Aerodynamic characterization of low-aspect-ratio swept wings at Re=400
11 Aerodynamic characterization oflow-aspect-ratio swept wings at Re = 400 Kai Zhang † and Kunihiko Taira Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA90095, USA
1. Introduction
The low-Reynolds-number aerodynamics have been extensively studied over the pastfew decades due the interests in designing small-scale air vehicles and understandingbiological flights. At these scales, flows over wings exhibit complex flow physics comprisedof unsteady separation, vortex formation, and wake interaction that are different fromthe high-Reynolds-number counterparts. In fact, the aerodynamic characteristics of low-Reynolds-number flows exhibit strong nonlinearity arising from the rich vortex dynamics.Liu et al. (2012) and Kurtulus (2015) numerically investigated two-dimensional unsteadyflows over a NACA 0012 airfoil at a Reynolds number of 1000. Both studies highlightedthe nonlinear behavior of the aerodynamic characteristics. Rossi et al. (2018) assessed theReynolds number effects ( Re = 100 − ◦ . The presence of multiplebifurcations in the flow behavior and aerodynamic characteristics was reported. Morerecently, Menon & Mittal (2019) conducted a comprehensive study on the effects of two-dimensional airfoil shapes and Reynolds number ( Re = 500 − et al. et al. b ).Such vastly different flow physics from the analogous two-dimensional flows suggeststhat the understanding of fully three-dimensional analysis is necessary for practical wingdesigns at low Reynolds numbers. In our previous study, the wake dynamics of a NACA0015 finite-aspect-ratio wings has been examined for a range of aspect ratios and anglesof attack at Re = 400 (Zhang et al. b ). The aerodynamic force coefficients of thefinite-aspect-ratio wings were observed to be sigificantly lower than those of the two-dimensional airfoils even for wings with large aspect ratios. As we recently studied theflow over finite-aspect-ratio swept wings (Zhang et al. a ), we observed that the sweep-induced midspan effects add another source of three dimensionality to the wake dynamics.However, the aerodynamic characteristics of the swept wings were not systematicallyreported in that study.In this Technical Note, we present a database of aerodynamic force coefficients forfinite-aspect-ratio swept wings at a low Reynolds number of 400. The aerodynamic dataare obtained from three-dimensional unsteady direct numerical simulations over a range † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] N ov K. Zhang & K. Taira
Figure 1.
Case setup for ( a ) unswept wing and ( b ) swept wing. ( c ) cross-sectional mesh atlocations indicated by dashed line in ( a ) and ( b ). of aspect ratios, angles of attack, and sweep angles. These data provides an improvedunderstanding of the low-Reynolds-number aerodynamic characteristics of the canonicalswept wings.
2. Computational methods
For the present aerodynamic characterization, we simulate incompressible flows overfinite-aspect-ratio swept wings with a NACA 0015 cross section. A schematic of the winggeometry is shown in figure 1. The wings are subjected to uniform flow with velocity U ∞ in the x direction. The z axis aligns with the spanwise direction of the unswept wing,and the y axis points in the lift direction. For the swept cases, the wings are shearedtowards the streamwise direction, and the sweep angle Λ is defined as the angle betweenthe z axis and leading edge of the wing. We consider a range of sweep angles from 0 ◦ to 45 ◦ . The symmetry boundary condition is prescribed along the midspan (wing root).Denoting the half wing span as b , the semi aspect ratio is defined as sAR = b/c , where c is the chord length, and is varied from 0.5 to 2. The Reynolds number, defined as Re ≡ U ∞ c/ν ( ν is the kinematic viscosity of the fluid), is fixed at 400, at which the flowremains laminar. The lift and drag coefficients are defined as C L = F L / ( ρU bc/
2) and C D = F D / ( ρU bc/ F L and F D are the aerodynamic forces in y and x directions,respectively, and ρ is the fluid density.The incompressible solver Cliff (in
CharLES software package, Cascade Technologies,Inc.) is used for simulating the flows over wings using direct numerical simulations. Thissolver employs a collocated, node-based finite-volume method to simulate the flows withsecond-order spatial and temporal accuracies (Ham & Iaccarino 2004; Ham et al. et al. b , a ), which have been extensively validated.
3. Results
Wake dynamics
We begin the discussions by presenting an overview of the wake dynamics of theswept finite-aspect-ratio wings. A classification of the wakes is presented in figure 2,with representative vortical structures shown for selected cases. For wings with α (cid:46) ◦ ,the wake over a NACA 0015 airfoil at Re = 400 remains stable. Steady flows withoutsignificant formation of tip vortices ( ), are observed regardless of the aspect ratio andsweep angle. For higher angles of attack, the wake dynamics are influenced by the complexinterplay between the tip effects and the midspan effects. The tip effects are responsiblefor the formation of steady wakes with low aspect ratios and low sweep angles ( ). Inthese cases, the downwash induced by the tip vortices suppresses the roll-up of the vortex ow- Re aerodynamics of finite-aspect-ratio swept wings Figure 2.
