10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV13 Delft, The Netherlands, July 1-3, 2013
Measurements of Vortex Stretching on Two-Dimensional Rotating Plates with
Varying Sweep
David E. Rival and Jaime G. Wong
Department of Mechanical Engineering, University of Calgary, Calgary, Canada derival@ucalgary.ca
ABSTRACT
The vortex growth around plunging and flapping profiles of varying sweep angle was studied with three-dimensional particle tracking velocimetry and direct-force measurements. For plunging kinematics all sweep angles tested resulted in the same force coefficient history and circulation history. Therefore, it was concluded that nominally two-dimensional spanwise flow has no effect on vortex growth or force history. However, when flapping kinematics were introduced, vortex growth was reduced due to vorticity convection. Vorticity convection for flapping cases could be modulated with the nominally two-dimensional spanwise flow. Reduced circulation for rotating cases corresponded to reduced force coefficient histories. No relationship between vortex stretching and vortex strength was observed at the Rossby number considered in this study. Through an analysis of the vorticity transport equation it was concluded that spanwise flow must be accompanied by gradients in vorticity magnitude in order to limit vortex growth.
1. INTRODUCTION
Natural fliers are exceptionally resilient to gusts and exhibit impressive manoeuvrability at low speeds. However, the vortex dynamics exploited in low-Reynolds number flight are still poorly understood. For instance, consider a simplified analogue to an animal wing shown in Figure 1, where a profile with some sweep angle Λ rotates around a fixed axis parallel to the freestream. Such a flapping arrangement would exhibit large spanwise gradients in both effective velocity and effective incidence, at near- and post-stall conditions. These spanwise gradients can couple with rapid flow separation to form highly three-dimensional vortex behaviour such as spanwise stretching and convection of vorticity. In this scenario, the low-momentum of separated flow is small relative to the impulse imparted by Coriolis and centripetal accelerations, which would act to drive flow away from the root [14]. The relationship between these two effects, i.e. the spanwise gradient of effective incidence and rotational (Coriolis and centripetal) accelerations, is not immediately obvious in the context of highly-separated flows as it is unclear how vorticity would be convected and stretched.
Of particular interest is how the leading-edge vorticity, often associated with a coherent leading-edge vortex (LEV), moves in the spanwise direction under the influence of swept planforms or rotational accelerations. For instance, it has been suggested that the LEV stability observed on insect wings is caused by spanwise flow resulting from large rotational accelerations [15]. However, it has been shown that even very large rotational accelerations do not result in LEV stability for waving kinematics, within the range of 10, 000≤ Re ≤ 60,000 [11]. A recent study further showed that nominally two-dimensional spanwise flow could not account for LEV stability, despite sweepback angles as high as 45◦[3].
Ω
Λ
U
1plane of interest
root-flapping axis
x
z
y
Figure 1: Profile-fixed coordinate system (x, y, z) defined relative to a profile of sweep Λ in a quasi two-dimensional arrangement, where the free surface and walls act as mirror planes. The plane of interest is far from either surface and is therefore essentially spanwise-periodic. The profile rotates around an axis at the lower wall at a rotational speed Ω.
As the nature of LEV growth and detachment remains poorly understood, low-cost computational models are currently being developed to account for LEV growth on two-dimensional geometries. These low-cost models are often based on potential flow [1, 8]. Discrete vortex methods have been extended to include quasi three-dimensional effects by including a potential sink to simulate spanwise flow [20]. However, such models cannot cope with spanwise accelerations or spanwise gradients in vorticity, which result in spanwise convection of vorticity and vortex stretching. For example, consider the z−component of the vorticity transport equation for an incompressible barotropic fluid with conservative body forces, neglecting vortex tilting and viscous diffusion:
∂ωz ∂t + u ∂ωz ∂x + v ∂ωz ∂y + w ∂ωz ∂z = ωz ∂w ∂z , (1)
where the terms from left to right represent the rate of change of vorticity due to unsteadiness, convection of vorticity in the x−,y−, and z−directions, and vortex stretching, respectively. The form of the vorticity transport equation in Equation (1) describes the case of a flapping profile with negligible tip effects such as that in Figure 1, given a sufficiently large Reynolds number to neglect viscous diffusion. Spanwise gradients in effective incidence on a flapping profile give rise to spanwise vorticity convection (w∂ωz
∂z ) and vortex
stretching (ωz∂w∂z) terms. The spanwise transport terms allow for a balance of leading-edge vorticity generation with a transport of
vorticity into a trailing vortex system, as illustrated in Figure 2. The ability to transport vorticity in the spanwise direction disappears in the two-dimensional limit of Equation (1), where the spanwise gradients reduce to zero:
∂ωz ∂t + u ∂ωz ∂x + v ∂ωz ∂y = 0 . (2)
x
y
z
U
∞U
∞v
1=
Ω
z
1v
2=
Ω
z
2w
w (∂ω
z/∂z)
!
