Preliminary Airfoil Design for Low Reynolds Numbers

PRELIMINARY AIRFOIL DESIGN FOR LOW REYNOLDS NUMBERS

Witold J. F. Koning , B. Natalia Perez Perez , and Haley V. Cummings Ames Research Center INTRODUCTION Advances in the fields of electronics, batteries, and electric motors have enabled broad adoption of small - scale aircraft for various terrestrial applications. Numerous Micro Aerial Vehicles (MAVs) and small Unmanned Aerial Vehicles (UAVs), both civil and military, are in use today.

Operation at their design flight condition often requires rotors, propellers, or wings to perform at Reynolds numbers significantly below 500,000 [1] .

Airfoil performance at low Reynolds numbers is generally characterized by low aerodynamic performance (i.e. the attainable lift - to - drag ratio) compared to performance of airfoils at higher Reynolds numbers. The criti cal chord - based Reynolds number for conventional airfoil lies around Re c = 100,000 to Re c = 500,000 [2] . Below this critical Reynolds number, laminar separation of the boundary layer is usually the cause for poor performance. Researchers have found that below the critical Reynolds number, flat and cambered plates can outperform smooth airfoils. Circular - arc cambered plates especially have remarkable performance advantages, compared to conventional airfoil shapes [3] .

AIRFOIL GEOMETRY A preliminary airfoil is designed to evaluate performance at low Reynolds numbers. The airfoil is presented in Appendix A.

The airfoil is represented as two - dimensional x,y - coordinates. The point density is increased near smaller features (i.e. the leadin g and trailing edges) to ensure any required splining of the geometry would follow naturally. Care is taken to yield a design which despite the thin profile, would still be manufacturable at most realistic chord lengths.

For analyses, care must be taken th at features of the airfoil might not be suitable for integral boundary layer analyses (viscous - inviscid interaction methods) due to the low thickness of the airfoil and the difficulty in ensuring accurate transition predictions. Higher - order analyses face similar problems, in that practical prediction of transition behavior is an on - going research field at the low Reynolds number range.

REFERENCES [1] Pines, D. J., and Bohorquez, F. “Challenges Facing Future Micro - Air - Vehicle Development.” Journal of Aircraft , Vol. 43, No. 2, 2006, pp. 290 – 305.

[2] Hoerner, S. F. Fluid - Dynamic Drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance . Hoerner Fluid Dynamics, 1965.

[3] Lissaman, P. B. S. “Low - Reynolds - Number Airfoils.” Annual Review of Fluid Mechanics , Vol. 15, No. 1, 1983, pp. 223 – 239.

APPENDIX A The preliminary airfoil coordinates are presented in Table 1 .

