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Effect of unequal flapping frequencies on flow structures
Aerospace Science and Technology 35 (2014) 39–53
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Effect of unequal flapping frequencies on flow structures Idil Fenercioglu ∗ , Oksan Cetiner Department of Astronautical Engineering, Istanbul Technical University, Maslak-Ayazaga, Istanbul 34469, Turkey
a r t i c l e
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Article history: Received 11 June 2013 Received in revised form 13 February 2014 Accepted 27 February 2014 Available online 15 March 2014 Keywords: Pitching and plunging airfoil PIV Unequal oscillating frequency
a b s t r a c t The effects of unequal pitching and plunging frequency on the flow structures around a flapping airfoil is investigated using the Digital Particle Image Velocimetry (DPIV) technique in the reduced frequency range of 0.31 k 6.26 corresponding to Strouhal number range of 0.1 St 1.0. The SD7003 airfoil model undergoes a combined flapping motion where the pitch leads the plunge with a phase angle of 90◦ in a steady current. The investigated cases are classified into five flow structure categories based on instantaneous and averaged vorticity patterns and velocity fields around and in the near-wake of the airfoil while the frequency of plunging motion was kept the same as the frequency of pitching motion. Example cases for each category were then investigated for unequal pitching and plunging frequencies and it is observed that employing unequal pitching and plunging frequencies for an oscillating airfoil may result in a change of flow structure category. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction Biological inspiration offers a means to enhance the performance of the next generation of small-scale air vehicles over existing fixed and rotary wing systems. Many insects carry large amounts of payload despite their small and fragile wings, yet they are capable of easily performing maneuvers with rapid accelerations and decelerations. There are an increasing number of researchers and groups building a flapper (Jones et al. [21], de Croon et al. [10], etc.) and attempting to quantify the relative performance merits of those Micro Air Vehicles (MAVs) (Groen et al. [15], Maniar et al. [33], Kim et al. [25]). Fundamental research in this area has received great interest in the past years and investigations are still widely under progress in order to find out if biomimetics is an effective solution to MAVs. The generation of thrust by an oscillating airfoil is known for quite a long time as summarized by Jones et al. [19] and although the basic mechanisms of thrust production are understood, the effectiveness of thrust production and its enhancement through several variants are still under investigation. The earliest theories concerning flapping wing flight were related to purely heaving airfoils. In their independent studies, Knoller [26] and Betz [3] were among the first to observe that a flapping wing generates an effective angle of attack, resulting in an aerodynamic force that includes both cross-stream and streamwise components and their theory was experimentally confirmed
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http://dx.doi.org/10.1016/j.ast.2014.02.007 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
by Katzmayer [24]. Birnbaum [4] applied Prandtl’s [41] unsteady thin airfoil theory to quantitatively predict the thrust generation of plunging airfoils. He was also the first to introduce a dimensionless similarity parameter named as the reduced frequency, which characterizes the oscillation behavior as the ratio between the flapping velocity and forward flight velocity [40]. In 1935, von Kármán and Burgers [49] experimentally observed the generation of a reverse vortex street and its corresponding jet like average velocity profile which have been associated with the thrust development on a flapping airfoil in an incompressible flow. Theodorsen [44] derived a theory for a thin flat plate with simple harmonic oscillations in incompressible uniform flow and showed that the circulatory lift was a function of the reduced frequency. Garrick [14] and von Kármán and Sears [50] were among the others to conduct theoretical analyses on pitching and plunging airfoils in forward flight. By the end of the century, the vortical signatures of oscillating airfoils had been investigated by many researchers (Jones et al. [22], Lai and Platzer [31] and Rival and Tropea [43]). Jones and Platzer [18] summarized the numerical and experimental investigations that led them to develop their flapping-wing micro air vehicle. Cleaver et al. [8] showed that a sinusoidally plunging airfoil can create lift improvement due to convection and that the interaction of the trailing edge vortices with the leading-edge vortex on the upper surface. Since pure pitching airfoils are also a popular topic for helicopter rotor performance, some studies describe dynamic features associated with incidence variations of sinusoidally pitching airfoils for prediction of dynamic stall (Kramer [29], McCroskey [34] and Carr [7]). Thrust generation by pure pitching was experimentally
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demonstrated by Koochesfahani [27] and it was shown that by controlling the frequency, amplitude and the shape of oscillation waveform, the structure of the wake can also be controlled. Panda and Zaman [38] investigated the wake surveys to estimate lift and observed that a vortex originates from the tailing edge just when dynamic stall vortex is shed and they referred to this kind of opposite signed vortex pair as a ‘mushroom’ wake. The vorticity convection and the shedding of vortices in the near wake were investigated by many others (Kuo and Hsieh [30] and Jung and Park [23]). In a review of experimental biomimetic studies by Triantafyllou et al. [46], the similar mechanisms of force production and flow manipulation in fish and birds were classified, noting that the conditions of swimming are different from flying. Pure heaving or plunging motion is mostly employed by large animals like birds and fish operating in the low frequency regime at larger Reynolds numbers, and insects typically use the combination of plunging and pitching motions in low Reynolds number regime. Wang’s [51] computational study also showed that pure plunging motion is not enough to generate thrust at low Reynolds numbers. Even though nearly all examples of flapping wing propulsions in nature combine both pitching and plunging, according to Jones et al. [20], the key parameter determining whether an airfoil creates thrust or extracts power from the flow is the effective angle of attack and its variation with the geometric angle of attack. The parameter space for combined plunging and pitching motion includes the frequencies and amplitudes of pitching and plunging motions and the phase angle, ψ , between pitch and plunge. Anderson et al. [1] defined the conditions for optimal thrust production to operate at 0.25 < St < 0.4; the values also match the range in which most fish operate at their maximum swimming velocities (Triantafyllou [45]), but the Strouhal number is also not enough to define the efficiency of oscillating foils [54]. Guglielmini and Blondeaux’s [16] vorticity equation computations showed good agreement with Anderson et al.’s [1] experiments and they concluded that the high efficiency and thrust coefficients are related with the formation of a jet of fluid leaving the foil from the trailing edge. Recently, Bohl and Koochesfahani [5] worked on force estimation, studied the effect of velocity fluctuations in the integration of the momentum theorem for pure pitch oscillations and showed that the switch in the vortex array orientation does not coincide with the condition for crossover from drag to thrust. The shedding of leading edge vortices and the use of dynamic stall process have shown to be key factors in lift generation of flapping wing mechanisms by Platzer and Jones [39]. The dynamic stall phenomenon for airfoils with combined pitching and plunging motions was also investigated by many researchers, for example, Isogai et al. [17] and Read et al. [42]. Recently, Baik et al. [2] studied experimentally the flow topology, leading-edge vortex dynamics and unsteady forces produced by pitching and plunging flat-plate aerofoils in forward flight. Periodic motions of the airfoils which are more applicable to forward flight of MAV applications are more systematically studied in literature [40]. Earlier studies on non-periodic motion of airfoils, such as the pitch-up problem, mainly focused on fixed wing aircraft maneuvering applications (Visbal and Shang [48], Visbal [47], Koochesfahani and Vanco [28], Conger and Ramaprian [9] and Oshima and Ramaprian [37]). The advance in MAVs resulted in investigations of transient problems with higher rates of pitch-up motions more relevant to maneuvering, perching and gust response (Ol et al. [36], Garmann and Visbal [13] and Yu et al. [55]). The same canonical pitch-hold-return problem was also experimentally investigated by Buchner et al. [6] using both stereoscopic and tomographic PIV methods to gain some understanding of how three-dimensionality develops in such flows.
