# Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number

^{*}

## Abstract

**:**

^{4}. Direct force/moment measurements and surface oil-flow visualizations were conducted in the wind-tunnel B at the Technical University of Munich. It was found that while the maximum lift at an aspect ratio of 2.03 remains unchanged, two other aspect ratios of 3.13 and 4.50 show a gradual increment in the maximum lift with an increasing sweptback angle. The largest leading-edge vortex contribution was found at the aspect ratio of 3.13, resulting in a superior lift production at a sufficient sweptback angle. This is similar to that of a revolving/flapping wing, where an aspect ratio around three shows a superior lift production. In the oil-flow patterns, it was observed that while the leading-edge vortices at aspect ratios of 2.03 and 3.13 fully covered the surfaces, the vortex at an aspect ratio of 4.50 only covered up the surface approximately three times the chord, similar to that of a revolving/flapping wing. Based on the pattern at the aspect ratio of 4.50, a critical length of the leading-edge vortex of a sweptback plate was measured as ~3.1 times the chord.

## 1. Introduction

^{4}were conducted. Three ARs of 2.03, 3.13, and 4.50, which cover the critical AR for the LEV of a revolving/flapping wing [15,16,17,18], were selected, and a sweptback angle was used to supply a spanwise flow. We found a resemblance in aerodynamic characteristics to that of a revolving/flapping wing, which is presumably due to an analogous behavior of the two LEVs. This suggests the applicability of an LEV of biological flyers for state-of-the-art aerial vehicle design.

## 2. Materials and Methods

^{®}) were mounted on the arm, and each encoder inside the motors provides an angular position with a resolution of 0.088 deg/step. The motors are in charge of the α, β, and a sweeping angle (fixed in this study), respectively, and we used two motors, Motor 1 and Motor 2, to adjust the two angles. This robotic arm was connected to a laptop computer, and was controlled by an in-house code written in LabVIEW™ via RS485 protocol. The baud rate was 2 Mbps, which was sufficient for high-speed motor control.

^{3}and 1.53 ± 0.9 × 10

^{−5}m

^{2}/s, respectively. The dynamic pressure was settled in the range of 26.4 ± 0.4 Pa, the freestream velocity was yielded as 6.82 ± 0.07 m/s; stall flutter appeared at AR = 4.50 beyond this speed. The corresponding Reynolds number based on the chord length is Re ≈ 2.85 × 10

^{4}.

_{1}and M

_{2}collected by each set of the straingauges can be expressed as:

^{−4}. Based on the maximum input voltages of the DAQ of ±5 V, the maximum range of the force and moment are ±4 N and ±120 N-mm, respectively. The loadcell was then mounted in between the end of the robotic arm and the model plates, and was connected to a straingauge amplifier (KWS 6A-5, HBM), a DAQ board (USB-6212, NI), and a PC (Figure 1b).

_{L}here is based on the mean C

_{L}in each case. The ΔC

_{L}was sufficiently converged over 7000 samples in all cases.

## 3. Results and Discussion

^{4}. Figure 3 shows lift and drag coefficients C

_{L}and C

_{D}for the three different ARs at β = 0 deg. One straight line on the left side indicates a theoretical value of a two-dimensional inviscid flow, i.e., C

_{Lα}= 2π. The two dashed lines following the slopes of the C

_{L}–α curves are the results of the lifting-line theory for a finite wing shown in Equation (4), where a

_{0}and τ denote the C

_{Lα}and shape factor, respectively (refer to [26] for detail). It was found that τ ≈ 0.5, which is pretty larger than the typical range of 0.05 to 0.25, showed a sufficient agreement with the C

_{Lα}in this study. This is one typical characteristic for low Reynolds number flow [27].

_{L}–α curves, such as linear increments in the pre-stall region and gradual reductions after the stall angles, are completely in line with the previous study on LAR wings at low Reynolds number [5]. No stall peaks in the C

_{L}–α curves also imply a laminar-dominant flow, which was insufficient to overcome the adverse pressure on the suction side of the model plates (refer to [27] for the disappearance of the stall peak as Re decreases). Decreasing AR relieved the C

_{Lα}in the pre-stall region (arrow (1) in Figure 3), further suggesting a gradual growth in an encroachment of the TiVs into the inboard region as the previous studies have shown [1,3].

