Changes between Version 34 and Version 35 of TestSuite/RadiativeInstability


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Timestamp:
10/11/11 14:06:17 (13 years ago)
Author:
blin
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  • TestSuite/RadiativeInstability

    v34 v35  
    1 {{{ #!latex
    2 {\bfseries{\Large Radiative Shocks \& the Oscillatory Instability }}
    3 }}}
     1= Radiative Shocks =
    42
    5 [[PageOutline]]
    6 = Introduction =
     3== Problem Background ==
     4This is a study of the oscillatory instabilities in radiative shocks. Radiatively cooled shocks in a stationary frame will sometimes exhibit oscillatory behavior as the sharp velocity gradient produces pressure imbalances that expand and contract. For more information, the references below detail the problem:
     5
    76Main references:
    87[[br]]
     
    1211[[br]]
    1312
    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 ==
     14This is a 1D test to verify that analytic cooling is working correctly.
    2215
    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}
    3316
    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 ==
     18The new chombo can be verified by comparing it to the density in the reference chombo.
    4919
    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)]] || ??? ||
    6722
    68 \noindent Since $ScaleCool=\frac{rScale*lScale}{m_H^2*VelScale^3}$, the cooling function will be stronger the bigger rScale and lScale values are. Also, note that $T_{shock} \propto v_0^2$:
    69 }}}
    70 [[Image(http://www.pas.rochester.edu/~blin/spr11/posttvel.png)]]
    7123
    72 == Numerical Results
    73 {{{ #!latex
    74 \noindent Using the equations and values outlined above, we have a post-shock temperature of $5\times10^8$ K. With this, we can find C for each $\alpha$ case.
    75 \\\\
    76 \noindent Having find the C for each case, we can then adjust the cooling parameters accordingly to have the numbers equal.
    77 }}}
    78 
    79 = Conclusion =
    80 {{{ #!latex
    81 Radiative shocks show general trend toward stability with increasing values of $\alpha$. We are currently not able to show this for the case of $\alpha=0$ due to scaling problems in Astrobear 1.0. We hope to have a solution for all three cases in Astrobear 2.0.
    82 }}} 
    83 ----
     24[http://clover.pas.rochester.edu/testresults/RadiativeShocks/testlog Test Log]