This chapter provides a brief discussion of several key aspects of aircraft configuration development. As aerodynamicists, we aspire to follow Küchemann, who explained:

[The] main task that remains is to establish enough confidence to believe that, for the type of aircraft and mission under consideration, there exists regions of no conflict between the various essential characteristics, with which a set of design requirements can be met naturally. What we are really seeking is probably that “harmony” between elements . . . So we are not out for “compromise” in the sense that we can achieve some desirable characteristic only by degrading another and where a “deal” is made at somebody else’s expense. We shall endeavor to explain what is meant by this by giving examples of good design concepts. . . . On the other hand such “good design” is not likely to be one where the overall result is an “optimum” with regard to any single parameter at just one point. Instead, all the significant parameters are in harmony and not in conflict for a set of design points and off design conditions, and the final solution is sound and healthy. . . . It was Prandtl who introduced the concept of healthy flows, and we are well-advised to follow him and to search for sound and healthy engineering solutions when designing aircraft.^[1]

Küchemann’s “healthy flows” include both attached flow and well behaved separated flows. He was apparently the originator of the notion that controlled leading-edge vortex flow could be used to obtain the required low-speed lift for the Concorde. Without exploiting vortex lift for takeoff and landing, the proposed Concorde concept would have been impractical. Thus, his words provide a high standard for us when we start an aerodynamic design.

Our approach in this course is more modest, but we continue to stress the importance of the underlying aerodynamic principles and show how they contribute to configuration development. The details require further study on the part of the reader, and key references are provided. We also emphasize the development of geometry associated with aerodynamic design, particularly in the exercises at the end of the chapter. Another objective is to illustrate the engineering aspects of the process. In terms of the use of computational aerodynamics, a key goal is to define a process for using software in aerodynamic design. Although the literature typically identifies a particular code as being used for the design, the reality is that the design is the result of work of the designer plus the code. Just having the code is not sufficient. Mark Maughmer, Professor of Aerospace Engineering at Pennsylvania State University, pointed out that the code isn’t everything in airfoil design, explaining that “just having a piano doesn’t mean you are a concert pianist.” Thus, some skill is required to use the available methods, and we describe a process that helps the aerodynamicist evaluate when a code is giving the “right” answer—the infamous “sanity check” identified by Waaland as an important part of today’s engineering using sophisticated codes.^[2]

As hard as it might be for the aerodynamicist to imagine, aircraft designs are the result of the integration of several technologies, of which aerodynamics, although key, is only one. Thus the aerodynamicist is responsible as a team member for the overall concept. Traditionally, the other technologies are propulsion, structures, and flight control. For many military aircraft, stealth has become a critical technology impacting the configuration shape. Some notes describing stealth issues for aerodynamic design are given in appendix B. Detailed examples of aerodynamic design are provided in appendix D using the software available on the website as described in appendix E.

To achieve a successful design, the aerodynamicist must be a strong advocate for effective aerodynamic design. If this isn’t the case, the team will likely not produce a winning design. If the aerodynamics group fails to persuade the company of the importance of aerodynamics, manufacturing costs will be reduced by eliminating wing camber and twist and the stealth group will be allowed to dominate the design at the expense of aerodynamic maneuvering performance. In one case, enormous development costs were incurred while still resulting in an aircraft with reduced performance. In another case, the proposed design lost to the competition. To be an effective advocate, the aerodynamicist must have a good understanding of several aspects of configuration development that are addressed in this chapter.

In section 4.1, we highlight the key considerations for aircraft configuration aerodynamic layout. This is followed by configuration architecture options and their aerodynamic implications in section 4.2. The discussion in section 4.2 might be characterized as addressing the question, Why do airplanes look the way they do? In section 4.3, we define the aerodynamic design problem in terms of wing loading (W/S) and thrust-to-weight (T/W). This is a key part of the initial sizing activity and is important in defining the cruise, takeoff, and landing lift coefficients that are some of the key requirements that aerodynamicists must meet. In section 4.4, we summarize the typical aerodynamic design tasks that occur during configuration design. This is followed in sections 4.5 and 4.6 by a discussion of the use of computational aerodynamics methods in design and explanations of the computational aerodynamic design methodology that has emerged as key to achieving improved designs in practical design cycle times. Section 4.7 provides a brief summary of the present status of aerodynamic design.

4.1 Configuration Aerodynamic Layout: Key Considerations

So where do we start when considering the aerodynamic layout of an airplane? In general, form follows function. We decide on candidate configurations (form) based on what the airplane is supposed to do (function). Generally, this starts with a decision on the type of payload and the mission the airplane is supposed to carry out with this payload. This is expressed generally in terms of the following questions that should be answered first:

• What does it carry?
• How far does it go?
• How fast is it supposed to fly?
• What are the field requirements? (e.g., runway length)
• Are there any maneuvering and/or acceleration requirements?

Another consideration is the specific safety-related regulatory requirements that must be satisfied. For commercial and civilian aircraft in the US, this means satisfying the Federal Aviation Regulations (FARs)* that are managed by the Federal Aviation Administration (FAA). These regulations (https://www.ecfr.gov/current/title-14) define the takeoff and landing distances, engine-out performance requirements, noise limits, icing performance, and emergency evacuation among many others. In the European Union (EU), Joint Aviation Requirements (JARs) were developed by the Joint Aviation Authorities (JAA) to harmonize aviation standards across participating European countries. The European Union Aviation Safety Agency (EASA), created in 2003 and fully functional since 2008, has taken over most of the JAA’s functions. JARs are now called JAA Certification Specifications (JAA CS), which form an acceptable basis for showing compliance with national airworthiness codes of any nation. Military aircraft also have numerous requirements which, much like the commercial requirements, emphasize safety, including handling qualities. Appendix C in this book provides a few examples of the regulatory requirements.

With this start, the aerodynamicist, reluctantly allowing other members of the design team and management to participate, develops a configuration concept and shape that responds to the mission. At the outset, the following list shows the considerations associated with defining a configuration concept. At this stage, we begin to see that configuration design resembles putting a puzzle together. These components all have to be completely integrated.

Configuration Concept

• Lifting surface arrangement
• Control surface(s) location
• Propulsion system selection
• Payload
• Landing gear

The components listed above must be coordinated in such a fashion that the airplane satisfies the requirements given in the following list. The configuration designer works to satisfy these requirements with inputs from the various team members. This is where the team member that dominates can distort the design in a way that keeps it from responding correctly to the design requirements. For the design to be successful, the following criteria must be met.

Good Aircraft

• Aerodynamically efficient, including propulsion integration (streamlining!)
• Must balance near stability level for minimum drag
• Landing gear must be located relative to CG to allow rotation at takeoff.
• Adequate control authority must be available throughout the flight envelope.
• Design to build easily (cheaply) and have low maintenance costs
• Today, commercial airplanes must be quiet and nonpolluting.

Two books do an especially good job of covering the aerodynamic layout issues, although their titles are slightly misleading. The first of the two books is by Ray Whitford,^[3] and the second is by Abzug and Larrabee.^[4] There are aircraft design books by Raymer^[5] and Roskam^[6] that discuss configuration options.

