In this chapter, we discuss subsonic aerodynamics, mainly in relation to airfoils and wings. We look at the basic aerodynamics mainly from an inviscid point of view. Generally, this is a reasonable starting point for thinking about aerodynamics in attached flows. Traditionally, the methods used to make these calculations are known as surface panel methods (or simply panel methods) and vortex lattice methods. The theory is described in Chapters 4 and 5 of the Applied Computational Aerodynamics book.^[1] Although we are primarily addressing the aerodynamic features, we provide brief overviews of the methods used to compute the results shown.

One of the key features of subsonic flow is the use of Laplace’s equation as the basis for the inviscid solution. Solutions of this linear equation can be obtained using superposition of “singularities” of unknown strength distributed over discretized portions of the surface (i.e., panels).* Hence the flow field solution is found by representing the surface by a number of panels and solving a linear set of algebraic equations to determine the unknown strengths of the singularities. An excellent entry into the panel method literature is available through two reviews by Hess.^[2],[3] Two other good sources are the survey by Erickson^[4] and the book by Katz and Plotkin.^[5]

A more approximate—yet physically appealing—approach is often used to compute wing aerodynamics. In this approach, the wing is represented by a system of horseshoe vortices placed on the mean surface. These methods are known as vortex lattice methods (VLMs). Generally, they ignore the thickness effects. This is possible because, for relatively thin surfaces, the effects of thickness cancel between the upper and lower surfaces, meaning that the lift is not influenced by the thickness to first order. We will illustrate wing aerodynamics using vortex lattice methods. A key reference for these methods is a workshop devoted to them at NASA in the mid-seventies.^[6] A nearly universal standard for vortex lattice predictions had been established by then in the form of a code developed at NASA Langley by Rich Margason, John Lamar, and their collaborators.^{[7],[8],[9],[10]} See section E.6.1 for more.

Chapter 5 of Applied Computational Aerodynamics^[11] provides complete details on the VLM method, together with references to other implementations of the method. Some of the most noteworthy variations on the basic method have been developed by Lan^[12] (developer of the quasi-vortex lattice method), Hough,^[13] DeJarnette,^[14] and Frink.^[15] Mook et al.^[16] at Virginia Tech have developed vortex lattice class methods that treat flow fields containing leading-edge vortex type separation and also handle general unsteady motions and deforming wakes. The book by Katz and Plotkin^[17] contains another variation. At Virginia Tech, Jacob Kay wrote a code using the method of Katz and Plotkin to estimate stability derivatives; see section E.9.1.^[18] Three other vortex lattice methods are worthy of note:

1. Tornado, a MATLAB code developed in Sweden^[19] (see section E.6.3)
2. AVL by Prof. Drela and Harold Youngren^[20] (see section E.6.4)
3. VSPAero by NASA (see section E.6.5)

5.1 Airfoils

In this section, we highlight basic ideas of airfoil subsonic aerodynamics. We discuss the accuracy and validation aspects of flow simulation computer programs in section 5.1.1. This is followed by a discussion of airfoil aerodynamics characteristics in section 5.1.2. We conclude by discussing issues related to airfoil selection in section 5.1.3.

The computed results presented in this section mainly use Jack Moran’s program PANEL.^[21] The flow is assumed to be inviscid and irrotational, governed by the Laplace’s equation or the potential flow equation. As the name of the program suggests, the body surface is approximated by a collection of panels, each with a distribution of singularities. It is important to emphasize that our choice of program PANEL should not be construed as an indication of any preference; it was more a matter of convenience. Once you understand the basic ideas of airfoil subsonic aerodynamics that we illustrate using PANEL, you will be well prepared to effectively use any other code of your choice. As a matter of fact, the current standard program for subsonic airfoil analysis and design is due to Mark Drela of MIT and is known as XFOIL.^[22] It includes viscous effects and is in the public domain for academic use; see section E.7.1 for the software website for accessing this code and more details.

5.1.1 PANEL and Other Flow Simulation Methods: Accuracy and Validation

PANEL was essentially created by Moran.^[23] PANEL’s node points are distributed employing the widely used cosine spacing function. This approach is used to provide a smoothly varying distribution of panel node points that concentrate points around the leading and trailing edges. An example of the accuracy of the program is given in figure 5-1, where the results from PANEL for the NACA 4412 airfoil are compared with results obtained from an exact conformal mapping of the airfoil (which can be considered to be an exact solution). The agreement is nearly perfect.

Most codes require that you select the number of panels to represent the surface. How many should you use? One of the first things the user should do is evaluate how many panels are needed to obtain the desired level of accuracy. As a matter of fact, this is the best practice for using any computer program. For a panel method, the user must determine the number of panels required to generate accurate results. It is best to examine forces, moments, and pressure distributions.

To check the sensitivity of the solution to the number of panels, compare force and moment results as well as pressure distributions as the number of panels increase. To estimate the limit for a large number of panels, the results can be plotted as a function of the reciprocal of the number of panels. Thus the limit result occurs as 1/n reaches zero. Figures 5-2 through 5-4 present the results in this manner for the case given above, including the pitching moment for examination in the analysis. We see in figure 5-2 that the drag is approaching zero, which is the correct result for a two-dimensional inviscid incompressible solution.

Figures 5-2 through 5-4 show that PANEL produces results that are relatively insensitive to the number of panels once fifty or sixty panels are used. By extrapolating to 1/n = 0, an estimate of the limiting value can be obtained.

In addition to forces and moments, the sensitivity of the pressure distributions to changes in panel density should also be investigated. Pressure distributions are shown in figures 5-5 and 5-6. Figure 5-5 contains a comparison between the cases of twenty panels and sixty panels. It appears that the pressure distribution is well defined with sixty panels. This is confirmed in figure 5-6, which demonstrates that it is almost impossible to identify the differences between the sixty and 100 panel cases. This type of study should (in fact, must) be conducted when using computational aerodynamics methods.

Having examined the convergence of the mathematical solution, we investigate the agreement with experimental data in a process called validation of the code. Figure 5-7 compares the lift coefficients from the inviscid solutions obtained from PANEL with experimental data from Abbott and von Doenhoff.^[24] Agreement is good at low angles of attack, where the flow is fully attached. The agreement deteriorates as the angle of attack increases. Viscous effects start to show up as a reduction in lift as angle of attack increases until, finally, the airfoil stalls. The inviscid solutions from PANEL cannot capture this part of the physics. There are significant differences in the airfoil behavior at stall between the cambered and uncambered airfoil. Essentially, the differences arise due to different flow separation locations on the different airfoils. The cambered airfoil separates at the trailing edge first. Stall occurs gradually as the separation point moves forward on the airfoil with increasing incidence. The uncambered airfoil stalls due to a sudden separation at the leading edge. Differences in pressure distributions can be studied next to see why this might be the case.

The pitching moment characteristics are also very important. Figure 5-8 provides a comparison of the PANEL pitching moment predictions (about the quarter chord point) with experimental data. In this case, the calculations indicate that the computed location of the aerodynamic center, dC_m / dC_l = 0, is not exactly at the quarter chord, although the experimental data is very close to this value. The uncambered NACA 0012 data shows nearly zero pitching moment until flow separation starts to occur. The cambered airfoil shows a significant pitching moment and a trend due to viscous effects that is exactly opposite the computed prediction.

