| 15 | | Where [[latex($\rho$)]] is mass density, v is velocity (only in x-direction for 1D), p is pressure, and E is total energy per unit volume. E is further defined as [[latex($E = \rho(\frac{1}{2}v^2 + e)$)]] where e is the specific internal energy. The specific internal energy depends on the equation of state. For an ideal gas [[latex($e = \frac{p}{\rho(\gamma - 1)}$)]] where [[latex($\gamma$)]] is the ratio of specific heats. Basically, these laws state that in a given volume, the change in a conserved quantity must be equal to the flux through the boundaries of that volume. In other words, the conserved quantity is in front of the time derivative and its flux is in front of the spatial derivative. |
| | 15 | Where [[latex($\rho$)]] is mass density, v is velocity (only in x-direction for 1D), p is pressure, and E is total energy per unit volume. E is further defined as [[latex($E = \rho (\frac{1}{2} v^2 + e)$)]] where e is the specific internal energy. The specific internal energy depends on the equation of state. For an ideal gas [[latex($e = \frac{p}{\rho(\gamma - 1)}$)]] where [[latex($\gamma$)]] is the ratio of specific heats. Basically, these laws state that in a given volume, the change in a conserved quantity must be equal to the flux through the boundaries of that volume. In other words, the conserved quantity is in front of the time derivative and its flux is in front of the spatial derivative. |
| 24 | | == Solving the Riemann Problem == |
| 25 | | The Riemann problem is essentially the Euler equations with discrete initial conditions. |
| | 24 | == The Riemann Problem == |
| | 25 | The Riemann problem is essentially the Euler equations with discrete initial conditions. The initial conditions consist of a left state and a right state separated by a discontinuity. 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: |
| 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: |
| | 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. Furthermore, velocity and pressure inside the star region is constant while density can be different in the star left and star right regions. Data across a shock wave has a discontinuous jump while data across a rarefaction has a smooth transition. This means that a rarefaction wave actually has some "thickness" to it. The region inside a rarefaction is known as the fan state. |
| 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. |
| | 33 | [[BR]] |
| | 34 | == The Exact Riemann Solver == |
| | 35 | A description of an exact Riemann solver can be quite tedious and lengthy, so I will leave out many details in this section. In solving the Riemann problem, we will use 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. Basically, the unknowns are in the star region, so if we can find these quantities then the problem is solved. |
| | 36 | === 1. Find the pressure and velocity in the star region === |
| | 37 | This pressure is computed through an iteration scheme. You start with a guess for the star pressure. This can be found using approximate Riemann solvers. Then, you calculate pressure functions to get a better value for the pressure. If the difference between the guess pressure and better pressure is below some tolerance then you have found the star pressure. If it is not below the tolerance, the better pressure becomes the new guess pressure and the process repeats. The star velocity is then very straightforward, as it is computed directly from the star pressure. |
| | 38 | === 2. Sampling the wave speed === |
| | 39 | Now that we have the solution for the star region, we can get the solution everywhere. This is done by sampling the wave speed S as opposed to sampling the position x. There are essentially 10 possible solutions for any given S: |
| | 40 | ||= =||= Left side of contact discontinuity =||= Right side of contact discontinuity =|| |
| | 41 | ||= Data =||= [[latex($W_L$)]] =||= [[latex($W_R$)]] =|| |
| | 42 | ||= Star (Shock) =||= [[latex($W_{\mathrm{*}L}^{SHO}$)]] =||= [[latex($W_{\mathrm{*}R}^{SHO}$)]] =|| |
| | 43 | ||= Star (Rarefaction) =||= [[latex($W_{\mathrm{*}L}^{RAR}$)]] =||= [[latex($W_{\mathrm{*}R}^{RAR}$)]] =|| |
| | 44 | ||= Rarefaction Fan =||= [[latex($W_{L}^{FAN}$)]] =||= [[latex($W_{R}^{FAN}$)]] =|| |
