2-D linearised Euler equation

(Difference between revisions)

Latest revision as of 12:31, 19 December 2008

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

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

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

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

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.

[-50,50]*[-50,50]

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

Characteristic Boundary Condition

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

Pressure

No mean flow

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

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.

@@ Line 8: / Line 8: @@
 [-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 ==
+th Order Compact scheme in space
+th 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}}