├── .gitignore ├── LICENSE ├── README.md ├── lessons ├── 00_Lesson00_QuickPythonIntro.ipynb ├── 01_Lesson01_sourceSink.ipynb ├── 02_Lesson02_sourceSinkFreestream.ipynb ├── 03_Lesson03_Assignment.ipynb ├── 03_Lesson03_doublet.ipynb ├── 04_Lesson04_vortex.ipynb ├── 05_Lesson05_InfiniteRowOfVortices.ipynb ├── 06_Lesson06_Assignment.ipynb ├── 06_Lesson06_vortexLift.ipynb ├── 07_Lesson07_methodOfImages.ipynb ├── 08_Lesson08_sourceSheet.ipynb ├── 09_Lesson09_flowOverCylinder.ipynb ├── 10_Lesson10_sourcePanelMethod.ipynb ├── 11_Lesson11_Assignment.ipynb ├── 11_Lesson11_Exercise.ipynb ├── 11_Lesson11_vortexSourcePanelMethod.ipynb └── resources │ ├── Cp_Flap_theoretical.csv │ ├── Cp_Main_theoretical.csv │ ├── Cylinder-Re=1dot54.png │ ├── FlapFoil_N=100.csv │ ├── FlapFoil_N=150.csv │ ├── FlapFoil_N=200.csv │ ├── FlapFoil_N=250.csv │ ├── FlapFoil_N=300.csv │ ├── FlapFoil_N=50.csv │ ├── Lesson11_Exercise_Fig.1.png │ ├── Lesson11_Exercise_Fig.1.svg │ ├── MainFoil_N=100.csv │ ├── MainFoil_N=150.csv │ ├── MainFoil_N=200.csv │ ├── MainFoil_N=250.csv │ ├── MainFoil_N=300.csv │ ├── MainFoil_N=50.csv │ ├── MultiElementAirfoil.png │ ├── NACA0012_sigma.txt │ ├── NACA0012_x.txt │ ├── NACA0012_y.txt │ ├── NACA23012_FlapFoil.csv │ ├── NACA23012_MainFoil.csv │ ├── NACA23012withFlap.png │ ├── StreamlinesTrailingEdge.png │ ├── assignment06-3_mesh.png │ ├── assignment06-4_cp.png │ ├── assignment06-4_streamlines.png │ ├── assignment06-4_velocity.png │ ├── assignment06-5_cp.png │ ├── assignment06-5_sketch.png │ ├── assignment06-5_streamlines.png │ ├── assignment06-5_velocity.png │ ├── assignment06-6_cp.png │ ├── assignment06-6_streamlines.png │ ├── assignment06-6_velocity.png │ ├── cylinder.dat │ ├── doubletSketch1.png │ ├── doubletSketch2.png │ ├── infiniteRowVortices.png │ ├── infiniteRowVorticesSketch.png │ ├── naca0012.dat │ ├── naca0012_discretized_in_circle.png │ ├── naca0012_in_circle.png │ └── panelSketch.png ├── paper.bib ├── paper.md └── styles └── custom.css /.gitignore: -------------------------------------------------------------------------------- 1 | .ipynb_checkpoints/ 2 | 3 | lessons/.ipynb_checkpoints/ 4 | 5 | **/*~ 6 | .DS_Store 7 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Copyright (c) 2018-2019 Lorena A. Barba, Olivier Mesnard 2 | 3 | 4 | Instructional Material 5 | ====================== 6 | 7 | All instructional material is made available under the Creative 8 | Commons Attribution license. You are free: 9 | 10 | * to Share---to copy, distribute and transmit the work 11 | * to Remix---to adapt the work 12 | 13 | Under the following conditions: 14 | 15 | * Attribution---You must attribute the work using "Copyright (c) 16 | Barba group" (but not in any way that suggests that we 17 | endorse you or your use of the work). Where practical, you must 18 | also include a hyperlink to https://github.com/barbagroup/AeroPython. 19 | 20 | With the understanding that: 21 | 22 | * Waiver---Any of the above conditions can be waived if you get 23 | permission from the copyright holder. 24 | * Other Rights---In no way are any of the following rights 25 | affected by the license: 26 | * Your fair dealing or fair use rights; 27 | * The author's moral rights; 28 | * Rights other persons may have either in the work itself or in 29 | how the work is used, such as publicity or privacy rights. * 30 | * Notice---For any reuse or distribution, you must make clear to 31 | others the license terms of this work. The best way to do this is 32 | with a link to http://creativecommons.org/licenses/by/4.0/. 33 | 34 | For the full legal text of this license, please see: 35 | http://creativecommons.org/licenses/by/4.0/legalcode 36 | 37 | 38 | 39 | Software 40 | ========= 41 | 42 | Except where otherwise noted, all software is made available under the 43 | OSI-approved BSD-3-Clause license (https://opensource.org/licenses/BSD-3-Clause): 44 | 45 | Redistribution and use in source and binary forms, with or without modification, 46 | are permitted provided that the following conditions are met: 47 | 48 | 1. Redistributions of source code must retain the above copyright notice, this 49 | list of conditions and the following disclaimer. 50 | 51 | 2. Redistributions in binary form must reproduce the above copyright notice, this 52 | list of conditions and the following disclaimer in the documentation and/or other 53 | materials provided with the distribution. 54 | 55 | 3. Neither the name of the copyright holder nor the names of its contributors may 56 | be used to endorse or promote products derived from this software without specific 57 | prior written permission. 58 | 59 | THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND 60 | ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED 61 | WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 62 | IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, 63 | INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, 64 | BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 65 | DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY 66 | OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE 67 | OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 68 | OF THE POSSIBILITY OF SUCH DAMAGE. 