The Topology of SymmetricTensor Fields

Combinatorial topology, also known as \rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of di erential equations. A link between topology and di erential equations is vector elds. Recent developments in scienti c visualization have shown that vector elds also play an important role in the analysis of second-order tensor elds. A second-order tensor eld can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines? reveals the structure of a tensor eld. To simplify an often complex tensor eld and to capture its important features, the tensor is decomposed into an isotropic tensor and a deviator. A tensor eld and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A deviator determines the properties of a tensor eld, while the isotropic part provides a uniform bias. Degenerate points are basic constituents of tensor elds. In 2-D tensor elds, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible ows share similar topological features due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the Graduate Student, Physics Department Graduate Student, Aeronautics and Astronautics Department Professor, Electrical Engineering and Aeronautics and Astronautics Departments, Member AIAA. Lecturer, Faculty of Aerospace Engineering, Member AIAA. Copyright c 1997 by Authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. area of the vortex core in the eld. In turbulent ows, the similarities and di erences of the topology of the deformation and the Reynolds stress tensors reveal that the basic eddie-viscosity assumptions have their validity in turbulence modeling under certain conditions.