BOUNDARY CONDITIONS IN AIRFOIL PROBLEMS

An indisputable boundary condition is that the fluid must not penetrate the solid body so that the flow must be tangent to the solid surface at all times.[34] The other boundary condition, which in the case of a fluid of infinite extent refers to the conditions of flow at infinitely large distance from the airfoil, requires a careful consideration.

In an incompressible fluid, the influence of any disturbance is instantly transmitted in all directions to infinity. Consider the motion of a fluid generated by the motion of an airfoil which begins at a time t0. During a finite time interval t tQ9 the airfoil moves about and sweeps out a region of space, every point of which is occupied by the airfoil at one time or another. Let this region be denoted by s4. Consider now a spherical surface with radius R0 so large that the entire region, я/ is enclosed in the sphere. Introduce the spherical polar coordinates, R, в, y>, where в and ip are the polar and azimuth angles, respectively. Let the resultant velocity at a point (R. в, ip) be V(R, в, ip). The total kinetic energy of the fluid in the region outside the sphere R0 is

і |*co |*it f*2it

K. E. = 2 Jb Jo Jo pr’:’Ri sin 6 dR d0 cbp

This must be a finite quantity because it is only part of the energy imparted to the fluid by the motion of the airfoil. Therefore the improper integral must be convergent, and it is seen that the velocity V must decrease to zero as R increases indefinitely at a rate faster than l/Jtk Thus the condition at infinity for an incompressible fluid is that the velocity dis­turbance decreases to zero, or that the velocity potential tends to a con­stant. The relation between the velocity and acceleration then shows that the acceleration of the fluid particles decreases to zero at infinity, while the acceleration potential and pressure tend to constant values at infinity.

If the airfoil is moving in a compressible fluid, the disturbance in the fluid due to the motion of the airfoil is propagated with the speed of sound. If the speed of motion of the airfoil (relative to a coordinate system which is at rest with respect to the fluid at infinity) is less than the speed of sound, the disturbance will be felt in all directions. After a sufficiently long period of time, the region of the fluid influenced by the motion of the airfoil will be much larger than the region sd swept out by the airfoil, and the argument of the last paragraph can be used again to conclude that the velocity disturbances decrease to zero at infinity.

If the motion of the airfoil is a small oscillation about a rectilinear translation, which has taken place for an indefinitely long time, the region ja/ named above consists of the volume occupied by the airfoil and a wake which extends indefinitely behind the airfoil. The conclusions reached above may be so stated that the velocity disturbances must vanish at least as fast as 1 /Rsl2 as R -» со, where R is the shortest distance between the point in question and the airfoil and its wake.

Consider finally the case in which the airfoil is moving at a supersonic speed. The speed of motion being higher than the speed of propagation of sound, the disturbances cannot be felt in front of the envelope of Mach waves generated by the leading edge of the airfoil. Hence, there is no disturbance upstream of the airfoil. On the other hand, at supersonic speeds, energy can be propagated to infinity in the form of shock waves. In a two-dimensional supersonic flow, the region influenced by the shock waves may be limited in extent, and the strength of disturbance does not necessarily diminish toward zero when the distance from the airfoil increases indefinitely. However, we may impose the condition that there are no sources of disturbance in the fluid other than the airfoil under consideration. This, together with the condition of no disturbance in front of the Mach-wave envelope from the leading edge, suffices to determine a unique solution in the supersonic case.

In the following discussions, we shall restrict ourselves to the linearized theory, which is valid only if the disturbance caused by the airfoil is infinitesimal. This implies that the wing is infinitely thin and of infinitesimal camber and has an infinitesimal angle of attack. Such a wing is said to be planar.

Let the free-stream direction be parallel to the mean position of the planar wing, which lies on the (хъ хг) plane. Then the distance from the wing surface to the (хъ x2) plane is an infinitesimal quantity of the first order. On the assumption that the quantities p’, p, щ, a{, etc., are continuous functions of space and time, it is easy to see (by expansion into power series) that the difference of any of these quantities on the surface of the airfoil from that on the (aq, ж2) plane is an infinitesimal of higher order than that quantity itself. Therefore, in the linearized theory it is permissible to apply the boundary conditions of a planar system in a region on the (xlt a:2) plane that represents the projection of the airfoil on that plane, instead of oh the actual airfoil surface.