Transonic flow occurs when there is mixed subsonic and supersonic local flow in the same flow field (typically with freestream Mach numbers from M = 0.6 or 0.7 to 1.2). Usually the supersonic region of the flow is terminated by a shock wave, allowing the flow to slow down to subsonic speeds. This complicates both computations and wind tunnel testing. There is also very little analytic theory available for guidance in designing for transonic flow conditions. Importantly, not only is the outer inviscid portion of the flow governed by nonlinear flow equations, but the nonlinear flow features typically require that viscous effects be included in the flow field analysis for accurate design and analysis work. Hypersonic vehicles with bow shocks necessarily have a region of subsonic flow behind the shock, so there is an element of transonic flow on those vehicles too.

In the days of propeller airplanes, the transonic flow limitations on the propeller mostly kept airplanes from flying fast enough to encounter transonic flow over the rest of the airplane. Since the propeller was moving much faster than the airplane, adverse transonic aerodynamic problems appeared on the propeller first, limiting the speed and thus transonic flow problems over the rest of the aircraft. However, WWII fighters could reach transonic speeds during dives, and major problems often arose. One notable example was the Lockheed P-38 Lightning. Transonic effects prevented the airplane from readily recovering from dives, and during one flight test, Lockheed test pilot Ralph Virden had a fatal accident. The pitching moment changing with the Mach number (Mach tuck) and Mach-induced changes in control effectiveness were major culprits identified in a paper by Foss.^[1] This paper also contains the estimated drag rise characteristics of the P-38.

The invention of the jet engine allowed aircraft to fly at much higher speeds. Recall that the Germans used the Me 262 at the end of WWII in 1944, and the Gloster Meteor was apparently the first operational jet fighter. Since the advent of the jet engine, virtually all commercial transports now cruise in the transonic speed range. However, as the Mach number increases, shock waves appear in the flow field, getting stronger as the speed increases. The shock waves lead to a rapid increase in drag, both due to the emergence of wave drag and also because the pressure rise from a shock wave thickens the boundary layer, leading to increased viscous drag. Thus cruise speed is limited by the rapid drag rise. To pick the value of the Mach number associated with the rapid increase in drag, we need to define the drag divergence Mach number, M_DD. Several definitions are available. The one used here is the Mach number at which dC_D/dM = 0.10. Review section 3.5.4 for more.

After WWII, it was found that the Germans were studying swept wings to delay the drag rise Mach number. Allied examination of German research led to both the North American F-86 and the Boeing B-47 designs being changed to swept-wing configurations. The idea of swept wings can be traced to Busemann’s paper at the fifth Volta Congress held in Rome, Italy, in 1935 on the topic of high velocities in aviation and to the wartime ideas of R. T. Jones at NACA. Ironically, the airplane used for the first manned supersonic flight (the X-1, in 1947) did not use a jet engine or wing sweep. It was a rocket-powered straight-wing airplane. However, shortly thereafter, the F-86, a swept-wing jet-powered fighter, went supersonic in a shallow dive.

Thus, advances in one technology, propulsion, had a major impact on another, aerodynamics, illustrating the need to carefully integrate various technologies to achieve the best total system performance. The formal process of performing this integration has become known as multidisciplinary design optimization (MDO).

6.1 Physical Aspects of Flowfield Development With Mach Number

Figure 6-1, taken from the classic training manual, Aerodynamics for Naval Aviators,^[2] shows how the flow develops with increasing Mach number, starting from subsonic speeds. At some freestream Mach number, the local flow becomes sonic at a single point on the upper surface where the flow reaches its highest local speed; this is known as the critical Mach number. As the freestream Mach number increases further, a region of supersonic flow develops. Normally the flow is brought back to subsonic speed by the occurrence of a shock wave in the flow. Although it is possible to design an airfoil to have a shock-free recompression, this situation is usually possible for only a single combination of Mach number and lift coefficient, with shock waves appearing if one deviates from this shock-free design condition. As the Mach number increases, the shock moves aft and becomes stronger. As the Mach number continues to increase, a supersonic region and shock wave also develop on the lower surface. As the Mach number approaches one, the shocks move all the way to the trailing edge. Finally, when the Mach number becomes slightly greater than one, a bow wave appears just ahead of the airfoil and the shocks at the trailing edge become oblique. These shock waves are the basis for the sonic boom. Many variations in the specific details of the flow field development are possible, depending on the specific geometry of the airfoil.

This typical progression of the flow pattern with increasing Mach number, shown in figure 6-1, leads to rapid variations in drag, lift, and pitching moment with change in Mach number. Today we can predict these variations computationally. However, when these changes were initially found in flight, they were dangerous and appeared mysterious to designers because there was little to no understanding of the fluid mechanics of the phenomena.

Note that problems with pitching moment variation with Mach number and the flow field over a control surface using a hinged deflection led to the introduction of the all-flying tail in the X-1 and later models of the F-86. Subsequently, all-flying tails became standard on most supersonic tactical aircraft. This was considered an important military advantage, and was classified for several years. Based on military experience, Lockheed used an all-flying tail on the L-1011, while the other transport manufacturers continued to use a horizontal tail and elevator.

