| 1 | | The current version of the code has changed the temperature from computational units to Kelvin or solves the energy equation instead of temperature. This note is to check the parameter and equations changes need to do to be consistent comparing the old ways of [https://astrobear.pas.rochester.edu/trac/wiki/u/bliu/AblativeRT/TempEquations thermal diffusion equation in Astrobear]. So lhs of the equation was replaced by the derivative of the energy. |
| | 1 | This try is to change the temperature from computational units to Kelvin or solves the energy equation instead of temperature. This note is to check the parameter and equations changes need to do to be consistent comparing the old way of [https://astrobear.pas.rochester.edu/trac/wiki/u/bliu/AblativeRT/TempEquations thermal diffusion equation in AstroBEAR since the old way works. This way all the parameters would be same as [https://astrobear.pas.rochester.edu/trac/wiki/u/bliu/AblativeRT/TempEquations the old way]. Only lhs of the equation was replaced by the derivative of the energy getdEdT. |
| | 2 | |
| | 3 | == 1. Equations and Parameters== |
| | 4 | Equation solved in Betti's code (SI units and Temperature in Joules) |
| | 5 | |
| | 6 | $$\rho C^{\prime}_{v}\frac{\partial K_{B}T}{\partial t}=\frac{\partial}{\partial x}\left[\kappa_{0}(K_{B}T)^{n}\frac{\partial (K_{B}T)}{\partial x}\right]$$ |
| | 7 | where $K_{B}$ is Boltzmann constant and $C^{\prime}_{v}=C_{v}/K_{B}$ as in [http://www.pas.rochester.edu/~bliu/AblativeRT/Docs/RBetti.pdf Betti's document] and $C_{v}$ is the normal specific heat capacity. |
| | 8 | And the flux is |
| | 9 | $$q_{0}=\kappa_{0}(K_{B}T)^{n}\frac{\partial (K_{B}T)}{\partial x_{SI}}$$ |
| | 10 | |
| | 11 | To convert this to cgs units we write |
| | 12 | |
| | 13 | $$10^{3}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\kappa_{0}K_{B}^{n}}{C^{\prime}_{v}}\frac{\partial}{10^{-2}\partial x_{cgs}}\left[T^{n}\frac{\partial T}{10^{-2}\partial x_{cgs}}\right]$$ |
| | 14 | |
| | 15 | That is |
| | 16 | |
| | 17 | $$\frac{1}{\gamma-1}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$ |
| | 18 | |
| | 19 | Put the equation in c.u., it will be |
| | 20 | |
| | 21 | $$\frac{1}{\gamma-1}\rho\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left[\kappa1 T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$ |
| | 22 | |
| | 23 | and the lhs is in c.u. of energy |
| | 24 | |
| | 25 | And the definition of $\kappa_{1}$ and $q^{*}_{cgs}$ would be same. |
| | 26 | |
| | 27 | $$\kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}}$$ |
| | 28 | and |
| | 29 | $$q^{*}_{cgs}=\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}} $$ |
| | 30 | |
| | 31 | Comparing the definition of $q_{0}$ and $q^{*}_{cgs}$ we have $$ q_{0}=10(\gamma-1)q^{*}_{cgs}C^{\prime}_{v}K_{B}$$ |
| | 32 | |
| | 33 | In [http://www.pas.rochester.edu/~bliu/AblativeRT/Docs/RBetti.pdf Betti's data], $C^{\prime}_{v}=7.816e26$ and $\gamma=5/3$ so |
| | 34 | $$q_{0}=\frac{2}{3}*7.816*1.38*10^{4}*q^{*}_{cgs}$$ |
| | 35 | |