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

Even though we don’t include predictions, we include the experimental drag polars corresponding to the lift and moment cases given above. Figure 5-9 contains the experimental data showing the effect of camber on the drag polar based on the 0012 and 4412 test cases. Note that the cambered airfoil has a higher drag than the uncambered airfoil at low lift coefficients but starts to show an advantage at a C_l above about 0.3 and is clearly superior at a C_l of around 0.6 to 0.9. The minimum drag is always higher for the cambered airfoil. The aerodynamicist has to decide on the appropriate amount of camber for each application. Keep in mind that the results given here are for old airfoils. The opportunity exists to design modern airfoils that operate over a wide range of lift coefficients without incurring drag penalties. The appeal of variable camber, either through leading-edge and trailing-edge device deflection schedules or smart structure “morphing,” is also evident.

In addition to force and moment comparisons, we need to compare the pressure distributions predicted with PANEL to experimental data. Figure 5-10 provides one example, comparing the NACA 4412 experimental pressure distribution to PANEL predictions. The agreement is generally very good. The primary disagreement is at the trailing edge. Here, viscous effects act to prevent the recovery of the experimental pressure to the levels predicted by the inviscid solution.

Finally, panel methods often have trouble in accurately simulating the flow at the trailing edge of thin airfoils with cusped trailing edges, where the included angle at the trailing edge is zero. The 6-series airfoils are an example. In those cases, PANEL may give poor results locally near the trailing edge. This problem is demonstrated in figure 5.21 in Chapter 5 of Applied Computational Aerodynamics.^[25]

While the 6-series airfoils were very difficult to use on operational aircraft because of their thin, cusped trailing edges, the subsequent 6A-series airfoils were introduced to remedy the problem. These airfoils have larger trailing-edge angles (approximately the same as the 4-digit series) and were made of nearly straight (or flat) surfaces over the last 20% of the airfoil. Most applications of 6-series airfoils today actually use the modified 6A-series thickness distribution.

5.1.2 Subsonic Airfoil Aerodynamics

“[M]ankind can be divided into two great classes: those who take airfoil selection seriously and those who don’t.”

—Peter Garrison, Flying Magazine, September 2002

In this section, we attempt to demonstrate how airfoil selection is related to subsonic aerodynamics. Using PANEL, we have a means of easily examining pressure distributions as well as forces and moments for different airfoil shapes. We present a discussion of airfoil characteristics using an inviscid analysis. In figure 5-11, we illustrate key areas to examine when studying airfoil pressure distributions using the NACA 0012 airfoil at 4° angle of attack as a typical example.

Several aspects of the pressure distribution shown in figure 5-11 should be discussed. First, for subsonic 2D flow, the stagnation pressure coefficient should be one. As the Mach number increases, the value will increase slightly. For swept wings, the stagnation point pressure coefficient will be less than one. The rapid flow acceleration near the leading edge and the subsequent recompression can suggest the possibility of flow separation. The abruptness depends on the airfoil shape close to the leading edge. Bigger leading edge radii soften this flow feature, as well as reducing sensitivity to angle of attack. Another relevant aspect of pressure distribution concerns the value of the pressure coefficient at the trailing edge. If the C_p seen in data or a viscous calculation is greater than about 0.20, the flow can be assumed to be attached, a desirable feature.

Remember that we are making an incompressible, inviscid analysis when we use the program PANEL. Thus, in this section, we examine the basic characteristics of airfoils from that point of view. We will examine viscous and compressibility effects later. However, the best way to understand airfoil characteristics from an engineering standpoint is to examine the inviscid properties and then consider changes in properties due to the effects of viscosity. The aerodynamicist controls the pressure distribution through selection of the geometry, controlling, or suppressing, adverse viscous effects. The mental concept of the flow best starts as a flow field driven by the pressure distribution that would exist if there were no viscous effects. The airfoil characteristics then change through the “relieving” effects of viscosity, where flow separation or boundary layer thickening reduces the degree of pressure recovery that would occur otherwise. For efficient airfoils, the viscous effects should be small at normal operating conditions.

5.1.2.1 Overview of Airfoil Characteristics: Good and Bad

We next illustrate the connection between the airfoil geometry and the airfoil pressure distribution, including ways to control the inviscid pressure distribution by changing the airfoil geometry. An aerodynamicist controls viscous effects by controlling the pressure distribution. Further discussion and examples providing insight into aerodynamic design are available in the excellent book by Jones.^[26] A book that captures much of the experience of the original designers of the NACA airfoils was written by aeronautical pioneer E. P. Warner.^[27]

Drag. We discussed the requirement that drag should be zero for this two-dimensional* inviscid incompressible irrotational prediction method when we studied the accuracy of the method in the previous section (section 5.1.1). At this point, we infer possible drag and adverse viscous effects by examining the effects of airfoil geometry and angle of attack on the pressure distribution.

Lift. Thin-airfoil theory predicts that the lift-curve slope should be 2π, and thick-airfoil theory says that it should be slightly greater than 2π, with 2π being the limit for zero thickness. You can easily determine how close PANEL comes to this value. These tests should give you confidence that the code is operating correctly. The other key parameter is α_ZL, the angle at which the airfoil produces zero lift; a related value is C_l0, the value of C_l at α = 0°.

Moment. Thin-airfoil theory predicts that subsonic airfoils have their aerodynamic centers at the quarter chord for attached flow. The value of C_m0 depends on the camber. We have seen in figure 5-8 that the computed aerodynamic center is not precisely located at the quarter chord. However, the slope of the moment curve in figure 5-8 corresponds to an aerodynamic center location of x/c = 0.2597, which is reasonably close to 0.2500.

Multi-element airfoils are also an important class of airfoils. However, their performance is so closely connected to the effects of viscosity that we must defer discussion of those airfoils until we address viscous effects at the first order in Chapter 7, “High-Lift Aerodynamics.”

Angle-of-attack effect

The starting place for understanding airfoil characteristics is an examination of the angle-of-attack effect on an uncambered airfoil. Figure 5-12 presents this effect for the NACA 0012 airfoil. Here, we see the progression from the symmetric zero angle-of-attack result. The α = 0° case produces a mild expansion around the leading edge followed by a monotonic recovery to the trailing-edge pressure. As the angle of attack increases, the pressure begins to expand rapidly around the leading edge, reaching a very low pressure and resulting in an increasingly steep pressure recovery at the leading edge.

Thickness effect

The next effect of interest is thickness. Figure 5-13 presents airfoil shapes for NACA 4-digit sections of 6%, 12%, and 18% thickness-to-chord ratio, t/c. The associated basic pressure distributions at zero angle of attack are shown in figure 5-14. Clearly, the thicker airfoil produces a larger disturbance and thus a lower minimum pressure. However, the 18-percent-thick airfoil produces a slightly milder expansion around the leading edge. The recompression near the trailing edge is also milder, and it extends further upstream than the thinner airfoils.