zFigure 2: Hypothesized LEV dynamics where the flux of vorticity into the LEV from the leading edge is balanced by a convection of vorticity away from the root. Due to the root-flapping kinematics, spanwise positions R= z1and z2see different effective plunge
velocities v1and v2. A gradient of LEV vorticity ωzresults from the spanwise variation in effective plunge velocity. A spanwise flow
winduced from a sweep angle Λ couples with a gradient of LEV vorticity to convect vorticity away from the root (w∂ωz ∂z).
Therefore, as two-dimensional models must necessarily neglect spanwise transport of vorticity, such models cannot capture all the features expected of a flow around a rotating profile including vortex stability, which is defined as an LEV convection speed of zero relative to the profile. In a recent investigation of a model undergoing combined pitching-plunging-flapping motions, it has been found that the three-dimensional behaviour of the laminar-separation bubble could not be reproduced from two-dimensional results at multiple spanwise locations [2]. The need for a fully three-dimensional description of LEV development was further demonstrated in a computational study that compared the LEV stability of a rotating wing to that of a plunging twisted wing, where it was demonstrated that LEV stability could not be attributed to a spanwise variation of effective incidence [7]. As neither spanwise variation in effective incidence nor nominally two-dimensional spanwise flow are responsible for LEV stability, two prominent hypotheses for the mechanism of LEV stability remain: either the downwash from the trailing vortex system may reduce effective incidence and limit vortex growth [4]; or spanwise flow arising from rotational accelerations transport vorticity into a trailing system [15]. Recent work on LEV topology has suggested that regardless of the mechanism that limits LEV growth, the LEV will detach when the dividing streamline reaches the profile trailing edge [17]. In a recent investigation the two hypotheses of downwash and vorticity convection were considered for a low aspect-ratio rotating profile in steady-state conditions [5]. It was determined that the magnitude of spanwise vorticity convection (w∂ωz
∂z) was small relative to vorticity convection in the remaining two directions (u ∂ωz
∂x and v ∂ωz
∂y), suggesting that spanwise vorticity
convection could not account for LEV stability. However, for low aspect-ratio profiles it is not immediately obvious how the effect of spanwise vorticity convection can be isolated from large tip effects and vorticity reorientation. In order to separate tip effects from the effects of rotation, a recent study compared the force histories of high-aspect ratio flapping and plunging profiles at several positive and negative sweep angles [18]. The study concluded that the magnitude of spanwise flow had no measurable effect on force coefficient magnitude or phase and speculated that the effect of vorticity convection was negligible. Therefore, the current study presents direct measurements of spanwise vorticity transport on high aspect ratio profiles undergoing plunging and flapping kinematics at varying sweep angles. By systematically varying sweep angle and kinematics the individual influence of vorticity convection and stretching on LEV circulation can be isolated. The measurement of individual vorticity transport terms and LEV circulation will directly test the hypothesis that spanwise vorticity convection and stretching can reduce the rate of LEV growth.
2. EXPERIMENTAL SETUP
All experiments were conducted in a free-surface water tunnel at the University of Calgary [18]. The models used in the experimental study were aluminium flat-plate profiles of 50mm chord and 3mm thickness and were manufactured with sweep angles of Λ= 0◦, +45◦,
and−45◦. The flat-plate models each pierced the free-surface, considered as a mirror plane. The chosen kinematics outlined below had negligible free-surface interaction. Furthermore, the tip gap between the models and the tunnel floor was maintained at less than 3mm and the boundary layer at the tunnel floor was measured to be 15mm thick. A similar arrangement was used to mitigate tip vortices by Bansmer and Radespiel [2]. The models were therefore assumed to have negligible tip effects and optical measurements outlined below were conducted three chord lengths from both the water-tunnel floor and free surface. All model kinematics (detailed below) were produced with a custom six degree-of-freedom hexapod manipulator. For all test cases, a six-component ATI Gamma force/moment sensor (1000Hz sample rate, 16-bit sample depth) was flanged to the hexapod, with the high aspect ratio flat-plate profiles supported below. The recorded force data was further post-processed by means of a Butterworth low-pass (5Hz cutoff) and a moving-average (200ms span) smoothing filter similar to that used by Jones and Babinsky [11]. The kinematics described below produced a negligible inertial contribution relative to the hydrodynamic forces, which was checked by reproducing model kinematics in air. Force history uncertainty was estimated as the standard deviation of an ensemble of 10 runs.
