

\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 be carried out 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 fluid is forced back around the leading
edge due to the pressure difference that exists across the tip. If the
leading edge is sharp then the flow is forced to detach at the leading edge.
Immediately after separation a shear layer forms that rapidly undergoes
transition into a turbulent shear layer provided the Reynolds number is not
unrealistically small (for creeping flows the flow may not separate). This
shear layer curves back towards the plate due to its high entrainment rate
(causing the layer to thicken) and suction on the upper surface near the
leading edge. For shallow angles of incidence (typically $<7%
%TCIMACRO{\UNICODE[m]{0xb0}}%
%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 where the boundary layer attempts to recover to a state of
turbulent equillibrium. This region 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. Consquently the
nature of the boundary layer after reattachment can have a significant
impact on the lift coefficient.\FRAME{ftbpFU}{381.9375pt}{122.4375pt}{0pt}{%
\Qcb{Schematic of the flow past a flat plate at shallow incidence.}}{\Qlb{%
plate}}{plate1.wmf}{\special{language "Scientific Word";type
"GRAPHIC";display "FULL";valid_file "F";width 381.9375pt;height
122.4375pt;depth 0pt;original-width 26.4027in;original-height
8.8609in;cropleft "0";croptop "1";cropright "1.0515";cropbottom "0";filename
'Plate1.wmf';file-properties "XNPEU";}}

A typical pressure coefficient distribution for a flat plate at shallow
incidence is illustrated in Figure \ref{plate_cp}. Also presented are
typical 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 this region. Reattachment
occurs approximately at the conclusion of the adverse pressure gradient that
exists over the aft half of the bubble.

\FRAME{ftbpFU}{4.0274in}{3.0277in}{0pt}{\Qcb{Typical pressure coefficient
plots.}}{\Qlb{plate_cp}}{plate_cp.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 4.0274in;height 3.0277in;depth 0pt;original-width
27.7778in;original-height 15.1524in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Plate_Cp.wmf';file-properties "XNPEU";}}

\subsection{The thin airfoil bubble}

The separation bubble that forms due to the inability of the boundary layer
to remain attached around the sharp leading edge is known as the
\textquotedblleft thin airfoil bubble\textquotedblright . 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 it is hypothesised that eddy viscosity models
(which effectively predict instantaneous transition to turbulence) can be
applied much more suitably to the thin airfoil bubble than to flows that
involve regions of laminar flow.

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. Crompton \cite{Crompton01} found Reynolds number
insensitivity of the reattchment length 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. 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 as shear layer enters a state of turbulent
equllibrium. 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 independent of
Reynolds number and fixed at approximately $2-2.5\%c$. This implies that the
mechanism initiating transition is not driven by the freestream flow (since
this is significantly dependent on Reynolds number) but instead transition
occurs very close to the leading edge due to a process called shear layer
flapping \cite{Driver87} where alternate vortices are shed from the origin
of the shear layer. Crompton \cite{Crompton01} detected low frequency
unsteadyness at too large a scale to be associated with the large eddies and
concluded that vortical structures, of the type associated with shear layer
flapping, must be present.

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 within the separation bubble
Reynolds number dependence can be expected. 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 length of the bubble.
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 bubble
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 velocity and hence
also the freestream Reynolds number.

\subsection{The short bubble}

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 near the leading edge. 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.

\FRAME{ftbpFU}{5.559in}{2.0141in}{0pt}{\Qcb{The short airfoil bubble (not to
scale).}}{\Qlb{short bubble}}{short_bubble1.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 5.559in;height 2.0141in;depth
0pt;original-width 26.3059in;original-height 9.9306in;cropleft "0";croptop
"1";cropright "1.0515";cropbottom "0";filename
'short_bubble1.wmf';file-properties "XNPEU";}}

Flows involving the short separation bubble pose several difficulties for
turbulence models. The boundary layer that exists between the forward
stagnation point and the separation bubble is laminar and generally
turbulence models predict early transition to turbulent flow in this region.
As an example, for flat plate boundary layers under zero pressure gradient,
turbulence 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 simulations involving short separation bubbles the
calculated boundary layer is likely to be turbulent upstream of the
separation point. Consequently the solution will exhibit an unrealistic
increase in skin friction due to the high shear of turbulent boundary layers
compared with laminar ones. 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 dettachment, upstream of the desired
location. 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 at predicting the correct
transition location 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 a difficult 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 additional difficulties to tuning the model since the
coefficients are usually calibrated using boundary layer flows. It is also
likely that within short separation bubbles some degree of relaminarisation
(reverse transition) will occur because the reversed flow is driven by a
favourable pressure gradient. This relaminarisation process is another
phenomenon that most turbulence models stuggle to predict correctly.

