Effect of the shape of flapping airfoils on aerodynamic forces

The rapid exhaustion of fossil fuels and the ozone depletion caused by the excessive usage of the fossil fuels has prompted researchers to look towards bioinspired designs for both propulsion and energy extraction purposes. Limited amount of work has been done to present the effects of airfoil shape on the aerodynamic forces on flapping foils. In this paper, we examine in detail the effect of airfoil camber and its position on flapping foil performance in both energy extraction and propulsion regimes. We also examine the effect of reflex camber on flapping foil performance in both flow regimes. In total, 42 airfoils are analyzed using the NACA 4 and 5-series cross-sections. The man objective of this research is to identify a trend, between airfoil shape and aerodynamic forces. The database created as a result will be used in the future work for designing a hydrokinetic turbine and a bio-inspired unmanned aerial vehicle. The results from the numerical simulations indicate that the airfoil shape has significant effects on the time averaged drag force on the airfoil in both flow regimes. However, the time averaged lift force remains negligible for all cases.


Introduction
A flapping mechanism combines the lift and thrust production, the two desired forces in aerodynamics.Flapping can be applied in various flow regimes to gain advantage in the overall efficiency of the wings and rotors of flying machines and turbine blades.The motion of birds and fish has been investigated in detail by previous researchers to make locomotion more efficient .Various researchers have studied the effects of airfoil shape on the aerodynamic performance of airfoils [10][11][12][13][14][15][16][17][18][19][20][21].A brief review of the previously published research work is presented in the following paragraphs.
A self-induced oscillating hydrofoil attached to an arm was numerically investigated at Reynolds number of 1100 using a commercial CFD code [10].The effects of camber were examined along with the effects of arm length and swing direction.In this study, the authors employed a C-shape hydrofoil.The results from the numerical simulations indicated the camber and pitch angle have significant effects on the aerodynamics forces on the airfoil, therefore on the power extraction as well.Quantitatively, a maximum efficiency of the turbine is at 0.285.The authors also studied various configurations of the arm -this portion of the study highlighted that the upwind and camber arm configuration achieves better performance over the downwind arm and camber configurations.An optimal pitch angle was found to be 143 • for upwind arm configuration [10].
The effects of large amplitude and non-sinusoidal motion on airfoil undergoing pitching motion was numerically analyzed using a commercially available code [11].The airflow was subjected to flow which corresponds to the propulsion regime.NACA 0012 airfoil was selected for this part of study.To study the effects of camber on airfoil performance, three cambered NACA sections were employed with different camber magnitudes and locations.The Reynolds number for this study was kept constant at 1.35 × 10 4 .This study revealed that adding camber to the airfoils has negligible effect on the thrust coefficient.The results for this study also indicated that the thrust coefficient increase with increase in reduced frequency while keeping the amplitude of oscillation fixed [11].
The effect of airfoil thickness and camber on airfoil propulsive performance was analyzed numerically at Reynolds number in the range of 200 and 2.0 × 10 6 [12].To study the effect of thickness on airfoil performance, the NACA 4-series symmetric airfoils with thickness varying from 0.06c-0.5cwere selected.The effect of camber was investigated using NACA 4-series airfoils with different camber magnitude and location.The airfoils considered include NACA 4215, 4415 and 4615.The results of this study revealed that adding camber to the airfoil shows a very small improvement in terms of the thrust coefficient in comparison to that of symmetric airfoils.It was also shown in this study that the thin airfoils outperform the thick airfoils at various Reynolds numbers [12].
In a relatively recent study, various types of airfoils including symmetric airfoils with varying thickness and maximum thickness locations; cambered airfoils with same thickness, thickness position and camber but different maximum camber positions; and cambered airfoils with same thickness, position and camber position but different maximum camber were analyzed [13].The airfoils investigated were NACA 4 and 6-series cross-sections.It was found in this study that efficiency of the hydrofoil first increases and subsequently decreases with an increase in maximum thickness while keeping maximum camber location constant.Furthermore, efficiency of the hydrofoil was found to increase and then decrease afterwards when the position of maximum thickness moves from the leading-edge towards the trailing-edge.It was also found that the efficiency for cambered airfoils with same thickness is lower than the airfoils with larger camber [13].
