**PANDA -- A Program for Analysis and Design
of Airfoils**

Version 1.4

by Ilan Kroo

© Desktop Aeronautics

P.O. Box A-L

Stanford, CA 94309

**Table of Contents**

1. Introduction | 3 | |

2. Basic Definitions and Methodology | ||

2.1 Terminology and Coordinates | 4 | |

2.2 Theory Summary | 5 | |

3. Using the Program | ||

3.1 Running PANDA |
8 | |

3.2 The Input File | 10 | |

3.3 The Menu Options | 12 | |

3.4 The Dialog Choices | 14 | |

3.5 Hidden Commands | 15 | |

4. Comparisons with Experiment and Theory | ||

4.1 Pressure Distributions | 16 | |

4.2 Lift and moment | 18 | |

4.3 Drag Polars | 20 | |

5. Hints and Limitations | ||

5.1 Using the Results -- Some Suggestions | 22 | |

5.2 Approximations | 23 | |

6. Other Programs | 24 | |

7. References& | 24 | |

8. Update Policy and Technical Support | 25 |

*PANDA* is a Program for Analysis and Design of Airfoils. It computes
and graphically displays the pressure distribution on airfoil sections in
subsonic flow. For a particular airfoil with coordinates stored in a standard
text file, the program calculates the inviscid pressure distribution over
the airfoil at a specified angle of attack and Mach number; lift and pitching
moment about the 1/4-chord point are also computed. The analysis is done
with remarkable speed (less than a second) so that the effects of changes
in angle of attack or airfoil geometry can be studied easily.

The program also computes the boundary layer properties based on this inviscid pressure distribution. The location of transition, laminar or turbulent separation, and total drag are computed based on integral boundary layer methods. It is possible to specify a position for "transition grit" on the upper and lower surfaces to force transition or model surface roughness.

A major feature of this program is its provision for rapidly changing
the airfoil geometry. This is done by positioning the cursor over the part
of the airfoil to be changed and clicking the mouse button. A smoothly-faired
bump (with specified but editable height and width) is added to the section
at this point and the new C_{p} distribution is quickly redrawn.
In this way the airfoil can be rapidly reshaped to produce a desirable C_{p}
distribution.

** PANDA** is based on a relatively simple theoretical foundation;
it does not substitute for wind tunnel studies, Navier-Stokes programs,
or other sophisticated analysis tools. It does, however, provide rapid answers
useful in initial design studies and in educational applications. Its accuracy
can best be assessed by comparing results with known (experimental) results.

This manual describes the basic use of the program, the theory on which the calculations are based, and several examples illustrating the accuracy of the method in various applications. The program is very easy to use, but the calculations are complex and there are many subtleties. Please read this manual before running the program. (The theory section is optional.)

**2.1. Terminology and Coordinates**

The following nomenclature is used throughout this manual and in the PANDA program:

**C**_{L} Lift coefficient = Section Lift / ( q_{0}
c_{ref})

C_{D} Drag coefficient = Section Drag / ( q_{0} c_{ref})

C_{m} Moment coefficient = Moment about quarter chord point /
( q_{0} c_{ref}^{2} )

C_{p} Pressure coefficient = (Local pressure
- Freestream static pressure ) / q _{0}

C_{p}^{*} Critical pressure coefficient, local Mach number
= 1.0

C_{pi} Incompressible pressure coefficient

c_{ref} Reference chord length (assumed 1.0 in ** PANDA**)

H Boundary layer shape factor =d^{*} / q

LE Leading edge

M_{0} Freestream Mach number = U_{0} / sound speed (Must
be < 1.0 in ** PANDA**.)

q_{0} Freestream dynamic pressure = .5 * r
* U_{0}^{2}

Re Reynolds number based on chord and freestream velocity = U_{0 }c_{ref}
/ n

Re_{q} Reynolds number based on momentum
thickness = Uq / n

TE Trailing edge

U_{0} Freestream velocity

U Velocity near the airfoil surface (but outside the boundary layer)

x Airfoil chordwise coordinate

y_{u}, y_{l} Airfoil upper and lower surface ordinates

a (or alpha) Angle of attack measured from chord line to freestream direction

d^{*} Boundary layer displacement
thickness

r Air density

n Fluid kinematic viscosity

q Boundary layer momentum thickness

The airfoil coordinates are measured from an origin at the leading edge and are scaled so that the trailing edge is located at x = 1.0. Within the program all quantities are dimensionless. Only when dimensional values, such as section lift, are computed from the program results does one need to specify a density or freestream velocity.

