On the behaviour of small disturbances to Poiseuille flow in a circular pipe

Abstract

A mathematical analysis of the behaviour of small disturbances to Poiseuille flow in a circular pipe can be tackled in two different ways. In the first of these, the disturbance velocity is expressed as a combination of terms, each of which represents a disturbance of a fixed wavelength that decays with time. In the second approach, each term which contributes to the disturbance velocity represents a disturbance of a fixed frequency that decays with downstream distance. The latter method is given more prominence in this paper, since, in experiments in which disturbances are introduced into the flow in a controlled manner, a constant frequency-generating device is commonly used.In the main, it is assumed not only that the Reynolds number of the Poiseuille flow is large, but that the frequency, f, of the disturbance is large compared with the frequency, v/a2, where a is the radius of the pipe and v the kinematic viscosity of the fluid. The mathematical problem is then of the singular perturbation type, and the disturbances with the smallest damping rates are confined to thin layers. A simple, but crude, analysis shows among other things that the radius a t which the disturbance velocity is a maximum is roughly that at which the velocity of the Poiseuille flow is equal to the frequency, f, times the disturbance wavelength. Eigenfunctions are found precisely for the two limiting cases in which, as fa2/v tends to infinity, the disturbance becomes confined to a thin layer situated (a) near the centre of the pipe, and (b) near the wall. The eigenfunctions are presented graphically in such a way that immediate comparison can be made with some of Leite's experimental results. Good agreement is found. A real disturbance, however, is made up of several modes, each of which is damped at a different rate with increasing downstream distance. Possible changes in the form and apparent damping rate of a disturbance are discussed in terms of particular case.Next, an asymptotic procedure is carried out, which proves to give a good approximation to the eigenvalues and eigenfunctions over a wide range of conditions. From the eigenvalue equation so obtained it is possible to calculate the wave-speed and damping rate for each mode as a function of the non-dimensional frequency, fa2/v, and the Reynolds number. For simplicity, the calculations are carried out for the case in which the Reynolds number is infinite, so that the eigenvalues depend only on fa2/v. For each mode it is found that the damping rate is an increasing function of the frequency for high frequencies, but the frequency is decreased the damping rate approaches a limiting value. These limiting values can be quite small for the Reynolds numbers, R, at which experiments are normally carried out, and for some low-frequency disturbances the distance required for the disturbance amplitude to be reduced by half is as much as R/1000 diameters.