\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, $X_{R}$, 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 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. 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 through small departures from
zero incidence.

\FRAME{ftbpFU}{5.0341in}{0.595in}{0pt}{\Qcb{Flat plate dimensions.}}{\Qlb{%
plate_dim}}{plate_crompton.bmp}{\special{language "Scientific Word";type
"GRAPHIC";display "USEDEF";valid_file "F";width 5.0341in;height
0.595in;depth 0pt;original-width 27.1248in;original-height 3.0969in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'plate_crompton1.bmp';file-properties "XNPEU";}}

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 CFX model}

Crompton's flate plate was modelled with thickness using the geometry
described in Figure \ref{plate_dim}. The thickness and asymmetry of the
plate influences the flow to a degree where it is necessary to use the true
geometry, rather than an infinitely thin plate for which grid generation
would be simpler. The asymmetry of the plate creates a lift force even at
zero degrees of incidence and which causes the flow to curve upwards (in the
direction of the lift axis) near the leading edge 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. 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 zero static pressure boundary condition was
imposed.

\FRAME{ftbpFU}{5.6394in}{3.0848in}{0pt}{\Qcb{Details of the computational
domain for the flat plate.}}{\Qlb{plate domain}}{plate_domain.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 5.6394in;height 3.0848in;depth
0pt;original-width 27.7778in;original-height 15.1524in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'plate_domain1.wmf';file-properties "XNPEU";}}

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 the wall and
at the trailing edge the aspect ratio reduces to 1:10.\FRAME{ftbpFU}{5.9586in%
}{3.7317in}{0pt}{\Qcb{Computational grid for the flat plate (medium
resolution).}}{\Qlb{plate_grid}}{plate_grids.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 5.9586in;height 3.7317in;depth
0pt;original-width 9.9998in;original-height 6.25in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'plate_grids1.wmf';file-properties
"XNPEU";}}

\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.
Therefore provided the grid independence is achieved for $\alpha =3$, the
same grid should be suitable for the $\alpha =1$ 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.

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%}}%
%BeginExpansion
\begin{figure}%
%
%EndExpansion
\begin{tabular}{ll}
\FRAME{itbpF}{2.687in}{1.7193in}{0in}{}{}{lift_gc1.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 2.687in;height 1.7193in;depth
0in;original-width 9.167in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'lift_GC1.wmf';file-properties
"XNPEU";}} & \FRAME{itbpF}{2.687in}{1.7193in}{0in}{}{}{drag_gc1.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.687in;height 1.7193in;depth
0in;original-width 9.167in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'drag_GC1.wmf';file-properties
"XNPEU";}}
\end{tabular}
\caption{Grid convergence of the lift and drag coefficients.}\label{plate_GC}%
%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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. Since forces are
integrated properties there is a possibility that the 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.

\FRAME{ftbpFU}{5.1552in}{3.0139in}{0pt}{\Qcb{Grid convergence of the surface
pressure coefficients $(\protect\alpha =3)$.}}{\Qlb{plate CP_GC}}{%
cp_grid_convergence.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
5.1552in;height 3.0139in;depth 0pt;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Cp_Grid_convergence1.wmf';file-properties "XNPEU";}}

The $C_{P}$ curves for the medium and fine grid lie almost directly upon one
anther, the fine grid solution has slightly less suction at the leading
edge, however this effect is minimal. Since the fine grid and medium grid
solutions are virtually identical the medium grid was chosen as the most
suitable grid, and has been 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}.

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%}}%
%BeginExpansion
\begin{figure}%
%
%EndExpansion
%TCIMACRO{
%\TeXButton{\begin{center}}{\begin{center}%
%} }%
%BeginExpansion
\begin{center}%
%
%EndExpansion
\begin{tabular}{ll}
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.031$}}{}{bl_s1.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_S1.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.125$}}{}{%
bl_s2.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S2.wmf';file-properties "XNPEU";}}
\end{tabular}
%TCIMACRO{
%\TeXButton{\end{center}}{\end{center}%
%}}%
%BeginExpansion
\end{center}%
%
%EndExpansion
\caption{Chordwise velocity profiles within the leading edge bubble
($\protect\alpha =1).$} \label{BL_S1-2} 
%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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.

