5. Classical Linear Theory

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Program PANEL

In this section we illustrate the results of the procedure outlined above. Program PANEL is an exact implementation of the analysis described above, and is essentially the program given by Moran.Error: Reference source not found Other panel method programs are available in the textbooks by Cebeci,Error: Reference source not found Houghton and Carpenter,13 and Kuethe and Chow.14 Two other similar programs are available. A MATLAB program, Pablo, written at KTH in Sweden is available on the web,15 as well as the program by Professor Drela at MIT, XFOIL.16 Moran’s program includes a subroutine to generate the ordinates for the NACA 4-digit and 5-digit airfoils (see Appendix A for a description of these airfoil sections). The main drawback is the requirement for a trailing edge thickness that is exactly zero. To accommodate this restriction, the ordinates generated internally have been altered slightly from the official ordinates. The extension of the program to handle arbitrary airfoils is an exercise. The freestream velocity in PANEL is assumed to be unity, since the inviscid solution in coefficient form is independent of scale.

PANEL’s node points are distributed employing the widely used cosine spacing function. The equation for this spacing is given by defining the points on the thickness distribution to be placed at:

. (5-31)

These locations are then altered when camber is added (see Eqns. A-1 and A-2 in App. A). This approach is used to provide a smoothly varying distribution of panel node points that concentrate points around the leading and trailing edges.

An example of the accuracy of program PANEL is illustrated in Fig. 5.10, where the results from PANEL for the NACA 4412 airfoil are compared with results obtained from an exact conformal mapping of the airfoil (Conformal mapping methods were described in Chapter 4. Conformal transformations can also be used to generate meshes of points for use in CFD methods). The agreement is nearly perfect.

Numerical studies need to be conducted to determine how many panels are required to obtain accurate results. Both forces and moments and pressure distributions should be examined. You can select the number of panels used to represent the surface. How many should you use? Most computational programs provide the user with freedom to decide how detailed (expensive - in dollars or time) the calculations should be. One of the first things the user should do is evaluate how detailed the calculation should be to obtain the level of accuracy desired. In the PANEL code your control is through the number of panels used.

Figure 5.10. Comparison of results from program PANEL with an essentially exact

mapping solution for the NACA 4412 airfoil at 6° angle-of-attack.

We check the sensitivity of the solution to the number of panels by comparing force and moment results and pressure distributions with increasing numbers of panels. This is done using two different methods. Figures 5.11 and 5.12 present the change of drag and lift, respectively, by varying the number of panels. For PANEL, which uses an inviscid incompressible flowfield model, the drag should be exactly zero. The drag coefficient found by integrating the pressures over the airfoil is an indication of the error in the numerical scheme. The drag obtained using a surface (or “nearfield”) pressure integration is a numerically sensitive calculation, and is a strict test of the method. The figures show the drag going to zero, and the lift becoming constant as the number of panels increase. In this style of presentation it is hard to see exactly how quickly the solution is converging to a fixed value.

The results given in Figs. 5.11 and 5.12 indicate that 60-80 panels (30 upper, 30 lower for example) should be enough panels. Note that the lift coefficient is presented in an extremely expanded scale, and the drag coefficient presented in Fig. 5.13 also uses an expanded scale. Because drag is typically a small number, it is frequently described in drag counts, where 1 drag count is a CD of 0.0001.

To estimate the limit for an infinitely large number of panels the results can be plotted as a function of the reciprocal of the number of panels. Thus the limit result occurs as 1/n goes to zero. Figures 5.13, 5.14, and 5.15 present the results in this manner for the case given above, and with the pitching moment included for examination in the analysis.

Figure 5.11. Change of drag with number of panels.

Figure 5.12. Change of lift with number of panels.

Figure 5.13. Change of drag with the inverse of the number of panels.

The results given in Figures 5.13 through 5.15 show that the program PANEL produces results that are relatively insensitive to the number of panels once fifty or sixty panels are used, and by extrapolating to 1/n = 0 an estimate of the limiting value can be obtained.

Figure 5.14. Change of lift with the inverse of the number of panels.

Figure 5.15. Change of pitching moment with the inverse of the number of panels.

In addition to forces and moments, the sensitivity of the pressure distributions to changes in panel density must also be investigated: pressure distributions are shown in Figs. 5.16 and 5.17. The 20 and 60 panel results are given in Fig. 5.16. In this case it appears that the pressure distribution is well defined with 60 panels. This is confirmed in Figure 5-17, which demonstrates that it is almost impossible to identify the differences between the 60 and 100 panel cases. This type of study should (in fact must) be conducted when using computational aerodynamics methods.

Figure 5.16. Pressure distribution from progrm PANEL, comparing results using 20 and 60 panels.

Figure 5.17. Pressure distribution from program PANEL, comparing results using 60 and 100 panels.

