1367:
2724:
2334:
529:
at high
Reynolds number, where it makes sense to scale the drag to the momentum flux into the frontal area of the object. But, there are other flow regimes. In particular at very low Reynolds number, it is more natural to write the drag force as being proportional to a drag coefficient multiplied by the speed of the object (rather than the square of the speed of the object). An example of such a regime is the study of the mobility of aerosol particulates, such as smoke particles. This leads to a different formal definition of the "drag coefficient," of course.
191:
3070:
2719:{\displaystyle {\begin{aligned}c_{\mathrm {d} }&={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\\&=c_{\mathrm {p} }+c_{\mathrm {f} }\\&=\underbrace {{\dfrac {2}{\rho v^{2}A}}\displaystyle \int _{S}\mathrm {d} S(p-p_{o})\left({\hat {\mathbf {n} }}\cdot {\hat {\mathbf {i} }}\right)} _{c_{\mathrm {p} }}+\underbrace {{\dfrac {2}{\rho v^{2}A}}\displaystyle \int _{S}\mathrm {d} S\left({\hat {\mathbf {t} }}\cdot {\hat {\mathbf {i} }}\right)T_{\rm {w}}} _{c_{\mathrm {f} }}\end{aligned}}}
752:
3108:, etc. or an airfoil with small angle of attack, whereas a blunt body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, a streamlined body will have lower resistance than a blunt body. Cylinders and spheres are taken as blunt bodies because the drag is dominated by the pressure component in the wake region at high
31:
1451:
is constant, but certainly is a function of
Reynolds number. At a low Reynolds number, the flow around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. At very low Reynolds numbers, without flow separation, the
528:
As a caution, note that although the above is the conventional definition for the drag coefficient, there are other definitions that one may encounter in the literature. The reason for this is that the conventional definition makes the most sense when one is in the Newton regime, such as what happens
2328:
The force between a fluid and a body, when there is relative motion, can only be transmitted by normal pressure and tangential friction stresses. So, for the whole body, the drag part of the force, which is in-line with the approaching fluid motion, is composed of frictional drag (viscous drag) and
1568:
over the whole front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the left of it shows equal pressure across the surface. In a real flat plate, the fluid must turn around the sides, and full stagnation pressure is found only
426:
The reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the projected frontal area of the vehicle. This may not necessarily be the cross-sectional area of the vehicle, depending on where the cross-section is taken.
3081:
produced is very small and drag is dominated by the friction component. Therefore, such a body (here an airfoil) is described as streamlined, whereas for bodies with fluid flow at high angles of attack, boundary layer separation takes place. This mainly occurs due to adverse
3115:
To reduce this drag, either the flow separation could be reduced or the surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices like cars, bicycle, etc. to avoid vibration and noise production.
1384:, as obtained from laboratory experiments. The dark line is for a sphere with a smooth surface, while the lighter line is for the case of a rough surface. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:
524:
Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients. Coefficients for unstreamlined objects can be 1 or more, for streamlined objects much less.
194:
Table of drag coefficients in increasing order, of 2D shapes between walls (right column) and 3D shapes (left column), depicted with the same projected frontal area, at
Reynolds numbers between 10 and 10 with flow from the left
623:
503:, the reference area is the nominal wing area. Since this tends to be large compared to the frontal area, the resulting drag coefficients tend to be low, much lower than for a car with the same drag, frontal area, and speed.
3062:. Thus, the shape of the body and the angle of attack determine the type of drag. For example, an airfoil is considered as a body with a small angle of attack by the fluid flowing across it. This means that it has attached
301:
1314:
within a practical range of interest is usually small, while for cars at highway speed and aircraft at cruising speed, the incoming flow direction is also more-or-less the same. Therefore, the drag coefficient
951:
1359:
results in a broad wake. The boundary layer will transition from laminar to turbulent if
Reynolds number of the flow around the body is sufficiently great. Larger velocities, larger objects, and lower
873:
2339:
803:
1892:
1846:
3044:
2911:
2874:
1984:
1938:
2985:
2946:
2837:
2802:
2159:
2122:
1753:
1696:
1654:
1625:
1596:
1562:
1479:
1449:
1342:
1212:
1152:
1098:
1029:
335:
226:
77:
1786:
1724:
1312:
1287:
1262:
1237:
1180:
1123:
1069:
461:
133:
3693:
494:
552:
1526:
704:
648:
365:
104:
993:
3007:
2771:
2749:
1499:
730:
678:
417:
391:
233:
3093:
Due to this, wake formation takes place, which consequently leads to eddy formation and pressure loss due to pressure drag. In such situations, the airfoil is
3655:
2128:
directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1
1598:
of a real flat plate would be less than 1; except that there will be suction on the backside: a negative pressure (relative to ambient). The overall
1412:•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic
3246:
3176:
1366:
3492:
884:
3639:
3600:
3221:
3537:
Basha, W. A. and Ghaly, W. S., "Drag
Prediction in Transitional Flow over Airfoils," Journal of Aircraft, Vol. 44, 2007, p. 824–32.
3517:
2285:
1031:
is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid
3688:
3572:
810:
3166:
3282:
3130:
2313:
1564:
equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up
1569:
at the center, dropping off toward the edges as in the lower figure and graph. Only considering the front side, the
3145:
182:. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.
547:
is referred to the transversal area (the area normal to the drag force, so the coefficient is locally defined as:
521:
of the airship volume (volume to the two-thirds power). Submerged streamlined bodies use the wetted surface area.
3476:
759:
1698:
is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with
3375:
3337:
538:
190:
1859:
1813:
3683:
3125:
3018:
2885:
2848:
3105:
1951:
1905:
1627:
of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore
136:
3193:
963:
on any object is proportional to the density of the fluid and proportional to the square of the relative
3135:
2064:
3388:
3350:
3069:
3097:
and has higher pressure drag than friction drag. In this case, the body is described as a blunt body.
2961:
2922:
2813:
2778:
2135:
2098:
1729:
1672:
1630:
1601:
1572:
1538:
1455:
1425:
1318:
1188:
1128:
1074:
1005:
311:
202:
53:
1351:
around the body must remain attached to the surface of the body for as long as possible, causing the
1766:
1704:
1565:
1292:
1267:
1242:
1217:
1160:
1103:
1049:
1036:
510:
2987:
is the pressure far away from the body (note that this constant does not affect the final result),
3392:
3354:
1422:
For other objects, such as small particles, one can no longer consider that the drag coefficient
430:
109:
3651:
Hoerner, Dr. Sighard F., Fluid-Dynamic Drag, Hoerner Fluid
Dynamics, Bricktown New Jersey, 1965.