69 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Classical Aerodynamics with Python 2 | 3 | > _Cite as_: Barba, Lorena A., and Mesnard, Olivier (2019). Aero Python: classical aerodynamics of potential flow using Python. Journal of Open Source Education, 2(15), 45, https://doi.org/10.21105/jose.00045 4 | 5 | [![Version](https://img.shields.io/badge/version-1.0-blue.svg)](None) 6 | [![License](https://img.shields.io/badge/license-BSD%203--Clause-blue.svg)](https://github.com/barbagroup/AeroPython/raw/master/LICENSE) 7 | [![License](https://img.shields.io/badge/license-CC--BY%204.0-lightgrey.svg)](https://github.com/barbagroup/AeroPython/raw/master/LICENSE) 8 | [![status](https://jose.theoj.org/papers/b679b34c976beec0bc64807bf087a468/status.svg)](http://jose.theoj.org/papers/b679b34c976beec0bc64807bf087a468) 9 | 10 | The _AeroPython_ series of lessons is the core of a university course (Aerodynamics-Hydrodynamics, MAE-6226) by Prof. Lorena A. Barba at the George Washington University. 11 | The first version ran in Spring 2014 and these Jupyter Notebooks were prepared for that class, with assistance from Barba-group PhD student Olivier Mesnard. 12 | In Spring 2015, we revised and extended the collection, adding student assignments to strengthen the learning experience. 13 | The course was also supported by an open learning space in the GW SEAS Open edX platform (taken offline in 2025). 14 | 15 | The materials are distributed publicly and openly under a Creative Commons Attribution license, [CC-BY 4.0](https://creativecommons.org/licenses/by/4.0/) 16 | 17 | ## Cite as: 18 | 19 | Barba, Lorena A., Mesnard, Olivier (2019). Aero Python: classical aerodynamics of potential flow using Python. Journal of Open Source Education, 2(15), 45, https://doi.org/10.21105/jose.00045 20 | 21 | ## Archive 22 | 23 | — Barba, Lorena A.; Mesnard, Olivier (2014): AeroPython. figshare. Code. 24 | DOI: [10.6084/m9.figshare.1004727.v5](https://doi.org/10.6084/m9.figshare.1004727.v5) 25 | 26 | ## List of notebooks 27 | 28 | ### 0. Getting Started 29 | 30 | * [Quick Python Intro](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/00_Lesson00_QuickPythonIntro.ipynb) 31 | 32 | ### Module 1. Building blocks of potential flow 33 | 34 | 1. [Source & Sink](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/01_Lesson01_sourceSink.ipynb) 35 | 2. [Source & Sink in a Freestream](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/02_Lesson02_sourceSinkFreestream.ipynb) 36 | 3. [Doublet](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/03_Lesson03_doublet.ipynb) 37 | 4. [Assignment: Source distribution on an airfoil](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/03_Lesson03_Assignment.ipynb) 38 | 39 | ### Module 2. Potential vortices and lift 40 | 41 | 1. [Vortex](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/04_Lesson04_vortex.ipynb) 42 | 2. [Infinite row of vortices](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/05_Lesson05_InfiniteRowOfVortices.ipynb) 43 | 3. [Vortex Lift on a cylinder](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/06_Lesson06_vortexLift.ipynb) 44 | 4. [Assignment: Joukowski transformation](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/06_Lesson06_Assignment.ipynb) 45 | 46 | ### Module 3. Source-panel method for non-lifting bodies 47 | 48 | 1. [Method of Images](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/07_Lesson07_methodOfImages.ipynb) 49 | 2. [Source Sheet](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/08_Lesson08_sourceSheet.ipynb) 50 | 3. [Flow over a cylinder with source panels](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/09_Lesson09_flowOverCylinder.ipynb) 51 | 4. [Source panel method](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/10_Lesson10_sourcePanelMethod.ipynb) 52 | 53 | ### Module 4. Vortex-source panel method for lifting bodies 54 | 55 | 1. [Vortex-source panel method](http://nbviewer.ipython.org/urls/github.com/barbagroup/AeroPython/blob/master/lessons/11_Lesson11_vortexSourcePanelMethod.ipynb) 56 | 2. [Exercise: Derivation of the vortex-source panel method](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/11_Lesson11_Exercise.ipynb) 57 | 3. [Assignment: 2D multi-component airfoil](http://nbviewer.ipython.org/github/barbagroup/AeroPython/blob/master/lessons/11_Lesson11_Assignment.ipynb) 58 | 59 | ## Statement of need 60 | 61 | Classical aerodynamics based on potential theory can be an arid subject when presented in the traditional "pen-and-paper" approach. It is a fact that the mathematical framework of potential flow was the only tractable way to apply theoretical calculations in aeronautics through all the early years of aviation, including the development of commercial aircraft into the 1980s and later. Yet, the only way to exercise the power of potential-flow aerodynamics is through numerical computation. Without computing, the student can explore only the simplest fundamental solutions of the potential equation: point sinks and sources, point vortex, doublet, uniform flow. 62 | 63 | The essential tool for applying this theoretical framework to aerodynamics is the panel method, which obtains the strength of a distribution of singularities on a body that makes the body a closed streamline. The addition of vortex singularities to satisfy a Kutta condition allows treating lifting bodies (like airfoils). The AeroPython series begins with simple point-singularity solutions of the potential equation, and applies the principle of superposition to show how to obtain streamline patterns corresponding to flow around objects. Around the half-way point, the module presents the learner with the fundamental relationship between circulation (via a point vortex) and the production of a lift force. Using a distribution of many point singularities on an airfoil, finally, the module shows how we can obtain pressure distributions, and the lift around an airfoil. With this foundation, the student is ready to apply the panel method in authentic engineering situations. 