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%} }%
%BeginExpansion
\begin{figure}%
%
%EndExpansion
\begin{tabular}{ll}
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.031$}}{}{bl_s1_close.wmf%
}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_S1_close.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.125$}}{}{%
bl_s2_close.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S2_close.wmf';file-properties "XNPEU";}}
\end{tabular}
\caption{ Near-wall chordwise velocity profiles within the leading edge
bubble ($\protect\alpha =1).$} \label{BL_S1-2 close} 
%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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}

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
difficult to quantify and it is hard to assess the structure of the
different boundary layers.

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%} }%
%BeginExpansion
\begin{figure}%
%
%EndExpansion
\begin{tabular}{cc}
\FRAME{itbpFU}{2.9283in}{1.7193in}{0.0104in}{\Qcb{$x/c=0.250$}}{}{bl_s3.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0.0104in;original-width 9.9998in;original-height 5.8332in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S3.wmf';file-properties "XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{%
0in}{\Qcb{$x/c=0.375$}}{}{bl_s4.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 2.9283in;height 1.7193in;depth 0in;original-width
9.9998in;original-height 5.8332in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'BL_S4.wmf';file-properties "XNPEU";}} \\ 
\FRAME{itbpFU}{2.9283in}{1.7193in}{0.0208in}{\Qcb{$x/c=0.500$}}{}{bl_s5.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0.0208in;original-width 9.9998in;original-height 5.8332in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S5.wmf';file-properties "XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{%
0in}{\Qcb{$x/c=0.625$}}{}{bl_s6.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 2.9283in;height 1.7193in;depth 0in;original-width
9.9998in;original-height 5.8332in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'BL_S6.wmf';file-properties "XNPEU";}} \\ 
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.750$}}{}{bl_s7.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_S7.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.875$}}{}{%
bl_s8.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S8.wmf';file-properties "XNPEU";}} \\ 
\multicolumn{2}{c}{\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=1.000$}%
}{}{bl_s9.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_S9.wmf';file-properties "XNPEU";}}}
\end{tabular}
\caption{Chordwise velocity profiles downstream of reattachment
($\protect\alpha =1).$} \label{BL S3-9} 
%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%} }%
%BeginExpansion
\begin{figure}%
%
%EndExpansion
%TCIMACRO{
%\TeXButton{\begin{center}}{\begin{center}%
%}}%
%BeginExpansion
\begin{center}%
%
%EndExpansion
\begin{tabular}{l}
\FRAME{itbpFU}{5.028in}{2.7371in}{0in}{\Qcb{$x/c=0.250$}}{}{logbl_s3.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 5.028in;height 2.7371in;depth
0in;original-width 9.9998in;original-height 5.4163in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'logBL_S3.wmf';file-properties
"XNPEU";}} \\ 
\FRAME{itbpFU}{5.028in}{2.7371in}{0.0104in}{\Qcb{$x/c=0.875$}}{}{logbl_s8.wmf%
}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 5.028in;height 2.7371in;depth
0.0104in;original-width 9.9998in;original-height 5.4163in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'logBL_S8.wmf';file-properties "XNPEU";}}
\end{tabular}
%TCIMACRO{
%\TeXButton{\end{center}}{\end{center}%
%}}%
%BeginExpansion
\end{center}%
%
%EndExpansion
\caption{Chordwise velocity profiles downstream of reattchment (log scale,
$\protect\alpha =1).$} \label{Log BL_S3,8} 
%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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).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 slight adverse pressure gradient that the boundary layer must
work against. However 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. At the reattachment point
the normal u-velocity gradient ($\partial u/\partial y)$ 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 near-wall
behaviour influences the entire velocity profile and at reattachment
velocity profiles have a more linear profile with signicifantly lower
velocities than for a typical turbulent boundary layer. A boundary layer
profile soon after reattachment 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 measurement staion was
positioned soon after reattachment.\FRAME{ftbpFU}{4.0274in}{2.3609in}{0pt}{%
\Qcb{Boundary layer profile just after reattachment (Crompton) at $\protect%
\alpha =3$ and $x/c=0.5.$}}{\Qlb{BL XR}}{bl_xr.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 4.0274in;height 2.3609in;depth
0pt;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_XR.wmf';file-properties
"XNPEU";}}

