D.1 Overview of Case Studies

This appendix provides examples of the procedures and computational aerodynamics tools for aerodynamic design. We include three case studies: (i) the B-2 in section D.2; (ii) comparisons of the Beech Starship and the Grumman X-29 in section D.3; and (iii) the YF-22 and YF-23 ATF candidate designs in section D.3 that were performed by student teams. The case studies illustrate many of the issues facing configuration aerodynamicists, tying together a number of aspects of aerodynamic theory and applications that are covered in the course. The results obtained in the term projects described in this appendix have been highly instructive to both student and teacher. They provided the students with an opportunity to examine real-world problems. Portions of these examples have been discussed previously in the AIAA paper, “Applied Computational Aerodynamics Case Studies.”^[1] A key component of these case studies is the need to gather information. Students must read the literature and get to know the sorts of reports that are available. This means using NASA and AIAA literature, as well as AGARD and news-type publications (Aviation Week, Interavia, Air International, etc.).

The examples are provided in considerable detail. We show use of the simple tools listed in appendix E because our focus is on aerodynamic analyses typical of early conceptual and preliminary design work. The input datasets for the simple computational tools almost always mimic the old “punched card” input styles, requiring that the values in the datasets be placed in specific fields. Although students aren’t used to this style, I don’t see a problem; it’s any easy adjustment. When making calculations, it is always important to assess whether the code is giving the “right” answer by conducting the infamous “sanity check.” These examples should help students examine results in their own work. Finally, many aerospace engineers are heavy users of computational methods. While some will modify existing codes, only a handful of graduates will develop entirely new algorithms and codes. The examples contained here depict the typical work of an aerodynamic engineer/designer.

The case studies presented in this appendix show that a significant number of issues associated with configuration aerodynamics can be resolved without expensive CFD calculations. Students can get considerable insight and make good sanity checks against much more sophisticated codes using a PC. However, other aspects of the problem require the use of sophisticated CFD methods. Still other aspects require wind tunnels or flight tests at present. The role of each is identified through the use of these projects. These projects require an assessment intended to improve the student’s ability for “critical thinking.”

D.2 The B-2 Study

The B-2 was unveiled in late November of 1988. This example was used as a class project in spring 1989. The statement of the assignment is given in table D-1.

Several other aspects of the design are required. For example, it also requires an estimate of the CG of the B-2, although this wasn’t explicitly stated. Some students were surprised that the static margin required both the neutral point and the CG, since they weren’t given the CG.

D.2.1 Develop a Geometry Model of the B-2 for Use in Analysis and Design

Initial specific sources of information included the Aviation Week story^[2] and the first three-view of the B-2 which appeared in Air International.^[3] The three-view was important. The side view allowed the students to estimate the CG by assuming a 15° angle from the landing gear ground contact to the CG location. Figure D-1 shows a three-view from Wikipedia. However, students used one from Janes All the World Aircraft that Jane’s had created based on the information available at that time. The lecture by Waaland^[4] and the Aerofax book^[5] were not yet available when this example was done.

The numerical values of the so-called corner points measured from the top view are shown in figure D-2. Many students can’t quite believe that engineers would be expected to scale a drawing to get quantitative values to develop a computational model. The integral properties were then found using the WingPlanAnal code. Table D-2 contains the input dataset, and table D-3 contains the output from the code.

Note that the results based on measuring the published drawing are very close to the values for leading- and trailing-edge sweep angles given in the literature, which are 35°. Recall that this is a stealth airplane using parallel edge alignment; see appendix B, section “Fifteen Minutes of Stealth.”

D.2.2 Estimate C_D0 for the B-2

To estimate C_D0, the planform is broken up into strips, as shown in figure D-3. The average chord of each strip and the associated wetted area is then estimated. To obtain this data, we can use the option in the WingPlanAnal computer program to get the leading- and trailing-edge x values at a prescribed value of the span (span locations not explicitly shown above in figure D-2).

As each of the strips is a trapezoid, some side calculations were made to find the area of each strip. This was then multiplied by four to include the top and bottom of the surface as well as the area of the “other” side of the planform. The results are contained in the input dataset for the FRICTION computer program, as shown in table D-4. It would be illustrative for students to compute the values contained in the table for themselves.

Using table D-4 as the input to the FRICTION program, table D-5 contains the output of the program.

The values of skin friction are relatively low, reflecting the small multiplier of wetted to reference area, and rather large Reynolds numbers. Note that the value changes with altitude, where the Reynolds number decreases as the altitude increases, so that the skin friction increases.