64 | 65 | ## Dependencies 66 | 67 | To use these lessons, you need Python 3, and the standard stack of scientific Python: NumPy, Matplotlib, SciPy. 68 | And of course, you need [Jupyter](http://jupyter.org)—an interactive computational environment that runs on a web browser. 69 | 70 | This mini-course is built as a set of [Jupyter notebooks](https://jupyter-notebook.readthedocs.org/en/latest/notebook.html) containing the written materials and worked-out solutions on Python code. 71 | To work with the material, we recommend that you start each lesson with a fresh new notebook, and follow along, typing each line of code (don't copy-and-paste!), and exploring by changing parameters and seeing what happens. 72 | 73 | ### Installing via Anaconda 74 | 75 | We *highly* recommend that you install the [Anaconda Python Distribution](https://docs.anaconda.com/anaconda/install/). 76 | It will make your life so much easier. 77 | You can download and install Anaconda on Windows, OSX, and Linux. 78 | 79 | After installing, to ensure that your packages are up to date, run the following commands in a terminal: 80 | 81 | ```shell 82 | conda update conda 83 | conda update jupyter numpy scipy matplotlib 84 | ``` 85 | 86 | If you prefer Miniconda (a mini version of Anaconda that saves you disk space), install all the necessary libraries to follow this course by running the following commands in a terminal: 87 | 88 | ```shell 89 | conda update conda 90 | conda install jupyter numpy scipy matplotlib 91 | ``` 92 | 93 | ### Without Anaconda 94 | 95 | If you already have Python installed on your machine, you can install Jupyter using pip: 96 | 97 | ```shell 98 | pip install jupyter 99 | ``` 100 | 101 | Please also make sure that you have the necessary libraries installed by running 102 | 103 | ```shell 104 | pip install numpy scipy matplotlib 105 | ``` 106 | 107 | ## Running the notebook server 108 | 109 | Once Jupyter is installed, open up a terminal and then run 110 | 111 | ```shell 112 | jupyter notebook 113 | ``` 114 | 115 | This will start up a Jupyter session in your browser! 116 | 117 | ## How to contribute to AeroPython 118 | 119 | We accept contributions via pull request. 120 | You can also open an issue if you find a bug, or have a suggestion. 121 | 122 | ## Copyright and License 123 | 124 | (c) 2017 Lorena A. Barba, Olivier Mesnard. All content is under Creative Commons Attribution [CC-BY 4.0](https://creativecommons.org/licenses/by/4.0/legalcode.txt), and all [code is under BSD-3 clause](https://github.com/barbagroup/AeroPython/blob/master/LICENSE) (previously under MIT, and changed on November 12, 2018). 125 | 126 | We are happy if you re-use the content in any way! 127 | 128 | ## Note 129 | 130 | Another Python course exists under the [AeroPython](https://github.com/AeroPython/Curso_AeroPython/) title with different content, and in the Spanish language. (See [tweet](https://twitter.com/LorenaABarba/status/464041427169583104) from 2014.) 131 | -------------------------------------------------------------------------------- /lessons/03_Lesson03_Assignment.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "###### Content provided under a Creative Commons Attribution license, CC-BY 4.0; code under BSD 3-Clause license. (c)2014 Lorena A. Barba, Pi-Yueh Chuang. Thanks: NSF for support via CAREER award #1149784." 8 | ] 9 | }, 10 | { 11 | "cell_type": "markdown", 12 | "metadata": {}, 13 | "source": [ 14 | "# Source Distribution on an Airfoil" 15 | ] 16 | }, 17 | { 18 | "cell_type": "markdown", 19 | "metadata": {}, 20 | "source": [ 21 | "In [Lesson 3](03_Lesson03_doublet.ipynb) of *AeroPython*, you learned that it is possible to represent potential flow around a circular cylinder using the superposition of a doublet singularity and a free stream. But potential flow is even more powerful: you can represent the flow around *any* shape. How is it possible, you might ask?\n", 22 | "\n", 23 | "For non-lifting bodies, you can use a source distribution on the body surface, superposed with a free stream. In this assignment, you will build the flow around a NACA0012 airfoil, using a set of sources.\n", 24 | "\n", 25 | "Before you start, take a moment to think: in flow around a symmetric airfoil at $0^{\\circ}$ angle of attack,\n", 26 | "\n", 27 | "* Where is the point of maximum pressure?\n", 28 | "* What do we call that point?\n", 29 | "* Will the airfoil generate any lift?\n", 30 | "\n", 31 | "At the end of this assignment, come back to these questions, and see if it all makes sense." 32 | ] 33 | }, 34 | { 35 | "cell_type": "markdown", 36 | "metadata": {}, 37 | "source": [ 38 | "## Problem Setup" 39 | ] 40 | }, 41 | { 42 | "cell_type": "markdown", 43 | "metadata": {}, 44 | "source": [ 45 | "You will read data files containing information about the location and the strength of a set of sources located on the surface of a NACA0012 airfoil. \n", 46 | "\n", 47 | "There are three data files: NACA0012_x.txt, NACA0012_y.txt, and NACA0012_sigma.txt. To load each file into a NumPy array, you need the function [`numpy.loadtxt`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.loadtxt.html). The files should be found in the `resources` folder of the `lessons`. " 48 | ] 49 | }, 50 | { 51 | "cell_type": "markdown", 52 | "metadata": {}, 53 | "source": [ 54 | "Using 51 mesh points in each direction, and a domain $[-1, 2]\\times[-0.5, 0.5]$, compute the velocity due to the set of sources plus a free stream in the $x$-direction with $U_{\\infty}=1$. Also compute the coefficient of pressure on your grid points." 