inlet conditions is studied for high Reynolds number, the only regime where the basic flow is nearly parallel. Even while the flow is still quasi-external dramatic alterations from the neutral stability criteria for external flow take place, being governed by five- and three- zone structures on the upper and lower branches respectively, from which the subsequent developments of the neutral stability criteria follow systematically. For the even neutral modes in developing channel flow, and for all the three-dimensional neutral modes in circular pipe flow, the parallel flow approximation eventually becomes invalid, despite the near-parallelism of the basic flows, so that the further development for such modes remains unknown. In contrast, the odd modes in developing channel flow allow a complete account of the behaviour of the neutral stability criteria, from the inlet to the fully developed stage of the basic flow. The structural approaches set out here should prove eminently suitable for further treatments of nonparallel and nonlinear stability features in a totally rational fashion.