Authors: M.A. Ashraf, J. Young and J.C.S. Lai
Received 8 April 2010;
accepted 10 November 2010.
Available online 11 January 2011.
Abstract
The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion at Reynolds number Re=200, 2000, 20 000 and 2×106 was studied by numerical simulations for fully laminar and fully turbulent flow regimes. The thickness study was performed on 2-D NACA symmetric airfoils with 6–50% thick sections undergoing pure plunging motion at reduced frequency k=2 and amplitudes h=0.25 and 0.5, and for combined pitching and plunging motion at k=2, h=0.5, phase 
=90°, pitch angle θo=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where
positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier–Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.
=90°, pitch angle θo=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier–Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.
Keywords: Flapping airfoil; Propulsion; Airfoil shape; Reynolds number; Navier–Stokes simulations
Article Outline
- 1.
- Introduction
- 2.
- Numerical method
- 2.1. Solver
- 2.2. Kinematics
- 3.
- Effect of thickness
- 3.1. Re=2000 and 20 000, plunging airfoil
- 3.2. Re=200, plunging airfoil
- 3.3. Re=2×106, plunging airfoil
- 3.4. Re=200, 20 000 and 2×106, combined motion
- 5.
- Conclusion
1. Introduction
Flapping wing aerodynamics has attracted significant attention recently not only because of its importance in understanding the fundamental science of flight in nature and in practical applications for the development of micro-aerial vehicles (MAVs) but also because of its potential to be employed for power generation. A detailed historical perspective of studies related to flapping wings is well covered in a numbers of papers ([Lai and Platzer, 1999], [Shyy et al., 1999], [Rozhdestvensky and Ryzhov, 2003], [Triantafyllou et al., 2004] and[Platzer et al., 2008]). Several analysis methods such as unsteady potential flow, Navier–Stokes (NS) computations and experiments have been employed to analyse the flow physics and to examine the effects of important flapping parameters on thrust generation and propulsive efficiency. Detailed experimental analysis related to the effects of kinematics as well as the wake visualisation behind the flapping airfoils can be found in Freymuth (1988), Koochesfahani (1989), Triantafyllou et al. (1993), Anderson et al. (1998), Lai and Platzer (1999), Read et al. (2003) and Schouveiler et al. (2005). Extensive computational work pertaining to oscillating foils can be found in Garrick (1937), Streitlien and Triantafyllou (1998), Isogai et al. (1999),Wang (2000) and Young and Lai (2004).
Most of these studies were focused on the influence of kinematic parameters such as flapping frequency, amplitude and phase difference between pure plunge and pitch motion on the thrust generation and propulsive efficiency. By comparison, systematic investigation of airfoil shape on thrust generation and propulsive efficiency has received much less attention. Freymuth (1988) used a NACA0015 airfoil for his wake visualisation experiments. Koochesfahani (1989), Anderson et al. (1998), Read et al. (2003),Schouveiler et al. (2005) and Triantafyllou et al. (1993) employed a NACA0012 section for their experiments of the flow over oscillating airfoil. Streitlien and Triantafyllou (1998) used a 12% thick Joukowski profile to numerically investigate the effect of frequency and Strouhal number on the thrust estimation of combined pitching and plunging. Isogai et al. (1999) and Young and Lai (2004) employed a NACA0012 airfoil for their NS simulations. Wang (2000) employed a fourth order finite difference NS solver to study the flow over 12.5% and 25% thick elliptic airfoil sections flapping at Re=192. Although this study was mainly focused on the optimal frequency selection, she also reported that in the case of thinner ellipse, more vortices were shed due to a sharper corner resulting in higher lift.
Using unsteady panel code simulations in comparison to Garrick's linear theory (Garrick, 1937) for NACA0003, NACA0009, NACA0012 and NACA0015 sections, Cebeci et al. (2004) found a negligible effect of increasing thickness on the propulsive efficiency of a plunging airfoil. Rozhdestvensky and Ryzhov (2003) briefly discussed the results of experiments conducted at the Institute of Theoretical and Applied Mechanics of the Academy of Sciences of the USSR for an isolated rigid vertically submerged wing having NACA sections with thickness between 6% and 21% undergoing lateral oscillations in water at rest. In contrast to the inviscid calculations of Cebeci et al. (2004), these results indicate that the thrust increases with airfoil thickness. Lentink and Gerritsma (2003) numerically studied the influence of airfoil shape with NS simulations on plunging airfoils at Re=150. They compared the performance of a blunt 10% ellipse and NACA0010 section with the thinner and sharper 2% ellipse, NACA0002 and NACA4702 airfoil sections. They reported that the NACA0010 section creates greater thrust than NACA0002 because of the higher frontal area against which low pressure leading edge vortices react. The NACA4702 section produced greater thrust than all the other airfoils tested, and they concluded that aft camber is highly important for high thrust and high efficiency in insect flight. However their study lacked the range of different thicknesses to conclude the optimal thickness for plunging airfoil propulsion.
