14 | | == Background |
15 | | {{{ #!latex |
16 | | \noindent Gas is heated when it goes through a shock, and as the heated gases radiate, energy is removed from the gas. Radiatively-cooled shocks in stationary frames will often exhibit oscillatory instabilities as the sharp velocity gradient produces pressure imbalances that expand and contract. |
17 | | \\\\ |
18 | | A shock is a propagating disturbance through a medium. The shock will cause an extremely rapid increase in pressure and temperature of the fluid, and a decrease in the density of the fluid. The module simulates a 1D oscillating shock using AstroBEAR 1.0 to test the analytically predicted behavior of an oscillating shock in the interstellar medium. Radiative shocks are an important aspect in the study of astrophysical fluid dynamics. Such shocks occur in supernova remnants, stellar and galatic jets, stellar wind bubbles, accreting white dwarfs, and galaxy formation. [1] By simulating radiative-cooling shocks, we can study the properties of the gases that are affected by such shocks in the interstellar medium. |
19 | | \\\\ |
20 | | The test is a 1D flow that collides into a wall. It consists of several simulations using different cooling parameters and scalings. From the simulations, the system’s density, temperature and velocity changes are observed and analyzed. Gravitational and magnetic field effects are ignored in this test module. |
21 | | }}} |
| 13 | == Reason for Test == |
| 14 | This is a 1D test to verify that analytic cooling is working correctly. |
23 | | = Problem Setup = |
24 | | {{{ #!latex |
25 | | \noindent The test module is a 1-dimensional Cartesian grid on the domain $0.0 \le x \le 1.0$. The choice of boundary conditions on the left and right edges of the grid has a critical effect on the behavior of the system. Since a flow into a hard wall is desired for this test, a uniform flow of gas with velocity $\emph{v}_0$ is initialized from an inflow/extrapolation boundary on the left. The gas will flow into a reflecting boundary at the right. From this, perturbations will naturally arise at the reflecting boundary, which will move toward the shock, perturbing it and causing oscillations. |
26 | | \\\\ |
27 | | The boundary conditions at the top and bottom are set up as periodic and because the flow only moves along the x-direction, these boundaries do not affect the system. A fixed course grid of 32x4 is specified with two levels of adaptive mesh refinement (AMR). |
28 | | \\\\ |
29 | | AstroBEAR solves the equations of radiative hydrodynamics in 1-D: |
30 | | \begin{equation}\frac{d\rho}{dt} + \rho\frac{dv}{dx} = 0, \end{equation} |
31 | | \begin{equation}\rho\frac{dv}{dt} + \frac{dP}{dx} = 0, \end{equation} |
32 | | \begin{equation}\frac{dP}{dt}-\frac{\gamma P}{\rho}\frac{d\rho}{dt} = -\frac{(\gamma-1)\rho^2}{\bar{m}^2}\Delta(T). \end{equation} |
34 | | The term: \begin{equation}\Delta(T)=C\rho^2T^\alpha \end{equation} |
35 | | represents gas cooling due to optically thin radiative losses via a power law cooling function. It has units of $ergs$ $s^{-1}$ $cm^{-3}$. $P$ is the gas pressure, $\rho$ is the mass density, $v$ is the velocity, and $T$ is the temperature. In our case, we apply a gas of Mach 40. $C$ is a physical constant that will be described in more details later, and $\alpha$ is the power law index we will explore to observe its effect on the oscillations of the system ($\alpha=\frac{1}{2}$ approximates bremsstrahlung cooling). Assuming an ideal gas equation of state, the specific heat ratio $\gamma$ takes a value of 5/3. |
36 | | \\\\ |
37 | | Our module tests $\alpha=0,\frac{1}{2},1$. We plot (not in real units) $\Delta(T) \propto T$: |
38 | | }}} |
39 | | [[Image(http://www.pas.rochester.edu/~blin/spr11/coolingfnc.png)]] |
40 | | {{{ #!latex |
41 | | \noindent $\alpha=0$ is plotted in blue. It is constant. In this case, we expect the gas to oscillate for a while. $\alpha=\frac{1}{2}$ is magenta. $\alpha=1$ is plotted in yellow. It is the strongest cooling. We expect the gas to cool the quickest in this case, i.e. rapid dampening of oscillations. |
42 | | \\\\ |
43 | | \noindent Note that changing the value of $\alpha$ changes the units of $\Delta(T)$. We compensate the change with the constant C. [2] Define: |
44 | | \begin{equation} C = \Lambda_{br}T_{shock}^{(1-2\alpha)/2}, \end{equation} |
45 | | where $\Lambda_{br}$ is the coefficient for nonrelativistic bremsstrahlung. For us, $\Lambda_{br} = 1$ $ergs$ $cm^3$ $s^{-1}$ $g^{-1}$ $K^{-1/2}$, and $T_{shock}$ is the postshock temperature, |
46 | | \begin{equation}T_{shock}=\frac{3}{16}\frac{\mu m_0}{k_B}\emph{v}_0^2, \end{equation} |
47 | | where $\mu$ is the mean molecular weight per atom, $m_0$ is the mass of one atomic mass unit, $k_B$ is Boltzmann's constant, and $v_0$ is the initial velocity. |
48 | | }}} |
| 17 | == Results == |
| 18 | The new chombo can be verified by comparing it to the density in the reference chombo. |
50 | | == Scaling |
51 | | {{{ #!latex |
52 | | \noindent Due to the sensitivity of the cooling function to scaling, it is important to take careful note of the scaling parameters we used. |
53 | | \begin{center} |
54 | | \begin{tabular}{ | l | l | l | p{2cm} |} |
55 | | \hline |
56 | | \textbf{Parameter} & \textbf{Variable} & \textbf{Value} & \textbf{Units} \\ \hline |
57 | | Number Density & nScale & $2.99\times10^{14}$ & $number/cm^3$ \\ \hline |
58 | | Mass Density & rScale & $ 5.00\times10^{-10}$ & $g/cm^3$ \\ \hline |
59 | | Temperature & TempScale & $2.50\times10^6$ & K \\ \hline |
60 | | Pressure & pScale & $1.03\times10^5$ & $dynes/cm^2$ \\ \hline |
61 | | Length & lScale & $5.56\times10^{10}$ & cm \\ \hline |
62 | | Velocity & VelScale & $1.44\times10^7$ & cm/s \\ \hline |
63 | | Cooling & ScaleCool & $3.35\times10^{27}$ & $\frac{g*s}{(cm/s)^5}$ \\ \hline |
64 | | Time & RunTimesc & $3.87\times10^3$ & s \\ \hline |
65 | | \end{tabular} |
66 | | \end{center} |
| 20 | || Reference Chombo || New Chombo || |
| 21 | || [[Image(Bx_ref.png, width=500)]] || ??? || |