Wing theory

The branch of aerodynamics concerned with the interaction between bodies and liquid or gas flows. The fundamental problem of wing theory is to determine the aerodynamic forces acting on the body, and to express the velocity field $ \mathbf u $ and the pressure $ p $ as functions of the time $ t $ and the Cartesian coordinates $ \mathbf x = ( x _ {1} \dots x _ {n} ) $, where $ n = 2 $( two-dimensional flows) or $ n = 3 $( three-dimensional flows).

In the case of irrotational barotropic flows, in the absence of viscous and body forces the density $ \rho $ of the gas is a known function of the pressure, $ \rho = \rho ( p) $, and the velocity components $ u _ {i} $ are the partial derivatives of a potential $ \phi $: $ u _ {i} = \partial \phi / \partial x _ {i} $. In the region occupied by the gas, $ \phi $ satisfies a quasi-linear equation:

$$ \tag{1 } \frac{1}{c ^ {2} } \frac{\partial ^ {2} \phi }{\partial t ^ {2} } + \frac{2}{c ^ {2} } \sum _ {i = 1 } ^ { n } \frac{\partial \phi }{\partial x _ {i} } \frac{\partial ^ {2} \phi }{\partial x _ {i} \partial t } = $$

$$ = \ \sum _ {i, j = 1 } ^ { n } \left ( \delta _ {ij} - \frac{\partial \phi }{\partial x _ {i} } \right ) \frac{\partial ^ {2} \phi }{\partial x _ {i} \partial x _ {j} } , $$

where $ c = ( d \rho /dp) ^ {-} 1/2 $ is the velocity of sound and $ \delta _ {ij} $ is the Kronecker symbol. The pressure $ p $ is determined by the potential with the aid of the Cauchy–Lagrange integral:

$$ \int\limits _ {p _ {0} } ^ { p } { \frac{dp} \rho } = - \frac{\partial \phi }{\partial t } - { \frac{1}{2} } | \nabla \phi | ^ {2} . $$

The boundary of the region of the flow is made up of the piecewise-smooth wing surface $ S $ and finitely many contact-discontinuity surfaces $ \Sigma _ {j} $, $ j = 1 \dots m $, which either intersect $ S $ along the sharp edges of the wing-tips or are tangent to $ S $. In two-dimensional flows, $ S $ and $ \Sigma _ {j} $ are piecewise-smooth curves, while the wing-tips are corner points of $ S $. The potential satisfies an impermeability condition on $ S $; on $ \Sigma _ {j} $ it satisfies contact-discontinuity conditions:

$$ \tag{2 } \frac{\partial F }{\partial t } + \nabla F \cdot \nabla \phi = 0 \ \mathop{\rm on} S, $$

$$ \tag{3 } \frac{\partial F _ {j} }{\partial t } + \nabla F _ {j} \nabla \phi ^ \pm = 0,\ p ^ {+} = p ^ {-} \ \mathop{\rm on} \Sigma _ {j} , $$

where $ F ( \mathbf x , t) = 0 $, $ F _ {j} ( \mathbf x , t) = 0 $ are the equations of the surfaces $ S $, $ \Sigma _ {j} $, and $ \phi ^ \pm $ are the limiting values of $ \phi $ when the surface $ \Sigma _ {j} $ is approached from two different sides. Along the lines of intersection of $ S $ with $ \Sigma _ {j} $ one has the Zhukovskii–Kutta–Chaplygin condition, according to which the pressure on the wing-tips is finite:

$$ \tag{4 } \lim\limits _ {x \rightarrow x _ {0} } | p ( x) | < \infty \ \ \textrm{ if } \mathbf x _ {0} \in S \cap \Sigma _ {j} . $$

In a steady flow, condition (4) is equivalent to the condition that the velocities at the points of $ S \cap \Sigma _ {j} $ be finite. The shape of the surfaces $ \Sigma _ {j} $ is unknown in the course of solving, and is determined together with the solution.

The surfaces $ \Sigma _ {j} $ model the vortex trail behind the body in a real flow (see Aerodynamics, mathematical problems of). This is in agreement with the fact that, if one assumes that the motion is irrotational, there exists no continuous general solution to the problem of flow around a wing with finite pressure at the sharp edges. In exceptional cases, e.g. in the case of steady two-dimensional flows with constant circulation around the wing profile, surfaces of discontinuity may be absent.

