1356:
2713:
180:
2323:
518:
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.
3059:
2708:{\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}}}
741:
20:
3097:, 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
1440:
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
517:
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
2317:
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
1557:
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
415:
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.
3070:
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
3104:
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.
1373:, 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:
513:
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.
183:
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
612:
492:, 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.
3051:. 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
290:
1303:
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
940:
1348:
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
862:
2328:
792:
1881:
1835:
3033:
2900:
2863:
1973:
1927:
2974:
2935:
2826:
2791:
2148:
2111:
1742:
1685:
1643:
1614:
1585:
1551:
1468:
1438:
1331:
1201:
1141:
1087:
1018:
324:
215:
66:
1775:
1713:
1301:
1276:
1251:
1226:
1169:
1112:
1058:
450:
122:
483:
541:
1515:
693:
637:
354:
93:
982:
2996:
2760:
2738:
1488:
719:
667:
406:
380:
222:
3082:
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
3644:
3682:
2117:
directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1
1587:
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
1401:•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic
3481:
873:
3589:
3526:
Basha, W. A. and Ghaly, W. S., "Drag
Prediction in Transitional Flow over Airfoils," Journal of Aircraft, Vol. 44, 2007, p. 824–32.
3506:
1020:
is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid
3235:
3165:
3561:
799:
3155:
3271:
527:
3628:
3210:
1553:
equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up
1558:
at the center, dropping off toward the edges as in the lower figure and graph. Only considering the front side, the
2274:
171:. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.
536:
is referred to the transversal area (the area normal to the drag force, so the coefficient is locally defined as:
510:
of the airship volume (volume to the two-thirds power). Submerged streamlined bodies use the wetted surface area.
3465:
1355:
748:
1687:
is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with
3677:
1848:
1802:
3007:
2874:
2837:
1940:
1894:
1616:
of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore
3182:
3119:
2302:
952:
on any object is proportional to the density of the fluid and proportional to the square of the relative
3134:
3086:
and has higher pressure drag than friction drag. In this case, the body is described as a blunt body.
2950:
2911:
2802:
2767:
2124:
2087:
1718:
1661:
1619:
1590:
1561:
1527:
1444:
1414:
1307:
1177:
1117:
1063:
994:
300:
191:
42:
1340:
around the body must remain attached to the surface of the body for as long as possible, causing the
3364:
3326:
1755:
1693:
1281:
1256:
1231:
1206:
1149:
1092:
1038:
2976:
is the pressure far away from the body (note that this constant does not affect the final result),
3672:
3114:
1411:
For other objects, such as small particles, one can no longer consider that the drag coefficient
419:
98:
3640:
Hoerner, Dr. Sighard F., Fluid-Dynamic Drag, Hoerner Fluid
Dynamics, Bricktown New Jersey, 1965.
3536:
3477:
2113:, while automobiles (and many other objects) use projected frontal area; thus, coefficients are
1521:. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.
455:
3308:
Fluid-Dynamic Drag : Practical
Information on Aerodynamic Drag and Hydrodynamic Resistance
3094:
2318:
pressure drag (form drag). The total drag and component drag forces can be related as follows:
1278:, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number
125:
3377:
3339:
3124:
2053:
1493:
678:
622:
339:
132:
or resistance of an object in a fluid environment, such as air or water. It is used in the
71:
3650:
3306:
532:
In the non dimensional form of the Cauchy momentum equation, the skin drag coefficient or
179:
147:
The drag coefficient of any object comprises the effects of the two basic contributors to
8:
3582:
1554:
1025:
959:
744:
Flow around a plate, showing stagnation. The force in the upper configuration is equal to
499:
3502:
3381:
3343:
2981:
2745:
2723:
1645:
for some shapes can change with the
Reynolds number and the roughness of the surfaces.
1473:
704:
652:
391:
365:
3039:
Therefore, when the drag is dominated by a frictional component, the body is called a
3624:
3385:
3347:
3231:
3206:
3161:
3072:
3058:
2288:
168:
2865:
is the unit vector in the direction of the shear stress acting on the body surface d
985:
670:
503:
2084:
As noted above, aircraft use their wing area as the reference area when computing
607:{\displaystyle c_{\mathrm {d} }={\dfrac {\tau }{q}}={\dfrac {2\tau }{\rho u^{2}}}}
3557:
3098:
1688:
1383:
1367:
1033:
24:
144:
drag. The drag coefficient is always associated with a particular surface area.
3083:
3067:
3052:
2178:
1749:
1518:
1392:
1341:
1337:
946:
327:
148:
129:
32:
3267:
3666:
3250:
1396:
735:
644:
331:
152:
141:
133:
3657:
The optimization of body details-A method for reducing the aerodynamics drag
643:, which is by definition the stress component in the direction of the local
3616:
2938:
2063:
1745:
1389:
1029:
696:
640:
357:
137:
285:{\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}}
3645:
http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf
3129:
2218:
1380:
1376:
1144:
27:
approximately 10 Shapes are depicted with the same projected frontal area
1032:
of the object are incorporated into a dimensionless quantity called the
502:
use the volumetric drag coefficient, in which the reference area is the
3360:
3322:
2902:
is the unit vector in the direction perpendicular to the body surface d
2261:
2248:
2235:
2231:
2205:
2192:
2118:
1402:
953:
722:
383:
3043:; whereas in the case of dominant pressure drag, the body is called a
136:
in which a lower drag coefficient indicates the object will have less
3637:. Dover Publications Inc., New York, Standard Book Number 486-60586-8
1349:
1021:
507:
330:, which is by definition the force component in the direction of the
164:
156:
3062:
Trade-off relationship between zero-lift drag and lift induced drag
2829:
2794:
1407:•5: post-critical separated flow, with a turbulent boundary layer.