The effect of thickness in softening the expansion and recompression around the leading edge is even more evident at an angle of attack. Figure 5-15 shows this effect at a lift coefficient of 0.48. The thinnest airfoil shows a dramatic expansion/recompression due to the location of the stagnation point below the leading-edge point, requiring a rapid flow expansion around the leading edge that has a very small radius of curvature. The thicker airfoil results in a significantly milder expansion and subsequent recompression due to the large radius of its leading edge. It is important to note that the pressure distributions shift “upward” as the thickness increases. However, the ΔC_p is nearly the same for each thickness, as predicted using the superposition of thickness effects and angle-of-attack effects arising in thin-airfoil theory.

Camber effect

The next effect to examine in understanding airfoil characteristics is camber. Figure 5-16 compares the shapes of the NACA 0012 and 4412 airfoils. The pressure distributions on the cambered airfoil for two different angles of attack are shown in figure 5-17. Note the role of camber in obtaining lift without producing the rapid leading-edge expansion and subsequent recompression that we’ve seen before. This reduces the possibility of leading-edge separation. Instead, the lift is distributed along the airfoil.

A comparison of the NACA 0012 and NACA 4412 airfoil pressure distributions at the same lift coefficient is presented for several values of lift in figures 5-18, 5-19, and 5-20. As the lift increases, the camber effects start to be overcome by the angle-of-attack effects, and the dramatic effects of camber are diminished until, at a lift coefficient of 1.43, the pressure distributions start to appear to be very similar.

Finally, we examine the effect of extreme aft camber, which was part of the design strategy of Richard Whitcomb when the NASA supercritical airfoils were developed. This effect can be simulated using the NACA 6712 airfoil shown in figure 5-21. The computed pressure distribution, given in figure 5-22, shows that the aft camber “opens up” the pressure distribution near the trailing edge. Two adverse properties of this type of pressure distribution are the large zero-lift pitching moment and the delayed-then-rapid pressure recovery on the upper surface. This type of pressure recovery is a very poor way to try to achieve a significant pressure recovery because the boundary layer will separate early. Whitcomb’s design work primarily improved the pressure recovery curve.

The airfoils used above to demonstrate geometry effects on pressure distributions use parametric geometry definition formulas developed in the 1930s. More modern airfoils are available to today’s aerodynamicist. Unfortunately, to obtain improved performance, the designs were developed without the use of simple geometric definitions and are available only as tables of coordinates. One modern airfoil that extends some of the previous shapes to obtain a high-performance airfoil is the GA(W)-1 airfoil.^[28] This 17-percent-thick airfoil designed by NASA’s Richard Whitcomb provides improved maximum lift and stall characteristics compared to previous NACA airfoils. Figure 5-23 shows the airfoil shape, and figure 5-24 shows the pressure distribution.

In this case, the upper-surface pressure distribution reaches a constant pressure plateau and then has a moderate pressure recovery. Aft camber is used to obtain lift on the lower surface and open up the airfoil pressure distribution near the trailing edge in a manner suggested previously in figure 5-22. The area of aft camber on the lower surface is known as the “cove” region. If the camber is too extreme here, the adverse pressure gradient will be too steep and the flow will separate on the lower surface before it separates on the upper surface. This type of pressure distribution has a significantly more negative C_m0 than conventional airfoil sections.

5.1.2.2 Comments on airfoil design

Having surveyed the broad effects of various shape changes on the airfoil pressure distribution, we now make a few comments on airfoil design. At subsonic speeds, the key considerations are low parasite (skin friction) drag at low lift coefficients, basic airfoil pitching moment, and maximum lift values. To get low skin friction drag, a significant portion of the boundary layer should be laminar. This is possible for relatively small airplanes (up to around regional jet size) and surfaces with small sweep angles. Modern composite manufacturing techniques, which can produce extremely smooth surfaces, are an enabling technology. Generally, this smoothness is achieved passively by delaying the onset of any adverse pressure gradients until aft on the airfoil. There are various possible combinations of active laminar flow control (boundary layer suction) and passive laminar flow control. A nice survey of laminar flow technology is available in an article by Joslin,^[29] as well as a chapter in a 1990 AIAA Progress in Astronautics and Aeronautics book.^[30]

The aft camber illustrated above provides extra lift, but with a possible penalty of a large C_m0. Finally, high maximum lift coefficients are desirable; ways of obtaining high lift are discussed in Chapter 7, “High-Lift Aerodynamics.” Essentially, achieving high lift depends on using a pressure distribution that maximizes the load-carrying capacity of the boundary layer. Note that most current airfoil design also actively addresses transonic flow effects, where the shock strength for a specified lift coefficient should be minimized.

Effects of shape changes on pressure distributions: So far, the examples have demonstrated global effects of camber and thickness. To develop an understanding of the typical effects of adding local modifications to the airfoil surface, one must carry out an investigation into the effects of local shaping on the airfoil characteristics. I have had good luck using a smooth “cubic bump” to modify surfaces. The key with those types of airfoil shape modifications is to distribute the bump over a large portion of the airfoil, say one half to nearly the entire chord, with relatively low amplitude changes. Short bumps lead to poor off-design performance. It is also worthwhile to investigate the very powerful effects that small deflections of the trailing edge can produce. This reveals the power of the Kutta condition and alerts the aerodynamicist to the importance of the viscous effects at the trailing edge.

Computational experimentation is extremely educational when implemented in an interactive computer program, where the aerodynamicist can make shape changes with a mouse and immediately see the effects on pressure distribution. Ilan Kroo created an outstanding code that does just this, called PANDA.^[31]

Shape for a specified pressure distribution: There is another way that aerodynamicists view the design problem. The local modification approach described above is useful to make minor changes in airfoil pressure distributions. Often, the aerodynamic designer wants to find the geometric shape corresponding to a prescribed pressure distribution. This problem is known as the inverse problem. It is more difficult than the analysis problem. In fact, it is possible to prescribe a pressure distribution for which no geometry exists! Even if the geometry exists, it may not be acceptable from a structural standpoint. For two-dimensional incompressible flow, it is possible to obtain conditions on the surface velocity distribution that ensure that a closed airfoil shape exists. Excellent discussions of this problem have been given by Volpe^[32] and Sloof.^[33] A two-dimensional panel method for airfoil analysis and design has been developed by Bristow.^[34] Numerical optimization can also be used to find the shape corresponding to a prescribed pressure distribution.^[35]

Camber line design and DesCam: Most of the discussion above corresponds to the analysis problem, and we’ve given only a very brief summary of the inverse problem for subsonic flows. However, the airfoil camber line for a prescribed loading, ΔC_p, can be found easily with vortex lattice methods, which are used often in three-dimensional flows (see section 5.2). Here, the chord load is specified and the camber shape required to produce the prescribed load is found. In this section, we provide an example. The program DesCam is used to make the calculation using a slight variation of the VLM method known as the quasi-vortex lattice method and developed by Prof. Edward Lan of the University of Kansas.^[36] His method involves mathematically selecting vortex and control point placements instead of the 1/4 - 3/4 rule typically used in VLM methods. One interesting aspect of the inverse problem is that it is a direct solution and does not require the solution of a system of equations. Instead, the shape is computed from a straightforward algebraic calculation to find camber line slopes. Once the slopes are known, they are integrated to obtain the camber line.

