Effect of Surface Morphing on the Wake Structure and Perforance of Pitching-rolling Plates Yan Ren 1, Geng Liu 2 and Haibo Dong 3 Departent of Mechanical & Aerospace Engineering University of Virginia, harlottesville, VA 22904 Most wings and fins found in nature are deforable during the flapping otion. In the current work, a finite-difference based iersed boundary solver is used for studying the wake structures and hydrodynaic perforance of elliptic plates with and without surface orphing under a pitching-rolling otion. Both chordwise and spanwise orphing are studied for understanding the propulsion perforance of the plates. he results indicate that pitching-rolling plates in general have unique wake topology coparing to the pitchingplunging plates. he wake structure and propulsion perforance of the plates can be significantly affected by the surface orphing. Noenclature Pitching angle Aplitude of plate pitching angle Bending angle D U, Bending aplitude at downstroke and upstroke, respectively Rolling angle Rolling aplitude wisting index D U, wisting aplitude at downstroke and upstroke, respectively c hord length hrust coefficient Mean thrust coefficient Power coefficient Mean power coefficient f Flapping frequency Re Reynolds nuber St Strouhal nuber Flapping period N 1. Introduction atural swiers/flyers, all equipped with deforable fins/wings, which can actively or passively produce deforations in both spanwise and chordwise [1-4]. It is reported that such deforations can provide extra hydrodynaic/aerodynaic benefits, including the increent in 1 Ph.D. andidate, AIAA student eber, yr5y@virginia.edu 2 Research Associate, AIAA eber, gl6d@virginia.edu 3 Associate Professor, AIAA Associate Fellow, haibo.dong@virginia.edu 1 Aerican Institute of Aeronautics and Astronautics
force production and the decreent in energy consuption [5-7]. he flexible propulsors are of great interest to both the scientific study [8-10] and the engineering design [11-13]. Flapping otion is widely adopted in the biological propulsion syste, such as fish pectoral fins and insect/bird wings. A lot of previous studies on the flapping otion are based on a siplified kineatic odel of pitching-plunging [14-16], which assues unifor otions in spanwise direction. However, the real fin/wing in nature presents pitching-rolling otion, in which the plunging otion is replaced by a rolling of the fin/wing about its root. he presence of rolling otion will enlarge the three-diensionality of the proble, which ay be iportant to the flow features or the perforance of the propulsors. Several experiental works are perfored previously to study the perforance of pitching-rolling plates with different shapes [17, 18]. hey found that twisting plays an iportant role in the perforance of the pitching-rolling plate. In this work, our purpose is to explore the effects of both spanwise and chordwise orphing in a flapping propulsor. A deforable plate odel in pitching-rolling otion, which is to iic the propulsor flapping around its root, is built. he perforance and wake structure related to the deforable otion are discussed in detail. he paper will be organized as follows. he ethodology applied by this work will be introduced first. Next, we will present results of rigid pitching-rolling plates with different pitching aplitudes. After that, the plate surface orphing effects will be discussed. At last, the conclusion of this work will be presented. 2. Methodology A atheatical odel for the pitching-rolling plate, which is inspired fro bio-propulsors, is built in this work. A second-order finite-difference based iersed-boundary solver [14, 19] is then used to obtain the corresponding aerodynaic perforances and flow features. he details of nuerical ethod, plate surface orphing odeling and siulation setup are introduced in the following sections. A. Nuerical ethod he in-house flow solver used in this work eploys a artesian grid ethod wherein flow past iersed coplex geoetries [14, 19, 20] can be siulated on non-body conforal artesian grids. he ethod eploys a second-order central difference schee in space and a second-order accurate fractional-step ethod for tie advanceent. ode validations and details of nuerical ethodologies can be found in Mittal et al. [19] B. Plate kineatics he plate kineatics is governed by the following equations: 1 t cos2 ft 2 (1) t sin 2 ft where is the rolling angle and is the plate pitching angle. and are the corresponding aplitudes of these two angles. In this work, is chosen to be 90 and varies between 15 and 60. Figure 1 shows the scheatic of this plate kineatics. he plate s aspect ratio applied in this work is AR 2. 2 Aerican Institute of Aeronautics and Astronautics