**2.2. Theory Summary**

*Inviscid Pressures:*

Many methods are available to compute the inviscid pressure distribution
over an airfoil. These include those based on complex variables such as
the conformal mapping method discussed by Jones^{1},
singularity panel methods such as that of Hess^{2}
or direct integration of the partial differential equation by finite difference
methods^{3}. ** PANDA** uses a method
based on superposition of sources and vortices. Like thin airfoil theory,
this method does not actually place singularities on the airfoil surface,
but nor does it linearize the boundary conditions in the usual way. The
method is attributed to Riegels and is described by Weber in references
4,5. This method achieves a computational efficiency similar to thin airfoil
theory, but does not produce (incorrect) singularities near the leading
edge. In fact, Riegels' method produces results that agree with exact solutions
in the case of ellipses and are very close to exact results for most airfoil
shapes. (See the discussion of the accuracy of the method in section 4.)
One limitation of the method is that it assumes that the airfoil leading
edge lies at x = 0, yu = 0, yl
= 0 and the trailing edge is at x = 1, yu = 0, yl = 0. In this version of the program, then, airfoils with
blunt trailing edges are not permitted.

Once the velocities, consistent with the differential equation for incompressible potential flow and the boundary and Kutta conditions, have been computed, the pressure coefficient is calculated from the Bernoulli relation:

where U is the local velocity near the airfoil surface.

The airfoil lift and moment coefficients are computed by integrating the pressure coefficient over the airfoil surfaces.

*Compressibility effects:*

The Karman-Tsien compressibility correction is a nonlinear approximation for Mach number effects which works quite well when the local velocities are subsonic. This expression relates the incompressible Cp values to those in compressible flow. The relation is:

This simple correction connot be expected to apply when the local velocities
approach sonic speeds. This situation will be apparent from the pressure
distribution plot since the value of C_{p}
corresponding to sonic velocity is drawn on the plot. This value is called
the critical pressure coefficient and is denoted C_{p}^{*}.

The boundary layer characteristics and drag computations are based on the incompressible velocities and are strictly incompressible. Future versions of the program may include compressibility effects on skin friction and transition. (Register for information on upgrades.)

*Boundary Layer Calculations:*

The boundary layer will have some effect on the pressure distribution
over the airfoil. While the boundary layer properties are computed in ** PANDA**,
the pressure distribution is not recalculated. The computed pressures are
thus inviscid and best representative of high Reynolds number flows. (See
section 4 for further discussion.)

The characteristics of the boundary layer are determined using so-called integral methods. The boundary layer begins as a laminar layer with properties determined by Thwaites' method. This method is discussed in reference 6 and involves a direct integration to obtain the momentum thickness together with empirical correlations for the displacement thickness and skin friction coefficient. The method also warns when laminar separation is predicted.

As the boundary layer thickens, it becomes less stable and eventually
transitions from laminar to turbulent flow. This point is either specified
by the user or is computed using Michel's transition criterion (Reference
7). This criterion is an empirical relationship between the Reynolds number
based on momentum thickness, Re_{q}
= Uq/n, and the Reynolds
number based on local position and velocity, Re_{x} =
U x / n. The relationship is:

and holds for values of Re_{x} between 10^{5} and 40 x 10^{6}. It
is therefore not suitable for use with very low Reynolds number airfoils
and other means should be used to determine the location of transition.

When the boundary layer has become turbulent, either because it has encountered transition "grit" or because it has naturally transitioned, the momentum thickness and displacement thickness are computed by another semi-empirical method known as Head's method. This method involves the solution of two simultaneous partial differential equations: the momentum equation, and an equation describing entrainment. The method is discussed in reference 8. It is a simple(istic?), but very fast approach. The boundary layer properties start with the values at the end of the laminar run and are computed by marching downstream to the trailing edge, or to the point where turbulent separation is predicted. The separation criterion used here is based on the value of the shape factor, H, the ratio of displacement thickness to momentum thickness: H = d* / q. When H exceeds 2.2, the flow is assumed to be separated and results should generally be regarded as meaningless.

