Computational Modeling of Circular Arc Airfoils at low Reynolds Number M.A.Mohamed, K. Fagbenro, and D.H.Wood Department of Mechanical and Manufacturing Engineering, The University Of Calgary, Alberta, 25 University Drive Northwest, Calgary, AB T2N N4 Email: kafagben@ucalgary.ca ABSTRACT Water-pumping is one of the oldest uses of wind power with the multi-bladed, high-solidity windmill still in widespread use. Windmills rotate slowly and have very high solidity, making their aerodynamic analysis fundamentally different to that for modern wind turbines. The main difference addressed here is high solidity effects on blade lift and drag in the expectation that windmills can analysed by a modification of standard blade element theory which assumes that the blades behave as airfoils. No cascade data is available for circular arc blades so the main aim of the present work is to compute the effects of solidity. Most calculations in this paper were done using the transition-sst model as implemented in ANSYS Fluent. The measured pressure distribution around a circular arc airfoil of 4% camber at Reynolds number of 62, was accurately reproduced. Calculations of the lift and drag at, of another airfoil of % camber also agreed with the experiments. The lift and drag were then computed for a range of solidity typical of windmills. Significant variation was found in the lift and drag.. contradiction over the location of the maximum is resolved when it is recognized that all small wind turbines have a lower stopping than starting wind speed, and that the conventionally-measured cut-in speed is some average of the two, eg Wood [2]. It is clear from fig. that a windmill has high solidity, σ = Nc/(2πr) where N is the number of blades of chord c at radius r. Solidity is well-known to affect the lift and drag of the blades in a similar manner to cascade effects on two-dimensional bodies. Standard blade element theory, in effect, assumes zero solidity by using airfoil data to calculate blade torque and thrust. This work was motivated by the only blade element studies of windmills, by Rijs & Smulders [3], Rijs et al. [4 and Islam and Islam [5], who all used airfoil data.. INTRODUCTION The traditional waterpumping windmill, shown in fig., has been used for well over a century and may well be experiencing a revival. For example, the Asian Development Bank has started a project to support the use of wind pumping and drip irrigation for cash crop production in solar greenhouses throughout Asia. For such an old technology it is surprising that very little is known about the detailed aerodynamic performance, and the available knowledge is not reassuring. Pinilla et al. [] showed that a typical windmill has its maximum overall efficiency of just over 2% at about 7% of the cut-in wind speed. The efficiency then drops off rapidly. The apparent Figure : A Kijito windmill (www.kijitowindpower.com)
Windmill blades are usually rolled from sheet steel to approximate thin, circular arc airfoils with a central circular spar near the midpoint of the pressure surface. There is considerable data available on the lift and drag of circular arc airfoils, eg Wallis [6], Pandey et al. [7], and Okamoto et al. [8], and even data on airfoils with a circular spar, Bruining [9]. However, we could find no lift and drag measurements of circular arc cascades from which to assess the effects of solidity. The measurements just referenced were all made on airfoils with camber of % or less. The aims of the work described here are to, first, accurately reproduce the lift and drag measurements with and without the spar, as well as the surface pressure measurements at Reynolds number, Re, of 62, from an ingenious experiment by Tezuka et al. [] who constructed their airfoil from rows of hypodermic tubes. These pressure distributions are particularly valuable because they delineate the separation regions which are not obvious from just lift and drag data. Then cascade effects will be simulated and lift and drag data computed as a function of Re, solidity and angle of attack, α. Figure 2: Close-up view of circular-arc airfoil mesh A grid-independence test was performed for the case of α = 6 and the results are shown in Figure 3. The test was carried out to see if there were significant variations to the C p on the pressure side of the airfoil. As depicted, the results are not mesh dependent. The finest mesh was used for all the calculations reported here. 2. COMPUTATIONAL DETAILS ANSYS Fluent was used for the calculations. Fluent uses finite volume discretization to solve the Reynolds Averaged Navier-Stokes (RANS) equations. The spatial discretization for gradients was solved using the least squares cell-based method. The governing equations were solved using the 2 nd order upwind discretizing scheme. The SIMPLE algorithm was used to couple velocity and pressure. Convergence was assumed when the residuals of the independent variables reduced to -5. The transition- SST turbulence model was used because it has the potential of capturing the important aspects of transition, including large laminar separation bubbles. It has been used previously for airfoil calculations (at much higher Re) with wind turbine applications, Sørensen []. The reader is referred to ANSYS Fluent theory manual for an explicit description of the turbulence model. The mesh for the calculations was a structured one of quadrilateral proportions. A C-grid mesh is normally used for airfoil calculations but given the geometry of the circular arc airfoil, an O-grid was chosen. Figure 2 shows the grid used for the airfoil calculations. Figure 3: Mesh-Independence Test The tuning of the empirical relations of the turbulence model was required to match the measured C p. Model constants c θt and σ θt were set to.2 and 3. respectively. For the case of α =, a code was written in the C language to change the F length in the model. The F length is used to control the length of the transition area. F length was set to. for α =. For the cascade calculations, periodic boundary conditions were imposed at a spacing ratio, s/c, where s is the spacing, equal to the required σ. No alterations were made to the default model constants. 3. RESULTS The first calculations were of the surface pressure measurements of Tezuka et al. []. The airfoil had 4% camber but the lift and drag were not measured.
