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\begin{document}

\title{Application of Computational Fluid Dynamics\ to Downwind Sail Flows}
\author{Stephen Collie}

\chapter{Parametric Study}

\section{Introduction}

\label{intro}In this chapter CFD is used to examine the performance of
different downwind sail designs in two dimensions. By varying draft and
camber a range of sail shapes are generated that cover the design space. In
order to reduce simulation times only a solitary downwind sail is used,
i.e., the influence of the mainsail is ignored. The base sail shape is also
tested with a mainsail included in order to establish the impact of the
mainsail presence and to determine whether downwind sails can be designed
(in two-dimensions) without considering the presence of the mainsail.

\subsection{Sail sections}

Figure \ref{section} illustrates the method for defining the sail sections
that is used by North Sails LTD. The shapes are described by the following
parameters: draft, camber, front percentage, back percentage, leading edge
slope and trailing edge slope which are defined as follows:

\begin{description}
\item \textbf{Camber} = $\frac{y_{\max }}{c},$ where $y_{\max }$ is the
greatest perpendicular distance between the sail and the chord line.

\item \textbf{Draft }= $\frac{x_{d}}{c},$ where $x_{d}$ is the chordwise
location of $y_{\max }.$

\item \textbf{Front percentage }= $\frac{y_{front}}{y_{\max }},$ where $%
y_{front}$ is the perpendicular distance between the sail and the chord line
halfway bewteen the leading edge and $x_{d}.$

\item \textbf{Back percentage} = $\frac{y_{back}}{y_{\max }},$ where $%
y_{back}$ is the perpendicular distance between the sail and the chord line
halfway bewteen $x_{d}$ and the trailing edge.

\item \textbf{Leading edge} \textbf{slope} is the slope of the tangent to
the sail at the leading edge.

\item \textbf{Trailing edge slope} is the slope of the tangent to the sail
at the trailing edge.
\end{description}

\FRAME{fhFU}{5.2395in}{2.1818in}{0pt}{\Qcb{A sail section with it's defining
geometry.}}{\Qlb{section}}{section.bmp}{\special{language "Scientific
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"XNPEU";}}Using these parameters the sail shape is generated using two
fourth order bezier curves. The two curves meet at the maximum draft
position ($x_{d}$), and at this point the curvature is forced to be
continuous.

\subsection{Design parameters}

Four different camber values were tested: 21\%, 23\%, 25\% and 27\%, for
each camber setting the draft was set at 40\%, 45\% and 50\%. In total this
gave 12 different section shapes. The design with 23\% camber and 45\% draft
was based upon a horizontal slice from a Team New Zealand gennaker under
load. The range of cambers investigated is illustrated in Figure \ref{camber}
where four sail shapes with 21\%, 23\%, 25\% and 27\% camber are plotted,
each sail has 45\% draft. In figure \ref{draft} three sail shapes are
plotted with 40\%, 45\% and 50\% draft, each sail has 23\% camber. For all
sails tested in this study the leading and trailing edge slopes were fixed
at 60 and 50 degrees respectively and the front and back precentages were
set at 82.45\% and 79.87\% respectively. The sails shapes are identified
using a naming convention of the format XXYY where XX is the camber and YY
is the draft, i.e., section 2345 has 23\% camber and 45\% draft.\FRAME{fhFU}{%
5.0286in}{1.3815in}{0pt}{\Qcb{Comparison of sails of varying camber (21\%,
23\%, 25\%, 27\%) with draft fixed at 45\%.}}{\Qlb{camber}}{camber.wmf}{%
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1.3815in}{0pt}{\Qcb{Comparison of sails of varying draft (40\%, 45\%, 50\%)
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\subsection{Analysis proceedure}

When sailing downwind IACC\ yachts experience an apparent wind direction
that can be anything from 60$%
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$ to 120$%
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.$ In average conditions the apparent wind angle is usually around 90$%
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$ and consequently the lift force is the primary contributor to the yachts
driving force and the drag force has more influence on the yachts heeling
moment and side force. Therefore downwind sails are designed and trimmed in
order to maximise lift while paying little consideration to drag. Even when
sailing at deep angles where the drag contributes to driving force (i.e
above 90$%
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$), the increase in drag that is experienced at angles above the maximum
lift angle (i.e post stall) does not succesfully compensate for the loss of
lift due to stall. Consequently downwind sails are always trimmed to maximum
lift.

As a yacht sails downwind the wind strength and direction is continually
changing - at times dramatically - and consequently both the yacht's
direction and the sail trim must be continually adjusted. However through
all these changes the sails are still trimmed to achieve maximum lift and
hense provided the section shapes of the sails remain constant (or a sail
change does not take place) then the angle of attack of the wind to the
chord line of the sail remains relatively constant. Consequently it is
possible to get a good idea of the performance of a particular sail shape
through all conditions by deteriming it's lift and drag coefficient at
maximum lift and resolving the drive and side forces across a range of
apparent wind angles. This does however neglect and Reynolds number effects
that may occur as the apparent wind speed changes.

For the current study the force coefficients were measured at four different
angles of attack around the angle of maximum lift. For most sail sections
the maximum lift angle was approximately 20$%
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$ $\pm 2%
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$ and suitably the forces were evaluated at $\alpha =$15$%
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$, 17.5$%
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$, 20$%
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$ and 22.5$%
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.$ The forces were then fitted with a cubic polynomial and the lift
coefficients at maximum lift were determined from this fit. In order to
evaluate the accuracy of this method simulations were preformed at several
other angles of attack for the base design (camber = 23\%, draft = 45\%).
The results of this investigation are presented in Section \ref{sect 2345}.

