Trans RINA, Vol 152, Part B2, Intl J Small Craft Tech, 2010 Jul-Dec


resulting n aerodynamic loads. These loads are then fed into a Velocity Prediction Program that predicts the speed of a yacht using those sails. The one sail


2. that


makes the yacht go fastest is then selected and built. Such an approach is very time-consuming because n sails need to be designed and analysed. In the end only one sail is the “optimum” design while all other sails are discarded. This procedure involves trial and error and is illustrated in figure 1.


2.1 THIN AEROFOIL THEORY 2.1 (a) Theory


n flow simulations


n vpp runs


Thin aerofoil theory constitutes one way of calculating the flow around an aerofoil. The derivation of this theory implies an aerofoil with an infinitesimally small thickness and a small camber with respect to the chord. Taking into account these assumptions, the aerofoil can be represented by a continuous sheet of vortices situated on the chord. This interesting idea was introduced by Birnbaum and Ackermann [6] and Glauert [7].


Best of the n sails n sail geometries n pressure distributions


Figure 1 - The analysis approach. A different, more scientific approach is


The camber-line must be a streamline of the flow, so the strength of the vortex sheet is determined such that the component of the velocity normal to the camber-line is zero along the camber-line. This leads to the fundamental equation of thin aerofoil


theory [8,9], given here as sketched in


Figure 2 and is based on combining aerodynamic optimisation techniques with an inverse design approach. The process starts with an aerodynamic optimisation that expresses the requirement of maximum boat speed in terms of an optimal pressure distribution over the sails. This part of the process is called the “optimisation”.


An inverse design algorithm then determines the geometry of the sail that realises this optimal pressure distribution. This is called the “inverse design” part of the project. The resulting sail is the “optimal” sail that will make the yacht go fastest. As opposed to the “analysis” method, the method from Figure 2 is a direct approach and guarantees that the best possible sail shape is found quickly.


Specification of maximum


possible boat speed


Speed prediction + Aerodynamic


optimisation to get the best pressure distribution


Inverse design


To get the best sail shape


Optimised pressure Figure 2 - The inverse sail design approach.


This paper only considers the inverse design part of the process, i.e.


the right-hand part of Figure


“optimisation” (left-hand side of the figure) is the subject of on going research and not considered here.


This paper extends previous research [5] and describes improvements to the inverse method for a single sail in an upwind condition.


The next two sections review the inverse design method in two-dimensions and in three-dimensions.


2. The Best of the n sails


equation (1). ()


2πη∫ x dU dz 1


c 2 −c 2 −


∞ ⎢ ⎣


The left-hand-side of


γη ηα ⎥=− ⎤ ⎡


dx⎦ (1) this equation represents the


downwash induced by the vortex sheet on the chord (and also approximately on the camber). The right-hand part of the equation is the velocity induced by the angle of attack and by the camber of the aerofoil. In order to make the camber a streamline both terms must be equal.


An elegant solution of this equation can be expressed using Fourier series. The transformation from cartesian to polar coordinates is made using:


xbco ( )s θ = (2)


Note that the representation of the aerofoil herein is slightly different from usual descriptions. In the present work, the x-coordinate varies from –c/2 to c/2.


Johnson [10] and Söhngen [11] have shown that, if the Kutta condition is satisfied, the result of substituting equation (2) into equation (1) is:


γ αθ ∞


=− +() (Ua an sin n )⎥ ⎣ ⎝⎠


2tan


∞ ⎢ ⎜⎟ 2


⎡ ⎛⎞ θ 0 ∑ n=1


⎤ ⎦


(3)


Where the a’s are the Fourier coefficients of the slope of the camber-line. As can be seen, this series satisfies the Kutta condition because the circulation at the trailing edge is zero. For a given camber-line (of known slope) the Fourier coefficients “a’s” can be determined from the theory of the Fourier series as described below.


TWO-DIMENSIONAL INVERSE PROCESS


B-108


©2010: The Royal Institution of Naval Architects


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