A KARHUNEN-LOÉVE ANALYSIS OF TURBULENT THERMAL CONVECTION

Abstract

SUMMARY The Karhunen-Ldve (K-L) procedure is applied to a turbulent thermal convection database which is generated numerically through integration of the Boussinesq equation in a periodic box with stress-free boundary conditions using a Fourier collocation spectral method. This procedure generates a complete set of mutually orthogonal functions in terms of which the turbulent flow fluctuation field is represented optimally in the mean square sense. A study is performed ranging from the direct projection of the database onto the set, resulting in a considerable data compression, to developing a system of dynamical equations employing the set as a basis for approximating the Boussinesq equation. In the latter a new strategy is proposed and tested for the treatment of the mean component of the turbulent flow. Finally, the direct projection and the dynamical equations are used to study the effects of truncation on the representation of the turbulent flow. A turbulent flow involves many scales of motion. This sets the basic requirement that a direct numerical simulation must meet if it is to represent turbulence. All scales must be adequately resolved by the computational mesh. This, together with the requirement that a large sample of the set of all possible motions allowed by the governing equation is necessary to provide an adequate statistical evaluation, results in a vast database to be generated through the simulation for subsequent analysis and assessment. A technique based on the Karhunen-Lokve (K-L) procedure’ has been used extensively in extracting essential features of the flow hidden within the database. These features are then used as a basis in subsequent data compression and model reduction schemes. Some applications range from data compression in turbulent thermal convection’ to model reduction in the Ginzburg-Landau (G-L) equation3 and in a turbulent boundary layer4 The application of this procedure to the stochastic turbulent field was proposed by Lumley’ as a rational and quantitative method, referred to as principal orthogonal decomposition (POD), for extraction of organized structures from the field. New perspectives on the applications of the technique are presented in Reference 6. The technique is based on the decomposition of the turbulent flow fluctuation field into a weighted sum of mutually orthogonal eigenfunctions of the two-point correlation tensor which meet the boundary conditions and appropriate side conditions. This decomposition is unique and optimal in the sense that the random variables appearing as weights in the representation are statistically orthogonal and the mean square error resulting from the truncated representation of the flow is minimized. The energy retained in the truncated representation of the flow is maximized by ordering the eigenfunctions in the representation based on the corresponding