Also evident in the log profiles of Figure \ref{Log BL_S3,8} is that the
near-wall velocities are much slower 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 has lower velocities than the CFD results. 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 flow variables 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 again at the results for $x/c=0.25$ (Figure \ref{Log BL_S3,8}) the
profiles predicted by the CFD also have lower velocities through the log-law
region than the experimental results. This is a combination of the lower
velocities that CFD predicts around reattachment and also because the CFD\
experiences 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 through the
log-region the boundary layer has recovered and is showing the typical
response to the favourable pressure gradient. Within the viscous sublayer
the velocities are still much lower than in the ZPG\ profile. In the log-law
region the CFD\ results show slightly sluggish boundary layers despite the
favourable pressure gradient, this would indicate that the influence of
reattachment point extends further downstream in the CFD than in the
experiments. 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.

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.

\subsubsection{Pressure distributions}

The pressure coefficient plot for the flat plate at $\alpha =1$ is presented
in Figure \ref{CP_1}. In the experimental data the pressures continue to
decrease downstream of the leading edge and 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 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
coefficient plot ($\protect\alpha =1)$.}}{\Qlb{CP_1}}{cp_1.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 4.0274in;height 2.194in;depth
0pt;original-width 9.9998in;original-height 5.4163in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'CP_1.wmf';file-properties
"XNPEU";}}\newpage

\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}
%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%}}%
%BeginExpansion
\begin{figure}%
%
%EndExpansion

\begin{tabular}{cc}
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.031$}}{}{bl_3s1.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_3S1.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.125$}}{}{%
bl_3s2.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_3S2.wmf';file-properties "XNPEU";}} \\ 
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.250$}}{}{bl_3s3.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'BL_3S3.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.375$}}{}{%
bl_3s4.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'BL_3S4.wmf';file-properties "XNPEU";}}
\end{tabular}
\caption{Chordwise velocity profiles within the leading edge
bubble($\protect\alpha =3).$}\label{BL 3 S1-4}

%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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 $(x/c=0.375),$ 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}{4.0274in}{2.3609in}{0pt%
}{\Qcb{Near wall chordwise velocity profiles at $x/c=0.031$ ($\protect\alpha %
=3).$}}{\Qlb{BL_3S1 close}}{bl_3s1_close.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 4.0274in;height 2.3609in;depth 0pt;original-width
9.9998in;original-height 5.8332in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'BL_3S1_close.wmf';file-properties "XNPEU";}}

\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 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. 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 secondary recirculation bubble
enhances the mean-shear in the leading edge region whichincreases 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 $\omega )$.

%TCIMACRO{
%\TeXButton{TeX field}{\begin{figure}%
%}}%
%BeginExpansion
\begin{figure}%
%
%EndExpansion

\begin{tabular}{cc}
\FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.031$}}{}{k_3s1.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0in;original-width 9.9998in;original-height 5.8332in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'k_3S1.wmf';file-properties
"XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{0in}{\Qcb{$x/c=0.125$}}{}{%
k_3s2.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.9283in;height 1.7193in;depth 0in;original-width 9.9998in;original-height
5.8332in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'k_3S2.wmf';file-properties "XNPEU";}} \\ 
\FRAME{itbpFU}{2.9283in}{1.7193in}{0.0311in}{\Qcb{$x/c=0.250$}}{}{k_3s3.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.9283in;height 1.7193in;depth
0.0311in;original-width 9.9998in;original-height 5.8332in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'k_3S3.wmf';file-properties "XNPEU";}} & \FRAME{itbpFU}{2.9283in}{1.7193in}{%
0in}{\Qcb{$x/c=0.375$}}{}{k_3s4.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 2.9283in;height 1.7193in;depth 0in;original-width
9.9998in;original-height 5.8332in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'k_3S4.wmf';file-properties "XNPEU";}}
\end{tabular}
\caption{Chordwise velocity profiles within the leading edge
bubble($\protect\alpha =3).$} \label{k 3S1-4}

%TCIMACRO{
%\TeXButton{TeX field}{\end{figure}%
%}}%
%BeginExpansion
\end{figure}%
%
%EndExpansion

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
all 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.

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 int he
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 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 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.

\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 $X_{R}$.

\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.
\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.
\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
result of the SST\ model blending to the $\epsilon $-equation through the
outer region of the boundary layer.
\end{itemize}
\end{enumerate}

\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.

\section{Conclusions}


