4656:, due to conservation of vorticity or Kelvin Theorem of Circulation Conservation. These streamwise vortices merge to two counter-rotating strong spirals separated by distance close to the wingspan and their cores may be visible if relative humidity is high. Treating the trailing vortices as a series of semi-infinite straight line vortices leads to the well-known lifting line theory. By this theory, the wing has a lift force smaller than that predicted by a purely two-dimensional theory using the Kutta–Joukowski theorem. This is due to the upstream effects of the trailing vortices' added downwash on the angle of attack of the wing. This reduces the wing's effective angle of attack, decreasing the amount of lift produced at a given angle of attack and requiring a higher angle of attack to recover this lost lift. At this new higher angle of attack, drag has also increased. Induced drag effectively reduces the slope of the lift curve of a 2-D airfoil and increases the angle of attack of
4615:
lift when they are convected to the trailing edge, inducing a new trailing edge vortex spiral moving in the lift decreasing direction. For this type of flow a vortex force line (VFL) map can be used to understand the effect of the different vortices in a variety of situations (including more situations than starting flow) and may be used to improve vortex control to enhance or reduce the lift. The vortex force line map is a two dimensional map on which vortex force lines are displayed. For a vortex at any point in the flow, its lift contribution is proportional to its speed, its circulation and the cosine of the angle between the streamline and the vortex force line. Hence the vortex force line map clearly shows whether a given vortex is lift producing or lift detrimental.
4636:
the distance between the vortex pair in production. With this approach, an explicit and algebraic force formula, taking into account of all causes (inner singularities, outside vortices and bodies, motion of all singularities and bodies, and vortex production) holds individually for each body with the role of other bodies represented by additional singularities. Hence a force decomposition according to bodies is possible.
1315:
25:
3775:
1039:
4629:
induced velocity at these singularities by all causes except those inside this body. The contribution due to each inner singularity sums up to give the total force. The motion of outside singularities also contributes to forces, and the force component due to this contribution is proportional to the speed of the singularity.
4454:
3430:
3981:
4635:
When in addition to multiple free vortices and multiple bodies, there are bound vortices and vortex production on the body surface, the generalized
Lagally theorem still holds, but a force due to vortex production exists. This vortex production force is proportional to the vortex production rate and
4614:
If, as for a flat plate, the leading edge is also sharp, then vortices also shed at the leading edge and the role of leading edge vortices is two-fold: 1) they are lift increasing when they are still close to the leading edge, so that they elevate the Wagner lift curve, and 2) they are detrimental to
4642:
For general three-dimensional, viscous and unsteady flow, force formulas are expressed in integral forms. The volume integration of certain flow quantities, such as vorticity moments, is related to forces. Various forms of integral approach are now available for unbounded domain and for artificially
4580:
The lift predicted by the Kutta-Joukowski theorem within the framework of inviscid potential flow theory is quite accurate, even for real viscous flow, provided the flow is steady and unseparated. In deriving the Kutta–Joukowski theorem, the assumption of irrotational flow was used. When there are
2731:
4621:
When a (mass) source is fixed outside the body, a force correction due to this source can be expressed as the product of the strength of outside source and the induced velocity at this source by all the causes except this source. This is known as the
Lagally theorem. For two-dimensional inviscid
4628:
For free vortices and other bodies outside one body without bound vorticity and without vortex production, a generalized
Lagally theorem holds, with which the forces are expressed as the products of strength of inner singularities (image vortices, sources and doublets inside each body) and the
2254:
2066:
4607:
When the angle of attack is high enough, the trailing edge vortex sheet is initially in a spiral shape and the lift is singular (infinitely large) at the initial time. The lift drops for a very short time period before the usually assumed monotonically increasing lift curve is
1310:{\displaystyle {\begin{aligned}{\frac {\rho }{2}}(V)^{2}+(P+\Delta P)&={\frac {\rho }{2}}(V+v)^{2}+P,\,\\{\frac {\rho }{2}}(V)^{2}+\Delta P&={\frac {\rho }{2}}(V^{2}+2Vv+v^{2}),\,\\\Delta P&=\rho Vv\qquad {\text{(ignoring }}{\frac {\rho }{2}}v^{2}),\,\end{aligned}}}
4569:
4581:
free vortices outside of the body, as may be the case for a large number of unsteady flows, the flow is rotational. When the flow is rotational, more complicated theories should be used to derive the lift forces. Below are several important examples.
4215:
797:
region outside. In applying the Kutta-Joukowski theorem, the loop must be chosen outside this boundary layer. (For example, the circulation calculated using the loop corresponding to the surface of the airfoil would be zero for a viscous fluid.)
205:
encircling the airfoil. At a large distance from the airfoil, the rotating flow may be regarded as induced by a line vortex (with the rotating line perpendicular to the two-dimensional plane). In the derivation of the Kutta–Joukowski theorem the
3770:{\displaystyle {\begin{aligned}\oint _{C}w'(z)\,dz&=\oint _{C}(v_{x}-iv_{y})(dx+idy)\\&=\oint _{C}(v_{x}\,dx+v_{y}\,dy)+i\oint _{C}(v_{x}\,dy-v_{y}\,dx)\\&=\oint _{C}\mathbf {v} \,{ds}+i\oint _{C}(v_{x}\,dy-v_{y}\,dx).\end{aligned}}}
801:
The sharp trailing edge requirement corresponds physically to a flow in which the fluid moving along the lower and upper surfaces of the airfoil meet smoothly, with no fluid moving around the trailing edge of the airfoil. This is known as the
3813:
1722:
3214:
2534:
486:
4204:
787:
2085:
3070:
1844:
1879:
4596:
as function of time is given by the Wagner function. In this case the initial lift is one half of the final lift given by the Kutta–Joukowski formula. The lift attains 90% of its steady state value when the
3419:
311:
1389:
184:
relates side force (called Magnus force) to rotation. However, the circulation here is not induced by rotation of the airfoil. The fluid flow in the presence of the airfoil can be considered to be the
1525:
2330:
4465:
4470:
3435:
2090:
1044:
3308:
2503:
667:
4592:
continuously shed at the trailing edge and the lift force is unsteady or time-dependent. For small angle of attack starting flow, the vortex sheet follows a planar path, and the curve of the
4097:
2848:
4622:
flow, the classical Kutta
Joukowski theorem predicts a zero drag. When, however, there is vortex outside the body, there is a vortex induced drag, in a form similar to the induced lift.
2985:
4652:
A wing has a finite span, and the circulation at any section of the wing varies with the spanwise direction. This variation is compensated by the release of streamwise vortices, called
1446:
First of all, the force exerted on each unit length of a cylinder of arbitrary cross section is calculated. Let this force per unit length (from now on referred to simply as force) be
1001:
2911:
4449:{\displaystyle {\bar {F}}={\frac {i\rho }{2}}\left=i\rho a_{0}\Gamma =i\rho \Gamma (v_{x\infty }-iv_{y\infty })=\rho \Gamma v_{y\infty }+i\rho \Gamma v_{x\infty }=F_{x}-iF_{y}.}
1422:
2339:
is used, in order to remove the pressure from the integral. Throughout the analysis it is assumed that there is no outer force field present. The mass density of the flow is
354:
2429:
1579:
4722:
4688:
546:
4643:
truncated domain. The Kutta
Joukowski theorem can be recovered from these approaches when applied to a two-dimensional airfoil and when the flow is steady and unseparated.
