Elucidation of the time scales of coherent structures in Newtonian turbulent channel flows through Karhunen-Loeve analysis
Date
2006
Authors
Oxberry, Geoffrey M.
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
First, the motivation for and the history behind Karhunen-Loeve (KL) Analysis (also called Karhunen-Loeve decomposition, proper orthogonal decomposition, empirical eigenfunction decomposition, etc.) is presented in the context of drag-reduction dynamics and turbulence modeling. Then, the theory and numerical methods of KL decomposition is discussed in detail, with the intent of summarizing the current body of literature presented on the topic of the numerical methods, as well as providing a reasonably self-contained introduction to the numerical methods needed to undertake this thesis. The theory is also extended slightly to clarify the concept of degeneracy when exploring the dynamics of the temporal eigenfunctions and their relation to the fluctuating kinetic energy of the flow. ☐ Next, the body of literature presenting results on static analyses of both Newtonian and viscoelastic turbulent channel flows, and some literature regarding dynamic analyses of Newtonian turbulent flows, is discussed. Here, the emphasis is on the classical interpretation of the results of KL analysis, and extracting physical meaning from the spatial and temporal eigenfunctions that are obtained. ☐ After that, the actual implementation of the theory is presented. A new method, related to the Direct Method of generating spatial eigenfunctions, is discussed, along with its implementation. Spatial eigenfunctions and eigenvalues obtained by this new method were exactly the same as those spatial eigenfunctions and eigenvalues obtained by Handler, who used the Method of Snapshots in a 2006 publication. ☐ Then, the standard methods used to generate temporal eigenfunctions is discussed. Two hundred unique temporal eigenfunctions, corresponding to the top 200 eigenvalues in the eigenvalue spectrum of the spatial eigenfunctions, were retained for analysis of Newtonian flow results of Housiadas and Beris at Re_0 generated via Direct Numerical Simulation. Temporal eigenfunctions were generated for 23240 computational time steps, corresponding to nearly 19 computational time units, or roughly 3347 plus (turbulent) time units. Statistical analyses of the temporal eigenfunctions were performed, along with phase plot analyses. ☐ Finally, cross-correlations were generated between temporal eigenfunctions as a way of elucidating the time scales over which energy and momentum transfer occurs between modes. Results indicate that the time scale of energy transfer is larger than previously suggested, with strong correlations persisting even after 11200 computational time steps, corresponding to approximately nine computational time units, or more than 1600 plus (turbulent) time units. This result suggests that further analysis should be done on the Newtonian case with data sets spanning longer time intervals. A convergence analysis was also done by comparing the Fourier spectra of cross-correlations sampled in the time domain versus the Fourier spectrum of the unsampled cross-correlations. Preliminary results suggest that sampling up to every 20 time steps does not decrease accuracy greatly.