55 | ] 56 | }, 57 | { 58 | "cell_type": "markdown", 59 | "metadata": {}, 60 | "source": [ 61 | "## Questions:" 62 | ] 63 | }, 64 | { 65 | "cell_type": "markdown", 66 | "metadata": {}, 67 | "source": [ 68 | "1. What is the value of maximum pressure coefficient, $C_p$?\n", 69 | "2. What are the array indices for the maximum value of $C_p$?\n", 70 | "\n", 71 | "Make the following plots to visualize and inspect the resulting flow pattern:\n", 72 | "\n", 73 | "* Stream lines in the domain and the profile of our NACA0012 airfoil, in one plot\n", 74 | "* Distribution of the pressure coefficient and a single marker on the location of the maximum pressure" 75 | ] 76 | }, 77 | { 78 | "cell_type": "markdown", 79 | "metadata": {}, 80 | "source": [ 81 | "**Hint**: You might use the following NumPy functions: [`numpy.unravel_index`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.unravel_index.html) and [`numpy.argmax`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html)" 82 | ] 83 | }, 84 | { 85 | "cell_type": "markdown", 86 | "metadata": {}, 87 | "source": [ 88 | "##### Think" 89 | ] 90 | }, 91 | { 92 | "cell_type": "markdown", 93 | "metadata": {}, 94 | "source": [ 95 | "1. Do the stream lines look like you expected?\n", 96 | "2. What does the distribution of pressure tell you about lift generated by the airfoil?\n", 97 | "3. Does the location of the point of maximum pressure seem right to you?" 98 | ] 99 | }, 100 | { 101 | "cell_type": "code", 102 | "execution_count": 1, 103 | "metadata": {}, 104 | "outputs": [ 105 | { 106 | "data": { 107 | "text/html": [ 108 | "\n", 109 | "\n", 110 | "\n", 111 | "\n", 235 | "\n" 251 | ], 252 | "text/plain": [ 253 | "" 254 | ] 255 | }, 256 | "execution_count": 1, 257 | "metadata": {}, 258 | "output_type": "execute_result" 259 | } 260 | ], 261 | "source": [ 262 | "from IPython.core.display import HTML\n", 263 | "def css_styling(filepath):\n", 264 | " styles = open(filepath, 'r').read()\n", 265 | " return HTML(styles)\n", 266 | "css_styling('../styles/custom.css')" 267 | ] 268 | } 269 | ], 270 | "metadata": { 271 | "kernelspec": { 272 | "display_name": "Python 3", 273 | "language": "python", 274 | "name": "python3" 275 | }, 276 | "language_info": { 277 | "codemirror_mode": { 278 | "name": "ipython", 279 | "version": 3 280 | }, 281 | "file_extension": ".py", 282 | "mimetype": "text/x-python", 283 | "name": "python", 284 | "nbconvert_exporter": "python", 285 | "pygments_lexer": "ipython3", 286 | "version": "3.7.1" 287 | } 288 | }, 289 | "nbformat": 4, 290 | "nbformat_minor": 1 291 | } 292 | -------------------------------------------------------------------------------- /lessons/11_Lesson11_Assignment.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "###### Content provided under a Creative Commons Attribution license CC-BY 4.0; code under BSD 3-Clause license. (c)2015 L.A. Barba, Pi-Yueh Chuang." 8 | ] 9 | }, 10 | { 11 | "cell_type": "markdown", 12 | "metadata": {}, 13 | "source": [ 14 | "# 2D Multi-element airfoil" 15 | ] 16 | }, 17 | { 18 | "cell_type": "markdown", 19 | "metadata": {}, 20 | "source": [ 21 | "This is your [_AeroPython_](https://github.com/barbagroup/AeroPython) assignment for the fourth course module, titled **\"Vortex-panel method for lifting bodies.\"** You will investigate a 2D multi-element airfoil, or wing section with extended flap, adapting the source-vortex panel method of [Lesson 11](11_Lesson11_vortexSourcePanelMethod.ipynb) to compute the flow around two airfoils. \n", 22 | "\n", 23 | "The key concepts for using the vortex-source panel method with a multi-element airfoil are:\n", 24 | "\n", 25 | "1. each airfoil has its own vortex strength, and\n", 26 | "2. the trailing edge at each airfoil must follow the Kutta condition.\n", 27 | "\n", 28 | "In the first part of the assignment, you will use a test for which we have a theoretical solution, due to Williams (1973). Have a look at the reference, and get a feel for its mathematical genius! In the second part, you will use a more popular airfoil (the NACA 23012) for both the main wing section and the flap. For this case, there is an old classic NACA report by Wenzinger (1938) giving data from experiments in a wind tunnel. This exercise will get you wondering about some very interesting aerodynamical questions!" 29 | ] 30 | }, 31 | { 32 | "cell_type": "markdown", 33 | "metadata": {}, 34 | "source": [ 35 | "## Part 1: Williams' test case" 36 | ] 37 | }, 38 | { 39 | "cell_type": "markdown", 40 | "metadata": {}, 41 | "source": [ 42 | "Your task is to calculate the pressure coefficient on a multi-element wing section, consisting of a *main airfoil* and an *external-airfoil flap*. This test is from Williams (1973), who obtained a beautiful theoretical solution using the techniques of conformal mapping.\n", 43 | "\n", 44 | "The profile of the wing section looks like this, with flap extended:\n", 45 | "
