The relationship between amplitude modulation, coherent structure and critical layers in wall turbulence

The importance of critical layers in determining aspects of the structure of wall turbulence is discussed. We have shown (Jacobi \& McKeon, 2013) that the amplitude modulation coefficient investigated most recently by Hutchins \& Marusic (2007) and co-authors, which describes the correlation between large scales above a (spatial) wavelength filter with the envelope of small scales below the filter, is dominated by very large scale motion (VLSM) at a single wavelength. The resolvent analysis of McKeon \& Sharma (2010) gives a suitable model for the three-dimensional, three-component form of the VLSM and energetic structure at other wavelengths. This model is used to identify the three-dimensional spatial variation of instantaneous critical layers in the presence of a mean velocity profile and to relate this to earlier observations of coherent structure in unperturbed flows (both experimental and via the resolvent model, Sharma \& McKeon, 2013); to the phase relationships between scales identified by Chung \& McKeon (2010, 2014); and to the structure of wall turbulence that has been modified by the addition of single synthetic scales, e.g. Jacobi \& McKeon (2011), Duvvuri \& McKeon (2015).