Temporally optimized spanwise vorticity sensor measurements in turbulent boundary layers

Abstract

Multi-element hot-wire anemometry was used to measure spanwise vorticity fluctuations in turbulent boundary layers. Smooth wall boundary layer profiles, with very good spatial and temporal resolution, were acquired over a Kármán number range of 2000–12,700 at the Melbourne Wind Tunnel at the University of Melbourne and the University of New Hampshire’s Flow Physics Facility. A custom hot-wire probe was necessary to simultaneously obtain velocity and spanwise vorticity measurements centered at a fixed point in space. A custom calibration/processing scheme was developed to utilize single-wall-parallel wires to optimize the accuracy of the measured wall-normal velocity fluctuations derived from the sensor’s $\times$-array.

Appendix: Comparison of \(\times\)-array data processing methods

Sensor drift is a major challenge when processing $\times$-array data. In most methods, such as look-up tables (LUT) the processing scheme depends on a unique relationship between the wire voltage. If wire drift is experienced, this relationship becomes unknown, and in general is impossible to rectify without performing a new calibration. Conversely, for a single wire normal to the flow, drift can be accounted for if the drift is in the form of a DC-offset (${\Delta } E$-drift)—see Talluru et al. (2014). To account for sensor drift in $\times$-arrays, an interpolation between the pre- and post-calibration is usually preformed (commonly linear with time or based on the temperature history). Inherently, this assumes the rate of response change for each wire. In thermally stable environments, sensors usually experience response changes differently making a simple interpolation a poor estimate of the unique relationship between the wires. In the present processing method, the $\times$-array sensors are considered separately and a relationship between the sensors is never needed. Since the processing scheme only uses voltage fluctuations ${\Delta } E$-drift is not present; i.e., the drift is removed as long as it is treated as a voltage shift. In fact, only profiles that were well-modeled by ${\Delta } E$-drift were considered. Therefore, the only drift that needs to be considered is that of the single parallel wires.

Fig. 12

Inner-normalized wall-normal velocity variance profile. Symbols are given in Table 1, and when a symbol is gray, it was processed with the LUT method. The black comparison profile is Sillero et al. (2013) DNS $\delta ^+ \approx 2000$

Fig. 13

Inner-normalized Reynolds stress profile. Symbols are given in Table 1, and when a symbol is gray it was processed with the LUT method. The black comparison profile is Sillero et al. (2013) DNS $\delta ^+ \approx 2000$

The LUT method is commonly found in the literature as a processing method for $\times$-arrays (Chew and Ha 1990; Bakken and Krogstad 2004; Burattini and Antonia 2005; Ebner 2014). The $\delta ^+ \simeq 1,951$ and $\delta ^+ \simeq 7,894$ $\times$-array data were processed with the present method and the LUT. For this comparison, the LUT surface was generated from the calibration using a modified ridge estimator (“gridfit” command in Matlab), and a linear interpolation was performed between the pre- and post-calibration. Figures 12 and 13 compare the wall-normal velocity variance and the Reynolds stress profiles. It is clear from both figures that profiles processed with the LUT show less consistent behavior between the two Reynolds numbers considered. Also, the LUT profile at $\delta ^+ \simeq 1951$ does not show the same level of agreement with the DNS as the present method. Peel-up is present in both methods suggesting that this phenomena is not a result of the processing scheme. These results provide evidence that the present method is more robust for these statistics. Note that the r.m.s. of vorticity was not considered here, since only a small percentage of its magnitude can be attributed to $\partial v/\partial x$ and becomes less so with increasing wall-normal position.