Richard L. Fearn

Professor Emeritus, Mechanical and Aerospace Engineering, University of Florida

e-mail: rlf@ufl.edu

February 23, 2018

Wolfram Demonstrations

Möbius Transformations on the Complex Plane and Riemann Sphere

Möbius transformations are considered on both the complex plane and the Riemann sphere. A reference point and adjustable grid are specified in polar coordinates, and the mapping can be varied using eight real parameters specifying a Möbius transformation. Bookmarks are used to save parameters illustrating some interesting properties of the transformation.

Joukowski Airfoil

This collection of three Wolfram Demonstrations are from the computational document Two Dimensional Potential-Flow Aerodynamics described below.

Joukowski Airfoil: Geometry

This Demonstration illustrates the mapping of a circle to a member of the family of Joukowski airfoils. The mapping is conformal except at critical points of the transformation where z'(ζ)=0. This occurs at ζ=±1 with image points at z=±2. The sharp trailing edge of the airfoil is obtained by forcing the circle to pass through the critical point at *ζ**=**1*. The trailing edge of the airfoil is located at *z**=**2*, and the leading edge is defined as the point where the airfoil contour crosses the *x* axis. This point varies with airfoil shape, and is computed numerically. The distance from the leading edge to the trailing edge of the airfoil is the chord, which the aerodynamics community uses as the characteristic length for dimensionless measures of lift and pitching moment per unit span. The shape of the airfoil is controlled by a reference triangle in the *ζ* plane defined by the origin, the center of the circle at and the point *ζ**=**1*.

Joukowski Airfoil: Aerodynamic Properties

The surface pressure distribution for potential flow over a member of the Joukowski family of airfoils is presented in the format conventional for airfoil aerodynamics; the characteristic length is the chord of the airfoil and pressure is presented as a dimensionless pressure coefficient. The shape of the airfoil is included as an insert. Slider controls are provided for the shape of the airfoil and the angle of attack.

For typical conditions, the upper part of the curve corresponds to the upper surface of the airfoil and the lower curve to the lower surface; color-coding is used to associate a part of the airfoil with the appropriate part of the pressure distribution. The area enclosed by the surface pressure coefficient curve is an estimate of the two-dimensional lift coefficient. Blasius’s equations are used to compute lift and pitching moment characteristics, which are summarized by lift coefficient, angle of attack at zero lift, slope of lift coefficient curve, pitching moment about the quarter-chord and location of the aerodynamic center.

Joukowski Airfoil: Flow Field. Potential flow over a Joukowski airfoil is one of the classical problems of aerodynamics. This Demonstration plots the flow field using complex analysis to map the simple known solution for potential flow over a circle to flow over an airfoil shape. Sliders control the angle of attack and the shape of the Joukowski airfoil. The flow field is illustrated by plotting streamlines and a color coded pressure field.

Potential Flow Over a NACA Four-Digit Airfoil

A panel method from the package Aerodynamics 1.2 is used to compute the flow field 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

A panel method from the package Aerodynamics 1.2 is used to compute the flow field and aerodynamic properties for steady two-dimensional potential flow over a selection of fifteen airfoils taken from the UIUC Airfoil Data Site.

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. Each document is available for free download by selecting the links in the headings below. You will need to download the free Wolfram CDF Player or have access to a Mathematica license to use these documents.

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

A list of documents and major sections are 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. Five interactive figures illustrate:

modular surfaces.

Pólya vector fields for a selection of complex functions.

Pólya vector field for a combined source and vortex.

complex mapping for a selection of complex functions.

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. Six interactive figures illustrate:

geometry of a Joukowski airfoil.

potential flow over a translated circle.

annotated free-body diagram for Joukowski airfoils.

surface pressure distribution for Joukowski airfoils.

region plots to guide the formulation of the inverse Joukowski transformation.

flow field for potential flow over Joukowski airfoils.

December, 2010

This collection of Mathematica packages 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 free download from the Wolfram Library Archive.

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). This chapter about the University of Florida was written in collaboration with Wei Shyy. Resource material for the chapter is located in University of Florida Archives.