3.9.1 Fixed bodies & translating bodies

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3.9.1 Fixed bodies & translating bodies - Galilean transformation

$\begin{figure} \begin{center} \epsfig{file=lfig116.eps,height=2in,clip=} \end{center} \end{figure}$

Reference system O: $\vec{v}, \phi, p$

$\begin{figure} \begin{center} \epsfig{file=lfig117.eps,height=1in,clip=} \end{center} \end{figure}$

$\begin{align}\nabla ^2\phi & = 0 \notag \\ \mathord{\buildrel{\lower3pt\hbox{$... ...i & \to 0 \mbox{\ as\ } \vert\vec{x}\vert \rightarrow \infty \notag \end{align}$

Reference system O': $\vec{v}', \phi', p'$

$\begin{figure} \begin{center} \epsfig{file=lfig118.eps,height=1in,clip=} \end{center} \end{figure}$

$\begin{align}\nabla ^2\phi' & = 0 \notag \\ \mathord{\buildrel{\lower3pt\hbox{... ...\to -Ux' \mbox{\ as\ } \vert\vec{x}'\vert \rightarrow \infty \notag \end{align}$

Galilean transform:

$\begin{align}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonu... ... \\ - Ux' + \phi (x = x' + Ut,y,z,t) & = \phi '(x',y,z,t) \notag \end{align}$

Pressure (no gravity)

$\begin{alignat}{4} p_\infty = & - \frac{1}{2}\rho v^2 + C_o & \hspace{1in} & p_\... ...}_o \notag \\ = & C_o & & & = {C'}_o - \frac{1}{2}\rho U^2 \notag \end{alignat}$

$\begin{displaymath}% latex2html id marker 1158 \therefore C_o = C'_o-\frac{1}{2}\rho U^{2} \notag \end{displaymath}$

In O: unsteady flow

$\begin{align}% latex2html id marker 1163 p_s = & - \rho \frac{\partial \phi }{\p... ...fty = & \frac{1}{2}\rho U^2 \mbox{\ stagnation pressure \ } \notag \end{align}$

In O`: steady flow

$\begin{align}p_s = & - \rho \underbrace {\frac{\partial \phi '}{\partial t}}_0 -... ...p_\infty = & \frac{1}{2}\rho U^2 \mbox{\ stagnation pressure} \notag \end{align}$

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