6.2 Technology Issues and Developments

6.2.1 The Slotted-Wall Wind Tunnel

Several advances in technology were key to our ability to design efficient transonic aircraft. Wind tunnel testing posed several problems, including the reflection of shocks from the tunnel walls and the tendency of the flow to choke as it passed the model. As a result, wind tunnel results were completely wrong. The invention of the slotted-wall wind tunnel at NACA Langley in the late 1940s allowed the tunnel interference effects to be significantly reduced, leading to practical wind tunnel testing methods. A chapter in Becker’s book^[3] describes how this capability was achieved. It’s a must read to get insight into the aerodynamic research and development process, as well as to get a physical understanding of how airfoils work and how the slotted-wall tunnel evolved. The slots are clearly shown in figure 6-2.

Using the eight-foot slotted-wall wind tunnel at NASA Langley, Whitcomb, who had earlier developed the area rule, made a breakthrough in airfoil design. His supercritical airfoil designs spurred renewed interest in airfoil design for increased efficiency at transonic speeds.^[4] His group also had to figure out how to simulate the full-scale Reynolds number at sub-scale conditions, as discussed by Blackwell,^[5] who worked for Whitcomb at NASA before going to work for Lockheed. Finally, in the early 1970s, breakthroughs in computational methods produced the first transonic airfoil analysis codes, which are described briefly in the next section.

6.2.2 Computational Challenges and Methods

The theoretical/computational transonic flow problem was very hard. Initially, very few analytic or numerical methods were available. By 1970, it seemed like everybody was trying to come up with a method to compute the transonic flow over an airfoil. What was known as the “blunt-body problem” had just been conquered (see Chapter 10), which was important in predicting the flow field at the nose of reentering ballistic missiles and the manned space program. The transonic problem is difficult because it is inherently nonlinear, and the steady flow assumption changes the mathematical type of the equation, being elliptic in the subsonic portion of the flow and hyperbolic in the supersonic part of the flow. Earll Murman and Julian Cole made the major breakthrough.^[6] Using transonic small-disturbance theory, they came up with a scheme that could be used to develop a practical computational method. Equation 6-1 illustrates how the nonlinear term in the equation allows for the equation to change type and indeed provides a way for the type to be found locally in a computational scheme.

$$\underset{\begin{array}{c}>\text{0 ellipitic PDE}\\ <\text{0 hyperbolic PDE}\end{array}}{\underbrace{[1-M_{\mathrm{\infty }}^2-(\gamma +1)M_{\mathrm{\infty }}^2{\varphi }_x]}}{\varphi }_{xx}+{\varphi }_{yy}=0\text{(6-1)}$$

In Murman and Cole’s scheme, shocks emerged naturally during the numerical solution of the equation. Essentially, they used finite-difference approximations for the partial derivatives in the transonic small-disturbance equation. The key to making the scheme work was to test the flow at each point to see if the flow was subsonic or supersonic. If it was subsonic, a central difference was used for the second derivative in the x-direction. If the flow was supersonic, they used an upwind difference to approximate this derivative. This allowed the numerical method to mimic the physical behavior of the flow field. The nonlinear coefficient of the ϕ_xx term in equation (6-1) includes a first derivative of potential, ϕ_x. A central difference approximation can be used for this term. Since the solution is found by iteration, old values can be used for ϕ_x. Their approach became known as “mixed differencing,” and it was a simple way to capture the physics of the mixed elliptic-hyperbolic type of the partial differential equation. Because shocks emerge during the solution process, the method is referred to as a “shock-capturing” method. It was much simpler for a general 3D method than competing methods at the time, in which shocks were located and the Rankine–Hugoniot conditions were satisfied analytically (called a “shock-fitting” method). Although several theoretical refinements were required, Murman and Cole’s scheme paved the way for today’s computational methods. Hall^[7] has described the circumstances under which this breakthrough took place. The code known as TSFOIL was the final development of small-disturbance theory methods for 2D.^[8] See section E.7.5 for pertinent software for such methods.

After hearing Earll Murman describe his new method at the AIAA Aerospace Sciences Meeting in New York City in January 1970, Antony Jameson, at the time working for Grumman, returned to Bethpage on Long Island, coded up the method himself, and then went on to extend the approach to solve the full-potential equation in body-fitted coordinates. This required several additional major contributions to the theory.^[9] The code he developed was known as FLO 6 and, after several more major methodology developments, resulted in the extremely efficient 2D full-potential flow code known as FLO 36.^[10] These were the first truly accurate and useful transonic airfoil analysis codes. Holst has published a survey of full-potential methods.^[11]

The next logical development was to add viscous effects to the inviscid calculations and then to switch to the Euler equations for the outer inviscid flow. By that point, many researchers were working on computational flow methodology, which had become an entire field known as CFD. An entrance to the literature on computational methodology is available in the survey article by Jameson.^[12] The Euler solution method used here is from a code known as MSES by Prof. Mark Drela of MIT.^[13]