α ( ) C l C d C l / C d.439.265 6.6 2 47.94 28.2 3.724.35 53.6 4.87.252 32. 5.88.364 24.2 6.34.529 9.5 8.26.63 2.9 Table : Lift and drag for circular arc airfoil with 4% camber Table shows the calculated lift and drag coefficients and the lift:drag ratio. The ratio is the key performance parameter for wind turbine airfoils and it is clear that the calculated range covers the optimum performance point. Figure 5 shows the computed and measured pressures for incidence angle, α, between 2 and 6 and the associated velocity contours are in fig 5. Figure 5a: Computed velocity contours for a 4% cambered circular arc airfoil, α = 2 Figure 5b: Computed velocity contours for a 4% cambered circular arc airfoil, α = 4 -.5 - - C p..2.3.4.6.7.8.9 Figure 4a: Computed and measured [] pressures for a 4% cambered circular arc airfoil, α = 2 Figure 5c: Computed velocity contours for a 4% cambered circular arc airfoil, α = 6 -.5 - C P -..2.3.4.6.7.8.9 Figure 4b: Computed and measured [] pressures for a 4% cambered circular arc airfoil, α = 4 C p -2 -.5 - -.5..2.3.4.6.7.8.9 Figure 4c Computed and measured [] pressures for a 4% cambered circular arc airfoil, α = 6 Figure 4 shows the computed and measured pressure distributions for the same experiments. It is clear that very good agreement was obtained for α at least as high as 4 after which significant separation occurs over the upper surface and the lift:drag ratio decreases. Figure 5d: Computed velocity contours for a 4% cambered circular arc airfoil, α = 8 The dark blue regions of negative velocity show there is localized separation near the leading edge for all angles and a recognizable laminar separation bubble has formed at 6. The large recirculation region at the rear of the airfoil at is absent at higher angles. By inspection of Figure 4c, it appears that the extent of the separation region is over-predicted for 8, as was found by Yuan et al. [2] in their calculations of the flow over the SD73 airfoil. The second calculations were of the measurements of Bruining [9]. The % cambered circular arc airfoil was tested at chord Re of 6,,,, and 2,. For the initial calculations described in this paper, we used the data at Re =, and (relative to the upstream flow) and performed simulations for an airfoil and a cascade. This angle is the only one for which the blade leading edge of the cascade representation does not change position with σ. For any other angle, a change in σ requires realigning the blades to match the position of the leading edge and, therefore, considerable re-
meshing.. In cascade terms, wind turbine rotors have no stagger. The lift and drag coefficients were calculated and compared with the experimental data in figure 6. The velocity contours for solidity.2 and.6 are shown in Figure 8a and b repsectively. Increasing solidity reduces the recirculation on the upper surface but increases that on the pressure surface. C l C d.8.6 Bruining (979).5 2 2.5 3 Calculated 3.5 result x 5 Figure 8a: Computed velocity contours, α = for the experiment of Bruining [9] on a % cambered circular arc airfoil.4.5 2 2.5 3 3.5 4 Reynolds number x 5 Figure 6: Computed and measured C l and C d α = for the experiment of Bruining [9] on a % cambered circular arc airfoil The calculated C l and C d did not closely match the experimental values. However, there is clearly a discrepancy in the pattern of the experimental lift with Re shown in Figure 6. The C P was computed and is shown in Figure 7. - Figure 8b: Computed velocity contours, α = σ =.2 for a % cambered circular arc airfoil - Figure 8c: Computed velocity contours, α = σ =.6 for a % cambered circular arc airfoil C P Figures 8 a and b show some evidence of the finite downwash velocities induced in the cascade. This is absent for the airfoil...2.3.4.6.7.8.9 Figure 7: Computed pressures, α = for the experiment of Bruining [9] on a % cambered circular arc airfoil Unfortunately, no experimental data for C P were available for validation.. Figure 8a displays the velocity