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# «by: Alexander M. Benoliel Thesis submitted to the faculty of the Virginia Polytechnic Institute & State University in partial fulfillment of the ...»

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The 2-D lift coefficient distribution for a McDonnell Douglas Supersonic Transport model (Fig.2) is shown in Fig. 16. The large 2-D lift coefficients at the inboard stations are due to the modeling of the fuselage with a large leading edge sweep angle. Note, from Fig.2, that the pitch-up occurs at about 6˚ angle of attack. If a 2-D sectional lift coefficient limit of 0.85 is chosen, it can be seen that part of the outboard sectional lift is in excess of this value at angles of attack beginning at about 6˚. This maximum lift value was picked because it is close to the actual airfoil maximum lift coefficient of the 3% thick airfoil section used for the outboard wing panel.

–  –  –

Figure 16. - 2-D sectional lift coefficient for a 71˚/57˚ sweep wing calculated with Aero2s.

Thus, it is proposed to use the equivalent 2-D section lift limit to model separated flow on the outboard panel. To estimate the outboard wing panel flow separation, a limit is imposed on the outboard section lift coefficient in the calculation of the total aircraft forces and pitching moment. This limit is chosen to be the maximum two-dimensional lift coefficient for the airfoil section of the outboard wing panel. The selection of this limit will be discussed later in this chapter. The sectional normal force on the outboard wing section is limited to a value such that it does not exceed the prescribed maximum 2-D lift coefficient, Clmax. Once the 2-D lift coefficient, as calculated in Eq. 1, exceeds the maximum 2-D airfoil lift coefficient, the correction to the 3-D sectional normal force

coefficient can be made with the following equation:

–  –  –

where N is the total number of spanwise stations, Cmj is the 3-D sectional pitching moment at section j (after the correction has been applied, if required), cj is the local chord at section j, and cave is the average chord over the span. Leading edge thrust and vortex lift effects are then added to this result to determine the final value of the pitching moment. The lift is calculated in a similar manner.

4.2 Results The cambered and twisted configuration presented in Fig. 2 is presented again in Fig.

17 with the results of the new Aerodynamic Pitch-up Estimation (APE) method. A maximum airfoil lift coefficient of 0.85 was chosen for the 3% thick outboard wing section. The results of the new method are shown with a solid line, labeled as Aero2s + APE. The new method estimates the pitch-up well, although it does not estimate the lift coefficient as accurately. At a lift coefficient of 0.6, the difference between experiment and the APE method is about 0.07.

–  –  –

Figure 17. - Comparison of lift and pitching moment estimation methods for a 71˚/57˚ sweep cambered and twisted cranked arrow wing (δTail = 0˚).

Page 20 When the method was applied to the aerodynamic assessment of a uncambered, untwisted wing configuration, the results were even more promising. The configuration shown in Fig. 18 is a flat, cranked arrow wing tested by Kevin Kjerstad (the data is from a yet to be published NASA TP). This model was part of a family of arrow wings tested by Kjerstad. The wings are flat plates of constant absolute thickness, with beveled leading and trailing edges. Inboard and outboard leading edge sweep angles are 74˚ and 48˚ respectively. A maximum airfoil lift coefficient of 0.75 was chosen.

–  –  –

Figure 18. - 74˚/48˚ sweep wing-body combination, comparison to experimental data.

The pitch-up of this configuration is not as severe. The APE method also does a better job at estimating the lift. At a lift coefficient of 0.6, the difference between the new method and experiment is about 0.03. The small non-zero pitching moment at zero lift was presumably due to a camber effect created by the beveled leading edge not modeled in the aerodynamic analysis.

The results for an F-16XL model tested by David Hahne (from an unpublished test) are shown in Fig. 19. This model incorporated a 70˚/50˚ sweep, cambered and twisted, cranked arrow wing. The data shown is for the model configured for an HSCT type planform. A maximum 2-D airfoil lift coefficient of 0.80 was chosen for the biconvex airfoil section of the outboard wing panel.

