Detection of Self-Similar Modes in Turbulence: Application to a Wake Flow

Abstract

By using the space-time scaling invariance of the Navier-Stokes equations, the spatio-temporal structure of fully developed turbulence, in the homogeneous or inhomogeneous cases, can be understood in terms of a spatio-temporal self-similarity. The latter consists in a hierarchy of spatial and temporal modes which are all similar, namely related to each other through a dilation symmetry group. Our analysis uses biorthogonal decompositions which decompose the flow into spatial and temporal orthogonal modes for which there is a one-to-one correspondence. It also permits the derivation of (exponential) spectrum laws which, for the kinetic energy of the flow, coincide with Kolmogorov’s k^−5/3 power law when the flow is both homogeneous and incompressible. Such self-similarity, or fractal structure in a spatio-temporal sense, originates in the space-time dynamics of the flow and is unlikely to be observed from the analysis of a one-dimensional signal (obtained by a single point measurement or an instantaneous picture of the flow). It is, however, recovered by the spatial and temporal two-point correlations and therefore should appear in the structure of the eigenvectors and eigenvalues of the Proper Orthogonal Decomposition in which the ensemble average is a time average. The analysis of spatial two-point correlations, experimentally measured in a turbulent far-wake flow, supports the previous theory.