We can translate the desirable properties of a good aircraft into specific aerodynamic characteristics. These desirable properties are outlined below.

Design for Performance

• Reduce minimum drag:
  • Minimize the wetted area (skin friction drag).
  • Streamline to reduce flow separation (pressure drag).
  • Distribute area smoothly, especially for supersonic aircraft (area ruling for wave drag).
  • Consider laminar flow (skin friction drag).
  • Emphasize clean design/manufacture with few protuberances, steps, or gaps (parasitic drag).

• Reduce drag due to lift:
  • Maximize span (must be traded against wing weight).
  • Tailor spanload to get a good span efficiency, e (twist).
  • Distribute lifting load longitudinally to reduce wave drag due to lift (a supersonic requirement according to R. T. Jones’ oblique-wing idea).
  • Consider camber as well as twist to integrate airfoils and maintain good 2D characteristics.

• Key constraints:
  • At cruise: buffet and overspeed constraints on the wing (preventing structural damage)
  • Adequate high lift for field performance (simpler is cheaper)
  • Alpha tailscrape: C_Lα goes down with sweep, AR.
  • Airport gate code: limitations on wingspans to fit existing airport gates—e.g., Code A: <15 m (49.2 ft); Code B: 15 m to <24 m (49.2 ft to <78.7 ft); Code C: 24 m to <36 m (78.7 ft to <118 ft); Code D: 36 m to <52 m (118 ft to <170.6 ft); Code E: 52 m to <65 m (170.6 ft to <213.3 ft); and Code F: 65 m to <80 m (213.3 ft to <262.5 ft)

Design for Handling Qualities

• Adequate control power is essential.
  • Nose-up pitching moment for stable vehicles
  • Nose-down pitching moment for unstable vehicles
  • Yawing moment, especially for flying wings and fighters at high-α (high angle of attack)
  • Consider the full range of center of gravity (CG) travel (ensure balanced flight operations).

• This implies you must properly balance the configuration around the CG.

Design for FAA and Military Requirements

• Safety: For the aerodynamic configuration, this means safe flying qualities.
  • FAR Part 25 and some of Part 121 for commercial transports
  • MIL STD-1797 for military airplanes
• Ability to use the airplane as a stable weapons platform
• Noise: Community noise, FAR Part 36, no sonic booms over land (High L/D in the takeoff configuration reduces thrust requirements, making the plane quieter.)

To begin discussion of the various configuration concepts, we use the successful transonic commercial transport configuration, exemplified by the Boeing 747 in figure 4-1, as a starting point. This type of configuration is mature. New commercial transports have almost uniformly adopted this configuration, and variations are minor. An interesting comparison of two different transport configuration development philosophies is available in the papers describing the development of the original Douglas DC-9 and Boeing 737 designs.^[7] Note that advances in performance and reductions in cost are typically obtained by improvements in the contributing technologies.

The Boeing 747 layout shown in figure 4-1 meets the criteria cited above. The payload is distributed around the CG. Longitudinal stability and control power comes from the horizontal tail and elevator, which has a very useful moment arm. The vertical tail provides directional stability, using the rudder for directional control. The wing-fuselage-landing gear setup allows the wing to provide its lift near the center of gravity and positions the landing gear so that the airplane can rotate at takeoff speed and also provides for adequate rotation without scraping the tail. This arrangement also results in low trimmed drag. The engines are located on pylons below the wing. This arrangement allows the engine weight to counteract the wing lift, reducing the wing root bending moment, resulting in a lighter wing. This engine location can also be designed so that there is essentially no adverse aerodynamic interference.

Having established the baseline, we next examine features of other configuration concepts that are often considered, giving a summary of the major options. Many, many other innovations have been tried, and we make no attempt to be comprehensive.

4.2 Aircraft Configuration Architecture Options

In this section, we look at a large number of options for architecting an aircraft configuration. In particular, we highlight the issues associated with sweep, including forward sweep, canards, flying wings, three-surface configurations, winglets, and variable sweep.

4.2.1 Why Swept Wings?

Many aircraft wings are swept. Just about all such wings are swept aft, but a small number have been swept forward. Thus aft sweep is part of the classical concept. Aft sweep integrates well into an aft tail configuration, as we saw above with the B-747. Generally the wing is swept to delay transonic drag rise. That’s because the compressibility effects can be associated with the Mach number normal to the leading edge on a well designed wing. The swept-wing concept was originated by German aerodynamicists prior to World War II and by R. T. Jones^[8] at the NACA during WWII. George Schairer^[9] has described the emergence of the swept-wing concept.

Below are the three key aspects of sweep.

• Subsonic (usually small sweep)
  • Adjust wing aerodynamic center relative to CG.
  • On a flying wing, get moment arm length for control.

• Transonic (significant, 20°–35°)
  • Delay drag rise Mach number (i.e., delay undesirable compressibility effects).

• Supersonic (large, 45°–70°)
  • Wing concept changes to slender wing or vortex flow wing—must distribute load longitudinally as well as laterally.
  • Reduce max cross-sectional area, and area variation occurs more gradually.

Sweeping the wing is not without drawbacks. Wing sweep increases the wing weight for a fixed span since the length of the wing spars increases with sweep to reach the specified wingspan. In addition, high-lift devices aren’t as effective when the trailing edge is swept. Wing sweep also tends to increase the likelihood of the outboard wing stalling first, leading to pitchup, a situation where the wing stops lifting well aft of the center of gravity while continuing to lift ahead of the center of gravity. This results in a sudden nose-up pitching moment and an unstable slope. Pitchup is discussed in detail in the paper by Shevell and Schaufele.^[10] This issue can be alleviated with the appropriate aerodynamic tailoring of the wing critical spanwise location. Finally, for aft sweep, the wing tends to be flutter critical.

4.2.2 Why Sweep the Wing Forward?

Several designs have adopted a forward-swept wing. It was known for some time that a forward-swept wing has the same delayed drag rise benefit as an aft-swept wing. However, forward-swept wings tend to suffer from aeroelastic divergence. This prevented them from being considered for most design concepts. One notable exception was a German bomber, the Junkers JU-287, used in WWII. The wing was swept forward so that the wing carry-through structure would be aft of the bomb that was carried on the CG. After the war, this design team produced a transonic business jet (the HFB-320 Hansa Jet) using the same logic to provide room for passengers.

With the advent of aeroelastic tailoring through the use of composites, forward-swept wing concepts were reconsidered. Studies at Grumman in the late 1970s and 1980s^[11],[12] identified numerous advantages of forward-swept wings in some applications, and the X-29 shown in figure 4-2 was built and flown.^[13]

Essentially, for transonic maneuvering using an airfoil concept–based wing, the highly loaded wing will produce a shock wave near the trailing edge. For fighters like the F-15 and F-16, the trailing edge is nearly unswept. However, for a tapered wing, the trailing edge can be highly swept if it is swept forward. This allows a shock near the trailing edge to be highly swept, which will reduce the wave drag. If the trailing-edge sweep is essentially the effective aerodynamic sweep, then the effective structural sweep will be less than the aerodynamic sweep, resulting in a structural weight advantage.