1466:
697:
381:
4043:
3976:{\displaystyle \oint _{C}(v_{x}\,dy-v_{y}\,dx)=\oint _{C}\left({\frac {\partial \psi }{\partial y}}dy+{\frac {\partial \psi }{\partial x}}dx\right)=\oint _{C}d\psi =0.}
250:
3805:
1027:
4005:
3336:
3245:
2772:
401:
2360:
1735:, with its first component equal to the real part and its second component equal to the imaginary part of the complex number. Then, the force can be represented as:
1607:
886:
813:
and small angle of attack, the flow can be assumed inviscid in the entire region outside the airfoil provided the Kutta condition is imposed. This is known as the
932:
589:
3108:
2526:
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1871:
1615:
1552:
906:
866:
609:
566:
509:
42:
4938:
2726:{\displaystyle {\bar {F}}=-ip_{0}\oint _{C}d{\bar {z}}+i{\frac {\rho }{2}}\oint _{C}|v|^{2}\,d{\bar {z}}={\frac {i\rho }{2}}\oint _{C}|v|^{2}\,d{\bar {z}}.}
5219:
Howe, M. S. (1995). "On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high
Reynolds numbers".
412:
2249:{\displaystyle {\begin{aligned}dz&=dx+idy=ds(\cos \phi +i\sin \phi )=ds\,e^{i\phi }\\{}\Rightarrow d{\bar {z}}&=e^{-i\phi }ds.\end{aligned}}}
3116:
732:
4108:
5313:
1741:
89:
61:
2061:{\displaystyle {\bar {F}}=-\oint _{C}p(\sin \phi +i\cos \phi )\,ds=-i\oint _{C}p(\cos \phi -i\sin \phi )\,ds=-i\oint _{C}pe^{-i\phi }\,ds.}
722:, the angle between the chord line and the fluid flow far upstream of the airfoil. Moreover, the airfoil must have a sharp trailing edge.
2993:
68:
4984:
4880:
4855:
4762:
260:
108:
75:
4821:
4588:
For an impulsively started flow such as obtained by suddenly accelerating an airfoil or setting an angle of attack, there is a
3348:
1474:
511:
enclosing the airfoil and followed in the negative (clockwise) direction. As explained below, this path must be in a region of
5043:
Li, J.; Wu, Z. N. (2015). "Unsteady lift for the Wagner problem in the presence of additional leading trailing edge vortices".
4564:{\displaystyle {\begin{aligned}F_{x}&=\rho \Gamma v_{y\infty }\,,&F_{y}&=-\rho \Gamma v_{x\infty }.\end{aligned}}}
5308:
46:
2265:
1326:
57:
5328:
166:
3781:
3250:
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725:
Any real fluid is viscous, which implies that the fluid velocity vanishes on the airfoil. Prandtl showed that for large
150:
5103:"Generalized two-dimensional Lagally theorem with free vortices and its application to fluid-body interaction problems"
632:
5153:"Generalized Kutta-Joukowski theorem for multi-vortex and multi-airfoil flow with vortex production — A general model"
4051:
210:
is usually mapped onto a circular cylinder. In many textbooks, the theorem is proved for a circular cylinder and the
4778:
Liu, L. Q.; Zhu, J. Y.; Wu, J. Z. (2015). "Lift and drag in two-dimensional steady viscous and compressible flow".
35:
5323:
4209:
Plugging this back into the
Blasius–Chaplygin formula, and performing the integration using the residue theorem:
2916:
2781:
3082:
To arrive at the
Joukowski formula, this integral has to be evaluated. From complex analysis it is known that a
789:, and small angle of attack, the flow around a thin airfoil is composed of a narrow viscous region called the
2336:
940:
82:
5318:
5246:
Wu, J. Z.; Lu, X. Y.; Zhuang, L. X. (2007). "Integral force acting on a body due to local flow structures".
1427:
1030:
186:
145:. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the
2861:
169:(or Joukowski) who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an
809:
Kutta and
Joukowski showed that for computing the pressure and lift of a thin airfoil for flow at large
190:
1397:
5255:
5193:
5117:
5052:
5009:
4946:
4787:
3083:
1609:
be the angle between the normal vector and the vertical. Then the components of the above force are:
332:
2385:
1561:
4693:
4659:
4646:
1431:
522:
3219:
The function does not contain higher order terms, since the velocity stays finite at infinity. So
1449:
672:
5271:
5133:
5068:
5025:
4803:
629:
The force per unit length acting on a right cylinder of any cross section whatsoever is equal to
359:
4918:. McGraw-Hill Series in Aeronautical and Aerospace Engineering. New York: McGraw-Hill Education.
4018:
829:
argument, based on physical insight. The second is a formal and technical one, requiring basic
5102:
4980:
4929:
4876:
4851:
4758:
4653:
4015:
border of the cylinder is a streamline itself, the stream function does not change on it, and
3787:
2775:
1850:
1009:
211:
3989:
3313:
3222:
3090:. From the physics of the problem it is deduced that the derivative of the complex potential
2739:
386:
222:
The theorem applies to two-dimensional flow around a fixed airfoil (or any shape of infinite
189:
of a translational flow and a rotating flow. This rotating flow is induced by the effects of
5263:
5228:
5201:
5164:
5125:
5060:
5017:
4954:
4896:
4795:
4734:
4593:
2342:
846:
834:
1592:
871:
4008:
3339:
3076:
1717:{\displaystyle F_{x}=-\oint _{C}p\sin \phi \,ds\,,\qquad F_{y}=\oint _{C}p\cos \phi \,ds.}
1555:
830:
810:
803:
726:
719:
709:
229:
194:
142:
911:
157:
around a closed loop enclosing the airfoil of the component of the velocity of the fluid
5259:
5197:
5121:
5056:
5013:
4950:
4791:
571:
3093:
3087:
2511:
2365:
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1537:
891:
851:
814:
790:
594:
551:
516:
512:
494:
141:
at a constant speed so large that the flow seen in the body-fixed frame is steady and
5302:
5275:
5137:
5072:
5029:
4825:
4807:
1728:
794:
404:
198:
181:
170:
154:
137:(and any two-dimensional body including circular cylinders) translating in a uniform
4589:
548:
is the component of the local fluid velocity in the direction tangent to the curve
162:
130:
126:
2850:
where the apostrophe denotes differentiation with respect to the complex variable
481:{\displaystyle \Gamma =\oint _{C}V\cdot d\mathbf {s} =\oint _{C}V\cos \theta \,ds}
1430:
version of this theorem applies on each element of the plate and is the basis of
1582:
24:
5267:
5184:
Wu, J. C. (1981). "Theory for aerodynamic force and moment in viscous flows".
5169:
5152:
5021:
3209:{\displaystyle w'(z)=a_{0}+{\frac {a_{1}}{z}}+{\frac {a_{2}}{z^{2}}}+\cdots .}
782:{\displaystyle {\mathord {\text{Re}}}={\frac {\rho V_{\infty }c_{A}}{\mu }}\,}
383:
are the fluid density and the fluid velocity far upstream of the airfoil, and
5232:
4958:
4753:
Anderson, J. D. Jr. (1989). "Pressure, Temperature, and
Density Altitudes".
826:
223:
16:
Formula relating lift on an airfoil to fluid speed, density, and circulation
4199:{\displaystyle w'^{2}(z)=a_{0}^{2}+{\frac {a_{0}\Gamma }{\pi iz}}+\cdots .}
888:. Let the airfoil be inclined to the oncoming flow to produce an air speed
5064:
4799:
4611:
Starting flow at large angle of attack for wings with sharp leading edges
5129:
1839:{\displaystyle F=F_{x}+iF_{y}=-\oint _{C}p(\sin \phi -i\cos \phi )\,ds.}
1727:
Now comes a crucial step: consider the used two-dimensional space as a
715:
207:
202:
174:
158:
146:
134:
5000:
Graham, J. M. R. (1983). "The lift on an aerofoil in starting flow".
