Changes between Version 1 and Version 2 of u/bliu/AblativeRT/TempEquations

Timestamp:

05/29/15 13:26:36 (11 years ago)

Author:

Baowei Liu

Comment:

—

u/bliu/AblativeRT/TempEquations

@@ -1 +1 @@
 * '''The following equations are based on blog:bliu01092014. and [http://www.pas.rochester.edu/~bliu/ConductFront/Finals/scales.pdf Scales in AstroBEAR]'''
 
+== 1. Equations and Parameters==
 Equation solved in Betti's code (SI units and Temperature in Joules)
 
@@ -16 +17 @@
 $$\frac{1}{\gamma-1}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]$$
 
+Here the term $\frac{1}{\gamma-1}$ was put for the convenience of scale convert in the code which will be explained in the part of '''computational scales'''.
+
 In AstroBEAR we define
 $$\kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}}$$
@@ -25 +28 @@
 In [http://www.pas.rochester.edu/~bliu/AblativeRT/Docs/RBetti.pdf  Betti's data], $C^{\prime}_{v}=7.816e26$ and $\gamma=5/3$ so
 $$q_{0}=\frac{2}{3}*7.816*1.38*10^{4}*q^{*}_{cgs}$$
+
+== 2. Computational Scales ==
+In AstroBEAR code, everything is in computational scales. In this part, I summarize the relations of scales for different physics quantities. The details can be found in [http://www.pas.rochester.edu/~bliu/ConductFront/Finals/scales.pdf Parameters and Scales for the Ablative RT problem (and in the code)].
+
+The following table gives scales the relationship to physics variables in C.U. in the code
+$$X_{cu} = X_{cgs}/xScale$$
+
+|| Variable || scale || relationship in cu ||
+|| length || lScale || ||
+|| time || timeScale || ||
+|| velocity || velScale || v=l/t ||
+|| density   || rScale || ||
+|| temperature || TempScale || ||
+|| pressure || pScale || $p=\rho T$ ||
+|| energy ||  || $E=\frac{1}{\gamma-1}(p+0.5*\rho *v^2)$ ||
+
+More details about physics relations in the code can be found in the [https://astrobear.pas.rochester.edu/trac/wiki/VisIt visit expressions]
+
+The above heat diffusion equation (everything is in C.U.) in AstroBEAR is
+ $$\frac{1}{\gamma-1}\rho\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left[\kappa_{1}T^{n}\frac{\partial T}{\partial x}\right]$$
+
+The reason the $\frac{1}{\gamma-1}$ is there because we want the left-hand-side to be energy where $\rho T$ gives pressure in C.U.
+
+