Having examined the convergence of the mathematical solution, we investigate the agreement with experimental data. Figure 5.18 compares the lift coefficients from the inviscid solutions obtained from PANEL with experimental data from Abbott and von Doenhof.17 Agreement is good at low angles of attack, where the flow is fully attached. The agreement deteriorates as the angle of attack increases, and viscous effects start to show up as a reduction in lift with increasing angle of attack, until, finally, the airfoil stalls. The inviscid solutions from PANEL cannot capture this part of the flow physics. The different stall character between the two airfoils arises due to different flow separation locations on the different airfoils. The cambered airfoil separates at the trailing edge first. Stall occurs gradually as the separation point moves forward on the airfoil with increasing incidence. The uncambered airfoil stalls due to a sudden separation at the leading edge. An examination of the difference in pressure distributions can be made to see why this might be the case.

Figure 5.18. Comparison of PANEL lift predictions with experimental data (Ref. Error: Reference source not found).

The pitching moment characteristics are also important. Figure 5.19 provides a comparison of the PANEL pitching moment predictions (taken about the quarter chord point) with experimental data. In this case the calculations indicate that the computed location of the aerodynamic center, , is not exactly at the quarter chord, although the experimental data is very close to this value. The uncambered NACA 0012 data shows nearly zero pitching moment until flow separation starts to occur. The cambered airfoil shows a significant nose down pitching moment, Cm0, and a trend with angle of attack due to viscous effects that is exactly opposite the inviscid prediction. This occurs because the separation is moving forward from the trailing edge of the airfoil and the load over the aft portion of the airfoil does not increase as fast as the forward loading. This leads to a nose up pitching moment until eventually the separation causes the airfoil to stall, resulting in a nose down pitching moment.

Figure 5.19 Comparison of PANEL moment predictions with experimental data, (Ref. Error: Reference source not found).

We do not compare the drag prediction from PANEL with experimental data. For two-dimensional incompressible inviscid flow the drag is theoretically zero. In the actual case, drag arises from skin friction effects, further additional form drag due to the small change of pressure on the body due to the boundary layer (which primarily prevents full pressure recovery at the trailing edge), and drag due to increasing viscous effects with increasing angle of attack. A well designed airfoil will have a drag value very nearly equal to the skin friction and nearly invariant with incidence until the maximum lift coefficient is approached.

In addition to the force and moment comparisons, we need to compare the pressure distributions predicted with PANEL to experimental data. Figure 5.20 provides one example. The NACA 4412 experimental pressure distribution is compared with PANEL predictions. In general the agreement is very good. The primary area of disagreement is at the trailing edge. Here viscous effects act to prevent the recovery of the experimental pressure to the levels predicted by the inviscid solution. The disagreement on the lower surface is a little surprising, and suggests that the angle of attack from the experiment may not be precise.

Figure 5.20. Comparison of pressure distribution from PANEL with data,18

Panel methods often have trouble with accuracy at the trailing edge of airfoils with cusped trailing edges, when the included angle at the trailing edge is zero. Figure 5.21 shows the predictions of program PANEL compared with an exact mapping solution (a FLO3619 run at low Mach number) for two cases. Figure 5.21a is for a case with a small trailing edge angle: the NACA 651-012, while Fig. 5.21b is for the more standard 6A version of the airfoil. The corresponding airfoil shapes are shown Fig. 5.22. The “loop” in the pressure distribution in Fig. 5.21a is an indication of a problem with the method.

Figure 5.21. PANEL Performance near the airfoil trailing edge

Figure 5.22. Comparison at the trailing edge of 6- and 6A-series airfoil geometries.

This case demonstrates a situation where this particular panel method is not accurate. Is this a practical consideration? Yes and no. The 6-series airfoils were theoretically derived by specifying a pressure distribution and determining the required shape. The small trailing edge angles (less than half those of the 4-digit series), cusped shape, and the unobtainable zero thickness specified at the trailing edge resulted in objections from the aircraft industry. These airfoils were very difficult to manufacture and use on operational aircraft. Subsequently, the 6A-series airfoils were introduced to remedy the problem. These airfoils had larger trailing edge angles (approximately the same as the 4-digit series), and were made up of nearly straight (or flat) surfaces over the last 20% of the airfoil. Most applications of 6-series airfoils today actually use the modified 6A-series thickness distribution. This is an area where the user should check the performance of a particular panel method.

5.2.3 Geometry and Design

Effects of Shape Changes on Pressure Distributions: So far we have been discussing aerodynamics from an analysis point of view. To develop an understanding of the typical effects of adding local modifications to the airfoil surface, Exercise 5 provides a framework for the reader to carry out an investigation to help understand what happens when changes are made to the airfoil shape. It is also worthwhile to investigate the very powerful effects that small deflections of the trailing edge can produce. This reveals the power of the Kutta condition, and alerts the aerodynamicist to the basis for the importance of viscous effects at the trailing edge.