Further, Vandenberghe et al. (2006) performed experiments on a rigid symmetric wing that flapped vertically and was free to rotate around a fixed vertical axis. Thrust produced by the wing drove rotation about the vertical axis, showing the onset of forward flight. They used a rectangular cross-section wing at Reynolds number based on vertical flapping velocity of around 200–2000. They reported that increasing the thickness (from 8.4% to 16.8%) decreased the resultant rotational speed of the wing, and further thickness increase (to 25.2%) produced no rotational motion, indicating no net thrust. More recently, An et al. (2009) used the lattice Boltzmann method to study plunging bodies of varying thickness from 5% (thin ellipse), 10% and then in steps of 10–100% thick (circular cylinder) at very low Reynolds number of (Re=50, 100 and 185) and Strouhal number of (St=0.2 and 0.4). Based on the wake vortex pattern and velocity profiles behind the plunging body, they identified that the thickness distribution of the airfoil is a crucial design parameter for thrust generation. Although they reported that a 10% thick elliptic airfoil produced maximum thrust at Re=185 and St=0.4, they did not explore the physics producing maximum thrust at this thickness as the main focus of their study was the aerodynamic transition from drag to thrust. Further, Dickinson and Götz (1993) performed an experimental study at Re=197 to study the effect of camber on unsteady aerodynamic forces. They compared a smooth 1 mm thick rounded leading edge and sharp trailing edge section wing with a cambered section. The camber was introduced by bending the profile by 12° about a point 0.3 chord from the leading edge. They reported that camber did not alter the performance of the wing, except for a slight increase in drag and greater instability at high angle of attack.
In summary, the results in Rozhdestvensky and Ryzhov (2003), Lentink and Gerritsma (2003) and An et al. (2009) indicate that thick airfoils can improve plunging airfoil performance, whereas Vandenberghe et al. (2006) and Wang (2000) suggest that thin airfoils perform better, and the inviscid analysis of Cebeci et al. (2004) concludes no influence of airfoil thickness on plunging airfoil propulsion. Also according to the literature ([Mueller and Batill, 1982] and [Kunz and Kroo, 2001]) for static (nonflapping) airfoils at low Reynolds number flows, thin and sharp airfoils perform better than the thick and blunt airfoils. These contradictory findings are a result of limited knowledge of low Reynolds number flows, and suggest that there exists an optimal thickness and shape of airfoil for the best flapping airfoil performance which is strongly influenced by the Reynolds number of the flow.
In this paper, we systematically evaluate and quantify the effect of varying the thickness and camber of a thrust producing harmonically pure plunging and combined pitching and plunging NACA airfoil on their propulsion performance. To elaborate the differences of thickness effects reported in literature, this study has been undertaken for different Reynolds numbers regimes Re=200, 2000, 20 000 and 2×106, where Re≤20 000 represents fully laminar regime and Re=2×106 represents fully turbulent regime. The current study employs a Navier–Stokes (NS) viscous flow solver which allows leading edge flow separation and vortex shedding effects to be evaluated. The physics behind the production of higher thrust and higher efficiency by varying thickness and camber is highlighted.
2. Numerical method
2.1. Solver
The unsteady flow field around the airfoil section was simulated using the commercially available CFD package Fluent version 6.3.26, with an unsteady incompressible solver and second-order upwind spatial discretization. The plunging motion of the airfoil was modelled by the introduction of a source term in the Navier–Stokes equations which allowed the unsteady formulation to be 2nd order accurate in time. For combined pitching and plunging motion, two zones of grid were used with a sliding grid interface. The inner zone employed a moving mesh for pitching motion and a source term for plunging whereas the outer zone was fixed with the source term only as employed by Kinsey and Dumas (2008).
The free-stream velocity of Uo=0.292 m/s was used and the kinematic viscosity was varied to achieve different Re. At Re=200 and 2000, the flow field was assumed to be laminar. At Re=20 000, the force coefficients of a plunging NACA0012 airfoil section at St=0−0.7 (St=kh/π, k=0–12.5) and h=0.175 computed by NS simulations for laminar flows agree very well with the experiments as shown by Young and Lai (2007) and Heathcote and Gursul (2007). Also the flow field results for similar Re range were validated using laminar flow assumption in Young and Lai (2004). However, recent work has been conducted by Visbal et al. (2009) and Visbal (2009) on the laminar-turbulent-transitional effects on flow over plunging a SD7003 airfoil at Re=10 000 and 40 000 with reduced frequency πfc/U∞=3.93 (or k=7.86) and plunge amplitude h=0.05. At Re=10 000, they show that the instantaneous and phase averaged flow structures agree well with each other. However, at Re=40 000, the flow became a complex mixed transitional-turbulent flow due to spanwise instabilities. The instabilities caused fine scale structures in instantaneous representation which disappeared in phase-averaged flow representation. In this paper, Re=20 000 is considered to be within the limits validated by (Young and Lai, 2004) and (Young and Lai, 2007). However, the test cases for pure plunge motion are conducted at h=0.5 (St=0.31, k=2) which causes a higher angle of attack and therefore there may be some transitional effects even at Re=20 000. To account for possible turbulent-transitional cases, a fully turbulent Spalart–Allmaras (SA) model (1992) is employed in addition to the laminar assumption, to establish the limiting behaviour of the flow at this Re.
