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Investigation of asymmetrically pitching airfoil at high reduced frequency

The expanding application in Micro-Air Vehicles has encouraged many researchers to understand the unsteady flow around a flapping foil at a low Reynolds number. We numerically investigate an incompressible unsteady flow around a two-dimensional pitching airfoil (SD7003) at high reduced frequency in the laminar regime. This study interrogates the effect of different unsteady parameters, namely amplitude (A), reduced frequency (k), Reynolds number (Re), and asymmetry parameter (S) for pitching motion on the force coefficients. The inviscid theoretical model is utilized to calculate the lift coefficient for sinusoidal motion in the viscous regime, and a comparison is made with the numerical results. The theoretical analysis identifies the influence of the non-circulatory lift over circulatory lift at a high reduced frequency. Further, the results indicate that the reduced frequency (k) and asymmetry parameter (S) have a significant impact on the instantaneous and time-averaged force coefficients as well as on the vortex structure in the wake. Finally, the Fast Fourier Transformation (FFT) analysis is carried out over a simulated case with fixed amplitude and Reynolds number for distinct k and S values. The findings confirm that the dominant frequency in the flow (k*) has a direct correlation to the airfoil pitching frequency (k).


INTRODUCTION
For the last couple of decades, the progress in the development of Micro-Air Vehicles (MAVs) and Unmanned Arial Vehicles (UAVs) has prompted scientists and engineers to understand the flow dynamics of a flapping wing at low Reynolds number. The aerodynamics characteristic of the flapping foil depends on the various unsteady parameters, namely amplitude, flapping frequency, type of pitching motion, Reynolds number. A few researchers extensively reviewed the influence of the unsteady parameters with low Reynolds numbers 1,2 . In addition to the unsteady parameters, McCroskey et al. 3, 4 experimentally showed the impact of the vortex structure around the airfoil (leading edge and trailing edge vortex) on the force coefficients. The reduced frequency (k = ꞷC/2U∞; ꞷ = angular pitching frequency, C = Chord length of the airfoil, U∞ = incoming flow velocity) is one of the significant parameters which governs the flow physics of the flapping foil. Many researchers have examined the low reduced frequency flapping airfoil at low Reynolds number (LRN). For example, Mehta 5 numerically showed that the lift coefficient contribution is due to the pressure at low Reynolds number (Re = 5000 and 10000) at low reduced frequency (k = 0.5 and 0.25). Whereas, Akbari and Price 6 reported delayed flow separation with the increase in reduced frequency (in turn affecting the force coefficients) while conducting the two-dimensional (2D) study of pitching airfoil (modified NACA0012) using the vortex method for LRN. In an experimental study at a relatively higher reduced frequency (k = 1), Ohmi et al. 7 qualitatively observed that the vortex wake pattern changes and depends on different unsteady parameters while investigating the flow field over NACA0012 as well as on an elliptical airfoil in the range of Re = 1500 to 10000.
The effect of unsteady parameters, Reynolds number and motion type, have been investigated by different researchers on the force coefficients. The significant studies are Kim and Chang 8 , Amiralaei et al. 9 , Schouveiler et al. 10 , Cleaver et al. 11 . The most notable one is the study of Kim and Chang 8 , where they investigated the effect of Reynolds number on sinusoidally pitching NACA0012 airfoil at a low reduced frequency (k = 0.1). They observed the non-linear behavior of the lift coefficient and pressure coefficient, which was primarily due to the nonlinear response of the unsteady boundary layer at high Re. For different Reynolds numbers, Amiralaei et al. 9 numerically and analytically studied 2D flow around the pitching airfoil in the LRN regime. Their theoretical results (Theodorsen's method) differed from the numerical results for high viscous flow (at low Re) with low reduced frequency (k < 0.1). On the other hand, the experimental investigation of Schouveiler et al. 10 reported the combined pitching and plunging motion for a fixed Re = 40000. Here, the authors concluded that a cosine motion generated the most efficient thrust, and the propulsive behavior of the flapping was due to jet type wake in the downstream. In addition to the plunging motion, for small amplitude and high Strouhal number, Cleaver et al. 11 experimentally indicated that the shear layer rolls into multiple coherent vortices for high pitching frequency at Re = 10000.
Moving forward to the high-frequency oscillations, both experimental as well as computational studies show that the flow physics completely changes at high reduced frequency pitching foil as compared to low frequency. The earlier work by Garrick 12 reported that the reverse von Karman vortex street is observed for a high k; as a result, the velocity deficit (wake-type, drag dominated) profile in the wake transforms to a velocity excess (jet-type, thrust generating) profile. Later, for the high reduced frequency (k > 2), Koochesfahani 13  The article is organized as follows: the numerical and theoretical models are discussed in section 2. Section 3 presents the computational details and validation studies. The final results are discussed in section 4, followed by the concluding remarks in section 5.

