Reissner’s Approach

The following assumptions are going to be made to simplify the integrals.

i) Similar to the lifting line theory, we assume the wing is loaded as quasi two dimensional at any spanwise station y.

ii) The chordwise wake vortex is projected forward from the trailing edge to a spanwise line passing through the point where the downwash is to be calculated.

iii) The spanwise vortex of the wake which deviates from two dimensional behavior can be projected up to a line passing through the calculation point.

Let us see now, the simplifications of the terms of Eq. 4.47 with following assumptions.

The integral in K(q) is named the Cicala function with its argument being

q = x(y – g)-

Let

We can see the difference between the two dimensional lifting pressure coef­ficient 3.23 in Chap. 3. Here, r is also a function of C(k) and shows us the spanwise variation of the circulation.

The aerodynamic coefficients can be calculated using the Reissner’s theory by the following steps.

For simple harmonic motion; (i) if only bending is considered: h(y*, t) = heix‘/h(y*), (ii) if torsion about an axis is considered: a(y*, t) = aeixt/a(y*), are employed.

1) Since the reduced frequency and the wing geometry is known l(k) and XX(2)(y*) are determined to solve 4.51 to find XX(y*).

2) 2) XX(2) (y*) and XX(y*) are known, r is determined.

3) At any station y* the aerodynamic coefficients are found using 2-D theory.

4) These coefficients are corrected with known values of r as the 3-D solution, as follows

ALh(y*, t) = 2npU2b0[ikrh(y*)]h(y*, t)/ba ALa(y*, t) = 2npU2b0[ik(1/2 – a)aa(y*)ha(y*, t).

Summary of the Reissner’s Theory:

i) Compared to a 2-D case, non circulatory term does not change

ii) At the wing tips non circulatory terms can contribute

iii) As compared with the experimental values for rectangular wings good agreement is observed for the aspect ratio values down to 2.

During experiments it is difficult to reduce the viscous effects on oscillating wings. However, at high reduced frequencies these effects are expected to be low. In their numerous experimental and computational work, Reissner and Stevens have shown that the finite wing effects can be neglected depending on the reduced frequency and the aspect ratio values. In summary:

1) For the wings with an aspect ratio around 6 if the reduced frequency is higher than 1, and for the wings with an aspect ratio around 3 if the reduced frequency is higher than 2, 3-D effects can be neglected.

2) For the wings with an aspect ratio around 6 if the reduced frequency is less than 0.5, and for the wings with an aspect ratio 3 if the reduced frequency is lees than 1, 3-D effects can not be neglected.