Table 1 . Preliminary A irfoil C oordinates x/c y/c x/c y/c x/c y/c x/c y/c 0.996799 0.002026 0.987195 0.004389 0.971188 0.008169 0.999999 0.000057 0.996669 0.002058 0.986937 0.004452 0.970803 0.008257 0.999995 0.000115 0.996537 0.002091 0.986678 0.004514 0.970415 0.008347 0.999988 0.000173 0.996403 0.002125 0.986415 0.004578 0.970024 0.008436 0.999980 0.000232 0.996266 0.002159 0.986150 0.004642 0.969631 0.008526 0.999968 0.000290 0.996126 0.002194 0.985882 0.004707 0.969236 0.008617 0.999954 0.000349 0.995984 0.002229 0.985612 0.004772 0.968837 0.008707 0.999937 0.000408 0.995840 0.002265 0.985339 0.004838 0.968437 0.008799 0.999918 0.000466 0.995692 0.002302 0.985064 0.004904 0.968033 0.008891 0.999896 0.000525 0.995543 0.002339 0.984786 0.004971 0.967627 0.008983 0.999872 0.000583 0.995390 0.002377 0.984506 0.005038 0.967219 0.009076 0.999845 0.000641 0.995235 0.002415 0.984223 0.005106 0.966808 0.009169 0.999816 0.000699 0.995078 0.002455 0.983937 0.005175 0.966394 0.009263 0.999784 0.000755 0.994918 0.002494 0.983649 0.005244 0.965978 0.009357 0.999749 0.000812 0.994755 0.002535 0.983358 0.005313 0.965559 0.009451 0.999712 0.000867 0.994590 0.002576 0.983065 0.005383 0.965138 0.009546 0.999672 0.000922 0.994422 0.002617 0.982769 0.005454 0.964714 0.009642 0.999630 0.000975 0.994252 0.002660 0.982471 0.005525 0.964288 0.009738 0.999585 0.001028 0.994079 0.002702 0.982170 0.005597 0.963859 0.009834 0.999538 0.001079 0.993903 0.002746 0.981867 0.005669 0.963427 0.009931 0.999488 0.001129 0.993725 0.002790 0.981561 0.005742 0.962993 0.010028 0.999435 0.001177 0.993545 0.002835 0.981252 0.005816 0.962557 0.010125 0.999380 0.001224 0.993362 0.002880 0.980941 0.005890 0.962117 0.010223 0.999323 0.001268 0.993176 0.002926 0.980627 0.005964 0.961676 0.010322 0.999262 0.001311 0.992988 0.002972 0.980311 0.006039 0.961231 0.010421 0.999200 0.001351 0.992797 0.003019 0.979992 0.006115 0.960784 0.010520 0.999134 0.001389 0.992604 0.003067 0.979671 0.006191 0.960335 0.010620 0.999067 0.001424 0.992408 0.003115 0.979347 0.006267 0.959883 0.010720 0.998996 0.001457 0.992209 0.003164 0.979020 0.006344 0.959428 0.010820 0.998923 0.001486 0.992008 0.003214 0.978691 0.006422 0.958971 0.010921 0.998848 0.001511 0.991805 0.003264 0.978359 0.006500 0.958511 0.011023 0.998769 0.001533 0.991599 0.003314 0.978025 0.006579 0.958049 0.011125 0.998689 0.001552 0.991390 0.003366 0.977688 0.006658 0.957584 0.011227 0.998606 0.001573 0.991179 0.003418 0.977349 0.006738 0.957117 0.011329 0.998520 0.001595 0.990965 0.003470 0.977007 0.006818 0.956647 0.011432 0.998431 0.001617 0.990748 0.003523 0.976663 0.006899 0.956174 0.011536 0.998340 0.001640 0.990529 0.003577 0.976316 0.006980 0.955699 0.011639 0.998247 0.001663 0.990308 0.003631 0.975966 0.007061 0.955222 0.011743 0.998151 0.001687 0.990084 0.003686 0.975614 0.007144 0.954742 0.011848 0.998052 0.001712 0.989857 0.003741 0.975259 0.007226 0.954259 0.011953 0.997951 0.001737 0.989628 0.003797 0.974902 0.007310 0.953774 0.012058 0.997847 0.001763 0.989396 0.003854 0.974542 0.007393 0.953286 0.012164 0.997741 0.001790 0.989162 0.003911 0.974180 0.007478 0.952795 0.012270 0.997632 0.001817 0.988925 0.003969 0.973815 0.007562 0.952302 0.012376 0.997521 0.001845 0.988685 0.004027 0.973447 0.007647 0.951807 0.012483 0.997407 0.001874 0.988443 0.004086 0.973077 0.007733 0.951309 0.012590 0.997290 0.001903 0.988199 0.004146 0.972705 0.007819 0.950808 0.012697 0.997171 0.001933 0.987952 0.004206 0.972329 0.007906 0.950305 