3184:
Dynamic response of pipe rack steel structures to explosion loads
3076:
935:{\displaystyle F_{\rm {d}}={\tfrac {1}{2}}\rho u^{2}c_{\rm {d}}A}
495:
489:
160:
3367:
for a possible force components transverse to the flow direction
3329:
for a possible force components transverse to the flow direction
740:
1336:
For a streamlined body to achieve a low drag coefficient, the
1114:. In a compressible flow, the speed of sound is relevant, and
2011:
2000:
949:
988:
of the fluid, which is equal to the kinetic energy density.
3090:
1752:. Although the drag coefficient decreases with increasing
409:
19:
3651:
http://www.princeton.edu/~asmits/Bicycle_web/blunt.html
893:
817:
3655:
Hucho, W.H., Janssen, L.J., Emmelmann, H.J. 6(1975):
3537:"Ask Us – Drag Coefficient & Lifting Line Theory"
3010:
2984:
2953:
2914:
2877:
2840:
2805:
2770:
2748:
2726:
2607:
2581:
2469:
2443:
2351:
2326:
2127:
2090:
1943:
1897:
1851:
1805:
1758:
1721:
1696:
1664:
1622:
1593:
1564:
1530:
1496:
1476:
1447:
1417:
1310:
1284:
1259:
1234:
1209:
1180:
1152:
1120:
1095:
1066:
1041:
997:
962:
876:
857:{\displaystyle F_{d}={\tfrac {1}{2}}\rho u^{2}c_{d}A}
802:
751:
707:
681:
655:
625:
576:
561:
544:
528:
Cauchy momentum equation § Nondimensionalisation
458:
422:
394:
368:
342:
303:
242:
225:
194:
101:
74:
45:
3633:
Abbott, Ira H., and Von
Doenhoff, Albert E. (1959):
3384:. Air is 1.293 kg/m at 0 °C (32 °F) and 1
3346:. Air is 1.293 kg/m at 0 °C (32 °F) and 1
3205:. New York: John Wiley & Sons, Inc. p. 24.
16:
Dimensionless parameter to quantify fluid resistance
3035:
is the unit vector in direction of free stream flow
2307:
1253:and the direction of the flow. For low Mach number
3027:
2990:
2968:
2929:
2894:
2857:
2820:
2785:
2754:
2732:
2707:
2142:
2105:
1967:
1921:
1875:
1829:
1769:
1736:
1707:
1679:
1637:
1608:
1579:
1545:
1509:
1482:
1462:
1432:
1325:
1295:
1270:
1245:
1220:
1195:
1163:
1135:
1106:
1081:
1052:
1012:
976:
934:
856:
786:
713:
687:
661:
631:
606:
477:
444:
400:
374:
348:
318:
284:
209:
116:
87:
60:
3664:
956:between the object and the fluid. The factor of
3203:Aerodynamics, Aeronautics, and Flight Mechanics
3409:
3407:
1174:For certain body shapes, the drag coefficient
3649:Drag of Blunt Bodies and Streamlined Bodies:
1648:
1395:upstream of the separation, and producing a
3404:
2073:Long flat plate perpendicular to flow (2D)
1845:Turbulent flat plate parallel to the flow (
787:{\displaystyle F={\frac {1}{2}}\rho u^{2}A}
521:
3157:Explosion Hazards and Evaluation, Volume 5
3380:, the air density can be found using the
3342:, the air density can be found using the
3200:
3055:, which produce much less pressure drag.
1799:Laminar flat plate parallel to the flow (
3683:Dimensionless numbers of fluid mechanics
3057:
1354:
739:
178:
18:
3558:"Boeing 787 Dreamliner : Analysis"
3304:
3180:
2906:, pointing from the fluid to the solid,
1876:{\displaystyle \mathrm {Re} >10^{6}}
1830:{\displaystyle \mathrm {Re} <10^{6}}
1715:). A smooth sphere, for example, has a
1352:contribute to larger Reynolds numbers.
3665:
3225:
3089:A streamlined body looks like a fish (
2045:Flat plate perpendicular to flow (3D)
1388:•3: separated unsteady flow, having a
3623:. Pitman Publishing Limited, London,
3153:
3028:{\displaystyle {\hat {\mathbf {i} }}}
2895:{\displaystyle {\hat {\mathbf {n} }}}
2858:{\displaystyle {\hat {\mathbf {t} }}}
452:(note this is not the surface area =
3470:
3426:Clift R., Grace J. R., Weber M. E.:
3190:. CHALMERS UNIVERSITY OF TECHNOLOGY.
1991:Trapeze with triangular basis (45°)
1968:{\displaystyle \mathrm {Re} =10^{6}}
1922:{\displaystyle \mathrm {Re} =10^{6}}
1333:can often be treated as a constant.
1203:only depends on the Reynolds number
945:is essentially a statement that the
386:of the object relative to the fluid;
159:. The drag coefficient of a lifting
13:
3560:. Lissys.demon.co.uk. 2006-06-21.
