Abstract The Jet Engine Flow
This lecture deals with the analysis and control of air-flow through a jet engine, or a compressor. This is a problem of great importance in modern technology as the jet engine manufactures are under pressure to produce lighter, more efficient and safer jet engines for travel in the 20th Century. The full details of the air-flow through the jet engine are very complex. The flow is described by the three-dimensional Navier-Stokes equations and these can only be approached numerically with current techniques. The three-dimensional simulation is a very computationally intensive and expensive problem. However, in 1986 Moore and Greitzer introduced a model of the flow through a turbomachine that has become very sucessful. This model consists of a partial differential equation (PDE) coupled to two ordinary differential equations (ODE) and has become known as the Moore-Greitzer equations. Adomaitis and Abed added a viscous term to the PDE and this term has been justified recently by Mezić in terms of eddy viscosity. The predictions of the Moore-Greitzer model are in good argreement with experiments done on compressors in laboratories. Over the last 12 years a number of authors have used low-order truncations of Fourier modes in the Moore-Greitzer equations. They obtain a low-dimensional system of ODE's and study bifurcations of solutions to the ODE's, as a models of changes taking place in the real flow, with varying flow-parameters.
The papers that this lecture is based on put the above methods on a rigorous footing and improve the qualitative results greatly, as well as making them quantitative. The theory of finite-dimensional attractors for dissipative PDE's is used to prove that the Moore-Greitzer equations have a finite dimensional attractor. This is a set of solutions that all initial data settle on in a relatively short time. Then the (fractal and Hausdorff) dimension of the attractor are estimated and shown to be finite. The dimension is a constant over the square root of the viscosity and can therefore be large. This frequently happens for such estimates and is the reason why they are not neccessarily useful. However, the theory of basic attractors, attractors that attract a prevalent set in infinite-dimensional phase space, is used to show that the basic attractor of the Moore-Greitzer equations is indeed low-dimensional. In practice, only this part of the attractor shows up, so this fully justifies the use of low-order truncations of Fourier modes, mentioned above. The concept of shy and prevalent in infinite-dimensional space were developed by Sauer, Hunt and Yorke. These generalizations of measure zero and almost every, were used by Birnir and a German postdoc Rainer Grauer to extend results of Milnor, to infinite-dimensional space. Milnor proved that in finite dimensions every compact attractor decomposes into a basic attractor and a remainder and that the basin of attraction of the basic attractor included almost every point, with regard to Lebesque measure. This implies that only the basic part of the attractor will be seen in experiments.
It was not clear how big a role the concept of a basic attractor would play in practical examples. This is completely clear up by showing that the basic attractor is crucial for the jet engine flow. An explicit qualitative description of the basic attractor of jet engine flow is given. It consists of a steady solution, called design flow, a periodic orbit that represents an instability called surge and last but not least, finitely many periodic orbits that are travelling waves and give instabilities called stall. As far as the spatial structure of the solutions is concerned, this description is complete.
This work also includes a complete stability analysis of the stall solutions as the flow-parameters in the Moore-Greitzer equations vary. All the eigenvalues and eigenfunctions of the equations linearized about the stall solutions are found and how these eigenfunctions affect the stall solutions. This result opens up the possibility of optimal nonlinear control of jet engine flow. It is known that these instabilites can be controlled by linear control. However, this may defeat the goal of the research because such control may be expensive to apply. Nonlinear control on the other hand has the advantage that it can lead to effecive control methods that are also cost effective. This theory is mostly open untill now. The jet engine flow problem is an ideal problem for which one may attempt to develop nonlinear control methods.