25 | | The Riemann problem is essentially the Euler equations with discrete initial conditions. Initial data has a left state and a right state, separated by a discontinuity. |
| 25 | The Riemann problem is essentially the Euler equations with discrete initial conditions. If you turned the differential form of the Euler equations into an eigenvalue problem, you would find three eigenvalues and three corresponding eigenvectors. These eigenvalues correspond to wave speeds of v, v + a, and v - a where v is the velocity and a is the sound speed. The sound speed is defined as: |
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| 27 | [[latex($a=\sqrt{\frac{\gamma p}{\rho}}$)]] |
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| 29 | An analysis of the eigenvectors would show that there are essentially three types of waves; contact, shock, and rarefaction. The contact wave is always in the middle, and the left or right waves can be either shock or rarefaction waves. The waves divide x-t space into four distinct regions: left data, star left, star right, and right data. Below is an image showing an example of this: |
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| 31 | [[Image(riemann_regions.gif, width= 600)]] |
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| 33 | Initial data has a left state and a right state, separated by a discontinuity. It is often easiest to solve the Riemann problem using what are known as primitive variables. The primitive variables are density, velocity, and pressure. It is important to keep in mind that, apart from density, these are different than the conserved variables. |