Figure 1: Scheatic of plate kineatics. Plate surface orphing odeling Plate surface orphing patterns, which are chordwise twisting and spanwise bending, are odeled in both tie and space. For twisting, as shown in the following equations, Fourier series with N 2 are used in the teporal odeling: N a0 2nt 2nt t ancos bnsin (2) 2 n1 where is twisting index. is the flapping period. a i and b i are the corresponding Fourier coefficients, which are deterined via solving the following syste of equations: D U 0 0; t1 ; t2 ; t1 0; t2 0 (3) where t 1 and t 2 denote the tiings of axiu twisting in downstroke and upstroke respectively. D U and denote twisting index aplitudes in downstroke and upstroke respectively, which are control paraeters of twisting. For the spatial odeling of plate twisting, local twisting angles follow a linear distribution. It is based on observations of the surface orphing of real insect wings and fish fins through highspeed caeras [4]. Figure 2(a) shows the scheatic plot of twisting odeling. he following equations show the spatial odeling of wing twisting: s s (4) 100 where s denotes the plate span location ( 0 s 100 ). 3 Aerican Institute of Aeronautics and Astronautics
(a) (b) Figure 2: Scheatic of (a) twisting and (b) bending odeling Figure 2(b) shows the scheatic plot of plate bending odeling. In this work, the inflexion ratio ps1 S1 S2 is fixed as 0.5, and the inflexion angle at every tie step is odeled using the equation below: N c0 2nt 2nt t cncos dnsin (5) 2 n1 Where N 2. c i and d i are the corresponding Fourier coefficients, which are deterined via solving the following syste of equations: D U 0 0; t ; t ; t 0; t 0 (6) 1 2 1 2 D U Siilar as the teporal odeling of twisting, here and denote inflexion angle aplitudes in downstroke and upstroke respectively, which are control paraeters of bending. For all cases discussed in the results part, the twisting index and inflexion angle aplitudes in downstroke and upstroke are set equal with each other, in order to reduce the paraeters involved in the paraetric study. In suary, we have two paraeters ( and ) to control the surface orphing of the pitching-rolling plates. We will discuss 7 cases in the following sections. he first 4 cases are rigid plate cases with different pitching aplitudes. Based on the results of those 4 cases, we choose one of the to be the baseline case and naed it as case R, which eans the baseline rigid plate case. After that, chordwise twisting, spanwise bending and the cobination of those two are added to the case R to study the effects of plate surface orphing. he 3 new cases are naed as R+, R+B and R++B, respectively. he detailed configurations of those cases are listed in table I. D. Siulation setup Figure 3 shows the stretching grids configuration of this work. he coputational doain has the diension of 2020 20, and the dense region has the diension of 8.0 6.0 2.2. he nuber of grids used in the dense region are 296 224 72. Along the plate chord, there are no less than 35 grids are used to resolve the oving boundary. he non-diensional incoing velocity U is fixed as 1, and zero gradient boundary condition is applied to the downstrea boundary. Noslip boundary condition is applied at the plate surface. As defined in the following equations, the Reynolds nuber and Strouhal nuber of all cases discussed below is fixed as 200 and 0.6, respectively. 4 Aerican Institute of Aeronautics and Astronautics
Uc fa Re, St (7) U Where c is id-chord length, denotes kineatic viscosity, f is flapping frequency and A stands for peak-to-peak aplitude easured at the id chord. ases able I: Pitching-rolling plate cases configurations 1 90 15 0 0 2: R 90 30 0 0 3 90 45 0 0 4 90 60 0 0 5: R+ 90 30 30 0 6: R+B 90 30 0 30 7: R++B 90 30 30 30 Figure 3: Stretching grids configuration 3. Results In this section, we first present the results of rigid pitching-rolling plates with different pitching aplitudes. And then, the effects of chordwise twisting and spanwise bending, which are added to a baseline rigid pitching-rolling plate with pitching aplitude equals to 30, will be exained. Finally, the echanis for the efficiency iproveent of twisting will be discussed. We use thrust coefficient ( ) and power coefficient ( ) to quantify the perforance of the pitching-rolling plate, which are defined as, Fx Pout, 2 (8) 3 0.5U S 0.5U S 5 Aerican Institute of Aeronautics and Astronautics