Figure 5-25 provides an example. We compare the results from DesCam with the analytic formula given in appendix A, “Geometry for Aerodynamicists,” for the NACA 6-series mean line with a = 0.4. The camber scale is greatly enlarged to demonstrate the excellent comparison. Even though the chord load is constructed by prescribing two straight-line segments, the resulting required camber line is highly curved over the forward portion of the airfoil. Additionally, thin-airfoil theory allows only two possible values for the pressure differential at the leading edge: zero or infinity. A finite load at the leading edge is not possible. A close examination of the camber line shape required to produce a finite load reveals a singularity. The camber slope is infinite at the leading edge. This feature is much easier to study using the analytic solution given in appendix A. This type of approach to obtaining the desired pressure loading can easily be extended to three dimensions.

Finally, to learn more about subsonic airfoil design, read the papers by Liebeck^[37] and Drela^[38] in the Applied Computational Aerodynamics volume of the AIAA Progress in Aeronautics and Astronautics series.

5.1.3 Airfoil Selection

Having covered some of the basic ideas of airfoil aerodynamics, we conclude our airfoil discussion by summarizing the issues associated with choosing an airfoil for a particular application. Even though we’ve discussed airfoils before planforms, the logical progression in airplane design is to start with the mission of the airplane (requirements), then select the appropriate planform(s) and finally the airfoil(s) that best match the wing planform to deliver the desired aerodynamic performance. The basic considerations are the cruise performance (cruise C_L) and the field performance (C_Lmax). When considering trade studies for various planform and airfoil combinations, recall that the objective is to minimize drag, not drag coefficient. So the aerodynamicist should be evaluating D/q = S_wngC_D.

5.1.3.1 Cruise Performance

In cruise, the C_L for L/D_max should be found from the planform selection and the estimated C_D0. In general, the higher the aspect ratio, the higher the C_L for L/D_max. The choice of a predominantly laminar flow or turbulent flow airfoil should be addressed at this point. In particular, the laminar flow airfoils tend to operate in a narrowly defined range of lift coefficients to achieve the design performance (drag). Note that the wing C_L and airfoil section C_l’s may be different. The airfoil C_l requirement should be used to select the camber required. In addition, transonic cruise speeds require special consideration (see the next chapter), and the 2D airfoil problem for an airfoil on a swept wing differs from just defining a streamwise cut for an airfoil on the wing. The relation between the airfoil problem and the swept-wing planform is given later in this chapter. In general, the thickness-to-chord ratio (t/c) should be as large as possible to allow the wing structural weight to be low and to provide internal volume.

5.1.3.2 Field performance

We start with the requirement for the airplane to meet a specified stall or approach speed. This translates to a required C_Lmax. The conflict in requirements between efficient cruise (small wing) and low landing speed (big wing) means that most airplanes will have some type of high-lift system, ranging from a simple flap to triple-slotted flap systems described in Chapter 7. Either a large leading-edge radius is selected or a separate leading-edge device is used to assist in obtaining the required lift. The high-lift system should be as simple as possible to reduce manufacturing and maintenance cost.

5.1.3.3 Tail sections

For a vertical tail, the section should be symmetric. For a horizontal tail, the section is often symmetric. Low-drag 6A-series airfoils are often selected for tail surfaces, but they have poor values for C_Lmax. The NACA 4-digit sections have higher values of C_Lmax but more zero-lift drag. If the horizontal tail may be required to generate large downloads, a cambered airfoil may be used. This is more likely to occur when the airplane has a very powerful high-lift system. B-52s have vortex generators on the bottom surface of the horizontal tail to allow the tail to achieve the downforce required to trim the plane in takeoff and landing.

Finally, some help is available in choosing an airfoil. Gil Crouse has developed a software package called Airfoil Optimizer.^[39] This program allows the user to specify design requirements, and the program will suggest an airfoil based on a large library of airfoils. This type of procedure falls under the category of an “expert system.” Peter Garrison devoted a column to this package along with an excellent discussion of airfoil selection issues in his Flying magazine column Technicalities in 2002.

5.2 Wings

In this section, we highlight basic ideas of the subsonic aerodynamics of 3D wings. Before discussing the aerodynamics of wings, we first provide examples of the typical use and accuracy of vortex lattice methods (VLMs) that are widely used for simulating wing aerodynamics in the early stages of conceptual design. Complete details of the VLMs (as well as surface panel methods) are available in Chapter 5 of the Applied Computational Aerodynamics book.^[40] The vortex lattice layout is clear for most wings and wing-tail or wing-canard configurations. The method can be used for wing-body cases by simply specifying the projected planform of the entire configuration as a flat lifting surface made up of a number of straight line segments. The exact origin of this somewhat surprising approach is unknown. Examples in the following sections will help illustrate the success of this approach.

5.2.1 Use and Accuracy of the VLM Method

To get good, consistent, reliable results, some simple rules for panel layout should be followed. This requires that a few common rules of thumb be used in selecting the planform break points: (i) the number of line segments should be minimized; (ii) breakpoints should line up streamwise* on front and rear portions of each planform and should line up between planforms; (iii) streamwise tips should be used; (iv) small spanwise distances should be avoided by making edges streamwise if they are actually very highly swept; and (v) trailing vortices from forward surfaces cannot hit the control point of an aft surface. Figure 5-26 illustrates these common best practices.

To illustrate typical results that can be expected from vortex lattice methods, we provide some examples from a study by John Koegler^[41] that illustrates the range of uses for the VLM method. This study provides specific applications of several methods for fighter airplanes. In addition to the vortex lattice method, he also used the PAN AIR and Woodward II panel methods (see Chapter 5 of Applied Computational Aerodynamics^[42] for a description of panel methods). Koegler compared his computed predictions with available data for the three-surface F-15, which became known as the STOL/Maneuver Demonstrator, and the F/A-18. These configurations are illustrated in figure 5-27.

As with panel method calculations for airfoils, there must be a study of the number of horseshoe vortices required to obtain good results before the actual calculations are made. The effects of the number of panels and the way they are distributed is presented in figure 5-28 for the F/A-18 configuration (see figure 5-27 for the geometry). In this case, the VLM method is seen to take between 130 to 220 panels to produce converged results. The change in neutral point is used because this is perhaps one of the most important uses of these methods in early aircraft design. For the vortex lattice method, it appears important to use a large number of spanwise rows and a relatively small number of chordwise panels (five or six appear to be enough).

The basic panel layout of the F-15 STOL/Maneuver Demonstrator is given in figure 5-29. Several computational models were studied. This shows how the aircraft was modeled as a flat planform, and the corner points of the projected configuration were used to represent the shape in the vortex lattice method. In this case, the rake of the wingtip was included in the computational model. In the vortex lattice model, the configuration was divided into three separate planforms, with divisions at the wing root leading and trailing edges. On this configuration, each surface was at a different height, and after some experimentation, the vertical distribution of surfaces shown in figure 5-30 was found to provide the best agreement with wind tunnel data.