\n", 46 | "\n", 47 | "We provide CSV files with the $x$ and $y$ coordinates for the end nodes of each panel; these files are in the `resources` directory of our GitHub repository. The files have the following naming convention, where `***` represents the value of `N`, the number of panels: `MainFoil_N=***.csv`, `FlapFoil_N=***.csv`.\n", 48 | "\n", 49 | "You can use fewer panels for debugging while developing your codes and then use more panels for more serious calculations when answering the questions in this assignment.\n", 50 | "\n", 51 | "There are also two files named `Cp_Main_theoretical.csv` and `Cp_Flap_theoretical.csv` containing the theoretical values of pressure coefficient versus $x$-coordinates, for the flap configuration we give you in the data files (a $30$º flap deflection, with the main airfoil at $0$º angle of attack).\n", 52 | "\n", 53 | "The theoretical lift and drag force for potential flow over this 2D wing section with zero angle of attack are $3.7386$ and $0$, respectively (non-dimensionalized by dynamic pressure).\n" 54 | ] 55 | }, 56 | { 57 | "cell_type": "markdown", 58 | "metadata": {}, 59 | "source": [ 60 | "### Hints" 61 | ] 62 | }, 63 | { 64 | "cell_type": "markdown", 65 | "metadata": {}, 66 | "source": [ 67 | "Recall the [exercise in Lesson 11](11_Lesson11_Exercise.ipynb), which asks you to derive the mathematical formulation of the vortex-source panel method. \n", 68 | "\n", 69 | "Now, for the 2-element airfoil, you need to derive the required mathematical expressions, following the same process as in the exercise, starting from the following expression for the potential:\n", 70 | "\n", 71 | "$$\n", 72 | "\\begin{split}\n", 73 | "\\phi(x, y) &= U_{\\infty}x\\cos\\alpha + U_{\\infty}y\\sin\\alpha \\\\\n", 74 | "& + \\int_{main} \\frac{1}{2\\pi} \\sigma(s) \\ln \\sqrt{(x-\\xi(s))^2+(y-\\eta(s))^2} ds \\\\\n", 75 | "& + \\int_{flap} \\frac{1}{2\\pi} \\sigma(s) \\ln \\sqrt{(x-\\xi(s))^2+(y-\\eta(s))^2} ds \\\\\n", 76 | "& - \\int_{main} \\frac{1}{2\\pi} \\gamma(s) \\tan^{-1} \\frac{y-\\eta(s)}{x-\\xi(s)} ds \\\\\n", 77 | "& - \\int_{flap} \\frac{1}{2\\pi} \\gamma(s) \\tan^{-1} \\frac{y-\\eta(s)}{x-\\xi(s)} ds\n", 78 | "\\end{split}\n", 79 | "$$\n", 80 | "\n", 81 | "Assume the following:\n", 82 | "\n", 83 | "1. $\\sigma(s)$ is constant on each panel, \n", 84 | "2. $\\gamma(s)$ is constant on each airfoil, and\n", 85 | "3. the 1st to $N_a$th panels are on the main airfoil and the $(N_a+1)$th to $N$th panels are on the flap (where $N=N_a+N_b$, and $N_b$ is the number of panels on the flap).\n", 86 | "\n", 87 | "There will be $N+2$ unknowns, that is, $\\sigma_1\\cdots\\sigma_N$, $\\gamma_a$, and $\\gamma_b$. The values of $\\gamma_a$, and $\\gamma_b$ represent the vortex strengths on the main and flap airfoils, respectively.\n", 88 | "\n", 89 | "You should be able to obtain the following matrix forms of normal and tangential velocity on the $i$th panel:\n", 90 | "\n", 91 | "$$\n", 92 | "U^n_i = b^n_i + \n", 93 | "\\left[\\begin{smallmatrix}A^n_{i1}\\cdots A^n_{iN}\\end{smallmatrix}, \\sum_{j=1}^{N_a}B^n_{ij}, \\sum_{j=N_a+1}^{N}B^n_{ij}\\right]\n", 94 | "\\left[\\begin{smallmatrix}\\sigma_1 \\\\ \\vdots \\\\ \\sigma_N \\\\ \\gamma_a \\\\ \\gamma_b\\end{smallmatrix}\\right]\n", 95 | "$$\n", 96 | "\n", 97 | "$$\n", 98 | "U^t_i = b^t_i + \n", 99 | "\\left[\\begin{smallmatrix}A^t_{i1}\\cdots A^t_{iN}\\end{smallmatrix}, \\sum_{j=1}^{N_a}B^t_{ij}, \\sum_{j=N_a+1}^{N}B^t_{ij}\\right]\n", 100 | "\\left[\\begin{smallmatrix}\\sigma_1 \\\\ \\vdots \\\\ \\sigma_N \\\\ \\gamma_a \\\\ \\gamma_b\\end{smallmatrix}\\right]\n", 101 | "$$\n", 102 | "\n", 103 | "Using the non-penetration conditions on the $1$st to $N$th panels, you will have $N$ linear equations. To solve for the $N+2$ unknowns, the extra two linear equations you need are the Kutta conditions on the two airfoils:\n", 104 | "\n", 105 | "$$\n", 106 | "U^t_{1} = U^t_{N_a} \\\\\n", 107 | "U^t_{N_a+1} = U^t_{N}\n", 108 | "$$\n", 109 | "\n", 110 | "And you are ready to solve for potential flow around a 2D multi-component wing!" 111 | ] 112 | }, 113 | { 114 | "cell_type": "markdown", 115 | "metadata": {}, 116 | "source": [ 117 | "##### Questions" 118 | ] 119 | }, 120 | { 121 | "cell_type": "markdown", 122 | "metadata": {}, 123 | "source": [ 124 | "Run your codes using 100 panels on both main and flap airfoils and answer the following numeric-value questions." 125 | ] 126 | }, 127 | { 128 | "cell_type": "markdown", 129 | "metadata": {}, 130 | "source": [ 131 | "### Q1." 132 | ] 133 | }, 134 | { 135 | "cell_type": "markdown", 136 | "metadata": {}, 137 | "source": [ 138 | "* What is the pressure coefficient on the $3$rd panel on the *main airfoil*?" 139 | ] 140 | }, 141 | { 142 | "cell_type": "markdown", 143 | "metadata": {}, 144 | "source": [ 145 | "### Q2." 146 | ] 147 | }, 148 | { 149 | "cell_type": "markdown", 150 | "metadata": {}, 151 | "source": [ 152 | "* What is the pressure coefficient on the $10$th panel on the *flap airfoil*?" 153 | ] 154 | }, 155 | { 156 | "cell_type": "markdown", 157 | "metadata": {}, 158 | "source": [ 159 | "### Q3." 160 | ] 161 | }, 162 | { 163 | "cell_type": "markdown", 164 | "metadata": {}, 165 | "source": [ 166 | "* What is the value of lift?\n", 167 | "\n", 168 | "**Hint:** $~L = - \\oint_{main}p\\vec{n}\\cdot\\vec{j}dl- \\oint_{flap}p\\vec{n}\\cdot\\vec{j}dl$" 169 | ] 170 | }, 171 | { 172 | "cell_type": "markdown", 173 | "metadata": {}, 174 | "source": [ 175 | "### Q4." 176 | ] 177 | }, 178 | { 179 | "cell_type": "markdown", 180 | "metadata": {}, 181 | "source": [ 182 | "* What is the value of drag? " 183 | ] 184 | }, 185 | { 186 | "cell_type": "markdown", 187 | "metadata": {}, 188 | "source": [ 189 | "### Q5." 190 | ] 191 | }, 192 | { 193 | "cell_type": "markdown", 194 | "metadata": {}, 195 | "source": [ 196 | "* Run the program again with 200 panels on both main and flap airfoils. What is the percentage of improvement of the absolute error in the value of lift?" 197 | ] 198 | }, 199 | { 200 | "cell_type": "markdown", 201 | "metadata": {}, 202 | "source": [ 203 | "## Part 2: Wenzinger's test case" 204 | ] 205 | }, 206 | { 207 | "cell_type": "markdown", 208 | "metadata": {}, 209 | "source": [ 210 | "You should now have a usable panel solver for 2-component airfoils. Next, you'll change the target airfoil to a more realistic profile. There are experimental results available (Wenzinger, 1938) for a NACA 23012 airfoil with a NACA23012 external-airfoil flap, which we'll compare with.\n", 211 | "\n", 212 | "The following figure shows the profile of this wing section with 0º flap angle:\n", 213 | "\n", 214 | "
