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\begin{document}
\title{Turbulence Modelling of The Flat Plate at Shallow Incidence}
\author{S. Collie}
\author{P. S. Jackson}
\address{Yacht Research Unit\\
University of Auckland\\
Private Bag 92019\\
Auckland\\
New Zealand}
\author{M. Gerritsen}

\begin{abstract}
This paper investigates the ability of turbulence models to reproduce the
flow over flat plates at shallow angles of incidence. Particular interest is
paid to leading edge bubble that forms as the flow separates from the sharp
leading edge. This type of leading edge bubble is most commonly found in
flows past yacht sails but is also present for some thin airfoils such as
those often used for turbine blades. Validation is carried out through
comparison to wind tunnel results compiled by Crompton \cite{Crompton01} and
particular attention is paid to the structure of the leading edge bubble and
the recovery of the turbulent boundary layer after reattachment.
\end{abstract}

\maketitle

\section{Introduction{}}

This paper is part of a larger research effort to assess the practicality of
using computational fluid dynamics (CFD) for reliable and accurate sail
design and performance analysis. As a step towards validation of sail flows
it is necessary to break down the problem into its components and to
understand the physics of the various flow features that are involved. One
particularly intersting feature of sail flows is the existance of thin
leading edge bubbles of the type that that is found for flat plates at
shallow incidence.

Sails are made of soft fabric and therefore in order to hold their shape the
forward stagnation point must rest on the pressure side of the sail.
Conseqently the flow must pass back around the front of the sail and is
forced to separate at the sharp leading edge. The shear layer that
subsequently develops rapidly undergoes transition and begins to curve back
torward the sail surface where it reattaches some distance, $X_{R},$
downstream. At the reattachment point the flow bifurcates with the majority
continuing along the length of the sail and a smaller fraction of the flow
is driven back towards the leading edge to form the leading edge bubble.
This bubble is known as the "thin airfoil bubble" and has significant
differences to "short bubbles" which can occur for some airfoils due to
laminar separation near the nose \cite{Crabtree57}. Most notably the shear
layer in the thin airfoil bubble is predominantly turbulent, whereas with
the short bubble the shear layer may be laminar over the first 60\% of the
length of the bubble. The boundary layer that forms downstream of the
reattachment point is embeded with turbulent structures that are a legacy of
the separated shear layer and consequently the layer is atypical of
turbulent boundary layers. Instead it is sluggish and prone to boundary
layer separation in adverse pressure gradient flows. Consequently,
especially for downwind sails, trailing edge separation frequenty occurs due
to the adverse pressure gradient accross the rear half of the sail.\FRAME{%
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The leading edge bubble has a critical influence on the pressure
distribution of downwind sails. The shape and structure of the separation
region itself has significant impact on the pressure peak at the leading
edge which is the primary contributor to the lift on a sail. The influence
of the leading edge bubble extends well downstream of reattachment the
leading edge the interaction between the leading edge bubble and the
trailing edge separation region is poorly understood. Whilst extensive CFD
validation studies have been carried out for trailing edge speparation there
have been no investigations into the performances of various turbulence
models for leading edge bubbles, nor has much thought been paid to the
suitability of CFD\ methods at predicting adverse pressure gradient driven
separation of a boundary layer that is in a mode of post-reattachment
recovery. By making comparison between CFD\ solutions and experimental data
for the inclined flat plate it is hoped thart a better understanding can be
gained on the performance of tubrulence models for flows involving leading
edge bubbles.

\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 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 of the separation bubbles. [Need to talk about how
Crompton discovered the secondary bubble and the question of whether this
exists in sail flows]

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 and to facilitate the pressure tappings, 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.\FRAME{ftbpFU}{3.2897in}{0.595in}{0pt}{%
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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 number 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 in length to chord ratio to leading edge bubbles 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 CFD model}

Crompton's flate plate was modelled with thickness using the geometry
described in Figure \ref{plate_dim} and the domain dimensions are
illustrated in Figure \ref{plate domain}. At the inlet cartesian velocity
components are specified according to the angle of incidence. Freestream
turbulence intensity is set at 0.05\% which was the maximum value measured
in the tunnel while the experiments were carried out \cite{Crompton01}. At
the outlet a zero static pressure gradient 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 $6.25\times 10^{-5}c$. Particular care was taken to
provide high 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
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\section{Results}

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 determine what influence the leading edge
bubble has on the accuracy of sail simulations.

At 1 degree of incidence the leading edge bubble is of similar length to
chord ratio to typical downwind sail flows and the pressure distribution was
more comparable to sail flows than the other cases. Therefore, for the
purpose of relevance, the validation study focuses on the case at 1 degree
of incidence. 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 fewer velocity measurements were taken within the
bubble than the cases at higher angles. Consequently it was decided to also
investigate the case of $\alpha =3$ in order to gain a better understanding
of the flow within the separation bubble. 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 $\alpha =3$ involves a larger separated
region than $\alpha =1$ it is likely to be more prone to grid dependency.
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.

