Baixe Aerofólio S809 - 1-s2.0-s0142727x10001529 (muito bom an?lise) e outras Notas de estudo em PDF para Engenharia Mecânica, somente na Docsity! International Journal of Heat and Fluid Flow 32 (2011) 329–339Contents lists available at ScienceDirect International Journal of Heat and Fluid Flow journal homepage: www.elsevier .com/ locate/ i jhf fFluid forces on a very low Reynolds number airfoil and their prediction Zhou Y. a,⇑, Md. Mahbub Alam a,b, Yang H.X. c, Guo H. d, Wood D.H. e,1 a Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, South Africa c Department of Building Services Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong d School of Aeronautical Science and Engineering, Beijing University of Aeronautics and Astronautics, Xue Yuan Road No. 37, HaiDian District, Beijing, China e School of Engineering, The University of Newcastle, Callaghan NSW 2308, Australia a r t i c l e i n f o a b s t r a c tArticle history: Received 9 April 2009 Received in revised form 26 July 2010 Accepted 28 July 2010 Available online 17 September 2010 Keywords: Low Reynolds number airfoil Stall of airfoil Aerodynamics of airfoil Reynolds number effect0142-727X/$ - see front matter 2010 Elsevier Inc. A doi:10.1016/j.ijheatfluidflow.2010.07.008 ⇑ Corresponding author. Fax: +852 2365 4703. E-mail address: mmyzhou@polyu.edu.hk (Y. Zhou) 1 Present address: Schulich School of Engineering, UThis paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (a) of attack (0–90) and low to small chord Reynolds numbers (Rec), 5.3 103–5.1 104, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both a and Rec is deter- mined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Rec P 1.05 104 but is absent at Rec = 5.3 103. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on a. 2010 Elsevier Inc. All rights reserved.1. Introduction The aerodynamic characteristics of airfoils at a chord Reynolds number (Rec = qcU1/l, where q and l are the density and viscosity of the fluid, respectively, U1 is the free-stream velocity and c is the chord length of an aerofoil) of less than 5 105 are becoming increasingly important from both fundamental and industrial point of view, due to recent developments in small wind turbines, small unmanned aerial vehicles (UAVs), micro-air vehicles (MAVs), as well as researches on bird/insect flying aerodynamics (Brendel and Mueller, 1988; Hsiao et al., 1989; Dovgal et al., 1994; Lin and Pauley, 1996). For example, at the starting stage of a 500 W wind turbine, the tip Rec increases from 1 104 to 1 105, and the angle (a) of attack reduces gradually from 86 to 20 (Ebert and Wood, 1997; Wright and Wood, 2004). A similar variation in a occurs dur- ing insect flight, but Rec may be even lower (e.g. Wang, 2005). For UAVs and MAVs, Rec is commonly in the range of 1 105 – 6 105. However, such low Rec problems have not been addressed sufficiently in the literature, let alone when combined with large angle of attack. General researches on airfoil aerodynamics have focused on conventional aircraft design with Rec beyond 5 105 and a below stall. Carmichael (1981), Lissaman (1983) and Mueller and DeLaurier (2003) reviewed the available low Rec studies, withll rights reserved. . niversity of Calgary, Canadaalmost all the measured Rec higher than the wind turbine values quoted above. The aerodynamics of hovering insect flight was explored (Ellington, 1984a–e). Usherhood and Ellington (2002a,b) investi- gated forces acting on hawkmoth and bumblebee wings in ‘propel- ler-like’ revolution at Rec = 1.1 103–2.6 104. The steadily revolving wings produced high lift and drag, which was ascribed to the formation of a leading-edge vortex. Miklosovic et al. (2004) measured in a wind tunnel the lift and drag on a flipper of a humpback whale (Rec = 5.05 105–5.2 105). They observed that the stall angle of a flipper with a