Spatial averaging of velocity measurements in wall-bounded turbulence: single hot-wires

Recent advancements in velocity measurements to understand high Reynolds number (Re) wall turbulence have pushed the boundaries of sensor size required to resolve the smallest scales. We present here a framework for studying the effect of finite sensor size on velocity measurements, and scrutinize in detail the behaviour of single-wire hot-wires. Starting with a general linear filter, expressions for the filtered correlation, spectrum and the corresponding variance are derived. Considering the special case of a box-type filter and a simple model for the two-point correlation, theoretical results are developed, which are favourably compared with the numerical simulation of hot-wires based on the turbulent channel flow direct numerical simulation databases. The results clarify the reason why previous studies found the approximate shape of the spectra not resolved by hot-wires as Gaussian. The length scale based on the correlation over the sensor length is found to be the appropriate length scale for characterizing averaging due to finite sensor size. The efficacy of the linear box filter is established by comparing the numerical simulation of hot-wires with experiments conducted at matched sensor lengths and Re in a channel flow, at least for hot-wire lengths of less than 40 in viscous scaling. Finally, a model of the streamwise two-point correlation is presented, which is employed to estimate the filtering effect on the peak of the streamwise velocity variances for a range of Re, and the model results compare favourably with that obtained from measurements. Even though the theoretical results are compared here in the case of wall turbulence, they are suitable for hot-wire measurements in turbulent flows in general.