# 2-D linearised Euler equation

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- | Problem Definition | + | == Problem Definition == |

:<math> \frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0 </math> | :<math> \frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0 </math> | ||

:<math> \frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0 </math> | :<math> \frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0 </math> | ||

+ | :<math> \frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial p}{\partial x}=0 </math> | ||

+ | :<math> \frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0 </math> | ||

+ | 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 == | ||

+ | :<math> p(x,0)=a*exp(-ln(2)*((x-xc)^2+(y-yc)^2)/b^2) </math> | ||

+ | == 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}} |

## Latest revision as of 12:31, 19 December 2008

## Contents |

## Problem Definition

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

## 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.