Trans RINA, Vol 152, Part B2, Intl J Small Craft Tech, 2010 Jul-Dec
It is assumed that the camber can be described by a 4th degree polynomial, i.e.
Z CCX + + + 4 01 2 =+ C X C X C X
23 4 3
(4)
A relationship between the Fourier coefficients “a’s”, and the polynomial coefficients “C’s”, can be found from Fourier Series theory and is given in [12].
The results for the Fourier coefficients in terms of the coefficients of a 4th degree polynomial describing the camber are:
aC b C abC
01 3 =+
3 2
ab C ab C
34
12 4 2
23 3
=+ = =
23 3 2
b C 3 (5) 0 -0.5 0 X [-] Figure 3 - Comparison of pressure distribution.
where the “C’s” are the coefficients of the polynomial which approximate the camber and the “a’s” are the coefficients of the Fourier series describing the slope camber-line shape.
From the bound circulation, the general expression for the pressure difference between the lower and upper surfaces of the aerofoil can be computed and is given by equation (6):
Δ= = U∞
Cp
2 4tan () ( ⎛⎞
γθα⎜⎟ − + 2
⎢⎥ ⎝⎠
⎡⎤3 a nθ)
⎣⎦ ∑
a0 n=1
Due to the nature of the first term in this series the pressure difference becomes singular at the leading edge where (θ⁄2)=π.
This suction peak can be avoided if α-a0=0. Under these conditions the aerofoil is said to operate at its ideal angle of attack. The ideal angle of attack is important in sail trimming because at this particular angle the stagnation point is right on the leading edge. Thus the flow does not have to negotiate this sharp edge and the risk of flow separation is minimised.
2.1 (b) Validation
The thin aerofoil theory described above has been compared to the results the two-dimensional panel code of Bailey [13]. The aerofoil is a section shape of an IACC headsail (1/4 height genoa) of San Diego vintage (1992, 1995) and was taken from Bailey’s thesis. The sail-section has a camber of 14.8 % of the chord length and a draft of 39,1% of the chord length. This camber-line can accurately be approximated using a 4th order polynomial (equation (4)) with the coefficients given in table 1. The angle of attack is 6 °.
n sin (6)
The agreement between the Bailey panel code and the present theory is good. However a marginal difference exists. This can be explained by the fact that the two- dimensional panel code involves a lattice of singularities situated on the camber-line [14,15,16], and thin aerofoil theory implies a vortex sheet placed on the chord. In addition it is known that panel codes give different results depending on where the control points are located.
2.2 INVERSE THIN AEROFOIL THEORY 2.2 (a) Theory In the inverse design process, the
0.5 2 2
Table 1 - Shape-coefficients for head sail. C0
C1 0.101 C2 C3 -0.082 -0.320 0.328
C4 -0.340
Figure 3 illustrates the pressure distribution along the Bailey sail. 4
Bailey Thin aerofoil theory
aerodynamic
characteristics are specified first, and the shape required to obtain them is computed. In this section the pressure distribution becomes the input and the camber-line shape is the result. In practice the desired pressure distribution is determined by some procedure and then different points on this pressure distribution are selected. The number of points corresponds to degree of the polynomial (“n”) describing the
future sail section.
Equation (6) is then applied to these “n” points. This generates a linear system of equations.
An analysis of this linear system shows that there is not a unique solution. From a given pressure distribution, the inverse thin aerofoil theory cannot compute both the camber of the aerofoil and its operating angle of attack. Thus, to ensure a result, either the angle of attack or the camber has to be specified. The inverse thin aerofoil process finds
the camber which generates a given
pressure distribution at a specified angle of attack. Or, alternatively, this method finds the angle of attack of a given aerofoil, which will produce the given pressure distribution. In the analytical approach the aerofoil is defined with zero camber at the leading and trailing edges. This boundary condition has to be taken into
©2010: The Royal Institution of Naval Architects B-109
dCp [-]
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