leading edge protuberance could be enlarged by approximately 40%, relatively to a flipper with a smooth leading edge, which led to increased lift and de- creased drag. In spite of their importance, the experimental lift and drag data for low Rec are only available for some airfoils, and seldom beyond stall angle of attack. Among others, Critzos et al. (1955), Sheldahl and Klimas (1981), Michos et al. (1983) and Devinant et al. (2002) presented the test data of NACA0012 airfoil for a = 0–90 at Rec = 3.6 105–1.8 106. Using force balance to measure lift and drag at Rec = 1 105–7 105 and a = 0–90, Devinant et al. (2002) showed that lift grew from zero to a maximum for increasing a be- tween zero and stall, and then tumbled suddenly at stall, which oc- curred at a = 8–20, depending on Rec. They further observed that lift grew with a and, after achieving the global maximum at a 45, dropped slowly from a = 45 to 90. On the other hand, drag increased monotonically with a, reaching a maximum at a 90. Laitone (1997) measured the mean drag and lift forces successfully Nomenclature AD area of the airfoil (of unit length) projected on the y–z plane, c sin a AL area of the airfoil (of unit length) projected on the x–z plane, c cos a C chord length of airfoil C, C0, C1, C2, C3 constants in Eqs. (2), (3a1), (3a2), (3b1), (3b2), (3c1), (3c2), (3d1), (3d2), (4a), (4b), (5), and (6). CD, CL time-averaged drag and lift coefficients, D=ðc 0:5qU21Þ; L=ðc 0:5qU 2 1Þ CDrms, CLrms fluctuating (root-mean-square) drag and lift coeffi- cients D, L mean drag and lift forces per unit length of airfoil EL power spectral density functions of the lift signal fn natural frequency of the airfoil-fluid system fv vortex shedding frequency K–H Kelvin–Helmholtz Pb base pressure Rec chord Reynolds number, qU1c=l S ratio to c of distance between the leading edge and flow separation point St Strouhal number, fvc/U1 U1 free-stream velocity U streamwise mean velocity urms, vrms streamwise (x-component) and lateral (y-component) rms velocities x, y, z Cartesian coordinates a angle of attack am a corresponding to the maximum CL l viscosity of fluid q density of fluid Superscript* denote normalization by c and/or U1 Flow x 2 3y 1 7 5 (a) Side view 2 5 3 4 1 8 7 9 6 5 4-2 4-1 4-3 4-1 6 330 Y. Zhou et al. / International Journal of Heat and Fluid Flow 32 (2011) 329–339at Rec = 2.07 104, though with a < 30. The mean drag and lift forces at the same range of a were investigated for wings with an aspect ratio of around four at Rec = 104 by Kesel (2000), and for 20 wings of higher aspect ratio by Sunada et al. (2002) at Rec = 4 103. Selig and his co-workers have made a highly influential contribu- tion to low-speed aerodynamics of airfoils (e.g. Selig et al., 1989, 1995, 1996; Selig and McGranahan, 2004). Selig et al. (1989) noted a peculiar drag increase at a lift coefficient of 0.5 (Rec = 6 104), where the drag coefficient reached a maximum of 0.032. They con- nected the observation to the laminar separation bubble, inferred from surface oil flow visualization, and referred to this drag increase as the ‘‘bubble drag”. Based on their DNS data, Hoarau et al. (2003) calculated the lift and drag coefficients of NACA0012 airfoil only at a = 20 and Rec = 0.8 103–1.0 104. Although measured at a < 30, the mean drag and lift force data is completely absent for higher a. Furthermore, studies pertaining to the fluctuating forces on an airfoil are very scant over the whole range of a, notwithstand- ing the fact that the forces cause vibrations on an airfoil and acoustic noise, even leading to structural fatigue failures. As a matter of fact, these forces have already been identified as the major cause for the relatively short life and damages that occur at the tip of wind tur- bine blades. Aerodynamics of an airfoil is dependent appreciably on the air- foil