Leading-edge flow reattachment and the lateral static stability of low-aspect-ratio rectangular wings

The relationship between lateral static stability derivative, $C_{l_{\beta }}$, lift coefficient, $C_L$, and angle of attack was investigated for rectangular wings of aspect ratio $A\mathrm{R}$ $=0.75,1,1.5,$ and 3 using Stereo-Digital Particle Image Velocimetry (S-DPIV) and direct force and moment measurements. When the product $C_{l_{\beta }}A\mathrm{R}$ is plotted with respect to $C_L$, the lateral stability curves of each wing collapse to a single line for $C_L<0.7$. For $C_L>0.7$, the linearity and scaling of $C_{l_{\beta }}$ with respect to $C_L$ is lost. S-DPIV is used to elucidate the flow physics in this nonlinear regime. At $\alpha ={10}^{\circ }$, the leading-edge separation region emerges on the leeward portion of the sideslipped wing by means of vortex shedding. For the $A\mathrm{R}$ $\le 1.5$ wings at $\alpha >{15}^{\circ }$, the tip vortex downwash is sufficient to restrict the shedding of leading-edge vorticity thereby sustaining the lift of the leading-edge separation region at high angles of attack. Concurrently, the windward tip vortex grows in size and strength with increasing angle of attack, displacing the leading-edge separation region further toward the leeward wing. This reorganization of lift-generating vorticity results in the initial nonlinearities between $C_{l_{\beta }}$ and $C_L$ at angles of attack for which $C_L$ is still increasing. At angles of attack near that of maximum lift for the $A\mathrm{R}$ $\le 1$ wings, the windward tip vortex lifts off the wing, decreasing the lateral static stability of the wing prior to lift stall. For the $A\mathrm{R}=3$ wing at $\alpha >{10}^{\circ }$, nonlinear trends in $C_{l_{\beta }}$ versus $C_L$ occur due to the spanwise evolution of stalled flow.

Figure 1

(Left) Roll moment and wind angle definitions for a rectangular wing of aspect ratio $A\mathrm{R}\equiv b^2/S=b/c$, where $b$ and $c$ are the span and chord, respectively. Roll moment, $l$, is defined positive clockwise about the vector ${\widehat{x}}_b$, where ${\widehat{x}}_b$ is fixed to the body and directed out the leading edge, parallel to the wing chord. (Right) The negative of the product of lateral static stability derivative and wing aspect ratio, $-C_{l_{\beta }}AR$, versus the lift coefficient, $C_L$, for rectangular flat-plate wings of various aspect ratio.

Figure 2

S-DPIV setup in the wind tunnel test section (left) and a close-up of the model adapter holding the $A\mathrm{R}=1$ wing (right). The laser sheet for both streamwise and cross-stream measurements are marked blue and green, respectively. The model adapter suspends each model upside down in the test section from two rods that are fastened to the pressure surface of the inclined wing.

Figure 3

Measurement planes and coordinate system for S-DPIV flow measurements. Coordinate vectors are defined with respect to the freestream: $\widehat{x}$ is defined positive downstream, $\widehat{z}$ positive toward zenith, and $\widehat{y}$ orthogonal to the freestream by $\widehat{z}\times \widehat{x}$. The coordinate system origin is fixed to the leeward corner of the sideslipped wing unless otherwise specified. Note that for $\beta =0^{\circ }$, the cross-stream measurement plane is coincident with the full trailing edge of the wing, and the streamwise measurement plane is oriented parallel to the chord.

Figure 4

Tip vortex (a) circulation, (b) horizontal spacing, and (c) vertical location each as a function angle of attack and wing aspect ratio, $\beta =0^{\circ }$. The measurement plane is at the trailing edge of the wing and is orientated normal to the freestream. The root-mean-square error of the instantaneous vortex parameters is plotted as error bars for the $A\mathrm{R}=1$ wing. The vortex properties are not plotted for cases where stalled flow is evinced in the measurement plane.

Figure 5

Time-averaged spanwise vorticity contours as a function of angle of attack, $\beta =0^{\circ }$. (Insets) Time-averaged streamwise vorticity at the trailing edge in a plane normal to freestream. (a) Lift ratio, $C_L/C_{L_{\alpha }}$, as a function of angle of attack. $C_{L_{\alpha }}$ is Helmbold's theoretical lift curve slope [38].