For turbulence modelling validation, two different turbulence models (SA (Spalart and Allmaras, 1992) andk−ω SST (Wilcox, 1998)) were tested and validated against the numerical results of Tuncer and Platzer (1996) for a plunging NACA0012 airfoil at h=0.4, Re=3×106 and different reduced frequencies with Baldwin–Lomax (BL) turbulence model. Fig. 1 shows the comparison of present NS simulation with the published results. Both SA and k–ω SST models agree very well with the published results. Therefore, for Re=2×106, the flow was treated as fully turbulent and the (SA) turbulence model was employed.
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Fig. 1.
Variation of CTmean against k for a plunging NACA0012 airfoil at Re=3×106.
Both the symmetric and cambered airfoil sections were modelled using a structured grid. A grid independence, time step refinement study and validation for laminar flow were also performed and the results were reported in Ashraf et al. (2007). The number of grid points used was 901×101 (streamwise×normal) with 401 points distributed on the airfoil surface. The first normal grid point was located at a distance of 0.0004c with a wall y+ value of order 1 and approximately 15 grid points in the boundary layer.
For large motion amplitudes and thick airfoils, results show some variability of lift and thrust from cycle to cycle as reported by Blondeaux et al. (2005), therefore simulations were run for 5 cycles (for h=0.25) or 10 cycles (for h=0.5) of the plunging motion, with 800 time steps per cycle. Fig. 2 shows the time history of thrust and lift coefficient of NACA0006 and NACA0030 airfoils undergoing plunging motion at k=2, h=0.5 and Re=20 000. It provides an example of cycle to cycle variation in CT and CL for NACA0030. Force and efficiency data presented for these cases are phase-averaged over the final four cycles of motion. The standard deviation of the forces is also provided in the tables where results are compared, and is shown in [Fig. 3] and [Fig. 4](b). Varying thickness cases are also simulated with an unsteady panel method (UPM) solver to eliminate the leading edge separation effects for comparison with the NS results. The details of the UPM solver can be found in Young and Lai (2004).
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Fig. 2.
Variation of CT and CL with time for plunging airfoils at k=2,h=0.5 and Re=20 000.
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Fig. 3.
Variation of CTmean and ηP with the thickness of symmetric plunging airfoil at: (a) k=2 and h=0.5 and (b) k=2, h=0.25 and Re=20 000.
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Fig. 4.
Comparison of variation of CTmean and ηP with the thickness of symmetric airfoil for pure plunging and combined pitch and plunging motion with =90° at k=2, h=0.5: (a) laminar, Re=200, (b) laminar, Re=20 000, and (c) fully turbulent, Re=2×106.
=90° at k=2, h=0.5: (a) laminar, Re=200, (b) laminar, Re=20 000, and (c) fully turbulent, Re=2×106.2.2. Kinematics
Two different plunge amplitudes h=0.25 and 0.5 at a plunging reduced frequency of k=2, at Re=20 000 were simulated. Further for h=0.5, simulations were run for Re=200, 2000 and 2×106. The sinusoidal pitching motion of the airfoil section is defined by the following equation:
(2) θ(t)=θosin(2πft+),where is the phase difference between pitching and plunging motion. For combined motion cases, two values of pitch angle θo=15° and 30° with =90° and the pitch axis is located at 1/3 of chord from leading edge are used. The values of kinematic parameters were chosen as they fall within the parameter range for the development of MAVs. The MAV developed by Jones et al. (2001) uses very similar motion operating at k≈0–2.5, h≈0.5.
),where is the phase difference between pitching and plunging motion. For combined motion cases, two values of pitch angle θo=15° and 30° with =90° and the pitch axis is located at 1/3 of chord from leading edge are used. The values of kinematic parameters were chosen as they fall within the parameter range for the development of MAVs. The MAV developed by Jones et al. (2001) uses very similar motion operating at k≈0–2.5, h≈0.5.
The effect of thickness on the propulsion performance was studied using seven symmetric airfoil sections: NACA0006, NACA0012, NACA0015, NACA0020, NACA0030, NACA0040 and NACA0050. The effect of camber on the propulsion performance was studied by using a fixed maximum camber of 4% at varying chordwise positions (20%, 40% and 60%) for the medium thickness airfoil sections (NACA4215, NACA4415 and NACA4615).
3. Effect of thickness
The time averaged thrust coefficient, CTmean and propulsive efficiency, ηP for all the tested airfoil thickness at different Reynolds numbers are presented in Fig. 3(a). Both UPM solver and NS code predictions are presented. The plot clearly shows that at all except the lowest Reynolds numbers considered (Re=2000, 20 000, 2×106), when a plunging airfoil operates in a thrust producing regime (k=2 and h=0.5), there exists an optimal range of airfoil thickness for maximum thrust. However, at very low Reynolds number (Re=200) where viscous forces dominate and there is no net thrust production from the plunging oscillations, drag is successively increased as the thickness increases. This latter behaviour is in agreement with previously reported results in the studies of Wang (2000), Vandenberghe et al. (2006) and An et al. (2009). The failure of the UPM code in predicting an optimum thickness, in agreement with the previous study of Cebeci et al. (2004), is due to the inviscid code's inability to capture the leading edge vortices produced (see discussion in Section 3.1). For Re=20 000, we also investigated the effect of thickness at a lower plunge amplitude ofh=0.25. The results, plotted in Fig. 3(b), show that the maximum thrust and efficiency is produced by the NACA0020 section compared to NACA0030 for the h=0.5 case (Fig. 3(a)) but the overall behaviour is quite similar in both cases. Further, considering that there may be some transitional effects at Re=20 000, as pointed out in Section 2.1, the results with a fully turbulent (SA) model (Spalart and Allmaras, 1992) are also shown in Fig. 3(a) and (b). While there are some differences in the magnitude of the predicted results for the thrust coefficient CTmean and propulsive efficiency ηP with fully laminar and fully turbulent models at Re=20 000, the effect of varying thickness on the thrust and propulsive efficiency of a plunging airfoil is almost identical for fully laminar and fully turbulent flows.