\n", 215 | "\n", 216 | "We provide CSV files for the endpoints of a panel discretization of this wing section, using 150 panels on each airfoil: `NACA23012_MainFoil.csv` and `NACA23012_FlapFoil.csv` in the `resources` folder.\n", 217 | "\n", 218 | "The flap has a rotating center (hinge) at the location $(1.03, -0.054)$ (see the configuration on Figure 1 of the reference). Using coordinate rotation, you can obtain different configurations with different flap deflection angles.\n", 219 | "\n", 220 | "Try to use different flap deflection angles and different angles of attack (of the main airfoil) and compare to Wenzinger's experimental results. Note that the definition of the total lift coefficient is $L/(l_{main}+l_{flap})$, where $L$ is the lift force (per unit span) like in the previous problem, and $l_{main}$ and $l_{flap}$ are the chord lengths of the two airfoils. **Think about what could be the sources of difference between your results and the experimental data**." 221 | ] 222 | }, 223 | { 224 | "cell_type": "markdown", 225 | "metadata": {}, 226 | "source": [ 227 | "##### Questions" 228 | ] 229 | }, 230 | { 231 | "cell_type": "markdown", 232 | "metadata": {}, 233 | "source": [ 234 | "### Q1." 235 | ] 236 | }, 237 | { 238 | "cell_type": "markdown", 239 | "metadata": {}, 240 | "source": [ 241 | "* Use the [bisection method](http://en.wikipedia.org/wiki/Bisection_method) to determine the angle of attack, $\\alpha$, that will cause zero lift when the flap angle is $20$º. Round your answer to the second digit after the floating point.\n", 242 | "\n", 243 | "**HINT**: you can start the bisection method within the range $-14\\le\\alpha\\le14$." 244 | ] 245 | }, 246 | { 247 | "cell_type": "markdown", 248 | "metadata": {}, 249 | "source": [ 250 | "### Q2." 251 | ] 252 | }, 253 | { 254 | "cell_type": "markdown", 255 | "metadata": {}, 256 | "source": [ 257 | "* Use the [bisection method](http://en.wikipedia.org/wiki/Bisection_method) to determine the flap angle that will cause zero lift when the angle of attack, $\\alpha$, is $0$º. Round your answer to the second digit after the floating point." 258 | ] 259 | }, 260 | { 261 | "cell_type": "markdown", 262 | "metadata": {}, 263 | "source": [ 264 | "### Q3." 265 | ] 266 | }, 267 | { 268 | "cell_type": "markdown", 269 | "metadata": {}, 270 | "source": [ 271 | "* Suppose that the main airfoil is attached to the fuselage of an airplane at an incidence angle of $4$º (that is, the built-in angle of attack of the wing when the airplane is in level flight is $4$º). Compute the lift coefficient with the flap at $0$º deflection, and at $5$, $10$ and $15$ degrees. How much slower (in percentage) can the airplane fly (in level flight) with the flaps extended at these settings, compared with $0$º?" 272 | ] 273 | }, 274 | { 275 | "cell_type": "markdown", 276 | "metadata": {}, 277 | "source": [ 278 | "##### Think" 279 | ] 280 | }, 281 | { 282 | "cell_type": "markdown", 283 | "metadata": {}, 284 | "source": [ 285 | "Think about the meaning of what you observed in **Q3**, above. What is the effect of using the flap?" 286 | ] 287 | }, 288 | { 289 | "cell_type": "markdown", 290 | "metadata": {}, 291 | "source": [ 292 | "## References" 293 | ] 294 | }, 295 | { 296 | "cell_type": "markdown", 297 | "metadata": {}, 298 | "source": [ 299 | "* B. R. Williams (1973), *An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils*, Reports & Memoranda No. 3717, Aeronautical Research Council of the United Kingdom // [PDF at Cranfield University](http://naca.central.cranfield.ac.uk/reports/arc/rm/3717.pdf)\n", 300 | "\n", 301 | "* C. J. Wenzinger (1938), *Pressure distribution over an NACA 23012 airfoil with an N.A.C.A. 23012 external-airfoil flap*, NACA Technical Report No.614 // [PDF at NASA](http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930091690.pdf)" 302 | ] 303 | }, 304 | { 305 | "cell_type": "markdown", 306 | "metadata": {}, 307 | "source": [ 308 | "-----------------------------" 309 | ] 310 | }, 311 | { 312 | "cell_type": "code", 313 | "execution_count": 1, 314 | "metadata": {}, 315 | "outputs": [ 316 | { 317 | "data": { 318 | "text/html": [ 319 | "\n", 320 | "\n", 321 | "\n", 322 | "\n", 446 | "\n" 462 | ], 463 | "text/plain": [ 464 | "" 465 | ] 466 | }, 467 | "execution_count": 1, 468 | "metadata": {}, 469 | "output_type": "execute_result" 470 | } 471 | ], 472 | "source": [ 473 | "from IPython.core.display import HTML\n", 474 | "def css_styling(filepath):\n", 475 | " styles = open(filepath, 'r').read()\n", 476 | " return HTML(styles)\n", 477 | "css_styling('../styles/custom.css')" 478 | ] 479 | } 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%% This BibTeX bibliography file was created using BibDesk. 