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\caption{Grid convergence of the lift and drag coefficients.}\label{plate_GC}
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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 is unlikely
to result in any increased accuracy of the solution. Therefore the medium
gird was used for all subsequent simulations.

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

\subsubsection{Boundary layer profiles in the leading edge bubble}

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 from the value determined by Crompton, the
SST model overpredicts $X_{R}$ by 5.8\% and the $k-\omega $ model
overpredicts $X_{R}$ by 24.0\%. 
\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 of the boundary layer profiles 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
slower velocities with the SST\ model at $0.290U_{\infty }$ and the $k-$ $%
\omega $ model at $0.284U_{\infty }.$ The reversed flow velocities are much
larger than usually experienced within separation bubbles, for example
within 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 experimental results
have very "laminar-like" profiles. In comparison, the CFD results cannot
emulate the relaminarisation process and conseqently\ predict much steeper
near-wall velocity gradients.

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\caption{ Near-wall chordwise velocity profiles within the leading edge
bubble ($\protect\alpha =1).$} \label{BL_S1-2 close} 
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One of the direct consequences of the more turbulent-like 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 its influence is likely to spread well downstream \cite%
{Crompton01}.

\subsubsection{Boundary layer profiles downstream of reattachment}

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\caption{Chordwise velocity profiles downstream of reattchment (log scale,
$\protect\alpha =1).$} \label{Log BL_S3,8} 
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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 and the viscous
sublayer profile. 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 slight adverse pressure gradient that
the boundary layer must work against. However the CFD results show much
lower velocities and this effect cannot be entirely due to the pressure
gradient. The low velocities in the log-law region must also be attributed
to the reattachment process and in both the experimental and CFD\ results
the boundary layer is in a recovery phase. Whilst the experiment results
seem to be only slightly sluggish the CFD is heavily influenced by the
behavoir upstream, the flow is taking longer to recover from reattachment.

As the turbulent boundary layer develops downstream of the reattachment
point it is the outer region that forms first and drives the development of
the inner structure through the inwards entrainment of momentum \cite%
{Crompton01, others?}. At the reattachment point the wall shear, $\tau _{w},$
is zero and conseqently the near wall profile does not have the
characteristic sharp increase of velocity known as the sublayer region. This
near-wall behaviour influences the entire velocity profile and at
reattachment the velocity profile has a very linear profile with
signicifantly lower velocities than for typical turbulent boundary layers. A
boundary layer profile soon after the reattachment point is presented in
Figure \ref{BL XR}, the profile is approximately linear over the lower 75\%
of the boundary layer height ($\delta )$. This profile is for the for the $%
\alpha =3$ case at $x/c=0.5$ which was the only angle of attack where a
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Downstream of reattachment the viscous sublayer in the CFD results has a
steeper velocity gradient than in the experiments. The CFD results
overpredict the velocity magnitude near the wall and produce sublayer
profiles more typical of equillibrium boundary layer flows. In the
development of these turbulence models the model constants were calibrated
to predict the correct asymptotic behaviour of both the velocities and the
turbulence parameters for simple homogenous boundary layers. Consequently
when applied to flows such as this - where the boundary layer structure is
quite different to simple boundary layer flows - they struggle to predict
the correct near-wall behaviour.

Looking now 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 through the
log-region the boundary layer has recovered and is showing the typical
response to the favourable pressure gradient. However within the viscous
sublayer the velocities are still much lower than in the ZPG\ profile,
suggesting that this region is still in a mode of recovery. In the log-law
region the CFD\ results show slower velocities compared with the experiments
which would indicate that the influence of reattachment point extends
further downstream in the CFD than in the experiments. However since the
peak velocities in the leading edge bubble were significantly higher in the
experiments (due to greater rate of entrainment) it is not surprising that
lower velocites are also experienced well downstream of the reattachment
point.

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 wall function based models were found to produce better results in
the recovery region than Low Reynolds Number models. This is because wall
functions, which are based on local-equilibrium arguments, force the flow to
asymtote to the log law-of-wall.

As a final note on the boundary layer profiles, 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. The SST model
seems to have a slower recovery from reattachment, which is believed to be
due to lower entrainment rates for the SST model due to the SST limiter on
the eddy viscosity.

