OSCILLATING FOILS WITH HIGH PITCH AMPLITUDES - ac

The 28th International Symposium on Transport Phenomena 22-24 September 2017, Peradeniya, Sri Lanka

OSCILLATING FOILS WITH HIGH PITCH AMPLITUDES

R. Zaman1, J. Young1 and J.C.S. Lai1

1 School of Engineering and Information Technology, The University of New South Wales Canberra, ACT 2600, Australia

ABSTRACT The aerodynamic performance of oscillating foils, analysed theoretically in the 1930s, has attracted considerable interest both computationally and experimentally in the last two decades because of its fundamental role in flights in nature and because of its many applications such as aeroelastic flutter and micro/nano air vehicles. The kinematics considered for flapping foils has been focused mainly on combined pitch and plunge motion and pure plunge motion; in contrast, pure pitch motion has not received as much attention and pure pitch motion with high pitch amplitudes θo (greater than 20°) has rarely been considered. Here, the flow periodicity of a NACA0012 foil undergoing sinusoidal pure pitch motion at a Reynolds number of 500 has been studied numerically for reduced frequency k = 8 to 16 and pitch amplitude θo = 10° - 58°. Results show that contrary to the potential analysis and previous findings that the mean thrust coefficient/θo

2 increases with θo for low θo (≤ 30°), it is found here that for a given k, the mean thrust coefficient/ θo

2 decreases with increasing θo when θo ≥ 6.28/k (approximately). This is because, for a given k, increasing θo increases the effective angle of attack and the trailing edge of the foil moves closer to the larger trailing edge vortices, causing a large suction pressure and hence higher drag. Furthermore, for a given k, the thrust coefficient has been found to change from periodic behavior to chaotic with increases in θo. For the chaotic cases, the thrust coefficient varies significantly from one flapping cycle to the next. INTRODUCTION Flapping wing aerodynamics has attracted considerable attention recently because of its potential practical applications such as propulsion of fish and marine animals[1-4]; dynamic stall phenomenon; and the design of Micro Air Vehicles (MAVs) . Thrust generation by an oscillating foil undergoing pure plunge, pure pitch or combined pitch and plunge motion is much dependent on the vortex dynamics. Andersen et al [5] found that the optimum efficiency occurs when leading edge vortices (LEVs) and trailing edge vortices (TEVs) interact constructively. A pure plunging NACA0012 foil has been shown to produce thrust above a critical non-dimensional plunge velocity kh of 0.4 [6-8], where k is the reduced frequency and h is the plunge amplitude. Lewin and Haj-Hariri found numerically the flow to be aperiodic at kh ≥ 0.8 for pure plunge motion at Reynolds number Re = 500. Numerical results of Ashraf et al [9] showed that at Re = 20000, the nature of the forces generated changed from periodic to quasi-periodic when kh exceeded 0.4 and to chaotic when kh exceeded 1.25. While natural flyers in