Although there are many experimental and numerical studies investigating flapping airfoils, effects of unequal oscillating frequencies of pitching and plunging motions received little attention and those unique investigations have not reached a conclusion as the problem becomes much more complex. Ol [35] briefly studied the oscillations of an SD7003 airfoil with different pitch and plunge frequencies. He considered a motion where the plunge frequency is half of the pitch frequency and observed that a separated leading edge vortex is formed every pitch period, unlike all the previously tested pure plunge cases. Webb et al. [52] also considered the effects of unequal pitch and plunge motion of an SD7003 airfoil, with various pitch pivot locations, to model the gust response with the lower-frequency motion where the higherfrequency motion is the kinematics of the flapping MAV. They considered results based on experiments at Re = 10 000 and computations for Re < 1000, stating a similar vortex shedding behavior. For the case where the pitch frequency was twice the plunge frequency and the pitch pivot point was at half chord, they pointed out a discrepancy between experiment and computational results. Their flow visualization experiments with dye injection showed that placing the pivot point closer to downstream location resulted in the formation of a stronger leading edge vortex and they observed the reverse Karman vortex street when they switched to the standard case of equal pitch and plunge frequencies. Both studies leave many aspects unmentioned, hence the role of unequal oscillation frequencies on the formation of vortex structures and the production of lift and thrust are yet to be explored. Motivated with practical applications to micro air vehicles with the size of small birds that flap wings, an experimental study is conducted using a pitching and plunging airfoil in steady water flow to observe the relationships between the flapping parameters and the related leading and trailing edge vortex formation. The investigated cases remain in the low Reynolds number range of O (103 )–O (104 ) which is of interest to the design and development of Micro Air Vehicle applications as also described in a recent study by Baik et al. [2]. The detailed quantitative evaluation of flow structures with instantaneous patterns of vorticity around and in the near-wake of a SD7003 airfoil model is previously obtained for a large parameter space for combined pitch and plunge motions in steady current (Fenercioglu and Cetiner [12] and Fenercioglu [11]) where the pitching frequency is set equal to the plunging frequency. The band of occurrence of flow structure categories on a k vs. hamp /c plot agreed well with wake classification types presented in Jones et al. [22,19] for pure plunge motion. The objective of this study is to explore the effect of using unequal oscillating frequencies for pitch and plunge motions in the near-wake of a two-dimensional wing. As the pitch amplitude and mean angle of attack are kept constant at moderate values as in the aforementioned baseline study, the plunge motion is expected to dictate the flow structures in the near-wake. 2. Experimental set-up Experiments were performed in the close-circuit, free-surface, large scale water channel located in the Trisonic Laboratories at the Faculty of Aeronautics and Astronautics of Istanbul Technical University. The cross-sectional dimensions of the main test section are 1010 mm × 790 mm. The experiments are conducted at Reynolds numbers of 2500 Re 13 700 which simulate forward flight of small birds. An SD7003 airfoil model, which is known to be optimized for low speed flows as indicated in studies of Windte et al. [53] and Lian and Shyy [32], is mounted in a vertical cantilevered arrangement in the water channel about its quarter chord. A rod connects the airfoil to the servo motor to provide a sinusoidal pitching
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where h(t ) is the linear plunge motion, transverse to the freestream velocity, α (t ) is the angular pitch motion, and ψ is the phase angle between pitch and plunge. h amp is the plunge amplitude, α amp is the pitch amplitude and α0 is the initial angle of attack. The frequency of plunging motion is represented by f 1 , and the frequency of pitching motion is f 2 . Considering the plunging frequency is equal to the pitching frequency, the airfoil undergoes combined pitching and plunging oscillations with a phase difference of ψ = π /2 where pitch leads the plunge motion. The initial angle of attack of the motion is α0 = 8◦ and the pitching amplitude is αamp = 8.6◦ . Fig. 2 shows pitch angle and plunge position as a function of time for the baseline and the unequal frequency cases investigated. The reduced frequency (k) and the Strouhal number (St) are defined as: Fig. 1. Experimental arrangement.
motion via a coupling system which itself is connected to a linear table. The chord length of the model is c = 10 cm with a span of s = 30 cm and it is manufactured of transparent Plexiglas to allow laser light to illuminate both the suction and pressure sides. End plates with a 30◦ outward bevel are suspended from the top of the channel above and below the fully submerged airfoil to reduce the end effects. The top end plate is painted mat-black to avoid background reflections and it has a slit to ensure the maximum plunging motion amplitude. The experimental arrangement is shown in Fig. 1. The vortex formation is recorded using the digital particle image velocimetry (DPIV) technique allowing detailed quantitative evolution of flow structures. The flow is illuminated at the airfoil’s mid-span plane by a dual cavity Nd:Yag laser (max. 120 mJ/pulse) and the water is seeded with silver coated glass spheres of 10 μm mean diameter. Two 10-bit cameras with 1600 × 1200 pixels resolution are positioned underneath the water channel. After obtaining the images to observe the flow structures around the airfoil, both cameras are shifted to the second location and the experimental cases are repeated to observe the flow structures in the near-wake. For each location, the two images from the two cameras are stitched before interrogation using two marker points in the illumination plane. Stitched PIV images are interrogated using a double frame, cross-correlation technique with a window size of 64 × 64 pixels and 50% overlapping in each direction. The final grid resolution of velocity vectors is 3.5 mm × 3.5 mm in the plane of the flow which corresponds to a resolution of 0.035c × 0.035c. The resulting measurement plane is represented with approximately 3600 velocity vectors. The maximum sampling rate for vector fields is 10 Hz which is used for a motion frequency of 0.1 Hz. Each data set includes 200 images which are acquired for at least two cycles of motion, whichever is slower in unequal frequency cases. The pitch and plunge motions of the airfoil are accomplished with two Kollmorgen/Danaher Motion AKM33E servo motors. The servo motors are connected to a computer via ServoSTAR S300 digital servo amplifier for pitching motion and ServoSTAR S600 for plunging motion, and motor motion profiles are generated by a signal generator Labview VI (Virtual Instrument) for the given amplitude and frequency. The same VI triggers the PIV system at the beginning of the third motion cycle of the airfoil and synchronization is achieved using a National Instruments PCI-6601 timer device. The kinematic motions of the airfoil considered in the current study are sinusoidal plunging and pitching and they are described with the following equations:
h(t ) = hamp cos(2π f 1 t + ψ)
α (t ) = α0 + αamp cos(2π f 2t )
k= St =
π fc U∞ 2 f hamp U∞
where U ∞ is the freestream velocity and f represents the frequency of motion. The investigated limits of non-dimensional parameters for the baseline cases of this study, when the plunging frequency is kept equal to the pitching frequency, are as follows: 0.1 St 1.0, 0.25 hamp /c 0.5, 0.31 k 6.26. 3. Results 3.1. Flow structure categories By investigating two-dimensional flow fields around and in the near-wake of a pitching and plunging airfoil using DPIV method, five main flow structure categories are identified depending on the role of separated vortex structures from the leading and trailing edges [12]. Flow structure categorization is summarized and sketched in Table 1 based on the five baseline cases. The first three columns show the instantaneous vorticity patterns around and in the nearwake of the airfoil at the beginning of downstroke, the next three columns show the vorticity patterns at the beginning of upstroke. A total of 200 instantaneous images were acquired for each case and data acquisition rates of the PIV system were chosen to record at least two periods of the airfoil motion for each frequency of individual cases. The last column of Table 1 shows the average of those 200 images in terms of combined vorticity and velocity fields. In Category A, two counter-rotating pairs of vortices are formed during an oscillation cycle. One pair appears on the lower side of the airfoil during the upstroke and one pair on the upper side during the downstroke and sheds into the wake. When the plunge amplitude is lower, the two counter rotating vorticity pairs are also formed. However, although the positive vortex which is generating from the leading edge during downstroke interacts with the negative signed vortex generating from the lower surface, it cannot connect with the positive vortex from the trailing edge, hence it loses strength and dissipates. This is a subset of Category A and this latter case is named A2 while the former one is named A1. For both subsets, averaged vorticity and velocity fields indicate a jet like flow. Similar to Category A, in Category B cases, a positive vortex is shed in the wake at the end of the downstroke while a negative vortex enlarges preserving its strength and detaches from the leading edge. However, in Category B cases, the process does not seem to be nearly reversed during the upstroke as in the Category A cases. The negative vortex is shed in the wake from the trailing
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Table 1 Summary of categories based on flow structures.
