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\begin{document}

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

\chapter{The Flat Plate at Shallow Incidence}

\section{Introduction}

\label{intro}

This chapter investigates ability of turbulence models to reproduce the flow
over flat plates at shallow angles of incidence. The flat plate can be
considered as a two-dimensional sail with zero camber. This case poses
significant relevance to sail flows due to the formation of a separation
bubble at the leading edge of the type that regularly appears in sail flows.
Such bubbles are always present for downwind sails and are common for genoas
with thin headfoils. This type of leading edge bubble may also be present
for various thin airfoils such as those often used for turbine blades.
Validation will carried through comparison to wind tunnel results compiled
by Crompton \cite{Crompton01} and particular attention will be paid to the
structure of the leading edge bubble and the recovery of the turbulent
boundary layer after reattachment.

\subsection{Flow structure}

The general structure of the flow past a flat plate at incidence is
illustrated in Figure \ref{plate}. At zero incidence the forward stagnation
point rests on the tip of the leading edge, the plate divides the flow
cleanly and laminar boundary layers develop immediately on either side of
the plate. As the angle of attack is increased the stagnation point moves to
the pressure side of the plate and the boundary layer is forced back around
the leading edge due to the pressure difference that exists at the tip. If
the leading edge is sharp then the boundary layer is forced to detach at the
leading edge due to the extreme curvature around the tip. Immediately after
separation a shear layer forms that rapidly undergoes transition into a
turbulent shear layer provided the Reynolds number is not extremely small.
This shear layer curves back towards the plate due to its high entrainment
rate (causing the layer to thicken) and the negative pressures on the upper
surface near the leading edge. For shallow angles of incidence (typically $<7%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$) the shear layer reattaches some distance, $X_{R}$, downstream of the
leading edge. At the reattachment point the shear layer bifurcates and a
portion of the flow is directed upstream to form the recirculation region.
The remainder (which is the majority) of the fluid in the shear layer
progresses downstream and gradually develops into a turbulent boundary
layer. This region downstream of the reattachment point is known as the
recovery region and is of particular significance to sail flows since the
structure of the boundary layer during recovery dictates not only the local
pressure coefficient but also the size and structure of any trailing edge
separation regions that may form downstream.

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A typical pressure coefficient distribution for a flat plate at shallow
incidence is illustrated in Figure \ref{plate_cp}. Also presented are the
pressure coefficient plots for a generic airfoil and an airfoil with a
leading edge separation bubble. The flat plate experiences a significantly
smaller suction peak at the leading edge compared with the airfoils due to
its inability to sustain any attached flow in the region. Reattachment
occurs approximately at the conclusion of the adverse pressure gradient that
exists over the aft half of the bubble.

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airfoils with rounded leading edges may also exhibit separation near\ the
leading edge (as in the airfoil case in Figure \ref{plate_cp}). These
bubbles form due to separation of the laminar boundary layer at the onset of
an adverse pressure gradient. Such bubbles are referred to as
\textquotedblleft short bubbles\textquotedblright\ and have been reported to
have distinctly different characteristics to separation bubbles of the type
illustrated in Figure \ref{plate} \cite{Crabtree57}. A typical short bubble
is illustrated in Figure \ref{short bubble}. As illustrated, transition
typically occurs at the onset of the adverse pressure gradient over the rear
portion of the bubble. Consequently a laminar shear layer develops that can
extend across up to 60\% of the bubble length. After transition the shear
layer thickens rapidly and curves back towards the airfoil surface where it
reattaches.