_{L}(arrow (2) in Figure 3; 15 < α < 25 deg). This is in line with one recent study [6], which revealed that TiVs of a lower AR wing suppress the vortex shedding and assist in establishing the Kutta condition around the midspan, thereby enhancing the lift. This also explains the slight bump in the C

_{D}at AR = 2.03 (α ≈ 20 deg), because such TiV effect would practically augment the net force acting on the surface. No boost or bump in the C

_{D}at AR = 4.5 further suggests that such an aerodynamic benefit from the TiV may be only effective at AR < 4.5.

_{L}–α and C

_{D}–α curves with respect to β. The dashed straight lines in each graph are the results of the lifting-line theory with τ = 0.5. In all cases, increasing β delayed growth of C

_{L}and C

_{D}(arrows (1) and (2) in Figure 4a). This is a typical feature of a sweptback plate as an increase in β gradually reduces the inflow velocity perpendicular to the wingspan as U

_{inflow}= U

_{∞}cosβ (as discussed later). The stall angles were gradually delayed (arrow (3) in Figure 4), also in line with previous studies on a sweptback angle (refer to [2] as an example).

_{L}at AR = 2.03 shows the prominent peaks at each stall α (Figure 4a). This feature, usually appears at a sufficiently high Reynolds number [27], is only found at this AR at β ≥ 24 deg. The maximum lift coefficient C

_{L,max}depending on β was also noticeable. At AR = 2.03, the C

_{L,max}was obtained at β = 0 deg already (arrow (3) in Figure 4a). This implies that supplying a spanwise flow and developing an LEV by using β may not be effective to enhance a lift force, at least for this AR. In other words, wings/plates with an AR of approximately two or below may be not able to produce an additional lift force supported by an LEV. At AR = 3.13 and 4.50, on the other hand, the C

_{L,max}was gradually improved with increasing β (arrow (4) in Figure 4b). The maximum value of C

_{L,max}appeared at β ≈ 30 deg for both cases. The rate of growth in C

_{L,max}was more conspicuous at AR = 3.13, implying that C

_{L,max}does not have simple monotonic relation with AR. This is similar to previous studies on a revolving/flapping wing (refer to [11] as an example).

_{L}–α curves in the pre-stall region, i.e., C

_{Lα}. These were extracted at α < 9 deg where the curves have a linear slope. The Helmbold equation, which is known to provide a closer fit at AR < 4 [26], are also added in Figure 5. The Helmbold equation is,

_{Lα}of the lifting-line theory is still higher than the measurement. The Helmbold equation gave a better fit than that of the theory with τ = 0.25, but also overestimates the C

_{Lα}. As Pelletier and Mueller [28] pointed out, these would be stemmed from the lower a

_{0}at the low Reynolds number, which had been assumed as 2π. The higher AR resulted in higher C

_{Lα}at each β. Increasing β resulted in a gradual reduction in C

_{Lα}. These were steadily converged to C

_{Lα}≈ 1.0 at β = 54 deg.

_{L,max}with respect to β. In addition, the chordwise component of C

_{L,max}at AR = 2.03, i.e., C

_{L,max@β=0}cos

^{2}β is given as the black dashed line (as discussed later). The C

_{L,max}at AR = 2.03 remained nearly unchanged at β < 36 deg. At AR = 3.13 and AR = 4.50, on the other hand, increasing β resulted in a gradual increment in the C

_{L,max}. This effect continued until the β reached 42 deg. The model plate of AR = 4.50 showed a higher rate of increment in the C

_{L,max}, but the largest C

_{L,max}appeared at AR = 3.13 with β = 36 deg. The AR = 3.13 also showed superior lift production at β > 36 deg.

_{L,max}at β = 0 deg can be the maximum value without support of the LEV, because there is no spanwise flow at β = 0 deg. The portion of this value in the other β cases becomes then C

_{L,max@β=0}cos

^{2}β as shown in the black dashed line in Figure 6a, because a chordwise component of a freestream is U

_{∞}cosβ. This approach, dividing a lift force into two components, obviously cannot decompose effect of TiVs, wake, or their interaction in practice. Other aerodynamic effects at β = 0 deg, such as the boost and bump of the C

_{L}as shown in Figure 3, also could be blended in this approach.