low Re regime are adaptive and responsive so that they might still perform well under chaotic conditions, the operation performance of MAVs would be degraded by chaotic force generation; therefore periodic flow would be desirable. By using inviscid flow theory, Garrick [10] showed analytically that for pure pitch motions, thrust increases with pitch amplitude θo for a given k. For a given θo, Koochesfahani [11] showed experimentally that k has to exceed a critical value for thrust to be produced; e.g. k > 6 for θo = 4°. Above the critical value, the mean thrust coefficient CTmean continues to increase with kθo, but the study was restricted to a maximum θo of 4° and k = 22. Numerical studies on pitch motion [12-14] confirmed that increasing k produces higher thrust for a given θo (ranging from 2.5° to 20°). Numerical analysis on pure pitch motion of a NACA0012 foil [15] showed that for a mean angle of attack αm = 15°– 20°, θo of 10° and k = 0.15 – 0. 5 at Re = 3000, increasing k delayed the dynamic stall to occur at larger angle of attack (25° at k = 0.5 compared to 21.5° at k = 0.3). By using molecular tagging velocimetry technique on a pitching NACA0012 foil with low θo = 2° at Re = 12600, Bohl and Koochesfahani [16] showed that the wake pattern changed from drag producing to thrust producing as k increased from 10 to 23. The transition from drag to thrust for a pitching NACA0012 foil was confirmed by Sarkar and Venkatraman [17] in their numerical investigations for k = 0 - 15 and θo = 2.5° - 5° at Re = 10000 with αm = 0° - 15°. They also observed that for a given k, CTmean dropped from maximum for a given θo when αm > 0°. Lee et al [12] found negative thrust (drag) at a low value of k = 1 and increase of CTmean with increasing k up to 12 for a pitching NACA0012 foil at θo = 20° and Re = 5000. Recent computational analysis by Lu et al [13] on a pitching NACA0012 foil with θo = 30° at Re = 13400 showed that CTmean increased with k from 6 to 18 except at k = 15 where there was a slight drop. However, the LEV formation, shedding and its interactions with the TEV were not discussed. Large amplitude flapping motions show significant effect on the wake pattern as seen from the pure plunging foil studies with high plunge amplitude (h > 2) [9, 18, 19]. In the numerical analysis of pure plunge motion by Ashraf et al [9], increasing h (≥ 2) shows large scale leading edge vortices which have strong influence on their subsequent interactions and trajectories, causing significant thrust coefficient variation. The experimental study of Lentink et al [18] showed that increasing k changed the wake topology as the vortices interacted strongly with each other and with the foil surface and those staying close enough in the wake merged or tore each other apart, thus resulting in an aperiodic wake. A number of studies [12-17] have shown


that increasing kθo produces higher thrust as higher kθo generates stronger thrust producing wake. Since for a given k, increasing θo increases CTmean, it is important to identify the limit of the values of important parameters for pitching foil propulsion such as pitch amplitude, flapping frequency beyond the range covered by the existing literature for the development of MAV. The complexities related to LE flow separation at large θo also require systematic investigations for better understanding. The objective of this study was, therefore, to examine the flow periodicity nature of sinusoidal pure pitch motion at higher amplitudes of θo = 10°- 58° for a NACA0012 foil at Re = 500 for k = 8 – 16. The range of reduced frequencies and pitch amplitudes is selected to identify their upper limits for MAV flights. The pitch axis is located at 0.25c from the LE similar to a number of experimental studies [11, 16] for better comparison of the simulation results with the experimental data. The flow Reynolds number considered, Re = 500 is representative of small MAV flights similar to small insects like the fruit fly studied by Sarkar and Venkatraman [20, 21] to understand the flow physics at this low Re range which is useful for MAV and NAV flights.

VelocityInlet

PressureOutlet

20c 20c

20c

20c

(a) Computational domain

c

(b) Grid around the NACA0012 foil

Figure 1 C-type grid with boundary conditions. NUMERICAL METHOD The unsteady incompressible laminar flow field around a NACA0012 foil was simulated using the commercially available CFD package ‘ANSYS Fluent14.5’. The Centre of Gravity (CG) motion was used to produce pitch motion of the foil through a User Defined Function (UDF) in the pre-processor. The Navier-Stokes (NS) equations were solved with the two dimensional double precision pressure based solver, SIMPLE pressure-velocity coupling, least squares cell based gradients and second order upwind spatial discretization with a two-dimensional structured C-

type grid with quadrilateral elements as shown in Figure 1. The velocity inlet boundary is at 20 chords upstream from the trailing edge (TE) of the airfoil and the pressure outlet boundary is 20 chords downstream from the TE. The upper and lower flow boundaries are placed 20c away from the TE. The first grid point from the foil surface is located at a normal distance of 0.0004c. The rigid NACA0012 foil undergoes sinusoidal pure pitch motion θ(t) given by:

θ (t) =θo sin(ωt) (1) where θo is the pitch amplitude, ω is the angular frequency and t is time.

(a) Thrust Coefficient CT.

(b) Lift Coefficient CL.

Figure 2 Grid Independence test for a sinusoidally pitching NACA0012 foil at k = 8, θo = 20° and Re=500 with five different grids and 800 time steps per cycle.

(a) coarse grid (b) medium grid

(c) fine grid (d) medium 2 grid Figure 3 Vorticity ωzU∞/c contours at t/T = 0.38 with 800

time steps per cycle.


