FOR AOE 3014
The following computer programs were
obtained from several sources and several have been modified to update
them and to make them work better on the PC. Most of the programs are
written in BASIC but a couple are written in FORTRAN. The user should note
that some variations in the program formats may be needed depending on the
versions of BASIC or FORTRAN installed in one's computer. Matlab versions
of THINFOIL, AIRFOIL, MONOPLEQ and PANEL are also provided (Matlab Version
These programs are all
fairly simple examples of the types of aerodynamics codes generally
employed in industry today. These are all inviscid codes, meaning that
boundary layer influences ( especially separation ) are not accounted for
in the programs. In general, adding a boundary layer solution to these
codes requires an iterative process in which the inviscid code is run, the
pressures from that run are used to compute the nature of the boundary
layer and the "momentum thickness" of the layer, which is then used to
find an artificially displaced "surface" for which a new inviscid solution
is determined. This process is repeated until the results of the momentum
thickness solution cease to change in successive calculations.
Most of these programs are
available in the College of Engineering computer lab in Randolph Hall and
may be copied from the disk there.
1. THINFOIL: A 2-D program in
BASIC based on classical thin airfoil theory but limited to a single,
pre-defined airfoil camber line equation.
2. PANEL: A "Smith-Hess"
type of 2-D panel code combining source panels and vortices for a
single-element, lifting airfoil in incompressible flow.
AIRFOIL: A FORTRAN program for a vortex panel method used for 2-D airfoils
in incompressible flows. Airfoil is available in both versions: F77 &
4. MONOPLEQN: A BASIC code which computes the lift and induced
drag on a 3-D wing using clasic lifting line theory. It is subject to the
normal sweep and aspect ratio limitations of lifting line theory.
5. VORLAT: A simple vortex lattice code in BASIC for a 3-D planar
wing of any sweep or aspect ratio.
6. DELTAWING: A very simple
BASIC program to evaluate the lift and dragt on a delta wing using
Polhamus' leading edge vortex theory data.
THINFOIL is a code based on
classical "thin-airfoil" theory derived from potential flow methods. This
theory assumes that the airfoil can be treated as a "vortex sheet" placed
on a camber or "mean" line through which no flow may pass. The theory
solves for a satisfactory distribution of vortex strength density along
the sheet using the no-flow criteria at the sheet itself and the Kutta
Condition which fixes the vortex strength or circulation as zero at the
The needed input to
the program is a definition of the camber line and the airfoil angle of
attack. Since all output is given in terms of coefficients, the velocity
is not needed.
programs use the camber line equations inherent in the NACA airfoil
designations. This program uses a camber line definition given by a cubic
equation which can approximate the NACA cambers with reasonable accuracy:
z = 4h [x - (k + 1) x^2 + kx^3]
In this equation all
dimensions are given as a fraction of the chord. Z is the height of the
camber line above the chord line and is zero at the leading and trailing
edges. X is the distance along the unit chord line ( zero at the
leading edge and unity at the trailing edge ). K determines the change in
curvature or slope of the camber line with a value of zero giving a
circular arc airfoil and positive values giving a reflexed trailing edge.
H determines the magnitude of the maximum camber as a fraction of
The program also allows
for the addition of a symmetrical trailing edge flap by specifying its
length as a fraction of the airfoil chord and its deflection angle
relative to the chord line.
program will present a plot of the defined camber line (with flap) and
values for the lift coefficient, the zero-lift angle of attack and the
pitching moment coefficient about the aerodynamic center ( always the
quarter-chord for this theory ) using simple equations which result from
using the above camber line formula in the classic thin airfoil solution:
Cl = 2*Pi*alpha + (4 -
3k)Pi*h alphaLo = - (2 - 3k/2) h
Cmac = (7k/8 - 1)
= 1/4 - Cmac/Cl
A second screen
will display the computed "pressure distribution" based on the difference
in the pressures above and below the camber line. This will always go to
infinity at the leading edge.
Program PANEL is based on the type
of two-dimensional airfoil code classed as a "Smith-Hess" panel method.
Smith-Hess codes utilize a combination of "source" panels and either
vortex panels or a system of discrete vortices placed at a critical place
on the airfoil. The method normally uses constant strength source panels
to define the shape of the airfoil; ie. to set the "no-flow" condition at
the desired airfoil surface. Vortex panels or individual vortices are
added to the equations to meet the Kutta Condition. When vortex panels are
used they are usually coincident with the source panels and are defined
such that all panels have the same vortex strength and the strengths sum
to that needed to place the rear stagnation point at the trailing edge by
requiring that the velocities tangent to the upper and lower rearmost
panels be equal. A simplier approach might use a single vortex of a
strength sufficient to make the rear panel tangential velocities equal at
the quarter chord. This code uses the latter approach.
