Computational Aerodynamics Questions & Answers  
Question by Prasanna
Professor, I am a bit confused about question #5 of Assignment 4. I have to find $$\frac{\partial F_3}{\partial U_4}=\frac{\partial (\rho u^2 +P)}{\partial U_4}=\frac{\partial (\rho u^2)}{\partial U_4}+\frac{\partial P}{\partial U_4}$$ and the difficult part seems to be in determining the $\frac{\partial P}{\partial U_4}$ term. The alternative method you taught involves using the chain rule, for example, $$\frac{\partial F_3}{\partial U_4}=\frac{\partial F_3}{\partial \rho_1} \frac{\partial \rho_1}{\partial U_4}+\frac{\partial F_3}{\partial \rho_2} \frac{\partial \rho_2}{\partial U_4}+\frac{\partial F_3}{\partial u} \frac{\partial u}{\partial U_4} +\frac{\partial F_3}{\partial \phi} \frac{\partial \phi}{\partial U_4} $$ where $F_3=F_3(\rho_1,\rho_2,u,\phi)$ and $\phi$ is some variable. For this problem, $\phi$ has to be a function of $U_1,U_2,U_3,U_4$ and also a function of P such that I can evaluate $\frac{\partial \phi}{\partial U_4}$. But then if I could express P in terms of $\phi$ which is a function of $U_1,U_2,U_3,U_4$, I would use the first method which you taught to evaluate the flux jacobian terms instead. I would like your comment regarding this.
04.16.19
You don't necessarily need to express $\phi$ as a function of $U$ to determine $\partial \phi/\partial U$ in the same way as you don't need to express $F$ as a function of $U$ to obtain $\partial F/\partial U$.
Question by Van Tien
Professor, in Assignment 5, Question #3, for the extrapolation, I am confused to use the 1D-Lagrange interpolation or 2D-Lagrange interpolation. In the case of 1D-Lagrange interpolation, I am not sure the polynomial function is based on x or y coordinate. In the case of 2D-Lagrange interpolation, I think I need more information from the other nodes.
05.08.19
Use a 1D extrapolation polynomial. 2D is too time consuming to compute.
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