Introduction to CFD Assignment 5 — Boundary Conditions  
Question #1
Consider the following domain:
Q1.png  ./download/file.php?id=3472&sid=9411c1e055522c73f1c47afb4fb18c86  ./download/file.php?id=3472&t=1&sid=9411c1e055522c73f1c47afb4fb18c86
The freestream properties correspond to $M_\infty=2$, $T_\infty=300$ K, $P_\infty=1$ atm, $\theta=30^\circ$ and the properties on the nodes at the iteration $n$ are as follows:
Node$P^n$, atm$T^n$, K$u^n$, m/s$v^n$, m/s
11.0300400-400
21.1310300-300
31.1310300-300
41.2320200-300
51.2330200-200
61.3340100-200
71.3340100-100
81.4350100-100
91.4360100-100
Knowing that the node spacing is constant, do the following:
(a)  Find the properties at iteration $n+1$ at node 1 using a 2nd degree polynomial to extrapolate the properties.
(b)  Find the properties at iteration $n+1$ at node 9 using a 2nd degree polynomial to extrapolate the properties.
Outline clearly what kind of boundary condition (subsonic inflow/outflow, supersonic inflow/outflow) you are choosing and why.
04.28.17
Question #2
Consider the following domain:
Q2.png  ./download/file.php?id=3473&sid=9411c1e055522c73f1c47afb4fb18c86  ./download/file.php?id=3473&t=1&sid=9411c1e055522c73f1c47afb4fb18c86
The freestream properties correspond to $M_\infty=0.7$, $T_\infty=300$ K, $P_\infty=1$ atm, $\theta=70^\circ$ and the properties on the nodes at the iteration $n$ are as follows:
Node$P^n$, atm$T^n$, K$u^n$, m/s$v^n$, m/s
11.0300100-80
21.1330130-100
31.1350150-130
41.2380170-160
Knowing that the node spacing is constant, do the following:
(a)  Find the properties at iteration $n+1$ at node 1 using a 1st degree polynomial to extrapolate the properties.
Outline clearly what kind of boundary condition (subsonic inflow/outflow, supersonic inflow/outflow) you are choosing and why.
Question #3
Consider a computational domain as follows:
Q5.png  ./download/file.php?id=4144&sid=9411c1e055522c73f1c47afb4fb18c86  ./download/file.php?id=4144&t=1&sid=9411c1e055522c73f1c47afb4fb18c86
with the gas constant $R=286~$J/kgK, the ratio of specific heats $\gamma=1.4$, $P_\infty=2~$atm, $T_\infty=300~$K, and $M_\infty=0.8$. Knowing the properties at nodes 1, 2, 3, and 4 at time level $n$:
Node$x$, m$y$, m$P^n$, Pa$T^n$, K$u^n$, m/s$v^n$, m/s
1 1.00 1.00 90000 310 400 30
2 0.99293 1.00707 90000 310 380 20
3 0.97879 1.02121 90000 350 350 10
4 0.96464 1.03535 90000 330 340 0
Do the following:
(a)  Determine what kind of boundary condition node 1 is.
(b)  Find the pressure and temperature at time level $n+1$ at node 1 using a 2nd degree polynomial to extrapolate the properties from the inner nodes 2, 3 and 4.
Hint: the spacing between the nodes 4, 3, 2, and 1 can not be assumed constant.
05.01.18
Answers
1.  
2.  
3.  267.5 K, 90 kPa.
Due on Thursday May 10th at 16:30. Do all questions.
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