A number of other advantages have been cited in the literature:

• Less twist to attain good spanload
• Wing stalls at the root (outboard ailerons remain effective)
• Better for reduced skin friction from the possibility of achieving some laminar flow because of reduced leading-edge sweep and no fuselage boundary layer contamination
• Integrates with a canard more naturally (less twist required)
• Good high-angle-of-attack performance (root stall, ailerons keep working)

However, there are some drawbacks. The first is the aeroelastic divergence penalty that remains to some degree, even when using composite materials. In addition, these configurations have difficulty in integrating landing gear relative to the center of gravity. They are also more unstable than aft-tail airplanes to achieve good performance (the X-29 was about 30%–35% unstable) and are not easily amenable to stealth because the leading-edge sweep tends to be low. Finally, the supersonic volumetric wave drag is high because the Mach cuts tend to foreshorten the design, resulting in a lower effective fineness ratio.

4.2.3 Why Canards?

Canard aircraft have been the subject of interest ever since the Wright Brothers’ first flight. The basic argument is that the trim surface carries positive load for positive g maneuvers, reducing the load required on the wing. Grumman considered a number of canard configurations,^[14],[15] including the Research Fighter Configuration (RFC) shown in figure 4-3. In general, I think canard configurations should only be considered for designs with stringent requirements for both supersonic cruise and transonic maneuvers. The recent generation of European fighters has adopted canard configurations. The SAAB JAS 39 Grippen is a good example.^[16]

Issues related to the use of canards include:

• Reduced subsonic-supersonic aerodynamic center shift
• The need to be balanced unstable to take full advantage of performance
• The downwash from the canard unloads the wing (for forward-swept wing concepts, this is good).
• If balanced stable, C_L on the canard is much higher than the wing.
• If balanced unstable, the control system design is expensive.
• Acceptable high-angle-of-attack lateral/directional characteristics are very hard to obtain.

Canard configurations are more difficult to develop than traditional aft-swept wing configurations. In figure 4-3, note that the inlets are behind the canards. Therefore, it is possible that under some circumstances the wake from the canard could enter the inlets. When balanced stable, the demands on the canard airfoil are quite high. When the airfoils used on the canards of early Rutan designs encountered rain, it appears that the boundary layer transitioned early and as a result didn’t produce the required lift to allow level flight. These airplanes had to get out of the rain before they encountered the ground! Garrison has given a particularly cogent discussion of canard configurations, primarily for general aviation designs.^[17]

4.2.4 Why a Flying Wing?

Another concept popular with aerodynamicists is the flying wing. The advantage is that eliminating the surface area of the fuselage and control surfaces reduces the parasite drag, hence improving the aerodynamic efficiency. You also eliminate the weight of the tail. This concept can work well if extreme maneuverability is not needed. The volume distribution can be an issue. Jack Northrop devoted much of his career to the development of flying wings, and the XB-35 (first flight, June 1946) and YB-49 (first flight, October 1947) bombers employed this concept. For a variety of reasons, these designs weren’t practical at the time. Jack Northrop’s excellent paper on flying wings that he presented as the 35th Wilbur Wright Memorial Lecture in London in 1947 is still worth reading.^[18]

One particularly important technology not available to Jack Northrop was the modern digital flight control system. The flying-wing concept makes extensive use of augmentation systems for stability and control. For a wing without a vertical tail and rudder, directional stability and control requires special attention. The usual method of producing yawing moments is to use the so-called “drag rudders,” which consist of split flaps on the outboard portion of the wings. The flying wing is also synergistic with relaxed static stability. This is because, when the airplane is unstable, the trailing edge will be deflected down to bring the nose up. This means that trailing-edge deflection for control and high lift are both in the same direction.^[19] This advantage of being balanced through being unstable was known to the designers of the XB-35, but flight control technology was not developed to the point where this could be achieved. However, the XB-35 did use stability in the form of a yaw damper to improve directional flying qualities.^[20]

Another flying wing, the B-2, shown in figure 4-4, was also developed at Northrop (for more on the B-2 development, see Waaland^[21]). The B-2 shows that this concept is also well suited for stealth. For use as a passenger transport, a large volume is required, and thus the flying-wing concept is most natural for large airplanes. More references and information on flying wings are available on one of my websites.^[22]

A variation of the flying-wing concept known as the blended-wing body was proposed by the McDonnell Douglas Aircraft design team located in Southern California.^[23] See section 1.2.1 for more details.

4.2.5 Why Three-Surface Configurations?

Some aircraft concepts employ a three-surface concept, which is the exact opposite of the flying wing. Since the flying wing doesn’t have a significant moment arm for control, the center of gravity must be tightly controlled. With three surfaces, you can trim the airplane with near-minimum drag over wide CG range.^[24],[25] Sometimes, efficient component integration leads to three surfaces to save weight. However, the argument will typically be made that, if you can do the job with two surfaces, why use three? The third adds cost, weight, and wetted area.

One very good airplane, the Piaggio Avanti shown in figure 4-5, has used this approach quite effectively.^[26] In this case, the three-surface concept allows the wing torque box, the aft pressure bulkhead, and the main landing gear to share the same fuselage bulkhead^[27] (see also Roskam’s Airplane War Stories.^[28]). Note that the X-29 should also be considered a three-surface configuration since it uses a flap at the end of the aft strake in the control system in addition to the canard.

4.2.6 Why Slender Wings?

Highly swept wings appropriate for supersonic flight represent a fundamentally different concept. The Concorde is the primary example. For these wings, the idea of an airfoil embedded in a wing is not appropriate. If anything, the spanwise cross section of the wing can be considered to be the relevant two-dimensional section. Two forces must be at work here to make highly swept slender wings viable. Considering supersonic flight, these wings allow for a smooth area distribution and low maximum cross-sectional area for a given wing area. Additionally, the longitudinal distribution of lift occurs over a long length. To be viable at low speeds, the wings get extra lift due to the presence of an additional low-pressure effect on the wing when a leading-edge vortex forms on the top of the wing at landing and while maneuvering angles of attack. In the US, Polhamus is usually associated with slender-wing concepts.^[29] We will discuss these types of vortex flows in a detail in a later chapter. Another key survey on the subject was conducted by Poisson-Quinton.^[30]

Figure 4-6 shows the F-16XL, which was an experimental version of the F-16 airplane that employed the highly swept slender-wing concept. This plane was used to study supersonic performance, vortex flow aerodynamics, and laminar flow and transition at supersonic speeds.