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Flows involving the short separation bubble pose significant difficulties
for turbulence models. The boundary layer that exists between the forward
stagnation point and the separation bubble is laminar and turbulence models
are notorious for predicting early transition to turbulent flow in this
region. For flat plate boundary layers under zero pressure gradient,
eddy-viscosity models will predict transition to occur at a Reynolds number
that is at least an order of magnitude lower than in reality \cite{Wilcox98}%
. Consequently for airfoil CFD simulations involving short separation
bubbles the calculated boundary layer is likely to be turbulent for a
significant distance upstream of the separation point. Consequently the
solution will exhibit a significant increase in skin friction due to the
high shear of turbulent boundary layers compared with laminar ones. Even
more troublesome is the fact that a turbulent boundary layer has much
steeper velocity gradients near the surface and is hence more resistant to
flow separation than a laminar layer. Therefore the short bubble is likely
to be delayed, if not omitted entirely, in CFD simulations. Should the
boundary layer in the CFD simulation remain laminar up until separation it
is likely that transition will occur in the shear layer almost immediately
after detachment. Consequently the shear layer will grow much faster which
will cause the shear layer to reattach much earlier than in reality where
the shear layer is laminar over the forward section of the bubble.
Transition models, such as the transition version of the $k-\omega $ model
designed by Wilcox \cite{Wilcox94}, have had moderate success for some
simple flows. Transition models typically use different model coefficients
for the turbulent transport equations in the three separate regions of the
flow; the laminar region, transition region and the turbulent region. Tuning
such a model is an arduous task and the position where transition occurs can
be quite unpredictable between different flows due to the influence of the
local pressure gradient. In the case of the short bubble transition actually
occurs within the shear layer which poses addition difficulties to tuning
the model. It is also likely that within short separation bubbles some
degree of relaminarisation (reverse transition) will occur due to the
favourable pressure gradient that drives the reversing flow. This
relaminarisatio nprocess is another potentially damaging obstacle for
turbulence models.

For the flat plate at incidence, leading edge separation is not caused by an
adverse pressure gradient as in short bubble separation but rather by the
inability of the boundary layer to accelerate around the infinitely sharp
leading edge. The resulting bubble that forms is known as the
\textquotedblleft thin airfoil bubble\textquotedblright . Unlike the short
bubble, the shear layer that develops in the thin airfoil bubble is
immediately unstable and turbulent transition occurs rapidly after
separation at the leading edge. Crompton \cite{Crompton01} reported that in
all cases transition occurred upstream of x/c = 0.025 and that intermittency
was observed well upstream of this point. Consequently eddy viscosity models
(which effectively predict instantaneous transition to turbulence) can be
applied much more suitably to the thin airfoil bubble than the short bubble.

Newman and Tse \cite{Newman92} found that above a Reynolds number of $%
2.5\times 10^{5}$ the reattachment length was insensitive to changes in
Reynolds number. \cite{Crompton01} found Reynolds number insensitivity above
a Reynolds number of approximately $10^{5}$. This limit increased slightly
as the incidence reduced due to greater viscous effects in the shallower
separation bubbles. The reattachment length is primarily dependent on the
rate of turbulent entrainment in the shear layer (turbulent entrainment
causes the layer to thicken and curve back towards the plate). At low
Reynolds numbers as the Reynolds number is increased so does the entrainment
rate and hence the reattachment length shortens. As the Reynolds number is
increased further the maximum entrainment rate is reached and the shear
layer becomes fully turbulent. At this point the reattachment length begins
to lengthen again due to a reduction of shear layer thickness associated
with increasing Reynolds number (due to lower diffusive effects resulting in
lower shear layer spreading rates). At approximately $10^{5}$ this thickness
reduction becomes negligible and the shear layer profile (and hence also the
reattachment length) becomes independent of Reynolds number.

Crompton's results also show that between a Reynolds number of $0.53\times
10^{5}$ and $2.13\times 10^{5}$ the transition point is also independent of
Reynolds number and fixed at approximately $2-2.5\%x/c$. This implies that
the mechanism initiating transition is not entrainment from the freestream
(since this is significantly dependent on Reynolds number) but instead
transition is triggered through turbulent entrainment from within the
separation bubble.

