/ CME306 CS205B Theory Homework 8
Euler equations
For incompressible flow the inviscid 1D Euler equations decouple to: t + ux = 0 px =0 ut + et + uex = 0 The 3D Euler equations are given by w v u uw uv u2 + p u v + uv + v 2 + p + vw = 0 w2 + p vw uw w (E + p)w z (E + p)v y (E + p)u x E t
(1)
where is the density, u = (u, v, w) are the velocities, E is the total energy per unit volume and p is the pressure. The total energy is the sum of the internal energy and the kinetic energy. E = e+ 1 u 2
2
= e + (u2 + v 2 + w2 )/2 where e is the internal energy per unit The mass. assumption of incompressiblity gives u = ux + vy + wz = 0, Show that in 3D the inviscid Euler equations with the assumption of incompressible flow decouple to: =0 px ut + u u + =0 py =0 vt + u v + pz wt + u w + =0 et + u e = 0 t + u (2)
1
Compressible Flow
Find the Jacobian and the right eigenvectors for Euler's equations in 1-D, (hint: it is ...