\input{aefm-macro}

\setcounter{chapter}{3}
\begin{center}{\large 
CHAPTER 3. HIGH-SPEED FLOWS} \\
AND BOUNDARY LAYERS \end{center}
3-1-invisc.tex\\
\today

In this chapter we examine high-speed flows of a viscous fluid. As a prelude, the limit of inviscid flows is breifly discussed. 


\setcounter{section}{0}
\section{Flow of invisid and homogeneous fluids} 
\setcounter{equation}{0}
\subsection{Irrotational flows}
  For an inviscid and incompressible fluid with constant density,
\[ \f{D\vec{\zeta}}{Dt} = \vec{\zeta} \cdot \nabla \vec{q} . \]
If $\vec{\zeta} = 0$ everywhere at $t = t_0$, then
\[ \f{D\vec{\zeta}}{Dt} = 0 \] 
at $t = t_0$  for all $ \vec{x}$. Therefore, at 
$t= t_0 + dt'$, $\vec{\zeta}   = 0   $ everywhere. 
Repeating the argument, $\vec{\zeta}$ remains zero at $t= t_0 + 2 dt, t_0+ 3 dt, \dots, $ for all $\vec{x}$. In other words the flow is irrotational at all times if it is so at the start.    A flow in which 
$ \vec{\zeta} = \nabla \times \vec{q} $ vanishes everywhere  is called an   irrotational flow.


It is a well known identity in vector analysis that an irrotational vector   can  be expressed as the gradient of a scalar potential. Thus we define the velocity potential $\phi$ by 
\be \vec{q} = \nabla \phi   \label{Eq:(8.1)} \ee
 An immediate consequence of continuity is that 
\be \nabla \cdot \vec{q} = \nabla \cdot \nabla \phi = \nabla^2 \phi =\f{\p^2 \phi}{\p x^2}+\f{\p^2 \phi}{\p y^2}+\f{\p^2 \phi}{\p z^2}= 0 \label{Eq:laplace}\ee
i.e., $ \phi$ is a harmonic function of $\vec{x}$. Note 
 that (\ref{Eq:laplace}) is the result of mass and momentum conservation. 
 
 If the motion is two-dimensional in the $x,y$ plane then continuity equation reads:
\be   \f{\p u}{\p x} + \f{\p v}{\p y} = 0 \label{Eq:9.1} \ee
and irrotationality requires
\be  \f{\p v}{\p x} - \f{\p u}{\p y} = 0 \label{Eq:9.2} \ee
These two equations are identical to the Cauchy-Riemann conditions relating    the real and imaginary parts  of an analytic function of the complex variable $z=x+iy$.  
The velocity components can be expressed as the gradient of a  two-dimensional  potential $\phi$
\be u = \f{\p \phi}{\p x} \qquad v = \f{\p \phi}{\p y} \label{Eq:9.3} \ee
 which  satifies the Laplace equation,
\be \f{\p^2 \phi}{\p x^2} + \f{\p^2 \phi}{\p y^2} = 0   \label{Eq:9.4} \ee
 It is  also useful to introduce another scalar function, the stream function $\psi$, defined by
\be u = \f{\p \psi}{\p y} \qquad v = - \f{\p \psi}{\p x} . \label{Eq:9.5} \ee
so that     (\ref{Eq:9.1}) is satisfied  automatically.
  Substituting Eqn. (\ref{Eq:9.5}) into Eqn. (\ref{Eq:9.2}), we find $\psi$ to be a harmonic function too.
\be \f{\p^2 \psi}{\p x^2} + \f{\p^2 \psi}{\p y^2} = 0 . \label{Eq:9.6} \ee

By definition, 
\be (u=) \f{\p \phi}{\p x}= \f{\p \psi }{\p y}, ~~~~ (v=)\f{\p \phi}{\p y}= -\f{\p \psi }{\p x}\ee
therefore $\phi$ and $\psi$ also satisfy Cauchy-Riemann conditons and are harmonic conjugates of each other.  This is why the theory of complex functions is an important tool in  two-dimensional potential flows. 

In the plane of $x,y$,
lines of constant $\phi$ are called  equipotential lines;  the velocity vector is normal to equipotential lines and is directed from lower to higher poteitials. 
Lines of constant $\psi$ are the  streamlines; the velocity vector is  tangential to the local streamline.  It follows that equi-potentials are perpendicular to streamlines.   As a formal proof we note that
\be \nabla \phi \cdot \nabla \psi = \phi_x \psi_x + \phi_y \psi_y = u(-v) + vu = 0  \label{Eq:9.7} \ee
 
 
\begin{figure} [h]\includegraphics[scale=0.75]{streamfn-1.eps}
\caption{Definition  of the stream function}. 
\label{fig:streamfunction-1}
\end{figure}