The new estimation method results indicate good agreement with the experimental data. The pitching moment curve slope was estimated with relatively good accuracy before

–  –  –

Figure 19. - Comparison of estimation methods for an F-16XL (70˚/50˚ sweep) model test.

4.2.1 Leading Edge Vortex Considerations The pitch-up of the cases presented thus far was due primarily to the flow separation on the outboard wing panel, outboard vortex breakdown, or a combination of the two effects. A case in which the pitch-up is due primarily to the strong inboard leading edge vortex is exemplified in a test by Coe11, shown in Fig. 20. The slender and highly swept, uncambered, untwisted, 74˚/70.5˚/60˚ sweep configuration promotes a strong inboard leading edge vortex which has little effect on the outboard wing section. A weak leading edge vortex also forms on the outboard section, although the flow on this section separates early.10 When this is the dominant flow mechanism, the original Carlson method (Aero2s) did a better job at estimating the pitching moment than the modified method. Now an investigation is conducted to explain these results.

–  –  –

Figure 20. - Comparison of lift and pitching moment estimation methods for a 74˚/70.

5˚/60˚ sweep uncambered and untwisted cranked arrow wing similar in planform to the AST-200.

–  –  –

where x is the chordwise position and xvor is the chordwise position of the vortex core, both measured aft of the leading edge. The value of x ranges from zero to two times the value of xvor. Thus, the vortex induced pressure distribution starts at the leading edge and peaks at the vortex core position. The value of k is such that the integrated area of the entire distribution is equal to the vortex lift calculated using the Polhamus suction analogy and the attainable thrust relations. If the local chord length is greater than 2 • xvor. then only part of the vortex force is applied (Fig. 21).

–  –  –

Figure 21. - Pressure distribution used to calculate the contribution of vortex lift.

The vortex placement estimates for the configuration shown in Fig. 20 are shown in Fig. 22, compared to the experimental results found by Coe.11 The code does not distinguish between two vortex systems and simply uses a continuous vortex whose position changes depending on the local sweep angle. For this configuration the inboard and outboard sweep angles are both large, thus the leading edge vortex, estimated by the code, has only a small effect on the outboard wing panel at high angles of attack. Note that the code extends the vortex to the nose of the aircraft. Aero2s assumes that vortex lift acts across the entire span, including the fuselage region. The vortex apex location specifier does not limit the vortex from acting inboard of that position.

The method can be refined by limiting the vortex effects to the wing only, and eliminating the contributions to the sectional characteristics of the fuselage. When this correction is applied a very different result for the pitching moment compared to Fig. 20 is found (Fig. 23). This refinement will be called the “limited vortex” modification. For the limited vortex modification shown in Fig. 23, the vortex is begun at the wing root.† Note that the new estimation method now becomes more accurate in predicting the pitching moment. This is because the long moment arm that the vortex force has in the fuselage region has been eliminated.

† A sensitivity of calculated vortex forces to the amount of grid points used was found.

This was a results of the increased resolution obtained with a larger number of spanwise stations. The increased number of grid points offered only a small change in the inviscid solution.

–  –  –

Figure 23. - Effects of limiting the vortex effects to the wing only for a 74˚/70.

5˚/60˚ sweep wing similar in planform to the AST-200. Limited vortex begins at wing root and does not extend into fuselage region.

When this correction to the vortex location was applied to the previously presented cases, only a small change in the results was found. This was due primarily to the fact that the pitch-up of these configurations was due to flow separation on the outboard panels rather than the strong leading edge vortex. The lower leading edge sweep angles of the Page 25 previously studied configurations allowed for the leading edge vortex to have a greater effect on the outboard wing panels. These cases had inboard sweep angles ranging from 68˚ to 71˚, and outboard sweep angles ranging from 48˚ to 57˚, compared to the 74˚/60˚ sweep angles of the AST-200 configuration in question.