The results from these models are compared with wind tunnel data in table 5-1.^[43] The vortex lattice method is seen to produce excellent agreement with the data for the neutral point location and for lift and moment curve slopes at Mach 0.2.

Table 5-1: Three-surface F-15 longitudinal derivatives

Subsonic Mach number effects are simulated in VLM methods by transforming the Prandtl–Glauert equation, which describes the linearized subsonic flow to Laplace’s equation using the Göthert transformation. However, this is only approximately correct, and the agreement with wind tunnel data is not as good at the transonic Mach number of 0.8. Nevertheless, when the VLM method is used in this manner, it is as good as PAN AIR, which is a high-order panel method. Note that the VLM method used in the three-surface F-15 study is not applicable at supersonic speeds. The wind tunnel data shows the shift in the neutral point between subsonic and supersonic flow. The Woodward method, as applied here, overpredicted the shift with Mach number. We can conclude that the three-surface F-15 configuration is neutral to slightly unstable subsonically, becoming stable at supersonic speeds.

Figure 5-31 provides an example of the use of the VLM method to study the effects of moving the canard with respect to the plane of the wing. Here, the wind tunnel test result is used to validate the method and to provide an “anchor” for the numerical study. (It would have been useful to have an experimental point at -15 inches.) This is typical of the use of the VLM method in aircraft design. When the canard is above the wing (positive values of canard height; see figure 5-31), the neutral point is essentially independent of the canard height. However, when the canard is below the wing (negative values of canard height), the neutral point varies rapidly with canard height.

Control effectiveness is also of interest in conceptual and preliminary design, and the VLM method can be used to provide estimates. Figure 5-32 provides an apparently accurate example of this capability for F-15 horizontal tail effectiveness. Change in lift coefficient and pitching moment coefficient with respect to horizontal tail deflection, C_Lδh and C_mδh, respectively, are presented. The VLM estimate is within 10% accuracy at both Mach 0.2 and 0.8. However, the F-15 has an all-moving horizontal tail to provide sufficient control power under both maneuvering and supersonic flight conditions. Thus, the tail effectiveness presented here is effectively a measure of the accuracy of the prediction of wing lift and moment change with angle of attack in a nonuniform flow field, rather than the effectiveness of a flap-type control surface. A flapped device such as a horizontal stabilizer and elevator combination will have significantly larger viscous effects, and the inviscid estimate from a vortex lattice or panel method (or any inviscid method) will overpredict the control effectiveness. This is shown next for an aileron.

The aileron effectiveness for the F-15 is presented in figure 5-33. It is more representative of classical elevator or flap effectiveness correlation between VLM estimates and experimental data. This figure presents the change in the airplane’s rolling moment coefficient due to aileron deflection. In this case, the device deflection is subject to significant viscous effects, and the figure shows that only a portion of the effectiveness predicted by the VLM method is realized in the actual data. The VLM method, or any method, should always be calibrated with experimental data close to the cases of interest to provide an indication of the agreement between theory and experiment. In this case, the actual results are found to be about 60% of the inviscid prediction at low speed.

5.2.2 VLMpc and the Warren-12 Test Case

The VLMpc vortex lattice method can be used on personal computers. The version of the Lamar program described in NASA TN D-7921^[44] works easily with personal computers and is available for student use as VLMpc (in fact, students typed this code in from the listing in the TN). See section E.6.1 for links to the code and input instructions. This code is still used in advanced design work and can be used to investigate many ideas in wing aerodynamics. As shown above, results can be obtained and used before the large time-consuming methods of CFD are used to examine a particular idea in detail.

Next, we define one reference wing case that is typically used to check the accuracy of vortex lattice codes. It provides a good test case for the evaluation of any new or modified code, as well as a check on the panel scheme layout. It is known as the Warren-12 planform and is defined, together with the “official” characteristics from previous calculations, in figure 5-34. For the results given in the figure, the reference chord used in the moment calculation is the average chord (which is slightly nonstandard, as the mean aerodynamic chord is normally used as the reference chord), and the moment reference point is located at the wing apex (which is also nonstandard). Any calculations made in checking your proficiency in making a VLM calculation should include this case.

5.2.3 Other VLM Programs

In addition to VLMpc, the MATLAB program Tornado is capable of handling very general geometries and flow conditions. The program, written by Tomas Melin at KTH in Stockholm,^[45] has been used by many students and may be replacing VLMpc. It is also worth noting that the AVL code, developed some years ago by Mark Drela and Harold Youngren at MIT, has been placed in the public domain. It is quite effective for the aerodynamic and flight-dynamic analysis of rigid aircraft of arbitrary configuration. It employs an extended vortex lattice model for the lifting surfaces, together with a slender-body model for fuselages and nacelles.

The NASA-developed VSPAERO is another popular VLM code which also has an option for using surface panels for computational simulations using linearized potential flow equations. VSPAERO is one of the many analysis tools bundled in OpenVSP (Open Vehicle Sketch Pad). OpenVSP^[46] is an open source parametric geometry tool for creating three-dimensional models that are well suited for conceptual design studies. VSPAERO is an easy-to-use, robust, and cost-effective tool with fast turnaround time.

5.2.4 Aerodynamics of High Aspect Ratio Wings

With cost-effective flow simulation methods available for three-dimensional geometries, we can examine the aerodynamics of wings. Most of the results presented in this section were computed using VLMpc. One key advantage of the vortex lattice method compared to lifting-line theory is the ability to treat swept wings. Classical Prandtl lifting-line theory is essentially correct for unswept wings, but it is completely erroneous for swept wings. Aerodynamics of unswept wings are closely related to the airfoil characteristics of the airfoil used in the wing. This relationship is less direct for swept wings. Many of the most important wing planform–oriented characteristics of wings arise when the planforms are swept. Even though sweep is used primarily to reduce compressibility effects, the important aerodynamic features of swept wings can be illustrated at subsonic speeds using the VLM method.

5.2.4.1 Basic requirements

Wings are designed to satisfy requirements related to stability and handling characteristics while achieving low drag at the design conditions (usually cruise and sustained maneuver). They must also attain high maximum lift coefficients to meet field performance and maneuver requirements. Although these requirements might at first appear overwhelming, a small number of key characteristics (e.g., aerodynamic center, spanload, taper, and sweep) can provide a basic physical understanding of the aerodynamics of wings. Table 5-2 provides some key characteristics of well known major transport aircraft with wings designed to emphasize efficient cruise while meeting takeoff and landing requirements.

Table 5-2: Typical planform characteristics of major transport aircraft (data courtesy of Nathan Kirschbaum*)

5.2.4.2 Key Characteristics

Aerodynamic center

The first key characteristic is the aerodynamic center of the wing, defined as the location at which dC_m/dC_L = 0. The neutral point of the configuration is the aerodynamic center for the entire configuration. The VLM method was shown to provide accurate predictions of the neutral point for many configurations in the previous sections. The location of the neutral point is important in initial configuration layout to position the wing and any longitudinal stability and control surfaces at the proper location on the aircraft. Subsequently, this information is fundamental in developing the control system. Wing planform shaping, as well as positioning, is used to control the location of the configuration’s neutral point.

