PLUTO Test Problems  4.4-patch2
Functions
init.c File Reference

Propagation of a conduction front. More...

Functions

void Analysis (const Data *d, Grid *grid)
 
void InitDomain (Data *d, Grid *grid)
 

Detailed Description

The problem sets the initial condition for a propagating conduction front for which an analytical solution exists (Reale 1995).

The equation

\[ \frac{\partial T}{\partial t} = a\frac{\partial}{\partial s} \left(T^n\frac{\partial T}{\partial s}\right) \]

has the solution

\[ T = T_c\left(1 - \frac{s^2}{s_f^2}\right)^{1/n} \]

where

\[ T_c = \left(\frac{Q^2}{at}\right)^{1/(n+2)} \left(\frac{n}{2(n+2)}\zeta_0^2\right)^{1/n}\,, \quad\quad\quad s_f = (aQ^nt)^{1/(n+2)}\zeta_0\,, \quad\quad\quad \zeta_0 = \left[\frac{(n+2)^{1+n}2^{1-n}}{n\pi^{n/2}}\right]^{1/(n+2)} \left[\frac{\Gamma(1/2 + 1/n)}{\Gamma(1/n)}\right]^{n/(n+2)}\,, \]

$Q$ is the integral over the whole space, and $\Gamma$ is the gamma function.

The setup is built to compare the numerical solution with the analytical one.

In order to solve just the internal energy equation, we force the velocity to be zero using the internal boundary. We then deal with the internal energy equation (in cgs units) in presence of conduction,

\[ \label{eq:internal_energy} \pd{(\rho\epsilon)}{t} = \pd{}{x}\left[\kappa T^{5/2}\pd{T}{x}\right] \]

where $\rho\epsilon$ is the gas internal energy (in cgs units), $T$ is the temperature (in K), $\kappa = 9.22\cdot 10^{-7}$ is the plasma thermal conductivity (in cgs units): The previous equation can be written in terms of the temperature variable only using

\[ \rho\epsilon = \frac{p}{\gamma-1} = \frac{2nk_BT}{\gamma-1} \qquad\qquad\Longrightarrow\qquad\qquad \pd{T}{x} = a \pd{}{x}\left(T^{5/2}\pd{T}{x}\right) \]

where $n=10^{10} \,{\rm cm}^{-3}$ ( $10^{9}$ is employed in 1D) is the plasma number density, $\gamma=5/3$ while

\[ a = \frac{(\gamma-1)\kappa}{2nk_B} \approx 0.226 \qquad (Titos\quad uses\quad 4.212 \quad ) \]

In code (non-dimensional) units, we adopt $t = t_0 \tilde{t}$, $x = L_0\tilde{x}$ and therefore Eq. (eq:internal_energy}) is solved as

\[\label{eq:internal_energy_code} \pd{(\widetilde{\rho\epsilon})}{\tilde{t}} = \left(\kappa \frac{t_0}{L_0^2}\frac{T_0^{5/2}}{2nk_B} \right) \pd{}{\tilde{x}}\left[\tilde{T}^{5/2} \pd{\tilde{T}}{\tilde{x}}\right] \]

Values with a tilde are dimensionless. Note that the reference velocity and temperature are computed directly from $ t_0 = 1\,{\rm s}$ and $L_0 = 10^8\,{\rm cm} $:

\[ v_0 = \frac{L_0}{t_0}\,,\qquad T_0 = \frac{m}{2k_B}v_0^2 \]

1D-hd-exp.jpg
Evolution of the TC front in the 1D case; numerical solution (points) and analytical (lines)

.

Configurations (EXPLICIT, STS, RKL):

Author
A. Mignone (migno.nosp@m.ne@p.nosp@m.h.uni.nosp@m.to.i.nosp@m.t)
Date
March 11, 2018

References:

Function Documentation

◆ Analysis()

void Analysis ( const Data *  d,
Grid *  grid 
)

Generate Analytical solution

◆ InitDomain()

void InitDomain ( Data *  d,
Grid *  grid 
)

Assign initial condition by looping over the computational domain. Called after the usual Init() function to assign initial conditions on primitive variables. Value assigned here will overwrite those prescribed during Init().