3547:
3488:
2124:, while automobiles (and many other objects) use projected frontal area; thus, coefficients are
1532:. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.
466:
3319:
Fluid-Dynamic Drag : Practical
Information on Aerodynamic Drag and Hydrodynamic Resistance
2329:
pressure drag (form drag). The total drag and component drag forces can be related as follows:
1289:, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number
3635:
3396:
3358:
3242:
3217:
3172:
3083:
2299:
179:
996:
681:
514:
1504:
689:
633:
350:
143:
or resistance of an object in a fluid environment, such as air or water. It is used in the
82:
3661:
3317:
3109:
1699:
1394:
1378:
1044:
543:
In the non dimensional form of the Cauchy momentum equation, the skin drag coefficient or
158:
The drag coefficient of any object comprises the effects of the two basic contributors to
35:
3593:
970:
755:
Flow around a plate, showing stagnation. The force in the upper configuration is equal to
3513:
3094:
3078:
3063:
2992:
2756:
2734:
2189:
1760:
1656:
for some shapes can change with the
Reynolds number and the roughness of the surfaces.
1529:
1484:
1403:
1352:
1348:
957:
715:
663:
402:
376:
338:
159:
140:
43:
3050:
Therefore, when the drag is dominated by a frictional component, the body is called a
3677:
3261:
1407:
746:
655:
342:
163:
152:
144:
2876:
is the unit vector in the direction of the shear stress acting on the body surface d
3627:
2949:
2074:
1756:
1400:
1040:
707:
651:
368:
148:
2095:
As noted above, aircraft use their wing area as the reference area when computing
618:{\displaystyle c_{\mathrm {d} }={\dfrac {\tau }{q}}={\dfrac {2\tau }{\rho u^{2}}}}
3568:
3140:
2229:
1391:
1387:
1155:
155:
drag. The drag coefficient is always associated with a particular surface area.
3371:
3333:
3278:
2272:
2259:
2246:
2242:
2216:
2203:
2129:
1413:
964:
733:
394:
3668:
The optimization of body details-A method for reducing the aerodynamics drag
1360:
1032:
654:, which is by definition the stress component in the direction of the local
518:
175:
167:
751:
296:{\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}}
17:
3656:
http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf
2840:
2805:
38:
approximately 10 Shapes are depicted with the same projected frontal area
1043:
of the object are incorporated into a dimensionless quantity called the
513:
use the volumetric drag coefficient, in which the reference area is the
3087:
2913:
is the unit vector in the direction perpendicular to the body surface d
506:
500:
171:
3054:; whereas in the case of dominant pressure drag, the body is called a
147:
in which a lower drag coefficient indicates the object will have less
3648:. Dover Publications Inc., New York, Standard Book Number 486-60586-8
341:, which is by definition the force component in the direction of the
3073:
Trade-off relationship between zero-lift drag and lift induced drag
30:
3068:
2022:
2011:
1418:•5: post-critical separated flow, with a turbulent boundary layer.
1365:
960:
189:
29:
3195:
Dynamic response of pipe rack steel structures to explosion loads
946:{\displaystyle F_{\rm {d}}={\tfrac {1}{2}}\rho u^{2}c_{\rm {d}}A}
3378:
for a possible force components transverse to the flow direction
3340:
for a possible force components transverse to the flow direction
3101:
420:
1347:
For a streamlined body to achieve a low drag coefficient, the
1125:. In a compressible flow, the speed of sound is relevant, and
999:
of the fluid, which is equal to the kinetic energy density.
1763:. Although the drag coefficient decreases with increasing
3662:
http://www.princeton.edu/~asmits/Bicycle_web/blunt.html
904:
828:
3666:
Hucho, W.H., Janssen, L.J., Emmelmann, H.J. 6(1975):
3548:"Ask Us – Drag Coefficient & Lifting Line Theory"
3021:
2995:
2964:
2925:
2888:
2851:
2816:
2781:
2759:
2737:
2618:
2592:
2480:
2454:
2362:
2337:
2138:
2101:
1954:
1908:
1862:
1816:
1769:
1732:
1707:
1675:
1633:
1604:
1575:
1541:
1507:
1487:
1458:
1428:
1321:
1295:
1270:
1245:
1220:
1191:
1163:
1131:
1106:
1077:
1052:
1008:
973:
887:
868:{\displaystyle F_{d}={\tfrac {1}{2}}\rho u^{2}c_{d}A}
813:
762:
718:
692:
666:
636:
587:
572:
555:
539:
Cauchy momentum equation § Nondimensionalisation
469:
433:
405:
379:
353:
314:
253:
236:
205:
112:
85:
56:
3644:
Abbott, Ira H., and Von
Doenhoff, Albert E. (1959):
3395:. Air is 1.293 kg/m at 0 °C (32 °F) and 1
3357:. Air is 1.293 kg/m at 0 °C (32 °F) and 1
3216:. New York: John Wiley & Sons, Inc. p. 24.
27:
Dimensionless parameter to quantify fluid resistance
3046:
is the unit vector in direction of free stream flow
1264:and the direction of the flow. For low Mach number
3038:
3001:
2979:
2940:
2905:
2868:
2831:
2796:
2765:
2743:
2718:
2153:
2116:
1978:
1932:
1886:
1840:
1780:
1747:
1718:
1690:
1648:
1619:
1590:
1556:
1520:
1493:
1473:
1443:
1336:
1306:
1281:
1256:
1231:
1206:
1174:
1146:
1117:
1092:
1063:
1023:
987:
945:
867:
797:
724:
698:
672:
642:
617:
488:
455:
411:
385:
359:
329:
295:
220:
127:
98:
71:
967:between the object and the fluid. The factor of
3214:Aerodynamics, Aeronautics, and Flight Mechanics
3420:
3418:
1185:For certain body shapes, the drag coefficient
3660:Drag of Blunt Bodies and Streamlined Bodies:
8:
1406:upstream of the separation, and producing a
2084:Long flat plate perpendicular to flow (2D)
1856:Turbulent flat plate parallel to the flow (
798:{\displaystyle F={\frac {1}{2}}\rho u^{2}A}
3168:Explosion Hazards and Evaluation, Volume 5
1790:
3391:, the air density can be found using the
3353:, the air density can be found using the
3066:, which produce much less pressure drag.