2960:
2921:
2812:
2777:
2693:
2673:
2619:
2565:
2481:
2422:
2407:
2363:
2337:
2134:
2097:
1948:
1945:
1902:
1899:
1856:
1853:
1810:
1807:
1763:
1760:
1728:
1701:
1698:
1671:
1629:
1600:
1571:
1537:
1454:
1424:
1317:
1289:
1286:
1264:
1261:
1239:
1236:
1214:
1211:
1187:
1157:
1154:
1127:
1100:
1097:
1073:
1046:
1043:
1004:
923:
883:
551:
310:
254:
232:
201:
108:
52:
14:
3694:
2740:is the planform area of the body,
1744:that varies from high values for
23:Drag coefficients in fluids with
3075:at the top and rear parts of an
3015:
2969:{\displaystyle p_{\mathrm {o} }}
2930:{\displaystyle T_{\mathrm {w} }}
2882:
2845:
2821:{\displaystyle c_{\mathrm {f} }}
2786:{\displaystyle c_{\mathrm {p} }}
2652:
2635:
2536:
2519:
2308:Blunt and streamlined body flows
2143:{\displaystyle c_{\mathrm {d} }}
2106:{\displaystyle c_{\mathrm {d} }}
1737:{\displaystyle c_{\mathrm {d} }}
1680:{\displaystyle c_{\mathrm {d} }}
1638:{\displaystyle c_{\mathrm {d} }}
1609:{\displaystyle c_{\mathrm {d} }}
1580:{\displaystyle c_{\mathrm {d} }}
1546:{\displaystyle c_{\mathrm {d} }}
1517:; for a sphere this is known as
1463:{\displaystyle F_{\mathrm {d} }}
1433:{\displaystyle c_{\mathrm {d} }}
1326:{\displaystyle c_{\mathrm {d} }}
1196:{\displaystyle c_{\mathrm {d} }}
1136:{\displaystyle c_{\mathrm {d} }}
1082:{\displaystyle c_{\mathrm {d} }}
1013:{\displaystyle c_{\mathrm {d} }}
319:{\displaystyle F_{\mathrm {d} }}
210:{\displaystyle c_{\mathrm {d} }}
61:{\displaystyle c_{\mathrm {d} }}
3595:from the original on 2015-09-23
3575:
3564:from the original on 2010-08-13
3550:
3529:
3520:
3509:from the original on 2013-02-16
3495:
3484:from the original on 2010-12-04
3459:
3446:
3433:
3420:
3391:
3370:
3353:
3332:
3274:from the original on 2011-03-02
3270:. Wright.nasa.gov. 2010-03-25.
3253:, and Von Doenhoff, Albert E.:
2762:is the wet surface of the body,
2029:Adult human (upright position)
3539:. Aerospaceweb.org. 2004-07-11
3315:
3298:
3285:
3260:
3244:
3226:Clancy, L. J. (1975). "5.18".
3219:
3194:
3174:
3147:
3019:
2886:
2849:
2656:
2639:
2540:
2523:
2507:
2488:
1366:for a sphere as a function of
795:and in the lower configuration
1:
3610:
3428:Bubbles, drops, and particles
3201:McCormick, Barnes W. (1979).
3181:AARØNÆS, ANTON STADE (2014).
2296:
1770:{\displaystyle \mathrm {Re} }
1708:{\displaystyle \mathrm {Re} }
1296:{\displaystyle \mathrm {Re} }
1271:{\displaystyle \mathrm {Ma} }
1246:{\displaystyle \mathrm {Ma} }
1221:{\displaystyle \mathrm {Re} }
1164:{\displaystyle \mathrm {Ma} }
1107:{\displaystyle \mathrm {Re} }
1053:{\displaystyle \mathrm {Re} }
729:
174:
167:also includes the effects of
128:that is used to quantify the
3305:Hoerner, Sighard F. (1965).
2941:acting on the body surface d
1777:, the drag force increases.
7:
3430:. Academic Press NY (1978).
3311:(2 ed.). p. 3–17.
3120:Automobile drag coefficient
3108:
2303:Automobile drag coefficient
2079:
445:{\displaystyle A=\pi r^{2}}
117:{\displaystyle c_{\rm {w}}}
10:
3699:
3480:. Engineeringtoolbox.com.
3452:Haider A., Levenspiel O.:
3135:Zero-lift drag coefficient
2312:
2300:
1653:
733:
525:
478:{\displaystyle 4\pi r^{2}}
416:For example, for a sphere
1983:Triangular trapeze (45°)
1649:Drag coefficient examples
534:skin friction coefficient
3365:vortex induced vibration
3327:vortex induced vibration
3140:
2998:is pressure at surface d
2003:, at subsonic velocity)
522:Cauchy momentum equation
3635:Theory of Wing Sections
3503:"Shape Effects on Drag"
3401:. Sections 4.15 and 5.4
3255:Theory of Wing Sections
3115:Automotive aerodynamics
3268:"Modern Drag Equation"
3257:. Sections 1.2 and 1.3
3063:
3029:
2992:
2970:
2931:
2896:
2859:
2822:
2787:
2756:
2734:
2709:
2144:
2107:
1969:
1923:
1877:
1831:
1771:
1738:
1709:
1681:
1639:
1610:
1581:
1547:
1511:
1484:
1464:
1434:
1408:
1327:
1297:
1272:
1247:
1222:
1197:
1165:
1143:is also a function of
1137:
1108:
1089:is thus a function of
1083:
1054:
1014:
978:
936:
864:
858:
788:
715:
689:
663:
633:
608:
479:
446:
402:
376:
350:
320:
286:
211:
185:
126:dimensionless quantity
118:
89:
62:
39:(commonly denoted as:
28:
3678:Aerospace engineering
3125:Ballistic coefficient
3061:
3030:
2993:
2971:
2932:
2897:
2860:
2823:
2788:
2757:
2735:
2710:
2145:
2108:
2054:Empire State Building
1970:
1924:
1878:
1832:
1772:
1739:
1710:
1682:
1640:
1611:
1582:
1548:
1512:
1510:{\displaystyle v^{2}}
1485:
1465:
1435:
1358:
1344:to be narrow. A high
1328:
1298:
1273:
1248:
1223:
1198:
1166:
1138:
1109:
1084:
1055:
1028:and a characteristic
1015:
979:
937:
859:
789:
743:
716:
690:
688:{\displaystyle \rho }
664:
634:
632:{\displaystyle \tau }
609:
480:
447:
403:
377:
351:
349:{\displaystyle \rho }
321:
287:
212:
188:The drag coefficient
182:
119:
90:
88:{\displaystyle c_{x}}
63:
22:
3160:. Elsevier Science.
3154:Baker, W.E. (1983).