4933:
5205:
3065:{\displaystyle {\bar {F}}={\frac {i\rho }{2}}\oint _{C}w'^{2}\,dz,}
1589:
is the arc element of the borderline of the cross section. Now let
624:
Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:
180:
Kutta–Joukowski theorem relates lift to circulation much like the
138:
4934:"Über die Entstehung des dynamischen Auftriebes von Tragflügeln"
4598:
4632:
Individual force of each body for multiple-body rotational flow
201:
of the airfoil. It should not be confused with a vortex like a
3807:
The second integral can be evaluated after some manipulation:
1029:
between the two sides of the airfoil can be found by applying
306:{\displaystyle L^{\prime }=\rho _{\infty }V_{\infty }\Gamma ,}
18:
3247:
represents the derivative the complex potential at infinity:
269:
2778:
of the flow. This is related to the velocity components as
3414:{\displaystyle a_{1}={\frac {1}{2\pi i}}\oint _{C}w'\,dz.}
1520:{\displaystyle \mathbf {F} =-\oint _{C}p\mathbf {n} \,ds,}
4757:(3rd ed.). New York: McGraw-Hill. pp. 100–103.
4824:. NASA Glenn Research Center. 2010-11-09. Archived from
5221:
Quarterly Journal of Mechanics and Applied Mathematics
2987:
and the desired expression for the force is obtained:
2325:{\displaystyle {\bar {F}}=-i\oint _{C}p\,d{\bar {z}}.}
1384:{\displaystyle L'=c\Delta P=\rho Vvc=-\rho V\Gamma \,}
153:
around the airfoil. The circulation is defined as the
4696:
4662:
4601:
has traveled a distance of about seven chord lengths.
4468:
4218:
4111:
4054:
4021:
3992:
3816:
3790:
3433:
3351:
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3253:
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2919:
2864:
2784:
2742:
2537:
2514:
2440:
2388:
2368:
2345:
2268:
2259:
Plugging this back into the integral, the result is:
2088:
1882:
1859:
1744:
1618:
1595:
1564:
1540:
1477:
1452:
1400:
1329:
1042:
1012:
943:
914:
894:
874:
854:
845:
For a heuristic argument, consider a thin airfoil of
735:
675:
635:
597:
574:
554:
525:
497:
415:
389:
362:
335:
263:
232:
1320:
so the downward force on the air, per unit span, is
825:
Two derivations are presented below. The first is a
49:. Unsourced material may be challenged and removed.
4716:
4682:
4563:
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4198:
4091:
4037:
3999:
3975:
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3769:
3413:
3330:
3303:{\displaystyle a_{0}=v_{x\infty }-iv_{y\infty }\,}
3302:
3239:
3208:
3102:
3064:
2979:
2905:
2842:
2766:
2725:
2520:
2498:{\displaystyle p=p_{0}-{\frac {\rho |v|^{2}}{2}}.}
2497:
2423:
2374:
2354:
2324:
2248:
2060:
1865:
1838:
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244:
4604:Impulsively started flow at large angle of attack
4585:Impulsively started flow at small angle of attack
4045:. Hence the above integral is zero. As a result:
868:and infinite span, moving through air of density
662:{\displaystyle \rho _{\infty }V_{\infty }\Gamma }
4875:. New York: John Wiley & Sons. Section 4.9.
4707:
4673:
4092:{\displaystyle a_{1}={\frac {\Gamma }{2\pi i}}.}
3310:. The next task is to find out the meaning of
5090:. Hong Kong: Macmillan Education. p. 226.
1394:and the upward force (lift) on the airfoil is
817:theory and works remarkably well in practice.
5101:Wu, C. T.; Yang, F. L.; Young, D. L. (2012).
4649:for wings, wing-tip vortices and induced drag
908:on one side of the airfoil, and an air speed
8:
4970:
4968:
2854:. The velocity is tangent to the borderline
1443:Formal derivation of Kutta–Joukowski theorem
718:either has camber or operates at a positive
2980:{\displaystyle v^{2}d{\bar {z}}=|v|^{2}dz,}
2843:{\displaystyle w'=v_{x}-iv_{y}={\bar {v}},}
934:on the other side. The circulation is then
177:flow in typical aerodynamic applications.
4903:. Cambridge University Press. p. 406.
1731:. So every vector can be represented as a
214:, but it holds true for general airfoils.
173:, but it is a good approximation for real
5168:
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2003:
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669:and is perpendicular to the direction of
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109:Learn how and when to remove this message
4979:. New York: Cambridge University Press.
3780:The first integral is recognized as the
1534:denotes the borderline of the cylinder,
996:{\displaystyle \Gamma =Vc-(V+v)c=-vc.\,}
591:is an infinitesimal length on the curve
4871:Kuethe, A. M.; Schetzer, J. D. (1959).
4745:
4576:Lift forces for more complex situations
4459:And so the Kutta–Joukowski formula is:
4117:
3042:
2736:Only one step is left to do: introduce
5151:Bai, C. Y.; Li, J.; Wu, Z. N. (2014).
4639:General three-dimensional viscous flow
7:
4690:(while also decreasing the value of
2906:{\displaystyle v=\pm |v|e^{i\phi }.}
254:
47:adding citations to reliable sources
3424:Now perform the above integration:
704:Circulation and the Kutta condition
5295:, Dover Publications Inc, New York
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4398:
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4326:
4314:
4276:
4170:
4070:
3928:
3920:
3899:
3891:
3791:
3294:
3275:
1407:
1377:
1344:
1248:
1176:
1085:
1013:
944:
757:
681:
656:
651:
641:
416:
403:is the circulation defined as the
390:
368:
341:
297:
292:
282:
14:
4901:An Introduction to Fluid Dynamics
3688:
1566:
1503:
1479:
1454:
1417:{\displaystyle \rho V\Gamma .\,}
442:
23:
4850:. London: Pitman. Section 4.5.
4102:Take the square of the series:
1668:
1270:
519:of the cylinder. The integrand
349:{\displaystyle \rho _{\infty }}
161:to the loop. It is named after
34:needs additional citations for
5157:Chinese Journal of Aeronautics
4364:
4329:
4225:
4133:
4127:
3867:
3827:
3757:
3717:
3664:
3624:
3605:
3565:
3542:
3521:
3518:
3489:
3462:
3456:
3134:
3128:
3003:
2958:
2949:
2939:
2883:
2875:
2831:
2758:
2752:
2714:
2694:
2685:
2650:
2630:
2621:
2588:
2544:
2476:
2467:
2424:{\displaystyle v=v_{x}+iv_{y}}
2313:
2275:
2204:
2192:
2160:
2133:
2000:
1973:
1941:
1914:
1889:
1823:
1796:
1574:{\displaystyle \mathbf {n} \,}
1468:. So then the total force is:
1296:
1237:
1199:
1164:
1157:
1124:
1111:
1091:
1076:
1064:
1057:
971:
959:
1:
5314:Eponymous theorems of physics
5086:Milne-Thomson, L. M. (1968).