Fig. 4(a)–(c) shows the effect of varying thickness for combined pitching and plunging motions in comparison to pure plunging airfoils for Re=200 (laminar), 20 000 (laminar) and 2×106 (fully turbulent), respectively. The phase difference between pitch and plunge motion was fixed to 90°. Two different pitch amplitudes θo=15° and 30° were selected. The reduced frequency and plunging amplitude were fixed atk=2 and h=0.5, respectively. For Re=200, compared to pure plunging mode, though the inclusion of the pitching mode of motion has enabled thinner airfoil sections to generate positive thrust, but at this low Re, there is a monotonic decline in mean thrust with the increase of the airfoil section thickness for both pitching amplitudes tested. For Re=20 000 and 2×106, the combined motion with θo=15° shows a similar trend in CTmean and ηP with varying thickness as for the pure plunging case; in contrast the combined motion with θo=30° shows superior performance with the thinnest airfoil tested. The reason for this transition in behaviour is the maximum angle of attack experienced by the airfoil during the flapping cycle, given by
(3) 
for a phase difference of =90°. At θo=15°,αmax reaches 30° which causes significant leading edge separation during the flapping cycle (see Fig. 14and discussion in Section 3.4). At θo=30°, αmax reaches only to 15° and does not cause significant leading edge separation and therefore the effect of varying the thickness of an airfoil is different for different pitching amplitudes. Table 1 summarises all the results of the thickness variation study at different Reynolds number and the mode of airfoil motion. It is clear that the optimum section changes with the Reynolds number and the mode of motion. In [3.1 Re=2000 and 20 000, plunging airfoil], [3.2 Re=200, plunging airfoil], [3.3 Re=2×106, plunging airfoil] and [3.4 Re=200, 20 000 and 2×106, combined motion]below, we discuss in detail that the main benefit in propulsion performance by varying thickness comes from the interaction of leading edge vortices with the frontal area and the point of maximum thickness of the section.
for a phase difference of =90°. At θo=15°,αmax reaches 30° which causes significant leading edge separation during the flapping cycle (see Fig. 14and discussion in Section 3.4). At θo=30°, αmax reaches only to 15° and does not cause significant leading edge separation and therefore the effect of varying the thickness of an airfoil is different for different pitching amplitudes. Table 1 summarises all the results of the thickness variation study at different Reynolds number and the mode of airfoil motion. It is clear that the optimum section changes with the Reynolds number and the mode of motion. In [3.1 Re=2000 and 20 000, plunging airfoil], [3.2 Re=200, plunging airfoil], [3.3 Re=2×106, plunging airfoil] and [3.4 Re=200, 20 000 and 2×106, combined motion]below, we discuss in detail that the main benefit in propulsion performance by varying thickness comes from the interaction of leading edge vortices with the frontal area and the point of maximum thickness of the section.
Table 1. Summary of the results of effect of thickness on flapping airfoil propulsion performance.
| Re | Mode of motion | Best performance thickness of symmetric airfoil |
|---|---|---|
| 200 | Plunging, k=2, h=0.5 | NACA0006 |
| 2000 | Plunging, k=2, h=0.5 | NACA0020 |
| 20 000 | Plunging, k=2, h=0.5 | NACA0030 |
| 20 000 | Plunging, k=2, h=0.25 | NACA0020 |
| 2×106 | Plunging, k=2, h=0.5 | NACA0020 |
| 200 | Combined, k=2, h=0.5, θo=15° | NACA0006 |
| 200 | Combined, k=2, h=0.5, θo=30° | NACA0006 |
| 20 000 | Combined, k=2, h=0.5, θo=15° | NACA0020 |
| 20 000 | Combined, k=2, h=0.5, θo=30° | NACA0006 |
| 2×106 | Combined, k=2, h=0.5, θo=15° | NACA0012 |
| 2×106 | Combined, k=2, h=0.5, θo=30° | NACA0006 |
3.1. Re=2000 and 20 000, plunging airfoil
For Reynolds number Re=2000 and 20 000 we observed similar behaviour of propulsion performance for a plunging airfoil with varying thickness, as shown in Fig. 3(a). Here it is evident that NS computations predict a significant increase in the thrust as the thickness of the airfoil is increased, whereas no significant benefit in forces is observed with the UPM simulations. For the NS results both CTmean and ηPincrease monotonically with varying thickness up to some critical thickness and then start decreasing. We found this critical thickness to be close to 20% and 30% for Re=2000 and 20 000, respectively.
Fig. 5 shows the instantaneous thrust coefficient CT and power coefficient CP at k=2 and h=0.5 from NS and UPM simulations. Most of the difference in the generated thrust between airfoils with different thickness occurs when the airfoil passes through the mean position (t/T=0, 0.5 and 1) where the plunging velocity is maximum.