Since the reattachment process is always a turbulent one, the downstream
recovery of the turbulent boundary layer can be expected to be only weakly
dependent of Reynolds number. However the portion of fluid that is forced
back to form the recirculation region will be governed by a degree of
Reynolds number dependence. As the shear layer bends towards the surface the
normal velocity fluctuations are damped by the impending wall. Also as the
shear layer bifurcates at the reattachment point the large eddies are split
into pairs of smaller eddies. These two effects contribute to a drop in
turbulent shear stress around the reattachment point. The reversing fluid is
subject to a strong favourable pressure gradient and subsequently it
accelerates and reaches a maximum velocity of approximately $0.4U_{\infty }$
at approximately half way along the bubbles length. This favourable pressure
gradient also has a stabilising effect on the reverse flow boundary layer
and a significant drop in turbulence intensity was recorded over this region 
\cite{Crompton01}. Corresponding to this drop in turbulence intensity the
velocity gradients near the surface reduce and become more laminar-like, the
favourable pressure gradient is initiating relaminarisation. Near the front
of the separation bubble the pressure gradient is adverse to the reversing
flow and the laminar-like boundary layer is prone to separate. Consequently
a small secondary separation bubbles develops near the leading edge. The
length of this separation bubble varies with both angle of incidence and
Reynolds number. The Reynolds number dependence is caused by the
relaminarisation process in the reversed flow boundary layer which itself is
dependent on the local velocities and hence also the freestream Reynolds
number.

\subsection{Experimental data}

Of the experimental results concerned with the flat plate at shallow
incidence the thesis by Crompton (2001), \textquotedblleft The Thin Aerofoil
Leading Edge Bubble\textquotedblright\ \cite{Crompton01}, is by far the most
detailed. Through the use of Laser Doppler Anemometry (LDA) insight into the
structure of the leading bubble was provided. Detailed LDA measurements were
also taken of the developing turbulent boundary layer in the recovery
region. As well as LDA measurements, surface pressures were recorded through
static pressure tappings and flow visualization was used to determine the
reattachment lengths, $X_{R}$, of the separation bubbles.

The wind tunnel model used in the experiments has a chord length of $160mm$
and a span of $800mm$ giving an aspect ratio of 5 which was sufficient to
provide nominally two-dimensional flow. In order to provide enough stiffness
to minimise deflection the plate was constructed out of $6mm$ steel plate,
giving a thickness to chord ratio of $3.75\%$. The leading edge of the plate
was chamfered at 20 degrees in order to provide a sharp leading edge. The
geometry of the leading edge can have significant influence on the separated
shear layer. Ideally for a sharp leading edge the flow will separate at any
departure from zero incidence. However in reality there must be some
roundedness to the leading edge which allows the boundary layer to remain
attached though small departures from zero incidence.

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The experiments were carried out over a range of Reynolds numbers in order
to investigate the influence of Reynolds number. The primary investigation
was at a Reynolds of $2.13\times 10^{5}$, it is at this Reynolds number that
comparison between the experiments and CFD have been made.

The experimental data is provided for angles of attack, $\alpha $ = 1 to 5,
in 1 degree intervals. At $\alpha $ = 1 the leading edge bubble is small and
similar to that experienced in downwind sail flows. At 5 degrees the flow is
separated for the majority of the length of the plate and at 7 degrees the
shear layer fails to reattach.

\subsection{The CFX model}

Cromptons flate plate was modelled with thickness using the geometry
described in Figure \ref{plate_dim}. It is believed that the thickness and
asymmetry of the plate has a significant influence on the flow and hence the
true geometry was used rather than using an infinitely thin plate. The
asymmetry of the plate creates a lift force even at zero degrees of
incidence and consequently there is an associated upwash upstream of the
plate and consequently a small leading edge separation bubble forms at zero
degrees. The so-called `ideal' angle of incidence, where laminar boundary
layers are able to develop on both surfaces without leading edge separation,
is likely to occur at a small negative angle.

The domain dimensions are illustrated in Figure \ref{plate domain}. At the
inlet cartesian velocity components are specified according to the angle of
incidence. Turbulence intensity is set at 0.05\% which is the maximum value
measured while the experiments were carried out. At the outlet zero static
pressure boundary condition was imposed.