3025:
3023:
3022:
3020:
2994:
2970:
2969:
2963:
2931:
2930:
2924:
2892:
2890:
2889:
2887:
2855:
2853:
2852:
2850:
2822:
2821:
2815:
2787:
2786:
2780:
2758:
2736:
2703:
2702:
2697:
2683:
2682:
2662:
2660:
2659:
2645:
2643:
2642:
2629:
2623:
2605:
2591:
2588:
2575:
2574:
2569:
2546:
2544:
2543:
2529:
2527:
2526:
2512:
2491:
2485:
2467:
2453:
2450:
2432:
2431:
2417:
2416:
2389:
2373:
2372:
2361:
2347:
2346:
2338:
2336:
2144:
2143:
2137:
2107:
2106:
2100:
1970:
1955:
1953:
1924:
1909:
1907:
1878:
1863:
1861:
1832:
1817:
1815:
1810:Laminar flat plate parallel to the flow (
1770:
1768:
1738:
1737:
1731:
1708:
1706:
1681:
1680:
1674:
1639:
1638:
1632:
1610:
1609:
1603:
1581:
1580:
1574:
1547:
1546:
1540:
1512:
1506:
1486:
1464:
1463:
1457:
1434:
1433:
1427:
1327:
1326:
1320:
1296:
1294:
1271:
1269:
1246:
1244:
1221:
1219:
1197:
1196:
1190:
1164:
1162:
1137:
1136:
1130:
1107:
1105:
1083:
1082:
1076:
1053:
1051:
1014:
1013:
1007:
977:
972:
933:
932:
922:
903:
893:
892:
886:
856:
846:
827:
818:
812:
786:
769:
761:
717:
691:
665:
635:
605:
586:
571:
561:
560:
554:
480:
468:
447:
432:
404:
378:
352:
320:
319:
313:
280:
264:
263:
252:
242:
241:
235:
211:
210:
204:
118:
117:
111:
90:
84:
62:
61:
55:
3694:Dimensionless numbers of fluid mechanics
2163:
750:
3569:"Boeing 787 Dreamliner : Analysis"
3157:
2917:, pointing from the fluid to the solid,
1887:{\displaystyle \mathrm {Re} >10^{6}}
1841:{\displaystyle \mathrm {Re} <10^{6}}
1801:
1726:). A smooth sphere, for example, has a
1363:contribute to larger Reynolds numbers.
3100:A streamlined body looks like a fish (
2056:Flat plate perpendicular to flow (3D)
1399:•3: separated unsteady flow, having a
3634:. Pitman Publishing Limited, London,
3039:{\displaystyle {\hat {\mathbf {i} }}}
2906:{\displaystyle {\hat {\mathbf {n} }}}
2869:{\displaystyle {\hat {\mathbf {t} }}}
463:(note this is not the surface area =
7:
3437:Clift R., Grace J. R., Weber M. E.:
3201:. CHALMERS UNIVERSITY OF TECHNOLOGY.
2002:Trapeze with triangular basis (45°)
1979:{\displaystyle \mathrm {Re} =10^{6}}
1933:{\displaystyle \mathrm {Re} =10^{6}}
1344:can often be treated as a constant.
1214:only depends on the Reynolds number
956:is essentially a statement that the
397:of the object relative to the fluid;
170:. The drag coefficient of a lifting
3571:. Lissys.demon.co.uk. 2006-06-21.
2971:
2932:
2823:
2788:
2704:
2684:
2630:
2576:
2492:
2433:
2418:
2374:
2348:
2145:
2108:
1959:
1956:
1913:
1910:
1867:
1864:
1821:
1818:
1774:
1771:
1739:
1712:
1709:
1682:
1640:
1611:
1582:
1548:
1465:
1435:
1328:
1300:
1297:
1275:
1272:
1250:
1247:
1225:
1222:
1198:
1168:
1165:
1138:
1111:
1108:
1084:
1057:
1054:
1015:
934:
894:
562:
321:
265:
243:
212:
119:
63:
25:
2751:is the planform area of the body,
1755:that varies from high values for
34:Drag coefficients in fluids with
3086:at the top and rear parts of an
3026:
2980:{\displaystyle p_{\mathrm {o} }}
2941:{\displaystyle T_{\mathrm {w} }}
2893:
2856:
2832:{\displaystyle c_{\mathrm {f} }}
2797:{\displaystyle c_{\mathrm {p} }}
2663:
2646:
2547:
2530:
2319:Blunt and streamlined body flows
2154:{\displaystyle c_{\mathrm {d} }}
2117:{\displaystyle c_{\mathrm {d} }}
1748:{\displaystyle c_{\mathrm {d} }}
1691:{\displaystyle c_{\mathrm {d} }}
1649:{\displaystyle c_{\mathrm {d} }}
1620:{\displaystyle c_{\mathrm {d} }}
1591:{\displaystyle c_{\mathrm {d} }}
1557:{\displaystyle c_{\mathrm {d} }}
1528:; for a sphere this is known as
1474:{\displaystyle F_{\mathrm {d} }}
1444:{\displaystyle c_{\mathrm {d} }}
1337:{\displaystyle c_{\mathrm {d} }}
1207:{\displaystyle c_{\mathrm {d} }}
1147:{\displaystyle c_{\mathrm {d} }}
1093:{\displaystyle c_{\mathrm {d} }}
1024:{\displaystyle c_{\mathrm {d} }}
330:{\displaystyle F_{\mathrm {d} }}
221:{\displaystyle c_{\mathrm {d} }}
72:{\displaystyle c_{\mathrm {d} }}
3606:from the original on 2015-09-23
3575:from the original on 2010-08-13
3520:from the original on 2013-02-16
3495:from the original on 2010-12-04
3285:from the original on 2011-03-02
3281:. Wright.nasa.gov. 2010-03-25.
3264:, and Von Doenhoff, Albert E.:
2773:is the wet surface of the body,
2040:Adult human (upright position)
3550:. Aerospaceweb.org. 2004-07-11
3237:Clancy, L. J. (1975). "5.18".
3030:
2897:
2860:
2667:
2650:
2551:
2534:
2518:
2499:
1377:for a sphere as a function of
806:and in the lower configuration
1:
3439:Bubbles, drops, and particles
3212:McCormick, Barnes W. (1979).