4717:{\displaystyle C_{L_{\max }}}
4683:{\displaystyle C_{L_{\max }}}
1849:The next step is to take the
541:{\displaystyle V\cos \theta }
4916:Fundamentals of Aerodynamics
4822:"Lift on rotating cylinders"
1585:normal to the cylinder, and
1461:{\displaystyle \mathbf {F} }
692:{\displaystyle V_{\infty }.}
129:used for the calculation of
125:is a fundamental theorem in
5291:Milne-Thomson, L.M. (1973)
4873:Foundations of Aerodynamics
4625:Generalized Lagally theorem
1006:The difference in pressure
613:
376:{\displaystyle V_{\infty }}
252:of the airfoil is given by
226:). The lift per unit span
5345:
5248:Journal of Fluid Mechanics
5110:Journal of Fluid Mechanics
5045:Journal of Fluid Mechanics
5002:Journal of Fluid Mechanics
4780:Journal of Fluid Mechanics
4038:{\displaystyle d\psi =0\,}
1873:and do some manipulation:
707:
5268:10.1017/S0022112006004551
5170:10.1016/j.cja.2014.03.014
5088:Theoretical Hydrodynamics
5022:10.1017/S0022112083001986
58:"Kutta–Joukowski theorem"
5293:Theoretical Aerodynamics
4959:10.1002/zamm.19250050103
3800:{\displaystyle \Gamma .}
1022:{\displaystyle \Delta P}
491:around a closed contour
4975:Saffman, P. G. (1992).
4000:{\displaystyle \psi \,}
3331:{\displaystyle a_{1}\,}
3240:{\displaystyle a_{0}\,}
2767:{\displaystyle w=f(z),}
2382:is related to velocity
2075:are related to changes
619:Kutta–Joukowski theorem
396:{\displaystyle \Gamma }
123:Kutta–Joukowski theorem
5233:10.1093/qjmam/48.3.401
4846:Clancy, L. J. (1975).
4755:Introduction to Flight
4718:
4684:
4565:
4450:
4200:
4093:
4039:
4001:
3977:
3801:
3771:
3415:
3332:
3304:
3241:
3210:
3104:
3086:can be presented as a
3066:
2981:
2907:
2844:
2768:
2727:
2522:
2499:
2425:
2376:
2356:
2355:{\displaystyle \rho .}
2326:
2250:
2062:
1867:
1840:
1718:
1603:
1575:
1548:
1521:
1462:
1418:
1385:
1311:
1023:
997:
928:
902:
882:
862:
783:
693:
663:
605:
585:
562:
542:
505:
482:
397:
377:
350:
307:
246:
5309:Aircraft aerodynamics
4939:Z. Angew. Math. Mech.
4914:Anderson, J. (2010).
4719:
4685:
4566:
4451:
4201:
4094:
4040:
4002:
3978:
3802:
3772:
3416:
3342:on the above series:
3333:
3305:
3242:
3211:
3105:
3067:
2982:
2908:
2858:, so this means that
2845:
2769:
2728:
2523:
2500:
2426:
2377:
2357:
2327:
2251:
2063:
1868:
1841:
1719:
1604:
1602:{\displaystyle \phi }
1576:
1549:
1522:
1463:
1419:
1386:
1312:
1024:
998:
929:
903:
883:
881:{\displaystyle \rho }
863:
793:near the body and an
784:
694:
664:
606:
586:
563:
543:
506:
483:
398:
378:
351:
308:
247:
149:of the fluid and the
5329:Aircraft wing design
5065:10.1017/jfm.2015.118
4800:10.1017/jfm.2015.584
4694:
4660:
4466:
4216:
4109:
4052:
4019:
3990:
3814:
3788:
3431:
3349:
3314:
3251:
3223:
3117:
3094:
3084:holomorphic function
3075:which is called the
2994:
2917:
2862:
2782:
2740:
2535:
2512:
2508:With this the force
2438:
2386:
2366:
2343:
2266:
2086:
1880:
1857:
1742:
1616:
1593:
1562:
1538:
1475:
1450:
1398:
1327:
1040:
1031:Bernoulli's equation
1010:
941:
912:
892:
872:
852:
733:
673:
633:
595:
572:
552:
523:
495:
413:
387:
360:
333:
261:
245:{\displaystyle L'\,}
230:
43:improve this article
5260:2007JFM...576..265W
5198:1981AIAAJ..19..432W
5130:10.1017/jfm.2012.45
5122:2012JFM...698...73W
5057:2015JFM...769..182L
5014:1983JFM...133..413G
4951:1925ZaMM....5...17W
4792:2015JFM...784..304L
4647:Lifting line theory
4153:
1432:thin-airfoil theory
927:{\displaystyle V+v}
4714:
4680:
4561:
4559:
4446:
4196:
4139:
4089:
4035:
3997:
3973:
3797:
3767:
3765:
3411:
3328:
3300:
3237:
3206:
3100:
3062:
2977:
2903:
2840:
2764:
2723:
2518:
2495:
2421:
2372:
2352:
2337:Bernoulli equation
2322:
2246:
2244:
2058:
1863:
1836:
1714:
1599:
1571:
1544:
1517:
1458:
1444:
1414:
1381:
1307:
1305:
1019:
993:
924:
898:
878:
858:
841:Heuristic argument
779:
689:
659:
601:
584:{\displaystyle ds}
581:
558:
538:
501:
478:
393:
373:
346:
303:
242:
218:Lift force formula
4654:trailing vortices
4288:
4247:
4228:
4185:
4084:
3935:
3906:
3381:
3195:
3168:
3103:{\displaystyle w}
3025:
3006:
2942:
2834:
2776:complex potential
2717:
2672:
2653:
2608:
2591:
2547:
2521:{\displaystyle F}
2490:
2375:{\displaystyle p}
2316:
2278:
2207:
2071:Surface segments
1892:
1866:{\displaystyle F}
1851:complex conjugate
1547:{\displaystyle p}
1442:
1438:Formal derivation
1284:
1274:
1197:
1155:
1109:
1055:
901:{\displaystyle V}
861:{\displaystyle c}
776:
740:
714:A lift-producing
617:is a form of the
604:{\displaystyle C}
561:{\displaystyle C}
504:{\displaystyle C}
327:
326:
212:Joukowski airfoil
167:Nikolai Zhukovsky
119:
118:
111:
93:
5336:
5324:Physics theorems
5280:
5279:
5243:
5237:
5236:
5216:
5210:
5209:
5181:
5175:
5174:
5172:
5163:(5): 1037–1050.
5148:
5142:
5141:
5107:
5098:
5092:
5091:
5083:
5077:
5076:
5040:
5034:
5033:
4997:
4991:
4990:
4972:
4963:
4962:
4926:
4920:
4919:
4911:
4905:
4904:
4897:Batchelor, G. K.
