The influence of wall oscillations, wall rotation, and entry eddies, on the breakdown of laminar flow in an annular pipe

Abstract 1. The problem of the breakdown of laminar flow has been much studied, both mathematically and experimentally. The mathematical method of approach is concerned with the prediction of the history of the deviant motion arising from an assumed initial disturbance, superimposed on a selected type of laminar motion and satisfying the equations of viscous fluid flow and the appropriate boundary conditions. A motion is considered to be stable if the deviant motion tends ultimately to vanish, and neutral or unstable if it persists or increases. In general, the disturbance assumed is small and of a form for which the deviant motion can be expressed in terms of linear differential equations; and these restrictions curtail the choice of problem amenable to mathematical treatment. A review of classical mathematical investigations on the stability of fluid motion has recently been published by Southwell and Chitty (1930). Of these investigations that made by Taylor (1923) for flow in the annular space between two coaxial rotating cylinders, gives an illustration of a successful prediction of instability, experimentally verified. More recent mathematical investigations are those on the stability of flow between parallel planes made by Prandtl (1921, 1931), Tietjens (1925), Tollmien (1929) and Schlichting (1932, 1933a, b, 1935). These investigations define the mathematical conditions under which the deviant motion grows, and give characteristic numbers, defining the wave-length and frequency detrimental to the maintenance of the laminar state: but these numbers do not necessarily indicate the onset of turbulence. A view commonly held is that breakdown of laminar flow is connected with adverse pressure gradients associated with velocity fluctuations, the magnitude of these gradients depending on the amplitude and frequency of the velocity fluctuations, and increasing with either (Dryden 1931, p. 559; Taylor 1936, p. 308). Further, it has been suggested that it is the amplitude, rather than the frequency, of velocity disturbances which is primarily responsible for the breakdown of laminar flow and the onset of turbulence; and this suggestion has received some support from experiments made by Nikuradse (1933).