0.012805 0.997050 0.001963 0.987702 0.004266 0.971952 0.007993 0.949799 0.012914 0.996925 0.001994 0.987450 0.004327 0.971571 0.008081 0.949290 0.013022 x/c y/c x/c y/c x/c y/c x/c y/c 0.948780 0.013131 0.919326 0.019081 0.883212 0.025513 0.840437 0.031941 0.948266 0.013240 0.918682 0.019204 0.882437 0.025641 0.839531 0.032064 0.947750 0.013350 0.918036 0.019327 0.881660 0.025769 0.838623 0.032186 0.947231 0.013460 0.917386 0.019450 0.880880 0.025896 0.837713 0.032308 0.946710 0.013570 0.916735 0.019574 0.880098 0.026024 0.836800 0.032430 0.946186 0.013681 0.916080 0.019697 0.879313 0.026152 0.835884 0.032551 0.945660 0.013792 0.915423 0.019821 0.878525 0.026279 0.834966 0.032673 0.945131 0.013903 0.914764 0.019945 0.877735 0.026407 0.834046 0.032794 0.944600 0.014015 0.914102 0.020069 0.876943 0.026535 0.833122 0.032914 0.944066 0.014126 0.913437 0.020194 0.876148 0.026662 0.832197 0.033035 0.943529 0.014239 0.912770 0.020318 0.875350 0.026790 0.831268 0.033155 0.942990 0.014351 0.912101 0.020443 0.874550 0.026918 0.830337 0.033275 0.942449 0.014464 0.911428 0.020568 0.873747 0.027045 0.829404 0.033395 0.941904 0.014577 0.910753 0.020693 0.872941 0.027173 0.828468 0.033515 0.941358 0.014691 0.910076 0.020818 0.872133 0.027300 0.827529 0.033634 0.940808 0.014805 0.909396 0.020943 0.871323 0.027427 0.826588 0.033753 0.940256 0.014919 0.908714 0.021068 0.870509 0.027555 0.825644 0.033872 0.939702 0.015033 0.908028 0.021194 0.869694 0.027682 0.824698 0.033991 0.939145 0.015148 0.907341 0.021319 0.868876 0.027809 0.823749 0.034109 0.938585 0.015263 0.906651 0.021445 0.868055 0.027936 0.822798 0.034227 0.938023 0.015378 0.905958 0.021571 0.867231 0.028063 0.821844 0.034344 0.937459 0.015494 0.905263 0.021697 0.866405 0.028190 0.820887 0.034462 0.936891 0.015610 0.904565 0.021823 0.865577 0.028317 0.819928 0.034579 0.936321 0.015726 0.903864 0.021949 0.864746 0.028444 0.818966 0.034695 0.935749 0.015842 0.903161 0.022076 0.863912 0.028571 0.818002 0.034812 0.935174 0.015959 0.902456 0.022202 0.863076 0.028697 0.817035 0.034928 0.934597 0.016076 0.901748 0.022329 0.862237 0.028824 0.816066 0.035044 0.934016 0.016193 0.901037 0.022455 0.861396 0.028950 0.815094 0.035159 0.933434 0.016311 0.900324 0.022582 0.860552 0.029076 0.814120 0.035274 0.932849 0.016429 0.899608 0.022709 0.859706 0.029202 0.813143 0.035389 0.932261 0.016547 0.898889 0.022836 0.858857 0.029329 0.812163 0.035503 0.931671 0.016665 0.898169 0.022963 0.858005 0.029454 0.811181 0.035618 0.931078 0.016784 0.897445 0.023090 0.857151 0.029580 0.810196 0.035731 0.930482 0.016903 0.896719 0.023217 0.856295 0.029706 0.809209 0.035845 0.929884 0.017022 0.895990 0.023344 0.855435 0.029831 0.808219 0.035958 0.929284 0.017141 0.895259 0.023471 0.854574 0.029957 0.807227 0.036071 0.928681 0.017261 0.894525 0.023599 0.853709 0.030082 0.806232 0.036183 0.928075 0.017380 0.893789 0.023726 0.852842 0.030207 0.805234 0.036295 0.927467 0.017500 0.893050 0.023853 0.851973 0.030332 0.804234 0.036406 0.926856 0.017621 0.892309 0.023981 0.851101 0.030457 0.803231 0.036518 0.926242 0.017741 0.891565 0.024108 0.850226 0.030582 0.802226 0.036629 0.925627 0.017862 0.890818 0.024236 0.849349 0.030706 0.801218 0.036739 0.925008 0.017983 0.890069 0.024363 0.848469 0.030831 0.800208 0.036849 0.924387 0.018104 0.889318 0.024491 0.847587 0.030955 0.799195 0.036959 0.923763 0.018225 0.888563 0.024619 0.846702 0.031079 0.798180 0.037068 0.923137 0.018347 0.887807 0.024746 0.845815 0.031202 0.797162 0.037178 0.922509 