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\begin{document}

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

\chapter{Preliminary Wind Tunnel and CFD\ Investigation}

\section{Introduction}

The review of sail flow research that was presented in\ Chapter \ref{intro}
highlighted the lack of available data for sail sections and the limited
understanding of the flow features. Moreover there have been no wind tunnel
tests conducted for downwind sail shapes (camber \TEXTsymbol{>} 20\%). This
chapter describes preliminary wind tunnel tests and CFD results for a
two-dimensional downwind sail section. The goal of the study was to obtain
insight into the behaviour of two-dimensional sail flows and to gain an
initial guage of the performance of different turbulence models. This
preliminary investigation was used to form the direction and goals of the
subsequent research.

\subsection{Wind tunnel setup}

Wind tunnel experiments were carried out at the De Bray wind tunnel in the
University of Auckland's Yacht Research Unit. This wind tunnel is a
low-speed, closed loop facility with a test section of dimensions $%
768mm\times 615mm$, a maximum speed of $60m.s^{-1}$ and a maximum turbulence
intensity of $1\%$. Figure \ref{tunnel_model} illustrates the model \ which
is positioned on aerodynamic struts above the force balance which itself is
mounted horizontally below the test section.

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From analysis of IACC\ sail shapes it was evident that a circular arc
represents a good approximation of a range of sail sections whilst providing
a shape that is easy to manufacture. Therefore for this initial study a
circular arc section was used with 24.7\% camber, the model has a radius of
200mm and a chord length of $319mm$. 1mm steel plate was rolled to the
desired shape with a radial tolerance of less than 3mm which is equivalent
to a camber variation of 0.6\%. Steel strengthening sections were soldered
to the underside of the model in order to minimize deflection under load and
to provide an attachment point for the struts. Pitch was controlled using
the rear strut and a rangle of different angles were investigated from 5%
%TCIMACRO{\U{b0} }%
%BeginExpansion
${{}^\circ}$
%EndExpansion
through to 30%
%TCIMACRO{\U{b0} }%
%BeginExpansion
${{}^\circ}$
%EndExpansion
at 2.5%
%TCIMACRO{\U{b0} }%
%BeginExpansion
${{}^\circ}$
%EndExpansion
intervals.

The arrangement of the wind tunnel model within the tunnel is illustrated in
Figure \ref{tunnel setup}. At each end of the model end plates were
positioned that span the height of the tunnel. These end plates were
positioned 165.5$mm$ inside the side walls of the tunnel to allow the
boundary layer on the tunnel walls to pass without influencing the model.
The leading edges of the end plates were fixed at $448mm$ in front of the
model and extend $711mm$ past the trailing edge. The model itself is $435mm$
($1.37c$) wide and small gaps ($<2mm$) exist between the tips of the model
and the end plates. Tape and petroleum jelly were used to limit tip leakage
through these gaps. Particular care was taken to ensure that friction
between the model and the side walls did not influence force measurements.
This was verified by measuring the forces when fixed weights were applied to
the model.\FRAME{ftbpFU}{5.6749in}{2.744in}{0pt}{\Qcb{The wind tunnel model
setup (from above, not to scale).}}{\Qlb{tunnel setup}}{tunnel_setup.bmp}{%
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The model was tested at an inlet speed of $25m.s^{-1}$ which corresponds to
a Reynolds number of 5.25$\times $10$^{5}$. This Reynolds number is
significantly lower than the typical Reynolds number range of $2-4$ million
for IACC sail flows. However at these Reynolds numbers it is believed that
the flow should be reasonably independent of Reynolds number and validation
of CFD models at the Reynolds number used in the wind tunnel should also
apply to full scale sail flows. The main reason that Reynolds number
insensitvity can be expected is the presence of the leading edge separation
bubble. As the shear layer separates at the leading edge transition occurs
rapidly and conequently the boundary layer reattachment process is always
turbulent and independent of Reynolds number. In the experimental study of
flat plate leading edge bubbles of Crompton and Barrett the shear layer was
observed to develop turbulent traits shortly after detachment at the leading
edge \cite{Crompton01}. For that study Reynolds number independent results
were obtained above a Reynolds number of 2.2$\times $10$^{5}$.

\subsection{The CFD Model}

A schematic of the domain is illustrated in Figure \ref{domain}. Due to the
size of the model in relation to the wind tunnel dimensions it was felt that
it was necessary to include the walls in the CFD\ model. Simulations were
carried out using both non-slip and free-slip boundary conditions on the
side walls and there was found to be no significant loss in accuracy when
free-slip conditions were used. Consequently free-slip boundary conditions
were used for all simulations presented in this chapter. At the inlet the
normal velocity was set to 5$m.s^{-1}$ in order to mimic the Reynolds number
of 5.25$\times 10^{5}$ that was used in the experiments. The tubulence
intensity and length scale were set at 1\% and 0.001$m$ respectively. At the
outlet a zero pressure gradient boundary condition was imposed.

