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Merge pull request #12 from barbagroup/master
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2 parents 6fecc15 + 6c03754 commit 394969f

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lessons/11_Lesson11_vortexSourcePanelMethod.ipynb

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{
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"metadata": {
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"name": "",
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"signature": "sha256:cf4c65f24e5cc8c0fb88e140e3a39bbaef85fa46059dd161bc48bdff73413944"
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"level": 5,
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"metadata": {},
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"source": [
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"Version 0.4 -- April 2015"
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"Version 0.5 -- April 2015"
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]
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},
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{
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"cell_type": "heading",
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"level": 1,
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"metadata": {},
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"source": [
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"Source-vortex panel method"
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"Vortex-source panel method"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In [Lesson 9](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/09_Lesson09_flowOverCylinder.ipynb) of _AeroPython_, you learned to use a _source panel method_ to represent a circular cylinder, and in [Lesson 10](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/10_Lesson10_sourcePanelMethod.ipynb) we used it for a symmetric airfoil at zero angle of attack. But what if we want the airfoil to generate some lift? If we place the airfoil at a non-zero angle of attack, we _should_ get lift, but will a source-panel representation be able to give you lift? Remember the _Kutta-Joukowski theorem_?\n",
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"In [Lesson 9](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/09_Lesson09_flowOverCylinder.ipynb) of _AeroPython_, you learned to use a _source panel method_ to represent a circular cylinder, and in [Lesson 10](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/10_Lesson10_sourcePanelMethod.ipynb) we used it for a symmetric airfoil at zero angle of attack. But what if we want the airfoil to generate some lift? If we place the airfoil at a non-zero angle of attack, we _should_ get lift, but will a source-panel representation be able to give you lift? Remember the [_Kutta-Joukowski theorem_](http://en.wikipedia.org/wiki/Kutta%E2%80%93Joukowski_theorem)?\n",
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"\n",
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"\n",
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"Historically, the first panel method ever developed was a source-sheet method. At the time, Douglas Aircraft Company was concerned with calculating the flow around bodies of revolution, and it was only later that the method was extended to lifting surfaces. (See the reference below for a nice historical account.)\n",
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"\n",
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"For example, using the source-sheet panel method of [Lesson 10](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/10_Lesson10_sourcePanelMethod.ipynb) with an angle of attack $\\alpha=4^\\circ$ (using 40 panels), and plotting the streamlines in an area close to the trailing edge, we get the following plot:\n",
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"\n",
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"<center><img src=\"./resources/StreamlinesTrailingEdge.png\"></center>"
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"<center><img src=\"./resources/StreamlinesTrailingEdge.png\" width=\"600\"></center>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"To enforce the *Kutta-condition*, we state that the pressure coefficient on the fisrt panel must be equal to that on the last panel:\n",
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"To enforce the *Kutta-condition*, we state that the pressure coefficient on the first panel must be equal to that on the last panel:\n",
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"\n",
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"$$C_{p_1} = C_{p_{N}}$$\n",
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"\n",
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# Data from Gregory and O'Reilly, NASA R&M 3726 (1970)
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# Reynolds number Re=2.88E+06
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# Angle of attack alpha=0deg
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# Data was digitized from a photocopy, hence is only approximate
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# Data for the upper airfoil surface only
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# [x/c] [Cp]
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0 1
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0.0023497 0.847673
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# Data from Gregory and O'Reilly, NASA R&M 3726 (1970)
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# Reynolds number Re=2.88E+06
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# Angle of attack alpha=10deg
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# Data was digitized from a photocopy, hence is only approximate
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# Data for the upper airfoil surface only
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# [x/c] [Cp]
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0 -3.66423
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0.00218341 -5.04375
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0.00873362 -5.24068
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# Data from Gregory and O'Reilly, NASA R&M 3726 (1970)
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# Reynolds number Re=2.88E+06
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# Angle of attack alpha=15deg
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# Data was digitized from a photocopy, hence is only approximate
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# Data for the upper airfoil surface only
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# [x/c] [Cp]
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-7.59438e-05 -8.65066
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0.0024302 -10.1789
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