\subsection{The CFX\ model}

In this study the sail sections were modelled as a infinately thin wall in
the same fashion as in Chapter \ref{Chapter validation} and the domain
dimensions are illustrated in Figure \ref{sail_domain}. At the inlet
cartesian velocity components are specified according to the angle of
incidence. For all simulations the chord lenth was set a 1$m$ and the
Reynolds number based on chord length was set at $3308706$ which is a
typical Reynolds number for IACC\ spinnakers and gennakers. For a typical
downwind sail (chord length $\simeq $ 14$m$ at mid grith) this Reynolds
number corrosponds to apparent wind speed of approximately $3.5m.s^{-1}$ or
7 knots. For IACC\ yachts 7 knots of apparent wind is obtained in 10 to 15
knots of true wind speed which is an average wind range for the Hauraki
Gulf. At the inlet the freestream turbulence intensity was set at 1\% with a
length scale of 0.001$m$. [mention earlier that solutions are independent of
freestream turbulence.] At the outlet a zero static pressure gradient
boundary condition was imposed.

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in ICEM-HEXA. Three levels of refinement were used and the grids are
referred to as coarse, medium and fine with $13200$, $55380 $ and $225940$
cells respectively. A grid convergence study is presented in Section \ref%
{sect GC}. At each refinement level the number of nodes along every block
boundary was doubled. The medium grid is illustrated in Figure \ref%
{design_grid}. In order to achieve a $y^{+}$ of approximately $1.0$ the near
wall spacing was set at $6.25\times 10^{-5}c$. Particular care was taken to
provide high quality cells around the leading and trailing edges and in
these regions the cells have an aspect ratio of $1:1$, the leading edge
region is illustrated in the close-up view in Figure \ref{design_grid}.
Further down the chord very high aspect ratio cells are used in order to
resolve the large flow gradients normal to the wall.\FRAME{ftbpFU}{6.0277in}{%
5.028in}{0pt}{\Qcb{Computational grid for the design study (section 2345)
(medium grid). }}{\Qlb{design_grid}}{grid.wmf}{\special{language "Scientific
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For all simulations the SST turbulence model was used. Time integration was
performed using second-order backward euler time stepping and interpolation
of the advection term was achieved using the high resolution upwind scheme.

\section{Results}

\subsection{Time step convergence study}

Time step convergence was investigated by starting with a time step of
0.00125$s$ and then repeating the simulations with the time step halved and
then halved again, i.e. the time step sizes used are 0.00125$s,$ 0.000625$s$
and 0.0003125$s.$ Simulations were performed at each angle of attack\ ($%
\alpha =15%
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,$ $17.5%
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,$ $20%
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$ and $22.5%
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),$ however only the results for $\alpha =15$ are presented since this is
the case with the fewest time steps per shedding cycle. For larger angles of
attack the time step convergence was better than the $\alpha =15$ case.\ In
Figure \ref{design_TC} the lift and drag coefficients are presented for the
sequence of time step sizes. 
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\caption{Time step convergence of the lift and drag coefficients
($\alpha=15$).}\label{design_TC} 
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As the timestep size is reduced both the lift and drag coefficients
level-off indicating satisfactory time step convergence. The lift for the
solution using the medium time step is within 0.075\% of the lift computed
using the short time step and the drag is within 0.19\%. The medium time
step size corresponds to approximately 33 time steps per shedding cycle for
the $\alpha =15$ case and 54 time steps per shedding cycle for the $\alpha
=22.5$ case.

\subsection{Grid convergence study}

\label{sect GC}

The grid convergence study was performed at $\alpha =20%
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$ which is approximately the angle where maximum lift occurs. Lift and drag
coefficients are presented in Figure \ref{design_GC} for the sequence of
grids. The force coefficients are plotted against $1/N[m]$ which is a
measure of the grid spacing where $N^{2}$ is the total number of cells in
the grid.%
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\caption{Grid convergence of the lift and drag coefficients.}\label%
{design_GC} 
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As the grid is refined there is a clear convergence of the lift and drag
coefficients as the grid is refined. The lift from the medium grid solution
is within 0.9\% of the fine grid solution and the drag is within 3.33\%. The
accuracy of the medium grid solution is not entirely pleasing with the drag
in particular showing considerable grid dependence. However the drag has
little influence on the performance of a sail section and many simulations
were required in this study and hense using the fine grid would have been
too time consuming. Hense it was decided that the medium grid was the most
suitable gird, and that it would at least provide sufficient accuracy for
qualitative analysis of the design space.

\subsection{Behaviour of the base section shape}

\label{sect 2345}

The lift versus angle of attack characteristics for the 2345 section is
presented in Figure \ref{forces_2345}. Also plotted in red is a polynomial
fit through the data points at 15$%
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,$ 17.5$%
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,$ 20$%
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$ and 22.5$%
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$ (highlighted in blue). The fitted curve estimates the maximum lift
coefficient to be 2.62 and to occur at 20.22$%
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$ and the fitted curve agrees well with the data points at 18.75$%
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$ and 21.25$%
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.$ The lift curve in Figure \ref{forces_2345} has the same general shape as
the was found in the validation study (Chapter \ref{chapter validation}). 
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\bigskip

\bibliographystyle{PHREPORT}
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\end{document}