Making ad hoc changes to an airfoil shape is extremely educational when implemented in an interactive computer program, where the aerodynamicist can easily make shape changes and see the effect on the pressure distribution immediately. An outstanding code that does this has been created by Ilan Kroo and is known as PANDA. 20 Strictly speaking, PANDA is not a panel method, but it is an accurate subsonic airfoil prediction method.

Shape for a specified pressure distribution: There is another way that aerodynamicists view the design problem. Although the local modification approach described above is useful to make minor changes in airfoil pressure distributions, often the aerodynamic designer wants to find the geometric shape corresponding to a prescribed pressure distribution from scratch. This problem is known as the inverse problem. This problem is more difficult than the analysis problem. It is possible to prescribe a pressure distribution for which no geometry exists. Even if the geometry exists, it may not be acceptable from a structural standpoint. For two-dimensional incompressible flow it is possible to obtain conditions on the surface velocity distribution that ensure that a closed airfoil shape exists. Excellent discussions of this problem have been given by Volpe21 and Sloof.22 A two-dimensional inverse panel method has been developed by Bristow.23 XFOIL also has an inverse design option.Error: Reference source not found Numerical optimization can also be used to find the shape corresponding to a prescribed pressure distribution.24

5.2.4 Issues in the Problem formulation for 3D potential flow over aircraft

The extension of panel methods to three dimensions leads to fundamental questions regarding the proper specification of the potential flow problem for flow over an aircraft. The main problem is how to model the wake coming from the fuselage aft of the wing and wing tips. The issue is how to specify the wake behind surfaces without sharp edges. The Kutta condition applies to distinct edges, and is not available if there are not well defined trailing edges.

In some methods wakes are handled automatically. In other methods the wakes must be specified by the user. This provides complete control over the simulation, but means that the user must understand precisely what the problem statement should be. The details of the wake specification often cause users difficulties in making panel models. Figure 5.23, from Erickson,Error: Reference source not found shows an example of a panel model including the details of the wakes. For high lift cases and for cases where wakes from one surface pass near another, wake deflection must be computed as part of the solution. Figure 5.24 comes from a one week “short” course that was given to prospective users of an advanced panel method, PAN AIR.25 Each surface has to have a wake, and the wakes need to be connected, as illustrated in Fig.5.24. The modeling can be complicated. Special attention to wake layout must be made by the user. To ensure that the problem is properly specified and to examine the entire flowfield in detail a complete graphics capability is required.

Hess26 provides an excellent discussion of these problems. Many different approaches have been used. Carmichael and Erickson27 also provide good insight into the requirements for a proper panel method formulation. Similarly, references Error: Reference source not found and Error: Reference source not found provide good overviews.

As illustrated above, a practical aspect of using panel methods is the need to pay attention to details (actually true for all engineering work). This includes making sure that the outward surface normal is oriented in the proper direction and that all surfaces are properly enclosed. Aerodynamics panel methods generally use quadrilateral panels to define the surface. Since three points determine a plane, the quadrilateral may not necessarily define a consistent flat surface. In practice, the methods actually divide panels into triangular elements to determine an estimate of the outward normal. It is also important that edges fit so that there is no leakage in the panel model representation of the surface. Nathman has recently extended a panel method to have panels include “warp”.28

Figure 5.23. Illustration of the panel model of an F-16XL,Error: Reference source not found including the wakes usually not shown in figures of panel models, but critical to the model.

Figure 5.24. Details of a panel model showing the wake model details and that the wakes are connected. (from a viewgraph presented at a PAN AIR user’s short course, Ref. Error: Reference source not found)

There is one other significant difference between two-dimensional and three-dimensional panel methods. Induced drag occurs even in inviscid, irrotational flow, and this component of drag can be computed by a panel model. However, its calculation by integration of pressures over the surface requires extreme accuracy, as we saw above for the two-dimensional example. The use of a farfield momentum approach is much more accurate. For drag this is known as a Trefftz plane analysis, see Katz and Plotkin.Error: Reference source not found

5.2.5 Example applications of panel methods

Many examples of panel methods have been presented in the literature. Figure 5.25 shows an example of the use of a panel model to evaluate the effect of the space shuttle on the Boeing 747. This is a classic example. Other uses include the simulation of wind tunnel walls, support interference, and ground effects. Panel methods are also used in ocean engineering. Recent America’s Cup designs have been dependent on panel methods for hull and keel design. The effects of the free surface can be treated using panel methods.

Figure 5.25. The space shuttle mounted on a Boeing 747.

(from the cover of an AIAA Short Course with the title Applied Computational Aerodynamics)

One example has been selected to present in some detail. It is an excellent illustration of how a panel method is used in design, and provides a realistic example of the typical agreement that can be expected between a panel method and experimental data in a demanding real world application. The work was done by Ed Tinoco and co-workers at Boeing.29 Figure 5.26 shows the modifications required to modify a Boeing 737-200 to the 737-300 configuration. The panel method was used to investigate the design of a new high lift system. They used PAN AIR, which is a Boeing developed advanced panel method.30 Figure 5.27 shows the panel method representation of the airplane.

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