Grid and Time step independence tests Grid independence and time step independence tests were carried out at k = 8, θo = 20° and Re = 500 for a sinusoidally pitching 2D NACA0012 foil. The grid independence test was performed with 800 time steps per cycle and 5 different grids: coarse - 325×150 grid (97500 cells with 250 points on the airfoil surface); medium - 650×150 grid (195000 cells with 500 points on the airfoil surface); medium 1- 650×75 grid (97500 cells with 500 points on the airfoil surface); medium 2 - 650×250 grid (325000 cells with 500 points on the airfoil surface); and fine - 1300×150 grid (390000 cells with 1000 points on the airfoil surface). The time history of the thrust coefficient CT and lift coefficient CL for five different grids in Figure 2 (a) and (b) respectively shows insignificant differences among the five different grids. Vorticity contours in Figure 3 show that the vortices are quite similar in all the cases, especially with the medium and high resolution grids. The mean thrust coefficient CTmean obtained by averaging CT over the last two cycles out of the five cycles calculated confirms that the medium grid is sufficiently fine to be used for all subsequent calculations as it is within 0.2 % of that for the fine grid. For time-step independence tests, 400, 800 and 1600 time steps per cycle were used with the medium grid of 195000 (650x150) cells. The time history of CT and CL in Figure 3 (a) and (b) respectively shows that there is very good agreement between the results for T/Δt = 800 and 1600 and the difference in CTmean is only 1.2% between the two cases. The vorticity contours for T/Δt = 800 and 1600 are virtually identical at t/T = 0.375 as shown in Figure 4, with negligible difference for the contours near the TE for T/Δt = 400 case. Hence, the medium grid with 195000 (650x150) cells and 800 time steps per cycle was used in all subsequent calculations.

(a) Thrust Coefficient CT.

(b) Lift Coefficient CL.

Figure 4 Time-step Independence test for k = 8, θo = 20° and Re = 500 with the medium grid (5th cycle).

(a) T/Δt = 400 (b) T/Δt = 800 (c) T/Δt = 1600

Figure 5 Vorticity ωzU∞/c contours at t/T = 0.38 for the medium grid.

Figure 6 Comparison of moment coefficient, CM between current simulation and Lee et al [12] at k = 8, θo = 20° and Re = 5000 for a sinusoidally pitching NACA0012 airfoil.

Validation For the validation of the Navier-Stokes calculations, the simulation results are compared with the experiments of Koochesfahani [11] who studied the wake of a sinusoidally pitching NACA0012 airfoil at small amplitudes (θo = 2° and 4°) in a Low Speed Water Channel at Re = 12000. The computed CTmean at k = 17.7 and θo = 2° is 0.054 within 5% of 0.057 measured by Koochesfahani [11]. As larger pitch amplitudes (up to θo = 58°) are considered here, comparisons are also made with the time history of the moment coefficient CM given by Lee et al [12] for a NACA0012 foil at Re = 5000 with k = 8 and θo = 20°. As shown in Figure 6, there is good agreement in the time history of the pitching moment coefficient CM between present calculations and the results of Lee et al [12] with less than 1% for the difference in CMmean. RESULTS Effect of θo on flow periodicity In order to illustrate the effect of large θo on the periodicity nature of the flow for a sinusoidally pitching NACA0012 foil, the CT history, the power spectral density (PSD) of CT and phase-space plots (CT vs. CL) are examined. Results are presented in Figure 7 for a sinusoidally pitching NACA0012 airfoil at Re = 500 and k = 8 with θo of (a) 20°, (b) 30°, (c) 38°, (d) 45°, (e) 50° and (f) 58°. To quantify the nature of flow periodicity of pure pitching, the time series plot of the variation of the cross correlation value between successive cycles is presented in Figure 8 for each θo case and the minimum cross correlation value (CCM) of the last 7 cycles out of a minimum of 20 cycles is used to set the boundary from periodic to quasi periodic and from quasi periodic to chaotic. The time series history plot gives an


indication of the fluctuations of the cross correlation value from one cycle to next and the CCM value gives a measure to quantify the specific regions (periodic, quasi periodic and chaotic). Here, CCM of 1 indicates the cycles are identical and the flow is periodic. The flow is considered quasi periodic when 0.96≤CCM <0.99 and chaotic when CCM < 0.96.