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Fig. 2. Motion kinematics for the baseline and the unequal frequency cases.
edge at the end of the upstroke. Category B also differs from Category A by having a slightly inclined downward trajectory of the shed vortices. As in Category A, depending on the amplitude of plunge motion, Category B presents two subsets with the same distinguishable differences. When the amplitude of plunge is equal or larger than 0.5c (for subsets denoted with 1), the roll-up radius of the vortex shedding is large at the trailing edge of the airfoil and the wake width of the jet like velocity profile is substantially increased. The presence of counter rotating pairs of vortices are still observed in Category B2; however, the vortices formed at the leading edge do not have enough time to evolve as they are swept away during
the following half cycle of the motion due to either lower plunge amplitude or larger free-stream velocity. Consequently, the roll-up radius of the vortex shedding is small for subsets denoted with 2. Category C is detected only for the cases where hamp /c = 0.25. The major characteristic of flow structures in this category is the shedding of two negative sign vortices per cycle of motion. In accordance, the average vorticity patterns clearly indicate the upward trajectory of additional negative vortex shed in the wake. In Category D, the negative signed vortex is generated from the leading edge during the downstroke motion of the airfoil. However, although the positive signed vortex is generated from the trailing edge during upstroke, the negative signed vortex is not
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formed. The near-wake vorticity patterns show that the negative vortex which detaches and sheds into the wake inclines downward and the positive vorticity layer at the trailing edge stretches in the wake and a trail of positive vortices lie along an upward arc. The near-wake patterns announce that the flapping motion will still induce a jet like velocity profile. In Category E, a negative vorticity layer on the upper surface and a positive vorticity layer on the lower surface of the airfoil are present during most of the motion stages and the wake also oscillates up and down with the airfoil motion. During the upstroke the vorticity layers are thin and attached on the surface; however, the negative vorticity layer detaches from the surface during the downstroke and exhibit localized cells of vorticity concentration, revealing Karman-like vortex formations in the near-wake at the beginning of the downstroke. The near-wake vorticity patterns announce a velocity deficit. Table 2, depicting the details of the parameter space includes all test cases of the aforementioned study. The table is color-coded in order to clearly distinguish the borders of the categories and the exemplified cases are boldfaced. The rows are grouped based on the plunge amplitude. For each row, the plunge amplitude and the frequency of motion for both pitch and plunge motions are the same. The flow speed increases from left to right across the table. The Strouhal number is fixed for each column and decreases from left to right; however, due to the difference in flapping motion frequency, the flow speed is not the same in a single column. The two letter notations in Table 2 indicate that these cases are identified as a transition from one category to the next one. The category notations with numbers indicate the subsets of the same category and the blanks indicate the absence of corresponding data acquisition. In this study, the effects due to unequal frequencies of the pitching and plunging motions are investigated for each experimental case representing the above mentioned flow structure categories [12]. 3.2. Effect of unequal frequency Table 3 and Table 4 show the averaged velocity and corresponding vorticity plots of unequal frequency cases along with the five baseline cases representing each flow structure category. The velocity fields are averaged over 200 images belonging at least to two periods of pitch or plunge motion, whichever is slower. Table 3 represents the unequal frequency cases where plunge motion frequency is kept the same as the baseline case and Table 4 represents those where pitch motion frequency is kept the same as the baseline case. For both tables, the first column of averaged images belongs to baseline cases with the frequency which is varied increasing from left to right in each row. Both tables show that the averaged vorticity patterns are mostly affected by the variation in plunge frequency. When the plunge frequency is kept the same as the baseline case (Table 3), major characteristics are very similar for f 1 > f 2 . Fig. 3 and Fig. 4 show the instantaneous vorticity patterns in the near-wake of the airfoil for the selected case exhibiting the flow structure characteristics of Category A1 for equal ( f 1 = f 2 = 0.2 Hz) and unequal motion frequencies ( f 1 = 0.2 Hz, f 2 = 0.1 Hz), respectively. A cycle of slower motion, namely pitch in this case, is selected for the representation of flow structure evolution. In Fig. 3 and Fig. 4, one period of plunge is represented by 8 images. The first image on the right hand side of the figure represents the airfoil at its mid-plunge position. As the flow is from left to right in the experiments, the figure simulates the forward motion of the airfoil from right to left. In comparison, the vorticity patterns in each plunge cycle is very similar to each other and to those obtained when the pitch and plunge frequencies are the same. Both positive and negative vortices are
shed twice in a pitch cycle locking on the plunge frequency. The reduction of pitch frequency with respect to the plunge frequency still results in slight differences. Both the positive vortex and the negative leading edge vortex are stretched more than those in the baseline case before they are shed from the trailing edge. The distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle is narrower compared to the observed ones at the leftmost and rightmost images which represent the beginning of the first and second pitch cycles respectively. As a final remark on differences, during the downstroke motion in the second plunge cycle, some remains of the shed negative vortex wanders close to the positive vortex which is rolling up and shedding. It should be noted that the freestream velocity for each row in Table 3 and Table 4, hence the convection of the vortices downstream, is the same. Based on a few averaged flow field results shown in Table 3, the flow structures are also very similar when the plunge frequency is kept same as the baseline case and the pitch frequency is increased, i.e., f 1 < f 2 . Fig. 5 exemplifies such a case exhibiting the flow structure characteristics of Category A1 when the plunge frequency is equal to pitch frequency and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of plunge motion when the pitch frequency is twice the plunge frequency ( f 1 = 0.2 Hz, f 2 = 0.4 Hz). In comparison with the baseline case, the increase in pitch frequency does not affect the major flow structures and the timing of vortex shedding. Both positive and negative vortices are shed once during two pitch cycles locking again on the plunge frequency. The slight difference with respect to the baseline case is related to the stretch of the negative leading edge vortex during the upstroke and to its convection and integrity when shed during the downstroke motion. As a result, during the upstroke motion in the second half of the second pitch cycle, some remains of the shed negative vortex wanders close to the positive vortex which is shed earlier. Table 4, representing the unequal frequency cases where the pitch frequency is kept the same with the baseline cases also shows that the averaged vorticity patterns are mostly affected by the variation in plunge frequency, whether increased or decreased. For all categories A through E, the use of a different and lower plunge motion frequency ( f 1 < f 2 ) generally results in a decrease in the averaged vorticity levels. The strength of vortical structures is mostly dictated by the plunge motion. The timing of the vortex shedding is also affected; vorticity patterns for Categories A1 and A2 actually show how employing unequal frequency has an effect on the vortex trajectories in the near-wake. The positive vortex shed from the trailing edge follows nearly a straight line; however there are two branches for the negative vortex, one parallel to the positive and the other shed downward in the freestream direction. Fig. 6 exemplifies the baseline case exhibiting the flow structure characteristics of Category A2 when pitch frequency is twice the plunge frequency ( f 1 = 0.3 Hz, f 2 = 0.6 Hz) and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of plunge motion. Although the positive vortex is shed once in a plunge cycle, the negative vortex is shed twice, both during the upstroke. A similar, however more pronounced trajectory change occurs for Category D. The negative vortex is shed upward parallel to the positive vortex. The instantaneous vorticity patterns for this case where the pitch frequency is kept the same and the plunge frequency is reduced by half show how the timing of vortex shedding is affected (Fig. 7). An early separation of the negative leading edge vortex causes the wake to exhibit a Karman-like shedding although the positive vortex is shed upward during the downstroke motion. Based on the few averaged flow field results shown in Table 4, the flow structures are also deeply affected when the pitch frequency is kept the same as the baseline case and the plunge
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Fig. 3. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 when the plunging frequency f 1 is equal to pitching frequency f 2 [12].
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Table 2 The details of the parameter space.
Fig. 4. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.2 Hz and f 2 = 0.1 Hz.
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Table 3 Averaged velocity and vorticity profiles for equal and unequal plunging ( f 1 ) and pitching ( f 2 ) frequencies for each category, when f 1 is kept the same as the baseline case.
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Table 4 Averaged velocity and vorticity profiles for equal and unequal plunging ( f 1 ) and pitching ( f 2 ) frequencies for each category, when f 2 is kept the same as the baseline case.
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Fig. 5. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.2 Hz and f 2 = 0.4 Hz.
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Fig. 6. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A2 for f 1 = 0.3 Hz and f 2 = 0.6 Hz.