*Drag:*

After the upper and lower surface boundary layer properties have been computed, the total drag is estimated by the Squire-Young formula. This formula relates the drag to the properties of the boundary layer (q, H, and U) at the airfoil trailing edge. It is discussed in detail in reference 9.

These methods are described in detail in several other textbooks. In
particular, this program was inspired by the wonderful collection of methods
and TIñ59 programs published by W.H. Mason, reference 10.

PANDA uses many elements of the standard Windows user interface. If you are familiar with other Windows programs you will have no trouble using this program. If you are not familiar with the concepts of selecting menu items, clicking on buttons, entering numbers in "dialog boxes", or editing files on the disk, this manual will not be sufficient to get you going -- please look over the manual that came with Windows or with other common utility software such as Paint or Write.

** **

Basic analysis procedure:

To run PANDA, start Windows and then click twice on the file PANDA.EXE just as you would any Windows program. A title and copyright notice will appear. Click with the mouse on the "OK" button or type Return. Another note will appear, informing you that the initial calculations are in progress. When the dialog box disappears in just a few seconds the program is ready to run. The first step is to select an airfoil to analyze. This is done by selecting Open... from the File menu. The standard file selection dialog box will appear and you should select the file with the airfoil coordinates of interest. (See the following section for a description of this file format.)

The calculations are done almost instantly and the airfoil shape and
pressure distribution are drawn on the screen. Note that the pressure distribution
is represented in the standard format -- the pressure coefficient,
Cp, is plotted on an inverted scale (more negative values of Cp appear higher
on the plot). On color monitors, the upper surface pressures are plotted
in blue, the lower surface in red. It is possible to save the numerical
values to a disk file for later examination or more detailed plotting. This
is done by selecting **Save** **Pressures...** from the **File**
menu. The resulting file is an ASCII text file which may be printed or read
with most any text editor program. It is written in a format compatible
with many plotting programs.

To see the effect of changes in angle of attack, select **Change Parameters
... **from the **Commands **menu. A dialog box appears which allows
you to change several parameters describing the airfoil and the flow conditions.
The angle of attack is selected initially and to change its value just type
in a new number (in degrees), then press the Return key or click the mouse
on the OK button. The pressures will be recomputed and replotted. Note that
you may change the angle of attack quickly by using the keyboard shortcuts
to select Change Parameters, typing the new angle, and then pressing Return.

Changes in the freestream Mach number are made similarly except that you must select the Mach number input field in the dialog box either by dragging the mouse over the current value or by pressing the Tab key which advances the currently selected field to the next position.

To compute the boundary layer properties on the upper and lower surfaces
of the airfoil, select **Boundary Layer** from the **Commands **menu.
The boundary layer equations are solved and the results are displayed as
follows: The letter 'T' stands for transition and is drawn above or below
the chord line (corresponding to upper and lower surfaces) at the location
of laminar to turbulent boundary layer transition. 'LS' and 'TS' may be
drawn similarly if laminar or turbulent separation is predicted. (Note that
if laminar separation is indicated and turbulent separation is not, this
implies that the flow has reattached but the accuracy of this prediction
is rather dubious due to the simplicity of the model.) The total drag is
calculated based on these properties and is shown in the form of a drag
coefficient.

The Reynolds number may be changed in the same way as Mach number and angle of attack, and the transition location on each surface may be changed by adding 'transition grit'. When the program begins, this grit is located at x/c = 1.0, the trailing edge. By moving the grit forward (by changing the number in the dialog box to a number less than 1.0) you can force transition to occur farther forward on the airfoil.

