Numerical Investigation of Non-Stationary Parameters on Effective Phenomena of a Pitching Airfoil at Low Reynolds Number

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1 Journal of Applie Flui Mechanics, Vol. 9, No. 2, pp , Available online at ISSN , EISSN Numerical Investigation of Non-Stationary Parameters on Effective Phenomena of a Pitching Airfoil at Low Reynols Number A. Naeri, M. Mojtahepoor an A. Beiki Aerospace Research Institute, Malek -Ashtar University of Technology, Tehran, Iran Corresponing Author Mojtahepoor@gmail.com (Receive December 23, 2014; accepte May 13, 2015) ABSTRACT Various applications of ornithopter have le to research interest in oscillation airfoils which affect on low Reynols number flight, like; pitching oscillation, heaving oscillation an flapping of a wing. The purpose of this stuy is investigation of aeroynamic characteristics of NACA0012 airfoil with a simple harmonic pitching oscillation at zero an 10 egrees of mean angle of attack. Therefore the effects of unstable parameters, incluing oscillation amplitue up to 10 egrees, reuce frequency up to 1.0, center of oscillation up to 6/8 chor length, an Reynols number up to 5000 have been stuie numerically. A pressure base algorithm using a finite volume element metho has been use to solve Navier-Stokes equations. Accoring to results, variation of each stuie parameters at mean angle of attack of 0 egree o not cause significant changes in flow phenomena on airfoil but at mean angle of attack of 10 egrees, changing in reuce frequency an specially Reynols number cause variations in flow phenomena. These variations are because of wake capturing an/or ae mass phenomena. Keywors: Oscillation amplitue; Reuce frequency; Center of oscillation; Pitching oscillation. 1. INTRODUCTION Appearances of flapping MAVs are the main reason for stuying of low Reynols flui ynamics. These evices are esigne in terms of non-stationary flow with moving wings. Due to small size an low spee of free-flow, the range of Reynols number is too low. The successful esign an moeling require the exact solution of the flow fiel which this solution totally relies on the knowlege of flow pattern an structure. Stuies of two-imensional unsteay low Reynols number flow is quite useful in perception the flow characteristics an it can be useful in preiction of aeroynamic efficiency epenence on various parameters such as oscillation amplitue, reuce frequency, Reynols number, an kinematic pattern. Also it reviews major impact on the comprehension of threeimensional phenomena such as wing vortices, flow in transverse irection an vortex interactions. Many researchers have been stuie low Reynols number flow aroun oscillating airfoils numerically, analytically an experimentally. The applications of analytical solution are limite to simplifie flows such as quasi-steay solutions or Theoorsen metho for two-imensional oscillation airfoils (Leishman 2006). For a pitching airfoil, which is consiere as a representative instance of a nonstationary flow, the most important components are oscillating amplitue, mean angle of attack, reuce frequency an Reynols number (Chang an Eun 2003). Dynamic stall is generally iscusse by researchers because wing motion amplitue is very great over the stall angle. Beyon the stall angle, oscillating amplitue an mean angle of attack are the ominant parameters because they affect leaing-ege vortex an trailing-ege vortex formations, which specify aeroynamic performance in ynamic stall (Ohmi et al. 1991). However, acceleration effect critically influences bounary-layer events within the stall angle, an it is ominantly epenent on pitching irection (Ericsson an Reing 1988). Kim measure aeroynamic coefficients for a pitching airfoil at 4 Re withinoors stall angle of 6 o. In accorance with their stuy, the unsteay pressure coefficient on the airfoil surface ramatically change with pitching irection (Kim et al. 2013). Jung an Park experimentally emonstrate that the vortices of an oscillating airfoil are strongly influence by reuce frequency (Jung an Park 2005). Experiments by Fuchiwaki an Tanaka on an oscillating airfoil at Reynols number of 4000 show that vortices an their sizes epen on reuce frequency. On the other sie, these structures are inepenent of airfoil shape an