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The grids were generated in ICEM-HEXA. Three levels of refinement were used
and the grids are referred to as coarse, medium and fine with $11920,49625$
and $202435$ 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
6. In order to achieve a $y+$ of approximately $1.0$ the near wall spacing
was set at $10mm$. Particular care was taken to provide good quality cells
around the leading edge and at the tip the cells have an aspect ratio of $%
1:1 $, the tip region is illustrated in the close-up view in Figure \ref%
{plate_grid}. Further down the chord very high aspect ratio cells are used
in order to resolve the large flow gradients normal to the wall and at the
trailing edge the aspect ratio reduces to 1:10.\FRAME{ftbpFU}{5.9586in}{%
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\section{Results}

\subsection{Introduction and goals}

The primary purpose of the validation study is to obtain an understanding of
the ability of turbulence models to reproduce the leading edge separation
bubbles of the type experienced in sail flows. It is hoped that through this
study it will be possible to establish the degree to which the leading edge
bubble contributes to the errors experienced in sail simulations. This study
is designed to enable a better understanding of the deficiencies that
turbulence models experience for flows involving leading edge separation.

The flow past the flat plate most closely resembles downwind sail flows at 1
degree of incidence and subsequently this angle of attack has been chosen as
the focus of the study. At this angle the leading edge separation bubble
stretches across the first 14\% of the chord length (i.e. $X_{R}/c=0.14$).
Since the bubble is quite small less velocity measurements were taken within
the bubble than the cases at higher angles of attack. Consequently it was
decided to also investigate the case of $\alpha =3$. At this higher angle of
incidence the reattachment length, $X_{R}/c=0.47$, i.e. the flow is
separated over almost half of the plate.

\subsection{Grid convergence study\label{sect GC}}

The grid convergence study was performed solely at $\alpha =3$ and only with
the SST turbulence model. Since $\alpha =3$ is a more demanding case than $%
\alpha =1$ it is believed that results should also apply to the lower angle
case. Lift and drag coefficients are presented in Figure \ref{plate_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. 
\begin{figure}[th]
$\underset{}{%
\begin{tabular}{ll}
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\caption{Grid convergence of the lift and drag coefficients.}
\label{plate_GC}
\end{figure}

As the grid is refined there is a clear convergence of the lift and drag
coefficients. The lift from the medium grid solution is within 0.015\% of
the fine grid solution and drag is within 0.0055\%, this indicates that the
solutions are independent of the grid and subsequent refinement will not
result in increased accuracy of the solution. Since forces are integrated
properties there is a possibility that the excellent grid convergence
illustrated in Figure \ref{plate_GC} could have occurred fortuitously. To
confirm that this is not the case the surface pressure distributions for the
three grids are presented in Figure \ref{plate CP_GC}. Note that the spike
in the pressure coefficient on the pressure side of the plate is caused by
the shoulder at the end of the chamfer.

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The $C_{P}$ curves for the medium and fine grid lie almost directly upon one
anther, the fine grid solution has slightly more suction at the leading
edge, however this effect is minimal. Grid independence has been achieved
between the medium and fine grids and therefore the medium grid was chosen
as the most suitable grid to be used for the remainder of the study.

\subsection{Comparison at $\protect\alpha =1$ \ }

\bigskip Boundary layer velocity profiles were measured by Crompton at 9
different chordwise locations along the plate using LDA. The measurement
stations are located at $x/c=\left\{
0.031,0.125,0.25,0.375,0.5,0.625,0.75,0.875,1.0\right\} $. As can be seen in
Table \ref{tab XR1} the reattachment lengths, $X_{R},$ computed using the
SST and $k-\omega $ models differ slightly from the value determined by
Crompton. The SST model overpredicts $X_{R}$ by 6.1\% and the $k-\omega $
model overpredicts $X_{R}$ by 31.5\%. 
\begin{table}[tbh]
\begin{center}
\begin{tabular}{|c|c|}
\hline
& $X_{R}$ \\ \hline\hline
Crompton & 0.14 \\ \hline
SST & 0.1486 \\ \hline
$k-\omega $ & 0.1841 \\ \hline
\end{tabular}%
\end{center}
\caption{Reattachment lengths for the flat plate at $\protect\alpha =1.$}
\label{tab XR1}
\end{table}

In order to make better comparison between the models and experiments the
measurement stations are scaled by $X_{R}$ so that for each model the
station is in the same position relative to the reattachment point. For $%
\alpha =1$ only the first two measurements stations are within the leading
edge bubble, these profiles are illustrated in Figure \ref{BL_S1-2}.