3008:
2982:
2951:
2912:
2875:
2838:
2803:
2768:
2746:
2724:
2324:
2125:
2088:
1941:
1895:
1849:
1803:
1756:
1719:
1694:
1662:
1620:
1591:
1562:
1528:
1494:
1474:
1445:
1415:
1308:
1282:
1257:
1232:
1207:
1178:
1150:
1118:
1093:
1064:
1039:
995:
960:
874:
800:
749:
705:
679:
653:
623:
542:
500:bodies of revolution
456:
420:
392:
366:
340:
301:
223:
192:
99:
72:
43:
1555:stagnation pressure
1470:is proportional to
1375:•2: attached flow (
1026:kinematic viscosity
977:{\displaystyle 1/2}
3478:"Drag Coefficient"
3456:. 58, 1989, 63-70.
3443:. 67, 1991, 87-91.
3382:barometric formula
3378:Earth's atmosphere
3376:Note that for the
3344:barometric formula
3340:Earth's atmosphere
3338:Note that for the
3073:pressure gradients
3064:
3025:
2988:
2966:
2927:
2892:
2855:
2818:
2783:
2752:
2730:
2705:
2703:
2700:
2684:
2679:
2605:
2572:
2556:
2551:
2467:
2389:
2140:
2103:
1965:
1919:
1873:
1827:
1767:
1734:
1705:
1677:
1635:
1606:
1577:
1543:
1507:
1480:
1460:
1430:
1409:
1323:
1293:
1268:
1243:
1218:
1193:
1161:
1133:
1104:
1079:
1050:
1010:
974:
932:
902:
867:The drag equation
865:
854:
826:
784:
711:
685:
659:
629:
604:
602:
570:
475:
442:
398:
372:
346:
316:
282:
280:
207:
186:
114:
85:
58:
29:
3454:Powder Technology
3441:Powder Technology
3237:978-0-470-15837-1
3167:978-0-444-59988-9
3022:
2991:{\displaystyle p}
2937:magnitude of the
2889:
2852:
2832:drag coefficient,
2797:drag coefficient,
2755:{\displaystyle S}
2733:{\displaystyle A}
2659:
2642:
2604:
2578:
2576:
2543:
2526:
2466:
2440:
2438:
2388:
2294:
2293:
2289:F-104 Starfighter
2077:
2076:
2021:Wires and cables
1483:{\displaystyle v}
1359:Drag coefficient
901:
825:
766:
714:{\displaystyle u}
662:{\displaystyle q}
601:
569:
408:is the reference
401:{\displaystyle A}
375:{\displaystyle u}
279:
169:lift-induced drag
3690:
3604:
3603:
3601:
3600:
3594:
3587:
3579:
3573:
3572:
3570:
3569:
3554:
3548:
3547:
3545:
3544:
3533:
3527:
3524:
3518:
3517:
3515:
3514:
3499:
3493:
3492:
3490:
3489:
3474:
3468:
3463:
3457:
3450:
3444:
3437:
3431:
3424:
3418:
3411:
3402:
3395:
3389:
3374:
3368:
3357:
3351:
3336:
3330:
3319:
3313:
3312:
3302:
3296:
3289:
3283:
3282:
3280:
3279:
3264:
3258:
3248:
3242:
3241:
3223:
3217:
3216:
3198:
3192:
3191:
3189:
3178:
3172:
3171:
3151:
3041:streamlined body
3034:
3032:
3031:
3026:
3024:
3023:
3018:
3013:
2997:
2995:
2994:
2989:
2975:
2973:
2972:
2967:
2965:
2964:
2963:
2936:
2934:
2933:
2928:
2926:
2925:
2924:
2901:
2899:
2898:
2893:
2891:
2890:
2885:
2880:
2864:
2862:
2861:
2856:
2854:
2853:
2848:
2843:
2827:
2825:
2824:
2819:
2817:
2816:
2815:
2792:
2790:
2789:
2784:
2782:
2781:
2780:
2761:
2759:
2758:
2753:
2739:
2737:
2736:
2731:
2714:
2712:
2711:
2706:
2704:
2699:
2698:
2697:
2696:
2685:
2680:
2678:
2677:
2676:
2666:
2662:
2661:
2660:
2655:
2650:
2644:
2643:
2638:
2633:
2622:
2617:
2616:
2606:
2603:
2599:
2598:
2582:
2571:
2570:
2569:
2568:
2557:
2552:
2550:
2546:
2545:
2544:
2539:
2534:
2528:
2527:
2522:
2517:
2506:
2505:
2484:
2479:
2478:
2468:
2465:
2461:
2460:
2444:
2431:
2427:
2426:
2425:
2412:
2411:
2410:
2394:
2390:
2387:
2383:
2382:
2369:
2368:
2367:
2366:
2352:
2342:
2341:
2340:
2153:
2152:
2149:
2147:
2146:
2141:
2139:
2138:
2137:
2112:
2110:
2109:
2104:
2102:
2101:
2100:
1974:
1972:
1971:
1966:
1964:
1963:
1951:
1928:
1926:
1925:
1920:
1918:
1917:
1905:
1882:
1880:
1879:
1874:
1872:
1871:
1859:
1836:
1834:
1833:
1828:
1826:
1825:
1813:
1780:
1779:
1776:
1774:
1773:
1768:
1766:
1743:
1741:
1740:
1735:
1733:
1732:
1731:
1714:
1712:
1711:
1706:
1704:
1686:
1684:
1683:
1678:
1676:
1675:
1674:
1644:
1642:
1641:
1636:
1634:
1633:
1632:
1615:
1613:
1612:
1607:
1605:
1604:
1603:
1586:
1584:
1583:
1578:
1576:
1575:
1574:
1552:
1550:
1549:
1544:
1542:
1541:
1540:
1516:
1514:
1513:
1508:
1506:
1505:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1459:
1458:
1457:
1439:
1437:
1436:
1431:
1429:
1428:
1427:
1332:
1330:
1329:
1324:
1322:
1321:
1320:
1302:
1300:
1299:
1294:
1292:
1277:
1275:
1274:
1269:
1267:
1252:
1250:
1249:
1244:
1242:
1227:
1225:
1224:
1219:
1217:
1202:
1200:
1199:
1194:
1192:
1191:
1190:
1170:
1168:
1167:
1162:
1160:
1142:
1140:
1139:
1134:
1132:
1131:
1130:
1113:
1111:
1110:
1105:
1103:
1088:
1086:
1085:
1080:
1078:
1077:
1076:
1059:
1057:
1056:
1051:
1049:
1019:
1017:
1016:
1011:
1009:
1008:
1007:
986:dynamic pressure
983:
981:
980:
975:
970:
941:
939:
938:
933:
928:
927:
926:
916:
915:
903:
894:
888:
887:
886:
863:
861:
860:
855:
850:
849:
840:
839:
827:
818:
812:
811:
793:
791:
790:
785:
780:
779:
767:
759:
720:
718:
717:
712:
694:
692:
691:
686:
671:dynamic pressure
668:
666:
665:
660:
638:
636:
635:
630:
613:
611:
610:
605:
603:
600:
599:
598:
585:
577:
571:
562:
556:
555:
554:
484:
482:
481:
476:
474:
473:
451:
449:
448:
443:
441:
440:
407:
405:
404:
399:
381:
379:
378:
373:
355:
353:
352:
347:
325:
323:
322:
317:
315:
314:
313:
291:
289:
288:
283:
281:
278:
274:
273:
260:
259:
258:
257:
243:
237:
236:
235:
216:
214:
213:
208:
206:
205:
204:
123:
121:
120:
115:
113:
112:
111:
94:
92:
91:
86:
84:
83:
67:
65:
64:
59:
57:
56:
55:
37:drag coefficient
3698:
3697:
3693:
3692:
3691:
3689:
3688:
3687:
3663:
3662:
3613:
3608:
3607:
3598:
3596:
3592:
3585:
3581:
3580:
3576:
3567:
3565:
3556:
3555:
3551:
3542:
3540:
3535:
3534:
3530:
3525:
3521:
3512:
3510:
3501:
3500:
3496:
3487:
3485:
3476:
3475:
3471:
3464:
3460:
3451:
3447:
3438:
3434:
3425:
3421:
3413:Clancy, L. J.:
3412:
3405:
3397:Clancy, L. J.:
3396:
3392:
3375:
3371:
3358:
3354:
3337:
3333:
3320:
3316:
3303:
3299:
3295:. Section 11.17
3291:Clancy, L. J.:
3290:
3286:
3277:
3275:
3266:
3265:
3261:
3249:
3245:
3238:
3224:
3220:
3213:
3199:
3195:
3187:
3179:
3175:
3168:
3152:
3148:
3143:
3111:
3099:Reynolds number
3053:boundary layers
3014:
3012:
3011:
3009:
3006:
3005:
2983:
2980:
2979:
2959:
2958:
2954:
2952:
2949:
2948:
2920:
2919:
2915:
2913:
2910:
2909:
2881:
2879:
2878:
2876:
2873:
2872:
2844:
2842:
2841:
2839:
2836:
2835:
2811:
2810:
2806:
2804:
2801:
2800:
2776:
2775:
2771:
2769:
2766:
2765:
2747:
2744:
2743:
2725:
2722:
2721:
2702:
2701:
2692:
2691:
2687:
2686:
2672:
2671:
2667:
2651:
2649:
2648:
2634:
2632:
2631:
2630:
2626:
2618:
2612:
2608:
2594:
2590:
2586:
2580:
2579:
2577:
2564:
2563:
2559:
2558:
2535:
2533:
2532:
2518:
2516:
2515:
2514:
2510:
2501:
2497:
2480:
2474:
2470:
2456:
2452:
2448:
2442:
2441:
2439:
2429:
2428:
2421:
2420:
2416:
2406:
2405:
2401:
2392:
2391:
2378:
2374:
2370:
2362:
2361:
2357:
2353:
2350:
2343:
2336:
2335:
2331:
2327:
2325:
2322:
2321:
2315:
2310:
2305:
2299:
2161:
2133:
2132:
2128:
2126:
2123:
2122:
2096:
2095:
2091:
2089:
2086:
2085:
2082:
1959:
1955:
1944:
1942:
1939:
1938:
1913:
1909:
1898:
1896:
1893:
1892:
1891:Smooth sphere (
1867:
1863:
1852:
1850:
1847:
1846:
1821:
1817:
1806:
1804:
1801:
1800:
1788:
1759:
1757:
1754:
1753:
1727:
1726:
1722:
1720:
1717:
1716:
1697:
1695:
1692:
1691:
1689:Reynolds number
1670:
1669:
1665:
1663:
1660:
1659:
1656:
1651:
1628:
1627:
1623:
1621:
1618:
1617:
1599:
1598:
1594:
1592:
1589:
1588:
1570:
1569:
1565:
1563:
1560:
1559:
1536:
1535:
1531:
1529:
1526:
1525:
1501:
1497:
1495:
1492:
1491:
1475:
1472:
1471:
1453:
1452:
1448:
1446:
1443:
1442:
1423:
1422:
1418:
1416:
1413:
1412:
1406:
1400:
1387:
1374:
1368:Reynolds number
1365:
1316:
1315:
1311:
1309:
1306:
1305:
1285:
1283:
1280:
1279:
1260:
1258:
1255:
1254:
1235:
1233:
1230:
1229:
1210:
1208:
1205:
1204:
1186:
1185:
1181:
1179:
1176:
1175:
1153:
1151:
1148:
1147:
1126:
1125:
1121:
1119:
1116:
1115:
1096:
1094:
1091:
1090:
1072:
1071:
1067:
1065:
1062:
1061:
1042:
1040:
1037:
1036:
1034:Reynolds number
1003:
1002:
998:
996:
993:
992:
984:comes from the
966:
961:
958:
957:
922:
921:
917:
911:
907:
892:
882:
881:
877:
875:
872:
871:
845:
841:
835:
831:
816:
807:
803:
801:
798:
797:
796:
794:
775:
771:
758:
750:
747:
746:
745:
738:
732:
706:
703:
702:
680:
677:
676:
654:
651:
650:
624:
621:
620:
594:
590:
586:
578:
575:
560:
550:
549:
545:
543:
540:
539:
530:
524:
469:
465:
457:
454:
453:
436:
432:
421:
418:
417:
393:
390:
389:
367:
364:
363:
341:
338:
337:
309:
308:
304:
302:
299:
298:
269:
265:
261:
253:
252:
248:
244:
241:
231:
230:
226:
224:
221:
220:
200:
199:
195:
193:
190:
189:
177:
107:
106:
102:
100:
97:
96:
79:
75:
73:
70:
69:
51:
50:
46:
44:
41:
40:
25:Reynolds number
17:
12:
11:
5:
3696:
3686:
3685:
3680:
3675:
3673:Drag (physics)
3661:
3660:
3653:
3647:
3641:
3638:
3631:
3612:
3609:
3606:
3605:
3588:. 2005-05-02.
