Abstract
In order to predict the axial development of the wingtip vortices strength, an accurate theoretical model is required. Several experimental techniques have been used to that end, e.g. PIV or hot-wire anemometry, but they imply a significant cost and effort. For this reason, we have performed experiments using the smoke-wire technique to visualize smoke streaks in six planes perpendicular to the main stream flow direction. Using this visualization technique, we obtained quantitative information regarding the vortex velocity field by means of Batchelor’s model for two chord-based Reynolds numbers, \(Re_c=3.33\times 10^4\) and \(10^5\). Therefore, this theoretical vortex model has been introduced in the integration of ordinary differential equations which describe the temporal evolution of streak lines as function of two parameters: the swirl number, S, and the virtual axial origin, \(\overline{z_0}\). We have applied two different procedures to minimize the distance between experimental and theoretical flow patterns: individual curve fitting at six different control planes in the streamwise direction and the global curve fitting which corresponds to all the control planes simultaneously. Both sets of results have been compared with those provided by del Pino et al. (Phys Fluids 23(013):602, 2011b. doi:10.1063/1.3537791), finding good agreement. Finally, we have observed a weak influence of the Reynolds number on the values S and \(\overline{z_0}\) at low-to-moderate \(Re_c\). This experimental technique is proposed as a low cost alternative to characterize wingtip vortices based on flow visualizations.
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Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments which helped to improve the manuscript. Also, the authors would like to thank Sergio Pinazo his technical support, and for doing the CAD of Fig. 1. This work has been supported by the Junta de Andalucía (Spain) Grant No. P11-TEP-7776.
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Appendix: Sensitivity to optimization parameters
Appendix: Sensitivity to optimization parameters
In this appendix, we study the influence of different parameters which are present in the optimization procedure. Recall that we solve Eq. 9 to find four parameters, S, \(\overline{z}_0\), \(x_0\) and \(y_0\). We impose two different numbers of points: N theoretical points to integrate the velocity field of the vortex and \(N_\text{exp}\) experimental points which describe the spiral, as shown in Figs. 1 and 3. We check the effect of the variation of these parameters for the case of \(Re=10^5\) and \(z=3c\). In Fig. 9a, we show the evolution of the two main parameters obtained from the optimization for \(N_{\mathrm{exp}}=107\), and a different number of theoretical points. It can be observed that there are small variations for a number of theoretical points \({>}250\), approximately. For this reason, we establish \(N=300\), so the variations in the parameter S are lower than 2.5 % with respect to the mean value for any value of N in the range 250–650. More fluctuations are observed in the case of \(\overline{z}_0\), though they are considered acceptable. In Fig. 9b, we show the effect of \(N_{\mathrm{exp}}\) using the same image, and \(N=300\). It is clear again that \(N_{\mathrm{exp}}\) has a weak influence on the variation of the main parameters in the optimization procedure, whenever this number is approximately \({>}100\). For example, if one compares the value of S with respect to the mean value for any \(N_{\mathrm{exp}}\), the error is lower than 3.8 %, which is considered very small. We set \(N_{\mathrm{exp}}=107\) in our computations.
Finally, we have modified the length of the numerical line used as a tracer (see the line from where streak lines departure in Fig. 2). We show the results in Fig. 10, and it can be observed that for lengths \({>}0.07\), the solution does not change significantly. We consider \(l_{\mathrm{tracer}}=0.1\) m (or \(\phi =l_{\mathrm{tracer}}/0.1=1\)) in our computations.
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Serrano-Aguilera, J.J., García-Ortiz, J.H., Gallardo-Claros, A. et al. Experimental characterization of wingtip vortices in the near field using smoke flow visualizations. Exp Fluids 57, 137 (2016). https://doi.org/10.1007/s00348-016-2222-9
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DOI: https://doi.org/10.1007/s00348-016-2222-9