In this chapter, our primary goal is to discuss drag in more depth than the discussions typical to a introduction course in aerodynamics. We believe drag to be of foundational significance for configuration aerodynamics, and it is an extremely important driver of the aerodynamic design of configuration. Therefore, we cover it before addressing other aerodynamic considerations.

Drag is one of the two components of the total aerodynamic force that acts on an aircraft that is moving through air. When the total force is resolved in the plane of symmetry of the aircraft, drag is one component and lift is the other. Lift is perpendicular to the flight path, and drag is parallel to the flight path. Drag opposes the motion of the aircraft, requiring a propulsion system to overcome its retarding effect.

In this chapter, we introduce the key aspects of aerodynamic drag. We first highlight, in section 3.1, some of the main reasons why drag is crucially important for aircraft design. The subsequent sections discuss the concepts and methodologies required to evaluate drag of aircraft configurations. Additional discussions are included in later chapters on subsonic, transonic, and supersonic aerodynamics.

3.1 The Importance of Drag

Drag is at the heart of aircraft aerodynamic design. There are many different contributors to the total drag of an airplane. In three-dimensional flows, drag occurs even when the flow is assumed to be inviscid. This is also true in two-dimensional flows when compressibility becomes important. Before discussing the aerodynamics of lifting systems, the fundamental aspects of aerodynamic drag need to be established.

The subject is complicated. All aerodynamicists secretly hope for negative drag! In practice, however, there is no escaping the downsides of drag. The value of drag is also very important. Even seemingly minor changes in drag can be critical. On the Concorde, a one-count drag increase (ΔC_D = 0.0001) required that two passengers out of the 90–100 passenger capacity be taken off the North Atlantic run.^[1] In design studies, a drag decrease is equated to the decrease in aircraft weight required to carry a specified payload a required distance. One advanced fighter study found the drag sensitivity in supersonic cruise to be 90 lb/drag count compared to 48 lb/ct for subsonic/transonic cruise. At the transonic maneuver design point, the sensitivity was 16 lb/ct (drag is very high here). In comparison, the growth factor was 4.1 pounds of takeoff gross weight for every 1 pound of fixed weight added. For one executive business jet, the range sensitivity is 17 miles/drag count. The large long-range advanced supersonic transports studied in the 1990s had range sensitivities of about 100 miles/drag count.

When new aircraft are sold, the sales contract stipulates numerous performance guarantees. One of the most important is range. The aircraft company guarantees a specified range before the aircraft is built and tested. The penalty for failure to meet the range guarantee is severe. Conservative drag projections aren’t allowed. The competition is so intense that, in the design stage, aerodynamicists are pressured to make optimistic estimates. In one briefing I attended in the early ’80s, an aerodynamicist for a major airframer said that his company was willing to invest $750,000 for each count of drag reduction! Under these conditions, the importance of designing for low drag and the ability to estimate drag can hardly be overstated.

The economic viability and future survival of an aircraft manufacturer depends on minimizing aerodynamic drag (together with incorporating advances in other key technologies of structures, propulsion, and control) while maintaining good handling qualities to ensure flight safety and ride comfort. New designs that employ advanced computational aerodynamics methods are expected to produce vehicles with less drag than the aircraft in service. Designs such as Boeing 777 and Airbus A340 already take advantage of computational aerodynamics, advanced experimental methods, and years of experience. Future improvements in aerodynamic performance present tough challenges requiring both innovative concepts and the very best methodologies possible.

In the early stages of a design effort, initial drag estimates can dictate the selection of a specific configuration concept after comparison with others. The drag predictions have a huge effect on the projected configuration size and cost and thus on the selection of a specific concept and the decision to proceed with the design.

There are two other key considerations in discussing drag. First, drag cannot yet be predicted accurately with high confidence levels^[2] (especially for nonconventional configuration concepts) without extensive testing. Second, no one is exactly sure of the minimum possible drag level that can be achieved for a practical configuration. To this extent, aerodynamic designers are the dreamers of the engineering profession.

Because of its importance, AGARD (the NATO Advisory Group for Aerospace Research & Development) has held numerous conferences devoted to drag and its reduction. AGARD publications include CP-124,^[3] CP-264,^[4] R-723,^[5] and R-786.^[6] These reports provide a wealth of information. An AIAA Progress Series book has also been devoted primarily to drag,^[7] including chapters that discuss the history of drag prediction, typical methods currently used to predict drag, and the intricacies of drag prediction for complete configurations. The most complete compilation of drag information is available due to Hoerner.^[8] Because of its importance, techical papers on drag prediction and reduction appear regularly. A good example is the survey by van Dam.^[9]

3.2 Drag—Numerous Viewpoints and Basic Concepts

In discussing drag, the numerous viewpoints that people use can create confusion. Here we illustrate the challenge of defining drag from several viewpoints. This also provides an opportunity to discuss various basic drag concepts.

3.2.1 Nearfield Drag Calculation Viewpoint

Consider the distribution of forces over a surface. The forces include a pressure force and a shear stress force due to the presence of viscosity. Many people assume that drag can be accurately calculated through an accurate integration of surface forces. This approach is known as a nearfield drag calculation. However, two problems exist:

• This integration requires extreme precision.
• The results are difficult to interpret for aerodynamic analysis. Exactly where is the drag coming from? Why does it exist, and how do you reduce it?

Thus in most cases, a simple integration over the surface is not satisfactory for use in aerodynamic design. Codes have only recently begun to be fairly reliable for nearfield drag estimation and then only for certain specific types of problems. Though the best success has been achieved for airfoils, even that situation isn’t perfect.

3.2.2 Fluid Mechanics Viewpoint

This viewpoint emphasizes the drag resulting from various fluid mechanics phenomena. This approach is important in conceiving a means to reduce drag. It also provides a means of computing drag contributions in a systematic manner. Thinking in terms of contributions from different physical effects in subsonic flow, a typical drag breakdown is shown in figure 3-1.

The key contributors are:

• Friction drag—caused by the action of viscosity in the boundary layer on the surface of a body; it’s commonly referred to as skin friction drag.
• Form drag—caused by the net difference between pressure forces acting on the forward-facing and the rearward-facing portions of a body. Note that viscous effects are the principal source of the different pressures that are highly dependent on the shape of the body.
• Wave drag—caused by the energy radiated away from the supersonic flow regions around a body in the form of pressure waves.
• Induced drag—caused by the vorticity shed behind a lifting surface, such as the vortex wake behind a wing.

Torenbeek’s book^[10] offers a good discussion of drag and its estimation. That discussion makes a convincing case that the subject can be complicated!

For a highly streamlined, aerodynamically clean shape, the zero-lift drag at subsonic speeds in the absence of shock waves should be mostly due to skin friction. Pressure drag arises from regions of separated flow, which need to be avoided. R. T. Jones^[11] has presented a striking figure, shown here as figure 3-2, comparing a 21-percent-thick NACA 64-421 airfoil to a circular wire of diameter equal to only about 0.006 times the airfoil thickness. It is hard to believe, but the two have the same drag at the same speed! It clearly demonstrates the importance of streamlining. Friction drag is the dominant contributor to the streamlined airfoil drag and form drag for the wire.

3.2.3 Aerodynamic Design Viewpoint

This viewpoint results from combining the fluid mechanics viewpoint with more practical considerations. From the perspective of the aerodynamic design of aircraft, it proves useful to think in terms of contributions from a variety of aircraft features. This includes effects due to the requirement to trim the aircraft and due to interactions between the aerodynamics of the vehicle and both propulsion-induced flow effects and structural deformation effects. Within this context, several other considerations are identified. The basic contributions from each component must be included.

This leads to a drag analysis based on typical configuration features itemized below:

• Individual component contributions to drag
• Base drag
• Inlet drag with spillage
• Boattail drag
• Camber drag
• Trim drag
• Thrust-drag bookkeeping
• Aeroelastic effects on drag

3.2.4 Aircraft Performance Viewpoint

To calculate the performance of an airplane, it is natural to define drag as the sum of the drag at zero lift and the drag due to lift. This is the approach that leads to the typical drag polar equation:

$$C_D=C_{D_0}+\frac{C_L^2}{\pi ARE}\text{(3-1)}$$

Here each term is a function of Mach number, Reynolds number (which, in practice, is given to the performance group in terms of Mach number and altitude), and the particular geometric configuration (flap deflection, wing sweep, etc.). The drag is not precisely a quadratic function of the lift, and the value of the Oswald efficiency factor, E, in equation (3-1) is a function of the lift coefficient and Mach number: E = E(C_L,M).