The effects of camber angle on the aerodynamic performance of a flapping wing micro air vehicle is examined in a recently published work [14].The effects of varying camber angle and flapping frequency were analyzed.It was found that the vortex structures are significantly affected by the frequency and magnitude of the twisting motion of the wing.The frequency and magnitude were found to be dependent on the camber angle.Quantitatively, on the recommended operating frequency of 24 Hz, a camber angle of 15 • has the highest propulsive efficiency.The propulsive efficiency was found to be 14 % more than the non-cambered wing [14].The maximum camber was varied from 0.0c to 0.3c to investigate the effect of maximum camber on lift for a flat plate [15].Numerical simulations were employed to study the lift characteristics of the wing.To change the camber of the wing, the chordwise curvature was applied to the cross-section, thereby resulting in various different wing shapes.It was found that the wings having higher camber result in flapping foil generating more lift.It was also reported that the most averaged lift is generated at the location of maximum camber which is nearest to the center of the mean average chord [15].
The aerodynamic effects of changes in the shape of a wing of a hovering hawkmoth were studied in detail [16].The wing shape was controlled by varying the twist, the camber and the spanwise bending.It was found that introducing curvature in the wing can increase the aerodynamic force with negligible effects on the aerodynamic performance.An optimum wing shape was found to increase lift or decrease drag by using numerical simulations [16].The effects of changing airfoil camber and wing twist investigated numerically [17].It was found that the aerodynamic forces on the flapping wing were more effected by the changing camber as compared to the wing twist.The wing deformation was shown to increase the maximum lift coefficient.Furthermore, it was suggested that the wing deformation can also result in the reduction of the power requirement for flight [17].
Four different cambers were created to study the effect of camber on the aerodynamic properties of airfoils [18].The cambers range from 0.125c to 0.25c, with increments of 0.042c.The study considered flat-plate based airfoils.The numerical simulations performed as a part of this study revealed an optimum camber of 0.167c to have the highest average lift and thrust values.The effect of camber on the lift was found to be more prominent than the effect of camber on the thrust produced by the airfoils [18].Various parameters of wings of micro-air vehicles such as airfoil camber, twist angle, and aspect ratio etc. were analyzed using CFD [19].The study also utilized flat-plate airfoils [19] with maximum camber and its location ranging from 0.05c -0.2c and 0.25c -0.75c.It was found that the amount of camber significantly changes the resultant aerodynamic force direction.Furthermore, the change in the location of the maximum camber was found to have less significant impact on either the lift coefficient or the rotary moment coefficient [19].
The effects of spanwise twist and chordwise camber for micro-air vehicles were studied in detail using numerical simulations [20].Quantitatively, the results indicated that the amount of camber plays a significant role in the improvement of both lift and L/D.An optimum configuration was found to improve lift force and L/D by up to 16.7 and 10.6 %, respectively.An experimental investigation was performed for flapping wings with and without camber to study the effect of camber amount on wing performance [21].The main goal of conducting the experiments is to study the aerodynamic effect of cambered wings for flapping-wing micro air vehicles.The study was conducted at a Reynolds number of 3600.For the amounts of cambers analyzed in this study, the C l, avg was reported to increase as the amount of camber increases while the C d, avg remained largely unchanged as compared to the flat wing [21].
The study [26] introduced a reactive control strategy using a Reinforcement Learning (RL) to govern the motion of a rotating flap on a NACA0012 airfoil for aerodynamic performance enhancement.CFD simulations were used to build data set, and Artificial Neural  Networks (ANNs) model aerodynamic coefficients.The trained RL agent effectively adapts to different angles of attack (AoA), achieving nearly optimal results.Rapid and accurate responses were observed for step, ramp, and random signals, demonstrating the neural network's versatility.The study was conducted at a Reynolds number of 1.9 × 10 6 .It is interesting to note neural networks being used for aerodynamic optimization [26].A research work compared the effect of various turbulence models on flapping foil.By comparing the results with both known experimental and calculated data, it was demonstrated that the flows having developed and significant flow separation, which are characterized by unsteady effects in turbulence, the SALSA turbulence model exhibited higher performance than the other tested models.By comparing the results with the established experimental data, the aerodynamic coefficients obtained with the SALSA turbulence model were found to be 10-15 % more accurate than those obtained with other turbulence models [27].
It can be clearly seen from the literature review presented that the airfoils selection is generally random when it comes to studies concerning the airfoil shape in flapping motion.A limited range of airfoil cambers and the location of maximum camber was investigated [10][11][12][13][14][15][16][17][18][19][20][21].Therefore, a more methodological and rigorous design space of 42 airfoils is created as a part of the present study, as explained in Section 2, Sub-Section B. Numerical methodology is presented in the "Methodology" section which is followed by the "Results and Discussion" section.

