This work is concerned with studying the effects of compressibility on the linear and nonlinear stability of compressible boundary layers. With the renewed interest in high speed flight vehicles an understanding of some of these effects may be important, especially as very little is known about the the nonlinear stability of such flows. Much of the work is based on using matched asymptotic expansions, and exploiting the asymptotic solution properties in the limit of large Reynolds numbers. A lot of the work also utilises the important scaling ideas of unsteady non-linear critical layer theory. Some results from this work may be found in publications 13, 17,18,23, 28.

Cross-flow instability plays an important role in the transition of a fully three-dimensional boundary layer, as for example in the flow past a swept wing, and as such it is important to understand the underlying mechanisms as well as possible interactions with other stabilities. Our effort in this area has been directed at trying to understand firstly, the linear stability properties from an analytical as well as numerical viewpoint, and secondly to study the nonlinear development of such instabilities. In the latter context, the research has led to studying the properties of some complicated novel partial-integro-differential equations which govern the nonlinear evolution of some of the modes, see publications 10,20,23,25,26,27,28. We are also investigating the absolute instabilities of three-dimensional boundary layer flows. One of my students Jeff Cole, was one of the first to demonstrate absolute instability in

R=10850 This work complements that of (1) and (2) above in trying to explore the flow properties when the analytical methods can no longer be used because of strong nonlinearities present in the problem. With the trends towards increased parallelism in computer architectures, we have concentrated on developing techniques making use of parallel algorithms. In this context some parallel multigrid techniques were successfully developed to study the flow in 2D and 3D lid-driven cavities. This is described in the M.Sc and Ph.D. dissertations of Turkyilmazoglu and Jackson. The work is being written up for submission to a journal.

The potential drag reducing properties of compliant surfaces has motivated much of this work. The underlying theme is again to develop suitable techniques for analysing the nonlinear properties either numerically or using asymptotic methods. Of several ongoing projects one is concerned with the full numerical solution of the triple-deck model. Another is aimed at understanding modal interactions using unsteady nonlinear critical theory. We have also extended some of the ideas to internal flows, as described in publication 24. Recently through 2 EPSRC grants we are looking at the stability of flow in compliant channels, pipes and boundary layers, to investigate the effects of heat transfer on the flow stability.

Recently we have spent considerable effort in devising techniques for studying absolute instabilities in two- and three-dimensional boundary layer flows, see for example publications 30-34,36. A number of significant results have emerged from these studies in particular for the wedge shaped and cusp shaped trailing edge configurations, where we have for the first time been able to look at the effects of separation and compressibility on the flow stability. One of our tentative suggestions is that for thicker aerofoils the flow upstream of the trailing-edge is absolutely unstable and if correct, this has enormous implications for laminar flow control on such aerofoils. Our current efforts are being directed at trying to compute much larger regions of separated flow over real aerofoil shapes and investigate their stability.