Tests were conducted at a freestream velocity of U∞= 0.1m/s, resulting in a chord-based Reynolds number of Re = 5, 000. Kinematics
consisted of a generic sinusoidal plunging motion ˙h, and a root flapping motion Ω, which were repeated for all sweep angles Λ= 0◦, +45◦, and−45◦. In the flapping cases, the axis of rotation was maintained at a virtual hinge at the tunnel floor using the hexapod
manipulator. For all tests, the reduced frequency k was maintained constant at: k= π f c
U∞
= 0.25 , (3)
as this reduced frequency k allowed for significant vortex growth [19]. The sinusoidal-plunging motion was therefore defined to be:
h(t) = h0cos(2π f t) , (4)
where h(t) is the plunge position and h0is the plunge amplitude. The plunge amplitude was defined as h0= c, resulting in a Strouhal
number of:
St= f(2h0) U∞
= 0.16 . (5)
In order to maintain the same effective velocity~ueffat a radius of R= 0.5S for the flapping cases, the flapping motion was defined as:
Ω= ˙h/2S = 0.21 sin(t) , (6)
where S is the span of the model (from tunnel floor to free surface). This point of equivalent velocity was chosen to correspond with the three-dimensional particle-tracking velocimetry (3D-PTV) measurement volume discussed below. The equivalent effective velocities, and therefore equivalent shear-layer characteristics, were chosen such that all PTV measurements captured regions of similar vorticity generation.
The Rossby number can be used to quantify the relative contribution of rotational accelerations, such as the flapping motion Ω. Previous work has defined the Rossby number with length scales chosen to be based on profile chord [15] or profile span [9]. However, all previous work has defined Rossby number geometrically. As the aspect ratio of the profile in the current study is nominally infinite, such geometric definitions are insufficient. Therefore, for the purposes of the current study we will consider a new formulation based on effective velocity: Ro= |~ueff| 2Ωc = q U2 ∞+ Ω2R2eq 2Ωc , (7)
where Reqis the spanwise position at which the point of equivalent velocity is defined. For the equivalent point defined above, this
results in a value of Ro= 5.2. 2.1 3D-PTV
The 3D-PTV technique, as described by Luethi et al. [16], was used to quantify the flowfield on the suction side of the pitching plates. This particle-tracking technique provides a Lagrangian view of the flowfield evolution. The accuracy of 3D-PTV is not affected by reflections from surfaces in the measurement volume [12]. In order to perform clear particle tracks the water tunnel was seeded with 100µm silver-coated, hollow-glass spheres. The seeding particles have a Stokes number of approximately 2.4x10−3and therefore will follow the fluid accelerations accurately. The 3D-PTV measurement volume was limited to a circular cylinder with an extent of l= 200mm in the streamwise direction and a diameter of d = 80mm, as shown in Figure 3(b). The centre of the measurement volume was evenly spaced between each of the water tunnel walls and was 250mm from the water tunnel floor. For illumination a columnated high-intensity discharge lamp was used and a mirror, placed approximately ten chord lengths downstream, reflected the light column into the streamwise direction. The effect of the mirror on the flow was checked with 3D-PTV in an otherwise empty test section and found to be negligible. The flat-plate profiles were painted black to prevent light scattering and together with precise alignment of the light source no digital masking was necessary at the image-processing stage.