2 set up angle (Fuchiwaki an Tanaka 2006). Koochesfahani have been one many researches in low Reynols number experimentally an showe frequency an oscillation amplitue have significant effect on vortices structure an propellant coefficient of an oscillating airfoil (Koochesfahani 1989). Shih in a similar stuy carrie the interactions of vortex-vortex an their effects on aeroynamic forces (Shih et al. 1995). Akbari an Price simulate flow aroun an oscillating airfoil at Re=10 4 by solving Navier- Stokes equations an reporte reuce frequency an Reynols number have highest an lowest impact on aeroynamic forces respectively (Akbari an Price 2003) Yung an Lai have stuie the effects of frequency an oscillations amplitue on flow fiel aroun a two-imensional airfoil. They showe propellant coefficients are more epenent on the multiplie of k. to or k singly where k is reuce frequency an is angle of attack (Yung an Lai 2004). The effect of Reynols 4 4 number in range of Re on aeroynamic characteristics of a pitching NACA0012 airfoil have been experimente (Kim an Chang 2014). In their stuy, reverse flow an near wake were visualize by smoke-wire flow visualization, an lift an pressure rag coefficients were estimate by measuring unsteay pressure through pressure istortion correction. They was use an airfoil sinusoi-pitche at quarter chor, an its mean angle of attack an oscillation amplitue were 0 egree an 6 egree respectively. The test Reynols numbers were Re , an with reuce frequency of k 1.0. In accorance with their stuy, the reverse flow visualization, the first an secon trailing-ege vortices an mushroom structure epening on the Reynols number. They showe, in lift an pressure rag coefficients, hysteresis loops comparatively varie with the Reynols number an so the phase angle, at which bounary-layer events occurre, is in inverse proportion to the increase in Reynols number. This result implies that the increase in Reynols number promotes the occurrence of bounary-layer events such as laminar separation an transition (Kim an Chang 2014). The investigations on oscillating airfoils show that the aeroynamic coefficients are affecte by oscillation function characteristics ramatically (Amiralaei et al. 2010). The aim of this stuy is investigation of aeroynamic characteristics of NACA0012 with harmonic pitching oscillation at lower Reynols numbers. Therefore the effects of unstable parameters, incluing oscillation amplitue, mean angle of attack, reuce frequency, an Reynols number have been stuie by solving the Navier- Stokes equations. The mean angles of attack were chosen equal to zero an 10 egrees, because at mean angle of attack of zero, the airfoil oes not meet separation but at mean angle of 10 egrees separation happens. Therefore, changing flow regime is stuie in non-separation an separation situations. We have use a pressure base algorithm using a finite volume element metho to solve Navier-Stokes equations. 2. GOVERNING EQUATIONS AND NUMERICAL MODELING The governing equations incluing the conservation equations of mass an momentum in Eulerian Lagrangian space is written as follows: V V S 0 t t t (1) V V V V S nˆ S t t S t S t (2) where V uiˆ vj ˆ is flow velocity vector, V ui ˆ vj ˆ is control volume surfaces velocity vector, S ( S )i ˆ ( S )j ˆ x y is normal vector to surface with magnitue of 2 2 1/2 S [( Sx) ( Sy ) ], nˆ ( n ) ˆ ( n ) ˆ x i y j is a unit normal vector to surface an is volume which is relate to control volume. is total stress tensor which inclues hyrostatic pressure p, an shear stress tensor σ = pδ +τ. ij Geometric conservation equation is in mass conservation equation. If the flui particle velocity is zero an ensity of the flui is one, the mass conservation equation for each swept surface of control volume will calculate by: V S m V (3) t m where m is surfaces counter of each finite volume an V is relate volume to each finite sub-volume. Figure 1 shows a control volume which inclues 5 sub control volumes (SCV) an the surroun consist of 10 surfaces (SS). S S SS9 SCV2 SCV1 SCV5 ij SCV3 ij A Control Volume SS10 SS1 SS8 SS7 SCV4 SS2 SS3 SS4 Fig. 1. A control volume consist of 5 SCV an 10 SS. SS6 SS5 644

