## A study by numerical methods of stability theory for a flat plate boundary layer of growing thickness

##### Abstract

The research, which is described in the following chapters
is designed to continue the studies made by Jordinson into the
behaviour of small disturbances of constant frequency in the
Blasius boundary layer over a flat plate. In his work a numerical
method due to Osborne was used to solve the problem of the nonlinearly
occurring eigenvalue which is associated with the Orr-
Sommerfeld equation in the case of space amplification. A
digital computer, the Edinburgh University KDF 9, performed the
calculations. The present thesis first of all extends the Orr-
Sommerfeld equation to include the effect of the growing thickness
of the boundary layer upon the eigenvalues, and then it discusses
a possible stage where the disturbance grows to such a size that
the linearised theory no longer completely represents its
behaviour.
Following a chapter of introduction a mathematical model
is devised which is based upon a Fourier Series expansion of the
flow stream function in terms of frequency harmonics. This
model is intended to represent the beginnings of finite disturbance development by showing the presence of second harmonic
oscillations and the effect of the transfer of energy and momentum
between the perturbation and the mean flow.
The third chapter analyses the importance of the normal
component terms of the Blasius mean flow from an empirical
viewpoint. These are the terms which are responsible for the
thickening of the boundary layer and it is shown, that in order to
be consistent with the assumptions of boundary layer theory their
retention is necessary. The topic of a linear disturbance is then
discussed and the Orr-Sommerfeld equation, together with the
contribution due to boundary layer thickening, is derived. In
addition a proof is sketched which demonstrates that the boundary
conditions of the differential equation are unaltered by the
presence of the extra terms.Since the eigenvalue character of the linearised problem is
of the same nature as Jordinson's problem, the same method of
solution is used and the details are given in the fourth chapter.
A rational discretisation of the differential equation is
performed in order to reduce the truncation error of the approximation. This is followed by a discussion of Osborne's
method. Tests carried out by varying the data parameters, the
discrete step length, and the range of integration demonstrate
remarkable stability.
The results given in the fifth chapter show that the
boundary layer is rendered slightly less stable if the effect
of growing thickness is included, and that the curve of neutral
stability is enlarged. This enlargement is in itself not sufficient to account for the differences existing between the
predictions of theory and the results of experiment, particularly
at lower Reynolds Numbers, but an argument later developed
explains why these differences occur.
The non-linear investigations ere discussed in the next two
chapters. In the first of these the Fourier-Series mathematical
model is studied more closely and details are given of the deter¬
mination of second harmonic oscillation values, distorted mean
flow components and Reynolds stress. The effect of finite dis¬
turbance development upon the first harmonic of a perturbation.
arises in the form of a set of coupled differential equations
which are solved by a straightforward iterative procedure. The
results of the investigations which are all given for a fixed fre¬
quency, show first of all that the Reynolds stress distribution
begins to oscillate as the Reynolds Number increases and that its
amplitude is increased. They also show that a perturbation of
fixed signal size has a far greater effect at higher Reynolds
Number upon mean flow and first harmonic component distortion
Results of the distribution of first and second harmonic values
are also given. The final chapter discusses a futuro research
topic and summarises the preceding work.
The computer programs for the calculation were developed
from Jordinson's program and were run initially on the KDF 9
computer and latterly on the IBM 360/5C and Systems 4/75 computers
in Edinburgh University. Details of the extensions to the
existing subroutines are given in the Appendix.