model, in particular, at a < 20, but very slightly or negligibly at a > 20. A symmetric NACA 0012 airfoil is used presently as a mod- el. This type of airfoil is used not only in low Re vehicles (Murthy, 2000) but also in large transport aircraft (Tan et al., 2005), yielding a large lift and having relatively high stability due to the symmet- rical shape about the centerline. Our measurements were per- formed at Rec = 5.3 103–5.1 104 and at a = 0–90 in a water tunnel. The work aims to document the lift and drag coefficients, using a highly sensitive force sensor, and to determine the depen- dence on a and Rec of the time-mean lift coefficient (CL), drag coef- ficient (CD), and root-mean-square (rms) values (CLrms and CDrms) of fluctuating lift and drag coefficients for a unit depth of the airfoil. Furthermore, a theoretical analysis is performed to predict CD and CL of an airfoil.(b) Front view (c) Zoom of the torque resisting system Fig. 1. Sketches of experimental setup: 1—airfoil, 2—end plates, 3—airfoil support, 4—torque-resisting system, 5—setup base, 6—connection pole, 7—load cell, 8— working section walls of water tunnel, 9—cover plate, 4–1—U-shaped connectors, 4–2—circular plate, 4–3—pins around which the connectors can turn freely.2. Experimental details 2.1. Test facility and setup Experiments were conducted in a closed-loop water tunnel, with a test section of 0.3 m (width) 0.6 m (height) 2.4 m(length), at The Hong Kong Polytechnic University. The flow speed in the test section ranges from 0.05 m/s to 4 m/s. NACA0012 airfoil was used as the test model with a chord length of c = 0.1 m and a span of 0.27 m. The tests were carried out at Rec = 5.3 103– 5.1 104, over which the free-stream turbulence level was -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0.20 0.40 0.60 0.80 1.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.20 0.40 0.60 0.80 1.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.20 0.40 0.60 0.80 1.00 0 0.1 0.2 0.3 0.4 0.5 0.6 α=50° α=30° α=10° * U * U * U * rmsu * rmsv * rmsu * rmsv * rmsu * rmsv y* (a) (b) (c) Fig. 3. LDA-measured velocities: (a) a = 10, (b) 30, (c) 50. s, U; urms; r, vrms . Rec = 1.05 104. 0 20 40 60 80 Re 1.0×104, Williamson et al. (1995) 5×103 7×103 1.0×104 4.2×104, Laitone (1997) Hoarau et al. (2003) } Re 5.3×103 1.05×104 5.3×103 1.05×104 } } Load-cell measurement Momentum method α (°) CD 0 0.5 1.0 1.5 Fig. 4. Comparison between the load-cell-measured CD and that estimated from the conservation of momentum. Y. Zhou et al. / International Journal of Heat and Fluid Flow 32 (2011) 329–339 333the load cell measurement. At a large a, the wake is broadened so that the measured velocity profiles, limited by the test section width, could not cover the entire wake width, thus resulting in an underestimated CD. Since the momentum conservation method is essentially two-dimensional, the good agreement between this estimate and the load cell measurement provides a strong evidence for the two-dimensionality of the flow. This agreement also pro- vides a validation for the load cell measurement of CD. CD measured by Williamson et al. (1995) was about 0.27 at a = 20 and Rec = 1.0 104 (see Hoarau et al., 2003), in good agree- ment with the present estimate (CD = 0.28, Rec = 1.05 104). At the same a, Hoarau et al.’s (2003) DNS calculation (smooth flow) pre- dicted CD = 0.29, 0.34 and 0.42 at Rec = 1.0 104, 7 103 and 5 103, respectively. The first is close to our measurement at Rec = 1.05 104, the second falls in the present range of 0.28– 0.35 obtained from the two different measurement methods, but the third is higher than the present estimate (0.35, Rec = 5.3 103) probably because of a difference in Rec and the turbulent intensity. Hoarau et al.’s data at a = 20 indicates a decrease in CD for higher 1.0 1.5 2.0 Measurement Rec=5.3×103 1.05×104 5.1×104 