Methodology
The flapping motion in airfoils is achieved by a combination of heaving and pitching motion.The motion of airfoil is illustrated in Figs.1-2.Within Fig. 2, the airfoil motion is from left to right of the viewer.The airfoil motion is viewed from the reference from of free stream velocity, U ∞ .The governing equation for the airfoil motion in the power extraction regime is given by equations ( 1) and ( 2).
While equations ( 3) and ( 4) present the equations of motion for the airfoil in the propulsion regime.The phase angle is kept at π/2 rad for all the cases.ϑ m is kept at 0 rad for the power propulsion cases [1].Air as an ideal gas is considered as a fluid for all the cases studied.The ambient pressure and temperature are kept at 101325 Pa and 293.15 K, respectively.Various parameters used in the propulsion and power extraction cases is mentioned in Table 1.
The aerodynamic characteristics of the selected airfoils are analyzed using a commercially available CFD code, ANSYS Fluent [22].Fluent employs the cell-centered finite volume method.We use the laminar model along with the SIMPLE solution algorithm.The discretization of the continuity, momentum and energy equations (equations ( 5)-( 7)) [23] is obtained by using the second-order accurate Upwind scheme.Time is discretized using the implicit second-order accurate scheme.Dynamic mesh with smoothing and remeshing options is used to simulate the flapping motion via "define_cg_motion" UDF [23].The UDF is coded as part of this research work.
F. Butt et al.

A. Validation and Verification
The accuracy of the present numerical simulations is ascertained by comparing the results with the published data [1,2].The data comparison with [1] is for the propulsion regime while the validation for the power extraction regime is done by comparing the present results with [2].A square computational domain is chosen for the present simulations.The size of the computational domain is kept at 40c in all directions, which is larger in size than the size used in the previously published literature [1,2].This is done to ensure that the walls of the computational domain have no effect on the simulation results.The mesh and computational domain are presented in Fig. 7 showing localized mesh refinement in various areas of interest.
The C d, avg from the present numerical simulations comes out to be 0.984 as compared to 0.996 from the published results [1], for the propulsion flow regime.The C d, avg from the present numerical simulations is 2.024 while the C d, avg from Ref. [2] is 2.017.Meanwhile, C l, avg from the present simulations is negligible as is the case in published literature [1,2].The C l, max for energy extraction regime in Ref. [2] is 1.902 in comparison to 2.024 from the present simulations.For the propulsion flow regime, the published C l, max [1] is 4.217 as compared with the C l, max from present simulations 4.148.The comparison of the results from the simulations performed as a part of this study with previously published literature is presented in Figs.3-4.The plots are for one complete flapping cycle.
Furthermore, mesh and time-step independence tests are carried out to ensure that the results from the numerical simulations depend on the boundary conditions alone and are not affected by these two properties.The results from the mesh and time-step independence studies are presented in Figs.5-6 for both flow regimes.It is clear that for both flow regimes, decreasing the time-step and cell size do not yield any accuracy improvements over the initial cell and time-step size.Therefore, the initial mesh and time-step size is selected for further simulations performed as a part of this study.It is important to note that the initial mesh and time-step size are used in the validation of our work with previously published data, as explained in Section 2, Sub-Section A. For clarity, only 1/10th of the data points are shown for the Case T1 and M2.The details about the cases M1, M2 and T1 are mentioned in Table 2.For the case M2, the mesh count is increased by 150 % as compared to the case M1.However, the observed difference in the results is only 0.5 %.The case T1 has time step reduced by 50 % as compared to the case M1 with negligible change in the results.It should be noted that local mesh controls are used to refine the mesh in areas of interest i.e. around and in the wake of the airfoil to ensure solution accuracy, as shown in Fig. 7.The mesh, made up of quadrilateral cells.The mesh for the case M1 is shown in Fig. 7.A closeup of the mesh at the trailing-edge and in the wake of the airfoil is also shown within Fig. 7 to clearly show localized grid refinement.Furthermore, 96 % of the cells in the mesh had orthogonal quality of unity.F. Butt et al.

