von Kármán swirling flow

This Fortran program solves the second-order, nonlinear ODEs that govern the self-similar von Kármán swirling flow, an interesting textbook exercise.

$$ 2F + H^\prime = 0, $$

$$ F^2 - G^2 + F^\prime H = F^{\prime\prime}, $$

$$ 2FG + G^\prime H = G^{\prime\prime}, $$

The code splits the governing equations into a system of first-order ODEs with the auxiliary variables $A=F^\prime$ and $B=G^\prime$. A block-elimination (Thomas) algorithm is then used to solve the system iteratively, the nonlinearity being treated with the Newton-Raphson method.

The main assumptions:

• steady state, incompressible flow
• infinite plane disk
• azimuthal symmetry
• no rotation at infinity

For Ubuntu/Linux: before running the provided bash script, first install gfortran and LAPACK by running the commands

sudo apt-get install gfortran
sudo apt-get install liblapack-dev

then navigate to the code's directory and allow your machine to execute the script

chmod +x deploy.sh

then run by typing

./deploy.sh

The input.in file is used to specify the number of points and domain width. A uniform grid is employed.

References

Aref, H., & Balachandar, S. (2018). A first course in computational fluid dynamics. Cambridge University Press. ISBN: 9781107178519.

Cebeci, T. (2002). Convective heat transfer. Horizons Pub. LCCN: 2002068512.