3192:AARØNÆS, ANTON STADE (2014).
1781:{\displaystyle \mathrm {Re} }
1719:{\displaystyle \mathrm {Re} }
1307:{\displaystyle \mathrm {Re} }
1282:{\displaystyle \mathrm {Ma} }
1257:{\displaystyle \mathrm {Ma} }
1232:{\displaystyle \mathrm {Re} }
1175:{\displaystyle \mathrm {Ma} }
1118:{\displaystyle \mathrm {Re} }
1064:{\displaystyle \mathrm {Re} }
178:also includes the effects of
139:that is used to quantify the
3316:Hoerner, Sighard F. (1965).
2952:acting on the body surface d
1788:, the drag force increases.
3441:. Academic Press NY (1978).
3322:(2 ed.). p. 3–17.
3131:Automobile drag coefficient
2314:Automobile drag coefficient
456:{\displaystyle A=\pi r^{2}}
128:{\displaystyle c_{\rm {w}}}
3710:
3491:. Engineeringtoolbox.com.
3463:Haider A., Levenspiel O.:
3146:Zero-lift drag coefficient
2311:
744:
536:
489:{\displaystyle 4\pi r^{2}}
427:For example, for a sphere
1994:Triangular trapeze (45°)
1660:Drag coefficient examples
545:skin friction coefficient
3376:vortex induced vibration
3338:vortex induced vibration
3009:is pressure at surface d
2014:, at subsonic velocity)
533:Cauchy momentum equation
3646:Theory of Wing Sections
3514:"Shape Effects on Drag"
3412:. Sections 4.15 and 5.4
3266:Theory of Wing Sections
3126:Automotive aerodynamics
3279:"Modern Drag Equation"
3268:. Sections 1.2 and 1.3
3074:
3040:
3003:
2981:
2942:
2907:
2870:
2833:
2798:
2767:
2745:
2720:
2155:
2118:
1980:
1934:
1888:
1842:
1782:
1749:
1720:
1692:
1650:
1621:
1592:
1558:
1522:
1495:
1475:
1445:
1419:
1338:
1308:
1283:
1258:
1233:
1208:
1176:
1154:is also a function of
1148:
1119:
1100:is thus a function of
1094:
1065:
1025:
989:
947:
875:
869:
799:
726:
700:
674:
644:
619:
490:
457:
413:
387:
361:
331:
297:
222:
196:
137:dimensionless quantity
129:
100:
73:
50:(commonly denoted as:
39:
3689:Aerospace engineering
3136:Ballistic coefficient
3072:
3041:
3004:
2982:
2943:
2908:
2871:
2834:
2799:
2768:
2746:
2721:
2156:
2119:
2065:Empire State Building
1981:
1935:
1889:
1843:
1783:
1750:
1721:
1693:
1651:
1622:
1593:
1559:
1523:
1521:{\displaystyle v^{2}}
1496:
1476:
1446:
1369:
1355:to be narrow. A high
1339:
1309:
1284:
1259:
1234:
1209:
1177:
1149:
1120:
1095:
1066:
1039:and a characteristic
1026:
990:
948:
870:
800:
754:
727:
701:
699:{\displaystyle \rho }
675:
645:
643:{\displaystyle \tau }
620:
491:
458:
414:
388:
362:
360:{\displaystyle \rho }
332:
298:
223:
199:The drag coefficient
193:
130:
101:
99:{\displaystyle c_{x}}
74:
33:
3171:. Elsevier Science.
3165:Baker, W.E. (1983).
3019:
2993:
2962:
2923:
2886:
2849:
2814:
2779:
2757:
2735:
2335:
2136:
2099:
1952:
1906:
1860:
1814:
1767:
1730:
1705:
1673:
1631:
1602:
1573:
1539:
1505:
1485:
1456:
1426:
1319:
1293:
1268:
1243:
1218:
1189:
1161:
1129:
1104:
1075:
1050:
1006:
971:
885:
811:
760:
716:
690:
664:
634:
553:
511:bodies of revolution
467:
431:
403:
377:
351:
312:
234:
203:
110:
83:
54:
1566:stagnation pressure
1481:is proportional to
1386:•2: attached flow (
1037:kinematic viscosity
988:{\displaystyle 1/2}
3489:"Drag Coefficient"
3467:. 58, 1989, 63-70.
3454:. 67, 1991, 87-91.
3393:barometric formula
3389:Earth's atmosphere
3387:Note that for the
3355:barometric formula
3351:Earth's atmosphere
3349:Note that for the
3084:pressure gradients
3075:
3036:
2999:
2977:
2938:
2903:
2866:
2829:
2794:
2763:
2741:
2716:
2714:
2711:
2695:
2690:
2616:
2583:
2567:
2562:
2478:
2400:
2151:
2114:
1976:
1930:
1884:
1838:
1778:
1745:
1716:
1688:
1646:
1617:
1588:
1554:
1518:
1491:
1471:
1441:
1420:
1334:
1304:
1279:
1254:
1229:
1204:
1172:
1144:
1115:
1090:
1061:
1021:
985:
943:
913:
878:The drag equation
876:
865:
837:
795:
722:
696:
670:
640:
615:
613:
581:
486:
453:
409:
383:
357:
327:
293:
291:
218:
197:
125:
96:
69:
40:
3465:Powder Technology
3452:Powder Technology
3248:978-0-470-15837-1
3178:978-0-444-59988-9
3033:
3002:{\displaystyle p}
2948:magnitude of the
2900:
2863:
2843:drag coefficient,
2808:drag coefficient,
2766:{\displaystyle S}
2744:{\displaystyle A}
2670:
2653:
2615:
2589:
2587:
2554:
2537:
2477:
2451:
2449:
2399:
2305:
2304:
2300:F-104 Starfighter
2088:
2087:
2032:Wires and cables
1494:{\displaystyle v}
1370:Drag coefficient
912:
836:
777:
725:{\displaystyle u}
673:{\displaystyle q}
612:
580:
419:is the reference
412:{\displaystyle A}
386:{\displaystyle u}
290:
180:lift-induced drag
16:(Redirected from
3701:
3615:
3614:
3612:
3611:
3605:
3598:
3590:
3584:
3583:
3581:
3580:
3565:
3559:
3558:
3556:
3555:
3544:
3538:
3535:
3529:
3528:
3526:
3525:
3510:
3504:
3503:
3501:
3500:
3485:
3479:
3474:
3468:
3461:
3455:
3448:
3442:
3435:
3429:
3422:
3413:
3406:
3400:
3385:
3379:
3368:
3362:
3347:
3341:
3330:
3324:
3323:
3313:
3307:
3300:
3294:
3293:
3291:
3290:
3275:
3269:
3259:
3253:
3252:
3234:
3228:
3227:
3209:
3203:
3202:
3200:
3189:
3183:
3182:
3162:
3052:streamlined body
3045:
3043:
3042:
3037:
3035:
3034:
3029:
3024:
3008:
3006:
3005:
3000:
2986:
2984:
2983:
2978:
2976:
2975:
2974:
2947:
2945:
2944:
2939:
2937:
2936:
2935:
2912:
2910:
2909:
2904:
2902:
2901:
2896:
2891:
2875:
2873:
2872:
2867:
2865:
2864:
2859:
2854:
2838:
2836:
2835:
2830:
2828:
2827:
2826:
2803:
2801:
2800:
2795:
2793:
2792:
2791:
2772:
2770:
2769:
2764:
2750:
2748:
2747:
2742:
2725:
2723:
2722:
2717:
2715:
2710:
2709:
2708:
2707:
2696:
2691:
2689:
2688:
2687:
2677:
2673:
2672:
2671:
2666:
2661:
2655:
2654:
2649:
2644:
2633:
2628:
2627:
2617:
2614:
2610:
2609:
2593:
2582:
2581:
2580:
2579:
2568:
2563:
2561:
2557:
2556:
2555:
2550:
2545:
2539:
2538:
2533:
2528:
2517:
2516:
2495:
2490:
2489:
2479:
2476:
2472:
2471:
2455:
2442:
2438:
2437:
2436:
2423:
2422:
2421:
2405:
2401:
2398:
2394:
2393:
2380:
2379:
2378:
2377:
2363:
2353:
2352:
2351:
2164:
2160:
2158:
2157:
2152:
2150:
2149:
2148:
2123:
2121:
2120:
2115:
2113:
2112:
2111:
1985:
1983:
1982:
1977:
1975:
1974:
1962:
1939:
1937:
1936:
1931:
1929:
1928:
1916:
1893:
1891:
1890:
1885:
1883:
1882:
1870:
1847:
1845:
1844:
1839:
1837:
1836:
1824:
1791:
1787:
1785:
1784:
1779:
1777:
1754:
1752:
1751:
1746:
1744:
1743:
1742:
1725:
1723:
1722:
1717:
1715:
1697:
1695:
1694:
1689:
1687:
1686:
1685:
1655:
1653:
1652:
1647:
1645:
1644:
1643:
1626:
1624:
1623:
1618:
1616:
1615:
1614:
1597:
1595:
1594:
1589:
1587:
1586:
1585:
1563:
1561:
1560:
1555:
1553:
1552:
1551:
1527:
1525:
1524:
1519:
1517:
1516:
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1470:
1469:
1468:
1450:
1448:
1447:
1442:
1440:
1439:
1438:
1343:
1341:
1340:
1335:
1333:
1332:
1331:
1313:
1311:
1310:
1305:
1303:
1288:
1286:
1285:
1280:
1278:
1263:
1261:
1260:
1255:
1253:
1238:
1236:
1235:
1230:
1228:
1213:
1211:
1210:
1205:
1203:
1202:
1201:
1181:
1179:
1178:
1173:
1171:
1153:
1151:
1150:
1145:
1143:
1142:
1141:
1124:
1122:
1121:
1116:
1114:
1099:
1097:
1096:
1091:
1089:
1088:
1087:
1070:
1068:
1067:
1062:
1060:
1030:
1028:
1027:
1022:
1020:
1019:
1018:
997:dynamic pressure
994:
992:
991:
986:
981:
952:
950:
949:
944:
939:
938:
937:
927:
926:
914:
905:
899:
898:
897:
874:
872:
871:
866:
861:
860:
851:
850:
838:
829:
823:
822:
804:
802:
801:
796:
791:
790:
778:
770:
731:
729:
728:
723:
705:
703:
702:
697:
682:dynamic pressure
679:
677:
676:
671:
649:
647:
646:
641:
624:
622:
621:
616:
614:
611:
610:
609:
596:
588:
582:
573:
567:
566:
565:
495:
493:
492:
487:
485:
484:
462:
460:
459:
454:
452:
451:
418:
416:
415:
410:
392:
390:
389:
384:
366:
364:
363:
358:
336:
334:
333:
328:
326:
325:
324:
302:
300:
299:
294:
292:
289:
285:
284:
271:
270:
269:
268:
254:
248:
247:
246:
227:
225:
224:
219:
217:
216:
215:
134:
132:
131:
126:
124:
123:
122:
105:
103:
102:
97:
95:
94:
78:
76:
75:
70:
68:
67:
66:
48:drag coefficient
21:
3709:
3708:
3704:
3703:
3702:
3700:
3699:
3698:
3674:
3673:
3624:
3619:
3618:
3609:
3607:
3603:
3596:
3592:
3591:
3587:
3578:
3576:
3567:
3566:
3562:
3553:
3551:
3546:
3545:
3541:
3536:
3532:
3523:
3521:
3512:
3511:
3507:
3498:
3496:
3487:
3486:
3482:
3475:
3471:
3462:
3458:
3449:
3445:
3436:
3432:
3424:Clancy, L. J.:
3423:
3416:
3408:Clancy, L. J.:
3407:
3403:
3386:
3382:
3369:
3365:
3348:
3344:
3331:
3327:
3315:
3314:
3310:
3306:. Section 11.17
3302:Clancy, L. J.:
3301:
3297:
3288:
3286:
3277:
3276:
3272:
3260:
3256:
3249:
3236:
3235:
3231:
3224:
3211:
3210:
3206:
3198:
3191:
3190:
3186:
3179:
3164:
3163:
3159:
3154:
3122:
3110:Reynolds number
3064:boundary layers
3017:
3016:
2991:
2990:
2965:
2960:
2959:
2926:
2921:
2920:
2884:
2883:
2847:
2846:
2817:
2812:
2811:
2782:
2777:
2776:
2755:
2754:
2733:
2732:
2713:
2712:
2698:
2678:
2641:
2637:
2619:
2601:
2597:
2590:
2570:
2525:
2521:
2508:
2481:
2463:
2459:
2452:
2440:
2439:
2427:
2412:
2403:
2402:
2385:
2381:
2368:
2364:
2354:
2342:
2333:
2332:
2326:
2321:
2316:
2310:
2172:
2139:
2134:
2133:
2102:
2097:
2096:
2093:
1966:
1950:
1949:
1920:
1904:
1903:
1902:Smooth sphere (
1874:
1858:
1857:
1828:
1812:
1811:
1799:
1765:
1764:
1733:
1728:
1727:
1703:
1702:
1700:Reynolds number
1676:
1671:
1670:
1667:
1662:
1634:
1629:
1628:
1605:
1600:
1599:
1576:
1571:
1570:
1542:
1537:
1536:
1508:
1503:
1502:
1483:
1482:
1459:
1454:
1453:
1429:
1424:
1423:
1417:
1411:
1398:
1385:
1379:Reynolds number
1376:
1322:
1317:
1316:
1291:
1290:
1266:
1265:
1241:
1240:
1216:
1215:
1192:
1187:
1186:
1159:
1158:
1132:
1127:
1126:
1102:
1101:
1078:
1073:
1072:
1048:
1047:
1045:Reynolds number
1009:
1004:
1003:
995:comes from the
969:
968:
928:
918:
888:
883:
882:
852:
842:
814:
809:
808:
807:
805:
782:
758:
757:
756:
749:
743:
714:
713:
688:
687:
662:
661:
632:
631:
601:
597:
589:
556:
551:
550:
541:
535:
476:
465:
464:
443:
429:
428:
401:
400:
375:
374:
349:
348:
315:
310:
309:
276:
272:
259:
255:
237:
232:
231:
206:
201:
200:
188:
113:
108:
107:
86:
81:
80:
57:
52:
51:
36:Reynolds number
28:
23:
22:
15:
12:
11:
5:
3707:
3705:
3697:
3696:
3691:
3686:
3684:Drag (physics)
3676:
3675:
3672:
3671:
3664:
3658:
3652:
3649:
3642:
3623:
3620:
3617:
3616:
3599:. 2005-05-02.
3585:
3560:
3539:
3530:
3505:
3480:
3469:
3456:
3450:Briens C. L.:
3443:
3430:
3428:. Section 4.17
3414:
3401:
3380:
3363:
3342:
3325:
3308:
3295:
3270:
3262:Abbott, Ira H.
3254:
3247:
3229:
3222:
3204:
3184:
3177:
3156:
3155:
3153:
3150:
3149:
3148:
3143:
3138:
3133:
3128:
3121:
3118:
3048:
3047:
3032:
3028:
3014:
2998:
2988:
2973:
2968:
2957:
2934:
2929:
2918:
2899:
2895:
2881:
2862:
2858:
2844:
2825:
2820:
2809:
2790:
2785:
2774:
2762:
2752:
2740:
2706:
2701:
2694:
2686:
2681:
2676:
2669:
2665:
2658:
2652:
2648:
2640:
2636:
2632:
2626:
2622:
2613:
2608:
2604:
2600:
2596:
2586:
2578:
2573:
2566:
2560:
2553:
2549:
2542:
2536:
2532:
2524:
2520:
2515:
2511:
2507:
2504:
2501:
2498:
2494:
2488:
2484:
2475:
2470:
2466:
2462:
2458:
2448:
2445:
2443:
2441:
2435:
2430:
2426:
2420:
2415:
2411:
2408:
2406:
2404:
2397:
2392:
2388:
2384:
2376:
2371:
2367:
2360:
2357:
2355:
2350:
2345:
2341:
2340:
2325:
2322:
2320:
2317:
2312:Main article:
2309:
2306:
2303:
2302:
2297:
2294:
2290:
2289:
2286:F-4 Phantom II
2283:
2280:
2276:
2275:
2270:
2267:
2263:
2262:
2257:
2254:
2250:
2249:
2240:
2237:
2233:
2232:
2227:
2224:
2220:
2219:
2214:
2211:
2207:
2206:
2201:
2198:
2194:
2193:
2190:F-4 Phantom II
2187:
2184:
2180:
2179:
2178:Aircraft type
2176:
2173:
2170:
2147:
2142:
2132:= 0.0001 of a
2110:
2105:
2092:
2089:
2086:
2085:
2082:
2078:
2077:
2072:
2068:
2067:
2062:
2058:
2057:
2054:
2050:
2049:
2046:
2042:
2041:
2038:
2034:
2033:
2030:
2026:
2025:
2020:
2016:
2015:
2008:
2004:
2003:
2000:
1996:
1995:
1992:
1988:
1987:
1973:
1969:
1965:
1961:
1958:
1948:Rough sphere (
1946:
1942:
1941:
1927:
1923:
1919:
1915:
1912:
1900:
1896:
1895:
1881:
1877:
1873:
1869:
1866:
1854:
1850:
1849:
1835:
1831:
1827:
1823:
1820:
1808:
1804:
1803:
1800:
1797:
1776:
1773:
1761:turbulent flow
1741:
1736:
1714:
1711:
1684:
1679:
1666:
1663:
1661:
1658:
1642:
1637:
1613:
1608:
1584:
1579:
1550:
1545:
1515:
1511:
1490:
1467:
1462:
1437:
1432:
1404:boundary layer
1395:separated flow
1374:
1349:boundary layer
1330:
1325:
1302:
1299:
1277:
1274:
1252:
1249:
1239:, Mach number
1227:
1224:
1200:
1195:
1170:
1167:
1140:
1135:
1113:
1110:
1086:
1081:
1059:
1056:
1017:
1012:
984:
980:
976:
954:
953:
942:
936:
931:
925:
921:
917:
911:
908:
902:
896:
891:
864:
859:
855:
849:
845:
841:
835:
832:
826:
821:
817:
794:
789:
785:
781:
776:
773:
768:
765:
745:Main article:
742:
739:
738:
737:
721:
711:
695:
685:
669:
659:
639:
608:
604:
600:
595:
592:
585:
579:
576:
570:
564:
559:
534:
531:
483:
479:
475:
472:
450:
446:
442:
439:
436:
424:
423:
408:
398:
382:
372:
356:
346:
323:
318:
288:
283:
279:
275:
267:
262:
258:
251:
245:
240:
228:is defined as
214:
209:
187:
184:
121:
116:
93:
89:
65:
60:
44:fluid dynamics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3706:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3679:
3670:. SAE 760185.