\subsubsection{Pressure distributions}

The pressure coefficient plot for the flat plate at $\alpha =1$ is presented
in Figure \ref{CP_1} (note here that the pressure coefficient axis is
plotted in reverse, i.e., a rise in the chart represents a drop in
pressure). In the experimental data the pressures continue to decrease
downstream of the leading edge and it has reached a minimum pressure at $%
x/c\simeq 0.045.$ Conversely the CFD\ results have a lower suction peak that
is flatter and does not experience the gradual decrease in pressure over the
forward half of the separation bubble that is seen in the experimental
results. The leading edge radius is known to have an 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) and the experimental data has
a shorter and fatter leading edge bubble which explains its higher suction
peak. Additionally the secondary separation bubble that is present only in
the experiments increases shear layer growth near the leading edge and hence
also 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, and the particulary low suction peak in the $k-\omega $
results. After reattachment the pressure coefficient in the experimental
results is slightly higher than the CFD results, this is directly related to
the greater velocities that were experienced in the leading edge bubble and
the boundary layer profiles downstream of reattachment. \FRAME{fhFU}{4.0274in%
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\subsection{Comparison at $\protect\alpha =3$ \ \label{sect a3}}

\subsubsection{Boundary layer profiles in the leading edge bubble}

The reattachment lengths for the $\alpha =3$ case are 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}
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\caption{Chordwise velocity profiles within the leading edge
bubble($\protect\alpha =3).$}\label{BL 3 S1-4}

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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 $($i.e., $x/c=0.375),$ CFD predicts much lower reversed
velocities than the experimental results. The measurement station at $%
x/c=0.25$ coincides with 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 $u=-0.393U_{\infty }$ for the experimental results whereas the
SST\ model predicts $u=-0.252U_{\infty }$ and the $k-\omega $ model predicts 
$u=-0.271U_{\infty }.$ At this measurement station the reversed boundary
layer in the experiments is begining to show laminar-like features and is
notably thicker than the CFD results. By $x/c=0.125$ there is much less
reversed flow which is 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 plays a role. At $x/c=0.031$ the secondary
separation bubble shows its presence in the experimental results close to
the wall. The CFD\ results do not have this feature and the near wall flow
is reversed all the way to the leading edge. A close-up of the near wall
region, showing the secondary bubble (in the experimental results), is
presented in figure \ref{BL_3S1 close}. \FRAME{ftFU}{3.2897in}{1.9303in}{0pt%
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\subsubsection{Turbulent kinetic energy profiles in the leading edge bubble}

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 where mean-shear is at its greatest [perhaps plot 
$du/dy]$. 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. This peak is much larger in the experimental results compared with
the CFD,\ with the maximum turbulent kinetic energy 2.5 times greater than
the SST\ model and 4.0 times greater than the $k-\omega $ model. It is
believed that the large peak in $k$ for the Crompton data is due to two
effects. Firstly the unstable shear layer flapping causes additional
entrainment of kinetic energy and is a source of intermittency, and secondly
the small secondary recirculation bubble at the leading edge enhances the
mean-shear in the leading edge region which increases the production of
turbulent kinetic energy. Both of these effects are not accounted for in the
CFD results. The SST model predicts more turbulent kinetic energy in the
shear layer than the $k-\omega $ model which is surprising since the SST
relation is designed to limit the production of $k$. The SST\ model uses the 
$k-\omega $ model across the inner portion of the boundary layer out to the
defect layer and the model blends to $k-\epsilon $ by the outer edge of the
boundary layer. Within the leading edge\ bubble the SST model blends to $%
k-\epsilon $ across the inner region of the bubble and so it is the $%
\epsilon $-equation that is solved across the shear layer. Therefore it
appears that the $\epsilon $-equation predicts a lower rate of dissipation
of turbulent kinetic energy ($\omega $ or $\epsilon /k)$ than the $\omega $%
-equation in the $k-\omega $ model (note that in the SST\ model the $%
\epsilon $-equation is written in terms of $\omega )$. [need to make this
point clearer - plots of $\omega $ across the leading edge bubble?]

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\begin{tabular}{cc}
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\caption{Chordwise velocity profiles within the leading edge
bubble($\protect\alpha =3).$} \label{k 3S1-4}

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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. Near the
wall the experimental results have lower levels of turbulent kinetic energy
than the CFD\ which is a result of the relamination process.

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 predicts the shear layer to reach local equibrium prematurely,
production of turbulent kinetic energy matches it's dissipation and the
levels of $k$ stop increasing. 

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 turbulent kinetic energy in the shear layer 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 and the decelleration of the shear layer as it approaches reattachment.
The experimetnal results show a similar drop in vertical velocity
fluctuations ($v^{\prime })$ as the shear layer bends towards the wall,
however this energy is converted to transverse ($w^{\prime })$ and
streamwise ($u^{\prime })$ velocity fluctuations and thus the kinetic energy
levels are retained. The SST and $k-\omega $ turbulence models cannot
emulate this transfer of energy from the vertical to transverse and
streamwise fluctuations because they assume isotropy of the Reynolds stress
tensor. In order to model this process a more advanced anisotropic
turbulence model is required (e.g. algebraic stress models or second-order
closures).

