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8.1 Compressible Flow without Viscosity



Example: Frictionless air flow in the vicinity of an aircraft. The flow equations in Eulerian formulation reduce to
$\displaystyle \frac{\partial \rho }{\partial t}+ \nabla \cdot \rho \mbox{$\bf v$}$ $\textstyle =$ $\displaystyle 0$ (8.5)
$\displaystyle \frac{\partial \rho \mbox{$\bf v$}}{\partial t}+ \nabla \cdot
\left[ \rho \mbox{$\bf v$} \mbox{$\bf v$} \right] + \nabla p$ $\textstyle =$ $\displaystyle 0$ (8.6)
$\displaystyle \frac{\partial e}{\partial t}+ \nabla \cdot \left[ (e+p) \mbox{$\bf v$} \right]$ $\textstyle =$ $\displaystyle 0$ (8.7)



Euler derivative: laboratory-fixed coordinate system; $\partial/\partial t$ at a fixed point in space

Lagrange derivative: properties of a volume element that is moving along with the flow; $
d /dt \equiv \partial / \partial t + \mbox{$\bf v$} \cdot \nabla
$

$\Longrightarrow$Lagrange form of the flow equations:
$\displaystyle \frac{d \rho}{dt}$ $\textstyle =$ $\displaystyle - \rho \nabla \cdot \mbox{$\bf v$}$ (8.8)
$\displaystyle \rho \frac{d \mbox{$\bf v$}}{dt}$ $\textstyle =$ $\displaystyle - \nabla p$ (8.9)
$\displaystyle \frac{d e}{dt}$ $\textstyle =$ $\displaystyle - (e+p) \nabla p -(\mbox{$\bf v$} \cdot \nabla)\, p$  
  $\textstyle =$ $\displaystyle -e (\nabla \cdot \mbox{$\bf v$}) - \nabla \cdot (p \mbox{$\bf v$})$ (8.10)

where the last equation may be written (see 8.4),
\begin{displaymath}
\frac{d \epsilon}{dt} = - \, \frac{p}{\rho} \,(\nabla \cdot \mbox{$\bf v$})
\end{displaymath} (8.11)



Subsections
next up previous
Next: 8.1.1 Explicit Eulerian Methods Up: 8. Hydrodynamics Previous: 8. Hydrodynamics
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001