3D-PTV data was recorded with four pco.Edge sCMOS cameras, as shown in Figure 3. The cameras have a resolution of 2560 x 1280 pixel2and were operated at a frame rate of 165Hz. The Lagrangian velocities and accelerations were derived by differentiation of the particle tracks. The inter-particle distance, which determines the accuracy of the spatial derivatives, was approximately 3mm during
(a) experimental setup
Ω
U
1d
l
Z Y XΩ
U
1d
d
d
l
l
Z Y XC
E
F
C
E
F
(b) schematic of control volume
Figure 3: (Left) An image of the experimental apparatus showing all system components, with flow moving from left to right. The various system components are: (A) 6 degree-of-freedom hexapod manipulator, (B) 6-component ATI Gamma force/moment sensor, (C) swept model in control volume, (D) pco.Edge 4-camera system, (E) mirror, and (F) HID lightsource. (Right) A schematic of the PTV control volume around a swept model (C) relative to the global coordinate frame (X,Y, Z), illuminated by mirror and lightsource (E,F). The control volume has a diameter of d= 80mm and a length of l = 200mm.
the experiments. The relative uncertainty of velocities determined with the 3D-PTV setup were estimated to be 1% of the freestream using the method described by Feng et al. [6]. Several Lagrangian data sets from repeated runs were superimposed and interpolated onto an Eulerian grid of uniform 4mm grid spacing with the method described in Hartloper et al. [10]. The velocity and velocity derivatives were obtained from weighted interpolations of the Lagrangian data in a sphere around each point [16, 13]. An estimate of the uncertainty in the velocity derivatives was made by comparing the Lagrangian acceleration at=Dui/Dt with the sum of the local
acceleration al=∂ui/∂t and convective acceleration ac=uj∂ui/∂xj. These normalized RMS uncertainties were calculated to be 1.5%,
2% and 1% of the x-, y-, and z-components of acceleration, respectively. 3. RESULTS AND DISCUSSION
In the following we will test the hypothesis that spanwise vorticity convection and vortex stretching can reduce the rate of LEV growth. First, we will test the argument outlined in Section that a nominally two-dimensional spanwise flow has no effect on LEV growth by applying a plunging motion. Second, we will introduce rotation (flapping motion) in order to analyse the effects of spanwise convection and stretching of vorticity on LEV growth in isolation from tip effects.
3.1 NOMINALLY TWO-DIMENSIONAL SPANWISE FLOW (PLUNGING MOTION)
In this section nominally two-dimensional spanwise flows were achieved by varying the sweep angle of plunging profiles with minimized tip effects. This nominally two-dimensional spanwise flow can be visualized with the Lagrangian particle tracks that are shown in the left column Figure 4 for the plunging case and at all three sweep angles Λ= 0◦, +45◦and−45◦. Particle tracks associated with the
LEV were isolated with a vorticity magnitude criterion and only tracks which intersect t/T = 0.2 are shown. The Lagrangian particle tracks in Figure 4 for the swept cases Λ= +45◦and−45◦show a helical pattern where particles are entrained through the leading-edge
shear layer and are then transported along the span of the profile within the LEV. The spanwise flow due to sweep acts co-linear with the LEV vortex line, as demonstrated by the isosurfaces in the right column of Figure 4. The isosurfaces in Figure 4 show the orientation of the LEV in the plate-fixed frame of reference by colouring an isosurface of vorticity magnitude by vorticity orientation. The uniform isosurface signifies that the only significant component of vorticity is that aligned with the spanwise direction, or in other words ~ω≈ ωz~ez.
In Section it was argued that the two-dimensional flow shown in Figure 4 would not affect the circulation of the LEV as the flow could not give rise to vorticity convection or vortex stretching. The independence of two-dimensional spanwise flow on LEV circulation is evidenced by the collapse of force histories with respect to sweep angle shown in Figure 5(a), in agreement with results in the literature [3]. LEV circulation measured with PTV is shown for each sweep angle Λ= 0◦, +45◦, and−45◦ in the first row of Figure 8 and once again there is no correlation between circulation and sweep angle. The plots in Figure 8 are truncated at t/T = 0.2 due to the LEVs leaving the PTV control volume shortly after this phase. Despite the large magnitudes of spanwise flow for the swept cases Λ= +45◦ and−45◦ demonstrated by the Lagrangian particle tracks, the magnitude of vorticity convection is zero throughout the measured phases, as shown in the second row of Figure 8. This result confirms the argument outlined in Section that LEV growth is unaffected by nominally two-dimensional spanwise flow.