3.6×105, Sheldahl & Klimas (1981) 7.6×105, Michos et al. (1983) CD Theoretical prediction from Eq. (4b) } Presen t 334 Y. Zhou et al. / International Journal of Heat and Fluid Flow 32 (2011) 329–339Rec for the range of Rec given. The present measurement shows that CD at Rec = 1.05 104 is always smaller than at Rec = 5.3 103 for a given a, re-confirming Hoarau et al.’s DNS calculation. The agree- ment between present and previous measurements provides an- other validation for the load cell measurement of CD. It is worth mentioning that Spedding and Hedenstrom (2009) estimated CD on a 2-D flat plate at Rec = 1.2 104 (a < 20) from the PIV mea- surement over x* = 2.0–3.2 based on the conservation of momen- tum. Their data displays a similar trend (not shown) to the present CD on an airfoil.0 10 20 30 40 50 60 70 80 90 0 0.5 α (°) – – – – – Rec = 5.3×103 — — — 1.05×104 ⎯ – ⎯ 7.6×105 Fig. 6. Dependence of CD on a.4. Measured mean drag and lift Figs. 5 and 6 present the blockage-corrected CL and CD as dis- cussed in Section 2.2, along with some published data at Rec = 3.6 105 and 7.6 105 measured by Sheldahl and Klimas (1981) and Michos et al. (1983), respectively. The smooth curve is a Spline curve fit to the measured data. Note that the present data at Rec = 5.1 104 is only shown for a 6 40 because the force at a > 40 is relatively large, exceeding the valid range of the load cell. Evidently, CL is dependent on Rec for all a, growing with higher Rec except near stall, consistent with previous reports, e.g., by Mas- sey (1979) and Laitone (1997) for a large change in Rec. The stall of an airfoil is characterized by a rapid drop in CL for a small increase in a and a burst of separation bubble, following a fully separated flow from the two edges of the airfoil. In general, the angle of attack at which stall occurs increases with Rec, though very slowly at Rec > 106 (Jacobs and Sherman, 1937; Marchman, 1987; Rusak et al., 2005). Fig. 5 shows the occurrence of the stall0 10 20 30 40 50 60 70 80 90 0 0.5 1.0 1.5 CL Rec 5.3×103 1.05×104 5.1×104 3.6×105, Sheldahl & Klimas (1981) 7.6×105, Michos et al. (1983) } Present (a) CL Prediction from Eq. (4a) – – – – – Rec = 5.3×103 — — — 1.05×104 ___ _ ___ 7.6×105 α (°) Measurement Rec = 5.3×103 1.05×104 7.6×105 0 10 20 30 40 50 60 70 80 90 0 0.5 1.0 1.5(b) Fig. 5. Dependence of CL on a (a) experimental measurement results, (b) compar- ison between experimentally obtained and theoretically predicted results.at a = 12 and 13 for Rec = 3.6 105 and 7.6 105, respectively, and about 10 presently for Rec = 1.05 104 and 5.1 104. The stall should occur slightly either before or beyond 10, which could not be accurately determined presently due to an increment of Da = 5. Interestingly, the stall is absent at Rec = 5.3 103, CL rising mono- tonically until a 45, without any appreciable drop as at higher Rec. The mechanism of stall has been previously reported at a Rec sufficiently high to lead to the stall (e.g., Devinant et al., 2002; Mueller and DeLaurier, 2003; Larsen et al., 2007; Yang et al., 2008; Raghunathan et al., 1988). As a increases from zero to stall, a number of phenomena can be seen: (i) the separation point on the suction side moves towards the leading edge; (ii) the separated boundary layer is laminar (e.g. Laitone, 1997), though transition to turbulence in the shear layer occurs initially at the tail of the sep- arated boundary layer and shifts towards the separation point; (iii) both CL and CD grow. With the stall a approached, transition to tur- bulence takes place near the separation point, and the separated boundary layer reattaches, forming a separation bubble. This bub- ble may suddenly burst, resulting in the occurrence of the stall. On the other hand, at a sufficiently low Rec, the transition to turbu- lence does not occur near the separation; the