2 | %% http://bibdesk.sourceforge.net/ 3 | 4 | %% Created for Lorena Barba at 2019-01-29 12:56:30 -0500 5 | 6 | 7 | %% Saved with string encoding Unicode (UTF-8) 8 | 9 | 10 | 11 | @misc{Barba2017, 12 | Author = {Barba, Lorena A.,}, 13 | Date-Added = {2019-01-28 20:57:11 -0500}, 14 | Date-Modified = {2019-01-29 12:56:30 -0500}, 15 | Howpublished = {figshare, \href{https://doi.org/10.6084/m9.figshare.4584328.v1}{doi:10.6084/m9.figshare.4584328.v1}}, 16 | Title = {MAE-6226: Aerodynamics Course Syllabus}, 17 | Year = {2017}} 18 | 19 | @article{margulieuxETal2016, 20 | Author = {Margulieux, Lauren E and Catrambone, Richard and Guzdial, Mark}, 21 | Date-Added = {2019-01-28 20:48:00 -0500}, 22 | Date-Modified = {2019-01-28 20:48:17 -0500}, 23 | Journal = {Computer Science Education}, 24 | Number = {1}, 25 | Pages = {44--67}, 26 | Publisher = {Taylor \& Francis}, 27 | Title = {Employing subgoals in computer programming education}, 28 | Volume = {26}, 29 | Year = {2016}, 30 | Doi = {10.1080/08993408.2016.1144429}} 31 | 32 | @article{catrambone1998, 33 | Author = {Catrambone, Richard}, 34 | Date-Added = {2019-01-28 20:44:43 -0500}, 35 | Date-Modified = {2019-01-28 20:45:28 -0500}, 36 | Journal = {Journal of Experimental Psychology: General}, 37 | Number = {4}, 38 | Pages = {355--376}, 39 | Publisher = {American Psychological Association}, 40 | Title = {The subgoal learning model: Creating better examples so that students can solve novel problems.}, 41 | Volume = {127}, 42 | Year = {1998}, 43 | Doi = {10.1037/0096-3445.127.4.355}} 44 | 45 | @article{chen2015worked, 46 | Author = {Chen, Ouhao and Kalyuga, Slava and Sweller, John}, 47 | Date-Added = {2019-01-28 20:36:39 -0500}, 48 | Date-Modified = {2019-01-28 20:36:39 -0500}, 49 | Doi = {10.1037/edu0000018}, 50 | Journal = {Journal of Educational Psychology}, 51 | Number = {3}, 52 | Pages = {689}, 53 | Publisher = {American Psychological Association}, 54 | Title = {The worked example effect, the generation effect, and element interactivity.}, 55 | Volume = {107}, 56 | Year = {2015}, 57 | Bdsk-Url-1 = {https://doi.org/10.1037/edu0000018}} 58 | 59 | @article{sweller2006worked, 60 | Author = {Sweller, John}, 61 | Date-Added = {2019-01-28 20:36:36 -0500}, 62 | Date-Modified = {2019-01-28 20:36:36 -0500}, 63 | Doi = {10.1016/j.learninstruc.2006.02.005}, 64 | Journal = {Learning and instruction}, 65 | Number = {2}, 66 | Pages = {165--169}, 67 | Publisher = {Elsevier Science}, 68 | Title = {The worked example effect and human cognition}, 69 | Volume = {16}, 70 | Year = {2006}, 71 | Bdsk-Url-1 = {https://doi.org/10.1016/j.learninstruc.2006.02.005}} 72 | 73 | @incollection{drela1989xfoil, 74 | Address = {Berlin, Heidelberg}, 75 | Author = {Drela, Mark}, 76 | Booktitle = {Low Reynolds number aerodynamics}, 77 | Date-Added = {2019-01-28 08:17:38 -0500}, 78 | Date-Modified = {2019-01-28 08:19:35 -0500}, 79 | Doi = {10.1007/978-3-642-84010-4_1}, 80 | Editor = {Mueller, T.J.}, 81 | Pages = {1--12}, 82 | Publisher = {Springer}, 83 | Series = {Lecture Notes in Engineering}, 84 | Title = {{XFOIL}: An analysis and design system for low {Reynolds} number airfoils}, 85 | Volume = {54}, 86 | Year = {1989}, 87 | Bdsk-Url-1 = {https://doi.org/10.1007/978-3-642-84010-4_1}} 88 | 89 | @article{BarbaForsyth2018, 90 | Author = {Barba, Lorena A., and Forsyth, Gilbert F.}, 91 | Date-Added = {2018-11-12 19:08:59 -0500}, 92 | Date-Modified = {2018-11-12 19:11:54 -0500}, 93 | Doi = {10.21105/jose.00021}, 94 | Journal = {Journal of Open Source Education}, 95 | Number = {9}, 96 | Pages = {21}, 97 | Title = {{CFD Python}: the 12 steps to {Navier--Stokes} equations}, 98 | Volume = {1}, 99 | Year = {2018}, 100 | Bdsk-Url-1 = {https://doi.org/10.21105/jose.00021}} 101 | -------------------------------------------------------------------------------- /paper.md: -------------------------------------------------------------------------------- 1 | --- 2 | title: 'Aero Python: classical aerodynamics of potential flow using Python' 3 | tags: 4 | - Python 5 | - aerodynamics 6 | - differential equations 7 | - numerical methods 8 | - finite differences 9 | authors: 10 | - name: Lorena A. Barba 11 | orcid: 0000-0001-5812-2711 12 | affiliation: 1 13 | - name: Olivier Mesnard 14 | orcid: 0000-0001-5335-7853 15 | affiliation: "1" 16 | affiliations: 17 | - name: The George Washington University (Washington DC, USA) 18 | index: 1 19 | date: 11 November 2018 20 | bibliography: paper.bib 21 | --- 22 | 23 | # Summary 24 | 25 | The **AeroPython** set of lessons present a computational approach to an engineering course in classical aerodynamics. 26 | The audience of such a course would be advanced undergraduate students in mechanical or aerospace engineering (or master's students), with a fluid mechanics pre-requisite. 27 | Classical aerodynamics is a broad theoretical subject—this learning module focuses on one topic: the use of potential flow for aerodynamic analysis, via the panel method. 28 | 29 | The series of lessons is the core of a university course (Aerodynamics-Hydrodynamics, MAE-6226) by Prof. Lorena A. Barba at the George Washington University. 30 | The first version ran in Spring 2014 and these Jupyter Notebooks were prepared for that class, with Olivier Mesnard. 31 | In Spring 2015, we revised and extended the collection, adding student assignments to strengthen the learning experience. 32 | With added discussions of theoretical topics, assigned readings, and assignments, the modules can span a full-semester course. 33 | 34 | **AeroPython** is a collection of Jupyter notebooks: one "lesson zero" introduction to Python and NumPy arrays; 11 lessons in potential flow using Python; three student assignments involving a coding mini-project; and one extra notebook with an exercise deriving the panel-method equations. 35 | 36 | The list of lessons is: 37 | 38 | * _Python crash course_: quick introduction to Python, NumPy arrays and plotting with Matplotlib. 39 | * _Source \& sink_: introduction to potential-flow theory; computing and plotting a source-sink pair. 40 | * _Source \& sink in a freestream_: adds a freestream to a source to get a Rankine half-body; then adds a freestream to a source-sink pair to get a Rankine oval; introduces Python functions. 