3574:
3549:
3528:
3519:
3494:
3469:
3458:
3445:
3439:Briens C. L.:
3432:
3419:
3417:. Section 4.17
3403:
3390:
3369:
3352:
3331:
3314:
3297:
3284:
3259:
3251:Abbott, Ira H.
3243:
3236:
3218:
3211:
3193:
3173:
3166:
3145:
3144:
3142:
3139:
3138:
3137:
3132:
3127:
3122:
3117:
3110:
3107:
3037:
3036:
3021:
3017:
3003:
2987:
2977:
2962:
2957:
2946:
2923:
2918:
2907:
2888:
2884:
2870:
2851:
2847:
2833:
2814:
2809:
2798:
2779:
2774:
2763:
2751:
2741:
2729:
2695:
2690:
2683:
2675:
2670:
2665:
2658:
2654:
2647:
2641:
2637:
2629:
2625:
2621:
2615:
2611:
2602:
2597:
2593:
2589:
2585:
2575:
2567:
2562:
2555:
2549:
2542:
2538:
2531:
2525:
2521:
2513:
2509:
2504:
2500:
2496:
2493:
2490:
2487:
2483:
2477:
2473:
2464:
2459:
2455:
2451:
2447:
2437:
2434:
2432:
2430:
2424:
2419:
2415:
2409:
2404:
2400:
2397:
2395:
2393:
2386:
2381:
2377:
2373:
2365:
2360:
2356:
2349:
2346:
2344:
2339:
2334:
2330:
2329:
2314:
2311:
2309:
2306:
2301:Main article:
2298:
2295:
2292:
2291:
2286:
2283:
2279:
2278:
2275:F-4 Phantom II
2272:
2269:
2265:
2264:
2259:
2256:
2252:
2251:
2246:
2243:
2239:
2238:
2229:
2226:
2222:
2221:
2216:
2213:
2209:
2208:
2203:
2200:
2196:
2195:
2190:
2187:
2183:
2182:
2179:F-4 Phantom II
2176:
2173:
2169:
2168:
2167:Aircraft type
2165:
2162:
2159:
2136:
2131:
2121:= 0.0001 of a
2099:
2094:
2081:
2078:
2075:
2074:
2071:
2067:
2066:
2061:
2057:
2056:
2051:
2047:
2046:
2043:
2039:
2038:
2035:
2031:
2030:
2027:
2023:
2022:
2019:
2015:
2014:
2009:
2005:
2004:
1997:
1993:
1992:
1989:
1985:
1984:
1981:
1977:
1976:
1962:
1958:
1954:
1950:
1947:
1937:Rough sphere (
1935:
1931:
1930:
1916:
1912:
1908:
1904:
1901:
1889:
1885:
1884:
1870:
1866:
1862:
1858:
1855:
1843:
1839:
1838:
1824:
1820:
1816:
1812:
1809:
1797:
1793:
1792:
1789:
1786:
1765:
1762:
1750:turbulent flow
1730:
1725:
1703:
1700:
1673:
1668:
1655:
1652:
1650:
1647:
1631:
1626:
1602:
1597:
1573:
1568:
1539:
1534:
1504:
1500:
1479:
1456:
1451:
1426:
1421:
1393:boundary layer
1384:separated flow
1363:
1338:boundary layer
1319:
1314:
1291:
1288:
1266:
1263:
1241:
1238:
1228:, Mach number
1216:
1213:
1189:
1184:
1159:
1156:
1129:
1124:
1102:
1099:
1075:
1070:
1048:
1045:
1006:
1001:
973:
969:
965:
943:
942:
931:
925:
920:
914:
910:
906:
900:
897:
891:
885:
880:
853:
848:
844:
838:
834:
830:
824:
821:
815:
810:
806:
783:
778:
774:
770:
765:
762:
757:
754:
734:Main article:
731:
728:
727:
726:
710:
700:
684:
674:
658:
648:
628:
597:
593:
589:
584:
581:
574:
568:
565:
559:
553:
548:
523:
520:
472:
468:
464:
461:
439:
435:
431:
428:
425:
413:
412:
397:
387:
371:
361:
345:
335:
312:
307:
277:
272:
268:
264:
256:
251:
247:
240:
234:
229:
217:is defined as
203:
198:
176:
173:
110:
105:
82:
78:
54:
49:
33:fluid dynamics
15:
9:
6:
4:
3:
2:
3695:
3684:
3681:
3679:
3676:
3674:
3671:
3670:
3668:
3659:. SAE 760185.