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Fig. 5.
Variation of CT and CP with time for plunging airfoil at k=2, h=0.5 and Re=20 000.
In order to understand the physics behind the increase in thrust due to increased airfoil thickness, the vorticity field and pressure coefficient plots are displayed in Fig. 6(a) and (b) at various instances during the down stroke (t/T=0.3–0.7). For t/T=0.4 the formation of the leading edge vortex (LEV) varies with the thickness of the airfoil section. For NACA0006, the separated flow area on the top surface of the airfoil is much larger compared with that for the NACA0030 airfoil at the same instant, because the thin airfoil has an effectively sharper leading edge compared to the rounder leading edge of the thicker airfoils. The pressure coefficient plots at t/T=0.4-0.5 show that once the vortex has been formed and starts to convect over the airfoil surface, the associated suction pressure peak is broader and weaker in the NACA0006 case than for the thicker airfoils, indicating a larger more diffuse vortex.
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Fig. 6.
(a) Vorticity field during last plunging cycle at t/T=0.3–0.7 for NACA0006, NACA0012, NACA0030 and NACA0050 and (b) corresponding phase-averaged pressure coefficients for NS (top) and UPM (bottom), at k=2, h=0.5 and Re=20 000.
Despite the apparent differences in the pressure distribution, the resulting integrated lift force (and hence the power coefficient) varies little between the three sections, as seen in Fig. 5. However, the frontal area available for the suction pressure to react upon is greater for the thick airfoil section and therefore more thrust is generated which is also reported by Lentink and Gerritsma (2003) for a 10% thick airfoil compared with a 2% thin airfoil. This is not the case with the calculations using the UPM solver, where the leading edge vortex generation is not possible, and the smaller frontal area of the NACA0006 airfoil is balanced by a stronger suction peak caused by a more rapid turning of the flow around the nose of the airfoil. Thus the leading edge separation is clearly shown to be the cause of variation of thrust with airfoil thickness.
Once the leading edge vortex has been convected past the point of maximum thickness, the associated suction reduces thrust. This is seen in Fig. 5 where the slightly stronger vortex in the NACA0030 case, coupled with the larger frontal area, produces slightly less thrust than the NACA0006 after t/T=0.7, but the difference in thrust between the two sections is much less pronounced at this stage in the flapping cycle due to diffusion of the vortex as it is convected over the airfoil (Fig. 6(a)). The mean thrust therefore remains highest for the NACA0030 section.
With further increase in the thickness of the section as shown for NACA0050 at t/T=0.4–0.5 (Fig. 6(a)), the LEV starts to form only beyond the maximum thickness point, which results in reduced thrust output throughout the cycle (Fig. 5). Since the power coefficient CP is not much altered due to varying airfoil section thickness, an optimally thick airfoil section without increase in weight would be more attractive for MAV design as it would deliver more propulsive power.
For a given airfoil section, the thrust output could also be improved by increasing the amplitude of plunge motion. However this is possible only at the expense of losing efficiency, as more input power is required to drive the airfoil at increased amplitude because of the higher plunge velocity (
). This can be seen in Table 2 for different airfoils operating at h=0.5 and 0.25. Table 2 also provides comparison of time averaged thrust coefficient and propulsive efficiency for the best performing airfoil section with the thin airfoil section tested. CTmean was increased by 372% and ηP was increased by 378% for NACA0030 airfoil when compared to NACA0006 airfoil plunging at k=2, h=0.5. These increases were 121% and 158%, respectively, for NACA0020 compared to NACA0006 when h was reduced to 0.25. Therefore it can also be concluded that for the airfoil sections tested here, a thick airfoil section is more beneficial when operating at higher plunging amplitudes.
). This can be seen in Table 2 for different airfoils operating at h=0.5 and 0.25. Table 2 also provides comparison of time averaged thrust coefficient and propulsive efficiency for the best performing airfoil section with the thin airfoil section tested. CTmean was increased by 372% and ηP was increased by 378% for NACA0030 airfoil when compared to NACA0006 airfoil plunging at k=2, h=0.5. These increases were 121% and 158%, respectively, for NACA0020 compared to NACA0006 when h was reduced to 0.25. Therefore it can also be concluded that for the airfoil sections tested here, a thick airfoil section is more beneficial when operating at higher plunging amplitudes.
Table 2. Effect of airfoil thickness on average thrust and efficiency of plunging airfoil at Re=20 000.
| k=2, h=0.5 | |||
| NACA0030 | NACA0006 | Difference (%) | |
| CTmean | 0.549±0.005 | 0.1163±0.0002 | 372 |
| ηP | 0.272±0.004 | 0.0568±0.0001 | 378 |
| k=2, h=0.25 | |||
| NACA0020 | NACA0006 | ||
| CTmean | 0.1217±0.0001 | 0.0551±0.00005 | 121 |
| ηP | 0.2980±0.0008 | 0.1155±0.0002 | 158 |
3.2. Re=200, plunging airfoil
For Re=200, a decline in both CTmean and ηP with increase in airfoil thickness is observed (Fig. 3). We do not observe any thrust production for pure plunging at Re=200, k=2 and h=0.5 with all airfoil thicknesses tested. This result is in accordance with the findings of Wang (2000) and Vandenberghe et al. (2006) for such low Reynolds number flow over flapping airfoils.