Figure 6-3 provides a comparison of the predictions for the transonic flow over an NACA 0012 airfoil at M = 0.75 and 2° angle of attack for three inviscid flow models (TSFOIL2, FLO 36, and MSES). In general, the results are in good agreement. However, the full-potential solution (FLO 36) predicts a shock that is too strong and too far aft. The solution based on small-disturbance theory (TSFOIL2) is in close agreement with the full-potential solution, while the Euler equation model (MSES), which is the most accurate of these flow models, has a weaker shock located further forward on the airfoil compared to the other two methods. This occurs for two reasons. First, the potential flow model does not contain the correct shock jump. Second, there is no loss in stagnation pressure across the shock in the potential flow models, making them insensitive to the “back pressure” downstream, to borrow terminology from nozzle flows.

Several key points need to be made while examining figure 6-3. First, the solutions for transonic flow are found from iterative solutions of a system of nonlinear algebraic equations. This is much more difficult than the subsonic case, where the equations for panel methods are linear. The codes require much more care in their operation to obtain good results. Students often ask for solutions for flow cases that are too difficult to solve. You can’t ask for a solution at M = 0.95 and 10° angle of attack and expect to get a result from most codes. Students should start with known cases that work and try to progress slowly to the more difficult cases. Another key point to consider is that the shock is typically smeared over several grid points in the numerical solution. The actual solution point symbols have been included in the plots in figure 6-3 to illustrate this. Finally, the region where the flow is locally supersonic can be observed by comparing the local value of the pressure coefficient to the critical value, shown on the figure as a dashed line labeled C_pcrit. If the pressure coefficient at a point on the airfoil is lower (more negative) than the critical value at that point, the flow is supersonic at that point. The critical value is the point on the airfoil where, assuming isentropic flow, the value of the pressure corresponds to a local Mach number of one. The derivation of C_pcrit is given in any good basic compressible flow textbook, and the formula is

$$C_{p_{\text{crit}}}=-\frac{2}{\gamma M_{\mathrm{\infty }}^2}[1-{(\frac{2}{\gamma +1}+\frac{\gamma -1}{\gamma +1}{M}_{\mathrm{\infty }}^2)}^{\frac{\gamma }{\gamma -1}}]\text{.}\text{(6-2)}$$

Dedicated airfoil pressure distribution plotting packages usually include a tick mark on the C_p scale to indicate the critical value.

Next, we show the full-potential equation solutions using FLO 36 to illustrate the development of the flow with increasing Mach number for the same NACA 0012 airfoil used in figure 6-3. Figure 6-4 shows how the pressure distribution changes from subcritical to supercritical as Mach number increases from 0.5 to 0.75. At M = 0.50, the flow expands around the leading edge and then starts to slow down. This is the typical subsonic flow behavior. At M = 0.70, the flow continues to expand after going around the leading edge, and it returns to subsonic speed through a shock wave, which is fairly weak. As the freestream Mach number increases further, the shock moves aft rapidly, becoming much stronger. In this case, we are looking at inviscid solutions, and this strong shock would likely separate the boundary layer, requiring the inclusion of viscous effects to get a solution that accurately models the real flow.

Figure 6-5 shows the effect of changing angle of attack from 0° to 2° on the pressure distribution using the NACA 0012 airfoil, the same one used in figures 6-3 and 6-4. The results are similar to what occurs when the Mach number increases. The solution changes rapidly with relatively small changes in angle of attack. The shock wave develops fast, with the strength increasing and the position moving aft rapidly.

6.3 Airfoils

6.3.1 NASA Supercritical Airfoils

As mentioned in section 6.2.1, in the late 1960s, Richard Whitcomb developed airfoils at NASA Langley that had significantly better transonic performance than previous airfoils. It was found that airfoils could be designed to have a drag rise Mach number much higher than previously obtained. To illustrate how this occurs, we will compare the typical transonic airfoils in use at the time, the NACA 6A-series foils, with one of the NASA supercritical airfoils. The differences in the solutions clearly illustrate the merits and drawbacks of the modern approach to transonic airfoil design.

Figure 6-6 contains a plot of the NACA 64A410 airfoil and its transonic pressure distribution at a Mach number of 0.72 and an angle of attack of 0°. The computed lift coefficient is 0.665.

Figure 6-7 contains similar data for a NASA supercritical airfoil, Foil 31, at a Mach number of 0.73 and an angle of attack of 0°. The computed lift coefficient is 1.04. The “jagged” pressure distribution in figure 6-7(b) is the result of the airfoil being defined by a set of discrete points which do not produce a smooth surface rather than the result of an analytical function which produces smooth surface.