3669:
3665:
3663:
3659:
3657:
3653:
3650:
3647:
3643:
3641:
3640:0-273-01120-0
3637:
3633:
3629:
3626:
3625:
3621:
3602:
3595:
3594:"Airbus A380"
3589:
3586:
3574:
3570:
3564:
3561:
3549:
3543:
3540:
3534:
3531:
3519:
3515:
3509:
3506:
3494:
3490:
3484:
3481:
3478:
3473:
3470:
3466:
3460:
3457:
3453:
3447:
3444:
3440:
3434:
3431:
3427:
3421:
3419:
3415:
3411:
3405:
3402:
3398:
3394:
3390:
3384:
3381:
3377:
3373:
3367:
3364:
3360:
3356:
3352:
3346:
3343:
3339:
3335:
3329:
3326:
3321:
3320:
3312:
3309:
3305:
3299:
3296:
3284:
3280:
3274:
3271:
3267:
3263:
3258:
3255:
3250:
3244:
3240:
3233:
3230:
3225:
3223:0-471-03032-5
3219:
3215:
3208:
3205:
3197:
3196:
3188:
3185:
3180:
3174:
3170:
3169:
3161:
3158:
3151:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3123:
3119:
3117:
3113:
3111:
3107:
3103:
3098:
3096:
3091:
3089:
3085:
3080:
3071:
3067:
3065:
3061:
3057:
3053:
3015:
3012:
2996:
2989:
2966:
2958:
2955:
2951:
2927:
2919:
2916:
2882:
2879:
2845:
2842:
2818:
2810:
2807:
2783:
2775:
2760:
2753:
2738:
2731:
2730:
2729:
2726:
2699:
2692:
2679:
2674:
2656:
2638:
2634:
2624:
2620:
2611:
2606:
2602:
2598:
2594:
2584:
2571:
2564:
2558:
2540:
2522:
2513:
2509:
2505:
2502:
2496:
2486:
2482:
2473:
2468:
2464:
2460:
2456:
2446:
2444:
2428:
2424:
2413:
2409:
2407:
2395:
2390:
2386:
2382:
2369:
2365:
2358:
2356:
2343:
2330:
2323:
2318:
2315:
2307:
2301:
2298:
2295:
2292:
2291:
2288:(supersonic)
2287:
2284:
2281:
2278:
2277:
2274:
2271:
2268:
2265:
2264:
2261:
2258:
2255:
2252:
2251:
2248:
2244:
2241:
2238:
2235:
2234:
2231:
2228:
2225:
2222:
2221:
2218:
2215:
2212:
2209:
2208:
2205:
2202:
2199:
2196:
2195:
2191:
2188:
2185:
2182:
2181:
2177:
2174:
2169:
2166:
2165:
2162:
2140:
2131:
2127:
2103:
2090:
2083:
2080:
2079:
2076:
2073:
2070:
2069:
2066:
2063:
2060:
2059:
2055:
2052:
2051:
2047:
2044:
2043:
2039:
2036:
2035:
2031:
2028:
2027:
2024:
2021:
2018:
2017:
2013:
2009:
2006:
2005:
2001:
1998:
1997:
1993:
1990:
1989:
1971:
1967:
1963:
1947:
1944:
1943:
1925:
1921:
1917:
1901:
1898:
1897:
1879:
1875:
1871:
1855:
1852:
1851:
1833:
1829:
1825:
1809:
1806:
1805:
1796:
1793:
1792:
1789:
1762:
1758:
1734:
1701:
1677:
1664:
1659:
1657:
1635:
1606:
1577:
1567:
1543:
1533:
1531:
1513:
1509:
1488:
1460:
1430:
1415:
1409:
1408:vortex street
1405:
1402:
1396:
1393:
1389:
1383:
1380:
1373:
1368:
1364:
1362:
1358:
1354:
1350:
1345:
1323:
1193:
1183:
1157:
1133:
1079:
1046:
1042:
1038:
1034:
1010:
1002:The value of
1000:
998:
982:
978:
974:
966:
962:
959:
940:
929:
923:
919:
915:
909:
906:
900:
889:
881:
880:
879:
862:
857:
853:
847:
843:
839:
833:
830:
824:
819:
815:
792:
787:
783:
779:
774:
771:
766:
763:
753:
748:
747:Drag equation
740:
735:
732:is the local
719:
712:
710:of the fluid;
709:
706:is the local
693:
686:
683:
680:is the local
667:
660:
657:
656:flow velocity
653:
650:is the local
637:
630:
629:
628:
625:
606:
602:
598:
593:
590:
583:
577:
574:
568:
557:
548:
546:
540:
532:
530:
526:
522:
520:
516:
512:
508:
504:
502:
497:
481:
477:
473:
470:
448:
444:
440:
437:
434:
422:
406:
399:
396:
380:
373:
371:of the fluid;
370:
354:
347:
344:
343:flow velocity
340:
316:
308:
307:
306:
303:
286:
281:
277:
273:
260:
256:
249:
238:
229:
207:
192:
185:
183:
181:
177:
173:
169:
165:
164:skin friction
161:
160:fluid dynamic
156:
154:
150:
146:
145:drag equation
142:
138:
114:
91:
87:
58:
49:
45:
37:
32:
19:
3667:
3654:Bluff Body:
3645:
3632:Aerodynamics
3631:
3628:L. J. Clancy
3608:. Retrieved
3588:
3577:. Retrieved
3563:
3552:. Retrieved
3542:
3533:
3522:. Retrieved
3508:
3497:. Retrieved
3483:
3472:
3464:
3459:
3451:
3446:
3438:
3433:
3426:Aerodynamics
3425:
3410:Aerodynamics
3409:
3404:
3383:
3366:
3345:
3328:
3318:
3311:
3304:Aerodynamics
3303:
3298:
3287:. Retrieved
3273:
3265:
3257:
3239:Aerodynamics
3238:
3232:
3213:
3207:
3194:
3187:
3167:
3160:
3114:
3099:
3092:
3076:
3059:
3055:
3051:
3049:
3010:
2953:
2950:shear stress
2914:
2877:
2727:
2331:
2327:
2167:
2125:
2094:
2075:Eiffel Tower
2010:Bullet (not
1794:
1759:to 0.47 for
1757:laminar flow
1669:In general,
1668:
1534:
1421:
1401:laminar flow
1381:
1371:
1356:
1346:
1184:
1041:length scale
1001:
955:
877:
736:of the fluid
708:mass density
684:of the fluid
652:shear stress
626:
549:
544:
542:
527:
523:
505:
498:
425:
369:mass density
304:
230:
198:
157:
153:hydrodynamic
47:
41:
3141:Drag crisis
2230:Airbus A380
2192:(subsonic)
2048:Ski jumper
1530:Stokes' law
1501:instead of
1452:drag force
1388:Stokes flow
1361:viscosities
1156:Mach number
149:aerodynamic
3678:Categories
3622:References
3610:2014-10-06
3579:2010-12-07
3554:2010-12-07
3524:2013-03-11
3499:2010-12-07
3397:atmosphere
3372:lift force
3359:atmosphere
3334:lift force
3289:2010-12-07
3060:bluff body
2308:Automobile
2273:Boeing 747
2260:Cessna 310
2243:Cessna 172
2217:Boeing 787
2204:Learjet 24
2175:Drag Count
2130:drag count
965:flow speed
741:Background
734:flow speed
537:See also:
395:flow speed
339:drag force
186:Definition
18:Bluff body
3241:. Wiley.