B. Airfoil Shapes
To study the effect of airfoil camber, NACA 4-series airfoils are considered.The amount of camber is changed from 0.02c to 0.08c, in increments of 0.02c.The camber position is varied by changing the camber location systematically from 0.2c to 0.8c, in increments of 0.2c.To study the effect of reflex camber on flapping foil performance, NACA 5-series airfoils are considered.The camber position and reflex are increased from 0.1c to 0.25c in increments of 0.05c and the design lift coefficient are changed from 0.2 to 0.8 in increments of 0.2.The details about equations used to plot NACA 4-series airfoils can be found in Ref. [24] while details about the NACA 5-series airfoils are available in Ref. [25].

Results and Discussion
The results and trends from the numerical simulations are presented and discussed in the following Sub-Sections.The quantitative results are tabulated in Tables 3-8.It is to be noted that for the airfoils operating in the energy extraction flow regime, the minimum C l, inst occurs near 0.5T (end of downstroke) while the maximum C l, inst exists near 1.0T (end of upstroke).Meanwhile, for the airfoils operating in the propulsion regime, minimum C l, inst occurs near 0.75T (middle of downstroke) and maximum C l, inst exists near 0.25T (middle of upstroke).The maximum C d, inst occurs near 0.25T and 0.75T.The minimum C d, inst exists near 0.0T, 0.5T and 1.0T for both flow regimes.

A. NACA 4-Series Airfoils
The discussion about NACA 4-series airfoils operating in the energy extraction regime is presented first.The maximum C d, inst increases as the amount of camber is increased.The minimum C d, inst also increases as the amount of camber is increased.C d, avg increases with an increase in the amount of maximum camber for airfoils having the location of maximum camber at the extreme ends of the airfoils i.e. near leading and trailing-edges.Meanwhile, C d, avg increases then subsequently decreases afterwards as the amount of camber is increased for the airfoils having maximum camber location near the middle of the chord length of the airfoils.The maximum C d, inst increases as the camber position is moved towards the leading-edge of the airfoils.The minimum C d, inst increases as the location of the maximum camber is moved away from the leading-edge of the airfoil.C d, avg increases as the camber location is moved towards the leading-edge of the airfoils.
The maximum C l, inst increases as the amount of maximum camber increases while the minimum C l, inst shows an opposite trend.The maximum C l, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoil.The minimum C l, inst increases as the location of the maximum camber is moved towards the leading-edge of the airfoil.C l, avg remains zero for all the cases while the airfoil is in the energy extraction flow regime.
The post-processed data from the NACA 2210 and 8810 airfoils is used to explain the change in C d and C l due to variations in the maximum camber and maximum camber location as shown in Fig. 8(a) and (b).The magnitude of vorticity around the axis perpendicular to both C l and C d during the downstroke and upstroke is shown in Figs.8-9.An anticlockwise vortex is observed at the leading-edge of the two airfoils during downstroke and vice-versa.A vortex forming at the trailing-edge with opposite direction of rotation to the leading-edge vortex is also observed as shown in Fig. 9(a) and (b).The trailing-edge vortex is stronger in magnitude for airfoils with high camber and the location of camber near the trailing-edge of the airfoils during the downstroke and vice-versa.
This vortex formation leads to the formation of corresponding zones of low pressure in the wake of the airfoils, resulting in a higher C d, avg for airfoils with high camber and maximum camber location near the trailing-edge.The low-pressure regions in the wake of  If the location of the maximum camber is moved towards the leading-edge of the airfoil, the maximum C l, inst increases.The minimum C l, inst also increases as the location of the maximum camber is moved towards the leading-edge.A reduction in the amount of maximum camber causes the maximum C l, inst to increase for the cases when the maximum camber location is far away from the   leading-edge and decreases when the location of the maximum camber is near the leading-edge of the airfoil.The minimum C l, inst increases as the amount of maximum camber decreases when location of maximum camber is far away from the leading-edge of the airfoil.The minimum C l, inst increases with increase in maximum camber when location of maximum camber is near the leading-edge of the airfoil.C l, avg remains negligible for all the cases investigated.
As was the case with the energy extraction regime for 4-series NACA airfoils, NACA 8810 and 2210 airfoils are employed to further explain the results.A vortex of higher strength and size is formed at the leading-edge of the airfoils with large camber and the location of maximum camber near the leading-edge of the airfoils.This vortex after being shed, leads to zones of relatively lower pressure in wake of the airfoils with high camber with location of maximum camber located near the trailing-edge of the airfoils.This particular vortex dynamics leads to more drag being produced by airfoils with less maximum camber and the location of maximum camber near the leading-edge of airfoils.The pressure distribution around and in the wake of the airfoils in shown in Fig. 11(a As is observed in the case of the energy extraction regime, the gain in C l, avg for downstroke is canceled out by an increased amount of negative lift in the upstroke.Therefore, the C l, avg for the complete cycle remains negligible for all the airfoils in the propulsion flow regime as well.

