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\begin{center}{\large
CHAPTER 3. HIGH-SPEED FLOWS} \\
AND BOUNDARY LAYERS \end{center}
3-1-invisc.tex\\
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In this chapter we examine high-speed flows of a viscous fluid. As a prelude, the limit of inviscid flows is breifly discussed.
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\section{Flow of invisid and homogeneous fluids}
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\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/
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