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Aero Python: classical aerodynamics of potential flow using Python |
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11 November 2018 |
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The AeroPython learning module 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 coding; and one extra notebook with an exercise deriving the panel-method equations.
The list of lessons is:
- Python crash course: quick introduction to Python, NumPy arrays and plotting with Matplotlib.
- Source & sink: introduction to potential-flow theory; computing and plotting a source-sink pair.
- 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.
- 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.
- Assignment 1: source distribution on an airfoil.
- Vortex: a potential vortex, a vortex and sink; idea of irrotational flow.
- Infinite row of vortices: superposition of many vortices to represent a vortex sheet.
- Lift on a cylinder: superposition of a doublet, a freestream, and a vortex; computing lift and drag.
- Assignment 2: the Joukowski transformation.
- 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.
- Source sheet: a finite row of sources, then an infinite row of sources along one line. Introduces SciPy for integration.
- Flow over a cylinder with source panels: calculates the source-strength distribution that can produce potential flow around a circular cylinder.
- Source panel method: solves for the source-sheet strengths to get flow around a NACA0012 airfoil.
- 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.
- Exercise: Derivation of the vortex-source panel method.
- Assignment: 2D multi-component airfoil.
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.
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 developmemt 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.
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.