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Fig. 7. Instantaneous vorticity patterns in the near-wake of the airfoil for Category D for (a) f 1 = f 2 = 0.6 Hz and (b) f 1 = 0.3 Hz and f 2 = 0.6 Hz.
frequency is increased, i.e., f 1 > f 2 . The use of a different and higher plunge motion frequency generally results in an increase in the averaged vorticity levels. The strength of vortical structures is mostly dictated by the plunge motion. Fig. 8 exemplifies such a case exhibiting the flow structure characteristics of Category A1 when f 1 = f 2 and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of pitch motion when the plunge frequency is twice the pitch frequency ( f 1 = 0.4 Hz, f 2 = 0.2 Hz). In comparison with the baseline case, the increase in plunge frequency does not affect the major flow structures and the timing of vortex shedding. There is high similarity in this in this respect between Figs. 4 and 8 representing unequal frequency cases where the plunge frequency is twice the pitch frequency. The vorticity patterns in Fig. 8 also exhibit slight differences with respect to the baseline case belonging to Category A1. The negative leading edge vortices are stretched more than those in the baseline case before they are shed from the trailing edge. On the other hand, the positive vortex is stretched more during the second plunge cycle; however, its roll-up is pronounced during the first plunge cycle. Similar to what is observed in Fig. 4, the distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle is narrower compared to the observed ones at the leftmost and rightmost images which represent the beginning of the first and second pitch cycles.
Interesting effects of having unequal frequencies are observed for Category E, when the pitch frequency is kept the same as the baseline case and the plunge frequency is increased distinctively. The use of a plunge frequency which is five times the pitch frequency results in a flow structure category change; i.e. the case where f 1 = 0.5, f 2 = 0.1 belongs to Category B rather than Category E. 4. Conclusions The investigation started from a flow structure categorization based on Strouhal numbers which are unique as the pitch and plunge frequencies are kept the same. Although the former Strouhal numbers are matched with either the pitch or plunge frequencies in the investigated cases of unequal frequencies, the kinematics in the current study are completely different. The brief overlook on the effect of unequal frequencies shows that the flapping motion becomes much more complicated and the physics underlying the effect of unequal frequencies deserves a more detailed investigation. However, a major conclusion still holds; for the parameters studied, it was found that the plunge motion influences more the near-wake flow structures as foreseen. Throughout this study the pitch amplitude and its initial value are kept constant and the variations in the maximum effective angle of attack are due to the variations in the Strouhal number, which in turn is
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Fig. 8. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.4 Hz and f 2 = 0.2 Hz.
related to the plunge amplitude. It is therefore not surprising that the plunge motion tends to dominate. The following summarized results fortify this conclusion: – When the plunge frequency is kept constant, the decrease in pitch frequency only results in slight differences with respect to the baseline cases. The differences include, firstly, the stretching of the vortices before they are shed from the trailing edge, secondly, the reduced distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle and finally, the near-wake exhibiting some remains of the shed negative vortex wandering close to the positive vortex which is rolling up and shedding.
– When the plunge frequency is kept constant, the increase in the pitch frequency does not affect the major flow structures and the timing of vortex shedding. Both positive and negative vortices are shed once during two pitch cycles locking again on the plunge frequency. – On the other hand, when the pitch frequency is kept constant, irrespective of the decrease or increase in the plunge frequency, the strength of vortical structures is mostly dictated by the plunge motion. The timing of the vortex shedding and the vortex trajectories in the near-wake are also affected when the plunge frequency is decreased. – The major flow structures and the timing of vortex shedding are highly similar when the plunge frequency is twice the pitch frequency, regardless of which one is kept constant as the baseline case. When the plunge frequency is increased
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significantly i.e., five times the pitch frequency, a change in flow structure category occurs; a jet like shedding takes over Karman-like shedding which is the major characteristic of the baseline case. In the light of the conclusions drawn from this investigation, the use of higher pitch amplitudes are yet to be explored to clarify the extent of plunge dominated flows in further studies on unequal frequencies. Acknowledgement The authors would like to thank Selin Alperer-Tatli, M.A., in Teaching English as a Foreign Language, for editing the manuscript. References [1] J.M. Anderson, K. Streitlien, D.S. Barrett, M.S. Triantafyllou, Oscillating foils of high propulsive efficiency, J. Fluid Mech. 360 (1998) 41–72. [2] Y.S. Baik, L.P. Bernal, K. Granlund, M.V. Ol, Unsteady force generation and vortex dynamics of pitching and plunging aerofoils, J. Fluid Mech. 709 (2012) 37–68. [3] A. Betz, Ein Beitrag zur Erklarung des Segelfluges, Z. Flugtech. Motorluftschif. 3 (1912) 269–272. [4] W. Birnbaum, Das ebene Problem des Schlagenden Fluegels, Z. Angew. Math. Mech. 4 (1924) 277–292. [5] D.G. Bohl, M.M. Koocheshahani, MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency, J. Fluid Mech. 620 (2009) 63–88. [6] A.-J. Buchner, N. Buchmann, K. Kilany, C. Atkinson, J. Soria, Steroscopic and tomographic PIV of a pitching plate, Exp. Fluids 52 (2012) 299–314. [7] L.W. Carr, Progress in analysis and prediction of dynamic stall, J. Aircr. 25 (1) (1988) 6–17. [8] D.J. Cleaver, Z. Wang, I. Gursul, Lift enhancement on oscillating airfoils, in: 39th AIAA Fluid Dynamics Conference, AIAA 2009-4028, 2009. [9] R.N. Conger, B.R. Ramaprian, Pressure measurements on a pitching airfoil in a water channel, AIAA J. 32 (1) (1994) 108–115. [10] G.C.H.E. de Croon, M.A. Groen, C. De Wagter, B.D.W. Remes, R. Ruijsink, B.W. van Oudheusden, Design, aerodynamics, and autonomy of the DelFly, Bioinspir. & Biomim. 7 (2) (2012) 025003. [11] I. Fenercioglu, Experimental investigation of flow structures around an oscillating airfoil in steady current, PhD thesis, Istanbul Technical University, 2010. [12] I. Fenercioglu, O. Cetiner, Categorization of flow structures around a pitching and plunging airfoil, J. Fluids Struct. 31 (2012) 92–102. [13] D.J. Garmann, M.R. Visbal, Numerical investigation of transitional flow over a rapidly pitching plate, Phys. Fluids 23 (2011) 094106. [14] I.E. Garrick, Propulsion of a flapping and oscillating airfoil, NACA Report No. 576, 1936. [15] M. Groen, B. Bruggeman, B. Remes, R. Ruijsink, B.W. van Oudheusden, H. Bijl, Improving flight performance of the flapping wing MAV DelFly II, in: International Micro Air Vehicle Conference and Competitions (IMAV 2010), 2010. [16] L. Guglielmini, P. Blondeaux, Propulsive efficiency of oscillating foils, Eur. J. Mech. B, Fluids 23 (2003) 255–278. [17] K. Isogai, Y. Shinmoto, Y. Watanabe, Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil, AIAA J. 37 (10) (1999) 1145–1151. [18] K.D. Jones, M.F. Platzer, Design and development considerations for biologically inspired flapping-wing micro air vehicles, Exp. Fluids 46 (5) (2009) 799–810. [19] K.D. Jones, C.M. Dohring, M.F. Platzer, Experimental and computational investigation of the Knoller–Betz effect, AIAA J. 36 (7) (1998) 1240–1246. [20] K.D. Jones, T.C. Lund, M.F. Platzer, Experimental and computational investigation of flapping wing propulsion for micro air vehicles, fixed and flapping wing aerodynamics for micro air vehicles, in: Progress in Astronautics and Aeronautics, AIAA, New York, 2001, p. 195, pp. 307–339, Chap. 16. [21] K.D. Jones, C.J. Bradshaw, J. Papadopoulos, M.F. Platzer, Bio-inspired design of flapping-wing micro air vehicles, Aeronaut. J. 109 (2005), 1098, pp. 385–393. [22] K.D. Jones, C.M. Dohring, M.F. Platzer, Wake structures behind plunging airfoils: a comparison of numerical and experimental results, AIAA 96-007, 1996. [23] Y.W. Jung, S.O. Park, Vortex-shedding characteristics in the wake of an oscillating airfoil at low Reynolds number, J. Fluids Struct. 20 (2005) 451–464. [24] R. Katzmayr, Effect of periodic changes of angle of attack on behavior of airfoils, NACA TM 147, 1922.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Effect of unequal flapping frequencies on flow structures Idil Fenercioglu ∗ , Oksan Cetiner Department of Astronautical Engineering, Istanbul Technical University, Maslak-Ayazaga, Istanbul 34469, Turkey
a r t i c l e
i n f o
Article history: Received 11 June 2013 Received in revised form 13 February 2014 Accepted 27 February 2014 Available online 15 March 2014 Keywords: Pitching and plunging airfoil PIV Unequal oscillating frequency
a b s t r a c t The effects of unequal pitching and plunging frequency on the flow structures around a flapping airfoil is investigated using the Digital Particle Image Velocimetry (DPIV) technique in the reduced frequency range of 0.31 k 6.26 corresponding to Strouhal number range of 0.1 St 1.0. The SD7003 airfoil model undergoes a combined flapping motion where the pitch leads the plunge with a phase angle of 90◦ in a steady current. The investigated cases are classified into five flow structure categories based on instantaneous and averaged vorticity patterns and velocity fields around and in the near-wake of the airfoil while the frequency of plunging motion was kept the same as the frequency of pitching motion. Example cases for each category were then investigated for unequal pitching and plunging frequencies and it is observed that employing unequal pitching and plunging frequencies for an oscillating airfoil may result in a change of flow structure category. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction Biological inspiration offers a means to enhance the performance of the next generation of small-scale air vehicles over existing fixed and rotary wing systems. Many insects carry large amounts of payload despite their small and fragile wings, yet they are capable of easily performing maneuvers with rapid accelerations and decelerations. There are an increasing number of researchers and groups building a flapper (Jones et al. [21], de Croon et al. [10], etc.) and attempting to quantify the relative performance merits of those Micro Air Vehicles (MAVs) (Groen et al. [15], Maniar et al. [33], Kim et al. [25]). Fundamental research in this area has received great interest in the past years and investigations are still widely under progress in order to find out if biomimetics is an effective solution to MAVs. The generation of thrust by an oscillating airfoil is known for quite a long time as summarized by Jones et al. [19] and although the basic mechanisms of thrust production are understood, the effectiveness of thrust production and its enhancement through several variants are still under investigation. The earliest theories concerning flapping wing flight were related to purely heaving airfoils. In their independent studies, Knoller [26] and Betz [3] were among the first to observe that a flapping wing generates an effective angle of attack, resulting in an aerodynamic force that includes both cross-stream and streamwise components and their theory was experimentally confirmed
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Corresponding author. Tel.: +90 (212) 285 3145. E-mail address: [email protected] (I. Fenercioglu).