*Airfoil Design:*

You may modify the airfoil geometry by editing the file containing the
coordinates, or by modifying an existing section as follows: Position the
cursor on the airfoil at a point where you wish to change the surface shape
(on either the upper or lower surface). Click the mouse button and a 'bump'
will be added to the section at that point. The bump is formed from two
cubics which fair smoothly into the existing airfoil. The height of the
bump and the half-width may be specified in the parameter dialog box. The
default height is .001 (.1% of the chord length) which causes a small change
in the pressures; for coarser changes increase this to .002 or even .005.
Every time that the mouse button is clicked, another bump is added to the
current section definition so that clicking twice with a height increment
of .002 produces the same effect as adding a single .004 bump. You may remove
a bump (add a dent) in three ways: by specifying a negative bump height,
by holding down the Shift key while clicking the left mouse button, or by
using the right mouse button. In general, adding a bump tends to reduce
the pressure (raise the Cp curve) locally with small effects on other parts
of the airfoil pressure distribution. An exception is near the nose where
small changes in geometry can have more complex effects on the pressures.
Experiment! You should not have to run the boundary layer calculation after
each change; you will soon learn how to change the pressures to improve
airfoil performance.

** **

You must begin with a table of airfoil coordinates or use PANDA to generate a starting section (see Menu Options). You may wish to analyze a particular section or to design a new one, but in either case the program needs a place to start. A sample table is provided in the included sample input files. The format of this input file is illustrated in the following example:

! Modified NACA 4412 ! Xu Yu Yl 0 0 0 .0125 .0244 -.0143 .025 .0339 -.0195 .05 .0473 -.0249 .075 .0576 -.0274 .1 .0659 -.0286 .15 .0789 -.0288 .20 .0880 -.0274 .25 .0941 -.0250 .3 .0976 -.0226 .4 .098 -.0180 .5 .0919 -.0140 .6 .0814 -.01 .7 .0669 -.0065 .8 .0489 -.0039 .9 .0271 -.0022 .95 .0147 -.0016 1.0 0 0 end

First, note that any blank line or line beginning with an exclaimation mark is regarded as a comment. The actual data must be specified in three columns. The first column is the chordwise position, x/c; the second is the upper surface coordinate, yu/c, at this x position; and the third column is the lower surface coordinate, yl/c, at the same x coordinate. The coordinates must start at the leading edge, x = 0, yu = 0, yl = 0, and end with the coordinates of the trailing edge: x = 1, yu = 0, yl = 0. Note that the columns may be separated by any number of spaces. It is important that a Return character is placed at the end of each line in the file. Finally, place the word END at the beginning of the the last line of the input file as shown above.

The input file is a standard text file and may be created using any of several text editors. The program NotePad, included with Windows, works well for this purpose.

**3.3 The Menu Options**

**File**

**New** Resets everything and lets you start from the beginning.

**Open...** Prompts for the name of a ** PANDA** input file
to be read from the disk.

**Save Coordinates...** Prompts for the name of a file in which to
save the current airfoil coordinates. The file may be edited later and
used as an input file for ** PANDA** or a plotting program.

**Save Pressures...** Prompts for the name of a file in which to
save the airfoil coordinates and pressure distribution for later reference
and plotting.

**Print** Sends the current drawing or text on the screen to the
printer.

**Quit** Terminates the program. Note that you will not be asked
if you want to save the results or the geometry before quitting.

**Commands**

**Change Parameters... **presents a dialog box which permits the
user to change angle of attack, Mach number, Reynolds number, position
of forced transition, and parameters related to interactive geometry modification.
(See discussion in the following section.)

**Compute Boundary Layer **computes the boundary layer properties
and displays the total drag and the location of transition and separation.

**Save Polar...** allows the user to specify a range of angle of
attack to be analyzed. After the minimum, maximum , and a
increment have been specified, the program will prompt for the name of
a file in which to save the results.

**Set Resolution...** The program fits the airfoil specified in the
input file with a spline and then computes the coordinates and pressures
at certain points. The number of points at which these computations are
made is set to 20 initially. This menu item enables the user to specify
the computational resolution as a number between 10 and 40. This might
be desirable if more accurate answers are needed.

**Erase Before Redrawing** Normally, the program erases the previous
pressure distribution plot whenever a change is made to the airfoil. It
is sometimes of interest to see how the change in the airfoil changes the
pressures. This option may be used to keep previous results from being
erased on the screen. (Try it.)

**Modify Thickness Only** When this item is checked, clicks on either
upper or lower surface add thickness (according to the user-specified bump
height and width) to both surfaces. In this way the camber distribution
is preserved. This is especially useful in the design of symmetrical airfoils.