Fig. 7 shows the instantaneous thrust coefficient, CT and power coefficient, CP for k=2 and h=0.5 NS simulations. Again, the instantaneous forces and pressure coefficients of three different thickness sections NACA0006, NACA0030 and NACA0050 out of seven tested are presented here. It can be seen that as the thickness of the airfoil is increased, the airfoil produces significant drag (negative thrust). At such low Reynolds number, viscous forces dominate the flow as evident in the vorticity contour and surface pressure plots in Fig. 8(a) and (b). Leading edge separation is evident for all three thicknesses, but the resultant vortex is increasingly diffuse as the thickness increases. NACA0006 and NACA0030 sections produce thrust only for a short time as the airfoil passes through the mean position (t/T=0, 0.5 and 1) but the average thrust remains negative. In the case of NACA0050 the leading edge suction is the weakest due to the rounder leading edge; this, coupled with the increased pressure drop on the lower surface of the airfoil near the trailing edge results in drag even when the airfoil is passing through the mean position (Fig. 8(b) t/T=0.5). Further, when the airfoil approaches the maximum excursion, the large slope from the maximum thickness point to the trailing edge in the thicker airfoil section creates more drag compared to the thin airfoil (Fig. 8(b) t/T=0.7). At this point in the cycle, the LEV has already convected past the maximum thickness point and contributes to the drag for all the airfoil sections.
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Fig. 7.
Variation of CT and CP with time for plunging airfoil at k=2, h=0.5 and Re=200.
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Fig. 8.
(a) Vorticity field during last plunging cycle at t/T=0.3–0.7 for NACA0006 (top), NACA0030 (middle) and NACA0050 (bottom) and (b) corresponding phase-averaged pressure coefficients, at k=2, h=0.5 and Re=200.
3.3. Re=2×106, plunging airfoil
In the case of fully turbulent flow Re=2×106 for the pure plunging airfoil case, we again observe an optimum thickness for propulsive performance, as for the laminar flow at Re=20 000 (Fig. 3). However the mechanism producing this behaviour is not the same as for the fully laminar case. The instantaneous CTand CP of three airfoil sections NACA0006, NACA0020 and NACA0050 are shown in Fig. 9. The difference in thrust production for the different sections is most prominent when the airfoil passes through the mean position (t/T=0, 0.5 and 1). The vorticity field and pressure coefficients at various instances during the down stroke (t/T=0.3–0.7) are shown in Fig. 10(a) and (b). As the flow is fully turbulent, the Spalart–Allmaras turbulence model restricts the leading edge separation compared with the fully laminar case in Fig. 4.
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Fig. 9.
Variation of CT and CP with time for plunging airfoil at k=2, h=0.5 and Re=2×106.
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Fig. 10.
(a) Vorticity field during last plunging cycle at t/T=0.3–0.7 for NACA0006 (top), NACA0020 (middle) and NACA0050 (bottom) and (b) corresponding phase-averaged pressure coefficients, at k=2, h=0.5 and Re=2×106.
For the NACA0006 section, a large leading edge vortex is formed due to the very sharp leading edge which causes significant drag just as for the lower Reynolds numbers. This can be seen in Fig. 10(b) at t/T=0.5 and 0.6, where the suction peak is beyond the point of maximum thickness of the airfoil. For the thicker airfoil sections there is no leading edge separation, which results in a larger suction peak. As for the inviscid analysis, as the thickness increases further, this suction peak is reduced and the thrust is reduced again (Fig. 10(b) t/T=0.5 and 0.6). There is also some separation towards the trailing edge for the thicker sections, reducing thrust still further for NACA0050 ([Fig. 9] and [Fig. 10](a) at t/T=0.7).
3.4. Re=200, 20 000 and 2×106, combined motion
At Re=200, Fig. 4(a) shows that for θo=15° and 30°, the propulsive performance of a combined pitching and plunging airfoil monotonically decreases with the thickness. To highlight the differences due to variation of the airfoil section, the instantaneous CT and CP of three airfoil sections NACA0006, NACA0015 and NACA0030 with θo=15° are plotted in Fig. 11, and the vorticity field and phase-averaged pressure coefficient at various instants in the downstroke in Fig. 12. Fig. 11 shows that all three sections generate thrust for some duration in the flapping cycle particularly when the airfoil passes through the mean position (t/T=0, 0.5 and 1) but only the thinner sections maintain the positive thrust for longer duration than the thicker section resulting in average positive thrust. Similar to pure plunging cases at Re=200, leading edge separation is evident in Fig. 12 for all sections but due to dominant viscous effects at such low Re, more diffuse vortex is formed. Also, as the thickness of the airfoil is increased, the more rounded leading edge of the airfoil weakens the suction peak (see pressure coefficients at t/T=0.4 and 0.5) causing lesser thrust force which leads to constant decline in the average thrust for thick airfoil sections at Re=200.
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Fig. 11.
Variation of CT and CP with time for airfoil with combined motion at θo=15°, k=2, h=0.5 and Re=200.
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Fig. 12.
(a) Vorticity field during last flapping cycle at t/T=0.3–0.7 for NACA0006 (top), NACA0030 (middle) and NACA0030 (bottom) and (b) corresponding phase-averaged pressure coefficients, θo=15°, k=2,h=0.5 and Re=200.