D.2.3 Find and Plot the Spanload Assuming an Untwisted Wing

To find the untwisted spanload and the span efficiency, e, for this case, we start by using the VLMpc program to compute the spanload. This calculation also provides other useful information. Table D-6 contains the input dataset. The output of this code is rather lengthy, and we will provide only the key parts of it in table D-7, which is split in three parts: Geometry Data (Part 1), Aerodynamic Data (Part 2), and Complete Configuration Characteristics (Part 3) .

The spanload results are shown in figure D-4; for comparison, the elliptic spanload at the same lift coefficient is included in the figure. A useful relation is

$$\frac{c{c}_l}{c_{\text{avg}}}=\frac{4}{\pi }\sqrt{1-{\eta }^2}{C}_L$$

With the basic spanload determined, we use the LIDRAG computer code to compute the span efficiency, e. This code does a Fourier series analysis. Table D-8 contains the input dataset, and table D-9 provides the output of the program.

Using the results for the spanload shown in figure D-4 that were obtained using the VLMpc vortex lattice code, a span e of 0.95 was found using LIDRAG (see table D-9). Considering the unusual planform and the non-elliptic shape of the spanload, this is a surprisingly high value. Figure D-4 also contains the minimum induced drag (elliptic) spanload.

D.2.4 Plot the Section C_l Distribution. Where will the wing stall first? Do you see a problem?

The output from VLMpc also provides the section lift coefficient distribution. This distribution is presented in figure D-5. This shows what happens when the planform has breaks leading to variations in the chord distribution. Because the spanload naturally tends toward a smooth distribution, the section lift coefficients vary to compensate for smaller chords by increasing C_l. In addition, a pointed tip, where the chord goes to zero, results in the local section lift coefficient becoming large. Locations where section C_l values are high would be locations where the wing would tend to stall first.

D.2.5 What would you do to improve the spanload? Plot and analyze a twist distribution that will improve e. Plot the new spanload and compute e.

The LAMDES program can be used to find the twist distribution required to improve the spanload. Table D-10 contains the LAMDES input dataset, which is quite similar to the VLMpc input.

Table D-11 contains the corresponding output. Once again, the output is a lengthy text file, and relatively unimportant portions have been deleted.

Figure D-6 shows the twist distribution required to obtain the minimum drag spanload. This was found using the constant chord-loading approach in LAMDES, which may not be a good assumption for this planform. However, the results are consistent with the changes in spanload required to achieve the e = 1 elliptic spanload shown in figure D-4.

Assuming that the small chord tip section C_l values shown in figure D-5 will be dominated by viscous effects in general, we see from figure D-6 that wing stall will also occur in the midspan area. To fill in the hole in the spanload for the untwisted wing, the optimized twist distribution actually increases the local lift coefficient. The potential stall problem would be a good reason to accept the e = .95 spanload without attempting to completely fill in the spanload distribution.

D.2.6  Estimate L/D_max and the C_L required to fly at L/D_max. Comment on the implications for the operation of the B-2. What can you say about the B-2 in comparison with conventional aircraft?

Using the results from FRICTION and the spanload data analysis results from LIDRAG, we can make an estimate of the L/D variation with altitude. First, we estimate the C_L requirement as the altitude increases as shown in figure D-7 for M = 0.78 and the weight corresponding to the published value of 336,000 lbs. The lift coefficients in the cruise altitude range of 30,000 to 40,000 feet are relatively low compared to typical commercial transports. This is typical of flying wing aircraft.

Figure D-8 shows the L/D variation with altitude or two different values of C_D0, 0.0060 and 0.0080.  These plots assume M = 0.78 and the weight of 336,000 pounds, which are the same values used for data in figure D-7. Based on the results presented in table D-5, the C_D0 value of eighty counts is likely close to the value for the B-2 and agrees with the published result of a cruise altitude of 37,000 feet. The value of (L/D)_max is slightly greater than twenty-one, which is higher than typical commercial transonic transports. Indeed the B-2 is a very efficient airplane.

D.2.7 Determine the neutral point of the B-2. Examine the available information, and estimate the static margin. Does your conclusion make sense?

Using the side view in figure D-1, the CG location was estimated to be between 32.25 and 36 feet aft of the nose. This was done assuming a 15° angle between the landing gear and the CG location. With the neutral point determined from the vortex lattice method to be 32.7 feet aft of the nose, the low-speed static margin ranges from 1.1% stable to 8.4% unstable. This is in the range that would be expected for an advanced flying wing design.