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Unfortunately since the model rotates in relation to the tunnel walls a new
grid was required for each new angle of attack. The stuctured grids used in
this study were generated using ICEM-HEXA. To ensure that the grids
themselves were not a source of error a grid refinement study was conducted
(Section \ref{sect_GC}).Three levels of refinement were used with the grids
referred to as coarse, medium and fine with 8496, 28912 and 120482 cells
respectively and the medium grid is illustrated in Figure \ref{grid}. At
each refinement level the number of nodes along every block boundary was
doubled. In order to achieve a $y^{+}$ of approximately $1.0$ the near wall
spacing was set at $1.276\times 10^{-5}c$. Particular care was taken to
provide high quality cells around the leading and trailing edges and at the
very ends of the sail the cells have an aspect ratio of $1:2.5$. Away from
the leading and trailing edges high aspect ratio cells are used in order to
resolve the large flow gradients normal to the wall.\FRAME{ftbpFU}{4.0274in}{%
3.0277in}{0pt}{\Qcb{The computational grid for the preliminary study.}}{\Qlb{%
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\section{Results}

\subsection{Convergence studies}

\label{sect_GC}

Figure \ref{prelim_GC} shows the time-averaged force coefficients for the
three grids at an angle of attack of 15 degrees which was felt to be a good
representation of downwind sail flows. 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. The medium grid produced a lift
coefficient that was 1.6\% smaller than the fine grid result. Unfortunately
halving the grid spacing again would result in a grid that would be
impractical with regards to memory requirements and solution time. However
for practical purposes the medium grid is acceptable and within 3\% of the
extrapolated value for lift.\FRAME{fhFU}{4.0274in}{3.0278in}{0pt}{\Qcb{Grid
convergence of the time-averaged lift and drag coefficients.}}{\Qlb{prelim_GC%
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A convergence study was also carried out on the time step. Solutions were
compared using time steps of 0.025s, 0.0125s and 0.00625s. The lift
coefficient calculated using the medium timestep was just 0.65\% Lower than
the lift coefficient calculated using the short time step. For all
simulations the 0.0125s was used for the timestep, this corresponds to
approximately 32 iterations per period of vortex shedding. In all cases the
simulations were run untill the solutions had settled into regular veroidic
vortex shedding, i.e., the forces had constant means and amplitudes and the
period of vortex shedding was was stable.

\subsection{Wind tunnel - CFD\ comparison}

CFD and wind tunnel time-averaged force coefficients are presented in Figure %
\ref{prelim_forces}. In the experimental data a peak in the lift coefficient
exists at 10 degrees which is associated with the so-called ideal angle of
incidence where there is no leading edge bubble disrupting the flow. There
is a small drop off in lift above 10 degrees associated with the formation
of the leading edge bubble. Below 10 degrees the flow stagnates on the
leeward side and a separation bubble develops on the windward side of the
sail, for flexible sails in this situation the luff would collapse. Above 15
degrees the lift starts to increase again. At 30 degrees the model is fully
stalled (totally separated flow) and above this angle the lift coefficient
is expected to drop off once more.\FRAME{fhFU}{6.3404in}{2.2773in}{0pt}{\Qcb{%
Force coefficients versus angle of attack ($\protect\alpha =15)$. \textbf{%
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The same trends are evident in the CFD results. The ideal angle of incidence
is slightly lower (around 8 degrees) in the CFD, and at 30 degrees the lift
curve for the SST model is starting to level off. At this angle the leading
edge bubble detaches and reattaches in a semi-chaotic manner. At lower
angles vortex shedding is regular and a vortex street exist in the wake (see
Figure \ref{vortex street}).\FRAME{ftbpFU}{3.5276in}{2.6524in}{0pt}{\Qcb{%
Velocity contours and streamlines ($\protect\alpha $ = 15$%
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%EndExpansion
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"XNPEU";}}

Downwind sails are trimmed to the lowest possible angle achievable without
the luff collapsing due to stagnation on the leeward side of the leading
edge. Due to constantly fluctuating wind directions it is difficult to trim
sails to ideal incidence and instead the sails are generally set at 5 -- 10
degrees above the ideal angle of attack so that a stable luff is maintained.
If we assume an apparent wind of 90%
%TCIMACRO{\U{b0} }%
%BeginExpansion
${{}^\circ}$
%EndExpansion
so that we are looking to maximise lift while paying little attention to
drag then Figure \ref{prelim_forces} would indicate that the sails should be
trimmed to much higher angles of attack. However for the three-dimensional
case there is considerable downwash due to tip vortices, consequently the
ideal angle of attack is much higher and is closer to the angle of attack
where real sails are fully separated (stalled). For real sails this means
that maximum lift occurs at the ideal angle of incidence.