Classification of flows around finite-aspect-ratio wings. : steady flows; : steadyflow due to tip effects; : unsteady shedding near midspan; : steady flow due to midspaneffects; : unsteady shedding near wing tip; : steady flow with streamwise vortices. Thedashed lines denote the approximate boundaries between steady (filled symbols) and unsteady(empty symbols) flows. The vortical structures are visualized by isosurfaces of Qc /U ∞ = 1 forrepresentative cases. sheet on the suction side of the wing (Taira & Colonius 2009; DeVoria & Mohseni 2017).With an increase in aspect ratio, the effects of the tip vortices become relatively weakeraway from the tip. This allows for the roll-up of the leading-edge vortex sheets, resultingin unsteady vortex shedding near the midspan ( ). Compared with sAR = 0 .
5, thestability boundaries of the wake for sAR = 1 and 2 shift toward lower angles of attackfor low-sweep wings.For wings with larger aspect ratios and larger sweep angles, the tip vortices are weakerthan those for lower sweep wings, and the midspan effects become profound in shapingthe wakes dynamics. The midspan effects are associated with the formation of a pairof vortical structures on the suction side of the midspan. These vortical structuresare aligned at an angle of 180 ◦ − Λ . For Λ (cid:54) = 0 ◦ , each of the vortical structures issubjected to the downward velocity induced by its symmetric peer on the other side ofthe midspan. Such mechanism stabilizes the wake over a considerable number of cases( ). The formation of the vortical structures near the midspan is also beneficial to theaerodynamic performance, as it will be discussed in detail in the following sections.The downward velocity described above is strong near the midspan and weak towardsthe outboard sections of the wing. For swept wings with large aspect ratios, unsteadyvortex shedding develops locally near the tip region, while the midspan region stillremains steady. The resulting flows resemble the “tip stall” phenomenon (Black 1956; K. Zhang & K. Taira
Figure 3.
Time-averaged lift coefficients for ( a ) sAR = 0 .
5, ( b ) sAR = 1 and ( c ) sAR = 2. Visbal & Garmann 2019), and prevail for swept wings of sAR = 1 − sAR = 2 with high sweep angles ( Λ = 37 . ◦ − ◦ ), the unsteadytip shedding further transitions to another type of steady flow, with the formation of thestreamwise vortices ( ). We refer the readers to our previous study (Zhang et al. a )for a thorough discussion on the wake dynamics of swept wings.3.2. Lift coefficients
The lift force coefficients of the wings with sAR = 0 .
5, 1 and 2 are presented in figure3. Also plotted are the inviscid limit for the lift of low-aspect-ratio unswept wings inincompressible flow (Helmbold 1942): C L = 2 πα (cid:112) /sAR ) + 1 /sAR . (3.1)The lift coefficients of the finite-aspect-ratios wings are significantly smaller than theinviscid limit for all three aspect ratios. Compared to a similar characterization (Taira &Colonius 2009) for flat-plate wings at Re = 300, we find that the airfoil shape is influentialeven at these low Reynolds numbers. For sAR = 0 .
5, the sweep has a positive effect onthe lift coefficients for α (cid:46) ◦ . The vortical lift enhancement with the sweep angle isdue to the vortical structures near the midspan, as discussed in section 3.1. As the flowtransitions to unsteady shedding at α = 30 ◦ , the lift coefficients undergo an abrupt jumpfor wings with low sweep. For these unsteady flows, the lift coefficients decrease slightlywith the sweep angle.As the aspect ratio increases to sAR = 1, for α (cid:46) ◦ , the lift coefficients acrossdifferent sweep angles remain close to each other, and increase almost linearly with theangle of attack with a steeper slope than that of the analogous cases with sAR = 0 . α ≈ ◦ − ◦ , the favorable effect of sweep on the lift coefficients becomes morenoticeable. The increase of C L with Λ saturates at high sweep angles. For higher anglesof attack ( α ≈ ◦ − ◦ ), the lift coefficients no longer exhibit a monotonic relationshipwith the sweep angle. Instead, high C L is observed for Λ = 15 ◦ at α = 26 ◦ , and Λ = 22 . ◦ at α = 30 ◦ .For sAR = 2, the lift coefficients decrease with increasing sweep angle for wings withlow angles of attack ( α (cid:46) ◦ ). The adverse effect of sweep on lift at sAR = 2 (contraryto the positive effect at sAR = 0 .