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\caption{Chordwise velocity profiles within the leading edge bubble
($\protect\alpha =1).$} \label{BL_S1-2} 

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The portion of the fluid that is forced upstream at the reattachment point
to form the recirculation region is accelerated by the favourable pressure
gradient and forms a reversed boundary layer. Crompton reported a maximum
reversed velocity of $-0.4U_{\infty },$ whereas the CFD results predict much
solwer velocities with the SST\ model at $-0.290U_{\infty }$ and the $k-$ $%
\omega $ model at $-0.284U_{\infty }.$ The reversed flow velocities are much
largeer than usually experienced within separation bubbles, for short
airfoil bubbles the reversed flow is typically below $0.2U_{\infty }$\cite%
{Crompton01}$.$ The reason for the high velocities in the flat plate case is
the high entrainment rate in the shear layer which in turn provides energy
to the reversed flow through momentum balance. The CFD\ results predict a
lower peak in the turbulent kinetic energy in the shear layer which is
analogous to a lower entrainment rate. This is discussed further in Section %
\ref{sect a3} where the kinetic energy profiles within the separation bubble
are presented.

Boundary layer profiles within the separation bubble are presented in
greater detail in Figure \ref{BL_S1-2 close}. The reversed boundary layer is
driven by a strong favourable pressure gradient which provides some degree
of stablisation and the shape of the boundary layer profiles in the
experimental results indicate that relaminarisation is occuring. This effect
is illustrated in Figure \ref{BL_S1-2 close}, where the experiment results
have very "laminar-like" profiles. In comparison the CFD results cannot
emulate the relaminarisation process and conseqently\ predict turbulent
boundary layers. 
\begin{figure}[h]
\begin{tabular}{ll}
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\end{tabular}%
\caption{ Near-wall chordwise velocity profiles within the leading edge
bubble ($\protect\alpha =1).$}
\label{BL_S1-2 close}
\end{figure}

One of the direct consequences of the turbulent boundary layer profiles
predicted by the CFD results is their resistance to separation. In the CFD
results the reversed boundary layer remains attached all the way to the
leading edge and there is no secondary separation of the type reported by
Crompton. Crompton suggests that the form of this bubble has significant
influence on the development of the shear layer near the leading edge and
consequently it's influence is likely to spread well downstream. 
\begin{figure}[p]
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\end{tabular}%
\caption{Chordwise velocity profiles downstream of reattachment ($\protect%
\alpha =1).$}
\label{BL S3-9}
\end{figure}

A comparison of the boundary layer profiles downstream of the reattachment
point is presented in Figure \ref{BL S3-9}. In general the CFD results
compare well with the boundary layer profiles measured by Crompton. From
Figure \ref{BL S3-9} it is evident that the wind tunnel results give
slightly fuller boundary layer results than the CFD however this effect is
dificult to quantify and it is hard to assess the structure of the different
boundary layers. In Figure \ref{Log BL_S3,8} the boundary layer profiles at $%
x/c=0.25$ and $0.875$ are presented using a log scale for the boundary layer
height. Also plotted is a typical turbulent boundary layer at zero pressure
gradient (ZPG) \cite{Osterlund99} as well as the log law of the wall
(reference equation in introduction) and the viscous sublayer profile
(reference equation in introduction). 
\begin{figure}[p]
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\caption{Chordwise velocity profiles downstream of reattchment (log scale, $%
\protect\alpha =1).$}
\label{Log BL_S3,8}
\end{figure}