3 If Eq. (3) always satisfie, the volume conservation of SCV will be establishe. In other wors, the swept surface of SCV shall be calculate that by using of Eq. (3), the exact amount of V / t is reache. We have use the secon-orer approach to approximate SCV erivation: t n 1 n n 1 V (3/2) V 2 V (1/2) V (4) Δt where n is counts time steps. By performing a series of arithmetic operations on the above equation it can be rewritten as follows: n+1 n V (3/ 2)( V ) (1/ 2)( V ) t t (5) n 1 n 1 n n n n 1 where V V V an V V V. If movement of surface causes increasing in volume SCV the amount of t become positive an vice versa. By using of this relation, one part of momentum equation which inclues the coefficient of gri movement effects can be rewritten for each swept surface as follow length scale an velocity scale in Reynols number an reuce frequency formulations are chor length an free stream velocity, respectively. Also, time is nonimensionalize with them. Variables Oscillation amplitue Reuce frequency Reynols number Reuce frequency Table 1 Case stuies Oscillation amplitue Reynols number x/c Ratio ~ / ~ an / an ~5000 1/4 x/c Ratio an ` 1/8~6/8 V V S m V m (6) Coupling of mass an momentum equations, an also, approximation of convection, iffusion an pressure terms performs by a finite volume element metho in Eulerian Lagrangian space such as in Eulerian space (Naeri et al. 2009). Fig. 2. Case characterization. 3. CASE AND PREPROCESSING STUDIES To investigate the effects of non-stationary parameters of pitching airfoil we have consiere NACA0012 airfoil at meant angle of attack of zero an 10 egrees. The chor length, c is unit an other lengths are nonimensionalize with it. NACA 0012 airfoil oscillating function is () t 0 1sin(2 f t), where 0 is mean angle of attack, 1 is oscillation amplitue an f is frequency which is nonimensionalize with reuce frequency, k as k fc U. Free stream velocity is U which has a unit magnitue an other velocities are nonimensionalize with it. The effects of oscillation amplitue, reuce frequency, Reynols number an center of oscillation location on aeroynamic coefficients have been investigate. For these simulations the efaults amounts have been set in table 1 an case characterization has been illustrate in Fig. 2. We have use unstructure gri for this stuy. Figure 3 shows some part of the gri before pitching movement. Also we have consiere tetraheral gri on airfoil an the unstructure triangular gir up to free bounaries, inlet an outlet. Our strategy of using hybri gris on internal an external flow was escribe exactly in reference (Darbani an Naeri 2006). The istance of center of airfoil from free bounaries an outlet is 30. The outlet pressure is also unit, Fig. 3. Unstructure gir aroun the airfoil. To investigate the sensitivity of solution to computing gri, three gris with ifferent noe number of 5989, 8130 an have been use. Figure 4 shows the sensitivity of solution to the gris in Reynols number of In accorance with Fig. 4, we can use the gri with number of 8130 noes in the next computations. Similarly, to etermine the sensitivity of the solution to the time steps, we have solve the flow in time steps of 0.025, 0.05 an 0.1 in Reynols number of 5000 separately. The obtaine results have been illustrate in Fig. 5. Accoring to Fig. 5, the time step in all other computations has been set to

4 with the computational results by other researcher very well. 4. RESULTS Fig. 4. Investigation of gri-inepenency in Reynols number of 5000, rag coefficient. 4.1 Effects of Oscillation Amplitue Effects of oscillation amplitue on aeroynamic coefficients in Reynols number of 1000 an reuce frequency of 0.5 have been illustrate in Figs. 7 an 8. In accorance with Fig. 7, the maximum lift is more at higher oscillation amplitues. Also the with of lift curve became greater for higher amplitues. The lift prouction at higher oscillation amplitue at upstroke in a same angle of attack than stationary airfoil- is more an also at ownstroke is less than stationary airfoil. Lift coefficient in upstroke is more than ownstroke. Figure 8 shows maximum of rag increase an its minimum ecrease. A minimum an maximum amount of rag coefficients will be appeare in upstroke an ownstroke that the maximum amounts are equal an minimums are equal also. The rag coefficient at zero angle of attack is not almost influence by oscillation amplitue. Fig. 5. Investigation of time-step-inepenency in Reynols number of 5000, rag coefficient. To further valiate our written coe, the pure pitching sinusoial motion of a NACA0012 airfoil with amplitue of 5 o an reuce frequency of 9 at Re= was simulate. The comparison of lift coefficient between our computational results an Lu (K. Lu et al. 2013) is shown in Fig. 6. Fig. 7. Effects of oscillation amplitue on lift coefficient at mean angle of attack of 0 egree Fig. 6. Variation of lift coefficient with time at reuce frequency of 9 an amplitue of 5 o for a pitching NACA0012 airfoil (Re=13500). Fig. 8. Effects of oscillation amplitue on rag coefficient at mean angle of attack of 0 egree As can be seen, our numerical preictions agree 646