The general structure of the two-dimensional flows - both in the wind tunnel
and from CFD - compares well with streamlines visualised on
three-dimensional model sails near mid-span. Figure \ref{Overlay} illustates
a streamlines computed using CFD\ overlayed on a photo of smoke-lines from a
three-dimensional wind tunnel test. Observations from smoke visualisation
indicate that three-dimensional downwind sails generally separate at
approximately 40-50\% along the chord and two-dimensional CFD results
(without mainsail) indicate that angles of attack of between 20 and 25
degrees are required in order to obtain this degree of separation.\FRAME{%
ftbpFU}{4.318in}{3.2448in}{0pt}{\Qcb{Wind tunnel flow visualisation
overlayed with streamlines computed using CFD.}}{\Qlb{Overlay}}{overlay.jpg}{%
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In general the lift forces are overpredicted by CFD which is a commonly
reported shortcoming of turbulence modelling. Overprediction of lift is due
to large values of the eddy viscosity causing the boundary layer to remain
attached in adverse pressure gradients that would normally cause separation.
This effect has been documented for many similar flows such as the NACA 4412
airfoil at maximum lift \cite{Menter96}. However in such high-lift
validation studies comparisons between CFD and experiment have shown much
better agreement than the current comparison.

In our case the primary source of the error is attributed to
three-dimensionality in the wind tunnel tests. Surface oil-flow
visualisation was used and spanwise variation of the separation point was
clearly apparent and surface tufts illustrated significant cross-flow in the
separated region near the trailing edge. Upstream of separation the flow was
nominally two-dimensional, although surface oil streams did bend slightly
immediately upstream of the separation line. The three-dimensional
structures present in the wake originate at the tunnel walls where the
boundary layer on the wind tunnel wall interacts with the boundary layer on
the sail creating a horseshoe vortex around the leading edge of the sail.
Consequently there is a greater degree of flow separation near the tunnel
walls. This effect influences the flow right across the span of the sail.

A basic sketch of the flow structure is illustrated in Figure \ref{wake
sketch}. On the near side of the foil the flow visualisation results are
illustrated with the arrows representing the directions of surface tufts.
Oil streaklines are also shown, upstream of the separation line. On the far
side of the foil a schematic of the wake structure is illustrated. At
approximately quarter and three-quarter span longitudinal vortices exist in
the wake. This vorticity develops from the second (lower) separation bubble
of the vortex pair.

This wake structure was confirmed using a three-dimensional CFD model and
the streamlines are illustrated in Figure \ref{CFD streamlines}. The grid
that was used was based upon the coarse grid from the two-dimensional study,
with 60 grid points in the spanwise direction and a symmetry plane was used
at the midspan of the model. The surface plots of pressure coefficient and
skin friction in Figure \ref{surface plots} illustrate spanwise variations
and the skin friction contours closely follow the separation line patterns
observed in the surface oil-flow visualisation.\FRAME{fpFU}{4.0274in}{%
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The presence of the longitudinal counter rotating vortex pair causes
downwash near mid span and upwash near the tunnel walls. Through examination
of the streamlines upstream of the foil it is clear that the net effect of
the three-dimensional wake is to reduce the effective angle of attack to the
leading edge across the majority of the span. The leading edge bubble in the
three-dimensional simulation is nominally two-dimensional and of almost
constant length across 90\% of the foil's span (the bubble breaks down near
the side walls). At this angle of attack (15%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
) the leading edge bubble reattaches at 0.033$c$, for the equivalent
two-dimensional simulation reattachment occurred at 0.077$c$. The smaller
leading edge separation bubble in the three-dimensional simulation is due to
the lower effective angle of attack imposed by the three-dimensionality of
the wake.

From Figure \ref{prelim_forces} it is evident that the wind tunnel predicts
the ideal angle of attack to be approximately 2 degrees higher than the CFD
results indicate. This higher ideal angle of attack is due to downwash
induced by the three-dimensional wake. Despite using tape to seal of the
tips between the model the side walls, there is still likely to be a small
amount of tip leakage. Flow visualisation was performed without tape sealing
the tips and considerably more three-dimensionality was observed. Force
measurements were also taken with the tape removed and the discrepancies
with the CFD results were greater. Notably the ideal angle of attack was
approximately 2.5 degrees higher than the previous wind tunnel experiments.

