# Wing theory

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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 and the pressure as functions of the time and the Cartesian coordinates , where (two-dimensional flows) or (three-dimensional flows).

In the case of irrotational barotropic flows, in the absence of viscous and body forces the density of the gas is a known function of the pressure, , and the velocity components are the partial derivatives of a potential : . In the region occupied by the gas, satisfies a quasi-linear equation:

 (1)

where is the velocity of sound and is the Kronecker symbol. The pressure is determined by the potential with the aid of the Cauchy–Lagrange integral:

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

 (2)
 (3)

where , are the equations of the surfaces , , and are the limiting values of when the surface is approached from two different sides. Along the lines of intersection of with one has the Zhukovskii–Kutta–Chaplygin condition, according to which the pressure on the wing-tips is finite:

 (4)

In a steady flow, condition (4) is equivalent to the condition that the velocities at the points of be finite. The shape of the surfaces is unknown in the course of solving, and is determined together with the solution.

The surfaces 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 . The type of the problem depends on the type of the flow and on the Mach number . For unsteady motion of a compressible fluid and steady supersonic flows, equation (1) is of hyperbolic type; for incompressible (, ) and steady subsonic flows, it is elliptic. In the latter case, if one assumes that is a piecewise-smooth curve with one corner point with angle , , the following proposition is true: For any vector , , there exists a such that if , the problem (1)–(2) has a unique solution satisfying the Zhukovskii–Kutta–Chaplygin condition at and the following condition at infinity:

moreover as and as , where 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 and its magnitude is given by

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 , the thickness of the wing and are small in comparison with the chord of the wing and the velocity of the unperturbed flow. In the theory of thin wings, the surface is simulated by its projection on the plane , and the contact-discontinuity surface by the half-plane , where is the union of all rays parallel to the axis and emanating from points of in the positive -direction. The function satisfies linearized equations and boundary conditions:

outside ;

where , are the constant velocity of sound and the Mach number corresponding to a uniform flow with velocity , the notation represents the jump in the value of across , and 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 () the condition is that the perturbations are damped out as ; in the case of small subsonic oscillations of a wing, one has the Sommerfeld radiation condition (see Radiation conditions); in a supersonic flow () the additional relation is for the front wave of the perturbations (the envelope of the characteristic cones with centres on ).

The basic method for solving problems in the theory of thin wings is to represent and 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 usually become infinite at points of not belonging to . 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 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)