**Modify Camber Only** This option is similar to the one above except
that clicks on the airfoil do not affect the thickness. This is very useful
when some part of the section has a thickness constraint and should not
be changed.

**Flap Deflection...** Selecting this item causes a dialog box to
be displayed so you may specify the flap chord (as a fraction of the total)
and deflection (in degrees). The program modifies the airfoil shape accordingly.
Note that a plain flap is assumed; the program simply modifies the section
coordinates.

**Generate NACA Section...** This option allows you to generate one
of the NACA four or five digit airfoils without typing in the coordinates
in an input file. Currently the program generates all of the 4-digit sections
and the 23xxx family of 5 digit sections.

** **

**3.4 The Change Parameters...Dialog Choices**

This dialog box allows you to change several parameters describing the airfoil and the flow. These include the following:

- Angle of attack (in degrees)
- Chord Reynolds number, Re, for boundary layer computations
- Location of transition grit which forces boundary layer transition to turbulent flow
- Half-width and height of bi-cubic "bumps" which may be placed on the airfoil to obtain desired changes in the pressure distribution
- Freestream Mach number

When a field is highlighted (as is the case for Angle of attack below) typing will replace the highlighted text. To advance to the next field, click the mouse button when the cursor is over that field, or press the tab key.

To continue place the cursor on the OK button and click the mouse button, or just type the return key. The values shown above are the default values.

**3.5 For "Power Users" -- Hidden Commands**

A few additional commands are available from the keyboard but not from the menus. Pressing these keys causes the command to be entered immediately. (You need not type Return.) These shortcuts include the following:

**a** Increase the angle of attack by 0.5 degrees.

**A** Decrease the angle of attack by 0.5 degrees.

**l,r,u,d** Move the plot of the airfoil and pressure distribution
left, right, up, or down by a small amount. Press these keys repeatedly
to maneuver the plot to a desired position on the screen.

**i,o** Zoom in or out. This is especially useful when making small
adjustments to the leading or trailing edges.

**n** Normal scaling returns from a zoom or shift in scale.

**B** Compute boundary layer on each analysis (typing this a second
time turns this feature off).

There are also a few tricks for the mouse:

- Click on the maximum or minimum Cp value shown in the axis label (the numbers on the left of the Cp plot) to change the scaling. Holding down the shift key while doing this (or using the right button) changes the value in the other direction. If things get confusing, remember to type an 'n' to restore the normal scaling.
- Hold down the Ctrl key while clicking somewhere on the Cp plot and PANDA will integrate the boundary layer equations to compute the Cp's associated with a constant shape factor pressure recovery. This will show the Cp distribution that can be tolerated without separation. It assumes a fully turbulent boundary layer up to the point at which you click.

** **

*PANDA* does a remarkably good job in predicting the pressure distribution
over airfoils. Although it is based on thin airfoil theory, the Riegels
correction permits accurate calculation of pressures even on a 100% thick
section (a circular cylinder). This test case enables us to assess the accuracy
of the program mathematically since the exact solution for inviscid flow
over a cylinder is known. Results are shown in the plot below. The small
disagreement near the trailing edge is due to numerical difficulties associated
with the spline fit of the coordinates there.

Results for a more conventional airfoil are compared in the plot below.
The airfoil was generated and analyzed by the method of conformal mappings
discussed in reference 1. The coordinates were then saved and used as an
input file for ** PANDA**.

The only descrepancy in these results occurs right at the trailing edge
where the conformal mapping method predicts a stagnation point. In real
flow no such stagnation point exists and ** PANDA** better approximates
the actual pressures.

In practice, results will not be this close because viscous effects modify
the flow field over the airfoil. ** PANDA** calculations are compared
with measured data and with the standard Theodorsen method

It may be seen that when the angle of attack is adjusted so that the
predicted and measured lift coefficients match, the predicted pressure distribution
is quite accurate. Without modification of the angle of attack to account
for viscous effects ** PANDA** overestimates the pressure variations,
especially near the nose.