Spanload

The next key consideration is the spanload distribution, cC_l/c_a, where c is the local chord, c_a is the average chord of the wing, and C_l is the local section lift coefficient. The spanload controls the location of the maximum section lift coefficient, the induced drag of the wing, and the magnitude of the wing’s root bending moment. The location and value of the maximum section lift coefficient provides a good initial estimate of where the wing will stall first.* If the wing stalls in front of a control surface, control will be poor at flight conditions just when control becomes very important. The shape of the spanload, together with the actual value of the wingspan, determines the value of the induced drag. For a specified span, the performance of the wing is evaluated by finding the value of the span efficiency factor, e, as described in Chapter 3. Finally, the wing root bending moment provides an indication of the structural loading requirements that the wing structure must be designed to accommodate. When considering the total system, the basic aerodynamic efficiency may be compromised to reduce structural wing weight. The shape of the spanload can be controlled through a combination of planform selection and wing twist. Typical twist distributions required to produce good wing characteristics are presented below.

The simplest example of planform shaping is the selection of wing aspect ratio (AR), wing taper (λ), and wing sweep (Λ), usually of the quarter-chord line or the wing leading edge. While the aerodynamicist would like to see high values for the aspect ratio, several considerations limit this factor. Perhaps the most important limitation is the increase of wing structural weight with increasing aspect ratio. In addition, the lift coefficient required to maximize the benefits of high-aspect-ratio wings increases with the square root of the aspect ratio. Hence, airfoil performance limits can restrict the usefulness of high aspect ratios, especially for highly swept wings based on airfoil concepts. In recent years, advances in both aerodynamics and structures have allowed aircraft to be designed with higher aspect ratios and reduced sweep.

Taper

Several considerations are used in selecting the wing taper. For a straight untwisted, unswept wing, the minimum induced drag corresponds to a taper ratio of about 0.4. However, a tapered wing is more difficult and therefore expensive to build than an untapered wing. Many general aviation aircraft wings are built with no taper; in this case, all ribs are the same, reducing fabrication cost, and the maximum section C_l occurs at the root, well away from the control surface, providing positive roll control at stall (a very good feature). To reduce structural weight, the wing should be highly tapered, with λ < 0.4. Although highly tapered wings are desirable structurally, the section lift coefficient near the tip may become quite high, which limits the amount of taper employed. Current jet transports use taper ratios in the range of 0.2 to 0.3, as well as progressively increasing twist upward from the tip. In one other example, the Aero Commander 500 has an aspect ratio of 9.5 and a taper ratio of 0.5; it also has -6.5° of twist and the quarter chord of the wing was swept forward 4°.

Sweep

Sweep is used primarily to delay the effects of compressibility and increase the drag divergence Mach number. The Mach number controlling these effects is approximately equal to the Mach number normal to the leading edge of the wing, M_eff = M_∞cosΛ. The treatise on swept planforms by Küchemann is very helpful in understanding swept-wing aerodynamics.^[47] Aerodynamic performance is based on the wingspan, b. For a fixed span, the structural span increases with sweep, b_s = b/cosΛ, resulting in a higher wing weight. Wing sweep also leads to aeroelastic problems. For aft-swept wings, flutter becomes an important consideration. If the wing is swept forward, divergence is a problem. Small changes in sweep can be used to control the aerodynamic center when it is not practical to adjust the wing position on the fuselage (the DC-3 is the most famous example of this approach).

To understand the effects of sweep, the Warren-12 wing is compared with wings of the same span and aspect ratio, but unswept and swept forward. The planforms are shown in figure 5-35. The wing leading-edge sweep of the aft-swept wing becomes the trailing-edge sweep of the forward-swept wing.

Figure 5-36(a) provides the spanload, cC_l/c_a, for the three wings in figure 5-35 from VLMpc. Here, c is the local chord, C_l is the local lift coefficient based on the local chord, and c_a is the average chord, S/b. Using this nomenclature, the area under the curve is the total wing lift coefficient. Sweeping the wing aft increases the spanload outboard, while sweeping the wing forward reduces the spanload outboard. This follows directly from a consideration of the vortex lattice model of the wing. In both cases, the portion of the wing aft on the planform is operating in the induced upwash flow field of the wing ahead of it, resulting in an increased spanload.

Figure 5-36(b) shows the corresponding value of the local lift coefficient for the three wings shown in figure 5-35. Here, the effect of sweep is more apparent. The forward-swept wing naturally results in a spanload with a nearly constant lift coefficient. This means that a comparatively higher wing lift coefficient can be achieved before the wing begins to stall. The program LIDRAG can be used to compare the span e’s associated with these spanloads.

Similar results are now presented for a series of wings with larger aspect ratios (AR = 8) than the wings used in the study given above. Figure 5-37 shows the three wing planforms. These results are similar to the previous results. However, the trends observed above are in fact exaggerated at the higher aspect ratio.

Figure 5-38(a) presents VLMpc-computed results for the spanwise distribution of spanload, cC_l/c_a, where c is the local chord, C_l is the local lift coefficient based on the local chord, and c_a is the average chord, S/b. Using this nomenclature, the area under the curve is the total wing lift coefficient. Sweeping the wing aft increases the spanload outboard, while sweeping the wing forward reduces the spanload outboard. This follows directly from a consideration of the vortex lattice model of the wing. In both cases, the portion of the wing aft on the planform is operating in the induced upwash flow field of the wing ahead of it, resulting in an increased spanload.

Figure 5-38(b) shows the corresponding value of the local lift coefficient for the three wings shown in figure 5-37. Here, the effect of sweep is more apparent. The forward-swept wing naturally results in a spanload with a nearly constant lift coefficient. This means that a comparatively higher wing lift coefficient can be achieved before the wing begins to stall. The program LIDRAG can be used to compare the span e’s associated with these spanloads.

Aerodynamic problems as well as structural penalties arise when using a swept wing. Because of the high section lift coefficient near the tip, aft-swept wings tend to stall near the tip first. Since the lift at the tip is generated well aft, the pitching moment characteristics change when the stall occurs. With the inboard wing continuing to lift, a large positive increase in pitching moment occurs when the wingtip stalls. This is known as pitchup and can be difficult to control, resulting in unsafe flight conditions. Frequently, the swept-wing pitching moment characteristics are compounded by the effects of flow separation on the outboard control surface. Figure 5-39 provides an example of the pitching moment characteristics of an isolated wing with an aspect ratio of 10 using experimental data.^[48] The figure also includes the predictions from VLMpc. The agreement is reasonably good at low angle of attack but deteriorates at high angle of attack as viscous effects become important. This is another reason that sweep is minimized.

Twist

To control the spanload, the wing can be twisted. Figure 5-40 shows typical twist distributions for aft-swept and forward-swept wings, obtained from John Lamar’s program LAMDES^[49] (see Chapter 3 for a description of this code). In each case, the twist is used to reduce the highly loaded areas and increase the loading on the lightly loaded portions of the wing, bringing the spanload to an elliptical shape. For an aft-swept wing, this means the geometric incidence is increased at the wing root, known as washin, and reduced at the wingtip, known as washout. Just the reverse is true for the forward-swept wing. The sudden drop in required twist at the tip for the forward-swept wing case is frequently seen in typical design methods; it is attributed to a weakness in the method and is accounted for by the aerodynamicist in giving the design to the lofting group.