http://dx.doi.org/10.1016/j.ast.2014.02.007 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
by Katzmayer [24]. Birnbaum [4] applied Prandtl’s [41] unsteady thin airfoil theory to quantitatively predict the thrust generation of plunging airfoils. He was also the first to introduce a dimensionless similarity parameter named as the reduced frequency, which characterizes the oscillation behavior as the ratio between the flapping velocity and forward flight velocity [40]. In 1935, von Kármán and Burgers [49] experimentally observed the generation of a reverse vortex street and its corresponding jet like average velocity profile which have been associated with the thrust development on a flapping airfoil in an incompressible flow. Theodorsen [44] derived a theory for a thin flat plate with simple harmonic oscillations in incompressible uniform flow and showed that the circulatory lift was a function of the reduced frequency. Garrick [14] and von Kármán and Sears [50] were among the others to conduct theoretical analyses on pitching and plunging airfoils in forward flight. By the end of the century, the vortical signatures of oscillating airfoils had been investigated by many researchers (Jones et al. [22], Lai and Platzer [31] and Rival and Tropea [43]). Jones and Platzer [18] summarized the numerical and experimental investigations that led them to develop their flapping-wing micro air vehicle. Cleaver et al. [8] showed that a sinusoidally plunging airfoil can create lift improvement due to convection and that the interaction of the trailing edge vortices with the leading-edge vortex on the upper surface. Since pure pitching airfoils are also a popular topic for helicopter rotor performance, some studies describe dynamic features associated with incidence variations of sinusoidally pitching airfoils for prediction of dynamic stall (Kramer [29], McCroskey [34] and Carr [7]). Thrust generation by pure pitching was experimentally
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demonstrated by Koochesfahani [27] and it was shown that by controlling the frequency, amplitude and the shape of oscillation waveform, the structure of the wake can also be controlled. Panda and Zaman [38] investigated the wake surveys to estimate lift and observed that a vortex originates from the tailing edge just when dynamic stall vortex is shed and they referred to this kind of opposite signed vortex pair as a ‘mushroom’ wake. The vorticity convection and the shedding of vortices in the near wake were investigated by many others (Kuo and Hsieh [30] and Jung and Park [23]). In a review of experimental biomimetic studies by Triantafyllou et al. [46], the similar mechanisms of force production and flow manipulation in fish and birds were classified, noting that the conditions of swimming are different from flying. Pure heaving or plunging motion is mostly employed by large animals like birds and fish operating in the low frequency regime at larger Reynolds numbers, and insects typically use the combination of plunging and pitching motions in low Reynolds number regime. Wang’s [51] computational study also showed that pure plunging motion is not enough to generate thrust at low Reynolds numbers. Even though nearly all examples of flapping wing propulsions in nature combine both pitching and plunging, according to Jones et al. [20], the key parameter determining whether an airfoil creates thrust or extracts power from the flow is the effective angle of attack and its variation with the geometric angle of attack. The parameter space for combined plunging and pitching motion includes the frequencies and amplitudes of pitching and plunging motions and the phase angle, ψ , between pitch and plunge. Anderson et al. [1] defined the conditions for optimal thrust production to operate at 0.25 < St < 0.4; the values also match the range in which most fish operate at their maximum swimming velocities (Triantafyllou [45]), but the Strouhal number is also not enough to define the efficiency of oscillating foils [54]. Guglielmini and Blondeaux’s [16] vorticity equation computations showed good agreement with Anderson et al.’s [1] experiments and they concluded that the high efficiency and thrust coefficients are related with the formation of a jet of fluid leaving the foil from the trailing edge. Recently, Bohl and Koochesfahani [5] worked on force estimation, studied the effect of velocity fluctuations in the integration of the momentum theorem for pure pitch oscillations and showed that the switch in the vortex array orientation does not coincide with the condition for crossover from drag to thrust. The shedding of leading edge vortices and the use of dynamic stall process have shown to be key factors in lift generation of flapping wing mechanisms by Platzer and Jones [39]. The dynamic stall phenomenon for airfoils with combined pitching and plunging motions was also investigated by many researchers, for example, Isogai et al. [17] and Read et al. [42]. Recently, Baik et al. [2] studied experimentally the flow topology, leading-edge vortex dynamics and unsteady forces produced by pitching and plunging flat-plate aerofoils in forward flight. Periodic motions of the airfoils which are more applicable to forward flight of MAV applications are more systematically studied in literature [40]. Earlier studies on non-periodic motion of airfoils, such as the pitch-up problem, mainly focused on fixed wing aircraft maneuvering applications (Visbal and Shang [48], Visbal [47], Koochesfahani and Vanco [28], Conger and Ramaprian [9] and Oshima and Ramaprian [37]). The advance in MAVs resulted in investigations of transient problems with higher rates of pitch-up motions more relevant to maneuvering, perching and gust response (Ol et al. [36], Garmann and Visbal [13] and Yu et al. [55]). The same canonical pitch-hold-return problem was also experimentally investigated by Buchner et al. [6] using both stereoscopic and tomographic PIV methods to gain some understanding of how three-dimensionality develops in such flows.