4893:
4887:
4886:
4868:
4862:
4861:
4843:
4837:
4836:
4834:
4833:
4818:
4812:
4811:
4775:
4769:
4768:
4750:
4735:Horseshoe vortex
4723:
4721:
4720:
4715:
4713:
4712:
4711:
4710:
4689:
4687:
4686:
4681:
4679:
4678:
4677:
4676:
4594:lift coefficient
4570:
4568:
4567:
4562:
4560:
4553:
4552:
4524:
4523:
4508:
4507:
4482:
4481:
4455:
4453:
4452:
4447:
4442:
4441:
4426:
4425:
4413:
4412:
4388:
4387:
4363:
4362:
4344:
4343:
4313:
4312:
4294:
4290:
4289:
4287:
4279:
4275:
4274:
4264:
4248:
4243:
4235:
4230:
4229:
4221:
4205:
4203:
4202:
4197:
4186:
4184:
4173:
4169:
4168:
4158:
4152:
4147:
4126:
4125:
4124:
4098:
4096:
4095:
4090:
4085:
4083:
4069:
4064:
4063:
4044:
4042:
4041:
4036:
4006:
4004:
4003:
3998:
3982:
3980:
3979:
3974:
3960:
3959:
3947:
3943:
3936:
3934:
3926:
3918:
3907:
3905:
3897:
3889:
3882:
3881:
3859:
3858:
3839:
3838:
3826:
3825:
3806:
3804:
3803:
3798:
3776:
3774:
3773:
3768:
3766:
3749:
3748:
3729:
3728:
3716:
3715:
3700:
3691:
3686:
3685:
3670:
3656:
3655:
3636:
3635:
3623:
3622:
3597:
3596:
3577:
3576:
3564:
3563:
3548:
3517:
3516:
3501:
3500:
3488:
3487:
3455:
3447:
3446:
3420:
3418:
3417:
3412:
3400:
3392:
3391:
3382:
3380:
3366:
3361:
3360:
3337:
3335:
3334:
3329:
3326:
3325:
3309:
3307:
3306:
3301:
3298:
3297:
3279:
3278:
3263:
3262:
3246:
3244:
3243:
3238:
3235:
3234:
3215:
3213:
3212:
3207:
3196:
3194:
3193:
3184:
3183:
3174:
3169:
3164:
3163:
3154:
3149:
3148:
3127:
3110:will look thus:
3109:
3107:
3106:
3101:
3071:
3069:
3068:
3063:
3051:
3050:
3049:
3036:
3035:
3026:
3021:
3013:
3008:
3007:
2999:
2986:
2984:
2983:
2978:
2967:
2966:
2961:
2952:
2944:
2943:
2935:
2929:
2928:
2912:
2910:
2909:
2904:
2899:
2898:
2886:
2878:
2849:
2847:
2846:
2841:
2836:
2835:
2827:
2821:
2820:
2805:
2804:
2792:
2773:
2771:
2770:
2765:
2732:
2730:
2729:
2724:
2719:
2718:
2710:
2703:
2702:
2697:
2688:
2683:
2682:
2673:
2668:
2660:
2655:
2654:
2646:
2639:
2638:
2633:
2624:
2619:
2618:
2609:
2601:
2593:
2592:
2584:
2578:
2577:
2568:
2567:
2549:
2548:
2540:
2527:
2525:
2524:
2519:
2504:
2502:
2501:
2496:
2491:
2486:
2485:
2484:
2479:
2470:
2461:
2456:
2455:
2430:
2428:
2427:
2422:
2420:
2419:
2404:
2403:
2381:
2379:
2378:
2373:
2361:
2359:
2358:
2353:
2331:
2329:
2328:
2323:
2318:
2317:
2309:
2299:
2298:
2280:
2279:
2271:
2255:
2253:
2252:
2247:
2245:
2232:
2231:
2209:
2208:
2200:
2191:
2185:
2184:
2067:
2065:
2064:
2059:
2047:
2046:
2028:
2027:
1969:
1968:
1910:
1909:
1894:
1893:
1885:
1872:
1870:
1869:
1864:
1845:
1843:
1842:
1837:
1792:
1791:
1776:
1775:
1760:
1759:
1723:
1721:
1720:
1715:
1691:
1690:
1678:
1677:
1644:
1643:
1628:
1627:
1608:
1606:
1605:
1600:
1580:
1578:
1577:
1572:
1569:
1553:
1551:
1550:
1545:
1526:
1524:
1523:
1518:
1506:
1498:
1497:
1482:
1467:
1465:
1464:
1459:
1457:
1423:
1421:
1420:
1415:
1390:
1388:
1387:
1382:
1337:
1316:
1314:
1313:
1308:
1306:
1295:
1294:
1285:
1277:
1275:
1272:
1236:
1235:
1211:
1210:
1198:
1190:
1172:
1171:
1156:
1148:
1132:
1131:
1110:
1102:
1072:
1071:
1056:
1048:
1028:
1026:
1025:
1020:
1002:
1000:
999:
994:
933:
931:
930:
925:
907:
905:
904:
899:
887:
885:
884:
879:
867:
865:
864:
859:
835:complex analysis
788:
786:
785:
780:
777:
772:
771:
770:
761:
760:
747:
742:
741:
738:
698:
696:
695:
690:
685:
684:
668:
666:
665:
660:
655:
654:
645:
644:
610:
608:
607:
602:
590:
588:
587:
582:
567:
565:
564:
559:
547:
545:
544:
539:
510:
508:
507:
502:
487:
485:
484:
479:
458:
457:
445:
431:
430:
402:
400:
399:
394:
382:
380:
379:
374:
372:
371:
355:
353:
352:
347:
345:
344:
321:
312:
310:
309:
304:
296:
295:
286:
285:
273:
272:
255:
251:
249:
248:
243:
240:
114:
107:
103:
100:
94:
92:
51:
27:
19:
5344:
5343:
5339:
5338:
5337:
5335:
5334:
5333:
5299:
5298:
5288:
5283:
5245:
5244:
5240:
5218:
5217:
5213:
5206:10.2514/3.50966
5183:
5182:
5178:
5150:
5149:
5145:
5105:
5100:
5099:
5095:
5085:
5084:
5080:
5042:
5041:
5037:
4999:
4998:
4994:
4987:
4977:Vortex Dynamics
4974:
4973:
4966:
4928:
4927:
4923:
4913:
4912:
4908:
4895:
4894:
4890:
4883:
4870:
4869:
4865:
4858:
4845:
4844:
4840:
4831:
4829:
4820:
4819:
4815:
4777:
4776:
4772:
4765:
4752:
4751:
4747:
4743:
4731:
4702:
4697:
4692:
4691:
4668:
4663:
4658:
4657:
4618:Lagally theorem
4578:
4573:
4558:
4557:
4541:
4525:
4515:
4513:
4496:
4483:
4473:
4464:
4463:
4433:
4417:
4401:
4376:
4351:
4332:
4304:
4280:
4266:
4265:
4253:
4249:
4236:
4214:
4213:
4174:
4160:
4159:
4116:
4112:
4107:
4106:
4073:
4055:
4050:
4049:
4017:
4016:
4009:stream function
3988:
3987:
3951:
3927:
3919:
3898:
3890:
3887:
3883:
3873:
3850:
3830:
3817:
3812:
3811:
3786:
3785:
3764:
3763:
3740:
3720:
3707:
3677:
3668:
3667:
3647:
3627:
3614:
3588:
3568:
3555:
3546:
3545:
3508:
3492:
3479:
3472:
3448:
3438:
3429:
3428:
3393:
3383:
3370:
3352:
3347:
3346:
3340:residue theorem
3317:
3312:
3311:
3286:
3267:
3254:
3249:
3248:
3226:
3221:
3220:
3185:
3175:
3155:
3140:
3120:
3115:
3114:
3092:
3091:
3077:Blasius theorem
3041:
3037:
3027:
3014:
2992:
2991:
2956:
2920:
2915:
2914:
2887:
2860:
2859:
2812:
2796:
2785:
2780:
2779:
2738:
2737:
2692:
2674:
2661:
2628:
2610:
2569:
2559:
2533:
2532:
2510:
2509:
2474:
2462:
2447:
2436:
2435:
2411:
2395:
2384:
2383:
2364:
2363:
2341:
2340:
2290:
2264:
2263:
2243:
2242:
2217:
2210:
2187:
2186:
2173:
2099:
2084:
2083:
2079:along them by:
2032:
2019:
1960:
1901:
1878:
1877:
1855:
1854:
1783:
1767:
1751:
1740:
1739:
1682:
1669:
1635:
1619:
1614:
1613:
1591:
1590:
1560:
1559:
1556:static pressure
1536:
1535:
1489:
1473:
1472:
1448:
1447:
1440:
1396:
1395:
1330:
1325:
1324:
1304:
1303:
1286:
1273:(ignoring
1254:
1245:
1244:
1227:
1202:
1182:
1163:
1144:
1143:
1123:
1094:
1063:
1038:
1037:
1008:
1007:
939:
938:
910:
909:
890:
889:
870:
869:
850:
849:
843:
831:vector analysis
823:
811:Reynolds number
804:Kutta condition
762:
752:
748:
731:
730:
727:Reynolds number
720:angle of attack
712:
710:Kutta condition
706:
676:
671:
670:
646:
636:
631:
630:
593:
592:
570:
569:
550:
549:
521:
520:
515:and not in the
493:
492:
449:
422:
411:
410:
385:
384:
363:
358:
357:
336:
331:
330:
319:
287:
277:
264:
259:
258:
233:
228:
227:
220:
195:angle of attack
171:inviscid theory
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
5342:
5340:
5332:
5331:
5326:
5321:
5319:Fluid dynamics
5316:
5311:
5301:
5300:
5297:
5296:
5287:
5284:
5282:
5281:
5238:
5227:(3): 401–425.