It can be seen that the shock wave on the supercritical airfoil is much weaker than the shock on the 64A410, even though the lift is significantly higher. This illustrates the advances made in airfoil design. Although the 6A-series airfoils were widely used in transonic and supersonic applications, they were actually designed during and just after WWII to attain laminar boundary layer flow over a portion of the airfoil. They were not designed for good transonic performance (no one knew how to do this at that time). The history and development of supercritical airfoils has been described by Harris^[14] using the written version of a talk given by Whitcomb and Harris at the NASA Langley Advanced Technology Airfoil Research Conference in March 1978. Most of the conference papers appeared in NASA CP-2046 in January 1979. However, publication of the Harris paper was delayed until 1990! The NASA TP-2969 by Harris explains the reasoning behind the concept development and the refinement in design. This is a must-read paper which also contains the coordinates for the entire family of airfoils and updates the research to 1990. In addition to this paper, we recommend two papers for readers to get the authentic description of the development of supercritical airfoils. The first is by Becker,^[15] and the other, by Whitcomb, is his first unclassified public release paper on the new airfoil concept presented in 1974.^[16] We give a very brief overview of the transonic supercritical airfoils here next.

Essentially, the key to transonic airfoil design is to control the expansion of the flow to supersonic speed and its subsequent recompression. It was remarkable that Whitcomb was able to do this using an experimental approach. Because his test-based approach was so difficult, he was a major proponent of developing computational methods for transonic airfoil design.

The following are four key elements of supercritical airfoils:

1. A relatively large leading-edge radius is used to expand the flow at the upper surface leading edge, thus obtaining more lift than obtained on airfoils like the 64A410 shown in figure 6-6.
2. To maintain the supersonic flow along a constant pressure plateau or to even have it slow down slightly approaching the shock, the upper surface is much flatter than previous airfoils. By slowing the flow going into the shock, a relatively weak shock (compared to the amount of lift generated) is used to bring the flow down to subsonic speed.
3. Another means of obtaining lift without strong shocks at transonic speed is to use aft camber. Note the amount of lift generated on the lower surface aft portion of the supercritical airfoil in figure 6-7(b) compared to the conventional airfoil in figure 6-6(b). One potential drawback to the use of aft camber is the large zero-lift pitching moment.
4. Finally, to avoid flow separation, the upper and lower surfaces at the trailing edge are nearly parallel, resulting in a finite-thickness trailing edge. The base drag is small at transonic speeds compared to the reduction in profile drag.

These are the essential ingredients in supercritical airfoil design, and modern aerodynamic designers pick the best aspect of these elements to fit their particular applications.

Whitcomb cited four design guidelines for airfoil development:

1. An off-design criterion is to have a well behaved sonic plateau at a Mach number that is 0.025 below the design Mach number.
2. The gradient of the aft pressure recovery should be gradual enough to avoid separation. This may mean a thick trailing-edge airfoil, typically 0.7% thick on a 10-percent- or 11-percent-thick airfoil.
3. The aft camber should ensure that, with α_des ≅ 0, the upper surface is not sloped aft.
4. Gradually decreasing supercritical velocity will result in a weak shock.

Aerodynamicists in the industry have also made significant contributions to transonic aerodynamic design. The best summary of transonic design for transport aircraft is by Lynch.^[17] Figure 6-8 contains a summary chart developed by Lynch to identify the issues associated with leading-edge radius and aft camber.

6.3.2 The Divergent Trailing-Edge Airfoil

Just when supercritical airfoil development appeared to be completed, a further development in airfoil design was made: the divergent trailing-edge (DTE) airfoil. Henne and Gregg ^[18] at McDonnell Douglas made further improvements in airfoil performance by using a trailing edge where the upper and lower surfaces are not just parallel to each other but diverging. Their paper takes airfoil design concepts one step further, describing the development and benefits of the DTE airfoil, which was discovered by the use of then-current computational aerodynamics methods.* Although the DTE concept is related to a simple triangular wedge that was added to the MD-11 outboard wing trailing edge for improved performance, the DTE airfoil is much more effective.

6.3.3 Transonic Airfoil Performance: The Korn Equation

Note: The material in this section has a lot of commonality with that in section 3.5.4. However, it is included here for the sake of completeness. Furthermore, the focus here is on the application of airfoil performance information for aircraft design.

Attempts have been made to estimate the capability of transonic airfoils for the purposes of design studies without performing wind tunnel tests or detailed computational design work. This is important in the initial stages of aircraft design, where airfoil performance needs to be estimated before the actual airfoil design has been done. Here we provide an approximate method for estimating the transonic performance of airfoils. It is based on the Korn equation, which was an empirical relation developed by Dave Korn at the NYU Courant Institute in the early 1970s and that was in use at Grumman when I arrived in 1974. Based on his experience, it appeared that airfoils could be designed for a variety of Mach numbers, thickness-to-chord ratios, and design lift coefficients, but in all cases, there seemed to be a limit to the combination of three parameters: drag divergence Mach number (M_dd), thickness-to-chord ratio (t/c), and lift coefficient (C_l). The limit is an airfoil technology factor, κ_A, and the Korn equation defines the relationship between the combination of these three parameters and the airfoil technology factor as follows:

$$M_{\mathrm{dd}}+\frac{C_l}{10}+\left(\frac{t}{c}\right)={\kappa }_A\text{(6-3)}$$

The airfoil technology factor on the right-hand side has a value of 0.87 for an NACA 6-series airfoil section and a value of 0.95 for a supercritical section. This relation provides a simple means of estimating the possible combination of Mach number, lift coefficient, and thickness ratio that can be obtained using modern airfoil design, and variations of it have been shown graphically by many authors, where the scales are often left off when presented for the public by aircraft companies. Note that the Korn equation is sensitive to the value of the technology factor.