Although there are many experimental and numerical studies investigating flapping airfoils, effects of unequal oscillating frequencies of pitching and plunging motions received little attention and those unique investigations have not reached a conclusion as the problem becomes much more complex. Ol [35] briefly studied the oscillations of an SD7003 airfoil with different pitch and plunge frequencies. He considered a motion where the plunge frequency is half of the pitch frequency and observed that a separated leading edge vortex is formed every pitch period, unlike all the previously tested pure plunge cases. Webb et al. [52] also considered the effects of unequal pitch and plunge motion of an SD7003 airfoil, with various pitch pivot locations, to model the gust response with the lower-frequency motion where the higherfrequency motion is the kinematics of the flapping MAV. They considered results based on experiments at Re = 10 000 and computations for Re < 1000, stating a similar vortex shedding behavior. For the case where the pitch frequency was twice the plunge frequency and the pitch pivot point was at half chord, they pointed out a discrepancy between experiment and computational results. Their flow visualization experiments with dye injection showed that placing the pivot point closer to downstream location resulted in the formation of a stronger leading edge vortex and they observed the reverse Karman vortex street when they switched to the standard case of equal pitch and plunge frequencies. Both studies leave many aspects unmentioned, hence the role of unequal oscillation frequencies on the formation of vortex structures and the production of lift and thrust are yet to be explored. Motivated with practical applications to micro air vehicles with the size of small birds that flap wings, an experimental study is conducted using a pitching and plunging airfoil in steady water flow to observe the relationships between the flapping parameters and the related leading and trailing edge vortex formation. The investigated cases remain in the low Reynolds number range of O (103 )–O (104 ) which is of interest to the design and development of Micro Air Vehicle applications as also described in a recent study by Baik et al. [2]. The detailed quantitative evaluation of flow structures with instantaneous patterns of vorticity around and in the near-wake of a SD7003 airfoil model is previously obtained for a large parameter space for combined pitch and plunge motions in steady current (Fenercioglu and Cetiner [12] and Fenercioglu [11]) where the pitching frequency is set equal to the plunging frequency. The band of occurrence of flow structure categories on a k vs. hamp /c plot agreed well with wake classification types presented in Jones et al. [22,19] for pure plunge motion. The objective of this study is to explore the effect of using unequal oscillating frequencies for pitch and plunge motions in the near-wake of a two-dimensional wing. As the pitch amplitude and mean angle of attack are kept constant at moderate values as in the aforementioned baseline study, the plunge motion is expected to dictate the flow structures in the near-wake. 2. Experimental set-up Experiments were performed in the close-circuit, free-surface, large scale water channel located in the Trisonic Laboratories at the Faculty of Aeronautics and Astronautics of Istanbul Technical University. The cross-sectional dimensions of the main test section are 1010 mm × 790 mm. The experiments are conducted at Reynolds numbers of 2500 Re 13 700 which simulate forward flight of small birds. An SD7003 airfoil model, which is known to be optimized for low speed flows as indicated in studies of Windte et al. [53] and Lian and Shyy [32], is mounted in a vertical cantilevered arrangement in the water channel about its quarter chord. A rod connects the airfoil to the servo motor to provide a sinusoidal pitching
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where h(t ) is the linear plunge motion, transverse to the freestream velocity, α (t ) is the angular pitch motion, and ψ is the phase angle between pitch and plunge. h amp is the plunge amplitude, α amp is the pitch amplitude and α0 is the initial angle of attack. The frequency of plunging motion is represented by f 1 , and the frequency of pitching motion is f 2 . Considering the plunging frequency is equal to the pitching frequency, the airfoil undergoes combined pitching and plunging oscillations with a phase difference of ψ = π /2 where pitch leads the plunge motion. The initial angle of attack of the motion is α0 = 8◦ and the pitching amplitude is αamp = 8.6◦ . Fig. 2 shows pitch angle and plunge position as a function of time for the baseline and the unequal frequency cases investigated. The reduced frequency (k) and the Strouhal number (St) are defined as: Fig. 1. Experimental arrangement.
motion via a coupling system which itself is connected to a linear table. The chord length of the model is c = 10 cm with a span of s = 30 cm and it is manufactured of transparent Plexiglas to allow laser light to illuminate both the suction and pressure sides. End plates with a 30◦ outward bevel are suspended from the top of the channel above and below the fully submerged airfoil to reduce the end effects. The top end plate is painted mat-black to avoid background reflections and it has a slit to ensure the maximum plunging motion amplitude. The experimental arrangement is shown in Fig. 1. The vortex formation is recorded using the digital particle image velocimetry (DPIV) technique allowing detailed quantitative evolution of flow structures. The flow is illuminated at the airfoil’s mid-span plane by a dual cavity Nd:Yag laser (max. 120 mJ/pulse) and the water is seeded with silver coated glass spheres of 10 μm mean diameter. Two 10-bit cameras with 1600 × 1200 pixels resolution are positioned underneath the water channel. After obtaining the images to observe the flow structures around the airfoil, both cameras are shifted to the second location and the experimental cases are repeated to observe the flow structures in the near-wake. For each location, the two images from the two cameras are stitched before interrogation using two marker points in the illumination plane. Stitched PIV images are interrogated using a double frame, cross-correlation technique with a window size of 64 × 64 pixels and 50% overlapping in each direction. The final grid resolution of velocity vectors is 3.5 mm × 3.5 mm in the plane of the flow which corresponds to a resolution of 0.035c × 0.035c. The resulting measurement plane is represented with approximately 3600 velocity vectors. The maximum sampling rate for vector fields is 10 Hz which is used for a motion frequency of 0.1 Hz. Each data set includes 200 images which are acquired for at least two cycles of motion, whichever is slower in unequal frequency cases. The pitch and plunge motions of the airfoil are accomplished with two Kollmorgen/Danaher Motion AKM33E servo motors. The servo motors are connected to a computer via ServoSTAR S300 digital servo amplifier for pitching motion and ServoSTAR S600 for plunging motion, and motor motion profiles are generated by a signal generator Labview VI (Virtual Instrument) for the given amplitude and frequency. The same VI triggers the PIV system at the beginning of the third motion cycle of the airfoil and synchronization is achieved using a National Instruments PCI-6601 timer device. The kinematic motions of the airfoil considered in the current study are sinusoidal plunging and pitching and they are described with the following equations:
h(t ) = hamp cos(2π f 1 t + ψ)
α (t ) = α0 + αamp cos(2π f 2t )
k= St =
π fc U∞ 2 f hamp U∞
where U ∞ is the freestream velocity and f represents the frequency of motion. The investigated limits of non-dimensional parameters for the baseline cases of this study, when the plunging frequency is kept equal to the pitching frequency, are as follows: 0.1 St 1.0, 0.25 hamp /c 0.5, 0.31 k 6.26. 3. Results 3.1. Flow structure categories By investigating two-dimensional flow fields around and in the near-wake of a pitching and plunging airfoil using DPIV method, five main flow structure categories are identified depending on the role of separated vortex structures from the leading and trailing edges [12]. Flow structure categorization is summarized and sketched in Table 1 based on the five baseline cases. The first three columns show the instantaneous vorticity patterns around and in the nearwake of the airfoil at the beginning of downstroke, the next three columns show the vorticity patterns at the beginning of upstroke. A total of 200 instantaneous images were acquired for each case and data acquisition rates of the PIV system were chosen to record at least two periods of the airfoil motion for each frequency of individual cases. The last column of Table 1 shows the average of those 200 images in terms of combined vorticity and velocity fields. In Category A, two counter-rotating pairs of vortices are formed during an oscillation cycle. One pair appears on the lower side of the airfoil during the upstroke and one pair on the upper side during the downstroke and sheds into the wake. When the plunge amplitude is lower, the two counter rotating vorticity pairs are also formed. However, although the positive vortex which is generating from the leading edge during downstroke interacts with the negative signed vortex generating from the lower surface, it cannot connect with the positive vortex from the trailing edge, hence it loses strength and dissipates. This is a subset of Category A and this latter case is named A2 while the former one is named A1. For both subsets, averaged vorticity and velocity fields indicate a jet like flow. Similar to Category A, in Category B cases, a positive vortex is shed in the wake at the end of the downstroke while a negative vortex enlarges preserving its strength and detaches from the leading edge. However, in Category B cases, the process does not seem to be nearly reversed during the upstroke as in the Category A cases. The negative vortex is shed in the wake from the trailing
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Table 1 Summary of categories based on flow structures.