Note that uppercase E in equation (3-1) accounts for additional losses due to the fuselage and viscous effects. We reserve the use of lowercase e for span efficiency factor, which is solely a function of the spanload estimated using inviscid flow theories, such as the lifting-line theory. This notation is the most prevalent in use in the US aircraft industry. However, other notations are frequently employed, and care must be taken when reading the literature to make sure that you understand the notation used. The drag also depends on the throttle setting, but that effect is usually included in the thrust table.

There is another drag polar approximation that is seen often. This approximation is more commonly used by aerodynamic designers trying to understand wing performance. It is used to consider the effect of wing camber and twist, which causes the drag polar to be displaced “upward,” becoming asymmetrical about the C_L = 0 axis, as shown in figure 3-3. It is given as equation (3-2) below.

$$C_D=C_{D_0}+\mathrm{\Delta }{C}_{D_m}+K{(C_L-C_{L_m})}^2\text{(3-2)}$$

In taking into account the effect of camber and twist on displacing, or “shifting,” the polar, the term ΔC_Dm represents a penalty associated with using twist and camber to achieve good performance at the design lift coefficient. This equation is for a fixed geometry. Note that the effect of ΔC_Dm is exaggerated for emphasis in figure 3-3. The value of K defines the shape of the polar. C_D0 represents the minimum drag of the configuration without camber and twist. The values of ΔC_Dm and C_Lm are functions of the design lift coefficient. Sometimes novice aerodynamicists fail to include ΔC_Dm properly and obtain incorrect values of E when evaluating published drag polars. This type of polar shape is discussed in more detail later in this chapter. Advanced aircraft concepts such as the X-29, as well as the F-16 and F-18, minimize this penalty by defining a device schedule to maximize performance across a broad range of lift coefficients.

3.2.5 Categorization of Drag Concepts

It is apparent that the types of viewpoints outlined above may be confusing, and sometimes they get in the way of technical discussions. To facilitate further discussion in the context of configuration aerodynamics, the chart in figure 3-4 provides a basic categorization of drag concepts. This aerodynamic configuration-focused approach to drag is not covered in fluid mechanics–oriented aerodynamics texts, but it is described in aircraft design books. Two good reference books are by Whitford^[12] and by Huenecke.^[13] An approach to the evaluation of drag performance, including the efficiency achieved on actual aircraft, was presented by Haines.^[14]

We next discuss refinements to several of these concepts in more detail. However, the most important overview of aerodynamic drag for design has been given by Küchemann^[15] and should be studied for a complete understanding of drag concepts.

3.2.5.1 A Fluid Mechanics Refinement: Transonic Wave Drag

The simplified picture of drag presented in figure 3-1 may suggest that wave drag appears suddenly at supersonic speeds. A more refined examination shows that wave drag arises at subsonic speeds when the flow accelerates locally to supersonic speeds and then returns to subsonic speed through a shock wave. This leads to the presence of wave drag at subsonic (actually, by definition, transonic) freestream speeds, as shown in figure 3-4. This initial drag increase, known as drag rise, is followed by a rapid increase in drag and is an important consideration in the design of wings and airfoils. The Mach number at which the rapid drag increase occurs is known as the drag divergence Mach number, M_DD. The increase in drag occurs directly because of the wave drag associated with the presence of shock waves. However, the drag also increases because the boundary layer thickness increases due to the sudden pressure rise on the surface due to the shock wave, which leads to increased profile drag. Lynch^[16] has estimated that, at drag divergence, the additional transonic drag is approximately evenly divided between the explicit shock drag and the shock-induced additional profile drag.

Several definitions of the drag rise Mach number are commonly used. The specific definition is usually not important because the drag rises very rapidly at drag divergence and different definitions all result in similar values of M_DD.

One standard definition of M_DD is the Mach number where

$${\frac{d{C}_D}{dM}|}_{C_L=\text{const.}}=\mathrm{0.1.}\text{(3-3)}$$

Another definition of M_DD is the Mach number at which

$$\mathrm{\Delta }{C}_D=0.0020\text{from the subsonic value.}\text{(3-4)}$$

Commercial transports fly at or close to M_DD, and the drag divergence Mach number is a key part of the performance guarantee.

Figure 3.5 (data from Shevell^[17]) illustrates the transonic wave drag, a fluid mechanics refinement, as additional drag, ΔC_Dcomp, due to compressibility as Mach number increases. Note that the M_DD values corresponding the criteria in equations (3-3) and (3-4) are quite close. Together with the definitions associated with the drag rise, this figure also illustrates a common characteristic known as drag creep, which occurs with many transonic designs. Drag creep is characterized by a slow rise in drag with Mach number increase or a small and roughly constant positive slope in the drag versus Mach curve starting well before M_DD. This drag increase is attributed to the increasing strength of the forward relatively benign shocks and to the gradual thickening of the boundary layer due to the shocks and the higher adverse pressure gradient from increasing C_p.^[18]

3.2.5.2 An Aerodynamics/Flight Mechanics Refinement: Trim Drag

A source of drag not directly related to fluid mechanics arises from the need to trim the vehicle (C_m = 0 about the center of gravity) for steady flight. This requirement can lead to control surface deflections that increase (or decrease) the drag. It can be especially important for supersonic aircraft because of the shift in the aerodynamic center location with changing Mach numbers. Other cases with significant trim drag may include configurations with variable wing sweep and the use of airfoils with large values of the zero-lift pitching moment about their aerodynamic center. Section 3.5.6 contains more details about trim drag.

3.2.5.3 An Aeropropulsion Integration Refinement: Thrust-Drag Bookkeeping

To determine aircraft performance, the key value is actually not drag but the balance between thrust and drag. The drag of the airframe is affected by the operation of the propulsion system, and care must be taken to understand and define these interactions. The amount of air used by the engine defines the size of the streamtube entering the inlet. If all the air in front of the inlet does not enter the inlet, a spillage drag will result. Similarly, the boattail drag over the external portion of the nozzle will depend on the nozzle setting in the case of engines with afterburners and the pressure of the nozzle flow. The definition of a system to properly account for aeropropulsion interactions on the specification of thrust minus drag values is known as thrust-drag bookkeeping. Since thrust is usually provided by the propulsion group, and drag is provided by the aerodynamics group, significant errors in the estimation of aircraft performance have occurred when the necessary coordination and adjustments were not made. The details of this procedure are described in the article by Rooney.^[19]

Generally, the aerodynamics group provides the performance group with a reference drag polar, and all thrust-dependent corrections to the drag polar are accounted for by making adjustments to the thrust values. This is done because it is natural to establish a performance calculation procedure using this approach. The precise details are not important as long as everyone involved in the performance prediction agrees to a specific approach. Usually this requires a specific document defining thrust-drag bookkeeping for each aircraft project.

3.2.5.4 An Aerostructural Interaction Refinement: Aeroelastic Effects on Drag

This issue is not strictly a drag consideration, but it can contribute to the drag if it is not addressed. Aircraft structures deform due to air loads. If the design is centered around a single design point, the aerodynamic shape at the design point can be defined, and the structural analysts will adjust for structural deformation, specifying a “jig shape” that will produce the desired aerodynamic shape at the design point and loading. This is harder to do if there are multiple design points. Deformation of wind tunnel models should also be considered when estimating drag.

3.3 Theoretical Analysis of Drag Components Based on Momentum Balance

In this section, we discuss the most accurate estimation of drag on a body by considering the overall momentum balance on a control volume surface well away from the body—a farfield calculation. This approach is particularly useful when our prediction methods are not exact. The results are also much less sensitive compared to the nearfield pressure integration approach, which requires detailed calculations of surface pressure and then the integration of the pressures over the surface to obtain the drag. It also allows the components to be found separately.

3.3.1 Farfield Drag Analysis

The farfield analysis makes use of the momentum theorem. References containing good derivations are Ashley and Landahl,^[20] sections 1.6, 6.6, 7.3, and 9.2, and Heaslet and Lomax.^[21]

For a surface S that encloses the volume containing the body, the force can be determined by balancing the momentum across S:

$$\mathbf{F}=-{\text{∯}}_S(p-p_{\mathrm{\infty }})d\mathbf{S}-{\text{∯}}_S\rho \mathbf{q}[({\mathbf{V}}_{\mathrm{\infty }}+\mathbf{q})\cdot d\mathbf{S}]\text{(3-5)}$$

where q is the disturbance velocity vector,

$$\mathbf{V}={\mathbf{V}}_{\mathrm{\infty }}+\mathbf{q}\text{(3-6)}$$

Define a control volume for use in equation (3-5), as shown in figure 3-6.