Figure 6-9, from Mason,^[19] compares the prediction from the Korn equation with other estimates. In figure 6-9(a), the estimates of older airfoils and Shevell’s estimates of modern supercritical airfoil performance^[20] are compared, and the agreement is good, with the exception of being overly pessimistic regarding older conventional airfoils at lower thickness ratios. Figure 6-9(b) compares the Korn equation with NASA projections^[21] for supercritical airfoils based on a wealth of data and experience. In this case, the Korn equation is extremely good at lift coefficients of 0.4 and 0.7 but overly optimistic at higher lift coefficients. This type of technology representation is important in developing integrated designs. Developing each point on this type of charge requires a large effort on the part of the designer (in this case, the aerodynamicist).

The extension of this method to swept wings using simple sweep theory and an approximate drag rise curve shape will be given in the section on wings below.

6.3.4 Design Methods

We gave some general approaches and guidelines for airfoil design for transonic flow above when describing Whitcomb’s supercritical airfoils. Although there are details beyond the scope (as they say!) of this chapter, we should point out that there are two distinct approaches available for airfoil design. Aerodynamicists frequently use inverse methods, where a target pressure distribution is specified and the required shape (which might not exist for an arbitrary pressure distribution prescription) is found. Alternatively, optimization methods may be used, where the shape is described by a set of design variables that are then used in an optimization routine, which can be computationally expensive. In addition, many optimization methods require gradients of the solution with respect to the design variables, and modern computational fluid dynamics methods have been developed to obtain the design’s sensitivity to geometric and flow perturbations as part of the solution. Finally, at transonic speed, we need to avoid undo sensitivity to the specified design conditions. This requires a statistical design approach. An important advance in this area has been made by Huyse.^[22]

6.4 Wings

We now turn our attention to wings. Today, at transonic speeds, wings are swept to delay drag rise. However, even though the Boeing B-47 had a swept wing, they weren’t adopted across the board immediately, even at Boeing. Initially, jet engines had poor fuel efficiency and weren’t considered appropriate for very long-range aircraft. In one famous instance, Boeing was working on a long-range turboprop bomber for the Air Force. When the Boeing team started to present their design to the Air Force at Wright-Patterson Air Force Base (WPAFB) in Dayton, Ohio, they were immediately told to switch to a swept-wing pure jet design. The team didn’t have time to return to Seattle, so they recruited other Boeing engineers already in Dayton and did the work there (with the help of phone calls back to Seattle). To show the Air Force the design, they made a model. Figure 6-10 shows the actual model on display at the Museum of Flight in Seattle. This design became the B-52. It is famous for having been designed in a Dayton, Ohio, hotel room!

Although swept wings delay drag rise, they are still associated with some problems, so that the aerodynamicist will want to use as little sweep as possible. Even at subsonic speed, as shown in the previous chapter, wing sweep will tend to shift the load outboard, leading to high section lift coefficients and the possibility of outboard stall, accompanied by pitchup. The wing is twisted (washed out) to unload the tip. The lift-curve slope also decreases. In addition, for a given span, the actual wing length is longer and therefore heavier. Another problem is that high-lift devices aren’t as effective if the trailing edge is swept. Finally, swept wings are prone to flutter. Thus, the total system design must be considered when selecting the wing sweep. One of the benefits of advanced airfoils is that they can achieve the same performance as a wing with a less-capable airfoil using less sweep. This explains the general trend to modern transports having less sweep than earlier transports.

6.4.1 Transonic Transport Wing Concepts

The transonic transport wings are generally high aspect ratio, swept, tapered wings that clearly have an airfoil embedded in them. Generally we consider aft-swept wings.

Cruise design. Normally, the aerodynamic designer is given the planform and maximum thickness and told to design the twist and camber, as well as shifting the thickness envelope slightly. He or she then tries to obtain “good” isobars on the wing. Since the natural tendency is for the flow to unsweep at the root and tip, the designer tries to reduce this tendency to obtain an effective aerodynamic sweep as large as the geometric sweep. If possible, they would actually like to make the effective aerodynamic sweep greater than the geometric sweep. This is unlikely to happen. Generally, the wing has a weak shock wave. Possibly the best tutorial paper on the problem of isobar unsweep is by Haines,^[23] who actually considers the thickness effects at zero lift. This is an old but important paper that explains how root and tip modifications are made to make the isobars swept on a swept wing.