4.2.7 Why Variable Sweep?

Variable-sweep wings have been used on a number of military aircraft. The F-14 is one such design and is shown in figure 4-7. Some of the pros and cons of variable-sweep wings are:

• Swept-back position (high-speed flight): low supersonic drag (low maximum cross-sectional area) and good “on-the-deck” ride quality (low C_Lα)
• Unswept position (low-speed flight): low landing speed (carrier suitability), efficient loiter
• Optimum sweep back available over transonic speed range
• Adds weight/complexity, currently unfashionable

Variable-sweep-wing aircraft design has been discussed by Kress^[31] and Poisson-Quinton.^[32] One of the issues in the design of variable-sweep-wing aircraft is the change of the aerodynamic center and thus the static margin as the wing sweeps.

4.2.8 Why Winglets?

Winglets have received a lot of attention since Whitcomb developed them. Figure 4-8 shows a typical winglet as used on a regional jet. Although they may not be a major architectural effect, they deserve a few comments. Below are some of the key considerations associated with winglets.

• Almost equivalent to a span extension without increased root bending moment
• Used where span limitations are important
• Good wingtip flow crucial to low drag
• The local flow field is extremely nonuniform, and for winglets to work, their design requires the use of advanced computational aerodynamics methods.

Essentially, the winglet operates in the highly nonuniform flow where the wingtip trailing vortex is forming. Two effects contribute to the reduction in drag. The local spanload is increased near the tip, reducing the downwash on the main wing near the tip and reducing the local induced drag. The other effect is that the local nonuniform flow rotates the normal force on the winglet so that it produces a thrust component relative to the freestream. We cite two references.^[33],[34] More references on winglets are available on one of my websites.^[35]

4.2.9 Propulsion System Integration Issues

Integrating the propulsion system into the configuration is always a key issue. For jet transports, the engines are either mounted on the aft fuselage or mounted on pylons under the wings. The first jet transport, the de Havilland Comet, which went into service in 1952, mounted the engines inside the wings very near the wing root. The inlets were mounted directly at the leading edge. This propulsion system installation can still be seen in the British Aerospace Nimrod, which is a direct descendant of the Comet. However, today, the preferred location for engines is under the wing, which was initially thought to result in a drag penalty until Boeing was able to demonstrate that this location could be used without penalty. Engines are still fuselage-mounted when the wings are positioned so close to the ground that there is no room for the engine. This is frequently the case for modern regional jets.

Advantages of wing-mounted engines are load relief, with the engine weight countering the lift force; good access to the engines for maintenance; and safety due to the distance from the passenger compartment. One drawback is the need to account for engine-out conditions when the thrust line is far from the centerline, as well as accounting for foreign object debris (FOD) damage when the inlets are so close to the ground. Swan and Sigalla^[36] have given an excellent overview of the issues for transport aircraft.

Other installations have also been used. Engines are placed above the wing on short takeoff and landing (STOL) airplanes when the high-lift system incorporates over-the-wing blowing, using the engine exhaust to flow over the deflected flaps, which leads to increased values of maximum lift. On most modern jet fighters, engine(s) are integrated within the fuselage. This type of installation reduces drag and contributes to improving the stealth characteristics.

4.2.10 Controlling the Airplane

Appropriate integration of control surfaces also requires careful consideration. Longitudinal (pitch) control is usually the starting point, with an aft-mounted tail being the standard configuration. Perhaps the chief consideration is controllability at high angles of attack. In particular, when the engines are fuselage-mounted, the horizontal stabilizer and elevator are typically attached to the top of the vertical stabilizer and rudder. This is known as a T-tail configuration. With a T-tail arrangement, the horizontal tail can become immersed in the wake of the wing at high angles of attack, resulting in the loss of effectiveness of the tail both as a stabilizing surface and as a control.

In the worst-case scenario for this pitchup problem, the plane can be caught in a high-angle-of-attack stable trim condition known as a “hung” or “deep stall.” The paper by Shevell and Schaufele^[37] describes the redesign of the DC-9 when the flight-test crash of a BAC-111 in October 1963 brought this problem to the attention of commercial transport designers. Military aerodynamicists were already familiar with the problem as a result of experiences with airplanes like the McDonnell F-101 Voodoo. Fighter airplanes employing strakes for high lift can also encounter hung stalls. One example is the F-16, which uses an alpha limiter (limiting the angle of attack, α) in the control system to avoid the problem. The C-17 also uses an alpha limiter in the control system.

Another aspect of the longitudinal control is the use of all-flying tails in place of the conventional horizontal stabilizer and elevator to increase the control effectiveness. This is especially important at high transonic and supersonic speeds. Yeager attributes the discovery of the all-flying-tail approach on the X-1 program to the eventual incorporation of the all-flying-tail approach on the F-86 and its subsequent outstanding high-speed performance.

The choice of the wing-planform sweep (Λ), taper ratio (λ), and aspect ratio is also critical in establishing the basic pitching moment characteristics. Typically, an aft-swept wing stalls near the tip and the forward inboard portion of the wing continues to lift, leading to a change in the slope of the pitching moment curve and to pitchup. The allowable boundaries for these parameters were developed experimentally in the 1950s and are summarized in figure 4-9, which was reproduced from a NACA report.^[38] This figure provides an indication of the allowable design space, although detailed design may allow acceptable characteristics to be obtained outside the region labeled “stable.”

The same report also includes data to show the influence of flaps, slats, and fences on expanding the stable region, which is also included in the USAF Stability and Control DATCOM.^[39]

Lateral control is normally achieved using ailerons. However, spoilers can also be used. Waaland^[40] makes the case for spoilers in his Wright Brothers Lecture. Directional stability and control is achieved using a conventional vertical tail and rudder when stealth considerations allow their use. Otherwise, nonconventional directional control must be used, with the drag rudders used on many flying airplanes being an example.

4.3 Configuration Sizing: Aerodynamic Considerations

Some of the key design characteristics can be defined with relatively little detailed information. One design consideration is wing loading (W/S), used to size the wing. The problem is multidisciplinary, and wing weight is also important in selecting the wing size. However, we can define the problem reasonably well.

The aerodynamic requirements are driven by two opposing conditions. To find the wing loading, we first consider how the wing characteristics affect the value of the specific range, sr, of the airplane (typically given in units of nautical miles of range per pound of fuel used). Equation (4-1) can be obtained (ignoring drag rise), showing how the various design characteristics affect the maximum sr, or sr_max.^[41]

$$s{r}_{max}=\frac{1.07}{sfc}{\left\{\frac{(W/S)}{\rho }\right\}}^{1/2}\frac{\{AR\cdot E{\}}^{1/4}}{{\left\{C_{D_0}\right\}}^{3/4}}\frac{1}{W}\text{(4-1)}$$

Here we see that a high value of W/S, high-altitude flight (low air density, ρ), high aspect ratio, AR, and planform efficiency, E, are desired. Similarly, specific range increases with low specific fuel consumption, sfc, drag coefficient at zero lift, C_D0, and aircraft weight, W.