separated boundary layer remains laminar for a rather long downstream distance and does not reattach. The stall will not occur without the separation bubble generated. The separated shear layer at Rec = 5.3 103 re- mains laminar for a longer distance and hence never reattaches on the surface for both a (Fig. 7a and c). On the other hand, the sep- arated shear layer at Rec = 1.05 104 becomes turbulent near sep- aration, reattaching on the surface at a = 10 but remaining separated at a = 15 (Fig. 7b and d). The result shows unambigu- ously that the separation bubble, which is evident at Rec = 1.05 104, is absent at Rec = 5.3 103, corroborating our assertion that the stall cannot occur without the formation of a separation bubble. Beyond the stall a, CL displays a maximum at a = am 45 (Fig. 5), regardless of Rec, and then drops to about 0.08 at a = 90. A similar observation was made previously, e.g., by Devinant et al. (2002) and Raghunathan et al. (1988), though without expla- nation. One begs the question that why CL reaches a maximum at a 45, which will be answered in Section 5. In the post-stall re- gion, fully separated flow prevails (e.g., Yang et al., 2008). CD increases monotonously with a (Fig. 6) and reaches the max- imum at a = 90. A sudden jump in CD at the stall a is evident at Rec = 3.6 105 or 7.6 105 but less so at Rec = 1.05 104 and 5.1 104 because of a relatively large increment in a in measure- ments. Below the stall a, say a < 10, CD largely results from flow separation on the upper (suction) surface of the airfoil, as noted α Rec=5.3×103 Rec=1.05×104 10° 15° (d) (c) (b) (a) Fig. 7. Typical photographs from the LIF flow visualization, which display the presence of the separation bubble at Rec = 1.05 104 but not at Rec = 5.3 103. Y. Zhou et al. / International Journal of Heat and Fluid Flow 32 (2011) 329–339 335in flow visualization (not shown here) and the skin friction at low- er Rec, and drops with increasing Rec. However, at a post-stall a, the effect of Rec on CD is not monotonic. CD decreases with increasing Rec for Rec P 1.05 104, but increases for Rec = 5.3 103– 1.05 104, which was also reported in Hoarau et al.’s (2003) DNS study (a = 20). The slope of CD, dCDda , may be approximated by DCd Da . Based on the measured data in Fig. 6, DCdDa increases from a = 0 to 45, and then declines till a = 90, that is, a = 45 is an inflection point of CD (a).5. Prediction of mean drag and lift A linear mathematical analysis is carried out in this section to predict CL and CD, along with the prominent features of their dependence on a: (i) CL reaches a maximum at a 45 and then drops to a very small value (0.08) at a = 90, (ii) CD is maximum at a 90, (iii) the inflection point of CD (a) occurs at a 45. Note that, as a increases from 0 to 90, the area AL of the airfoil projected on the x-z plane shrinks following AL = ccos a given a unit spanwise length and a negligible thickness (only 12% of the chord for the NACA 0012 airfoil). The thickness may affect AL appreciably only at a 90. AL is directly linked with the magnitude of the lift force. Similarly, the area AD projected on the y–z plane could be ex- pressed as AD = csin a, which may be connected with the magni- tude of the drag force. With a increasing from 0 to 90, the bluffness of the airfoil changes from a streamline to maximum, where bluffness is defined as the body height, i.e., c sin a, projected in the y–z plane. It is plausible to assume that the base pressure (Pb), defined as the pressure at the midpoint of the suction surface, increases with a and its increase, i.e., dPb, is directly proportional to the increase in the ratio of bluffness to c, i.e., d{(c sin a)/c}, viz. dPb / d{(c sin a)/c} or Pb ¼ C1 sin aþ C2 ð2Þ where C1 and C2 are two constants. Pb is directly linked with CL or CD and could be assumed to be the representative