Equations (1)–(4), together with the initial data, constitute a boundary value problem for the determination of $ \phi , \Sigma _ {j} $. The type of the problem depends on the type of the flow and on the Mach number $ M = | \nabla \phi | c ^ {-} 1 $. For unsteady motion of a compressible fluid and steady $ ( \partial \phi / \partial t = 0) $ supersonic $ ( M > 1) $ flows, equation (1) is of hyperbolic type; for incompressible ( $ \rho = \textrm{ const } $, $ c = \infty $) and steady subsonic $ ( M < 1) $ flows, it is elliptic. In the latter case, if one assumes that $ S $ is a piecewise-smooth curve with one corner point $ \mathbf x _ {0} $ with angle $ \alpha \pi $, $ \alpha \in [ 0, 1] $, the following proposition is true: For any vector $ \mathbf k $, $ | \mathbf k | = 1 $, there exists a $ \lambda > 0 $ such that if $ q \in [ 0, \lambda ) $, the problem (1)–(2) has a unique solution satisfying the Zhukovskii–Kutta–Chaplygin condition at $ \mathbf x _ {0} $ and the following condition at infinity:

$$ \overline{\lim\limits}\; _ {\mathbf x \rightarrow \mathbf x _ {0} } | \nabla \phi | < \infty ,\ \ \lim\limits _ {| \mathbf x | \rightarrow \infty } \nabla \phi ( \mathbf x ) = q \mathbf k ; $$

moreover $ M ( q) \rightarrow 0 $ as $ q \rightarrow 0 $ and $ M ( q) \rightarrow 1 $ as $ q \rightarrow \lambda $, where $ M ( q) = \sup _ {\mathbf x } M ( \mathbf x ) $ is the Mach number of the flow.

In steady subsonic two-dimensional flows, one has the fundamental theorem of Zhukovskii (see [1]–[3]): In a flow around a profile, the total force exerted on the profile from the fluid is normal to $ \mathbf k $ and its magnitude $ R $ is given by

$$ R = q \rho _ \infty \oint _ { S } \frac{\partial \phi }{\partial s } ds,\ \ \rho _ \infty = \lim\limits _ {| \mathbf x | \rightarrow \infty } \rho ( \mathbf x ). $$

For such flows it has been proved that the following more general problems are mathematically well-posed: simultaneous flow around several profiles; flow around a wing with separation of the jets and with formation of a stagnation zone (jet flows); and converse problems — to determine the shape of the wing and its parts given the pressure curve [4].

Since the solution of problems in wing theory in their exact formulations is difficult, much importance attaches to approximate models: the theory of thin wings, the theory of wings of small elongation, etc. The most widely used model is that of the linear theory of a weakly curved thin wing (see [1], [5]–[11]). This model is based on the following assumptions: The potential of the flow is given by $ \phi = qx _ {1} + \Phi $, the thickness of the wing and $ \nabla \Phi $ are small in comparison with the chord of the wing and the velocity $ q > 0 $ of the unperturbed flow. In the theory of thin wings, the surface $ S $ is simulated by its projection $ S _ {0} $ on the plane $ x _ {n} = 0 $, and the contact-discontinuity surface $ \Sigma $ by the half-plane $ \Sigma _ {0} = \Omega \setminus S _ {0} $, where $ \Omega $ is the union of all rays parallel to the axis $ 0x _ {1} $ and emanating from points of $ S _ {0} $ in the positive $ x _ {1} $- direction. The function $ \Phi ( \mathbf x , t) $ satisfies linearized equations and boundary conditions:

$$ \frac{1}{c _ \infty ^ {2} } \frac{\partial ^ {2} \Phi }{\partial t ^ {2} } + 2 \frac{M _ \infty }{c _ \infty } \frac{\partial ^ {2} \Phi }{\partial x _ {1} \partial t } = \ ( 1 - M _ \infty ^ {2} ) \frac{\partial ^ {2} \Phi }{\partial x _ {1} ^ {2} } + \sum _ {i = 2 } ^ { n } \frac{\partial ^ {2} \Phi }{\partial x _ {i} ^ {2} } $$

outside $ \Omega $;

$$ \lim\limits _ {x _ {n} \rightarrow \pm 0 } \frac{\partial \Phi }{\partial x _ {n} } = v ^ \pm \ \mathop{\rm on} S _ {0} ; $$

$$ \left [ \frac{\partial \Phi }{\partial x _ {n} } \right ] = \left [ \frac{\partial \Phi }{\partial t } + q \frac{\partial \Phi }{\partial x _ {1} } \right ] = 0 \ \mathop{\rm on} \Sigma _ {0} ; $$

$$ \overline{\lim\limits}\; _ {x \rightarrow x _ {0} } \left | \frac{\partial \Phi }{\partial t } + q \frac{\partial \Phi }{\partial x _ {1} } \right | < \infty \ \textrm{ for } x _ {0} \in S _ {0} \cap \Sigma _ {0} , $$

where $ c _ \infty $, $ M _ \infty $ are the constant velocity of sound and the Mach number corresponding to a uniform flow with velocity $ q $, the notation $ [ f ] $ represents the jump in the value of $ f $ across $ \Sigma _ {0} $, and $ v ^ \pm $ are given functions that define the shape and motion conditions of the wing.

These equations are augmented by further relations that determine the behaviour of the solutions at infinity: In a steady subsonic flow ( $ M _ \infty < 1 $) the condition is that the perturbations are damped out as $ | \mathbf x | \rightarrow \infty $; in the case of small subsonic oscillations of a wing, one has the Sommerfeld radiation condition (see Radiation conditions); in a supersonic flow ( $ M _ \infty > 1 $) the additional relation is $ \Phi = 0 $ for the front wave of the perturbations (the envelope of the characteristic cones with centres on $ S _ {0} $).