Figure 4: (Left) The particle tracks that intersect t/T = 0.2 coloured by time for the plunging case. (Row 1: Λ = 0◦, row 2: Λ= +45◦, row 3: Λ=−45◦; minimum length Nmin= 100 frames, minimum vorticity (ωc/U
∞)min= 5). For the swept cases the particles that
are associated with the LEV show a helical pattern associated with strong spanwise flow. Note that the wing is shown for t/T = 0.2. (Right) Isosurfaces of vorticity magnitude coloured by vorticity orientation for the plunging case (at t/T = 0.2) shows near-perfect alignment with the spanwise direction for all sweep angles (ωxin red, ωyin green, ωzin blue).
0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 t/T CL Λ = 0◦ Λ = +45◦ Λ =−45◦
(a) plunge force
0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 t/T CL Λ = 0◦ Λ = +45◦ Λ =−45◦ (b) flapping force
Figure 5: (a) Force histories for plunging kinematics show no significant variation with sweep angle. The independence of force history to levels of spanwise flow is in agreement with results in the literature [3]. (b) Force histories for flapping kinematics show a dependence on sweep angle, with higher sweep angle producing lower force coefficients.
3.2 EFFECTS OF ROTATION ON VORTICITY CONVECTION (FLAPPING MOTION)
Lagrangian particle tracks as well as isosurfaces of vorticity magnitude for flapping kinematics are shown in Figure 6 for all sweep angles Λ= 0◦, +45◦and−45◦. Particle tracks, vortex size and vorticity orientation for the flapping kinematics appear qualitatively similar to the corresponding plunging cases shown in Figure 4 with no obvious variation in vortex size along the spanwise direction. However, as opposed to the plunging kinematics discussed in Section , flapping kinematics result in spanwise gradients in effective incidence as well as rotational accelerations. Observing the force histories in Figure 5(b) it is clear that force histories are dependent on sweep angle for the flapping case, with peak force coefficient decreasing with increased sweep angle. The dependence of lift on sweep angle appears to be due to LEV strength. The spanwise mean circulation of each flapping case is shown in the first row of Figure 8 and also correlates well with sweep angle. Peak circulation decreases with increased sweep angle, just as peak force coefficient decreases with increased sweep angle as well.
Figure 6: (Left) The particle tracks that intersect t/T = 0.2 coloured by time for the flapping case. (Row 1: Λ = 0◦, row 2: Λ= +45◦,
row 3: Λ=−45◦; minimum length Nmin= 100 frames, minimum vorticity (ωc/U∞)min= 5). For the swept cases the particles that
are associated with the LEV show a helical pattern associated with strong spanwise flow. Note that the wing is shown for t/T = 0.2. (Right) Isosurfaces of vorticity magnitude coloured by vorticity orientation for the flapping case (at t/T = 0.2) shows near-perfect alignment with the spanwise direction for all sweep angles, as was seen in the plunging case (ωxin red, ωyin green, ωzin blue).
As it was hypothesized that spanwise vorticity transport would limit LEV growth in the flapping case, a comparison of vorticity transport between plunging and flapping kinematics was necessary. Therefore, the values of vorticity convection, vortex stretching and circulation were sampled across 18 planes at different spanwise coordinates and for each test case. The 18 sample planes are shown in Figure 7, where the sample planes are spaced one PTV grid spacing (4mm) apart and oriented normal to the spanwise direction. Furthermore, in order to facilitate direct comparisons of vorticity transport to circulation history the volumetric data for vorticity convection and vortex stretching had to be reduced to a single value per timestep. Therefore, for each test case spanwise vorticity convection and vortex stretching were averaged with area across each spanwise plane and the mean value across all spanwise planes was plotted in Figure 8. The second and third rows of Figure 8 show vorticity convection and vortex stretching, respectively. Considering the unswept Λ= 0◦ case, the flapping kinematics produce a positive vorticity convection that limits circulation growth. In the swept-back Λ= +45◦case the rate of vorticity convection is enhanced by increased spanwise flow away from the root relative to the unswept case. As a result the swept-back Λ= +45◦ case has lower overall circulation relative to the unswept case. Similarly the swept-forward Λ=−45◦ case produces a negative vorticity convection due to the spanwise flow towards the root associated with the negative sweep angle. The negative vorticity convection of the swept-forward Λ=−45◦case results in a larger circulation relative to the unswept case. Meanwhile vortex stretching does not appear to affect LEV strength at the tested Rossby number or Ro= 5.2. It is speculated that the effect of vortex stretching is a local phenomenon, changing local values of vorticity rather than integral values of circulation. Therefore, at lower Rossby numbers large values of vortex stretching may still enhance LEV stability by limiting vortex size [17], although as of yet this effect has not been measured. Nevertheless, the superposition of nominally two-dimensional spanwise flows from sweep appears to modulate vorticity convection for flapping kinematics. Furthermore, the values of vorticity convection correlate with LEV circulation. This confirms the original hypothesis that vorticity convection can limit LEV growth.