Although geometric sweep is used to reduce the effective Mach number of the airfoil, the geometric sweep is not completely effective. The flow field resists the sweep. In particular, the wing root and tip regions tend to effectively unsweep the wing. Aerodynamicists study lines of constant pressure on the wing planform known as isobars to investigate this phenomenon. Figure 5-41 presents the computed isobars for a typical swept wing,^[50] using a transonic small-disturbance method.^[51] The effect is dramatic. The effective sweep may actually correspond to the isobar line from the wing root trailing edge to the leading edge at the wingtip. To increase the isobar sweep, in addition to geometric sweep and twist, the camber surface and thickness are typically adjusted to move the isobars forward at the wing root and aft at the wingtip. This is a key part of the aerodynamic wing design job, regardless of the computational methodology used to obtain the predicted isobar pattern.

Using the wing planform and twist, together with a constant chord loading, figure 5-42 provides the camber lines required to support the load near the root, the midspan, and the wingtip. These results were also computed using LAMDES.^[52] At each station, a similar chord load is specified. We can easily see the differences in the camber required. This is an explicit illustration of the modification to an airfoil camber line required to maintain two-dimensional airfoil-type performance when the airfoil is placed in a swept wing. These modifications represent the explicit effects of the three dimensionality of the flow field.

Many other refinements are available to the aerodynamic designer. Insight into both the human and technical aspects of wing design prior to the introduction of computational aerodynamics is available in two recent books describing the evolution of the Boeing series of jet transports.^[53],[54]

Yehudi flap

One interesting refinement of swept wings has been the addition of trailing-edge area at the wing root. Generally known as a Yehudi flap, this additional area arises for at least two reasons. The reason cited most frequently is the need to provide structure to attach the landing gear at the proper location. However, the additional chord lowers the section lift coefficient at the root, where wing-fuselage interference can be a problem, and the lower required section lift makes the design job easier.

Douglas introduced this planform modification for swept wings on the DC-8, while Boeing did not incorporate it until the 320 model of the 707. However, as retired Boeing engineer William Cook writes on page 83 of his book,^[55] it was first introduced on the B-29 to solve an interference problem between the inboard nacelle and the fuselage. The aerodynamic benefit to the B-29 can be found in the paper by Snyder.^[56] In a letter to me, Cook wrote that the device got its name because each wind tunnel part needed a name and there was a popular radio show at the time that featured the continuing punchline “Who’s Yehudi?” (Note: This was the Bob Hope Radio show featuring Jerry Colonna, who had the line.) Thus, a Boeing engineer decided to call it a Yehudi flap. This slight extra chord is readily apparent when examining the B-29, though it is very difficult to photograph.

5.2.5 The Relation Between Airfoils and Swept Wings

In section 5.1, we examined the basic aerodynamics of airfoils using panel methods. So far in section 5.2, we have emphasized the planform shape and its analysis using vortex lattice methods. The connection between the airfoil and planform is important. In most cases, the integration of the airfoil concept and the wing planform concept is crucial to the development of a successful configuration. Simple sweep theory can be used to provide, at least approximately, the connection between the airfoil and the planform for swept wings. The typical aerodynamic design problem for an airfoil in a wing is defined by specifying the streamwise thickness-to-chord ratio, t/c, the local section lift coefficient, C_ldes, at design conditions, and the Mach number. This three-dimensional problem is then converted to a corresponding two-dimensional problem. The desired two-dimensional airfoils are then designed and transformed back to the streamwise section to be used as the wing airfoil section. Examples of the validity of this technique, together with details on other properties (including the “cosine cubed” law for profile drag due to lift) are available in the NACA report by Hunton.^[57] Figure 5-43 shows the relationships between the streamwise airfoil properties and the chordwise properties (values normal to the leading edge).

These relations demonstrate that the equivalent two-dimensional airfoil is thicker and operates at a lower Mach number and at a higher lift coefficient than the related three-dimensional wing airfoil section. Taper effects on real wings require the selection of an effective sweep angle. Numerous approaches have been used to determine the effective angle, where guidance has been obtained by examining experimental data. The quarter-chord sweep or shock sweep are typical choices.

One good example of airfoil/planform matching is the Grumman X-29. In that case, wind tunnel testing of advanced transonic maneuver airfoil sections on aft-swept wing configurations led the aerodynamicists (Glenn Spacht in particular) to conclude that the proper planform to take advantage of the advanced airfoil section performance should be swept forward.

5.2.6 Wing-Tail and Canard-Wing Aerodynamics

Additional lifting surfaces are used to provide control of aircraft configurations over a wide range of flight conditions. If modern advanced control systems are not used, the extra surface is also designed, together with the rest of the configuration, to produce a stable design. In considering aft-tail configurations, the problem of pitchup described above for isolated wings must be reconsidered. In particular, T-tail aircraft can encounter problems when the horizontal tail interacts with the wake of the wing at stall. The pitching moment characteristics of the DC-9^[58] show that an initial abrupt nose-down characteristic is the result of careful design, occurring before a large pitchup develops. Even though pitchup is a viscous effect, inviscid calculations clearly show why it happens and can provide valuable information. On the DC-9, a stable trim condition occurs at an angle of attack of 43°. This is an undesirable equilibrium condition, which could result in the vehicle actually trying to fly at this angle of attack. If adequate control power is not available, it may even be difficult to dislodge the vehicle from this condition, which is commonly known as a deep or hung stall. This will result in a rapid loss of altitude due to the very high drag. For this configuration, full down elevator eliminates the possibility of getting trapped in a trimmed flight condition at this angle of attack, but the amount of pitching moment available may not be sufficient to affect a rapid recovery from this condition. Examples of pitchup characteristics are not readily available. Aerodynamic designers do not like to admit that their configurations might have this characteristic. This aspect of swept-wing and wing-tail aerodynamics is an important part of aerodynamic configuration development.

Even low tail placement cannot guarantee that there will not be a problem. The pitching moment characteristics for an F-16 wind tunnel model^[59] showed a deep stall. In fact, the allowable angle of attack on the F-16 is limited by the control system to prevent the airplane from encountering this problem. In this case, the pitchup arises because of powerful vortices generated by the strakes, which continue to provide lift as the wing stalls.

Canard configurations provide another interesting example of multiple lifting surface interaction. The downwash from the canard wake as it streams over the wing reduces the effective angle of attack on the wing locally and the local lift on the wing behind the canard. Wing twist is used to counteract this effect. Figure 5-44 illustrates how this interaction occurs. The relative loading of the surfaces (how much lift is carried by the canard and how much is carried by the wing) is an important consideration in configuration aerodynamics. The induced drag is highly dependent on the relative wing loading, which is determined by the selection of the configuration stability level and the requirement to trim about the center of gravity.

Figure 5-45 shows a wing-canard combination that can be used to illustrate the strong effect of CG position on induced drag. This figure shows the plan view of the configuration (bottom) and notional chordwise loadings at a spanwise station on the canard and the wing (top).