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Fig. 2. Motion kinematics for the baseline and the unequal frequency cases.
edge at the end of the upstroke. Category B also differs from Category A by having a slightly inclined downward trajectory of the shed vortices. As in Category A, depending on the amplitude of plunge motion, Category B presents two subsets with the same distinguishable differences. When the amplitude of plunge is equal or larger than 0.5c (for subsets denoted with 1), the roll-up radius of the vortex shedding is large at the trailing edge of the airfoil and the wake width of the jet like velocity profile is substantially increased. The presence of counter rotating pairs of vortices are still observed in Category B2; however, the vortices formed at the leading edge do not have enough time to evolve as they are swept away during
the following half cycle of the motion due to either lower plunge amplitude or larger free-stream velocity. Consequently, the roll-up radius of the vortex shedding is small for subsets denoted with 2. Category C is detected only for the cases where hamp /c = 0.25. The major characteristic of flow structures in this category is the shedding of two negative sign vortices per cycle of motion. In accordance, the average vorticity patterns clearly indicate the upward trajectory of additional negative vortex shed in the wake. In Category D, the negative signed vortex is generated from the leading edge during the downstroke motion of the airfoil. However, although the positive signed vortex is generated from the trailing edge during upstroke, the negative signed vortex is not
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formed. The near-wake vorticity patterns show that the negative vortex which detaches and sheds into the wake inclines downward and the positive vorticity layer at the trailing edge stretches in the wake and a trail of positive vortices lie along an upward arc. The near-wake patterns announce that the flapping motion will still induce a jet like velocity profile. In Category E, a negative vorticity layer on the upper surface and a positive vorticity layer on the lower surface of the airfoil are present during most of the motion stages and the wake also oscillates up and down with the airfoil motion. During the upstroke the vorticity layers are thin and attached on the surface; however, the negative vorticity layer detaches from the surface during the downstroke and exhibit localized cells of vorticity concentration, revealing Karman-like vortex formations in the near-wake at the beginning of the downstroke. The near-wake vorticity patterns announce a velocity deficit. Table 2, depicting the details of the parameter space includes all test cases of the aforementioned study. The table is color-coded in order to clearly distinguish the borders of the categories and the exemplified cases are boldfaced. The rows are grouped based on the plunge amplitude. For each row, the plunge amplitude and the frequency of motion for both pitch and plunge motions are the same. The flow speed increases from left to right across the table. The Strouhal number is fixed for each column and decreases from left to right; however, due to the difference in flapping motion frequency, the flow speed is not the same in a single column. The two letter notations in Table 2 indicate that these cases are identified as a transition from one category to the next one. The category notations with numbers indicate the subsets of the same category and the blanks indicate the absence of corresponding data acquisition. In this study, the effects due to unequal frequencies of the pitching and plunging motions are investigated for each experimental case representing the above mentioned flow structure categories [12]. 3.2. Effect of unequal frequency Table 3 and Table 4 show the averaged velocity and corresponding vorticity plots of unequal frequency cases along with the five baseline cases representing each flow structure category. The velocity fields are averaged over 200 images belonging at least to two periods of pitch or plunge motion, whichever is slower. Table 3 represents the unequal frequency cases where plunge motion frequency is kept the same as the baseline case and Table 4 represents those where pitch motion frequency is kept the same as the baseline case. For both tables, the first column of averaged images belongs to baseline cases with the frequency which is varied increasing from left to right in each row. Both tables show that the averaged vorticity patterns are mostly affected by the variation in plunge frequency. When the plunge frequency is kept the same as the baseline case (Table 3), major characteristics are very similar for f 1 > f 2 . Fig. 3 and Fig. 4 show the instantaneous vorticity patterns in the near-wake of the airfoil for the selected case exhibiting the flow structure characteristics of Category A1 for equal ( f 1 = f 2 = 0.2 Hz) and unequal motion frequencies ( f 1 = 0.2 Hz, f 2 = 0.1 Hz), respectively. A cycle of slower motion, namely pitch in this case, is selected for the representation of flow structure evolution. In Fig. 3 and Fig. 4, one period of plunge is represented by 8 images. The first image on the right hand side of the figure represents the airfoil at its mid-plunge position. As the flow is from left to right in the experiments, the figure simulates the forward motion of the airfoil from right to left. In comparison, the vorticity patterns in each plunge cycle is very similar to each other and to those obtained when the pitch and plunge frequencies are the same. Both positive and negative vortices are
shed twice in a pitch cycle locking on the plunge frequency. The reduction of pitch frequency with respect to the plunge frequency still results in slight differences. Both the positive vortex and the negative leading edge vortex are stretched more than those in the baseline case before they are shed from the trailing edge. The distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle is narrower compared to the observed ones at the leftmost and rightmost images which represent the beginning of the first and second pitch cycles respectively. As a final remark on differences, during the downstroke motion in the second plunge cycle, some remains of the shed negative vortex wanders close to the positive vortex which is rolling up and shedding. It should be noted that the freestream velocity for each row in Table 3 and Table 4, hence the convection of the vortices downstream, is the same. Based on a few averaged flow field results shown in Table 3, the flow structures are also very similar when the plunge frequency is kept same as the baseline case and the pitch frequency is increased, i.e., f 1 < f 2 . Fig. 5 exemplifies such a case exhibiting the flow structure characteristics of Category A1 when the plunge frequency is equal to pitch frequency and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of plunge motion when the pitch frequency is twice the plunge frequency ( f 1 = 0.2 Hz, f 2 = 0.4 Hz). In comparison with the baseline case, the increase in pitch frequency does not affect the major flow structures and the timing of vortex shedding. Both positive and negative vortices are shed once during two pitch cycles locking again on the plunge frequency. The slight difference with respect to the baseline case is related to the stretch of the negative leading edge vortex during the upstroke and to its convection and integrity when shed during the downstroke motion. As a result, during the upstroke motion in the second half of the second pitch cycle, some remains of the shed negative vortex wanders close to the positive vortex which is shed earlier. Table 4, representing the unequal frequency cases where the pitch frequency is kept the same with the baseline cases also shows that the averaged vorticity patterns are mostly affected by the variation in plunge frequency, whether increased or decreased. For all categories A through E, the use of a different and lower plunge motion frequency ( f 1 < f 2 ) generally results in a decrease in the averaged vorticity levels. The strength of vortical structures is mostly dictated by the plunge motion. The timing of the vortex shedding is also affected; vorticity patterns for Categories A1 and A2 actually show how employing unequal frequency has an effect on the vortex trajectories in the near-wake. The positive vortex shed from the trailing edge follows nearly a straight line; however there are two branches for the negative vortex, one parallel to the positive and the other shed downward in the freestream direction. Fig. 6 exemplifies the baseline case exhibiting the flow structure characteristics of Category A2 when pitch frequency is twice the plunge frequency ( f 1 = 0.3 Hz, f 2 = 0.6 Hz) and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of plunge motion. Although the positive vortex is shed once in a plunge cycle, the negative vortex is shed twice, both during the upstroke. A similar, however more pronounced trajectory change occurs for Category D. The negative vortex is shed upward parallel to the positive vortex. The instantaneous vorticity patterns for this case where the pitch frequency is kept the same and the plunge frequency is reduced by half show how the timing of vortex shedding is affected (Fig. 7). An early separation of the negative leading edge vortex causes the wake to exhibit a Karman-like shedding although the positive vortex is shed upward during the downstroke motion. Based on the few averaged flow field results shown in Table 4, the flow structures are also deeply affected when the pitch frequency is kept the same as the baseline case and the plunge
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Fig. 3. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 when the plunging frequency f 1 is equal to pitching frequency f 2 [12].
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Table 2 The details of the parameter space.
Fig. 4. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.2 Hz and f 2 = 0.1 Hz.
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Table 3 Averaged velocity and vorticity profiles for equal and unequal plunging ( f 1 ) and pitching ( f 2 ) frequencies for each category, when f 1 is kept the same as the baseline case.
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Table 4 Averaged velocity and vorticity profiles for equal and unequal plunging ( f 1 ) and pitching ( f 2 ) frequencies for each category, when f 2 is kept the same as the baseline case.
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Fig. 5. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.2 Hz and f 2 = 0.4 Hz.
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Fig. 6. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A2 for f 1 = 0.3 Hz and f 2 = 0.6 Hz.
