Aerofoil Section Analysis using 2D panel methods,
incorporating 1D corrections for boundary layer flow

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The prediction of aerodynamic properties of most aerofoil sections can be obtained relatively accurately using two dimensional panel method analysis. The solutions will be primarily inviscid flow predictions. However, with the introduction of some simple one-dimensional boundary layer theory, the inviscid solutions can be corrected due to small viscosity effects. This allows estimation of lift, drag and pitching moment coefficients for sections were there are only small effects due to flow separation or friction.

The solution is obtained by two separate calculations,

It is possible to iterate between the results of these two solutions until a final converged solution is obtained but in many cases problems may arise due to the number of iterations required and the possibility of an unstable iteration. A reasonable result is generally obtained by just using a single pass of the solution parts.

PART 1 : 2D Inviscid panel method

A potential flow solution of any general aerofoil section can be modelled by descretising the surface contour using singularity panels. Many different techniques are possible but for the program used here, the following configuration has been employed for the panel modelling,




Each panel ( j ) is a straight line segment between surface contour points (j and j+1). Along the panel, a source distribution of constant strength () is applied. This distribution strength varies from panel to panel. As well, along each panel is a constant vorticity distribution (). The vorticity is the same on each panel around the contour and produces the required circulation for the lifting section.

As the geometry of the section and the freestream flow conditions ( -- velocity , -- angle of attack ) are set, the requirement will be to define boundary condition equations in order to determine the necessary distribution strengths ( and , j = 1 to N (number of panels) ), for an accurate model of the problem.


A boundary condition of no flow through surface ( ) can be applied at the center of each panel. This produces N equations in N+1 unknowns. In order to correctly solve for the extra unknown vorticity, a Kutta condition must be applied at the trailing edge.




For a single panel (i) the boundary condition will be applied as,

at Panel (i)

where the coefficient,,represents the influence of the source component on panel (j) on the control point on panel (i) and , , represents the influence of panel (j) vortex component on the control point of panel (i).represents the freestream influence. All coefficients are functions of the geometry of the section, function (x,y), due to orientation and spacing of panels.

The Kutta condition, equation N+1, can be applied in terms of trailing edge tangential velocities,

thus .

Written interms of influence coefficients contributing to the sum of trailing edge tangential velocities, this becomes,

.

This gives a system of linear equations which allow the solution for the required distribution strengths to be found.

Once the distribution strengths ( ) have been calculated, surface tangential velocities at the center of each panel can be calculated ( V ) and then surface pressure coefficients,

The lift coefficient can be calculated assuming a small angle of attack as the integration of surface pressure coefficient acting in the y-direction, ie. projected on the x axis.


Solutions only need to be calculated for one or two angles of attack as the lift curve will be linear. Stall and boundary layer effects are not predicted by the first part of the process.

PART 2 : 1D Boundary Layer Theory.

Once the surface velocities have been predicted, it is possible to start some simple calculations for the viscous surface effects and drag cofficient.

APPLICATION : 2D Panel Code Computer Program.

The following program accepts ASCII data files which consist of a list 2-D aerofoil section coordinates. The format of these aerofoil input data files is the same as that produced by the NACA section generation program.

 
 -------------------------------------------------
|Header Line                                      |
|Number of Data Points (N)                        |
|   x(1)        y(1)                              |
|   x(2)        y(2)                              |
|   x(3)        y(3)                              |
|                                                 |
|   ...         ...                               |
|   x(3)        y(3)                              |
|                                                 |
|   x(N)        y(N)                              |
 -------------------------------------------------


There is an initial header line, followed by a line giving the number of data points used to describe the aerofoil and then pairs of surface coordinate points (x,y). The order of surface points is anti-clockwise, starting at the trailing edge, going back over the upper surface around the nose and then forward along the underside back to the trailing edge. Data files for other aerofoil sections can be created using Excel or similar spreadsheet to give a data file in the correct format.

Some sample section files are NACA 0012, NACA 4412 and NACA 64-012.

From the surface coordinate data file, the program calculates an inviscid flow solution. The 1-D momentum integration is run to predict the boundary layer near the aerofoil surface.

The program can predict CL for a given angle, Cm(1/4c) and CD for the specified aerofoil section.

SOFTWARE

ExecutableProgram : 2-D Panel Method Solution (260k)

Executable Program : 2-D NACA Aerofoil Section Generator (286k)

Executable Program : XFOIL v6.94 Aerofoil Development System 2-D (1260k)

For Full Documentation and latest solution software visit the Marc Drela GPL Aerofoil Solution System site at MIT XFOIL site..

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