Investigation of static wings interacting with vertical gusts of indefinite length at low Reynolds numbers

Abstract

A novel gust generator was developed to study the interaction of wings with vertical gusts of controllable interaction time with magnitudes up to 30% of the freestream velocity at low Reynolds numbers. This work focuses on the collection of force and flow field data for three airfoils entering vertical gusts. Full-span NACA 0012, Eppler 387, and SD 5060 airfoils were tested with gusts of 20 and 30% of the freestream velocity at a Reynolds number of $12\times 10^{3}$ and 20% of the freestream at a Reynolds number of $54.4\times 10^{3}$. The lift generated during the gust interactions showed rapid increases in lift, often with a short-lived overshoot above the steady-state lift at the lower Reynolds number. Accompanying flow field data showed that the overshoot in lift was caused by a reattachment of the flow at the trailing edge of the symmetric NACA 0012, which was observed as a deflection of the streamlines in the freestream flow direction for the cambered airfoils. A new model was created to predict the lift during the interaction, which uses the static lift curve to predict the lift at a given effective flow angle, with corrections to account for the rate of change of the flow angle and camber of the airfoil. This model was able to more accurately predict the lift during gust interactions than the currently used modified Goman–Khrabrov model, while remaining a simple model to implement.

Appendices

Appendix A Unique stall instability of the Eppler 387 during gust encounters

Investigation of the collected lift force data for the Eppler 387 at two angles, $\alpha = 5^\circ$ and $10^\circ$, during the GR 0.2 interaction at Re = $54.4\times 10^{3}$ demonstrated interesting instabilities. The $\alpha =5^\circ$ case, Fig. 16a, appears to show the airfoil experiencing a stall-like drop in lift at random times within a window of $\approx \!200$ convective times, all after the steady-gust state has been reached. This instability was not seen in any other cases in this work. This data suggests that the airfoil was likely very close to the static stall angle during this time and thus a stall event was triggered by some instability in the flow, likely freestream turbulence. The $\alpha =10^\circ$ case, Fig. 16b, shows a bi-stability in the pre- and post-gust states. Likely because the model angle of $10^\circ$ is very close to the static stall angle for the Eppler 387 at this Reynolds number. This bi-stability serves as a good illustrator of the sensitivity of this airfoil to flow disturbances near stall.

Fig. 16

Examples of Eppler 387’s sensitivity to flow angle for GR 0.2 at Re = $54.4\times 10^{3}$ at model angles of a 5$^\circ$ and b 10$^\circ$. Note these $C_L$ values are calculated with $U_{\infty }$

Appendix B method for calculating uncertainty

The method for calculating the uncertainty of the measured force data was taken from Figliola and Beasley (2012). This method accounts for the random uncertainty, s, and bias uncertainty, b, inherent in data collection. The random uncertainty is a measure quantifying the variability of each input, whereas the bias uncertainty covers measurement accuracy and error. The equations used on the data from the force balance are given below, where Eqs. 6–8 are applied to the normal and axial force measurements. The outputs of Eqs. 7 and 8 are used in 10 and 11 with subscripts denoting the normal and axial directions. The random and bias uncertainty values used in the equations, along with how they were determined, are given in Table 3. Note that there is assumed to be no random uncertainty for air density, model planform area, or model angle as these values should not fluctuate. The measured static lift curves and the corresponding uncertainty values are tabulated in Tables 4 and 5.

$$\begin{aligned} C_F & = \frac{F}{\frac{1}{2}\rho |V |^2 S}\,, \quad where \; F = \left\{ \begin{array}{cc} N, &{} \; Normal \, Force \\ A, &{} \; Axial \, Force \end{array} \right. \end{aligned}$$

(6)

$$\begin{aligned} s_R &= \sqrt{\left[ \frac{\partial C_F}{\partial F} s_{F}\right] ^2 + \left[ \frac{\partial C_F}{\partial \rho } s_{\rho }\right] ^2 + \left[ \frac{\partial C_F}{\partial |V |} s_{|V |}\right] ^2 + \left[ \frac{\partial C_F}{\partial S} s_{S}\right] ^2} \end{aligned}$$

(7)

$$\begin{aligned} b_R &= \sqrt{\left[ \frac{\partial C_F}{\partial F} b_{F}\right] ^2 + \left[ \frac{\partial C_F}{\partial \rho } b_{\rho }\right] ^2 + \left[ \frac{\partial C_F}{\partial |V |} b_{|V |}\right] ^2 + \left[ \frac{\partial C_F}{\partial S} b_{S}\right] ^2} \end{aligned}$$

(8)

$$\begin{aligned} C_L= & {} C_N \cos {\alpha } + C_A \sin {\alpha } \end{aligned}$$

(9)

$$\begin{aligned} s_L & = \sqrt{\left[ \frac{\partial C_L}{\partial C_N} s_{R_N}\right] ^2 + \left[ \frac{\partial C_L}{\partial C_A} s_{R_A}\right] ^2 + \left[ \frac{\partial C_L}{\partial \alpha } s_{\alpha }\right] ^2 } \end{aligned}$$

(10)

$$\begin{aligned} b_L& = \sqrt{\left[ \frac{\partial C_L}{\partial C_N} b_{R_N}\right] ^2 + \left[ \frac{\partial C_L}{\partial C_A} b_{R_A}\right] ^2 + \left[ \frac{\partial C_L}{\partial \alpha } b_{\alpha }\right] ^2 } \end{aligned}$$

(11)

$$\begin{aligned} u_{C_L} & = \pm \sqrt{s_{L}^2 + b_{L}^2} \end{aligned}$$

(12)

Table 3 Uncertainty calculation inputs

Table 4 Tabulated static lift coefficients at Re=$12\times 10^{3}$

Table 5 Tabulated static lift coefficients at Re=$54.4\times 10^{3}$