Triadic scale interactions in a turbulent boundary layer

Abstract

A formal relationship between the skewness and the correlation coefficient of large and small scales, termed the amplitude modulation coefficient, is established for a general statistically stationary signal and is analysed in the context of a turbulent velocity signal. Both the quantities are seen to be measures of phase in triadically consistent interactions between scales of turbulence. The naturally existing phase relationships between large and small scales in a turbulent boundary layer are then manipulated by exciting a synthetic large-scale motion in the flow using a spatially impulsive dynamic wall roughness perturbation. The synthetic scale is seen to alter the phase relationships, or the degree of modulation, in a quasi-deterministic manner by exhibiting a phase-organizing influence on the small scales. The results presented provide encouragement for the development of a practical framework for favourable manipulation of energetic small-scale turbulence through large-scale inputs in a wall-bounded turbulent flow.

Phase relations in a forced turbulent boundary layer: implications for modelling of high Reynolds number wall turbulence

“…The full amplitude modulation coefficient was investigated in Duvvuri & McKeon [14] and shown to deviate from the variation in the unperturbed boundary layer only in the wall-normal region where the synthetic scale was energetic, consistent with triadic interaction arguments made therein. The phase relationship between a single synthetic large scale and the envelope of the small scales has been previously identified [13,14].…”

“…Recollect that external forcing was applied in a spatially impulsive manner, and hence the disturbance decays with streamwise distance downstream of the perturbation. The complex components of the streamwise wavenumbers k x associated with exponential decay can be easily calculated (see [19]); however, for clarity of the presentation the streamwise decay in mode amplitudes has not been applied to the spatial mode shapes. Clearly, the directly excited modes, u 1 andũ 2 , are energetically dominant, but the modes generated by nonlinear interaction of these modes interrupt the otherwise somewhat clean beating between them.…”

“…The phase relationship between a single synthetic large scale and the envelope of the small scales has been previously identified [13,14]. A correlation coefficient, Ψ , was introduced to quantify this relationship [14,15], which is effectively the amplitude modulation coefficient associated with a large-scale signal consisting only of the synthetic scale…”

“…Further, the resolvent formulation of equation (3.2) offers a rigorous analytical basis by which to analyse these interactions. Preliminary studies have shown that the synthetic modes may be reasonably approximated by passing the stress, R jk-i , calculated from experiments through the resolvent operator, or by a low-order assembly of the singular functions determined from a singular value decomposition of the resolvent [13,19]. Work on these developments is ongoing.…”

“…This experimental technique was adapted by the authors in previous work [14] to effect targeted forcing in a turbulent boundary layer at a single large spatio-temporal scale and study the modified triadic phase relations in the flow. A striking phase-locking effect was observed among triadic small scales directly coupled to the synthetic large scale, revealing organization of a significant degree built into the dynamics of the governing equations [14,15]. An extension of the wall perturbation technique is implemented to simultaneously force two distinct large-scale modes in the flow and highlight their nonlinear interactions.…”