B. NACA 5-Series Airfoils
In the proceeding section, the results for NACA 5-series airfoils operating in the energy extraction flow regime are presented.For NACA 5-series airfoils with reflex camber cross-section, C d, avg increases 2.054 and achieves a maximum value of 2.112 as the amount of maximum camber increases as presented in Table 5.The maximum C d, inst increases and achieves a maximum value of 0.114 then it starts to subsequently decrease as the amount of maximum camber is increased for airfoils having the maximum camber location near the leading-edge of the airfoils.The maximum C d, inst increases as the amount of maximum camber increases for airfoils having the maximum camber location away from the leading-edge of the airfoils.C d, avg increases as the location of maximum camber is moved away from the leading-edge of the airfoil.The maximum C d, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoil for airfoils having low maximum camber.The maximum C d, inst increases then decreases for airfoils having The maximum C l, inst first increases then decreases as the amount of camber decreases when the location of maximum camber is near the leading-edge of the airfoil.The maximum C l, inst increases as the amount of camber increases when the location of maximum camber is away from the leading-edge of the airfoil.The maximum C l, inst first increases and then decreases as the location of maximum camber is moved away from the leading-edge of the airfoil for the airfoils with lower maximum camber.The maximum C l, inst increases as the location of location of maximum camber is moved away from the leading-edge of the airfoil for the airfoils with higher maximum camber.C l, avg remains negligible for all cases.
For NACA 5-series airfoils without reflex camber cross-section, C d, avg increases as the amount of camber increases.The maximum C d, inst increases then decreases as the amount of camber increases for airfoils having the location of maximum camber near the leading-edge of the airfoils.The maximum C d, inst increases as the amount of camber increases for the airfoils having the location of maximum camber away from the leading-edge of the airfoils.The C d, avg first increases then decreases when location of maximum camber is moved away from the leading-edge of the airfoil for the airfoils with lower maximum camber.C d, avg increases as the location of maximum camber is moved away from the leading-edge of the airfoil for the airfoils with higher maximum camber.The maximum C d, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoils.The minimum C d, inst remains negligible for all the cases in this flow regime.
The maximum C l, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoil for the cases when there is low to moderate maximum camber.The maximum C l, inst increases as the location of maximum camber is moved away from the leading-edge of the airfoil for airfoils with high maximum camber.The minimum C l, inst increases as the amount of maximum camber is moved away from the leading-edge of the airfoils.The maximum C l, inst increases as the amount of maximum camber is decreased for airfoils with location of maximum camber near the leading-edge of the airfoils.The maximum C l, inst increases as the amount of camber is increased when the location of maximum camber is away from the leading-edge of the airfoils.The minimum C l, inst increases with increase with the increase in the maximum camber.C l, avg remains negligible for all the cases in this flow regime.
In the following paragraphs, the results for NACA 5-series airfoils operating in the propulsion regime are presented.For NACA 5series airfoils with reflex camber cross-section, C d, avg increases as the location of maximum camber is moved towards the leading-edge of the airfoils for airfoils having moderate to high maximum cambers.The minimum C d, inst increases as the location of maximum camber is moved away from the leading-edge of airfoil.As the maximum camber of the airfoils in decreased, an increase in C d, avg is observed.The minimum C d, inst increases as the amount of maximum camber is increased.The maximum C d, inst remains negligible for all cases in this flow regime.
The maximum C l, inst increases as the location of maximum camber is moved away from the leading-edge of the airfoil for airfoils having lower maximum camber.The maximum C l, inst first increases then decreases as the location of camber is moved away from the leading-edge for airfoils with higher maximum camber.The minimum C l, inst increases as the location of maximum camber is moved near the leading-edge of the airfoils for airfoils with lower maximum camber.The minimum C l, inst increases then decreases as the location of maximum camber is moved away from the leading-edge of the airfoils for the airfoils with higher maximum camber.The minimum C l, inst increases as the amount of maximum camber is decreased.C l, avg remain negligible for all the airfoils in the propulsion regime.
For NACA 5-series airfoils without reflex camber cross-section, C d, avg increases as the amount of camber is reduced.The minimum C d, inst increases as the amount of maximum camber in increased.C d, avg first increases then decreases as the location of the maximum camber is moved towards the leading-edge of the airfoils for airfoils having low maximum camber.For airfoils having more maximum camber, the C d, avg increases as the maximum camber location of moved towards the leading-edge of the airfoils.The maximum and minimum C d, inst increases as the location of maximum camber is moved away from the leading-edge of the airfoils.
The maximum C l, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoils.The minimum C l, inst increases as the location of maximum camber is moved away from the leading-edge of the airfoils for airfoils having low maximum camber.The minimum C l, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoils for airfoils having high maximum camber.The maximum C l, inst increases as the amount of maximum camber is increased.The minimum C l, inst increases as the amount of maximum camber is decreased.C l, avg remains negligible for all airfoils in this regime.
The observed trends reported in Tables 5-6 are further explained using the pressure contours for flapping motion of NACA 22010 and 54010 airfoils.The downstroke at various time instances is shown in Fig. 12(a) and (b) while the upstroke is shown in Fig. 13(a) and (b).Within Figs.12-13, the dashed line, "-", represents C d, inst while the solid line "__" represents C l, inst .It is observed that there is negligible difference between C d, inst during the downstroke between the two airfoil types.The C d, avg for downstroke is 1.94 for NACA 54010 as compared to 1.92 for NACA 22010.This negligible difference in C d, avg can be explained by almost the same pressure values at the aft and fore of the airfoils.The NACA 22010 is observed to have slightly more negative C l, inst between ~0.1T and ~0.3T.The more negative lift is due to the lower pressure at the leading-edge of the airfoil which in turn is due to the formation of the leading-edge vortex.The vortex formation takes place earlier in the cycle for airfoils with low maximum camber and location of camber near the leading-edge of the airfoils.As a result, the C l, avg for the downstroke is around 3 % more NACA 22010 as compared to NACA 54010.The C l, avg comes out to be − 1.174 for NACA 54010 as compared to − 1.214 for NACA 22010.The C d, avg for NACA 54010 during upstroke comes out to be 2.236 as compared to 2.183.The higher C d, avg is due to higher pressure difference between aft and fore of the NACA 54010 as compared to NACA 22010, specially at ~0.55-0.65Tand ~0.75T-0.9T,as visible in Fig. 13.At ~0.55T for example, the higher pressure in the concave region near the leading-edge, on the pressure side of the NACA 54010 airfoil is the cause for higher C d, avg .Meanwhile, the relatively lower pressure in the wake of the NACA 54010 airfoil is responsible for a higher C d, avg at ~0.85T.The NACA 54010 has more positive C l, inst in the at ~0.68-0.91T.This can be explained by the lower pressure at the leading-edge of the NACA 54010 airfoil, caused by the leading-edge vortex as the vortex sheds.The C d, avg during upstroke for NACA 54010 comes out to be 1.503 as compared to 1.462 for the NACA 22010.It is to be noted that the in the contours shown throughout the manuscript, the red  The observed trends of maximum and average force coefficients are elaborated using comparison of the results for NACA 45110 and 45010 airfoils.The results can be explained by closely looking at the C p plots at 0.8T as shown in Fig. 16.The pressure contours are also shown in Fig. 16(a) and (b).There is a relatively large pressure difference between the leading-and the trailing-edge of the NACA 45010 airfoil as compared to the NACA 45110 airfoil, resulting in more C d, avg in the airfoil without reflex camber.It is also evident from the C p plot and as well as from the pressure contours that a larger pressure difference exists between the suction and pressure sides of the NACA 45110 as compared to NACA 45010, resulting in higher C l, avg for NACA 45110.This pressure difference is more evident from mid-chord towards the trailing-edge of the airfoils.It is also noticed that the higher-pressure region is confined within the concave region on the pressure side of the airfoil with reflex camber cross-section.
Furthermore, it is evident from Fig. 17(a) and (b) that stronger vortex formation takes place at the leading-edge of the airfoil with reflex camber cross-section.This vortex results in the lower pressure zones in the wake of the NACA 45110 airfoil.It should also be noted that due to a higher curvature on the pressure side of the airfoils with reflex camber cross-section, a relatively higher pressure is      developed on the pressure side of the airfoils with reflex camber.This along with lower pressure due to the stronger leading-edge vortex contributes to a higher C d, avg and C l, avg for airfoils with reflex camber.

