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


method presented here is to combine the inverse 2D thin aerofoil theory, described in section 2, and the successful vortex lattice method analysis dimensional


to tackle the three- inverse problem. An approach was


envisaged that uses an iterative inverse process where the sail shape is modified in such a way that with each iteration, successive pressure maps becomes closer and closer to the target pressure distribution.


The way the sail shape is modified is based on thin aerofoil theory. Consider a comparison of two aerofoils with the same chord but at different angles of attack and with different cambers. For each aerofoil, equation (6) is applied.


Aerofoil A : Cp


Aerofoil B : Cp


Δ= ⎛⎞ − +() () n=1


AA A,0


4tan⎜⎟ 2


⎡⎤ θ α a


⎣⎦ ∑


n


⎢⎥ ⎝⎠


a ,A n sin nθ


Δ= ⎛⎞ − +() () n=1


BB B,0


4tan⎜⎟ 2


⎡⎤ θ α a


⎣⎦ ∑


n


⎢⎥ ⎝⎠


a ,B n sin nθ


Now subtract equation (8b) from (8a) which gives: ()


⎡⎤ α−


delta Cp


Δ=⎢⎥ ⎢⎥ +


4


⎢⎥ ⎝⎠


tan 2


⎜⎟ delta delta a n


⎛⎞ θ


⎢⎥ ⎣⎦


∑delta a nθ n=1


n sin


() 0


This approach gives a relationship between the pressure distribution difference (delta ∆Cp), the difference in angle of attack (delta α) and the difference in camber (delta a’s) of the two aerofoils (A and B). Equation (8c) is a re-expression of (6).


For easier understanding equation (8c) is simplified to : [


d ** ]' elta Cp P delta Q delta a s Δ= +]α [ (9)


where P and Q correspond to the trigonometric terms of equation (8c).


As discussed earlier, the inverse process does not


produce a unique solution. Either the camber or the angle of attack (or twist of the sail) has to be specified. Then the solution obtained is unique.


3.2 INVERSE PROCESS WITH FIXED TWIST AND SPECIFIED PLANFORM


3.2 (a) Solution procedure to find camber If the twist is fixed, equation (9) becomes: d *'


elta Cp Q delta a s Δ= (8c)


Does the pressure


distribution of the updated sail shape match the target?


Yes Output: The sail shape.


Figure 6 - Flow diagram for the 3D inverse sail design method.


This approach is exactly the same as described in [5]. Note that the inverse thin aerofoil theory which is embedded in this three-dimensional approach is two- dimensional in nature, which means that this theory cannot account for any spanwise flow components due to tip vortices and heel. Thus some errors where three- dimensional effects are strong, e.g. at the head and the foot of the sail, can be expected. To avoid this problem, the inverse thin aerofoil theory can be applied to only the central 90% of the sail and interpolation can been used at the head and foot to determine the sail shape there. This procedure was found to be numerically stable.


(10)


Any CFD code can be used in the proposed method shown in figure 6. In the present work, a vortex lattice method code has been used. This vortex lattice method code has been developed by the authors and is based on


©2010: The Royal Institution of Naval Architects B-111 No (8b)


Apply equation (9) to get the camber difference.


Update the sail shape. (8a)


In the 3-D inverse design process, equation (10) is applied to the difference between the desired pressure map and the pressure of an initial sail shape at different sail sections along the mast. The calculated camber difference from equation (10) is then added to the initial shape to give an improved shape whose pressure distribution is closer to the desired one. This process is repeated until the generated sail produces the desired pressure map. The process is illustrated in figure 6.


Inputs: Initial shape, Initial pressure distribution, Target pressure distribution.


Subtract the pressure distribution of the initial sail from the target pressure distribution to get the delta ∆Cp.


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