Symmetry, spatio-temporal self-similarity and scaling in wall turbulence*

Self-similarity of the statistics of wall turbulence has been well-studied. The properties of the mean velocity profile and a description of the inertial (logarithmic) region are well-documented, while work in the last two decades has unraveled some of the complexities of the scaling of the Reynolds stresses and spectral representations of the fluctuations. An incomplete list of recent models for the scaling behavior for these flows include the Lie group symmetry analysis of Oberlack (1999), the work of She et al. (2017), and the mean momentum balance (MMB) approach of Klewicki et al. (2007). The attached eddy hypothesis (AEH), e.g. Perry & Chong (1982), and recent extensions (Marusic & Monty, 2019) provide a static structural analog replicating much of the observed behavior. Recently explored equation- and data-driven analysis techniques such as resolvent analysis (McKeon & Sharma, 2010; Hwang & Cossu, 2010; McKeon, 2017; Jovanovic, 2021) and spectral proper orthogonal decomposition (Towne et al, 2018) permit the investigation of instantaneous coherent structure.

I will discuss symmetry in the context of spatio-temporal coherent structure, as admitted by the Navier-Stokes equations and the ``anatomy” of the turbulent boundary layer, including some associated analytical developments with regards to the linear and nonlinear terms in the Navier-Stokes equations for flows over a range of Mach number and various boundary conditions.