Figure 5-46 presents the induced drag computed using LAMDES^[60] and shows the variation of trimmed drag changes with CG position. Anything above the minimum drag value should be considered the trimmed drag penalty. Three different canard heights are shown for a range of CG positions, which is equivalent to varying the stability level.

Figure 5-47 provides an example of the wing twist required to account for the effect of the canard downwash. Results depend on configuration details, balance, and the numerics that lead to sudden drop in incidence at the wingtip of the forward-swept wing, all of which may or may not be accurate. Due to the canard, the forward-swept wing twist increment acts to reduce the twist required, which is exactly opposite of the canard effect for the aft-swept wing.

5.2.7 Ground Effects Using a VLM Code

We complete our discussion of wings at subsonic speeds with an example of ground effects, computed using VLM codes. Figure 5-48 shows the effects of the presence of the ground on the aerodynamics of simple unswept rectangular wings. The lift and pitching moment slopes are presented for calculations made using JKayVLM^[61] and compared with the results published by Kalman et al.^[62] and with experimental data. The agreement between the data and calculations is excellent for the lift-curve slope. The AR = 1 wing shows the least effects of ground proximity because of the three-dimensional relief provided around the wingtips. As the aspect ratio increases, the magnitude of the ground effects increases. The lift-curve slope starts to increase rapidly as the ground is approached.

As shown in the image, the wings also experience a significant change in the pitching moment slope (aerodynamic center shift). Note that the predictions start to differ as the ground is approached. JKayVLM actually rotates the entire surface to obtain another solution to use in estimating the lift-curve slope. The standard procedure used by most methods is to simply change the slope condition at the mean line location. Because of the proximity to the ground, this approach might be a case where the application of the boundary condition on the mean line may not be accurate.

Figure 5-49 presents similar information for the effect of dihedral angle, Γ, on a wing with an aspect ratio of 4. In this case, the effects of anhedral changes (negative Γ), where the wingtip approaches the ground, are extremely great. The results of dihedral changes for a wing out of ground effect (h/c = ∞) are shown for comparison. The figure compares computed results from JKayVLM^[63] and published results of Kalman et al.^[64] Both methods agree well with each other, with differences appearing only as the wingtips approach the ground. Here again, JKayVLM actually rotates the entire geometry, apparently resulting in an increase in the effects as the tips nearly contact the ground. It also prevents calculations from being obtained as close to the ground as the published results. In making these calculations, it was discovered that the wing panel was rotated and not sheared, so that the projected span decreases as the dihedral increases, and this produces much more pronounced changes in the lift-curve slope due to the reduction in projected span. Most of the decrease in lift-curve slope values for the wing out of ground effect is due to the reduced span as the wing rotates from the zero-dihedral condition.

Ground effects shown here arise because of the nonpenetration condition at the ground, resulting in generally higher lift slopes. The effect is associated with three-dimensional flow. However, ground effects are considerably more complicated than suggested here. See Torenbeek^[65] for a more complete discussion.

5.2.8 Low Aspect Ratio “Slender Wings”

Vortex flow effects: For highly swept wings at even moderate angles of attack, the classical attached flow/trailing-edge Kutta condition flow model we’ve adopted is wrong. Instead of the flow remaining attached on the wing and leaving the trailing edge smoothly, the flow separates at the leading edge, forming a well defined vortex. This vortex plays an important role in the design of highly swept, or “slender wing,” aircraft. The most notable example of this type of configuration is the Concorde. Sharp leading edges promote this flow phenomena. The basic idea is illustrated in the sketch from Payne and Nelson^[66] given here in figure 5-50.

An important consequence of this phenomena is the change in the characteristics of the lift generation as the wing angle of attack increases. The vortex that forms above the wing provides an additional low pressure force due to the strongly spiraling vortex flow. The low pressure associated with the centrifugal force due to the vortex leads to lower pressure on the wing. As the wing increases its angle of attack, the vortex gets stronger, further reducing the pressure on the wing. The resulting increase in lift due to the vortex can be large, as Polhamus^[67] shows in figure 5-51.

This is an important flow feature. Slender wings have very low attached-flow lift-curve slopes, and without the additional vortex lift, it would be impractical to build configurations with low aspect ratio wings. The low attached-flow lift-curve slope would prevent them from being able to land at acceptable speeds or angle of attack. Vortex lift made the Concorde possible. Another feature of the flow is the high angle of attack at which maximum lift occurs; it also typically suffers only a very mild lift loss past maximum lift. These features are a direct result of the leading-edge vortex flow structure that occurs on slender wings.

Leading-edge suction analogy: Although the vortex lattice method formulation presented above does not include this leading-edge vortex effect, VLMs are often used as the basis for extensions that do include the leading-edge vortex effects. A remarkable, reasonably accurate flow model for leading-edge vortex flows was introduced by Polhamus^[68],[69] at NASA Langley in 1966 after examining lots of data. This flow model is known as the leading-edge suction analogy. The concept is quite simple and was invented for sharp-edged wings. The leading-edge suction that should exist according to attached flow theory is assumed to rotate 90° and generate a vortex-induced force instead of a suction when leading-edge vortex flow exists. Thus, the vortex flow force is assumed to be equal to the leading-edge suction force. However, the force now acts in the direction normal to the wing surface in the direction of lift rather than in the plane of the wing’s leading edge. The concept is shown in figure 5-52 from the original Polhamus NASA report.^[70] Further details on the effects of vortex flow effects are also available in reports by Kulfan.^[71],[72]

Polhamus developed charts to compute the suction force for simple wing shapes. For a delta wing with a sharp leading edge, the prediction using the suction analogy is compared with the experimental data of Bartlett and Vidal^[73] in figure 5-53. The agreement is quite good. Though my reconstruction doesn’t show agreement as good as that presented by Polhamus,^[74] it is still impressive. The figure also shows the large amount of the vortex lift and the nonlinear shape of the lift curve when large angles are considered. This characteristic was exploited in the design of the Concorde.

To find the vortex lift using the leading-edge suction analogy, an estimate of the leading-edge suction distribution is required. However, the suction analogy does not result in an actual flow field analysis that includes leading-edge vortices. The VLMpc computer program optionally includes a fully developed suction analogy based on Polhamus’ ideas, with extensions to treat side-edge suction that can be found in a NASA report by Lamar.^[75]

Other approaches have been developed to compute leading-edge vortex flows in more detail. Many of these methods allow vortex filaments, simulating the leading-edge vortices, to leave the leading edge. The location of these vortices are explicitly computed, as are their effect on the wing aerodynamics as they roll up. Mook et al.^[76] are leaders in this methodology.

The area of vortex flows in configuration aerodynamics is so fascinating that, in 1985, an entire conference at NASA Langley^[77] was devoted to this topic. The references cited above were selected to provide an entry to the literature on these flows. Interest in the area remains strong. The effects of round leading edges have been investigated by Ericsson and Reding^[78] and by Kulfan.^[79] The relationship between sweep, vortex lift, and vortex strength has been explored by Hemsch and Luckring.^[80]

Figure References