Consider flows far enough away from the body such that linearized flow relations are valid. Use the small disturbance relations:

$$\rho \cong {\rho }_{\mathrm{\infty }}(1-M_{\mathrm{\infty }}\frac{u}{V_{\mathrm{\infty }}})\text{(3-7)}$$

and

$$(p-p_{\mathrm{\infty }})\cong -{\rho }_{\mathrm{\infty }}[V_{\mathrm{\infty }}u+\frac{1}{2}(u^2+v^2+w^2)]+\frac{1}{2}{\rho }_{\mathrm{\infty }}{M}_{\mathrm{\infty }}^2{u}^2\text{(3-8)}$$

Now, consider the drag component of equation (3-5), making use of equation (3-7) and equation (3-8):

$$D=\frac{1}{2}{\rho }_{\mathrm{\infty }}{\iint }_{I+II}[(M_{\mathrm{\infty }}^2-1)u^2+v^2+w^2]dydz-{\rho }_{\mathrm{\infty }}{\iint }_{III}u{v}_{r}rd\theta dx\text{(3-9)}$$

The radial component is v_r,

$$v_r^2=v^2+w^2,$$

and

$$r^2=x^2+y^2\text{.}$$

Considering the control volume shown in figure 3-6, place I and II far upstream and downstream, respectively, and make r large. Then, the integral over I is zero as x → −∞ . The integral over II as x → ∞ , corresponds to the so-called Trefftz plane (see section 3.3.2). The integral over III is the wave drag integral, which is zero for subsonic flow and when any embedded shock waves do not reach III.

3.3.2 Lift-Induced Drag: The Trefftz plane

Let us first outline the basic concept of the Trefftz Plane (named after a German mathematician and aerodynamicist):

1. Far downstream, the motion produced by the trailing vortices becomes 2D in the y-z plane (no induced velocity in the x-direction).
2. For a wing moving at a speed U_∞ through the fluid at rest, an amount of mechanical work D_iU_∞ is done on the fluid per unit of time. Since the fluid is nondissipative (potential flow), it can store energy in kinetic form only. Therefore, the work D_iU_∞ must show up as the value of kinetic energy contained in a length U_∞ of the distant wake.
3. The vortices in the trailing vortex system far downstream can be used to find the induced drag.

Consider the integral over II in figure 3-6. This is the first integral in equation (3-9), the induced drag integral:

$$D_i=\frac{1}{2}{\rho }_{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[(M_{\mathrm{\infty }}^2-1)u^2+v^2+w^2]dydz\text{(3-10)}$$

Note that many supersonic aerodynamicists call this the vortex drag, D_v, since it is associated with the trailing vortex system. However, it is in fact the induced drag. The term vortex drag is confusing in view of the current use of the term “vortex” to denote effects associated with other vortex flow effects. Far downstream, at u → 0, we are left with the v- and w-components of velocity induced by the trailing vortex system. The trailing vortex sheet can be thought of as legs of a horseshoe vortex. Thus the integral becomes

$$D_i=\frac{1}{2}{\rho }_{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(v^2+w^2)dydz\text{(3-11)}$$

and relates the induced drag to the kinetic energy of the trailing vortex system.

Now, the flow is governed downstream by the Prandtl–Glauert equation (even if the flow at the vehicle has large disturbances, the perturbations decay downstream):

$$(1-M_{\mathrm{\infty }}^2){\varphi }_{xx}+{\varphi }_{yy}+{\varphi }_{zz}=0\text{(3-12)}$$

As x → ∞, then u = 0 and u_x = ϕ_xx = 0. As a result, the governing equation for the disturbance velocities is Laplace’s equation for the crossflow velocity:

$${\varphi }_{yy}+{\varphi }_{zz}=0\text{(3-13)}$$

An interesting result arises here. The induced drag is explicitly independent of Mach number! The analysis is valid for subsonic, transonic, and supersonic flows. The Mach number only enters the problem in an indirect manner through the boundary conditions, as we will see.

We now use Green’s theorem to convert the area integral in equation (3-11) to a contour integral. Applying the theorem to the drag integral, we arrive at equation (3-14).

$${\iint }_{II}(v^2+w^2)dS=-{\int }_c\varphi \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }n}dc\text{(3-14)}$$

This is a general relation that converts the integral over the entire crossplane into an integral along the contour. It applies to multiple lifting surfaces. To illustrate the application of the integral to the determination of the induced drag, we consider the special case of a planar lifting surface. Here the contour integral is taken over the surface shown in figure 3-7, where the trace of the trailing vortices shed from the wing are contained in the slit between -b/2 and b/2.

In this Trefftz plane, the integral vanishes around the outside contour as R → ∞ and the integrals along AB and CD cancel. Thus, the only contribution comes from the slit containing the trace of vorticity shed from the wing. The value of ϕ is equal and opposite above and below the vortex sheet. On the sheet, ∂ϕ/∂n = w, the downwash velocity.

Thus the integral for a single flat lifting surface can be rewritten as

$$D_i=-\frac{1}{2}{\rho }_{\mathrm{\infty }}{\int }_{-b/2}^{b/2}(\mathrm{\Delta }\varphi {)}_{x=\mathrm{\infty }}{w}_{x=\mathrm{\infty }}dy\text{(3-15)}$$

where w is the velocity induced by the trailing vortex system. The jump in the potential on the slit at infinity can be related to the jump in potential at the trailing edge. To see this, first consider the jump in the potential at the trailing edge. Recall that the circulation is given by the contour integral:

$$\mathrm{\Gamma }=\text{∮}\mathbf{V}\cdot d\mathbf{s}\text{(3-16)}$$

For an airfoil, we illustrate the concept by considering a small disturbance–based argument. However, the results hold regardless of the small disturbance–based illustration. Consider the airfoil given in figure 3-8.

The dominant velocity is in the x-direction, u = ϕ_x , and the integral around the airfoil, equation (3-16), can be seen to be essentially:

$$\begin{array}{rl}\mathrm{\Gamma }& ={\int }_{T{E}_{\text{lower}}}^{LE}{\varphi }_{x}dx+{\int }_{LE}^{T{E}_{upper}}{\varphi }_{x}dx\\ & ={\varphi |}_{T{E}_{\text{lower}}^{LE}}+{\varphi |}_{LE}^{T{E}_{upper}}\\ & ={\varphi }_{LE}-{\varphi }_{T{E}_{\text{lower}}}+{\varphi }_{T{E}_{upper}}-{\varphi }_{LE}\\ & ={\varphi }_{T{E}_{\text{upper}}}-{\varphi }_{T{E}_{\text{lower}}}\\ & =\mathrm{\Delta }{\varphi }_{TE}\end{array}\text{(3-17)}$$

The value of the potential jump at infinity can be found by realizing that the circulation is created by the wing, and any increase in the contour of integration will produce the same result. Therefore,

$$\mathrm{\Delta }{\varphi }_{x=\mathrm{\infty }}=\mathrm{\Delta }{\varphi }_{TE}=\mathrm{\Gamma }(y).\text{(3-18)}$$

Next, the induced velocity required in equation (3-15) is found from the distribution of vorticity in the trailing vortex sheet by considering the slit to be a sheet of vorticity and using the velocity induced by a distribution of vorticity from the following integral, which is a form of the so-called thin airfoil theory integral shown in equation (3-19).

$$w_{x=\mathrm{\infty }}(y)=\frac{1}{2\pi }{\int }_{-b/2}^{b/2}\frac{\gamma (\eta )}{y-\eta }d\eta \text{(3-19)}$$

To complete the derivation, we have to connect the distribution of vorticity in the trailing vortex sheet to the circulation on the wing. To do this, consider the sketch of the circulation distribution given in figure 3-9.