where F x is the thrust force; P out is the power output; is the density of the fluid and S is the area of the plate. he tie averaged and over one flap cycle during the steady state are denoted by and, respectively. he propulsive efficiency is quantified by in this paper. A. Pitching-rolling plate without surface orphing Figure 4 shows the tie variation of the thrust and power coefficients for various pitching aplitudes with St 0.6 and Re 200. he plots show the fifth cycle in the siulations, by which tie the flow has reached a stationary state. For all cases except for the case with 60, the thrust peaks twice in each cycle at the tie instant in the cycle when the foil is near the center of its trajectory, and so as the power coefficients peaks. In the 60 case, double peaks of thrust and power coefficients appears in each half stoke, which ay be the results of wake capture in higher pitching aplitude cases, while lower pitching aplitude cases cannot interact with the wake of previously shed vortices due to liited otions. (a) (b) Figure 4: hrust (a) and power (b) coefficient of the pitching-rolling plate with different pitching aplitudes (fro 15 to 60 ). able III: ie averaged value of plate during the fifth flapping cycle., and of rigid pitching-rolling 15 30 45 60 1.98 2.78 2.34 0.86 21.37 13.65 7.86 5.45 0.093 0.204 0.298 0.158 6 Aerican Institute of Aeronautics and Astronautics
(a) (b) R2 EV V R1 RV LEV Figure 5: Flow structure of a rigid pitching-rolling plate (pitching aplitude is 30 ) at iddownstroke. (a) contour of z on the vertical cut slice at 2/3 span away fro the root; (b) perspective view of the 3D flow. he vortical structures are identified by the iso-surface of Q- criteria (Q=1.0). able II listed cycle averaged values of thrust and power coefficients, and also the propulsive efficiencies, which defined as thrust over power ratio. It is found that the case with 30 has the highest thrust production while the propulsive efficiency is relatively high aong the 4 cases. Previous study [17] on a pitching-rolling plate at high Reynolds nuber also found that the pitching aplitude variation can change the propulsive perforance in ters of both the thrust and the propulsive efficiency. Figure 5 shows the flow features of the case with 30. A thrust producing wake can be identified in Figure 5(a). Vortex pairs with different directions, or say, vortex rings, can be observed, which will induce local jets (red arrow) to the direction opposite to the thrust direction. Figure 5(b) shows vortex structures in three-diensional space. he shed trailing edge vortex (EV), along with the tip vortex (V), connect with the newly developing leading edge vortex (LEV) and root vortex (RV) to for a vortex ring, which will shed right after the stroke reversal. hose vortex rings will induce flow jets as shown in Figure 5(a). he flow structures of a typical pitching-rolling plate is siilar to that of a pitching-plunging plate [14]. B. Pitching-rolling plate with surface orphing In the previous sub-section, we copared the perforance of rigid pitching-rolling plates with different pitching aplitude ( ) and found that the case with 30 has the highest thrust. In this sub-section, we use this case as the baseline case, and add the chordwise and spanwise orphing to it to exaine the effects. he aplitudes of both the chordwise and spanwise orphing are 30. 7 Aerican Institute of Aeronautics and Astronautics
(a) (b) Figure 6: hrust (a) and power (b) coefficient history of the pitching-rolling plate (pitching aplitude = 30 ) with different cobinations of surface orphing patterns. R: baseline rigid plate case; R+: chordwise twisting is added to the baseline case; R+B: spanwise bending is added to the baseline case; R++B: both the chordwise twisting and spanwise bending are added to the baseline case. able III. ie averaged values of, and of pitching-rolling plate with different cobinations of surface orphing patterns. he pitching aplitude of the baseline rigid plate case is 30. ases R R+ R+B R++B 2.84 1.97 2.85 2.53 14.10 6.41 14.60 8.32 0.202 0.307 0.196 0.304 Figure 6 and able III shows the perforance coparison of the rigid plate and deforable plates. First, we exaine the effect of the bending otion. According to able III, by coparing the