_{L,LEV}, which is the remaining part of the C

_{L,max}after extracting the C

_{L,max@β=0}cos

^{2}β. At β < 30 deg, the C

_{L,LEV}showed a rapid increment with increasing β. The increment in the C

_{L,LEV}even appeared at AR = 2.03, implying that the plateau of the C

_{L,max}at AR = 2.30 at β < 30 deg in Figure 6a originally resulted from the LEV that compensated a gradual loss in the lift force with increasing β (refer to the black dashed line in Figure 6a for the loss in the lift force). At β > 36 deg, The C

_{L,LEV}of the AR = 3.13 plate resulted in higher values. This suggests that an LEV would be most effective at an AR of approximately three with a sufficient sweptback angle. All the C

_{L,LEV}then decreased at β > 45 deg, where the spanwise component of the freestream is larger than the chordwise one. The C

_{L,LEV}gradually reduced from this point; the C

_{L,LEV}will eventually approach zero with increasing β as the frontal area of the model plates approach zero.

_{L,max}at AR = 2.03 was achieved by the two strong TiVs at β = 0 deg already. Developing an LEV with increasing β at AR = 2.03, therefore, could not be effective to obtain an additional lift, because the Kutta condition at the trailing edge, which is expected as a result of the LEV attachment, was already achieved by the two TiVs (refer to Kweon and Choi [29] for the LEV system of a flapping plate in detail). This suggests that from this AR, supplying a spanwise flow using a sweptback angle for a lower AR plate would not be beneficial for lift production.

_{L,max}as shown in Figure 6a. The relatively wider inboard region of the AR = 4.50 plate would result in both the lowest C

_{L,max}at β = 0 deg and the rapid rate of growth of the C

_{L,max}with the slightly extended LEV (as discussed later). This manner, i.e., increasing β to develop an LEV and to enhance a lift, however, seems to need at least a certain higher β because there were also slight losses in the lift at β = 6 deg at the two higher ARs (Figure 6a). The C

_{L,LEV}at β = 6 deg at the two ARs remained near zero (Figure 6b), also implying an insiginifcant LEV contribution.

_{∞}with the chordwise component of the freestream, i.e., U

_{∞}cosβ. The related lift coefficient C

_{L,cw}is then,

_{∞}sinβ is added to the model plates that were placed in a freestream of U

_{∞}cosβ. Note that in this approach, the C

_{L,cw}does not directly reflect the level of the lift force; the C

_{L,cw}at higher β would result in a smaller lift.

_{L,cw}for the three AR cases. We noticed that the slopes of C

_{L,cw}–α curves in a pre-stall region eventually show consistency with the other β cases at each AR. This indicates that the reduction in C

_{Lα}with increasing β (Figure 4) was mainly stemmed from the reduction in chordwise inflow speed. Increasing β enhanced both the maximum C

_{L,cw}and stall α, implying that the spanwise flow stabilized the LEV and flow over the model plates at the higher α, as a spanwise flow on a revolving/flapping wing stabilizes an LEV. The identical slopes at each AR further suggest satisfaction of the Kutta condition, as an LEV on a flapping wing does [16].

_{L,cw}with respect to β, which is in rough proportion to β. It was found that the C

