Unsteady Boundary-Element Method in 2-D (UBEM2D)

This is a Python-3 implementation of the unsteady boundary-element method of Basu & Hancock for the flow of an inviscid incompressible fluid past a moving airfoil. The method is described in the following paper:

B.C. Basu and G.J. Hancock, The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow, Journal of Fluid Mechanics, 87 (1), pp. 159-178, 1978.

Last maintenance date

This project was most recently validated on 09-Jan-2025 using an iMac M4 (Apple Silicon) and the following software verions: python 3.9.6 pip 24.3.1 matplotlib 3.9.4 numpy 2.0.2 scipy 1.13.1

Authors of the Code

Michael J. Fairchild and Clancy W. Rowley, Princeton University.

License

This code may be freely used and modified, and it is distributed under no particular license. Instead, users of this code are merely asked to acknowledge the authors in any scientific talk or publication whose results make use of this code.

Dependencies

This code has a few standard dependencies, namely numpy, matplotlib, scipy, and ffmpeg (only needed when saving movies to filesystem). To install the dependent packages (if missing), use Python's built-in pip module, which should be brought up to date with: python3 -m pip install --upgrade pip:w

After updating the pip module, install the dependent packages as follows: python3 -m pip install numpy python3 -m pip install matplotlib python3 -m pip install scipy python3 -m pip install ffmpeg

For more information on the ffmpeg dependency, see the ffmpeg homepage).

Documentation

LaTeX documentation of the underlying fluid dynamics, as well as details of the Python implementation, may be found in the docs subdirectory.

Installation and Verification

If the user does not wish to run all the examples and validation code from the ubem2d_python home directory, then append that directory to the PYTHONPATH environment variable. The following examples and test cases may be used to verify the installation and the validity of the code.

Pre-defined examples

Several examples of increasing complexity are provided in order to demonstrate the basic usage of the library, including:

• Test steady flow past a NACA airfoil: python3 examples/naca.py
• Test steady flow past several NACA airfoils: python3 examples/naca_multiple.py
• Test unsteady flow past a moving NACA airfoil: python3 examples/unsteady.py
• Test kinematics and animation: python3 examples/animate_kinematics.py

Validation

Several validation cases are provided:

• Validate the steady flow past a cylinder with arbitrary circulation vs. the analytical solution: python3 validation/cylinder/cylinder.py
• Validate the steady flow past a NACA 2412 cambered airfoil with 120 panels at 8-degrees angle of attack against the linear-strength vortex-panel method digitized from Figure 5.25 on p.164 of the book Foundations of Aerodynamics, 5th Ed., by Kuethe and Chow: python3 validation/naca_kc/naca_kc.py
• Validate the lift curve (i.e. lift coefficient vs. angle of attack) for steady flow past symmetric and cambered NACA airfoils against experimental data and thin-airfoil theory: python3 validation/naca_avd/naca_avd.py
• Validate rapid pitching motion of an 8.4%-thick von Mises airfoil vs. the U2DIIF Fortran code of the Naval Postgraduate School: python3 validation/u2diif_pitch/pitch.py
• Validate rapid plunging motion of a NACA 0015 airfoil vs. the U2DIIF Fortran code of the Naval Postgraduate School: python3 validation/u2diif_plunge/plunge.py
• Validate a pitch-up maneuver for an 8.4%-thick von Mises airfoil vs. the U2DIIF Fortran code of the Naval Postgraduate School: python3 validation/u2diif_ramp/ramp.py

The experimental data in the lift-curve validation case were digitized from images taken of pages in the book Theory of Wing Sections, by Ira H. Abbott and Albert E. von Doenhoff, Dover Publications, 1959.

The U2DIIF Fortran code referenced in several of the above cases is described in the following thesis, which may be downloaded from the Naval Postgraduate School's website:

Ngai-Huat Teng, THE DEVELOPMENT OF A COMPUTER CODE (U2DIIF) FOR THE NUMERICAL SOLUTION OF UNSTEADY, INVISCID AND INCOMPRESSIBLE FLOW OVER AN AIRFOIL, Naval Postgraduate School, 1987.

Unit tests

Several unit tests are available in the ubem2d/tests directory, which the user may invoke with Python's unittest module in the usual manner. For example:

• python3 -m unittest ubem2d/tests/test_cylinder.py
• python3 -m unittest ubem2d/tests/test_hess_smith.py
• python3 -m unittest ubem2d/tests/test_steady_lift_model.py

In addition, many modules have a __main__ method which may be used to demonstrate their basic usage.