Unfortunately the errors caused by the three-dimensionality of the wind
tunnel results are too large to allow reliable evaluation of the relative
performances of the turbulence models. The forces from the three-dimensional
CFD results (at 15%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
) were in better agreement with the wind tunnel experiments than the
two-dimensional simulations. The lift was just 2.5\% lower than the
experimental results and the drag was 9.3\% lower. However the grid was
relatively coarse and no refinement studies were performed. The grid
refinement study in Figure \ref{prelim_GC} indicates that the forces can be
expected to rise with grid refinement suggesting that a grid independent
solution is likely to be even closer to the force values measured in the
experiments. The three-dimensional simulations were performed primarily for
visualisation and to aid understanding of this complex three-dimensional
flow. To draw stronger conclusions would require a more in-depth study of
the three-dimensional flow.

An attempt was made to record the Strohal number in the wind tunnel using a
hot-wire anemometer positioned in the wake. However fourier analysis showed
only low frequency modes which were attibuted to fluctuations of
three-dimensional wake structures rather than unsteady vortex shedding. In
order to determine the Strohal number - or even prove that unsteady vortex
shedding is present for this flow case - it is necessary to haev nominally
two-dimensional flow.

\section{Conclusions}

The main goal of the preliminary study was to was to obtain insight into the
behaviour of two-dimensional sail flows and to gain an initial guage of the
performance of different turbulence models. Direct comparison between CFD\
and the wind tunnel tests was comprimised by the presence of significant
crossflow in the separated region of the wind tunnel model. Visualisation
techniques were used to show that three-dimensional structures in the wake
were dramatically altering the flow and inducing downwash across the span of
the model. Consequently the drag in the wind tunnel tests was increased and
the lift decreased. A three-dimensional CFD simulation was performed in
order to help visualise the wake structure and the computational model
agreed well with the flow visualisation performed in the wind tunnel. CFD
proved to be a useful tool in gaining understanding of a particularily
complicated flow problem.

Due to an inability to produce nominally two-dimensional flow the
preliminary wind tunnel tests proved to be a poor test case for the
validation of turbulence models. The error associated with three-dimensional
effects in the wind tunnel outweighed the differences between the CFD\
results making it impossible to discern between the performances of the
turbulence models tested. While the SST model is perceived to be the most
suitable turbulence model for flows with mild separation, little can be said
here about its suitability for highly separated flows.

It is highly likely that the sail section data presented by Milgram (1971,
1978) is affected by similar three-dimensional effects. Milgram notes that
wall effects are neglected and that it was impossible to estimate the
influence that non-uniform flow separation would have on the results.
Milgram's model had an aspect ratio of just 2.2 and there were 1.6$mm$ gaps
at the tips. The models tested by Wilkinson (1984, 1989, 1990) had an aspect
ratio of 3.0 and hense, for the high camber sections (up to 20\% camber) at
least, some degree of three-dimensionality must have exisited in the wake.
However Wilkinson's data was in the form of pressure coefficient
measurements and boundary layer profiles at the midspan of the model and it
is possible that three-dimesional effects are isolated near the walls and
the flow at midspan was nominally two-dimensional. Wilkinson confirmed this
by comparing pressure distributions three different spanwise locations.
Unfortunately the data recorded by Wilkinson does not include velocity and
turbulence measurements within the leading and trailing edge separation
regions. Without such data it is difficult to compare turbulence models and
to understand the physical and mathematical reasons for their relative
performance. Also the models tested by Wilkinson were derived from upwind
sail designs and have less camber than downwind sails, consequently the data
compiled by Wilkinson is not ideal for the validation of turbulence models
for downwind sail design.

Clearly the wind tunnel tests presented in this chapter are insufficient for
validation of CFD results and it was evident at the completion of the
project that further wind tunnel tests were required. From the results of
this preliminary study it was concluded that significant changes to the
setup and measurement techniques would be necessary for subsequent wind
tunnel tests; they are:

\begin{enumerate}
\item High aspect ratio sections are required in order to minimise
three-dimensional effects. CFD simulations can be used to determine suitable
aspect ratios.

\item A larger wind tunnel was required in order to obtain higher aspect
ratios whilst maintaining practical chord lengths for resolution of
measurement techniques and to maintain a convenient Reynolds number.

\item Measurements need to be conducted at midspan in order to remove errors
associated with three-dimensional effects at the model/wall junction, i.e.,
no span-averaged measurements of lift, drag etc.

\item Velocity and turbulent stress measurements are required in order to
provide a more detailed understanding of sail flows. In particular the
physics of the leading and trailing edge separation regions as well as the
boundary layer leading up to the trailing edge separation point need to be
understood better.
\end{enumerate}

These conclusions were used to form the strategy for the wind tunnel tests
carried out at NASA\ Ames in conjunction with Stanford Yacht Research and
the Center for Turbulence Reseach at Stanford University. The results of
these tests are presented in the next chapter.

\bibliographystyle{ieee}
\bibliography{INTRODUCTION,PLATE,turbulence}

\end{document}