As the circulation on the wing, Γ, changes across the span, circulation is conserved by shedding an amount equal to the local change into the wake. Thus, the trailing vorticity strength is related to the change in circulation on the wing by

$$\gamma (\eta )=-d\mathrm{\Gamma }/d\mathrm{y.}\text{(3-20)}$$

Substituting this into equation (3-19), we obtain

$$w_{x=\mathrm{\infty }}=-\frac{1}{2\pi }{\int }_{-b/2}^{b/2}\frac{d\mathrm{\Gamma }/dy}{y-\eta }d\eta \text{.}\text{(3-21)}$$

Substitute equations (3-18) and (3-21) into equation (3-15) and integrate by parts using the conditions that Γ(-b/2) = Γ(b/2) = 0 (which simply states that the load distribution drops to zero at the tip). This results in equation (3-22).

$$D_i=-\frac{{\rho }_{\mathrm{\infty }}}{4\pi }{\int }_{-b/2}^{b/2}{\int }_{-b/2}^{b/2}\frac{d\mathrm{\Gamma }\left(y_1\right)}{dy}\frac{d\mathrm{\Gamma }(y_2)}{dy}\ln |y_1-y_2|d{y}_{1}d{y}_2\text{(3-22)}$$

Equation (3-22) shows how the spanload distribution is related to the induced drag. Because of the double integral, we can get the total induced drag, but we have lost the ability to get detailed distributions of the induced drag on the body. This is the price we pay for the use of the farfield analysis.

Finally, this result shows that the induced drag is a function of the Γ distribution (spanload) alone. The only impact made by the Mach number is its effect on the circulation distribution on the wing. We will show later how equation (3-22) can be used to obtain the classical result that an elliptic spanload minimizes the induced drag for a planar wing with span b.

3.3.3 Supersonic Wave Drag: The Farfield Wave Drag Integral

Consider the integral over III in figure 3-6. This is the farfield wave drag integral. The integral corresponds to the last term on the right-hand side of equation (3-9) and can be written as

$$D_w=\underset{r\to \mathrm{\infty }}{lim}(-{\rho }_{\mathrm{\infty }}r{\int }_0^{2\pi }d\theta {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}u{v}_{r}dx)\text{.}\text{(3-23)}$$

If u,v_r → 0 as r → ∞, then D_w = 0 . Thus, when the flow is subsonic, there is no wave drag, as we already know. However, if the flow is supersonic and shock waves are generated, the integral is not zero. This integral can be calculated for any solution of the flow field. In this analysis, we assume that the flow is governed by the Prandtl–Glauert equation. Equation (2-135) is reproduced here from Chapter 2 for convenience.

$$(1-M_{\mathrm{\infty }}^2){\varphi }_{xx}+{\varphi }_{yy}+{\varphi }_{zz}=0$$

This equation implies small disturbance flow. This is valid if the vehicle is highly streamlined, as any supersonic vehicle must be. However, this is not a significant restriction because, far from the body causing the disturbance, this equation can model flows from any vehicle.

To obtain an expression for ϕ that can be used to calculate the farfield integral, assume that the body can be represented by a distribution of sources on the x-axis (making the aircraft look very “slender” from far away). To illustrate the analysis, assume that the body is axisymmetric. Recall that there are different forms for the subsonic and supersonic sources:

$$\varphi =\underset{\begin{array}{c}\text{subsonic source}\\ \downarrow \\ \varphi \to 0\text{as}r\to \mathrm{\infty }\end{array}}{\underbrace{-\frac{1}{4\pi }\frac{1}{\sqrt{x^2+{\beta }^2{r}^2}}}},\varphi =\underset{\begin{array}{c}\text{supersonic source}\\ \downarrow \\ \varphi \to 0\text{as}r\to \mathrm{\infty }\\ \text{except as}r\to \frac{x}{\beta }\end{array}}{\underbrace{-\frac{1}{2\pi }\frac{1}{\sqrt{x^2-{\beta }^2{r}^2}}}}\text{(3-24)}$$

This means that the integral will have a contribution along the Mach wave regardless of how far away the outer control volume is taken. Figure 3-10 illustrates this effect. The resulting force is exactly what is expected in regards to the shock wave contribution to drag—the wave drag.

The farfield behavior of the source singularity given in equation (3-24) can be used to obtain an expression for the farfield integral in terms of geometric properties of the aircraft. A complete analysis is given in Ashley and Landahl,^[22] sections 1.6, 6.6, 7.3, and 9.2, and in Liepmann and Roshko.^[23] The key connection is the assumption relating the supersonic source strength and aircraft geometry. The approximate boundary conditions on the surface equate the supersonic source strength to longitudinal change of cross-sectional area: σ(x) = S’(x). One required assumption is that the cross-sectional area distribution, S(x), satisfies S’(0) = S’(l) = 0. After some algebra, the following wave drag integral is obtained:

$$\overline{D}(\theta {)}_w=-\frac{{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2}{4\pi }{\int }_0^l{\int }_0^l{S}^{\mathrm{\prime }\mathrm{\prime }}\left(x_1\right)S^{\mathrm{\prime }\mathrm{\prime }}\left(x_2\right)\ln |x_1-x_2|d{x}_{1}d{x}_2\text{(3-25)}$$

Note that it is in the same mathematical form as the induced drag integral in equation (3-22). While the spanload distribution is the key contributor to the induced drag, longitudinal variation of the cross-sectional area distribution controls the volumetric wave drag.

The standard method for evaluation of this integral is available in a program known as the Harris wave drag program.^[24] This program determines the cross-sectional area distribution of the aircraft and then evaluates the integral numerically. Note that as given above, the Mach number doesn’t appear explicitly. A refined analysis^[25] for bodies that aren’t extremely slender extends this approach by taking slices, or Mach cuts, of the area through the body at the Mach angle. This is how the Mach number dependence enters the analysis. Finally, for nonaxisymmetric bodies, the area associated with the Mach cuts changes for each angle around the circumferential integral for the cylindrical integration over eegion III in figure 3-6. Thus the area distribution must be computed for each angle. The total wave drag is then found from

$$D_w=\frac{1}{2\pi }{\int }_0^{2\pi }{\overline{D}}_w(\theta )d\theta \text{.}\text{(3-26)}$$

Examples of the results obtained using this computational method are given in section 3.4.5 and in Chapter 9.

3.4 Important Aspects of Lift-Induced Drag

To establish a technical basis for understanding the drag due to lift of wings, singly and in combination, four important aspects must be discussed: (a) planar wing induced drag in section 3.4.1; (b) nonplanar wing induced drag in section 3.4.2; (c) induced drag of multiple lifting surfaces and Munk’s Stagger Theorem in section 3.4.3; and (d) leading edge suction concept in section 3.4.4 that helps us understand additional induced drag due to span efficiency factor “e” (see section 3.2.4) being less than 1, and typical variation of E. In addition, supersonic wave drag is discussed in section 3.4.5 where one of the most important concepts in wave drag is addressed: Whitcomb’s Area Rule.

3.4.1 Induced Drag—Planar Surfaces

The three-dimensional flow field over a lifting surface (for which a horseshoe vortex system is a very good conceptual model) produces a drag, even if the flow is inviscid. (Before proceeding, it might be worthwhile for the reader to review the Trefftz plane analysis in section 3.3.2.) At the wing, the induced drag can be visualized as an effective change in the angle of attack along the wing span induced by the trailing vortex system. This induced change of angle results in a local inclination of the force vector relative to the freestream, and thus it produces an “induced” drag. It is one part of the total drag due to lift, and its form typically arises naturally, as shown in equation (3-27) below.

$$C_{D_i}=\frac{C_L^2}{\pi ARe}\text{(3-27)}$$

The small e in this equation is known as the span efficiency. In this case, the induced drag is assumed to be only a function of the spanload. Additional losses due to the fuselage and viscous effects are included when a capital E, known as Oswald’s E, is used in this expression, as mentioned in section 3.2.4. It is worth reiterating that this notation is the most prevalent in use in the US aircraft industry. However, other notations are frequently employed, and care must be taken when reading the literature to make sure that you understand the notation used.

When designing and evaluating wings, the question becomes What is e, and how large can we make it? The “conventional wisdom” is that, for a planar surface, e_max = 1, and for a nonplanar surface or a combination of lifting surfaces, e_max > 1, where the aspect ratio, AR, is based on the projected span of the wing with the largest span.* However, studies searching for higher e’s abound. The quest of the aerodynamicist is to find a fundamental way to increase aerodynamic efficiency.

In the 1970s, increased aerodynamic efficiency, e, was sought by exploiting nonplanar surface concepts such as winglets and canard configurations. Indeed, these concepts are now commonly employed on aircraft configurations in service. In the 1980s, a great deal of attention was devoted to the use of advanced wingtip shapes on nominally planar configurations. But it is not clear that the advanced wingtips resulted in theoretical e’s above unity. However, in practice, these improved tip shapes help clean up the flow field at the wingtip, reducing viscous effects and resulting in a reduction in drag. Kroo conducted an excellent survey of induced drag and reduction prospects in the Annual Review of Fluid Mechanics,^[26] which is electronically available for no charge to Virginia Tech students through the library website.