0.018469 0.887047 0.024874 0.844925 0.031326 0.796141 0.037287 0.921877 0.018591 0.886285 0.025002 0.844032 0.031450 0.795118 0.037396 0.921243 0.018713 0.885521 0.025130 0.843137 0.031573 0.794092 0.037504 0.920607 0.018835 0.884754 0.025257 0.842240 0.031696 0.793064 0.037613 0.919968 0.018958 0.883984 0.025385 0.841339 0.031819 0.792033 0.037721 x/c y/c x/c y/c x/c y/c x/c y/c 0.791000 0.037829 0.734902 0.042957 0.672143 0.047055 0.602722 0.049695 0.789964 0.037937 0.733735 0.043049 0.670846 0.047122 0.601295 0.049729 0.788925 0.038044 0.732566 0.043140 0.669546 0.047189 0.599864 0.049763 0.787884 0.038151 0.731395 0.043231 0.668244 0.047255 0.598431 0.049796 0.786841 0.038258 0.730220 0.043321 0.666939 0.047320 0.596996 0.049828 0.785795 0.038365 0.729044 0.043411 0.665631 0.047385 0.595558 0.049859 0.784746 0.038472 0.727864 0.043501 0.664321 0.047449 0.594117 0.049890 0.783695 0.038578 0.726682 0.043590 0.663009 0.047513 0.592674 0.049920 0.782641 0.038684 0.725498 0.043679 0.661694 0.047576 0.591229 0.049949 0.781584 0.038790 0.724311 0.043767 0.660376 0.047639 0.589780 0.049978 0.780525 0.038895 0.723121 0.043855 0.659056 0.047701 0.588329 0.050005 0.779464 0.039001 0.721929 0.043942 0.657733 0.047762 0.586876 0.050032 0.778400 0.039106 0.720734 0.044030 0.656408 0.047823 0.585420 0.050058 0.777333 0.039210 0.719537 0.044116 0.655080 0.047883 0.583962 0.050084 0.776264 0.039315 0.718337 0.044202 0.653750 0.047943 0.582501 0.050108 0.775192 0.039419 0.717135 0.044288 0.652417 0.048002 0.581037 0.050132 0.774118 0.039523 0.715930 0.044374 0.651081 0.048060 0.579571 0.050155 0.773041 0.039626 0.714722 0.044458 0.649743 0.048118 0.578102 0.050178 0.771961 0.039730 0.713512 0.044543 0.648402 0.048175 0.576631 0.050199 0.770879 0.039833 0.712300 0.044627 0.647059 0.048232 0.575157 0.050220 0.769795 0.039935 0.711084 0.044710 0.645713 0.048288 0.573680 0.050240 0.768707 0.040038 0.709867 0.044793 0.644365 0.048343 0.572201 0.050259 0.767618 0.040140 0.708646 0.044876 0.643014 0.048397 0.570720 0.050278 0.766525 0.040242 0.707423 0.044958 0.641660 0.048452 0.569236 0.050295 0.765431 0.040343 0.706198 0.045040 0.640304 0.048505 0.567749 0.050312 0.764333 0.040445 0.704970 0.045121 0.638946 0.048558 0.566260 0.050328 0.763233 0.040545 0.703739 0.045202 0.637584 0.048610 0.564768 0.050343 0.762131 0.040646 0.702506 0.045282 0.636221 0.048661 0.563274 0.050358 0.761026 0.040746 0.701271 0.045362 0.634854 0.048712 0.561777 0.050371 0.759918 0.040846 0.700032 0.045441 0.633485 0.048762 0.560277 0.050384 0.758808 0.040946 0.698792 0.045520 0.632114 0.048812 0.558775 0.050396 0.757695 0.041045 0.697548 0.045598 0.630740 0.048861 0.557271 0.050407 0.756580 0.041144 0.696302 0.045676 0.629363 0.048909 0.555763 0.050417 0.755462 0.041243 0.695054 0.045753 0.627984 0.048956 0.554254 0.050427 0.754341 0.041341 0.693803 0.045830 0.626603 0.049003 0.552741 0.050436 0.753218 0.041439 0.692549 0.045906 0.625218 0.049049 0.551227 0.050443 0.752093 0.041536 0.691293 0.045982 0.623832 0.049095 0.549709 0.050450 0.750965 0.041634 0.690034 0.046057 0.622442 0.049140 0.548189 0.050457 0.749834 0.041731 0.688773 0.046132 0.621050 0.049184 0.546667 0.050462 0.748701 0.041827 0.687509 0.046206 0.619656 0.049228 0.545142 0.050466 0.747565 0.041923 0.686243 0.046280 0.618259 0.049270 0.543614 0.050470 0.746427 0.042019 0.684974 0.046353 0.616859 0.049312 0.542084 0.050473 0.745286 0.042115 0.683702 0.046425 0.615457 0.049354 0.540551 0.050475 0.744142 0.042210 0.682428 0.046497 