Boundary conditions

Upstream: Symmetry

Symmetry boundary conditions are applied at the upstream side, corresponding to zero flow through the boundary.

\[\partial_{||}n = 0 \qquad \partial_{||}p = 0 \qquad \partial_{||}T_e = 0 \qquad V_{||} = 0 \qquad nV_{||} = 0\]

Since the boundary is half-way between grid points, this is implemented as

\[\begin{split}\begin{aligned} n_0 &=& n_1 \\ p_0 &=& p_1 \\ T_{e,0} &=& T_{e,1} \\ V_{||,0} &=& -V_{||,1} \\ nV_{||,0} &=& -nV_{||,1}\end{aligned}\end{split}\]

Downstream: Sheath

Boundary conditions are applied to the velocity and the heat flux:

  • At the left boundary a no-flow condition is applied:

    \[\begin{split}\begin{aligned} V_{||} &=& 0 \nonumber \\ \partial_{||}T_e &=& 0 \nonumber \end{aligned}\end{split}\]

  • At the right boundary is a sheath boundary:

    \[\begin{split}\begin{aligned} V_{||} &\ge& v_s \nonumber \\ \partial_{||}T_e &=& 0 \nonumber \end{aligned}\end{split}\]

    where the inequality is implemented by switching from a Dirichlet to a Neumann boundary if \(V_{||} > v_s\) in front of the boundary.

    The critical speed into the sheath, \(v_s\) is sensitive to assumptions on the thermodynamics of the sheath, taking the form: [1]_

    \[v_s = \left( \frac{e\left(T_e + \gamma T_i\right)}{m_i}\right)^{1/2}\]

    where \(T_e\) is the electron temperature (in eV), \(T_i\) is the ion temperature, \(\gamma\) is the ratio of specific heats. For isothermal flow \(\gamma=1\), for adiabatic flow with isotropic pressure \(\gamma=5/3\), and for one-dimensional adiabatic flow \(\gamma=3\). Here we are assuming \(T_e = T_i\) and \(\partial_{||}T_e\) so take the isothermal case. This therefore becomes:

    \[v_s = \left( \frac{p}{n}\right)^{1/2} \label{eq:sheath_speed}\]

Note: If the sheath velocity is subsonic, then waves can propagate in from the boundary. Their domain of dependence is outside the simulation domain, so these waves can cause numerical instabilities.

Several boundary conditions are available for the density and pressure, including free boundaries and Neumann (zero gradient). These are controlled by settings density_sheath and pressure_sheath. Density can have the following values:

  1. Free boundary, linearly extrapolating the value from inside the domain

    \[n_{-1} = 2n_{-2} - n_{-3}\]

  2. Neumann (zero gradient)

    \[n_{-1} = n_{-2}\]

  3. Constant flux

    \[n_{-1/2} = n_{-2}v_{-2}J_{-2} / \left( v_s J_{-1/2} \right)\]

    where the Jacobian factors \(J\) account for a changing flux tube cross-section area.

Pressure can have the following values:

  1. Free boundary, linearly extrapolating the value from inside the domain

    \[p_{-1} = 2p_{-2} - p_{-3}\]

  2. Neumann (zero gradient)

    \[p_{-1} = p_{-2}\]

  3. Constant energy flux \(\frac{5}{2}pv + \frac{1}{2}nv^3\)

    \[5 p_{-1/2} = \left( 5p_{-2}v_{-2} + n_{-2}v_{-2}^3\right) / v_s - n_{-1/2}v_s^2\]