City Research Online

Abstract

This investigation was initiated with the view of studying the stability of flows in symmetric curved walled channels, by essentially combining Fraenkel's small wall curvature theory, with the multiple scaling (or WKB) method.

The overall scheme is designed so that the straight walled channel problem can be retrieved to any relevant order of accuracy. In this way the higher order and curvature effects can be described separately or together. We rewrite the Navier-Stokes equations for an incompressible fluid in terms of Fraenkel's generalized orthogonal coordinates (mathematical equation refer to abstract in thesis) allowing for wall curvature. Then, by assuming a steady state slowly varying basic flow in the (mathematical equation refer to abstract in thesis) direction, and posing an asymptotic expansion in powers (mathematical equation refer to abstract in thesis) of for the steady state stream function (mathematical equation refer to abstract in thesis), we ensure the leading term is characterised by the non-linear Jeffery-Hamel profiles. These non-linear profiles are linearized by perturbing about v (the parameter defining the profiles), and then solving the resulting set of linear differential equations. These equations, and the linear higher order equations necessary to make the asymptotic expansion plausible, are solved directly by using central difference formulae to express and solve the equations in matrix form.

We are now able to develop the stability analysis by superimposing an infinitesimal disturbance (mathematical equation refer to abstract in thesis) to the basic flow, and to obtain the linearized disturbance equation. The coefficients of this equation are slowly varying with £ and independent of the time t, so constant frequency disturbances with wave number slowly varying with (mathematical equation refer to abstract in thesis) are appropriate. An asymptotic expansion for (mathematical equation refer to abstract in thesis) in powers of (mathematical equation refer to abstract in thesis) yields the well known Orr-Sommerfeld problem at lowest order. The coefficients here however, are functions of the slow variable oi = E?2£. By constructing an analytical approximation to the eigenrelation, a good initial guess is predicted by which a modified Newton Raphson is used to converge to the correct eigenvalue. The eigenfunction and higher order disturbance equations can now be solved using Runga-Kutta methods.

Spatially dependent growth rates are defined, and the "true" measure of the growth of the disturbance is taken to be the mean kinetic energy density of the disturbance relative to the mean kinetic energy density of the basic flow. Different flow quantities are found to have different growth rates, where the quasi-parallel prediction appears at lowest order, and is common to all flow quantities. By considering neutral stability curves of the relative energy growth rates, we are able to consider the separate effects of higher order corrections and curvature.

The higher order corrections to the straight walled channel do produce shifts in the stability curves, but these shifts are small. However, a constant positive curvature, produces a marked stabilizing effect, whereas a constant negative curvature produces a marked destabilizing effect, at a position in the channel where the angle of divergence is the same as the straight walled case.

A related problem in which curvature is allowed to vary in sign is also considered. This particular channel exhibits a bottle-neck effect, and the flow becomes more like Pois euille far upstream and downstream. R-crits are found at different positions in the streamwise direction, and these R-crits decrease or increase according to whether the angle of divergence is increasing or decreasing respectively. Thus, minimum R-crits can be found for different channels. This type of channel may be more suitable for experimentation than the previous idealised constant curvature channel. A further channel problem with varying curvature is suggested, which would also exibit Pois ueille flow far upstream and downstream.

Finally, the divergent straight walled channel is considered once more and a model of a wave maker producing an impulsive type disturbance at some suitable position is constructed. This isolated disturbance is shown to produce a wave packet type disturbance, which, according to quasi-parallel theory will grow or decay downstream, depending on whether R > R-crit or R < R-crit respectively. This idea is extended to the non-parallel case by superposing slowly varying, fixed frequency modes, which satisfy the linear disturbance equations. In this case, the isolated disturbance still produces a wave packet type disturbance, but any growth that appears is limited in the streamwise direction, and is restricted to some interval of time. All the cases considered show that the disturbance eventually decays downstream according to this linearized slowly varying approximation. Nevertheless, the results do suggest that when the dominant terms measuring the growth of the disturbance grow, (even if only for a very small range downstream, and a very small time interval)for some R> R-crit, then the disturbance might also be expected to grow for this R > R-crit.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

Export

Downloads