In section 3.3, we derived the expression for the drag due to the trailing vortex system in the Trefftz plane. Here we explain the physical basis of the idea of the Trefftz plane following sections 1.6, 6.6, 7.3, and 9.2 in Ashley and Landahl.^[27] An alternate and valuable procedure has been described by Sears.^[28]

Here we repeat the expression for induced drag, equation (3-22), for convenience:

$$D_i=-\frac{{\rho }_{\mathrm{\infty }}}{4\pi }{\int }_{-b/2}^{b/2}{\int }_{-b/2}^{b/2}\frac{d\mathrm{\Gamma }\left(y_1\right)}{dy}\frac{d\mathrm{\Gamma }(y_2)}{dy}\ln |y_1-y_2|d{y}_{1}d{y}_2.$$

The usual means of evaluating the integral is to represent Γ as a Fourier series, as shown in equation (3-28).

$$\mathrm{\Gamma }=U_{\mathrm{\infty }}b\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin n\theta \text{(3-28)}$$

The unknown values of the A_n’s are found from a Fourier series analysis, where Γ(y) is known from an analysis of the configuration. Panel or vortex lattice methods can be used to find Γ(y). (See Chapter 5 for a brief description of vortex lattice methods.) Integration of the drag integral using equation (3-28) results in

$$D_i=\frac{\pi {\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2{b}^2}{8}\sum _{n=1}^{\mathrm{\infty }}n{A}_n^2\text{(3-29)}$$

and

$$L=\frac{\pi }{4}{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2{b}^2{A}_1\text{(3-30)}$$

which are the classical results frequently derived using lifting-line theory. Note that the lift depends only on the first term of the series, whereas all of the components contribute to the drag. Putting the expressions for lift and drag into coefficient form and then replacing the A₁ term in the drag integral by its definition in terms of the lift coefficient leads to the classical result given above by equation (3-27), which is reproduced here:

$$C_{D_i}=\frac{C_L^2}{\pi ARe}$$

In this case,

$$e=\frac{1}{[1+\sum _{n=2}^{\mathrm{\infty }}n{\left(\frac{A_n}{A_1}\right)}^2]}\text{.}\text{(3-31)}$$

It is important to understand that the contribution of induced drag to the drag due to lift assumes that the airfoil sections in the wing are operating perfectly, as if in a two-dimensional potential flow that has been reoriented relative to the freestream velocity at the angle associated with the effects of the trailing vortex system. Wings can be designed to operate close to these conditions. It is also noteworthy that, if the wing is twisted and the shape of the spanload changes as the lift changes, then e is not a constant, independent of the lift coefficient.

We conclude from this discussion:

1. Regardless of the wing planform(s), induced drag is a function of circulation distribution alone, independent of Mach number except in the manner which Mach number influences the circulation distribution (a minor effect in subsonic and transonic flows).
2. Given Γ, e can be determined by finding the A_n’s of the Fourier series for the simple planar wing case. Other methods are required for nonplanar systems.
3. Extra drag due to the airfoil’s inability to create lift ideally must be added over and above the induced drag. Our analysis here assumes that the airfoils operate perfectly in a two-dimensional sense; there is no drag due to lift in two-dimensional flow.

3.4.2 Nonplanar Lifting Surfaces Including Winglets

The expressions given in section 3.4.1 show that e_max = 1 for a planar lifting surface. However, if the slit representing the trailing vortex system is not a simple flat surface and C_Di is based on the projected span, a nonplanar or multiple lifting surface system can result in values of e > 1. In this section, we look at the case of nonplanar wings. First, we consider the case of a wing with dihedral. The effect of dihedral on the maximum e is given in figure 3-11, which is recreated from a figure in one of my papers.^[29] The symbols (labeled Numerical Result, Ref. 6) in the figure are numerical results by Lamar^[30] and the curve (labeled Analytical Result, Ref. 23) is by Letcher.^[31] The figure shows that modest amounts of dihedral lead to modest increases in e_max using the projected wingspan. If the running length were used instead of the projected span, the e would not exceed one.

The other case of considerable interest is the use of winglets to reduce induced drag. Figure 3-12 shows the e that can be expected from a winglet that obtains its maximum performance. This figure is recreated from one in a paper by Feifel.^[32] The figure shows good agreement between VLM estimates using 25 and 50 spanwise panels with the curve labeled Exact Solution (Lundry Ref. 8) from Lundry^[33]

3.4.3 Multiple Lifting Surfaces and Munk’s Stagger Theorem

Biplane theory can be used to investigate the multiple lifting surface case. Thwaites^[34] provides a detailed discussion. Figure 3-13, adapted from Thwaites, shows the potential benefit of biplanes obtaining performance improvements in e for a given span using two lifting surfaces of spans b₁ and b₂ separated by height h. The parameter κ is actually 1/e. The figure also contains the associated distribution of lift between the two lifting surfaces using the parameter λ, which is the lift on the smaller span surface as a fraction of the total lift. Two limiting cases can be considered. If the lifting elements are in the same plane, then the sum of the spanloads should be elliptic for minimum drag. If the elements are vertically separated by a large distance, then each component individually needs an elliptic spanload to obtain minimum induced drag.

An important result in the consideration of multiple lifting surfaces is Munk’s stagger theorem. You can find a proof of the theorem in Milne-Thomson.^[35] The theorem states that the total induced drag of a multisurface system does not change when the elements of the system are translated parallel to the direction of the flow (as illustrated in the sketch in figure 3-14) provided that the circulation distributions on the elements are left unchanged. Thus the drag depends only on the projection of the system in the Trefftz plane. This means that, given the circulation distributions, the Trefftz plane analysis can be used to find the induced drag. This is consistent with the analysis given for the Trefftz plane above and reinforces the concept of using the farfield analysis to determine the induced drag. Naturally, to maintain the circulation distribution of the elements when they are repositioned, their geometric incidence and twist or camber have to be changed.

We complete this section by presenting a classic table showing the types of values of e that can be obtained for a variety of multiple and nonplanar lifting surfaces. Figure 3.15 shows the Trefftz plane trace of the surfaces and the associated values of e.^[36] In each case, the wing(s) will have to be twisted and cambered to achieve the spanload distribution required to achieve the e given in the figure. In terms of analytic estimation techniques for unusual configurations, the NASA CRs by John deYoung^[37],[38] should be studied. Numerical methods for computing span e’s and the associated spanloads are described in section 3.5.3, where several different codes are available for the calculation.

3.4.4 The Leading-Edge Suction Concept

Aerodynamicists often evaluate the performance of configurations in term of what is called leading-edge suction. The concept can be explained by considering the inviscid flow over the proverbial thin (zero thickness) flat plate at angle of attack in an incompressible, inviscid flow, as shown in figure 3-16.

What is the drag? According to theory, it must be zero. In the sketch, we see that the force acts in a direction perpendicular to the plate, and this clearly leads to a force component in the drag direction. What’s the explanation for the paradox? Consider the sketch of the front portion or leading edge of the thin flat plate in figure 3-17.

There is a low pressure region over the front edge face due to the expansion of the flow around the leading edge. The expansion becomes stronger as the thickness decreases, so that the force on the front face of the plate due to the product of the pressure and plate thickness is

$$F_s=\underset{\delta \to 0}{lim}(\delta \cdot C_{p_s}{q}_{\mathrm{\infty }})=\text{finite.}\text{(3-32)}$$

The value of F_s is just the amount for the drag of the plate to be zero. Thus the correct model of the flow over the flat plate is actually modified from the sketch given above to include an edge force, as shown in figure 3-18.

Of course, a very thin flat plate will realize almost none of the suction force and hence will have a drag component. However, an airfoil section (even a fairly thin one) with a smooth, round nose may in fact achieve nearly all of the suction force, at least at small angles of attack. If the airfoil section in the wing does not achieve the full suction performance, the resulting drag must be added to the induced drag.

The drag due to lift is thus broken up into induced drag and additional profile drag. As described previously, the induced drag is a function of the wing spanload only and is independent of the details of the particular airfoil used in the wing. The additional profile drag is associated with the airfoil used in the wing. At low lift coefficients, this drag should be small, only becoming important as flow separation starts to develop on the airfoil section. The additional profile drag becomes large as wing stall is approached.