Conclusion
The main objective of the present research is to conduct a systematic study to analyze the effect of the airfoil shape on the aerodynamics forces for the flapping motion of an airfoil.By analyzing the results, it can be clearly seen that airfoils with high maximum camber located near the leading-edge of the airfoils generally have maximum drag and lift for energy extraction regime.For NACA 4series airfoils operating in the propulsion regime, a trade-off is noticed between C l, avg and C d, avg .Generally, airfoils with maximum camber location near the leading-edge of the airfoil have the most C l, avg and C d, avg , while the airfoils with higher camber have more C l, avg and vice-versa.For airfoils operating in the energy extraction regime, the airfoil having the most C d, avg is identified as NACA 8210.Meanwhile, the NACA 4-series airfoil with most C l, avg is the NACA 6210.For NACA 4-series airfoils operating in the propulsion regime, the airfoil having the most C d, avg is identified as the NACA 2210 while the airfoil with most C l, avg is identified as the NACA 8210.Therefore, the NACA 8210 can be described as the best performing NACA 4-series airfoil, in terms of aerodynamics forces.For NACA 5series airfoils, it is found that a higher C l, avg is shown by airfoils with a high maximum camber and location of the maximum camber located away from the leading-edge of the airfoils for both flow regimes.This phenomenon also holds true for C d, avg for airfoils operating in the energy extraction regime.For the propulsion regime, a higher C d, avg value is achieved by airfoils with low maximum camber, having the maximum camber near the leading-edge of the airfoils.For the NACA 5-series airfoil without reflex camber crosssection, operating in the energy extraction flow regime with most C d, avg and C l, avg is shown by 55010.For the NACA 5-series airfoil with reflex camber cross-section, the most C d, avg is 55110 and the most C l, avg is shown by 54110.For the NACA 5-series airfoil without reflex camber cross-section, operating in the propulsion regime, most C d, avg is shown by 24010 and the most C l, avg is shown by 54010.For the NACA 5-series airfoil with reflex camber cross-section, the most C d, avg is 23110 and the most C l, avg is shown by 54110.This study will facilitate the selection of the best airfoil for the development of a hydrokinetic turbine and a micro air vehicle.The research work on these projects in on going.

