Changes between Version 2 and Version 3 of u/bliu/AblativeRT/Equations

Timestamp:

05/22/15 13:22:23 (10 years ago)

Author:

Baowei Liu

Comment:

—

u/bliu/AblativeRT/Equations

@@ -2 +2 @@
 
 1. Equations
+$$\frac{1}{\gamma-1}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$
 
-2. subroutine getdEdT in EOS.f90
+where T is in cgs unit or Kelvin.
+
+In current version of AstroBEAR, we have
+$$\frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$
+
+And same as blog:bliu01092014, we define
+
+$$ \kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v} } $$
+$$ q^{*}_{cgs}=\kappa_{1}T^{*}\frac{\partial T}{\partial x_{cgs}}$$
+
+And so same as blog:bliu01092014, we have
+
+In Betti's data, $C_{v}=7.816e26$ and $\gamma=5/3$ so
+$$q_{0}=\frac{2}{3}*7.816*1.38*10^{4}*q^{*}_{cgs}$$
+
+2. Scales for converting from cgs to cu
+ ScaleFlux is the scale which converts the heat flux from cgs units to computational units in AstroBEAR. Since 
+$$\frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{\partial Q}{\partial x}$$
+The new scaleFlux is $EScale*velScale$ or $pScale*velScale$ as the temperature is in Kelvin now.
+While in the older version where the temperature is in c.u. scale,
+$$\frac{1}{\gamma-1}\rho\frac{\partial T}{\partial t}=\frac{\partial Q}{\partial x}$$
+So $scaleFlux is rScale*TempScale*velScale$ instead...
+
+scaleDiff is the scale that converts the kappa1 from cgs units to computational units. Since
+$$ q^{*}_{cgs}=\kappa_{1}T^{*}\frac{\partial T}{\partial x_{cgs}}$$
+So
+$$scaleDiff=scaleFlux*lScale$$
+
+3. subroutine getdEdT in EOS.f90
 This subroutine will return the value of dE/dT which has a unit as C.U. Energy/Kelvin
 
-3. source and stencil_fix(0) in diffusion.f90 will be modified as
+4. source and stencil_fix(0) in diffusion.f90 will be modified as
 
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