(a) unswept (b) swept
Figure 7: Eulerian PTV data was sampled on 18 planes spaced c/10 apart (equivalent to the Eulerian grid spacing). The sampled planes were oriented in the plate-fixed frame of reference as shown here, in order to facilitate spanwise-averaging to increase the signal to noise ratio. 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 t/T Γ / U∞ c Plunging Flapping 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 t/T Γ / U∞ c Plunging Flapping 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 t/T Γ / U∞ c Plunging Flapping 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T w ∂ ωz ∂z c 2 U 2 ∞ 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T w ∂ ωz ∂z c 2 U 2 ∞ 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T w ∂ ωz ∂z c 2 U 2 ∞ 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T ωz ∂ w ∂z c 2 U 2 ∞ 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T ωz ∂ w ∂z c 2 U 2 ∞ 0 0.05 0.1 0.15 0.2 −0.1 −0.05 0 0.05 0.1 t/T ωz ∂ w ∂z c 2 U 2 ∞
Figure 8: Spanwise-averaged circulation, spanwise flow, and vorticity transport parameters for all test cases. Column 1: no sweep Λ= 0◦; Column 2: sweep back Λ= +45◦; Column 3: sweep forward Λ=−45◦. Row 1: circulation; Row 2: vorticity convection; Row 3: vortex stretching.
4. CONCLUSIONS
The current study investigated LEV growth on swept profiles undergoing plunging and flapping kinematics with 3D-PTV and direct-force measurements. By inspecting the vorticity transport equation it was expected that two-dimensional spanwise flow alone was not sufficient to limit LEV growth as nominally two-dimensional spanwise flow did not result in vorticity convection or vortex stretching. Rather, it was hypothesized that spanwise gradients in effective incidence and spanwise flow were both necessary in order for vorticity convection to occur. In order to test this hypothesis a plunging motion with no spanwise gradient in effective incidence was compared to a flapping motion with a linear gradient in effective incidence. The plunging and flapping motions were repeated at multiple sweep angles in order to superimpose spanwise flow magnitudes that were small (Λ= 0◦), large and positive (Λ= +45◦), and large and negative (Λ=−45◦). The Reynolds number Re= 5, 000 and reduced frequency k = 0.25 were held constant for all test cases. Three primary conclusions were drawn from this study. First, it was shown that nominally two-dimensional spanwise flow does not result in any change to LEV circulation or force history, in agreement with previous work [3, 18]. Second, vortex stretching had no measurable relationship with circulation or force history. It is speculated that vortex stretching may affect local values of vorticity rather than integral values such as circulation. Finally, positive spanwise vorticity convection was observed to mediate LEV circulation, and therefore also force history. Furthermore, the vorticity convection that resulted from root-flapping kinematics could be modulated by superimposing a nominally two-dimensional spanwise flow, including convection towards the root in the swept-forward Λ=−45◦
case.
The chosen PTV measurement volume did not allow for measurements of vortex detachment later in the plate motion towards t/T = 0.2, and so the effect of mediated LEV growth on vortex stability remains uncertain. As the LEV quickly grows to the topologically-limited size one chord length [17], lower Rossby numbers may be required to observe enhanced LEV stability [15]. Furthermore, if vortex stretching can reduce vortex size (and therefore enhance LEV stability), this effect may be easier to detect at lower Rossby numbers as well.
5. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of the U.S. Air Force Office of Scientific Research under grant number FA9550-13-1-0117, monitored by Dr. Douglas Smith.
REFERENCES
[1] S. A. Ansari, R. ˙Zbikowski, and K. Knowles. Non-linear Unsteady Aerodynamic Model for Insect-Like Flapping Wings in the Hover. Part 1: Methodology and Analysis. Journal of Aerospace Engineering, 220(2):61–83, 2006.
[2] Stephan Bansmer and Rolf Radespiel. Validation of Unsteady Reynolds-Averaged Navier-Stokes Simulations on Three-Dimensional Flapping Wings. AIAA Journal, 50(1):190–202, 2012.