We illustrate the problem of isobar unsweep with an example taken from work at Grumman to design the initial G-III wing (though the G-III doesn’t actually have this wing, as it was considered too expensive; the actual G-III has a highly modified version of the G-II wing). Figure 6-11 from a paper by Mason et al.^[24] illustrates the situation.

In figure 6-11, we see the isobar pattern at transonic speed on the planform in the upper left-hand side of the figure. This wing was designed with subsonic methods, which were essentially all that was available at the time. Note that the isobars are tending to unsweep. On the upper right-hand side of the figure, we see that at subsonic speed the pressure distributions at the 30% and 70% span stations lie on top of each other, the isobars are good. The lower left-hand side of the figure shows the predicted pressure distributions at these same two stations when the Mach number is increased to the transonic cruise Mach. There has been a large change in the distributions at the two stations, with the outboard shock ahead of the inboard shock. Finally, on the lower right-hand side, we see the same predictions but with wind tunnel data included. Clearly, the prediction and the test results agree well and show that extra effort is required to design the wing when the flow is transonic.

Although the aerodynamicist generally prefers an elliptic spanload for a given span, it might be better for the design if the load is shifted inboard slightly, reducing the root bending moment and therefore wing structural weight.^[25] The optimum spanload with a winglet present is not elliptic either.

Essentially, the twist distribution is found to generate the design spanload. Spanloads are predicted fairly well using linear theory codes; it is primarily the chord load that reflects the nonlinearity of transonic flow. Once the basic twist is found, root and tip mods are developed to maintain the isobar pattern. Without special effort, the chord load is drawn aft at the root and shifts forward at the tip. Changes in camber and thickness are introduced to counter these effects. The planform may deviate from pure trapezoidal. In all likelihood, there will be a Yehudi (see Chapter 5) at the inboard trailing edge to house the landing gear. The planform in figure 6-11 also has a leading-edge glove inboard. This allows the t/c to be lower for the same t, and increasing the chord lowers the section C_l required to obtain the spanload required, as well as helping maintain isobar sweep. Breaks in the planform chord distribution produce rapid variations in the section lift distribution because the spanload will tend to remain smooth. The section lift distribution may be smoothed out by using several smaller spanwise breaks. This has been used in modern Boeing and Airbus designs.

The designer also has to consider buffet margins. This means the wing C_L has to be capable of a 1.3 g turn at the highest cruise Mach number without predicting any significant flow separation. The relevant considerations are presented in section 6.4.4.

Other important details include nacelle/pylon interference and the resulting detailed shaping, as well as manufacturing constraints. This means considering the limits to curvature and the manufacturing department’s desire for straight-line wrap or ruled surfaces.

Once the design starts to get close to the desired properties, local inverse methods can be applied to achieve the target pressure distributions.

Transonic configuration design. Finally, a review of the design process by Jameson is worth reading to get some idea of the design process and possibilities of improving it through the use of computations.^[26] This paper contains a good description of the transport wing design problem. Jameson clearly articulates the process for the nonexpert wing designer, something the company experts haven’t done often (possibly because they consider the process to be competition sensitive).

6.4.2 The Korn Equation Applied to Drag Prediction on Swept Wings

Note: The material in this section has a lot of commonality with that in section 3.5.4. However, it is included here for the sake of completeness. Also, the focus here is on the application of transonic drag prediction methodology to aircraft design.

As described in section 3.5.4, the Korn equation has been extended to estimate the drag divergence Mach number for a 3D swept wing by including sweep using simple sweep theory.^[27],[28] The result for a 3D wing is given by

$$M_{DD}=\frac{{\kappa }_A}{\cos \mathrm{\Lambda }}-\frac{(t/c)}{{\cos }^2\mathrm{\Lambda }}-\frac{C_L}{10{\cos }^3\mathrm{\Lambda }}\text{.}\text{(6-4)}$$

This model estimates the drag divergence Mach number, M_DD, as a function of an airfoil technology factor (κ_A), the thickness-to-chord ratio (t/c), the lift coefficient (C_L), and the sweep angle (Λ). Recall that the airfoil technology factor has a value of 0.87 for a NACA 6-series airfoil section and a value of 0.95 for a supercritical section.

With this approximation for the drag divergence Mach number, we can now calculate the critical Mach number. The definition of the drag divergence Mach number is given in equation (6-5).

$$\frac{\mathrm{\partial }{C}_D}{\mathrm{\partial }M}=0.1\text{(6-5)}$$

Next, we make use of Lock’s proposed empirically-derived shape of the drag rise.^[29]

$$C_D=20{(M-M_{crit})}^4\text{(6-6)}$$

The definition of the drag divergence Mach number is equated to the derivative of the drag rise formula given above to produce the following equation:

$$\frac{\mathrm{\partial }{C}_D}{\mathrm{\partial }M}=0.1=80{(M-M_{crit})}^3\text{(6-7)}$$

We can then solve this equation for the critical Mach number:

$$M_{crit}=M_{DD}-{\left(\frac{0.1}{80}\right)}^{1/3}\text{(6-8)}$$

where the drag divergence Mach number is given by the extended Korn equation.

