DNS data for channel flow at Re τ = 1025 are used to analyse the interaction between large outer scales in the log-law region -referred to as super-streaks -and the small-scale, streaky, streamwisevelocity fluctuations in the viscosity-affected near-wall layer. The study is inspired by extensive experimental investigations by Mathis, Marusic and Hutchins, culminating in a predictive model that describes, in a supposedly universal manner, the "footprinting" and "modulating" effects of the outer structures on the small-scale near-wall motions. The approach used herein is based on the examination of joint PDFs for the small-scale fluctuations, conditioned on regions of large-scale footprints. The large and small scales are separated by means of the Huang-Hilbert empirical-mode decomposition, the validity of which is demonstrated by way of pre-multiplied energy spectra, correlation maps and energy profiles for both scales. Observations derived from the PDFs then form the basis of assessing the validity of the assumptions underlying the model. Although the present observations support some elements of the model, the results imply that modulation by negative and positive large-scale fluctuations differ greatly -an asymmetric response that is not compatible with the model. The study is thus extended to examining the validity of an alternative proposal, which is based on the assumption that a universal description of the small-scale response to the large-scale motions has to rely on the velocity fluctuations being scaled with the largescales-modified local friction velocity, rather than with the mean value. This proposal is partially supported by the present analysis. Finally, an alternative, new phenomenological model is proposed and examined.

A DNS-based study is presented, which focuses on the response of near-wall turbulence and skin friction to the imposition of an oscillatory spanwise wall motion in channel flow.One point of contrast to earlier studies is the relatively high Reynolds number of the flow, namely Re τ =1000. Another is the focus on transients in the drag that are in the form of moderate oscillatory variations in the skin friction and near-wall turbulence around the low-drag state at a sub-optimal actuation period. These conditions allow phase-averaged statistics to be extracted, during the periodic drag decline and rise, that shed light on the interaction between turbulence and the unsteady Stokes strain. Results are presented for, among others, the phase-averaged second-moments of stochastic fluctuations and their budgets, enstrophy components and joint PDFs. The study identifies velocity skewnessthe wall-normal derivative of the angle of the velocity vector -as playing a significant role † Email address for correspondence: l.agostini@imperial.ac.uk 2 L. Agostini, E. Touber and M. A. Leschzinerin the streak-damping process during the drag-reduction phase. Furthermore, the phasewise asymmetry in the skewness is identified as the source of a distinctive hysteresis in all properties, wherein the drag decline progresses over a longer proportion of the actuation cycle than the drag increase. This feature, coupled with the fact that the streakgeneration time scale limits the ability of the streaks to re-establish themselves during the low-skewness phase when the actuation period is sufficiently short, is proposed to drive the drag-reduction process. The observations in the study thus augment a previously identified mechanism proposed by two of the present authors, in which the drag-reduction process was linked to the rate-of-change in the Stokes strain in the upper region of the viscous sublayer where the streaks are strongest. Furthermore, an examination of the stochastic-stress budgets and the enstrophy lead to conclusions contrasting those recently proposed by other authors, according to which the drag-reduction process is linked to increases in enstrophy and turbulence-energy dissipation. It is shown, both for the transient drag-reduction phase and the periodic drag fluctuations around the lowdrag state, that the drag decrease/increase phases are correlated with decreases/increases in both enstrophy and dissipation.

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Data from a direct numerical simulation for channel flow at a friction Reynolds number of 1000 are analysed to derive statistical properties that offer insight into the mechanisms by which large-scale structures in the log-law region affect the small-scale turbulence field close to the wall and the statistical skin-friction properties. The data comprise full-volume velocity fields at 150 time levels separated by 50 wall-scaled viscous time units. The scales are separated into wavelength bands by means of the ‘empirical mode decomposition’, of which the two lowest modes are considered to represent the small scales and three upper modes to represent the large scales. Joint and conditional probability density functions are then derived for various scale-specific statistics, with particular emphasis placed on the streamwise and shear stresses conditional on the large-scale fluctuations of the skin friction, generally referred to as ‘footprinting’. Statistics for the small-scale stresses, conditional on the footprints, allow the amplification and attenuation of the small-scale skin friction, generally referred to as ‘modulation’, to be quantified in dependence on the footprints. The analysis leads to the conclusion that modulation does not reflect a direct interaction between small scales and large scales, but arises from variations in shear-induced production that arise from corresponding changes in the conditional velocity profile. This causal relationship also explains the wall-normal change in sign in the correlation between large scales and small scales at a wall-scaled wall distance of approximately 100. The effects of different scales on the skin friction are investigated by means of two identities that describe the relationship between the shear-stress components and the skin friction, one identity based on integral momentum and the other on energy production/dissipation. The two identities yield significant differences in the balance of scale-specific contributions, and the origins of these differences are discussed.