Low-dimensional models for flows with density fluctuations

A low-dimensional model, using the proper orthogonal, or Karhunen-Loeve decomposition, has been remarkably successful in representing the behavior of the wall region of a turbulent boundary layer. We briefly summarize this work. We may hope for similar success in other flows in which coherent structures play an important role, in particular flows with density fluctuations. We sketch an approach to such a decomposition for flows with density fluctuations, suggesting various alternatives which weigh the available information differently. In such a low-dimensional model, obtaining the empirical eigenfunctions poses a problem, since they can usually be determined only from extensive measurements or direct numerical simulations. However, recent work with energy method stability theory (modified by use of an anisotropic eddy viscosity and feedback to the mean profile) has been remarkably successful in predicting the form of the empirical eigenfunctions in the isothermal boundary layer. We present here preliminary results for sheared Rayleigh-Benard convection; these results do not include anisotropic eddy viscosities and feedback, and do not predict directly the form of the POD eigenfunctions; however, a very satisfactory comparison can be made with the second order moments obtained from a DNS. (C) 1997 American Institute of Physics.