   
Indeed the difference of the stream functions at two points is just the volume flux rate between   the two points. 
 This can be seen by using the definitions (\ref{Eq:9.5}).  First $\psi (x,y)$ has the dimension of volume flux rate : $UL = L^2/T$. With reference to  Figure (\ref{fig:streamfunction}), 
the  flux between two streamlines can be calculated in two equivalent ways 
\[
  u\delta y \,  (\mbox{along} \; x = \,\mbox{constant})= -v\delta x \,  (\mbox{along} \; y = \,\mbox{constant})
\]
In view of (\ref{Eq:9.5}),  
\[
u = \f{\p \psi}{\p y}=\f{\delta \psi}{\delta y} \mid_{x = \, \mbox{const.}}, ~~~
v = - \f{\p \psi}{\p x}= - \f{\delta \psi}{\p x} \mid_{y = \, \mbox{const.}},
\]
hence 
\[ u\delta y=\f{\delta \psi}{\p y}\delta y = \delta \psi, ~~~-v\delta x=\f{\delta \psi}{\p x}\delta x = \delta \psi.\] 
where $\delta \psi=\psi_2-\psi_1$.
simple observations will confirm that the stream funciton  has   all the features  
of  the rate of volume flux.
\begin{figure} \includegraphics[scale=0.75]{streamfn-2.eps}
\caption{Physical meaning of the stream function}. 
\label{fig:streamfunction-2}
\end{figure}
From the theory of complex functions, the following complex potential 
\be w = \phi (x,y) + i\psi (x,y) \qquad i = \sqrt{-1} . \label{Eq:9.9} \ee
is  analytic in $z = x+iy$, except at singular points.  In particular the derivative   is independent of direction. Indeed, 
\[ \f{d w}{d z}= \f{\p w}{\p x}= \f{\p w }{i\p y}\]
since \[\f{\p w}{\p x}= \f{\p \phi}{\p x}+i\f{\p \psi}{\p x}=u-iv,~~\mbox{and}~~\f{\p w }{i\p y}=
 -i \f{\p \phi }{\p y}+\f{\p \psi }{\p y}=u-iv.\]
Because of these connections to complex variables, the theory of analytical functions has been a powerful tool for solving 2D irrotational flow problems for a long time. Its luster has  faded somewhat only after  the advent of computers. 

\subsection{Bernoulli theorems of homogeneous fluids}   

\underline{Unsteady and irrotational  flows}

From the momentum equation, 
\be \f{\p \vec{q}}{\p t} + \nabla \, \f{\vec{q}\,^2}{2} - \vec{q}  \times \lp \nabla \times \vec{q} \rp = - \f{1}{\rho} \, \nabla p + \vec{f} . \label{Eq:(8.2)} \ee
If the body force is conservative and the flow irrotational, i.e., $\vec{f} = - \nabla \Gamma$   and  $\vec{q} = \nabla \phi$, then
\[ \nabla \lb \f{\p \phi}{\p t} + \f{\vec{q}\,^2}{2} + \f{p}{\rho} + \Gamma \rb = 0 \]
which can be integrated in space to give
\be \f{\p \phi}{\p t} + \f{\vec{q}\,^2}{2} + \f{p}{\rho} + \Gamma = C(t)  \label{Eq:(8.3)} \ee
for all $\vec{x}$. This Bernoulli law is useful in the theory of surface waves. 

\underline{Steady but rotational flows}    
The momentum equation reads:
\[ \vec{q} \cdot \nabla \vec{q} = - \f{1}{\rho} \, \nabla p + \vec{f} \]
still  applies. If $\rho = $ constant and $\vec{f} = - \nabla \Gamma$, we have
\[ q_j \, \f{\p q_i}{\p x_j} = - \f{1}{\rho} \, \f{\p p}{\p x_i} - \f{\p \Gamma}{\p x_i} \]
and, by scalar multiplication with $q_i$, 
\[ q_i \lp q_j \, \f{\p q_i}{\p x_j} \rp = q_i \, \f{\p}{\p x_i} \, \lp - \f{p}{\rho} - \Gamma \rp \]
Now the left-hand side can be written as
\[ q_j \, \f{\p}{\p x_j} \, \f{q_iq_j}{2} \quad \mbox{since} \quad \f{\p q_j}{\p x_j} = 0 \]
Therefore,
\[ q_i \, \f{\p}{\p x_i} \, \lb \f{\vec{q}\,^2}{2} + \f{p}{\rho} + \Gamma \rb = 0 .  \]
and
\be \f{\vec{q}\,^2}{2} + \f{p}{\rho} + \Gamma = \mbox{constant along a streamline.} \label{Eq:(8.4)} \ee
A streamline is a curve along which the velocity vectors are tangent to the line. It is importatn that the constant may be different for different streamlines, hence (\ref{Eq:(8.4)}) is different from (\ref{Eq:(8.3)}). 


Most of the wave phenonmena in fliuds can be described by an  inviscid  theory. The interested reader should visit the website for WAVES:

http://web.mit.edu/fluids-modules/waves/www/

\end{document}