contours. The higher camber has caused significant separation on the pressure surface. The predicted upper surface boundary layer is laminar up till separation. The low computed C d suggests that this is incorrect which in turn implies that the extent of the rear separation region is over-predicted. The effects of solidity on the lift and drag are shown in Table 2. A decrease in lift is to be expected but the magnitude of it is surprising. Solidity C l C d C l / C d 29.62 8.64.2.482.522 9.23.6.384.7 5.4 Table : Lift and drag for circular arc airfoil The flow over a circular arc airfoil with a spar for Re = 6, at α = was also computed but has not been fully analysed. Using the Spalart-Allmaras turbulence model and an unstructured mesh gave 7 and.487 for C l and C d respectively for the airfoil and spar. The measured values were 66 and.96 respectively. Figure 9: Computed velocity contours, α = for the experiment of Bruining [9] on a % cambered circular arc airfoil with a spar at mid-chord offset by 4 mm CONCLUSIONS AND FURTHER WORK The transition-sst turbulence model appears capable of predicting of the complex flow over circular arc
airfoils where separation occurred somewhere on the airfoil for all cases examined here. There is evidence that the computation of the % cambered airfoil gives an unrealistic extensive region of laminar flow on the upper surface. As a consequence, the separation region is over-predicted and it is clear that additional experimental data will be required to validate the computations. The initial calculations show that the effects of finite solidity on the lift and drag are substantial. Thus a complete map of lift and drag as a function of solidity and angle of attack will be required for detailed windmill modeling. This work is currently underway. The preliminary results for the airfoil and spar are encouraging but appear to suffer the same underprediction of the drag. +9 degrees, Delft University of Technology, Rept LR-28. Can be downloaded from: http://repository.tudelft.nl/view/ir/uuid:6b92442a- f7-4b7c-8d53-c4f72ff3e/. Sørensen NN (29) CFD modelling of laminarturbulent transition for airfoils and rotors using the γ - R eθ model, Wind Energy, 2; 75 733.. Tezuka A, Sunada Y, Rinoie K (27) Surface pressure measurements on 4% circular arc airfoil at low Reynolds number, J. Aircraft, 45; 264-267. 2. Yuan, W., Khalid, M., Windte, J., Scholz, U., Radespiel, R. (27) Computational and experimental investigation of low-reynolds-number flow past an aerofoil, The Aero. J., 3: 7-29. ACKNOWLEDGMENTS We thank NSERC for support under the Industrial Research Chairs program. Professor K Rinoie provided tabulations of the data from ref. [] REFERENCES. Pinilla, A.E., Burton, J.D., Dunn, P.D. (984), Wind energy to water pumped: conversion efficiency limits using single-acting lift pumps, Proc. 984 BWEA Conf. 2. Wood, D.H. (2) Small Wind Turbines: Analysis, Design, and Application, Srpinger, London 3. Rijs RPP, Smulders PT (99) Blade element theory for performance analysis of slow running rotors, Wind Engineering, 4; 62 79. 4. Rijs RPP, Jacobs P, Smulders PT (992) Parameter study of the performance of slow running rotors, J of Wind Engg & Industrial Aerodyn., 39; 95 3. 5. Islam M Q, Islam AKMS (994) The aerodynamic performance of a horizontal-axis wind turbine calculated by strip theory and cascade theory, JSME International Journal Series B, 37; 87-877. 6. Wallis RA (946) Wind tunnel tests on a series of circular arc plate aerofoils, Aerodynamics Note 74, Divn Aeronautics, CSIRO, Australia. 7. Pandey MM, Pandey KP, Ojha TP (988) Aerodynamic characteristics of cambered steel plates in relation to their use in wind energy conversion systems, Wind Engineering, 2; 9 3. 8. Okamoto M, Azuma A (25) Experimental Study on Aerodynamic Characteristics of Unsteady Wings at Low Reynolds Number, AIAA J, 43; 2526 2536 9. Bruining A (979) Aerodynamic Characteristics of a Curved Plate Airfoil Section at Reynolds Numbers 6, and, and angles of attack from - to