Numerical Method
The governing equations for unsteady, two-dimensional, incompressible, laminar flow around SD7003 are listed below in the non-dimensional form The above equations are solved using the finite volume method in the OpenFOAM framework

Theoretical Formulation (inviscid assumption)
The analytical model, extended ONERA(EDLin) 28 dynamic stall model, is invoked to calculate the unsteady aerodynamic lift over the sinusoidally pitching airfoil. Venkatesan and Friedmannn 30 modified model for higher-order (second-order) approximation for Theodorsen lift deficiency function. The stall model assumes that forces and moment act at the quarter chord point. The unsteady lift acting on the airfoil is given by The various coefficients of the equations (3, 4, and 5) can be found from ref. 29 and the coefficient for is taken from the static data presented by Anyoji et al. 32 for SD7003.
In the present study, the maximum amplitude (Amax) is 6°, which is an unstalled condition.
Therefore, in the equation (3), Г 2 is dropped for the unstalled condition and ∆ is taken as zero 29 . The stall model is furthermore applied only to the harmonic motion, α = α 0 + A e iωt , of the airfoil. With these considerations, the final form of equation (3) is listed below. The Lift force is expressed in two terms namely circulatory lift (LC) and non-circulatory lift (LNC) which are given by , second order approximation 30  The total lift coefficient is given by equation (9), which is a summation of the non-circulatory lift coefficient (CL,NC) and circulatory lift coefficient (CL,C).

Kinematics of the airfoil motion
The extensive study exists for the typical sinusoidal motion of the airfoil with higher reduced frequency, numerically as well as experimentally. The typical sinusoidal motion of the airfoil is given by Here 0 is the mean angle of attack (AoA), and A is the maximum angle of attack. In the present study, the sinusoidal motion is compared with the asymmetric sinusoidal motion and the mean 0 is taken as zero. The asymmetry parameter S is introduced into the equation of motion to achieve asymmetric cosine wave, as shown in Fig. 1(a). The parameter S is defined as the fraction of the time spent from one trough to the next crest with respect to the time period of one cycle. Hence, for S = 0.5, relation 11 gives the normal sinusoidal motion. For the smaller values of S, i.e., S = 0.3 and 0.4, the pitching upcycle takes less time as compared to pitching down cycle implies that the pitch-up cycle becomes faster, and the pitch-down cycle becomes slower. The asymmetric sinusoidal motion is represented by Where T is the time period for one cycle. Initially, the airfoil is at the minimum amplitude (Amin), and the pitching axis for all cases is at quarter chord from the leading edge. The effect of asymmetry parameter S on the airfoil's angular acceleration (α̈) is shown in Fig. 1(b) for A = 2°. The normalized time (τ) is given by τ = t / T. All the cases considered for the present study are listed in Table I.   Figure 2 depicts the simulation domain for the present study with boundary conditions. A highquality structured, multiblock grid is prepared by using ANSYS ICEM CFD ® with two mesh regions, one is the inner rotating grid, and the other is the outer stationary grid. The sliding interference is accommodated by cyclic-AMI boundary condition, which gives freedom to provide unequal grid-points over the interface of the inner and outer circle grid. Thus, a coarse grid can be used at far-field to reduce the computational shown for three different reduced frequencies with the percentage change over coarse G1 to medium G2

BOUNDARY CONDITIONS AND VALIDATION STUDY
and medium G2 to fine G3 grid (Table II). As referred from  (Table III). Table III shows that for the present study, the mean drag

RESULTS AND DISCUSSION
This section elucidates the simulation results for pitching SD7003 airfoil at a high reduced frequency and low Reynolds number. The different unsteady parameters are analyzed and aim to interrogate the influence of an asymmetry sinusoidal motion of the airfoil. The results of the present study are given into two major parts: first part illustrates the comparison between the numerical and the theoretical results