5 Fig. 9. Effects of oscillation amplitue on lift coefficient at mean angle of attack of 10egree Copyright. Fig. 11. Effects of reuce frequency on lift coefficient at mean angle of attack of 0 egree Figures 9 an 10 show, effects of oscillation amplitue on lift an rag coefficients at mean angle of attack of 10 egrees. In accorance with Figs. 9 an 10, by increasing of oscillation amplitue, minimum amounts of lift an rag a little varie at higher amplitues but the maximum amounts increase graually. Lift an rag coefficients curves treat as expecte by increasing amplitue. Fig. 12. Effects of reuce frequency on rag coefficient at mean angle of attack of 0 egree Fig. 10. Effects of oscillation amplitue on rag coefficient at mean angle of attack of 10egree 4.2 Effects of Reuce Frequency Effects of reuce frequency on the aeroynamic coefficients in oscillation amplitue of 10 egrees, mean angle of attack of 0 egree an Reynols number of 1000 have been shown in Figs. 11 an 12. Accoring to Fig. 11, the with of oscillation curve increases by growth of reuce frequency, inee, in reuce frequency of 1.0 the growth in oscillation curve with is more than other frequencies. Accoring to Fig. 12, maximum of rag coefficient increases, an minimum rag coefficient ecreases an oscillation curve with has increase by increasing of reuce frequency. At k 1, the minimum an maximum amounts of rag coefficient in upstroke an ownstroke are happene at about ~ 8 egree. Also, the rag coefficient at zero angle of attack has increase by increasing of reuce frequency. By increasing of k significant changes has not been seen in behavior of coefficients curves. Fig. 13. Effects of reuce frequency on lift coefficient at mean angle of attack of 10 egree 647

6 Fig. 14. Effects of reuce frequency on rag coefficient at mean angle of attack of 10 egrees Figures 13 an 14 show, effects of reuce frequency at mean angle of attack of 10 egrees. Accorance with these figures, by increasing of reuce frequency, the behavior of the curves are changing an this means some ifferent phenomena affect on airfoil by increasing reuce frequency. Accoring to (Anro an Jacquin 2009) at low frequencies, aeroynamic forces result from the quasi-steay plus leaing ege vortex an by increasing of frequency, they are results from wakecapture an ae mass phenomena. 4.3 Effects of Reynols Number Effects of Reynols number on the aeroynamic coefficients in reuce frequency of 0.5 an oscillation amplitue of 10 egrees have been shown in Figs. 15 an 16. Lift coefficients change a little with increasing Reynols number up to Re=5000 where it seems the flow effects is change. The rag coefficient is ecrease ramatically by increasing of Reynols number. Lift an rag coefficients are preictable at each Reynols number, except at Re=5000. Fig. 16. Effects of Reynols number on rag coefficient at mean angle of attack of 0 egree. In accorance with Fig 17.a an 17.b, there are two light separation at the trailing ege. However at Re=5000, a fluctuation wave is prouce in the wake region. This fluctuation comes from a vortex generation an breakown at the trailing ege. This phenomena affects on lift an rag coefficients where the airfoil is locate on its maximum an minimum positions. Fig. 17.a. Streamlines at Re=500; α= 10 egrees. Fig. 15. Effects of Reynols number on lift coefficient at mean angle of attack of 0 egree. Streamlines comparisons at Re=500 an Re=5000 at the situation that the airfoil is oscillating with mean angle of attack of zero egree an it is locate at its maximum angle of attack have been illustrate in Figs. 17.a an 17.b, respectively. Fig. 17.b. Streamlines at Re=5000; α= 10 egrees. Effects of Reynols number on lift an rag coefficients at mean angle of attack of 10 egrees have been shown in Figs. 18 an 19. Accoring to Fig. 18, by increasing of Reynols number from 648

7 500 to 1000, 2000 an 5000 some complexity will be appeare. This complexity causes that the perioic effect of c l an c was occurre through two stroke of oscillation at Re=5000 while in other Reynols numbers the perioic effects were occurre through one stroke. Fig. 18. Effects of Reynols number on lift coefficient at mean angle of attack of 10 egree. 2 proportional to A0 an varies linearly as the square of frequency which A 0 is amplitue an is angular frequency (Anro an Jacquin 2009). Also Birch an Dickinson expresse when vortex capture on the rear sie of airfoil, the unsteay mechanism which calle wake capture will be happene (Birch an Dickinson 2003). This force strongly epens on both kinematics an frequency. Reynols number is the main scaling parameters here, because it fixes the maximum aeroynamic incience which itself strongly etermines the strength of leaing ege vortex. From Figs. 18 an 19, at Re=5000, the main peak of c l an c is occurre at 10 egree, the beginning of the stroke when acceleration reaches its maximum value. The seconary peak is occurre at 15 egree, the en of stroke when wake sticks severely rear at a part of airfoil. In accorance with mean angle of attack of 10 egrees, Reynols number of 5000, c p an c f graphs, it seems the complexity in Figs. 18 an 19 is because of wake capture an/or ae mass forces. Fig. 19. Effects of Reynols number on rag coefficient at mean angle of attack of 10 egree. For instance, we have use FFT (Fast Fourier Transform) graph of c l to show ominant frequency an other frequencies, at ifferent Reynols number. Accoring to Fig. 20, by increasing of Reynols number, the numbers of picks have increase graually, but in all Reynols number the ominant frequency is about 0.15 which is oscillation frequency of pitching airfoil. Also Fig. 21 shows, wall friction an pressure coefficient on airfoil at ifferent Reynols number where airfoil is locate at its maximum angle of attack. As it is obvious from Fig. 21, c f an c p are change smoothly at Re=500, 1000, 2000 while at Re=5000 some eviations are appeare. Anro analyze the three funamental mechanisms that govern aeroynamic efforts acting on pitching airfoil which two important of them at high angle of attack an high frequencies are ae mass reaction an wake capture (Anro an Jacquin 2009). Ae mass force explains the mass of flui isplace with the airfoil an it is maximum when acceleration reaches its maximum. This force is Fig. 20. FFT graph at Reynols number of 5000 an mean angle of attack of 10. Fig. 21. Wall friction an pressure coefficient graph at ifferent Reynols number an mean angle of attack of 10 egrees. 4.4 Effects of Center of Oscillation Effects of x/c ratio, the center of oscillation have 649