3031:^
2898:^
2861:^
2693:⏟
2668:^
2657:⋅
2651:^
2621:∫
2599:ρ
2565:⏟
2552:^
2541:⋅
2535:^
2506:−
2483:∫
2461:ρ
2383:ρ
2081:1.98–2.05
1414:turbulent
1357:form drag
1035:. Speed,
1033:viscosity
916:ρ
840:ρ
780:ρ
694:ρ
638:τ
599:ρ
594:τ
575:τ
519:cube root
509:and some
474:π
441:π
355:ρ
274:ρ
176:hydrofoil
168:form drag
3630:(1975):
3601:Archived
3573:Archived
3518:Archived
3516:. NASA.
3493:Archived
3283:Archived
3120:See also
2841:friction
2806:pressure
2091:Aircraft
1999:0.9-1.7
507:Airships
501:airfoils
3106:Oropesa
3095:stalled
3088:airfoil
2839:is the
2804:is the
2728:where:
2324:Concept
2223:0.0265
2071:1.8–2.0
2061:1.3–1.5
2045:1.1-1.3
2037:1.0–1.3
2029:1.0–1.3
2019:1.0–1.1
1665:General
627:where:
517:of the
393:is the
367:is the
337:is the
305:where:
172:airfoil
135:) is a
3638:
3477:Shapes
3245:
3220:
3175:
2293:0.048
2279:0.044
2266:0.031
2253:0.027
2236:0.027
2210:0.024
2197:0.022
2183:0.021
1392:steady
1390:) and
515:square
162:drag:
46:, the
3604:(PDF)
3597:(PDF)
3199:(PDF)
3152:Notes
3056:blunt
2023:Skier
2012:ogive
2007:0.295
1991:0.81
1853:0.005
1807:0.001
1802:Item
1416:wake,
961:force
3636:ISBN
3374:and
3370:See
3336:and
3332:See
3243:ISBN
3218:ISBN
3173:ISBN
3102:tuna
3079:wake
3077:The
2053:1.28
1945:0.47
1872:>
1826:<
1353:wake
958:drag
499:For
421:area
166:and
141:drag
3104:),
3058:or
2296:480
2282:440
2269:310
2256:270
2247:182
2239:270
2226:265
2213:240
2200:220
2186:210
2126:not
1899:0.1
496:).
174:or
151:or
106:or
42:In
3680::
3417:^
3112:.
3090:.
2161:.
1986:)
1968:10
1940:)
1922:10
1894:)
1876:10
1848:)
1830:10
1535:A
1382:Re
1182:.
1071:.
79:,
3613:.
3582:.
3557:.
3527:.
3502:.
3399:.
3361:.
3292:.
3251:.
3226:.
3181:.
3027:i
3013:,
3011:S
2997:p
2972:o
2967:p
2956:,
2954:S
2933:w
2928:T
2915:S
2894:n
2880:,
2878:S
2857:t
2824:f
2819:c
2789:p
2784:c
2761:S
2739:A
2705:f
2700:c
2685:w
2680:T
2675:)
2664:i
2647:t
2639:(
2635:S
2631:d
2625:S
2612:A
2607:2
2603:v
2595:2
2585:+
2577:p
2572:c
2559:)
2548:i
2531:n
2523:(
2519:)
2514:o
2510:p
2503:p
2500:(
2497:S
2493:d
2487:S
2474:A
2469:2
2465:v
2457:2
2447:=
2434:f
2429:c
2425:+
2419:p
2414:c
2410:=
2396:A
2391:2
2387:v
2375:d
2370:F
2366:2
2359:=
2349:d
2344:c
2245:/
2171:d
2168:c
2146:d
2141:c
2109:d
2104:c
1972:6
1964:=
1960:e
1957:R
1926:6
1918:=
1914:e
1911:R
1880:6
1868:e
1865:R
1834:6
1822:e
1819:R
1798:d
1795:c
1775:e
1772:R
1740:d
1735:c
1713:e
1710:R
1683:d
1678:c
1641:d
1636:c
1612:d
1607:c
1583:d
1578:c
1549:d
1544:c
1514:2
1510:v
1489:v
1466:d
1461:F
1436:d
1431:c
1410:,
1397:,
1375:d
1372:C
1329:d
1324:c
1301:e
1298:R
1276:a
1273:M
1251:a
1248:M
1226:e
1223:R
1199:d
1194:c
1169:a
1166:M
1139:d
1134:c
1112:e
1109:R
1085:d
1080:c
1058:e
1055:R
1016:d
1011:c
983:2
979:/
975:1
941:A
935:d
930:c
924:2
920:u
910:2
907:1
901:=
895:d
890:F
863:A
858:d
854:c
848:2
844:u
834:2
831:1
825:=
820:d
816:F
793:A
788:2
784:u
775:2
772:1
767:=
764:F
720:u
668:q
658:;
607:2
603:u
591:2
584:=
578:q
569:=
563:d
558:c
482:2
478:r
471:4
449:2
445:r
438:=
435:A
407:A
381:u
345:;
322:d
317:F
287:A
282:2
278:u
266:d
261:F
257:2
250:=
244:d
239:c
213:d
208:c
120:w
115:c
92:x
88:c
64:d
59:c
20:)
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