5) is due the fact that the additional generation ofvortical lift is limited to the midspan region, while the elongated outboard region isfeatured by lower sectional lift. However, at higher angles of attack, the lift coefficients ow- Re aerodynamics of finite-aspect-ratio swept wings Figure 4.
Time-averaged drag coefficients for ( a ) sAR = 0 .
5, ( b ) sAR = 1 and ( c ) sAR = 2. for Λ = 45 ◦ surpass those of moderate sweep angles ( Λ = 15 ◦ − . ◦ ), although they aresignificantly smaller than those for Λ = 0 ◦ − . ◦ . Compared with sAR = 0 . sAR = 2 wings are generally higher. An exception of the this trend isobserved for Λ = 45 ◦ wings for α = 0 ◦ − ◦ , where the lift coefficients remain almostthe same with those of the sAR = 1 wings.3.3. Drag coefficients
The drag coefficients of the wings exhibit an quadratic growth with angle of attackover the studied range, as shown in figure 4. For the low aspect ratio of sAR = 0 .
5, thedifference in drag coefficients among cases with different sweep angles is not noticeableuntil α = 20 ◦ . At α = 26 ◦ , C D increases with the sweep angle. As the flow destabilizesat α = 30 ◦ , similar to the lift coefficients shown in figure 3( a ), the drag coefficientbecomes negatively affected by the sweep angle. Compared to the drag coefficients at sAR = 0 .
5, those at sAR = 1 are generally smaller for α (cid:46) ◦ . For these cases, thedrag decreases with increasing Λ , although the difference among different sweep anglesremains small. At higher angles of attack ( α = 26 ◦ − ◦ ), the drag coefficients of wingswith Λ = 0 ◦ − . ◦ are significantly higher than those with Λ = 30 ◦ − ◦ . Similar to sAR = 1, the drag coefficients at sAR = 2 also decreases with increasing sweep angle.However, the difference in C D among different sweep angles becomes larger even at lowangles of attack. At higher angles of attack, the drag coefficients of wings with low sweepangles grow much faster with α than those with high sweep angles.3.4. Lift-to-drag ratios
The time-averaged lift-to-drag ratios are compiled in figure 5. The C L /C D generallyimproves with increasing aspect ratio. However, due to the low-Reynolds-number natureof the flow, the lift-to-drag ratio remains below 1.5 for the cases considered herein. At sAR = 0 .
5, the lift-to-drag ratio increases with the angle of attack up to α ≈ ◦ , atwhich the maximum C L /C D is achieved. The lift-to-drag ratio increases with the sweepangle, due to the lift enhancement mechanism of the midspan effects discussed in § C L /C D is also observed for wings with sAR = 1.The maximum C L /C D for sAR = 1 is achieved at α ≈ ◦ for Λ = 0 ◦ − . ◦ , and at α ≈ ◦ for Λ = 45 ◦ . For sAR = 2 at low angles of attack ( α (cid:46) ◦ ), the lift-to-dragratios of the Λ = 45 ◦ wings are significantly lower than those with lower sweep angles.At higher angles of attack ( α = 20 ◦ − ◦ ), C L /C D of the Λ = 45 ◦ wing are only slightly K. Zhang & K. Taira
Figure 5.
Time-averaged lift-to-drag ratio for ( a ) sAR = 0 .
5, ( b ) sAR = 1 and ( c ) sAR = 2. higher than the rest of the cases. This suggests that care should be taken in selectingthe right type of wings if midspan lift enhancement is to be taken advantage of withfinite-aspect-ratio swept wings.
4. Conclusions
We have performed unsteady three-dimensional direct numerical simulations to studythe aerodynamic characteristics of finite-aspect-ratio swept wings with a NACA 0015cross-section at a chord-based Reynolds number of 400. The effects of the sweep angle( Λ = 0 ◦ − ◦ ) on the aerodynamic force coefficients were examined for finite-aspect-ratio wings ( sAR = 0 .
5, 1, and 2) over a wide range of angles of attack ( α = 0 ◦ − ◦ ).The unsteady laminar separated flows exhibit complex aerodynamic characteristics withrespect to these parameters. The introduction of sweep enhances lift for wings with lowaspect ratios of sAR = 0 . Acknowledgments
We acknowledge the US Air Force Office of Scientific Research (Program Managers:Dr. Gregg Abate and Dr. Douglas Smith, Grant number: FA9550-17-1-0222) for fund-ing this project. We thank Ms. Shelby Hayostek, Prof. Michael Amitay, Dr. Wei He,Mr. Anton Burtsev and Prof. Vassilios Theofilis for insightful discussions.
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