(a) θo = 20°, (periodic)

(b) θo = 30°, (periodic)

(c) θo = 38°, (periodic)

(d) θo = 45°, (periodic)

(e) θo = 50°, (quasi periodic)

(f) θo = 58°, (chaotic)

Figure 7 Time history plot of CT, phase-space plot (CT vs. CL) and power spectral density (PSD) of CT for a

sinusoidally pitching NACA0012 foil at k = 8.

Figure 8 Time history plot of cross correlations of CT for

consecutive cycles at k=8.


The time variation of CT changes from periodic to chaotic as θo increases from 20° to 58° for k = 8. At θo = 20° and 30°, the phase-space plot in Figure 7 (a) and (b) respectively shows no variation in the trajectories of CT and CL from one cycle to another with consecutive cycles overlapping well on top of one another and one dominant peak at the driving frequency (k = 8) in the PSD plot for CT. As θo increases to 38°, the ‘figure of eight’ pattern in the phase-space plot in Figure 7(c) is maintained with the presence of a small second peak at k = 16 (the first harmonic of k = 8) in the PSD plot for CT along with the first sub-harmonic (k = 4). As θo further increases up to 45° in Figure 7(d), the first harmonic (k = 16) and the first sub-harmonic (k = 4) become more prominent. Although the CT history is periodic, the difference in the CT peaks in the upstroke and the downstroke of a cycle increases with increasing θo. This has also been observed by Lewin and Haj-Hariri [19] and Baik et al [22]. It has been observed that at lower amplitudes θo (<30°), the LEV size is very small and the effect of the LEV on CT is negligible. So, the CT history is dominated by the TEV only. As the amplitude θo increases (>30°), the LEV grows larger and its effect on CT is much more prominent compared to low amplitude cases. Since at higher amplitudes, the LEV sheds either during the upstroke or downstroke, the CT peaks in the upstroke and downstroke are different in a cycle. Baik et al [22] mentioned that the variation in the CT peaks in a cycle is due to the increase in the normal force produced by the LEVs. The CCM of CT value obtained from Figure 8 remains 1 for k = 8 and θo = 20° - 45° which confirms that the flow is periodic. At θo = 50°, the PSD plot for CT in Figure 7(e) shows that consecutive cycles in the CT history are not identical and the trajectories in the phase-space plot (CT vs. CL) are not overlapping well, indicating that CT is not periodic and lies in the quasi periodic regime with CCM of 0.96. For θo = 58°, the CT history in Figure 7(f) does not repeat from cycle to cycle and the PSD plot for CT shows multiple peaks with a more broadband pattern while still being dominated by the driving frequency at k = 8. The trajectories in the phase-space plot do not repeat themselves at all, indicating that the force generation is chaotic. The corresponding CCM of CT reduced to 0.94.

Periodic zone

Quasi periodiczone

Chaotic zone

Figure 9 Periodicity analysis for a NACA0012 foil

undergoing sinusoidal pitch motion at Re = 500.

Results for k=12 and 16 are similar to those for k=8 except that the transition from periodic to quasi-periodic and chaotic flow occurs at lower θo as k increases. The results of the analysis of the periodicity of a pitching NACA0012 foil for k = 8, 12, 16 and θo = 20°, 30°, 28° 45°, 50° and 58° at Re = 500 are summarised in Figure 9. These results show that for a given k, a minimum value of θo is required to generate thrust from a sinusoidally pitching airfoil. The minimum value of θo changes with k which is in agreement with the experimental findings of Koochesfahani [11]. For a given θo = 10°, k = 16 produces thrust but k = 8 produces negative thrust. This is because although the potential flow analysis of Garrick [10] for a sinusoidally pitching airfoil shows that for a given k, thrust increases with θo. his analysis assumed that the flow was inviscid, smooth flow separation at the trailing edge only and no self- induced wake evolution occurred. For Navier-stokes (NS) calculations which account for viscous effects, it is found that for k = 8 and θo = 10°, the viscous drag dominates and the NACA0012 foil produces drag. As shown in Figure 10, the positive pressure component (CT,pressure), hence thrust, increases with k while the negative viscous component (CT,viscous), hence drag, also increases with k but at a slower rate than CT,pressure. Hence, as k increases beyond 9, there is net thrust generation CT,total (=CT,pressure + CT,viscous) which increases with k. At or below kθo = 1.39 (dashed line in Figure 9), drag is produced. The continuous line separates the region from periodic to quasi periodic and the dash dot line separates the region from quasi periodic to chaotic.

