Introduction to Finite Difference Methods for Numerical Fluid Dynamics


Evan Scannapieco and Francis H. Harlow

This work grew out of a series of exercises that Frank Harlow, a senior fellow in the Fluid Dynamics Group (T-3) at Los Alamos National Laboratory developed to train undergraduate students in the basics of numerical fluid dynamics. My job during my year at LANL was rather different than his other students, as instead of using these exercises as a launching point for further research, he and I developed them into a self-contained textbook that could be used by students outside of T-3.

The work is written for a student level ranging from high-school senior to university senior. Equations are derived from basic principles using algebra. Some discussion of partial differential equations is included, but knowledge of calculus is not essential. The work does assume, however, some familiarity with the FORTRAN computing language.

Topics examined include one-dimensional heat flow, one-dimensional compressible fluid flow, two-dimensional compressible fluid flow, and two dimensional incompressible fluid flow with additions of the equations of heat flow and the K-epsilon model for turbulence transport. Emphasis is placed on numerical instabilities and methods by which they can be avoided, techniques that can be used to evaluate the accuracy of finite-difference approximations, and the writing of the finite-difference codes themselves.

Concepts introduced include flux and conservation, implicit and explicit methods, Lagrangian and Eulerian methods, shocks and rarefactions, donor-cell and cell-centered advective fluxes, compressible and incompressible fluids, the Boussinesq approximation for heat flow, Cartesian tensor notation, the Boussinesq approximation for the Reynolds stress tensors, and the modelling of transport equations. A glossary is provided that defines these and other terms.

This work was published locally as Los Alamos National Lab Report 12984 and a pdf file version is available here (part 1) and here (part 2) in order to make this document accessible to a wider audience.

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Background: Painting of Santa Fe by Dr. Harlow