_{L,cw}at AR = 3.13 eventually reached the superior lift force at β > 36 deg. This, once again, suggests that an AR of approximately three would be most effective for the lift enhancement with a sufficient sweptback angle.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Torres, G.E.; Mueller, T.J. Low aspect ratio aerodynamics at low Reynolds numbers. AIAA J.
**2004**, 42, 865–873. [Google Scholar] [CrossRef] - Polhamus, E.C. Predictions of vortex-lift characteristics by a leading-edge suctionanalogy. J. Aircr.
**1971**, 8, 193–199. [Google Scholar] [CrossRef] - Shields, M.; Mohseni, K. Effects of sideslip on the aerodynamics of low-aspect-ratio low-Reynolds-number wings. AIAA J.
**2012**, 50, 85–99. [Google Scholar] [CrossRef] [Green Version] - Ananda, G.K.; Sukumar, P.P.; Selig, M.S. Measured aerodynamic characteristics of wings at low Reynolds numbers. Aerosp. Sci. Technol.
**2015**, 42, 392–406. [Google Scholar] [CrossRef] - Taira, K.; Colonius, T.I.M. Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J. Fluid Mech.
**2009**, 623, 187–207. [Google Scholar] [CrossRef] [Green Version] - DeVoria, A.C.; Mohseni, K. On the mechanism of high-incidence lift generation for steadily translating low-aspect-ratio wings. J. Fluid Mech.
**2017**, 813, 110–126. [Google Scholar] [CrossRef] [Green Version] - Zhang, K.; Hayostek, S.; Amitay, M.; He, W.; Theofilis, V.; Taira, K. On the formation of three-dimensional separated flows over wings under tip effects. J. Fluid Mech.
**2020**, 895, A9. [Google Scholar] [CrossRef] - Ellington, C.P.; Van Den Berg, C.; Willmott, A.P.; Thomas, A.L. Leading-edge vortices in insect flight. Nature
**1996**, 384, 626–630. [Google Scholar] [CrossRef] - Lentink, D.; Dickinson, M.H. Biofluiddynamic scaling of flapping, spinning and translating fins and wings. J. Exp. Biol.
**2009**, 212, 2691–2704. [Google Scholar] [CrossRef] [Green Version] - Lentink, D.; Dickinson, M.H. Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Exp. Biol.
**2009**, 212, 2705–2719. [Google Scholar] [CrossRef] [Green Version] - Bhat, S.S.; Zhao, J.; Sheridan, J.; Hourigan, K.; Thompson, M.C. Aspect ratio studies on insect wings. Phys. Fluids
**2019**, 31, 121301. [Google Scholar] [CrossRef] - Harbig, R.R.; Sheridan, J.; Thompson, M.C. The role of advance ratio and aspect ratio in determining leading-edge vortex stability for flapping flight. J. Fluid Mech.
**2014**, 751, 71–105. [Google Scholar] [CrossRef] - Han, J.-S.; Chang, J.W.; Han, J.-H. The advance ratio effect on the lift augmentations of an insect-like flapping wing in forward flight. J. Fluid Mech.
**2016**, 808, 485–510. [Google Scholar] [CrossRef] - Jardin, T. Coriolis effect and the attachment of the leading edge vortex. J. Fluid Mech.
**2017**, 820, 312–340. [Google Scholar] [CrossRef] [Green Version] - Kruyt, J.W.; van Heijst, G.F.; Altshuler, D.L.; Lentink, D. Power reduction and the radial limit of stall delay in revolving wings of different aspect ratio. J. R. Soc. Interface
**2015**, 12, 20150051. [Google Scholar] [CrossRef] [Green Version] - Han, J.-S.; Chang, J.W.; Cho, H.K. Vortices behavior depending on the aspect ratio of an insect-like flapping wing in hover. Exp. Fluids
**2015**, 56, 181. [Google Scholar] [CrossRef] - Lee, Y.J.; Lua, K.B.; Lim, T.T. Aspect ratio effects on revolving wings with Rossby number consideration. Bioinspir. Biomim.
**2016**, 11, 056013. [Google Scholar] [CrossRef] - Jardin, T.; Colonius, T. On the lift-optimal aspect ratio of a revolving wing at low Reynolds number. J. R. Soc. Interface
**2018**, 15, 20170933. [Google Scholar] [CrossRef] [PubMed] - Breitsamter, C. Unsteady flow phenomena associated with leading-edge vortices. Prog. Aerosp. Sci.
**2008**, 44, 48–65. [Google Scholar] [CrossRef] - Willmott, A.P.; Ellington, C.P. The mechanics of flight in the hawkmoth Manduca sexta. I. Kinematics of hovering and forward flight. J. Exp. Biol.
**1997**, 200, 2705–2722. [Google Scholar] [PubMed] - Meng, X.G.; Sun, M. Wing and body kinematics of forward flight in drone-flies. Bioinspir. Biomim.
**2016**, 11, 056002. [Google Scholar] [CrossRef] [PubMed] - Tobalske, B.W.; Warrick, D.R.; Clark, C.J.; Powers, D.R.; Hedrick, T.L.; Hyder, G.A.; Biewener, A.A. Three-dimensional kinematics of hummingbird flight. J. Exp. Biol.
**2007**, 210, 2368–2382. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yamamoto, M.; Isogai, K. Measurement of unsteady fluid dynamic forces for a mechanical dragonfly model. AIAA J.
**2005**, 43, 2475–2480. [Google Scholar] [CrossRef] - Han, J.-S.; Chang, J.W.; Kang, I.-M.; Kim, S.-T. Flow visualization and force measurement of an insect-based flapping wing. In Proceedings of the 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, USA, 4–7 January 2010. AIAA paper 2010-66. [Google Scholar]
- Buzica, A.; Breitsamter, C. Turbulent and transitional flow around the AVT-183 diamond wing. Aerosp. Sci. Technol.
**2019**, 92, 520–535. [Google Scholar] [CrossRef] - Anderson, J.D. Fundamentals of Aerodynamics; McGraw–Hill: New York, NY, USA, 2007. [Google Scholar]
- Alam, M.M.; Zhou, Y.; Yang, H.X.; Guo, H.; Mi, J. The ultra-low Reynolds number airfoil wake. Exp. Fluids
**2010**, 48, 81–103. [Google Scholar] [CrossRef] - Pelletier, A.; Mueller, T.J. Low Reynolds number aerodynamics of low-aspect-ratio, thin/flat/cambered-plate wings. J. Aircr.
**2000**, 37, 825–832. [Google Scholar] [CrossRef] - Kweon, J.; Choi, H. Sectional lift coefficient of a flapping wing in hovering motion. Phys. Fluids
**2010**, 22, 071703. [Google Scholar] [CrossRef] - Yen, S.C.; Huang, L.C. Flow patterns and aerodynamic performance of unswept and swept-back wings. J. Fluids Eng.
**2009**, 131, 111101. [Google Scholar] [CrossRef] - Lu, Y.; Shen, G.X.; Lai, G.J. Dual leading-edge vortices on flapping wings. J. Exp. Biol.
**2006**, 209, 5005–5016. [Google Scholar] [CrossRef] [Green Version] - Gordnier, R.; Visbal, M. Higher-Order Compact Difference Scheme Applied to Low Sweep Delta Wing Flow. In Proceedings of the 41st Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 6–9 January 2003. AIAA paper 2003-620. [Google Scholar]
- Taylor, G.S.; Gursul, I. Buffeting flows over a low-sweep delta wing. AIAA J.
**2004**, 42, 1737–1745. [Google Scholar] [CrossRef] - Lentink, D.; Müller, U.K.; Stamhuis, E.J.; De Kat, R.; Van Gestel, W.; Veldhuis, L.L.M.; Henningsson, P.; Hedenström, A.; Videler, J.J.; Van Leeuwen, J.L. How swifts control their glide performance with morphing wings. Nature
**2007**, 446, 1082–1085. [Google Scholar] [CrossRef] - Chellapurath, M.; Noble, S.; Sreejalekshmi, K.G. Design and kinematic analysis of flapping wing mechanism for common swift inspired micro aerial vehicle. Proc. Inst. Mech. Eng. Part C
**2020**, 0954406220974046. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setup. (