The comparison of pressure distributions with other theories and experiments
suggests that the integrated values of lift and moment will agree well with
inviscid theory but will be larger than the measured values. This is indeed
the case and the following plots illustrate the degree to which viscous
effects change the predicted lift vs. angle of attack. It should be noted
that the values of lift at zero angle of attack and the moment at a specified
lift are predicted quite well by this method. Viscosity manifests itself
mostly through an effective change in angle of attack. The plots on the
following page show results for the NACA 65_{2}-015
airfoil from Abbott and von Doenhoff^{11}.

Drag is a difficult quantity to compute. It is generally two orders of
magnitude smaller than the lift and is very sensitive to boundary layer
properties, especially near the trailing edge. For this reason, computed
drag polars from different programs often disagree. Results from ** PANDA**
compare favorably with much larger programs such as LBAUER (Ref. 12). The
only means of assessing a program's accuracy in this respect is by comparison
with experimental results. Unfortuately, it is also difficult to obtain
accurate measures of drag even in a wind tunnel and one must be careful
in accepting experimental values. With these caveats, we compare drag polars
predicted by

Results agree very well, including the laminar drag bucket at low Cl and the onset of separation at a Cl of about 1.4. The width of the drag bucket is underestimated a bit, but the drag levels are close in both regions. This calculation was made with a resolution of 40 points, with free transition.

The second example shows one of the poorer comparisons and illustrates that one must be careful about applying these results with too much confidence. This section is the NACA 23015. The not-so-good agreement is attributed to several possible factors including:

1. The tunnel turbulence level or model construction may have been such that the model did not achieve the extent of laminar flow that was predicted and might otherwise have been possible.

2. The abrupt closure at the trailing edge causes problems for ** PANDA**
since the boundary layer properties change quickly in this region and small
amounts of trailing edge separation are predicted over much of the polar.
Drag results are printed even when some separation is predicted near the
trailing edge, but the results must be regarded with great skepticism.

Results were obtained with 40 point resolution. Free transition (laminar) and forced transition (grit at x/c = .02) cases are shown.

** **

PANDA may be used to analyze a particular airfoil whose coordinates are published in an airfoil catalog or to design a new section. Rarely will an airfoil section be available that exactly meets the requirements of a new design; the best approach may be to start with a section that is close and make minor modifications with this program. The design of good airfoil sections is an involved topic and this section will not show you how to design airfoils (see reference 13 for an excellent discussion), but here are a few fundamentals.

Adverse pressure gradients cause transition and, eventually, separation. Adverse gradients are regions of increasing pressure represented by a downward sloping curve in the Cp plot. Favorable gradients, on the other hand, stabilize the boundary layer and promote low drag, laminar flow. Note that the farther a favorable gradient extends along the airfoil, the more severe the adverse gradient near the trailing edge. The maximum extent of favorable gradient can be achieved with a concave presseure recovery region (an adverse gradient which begins with a steep slope and gradually flattens out near the trailing edge.) A disadvantage with this sort of pressure recovery is that separation quickly spreads forward on the airfoil. Convex recovery sections may separate near the trailing edge with the separated region slowly increasing with angle of attack. This leads to a much more gradual trailing edge stall.

One must be careful not to depend too strongly on laminar flow. Some experimentation will show that laminar flow over much of the airfoil not only reduces drag, but increases the maximum lift that can be obtained before separation. If flight conditions prevent laminar flow from occuring (bugs or rain work well as transition grit), the boundary layer will not be able to negotiate adverse gradients as well. Try placing transition grit near the nose (say at x/c = 0.02) to be sure that the airfoil performance is not too badly degraded under such conditions. Transition grit can also be used to advantage in avoiding laminar separation. As discussed earlier, it is difficult to predict the location of transition accurately and forcing transition to occur before the laminar boundary layer is in danger of separating is often a good idea.

Although ** PANDA** is a subsonic program it may be used to
estimate, with surprising accuracy, the Mach number at which transonic drag
rise occurs. This is done by selecting the desired lift coefficient and
varying Mach number and angle of attack to maintain this CL.
The drag divergence Mach number is approximately 2% to 3% higher than the
Mach number at which the Cp at the airfoil crest reaches
Cp*. The airfoil crest is the point on the airfoil
at which the surface is parallel to the freestream. This method is discussed
in detail in reference 14.

Remember that ** PANDA** is solving the equation for irrotational,
inviscid, linearized flow. It is quite willing to compute answers in situations
where these assumptions are not likely to hold. High angles of attack, transonic
or supersonic Mach numbers, or very low Reynolds numbers can and will give
results which bear little resemblance to reality.

