Radiative Transfer MS Project

Blog for step by step updates of this project that include useful youtube links, websites, my figures, equations, etc. The link for updates on the code can be found here

Code
github.io page

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Learning Radiative Transfer

Useful YouTube videos that physically explain radiative transfer.

Radiative Transport Equation

Explaining Specific Intensity

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1D sod test

Here is the github repository for Cholla.
Cholla

I have started by running through the 1D example given on the main page (linked above). Just using the given parameter file here are the plots you end up getting.
Fig 1

The results shown in Fig 1 have the initial values found in Table 1 below.

Before making a time series, I am going to practice and explore this Reimann problem. By changing the initial values of the velocity and density I will explore what happens with the original problem (above) only with swapped the initial densities and pressures, converging initial velocities, and diverging initial velocities.

For all of the following tests the location of the initial discontinuity will be: diaph=0.5

Original sod.txt
Table 1 – (corresponding to Fig 1)

Dens_L	Dens_R	Press_L	Press_R	Vel_L	Vel_R
1.0	0.1	1.0	0.1	0.0	0.0

Swapped Densities and Pressures
Table 2 – (corresponding to Fig 2)

Dens_L	Dens_R	Press_L	Press_R	Vel_L	Vel_R
1.0	0.1	1.0	0.1	0.0	0.0
0.1	1.0	0.1	1.0	0.0	0.0
1.0	2.0	1.0	2.0	0.0	0.0

Table 2 rows list the initial values for the rows in Fig 2

Fig 2

When you flip the pressures and densities the results flip symmetrically about the y-axis. Aka the shock seen in the first row is traveling in the opposite direction in the second row (as expected with flipped IC's).

In the third row the densities and pressures were both increased. However, the box with the higher initial density now has greater pressure (this is different than in the first two cases). Even with the flipped pressure gradient, the density is the dominating term forcing the shock toward the side with the lower initial density.

Converging Velocities
Table 3

Dens_L	Dens_R	Press_L	Press_R	Vel_L	Vel_R
1.0	0.1	1.0	0.1	1.0	-1.0
1.0	0.1	1.0	0.1	3.0	-3.0

Fig 3

In both of these cases (the two rows) the density and pressure are consistent. Although the initial velocities weren't changed by much, the effect on the momentum of each fluid caused a dramatic change in the result of the system.

The overall increase of momentum in the system (due to increasing each velocity by a factor of three) could cause the system's density and pressure profiles to be on the verge of a delta function which is much more dramatic than the 'flat' pieces of the profiles in the first case. The increased momentum could allow the second system to change more quickly than the first.

Diverging Velocities
Table 4

Dens_L	Dens_R	Press_L	Press_R	Vel_L	Vel_R
1.0	0.1	1.0	0.1	-1.0	1.0
1.0	0.1	1.0	0.1	-3.0	3.0

Fig 4

The diverging cases show a similar trend to the converging cases in regards to when the momentum is increased, for each fluid, the system changes more rapidly.

In the second case, the fluids are moving away from each other much more rapidly. Hence why the second case is going to have a much more flat density and pressure profile than the first case. The velocity profile in the second case looks like it has a more continuous profile than in the first case. However, the internal energy makes it look like there's a shock in the middle of the simulation where there physically shouldn't be one…

Understanding MUSCL-Hancock Method

Useful papers

https://arxiv.org/pdf/1803.01008.pdf

http://bulletin.pan.pl/(60-1)45.pdf

How Cholla Works

Cholla 2015

Kannan RT Implimentation

Paper: Kannan et al 2018

1D Radial Advection Test

Section 4.1 Kannan et al 2018

A 2D vacuum box of length 6 contains a circular radiation pulse oriented around the center of the box. At t=0 the pulse has a radius of 1.2, the box has an initial gaussian-shaped photon number density with a spread of 0.1 and maximum amplitude of 40, and the direction of radiation flux is in +r-hat.

As time advances, the pulse advects radially outward at speed c=1. At t=0.8, the pulse has reached a new radius of r=2, maintained a constant ring width, and decreased the maximum photon number density.

Analytic solution

Figure 1 in Kannan et al 2018 compares the results of different functions, schemes, and meshes (regular vs irregular). However, we started with directly computing the analytic solution to the problem, Fig 5 below. This result is most accurately represented by their higher order gradient extrapolated scheme on a regular mesh.

Fig 5

Compressing this 2D map of the photon energy density onto a 1D function of radius reveals it's gaussian nature. Fig 6 below is directly comparable with the analytic solution (black line) from Figure 2 of Kannan et al 2018.

Fig 6

This first calculation did not involve any mesh or analytic scheme, this is the next step of the project.

Python mesh grid solution