Using these values, the C_m-C_L curves presented in figures D-9 and D-10 were used to illustrate the setup and advantages of near-neutral or negative static margins compared to classically stable designs. The figures show that the use of modern control system technology, allowing an unstable airplane, plays an important part in the reemergence of the flying wing concept. These figures are based on the paper by Sears.^[6]

Figure D-9 illustrates pitching moment trim for a classical stable airplane. We can see that (i) the trim requirements are highly restrictive on C_Lmax; (ii) reflexed trailing edges and upward deflection of devices to trim with increasing lift are inefficient; and (iii) sweep with washout of tips is good for both stability and aerodynamic characteristics.

Figure D-10 illustrates pitching moment trim for a neutral-to-unstable airplane, like a flying wing. Note that (i) near-neutral stability means almost self-trimming; (ii) we can use airfoils with negative C_m0; and (iii) unstable balance (relaxed static stability) and flying wing concepts are highly complementary, as the trailing edge devices work in the “natural” direction.

The important outcome of studying figures D-9 and D-10 is that a slightly unstable flying wing can trim at higher lift coefficients by deflecting the trailing edges down. This is in the right direction for achieving high lift for takeoff and landing. Thus relaxed static stability plays an important role in making the flying wing a practical concept.

D.3 Comparison of the Beech Starship and X-29

During the period from the late 1970s through the 1990s, canard concepts were popular. Burt Rutan was involved with Beech in developing the Starship. It had been certified when this project was carried out by the students. The objective was to try to understand the configuration concept and canard concepts in general. The Grumman X-29 was a good example of a potential military canard configuration and was used for comparison. The tools used for the B-2 study could be applied to these configurations. Table D-12 summarizes the work. Students were expected to consider available sources of information as needed. Item 6 is representative of a question that your boss might ask, expecting an answer in “a day or so.”

The format is similar to that of the B-2 project, but the configuration now contains two lifting surfaces. In this case, the key resource for geometry was Jane’s.^[11] The students were able to conduct their investigation without any additional information. The schematic of the Starship planform used for analysis in this project is shown in figure D-11. The estimates of center of gravity and neutral point are included. In the case of the Starship, the basic configuration was estimated to be about 10% stable. The aerodynamic analysis result is consistent with the operation of this aircraft. The Starship does not use an advanced fly-by-wire flight control system, and it is statically stable.

Figure D-12 shows the schematic of the X-29 planform used for analysis in this project. The X-29 was found to be about 32% unstable. Thus the aerodynamic analysis result is consistent with the operation of this aircraft. In contrast to the Starship, the X-29 exploits the advantages of an advanced flight control system to balance the aircraft at a significant level of static instability.

Table D-13 contains the VLMpc dataset developed using the information in figure D-11. Table D-14 is the VLMpc dataset for the X-29, created using figure D-12.

The forward position of the Starship canard, or foreplane, is connected to the extension of the Fowler flaps. The additional area of the flaps was not estimated by students in this project, and the forward position leads to an approximately neutral static margin. According to Swanborough,^[12] the area of the Fowler flap ensures the stability level stays the same as the canard moves forward. This feature illustrates the sophistication required to develop an aircraft concept. Figure D-13 show a photo of the Starship taken by Mason at the Virginia Tech airport in the early 1990s.

Figures D-14(a) and D-14(b) provide the load-sharing requirements for trim between the canard and the wing for Starship and X-29, respectively. The results change with the center of gravity position. In the case of the Beech Starship, the requirement for a stable aircraft means the canard must always operate at a lift coefficient higher than the wing. When designed properly, this results in an airplane where the canard will always stall before the main wing. In the case of the X-29, the canard is at a significantly lower lift coefficient than the wing.

The choice of center of gravity location is important in obtaining the minimum trimmed induced drag. In figure D-15(a), the Starship is shown to be limited by stability requirements from reaching the highest cruise efficiency available for the configuration. Figure D-15(b) shows the benefit of relaxed static stability technology. The center of gravity for minimum induced drag corresponds to a negative static margin. The X-29 is balanced at the edge of the minimum drag bucket. These approximate calculations were made using Lamar’s design code LAMDES,^[13] ignoring the limits to airfoil performance, which are also important.^[14] This example requires that the induced drag be calculated considering the multiple lifting surfaces and nonplanar effects. LAMDES can be used as an extended version of LIDRAG to account for these effects.

Canard effectiveness as a control is slightly unusual. The canard is an effective moment generator, but increasing the canard incidence does not produce an equivalent increase in configuration lift. In cases where linear aerodynamic theory is valid, the increased lift on the canard produces additional downwash on the wing. The result is a loss of lift on the wing roughly equal to the canard lift. In transonic and separated flow situations, the linear aerodynamic flow field model is not valid and improved calculations or testing are required. Figure D-16 shows the X-29 in high-angle-of-attack research flight on September 10, 1991, by NASA research pilot Rogers Smith  (https://www.nasa.gov/image-detail/amf-ec91-491-07). Smoke generators in the nose of the aircraft were used to help researchers see the behavior of the air flowing over the aircraft. The smoke here is demonstrating forebody vortex flow.