Wing performance is evaluated based on the ability to obtain a high value of the lift-to-drag ratio, L/D, relative to the maximum possible for that planform and based on the ability to achieve a high maximum lift coefficient. Essentially, the wing is designed to allow the airfoil to achieve its full performance. Recalling that a two-dimensional airfoil under the assumption of inviscid subsonic flow has no drag due to lift, the maximum performance should occur by adding the induced drag—assuming an elliptic spanload—to the zero-lift drag. This is known as the 100% suction polar since the airfoil section has no additional profile drag due to lift and is thus achieving 100% of the leading-edge suction required to eliminate the drag force in a two-dimensional flow. The drag due to lift may be expressed as

$$C_{D_{L_{100\mathrm{\%}}}}\cong C_L^{2/}\pi \mathrm{A}\mathrm{R.}\text{(3-33)}$$

At the other extreme is the 0% leading-edge suction case, in which the airfoil fails to produce any efficient lift, such that the only force is normal to the surface and there is no edge or suction* force. In this case, the entire lifting force on the wing is the normal force, and the polar can be determined by resolving that force into lift and drag components. The equation for the 0% suction drag can be expressed in a variety of forms, starting with

$$C_{D_{L_{0\mathrm{\%}}}}\cong C_L\tan (\alpha -{\alpha }_0)\text{(3-34)}$$

where a₀ is the zero-lift angle of attack. We also use the linear aerodynamic relation,

$$C_L=C_{L_{\alpha }}(\alpha -{\alpha }_0)\text{,}\text{(3-35)}$$

which can be solved for the angle of attack:

$$(\alpha -{\alpha }_0)=\frac{C_L}{C_{L_{\alpha }}}\text{.}\text{(3-36)}$$

Finally, substitute equation (3-36) into equation (3-34) for the angle of attack as follows:

$$\begin{array}{rl}{C}_{D_{L_{0\mathrm{\%}}}}& =C_L\tan (\alpha -{\alpha }_0)\cong C_L(\alpha -{\alpha }_0)\\ & \cong C_L\frac{C_L}{C_{L_{\alpha }}}\end{array}$$

or

$$C_{D_{L_{0\mathrm{\%}}}}\cong \frac{C_L^2}{C_{L_{\alpha }}}\text{(3-37)}$$

This equation for the 0% suction polar shows why this polar is often referred to as the “1/C_Lα” polar by aerodynamicists. Using this approach, effective wing performance is quoted in terms of the fraction of suction achieved, based on the difference between the actual drag and the 100% and 0% suction values, as shown in figure 3-19. This figure illustrates how wings typically perform. The wing will approach the 100% level at low lift coefficients, and then as flow separation starts to develop, the performance deteriorates. Eventually, the wing may have a drag substantially higher than the 0% suction value.

The value of E for this level of performance can be found by equating equation (3-37) to the standard form,

$$C_{D_L}=\frac{C_L^2}{\pi ARE}\text{,}\text{(3-38)}$$

which leads to

$$E_{0\mathrm{\%}}=\frac{C_{L_{\alpha }}}{\pi AR}\text{.}\text{(3-39)}$$

Typically, the value of E varies with the lift coefficient. By plotting experimental data, typical variations can be obtained for various classes of wings. Figure 3-20 shows a typical variation. This relation was shown in general by McKinney and Dollyhigh.^[39]

Alternately, in supersonic flow, the drag due to lift relation is frequently written as

$$C_{D_L}=K{C}_L^2\text{(3-40)}$$

for uncambered airfoils. For cambered and twisted wings, the polar is shifted, and the minimum drag occurs at a C_L other than zero, as shown previously in figure 3-2 and described by equation (3-2). In practice, we expect the wing to achieve a performance level between the K_100% and K_0% limits. This approach is described in detail by Raymer.^[40]

In considering the shift of the polar, a few comments are warranted. First, the wing performance cannot exceed the optimum value, which is E = 1 for subsonic flow over a single planar lifting surface. Especially for wings in supersonic flow, it is hard to get 100% of the leading-edge suction. In that case, the approach is to camber the wing to make the drag performance of a wing with less than 100% suction attain the 100% suction level at a specified value of design lift coefficient, C_Ld. This uses the polar definition shown in equation (3-41).

$$C_D=\mathrm{\Delta }{C}_{D_m}+K_{xx\mathrm{\%}}{(C_L-C_{L_m})}^2\text{(3-41)}$$

where the value of K corresponds to the performance of the wing in terms of leading-edge suction (i.e., xx is a number between 0 and 100). We find the values of ΔC_Dm and C_Lm in terms of the design lift coefficient, C_Ld. To do this, equate the polar to the 100% suction value at the design lift. This polar must also be tangent to the 100% polar at this point so that the polar will not predict better performance than the optimum at other values of the lift. If we use a 0% leading-edge suction wing as an example, we get equations (3-42) and (3-43).

$$C_D(100\mathrm{\%}LES)={C_D(0\mathrm{\%}LES)|}_{C_L=C_{Ld}}\text{(3-42)}$$

$$\frac{d{C}_D(100\mathrm{\%}LES)}{d{C}_L}={\frac{d{C}_D(0\mathrm{\%}LES)}{d{C}_L}|}_{C_L=C_{L_d}}\text{(3-43)}$$

The unknown values of ΔC_Dm and C_Lm in equation (3-41) are

$$C_{L_m}=(1-\frac{K_{100\mathrm{\%}}}{K_{0\mathrm{\%}}})C_{L_d}\text{(3-44)}$$

and

$$\mathrm{\Delta }{C}_{D_m}=K_{100\mathrm{\%}}{C}_{L_d}^2-K_{0\mathrm{\%}}{(C_{L_d}-C_{L_m})}^2\text{.}\text{(3-45)}$$

In any experimental evaluation of wing performance, both the 100% and 0% theoretical polars should be constructed and used to establish bounds on the experimental polar. Thus a typical drag polar would include the 100% and 0% suction polars as well as the predicted or measured performance to establish a basis for evaluating a wing’s efficiency. Figure 3-21 presents the actual performance of an unswept rectangular wing at subsonic speed. Here, the performance is very close to the lower drag limit until the wing stalls.

It is difficult to identify the initial flow breakdown using the drag polar. Often you can identify flow breakdown more clearly by plotting the axial force as a function of normal force. In such a plot, the axial force should initially decrease, as described above. When the airfoil section starts to lose leading-edge suction, the data displays a sharp “break.” Figure 3-22 illustrates this approach to the examination of wing efficiency.

For configurations with very poor aerodynamic efficiency, the 0% suction force provides a good estimate of the vehicle drag. However, 0% suction levels are so inefficient that this level of performance would be unacceptable and not competitive for most aircraft designs.

To estimate the performance of real configurations, which operate between the two limits, Harry Carlson at NASA’s Langley Research Center established the notion of “attainable” leading-edge suction.^[41],[42] Based on an extensive analysis of 2D airfoil data, Carlson established an empirical correlation which is used to estimate the fraction of the full suction that should be attained for the specified airfoil, planform, and flight condition. Carlson’s concepts are based on linear theory.

Nonlinear effects can be important and can be exploited. Although the linear theory–based concepts described here provide a valuable way of looking at wing designs, nonlinear effects can provide a means of improving performance. In considering nonlinear effects, interactions between thickness and lifting effects can be exploited.^[43]

3.4.5 Supersonic Wave Drag: The Farfield Wave Drag Integral and Whitcomb’s Area Rule

The farfield analysis (see section 3.3) showed us that an aircraft in supersonic flight has wave drag. Not surprisingly, the supersonic wave drag has played a key role in the aerodynamic design of supersonic aircraft.

The equations for estimating the total wave drag, equations (3-25) and (3-26), are repeated here for convenience:

$${\overline{D}}_w(\theta )=-\frac{{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2}{4\pi }{\int }_0^l{\int }_0^l{S}^{\mathrm{\prime }\mathrm{\prime }}\left(x_1\right)S^{\mathrm{\prime }\mathrm{\prime }}\left(x_2\right)\ln |x_1-x_2|d{x}_{1}d{x}_2$$

and

$$D_w=\frac{1}{2\pi }{\int }_0^{2\pi }{\overline{D}}_w(\theta )d\theta$$

where the S(x) values represent the area from an oblique (Mach angle) cut to find the cross-sectional area of the aircraft at a specific roll angle, θ.

The importance of the distribution of the cross-sectional area on the value of wave drag is clear in the integral. To minimize the integral, the area change should be very smooth so that the S”(x) is minimized. Thus, the shaping of the design geometry plays a major role in the value of the integral. It has been shown that low drag is achieved by minimizing the maximum cross-sectional area of the design. The key parameter is the fineness ratio, which is the length divided by the maximum diameter. Increasing the fineness ratio decreases the wave drag. A number of minimum drag bodies of revolution have been derived using equation (3-25). The geometric details of these shapes are given in appendix A.

