35.8 3T Shock Simulations

FLASH has the ability to simulate plasmas which have separate ion, electron, and radiation temperatures (see Chp:HEDP). Usually, simulations which multiple temperatures have several physics models active including:

• Electron thermal conduction
• Ion/Electron equilibration
• Radiation emission, absorption, and diffusion

This section contains a series of simulations which verify FLASH through comparisons with analytic solutions of steady shocks where various assumptions are active. Unfortunately, no single analytic solution contains three distinct temperatures with realistic physical coefficients. Thus, each simulation is performed with a different set of assumptions active. Taken together, they adequately exercise the 3T capabilities in FLASH.

35.8.1 Shafranov Shock

The Shafranov problem (Shafranov, 1957) is a one-dimensional problem that provides a good verification for structure of 1D shock waves in a two-temperature plasma with separate ion and electron temperatures. The Shafranov shock solutions takes as input a given upstream condition and shock speed. It then computes the downstream conditions and a shock profile. The solution is fairly sophisticated in that it takes into account electron thermal conduction and ion/electron equilibration. An assumption is made that the electron entropy is continuous across the shock. Thus, immediately downstream of the shock, the ion-temperature is substantially higher than the electron temperature. Far downstream, the temperature equilibrate. Electron conduction creates a preheat region upstream of the shock. An gamma-law EOS is used (typically with $\gamma=5/3$).

Unfortunately, the Shafranov shock solution can only be simplified to an ODE which must be numerically integrated. The ShafranovShock simulation directory includes analytic solutions for several materials including Hydrogen, Helium, and Xenon in the files plasma_shock.out, plasma_shock_Z2.out, and plasma_shock_Z54.out respectively.

Several solutions are compared here for the fully-ionized Helium case with the following initial/boundary conditions:

• Upstream Ion/Electron Temperature: 5 eV
• Upstream Density: 0.0018 g/cc
• Upstream Velocity: 0.0 cm/s
• Shock Speed: 1.623415272E+07 cm/s

Figure 35.82, Figure 35.83, and Figure 35.84, shows the electron/ion temperature, density, and velocity at 0.15 ns for three cases:

• Analytic Solution: This is the analytic solution for the steady shock. This solution is used to initialize the FLASH simulations. The simulations are correct to the extent that they are able to maintain this shock profile.
• Entropy Advection Solution: This is a 1D FLASH simulation using the split hydrodynamics solver in “entropy advection” mode as described in Sec:3TEntropyAdvection.
• RAGE-like Solution: This is a 1D FLASH simulation using the split hydrodynamics solver in “RAGE-like” mode as described in Sec:3TRageLikeHydro.

The figures show that the both the entropy advection and RAGE-like FLASH simulation are able to maintain the correct shock speeds. However, the entropy advection approach closely matches the correct analytic ion temperature profile while the RAGE-like simulation a peak ion temperature that is too low. This is the expected behavior since the RAGE-like mode does not attempt to ensure that electrons are adiabatically compressed by the shock.

This simulation can be setup with the following setup command:  

# For the Entropy Advection Case:
./setup ShafranovShock -auto -1d +pm4dev +3t -parfile=flash_Z2.par


# For the RAGE-like Case:
./setup ShafranovShock -auto -1d +pm4dev +3t -parfile=ragelike_Z2.par

Figure 35.82: Electron and ion temperatures from Shafranov shock simulation

Figure 35.83: Mass density from Shafranov shock simulation

Figure 35.84: Velocity from Shafranov shock simulation

35.8.2 Non-Equilibrium Radiative Shock

The non-equilibrium radiative shock solution is an analytical solution to a steady, 1D, radiative shock where $T_e = T_i$ but $T_e \ne T_r$. It is presented in (Lowrie, 2008). A constant opacity is assumed, making this a gray simulation. The “analytic” solution is fairly complex and reduces to an ODE which must be evaluated numerically. This ODE is evaluated for a given set of upstream conditions and a given Mach number (evaluated relative to the upstream sound speed). A gamma-law equation of state is assumed.

The FLASH implementation of this simulation resides in the GrayDiffRadShock simulation directory. The simulation can be set up using the following command:  

./setup -auto GrayDiffRadShock -1d +pm4dev +splitHydro +3t mgd_meshgroups=1

The simulation is performed using 3T hydrodynamics with the multigroup radiation diffusion (MGD) unit (see chp:RadTrans). A single radiation energy group is used with opacities set to $\sigma_a = \sigma_e = 423\;\mathrm{cm}^{-1}$ and $\sigma_t = 788\;\mathrm{cm}^{-1}$ (see Sec:OpacityConstant). A $\gamma=5/3$ gamma-law EOS is used with $Z = 1$ and $A = 2$. The electron and ion temperatures are forced to equilibrium by using the Spitzer Heatexchange implementation (see: Sec:HeatexchangeSpitzer) where the ion/electron equilibration time has been reduced by a factor of $10^6$ by setting the hx_ieTimeCoef runtime parameter.

The initial conditions are defined by a step function where the jump occurs at $x = 0$. The upstream ($x < 0$) and downstream ($x > 0$) conditions are chosen so that the shock remains stationary. The correct jump conditions are to maintain a stationary mach two shock are:

• $\rho_0 = 1.0$ g/cc
• $T_0 = 100$ eV
• $u_0 = 2.536e+07$ cm/s
• $\rho_1 = 2.286$ g/cc
• $T_1 = 2.078$ eV
• $u_1 = 1.11e+07$ cm/s

where the subscript 1 represents downstream conditions and the subscript 0 represents upstream conditions. The runtime parameter sim_P0 representing the ratio of radiation to matter pressure is set to $10^{-4}$.

The verification test is successful to the extent that FLASH is able to maintain a stationary shock with the correct steady state spatial profile as shown in Figure 8 of (Lowrie, 2008). This is an excellent verification test in that no special modifications to the FLASH code are needed to perform this test. The simulation is run for 4.25 ns which is enough time for the initial step function profile to reach a steady state solution. Figure 35.85 compares the temperatures in the FLASH simulation to the analytic solution. Figure 35.86 compares density to the analytic solution. Excellent agreement is obtained.

Figure 35.85: Temperatures from non-equilibrium radiative shock simulation. The FLASH results are compared to analytic solution.

Figure 35.86: Mass density from non-equilibrium radiative shock simulation. The FLASH result is compared to the analytic solution.

35.8.3 Blast Wave with Thermal Conduction

The ReinickeMeyer blast wave solution (Reinicke, 1991) models a blast wave in a single temperature fluid with thermal conduction. The semi-analytic solution reduces to an ODE which must be integrated numerically. Figure 35.87 compares a FLASH simulation to the analytic solution (obtained from cococubed.asu.edu/code_pages/vv.shtml after 0.3242 ns for a particular set of initial conditions. Excellent agreement is obtained with the analytic solution. The image shows that at this time the blast wave is at approximately 0.45 cm and the conduction front is at 0.9 cm. The simulation can be set up using the following command:  

./setup -auto ReinickeMeyer -1d +pm4dev +spherical -parfile=flash_SPH1D.par

The shortcut +splitHydro may have to be added to reproduce the presented results.

Figure 35.87: Mass density and temperature from ReinickeMeyer blast wave FLASH simulation compared to analytic solution.