Figure 6

Circulation associated with time-averaged positive and negative spanwise vorticity, ${\mathrm{\Gamma }}_y^{+}$ and ${\mathrm{\Gamma }}_y^{-}$, as a function of spanwise location and angle of attack for (a)–(d) $A\mathrm{R}=0.75$–3 rectangular wings at $\beta =0^{\circ }$. A surface plot is superimposed to highlight low magnitudes of ${\mathrm{\Gamma }}_y^{-}$. Red markers indicate local maxima.

Figure 7

TKE with superimposed time-averaged velocity vectors in a measurement plane orientated normal to the freestream at a streamwise distance coincident with the trailing edge of inclined rectangular flat plate wings of various aspect ratio, $\beta =0^{\circ }$. Blue dots mark the spanwise location at which streamwise flow-field measurements were taken.

Figure 8

Snapshots of instantaneous vorticity at (a) $y/b=0$, (b) $y/b=0.15$, and (c) $y/b=0.30$ on an $A\mathrm{R}=1$ wing at $\alpha ={20}^{\circ }$ at zero sideslip. $y/b$ is measured with respect to the midspan of the wing. The time between snapshots is $t^{*}=T^{*}/6$ where $T^{*}=T{U}_{\infty }/c=2.13$ is the nondimensional shedding period at $y/b=0$.

Figure 9

Vorticity isosurfaces of $A\mathrm{R}=1$ wing at (a) $\beta =0^{\circ }$ and (b) $\beta =-{10}^{\circ },\alpha ={35}^{\circ }$. Replotted from DeVoria and Mohseni [25]. The spanwise- and streamwise-orientated vortex structures are marked green and red/blue, respectively.

Figure 10

Tip vortex (a) circulation and (b) vertical location as a function angle of attack and wing aspect ratio, $\beta =-{10}^{\circ }$. The measurement plane is orientated normal to the freestream at a streamwise distance coincident with the leeward trailing corner. The vortex properties are not plotted for cases where stalled flow is evinced in the measurement plane.

Figure 11

Time-averaged cross-stream vorticity contours as a function of angle of attack, $\beta =-{10}^{\circ }$. (Insets) Time-averaged streamwise vorticity in a plane normal to freestream at streamwise location marked by the dotted line. (a) Lift (dashed markers), $C_L$, and roll moment (markers only), $C_l$, versus angle of attack, $\beta =-{10}^{\circ }$. $C_L$ at $\beta =0^{\circ }$ (solid line).

Figure 12

Circulation associated with time-averaged positive and negative cross-stream vorticity, ${\mathrm{\Gamma }}_y^{+}$ and ${\mathrm{\Gamma }}_y^{-}$, as a function of cross-stream location and angle of attack for (a)–(d) $A\mathrm{R}=0.75$–3 rectangular wings at $\beta =-{10}^{\circ }$. A surface plot is superimposed to highlight low magnitudes of ${\mathrm{\Gamma }}_y^{-}$. Red markers indicate local maxima.

Figure 13

TKE with superimposed time-averaged velocity vectors in a measurement plane orientated normal to the freestream at a streamwise distance coincident with the leeward trailing edge of inclined rectangular flat plate wings of various aspect ratio in sideslip, $\beta =-{10}^{\circ }$. Blue dots mark the spanwise location at which streamwise flow-field measurements were taken.

Figure 14

Snapshots of instantaneous vorticity at (a) $y/b=-0.25$ and (b) $y/b=-0.55$ on an $A\mathrm{R}=1$ wing at $\alpha ={20}^{\circ }$ at $\beta =-{10}^{\circ }$ with $y/b$ measured with respect to the origin at the leeward trailing edge. The time between snapshots is $t^{*}=T^{*}/6$ where $T^{*}=T{U}_{\infty }/c=2.13$ is the nondimensional shedding period at the midspan of the $A\mathrm{R}=1$ wing at $\alpha ={20}^{\circ },\beta =0^{\circ }$.

Figure 15

The negative of the product of lateral static stability derivative and wing aspect ratio, $-C_{l_{\beta }}AR$, versus the lift coefficient of the wing at zero sideslip, $C_L$. Roman numerals, I, IIa, IIb, and III, mark the approximate lift coefficients corresponding to specific angles of attack for the $A\mathrm{R}=0.75$ (dotted), 1 (solid), 1.5 (dashed), and 3 (dotted-dashed) wings.