Joel Grasmeyer et al.^[30] then developed a method to compute the wave drag coefficient for use in MDO studies of a transonic strut-braced wing concept using the following relation:

$$C_{d_{\text{wave}}}=20{(M-M_{crit})}^4\frac{S_{\text{strip}}}{S_{\text{ref}}}\text{for}M>M_{crit,}\text{(6-9)}$$

where the local t/c, C_l, and half-chord sweep angle are specified for a number of spanwise strips along the wing and the drag of each strip is combined to form the total wave drag. In the equation above, the wave drag for each strip is multiplied by the ratio of the strip area (S_strip) to the reference area (S_ref).

This method has been validated with the Boeing 747-100 using eight spanwise strips. The results are shown in figure 6-12. The curved lines represent the current model predictions, and the discrete data points represent the Boeing 747 flight test data from Mair and Birdsall.^[31] The predictions show good agreement with the data over a wide range of Mach numbers and lift coefficients. We reemphasize that the results are sensitive to the value of the airfoil technology factor. A value of 0.89 was used for the Boeing 747 results in figure 6-12. Based on an analysis of the Boeing 777, a value of 0.955 was used to simulate that aircraft’s wave drag characteristics.

6.4.3 Fighter Wing Concepts and Issues

In this section, we provide a few comments on fighter wing concepts and issues in transonic flight. A good survey of the issues for transonic aerodynamic design of fighters has been given by Bradley.^[32]

Attached flow maneuver wing design. To push performance past the cruise lift condition, the situation changes. If the goal is to obtain efficient lift at high lift coefficients using attached flow design, the emphasis switches from an elliptic loading to a span loading that pushes each section lift coefficient to its limit. Thus, if the planform is a simple trapezoidal planform with a single airfoil section, the goal is to attain a constant section C_L across the wing.^[33] The penalty for a nonelliptic spanload is small compared to the additional profile drag for airfoils operating past their attached flow condition on portions of the wing. This is essentially what was done on the X-29. The so-called Grumman K airfoil was used on the X-29.

Two other considerations need to be addressed. Wings designed to operate over a wide range of conditions can use the leading- and trailing-edge devices to approximate the optimum wing shape by using a deflection schedule to automatically deflect to the best shape. Although research has been done on smooth surfaces to do this, in most cases, the devices are simply flap deflections. In the case of the X-29, the airfoil was shaped for the maneuver design point, and the devices were used to reduce the trailing-edge camber at lower lift coefficients.

The second consideration is airfoil-planform integration. If the airfoil is designed to be heavily loaded, there is likely to be a fairly strong shock well aft on the wing. To obtain low drag, this shock should be highly swept. This means that the trailing edge of the wing should be highly swept. This can be done using a wing with inverse taper or a forward-swept wing. This is one reason to consider a forward-swept wing concept. However, a forward-swept wing with a canard must be balanced with a large negative static margin to gain the full benefit of the concept. The X-29 is about 32% to 35% unstable for this reason.

Finally, when the airfoils are being pushed to their limits, planform kinks are a very poor idea. The tendency of the spanload to remain smooth means that the local lift coefficients change rapidly in the kink region, and local lift coefficients often become excessively large.

Another alternative is to include a canard in the configuration. A canard can be used to carry additional load at extreme maneuver conditions.

Vortex flow/strake maneuver wing design. Another method of obtaining high-maneuver lift has proven effective on the F-16 and F-18 aircraft. In this case, inboard strakes are used (Northrop called theirs a LEX, leading-edge extension, dating back to the F-5 days). The strakes produce a strong vortex at high angles of attack. The vortices flow over the aircraft surfaces and, as a result of the low-pressure field, create additional lift. Careful shaping of the strake is required, but good performance can be obtained. Note that these airplanes also use leading- and trailing-edge wing device scheduling to achieve optimum performance.

Figure 6-13 shows, in a rough sense, how the two concepts compare. Here, E is the efficiency factor in the drag due to lift term of the classic drag polar:

$$C_{D_{\mathrm{L}}}=\frac{C_L^2}{\pi ARE}\text{(6-10)}$$

6.4.4 Buffeting Considerations

Buffeting* is defined as a response of the structural modes to the aerodynamic excitation imposed as pressure fluctuations produced by separated flow.^[34],[35] The pressure fluctuations excite the flexible modes, resulting in structural vibrations that can have a strong influence on aircraft aerodynamic performance and, when left unchecked, can lead to structural damage or even catastrophic failure.^[36] The fluctuations may excite rigid body modes, such as “wing rocking,” “wing dropping,” or “nose slicing,” at much lower frequencies, but these modes can be regarded as aircraft handling problems.

