The unsteady ow around an aerofoil placed in a uniform ow stream with an angle of attack is investigated, under the assumption of inviscid, incompressible, two-dimensional flow. In particular, a function of the velocity jump over the wake is achieved, where this function depends on the horizontal displacement and time.

The aerofoil geometry is represented by two arbitrary functions, one for the upper and one for the lower side of the aerofoil. These functions are dependent on time, hence the aerofoil can perform oscillating movement, which is the case when subjected to utter. The governing equations for the ow are the Euler equations. By assuming thin aerofoil, small angle of attack and that the perturbation of the wake is small, the problem is linearised.

It is shown that the linearised Euler equations can be rewritten as the Cauchy-Riemann equations, and an analytic function exists where its real part is the horizontal velocity component and its imaginary part is the vertical velocity component with opposite sign. The ow eld is then investigated in the complex plane by making an appropriate branch cut removing all discontinuities, and with restrictions on the analytic function such that the kinematic and boundary conditions are satis ed.

By using Cauchy’s integral formula an expression for the anti-symmetric part of the analytic function is achieved. A general expression for the velocity jump over the wake is obtained, which is applied to the speci c case of harmonic oscillations for a symmetric aerofoil. In the end three types of utter is investigated; twisting oscillations around the centre of stiness, vertical oscillation, and aileron flutter.

Source: KTH

Author: Barman, Emelie

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