Introduction to CFD Questions & Answers  
Question by Student 201527110
Professor, I don't understant what does 'Semi Disc. equation' exactly what we do last lecture. Usually, scalar equation can be denoted as $$\frac{\partial u} {\partial t} + a \frac{\partial u}{\partial x}=0$$, but I wonder why we didn't considered about wave speed 'a' in 'Semi Disc. equation'. (I think you noted semi disc eq as $\frac{u^{n+1}_i-u_i} {\delta t} + \frac{u_{i+1/2}-u_{i-1/2}}{\delta x}=0$)
05.25.17
That's a mistake: I should have added the sound speed of course. Please correct your notes. Because it is constant, an easy way to add the sound speed is to change $\Delta t$ to $a\Delta t$ everywhere $\Delta t$ appears. This will make more sense! 2 points bonus.
Question by Student 201427564
Professor, when you explain about 'Roe Average', you wrote like this. $ { A }_{i+ \frac {1} {2}} ({U}_{i+1} - {U}_{i}) = {F}_{i+1} - {F}_{i} $. I can not understand that how can it be possible that $ {A}_{i+ \frac {1} {2}} {U}_{i+1} = {F}_{i+1} $ and $ {A}_{i+ \frac {1} {2}} {U}_{i} = {F}_{i} $ . Because subscript of A and U are different each other.
05.29.17
Hm, I don't understand the question. You need to explain better what you don't understand..
Question by Student 201427564
Oh.. I mean How can it be possible that $ {A}_{i+ \frac {1} {2}} {U}_{i+1} = {F}_{i+1} $ rather than $ {F}_{i+\frac {1} {2}} $ . Are there any rules?
05.30.17
I'm not sure what you mean. But it is not correct that $A_{i+1/2}U_{i+1}=F_{i+1}$. This is not the Roe average. The Roe average if $\Delta F= A_{i+1/2}\Delta U$. I'll give you 0.5 point for the effort.
Question by Student 201427102
Professor, I'm confused. For finding ${F^{-}_{i+1/2}}$, you used below form. $$r^{-}_{i}=\frac {F^{-}_{i}-F^{-}_{i+1}}{F^{-}_{i+1}-F^{-}_{i+2}}$$ According to table, this form for $alpha < 0 $. But $alpha > 0 $ at Q#2 of assign.#8.
05.31.17
What does “alpha” mean? I didn't use this in class. You need to rephrase your question and use the same symbols as used in class. Or, define clearly a new symbol you are introducing.
Question by Student 201427102
Professor, I'm confused. For finding ${F^{-}}_{i+1/2}$, you used below form. $$r^{−}_{i} = \frac{{F^{−}}_{i}−{F^{−}}_{i+1}}{{F^{−}}_{i+1}−{F^{−}}_{i+2}}$$ According to table, this form for a< 0. But a>0 at Q#2 of assign .
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Now I see what you mean. When dealing with a system of equations, the wave speeds are not necessarily $a$ or $u$. Rather, the wave speeds are the eigenvalues. So, when determining $F^\pm_{i+1/2}$, the wave speeds are within $\Lambda^\pm$ (those play the same role as the “$a$” does for a scalar equation). 1.5 point bonus.
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