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.Butt et al.

Fig. 3 .
Fig. 3. Comparison of the results of present simulations with previously published results, in the propulsion regime [1].

Fig. 4 .
Fig. 4. Comparison of the results of present simulations with previously published results, in the energy extraction regime [2].

Fig. 5 .
Fig. 5. Comparison of the results for various verification cases, energy extraction regime.

Fig. 6 .
Fig. 6.Comparison of the results for various verification cases, propulsion regime.

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Fig. 7 .
Fig. 7.The computational domain and the mesh.
airfoils is clearly visible in Fig.10(a) and (b).The C p is plotted along the yellow line which is visible in the wake of airfoils.It should be noted that the diameter of the vortex for NACA 8810 airfoil is 0.12c while the diameter of the vortex in the wake of NACA 2210 comes out to be 0.069c.In terms of C l, avg , it is observed that the gain in C l, avg for upstroke due to the leading-edge vortex is canceled out by additional negative lift generated during the downstroke.Hence, the C l, avg for the complete cycle remains zero.For the NACA 4-series airfoils operating in the propulsion regime, C d, avg increases as the amount of maximum camber decreases.The minimum C d, inst increases as the amount of camber increases.C d, avg at first decreases then increases as the location of the maximum camber is moved towards the leading-edge of the airfoil.The minimum C d, inst increases as the location of maximum camber is moved towards the leading-edge of the airfoil.The maximum C d, inst remains negligible for all cases.

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).The C p values shown in Fig. 11(b) are extracted along surface of the airfoils.The diameter of the vortex for NACA 2210 airfoil is observed to be 0.31c while the vortex diameter for the NACA 8810 is 0.58c.The diameter is clearly indicated by the yellow lines visible towards the top right corner of Fig. 11.

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Table 1
Parameters of flow for each regime.

Table
Comparison of various cases.
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