Trans RINA, Vol 152, Part B2, Intl J Small Craft Tech, 2010 Jul-Dec
account in the linear system and ensures a realistic aerofoil is obtained
The zero camber condition at the leading and trailing edges when an aerofoil is approximated by a polynomial of degree “m” is given by
m
∑ ∑
2
n=0 m+1
2 n=0
where “C” ’s are the coefficients of the polynomial. The linear system is thus over-determined and can be solved using a least squares approach.
An advantage of the inverse thin aerofoil theory over an inverse process using a panel code is that the method is direct. In the panel code, the singularities are situated on the camber which is unknown in the inverse method. Thus an iterative approach has to be employed. As explained by O’Brien [17], a pressure distribution on an arbitrary aerofoil is calculated and then the geometry is modified in a systematic manner until the specified pressure is obtained. This inverse method involves a larger time computation than the inverse thin aerofoil theory.
2.2 (b) Validation
The following example deals with a sail at a fixed angle of attack (6°) and unknown camber. As mentioned in section 2.1, a 4th degree polynomial
accurate enough to describe its camber. The
is assumed to be user
specifies a pressure distribution at a given position along the chord. It should be noted that the theory works for any specified pressure distribution even if unrealistic. Thus the user has to know if the specified pressure distribution leads to flow separation or not. If it does the current approach is
not suitable to generate the
corresponding sail shape because it is based on potential flow theory. To avoid that problem in this example, the given pressure distribution comes from the analysis of an existing sail shape and is thus a realistic, separation-free pressure distribution. This pressure distribution is shown in Figure 4 along with the points selected for input to the inverse process. 4
Thin aerofoil theory Desired pressure distribution
C C
2n+1 2n+1
() ()
0.5 0.5 21 n+ 21 n+
= =
0 (m even) (7) 0 (m odd) 8% 12%
Thin aerofoil theory Inverse thin aerofoil theory
The result from the inverse thin aerofoil theory is compared to the camber used in the forward analysis in figure 5. As it can be seen, the match between the camber-lines is perfect and thus. the inverse process is able to replicate the original geometry.
4%
0% -0.5 0 X [-]
Figure 5 - Comparison of initial camber and that generated by the inverse method described.
In practice the user will not specify a pressure
distribution that was the result of the thin aerofoil theory. In such cases, the inverse method developed calculates the camber-line which will give the closest pressure distribution to the one specified. This is accomplished by using equation (6) and a least squares approach.
3.
THREE-DIMENSIONAL INVERSE PROCESS
As stated previously, the solution of the inverse problem is not unique. There are an infinite number of sail shapes which can generate a given pressure distribution. For this reason, some sail parameters have to be fixed in order to generate a unique solution. Furthermore, this allows the designer to control the process. In sail design, three variables are needed to define a sail shape:
• planform • camber • twist
In this paper the planform is always specified as an input, and this is justified as the planform is usually determined by the class-rules for a yacht.
2
3.1 OUTLINE OF THE INVERSE DESIGN METHOD
0 -0.5 0 X [-] Figure 4 - Desired pressure distribution across the sail. B-110 0.5
The goal of the inverse design process is to find a sail shape which will produce a given three-dimensional pressure distribution. Codes based on the Vortex Lattice Method have proven to be powerful tools in upwind conditions [19]. Those codes are based on the same hypotheses as thin aerofoil theory, i.e. on potential flow. The idea behind the three-dimensional inverse design
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