On Three and Two-Dimensional Disturbances of Pipe Flow

Abstract

A system of ordinary, coupled differential equations is set up for three-dimensional disturbances of Poiseuille flow in a straight pipe of circular cross section. The commonly treated equations are shown to be special cases arising from particular assumptions. It is shown that in the nonviscous, and therefore also in the general case, there exists, in contrast to the analogous problem in Cartesian co-ordinates, no transformation reducing the given problem to a two-dimensional one. A fourth-order differential equation is derived for disturbances independent of the direction of the main flow. The solutions, which are obtained, show that those two-dimensional disturbances, termed cross disturbances, decay with time and do therefore not disturb the stability of the main flow. Explicit expressions for the cross disturbances are obtained and a discussion of their nature is given.

Transitional Flow in Isosceles Triangular Ducts

“…This result suggests strongly that inferences drawn from simple one-dimensional flows (and theoretical analyses of the same) involving only two-dimensional disturbances may provide misleading guides for more complex threedimensional phenomena. A similar conclusion follows from the work of Spielberg and Timan (1960), who concluded that an essential difference exists between twoand three-dimensional stability characteristics of classical Poiseuille pipe flow. Therefore, it appears that further study involving complex three-dimensional flows is required before a complete understanding of stability of laminar flow and transition therefrom is to be had.…”

Hydrodynamic Stability of a Laminar Boundary Layer with Transverse Curvature Effect

“…G, are the perturbation velocities, which are accompanied by the perturbation pressure p . Both the mean mo--"I ( y 7 _ _ _ _ _ ___c @ tion and the perturbation motion are assumed t o + satisfy the NAVIER-STOKES equations so that subtracting the former equations from the latter, and linearizing them in the usual manner, one has c 2a * Sketch of the roordinste system nsed In view of the fact that SQUIRE'S theorem is not valid ( [13], [16]), it is necessary to consider all the four equations above in general. For this purpose i t is useful to introduce the generalized stream functions (MOORE [lo], calls them vector potentials) and 5 for the perturbed threedimensional motion, such t h a t --a3…”

The stability of pipe flow Part 1. Asymptotic analysis for small wave-numbers

“…Pretch (1941)) Pekeris (1948)) Sex1 & Spielberg (1958)) and Corcos & Sellars (1959) have all considered axially symmetric disturbances and found them to be stable. Spielberg & Timan (1960) have studied the case of disturbances independent of the direction of the primary flow and have concluded that such disturbances are also stable. Lessen, Sadler & Liu (1968) have considered non-symmetric disturbances, but limited their attention to the case n = 1, where n is the angular wave-number.…”