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  


1. Introduction

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 Cp distribution is quickly redrawn. In this way the airfoil can be rapidly reshaped to produce a desirable Cp 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. Basic Definitions and Methodology

2.1. Terminology and Coordinates

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

CL Lift coefficient = Section Lift / ( q0 cref)

CD Drag coefficient = Section Drag / ( q0 cref)

Cm Moment coefficient = Moment about quarter chord point / ( q0 cref2 )

Cp Pressure coefficient = (Local pressure - Freestream static pressure ) / q 0

Cp* Critical pressure coefficient, local Mach number = 1.0

Cpi Incompressible pressure coefficient

cref Reference chord length (assumed 1.0 in PANDA)

H Boundary layer shape factor =d* / q

LE Leading edge

M0 Freestream Mach number = U0 / sound speed (Must be < 1.0 in PANDA.)

q0 Freestream dynamic pressure = .5 * r * U02

Re Reynolds number based on chord and freestream velocity = U0 cref / n

Req Reynolds number based on momentum thickness = Uq / n

TE Trailing edge

U0 Freestream velocity

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

x Airfoil chordwise coordinate

yu, yl 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 Jones1, singularity panel methods such as that of Hess2 or direct integration of the partial differential equation by finite difference methods3. 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 Cp corresponding to sonic velocity is drawn on the plot. This value is called the critical pressure coefficient and is denoted Cp*.

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, Req = Uq/n, and the Reynolds number based on local position and velocity, Rex = U x / n. The relationship is:

and holds for values of Rex between 105 and 40 x 106. 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.


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.

3. Using the Program

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.


3.1 Running PANDA

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.


3.2 The Input File

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

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


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.


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:

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:




4. Examples -- Comparisons with Experiment and Theory


4.1 Pressure Distribution

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 method11 in the plot below. It should be noted that the present method yields results in a fraction of the time required by the Theodorsen method and does not experience the convergence difficulties sometimes encountered by the latter.


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.


4.2 Lift and Moment

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 652-015 airfoil from Abbott and von Doenhoff11.

4.3 Drag Polars

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 PANDA with the data from reference 11 in two cases. The first case represents a laminar flow section, the NACA 652-015, with extensive runs of laminar flow. The comparison is shown below.

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.

5. Hints and Limitations



5.1 Using the Results -- Some Suggestions

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.


5.2 Approximations

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.


6. Other Programs

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.


7. References

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.


8. Update Policy and Technical Support

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.

Registered users may also call to request technical support. This includes questions about the program and its use, but does not include aerodynamic consulting services -- you are on your own as far as figuring out what the answers mean or how best to design a particular section.

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