THE PROPER ORTHOGONAL DECOMOPOSTION, WAVELETS AND MODAL APPROACHES TOTHE DYNAMICS OF COHERENT STRUCTURES IN TURBULENCE

We present brief precis of three related investigations. Fuller accoun ts can be found elsewhere. The investigations bear on the identificati on and prediction of coherent structures in turbulent shear flows. A s econd unifying thread is the Proper Orthogonal Decomposition (POD), or Karhunen-Loeve expansion, which appears in all three investigations d escribed. The first investigation demonstrates a close connection betw een the coherent structures obtained using linear stochastic estimatio n, and those obtained from the POD. Linear stochastic estimation is of ten used for the identification of coherent structures. The second inv estigation explores the use (in homogeneous directions) of wavelets in stead of Fourier modes, in the construction of dynamical models; the p articular problem considered here is the Kuramoto-Sivashinsky equation . The POD eigenfunctions, of course, reduce to Fourier modes in homoge neous situations, and either can be shown to converge optimally fast; we address the question of how rapidly (by comparison) a wavelet repre sentation converges, and how the wavelet-wavelet interactions can be h andled to construct a simple model. The third investigation deals with the prediction of POD eigenfunctions in a turbulent shear flow. We sh ow that energy-method stability theory, combined with an anisotropic e ddy viscosity, and erosion of the mean velocity profile by the growing eigenfunctions, produces eigenfunctions very close to those of the PO D, and the same eigenvalue spectrum at low wavenumbers.