5211:
5192:(4): 432–441.
5176:
5143:
5093:
5078:
5035:
4992:
4985:
4964:
4921:
4906:
4888:
4881:
4863:
4856:
4838:
4813:
4770:
4763:
4744:
4742:
4739:
4738:
4737:
4730:
4727:
4726:
4725:
4709:
4705:
4700:
4675:
4671:
4666:
4650:
4644:
4640:
4637:
4633:
4630:
4626:
4623:
4619:
4616:
4612:
4609:
4605:
4602:
4586:
4577:
4574:
4572:
4571:
4556:
4551:
4548:
4544:
4540:
4537:
4534:
4531:
4528:
4526:
4522:
4518:
4514:
4512:
4506:
4503:
4499:
4495:
4492:
4489:
4486:
4484:
4480:
4476:
4472:
4471:
4457:
4456:
4445:
4440:
4436:
4432:
4429:
4424:
4420:
4416:
4411:
4408:
4404:
4400:
4397:
4394:
4391:
4386:
4383:
4379:
4375:
4372:
4369:
4366:
4361:
4358:
4354:
4350:
4347:
4342:
4339:
4335:
4331:
4328:
4325:
4322:
4319:
4316:
4311:
4307:
4303:
4300:
4297:
4293:
4286:
4283:
4278:
4273:
4269:
4262:
4259:
4256:
4252:
4246:
4242:
4239:
4233:
4227:
4224:
4207:
4206:
4195:
4192:
4189:
4183:
4180:
4177:
4172:
4167:
4163:
4156:
4151:
4146:
4142:
4138:
4135:
4132:
4129:
4123:
4119:
4115:
4100:
4099:
4088:
4082:
4079:
4076:
4072:
4067:
4062:
4058:
4033:
4030:
4027:
4024:
3995:
3984:
3983:
3972:
3969:
3966:
3963:
3958:
3954:
3950:
3946:
3942:
3939:
3933:
3930:
3925:
3922:
3916:
3913:
3910:
3904:
3901:
3896:
3893:
3886:
3880:
3876:
3872:
3869:
3866:
3863:
3857:
3853:
3849:
3846:
3843:
3837:
3833:
3829:
3824:
3820:
3796:
3793:
3778:
3777:
3762:
3759:
3756:
3753:
3747:
3743:
3739:
3736:
3733:
3727:
3723:
3719:
3714:
3710:
3706:
3703:
3699:
3696:
3690:
3684:
3680:
3676:
3673:
3671:
3669:
3666:
3663:
3660:
3654:
3650:
3646:
3643:
3640:
3634:
3630:
3626:
3621:
3617:
3613:
3610:
3607:
3604:
3601:
3595:
3591:
3587:
3584:
3581:
3575:
3571:
3567:
3562:
3558:
3554:
3551:
3549:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3515:
3511:
3507:
3504:
3499:
3495:
3491:
3486:
3482:
3478:
3475:
3473:
3471:
3468:
3464:
3461:
3458:
3454:
3451:
3445:
3441:
3437:
3436:
3422:
3421:
3410:
3407:
3404:
3399:
3396:
3390:
3386:
3379:
3376:
3373:
3369:
3364:
3359:
3355:
3324:
3320:
3296:
3293:
3289:
3285:
3282:
3277:
3274:
3270:
3266:
3261:
3257:
3233:
3229:
3217:
3216:
3205:
3202:
3199:
3192:
3188:
3182:
3178:
3172:
3167:
3162:
3158:
3152:
3147:
3143:
3139:
3136:
3133:
3130:
3126:
3123:
3099:
3088:Laurent series
3073:
3072:
3061:
3058:
3055:
3048:
3044:
3040:
3034:
3030:
3024:
3020:
3017:
3011:
3005:
3002:
2976:
2973:
2970:
2965:
2960:
2955:
2951:
2947:
2941:
2938:
2932:
2927:
2923:
2902:
2897:
2894:
2890:
2885:
2881:
2877:
2873:
2870:
2867:
2839:
2833:
2830:
2824:
2819:
2815:
2811:
2808:
2803:
2799:
2795:
2791:
2788:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2734:
2733:
2722:
2716:
2713:
2707:
2701:
2696:
2691:
2687:
2681:
2677:
2671:
2667:
2664:
2658:
2652:
2649:
2643:
2637:
2632:
2627:
2623:
2617:
2613:
2607:
2604:
2599:
2596:
2590:
2587:
2581:
2576:
2572:
2566:
2562:
2558:
2555:
2552:
2546:
2543:
2517:
2506:
2505:
2494:
2489:
2483:
2478:
2473:
2469:
2465:
2459:
2454:
2450:
2446:
2443:
2418:
2414:
2410:
2407:
2402:
2398:
2394:
2391:
2371:
2362:Then pressure
2351:
2348:
2333:
2332:
2321:
2315:
2312:
2306:
2302:
2297:
2293:
2289:
2286:
2283:
2277:
2274:
2257:
2256:
2241:
2238:
2235:
2230:
2227:
2224:
2220:
2216:
2213:
2211:
2206:
2203:
2197:
2194:
2189:
2188:
2183:
2180:
2176:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2100:
2098:
2095:
2092:
2091:
2069:
2068:
2057:
2054:
2051:
2045:
2042:
2039:
2035:
2031:
2026:
2022:
2018:
2015:
2012:
2009:
2006:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1967:
1963:
1959:
1956:
1953:
1950:
1947:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1908:
1904:
1900:
1897:
1891:
1888:
1862:
1847:
1846:
1835:
1832:
1829:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1790:
1786:
1782:
1779:
1774:
1770:
1766:
1763:
1758:
1754:
1750:
1747:
1733:complex number
1725:
1724:
1713:
1710:
1707:
1703:
1700:
1697:
1694:
1689:
1685:
1681:
1676:
1672:
1667:
1663:
1660:
1656:
1653:
1650:
1647:
1642:
1638:
1634:
1631:
1626:
1622:
1598:
1568:
1558:of the fluid,
1543:
1528:
1527:
1516:
1513:
1510:
1505:
1501:
1496:
1492:
1488:
1485:
1481:
1456:
1441:
1439:
1436:
1412:
1409:
1406:
1403:
1392:
1391:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1336:
1333:
1318:
1317:
1301:
1298:
1293:
1289:
1283:
1280:
1269:
1266:
1263:
1260:
1257:
1255:
1253:
1250:
1247:
1246:
1242:
1239:
1234:
1230:
1226:
1223:
1220:
1217:
1214:
1209:
1205:
1201:
1196:
1193:
1188:
1185:
1183:
1181:
1178:
1175:
1170:
1166:
1162:
1159:
1154:
1151:
1146:
1145:
1141:
1138:
1135:
1130:
1126:
1122:
1119:
1116:
1113:
1108:
1105:
1100:
1097:
1095:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1070:
1066:
1062:
1059:
1054:
1051:
1046:
1045:
1018:
1015:
1004:
1003:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
923:
920:
917:
897:
877:
857:
842:
839:
822:
819:
815:potential flow
791:boundary layer
775:
769:
765:
759:
755:
751:
745:
729:, defined as
708:Main article:
705:
702:
701:
700:
688:
683:
679:
658:
653:
649:
643:
639:
600:
580:
577:
557:
537:
534:
531:
528:
517:boundary layer
513:potential flow
500:
489:
488:
477:
474:
470:
467:
464:
461:
456:
452:
448:
444:
440:
437:
434:
429:
425:
421:
418:
392:
370:
366:
343:
339:
325:
324:
315:
313:
302:
299:
294:
290:
284:
280:
276:
271:
267:
239:
236:
219:
216:
197:and the sharp
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
5341:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5306:
5304:
5294:
5290:
5289:
5285:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5242:
5239:
5234:
5230:
5226:
5222:
5215:
5212:
5207:
5203:
5199:
5195:
5191:
5187:
5180:
5177:
5171:
5166:
5162:
5158:
5154:
5147:
5144:
5139:
5135:
5131:
5127:
5123:
5119:
5115:
5111:
5104:
5097:
5094:
5089:
5082:
5079:
5074:
5070:
5066:
5062:
5058:
5054:
5050:
5046:
5039:
5036:
5031:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
4996:
4993:
4988:
4986:0-521-42058-X
4982:
4978:
4971:
4969:
4965:
4960:
4956:
4952:
4948:
4944:
4941:
4940:
4935:
4931:
4925:
4922:
4917:
4910:
4907:
4902:
4898:
4892:
4889:
4884:
4882:0-471-50952-3
4878:
4874:
4867:
4864:
4859:
4857:0-273-01120-0
4853:
4849:
4842:
4839:
4828:on 2014-01-11
4827:
4823:
4817:
4814:
4809:
4805:
4801:
4797:
4793:
4789:
4785:
4781:
4774:
4771:
4766:
4764:0-07-001641-0
4760:
4756:
4749:
4746:
4740:
4736:
4733:
4732:
4728:
4703:
4698:
4669:
4664:
4655:
4651:
4648:
4645:
4641:
4638:
4634:
4631:
4627:
4624:
4620:
4617:
4613:
4610:
4606:
4603:
4600:
4595:
4591:
4587:
4584:
4583:
4582:
4575:
4554:
4546:
4542:
4535:
4532:
4529:
4527:
4520:
4516:
4510:
4501:
4497:
4490:
4487:
4485:
4478:
4474:
4462:
4461:
4460:
4443:
4438:
4434:
4430:
4427:
4422:
4418:
4414:
4406:
4402:
4395:
4392:
4389:
4381:
4377:
4370:
4367:
4356:
4352:
4348:
4345:
4337:
4333:
4323:
4320:
4317:
4309:
4305:
4301:
4298:
4295:
4291:
4284:
4281:
4271:
4267:
4260:
4257:
4254:
4250:
4244:
4240:
4237:
4231:
4222:
4212:
4211:
4210:
4193:
4190:
4187:
4181:
4178:
4175:
4165:
4161:
4154:
4149:
4144:
4140:
4136:
4130:
4121:
4113:
4105:
4104:
4103:
4086:
4080:
4077:
4074:
4065:
4060:
4056:
4048:
4047:
4046:
4031:
4028:
4025:
4022:
4014:
4010:
3993:
3970:
3967:
3964:
3961:
3956:
3952:
3948:
3944:
3940:
3937:
3931:
3923:
3914:
3911:
3908:
3902:
3894:
3884:
3878:
3874:
3870:
3864:
3861:
3855:
3851:
3847:
3844:
3841:
3835:
3831:
3822:
3818:
3810:
3809:
3808:
3794:
3783:
3760:
3754:
3751:
3745:
3741:
3737:
3734:
3731:
3725:
3721:
3712:
3708:
3704:
3701:
3697:
3694:
3682:
3678:
3674:
3672:
3661:
3658:
3652:
3648:
3644:
3641:
3638:
3632:
3628:
3619:
3615:
3611:
3608:
3602:
3599:
3593:
3589:
3585:
3582:
3579:
3573:
3569:
3560:
3556:
3552:
3550:
3539:
3536:
3533:
3530:
3527:
3524:
3513:
3509:
3505:
3502:
3497:
3493:
3484:
3480:
3476:
3474:
3469:
3466:
3459:
3452:
3449:
3443:
3439:
3427:
3426:
3425:
3408:
3405:
3402:
3397:
3394:
3388:
3384:
3377:
3374:
3371:
3367:
3362:
3357:
3353:
3345:
3344:
3343:
3341:
3322:
3318:
3291:
3287:
3283:
3280:
3272:
3268:
3264:
3259:
3255:
3231:
3227:
3203:
3200:
3197:
3190:
3186:
3180:
3176:
3170:
3165:
3160:
3156:
3150:
3145:
3141:
3137:
3131:
3124:
3121:
3113:
3112:
3111:
3097:
3089:
3085:
3080:
3078:
3059:
3056:
3053:
3046:
3038:
3032:
3028:
3022:
3018:
3015:
3009:
3000:
2990:
2989:
2988:
2974:
2971:
2968:
2963:
2953:
2945:
2936:
2930:
2925:
2921:
2900:
2895:
2892:
2888:
2879:
2871:
2868:
2865:
2857:
2853:
2837:
2828:
2822:
2817:
2813:
2809:
2806:
2801:
2797:
2793:
2789:
2786:
2777:
2761:
2755:
2749:
2746:
2743:
2720:
2711:
2705:
2699:
2689:
2679:
2675:
2669:
2665:
2662:
2656:
2647:
2641:
2635:
2625:
2615:
2611:
2605:
2602:
2597:
2594:
2585:
2579:
2574:
2570:
2564:
2560:
2556:
2553:
2550:
2541:
2531:
2530:
2529:
2515:
2492:
2487:
2481:
2471:
2463:
2457:
2452:
2448:
2444:
2441:
2434:
2433:
2432:
2416:
2412:
2408:
2405:
2400:
2396:
2392:
2389:
2369:
2349:
2346:
2338:
2319:
2310:
2304:
2300:
2295:
2291:
2287:
2284:
2281:
2272:
2262:
2261:
2260:
2239:
2236:
2233:
2228:
2225:
2222:
2218:
2214:
2212:
2201:
2195:
2181:
2178:
2174:
2169:
2166:
2163:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2101:
2096:
2093:
2082:
2081:
2080:
2078:
2074:
2055:
2052:
2049:
2043:
2040:
2037:
2033:
2029:
2024:
2020:
2016:
2013:
2010:
2007:
2004:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1970:
1965:
1961:
1957:
1954:
1951:
1948:
1945:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1911:
1906:
1902:
1898:
1895:
1886:
1876:
1875:
1874:
1860:
1853:of the force
1852:
1833:
1830:
1827:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1793:
1788:
1784:
1780:
1777:
1772:
1768:
1764:
1761:
1756:
1752:
1748:
1745:
1738:
1737:
1736:
1734:
1730:
1729:complex plane
1711:
1708:
1705:
1701:
1698:
1695:
1692:
1687:
1683:
1679:
1674:
1670:
1665:
1661:
1658:
1654:
1651:
1648:
1645:
1640:
1636:
1632:
1629:
1624:
1620:
1612:
1611:
1610:
1596:
1588:
1584:
1557:
1541:
1533:
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1499:
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1490:
1486:
1483:
1471:
1470:
1469:
1437:
1435:
1433:
1429:
1424:
1410:
1404:
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1374:
1371:
1368:
1365:
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1338:
1334:
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1323:
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1299:
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1251:
1240:
1232:
1228:
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1218:
1215:
1212:
1207:
1203:
1194:
1191:
1186:
1184:
1179:
1173:
1168:
1160:
1152:
1149:
1139:
1136:
1133:
1128:
1120:
1117:
1114:
1106:
1103:
1098:
1096:
1088:
1082:
1079:
1073:
1068:
1060:
1052:
1049:
1036:
1035:
1034:
1032:
1016:
989:
986:
983:
980:
977:
974:
968:
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962:
956:
953:
950:
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937:
936:
935:
921:
918:
915:
895:
875:
855:
848:
840:
838:
836:
832:
828:
820:
818:
816:
812:
807:
805:
799:
796:
795:inviscid flow
792:
773:
767:
763:
753:
749:
743:
728:
723:
721:
717:
711:
703:
699:
686:
677:
647:
637:
627:
626:
625:
622:
620:
616:
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498:
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472:
468:
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462:
459:
454:
450:
446:
438:
435:
432:
427:
423:
419:
409:
408:
407:
406:
405:line integral
364:
337:
323:
316:
314:
300:
288:
278:
274:
265:
257:
256:
253:
237:
234:
225:
217:
215:
213:
209:
204:
200:
199:trailing edge
196:
192:
188:
187:superposition
183:
182:Magnus effect
178:
176:
172:
168:
164:
160:
156:
155:line integral
152:
148:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
5292:
5286:Bibliography
5251:
5247:
5241:
5224:
5220:
5214:
5189:
5186:AIAA Journal
5185:
5179:
5160:
5156:
5146:
5113:
5109:
5096:
5087:
5081:
5048:
5044:
5038:
5005:
5001:
4995:
4976:
4945:(1): 17–35.
4942:
4937:
4924:
4915:
4909:
4900:
4891:
4872:
4866:
4848:Aerodynamics
4847:
4841:
4830:. Retrieved
4826:the original
4816:
4783:
4779:
4773:
4754:
4748:
4590:vortex sheet
4579:
4458:
4208:
4101:
4012:
4011:. Since the
3985:
3779:
3423:
3338:. Using the
3218:
3081:
3074:
2855:
2851:
2735:
2507:
2334:
2258:
2076:
2072:
2070:
1848:
1726:
1586:
1531:
1529:
1445:
1428:differential
1425:
1393:
1319:
1005:
844:
824:
808:
800:
724:
713:
628:
623:
618:
612:
611:. Equation
490:
328:
317:
221:
179:
163:Martin Kutta
127:aerodynamics
122:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
5254:: 265–286.
5051:: 182–217.
5008:: 413–425.
4786:: 304–341.
3784:denoted by
3782:circulation
2913:Therefore,
1583:unit vector
151:circulation
143:unseparated
5303:Categories
4930:Wagner, H.
4832:2013-11-07
4741:References
821:Derivation
69:newspapers
5276:122293574
5138:120656935
5116:: 73–92.
5073:121892071
5030:123501457
4808:125643946
4550:∞
4539:Γ
4536:ρ
4533:−
4505:∞
4494:Γ
4491:ρ
4428:−
4410:∞
4399:Γ
4396:ρ
4385:∞
4374:Γ
4371:ρ
4360:∞
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4327:Γ
4324:ρ
4315:Γ
4302:ρ
4282:π
4277:Γ
4258:π
4241:ρ
4226:¯
4191:⋯
4176:π
4171:Γ
4078:π
4071:Γ
4026:ψ
3994:ψ
3965:ψ
3953:∮
3929:∂
3924:ψ
3921:∂
3900:∂
3895:ψ
3892:∂
3875:∮
3848:−
3819:∮
3792:Γ
3738:−
3709:∮
3679:∮
3645:−
3616:∮
3557:∮
3503:−
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3440:∮
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3201:⋯
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3019:ρ
3004:¯
2940:¯
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2872:±
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2715:¯
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2528:becomes:
2464:ρ
2458:−
2347:ρ
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2292:∮
2285:−
2276:¯
2229:ϕ
2223:−
2205:¯
2193:⇒
2182:ϕ
2158:ϕ
2155:
2143:ϕ
2140:
2044:ϕ
2038:−
2021:∮
2014:−
1998:ϕ
1995:
1986:−
1983:ϕ
1980:
1962:∮
1955:−
1939:ϕ
1936:
1924:ϕ
1921:
1903:∮
1899:−
1890:¯
1821:ϕ
1818:
1809:−
1806:ϕ
1803:
1785:∮
1781:−
1702:ϕ
1699:
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1655:ϕ
1652:
1637:∮
1633:−
1597:ϕ
1491:∮
1487:−
1408:Γ
1402:ρ
1378:Γ
1372:ρ
1369:−
1354:ρ
1345:Δ
1279:ρ
1262:ρ
1249:Δ
1192:ρ
1177:Δ
1150:ρ
1104:ρ
1086:Δ
1050:ρ
1014:Δ
981:−
957:−
945:Γ
876:ρ
827:heuristic
774:μ
758:∞
750:ρ
682:∞
657:Γ
652:∞
642:∞
638:ρ
536:θ
533:
469:θ
466:
451:∮
436:⋅
424:∮
417:Γ
391:Γ
369:∞
342:∞
338:ρ
298:Γ
293:∞
283:∞
279:ρ
270:′
4932:(1925).
4899:(1967).
4729:See also
4608:reached.
4118:′
3453:′
3398:′
3125:′
3043:′
2790:′
2335:Now the
1335:′
238:′
99:May 2015
5256:Bibcode
5194:Bibcode
5118:Bibcode
5053:Bibcode
5010:Bibcode
4947:Bibcode
4788:Bibcode
4007:is the
1581:is the
1554:is the
716:airfoil
208:airfoil
203:tornado
175:viscous
159:tangent
147:density
135:airfoil
83:scholar
5274:
5136:
5071:
5028:
4983:
4879:
4854:
4806:
4761:
1530:where
568:, and
329:where
191:camber
133:of an
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5272:S2CID
5134:S2CID
5106:(PDF)
5069:S2CID
5026:S2CID
4804:S2CID
3986:Here
847:chord
139:fluid
90:JSTOR
76:books
4981:ISBN
4877:ISBN
4852:ISBN
4759:ISBN
4599:wing
2774:the
2431:by:
833:and
356:and
224:span
165:and
131:lift
121:The
62:news
5264:doi
5252:576
5229:doi
5202:doi
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5126:doi
5114:698
5061:doi
5049:769
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5006:133
4955:doi
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4708:max
4674:max
2152:sin
2137:cos
1992:sin
1977:cos
1933:cos
1918:sin
1815:cos
1800:sin
1696:cos
1649:sin
614:(1)
530:cos
463:cos
45:by
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