The profiles in Figure \ref{Log BL_S3,8} illustrate the recovery of the
boundary layer. At $x/c=0.25$ the experimental and CFD results have boundary
layer profiles that lie underneath both the log-law of the wall and the ZPG\
profile, i.e. the velocities in the sublayer and log-law regions are slower
than would be expected for a typical turbulent boundary layer.\ This effect
is partially due to the the slight adverse pressure gradient that the
boundary layer has too work against. However this slow nature of the
boundary layer must also be attributed to the reattachment process and in
both the experimental and CFD\ results the boundary layer is in a recovery
phase. At the reattachment point the velocity gradient is zero at the wall
(zero shear) and conseqently the near wall does not have the characteristic
sharp increase of velocity known as the sublayer region. This effect
influences the entire velocity profile and at reattachment velocity profiles
are typically slow and have a more linear profile with signicifantly lower
velocities than for typical a turbulent boundary layer. This shape is
illustrated in Figure \ref{BL XR}, which is the boundary layer profile for
the $\alpha =3$ case at $x/c=0.5.$ This measurement station was very close
to the reattachment point which for this case was $X_{R}=0.47.$

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\bigskip Looking again at the results for $x/c=0.25$ (Figure \ref{Log
BL_S3,8}) the profiles predicted by the CFD have lower velocities through
the log-law region than the experimental results. This would indicate that
the CFD\ predicts a slower recovery after reattachment. At $x/c=0.875$ there
is a slightly favourable pressure gradient and as expected the experimental
results show an accelerated boundary layer (greater velocities than in the
ZPG\ profile). It appears that the boundary layer has fully recovered and is
showing typical response to the favourable pressure gradient. Conversely the
CFD\ results show slightly retarded boundary layers indicating that they
have not quite yet recovered fully. Slow post reattachment recovery of
boundary layers is a common trait of turbulence models. So. Lai and Yoo \cite%
{So88} observed similar post-reattachment behaviour for the flow past a
backward facing step. For this case several Low Reynolds Number models were
found to produce worse results than wall functions in the recovery region.
This is because wall functions, (which based on local-equilibrium arguments)
force the flow to asymtote to the log law-of-wall.

Also evident in the log profiles of Figure \ref{Log BL_S3,8} is that the
sublayer profiles in the recovery region are thin compared with typical
turbulent boundary layer profiles. This is a lag effect from the
reattachment process. As the boundary layer develops downstream it is the
outer region that forms first and drives the development of the inner
structure through the inwards entrainment of momentum. Unfortunately the
experimental results do not have enough resolution to extend down throughout
the viscous sublayer. However from the data that is available it appears
that the sublayer in the experiments is also thinner and slower than the CFD
results and has a long blending region between the sublayer and logarithmic
profile. The CFD results overpredict the velocity magnitude near that wall
and produce sublayer profiles more typical of fully turbulent boundary
layers. In the development of these turbulence models the model constants
were calibrated to predict the correct asymptotic behaviour of both the flow
variables for simple homogenous boundary layers. Consequently they struggle
predict flows that differ from the typical turbulent boundary layer
behaviour.

Figure \ref{Log BL_S3,8} shows that the $k-\omega $ model predicts profiles
that are slightly closer to the profiles measured by Crompton than the SST\
model which seems to have a slower recovery from reattachment. However the
differences are small. [Say more with $k$ profiles]

The pressure coefficient plot for the flat plate at $\alpha =1$ is presented
in Figure \ref{CP_1}. The CFD\ simulations have a smaller suction peak with
a significant increase in pressure immediately after the leading edge. In
the experimental data the pressures continue to decrease downstream of the
leading edge and has a minimum pressure at $x/c\simeq 0.045.$ The radius of
the leading edge is likely to have a significant effect on the local
pressure coefficient with more rounded tips enabling the flow to remain
attached longer resulting in greater suction peaks. However Crompton paid
special care to get a sharp leading edge and furthermore roundedness does
not explain the differences between the two turbulence models. The pressures
within the leading bubble are related to the curvature of the shear layer
(higher streamline curvature = lower pressure). The Experimental data has a
shorter and fatter leading edge bubble which explains it's higher suciton
peak. Additionally the secondary separation bubble that is present only in
the experiments increases shear layer growth near the leading edge and hense
contributes to the suction peak.

