# 2-D linearised Euler equation

(Difference between revisions)
 Revision as of 02:03, 8 October 2005 (view source)← Older edit Latest revision as of 12:31, 19 December 2008 (view source)Peter (Talk | contribs) m (Reverted edits by DarpaSbotr (Talk) to last version by Harish gopalan) (11 intermediate revisions not shown) Line 1: Line 1: - Problem Definition + == Problem Definition == :$\frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$ :$\frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$ + :$\frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0$ + :$\frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial p}{\partial x}=0$ + :$\frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0$ + 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 + [[Image:Nomeanflow.jpg]] + :Mean Flow to left at U=0.5 (c assumed to be 1 m/s) + [[Image:Meanflow.jpg]] + + + ==  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]

## 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

Mean Flow to left at U=0.5 (c assumed to be 1 m/s)

## 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.