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Fig. 7. Instantaneous vorticity patterns in the near-wake of the airfoil for Category D for (a) f 1 = f 2 = 0.6 Hz and (b) f 1 = 0.3 Hz and f 2 = 0.6 Hz.
frequency is increased, i.e., f 1 > f 2 . The use of a different and higher plunge motion frequency generally results in an increase in the averaged vorticity levels. The strength of vortical structures is mostly dictated by the plunge motion. Fig. 8 exemplifies such a case exhibiting the flow structure characteristics of Category A1 when f 1 = f 2 and shows the instantaneous vorticity patterns in the near-wake of the airfoil for one cycle of pitch motion when the plunge frequency is twice the pitch frequency ( f 1 = 0.4 Hz, f 2 = 0.2 Hz). In comparison with the baseline case, the increase in plunge frequency does not affect the major flow structures and the timing of vortex shedding. There is high similarity in this in this respect between Figs. 4 and 8 representing unequal frequency cases where the plunge frequency is twice the pitch frequency. The vorticity patterns in Fig. 8 also exhibit slight differences with respect to the baseline case belonging to Category A1. The negative leading edge vortices are stretched more than those in the baseline case before they are shed from the trailing edge. On the other hand, the positive vortex is stretched more during the second plunge cycle; however, its roll-up is pronounced during the first plunge cycle. Similar to what is observed in Fig. 4, the distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle is narrower compared to the observed ones at the leftmost and rightmost images which represent the beginning of the first and second pitch cycles.
Interesting effects of having unequal frequencies are observed for Category E, when the pitch frequency is kept the same as the baseline case and the plunge frequency is increased distinctively. The use of a plunge frequency which is five times the pitch frequency results in a flow structure category change; i.e. the case where f 1 = 0.5, f 2 = 0.1 belongs to Category B rather than Category E. 4. Conclusions The investigation started from a flow structure categorization based on Strouhal numbers which are unique as the pitch and plunge frequencies are kept the same. Although the former Strouhal numbers are matched with either the pitch or plunge frequencies in the investigated cases of unequal frequencies, the kinematics in the current study are completely different. The brief overlook on the effect of unequal frequencies shows that the flapping motion becomes much more complicated and the physics underlying the effect of unequal frequencies deserves a more detailed investigation. However, a major conclusion still holds; for the parameters studied, it was found that the plunge motion influences more the near-wake flow structures as foreseen. Throughout this study the pitch amplitude and its initial value are kept constant and the variations in the maximum effective angle of attack are due to the variations in the Strouhal number, which in turn is
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Fig. 8. Instantaneous vorticity patterns in the near-wake of the airfoil for Category A1 for f 1 = 0.4 Hz and f 2 = 0.2 Hz.
related to the plunge amplitude. It is therefore not surprising that the plunge motion tends to dominate. The following summarized results fortify this conclusion: – When the plunge frequency is kept constant, the decrease in pitch frequency only results in slight differences with respect to the baseline cases. The differences include, firstly, the stretching of the vortices before they are shed from the trailing edge, secondly, the reduced distance in between the positive shed vortex and the stretched negative leading edge vortex at the beginning of the second plunge cycle and finally, the near-wake exhibiting some remains of the shed negative vortex wandering close to the positive vortex which is rolling up and shedding.
– When the plunge frequency is kept constant, the increase in the pitch frequency does not affect the major flow structures and the timing of vortex shedding. Both positive and negative vortices are shed once during two pitch cycles locking again on the plunge frequency. – On the other hand, when the pitch frequency is kept constant, irrespective of the decrease or increase in the plunge frequency, the strength of vortical structures is mostly dictated by the plunge motion. The timing of the vortex shedding and the vortex trajectories in the near-wake are also affected when the plunge frequency is decreased. – The major flow structures and the timing of vortex shedding are highly similar when the plunge frequency is twice the pitch frequency, regardless of which one is kept constant as the baseline case. When the plunge frequency is increased
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significantly i.e., five times the pitch frequency, a change in flow structure category occurs; a jet like shedding takes over Karman-like shedding which is the major characteristic of the baseline case. In the light of the conclusions drawn from this investigation, the use of higher pitch amplitudes are yet to be explored to clarify the extent of plunge dominated flows in further studies on unequal frequencies. Acknowledgement The authors would like to thank Selin Alperer-Tatli, M.A., in Teaching English as a Foreign Language, for editing the manuscript. References [1] J.M. Anderson, K. Streitlien, D.S. Barrett, M.S. Triantafyllou, Oscillating foils of high propulsive efficiency, J. Fluid Mech. 360 (1998) 41–72. [2] Y.S. Baik, L.P. Bernal, K. Granlund, M.V. Ol, Unsteady force generation and vortex dynamics of pitching and plunging aerofoils, J. Fluid Mech. 709 (2012) 37–68. [3] A. Betz, Ein Beitrag zur Erklarung des Segelfluges, Z. Flugtech. Motorluftschif. 3 (1912) 269–272. [4] W. Birnbaum, Das ebene Problem des Schlagenden Fluegels, Z. Angew. Math. Mech. 4 (1924) 277–292. [5] D.G. Bohl, M.M. Koocheshahani, MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency, J. Fluid Mech. 620 (2009) 63–88. [6] A.-J. Buchner, N. Buchmann, K. Kilany, C. Atkinson, J. Soria, Steroscopic and tomographic PIV of a pitching plate, Exp. Fluids 52 (2012) 299–314. [7] L.W. Carr, Progress in analysis and prediction of dynamic stall, J. Aircr. 25 (1) (1988) 6–17. [8] D.J. Cleaver, Z. Wang, I. Gursul, Lift enhancement on oscillating airfoils, in: 39th AIAA Fluid Dynamics Conference, AIAA 2009-4028, 2009. [9] R.N. Conger, B.R. Ramaprian, Pressure measurements on a pitching airfoil in a water channel, AIAA J. 32 (1) (1994) 108–115. [10] G.C.H.E. de Croon, M.A. Groen, C. De Wagter, B.D.W. Remes, R. Ruijsink, B.W. van Oudheusden, Design, aerodynamics, and autonomy of the DelFly, Bioinspir. & Biomim. 7 (2) (2012) 025003. [11] I. Fenercioglu, Experimental investigation of flow structures around an oscillating airfoil in steady current, PhD thesis, Istanbul Technical University, 2010. [12] I. Fenercioglu, O. Cetiner, Categorization of flow structures around a pitching and plunging airfoil, J. Fluids Struct. 31 (2012) 92–102. [13] D.J. Garmann, M.R. Visbal, Numerical investigation of transitional flow over a rapidly pitching plate, Phys. Fluids 23 (2011) 094106. [14] I.E. Garrick, Propulsion of a flapping and oscillating airfoil, NACA Report No. 576, 1936. [15] M. Groen, B. Bruggeman, B. Remes, R. Ruijsink, B.W. van Oudheusden, H. Bijl, Improving flight performance of the flapping wing MAV DelFly II, in: International Micro Air Vehicle Conference and Competitions (IMAV 2010), 2010. [16] L. Guglielmini, P. Blondeaux, Propulsive efficiency of oscillating foils, Eur. J. Mech. B, Fluids 23 (2003) 255–278. [17] K. Isogai, Y. Shinmoto, Y. Watanabe, Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil, AIAA J. 37 (10) (1999) 1145–1151. [18] K.D. Jones, M.F. Platzer, Design and development considerations for biologically inspired flapping-wing micro air vehicles, Exp. Fluids 46 (5) (2009) 799–810. [19] K.D. Jones, C.M. Dohring, M.F. Platzer, Experimental and computational investigation of the Knoller–Betz effect, AIAA J. 36 (7) (1998) 1240–1246. [20] K.D. Jones, T.C. Lund, M.F. Platzer, Experimental and computational investigation of flapping wing propulsion for micro air vehicles, fixed and flapping wing aerodynamics for micro air vehicles, in: Progress in Astronautics and Aeronautics, AIAA, New York, 2001, p. 195, pp. 307–339, Chap. 16. [21] K.D. Jones, C.J. Bradshaw, J. Papadopoulos, M.F. Platzer, Bio-inspired design of flapping-wing micro air vehicles, Aeronaut. J. 109 (2005), 1098, pp. 385–393. [22] K.D. Jones, C.M. Dohring, M.F. Platzer, Wake structures behind plunging airfoils: a comparison of numerical and experimental results, AIAA 96-007, 1996. [23] Y.W. Jung, S.O. Park, Vortex-shedding characteristics in the wake of an oscillating airfoil at low Reynolds number, J. Fluids Struct. 20 (2005) 451–464. [24] R. Katzmayr, Effect of periodic changes of angle of attack on behavior of airfoils, NACA TM 147, 1922.
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