To estimate the wave drag, a theoretical analysis of the integral is available.* Since the integrand is proportional to the second derivative of the area distribution, it is clear even without an analysis that the lowest drag occurs when the distribution is made as smooth as possible. Eminton^[44] devised the standard method for the numerical evaluation of the integral. The difficulty in evaluating the integral is that the result depends on the second derivative of the area distribution. This distribution is made up of contributions from numerous components, and it is not known with great precision. Polynomials or other interpolation schemes used to perform the quadrature (i.e., calculate the definite integral) may amplify any imprecision in the data and produce unreasonably high drag predictions. Eminton used a Fourier series for the distribution of the gradient of the area. The coefficients are then found by solving an optimization problem that determines the coefficients that will produce the curve passing through the known values of the area with the least drag. In this sense, the method is also a design method. By specifying a small number of control stations (say, from a designer’s configuration layout) with a specified area distribution, the method will provide the complete distribution of area required for minimum drag and to satisfy the imposed control station constraints.

3.4.5.1 Whitcomb’s Area Rule

The principle that aerodynamicists use to achieve low values of wave drag is known as the area rule. Proposed by Richard Whitcomb* at the NACA’s Langley Field (now NASA Langley Research Center), the area rule states that the air displaced by the body should develop in a smooth fashion as it moves around and along the body, with no sudden discontinuities. Thus the total aircraft area distribution should form a smooth progression. In particular, when the wing becomes part of the cross-sectional area, the adjacent fuselage area should be reduced to make the total area distribution smooth. This results in the distinctive area ruled, or “coke bottle,” fuselage shape.

Whitcomb, who won the National Aeronautics Association’s Collier trophy for his work on area rule in 1954, obtained evidence for the validity of this rule experimentally (the computer had not yet become a practical design tool). Figure 3-23 shows the key result obtained by Whitcomb.^[45] The increase in drag with increasing transonic Mach number is almost identical for a wing-body combination and a body of revolution with the same cross-sectional area distribution. The wing-body combination has significantly higher subsonic drag because of the increased surface area compared to the body-alone case. All the cases Whitcomb presented weren’t as dramatic, but similar trends were found for a number of shapes. Whitcomb’s original idea addressed transonic speeds, and the normal area distribution (the area in the plane perpendicular to the flow) was made smooth to obtain low drag. At supersonic speeds, the problem is more complicated. Instead of using the normal area distribution, the supersonic area rule requires smoothness for the area of the so-called Mach cuts that correspond to the area distribution along the Mach angle for each θ angle around the circumference (see equation 3-26 in section 3.3.3).

3.4.5.2 Practical Applications of Area Rule

Convair F-102 and F-106

The most famous application of the area rule occurred on the Convair F-102 aircraft program.* This airplane was supposed to be supersonic in level flight. When it first flew, the original prototype YF-102 was unable to break the sound barrier and fly supersonically. The nose was lengthened approximately five feet and area was added (with the plane already completed, it was impossible to reduce area) to the fuselage via faired bulges—or bustles—at the intersection of the fuselage and the wing trailing edge. The bulges were faired beyond the engine exhaust nozzle to improve the fineness ratio and area distribution. After these modifications, the prototype YF-102, shown in figure 3-24(a),^[46] was capable of penetrating deeper into the transonic region. However, it was still not capable of exceeding Mach 1.0 in level flight. A complete redesign was necessary to continue the contract.

One hundred and seventeen working days later(!), a new, completely redesigned F-102 was ready to fly. The fuselage fineness ratio and area distribution had been increased and refined. The fuselage midsection cross-sectional area had been reduced (cinched-up, wasp-waisted, or coke-bottled) as much as structure and component integration would permit. It was lengthened eleven feet, three inches, with most of the increased length added ahead of the wing. The cockpit canopy was reduced in cross section with a near-triangular cross section and headed by a flat plate, highly swept V-shaped windshield. The cockpit and the side-mounted engine inlets were moved forward to reduce their sudden area buildup, or impact on the fuselage area. The aft fuselage bustles were retained to avoid the rapid collapse of the cross-sectional area at the delta wing trailing edge. Reconfigured in this way, the airplane was able to fly at low supersonic speeds (M = 1.2). Figure 3-24(b) shows the reconfigured F-102A as produced for service use.^[47]

The resulting change in drag from the YF-102 to the F-102A was about twenty-five counts and is shown in figure 3-24(c), recreated from an original Convair plot. Although the change might not appear dramatic, the reduction in wave drag was sufficient to allow the plane to fly faster then the speed of sound. Notice that the use of conical camber (discussed later), introduced to improve the lift and drag-due-to-lift characteristics of the delta wing, added a significant penalty (camber drag) to the minimum drag.

Subsequently, the configuration was completely redesigned to incorporate a more refined, integrated area rule. Further slimmed down by a reduced weapon bay capacity and by shortened and repositioned engine air intake ducts, and powered by a 50% more powerful engine, it was capable of routine Mach 2+ speeds. The designation was then changed to F-106A. This design is shown in figure 3-25. The volume of the increased area of the vertical tail on the F-106A, required to counteract the loss of tail surface effectiveness at the increased operational Mach number, replaced the aft side bustles on the F-102.

When considering the area rule, remember that this is only one part of successful airplane design.^[50] Moreover, extreme area ruling for a specific Mach number may significantly degrade the performance of the design at other Mach numbers.

3.5 Predicting Drag

Accurate prediction of drag is one of the most important yet most challenging tasks of aircraft design projects for aerodynamicists. Actual flight performance is heavily dependent on the accurate value of drag. For example, if drag is underpredicted (predicted value is less than the actual value), then the actual fuel consumption will be higher than the predicted one. Similarly, if the drag is overpredicted (predicted value is more than the actural value), the aircraft might end up being overdesigned (for example, featuring a higher gross weight due to higher predicted fuel consumption). For the challenges associated with accurate drag prediction, Jobe^[51] is a good source. In this section, we discuss the relevant drag prediction methodologies.

3.5.1 Overview of Methodologies

We group aerodynamic drag prediction methodologies into three categories:

1. semi-empirical correlation curves based on data from past and present aircraft
2. numerical simulations of flow about an aircraft using computational fluid dynamics (CFD) methods
3. physical simulations of flow about a scale model of an aircraft using ground test facilities, such as the wind tunnels

All three approaches have their own unique strengths and weaknesses. Selection of the best approach(es) depends on the aircraft design stage (see Chapter 4) and the nature and scope of the required drag data. In this book, our focus is on the early stages of design, which are characterized by the evolution of configuration geometry, also known as outer mold line (OML), through frequent modifications. Designers receive inputs from teams of experts from various disciplines such as aerodynamics, structures, stability and control, cost, and manufacturing. The OML is frequently modified to accommodate those inputs. One of the major tasks of the aerodynamics experts on the team is to predict the drag of constantly changing OMLs and to find ways of keeping it as low as possible. Aerodynamics experts are expected to provide accurate drag data at the right time and the right cost to meet the strict scope, cost, and schedule specifications of the design project. The use of semi-empirical correlation curves and numerical simulations with rapid turnaround times provides the right balance between the required accuracy and tight schedules. Physical simulations (wind tunnel testing) are typically too costly and take too long to be effective in the early stages of design. They are better suited for later stages, where they are extensively used. Readers interested in learning more about the semi-empirical correlations can find them in just about any aircraft design book. For state of the art numerical simulations, the Applied Computational Aerodynamics book by Cummings et al.^[52] is an excellent source of information.

It is worth mentioning that you may find drag components being characterized as “inviscid” or “viscous” drag in the literature. This characterization reflects the fluid mechanics viewpoint discussed in section 3.2.2. For example, lift-induced drag and wave drag are considered inviscid drags, whereas friction drag and form drag are viscous. It is straightforward to see why friction drag is considered viscous. After all, it represents the total force on a body due to surface shear forces caused by viscosity. However, form drag is a bit more nuanced since it represents the net force due to surface pressures. We know from d’Alembert’s paradox that the net force on a body is zero in steady, incompressible, inviscid flow. However, when viscous effects modify the surface pressure distributions, the net force is no longer zero. This non-zero net force is the form drag. Friction drag and form drag contribute to the total drag even when a body produces no lift. Methods to predict them are discussed in section 3.5.2. We can estimate lift-induced drag using methods based on inviscid flow approximations, as highlighted in section 3.5.3. Wave drag can also be estimated using inviscid flow methods, as discussed in sections 3.5.5 and 3.5.6.

