A study by numerical methods of stability theory for a flat plate boundary layer of growing thickness
Barry, Michael David John
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.