41 | * _Doublet_: develops a doublet singularity from a source-sink pair, at the limit of zero distance; adds a freestream to get flow around a cylinder. 42 | * _Assignment 1: source distribution on an airfoil_. 43 | * _Vortex_: a potential vortex, a vortex and sink; idea of irrotational flow. 44 | * _Infinite row of vortices_: superposition of many vortices to represent a vortex sheet. 45 | * _Lift on a cylinder_: superposition of a doublet, a freestream, and a vortex; computing lift and drag. 46 | * _Assignment 2: the Joukowski transformation_. 47 | * _Method of images_: source near a plane wall; vortex near a wall; vortex pair near a wall; doublet near a wall parallel to a uniform flow. Introduces Python classes. 48 | * _Source sheet_: a finite row of sources, then an infinite row of sources along one line. Introduces SciPy for integration. 49 | * _Flow over a cylinder with source panels_: calculates the source-strength distribution that can produce potential flow around a circular cylinder. 50 | * _Source panel method_: solves for the source-sheet strengths to get flow around a NACA0012 airfoil. 51 | * _Vortex-source panel method_: start with the source panel method of the previous lesson, and add circulation to get a lift force. Introduces the idea of the Kutta condition. 52 | * _Exercise: Derivation of the vortex-source panel method_. 53 | * _Assignment 3: 2D multi-component airfoil_. 54 | 55 | The design of the lessons follows the pattern of the **CFD Python** collection [@BarbaForsyth2018], with step-by-step, incrementally worked-out examples, leading to a more complex computational solution. In this case, the final product is a panel-method solution of potential flow around a lifting airfoil. 56 | 57 | # Statement of need 58 | 59 | Classical aerodynamics based on potential theory can be an arid subject when presented in the traditional "pen-and-paper" approach. 60 | It is a fact that the mathematical framework of potential flow was the only tractable way to apply theoretical calculations in aeronautics through all the early years of aviation, including the development of commercial aircraft into the 1980s and later. 61 | Yet, the only way to exercise the power of potential-flow aerodynamics is through numerical computation. Without computing, the student can explore only the simplest fundamental solutions of the potential equation: point sinks and sources, point vortex, doublet, uniform flow. 62 | 63 | The essential tool for applying this theoretical framework to aerodynamics is the panel method, which obtains the strength of a distribution of singularities on a body that makes the body a closed streamline. The addition of vortex singularities to satisfy a Kutta condition allows treating lifting bodies (like airfoils). The AeroPython series begins with simple point-singularity solutions of the potential equation, and applies the principle of superposition to show how to obtain streamline patterns corresponding to flow around objects. Around the half-way point, the module presents the learner with the fundamental relationship between circulation (via a point vortex) and the production of a lift force. Using a distribution of many point singularities on an airfoil, finally, the module shows how we can obtain pressure distributions, and the lift around an airfoil. With this foundation, the student is ready to apply the panel method in authentic engineering situations. 64 | 65 | Exercising the power of superposition—the hallmark of the linear potential equation—via numerical computing with Python, and immediately visualizing the resulting flow patterns, the student begins to quickly interpret and imagine the real applications of this theory. 66 | At the end of the course, the student is ready to use panel methods for aerodynamic design, and knowingly use other software packages such as XFOIL (http://web.mit.edu/drela/Public/web/xfoil/) [@drela1989xfoil] and XFLR5 (http://www.xflr5.com). 67 | 68 | # Pedagogy and instructional design 69 | 70 | **AeroPython** adopts the instructional design patterns used in the **CFD Python** learning module [@BarbaForsyth2018]. 71 | It breaks down a rather complex numerical problem—in this case, the panel method for lifting bodies—into small steps, it chunks these steps together logically, and adds narrative and context. 72 | The pedagogical strategy relies on the worked-example effect [@sweller2006worked; @chen2015worked] and sub-goals [@catrambone1998; @margulieuxETal2016]. 73 | In the classroom, students are guided to re-write the worked examples in their own Jupyter notebooks, and annotate with their own reflections during class discussions. 74 | These discussions may take the form of short introductions to a new section of the course, or a debrief after completing a sub-goal. 75 | Classroom activity may also be peppered with "pair-share" exercises, and individual inspection of student work by the instructor. 76 | Some assigned readings can also be selected; see some examples in the syllabus for a course at the George Washington University using these lessons [@Barba2017]. 77 | Barba has taught the semester-long course there four times since 2014, and Mesnard is currently teaching its fifth iteration. 78 | 79 | # References 80 | -------------------------------------------------------------------------------- /styles/custom.css: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 125 | 126 | 147 | --------------------------------------------------------------------------------