The coupling between inner and outer scales in a zero pressure boundary layer evaluated using a Hölder exponent framework

This work considers the connectivity between large and small scales in boundary-layer turbulence by formalizing the modulation effect of the small scales by the large in terms of the pointwise Hölder condition for the small scales. We re-investigate a previously published dataset from this perspective and are able to characterize the coupling effectively using the (cross-)correlative relations between the large scale velocity and the small scale Hölder exponents. The nature of this coupling varies as a function of dimensionless distance from the wall based on inner-scaling, ${y}^{+}$, as well as on the boundary-layer height, δ. In terms of the fundamental change in the sign of the coupling between large and small scales, the critical height appears to be ${y}^{+}\sim 1000$. Below this height, small scale structures are associated with (and occur earlier than) maxima in the large scale velocity. Above this height, while the lag is similar in magnitude, the small scale structures are associated with minima in the large scale velocity. To consider these results further, we introduce a modified quadrant analysis and show that it is the coupling to the large scale low velocity state that is critical for the dynamics.