Figures D-17(a) and D-17(b) show the spanloads that correspond to the operation of the Starship and X-29 at their design points, respectively. Because the canard and wing are nearly coplanar, it is appropriate to combine them. Essentially, the vertical separation of the surfaces results in two distinct limits. In the first, the canard is coplanar with the wing, and the sum of the spanloads should be elliptic. As the vertical separation grows larger, individual spanloads should become ellipctic. Most canard designs correspond to the first case, and this is evidenced in the results of an optimization, as shown in figures D-17(a) and D-17(b).

Figures D-18(a) and D-18(b) show the wing incidence distribution required to achieve the spanloads shown in figures D-17(a) and D-17(b), respectively. This includes the basic angle of attack and additional twist. The design wing twist will change when the wing is in the wake of the canard. In this case, the canard wake is held flat and fixed, resulting in the most extreme condition. This shows how you need to compensate for flow nonuniformity in interacting flow fields. It is also noteworthy that the trends in twist between aft- and forward-swept wings are exactly opposite. In particular, the presence of the canard reduces the twist variation required across the wing in the case of the X-29.

Additional information on the X-29 aerodynamic design is given in several papers.^{[15],[16],[17],[18]} Many comparisons of forward/aft-swept wings and canard/tail configurations have also been published. Key reading should include McKinney and Dollyhigh,^[19] Landfield and Rajkovic,^[20] and McGeer and Kroo.^[21]

For this case study, the assessment also required consideration of a competitor aircraft, the Piaggio Avanti. In this case, the Aviation Week^[22] article provided a useful analysis of the Starship and Avanti. Some confusion exists within the literature on the aerodynamics of three-surface configurations. The definitive analysis has been given in a NASA TP,^[23] and the code is now available to students for future projects. The key benefit of a three-surface configuration is the reduction in the trim drag variation with balance location.

D.4 Term Project: Examine the YF-22 and YF-23

The US Air Force was in the process of selecting its new Advanced Tactical Fighter (ATF) when this project was assigned. Therefore, this was a timely project. The selection was scheduled to be announced about the time the assignment was due. As luck would have it, the announcement was made on the exact day that the assignment was due. The objective was to make an assessment of the aerodynamic design of the YF-22 and YF-23.

D.4.1 The Assignment

Use recent aviation magazines to establish a geometric model of each aircraft. Aviation Week & Space Technology magazine articles during Fall 1990 are a good source. Turn in an engineering report, including copies of input data sets as appendices. Use all the tools we have from class, and explain how you used them. Reference other data sources used.

1. Compare your estimate of the low-speed static margins for both candidates. (Review your notes from stability and control to recall definitions of SM and aircraft trim requirements.)
2. Compare the load sharing between the tail and wing for the YF-22 and YF-23. Do this for both up and away flight and the approach condition. Consider a range of CGs. What are the implications for selection of wing and tail airfoils?
3. Examine the control effectiveness of the tail. What is C_mδt, C_Lδt?
4. What is the span e for each plane with an untwisted wing?
5. What is the twist distribution required to obtain an elliptic spanload for each airplane? Consider both the isolated wing case and the wing-tail case.
6. Using the analysis performed above, examine and discuss the trim drag issues. What if you used thrust vectoring to help trim?
7. Estimate the skin friction drag on each airplane.
8. Make your assessment. Would you pick the YF-22 or YF-23? Explain your choice, and comment on any refinements you would make.

The students used the Aviation Week & Space Technology^[24] and Air International^[25] analyses and the book by Sweetman.^[26] Again, the requirements were similar to the previous projects, with the addition of a requirement to consider the estimation of friction drag. This allowed the students to estimate the L/D of the airplanes. Although interesting, without explicit requests, this had not been done previously.

Since the due date coincided with the Air Force announcement of the selection, student interest was extreme. Interestingly enough, a number of their parents turned out to be employed by the DOD and were able to supply an extraordinary amount of propaganda that was distributed by lobbyists.

D.5 Discussion

Let me reiterate that the case studies presented in this appendix show that a significant number of issues associated with configuration aerodynamics can be resolved without expensive CFD calculations. Students can get considerable insight and conduct good sanity checks against much more sophisticated codes using a PC. However, other aspects of the problem require the use of sophisticated CFD methods. Still other aspects require wind tunnels or flight tests at present. The role of each is identified with the use of these projects. These projects require an assessment intended to improve the student’s ability for “critical thinking.”

Figure References