At $x/c=0.375$ 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 entrainment rate
is much greater in the experimental results.

In the CFD\ turbulent shear stress follows the same trends as the $k$
profiles, unfortunately turbulent shear stress data was not recorded in the
experiments and so no comparison could be made.

\subsubsection{Downstream of reattachment}

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 and larger velocities. Similarily downstream of
reattachment the boundary layers predicted by the CFD\ results are slower
than in the experiments. This is partially a lag effect from the lower
velocities predicted by the CFD\ in the shear layer upstream of
reattachment, but it is also apparent that the CFD\ results recover slowly
from 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 [check this, I think that only
the log-layer recovers rapidly and the sublayer is actually slower in the
experiments].

\subsubsection{Pressure distributions? }

\subsection{Summary of Results}

\subsubsection{CFD\ vs Experiments}

\begin{enumerate}
\item \textit{The experiments experience greater growth and curvature of the
shear layer which bounds the leading edge bubble.}

\ \ \ \ \ \ REASON

\begin{itemize}
\item Increased entrainment at the leading edge due to the secondary
separation bubble and shear layer flapping.

\item Transport and production of turbulent kinetic energy dominates
dissapation and hence $k$ continues to increase along the length of the
shear layer. CFD\ has difficulities modelling non-equilibrium flows due to
the basic assumptions made in the derivation of the turbulence models.
\end{itemize}

\ \ \ \ \ \ \ CONSEQUENCES

\begin{itemize}
\item Reattachment is delayed in the CFD

\item Lower suction peak in the CFD
\end{itemize}

\item \textit{The reversed flow in the leading edge bubble is subject to a
favourable pressure gradient which causes relaminarisation. This effect is
not present in the CFD results.}

\ \ \ \ \ \ REASON

\begin{itemize}
\item The SST and $k-\omega $ models are designed for fully turbulent flows
and are not capable of predicting relaminarisation.
\end{itemize}

\ \ \ \ \ \ CONSEQUENCE

\begin{itemize}
\item CFD fails to predict the secondary bubble at the leading edge.
\end{itemize}

\item \textit{Downstream of reattachment the CFD\ profiles recover towards
turbulent equillibrium slowly compared with the experiments.}

\ \ \ \ \ \ \ REASON

\begin{itemize}
\item The developing boundary layer is dominated by two distinctly different
spectrums of eddy scales; the scales in the outer region of the near wall
that have their origin in the separated shear layer, and the scales in the
inner region associated with the development of the new boundary layer.
Turbulence models are based on a single spectrum of eddy scales and
therefore cannot be expected to emulate the complicated recovery process.
\end{itemize}

\ \ \ \ \ \ \ CONSEQUENCES

\begin{itemize}
\item Velocities in the CFD\ boundary layer\ profiles are slower than in the
experiments and are therefore more subseptable to separation

\item Higher pressure coefficients in the CFD\ results and hense potentially
lower lift predictions.
\end{itemize}
\end{enumerate}

\subsubsection{SST vs $k-\protect\omega $}

\begin{enumerate}
\item The SST\ model captures the leading edge bubble more accurately than $%
k-\omega ,$ it predicts a reattachment length that is closer to the
experimental value and does a better job of predicting the pressure
coefficient in the vicinity of the leading edge bubble.

\ \ \ \ \ \ REASON \ \ \ \ \ \ 

\begin{itemize}
\item The SST model predicts greater turbulent kinetic energy near the
leading edge and therefore predicts greater shear layer growth. As a results
the shear layer has more curcature and bends back towards the plate earlier
than the $k-\omega $ model.
\end{itemize}

\item Downstream of reattachment the $k-\omega $ model predicts a more rapid
boundary layer recovery than the SST\ model, however the recovery is still
slower than the experimental results.

\ \ \ \ \ \ REASON

\begin{itemize}
\item Since there are two distinctly different length scales interacting in
the recovery region a two-equation model, that operates using just one
length scale ($k^{1/2}/\omega ),$ cannot be expected to perform well. The
diffence between the velocity profiles are small and are likely to be a
consequence of the SST\ model blending to the $\epsilon $-equation through
the outer region of the boundary layer.
\end{itemize}
\end{enumerate}

\section{Conclusions}

\bigskip 1. Transition is important - particulary reverse transition in the
bubble.

\bigskip \bigskip 
\bibliographystyle{PHREPORT}
\bibliography{PLATE}

\end{document}
