Computing the Karhunen–Loève dimension of an extensively chaotic flow field given a finite amount of data

Abstract

The use of Karhunen–Loève decomposition (KLD) to explore the complex fluid flows that are common in engineering applications is increasing and has yielded new physical insights. However, for most engineering systems the dimension of the dynamics is expected to be very large yet the flow field data is available only for a finite time. In this context, it is important to establish the amount of data required to compute the asymptotic value of the Karhunen–Loève dimension given a finite amount of data. Using direct numerical simulations of Rayleigh–Bénard convection in a finite cylindrical geometry we compute the asymptotic value of the Karhunen–Loève dimension. The amount of time required for the Karhunen–Loève dimension to reach a steady value is very slow in comparison with the time scale of the convection rolls. We show that the asymptotic value of the Karhunen–Loève dimension can be determined using much less data if one uses the azimuthal symmetry of the governing equations prior to performing a KLD. The Karhunen–Loève dimension is found to be extensive as the system size is increased and for a dimension measurement that captures 90% of the variance in the data the Karhunen–Loève dimension is approximately 20 times larger than the Lyapunov dimension.