At Re=20 000, Fig. 4(b) shows that for θo=15°, the propulsive performance of a combined pitching and plunging airfoil increases with the thickness up to NACA0020 and then it monotonically decreases, as observed for a pure plunging airfoil. For θo=30°, the propulsive performance steadily decreases with thickness. To examine these results in detail, the instantaneous CT and CP of two airfoil sections NACA0006 and NACA0020 with θo=15° and 30° are plotted in Fig. 13, and the vorticity field and pressure coefficient at various instants in the downstroke in Fig. 14.
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Fig. 13.
Variation of CT and CP with time for airfoil with combined motion at k=2, h=0.5 and Re=20 000.
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Fig. 14.
(a) Vorticity field during last flapping cycle at t/T=0.3–0.7 for NACA0006 (top) and NACA0020 (bottom) and (b) corresponding phase-averaged pressure coefficients, θo=15°, k=2, h=0.5 and Re=20 000.
For θo=15°, leading edge separation is observed for all the thickness tested. In comparison to the pure plunging case, the pitch angle of the airfoil acts to increase the frontal area against which the low pressure leading edge vortex can react. Thus the thin sections do not suffer as much reduction in thrust as in the plunging case (Fig. 13).
For θo=30°, the frontal area is greater still, but the high pitch amplitude keeps the angle of attack low which lessens leading edge flow separation as well as reducing aerodynamic loads in general. The vorticity field and pressure coefficients at various instances during the down stroke (t/T=0.3–0.7) are shown in Fig. 15. At t/T=0.4–0.6, leading edge separation is much reduced for both sections, and the behaviour is closer to that seen for the inviscid analysis, whereby thin sections produce greater thrust.
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Fig. 15.
(a) Vorticity field during last flapping cycle at t/T=0.3–0.7 for NACA0006 (top) and NACA0020 (bottom) and (b) corresponding phase-averaged pressure coefficients, θo=30°, k=2, h=0.5 and Re=20 000.
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Fig. 16.
Variation of CT and CP with time for plunging airfoil at k=2, h=0.5 and Re=2×106.
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Fig. 17.
(a) Vorticity field during last flapping cycle at t/T=0.3–0.7 for NACA0006 (top) and NACA0020 (bottom) and (b) corresponding phase-averaged pressure coefficients, θo=15°, k=2, h=0.5 and Re=2×106.
For θo=30°, fully turbulent case at Re=2×106 as shown in Fig. 4, no loss in thrust due to leading edge separation is observed. However, for thicker airfoil sections (NACA0030, NACA0040 and NACA0050), the UPM results predicts more drag on the airfoil when the airfoil comes near the mean position (at t/T=0.4Fig. 18(a)). At this instant, NS simulations show trailing edge separation (Fig. 18(b)) on the lower surface of the airfoil which is suppressed in inviscid calculations and as a result overall low surface pressure is produced on the lower surface of the airfoil which results in increased drag (Fig. 18(c)) for the UPM case.
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Fig. 18.
(a) Variation of CT with time for NACA0040, (b) vorticity field during last flapping cycle at t/T=0.4 and (c) corresponding phase-averaged pressure coefficient, θo=30°, k=2, h=0.5 and Re=2×106.
4. Effect of camber, plunging airfoil
The instantaneous thrust coefficient CT and power coefficient CP results for a 4% cambered section with varying camber location (NACA4215, NACA4415 and NACA4615) are compared with that of the NACA0015 (i.e. same thickness of 0.15c but with no camber) in Fig. 19 for k=2 and h=0.5 at Re=20 000. In contrast to the effect of thickness, varying the camber of airfoil sections offers very little or no benefit in thrust over symmetric airfoil sections. As expected, the only apparent difference in the results of a plunging cambered airfoil to that of a plunging symmetric airfoil is that the cambered airfoil produces an asymmetric thrust coefficient during a plunging cycle. More thrust is generated in the down stroke and approximately the same amount less is generated in the upstroke, so that the overall time averaged thrust remains almost the same. The shape of the airfoil again plays an important role for this asymmetric behaviour in thrust production. As shown in Fig. 20(a), during the downstroke (t/T=0.5–0.7) the main LEV forms over the upper surface of the airfoil and due to camber, the frontal area is slightly larger than that of a symmetric airfoil. During the upstroke (t/T=1.0–0.1), the LEV forms on the lower surface of the foil and due to camber, the lower surface of the cambered foil is relatively flat compared to its symmetric counter part and the LEV has less frontal area to react against. Thus less thrust is generated during the upstroke as shown in the surface pressure plots in Fig. 20(b) at t/T=1.0. Table 3 presents the comparison of time averaged thrust coefficient and propulsive efficiency of symmetric and cambered airfoils. The results show around 11–12% loss in thrust for the maximum camber NACA4615 section compared to NACA0015 over the range of k=2,h=0.5 and 0.25. A very slight gain in CTmean (
2%) and ηP (4%) is observed for NACA4215 compared with NACA0015 for k=2 and h=0.5.
2%) and ηP (4%) is observed for NACA4215 compared with NACA0015 for k=2 and h=0.5. | Full-size image (36K) |
Fig. 19.
