\chapter{Background}
\section{Introduction and background}
This thesis deals with the design of mean camber lines and surfaces 
of lifting foils described in 
terms of B-splines.  Before the actual research can be presented, 
some background material must be reviewed.  First, the physics behind how 
lift is generated will be discussed.  Potential flow will then be introduced
as a means to represent ideal fluid flow passed a foil.  Next, the
geometric representation used to represent and manipulate the shape of the 
foil will be introduced.  Finally, the inverse design process will be 
described, through which one designs the shape of a foil by specifying the 
desired lift distribution over the foil's geometry.

\section{The physics behind the generation of lift}
%%  Talk about generation of lift as pressure difference across body in fluid
As a body moves through a fluid, it experiences forces generated by the
pressure of the fluid as it flows around the body.  The force exerted 
on a body can be resolved into two components, lift and drag. 
Lift is defined as the force 
acting normal to the body's direction of travel while drag is the 
force acting parallel and opposite to its
motion.  As the object moves
through a fluid, the fluid's motion (flow) is dictated by the laws
of physics and the geometry of the body.  By creating an 
asymmetric flow as it travels, unbalanced regions of high pressure 
and low pressure
will occur over the body creating a pressure difference across it.  
This pressure difference acts over the surfaces
of the body to generate a net force.  The generation of 
force due to a pressure imbalance is illustrated in 
Figure~\ref{fig:pressDiff}a.


%%  Discuss Bernoulli and relationship between pressure and velocity
In the early eighteenth century,
Bernoulli postulated an equation relating the total pressure exerted by 
an inviscid, incompressible, steady fluid flow on its surroundings to the 
velocity 
at which it is flowing~\cite[pp. 107-8]{Newman86}.  This 
relation 
has become known as Bernoulli's 
equation and relates the pressure in an inviscid, incompressible, 
steady flow
to the local velocity as given by
\begin{equation} \label{eqn:bern}
p + \frac{1}{2} \rho V^2 = constant,
\end{equation}
where $ V $ is the magnitude of the local flow velocity.
Therefore, to generate a given pressure distribution, one can expect a 
corresponding velocity field as defined by Equation~\ref{eqn:bern}.  
\begin{figure} 
	\centering
	\psfig{figure=/usr/deslab7/trjackso/Documents/MastersThesis/Figures/curve1.PS,width=2.7in}
\caption{Example of B-spline curve.}
\label{fig:bspCrvEx}
\end{figure}