It is also possible that the vortex system developed is such that it does not promote pitch-up at low angles of attack. The theoretical and experimental results for a 70˚/48.8˚ sweep configuration tested by Quinto and Paulson51 are shown in Fig. 24 without the correction to the leading edge vortex. This is a flat wing configuration which uses an NACA 0004 airfoil section for the entire wing. A maximum airfoil lift coefficient of 0.90 was chosen. For this case, pitch-up, due to the flow separation on the outboard wing panel, did not occur until about 18˚ angle of attack. Neither the original Aero2s nor the pitch-up estimation method (which is tied to the baseline method) predict the pitching moment characteristics well. Note that the sweep angles of this configuration differ little from the F-16XL planform.

–  –  –

0.2 Figure 24. - Comparison of lift and pitching moment estimation methods for a 70˚/48.8˚ sweep uncambered and untwisted cranked arrow wing.

Estimates made before “limited vortex” modification.

Here again, if the vortex placement is “limited” for this case, an improvement in the results is found, as shown in Fig. 25. The location of the pitch-up estimation is indicated by the location at which the two curves representing the limited vortex case diverge. This is also close to where there is an initial inflection in the experimental curve. The method fails to estimate the second, and much larger, pitch-up at 20˚ angle of attack.

Note: All subsequent analysis results incorporate the limited vortex modification.

–  –  –

Figure 25. - Effects of limiting the vortex effects to the wing only for a 70˚/48.

8˚ sweep uncambered and untwisted cranked arrow wing. Limited vortex begins at wing root and does not to extend into fuselage region.

4.2.2 Horizontal Tail and Flap Effect Analysis An analysis of the McDonnell Douglas 71˚/57˚ sweep configuration, shown in Fig.

17, was conducted for the configuration with flaps deflected (δTE = 30˚, δTail = 0.0˚, δLE = 13˚/34˚/35˚/35˚/19˚/29˚). A comparison between the new method and experiment is shown in Fig. 26. A maximum lift coefficient of 1.80 was used for the outboard flapped wing section. The APE method estimate for the lift agrees well throughout the angle of attack range. The estimate of the pitching moment agrees well with the experimental data after the pitch-up, but does not accurately estimate ∂CM/∂CL before the pitch-up. At zero degrees angle of attack, the average slope of the experimentally derived pitching moment curve, ∂CM/∂CL, is equal to -0.0855. The corresponding, estimated value for this slope is

0.0623. If the horizontal tail is removed, as shown in Fig. 27, the experimental and estimated values of ∂CM/∂CL are -0.0085 and 0.08688, respectively. The improvement of the estimation suggests the indication of the code’s lack of accuracy when predicting the characteristics for the second surface in the presence of the wing wake. The estimation of ∂CM/∂CL for two surface aircraft was shown to improve as the distance between the two surfaces was decreased29.

–  –  –

Figure 26. - Comparison of lift and pitching moment estimation methods for a 71˚/57˚ sweep cambered and twisted cranked arrow wing with flaps deflected (δTail = 0˚, δTE = 30˚, δLE = 13˚/34˚/35˚/35˚/19˚/29˚).

–  –  –

Figure 27. - Comparison of lift and pitching moment estimation methods for a 71˚/57˚ sweep cambered and twisted cranked arrow wing with flaps deflected and tail removed (δTE = 30˚, δLE = 13˚/34˚/35˚/35˚/19˚/29˚).

–  –  –

Figure 28. - Comparison of lift and pitching moment estimation methods for a 71˚/57˚ sweep cambered and twisted cranked arrow wing without flaps and horizontal tail removed.

Although the low angle of attack ∂CM/∂CL is not well predicted, the estimated effects of tail deflection correlate well with the experimental data. Figure 29 shows the experimental and theoretically estimated data for the 71˚/57˚ configuration with flaps deflected and tail deflected -10˚. The correlation between the data is similar to that found in Fig. 26 for a zero degrees tail deflection, thus indicating a good prediction of the tail effectiveness (the difference in aerodynamic characteristics between 0˚ and -10˚ tail deflection).

–  –  –

Figure 29. - Comparison of lift and pitching moment estimation methods for a 71˚/57˚ sweep cambered and twisted cranked arrow wing with flaps deflected (δTail = -10˚, δTE = 30˚, δLE = 13˚/34˚/35˚/35˚/19˚/29˚).

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