The flow separation over the wing is usually the primary cause of buffet, and three factors that strongly influence flow separation are (i) Mach number, (ii) angle of attack, and (iii) wing geometry. (Although localized flow separation may occur on other parts of the aircraft, such as fuselage, it rarely causes the buffet phenomenon.) For a given wing geometry, the 1.0 g buffet onset boundary is typically represented by a C_LB-vs-M curve shown in figure 6-14, which implicitly accounts for the relative influence of the angle of attack.^[37] At subsonic speeds, buffet is commonly caused by vortex shedding or turbulent wake regions behind the wing at angles of attack near stall. A well known example is twin-tail buffet of the F/A-18 aircraft caused by bursting of vortices emanating from the leading-edge extensions (LEX) under high-angle-of-attack maneuvering conditions. At transonic speeds, buffet is caused by self-sustained, low-frequency oscillations of the shock on the upper surface of the wing due to shock-induced boundary layer separation, which imposes unsteady loading on the aircraft wings.

For a given aircraft, a C_LM² is a constant curve, representing flying at a certain pressure altitude. The point where such a curve becomes tangent to the buffet boundary represents the highest achievable altitude regardless of the available thrust. This altitude is called the aerodynamic ceiling. Flying at this altitude is problematic because any change in M will lead to buffeting and/or stalling. Pilots call it the coffin corner.

Airplanes are typically constrained to fly in the region to the left of the buffet onset boundary shown in figure 6-14 to avoid potentially damaging effects of buffet. Regulatory agencies require that C_L in operational conditions be limited to ensure that the airplane can fly at a load factor (n = L/W) of 1.3 without encountering buffet. This load factor is equivalent to a 40° bank angle while maintaining level flight. For aircraft that cruise at transonic Mach numbers, a strong gust can increase the incidence to a level where flow separation occurs. To meet the regulatory requirements, operational aircraft buffet envelope is defined by flight test. The 1.0 g onset is typically identified as the speed at which the vibration reaches ±0.050 g at the pilot’s seat position while the pilot executes windup turns at a constant Mach number. This means that the certified operational constraint of the 1.0 g buffet onset is not to be taken as a strict physical limit to the actual flight envelope of the aircraft; it basically sets a boundary between a safe flight domain and a flight domain in which the flight crew may encounter serious control problems and/or the aircraft is prone to severe instantaneous or fatigue loads.^[38] Additional operator specific requirements would also include maneuver capability through moderate turbulence (e.g., penetration with 0.50 g margin).

Clearly, the buffet boundary has a strong influence on the choice of the design lift coefficient, C_L, at the design Mach number, M. Since M and C_L are chosen during the early conceptual design phase, it is imperative to consider the influence of buffet boundary. Otherwise, designers run the risk of having the design C_L limited by buffet in later stages of design. This may adversely impact flight performance by limiting the maximum lift-to-drag ratio and operational ceiling.^[39] However, the overarching challenge for the designers during the conceptual design phase is to accurately estimate the buffet boundary when the wing geometry is not fully defined. As a matter of fact, one of the key outputs of the early design efforts is an efficient wing geometry and associated parameters such as planform area, aspect ratio, sweep, taper, camber, and thickness. Of course, once the wing geometry is specified, we can use buffet prediction methods based on data from wind tunnel tests^[40],[41] or computational fluid dynamics (CFD) simulations.^[42],[43]

6.4.4.1 Buffet Boundary Prediction Methods

Berard and Isikveren^[44] and Eijndhoven^[45] present a relatively simple, economical, flexible, and robust semi-empirical method that is well suited for conceptual design of new transport aircraft. Berard and Isikveren identify six wing geometry parameters that have functional sensitivity with the maximum attainable lift coefficient for buffet onset, and they provide mathematical models of C_L as a function of wing geometric parameters for a given M. Figure 6-15 illustrates the effect of wing geometric parameters—namely, leading edge sweep (Λ), thickness-to-chord ratio (t/c), and camber—in a mutually exclusive sense. Two variants of the method are discussed in their paper. The first uses a generic reference curve for a generic reference wing to evaluate the buffet envelope of the target aircraft. The second variant starts from the buffet onset of a known seed aircraft and uses fractional change theory to predict buffet onset boundary of the new wing geometry.

Segee et al.^[46] developed a surrogate model of the maximum allowable lift coefficient before buffet offset for multidisciplinary design optimization (MDO) applications for transonic transport aircraft studies. The authors generated aerodynamic data for the BACJ airfoil using steady RANS (Reynolds-averaged Navier–Stokes) CFD for a series of thickness ratios, and they developed C_LB-vs-M models for different thickness ratios using correlation of “shock position” and “local Mach number ahead of shock.” When performing MDO studies, this model was applied in a stripwise manner. That is, the 2D lift coefficient for each wing strip was kept below 0.8. Kenway and Martins^[47] present a new formulation of 3D wing buffet-onset prediction that is well suited for MDO studies. Their method is based on the integration of a flow separation sensor along with a cutoff value. It can determine buffet onset using data from one steady RANS CFD simulation.

Figure References