Theoretical result (Effect of pitching amplitude)
For the steady-state response, equation (8) is converted to the state-space form, and the fourth-order Runge-Kutta time integration scheme has been used 29 . The initial aerodynamic states are assumed to be zero. The brackets < > shows the force coefficients over the fourth cycle. The theoretical result is obtained for the symmetrically pitching airfoil.
The simulations are conducted for all the combinations of parameters, which are mentioned in Table I. Only the most significant results are shown and discussed here. The total theoretical lift coefficient is compared with the numerical result for the parameters, A = 2°, and 6°, Re = 3000 and 10000, and pitching frequency k = 9 (Fig. 4). The comparison of the lift coefficient components (refer equation 9), non-circulatory and circulatory lift coefficient, entails that the non-circulatory lift coefficient is preeminent over the circulatory lift coefficient at the higher reduced frequency pitching motion. The reason behind such domination is that the non-circulatory lift is directly proportional to the angular acceleration of the airfoil (refer equation 7), and the angular acceleration varies quadratically with the angular frequency (⁓ k 2 ), implying that as reduced frequency (k) increases, the acceleration of the airfoil increases tremendously.
Hence the contribution of the non-circulatory lift is much higher than the circulatory lift. At the starting of the cycle (at α = Amin), the acceleration of the airfoil is maximum (Fig. 1(b)), so at that instant, the CL,NC will be maximum. As airfoil is going from Amin to Amax, the acceleration decreases; the effect can be seen in CL,NC which also decreases. After reaching the max AoA, the airfoil starts pitching down from where acceleration will increase so that CL,NC increases. On the other hand, the circulatory lift (CL,C) has a nominal value, and as typically observed for the low reduced frequency (k < 1) pitching airfoil, it increases with increasing AoA. The non-circulatory lift coefficient even dominates the shape of the total lift coefficient hysteresis curve. The results are similar for the other parameters combination for both Re 3000 and 10000.
The theoretical total lift coefficients are in good agreement with the numerical result for low amplitude A = 2° and high reduced frequency k = 9 for both Re 3000 and 10000. However, at the high amplitude (i.e., A = 6°), the theoretical method underpredicts the total lift coefficient compared to the numerical study for both the Reynolds numbers 3000 and 10000 (Fig. 4). The cause of a difference is that the analytical model

Reynolds number effect
Here the analysis is extended for two different Reynolds numbers, i.e., 3000 and 10000, to address the impact of Reynolds number on the force coefficients at a higher reduced frequency (k ≥ 3) for the pitching airfoil. Figure Table IV. The CL,max, and CL,min have a comparatively moderate difference over Re 3000 and 10000 at the lower pitching reduced frequency (k = 3), but with increasing the k value, the percentage difference between the peaks of the lift coefficients diminishes (Table IV). These observations indicate that at a higher reduced frequency (k = 9), the maximum and minimum lift generation is mostly dependent on the inertial forces due to the acceleration of the airfoil, and Reynolds number has a negligible effect on the peaks of the CL. This is also in agreement with the discussion in the previous section, the lift generation for high pitching rate airfoil is mainly due to the non-circulatory component where the viscous wake effect (circulatory lift) on the lift force is negligible. However, the Reynolds number effect is more significant on the drag coefficient (CD). At low reduced frequency (k = 3), the drag coefficient is higher for Re = 3000 as compared to Re = 10000, due to higher viscous effect at low Reynolds number.
Increasing reduced frequency only makes inertial forces stronger, resulting in viscous effect becoming negligible compared to inertial forces at the k = 9.
Therefore, CD varies approximately the same for both Re, as shown in Fig. 5(b). In Fig. 5(c), this same feature is observed for the force coefficients over the Reynolds numbers at k = 9 despite the asymmetric pitching motion with S = 0.3 (see Table V Fig. 6(a)) for A = 2° for distinct S values is similar in appearance with the angular acceleration ( Fig. 1(b)), which is consistent with the previous discussion that the lift coefficient depends on the angular acceleration of the airfoil at a high reduced frequency. For the same parameters with k = 9, a similar trend is observed, although the magnitude of the peaks is considerably higher than k = 3. The pressure coefficient (Cp) over the airfoil surface is additionally investigated for the case A = 2°, Re = 10000, and k = 3 for distinct S values. Fig. 7 reports the pressure coefficient for three different airfoil positions: (a) 0° upstroke, (b) 1.73° upstroke, and (c) 0° downstroke. Notably, the plot indicates the considerable effect of S on the pressure coefficient, which is useful to understand the instantaneous force coefficient behavior. As mentioned in Fig. 7(a), at 0° upstroke, the pressure coefficient on the airfoil upper side is negative, and for S = 0.3, it has minimum value as compared to S = 0.5. For the lower surface of the airfoil, the Cp value for S = 0.3 is maximum compared to S = 0.5. Due to the pressure distribution over the airfoil surface, at 0° upstroke, the instantaneous lift and drag coefficient for S = 0.3 has a higher value than S = 0.5, which can be seen from Fig. 6(c). The higher pitching rate during the pitching up motion for S = 0.3 is the reason for the higher surface pressure value. Similar pressure distribution is observed at 0° downstroke ( Fig. 7(c)) besides the lower surface attains the negative pressure distribution that explains the negative instantaneous lift coefficient at that instance (Fig. 6(c)). The value of Cp for distinct S values is pretty much close, which also reflects from Fig. 6(c) that <CL> is almost similar at 0° downstroke. Fig. 7(b) shows the pressure coefficient over the airfoil at 1. parameters also exhibit similar pressure distribution, as reflected from the identical force coefficient.