Variation of CT and CP with time for plunging airfoil at k=2, h=0.5 and Re=20 000.
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Fig. 20.
(a) Vorticity field during last plunging cycle at t/T=0.3–0.7 for NACA0015 (top) and NACA4615 (bottom) and (b) corresponding phase-averaged pressure coefficients, at k=2, h=0.5 and Re=20 000.
Table 3. Effect of airfoil camber on average thrust and efficiency of plunging airfoil at Re=20 000.
| NACA0015 | NACA4215 | NACA4415 | NACA4615 | Difference (best cambered vs. symmetric) (%) | |
| k=2, h=0.5 | |||||
| CTmean | 0.382±0.007 | 0.3881±0.0005 | 0.374±0.003 | 0.351±0.006 | 2 |
| ηP | 0.172±0.006 | 0.1785±0.0005 | 0.173±0.001 | 0.169±0.004 | 4 |
| k=2, h=0.25 | |||||
| CTmean | 0.123±0.001 | 0.100±0.001 | 0.108±0.002 | 0.110±0.001 | −12 |
| ηP | 0.297±0.004 | 0.232±0.002 | 0.256±0.005 | 0.267±0.005 | −11 |
5. Conclusion
The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion in the Reynolds number range 200–2×106 was numerically studied. The thickness study was performed on 2-D NACA symmetric airfoils with 6%, 12%, 15%, 20%, 30%, 40% and 50% thick sections undergoing pure plunging motion at k=2, h=0.25 and 0.5 and combined pitching and plunging motion atk=2, h=0.5, =90°and θo=15° and 30°. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils because the more rounded leading edge of the thicker airfoil weakens the suction peak. At higher Re, it was found that in both the fully laminar and fully turbulent flow regimes, significant gains in thrust generation and propulsive efficiency could be achieved by using an optimum thickness airfoil section for plunging and combined motion with low pitch amplitude. At Re=20000 where the flow is fully laminar, the time-averaged thrust coefficient and propulsive efficiency was increased by 372% and 378%, respectively, for a NACA0030 airfoil undergoing pure plunging motion when compared to NACA0006 airfoil for k=2, h=0.5; this increment was reduced to 121% and 158% when h was reduced to 0.25 for NACA0020 compared to NACA0006. This benefit in the fully laminar flow regime is due to the stronger suction and larger separation zone produced by the leading edge vortex as it interacts with the increased frontal area and point of maximum thickness of a thicker airfoil, and has not been observed by previous researchers using attached flow assumptions inherent in inviscid codes and linear analysis methods. On the other hand, in the fully turbulent flow regime, the mechanism for optimum performance due to thickness is different. For the thin airfoil section NACA0006, a large leading edge vortex is produced because of the very sharp leading edge, whereas for thicker sections, leading edge separation is suppressed by turbulence, thus resulting in larger suction peak which decreases with thickness.
=90°and θo=15° and 30°. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils because the more rounded leading edge of the thicker airfoil weakens the suction peak. At higher Re, it was found that in both the fully laminar and fully turbulent flow regimes, significant gains in thrust generation and propulsive efficiency could be achieved by using an optimum thickness airfoil section for plunging and combined motion with low pitch amplitude. At Re=20000 where the flow is fully laminar, the time-averaged thrust coefficient and propulsive efficiency was increased by 372% and 378%, respectively, for a NACA0030 airfoil undergoing pure plunging motion when compared to NACA0006 airfoil for k=2, h=0.5; this increment was reduced to 121% and 158% when h was reduced to 0.25 for NACA0020 compared to NACA0006. This benefit in the fully laminar flow regime is due to the stronger suction and larger separation zone produced by the leading edge vortex as it interacts with the increased frontal area and point of maximum thickness of a thicker airfoil, and has not been observed by previous researchers using attached flow assumptions inherent in inviscid codes and linear analysis methods. On the other hand, in the fully turbulent flow regime, the mechanism for optimum performance due to thickness is different. For the thin airfoil section NACA0006, a large leading edge vortex is produced because of the very sharp leading edge, whereas for thicker sections, leading edge separation is suppressed by turbulence, thus resulting in larger suction peak which decreases with thickness.
Irrespective of whether it is in the fully laminar or turbulent flow regime, an optimally thick airfoil produces larger thrust and higher efficiency in combined pitch/plunge motion than in pure plunge motion.
It has also been shown that a cambered airfoil offers little to no benefit over symmetric airfoils in terms of the time averaged thrust coefficient and propulsive efficiency. This is because more thrust is generated in the downstroke but approximately the same amount less is generated in the upstroke due to the effective increase in thickness of upper surface and decrease in thickness on lower surface of the airfoil caused by the camber.
All the results presented in this paper are two-dimensional. Visbal (2009) found for low amplitude high frequency motions that the LEV breaks down into small-scale three-dimensional structures for Re=40 000 and above. For high amplitude cases similar to the conditions used in the present study, the LEV has been shown to remain coherent and stable (Ashraf, 2010).
Acknowledgements
Navier–Stokes calculations were performed in part with the support of a grant under the Merit Allocation Scheme of the Australian National Computational Infrastructure National Facility (NCI-NF). The first author (MAA) acknowledges receipt of a University College Postgraduate Research Scholarship for the pursuit of this study.
