Ablative RT in General

This study focus on the nonlinear regime of RT growth when a strong heat flux is applied at the bottom of the interface. The initial RT profile are plotted below:

density and temperature

vx and vy speed

In order to suppress the RT instability from growing on the target surface, one assumption is that applying heat flux will help. Normally, this situation involves strong Spitzer heat conduction (kTⁿ) at the target surface where the cold target and the hot environment are separated. The time scale for the heat to transfer through the temperature length scale can be expressed as: l²/k, whereas the sound crossing time for the temperature length scale is l/c, where l is the minimum temperature length scale, c is the sound speed.

Because of the high heat conductivity, the heat conduction time scale is usually much faster compared to the sound crossing time scale. This fact results in difficulties when one tries to establish the numerical solver based on explicit time stepping: the heating process is so fast that the time step required for stability is thousands of times smaller comparing to the hydrodynamic time scale. The algorithm has to spend 99% of the steps solving the heat equation and is extremely ineffective. In fact, Using the data from LLE, the actual thermal conduction time step is about 500 times smaller. Therefore an implicit algorithm is required to solve the problem.

In AstroBEAR, we build the implicit diffusion solver which converts the differential equation into a linear system. Then we use Hypre to solve this linear system and obtain the correct solution. This method has loose requirement on the time stepping and therefore greatly increases the efficiency. We also incorporate the nonlinear property of the thermal conduction by using the Crank-Nicholson scheme. Following is a detailed document discussing how to build the linear system using CR scheme.
​open ppt

The boundary condition of this problem is: periodic on x direction, and hydrostatic at y up. The y bottom boundary condition is much trickier because we need to fix the heat flux. This condition involves first solve the boundary temperature using the nonlinear diffusion equation, then using this temperature to find out the density that satisfies the quasi-hydrostatic requirement. Following is a more detailed discussion on the RT boundary condition.
​open pdf

Ablative RT in General

Data files used (the three columns: x position, y position, physical value (SI units)):
Density:
​http://www.pas.rochester.edu/~shuleli/ablativeRT/density.data
Pressure:
​http://www.pas.rochester.edu/~shuleli/ablativeRT/pressure.data
Vx:
​http://www.pas.rochester.edu/~shuleli/ablativeRT/vx.data
Vy:
​http://www.pas.rochester.edu/~shuleli/ablativeRT/vy.data

Treatment of Boundary condition:
​http://www.pas.rochester.edu/~shuleli/ablativeRT/Readme1(1).doc

From the above setup, the density, temperature, velocity and pressure evolution for several instability growth time are plotted here (no perturbation):
​http://www.pas.rochester.edu/~shuleli/frame_0615.htm

Scaling ablative RT

​open pdf

The ablative RT problem requires carefully scaling the physical parameters, especially the thermal conductivity and the bottom heat flux into computational units. Following is a document discussing the scaling of RT and the conduction parameters.

Ablative RT in Equilibrium

To tackle the ablative RT problem, it is important to have an initial equilibrium. In our problem, the heat flux applied and the conductivity are often related. The combination of them should be able to give the correct initial temperature gradient at the bottom, otherwise the temperature will fluctuate at the bottom from the start since we are fixing the flux. This work requires fine tuning the heat flux applied at the bottom and the conductivity. The sound crossing time for the is expressed as y/c, the instability growing time can be approximated as (x/g)^½, where x and y are the domain lengths along x and y directions respectively. The sound crossing time is about 10 times longer than the instability growing time: 10^-2 for sound crossing, 10^-3 for instability growth. The following plots are for the different bottom fluxes applied: red: 8, blue: 10, green: 12, at time 2*10^-4. Black is the initial profile.

density and temperature