The basic method for solving problems in the theory of thin wings is to represent $ S _ {0} $ and $ \Sigma _ {0} $ as vortex surfaces and to reduce the boundary value problems to singular integral equations for the vortex density. When this is done, the derivatives of $ \Phi $ usually become infinite at points of $ S _ {0} $ not belonging to $ \Sigma _ {0} $. The linear theory is suitable for describing real flows only outside a certain neighbourhood of the leading wing-tip.

In the linear theory of thin wings, solutions can be expressed as infinite series in special functions, in the case of problems such as the two-dimensional problem of small harmonic perturbations of a wing profile and the problem of three-dimensional steady flow in case $ S $ is an ellipse (see [1], [5]–[9]). Numerical methods have been developed for the computation of wings of arbitrary shape (see [10], [11]).

References

[1]	L.J. [L.I. Sedov] Sedov, "Two-dimensional problems in hydrodynamics and aerodynamics" , Acad. Press (1965) (Translated from Russian)
[2]	N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , Interscience (1964) (Translated from Russian)
[3]	L. Bers, "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley (1958)
[4]	V.N. Monakhov, "Boundary value problems with free boundaries for elliptic systems of equations" , Amer. Math. Soc. (1983) (Translated from Russian)
[5]	A.I. Nekrasov, "Wing theory in a non-steady flow" , Moscow-Leningrad (1947) (In Russian)
[6]	D.N. Gorelov, "Wing theory in a non-steady flow" , Novosibirsk (1975) (In Russian)
[7]	J. Miles, "The potential theory of unsteady supersonic flow" , Cambridge Univ. Press (1959)
[8]	V.V. Golubev, "Works in aerodynamics" , Moscow-Leningrad (1957) (In Russian)
[9]	N.E. Kochin, Prikl. Mat. i Mekh. , 9 : 1 (1945) pp. 13–66
[10]	E.A. Krasil'shchikova, "Wings of finite size in compressible fows" , Moscow-Leningrad (1952) (In Russian)
[11]	O.M. Belotserkovskii, M.I. Nisht, "Discontinuous and continuous flow of an ideal fluid around thin wings" , Moscow (1978) (In Russian)

Comments

The part of wing theory as described above is mainly restricted to the influence of compressibility in unsteady flows. There is an easier part of the theory, which deals with steady incompressible flows. The basic equation here is the Laplace equation, so that all the tools of potential theory may applied. In particular, the theory for thin aerofoils (in two dimensions) lends itself to a completely analytical treatment. Its main practical result refers to the lift coefficient

$$ C _ {L} = \frac{2L}{\rho V _ \infty ^ { 2 } A } = 2 \pi ( \alpha - \alpha _ {0} ) , $$

where $ L $ is the lift, $ \rho $ the fluid density, $ V _ \infty $ the fluid speed far away from the airofoil, $ A $ the area of the wing, $ \alpha $ the angle of attack, and $ \alpha _ {0} $ a constant depending on the shape of the mean line of the airofoil. A similar formula is obtained for the pitching moment coefficient. Another important result of the theory is the distribution of pressure along the aerofoil. The pressure at the leading edge turns out to be finite if and only if the attack angle has a certain value ( "ideal" or "design angle of attack" ), as first shown by T. Theodorsen. These results apply to the linearized theory for subsonic compressible flows ( $ M _ \infty < 1 $) as well, as can be shown by a similarity transformation.

Note also that the Zhukovskii–Kutta–Chaplygin condition is usually called the Kutta condition, and Zhukovskii's theorem is usually referred to as the Kutta–Zhukovskii theorem in the Western literature. Finally, Zhukovskii is often rendered as Joukowski.

References

[a1]	H.W. Liepmann, A. Roshko, "Elements of gas dynamics" , Wiley (1957)
[a2]	L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Addison-Wesley (1959) (Translated from Russian)
[a3]	H. Schlichting, "Boundary layer theory" , McGraw-Hill (1955) (Translated from German)
[a4]	G. Birhoff, "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press (1960) pp. Chapt. IV
[a5]	H. Lamb, "Hydrodynamics" , Cambridge Univ. Press (1932)
[a6]	L.M. Milne-Thompson, "Theoretical hydrodynamics" , Macmillan (1957)
[a7]	L. Prandtl, O.G. Tietjens, "Applied hydro- & aeromechanics" , Dover, reprint (1934)
[a8]	L. Prandtl, O.G. Tietjens, "Applied hydro- & aeromechanics" , Dover, reprint (1934)
[a9]	S. Goldstein (ed.) , Modern developments in fluid mechanics , 1–2 , Dover, reprint (1965)
[a10]	R. von Mises, "Theory of flight" , Dover, reprint (1959)
[a11]	J. Lighthill, "An informal introduction to theoretical fluid mechanics" , Clarendon Press (1986)