**a**) A robotic arm consisting of three servo motors; (

**b**) The test setup in the wind tunnel B at the Technical University of Munich; (

**c**) The gram-scale force/moment bending-beam loadcell; (

**d**) A relation of the measured moments and force on the loadcell.

**Figure 4.**C

_{L}–α and C

_{D}–α curves with respect to β. (

**a**) AR = 2.03; (

**b**) AR = 3.13; (

**c**) AR = 4.50.

**Figure 6.**The maximum C

_{L}and the contribution of the leading-edge vortex. (

**a**) The maximum lift coefficients C

_{L,max}with respect to β; (

**b**) The contribution of the leading-edge vortex C

_{L,LEV}with respect to β.

**Figure 7.**Lift coefficients based on the chordwise component of the freestream C

_{L,cw}. (

**a**) AR = 2.03; (

**b**) AR = 3.13; (

**c**) AR = 4.50.

**Figure 9.**Surface oil-flow pattern on the AR = 3.13 plate. (

**a**) β = 0 deg; (

**b**) β = 24 deg; (

**c**) β = 36 deg.

**Figure 10.**Surface oil-flow pattern near stall α at β = 36 deg. (

**a**) AR = 2.03; (

**b**) AR = 3.13; (

**c**) AR = 4.50.

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**MDPI and ACS Style**

Han, J.-S.; Breitsamter, C.
Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number. *Appl. Sci.* **2021**, *11*, 2450.
https://doi.org/10.3390/app11062450

**AMA Style**

Han J-S, Breitsamter C.
Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number. *Applied Sciences*. 2021; 11(6):2450.
https://doi.org/10.3390/app11062450

**Chicago/Turabian Style**

Han, Jong-Seob, and Christian Breitsamter.
2021. "Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number" *Applied Sciences* 11, no. 6: 2450.
https://doi.org/10.3390/app11062450