8 been investigate in this part. Figure 22 shows the effects of center of oscillation on lift coefficient. Fig. 22. Effects of center of oscillation on lift coefficient Fig. 23. Effects of x/c ratio on rag coefficient In accorance with Fig. 22, by increasing of x/c ratio from 1/8 to 6/8, with of lift coefficient curve ecreases an its minimum an maximum o not change significantly. Accoring to Fig. 23, the with of rag coefficient curve ecreases when x/c increases. There is no significant change in the maximum an minimum amounts also. Drag coefficient has ecrease at angle of attack of 0 egree, by increasing of x/c ratio. The behaviors of the curves o not change with x/c ratio changing an they are preictable. 5. CONCLUSION The purpose of this stuy is investigation of aeroynamic characteristics of NACA0012 with harmonic pitching oscillation at zero an 10 egrees mean angles of attack. Therefore the effects of unstable parameters, incluing oscillation amplitue, reuce frequency, Reynols number an center of oscillation have been stuie. A pressure base algorithm using a finite volume element metho to solve Navier-Stokes equations has been use. Results show, by increasing of oscillation amplitue at mean angle of attack of 0 egree, the with of lift coefficient curve has increase graually. Also by increasing of oscillation amplitue the minimum an maximum amount of rag an lift coefficients have increase sharply. Also, this behavior has been seen at mean angle of attack of 10 egrees an lift an rag coefficients curves have varie as expecte. By increasing of reuce frequency, the maximum values of lift an rag coefficients have increase, their minimums have ecrease an the with of oscillation curves increase. Also, the rag coefficient at zero angle of attack has increase by increasing of reuce frequency. By changing of reuce frequency at mean angle of attack of 10 egrees, the behaviors of the curves have been varie. This variation is because of wake capture an ae mass phenomena. By increasing of Reynols number at mean angle of 0 egree, significant changes have not seen in lift curves except at Reynols number of 5000 where it seems the flow regime is change, but the maximum an minimum amounts of rag coefficients at zero angle of attack have ecrease acutely. At mean angle of attack of 10 egrees, ominant frequency in ifferent Reynols numbers was observe 0.15 Hz which is oscillation frequency. At Reynols number of 5000 an at high angle of attacks, some complexity will be appeare in lift an rag coefficients because wake capturing an/or ae mass phenomena which are ominant effects. By increasing of center of oscillation up to 6/8 chor length (x/c ratio), lift coefficient curve has ecrease an its minimum an maximum have not change excessively. Drag coefficient has ecrease at angle of attack of 0 egree, by increasing of x/c ratio. The manners of the curves o not change with x/c ratio changing an they varie as expecte. REFERENCES Akbari, M. H. an S. J. Price (2003). Simulation of ynamic stall for a NACA0012 airfoil using a vortex metho. Journal of Flui an Structures 17(6), Amiralaei, M. R., H. Alighanbari an S. M. Hashemi (2010). An investigation into the effects of unsteay parameters on the aeroynamics of a low Reynols number pitching airfoil. Journal of Flui an Structures 26(6), Anro, J. Y. an L. Jacquin (2000). Frequency effects on the aeroynamic mechanisms of a heaving airfoil in a forwar flight configuration. Aerospace Science an Technology 13(1), Birch, J. M. an M. H. Dickinson (2003). The influence of wing-wake interactions on the prouction of aeroynamic forces on prouction of in flapping flight. Journal of 650

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