Spatial averaging effects on the streamwise and wall-normal velocity measurements in a wall-bounded turbulence using a cross-wire probe

Abstract

The spatial averaging effects due to a cross-wire probe on the measured turbulence statistics in a wall-bounded flow are investigated using a combined approach of Direct Numerical Simulation data, theoretical methods and experiments. In particular, the wire length (l), spacing (Δs<sub>y</sub>) and angle (θ) of a cross-wire probe configured to measure the streamwise and wall-normal velocities are systematically varied to isolate effects of each parameter. The measured streamwise velocity from a cross-wire probe is found to be an average of the filtered velocities sensed by the two wires. Thus, in general, an increase in the sensor dimensions when normalised by viscous units leads to an attenuated variance for the streamwise velocity (〈u<sup>2</sup>〉), resulting from a larger contribution to the spatial averaging process from poorly correlated velocities. In contrast, the variance for the wall-normal velocity (〈w<sup>2</sup>〉) can be amplified, and this is shown to be the result of an additional contributing term (compared to 〈u<sup>2</sup>〉) due to differences in the filtered wire-normal velocity between the two wires. This additional term leads to a spurious wallnormal velocity signal, resulting in an amplified variance recorded by the cross-wire probe. Compared to the streamwise and wall-normal velocity variances, the Reynolds shear stress (〈-uw〉) perhaps surprisingly shows less variation when l, Δs<sub>y</sub> and θ are varied. The robustness of Reynolds shear stress to the finite sensor size is due to two effects. (I): Reynolds shear stress is devoid of energetic contributions from the near-isotropic fine scales unlike the〈u<sup>2</sup>〉and〈w<sup>2</sup>〉statistics, hence cross-wire probe dimensions are typically sufficiently small in terms of viscous unit to adequately capture the uw statistics for a range of l and θ investigated. (II) The Δs<sub>y</sub> dependency arises due to cross terms between the filtered velocities from two wires, however, it turns out that these terms cancel one another in the case of Reynolds shear stress, but not for the〈u<sup>2</sup>〉and〈w<sup>2</sup>〉statistics