Small-scale Phase Organization through Large-scale Inputs in a Turbulent Boundary Layer

A synthetic large-scale motion is excited in a flat plate turbulent boundary layer experiment and its influence on small-scale turbulence is studied. The synthetic scale is seen to alter the average natural phase relationships in a quasi-deterministic manner, and exhibit a phase-organizing influence on the directly coupled small-scales. The results and analysis presented here are of interest from a scientific perspective, and also suggest the possibility of engineering schemes for favorable manipulation of energetic small-scale turbulence through practical large-scale inputs. BACKGROUND AND INTRODUCTION Phase relationships in turbulent shear flows have been an area of research interest since the pioneering works of Brown & Thomas (1977) and Bandyopadhyay & Hussain (1984) on the correlations between large and small scales. This area has seen a recent surge in activity with several investigations, experimental and numerical, over the past two decades confirming the presence of very-large-scale motions (VLSM) in wall-bounded turbulent flows (see Smits et al. 2011 and reference therein). Of particular interest, from a scientific and an engineering perspective, is the influence of large scales on small-scale turbulence through nonlinear coupling. The experimental study of Rao et al. (1971) showing the outer scaling of turbulent bursts in the inner region of a boundary layer provided the first clear evidence of such scale coupling, and emphasized the significance of inner-outer interactions in wall turbulence. More recently, through a careful analysis of highReynolds number boundary layer data, Hutchins & Marusic (2007) suggested an amplitude modulation influence on near-wall small-scale turbulence by large-scale motions centered in the log region. The modulation effect was later quantified by Mathis et al. (2009) through a demodulation scheme in which a correlation coefficient (termed amplitude modulation coefficient R) between the large-scale velocity signal and an envelope of the small-scale velocity signal from a turbulent boundary layer was taken to be a measure of amplitude modulation. Jacobi & McKeon (2013) using a co-spectral technique demonstrated that the strongest modulating influence in the large-scale signal comes from a wavenumber that matches the VLSM. Mathis et al. (2009) also noted an interesting similarity between the behavior of the amplitude modulation coefficient and skewness (S) of the turbulence signal with wall-normal distance. Schlatter & Örlu (2010) suggested that the amplitude modulation coefficient is to a large extent another representation of a cross term in the scale-decomposed skewness factor. A clear connection was established empirically by Mathis et al. (2011) between the cross term 3uLuS of the skewness, obtained by a scale-decomposition of the velocity signal u into largeand small-scale components (u “ uL ` uS), and the amplitude modulation coefficient across a range of Reynolds numbers in a turbulent boundary layer. Bernardini & Pirozzoli (2011) studied two-point velocity correlations obtained from DNS data of a compressible turbulent boundary layer to show clear evidence of top-down influence of large-scale outer events on the small-scales in the inner part of the boundary layer, and the same was interpreted as amplitude modulation. Chung & McKeon (2010) note that the amplitude modulation coefficient can also be interpreted as a phase relationship between the large scales and the smallscale envelope. It is clear from the correlation studies thus far that a definitive phase relationship exists between largeand small-scale activity in wall-bounded turbulent flows. A formal relationship between the amplitude modulation coefficient and the skewness for a general statistically stationary signal was established recently by the authors (Duvvuri & McKeon 2015, henceforth referred to as DM15) through a simple multi-scale analysis. Both the quantities were shown to be fundamentally a measure of phase in interactions between triadically consistent scales, i.e. sets of wavenumbers tkl ,km,knu such that kl “ kn ́ km. Note that triadic interactions assume physical significance given the quadratic nature of non-linearity that governs coupling between scales. More interestingly, DM15 describes an experiment in which a synthetic large-scale motion was excited in a turbulent boundary layer to generalize and study the influence of large-scale motions on small-scale turbulence. It was shown that the naturally existing phase relationships in the flow can be manipulated in a quasi-deterministic manner. The influence of the synthetic scale is felt strongly by directly coupled pairs of small-scale wavenumbers; a clear phase-organization is seen among the triadically coupled