If we consider maneuvering flight as well as takeoff and landing requirements, the demands on W/S are reversed. Consider the requirement for obtaining a sustained maneuver load factor:

$$n_{\text{sustained}}=\frac{q}{(W/S)}\sqrt{\pi ARE(\frac{1}{q}\left(\frac{T}{W}\right)\left(\frac{W}{S}\right)-C_{D_0})}\text{(4-2)}$$

Clearly a low W/S is a key requirement for achieving a high value of n_sustained. Similarly, the takeoff distance is related to the so-called takeoff parameter, TOP, which also includes C_Lmax, which is another key aerodynamic parameter:

$$TOP=\frac{(W/S)}{C_{L_{max}}(T/W)}\text{.}\text{(4-3)}$$

The landing distance can be related to the approach speed:

$$V_{APP}=17.15\sqrt{\frac{W/S}{\sigma \cdot C_{L_{APP}}}}(\text{knots})\text{(4-4)}$$

In each of these cases, a low W/S is desired (σ is the ratio of air density at a specific altitude to the density at sea level). For the takeoff and landing cases, the need for a low W/S and high lift coefficient must be considered in a trade study.

So how is the selection of wing loading made? A constraint diagram is used to help the design team understand the requirements. Figure 4-10 is a notional constraint diagram adapted by McDonald^[42] from Loftin,^[43] which illustrates the situation. Typical constraints for a transport aircraft are:

• Cruise
• Takeoff field length
• Landing field length
• Second segment climb gradient
• Missed approach
• Top of climb rate of climb

See appendix C for the relevant specifications of each of these requirements. They are defined very precisely by FAA and military specifications.

For some cases involving point performance (local quantities), the trade between aerodynamics and structures can be found analytically. As an example, consider the transonic maneuver–dominated fighter. Minimizing the sum of the wing and engine weight, the maneuver lift coefficient can be found to be

$$C_{L_{MD{P}_{opt}}}=\sqrt{\pi ARE(C_{D_{0_W}}+\frac{1}{q}\frac{d{W}_{WNG}/d{S}_W}{d{W}_{ENG}/d{T}_{RE{Q}^{\mathrm{\prime }}D}})}\text{(4-5)}$$

We see the connection between the aerodynamic, structural, and propulsion characteristics. For the derivation of this equation and example applications, see the paper by Mason.^[44]

4.4 Overview of the Specific Aerodynamic Design Tasks

Initially, the aerodynamicist works with the design team to establish the appropriate concept(s) for meeting the design requirements. The wing planform as well as the control approach should be chosen, and the wing loading and thrust-to-weight should be estimated, as described in the preceding sections above. Targets are set for the design lift coefficients at key points in the mission, such as cruise, takeoff, landing, and any sustained and instantaneous maneuver requirements. The wing sweep and maximum t/c are naturally part of a design tradeoff between the structural and aerodynamic requirements. Another consideration may also be the wing volume available for fuel. Once the wing sweep and thickness distribution are selected, it will be very difficult to change them because of their pervasive impact on many other parameters.

The baseline configuration geometry is obtained from the configuration designer. The aerodynamic design job starts with an analysis of the baseline geometry to establish the performance relative to the requirements. Then the aerodynamic designer modifies the geometry to improve the aerodynamic performance. Remember that one definition of aerodynamics is “50% flow field and 50% geometry” (though actually, geometry is much more than 50% of the day-to-day work of the aerodynamics designer). The aerodynamicist controls the flow field by manipulating the geometry. Thus geometric modeling is a key aspect of an aerodynamicists’ job. Appendix A provides the geometry definitions of commonly used airfoils and bodies of revolution.

The next item of business is to obtain the neutral point* of the configuration and work with the configuration designer to ensure that the wing is placed longitudinally on the fuselage to obtain the desired stability level. To do this, you may need to do a minimum trimmed drag analysis to establish the desired stability level. The weights engineer (historically referred as the weights “guy”) defines the center of gravity. If the airplane is to fly at supersonic speeds, the volumetric wave drag analysis should begin immediately and the cross-sectional area distribution should be developed to minimize the wave drag. An important aspect of this work is to ensure that the maximum cross-sectional area is minimized. The other initial aerodynamic task is to estimate the parasite drag. This requires the wetted area of the configuration at a component-by-component level.

Once this work is done, the detailed aerodynamic design can begin. The nominal t/c distribution is typically defined during the initial studies, and once specified, an appropriate airfoil can be either picked or designed. Given the wing planform and thickness, the wing camber and twist are found. This is a major part of the detailed design effort. In addition, at this time, the high lift system requirements are defined and a high lift system design is selected to meet the requirements. Although the performance can be predicted with some certainty at key design conditions, issues with the airplane’s handling qualities are associated with the boundaries of the flight envelope, where significant separated flow exists; they are also associated with flight with unusual combinations of controls, engine thrust, etc. As such, once the basic aerodynamic design is done, much of the remaining effort, involving wind tunnel and flight testing, will be devoted to “fine tuning” the shape to obtain the desired handling qualities.

Another consideration in defining the aerodynamic shape is the difficulty of manufacturing complicated shapes. Ultimately, the master dimensions, created by the lofting group, controls the contour, and this group may change the shape specified by the aerodynamicist. If the aerodynamicist specifies the shape at only a few span stations (e.g., the root and tip airfoils and wing root incidence and wingtip washout), the contours between these control stations may not be the contours expected by the aerodynamicist. Work the exercises at the end of this chapter to derive the details substantiating this statement.

4.5 Use of Computational Aerodynamics in Aerodynamic Design

Today, computational aerodynamics plays a key role in aerodynamic design, and we start with some sage words from one of the most inventive aerodynamicists in US history, R. T. Jones, whose book highlights his concerns about the use of computational aerodynamics.

Aeronautical calculations today rely on the awesome power of the computer. However, as has been observed, power can corrupt. Equipped with an appropriate address book, giving the location and availability of various programs, the aeronautical engineer can now command the solution of a great variety of aerodynamics problems. Moreover, the capacity of the computer has made possible the inclusion of many small physical influences that until now had to be neglected but sometimes create a false impression of high accuracy. However, the basic physical assumptions of calculations, if they are discussed at all, are often not given adequate treatment. If ‘computer aerodynamics’ is to realize its full potential, then more attention must be devoted to these underlying principles.^[45]

Although the powerful software described by Jones can, in many cases, be used on a standard laptop computer, the user must acquire experience with it before using it to make design decisions. In fact, experts are continually evaluating the accuracy of their methods. A set of test cases was developed by the Fluid Dynamics Panel Working Group, and selected test results and detailed geometric descriptions were published in advisory reports by AGARD.^[46],[47]

A series of CFD Drag Prediction Workshops (DPWs) has been held in the US since 2001 to help the developers and users of computational aerodynamics to better understand the accuracy of the computational methods. For additional information, see the references cited in section 3.6.1. In 2003, the prediction of stability and control characteristics using CFD was the topic of a symposium sponsored by NASA.^[48] In the following sections, we list a few steps that should be taken when using computational aerodynamics programs.

4.5.1 Steps to Take When Using a Program for the First Time and on a New Configuration

The checklist that we provide is perhaps obvious. In fact, it is sometimes apparently so trivial, we are tempted to skip some of the steps. Speaking from personal experience, skipping steps is always a big mistake.