0.614052 0.049395 0.539016 0.050476 0.742996 0.042304 0.681151 0.046569 0.612645 0.049435 0.537478 0.050476 0.741847 0.042399 0.679872 0.046640 0.611235 0.049474 0.535937 0.050475 0.740696 0.042493 0.678590 0.046711 0.609823 0.049513 0.534394 0.050474 0.739543 0.042586 0.677306 0.046781 0.608408 0.049551 0.532849 0.050472 0.738386 0.042680 0.676019 0.046850 0.606990 0.049588 0.531301 0.050468 0.737227 0.042772 0.674729 0.046919 0.605570 0.049624 0.529750 0.050464 0.736066 0.042865 0.673437 0.046987 0.604148 0.049660 0.528197 0.050459 x/c y/c x/c y/c x/c y/c x/c y/c 0.526641 0.050453 0.446880 0.049011 0.373052 0.045768 0.305884 0.041321 0.525082 0.050447 0.445369 0.048963 0.371671 0.045691 0.304634 0.041225 0.523521 0.050439 0.443860 0.048913 0.370292 0.045613 0.303386 0.041129 0.521958 0.050430 0.442353 0.048864 0.368916 0.045535 0.302141 0.041033 0.520392 0.050421 0.440849 0.048813 0.367543 0.045456 0.300898 0.040936 0.518823 0.050411 0.439348 0.048762 0.366172 0.045377 0.299658 0.040839 0.517252 0.050399 0.437849 0.048710 0.364804 0.045297 0.298420 0.040742 0.515678 0.050387 0.436353 0.048658 0.363438 0.045217 0.297185 0.040645 0.514102 0.050374 0.434859 0.048605 0.362075 0.045137 0.295953 0.040547 0.512523 0.050360 0.433368 0.048551 0.360715 0.045056 0.294723 0.040449 0.510942 0.050346 0.431879 0.048497 0.359357 0.044974 0.293495 0.040351 0.509358 0.050330 0.430393 0.048441 0.358001 0.044893 0.292271 0.040253 0.507771 0.050313 0.428910 0.048386 0.356648 0.044810 0.291048 0.040154 0.506182 0.050295 0.427429 0.048330 0.355298 0.044728 0.289829 0.040055 0.504590 0.050277 0.425950 0.048273 0.353950 0.044645 0.288612 0.039956 0.502996 0.050257 0.424474 0.048215 0.352605 0.044562 0.287397 0.039857 0.501399 0.050237 0.423001 0.048157 0.351262 0.044478 0.286185 0.039758 0.499800 0.050216 0.421531 0.048098 0.349922 0.044394 0.284975 0.039658 0.498201 0.050194 0.420062 0.048039 0.348585 0.044309 0.283769 0.039558 0.496605 0.050171 0.418597 0.047979 0.347250 0.044224 0.282564 0.039458 0.495012 0.050147 0.417134 0.047918 0.345918 0.044139 0.281363 0.039358 0.493420 0.050122 0.415674 0.047857 0.344588 0.044053 0.280163 0.039257 0.491832 0.050096 0.414216 0.047795 0.343261 0.043967 0.278967 0.039156 0.490246 0.050070 0.412760 0.047733 0.341936 0.043880 0.277773 0.039055 0.488663 0.050043 0.411308 0.047670 0.340614 0.043793 0.276581 0.038954 0.487082 0.050014 0.409857 0.047606 0.339294 0.043706 0.275392 0.038853 0.485504 0.049985 0.408410 0.047542 0.337977 0.043619 0.274206 0.038751 0.483928 0.049956 0.406965 0.047478 0.336663 0.043531 0.273022 0.038649 0.482355 0.049925 0.405522 0.047413 0.335351 0.043442 0.271841 0.038548 0.480784 0.049894 0.404082 0.047347 0.334042 0.043354 0.270662 0.038445 0.479216 0.049862 0.402645 0.047281 0.332735 0.043265 0.269486 0.038343 0.477651 0.049829 0.401210 0.047214 0.331431 0.043175 0.268312 0.038241 0.476088 0.049795 0.399778 0.047146 0.330129 0.043085 0.267141 0.038138 0.474528 0.049760 0.398348 0.047078 0.328830 0.042995 0.265973 0.038035 0.472970 0.049725 0.396921 0.047010 0.327533 0.042905 0.264807 0.037932 0.471415 0.049689 0.395497 0.046941 0.326239 0.042814 0.263643 0.037829 0.469862 0.049652 0.394074 0.046871 0.324948 0.042723 0.262483 0.037726 0.468312 0.049615 0.392655 0.046801 0.323659 0.042632 0.261324 0.037622 0.466765 0.049576 0.391238 0.046731 0.322373 0.042540 0.260169 0.037519 0.465220 0.049537 0.389824 0.046660 0.321089 0.042448 0.259016 0.037415 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