From an observation of the brief history of potential flow solutions, and the methodology presented in Chapters 3-5, it is clear that the trend is toward using distributions of elementary solutions with gradually increasing com­plexity and determining their strength via the boundary conditions. So in principle, if a problem can be solved by distributing the unknown quantity on the boundary surface rather than in the entire volume surrounding the body (as in finite-difference methods), then a faster numerical solution is obtainable. This observation is true for most practical in viscid flow problems (e. g., lift of wings in attached flows, etc.).

This reduction of the three-dimensional computational domain to a two-dimensional one (on a three-dimensional boundary) led to the rapid development of computer codes for the implementation of panel methods and some of them are listed in Table 12.1. Probably the first successful three – dimensional panel code is known as the Hess code121 (or Douglas-Neumann), which was developed by the Douglas Aircraft Company and used a Neumann

TABLE 12.1

Chronological list of some panel methods and their main features


Geometry of panel






1962, Douglas – Neumann12 1


Constant source


1966, Woodward I12 2


Linear sources Constant vortex


M> 1

1973, USSAERO123


Linear sources Linear vortex


M >1

1972, Hess I12 4


Constant source Constant doublet


1980, MCAIR12 5


Constant source Quadratic doublet


Coupling with B. L.

design mode

1980, SOUSSA126


Constant source Constant doublet




1981, Hess II12 7


Linear source Quadratic doublet


1981, PAN AIR12 8’12 9



Linear source Quadratic doublet


M> 1

1982, VSAERO1210,1211


Constant source and doublet


Coupling with B. L., wake rollup

1983, QUADPAN12 12


Constant source and doublet


1987, PMARC12 13,12 14


Constant source Constant doublet


Unsteady wake rollup

velocity boundary condition. This method was based on flat source panels, and had a true three-dimensional capability for nonlifting potential flows.

The Woodward I code,122 which originated in the Seattle area, was capable of solving lifting flows for thick airplane-like configurations. This code also had a supersonic potential flow solution option that increased its applicability. The method was later improved and was released as the USSAERO code12 3 (or the Woodward II code). At about the same time Hess added doublet elements to his nonlifting method so that he could solve for flow with lift; this code12 4 was widely used by the industry and was called the Hess I code.

All of the computer codes listed in Table 12.1 had the capability to correct for low-speed compressibility effects by using the Prandtl-Glauert transformation (as in Section 4.8).

The above computer codes were considered to be the first-generation panel programs, but as the computer technology evolved, more complex algorithms could be developed based on higher-order approximations to the panel surface and singularity distribution. For example, the MCAIR code,12 5 which evolved into a high-order singularity method, had two new interesting features. One was an inverse two-dimensional solution for multielement airfoils with prescribed pressure distribution. The second option was an iterative coupling with a boundary layer procedure. Pressure and velocity data from the potential flow solution were fed into a boundary layer analysis that estimated the displacement thickness and surface friction. During the next iteration of the potential solver the three-dimensional panel geometry was modified to include the added displacement thickness of the boundary layer.

At about the same time the SOUSSA code12 6 was developed and it used the Dirichlet boundary condition (as did MCAIR) and had the additional feature of an unsteady oscillatory mode. Also, John Hess of the McDonnell Douglas Aircraft Co. had updated the Hess I code to the Hess II code,12 7 which now had parabolic panel shape and higher-order singularity distributions.

During this second-generation panel code development period, the largest effort was invested in the PAN AIR code12 8,12 9 which was developed for NASA by the Boeing Co. The code had a five flat, subelement panel with higher-order singularity distribution and boundary conditions were usually Dirichlet, but on selected areas the Neumann condition could be used as well. This code also had the capability for solving the supersonic potential flow equations.

Until the early 1980s most panel codes were limited (along with the availability of mainframe computers) to the larger aerospace companies. However, computer technology rapidly evolved and cost decreased in these years, so that it was economically logical for smaller companies (e. g., general aviation contractors, boat builders, race-car teams, etc.) to use this technology. The first panel code commercially available to the smaller industries was VSAERO12 101211 (which was developed under a grant from NASA Ames Research Center). This code can be viewed as the beginning of a third period in the development of panel codes, since it returned to simpler, first-order panel and singularity elements. This code used the Dirichlet boundary condition for thick bodies and the Neumann condition for thin vortex-lattice panels. Interaction with several methods of boundary layer solutions along streamlines was used, but the displacement thickness effect was corrected by adding sources (blowing or transpiration), rather than adjusting the panel geometry (as in MCAIR). Also, a wake rollup routine was added that computed the induced velocity on the wake and moved the wake vortices to a new “force free” position. Following the success of this code (due to computational economy) the Lockheed company developed a similar method, called QUADPAN.1212

At this point it seems that the theory of panel methods has matured and most of the effort is invested in pre – and postprocessing (automatic generation of surface grids and graphical representation of results). Also, interactive airfoil and wing design is being developed where the designer can modify interactively the body’s geometry in order to obtain a desirable pressure distribution.

Some of the other improvements of these methods, during the second

TABLE 12.2

Claimed advantages of low – and high-order panel codes

Low-order methods

High-order methods

Derivation of

Simple derivation

More complex derivation

influence coefficients

Computer programming

Relatively simple coding

Requires more coding effort

Program size

Short (fits minicomputers)

Longer (will run on mainframes only)

Run cost


Considerably higher


Less—for same number of panels

(but more accurate for same run time)

Higher accuracy for a given number of panels

Sensitivity to

Not very sensitive*

Not allowed

gaps in paneling

Extension to M > 1


Simple (for arbitrary geometry)

* This is a major advantage for the comparatively untrained user. Also this feature allows for an easy treatment of very narrow gaps where viscous effects control the otherwise high-speed inviscid flow (see Example 6 in

Section 12.7).

half of the 1980s, was the addition of an unsteady motion option,12 13 and an overall numeric optimization of the method (in terms of computer memory requirement and efficiency of matrix solver). Such a code is PM ARC1214 (Panel Method Ames Research Center) which was developed at NASA Ames and is now suitable for personal computers.

The recent trend of some code developers toward the use of low-order methods, and the fact that many different methods are now being used, led to several comparison studies (such as in Ref. 12.15). This particular study indicates that low-order methods are clearly faster and cheaper to operate. Some of the claimed advantages of each of the methods are listed in Table 12.2 and the decision of which one to choose for a particular application is not obvious. It is important to point out that “any method will provide good results after validating it through a large number of test cases” (free quote of Dr. John Hess).