results of case R and R+B, it is found that the thrust increent is less than 1% and the efficiency decreased about 3%. Both of the are negligible. his result deonstrates that the effect of adding only the spanwise bending to a rigid pitching-rolling plate is relatively sall. he effect of the chordwise twisting is uch ore significant than that of the spanwise bending. he results of case R and R+ show that by adding the chordwise twisting to the rigid plate, the thrust decreases by 30.6%. However, the power output decreases uch ore than this, which is 54.5%. And this results in a rearkable efficiency iproveent (about 52%). he results deonstrate that adding the chordwise twisting is an effective way of iproving efficiency in a pitching-rolling plate. It is worth noting that in case R+, the tie interval where shows significant decreent of thrust coefficient coparing to case R, is quite different fro that of power coefficient. he thrust decreent happens in only two narrow tie intervals ( 4.1 t 4.35 in the first half stroke and 4.6 t 4.85 in the second half stroke) during one flapping cycle. However, the tie interval of power decreasing ( 4.05 t 4.40 in the first half stroke and 4.55 t 4.90 in the second half stroke) is wider than forer. hese features is highly related the unsteady flows around the plates, which will be discussed in the future. 8 Aerican Institute of Aeronautics and Astronautics
(a) (b) (c) (d) u U u U Figure 7: Side view of the flow structures of (a) baseline rigid plate (case R) with 30 and (b) plate with both chordwise and spanwise orphing (case R++B). Mean flow (incoing flow subtracted) contour of (c) case R and (d) case R++B. Finally, the effect of adding both the twisting and bending to the baseline rigid plate is studied. he coparison of the results for case R+ and R++B indicates that spanwise bending will be ore effective if it is cobined with chordwise twisting. he thrust in case R++B is about 28.4% higher than that in case R+, while the efficiency only drops about 1%. Actually, spanwise bending is usually observed accopanied with the chordwise twisting in biological propulsors such as fish pectoral fins and insect wings. Our results deonstrate that one of the advantages of the cobination of spanwise and chordwise orphing is to enhance the thrust production while keep the efficiency the sae. he perforance of the plate is directly related to the induced flow structures. Here, we copare the flow features of the baseline rigid plate case (R) to the case with surface orphing (R++S). As shown in figure 7, it is found that the surface orphing of the plate do not change the ain flow features of pitching-rolling plate. wo sets of inclined vortex ring loops can be observed in both cases. he ean flows of both cases show the sae feature, i.e., two horn-like backward jets is induced by the vortex rings in the downstrea wake. However, the difference is also obvious. he inclination angles ( in figure 7(b)) of the wake in these two cases are quite different. he value of in deforable plate (case R++B) is only 25 which is only 50% of that in rigid plate (case R). his leads to ore concentrated backward jets in the case with surface orphing than that in baseline rigid plate case, which is ainly responsible for the efficiency iproveent. 4. onclusion In this work, we ai at exaining the role of chordwise and spanwise orphing in pitchingrolling plates. First of all, we build a orphing plate odel, which includes chordwise twisting and 9 Aerican Institute of Aeronautics and Astronautics
spanwise bending in pitching-rolling otion inspired fro bio-propulsors. Force production, power consuption and flow structures of the odeled plate are obtained through an in-house iersed boundary ethod based FD solver. Rigid pitching-rolling plates with different pitching aplitudes, varied fro 15 to 60, are studied. he results show that the case with pitching aplitude of 30 generates the highest thrust. Using this case as the baseline case, the roles of the surface orphing are exained by adding the spanwise bending and chordwise twisting to it. he siulation results show that there is no big variation of the perforance by adding only the spanwise bending to a rigid plate. However, by adding the chordwise twisting, the efficiency (quantified by averaged thrust over power) of the pitching-rolling plate is increased fro 0.20 to 0.31, while the thrust production is decreased fro 2.84 to 1.97. Although adding spanwise