Trans RINA, Vol 153, Part B2, Intl J Small Craft Tech, 2011 Jul-Dec
to use second-order numerical schemes rather than first- order ones. For more details on best practices to predict the right qualitative behaviour of lift and drag force coefficients as functions of sail camber, angle of attack, mast diameter, see following references [19, 20]. Best practices should also be considered when conducting more complex simulations around three-dimensional sails or rigs.
Clmax = f(f/c)
2,5 3
1,5 2
0,5 1
0 5% 10% 15% 20% 25% 30%
Figure 4: Clmax versus f/c ranging from 5% to 30%. (O1) first-order scheme, (O2) second-order scheme
fmax = f(f/c)
100 120 140
20 40 60 80
0 5% 10% 15% 20% 25% 30%
Figure 5: (L/D)max versus f/c ranging from 5% to 30%. (O1) first-order scheme, (O2) second-order scheme
7.2 OPTIMAL SINGLE SAIL
From Bethwaite [39], it is known that there exists an optimum sail camber for a given mast. This observed fact has been chosen as a test case to validate ADONF and the implemented optimization algorithms for sail design questions. For a single sail, the optimization problem may be formulated as follows: for a given apparent wind angle, what is the optimal camber and related trim angle which maximize the driving force Fr? The apparent wind angle chosen was = 30° a typical
O1 O2
O1 O2
value for upwind conditions. Other sail parameters are listed in the following table:
Table 1: sail parameters of the optimization problem xf
C (f/c)0 0 30% 6500 7% 13°
A gradient based algorithm, known as the Simplex method, is used. The optimization problem has been resolved by computing solutions based on the RANS model. The computation of twenty designs has been sufficient to obtain a good convergence of the sail camber, f/c, and trim angle, (Figure 6a, 6b, 7). The following optimal solutions for maximum driving force and maximum lift-to-drag ratio have been found:
Table 2: results of the 2 optimization problem resolved Objective (f/c)*
*
Max(Fr) 18% 22° Max(L/D) 8% 27°
The number of RANS simulations needed to determine the optimal solution is dependent on the number of variables. In this example with only two variables, the convergence to the optimal solution is fast. It has been verified, by changing the initial condition,
that the
optimal solution was independent to the initial condition. An example of algorithm convergence is given in Figures 6 & 7 for the two parameters: camber, trim angle and the objective function driving force.
It is interesting to note that the optimal solution, maximizing the driving force, presents a separation point near the trailing-edge on the suction surface (Figure 8). This clearly illustrates the ability of viscous CFD to make a trade-off, between high lift by high camber and high angle of attack and low drag by low flow separation on the suction side of the sail, through RANS simulations.
Figure 6a: convergence of the trim angle versus the number of explored design
©2011: The Royal Institution of Naval Architects B-109
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