**2.1 Unsteady motion of two dimensional airfoil in incompressible inviscid flow**

In this subchapter we shall be looking at many ways in which to solve the problem of unsteady incompressible flow over an aerofoil. The flow being incompressible is a great simplifier to the problem, this allows to take many of the results of steady flow as read. It is still however, not a trivial problem.

Solution of flow over airfoil (moving arbitrary), depending on time and starting from time t=0 is calculated in successive time intervals tk (to= 0, k=1,2,3...). Fig. 1 shows model for the time tk.

the free stream direction) is attached to the trailing edge. Length k and angle

arbitrary in first iteration. Their values will be determined as the part of the solution.

*kw k k* <sup>1</sup> *<sup>k</sup>*

Hence, circulation on the element is equal to the difference between circulations around airfoil in times tk-1 and tk, assuming that k-1 has been already determined. Vortex wake consist of concentrated vortices formed by vorticity shed at earlier times, which is assumed to be transformed into discrete vortices. Concentrated vortices is moving with resulting velocity calculated in the center of each vortex at each successive time interval. Therefore, strength and positions of discrete vortices are regarded as known at time tk, and there are

k is the same for each element on airfoil and k denotes time tk. Total circulation

k at the time tk,.

w)k, where:

i)k and 

(1)

( ) 0 *Vnj k* (2)

k are uniform source

k are

i)k varies from one element

k in regard to the x-axis (to

**2.1.3 Discretization and numerical solution procedure** 

Airfoil contour in time tk is replaced by N linear elements. (

<sup>k</sup> (airfoil perimeter).

Circulation in trailing edge vortex wake element is (

i)k (i=1,2,...,N),

They are determined by satisfying following conditions:

k, k and

N conditions of zero normal velocity component in external middle point at each

to another,

k is given as

N+3 unknowns (

Fig. 1. Solution at time tk

segment of airfoil

and circulation distributions on i-th element (i=1,2,...,N), where (

An elementary vortex wake with length of k and pitch angle of
