Richard L. Fearn

Professor Emeritus, Mechanical and Aerospace Engineering, University of Florida

e-mail: rlf@ufl.edu

Last update: March 13, 2017

Two-Dimensional Potential-Flow Aerodynamics

January 15, 2017

This collection of documents uses interactive graphics powered by Mathematica to illustrate the use of complex analysis to solve the classical problem of potential flow over Joukowski airfoils. Documents are available for free download by selecting hyperlinks below. You will need the free Wolfram CDF Player or have access to a Mathematica license to use these documents.

The complete collection is available from this site as a compressed file, or you can download each document individually by using the links in the headings below. This collection of documents is also available from the Wolfram Library Archive.

Some conventions of Mathematica and assumptions about the reader’s background are discussed.

A list of documents and major sections is provided with hyperlinks to serve as a roadmap.

A brief and elementary review of vector fields includes illustrations using Mathematica functions.

2 Introduction to Complex Analysis

The basic tools of complex analysis used in airfoil aerodynamics are discussed. Interactive figures illustrate some of the primary concepts:

1. modular surfaces

2. Pólya vector fields for a selection of complex functions

3. Pólya vector field for a combined source and vortex

4. complex mapping for a selection of complex functions

5. mapping at critical points

3 Review of 2-D Potential Flow

A brief review of potential flow emphasizes the use of complex analysis and includes an interactive graph for potential flow over a centered unit circle with circulation.

4 Potential Flow Over Joukowski Airfoils

The tools of the previous three sections are applied to the problem of potential flow over Joukowski airfoils. Interactive figures illustrate some of the major steps in the solution.

1. geometry of a Joukowski airfoil

2. potential flow over a translated circle

3. annotated free-body diagram for Joukowski airfoils

4. surface pressure distribution for Joukowski airfoils

5. region plots guiding formulation of the inverse Joukowski transformation

6. streamlines and pressure field for potential flow over Joukowski airfoils

Aerodynamics 1.2

December, 2010

Aerodynamics 1.2 provides a collection of computational tools written in the Wolfram Language for solving steady two-dimensional potential-flow over an airfoil using panel methods. Over thirty functions are defined to facilitate the computation of airfoil geometry, influence coefficients, aerodynamic coefficients and presentation graphics. Additionally, three panel methods are fully implemented. Documentation is provided for defined functions along with a tutorial. Users have free access to the Mathematica packages and can modify the programs. This collection of Mathematica packages is available for download from the Wolfram Library Archive.

Wolfram Demonstrations Project

Joukowski Airfoil: Geometry is the first of three demonstrations on the family of Joukowski airfoils. These three demonstrations provide a preview of the types of interactive figures in the collection of documents Two-Dimensional Potential Flow Aerodynamics described above. This demonstration illustrates the mapping of a circle to a member of the family of Joukowski airfoils, and allows you to control the two parameters for the shape of the airfoil.

Joukowski Airfoil: Aerodynamic Properties shows the surface pressure coefficient with controls for the shape of the airfoil and angle of attack.

Joukowski Airfoil: Flow Field shows the flow field in terms of streamlines and color coded pressure coefficient with controls for the shape of the airfoil and angle of attack.

Potential Flow Over NACA Four-Digit Airfoil is the first of two demonstrations illustrating the capabilities of the packages in Aerodynamics 1.2. This demonstration computes flow visualization and aerodynamic properties for steady two-dimensional potential flow over a member of the family of NACA four-digit airfoils.

Potential Flow Over an Airfoil Specified by Numerical Data File computes flow visualization and aerodynamic properties for steady two-dimensional potential flow over a selection of fifteen airfoils included on the UIUC Airfoil Data Site.

Aeronautical and Aerospace Engineering at the University of Florida

Book Chapter, 2003

This is a chapter in the book Aerospace Engineering Education During the First Century of Flight edited by Barnes McCormick, Conrad Newberry and Eric Jumper (2004). The chapter Aeronautical and Aerospace Engineering at the University of Florida was written in collaboration with Wei Shyy. Resource material for this chapter is located in University of Florida Archives.