Another notable feature of Figure \ref{CP_1} is that between $x/c=0.1$ and $%
x/c=0.2$ the \thinspace $k-\omega $ model predicts lower pressures than both
the SST model and the experimental data. This is due to model overpredicting
the length of the separation bubble. When the $C_{P}$ curves are normalised
with the reattachment length the comparison is good with the only
significant difference being the large suction at the leading edge in the
Crompton data. \FRAME{fhFU}{4.0274in}{2.194in}{0pt}{\Qcb{Pressure
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\subsection{Comparison at $\protect\alpha =3$ \ }

The reattachment lengths for the $\alpha =3$ case is presented in table \ref%
{tab XR3}. The SST model underpredicts $X_{R}$ by 6.4\% and the $k-\omega $
model overpredicts $X_{R}$ by 8.4\%. Once again the chordwise positioning of
the profiles presented in this chapter are scaled by $X_{R}.$%
\begin{table}[tbh]
\begin{center}
\begin{tabular}{|c|c|}
\hline
& $X_{R}$ \\ \hline\hline
Crompton & 0.47 \\ \hline
SST & 0.4374 \\ \hline
$k-\omega $ & 0.5096 \\ \hline
\end{tabular}%
\end{center}
\caption{Reattachment lengths for the flat plate at $\protect\alpha =1.$}
\label{tab XR3}
\end{table}

Velocity Profiles within the separation bubble are presented in Figure \ref%
{BL 3 S1-4}. Looking first at the measurement station closest to the
reattachment point ($x/c=0.375),$ the CFD results predict much lower
reversed velocities than the experimental results. The greater velocities in
the experimental results are caused by the large entrainment rate at the
shear layer edge. Accordingly the velocities at the shear layer edge are
also higher in the experimental results and the leading edge bubble is
taller and shorted, i.e. the shear layer has higher curvature. The
measurement station at $x/c=0.25$ coinsides to approximately the mid length
of the leading edge bubble, here the velocities are at their greatest, both
at the outside edge of the shear layer and also in the reversed flow region.
In the reversed flow region $u=-0.393U_{\max }$ for the experimental resutls
whereas the SST\ model predicts $u=-0.252U_{\max }$ and the $k-\omega $
model predicts $u=-0.271U_{\max }.$ At this measurement station the reversed
boundary layer is begining to show laminar-like features and is notably
thicker than the CFD results. By $x/c=0.125$ the reversed boundary layer has
slowed considerably primarily due to mass flow in the direction normal to
the plate but in the case of the experimental results the presence of an
adverse pressure gradient also is plays a role. At $x/c=0.031$ the secondary
separation bubble is present in the experimental results close to the wall.
The CFD\ results do not have this feature and the near wall flow is reverse
and the has a shape typical of a turbulent boundary layer. A close up of the
near wall region is presented in figure \ref{BL_3S1 close}. 
\begin{figure}[t]
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\end{tabular}%
\caption{Chordwise velocity profiles within the leading edge bubble($\protect%
\alpha =3).$}
\label{BL 3 S1-4}
\end{figure}
\FRAME{ftFU}{4.0274in}{2.3609in}{0pt}{\Qcb{Near wall chordwise velocity
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\begin{figure}[t]
\begin{tabular}{cc}
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\end{tabular}%
\caption{Chordwise velocity profiles within the leading edge bubble($\protect%
\alpha =3).$}
\label{k 3S1-4}
\end{figure}

Further insight into the structure of the leading edge bubble can be gained
from the profiles of turbulent kinetic energy, $k,$ that are presented in
Figure \ref{k 3S1-4}. At $x/c=0.031$ the experimental results show two peaks
in the turbulent kinetic energy profile. The lower peak is associated with
the secondary separation bubble that is not present in the CFD results. At
the outer edge of the secondary separation bubble there is a region of high
mean shear which increases production of turbulent kinetic energy. The
second peak occurs at the point of inflection in the velocity profile in the
outer shear layer which is also a region of high mean shear. This second
peak in kinetic energy is much stronger than the first as it is driven by
the large velocities around the leading edge of the plate. Additionally the
experimental results have a much larger peak than the CFD\ results with the
maximum turbulent kinetic energy 2.5 times greater than the SST\ model and
4.0 times greater than the $k-\omega $ model. This large peak in $k$ for the
Crompton data is a consequence of the secondary recirculation bubble that
not resolved in the CFD results. As well as amplifing production of $k$ near
the wall the secondary separation bubble forces the reversed flow region
further from the wall which initiates particularly steep velocity gradients
accross the outer shear layer.