The program solves a linear partial differential equation to obtain the pressures and then corrects these for compressibility effects using the nonlinear Karman - Tsien compressibility correction. This means that when the local surface velocities approach the speed of sound results will be in error. This condition becomes apparent in the Cp plot when the local Cp extends above the value of Cp* (Cp for sonic velocity) which is drawn on the plot. No compressibility effects on the boundary layer are modeled. The local pressures, corrected for compressibility are used but the boundary layer equations are solved based on incompressible flow.

The predicted drag values will not be reliable when the program predicts separation of any kind. No attempt has been made to model separated flow and the boundary layer computation is stopped when turbulent separation is encountered.

The presence of a boundary layer changes the pressure distribution on an airfoil and the pressure distribution affects the boundary layer. The program does not iterate to find the effect of boundary layer displacement thickness on the pressures. Thus, the lift curve slope will be somewhat overpredicted and the aerodynamic center position will be farther aft than would be measured in flight. These effects are not large, however, and the calculated properties should be close enough to be indicative of the basic airfoil performance.

Custom versions of ** PANDA** have been developed with additional
features including trailing edge flap models, alternative boundary layer
methods, and boundary layer iteration. We would happy to discuss your requirements
and develop a program which is tailored for your application. Call or write:
Desktop Aeronautics, P.O. Box A-L, Stanford, CA 94309. (650) 424-8588.

1. Jones, R.T., McWilliams, R., "The Oshkosh Airfoil Program," July 1983.

2. Hess, J.L., Smith, A.M.O., "Calculation of Potential Flow About Arbitrary Bodies," Progress in Aero. Sci., Vol. 8, Pergamon Press, NY, 1966.

3. Jameson, A., "Transonic Flow Calculations," in **Numerical
Methods in Fluid Dynamics**, Wirz, J., Smolderen, J., ed., Hemisphere
Pub. Corp., Washington, 1978.

4. Weber, J., "The Calculation of the Pressure Distribution over the Surface of Two-Dimensional and Swept Wings with Symmetrical Aerofoil Sections," Reports and Memoranda No. 2918, July 1953.

5. Weber, J., "The Calculation of the Pressure Distribution on the Surface of Thick Cambered Wings and the Design of Wings with Given Pressure Distribution," Reports and Memoranda No. 3026, June 1955.

6. Cebeci, T., Bradshaw, P., **Momentum Transfer in Boundary Layers**,
Hemisphere Pub. Corp., Washington 1977.

7. Cebeci, T., Smith, A.M.O., **Analysis of Turbulent Boundary Layers**,
Academic Press, NY, 1974.

8. Reynolds, W.C., Cebeci, T., "Calculation of Turbulent Flows,"
in **Turbulence**, Bradshaw, P., ed., Springer-Verlag, Topics in Applied
Physics Series, Vol. 12, 1978

9. Thwaites, B., ed., **Incompressible Aerodynamics**, Oxford, 1960.

10. Mason, W.H., **Aerodynamic Calculation Methods for Programmable
Calculators and Personal Computers**, AEROCAL, Huntington, NY, 1981.

11. Abbott, I., Von Doenhoff, A., **Theory of Wing Sections**, McGraw
Hill, 1949, Dover Edition, 1959.

12. Bauer, F., Garabedian, P., Korn, D., Jameson, A., **Supercritical
Wing Sections II**, Sringer-Verlag, Berlin, 1975.

13. Smith, A.M.O., "High-Lift Aerodynamics," AIAA Paper No. 74-939, Wright Brothers Lecture, August 1974.

14. McGeer, T., Shevell. R.S., "A Method for Estimating the Compressibility Drag of an Airplane," Stanford University Dept. of Aeronautics and Astronautics Rept. 535.

Registered users of ** PANDA** may receive updates to the program
including any bug fixes or compatibility improvements without charge. Substantial
revisions of the program may be offered at a very reasonable upgrade price.

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Please return the enclosed registration card so that we may keep you
informed of new developments. We also would appreciate letters with comments
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If you design a new airfoil section and obtain test results, let us know.
Thanks for your interest.