# 2-D linearised Euler equation

(Difference between revisions)
 Revision as of 07:24, 12 November 2005 (view source)← Older edit Revision as of 07:37, 12 November 2005 (view source)Newer edit → Line 15: Line 15: 4th order low storage RK in time 4th order low storage RK in time == Results == == Results == + Pressure + No mean flow + [[Image:Nomeanflow.jpg]] + Uniform Mean flow to the left at U=0.5 (speed of sound assumed to be 1) + ==  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=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}} *{{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]

## Initial Condition

$p(x,0)=a*exp(-ln(2)*((x-xc)^2+(y-yc)^2)/b^2)$

## Boundary Condition

Characteristic Boundary Condition

## Numerical Method

4th Order Compact scheme in space 4th order low storage RK in time

## Results

Pressure No mean flow Uniform Mean flow to the left at U=0.5 (speed of sound assumed to be 1)

## 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.