eulerThreeTemp

Three temperature fluid model. Heavy particle translational temperature, heavy particle vibrational temperature, and electron translational temperature. The equations are solved in conservative form.

\[\notag \begin{align} \frac{\partial}{\partial t}\left( \begin{array}{c} \rho_{l} \\ \rho\,u_{x} \\ \rho\,u_{y} \\ \rho\,u_{z} \\ e_{t} \\ e_{e} \\ e_{v} \\ \end{array} \right) + \nabla\cdot\left( \begin{array}{ccc} \rho_{l}\,u_{x} & \rho_{l}\,u_{y} & \rho_{l}\,u_{z} \\ \rho\,u_{x}^{2}+P & \rho\,u_{x}\,u_{y} & \rho\,u_{x}\,u_{z} \\ \rho\,u_{y}\,u_{x} & \rho\,u_{y}\,u_{y} + P & \rho\,u_{y}\,u_{z} \\ \rho\,u_{z}\,u_{x} & \rho\,u_{z}\,u_{y} & \rho\,u_{z}\,u_{z} + P \\ u_{x}\left(e_{t}+P\right) & u_{y}\left(e_{t}+P\right) & u_{z}\left(e_{t}+P\right)\\ u_{x}\left(e_{e}+P_{e}\right) & u_{y}\left(e_{e}+P_{e}\right) & u_{z}\left(e_{e}+P_{e}\right)\\ u_{x}\left(e_{v}\right) & u_{y}\left(e_{v}\right) & u_{z}\left(e_{v}\right) \end{array} \right) = \left( \begin{array}{c} s_{l}\\ 0\\ 0\\ 0\\ \mathbf{E}\cdot \mathbf{J} + Q_{v-t}\\ \mathbf{E}\cdot \mathbf{J_e}\\ -Q_{v-t} \end{array} \right) \end{align}\]
\[\notag \begin{align} \begin{array}{l} P = P_{l} + P_{e}\\ Q_{v-t} = \sum_{l}\rho_{l}\left(\frac{e_{v,l}^{\ast} - e_{v,l}}{\tau_{l}}\right)\\ e_{v,l} = \frac{R}{M_{l}}\left[\frac{\theta_{v,l}}{exp(\theta/T_{v})-1}\right]\\ e_{v,l}^{\ast} = \frac{R}{M_{l}}\left[\frac{\theta_{v,l}}{exp(\theta/T)-1}\right]\\ J_{e,x} = n_{e}e^{-}u_{x}\\ J_{x} = \left(n_{i}-n_{e}\right)e^{-}u_{x}\\ E_{x} = -\frac{\partial{P_{e}}}{\partial{x}}\frac{1}{n_{e}e^{-}} \end{array} \end{align}\]

Primary Variables (7)

  1. \(\rho_{l}\) mass density of a species “l”
  2. \(\rho\,u_{x}\) x momentum density
  3. \(\rho\,u_{y}\) y momentum density
  4. \(\rho\,u_{z}\) z momentum density
  5. \(e_{t}\) total energy density
  6. \(e_{e}\) electron energy density
  7. \(e_{v}\) vibrational energy density

Auxiliary Variables (7)

1st auxiliary variable is the total mass density

  1. \(\rho\) total mass density

2nd auxiliary variable is the electric field (3 components)

  1. \(E_{x}\) x electric field
  2. \(E_{y}\) y electric field
  3. \(E_{z}\) z electric field

3rd auxiliary variable is the current density (3 components)

  1. \(J_{x}\) x current density
  2. \(J_{y}\) y current density
  3. \(J_{z}\) z current density

4th auxiliary variable is the electron current density (3 components)

  1. \(J_{e,x}\) x current density
  2. \(J_{e,y}\) y current density
  3. \(J_{e,z}\) z current density

5th auxiliary variable

  1. \(\tau_{l}\) vibrational-translational energy relaxtion time of a species “l”

6th auxiliary variable

  1. \(\theta_{v,l}\) characteristic vibrational temperature of a species “l”

7th auxiliary variable

  1. \(D_{l}\)

Parameters

basementPressure (float) [0.0]

The minimum pressure allowed

basementDensity (float) [0.0]

The minimum density allowed

Note

basementPressure and basementDensity are only used if correct=true

correct (boolean) [true]

Tells whether or not densities or pressures should be corrected when the fall below basement pressures or basement densities. When set to true pressure=max(basementPressure, pressure) and density = max(basementDensity, density)

Note

Setting correctNans or correct to true can lead to energy conservation errors

Example

An example eulerThreeTemp equation block is given below::

equations = [euler]
<Equation euler>
  kind = eulerThreeTemp
  correct = false
</Equation>