[3] H. Beem, D.E. Rival, and M.S. Triantafyllou. On the Stabilization of Leading-Edge Vortices with Spanwise Flow. Experiments in Fluids, 51:511–517, 2012.
[4] James M. Birch and Michael H. Dickinson. Spanwise Flow and the Attachment of the Leading-Edge Vortex on Insect Wings. Nature, 412:729–733, 2001.
[5] Bo Cheng, Sanjay P. Sane, Giovanni Barbera, Daniel R. Troolin, Tyson Strand, and Xinyan Deng. Three-Dimensional Flow Visualization and Vorticity Dynamics in Revolving Wings. Experiments in Fluids, 54(1):1–12, 2013.
[6] Y. Feng, J. Goree, and B. Liu. Errors in Particle Tracking Velocimetry with High-Speed Cameras. Review of Scientific Instruments, 82:053707, 2011.
[7] D. Garmann, M. Visbal, and P. Orkwis. Three-Dimensional Flow Structure and Aerodynamic Loading on a Low Aspect Ratio, Revolving Wing. In 42nd AIAA Fluid Dynamics Conference and Exhibit; New Orleans, Louisiana, USA. AIAA 2012-3277, 2012. [8] Patrick Hammer, Arron Altman, and Frank Eastep. A discrete vortex method for application to low reynolds number unsteady
aerodynamics. In 28th AIAA Applied Aerodynamics Conference; Miami, Florida, USA. AIAA 2010-4391., 2010.
[9] R. R. Harbig, J. Sheridan, and M. C. Thompson. Reynolds Number and Aspect Ratio Effects on the Leading-Edge Vortex for Rotating Insect Wing Planforms. Journal of Fluid Mechanics, 717:166–192, 2013.
[10] Colin Hartloper, Matthias Kinzel, and David E. Rival. On the Competition between Leading-Edge and Tip-Vortex Growth for a Pitching Plate. Experiments in Fluids, 54:1447, 2013.
[11] A. R. Jones and H. Babinsky. Reynolds Number Effects on Leading Edge Vortex Development on a Waving Wing. Experiments in Fluids, 51:197–210, 2011.
[12] Christian J. K¨ahler, Sven Scharnowski, and Christian Cierpka. On the Uncertainty of Digital PIV and PTV Near Walls. Experiments in Fluids, 52(6):1641–1656, 2012.
[13] M. Kinzel, M. Wolf, M. Holzner, B. L¨uthi, C. Tropea, and W. Kinzelbach. Simultaneous Two-Scale 3D-PTV Measurements in Turbulence under the Influence of System Rotation. Experiments in Fluids, 51:75–82, 2011.
[14] D. Lentink and M. H. Dickinson. Biofluiddynamic Scaling of Flapping, Spinning and Translating Fins and Wings. The Journal of Experimental Biology, 212:2691–2704, 2009.
[15] D. Lentink and M. H. Dickinson. Rotational Accelerations Stabilize Leading Edge Vortices on Revolving Fly Wings. The Journal of Experimental Biology, 212:2705–2719, 2009.
[16] B. Luethi, A. Tsinober, and W. Kinzelbach. Lagrangian Measurement of Vorticity Dynamics in Turbulent Flow. Journal of Fluid Mechanics, 528:87–118, 2005.
[17] David E. Rival, Jochen Kriegseis, Pascal Schaub, Alex Wildmann, and Cameron Tropea. On the Viscous-Inviscid Interaction of Vortex Detachment from Unsteady Aerodynamic Profiles. Physics of Fluids, (in revision).
[18] Jaime G. Wong, Jochen Kriegseis, and David E. Rival. An investigation into vortex growth and stabilization for two-dimensional plunging and flapping plates with varying sweep. Journal of Fluids and Structures, (In Press).
[19] Jaime G. Wong, Ali Mohebbian, Jochen Kriegseis, and David E. Rival. Rapid flow separation for transient inflow conditions versus accelerating bodies: An investigation into their equivalency. Journal of Fluids and Structures, 2013. http://dx.doi.org/10.1016/j.jfluidstructs.2013.04.010.
[20] Xi Xia and Kamran Mohseni. Trapped vortex on a flat plate: Equilibrium and stability. In 42nd AIAA Fluid Dynamics Conference and Exhibit; New Orleans, Louisiana, USA. AIAA 2012-3156, 2012.