3658:
3654:
3652:
3648:
3646:
3642:
3639:
3636:
3632:
3630:
3629:0-273-01120-0
3626:
3622:
3618:
3615:
3614:
3591:
3584:
3583:"Airbus A380"
3578:
3563:
3559:
3553:
3538:
3532:
3523:
3508:
3504:
3498:
3483:
3479:
3473:
3467:
3462:
3455:
3449:
3442:
3436:
3429:
3423:
3416:
3410:
3408:
3400:
3394:
3387:
3383:
3379:
3373:
3366:
3362:
3356:
3349:
3345:
3341:
3335:
3328:
3324:
3318:
3310:
3309:
3301:
3294:
3288:
3273:
3269:
3263:
3256:
3252:
3247:
3239:
3233:
3229:
3222:
3214:
3212:0-471-03032-5
3208:
3204:
3197:
3186:
3185:
3177:
3169:
3163:
3159:
3158:
3150:
3146:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3112:
3106:
3102:
3100:
3096:
3092:
3087:
3085:
3080:
3078:
3074:
3069:
3060:
3056:
3054:
3050:
3046:
3042:
3004:
3001:
2985:
2978:
2955:
2947:
2944:
2940:
2916:
2908:
2905:
2871:
2868:
2834:
2831:
2807:
2799:
2796:
2772:
2764:
2749:
2742:
2727:
2720:
2719:
2718:
2715:
2688:
2681:
2668:
2663:
2645:
2627:
2623:
2613:
2609:
2600:
2595:
2591:
2587:
2583:
2573:
2560:
2553:
2547:
2529:
2511:
2502:
2498:
2494:
2491:
2485:
2475:
2471:
2462:
2457:
2453:
2449:
2445:
2435:
2433:
2417:
2413:
2402:
2398:
2396:
2384:
2379:
2375:
2371:
2358:
2354:
2347:
2345:
2332:
2319:
2304:
2290:
2287:
2284:
2281:
2280:
2277:(supersonic)
2276:
2273:
2270:
2267:
2266:
2263:
2260:
2257:
2254:
2253:
2250:
2247:
2244:
2241:
2240:
2237:
2233:
2230:
2227:
2224:
2223:
2220:
2217:
2214:
2211:
2210:
2207:
2204:
2201:
2198:
2197:
2194:
2191:
2188:
2185:
2184:
2180:
2177:
2174:
2171:
2170:
2166:
2163:
2158:
2155:
2154:
2151:
2129:
2120:
2116:
2092:
2072:
2069:
2068:
2065:
2062:
2059:
2058:
2055:
2052:
2049:
2048:
2044:
2041:
2040:
2036:
2033:
2032:
2028:
2025:
2024:
2020:
2017:
2016:
2013:
2010:
2007:
2006:
2002:
1998:
1995:
1994:
1990:
1987:
1986:
1982:
1979:
1978:
1960:
1956:
1952:
1936:
1933:
1932:
1914:
1910:
1906:
1890:
1887:
1886:
1868:
1864:
1860:
1844:
1841:
1840:
1822:
1818:
1814:
1798:
1795:
1794:
1790:
1785:
1782:
1781:
1778:
1751:
1747:
1723:
1690:
1666:
1646:
1624:
1595:
1566:
1556:
1532:
1522:
1520:
1502:
1498:
1477:
1449:
1419:
1404:
1398:
1397:vortex street
1394:
1391:
1385:
1382:
1378:
1372:
1369:
1362:
1357:
1353:
1351:
1347:
1343:
1339:
1334:
1312:
1182:
1172:
1146:
1122:
1068:
1035:
1031:
1027:
1023:
999:
991:The value of
989:
987:
971:
967:
963:
955:
951:
948:
929:
918:
912:
908:
904:
898:
895:
889:
878:
870:
869:
868:
851:
846:
842:
836:
832:
828:
822:
819:
813:
808:
804:
781:
776:
772:
768:
763:
760:
755:
752:
742:
737:
736:Drag equation
724:
721:is the local
708:
701:
699:of the fluid;
698:
695:is the local
682:
675:
672:
669:is the local
656:
649:
646:
645:flow velocity
642:
639:is the local
626:
619:
618:
617:
614:
595:
591:
587:
582:
579:
572:
566:
563:
557:
546:
537:
535:
529:
519:
515:
511:
509:
505:
501:
497:
493:
491:
486:
470:
466:
462:
459:
437:
433:
429:
426:
423:
411:
395:
388:
385:
369:
362:
360:of the fluid;
359:
343:
336:
333:
332:flow velocity
329:
305:
297:
296:
295:
292:
275:
270:
266:
262:
249:
245:
238:
227:
218:
196:
181:
172:
170:
166:
162:
158:
154:
153:skin friction
150:
149:fluid dynamic
145:
143:
139:
135:
134:drag equation
131:
127:
103:
80:
76:
47:
38:
34:
26:
21:
3656:
3643:Bluff Body:
3634:
3621:Aerodynamics
3620:
3617:L. J. Clancy
3597:. Retrieved
3577:
3566:. Retrieved
3552:
3541:. Retrieved
3531:
3522:
3511:. Retrieved
3497:
3486:. Retrieved
3472:
3461:
3453:
3448:
3440:
3435:
3427:
3422:
3415:Aerodynamics
3414:
3399:Aerodynamics
3398:
3393:
3372:
3355:
3334:
3317:
3307:
3300:
3293:Aerodynamics
3292:
3287:
3276:. Retrieved
3262:
3254:
3246:
3228:Aerodynamics
3227:
3221:
3202:
3196:
3183:
3176:
3156:
3149:
3103:
3088:
3081:
3065:
3048:
3044:
3040:
3038:
2999:
2942:
2939:shear stress
2903:
2866:
2716:
2320:
2316:
2156:
2114:
2083:
2064:Eiffel Tower
1999:Bullet (not
1783:
1748:to 0.47 for
1746:laminar flow
1658:In general,
1657:
1523:
1410:
1390:laminar flow
1370:
1360:
1345:
1335:
1173:
1030:length scale
990:
944:
866:
725:of the fluid
697:mass density
673:of the fluid
641:shear stress
615:
538:
533:
531:
516:
512:
494:
487:
414:
358:mass density
293:
219:
187:
146:
142:hydrodynamic
36:
30:
3130:Drag crisis
2219:Airbus A380
2181:(subsonic)
2037:Ski jumper
1519:Stokes' law
1490:instead of
1441:drag force
1377:Stokes flow
1350:viscosities
1145:Mach number
138:aerodynamic
3667:Categories
3611:References
3599:2014-10-06
3568:2010-12-07
3543:2010-12-07
3513:2013-03-11
3488:2010-12-07
3386:atmosphere
3361:lift force
3348:atmosphere
3323:lift force
3278:2010-12-07
3049:bluff body
2297:Automobile
2262:Boeing 747
2249:Cessna 310
2232:Cessna 172
2206:Boeing 787
2193:Learjet 24
2164:Drag Count
2119:drag count
954:flow speed
730:Background
723:flow speed
526:See also:
384:flow speed
328:drag force
175:Definition
3230:. Wiley.