Mean force coefficients
In continuation to the earlier discussion, one can notice that the force coefficients are periodic after the first cycle and approximately, the phase values of CL and CD are duplicated during each cycle; hence, the time-averaged mean force coefficients (C L ̅̅̅ and C T ̅̅̅ ) are calculated based on only the fourth cycle. The The asymmetric sinusoidal motion (S = 0.3 and 0.4) has a remarkable effect on the C L ̅̅̅ and C T ̅̅̅ for both Re 10000 and 3000 (Figs. 8 and 9). The C T ̅̅̅ raises significantly, two-three times for S = 0.3 as compared to S = 0.5 for all the amplitudes and the Reynolds numbers. An identical increment is observed in the C L ̅̅̅ for the asymmetric pitching motion. The significant increment in C L ̅̅̅ and C T ̅̅̅ is due to the drastic increase in peaks of force coefficients as the airfoil goes under asymmetric sinusoidal motion (Fig. 6). The maximum C L ̅̅̅ = 5.4385 and C T ̅̅̅ = 0.8911 occur for the parameters combination with Reynolds number 10000, amplitude A = 6°, reduced frequency k = 9 and asymmetric parameter S = 0.3. So, it may be concluded that the asymmetric sinusoidal motion enhances the mean force coefficients over the pitching airfoil for the same amplitude (A) and the reduced frequency (k). Above all discussions talk about the quantitative analysis which shows the impact of the unsteady parameters on the force coefficients. The effect of the amplitude and the reduced frequency on the mean thrust coefficient is similar to the NACA0012 for the same range of parameters, although the magnitudes are different 13,14,15 . Furthermore, the vorticity plot is shown in the next section to examine the influence of the unsteady parameters on the flow structures.

Fast Fourier Transformation
The previous discussion supports the fact that the force coefficients and the flow structure are periodic for high reduced frequency (k ≥ 3) and small amplitude (A ≤ 6°) pitching motion for LRN. The Fast Fourier Transformation (FFT) is an efficient tool to investigate the frequency of the periodic flow. The Fast Fourier Transformation is performed for the case A = 4° and Re 10000; for k = 3, 6, and 9 with symmetric sinusoidal motion, and for k = 6 with asymmetric sinusoidal motion. As shown in Fig. 2(b), four probes are located (also see Fig. 10   Based on the inlet velocity U∞, the FFT frequency at the P1 location is converted into the FFT reduced frequency (k*), which is defined as k* = 2πfC / 2U∞, where f is the FFT frequency, and C is the chord length of the airfoil. The FFT frequencies are 0.145, 0.2899, and 0.4272 Hz for the pitching reduced frequency (k) 3, 6, and 9, respectively (Fig. 12) at the location P1. The k* based on the FFT and corresponding reduced frequencies (k) of the pitching airfoil are listed in Table VI. It is seen that for all reduced frequency cases, at location P1, the FFT reduced frequency (k*) is approximately the same with the pitching reduced frequency (k). These observations have contributed considerably towards the conclusion that in the flow domain which is not affected by the viscous boundary layer or any vortex structure, the flow property is governed by the airfoil pitching frequency. As a result, the lift coefficient for a higher reduced frequency pitching airfoil is dominated by the frequency (angular acceleration ⁓ k 2 ) of the airfoil's pitching motion. For the asymmetric sinusoidal motion (S = 0.3 and 0.4), Figure 13 reports the FFT for the same probe locations, as mentioned earlier with k = 6. It is observed that at P1 location for S = 0.3 ( Fig. 13(a)), there is more than one dominant frequency, i.e., second, third, and fourth harmonic frequencies, present due to the asymmetric pitching motion of the airfoil. Whereas for the asymmetric motion S = 0.4, only second harmonic frequency has a significant impact at P1 (Fig. 13(b)). However, for both cases, i.e. S = 0.3 and S = 0.4, the dominating frequency is 0.2899 Hz (k* = 6.07), and as mentioned above, it is equivalent to the airfoil's pitching reduced frequency (k = 6). As moving to the downstream, near the airfoil trailing edge (at P2), the similar frequencies are observed for both S values 0.3 and 0.4, which shows that near the airfoil, the flow dynamics are dominated by the airfoil motion at a higher reduced frequency. In the wake, at the location P3 and P4, there is only one dominant frequency present f = 0.2899 Hz (k* = 6.07), and the higher harmonic frequencies are insignificant.

Conclusion
The