Interactions between scales in wall turbulence: phase relationships, amplitude modulation and the importance of critical layers
We present a framework for predicting the interactions between motion at a single scale and the underlying stress fluctuations in wall turbulence, derived from approximations to the Navier–Stokes equations. The dynamical equations for an isolated scale and stress fluctuations at the same scale are obtained from a decomposition of the governing equations and formulated in terms of a transfer function between them. This transfer function is closely related to the direct correlation coefficient of Duvvuri & McKeon (J. Fluid Mech., vol. 767, 2015, R4), and approximately to the amplitude modulation coefficient described in Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337), by consideration of interactions between triadically consistent scales. In light of the agreement between analysis and observations, the modelling approach is extended to make predictions concerning the relationship between very-large motions and small-scale stress in the logarithmic region of the mean velocity. Consistent with experiments, the model predicts that the zero-crossing height of the amplitude modulation statistic coincides with the wall-normal location of the very large-scale peak in the one-dimensional premultiplied spectrum of streamwise velocity fluctuations, the critical layer location for the very large-scale motion. Implications of fixed phase relationships between small-scale stresses and larger isolated scales for closure schemes are briefly discussed.
© The Author(s), 2021. Published by Cambridge University Press. Received 20 March 2020; revised 23 July 2020; accepted 7 September 2020. Published online by Cambridge University Press: 05 March 2021. This research was carried out over a period of years. We gratefully acknowledge support from (D.C.) a National Aeronautics and Space Administration contract at the Jet Propulsion Laboratory, California Institute of Technology, and from (B.J.M.) the Air Force Office of Scientific Research (grants FA9550-09-1-0701 and FA9550-12-1-0469) and Office of Naval Research (grants N00014-17-1-2307 and N00014-17-1-3022). We also thank the anonymous referees for insightful suggestions. The authors report no conflict of interest.