bending does not alter the perforance uch, the cobination of the spanwise bending and chordwise twisting can further iprove the thrust production while keep the propulsive efficiency the sae. he plate with both spanwise and chordwise orphing has a narrower wake and this ake the backward jet ore concentrated, which is responsible for the efficiency iproveent. Acknowledgeent his work is supported by AFOSR FA9550-12-1-0071 onitored by Dr. Douglas Sith and ONR-MURI Grant N00014-14-1-0533 onitored by Dr. Robert Brizzolara. References [1] G. V. Lauder, P. G. Madden, R. Mittal, H. Dong, and M. Bozkurttas, "Locootion with flexible propulsors: I. Experiental analysis of pectoral fin swiing in sunfish," Bioinspiration & Bioietics, vol. 1, p. S25, 2006. [2] Z. J. Wang, "Dissecting insect flight," Annu. Rev. Fluid Mech., vol. 37, pp. 183-210, 2005. [3] S. M. Walker, A. L. hoas, and G. K. aylor, "Photograetric reconstruction of highresolution surface topographies and deforable wing kineatics of tethered locusts and free-flying hoverflies," Journal of he Royal Society Interface, vol. 6, pp. 351-366, 2009. [4]. Koehler, Z. Liang, Z. Gaston, H. Wan, and H. Dong, "3D reconstruction and analysis of wing deforation in free-flying dragonflies," he Journal of experiental biology, vol. 215, pp. 3018-3027, 2012. [5] G. Lauder and P. Madden, "Fish locootion: kineatics and hydrodynaics of flexible foil-like fins," Experients in Fluids, vol. 43, pp. 641-653, 2007. [6] J. Young, S. M. Walker, R. J. Bophrey, G. K. aylor, and A. L. hoas, "Details of insect wing design and deforation enhance aerodynaic function and flight efficiency," Science, vol. 325, pp. 1549-1552, 2009. [7] S. Alben, "Optial flexibility of a flapping appendage at high Reynolds nuber," J. Fluid Mech, vol. 614, pp. 355-380, 2008. [8] P. Liu and N. Bose, "Propulsive perforance fro oscillating propulsors with spanwise flexibility," Proceedings of the Royal Society of London. Series A: Matheatical, Physical and Engineering Sciences, vol. 453, pp. 1763-1770, 1997. [9] M. B. H. DONG, R. MIAL, and P. M. A. G. V. LAUDER, "oputational odelling and analysis of the hydrodynaics of a highly deforable fish pectoral fin," J. Fluid Mech, vol. 645, 2010. [10] Y. Ren and H. Dong, "oputational Optiization of Flexible Wing Aerodynaic Perforance in Hover," in the 30th AIAA Applied Aerodynaics onference, 2012. 10 Aerican Institute of Aeronautics and Astronautics
[11] J. angorra, P. Anquetil,. Fofonoff, A. hen, M. Del Zio, and I. Hunter, "he application of conducting polyers to a biorobotic fin propulsor," Bioinspiration & bioietics, vol. 2, p. S6, 2007. [12] J. L. angorra, G. V. Lauder, I. W. Hunter, R. Mittal, P. G. Madden, and M. Bozkurttas, "he effect of fin ray flexural rigidity on the propulsive forces generated by a biorobotic fish pectoral fin," he Journal of Experiental Biology, vol. 213, pp. 4043-4054, 2010. [13]. J. Esposito, J. L. angorra, B. E. Flaang, and G. V. Lauder, "A robotic fish caudal fin: effects of stiffness and otor progra on locootor perforance," he Journal of experiental biology, vol. 215, pp. 56-67, 2012. [14] H. Dong, R. Mittal, and F. Najjar, "Wake topology and hydrodynaic perforance of lowaspect-ratio flapping foils," Journal of Fluid Mechanics, vol. 566, pp. 309-344, 2006. [15] J. M. Anderson, K. Streitlien, D. S. Barrett, and M. S. riantafyllou, "Oscillating foils of high propulsive efficiency," Journal of Fluid Mechanics, vol. 360, pp. 41-72, Apr 10 1998. [16] D. A. Read, F. S. Hover, and M. S. riantafyllou, "Forces on oscillating foils for propulsion and aneuvering," Journal of Fluids and Structures, vol. 17, pp. 163-183, Jan 2003. [17] A. H. echet, "Propulsive perforance of biologically inspired flapping foils at high Reynolds nubers," Journal of Experiental Biology, vol. 211, pp. 274-279, Jan 15 2008. [18] P. R. Bandyopadhyay, D. N. Beal, J. D. Hrubes, and A. Mangala, "Relationship of roll and pitch oscillations in a fin flapping at transitional to high Reynolds nuber," Journal of Fluid Mechanics, vol. 702, pp. 298-331, Jul 10 2012. [19] R. Mittal, H. Dong, M. Bozkurttas, F. Najjar, A. Vargas, and A. Von Loebbecke, "A versatile sharp interface iersed boundary ethod for incopressible flows with coplex boundaries," Journal of oputational Physics, vol. 227, pp. 4825-4852, 2008. [20] R. Mittal and G. Iaccarino, "Iersed boundary ethods," Annu. Rev. Fluid Mech., vol. 37, pp. 239-261, 2005. 11 Aerican Institute of Aeronautics and Astronautics