BA Mihály Török: Difference between revisions
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= Grid = | = Grid = | ||
During the project we want to test if the boundary condition set at the edge of the accretor has any effect on the flow morphology. To do that we need the dependence of the number of cells in radial direction \(\text{N}_r\) on \(\text{R}\) and \(\text{R}_{max}\). For the radial component it is true, that \(r\in[R,R_\text{max}]\) and the coordinate of the inner cell wall of cell number \(k\) is given as follows: | |||
\[ | |||
r_k = R\left(\frac{R_\text{max}}{R}\right)^{k/N_r} | r_k = R\left(\frac{R_\text{max}}{R}\right)^{k/N_r} | ||
\]. | \] | ||
The value of \(k\) starts at 0 and ends at \(N_r - 1 \) <br> | |||
With this and with the condition for quadratic cells we can calculate the dependence. The condition for quadratic cells is: \(\Delta \Theta \cdot r_k = \Delta r_k\). With \(\Delta r_k = r_{k+1} - r_k\) follows: \(\Delta \Theta = \frac{r_{k+1}}{r_k}-1\). Using the equation for \(r_k\) we get the following equation: | |||
\[ | |||
\left(\frac{R_\text{max}}{R}\right)^{1/N_r} = \Delta \Theta + 1 = \frac{\pi}{N_{\Theta}} + 1 | |||
\] | |||
After taking the natural logarithm of the equation and simplifying it we get: | |||
\[ | |||
N_{r} = \frac{ln(R_{max}/R)}{ln(1 + \pi / N_{\Theta})} | |||
\] | |||
= Steady State? = | = Steady State? = | ||
Revision as of 17:24, 12 June 2025
Natural Units
Originally the CGS system of units was used for the simulation. Instead I will use natural units. The advantage of the natural units is, that the most relevant physical constants or parameters can be used to express other physical quantities. For example if I would take the accretion radius \(\text{R}_\text{HL}\) as the natural unit for length then the radius of the accretor could be given as \(\text{R} = 0.1 \cdot \text{R}_\text{HL}\). This of course only makes sense if \(\text{R}_\text{HL}\) doesn't change during the project. And it also means that \(\text{R}_\text{HL}\) is given as \(\text{R}_\text{HL}\) = \(\text{R}_\text{HL}\cdot 1 \).
The most important physical quantities for the project are: Length, Velocity and Mass. The natural units for each of these are defined as follows:
The most important length scale for the simulation is the Hoyle-Lyttleton radius or accretion radius. Therefore it is reasonable to use it as the natural unit for the Length. It also doesn't change during the project.
The natural unit of the velocity will be the external velocity, the velocity of the gas defined at a point where the gas is unperturbed. This also won't change during the project, and is used to calculate the Mach number so it is also a reasonable choice for a natural unit.
For the mass the natural unit is defined through \(\rho_{ext}\), the density of the gas at a point where it is unperturbed. With that the unit for mass is given as \(\rho_\text{ext} \cdot R_\text{HL}^3 = 1\). This is also reasonable since \(\rho_{ext}\) doesn't change during the project.
We also set \(\text{G}\cdot \text{M} = \frac{1}{2}\). G is the gravitational constant and M is the mass of the black hole.
Grid
During the project we want to test if the boundary condition set at the edge of the accretor has any effect on the flow morphology. To do that we need the dependence of the number of cells in radial direction \(\text{N}_r\) on \(\text{R}\) and \(\text{R}_{max}\). For the radial component it is true, that \(r\in[R,R_\text{max}]\) and the coordinate of the inner cell wall of cell number \(k\) is given as follows:
\[
r_k = R\left(\frac{R_\text{max}}{R}\right)^{k/N_r}
\]
The value of \(k\) starts at 0 and ends at \(N_r - 1 \)
With this and with the condition for quadratic cells we can calculate the dependence. The condition for quadratic cells is: \(\Delta \Theta \cdot r_k = \Delta r_k\). With \(\Delta r_k = r_{k+1} - r_k\) follows: \(\Delta \Theta = \frac{r_{k+1}}{r_k}-1\). Using the equation for \(r_k\) we get the following equation:
\[
\left(\frac{R_\text{max}}{R}\right)^{1/N_r} = \Delta \Theta + 1 = \frac{\pi}{N_{\Theta}} + 1
\]
After taking the natural logarithm of the equation and simplifying it we get:
\[
N_{r} = \frac{ln(R_{max}/R)}{ln(1 + \pi / N_{\Theta})}
\]
Steady State?
Fourier-Zerlegung der Massenstromdichte
Die Fourier-Zerlegung setzt einen periodischen Funktionsverlauf voraus. Eine Funktion des Polarwinkels \(\theta\in[0,\pi]\) ist i.A. nicht periodisch, aber wenn man in einem System mit Zylindersymmetrie vom Südpol aus "auf der Rückseite" wieder zum Nordpol läuft (\(\theta\in[\pi,2\pi]\)), dann ist die Gesamtfunktion periodisch (und spiegelsymmetrisch bzgl. \(\pi\)).