Figure 10 Variation of thrust coefficient components (CT,pressure and CT,viscous) and CT,total with k for θo =10°.

Effect of θo on CTmean Figure 11(a) shows that for a given θo, CTmean/θo

2 calculated by Garrick’s analysis increases with increasing

k, but is independent of θo for a given k Figure (see Figure 11(b)). By comparisons, NS results in Figure 11(b) show that for a given k, CTmean/θo

2 increases with θo for small θo but decreases with θo for large θo. This is because at small θo (≤ 20°), the viscous drag dominates but is not calculated in Garrick’s results. At larger θo (≥ 38°), NS calculations take into account both the formation and interactions between the leading and trailing edge vortices and their interactions with the airfoil surface. The standard deviations calculated for CTmean/θo

2 based on the

Thru

st C

oeff

icie

nt


last 7 out of 20 simulated cycles are marked in Figure 11(b) to reflect the quasi-periodic/chaotic behaviour of CTmean/θo

2 with increasing k. The results in Figure 11(b) indicate a critical θo value below which CTmean/θo

2

increases with θo and above which CTmean/θo2 decreases

with θo. The critical value of θo is found to be dependent

on k and can be determined from approximately kθo = 6.28 rad. These results indicate that for a given k, the drop of CTmean/θo

2 with θo is not only dependent on k or

θo individually but is a function of both.

(a)

(b)

Figure 11 Variation of CTmean/θo2 with (a) k and (b) θo.

Lines with triangles (∇) are Garrick’s results. CONCLUSIONS The flow periodicity of a rigid two-dimensional NACA0012 undergoing sinusoidal pure pitch motion at a Reynolds number of 500 has been studied numerically for high reduced frequencies k=8, 12 and 16 and high pitch amplitudes θo = 10° - 58°. Results show that kθo has to be greater than 1.39 for thrust to be produced. In the thrust producing regime, for a given k, the thrust will transit from periodic behavior to quasi-periodic and chaotic as θo increases and the transition behaviour occurs at a lower θo as k increases. According to potential flow analysis, the mean thrust coefficient CTmean/θo

2 increases as k increases but is independent of θo for a given k. Results here show that CTmean/θo

2 does increase as k increases but it is also dependent on θo. In fact, for a given k, CTmean/θo

2 increases with θo until a critical value of θo determined approximately from kθo = 6.28 above which it will decrease. These results will have strong implications for MAVs operating at high k and θo .

NOMENCLATURE A wake width c chord length (m)

CL instantaneous lift coefficient, L / 12ρU

∞2c

CM instantaneous moment coefficient, M / 12ρU

∞2cA

CT instantaneous thrust coefficient, Th /12ρU

∞2c

CTmean mean thrust coefficient, CTmean =1T

CT (t)dtt

t+T∫

f pitching frequency, 1/T (s-1) ho plunge amplitude (m) h non-dimensional plunge amplitude, ho/c k reduced frequency, 2πfc/U∞ L lift (N) M moment (N.m) Re Reynolds number, U∞c/ν t time (s) T period of oscillation, 2π/ω (s) Th thrust (N) ν kinematic viscosity (m2.s−2) θ instantaneous pitch amplitude (°) θo sinusoidal pitch amplitude (°) ρ density, kg/m3 ω angular velocity, 2πf (rad.s−1) ACKNOWLEDGEMENT The first author acknowledges receipt of a University College Postgraduate Research Scholarship for the pursuit of this study. This research was undertaken with the assistance of resources provided at the NCI National Facility systems at the Australian National University through the National Computational Merit Allocation Scheme supported by the Australian Government. This work was supported under the Australian Research Council’s Discovery Projects funding scheme (project number DP130103850). REFERENCES 1. Bandyopadhyay, P.R., Castano, J.M., Nedderman,

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OSCILLATING FOILS WITH HIGH PITCH AMPLITUDES - ac