3020:^
2887:^
2850:^
2682:⏟
2657:^
2646:⋅
2640:^
2610:∫
2588:ρ
2554:⏟
2541:^
2530:⋅
2524:^
2495:−
2472:∫
2450:ρ
2372:ρ
2070:1.98–2.05
1403:turbulent
1346:form drag
1024:. Speed,
1022:viscosity
905:ρ
829:ρ
769:ρ
683:ρ
627:τ
588:ρ
583:τ
564:τ
508:cube root
498:and some
463:π
430:π
344:ρ
263:ρ
165:hydrofoil
157:form drag
3619:(1975):
3590:Archived
3562:Archived
3507:Archived
3505:. NASA.
3482:Archived
3272:Archived
3109:See also
2830:friction
2795:pressure
2080:Aircraft
1988:0.9-1.7
496:Airships
490:airfoils
3095:Oropesa
3084:stalled
3077:airfoil
2828:is the
2793:is the
2717:where:
2313:Concept
2212:0.0265
2060:1.8–2.0
2050:1.3–1.5
2034:1.1-1.3
2026:1.0–1.3
2018:1.0–1.3
2008:1.0–1.1
1654:General
616:where:
506:of the
382:is the
356:is the
326:is the
294:where:
161:airfoil
124:) is a
3627:
3466:Shapes
3234:
3209:
3164:
2282:0.048
2268:0.044
2255:0.031
2242:0.027
2225:0.027
2199:0.024
2186:0.022
2172:0.021
1381:steady
1379:) and
504:square
151:drag:
35:, the
3593:(PDF)
3586:(PDF)
3188:(PDF)
3141:Notes
3045:blunt
2012:Skier
2001:ogive
1996:0.295
1980:0.81
1842:0.005
1796:0.001
1791:Item
1405:wake,
950:force
3625:ISBN
3363:and
3359:See
3325:and
3321:See
3232:ISBN
3207:ISBN
3162:ISBN
3091:tuna
3068:wake
3066:The
2042:1.28
1934:0.47
1861:>
1815:<
1342:wake
947:drag
488:For
410:area
155:and
130:drag
3093:),
3047:or
2285:480
2271:440
2258:310
2245:270
2236:182
2228:270
2215:265
2202:240
2189:220
2175:210
2115:not
1888:0.1
485:).
163:or
140:or
95:or
31:In
3669::
3406:^
3101:.
3079:.
2150:.
1975:)
1957:10
1929:)
1911:10
1883:)
1865:10
1837:)
1819:10
1524:A
1371:Re
1171:.
1060:.
68:,
3602:.
3571:.
3546:.
3516:.
3491:.
3388:.
3350:.
3281:.
3240:.
3215:.
3170:.
3016:i
3002:,
3000:S
2986:p
2961:o
2956:p
2945:,
2943:S
2922:w
2917:T
2904:S
2883:n
2869:,
2867:S
2846:t
2813:f
2808:c
2778:p
2773:c
2750:S
2728:A
2694:f
2689:c
2674:w
2669:T
2664:)
2653:i
2636:t
2628:(
2624:S
2620:d
2614:S
2601:A
2596:2
2592:v
2584:2
2574:+
2566:p
2561:c
2548:)
2537:i
2520:n
2512:(
2508:)
2503:o
2499:p
2492:p
2489:(
2486:S
2482:d
2476:S
2463:A
2458:2
2454:v
2446:2
2436:=
2423:f
2418:c
2414:+
2408:p
2403:c
2399:=
2385:A
2380:2
2376:v
2364:d
2359:F
2355:2
2348:=
2338:d
2333:c
2234:/
2160:d
2157:c
2135:d
2130:c
2098:d
2093:c
1961:6
1953:=
1949:e
1946:R
1915:6
1907:=
1903:e
1900:R
1869:6
1857:e
1854:R
1823:6
1811:e
1808:R
1787:d
1784:c
1764:e
1761:R
1729:d
1724:c
1702:e
1699:R
1672:d
1667:c
1630:d
1625:c
1601:d
1596:c
1572:d
1567:c
1538:d
1533:c
1503:2
1499:v
1478:v
1455:d
1450:F
1425:d
1420:c
1399:,
1386:,
1364:d
1361:C
1318:d
1313:c
1290:e
1287:R
1265:a
1262:M
1240:a
1237:M
1215:e
1212:R
1188:d
1183:c
1158:a
1155:M
1128:d
1123:c
1101:e
1098:R
1074:d
1069:c
1047:e
1044:R
1005:d
1000:c
972:2
968:/
964:1
930:A
924:d
919:c
913:2
909:u
899:2
896:1
890:=
884:d
879:F
852:A
847:d
843:c
837:2
833:u
823:2
820:1
814:=
809:d
805:F
782:A
777:2
773:u
764:2
761:1
756:=
753:F
709:u
657:q
647:;
596:2
592:u
580:2
573:=
567:q
558:=
552:d
547:c
471:2
467:r
460:4
438:2
434:r
427:=
424:A
396:A
370:u
334:;
311:d
306:F
276:A
271:2
267:u
255:d
250:F
246:2
239:=
233:d
228:c
202:d
197:c
109:w
104:c
81:x
77:c
53:d
48:c
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