The Karhunen-Loève Decomposition of the Autonomous Minimal Flow Unit

Abstract

We propose to examine the effects of restricting the interaction between nearwall turbulence and the outer flow on the structure and dynamics of the empirical eigenfunctions functions determined using the proper orthogonal decomposition (POD). This research is motivated by the fact that standard POD-based low-dimensional models for the near-wall region have been derived for empirical eigenfunctions computed for an unbiased channel flow. However, under the present truncation of the flow dynamics, the POD basis may be significantly affected so that the common assumption that effective reduced-order models can be constructed from the POD basis of an unaltered flow may be suspect. This issue is explored for plane, incompressible, turbulent channel flow at Reynolds number, Re_τ = 180. Based on direct numerical simulations, POD basis functions are constructed for an unbiased and four truncated minimal channel flows. The POD eigenfunctions which characterize these modified flows are associated to a travelling-wave solution which undergoes a series of bifurcation until settling into a turbulent regime. A POD-based two-mode model is also derived for the near-wall layer and is evaluated. It is shown that travelling-wave solutions appear as a backbone to the low-dimensional dynamics of the autonomous near-wall region.