pressure for the en- tire base (suction) surface. As such, the mean lift L and drag D on a spanwise unit length of the airfoil could be written as L / ALPb and D / ADPb, viz. L ¼ C3ALPb ð3a1Þ D ¼ C3ADPb ð3a2Þ where C3 – 0 is a proportionality constant, relating Pb and the forces. Plugging the expressions for AD, AL and Pb in Eqs. (3a1) and (3a2) yields L ¼ cC1C3 cos a sinaþ cC2C3 cos aD ¼ cC1C3 sin a sinaþ cC2C3 sin a Transform L and D to CL and CD, respectively, CL ¼ C1C3 2c sin a cos a qU21c þ C2C3 2c cos a qU21c ð3b1Þ CD ¼ C1C3 c2 sin a sina qU21c þ C2C3 2c sin a qU21c ð3b2Þ Eq. (3b) presents a general relationship of CL or CD with a and U1. For a given U1 or Rec, Eq. (3b) could be rewritten as CL ¼ C sin 2aþ C0 cos a ð3c1Þ CD ¼ 2C sin2 aþ C02 sina ð3c2Þ where C ¼ C1C3 qU21 ð3d1Þ and C0 ¼ C2C3 q21 are constants: ð3d2Þ For a non-cambered (symmetric) airfoil such as NACA 0012, CL = 0 at a = 0. Then C0 = 0 from Eq. (3c1), and C2 = 0 from Eq. (3d2). Eq. (3c) could be reduced to CL ¼ C sin 2a ð4aÞ CD ¼ 2C sin2 a ð4bÞ The constant C may be estimated from CL or CD measured at a post-stall a such as a = 45. C is presently 0.83 at Rec = 5.3 103 and 0.98 and at Rec = 1.05 104. Eq. (4a) articulates that CL is a sine function of a. It is likely that CL in Fig. 5 follows a sine curve except near the stall region, where the separation bubble bursts. The burst of a bubble always occurs in a discontinuous manner, resulting in a drastic change in the force coefficients (Alam et al., 2005), which is not considered in this analysis. Thus, CL calculated from Eq. (4a) conforms well to the data at a small Rec, i.e., 5.3 103, when the stall is absent. In order to derive the a, at which a maximum or minimum CL occurs, we differentiate Eq. (4a) with respect to a: dCL da ¼ 2C cos 2a ð5Þ Let dCLda ¼ 0, viz. 2C cos 2a ¼ 0 ð6Þ The solution to Eq. (6) is a = am = ±45. The positive and nega- tive values of am correspond to the maximum and minimum CL, respectively, which may be confirmed from the second derivative 338 Y. Zhou et al. / International Journal of Heat and Fluid Flow 32 (2011) 329–339a = 0–90 and Rec = 5.3 103–5.1 104. The dependence of the forces on a and Rec has been examined. The following conclusions may be drawn based on present measurements. (1) At the small Rec, i.e. 5.3 103, there is no rapid drop in CL nor a jump in CD, suggesting the absence of the stall that is asso- ciated with an airfoil wake of Rec P 1.0 104. (2) CD and CL display a strong dependence on a, as expected. CD increases monotonically from a = 0 to 90, whilst CL grows from 0 to its maximum at a 45 and then declines. The increase in CD is rather rapid up to a = 45 and less so beyond a = 45. Both CLrms and CDrms increase from a = 0 to 90, with a local maximum at a 45. (3) A linear theoretical analysis is developed to predict the dependence of CD and CL on a. The analysis is consistent with the measured CD and CL and explains why CL and CD reach the maximum at a = 45 and 90, respectively. With a increasing from 0 to 90, the airfoil changes from a stream- lined body to a bluff body (like a normal plate). Accordingly, Pb grows from zero to the maximum and AL = c cos a retreats from the maximum to zero. As such, CL reaches its maximum at an intermediate a value between 0 and 90. On the other hand, both Pb and AD = c sin a grow with increasing a. Thus, CD displays its maximum at a = 90. (4) The Rec effect on CD and CL depends on a. As Rec increases from 5.3 103 to 1.05 104, CL displays an appreciable increase, except at a = 90 where the increase is rather mild; meanwhile, CD decreases since transition to turbulence in the shear layer moves towards the separation point. CL and CD vary little for Rec = 1.05 104–5.1 104 because of a neg- ligibly small variation in the flow separation point. 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