This code includes a subroutine
defining the surface for NACA four or five digit airfoil shapes and
automatically placing panels on that surface. The program will plot the
chosen airfoil shape and the pressure coefficient distribution.
AIRFOIL is a very versatile vortex panel code which forms the basis for
numerous 2-D aerodynamics programs used in industry and research. It uses
vortex panels on which the vortex strength varies linearly from one end of
the panel to the other. The code solves for the values of the vortex
strength at the panel end points using the "no-flow" through the panel
requirement and the Kutta Condition. The Kutta Condition is met by
requiring the vortex strengths at the end of the first and last panels (
at the lower and upper part of the trailing edge ) to sum to zero.
This is a FORTRAN code which
computes the velocity and pressure coefficient values at the center point
of each panel. The data printout gives the x and y location of each panel
center or control point, the slope of each panel, the length of each
panel, the values of the vortex strength at each panel endpoint, and the
velocity and pressure at the control point. For an airfoil with n panels
there are n values of all parameters printed ( at the panel control points
or centers ) and n + 1 values of gamma printed ( one at eacn panel end
point [ not at the centers ] ).
Care should be taken in the interpretation of the printed results. As
usual, all lengths and distances are given as a fraction of the chord.
It is particularly important to note that the GAMMA value given is
not the actual vortex strength but is the vortex strength divided by
2*Pi*Vinf. The velocity values given are also normalized by the free
airfoil solutions are given, one for the NACA 4412 airfoil and the other
for a Wortmann airfoil ( a very good low speed shape ). The sample case
shown in the program itself is for a 12 panel airfoil but the sample
pressure plots given show the effects of increasing the number of panels.
In redimensioning the program to accomodate a wing with more than 12
panels remember that there will always be one more gamma value solved for
than the number of panels used.
This BASIC computer program uses
the classical monoplane equation from lifting line theory to solve for the
lift and induced drag coefficients and the spanwise load distribution on a
specified wing ( 3-D ). Lifting line theory assumes that a wing's behavior
at any spanwise location where the equations are solved is essentially
two-dimensional ( no spanwise flow ). It is also based on a model using a
bundle of lift inducing vortices placed at the unswept quarter chord of
the wing. The method is, therefore, only valid for wings with unswept
quarter-chord lines and with moderate-to-high aspect ratio.
The program is set up for the
fairly standard four term solution ( solving the monoplane equation at
four points along the span ) at values of f of 22.5, 45, 67.5 and 90o . If
more accuracy is needed the program may be modified fairly easily to solve
a N x N matrix instead of a 4 x 4.
The program allows specification of
the wing span, root chord and tip chord, the twist of the wing tip
relative to the root ( in degrees ), the airfoil section lift curve slope
and the section zero-lift angle of attack. The wing angle of attack is
defined as relative to the root chord. The wing twist is assumed to be
linear from root to tip with a "nose-up" twist considered positive.
This is a
simple version of a vortex lattice code which assumes a planar wing. The
code essentially determines the effect of altering a wing's planform and
can handle any type of linear wing taper and sweep. This code does not
account for wing camber or twist or other geometric effects. It; therefore
represents the simpliest possible example of a vortex lattice program but
is, nonetheless, useful for demonstrating the ability of the vortex
lattice method to handle the effects of sweep and low aspect ratio which
cannot be determined from the classical lifting line approach.
The code assumes a wing similar to the
one in the figure below using 12 panels and 12 control points. Using
symmetry, the program solves for only six unknown vortex strengths. Using
the input data consisting of the leading edge sweep and the tip chord and
root chord as fractions of the span, the program divides the wing into 12
panels, defines the panel coordinates and the coordinates for the control
points at the three-quarter chord point at the center of each panel. It
also determines the coordinates of the end points for the vortex along the
quarter chord of each panel.
calculations are for the left wing only and the normalized vortex
strengths are for panels 1 - 6 on that wing. Note that the vortex
"strengths" are really gamma divided by the product of four Pi, the span,
the free stream velotity and the angle of attack in radians. A rough
spanwise load distribution can be found by using the calculated vortex
strengths at the spanwise center of each of the three pairs of panels (
1&4, 2&5, 3&6 ) and finding the "local" lift from:
modifications, the program will handle different arrays of panels from the
2x6 used here. The limit depends on the computer's ability to invert the
matrix. Changes must be made in lines 100, 110, 140, 640 and 970 of the
is a very simple code for calculation of the lift and drag coefficients of
a delta wing at low Mach number. It is a direct application of the
Polhamus leading edge suction analogy which is described in Chapter 7 of
Bertin and Smith.
Send comments and
suggestions to Joe Honaker