Incompressible Flow About an Airfoil

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Two dimensional incompressible flows past thin lifting surfaces are studied. The lifting surface is modeled with a vortex sheet. The Biot–Savart law is used to establish the relation between the vortex sheet strength and the downwash w(x,t). The strength of the vortex sheet is calculated via the inversion of a singular integral equation given in terms of the downwash at the surface as a boundary condition. First, for an airfoil steady state solution after the impulsive start in a uniform stream is considered. Carleman’s formula is used to invert the integral relation which explicitly gives the vortex sheet strength satisfying the well known Kutta condition. The lifting pressure distribution and the sectional lift and the moment coefficients are found with integration of lifting pressure along the chord. Furthermore, the concept of center of pressure and aerodynamic center are introduced. The locations of both centers on the chord are then calculated. Unsteady flow case is studied with distribution of a vortex sheet in the wake as well. The wake vorticity and the bound vorticity are tied together with the unsteady Kutta condition which is nothing but imposing zero lifting pressure at the wake region. The Laplace transform technique is employed to establish the relation between the total bound circulation and the downwash. Then the transformed form of the bound vortex sheet strength and the downwash relation is used to obtain the expression for the lifting surface pressure in the Laplace domain. Since the inverse Laplace transform of the pressure expression is quite complex, the inversion is performed for the simple harmonic pressure variation. The lifting surface pressure has three terms each signifying different aspects of aerodynamic phenomena. The first term is the quasi steady term which is identical with the steady pressure term, the second term accounts for the contribution of the wake vorticity and, finally the third term is the ‘apparent mass’ term which is responsible for the non circulatory lifting pressure without the presence of free stream. Depending on the type of aerodynamics we use, the relevant terms are kept in the expression given for the pressure. Accordingly, (i) if all three terms are retained then the approach is called ‘unsteady aerodynamics’ and it is used for the problems having oscillations of order of 40 Hz, (ii) for the case of ‘quasi unsteady aerodynamics’ we neglect the apparent mass term where the approach is applicable for the 5–15 Hz range, and finally, (iii) ‘quasi steady aerodynamics’ requires retaining the quasi steady term only where the approach is good for the frequencies of 1 Hz or less. As a special type of an unsteady flow, Simple Harmonic Motion of an airfoil in pitch and/or in vertical translation is considered. The Theodorsen function here is indicative of the effect of the wake vorticity on the circulatory term in the lift expression. The effect of the wake vorticity on the profile shows itself as reduction of the magnitude of the lift coefficient, and the lagging of the response of the airfoil with the angle of attack change in time. In order to demonstrate this several basic examples of SHM are provided, wherein the hysteresis under the lift versus angle of attack curve is accounted for. In addition, returning wake problem of Loewy is studied with the help of a new function which depends on the wake spacing, reduced frequency and the rotational frequency of the blade, replace the Theodorsen function. Arbitrary unsteady motions are analyzed with the concept of the ‘indicial admittance’ function applicable to linear systems. First, the Wagner function as the indicial admittance to the arbitrary angle of attack change of an airfoil in a free stream is considered. Then, the Kussner function of the sharp edged gust impingement is obtained as the indicial admittance. In analyzing the response of an airfoil to an arbitrary motion the integral Fourier Transform Technique is utilized. The sinusoidal gust problem is studied by establishing the Sears function as the indicial admittance. For the analysis of the moving gust problem the concept of the Miles function is introduced. The Miles function is generally utilized in rotor aerodynamics. However, when the moving gust velocity becomes zero the Miles function transforms itself into the Kussner function. Finally, as an application of Wagner function an airfoil immersed in a sinusoidally varying free stream velocity is considered. This problem can be utilized in obtaining the estimate of the total lift coefficient of a blade in a forward flight from two dimensional considerations only.