4.5.1.1 Initial Validation

1. Demand that you be provided a sample input and output.
   It is impossible to use a code obtained from any source without checking that you have a version that actually works properly. Together with the user’s manual, you must also obtain a sample input and corresponding output. Without these files, the code is unlikely to be worth your time to try and use.
2. Run the code yourself using the provided sample input.
   The next step is to run the code on the platform you intend to use with the input sample dataset you were provided. Often, the code won’t run on a system even slightly different than the one on which it was developed. At this point, some interaction with the provider of the code is typically required. This should be done the same day you obtain the code!*
3. Carefully study the output ,and compare with the sample output provided.
   You need to examine the input and the output obtained on your system with the sample output provided. There are two reasons for this. First, you need to make sure you get the same answers. Often you won’t. When this happens, you need to try to understand why the answers are different. Often, the sample input you were provided doesn’t correspond exactly to the one used to create the sample output. I’ve been guilty of doing this. The second reason to study the sample input and output in detail is to learn the details of the code and its capabilities. At this point, make sure you understand factors such as the nondimensionalizations, the detailed definition of the reference area, and the exact coordinate system used, among others. Collect your questions, and contact the person that sent you the code. However, don’t call too quickly. Review your issues, making sure you really need to ask the question. You need to establish that you are a serious and informed user if you expect to get support. This is especially true if you are not paying for support.
4. Investigate sensitivity to various parameters.
   Today’s aerodynamics codes come with numerous options. You will never be able to test every combination. However, establish which ones are key for your problem, and investigate their effects. The options typically come in two classes. One class will be associated with obtaining the numerical solution. This includes convergence criteria, numerical step sizes, the number of panels, and the number of grid points, among other options. Make sure you understand how to exercise the code to obtain a solution that is converged with respect to these factors. The other class of options is associated with the flow physics. This group includes the options for turbulence models, boundary condition treatment, and possibly differences in behavior, depending on Mach and Reynolds numbers. This type of study is another important step in developing experience using a particular code.

4.5.1.2 Configuration Buildup Approach

Once you have performed the steps outlined above, you are ready to start using the code for your own work.

1. Make up your own test case.
   Pick a case “close” to the one you are trying to solve and for which there is an analytic solution available, whether it’s in the form of a published numerical solution or experimental data. Run the code to compare to the other results. See how closely you match this result, and try to understand the reason for any differences.
2. Finally, start to use the code for the configuration you are interested in investigating.
   1. Start with the simplest possible model. This is probably an isolated airfoil or wing. Investigate the solution and its convergence process. Study the physics.
   2. Add the tail and/or canard to the isolated wing case. Does the code still work? What is the effect of the added component?
   3. Add the fuselage to the isolated wing case. What are the fuselage effects on the results?
   4. Finally, run configuration with the full level of geometric complexity. Following this procedure, you will gain confidence in the results and will be able to identify the contributions of the components to the complete results.

4.6 A Review of Detailed Computational Aerodynamic Design Approaches

This section originated in a three-volume report written a few decades ago.^[49] Revisions have been made to the software program to reflect current practices. Surveys of the use of computational aerodynamics have appeared regularly since the beginning of computational aerodynamics’ extensive use in aerodynamic design. Among the many reviews, we cite two surveys. Jameson^[50] discusses the role of CFD in the design of aircraft, providing an overview of the design process in a multidisciplinary environment. Johnson, Tinoco, and Yu^[51] provide specific examples of CFD application to Boeing designs in their Seattle facility.

4.6.1 Introduction: Analysis Versus Design

Although the use of the computer to simulate the flow field about a vehicle with a specific geometric configuration is an extremely useful and important capability, it is an indirect response to the aerodynamic design question. The aerodynamic design question is typically posed at several levels, starting with some vague and general question about the “best” shape of the airplane for a particular mission, before proceeding to more specific and detailed questions concerning the actual wing (and fuselage) lines, which are subject to a large variety of constraints. In the old paradigm, the aerodynamicist designs a wing using the methods of computational aerodynamics, the lines are given to the contour development group, and a wind tunnel model is built and tested. In the analysis mode, aerodynamic computer programs are used to simulate a wind tunnel. Of course, the computer simulation can be used much sooner in the design cycle than a wind tunnel test, and this strategy should produce an improved final design at a reduced cost in a shorter time period. This was the proper initial introduction of computer simulations into the wing design process. Indeed, this technique was used for subsonic and supersonic wing design since the 1960s, employing linear aerodynamics methods. Subsequently, transonic wing design using fully transonic three-dimensional wing-body computer methods was done in a similar manner.

Once the computer is introduced into the design cycle, it becomes evident that it can be used in a fundamentally different mode than to simply supplement wind tunnel testing. The use of flow field simulation in this manner is naturally referred to as the “design mode,” as opposed to the “analysis mode” of operation. A design mode has been available for linearized subsonic and supersonic flow fields since shortly after the analysis codes became available. The most extensive use of a design mode appears to have been the elaborate system of linear aerodynamics supersonic wing design codes that evolved from the work of Carlson and Middleton,^[52] developed for the US SST program.

After a brief review of the design problem and some of the methods used over the years, we describe the current approach to aerodynamic design. We include a few illustrations of very simple problems to provide some insight. Specific codes will be discussed in more detail in subsequent chapters.

4.6.2 Review of the Computational Design Process

A variety of possibilities emerge when the problem formulation for a design mode of operation is explored. The reason for this range of possibilities can be attributed to the manner in which the design problem is posed, as noted above. Ideally, the aircraft designer would specify the aircraft mission (or missions) and a computer program would provide the detailed lines of the optimum aircraft, but such a smart computer program will not exist for some time. However, most aircraft companies and governmental agencies routinely employ programs that predict the gross features of an optimum aircraft for a particular mission with some assumption regarding the rate of development of various technologies. These programs use low-fidelity models of the various disciplines, as well as databases developed from previous aircraft designs. Typical aerodynamic outputs from the programs are takeoff gross weight, wing area, wing loading, and planform details such as aspect ratio (AR), taper ratio (λ), sweep (Λ), and thickness ratio (t/c). Usually, a target/assumed drag level for the configuration is also specified. Examples of this type of program are the NASA ACSYNT program^[53] and the NASA FLOPS program.^[54]

Hence, the computer is used to determine the overall features of the required airplane. The typical aerodynamic design problem thus becomes less vague and more manageable, with the statement being reduced to something along the following process.

Given:

• AR, λ, Λ, t/c (basic geometric parameters)
• M, Re (flight conditions)
• C_Lcruise or C_{Dmax allowable} (design goals)

Find:

• C_Dmin for C_Lcruise or C_Lmax for C_{Dmax allowable}
  
• Detailed geometry (i.e., detailed aerodynamic aircraft design definition), subject to geometric constraints on twist, camber, root bending moment, etc., and subject to aerodynamic requirements on performance at other flight conditions.