For the experimental results at $x/c=0.125$ the magnitude of the turbulent
kinetic energy in the shear layer is slightly lower than at the upstream
station due to the diminished effect of the secondary separation bubble
which itself extends no further back than $x/c=0.045$. The turbulent kinetic
energy \ in the CFD\ results has increased for both the SST\ and $k-\omega $
model and the SST model is in better agreement with the experimental
results. In both the CFD\ and the experimental results the peak in turbulent
kinetc energy has widened due to the thickening of the shear layer.

Between $x/c=0.125$ and $x/c=0.25$ the turbulent kinetic energy keeps a
relatively constant profile for the SST\ model. Conversely the experimental
results indicate that the kinetic energy in the shear layer is still
increasing and the maximum entrainment rate has not yet been reached. The
SST\ model clearly predicts the shear layer to reach local equibrium
prematurely. Between $x/c=0.250$ and $x/c=0.375$ both the SST and $k-\omega $
models predict a decrease in turbulent kinetic energy whereas in the
experimental results $k$ continues to increase. By this stage the $k-\omega $
model has also reached local equilibrium in the shear layer, whereas the
experimental results indicate the entrainment rate is continuing to
increase. The decrease in turbulent kinetic energy predicted by the CFD
results is caused by the shear layer curving back towards the wall. In the
experimetnal results show a similar drop in vertical fluctuations as the
shear layer bends towards the wall, however this energy is largely converted
to transverse fluctuations and thus the kinetic energy levels are retained.

By $x/c=0.375$ both the SST\ model and the $k-\omega $ model predict peaks
in turbulent kinetic energy that are approximately 1.7 times smaller than in
the experimental results. In the experimental results the kinetic energy
peak is wider than the CFD\ indicating that the CFD results underpredict the
shear layer thickness. This not surprising considering the high entrainment
rate found in the experimental results.

The low turbulent kinetic energy across the rear half of the leading edge
bubble in the CFD\ results has considerable effects well downstream of
reattachment and also within the leading edge bubble itself. As mentioned
earlier the reversed boundary layer reaches much greater velocities in the
experimental results this is caused by greater entrainment which allows for
higher shear layer curvature which leads to greater velocities. Similarily
downstream of reattachment the boundary layers predicted by the CFD\ results
are slower than in the experiments. This is a combination of a lag effect
from the lower velocities predicted by the CFD\ in the shear layer upstream
of reattachment due to low turbulent kinetic energy, and also due to the the
CFD\ results being slow to recover from the reattachment process. Downstream
of the reattachment point the kinetic energy values are again lower in the
experimental results which is a legacy effect from the low values in the
leading edge bubble. The higher levels of $k$ in the experimental results
enable much steeper gradients near the wall and conseqently the sublayer
develops more rapidly enabling the boundary layer to regain a state of local
equilibrium.

\subsection{\protect\bigskip \protect\bigskip}

\subsection{Effect of angle of attack}

Look at plots of reattachment length with angle of attack and also the
pressure coefficient plots. Also look at the pressure coeffcient plots
normalised to $X_{R}$.

\bigskip

\bigskip

\bigskip

\section{Comparison with other flow fields}

General dissucusion of the significance of this test case with reference to
flows that posess similar traits. Esp Sail flows. Disscussion about similar
test cases e.g. backward facing step.

\bigskip

\bigskip

\bigskip

\section{Conclusions}

\bigskip

\bigskip

\bigskip

\bigskip 
\bibliographystyle{ieee}
\bibliography{phD}

\bigskip

\end{document}
