# 2-D linearised Euler equation

(Difference between revisions)
 Revision as of 02:08, 8 October 2005 (view source)← Older edit Revision as of 02:10, 8 October 2005 (view source)Newer edit → Line 1: Line 1: - Problem Definition + == Problem Definition == :$\frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$ :$\frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$ :$\frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0$ :$\frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0$ Line 5: Line 5: :$\frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0$ :$\frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0$ where  M is the mach number , speed of sound is assumed to be 1, all the variabled refer to acoustic perturbations over the mean flow. where  M is the mach number , speed of sound is assumed to be 1, all the variabled refer to acoustic perturbations over the mean flow. - :Domain [-50,50]*[-50,50] + == Domain == [-50,50]*[-50,50] - :Initial Condition + == Initial Condition == - :Boundary Condition + == Boundary Condition == - :Numerical Method + == Numerical Method == - :Results + == Results == - :Reference + ==  Reference == + *{{reference-paper|author=Williamson, Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}} + + *{{reference-paper|author=Lele, Lele, S. K.|year=1992|title=Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics|rest=Journal of Computational Physics, Vol. 103, pp 16–42}}

## Problem Definition

$\frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$
$\frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0$
$\frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial p}{\partial x}=0$
$\frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0$

where M is the mach number , speed of sound is assumed to be 1, all the variabled refer to acoustic perturbations over the mean flow. == Domain == [-50,50]*[-50,50]

## Reference

• Williamson, Williamson (1980), "Low Storage Runge-Kutta Schemes", Journal of Computational Physics, Vol.35, pp.48–56.
• Lele, Lele, S. K. (1992), "Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics", Journal of Computational Physics, Vol. 103, pp 16–42.