At this point, we could begin to consider the direct use of a computer code to help determine the optimum aerodynamic shape and performance that can be obtained for the specified problem. More typically, the aerodynamic designer employs his experience and judgment to specify a desired pressure distribution (unfortunately, it appears that designers with this ability are becoming increasingly rare). This type of program is usually described as an “inverse method,” while a program that attempts to address the problem more directly is usually termed an “optimization method.” The “classical optimization” approach uses well established numerical optimization methods to find the aerodynamic shape. Each of these approaches has its own strengths and weaknesses. A contrast between optimization and inverse methods is summarized below.

Classical Optimization

• Requires many analysis submissions for a single design case
• Solution depends critically on the user-assumed form of the answer.
• Can handle a variety of geometric and off-design constraints
• If performed through a large optimization code, solution is not obtained from “aerodynamic thinking.”

Inverse

• Generally almost as fast as a single analysis
• The geometry may not always exist for a given pressure distribution.
• Difficult to treat off-design and geometric constraints
• Solution is a direct result of best current “aerodynamic thinking.”

Another drawback of the optimization approach is that the path taken to the final result is often rather obscure, ensuring that the relative importance of the various aspects of the final design produced in this manner are not readily apparent.

The original numerical optimization techniques employed in the design methods were of the search type and did not employ any of the elements of calculus of variations to obtain the maxima. More importantly, in fluid mechanics, it was not clear how to find the aerodynamic gradients of design variables without using simple finite-difference approaches. This meant that many additional calculations had to be made at each optimization iteration. Although an entire book by Angelo Miele^[55] had been devoted to aerodynamic optimization using calculus of variations, these concepts were not used until Antony Jameson introduced the current modern methods for aerodynamic design. His adjoint methods are closely connected to classical calculus of variations and control theory.^[56] Jameson’s 1997 paper contains numerous references to his design research. The advantage of current modern methods is that the gradients of the solutions can be obtained very efficiently.

4.6.3 Examples of Design Methods and Issues Drawn From Two-Dimensional Studies

A variety of numerical approaches have been used to design transonic airfoils. The book edited by Thwaites^[57] discusses the classical approaches to the incompressible inverse methods and points out that some judgment must be used by the designer in specifying the desired pressure distribution. A solution does not necessarily exist. You cannot specify any arbitrary pressure distribution and obtain a real geometry. Inverse methods for transonic speed airfoil design have to contend with this same problem. However, in practice, aerodynamicists have been able to use inverse methods without any undue hardship. A review of inverse methods is available in the AIAA book edited by Henne,^[58] in articles by Drela,^[59] and in Volpe’s writing on transonic flow.^[60] The computer programs have proven to be very useful.

Another approach to transonic airfoil design must be mentioned in any review, although it isn’t used today. Hodograph methods were used to design some very good airfoil sections. The method worked well in the hands of the skilled users at the Courant Institute.^[61] One of the main problems with the method was extending it to three dimensions.

The numerical optimization approach to airfoil design is more recent, unlike the inverse methods, which were available in the 1940s for subsonic flows. However, like many of the currently used aerodynamics methods, inverse methods for detailed aircraft work were not routine engineering tools until the widespread availability of computers. An initial study of numerical optimization applied to aerodynamic design was presented in 1974 by Hicks, Murman, and Vanderplaats.^[62] The underlying idea in this approach is to couple a modern numerical optimization code with an aerodynamic analysis code. The airfoil design problem is then cast as an optimization problem, and the entire apparatus associated with optimization methods is brought to bear on the problem. The most attractive aspect of the optimization method is its ability to handle design constraints. These constraints include both off-design performance requirements and design-point geometry restrictions. The report by Vanderplaats and Hicks^[63] provides a detailed description of the techniques used to formulate the design problem as an optimization problem.

The optimization method has become the standard approach to aerodynamic computational design. However, there are some drawbacks that need to be addressed. These drawbacks are in part related to computer run times. In optimization methods jargon, optimization methods minimize an “objective function,” which is a function of a set of design variables subject to a set of constraints. The objective function could be drag, for example, while the design variables are typically the variables used to specify the shape of the airfoil. The constraints could be a minimum lift coefficient, a prescribed pitching moment, airfoil thickness, off-design drag values, or virtually any other requirement that might arise in practice. The selection of the appropriate objective function and design variables is crucial to the success of optimization methods.

The design variable specification is perhaps the biggest challenge in the application of optimization methods. In principle, the number of airfoil ordinates used to specify the shape could each be used as design variables, however, if sixty upper-surface and forty lower-surface points (a typical number of ordinates) are used, then there are 100 design variables. In practice, no more than about ten independent design variables can be treated reliably. Thus the airfoil shape must be constructed from shape functions that describe more than a single ordinate (i.e., coefficients of polynomials used to approximate airfoil shapes).

Experience led to the realization that polynomials were not appropriate shape functions, and schemes that use linear combinations of present supercritical shapes and local geometric perturbations to these shapes appear to be the most practical method to obtain useful results with a small number of design variables. Thus the linear combination of known airfoil shapes, as used by Vanderplaats and Hicks,^[64] and the use of shape functions obtained using inverse methods proved very effective.^[65] This approach also proved effective in three dimensions, although it requires some modifications. It is important to realize that the optimization method will only identify the best of a particular set of possible airfoil shapes arising from the shape functions. If the actual optimum airfoil is not among this set of shapes, the method cannot find this shape. Hence, the optimization methods also require the user to apply insight into the problem.

Three key issues are worth mentioning. Jameson and his coworkers have addressed the issue of low-cost gradient calculations.^[66] They combined the efficient calculation of gradients with a numerical optimization procedure to obtain an aerodynamic design procedure. Their work has been demonstrated in numerous applications.^[67] Another issue is the need to avoid designs that are too narrowly optimized. Any practical design must be efficient over a range of flight conditions and in the presence of possible uncertainty in the shape specification. The work of Huyse and his collaborators^{[68],[69],[70]} provides practical methods of addressing these issues. The third key area of concern is reflected in the work carried out at Virginia Tech^[71] addressing the issue of using high-fidelity aerodynamics in the early stages of design by exploiting parallel computing to precompute aerodynamic results for the specific design space and using statistical methods to interpolate this “data base” during optimization studies. This approach is tailored to multidisciplinary design since other disciplines are also involved in optimizing an entire system.

The comments concerning inverse and optimization methods in 2D in the previous section carry over to the 3D design case. One curious aspect of the 3D inverse and optimization methods is that the solution may be non-unique near the wing root, a result that was reported by Sloof.^[72] This occurs because the same pressure distribution can be obtained by shaping the surface on either side of the junction. Although complicating the design method, the result is more freedom available to the designer.

In the next section, we illustrate the possible use of the three-dimensional transonic methodology in a design environment by applying it to two model problems.

4.6.4 Application of the 3D Transonic Program to Wing Design Problems

The feasibility of using the present computer program in transonic wing design as more than a straightforward analysis tool was investigated through two model problems. The first model problem was conducted making use of the NASA optimization program CONMIN.^[73] The main purpose of the exercise was to gain familiarity with the use of optimization codes in aerodynamic applications. The second model problem was undertaken to assess the effort required to introduce an automatic geometry alteration loop driven by the results of a previous iteration into the code. The stability of this type of iterative procedure is also of interest.

4.6.4.1 First Model Problem

The first model problem provided an opportunity to obtain experience using CONMIN. The problem was specified simply as follows: Using lifting-line theory for the aerodynamic representation of the finite wing, have CONMIN determine the twist distribution required to minimize the induced drag. In this case, the exact solution can be found for straight tapered wings to be

$${\alpha }_g(\eta )=\frac{C_L}{\pi AR}\{1+\frac{AR(1+\lambda )}{\pi }\cdot \frac{{(1-{\eta }^2)}^{1/2}}{[1-(1-\lambda )\eta ]}\}\text{.}\text{(4-6)}$$

For an untapered wing, equation (4-6) for α_g shows that the basic incidence variation along the span is elliptic. Observing the functional form of the exact solution, we note that this particular ratio of the root of a second-order polynomial to a first-order polynomial would have been an unlikely selection for the assumed variation of spanwise twist. To repeat, unless equation (4-6) was contained as a subset of the functional forms selected for the optimization study, the true optimum twist distribution would not have been found. This fact serves to demonstrate the importance of using the insight gained from analytical theories to maximize the benefits of numerical solutions.

Indeed, initial efforts to obtain the minimum solution using a cubic polynomial for the span variation of twist were not particularly satisfying. The results never approached the true minimum, and apparently there were several combinations of coefficient values that were equally close to true minimum, such that several substantially different answers for the twist variation could be obtained depending on the initial guess supplied to the program. These calculations typically took on the order of ten iterations, each of which required a number of function evaluations to obtain the local gradient of the objective function. In aerodynamic terms, this means that there were ten main executions of the aerodynamic program and a number of “small” executions that were required to be run long enough to provide the local gradient of the solution with respect to each design variable. It is clear that this can quickly lead to an immense amount of computational effort.

Finally, the optimization scheme was run with the design variables consisting of a coefficient to equation (4-6) and the coefficient of an additional term added to equation (4-6). Figure 4-11^[74] shows the path through design space for this two-parameter optimization run. Note that the minimum occurs when β₂ = 0, and β₁ ≈ 1 (β₁ is not exactly because a lift-curve slope slightly different than 2π was employed). The run terminated after eight iterations, with the numerical solution predicting that the optimum had been achieved.

Figure 4-12 shows a close-up view of the last iterations of the path through the design space. The result demonstrated that the program could in fact select the true optimum if it was embedded in the design variable space. This effort demonstrated both the difficulties and possibilities associated with the use of optimization methods.

4.6.4.2 Second Model Problem

The second model problem is considerably different in concept. For this problem, the question posed was simple: For a given planform and spanload, determine the twist required to produce the spanload. Initially, lifting-line theory was employed to verify that the basic iteration scheme adopted would converge for a simple aerodynamics model before attempting to incorporate the iteration into a transonic computational method. The twist was determined by adjusting the section incidence at the finite set of span stations at which the computation provided results, without making any assumption concerning the functional relationship between the incidence at adjacent span stations. The basic iteration tested was

$${\alpha }_{D_j}={\alpha }_j^K+\left(\frac{C_{l_{D_j}}-C_{l_j^K}}{C_{l_{{\alpha }_j}}}\right)\text{(4-7)}$$

where j denotes the particular span station, D denotes the design condition, and K indicates the iteration number. C_lα is approximated by

$$C_{l\alpha }^k=\frac{C_l^k-C_l^{k-1}}{{\alpha }^k-{\alpha }^{k-1}}\text{.}\text{(4-8)}$$

For the lifting-line simulation, this iteration procedure converged to the exact solution given by equation (4-6) in about four or five iterations. This result was obtained without difficulty even though the approximation given in equation (4-8) is poor for numerical computation due to the progressively smaller differences between the values as the iteration converges.

Equation (4-7) is equivalent to a more general form,

$$\left\{{\alpha }_D\right\}=\left\{{\alpha }^K\right\}+{\left[{\tilde{A}}^{K-1}\right]}^{-1}\{C_{l_D}-C_l^K\}\text{(4-9)}$$

where [A] with a tilde on top in equation (4-9) is an approximation to the actual influence coefficient matrix [A], which relates C₁ and α:

$$\left\{C_l\right\}=[A]\{\alpha \}\text{.}\text{(4-10)}$$

In the present method, [A] with a tilde on top in equation (4-9) has been given by the extremely simplified relationship in equation (4-8) for the diagonal terms, with the off-diagonal terms assumed to be zero. This result shows that [A] in equation (4-10) can be crudely approximated if an iteration is allowed to determine the final result. Naturally, as the approximation to the matrix [A] improves, the number of iterations required is reduced.

The basic inviscid program was modified to incorporate this type of iteration scheme without difficulty. It was found that a relatively fine grid was required to obtain the straight-wing result computed previously using the lifting-line aerodynamic model. Refinements to the iteration included the use of underrelaxation of the twist increment and the use of the initial C_lα value for all iterations. These refinements led to a smoothly converging solution that took about 50% longer than the basic solution. The method was then applied to a 45° swept untapered wing. The refined procedure led to a solution with the results obtained shown in figure 4-13, which also contains the straight-wing results for comparison. In this case, attempts to compute the result with C_lα changing from iteration to iteration led to a diverging result at the point where no shift in angle was required (about 45% semispan), and this experience showed that, in an actual production program, an improved approximation to [A] should be used. However, this improved approximation can be constructed without difficulty so that a design option of the type described above could be included in the basic analysis program.

In this section, we have demonstrated the variety of possibilities that arise when incorporation of design options is suggested. One of the options would provide immediate benefits to the designer, allowing analysis codes to be easily modified to provide design options.

4.6.5 Design Within the Context of Multidisciplinary Design Optimization

Broader issues related to aerodynamic design within the MDO context, which considers other disciplines simultaneously, have been the subject of extensive research at Virginia Tech since the 1990s. An overview of our thinking is given in a paper by Giunta, et al.^[75] Another source is the MAD Center Report 96-06-01 (Virginia Tech, AOE Dept., Blacksburg, VA, June 1996). The ability to combine high-fidelity results from numerous disciplines in early design is an important area of research for configuration aerodynamics.

4.7 Summary of the Status of Aerodynamic Design

Aerodynamic optimization has become practical using CFD. The ability to use it in conjuncture with other disciplines simultaneously is currently being addressed. We conclude with a quote from a 2004 paper by Jameson:

The accumulated experience in the last decade suggests that most existing aircraft which cruise at transonic speeds are amenable to a drag reduction of the order of 3 to 5 percent, or an increase in the drag rise Mach number of at least 0.02. These improvements can be achieved by very small shape modifications, which are too subtle to allow their determination by trial and error methods. When larger scale modifications such as planform variations or new wing sections are allowed, larger gains in the range of 5-10 percent are attainable.^[76]

Figure References