3.5.2 Zero Lift Drag: Skin Friction and Form Drag Estimation

Estimates of skin friction based on classical flat plate skin friction formulas can be used to provide initial estimates of the friction and form drag portion of the zero-lift drag. These are required for aerodynamic design studies using the rest of the methods described here. These simple formulas are used in conceptual design in place of detailed boundary layer calculations, and they provide good initial estimates until more detailed calculations are made using boundary layer calculations or other methods. They are included here because they appear to have been omitted from current basic aerodynamics textbooks.* An excellent examination of the methods and accuracy of the approach described here was given by Paterson, MacWilkinson, and Blackerby of Lockheed.^[53]

For a highly streamlined aerodynamically clean shape, the zero-lift drag should be mostly due to friction and form drag at subsonic speeds where there are no shock waves and can be estimated using skin friction formulas. However, table 3-1 shows that, for a typical military attack airplane, about two-thirds of the zero-lift drag is associated with skin friction and form drag. This illustrates the serious performance penalties associated with seemingly small details.

Until recently, aerodynamicists assumed the flow on actual airplanes was completely turbulent. However, as a result of work at NASA, some configurations can now take advantage of at least some laminar flow, showing significant reduction in friction drag. Advanced airfoils, such as a natural laminar flow (NLF) airfoil, can have as much as 30% to 50% laminar flow. As an example of this approach, consider a typical turbulent flow skin friction formula (for one side of a flat plate surface only):

$$C_F=\frac{0.455}{{[\log {\mathrm{R}\mathrm{e}}_c]}^{2.58}}\text{(3-46)}$$

where “log” means logarithm to the base 10. Note also that the capital C_F denotes an integrated value. Formulas for the local skin friction coefficient customarily use a lowercase f subscript.

Numerous form factors are available to help account for effects due to thickness and additional trailing-edge pressure drag; see Hoerner^[54] and Covert^[55] for summaries. For planar surfaces, one form factor is

$$FF=1+2.7\left(\frac{t}{c}\right)+100{\left(\frac{t}{c}\right)}^4\text{(3-47)}$$

where t/c is the maximum thickness-to-chord ratio.

For bodies, the form factor would be

$$FF=1+1.5{\left(\frac{d}{l}\right)}^{1.5}+7{\left(\frac{d}{l}\right)}^3\text{(3-48)}$$

where d/l is the diameter-to-length ratio.

The skin friction coefficient estimate is then converted to aircraft drag coefficient form through equation (3-49) below.

$$C_{D_0}\cong C_F\frac{S_{\text{wet}}}{S_{\text{ref}}}FF\text{(3-49)}$$

Here S_wet is the total area scrubbed by the flow, and S_ref is the reference area used in the definition of the force coefficients. For a thin wing, the reference area is usually the planform area and the wetted area is approximately twice the planform area (including the upper and lower surface of the wing).

A computer program FRICTION automates this procedure using slightly improved skin friction formulas that include compressibility effects. The program computes the skin friction and form drag over each component, including regions of laminar and turbulent flow. The user can input either the Mach and Reynolds numbers or the Mach number and altitude. The program and its use are described in an online manual.^[56] This analysis assumes that the aircraft is highly streamlined. For many aircraft, this is not the case. As mentioned in section 3.2, table 3-1 provides an example of the significantly increased drag that results when developing an aircraft for operational use. Note that the parasite drag coefficient based on a total wetted area of 1119 ft² is 0.00495, whereas it is 0.0213 based on the reference area of 260 ft².

Table 3-1: Drag breakdown on a "dirty" military airplane. Total speed minimum parasite drag breakdown: M < .65, C_L = 0.0.

3.5.3 Computer Programs for Lift-Induced Drag Estimation

The lift-induced drag, as discussed in section 3.4, can be estimated using many computer programs; three are highlighted in this section. Readers should see appendix E for these and many other computer programs that are useful for aerodynamics and aircraft design efforts.

3.5.3.1 Program LIDRAG: Induced Drag of a Single Planar Wing

The LIDRAG program computes the span efficiency factor e for a single planar lifting surface given the spanload using a fast Fourier transform. For reference, note that the e for an elliptic spanload is 1.0 and the e for a triangular spanload is about 0.72. LIDRAG was originally written by Dave Ives, who was a research scientist at Pratt & Whitney, and the program is employed in numerous aerodynamics codes.^[57]

3.5.3.2 Program LAMDES: Induced Drag of Simple Nonplanar Lifting Systems with Camber Line Design

When the lifting system is not limited to a single lifting component, LIDRAG cannot be used to find the span efficiency factor e. When the system is composed of two lifting surfaces or a lifting surface with dihedral breaks, including winglets, then the induced drag can be estimated using a code known as LAMDES that was originally developed by John Lamar,^[58] who served as an aerospace research scientist at NASA-Langley. As originally developed, this code finds the minimum induced drag and the required spanloads for a prescribed lift and pitching moment constraint. This program is much more elaborate than LIDRAG. For subsonic flow, the program can also estimate the camber and twist of the lifting surfaces required to achieve the spanload for minimum drag. In my work, I extended this code to incorporate, approximately, the effects of viscosity and find the system e for a user-supplied spanload distribution.^[59]

3.5.3.3 Program IDRAG: Induced Drag of Nonplanar Lifting Systems

The IDRAG program can be used to find the induced drag of a system of nonplanar lifting elements. It was written by Joel Grasmeyer while he was a master’s student at Virginia Tech. It has both design and analysis capabilities. This means that you either find the spanload required to obtain the minimum induced drag or you can input a spanload and find the induced drag. The program also provides the span efficiency factor e. This program does not give you the twist and camber required to generate the spanloads. Three FORTRAN programs are required and must be linked to run the IDRAG program.

3.5.4 Transonic Drag Rise: The Korn Equation

Attempts have been made to estimate the capability of transonic airfoils for the purposes of design studies without performing wind tunnel 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 is completed. Here, we provide an approximate method for estimating the transonic performance of airfoils. It is based on the Korn equation—an empirical relation developed by Dave Korn, who earned a PhD in applied mathematics from the New York University Courant Institute of Mathematical Sciences and contributed to computer simulations of transonic airfoils in the early 1970s. The Korn equation was in use at Grumman when I arrived in 1974.

Based on Dave Korn’s experience, it appeared that airfoils could be designed for a variety of Mach numbers, thickness-to-chord ratios, and design lift coefficients. However, 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). In particular, the Korn equation for a 2D airfoil is

$$M_{\mathrm{dd}}+\frac{C_l}{10}+\left(\frac{t}{c}\right)={\kappa }_A\text{(3-50)}$$

where κ_A is an airfoil technology factor with a value of 0.87 for an NACA 6-series airfoil section and a value of 0.95 for a supercritical section. Equation (3-50) 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 explored graphically by many authors; the scales are often left out of public presentations by aircraft companies. Note that the Korn equation is sensitive to the value of the airfoil technology factor.

Mason^[60] compares the prediction from the Korn equation with other estimates. Figure 3-27(a) shows a comparison of the performance estimates of both older airfoils and modern supercritical airfoil presented by Shevell.^[61] The agreement is good with the exception of being overly pessimistic regarding older conventional airfoils at lower thickness ratios.

Figure 3-27(b) compares the Korn equation with NASA projections^[62] 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 it is overly optimistic at higher lift coefficients. This type of technology representation is important in developing integrated designs. To develop each point on this type of chart represents a large effort on the part of the designer (in this case, the aerodynamicist).

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.^[63],[64] The result for a 3D wing is given by

$$M_{\mathrm{DD}}=\frac{{\kappa }_A}{\cos \mathrm{\Lambda }}-\frac{(t/c)}{{\cos }^2\mathrm{\Lambda }}-\frac{C_L}{10{\cos }^3\mathrm{\Lambda }}\text{.}\text{(3-51)}$$

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 wing lift coefficient (C_L), and the effective 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 taken to be

$$\frac{\mathrm{\partial }{C}_D}{\mathrm{\partial }M}=\mathrm{0.1.}\text{(3-52)}$$

(Note that C_d should replace C_D for 2D airfoils.) Next, we make use of Lock’s proposed empirically derived shape of the drag rise, resulting in equation (3-53).^[65]

$$C_D=20{(M-M_{crit})}^4\text{(3-53)}$$

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:

$$\frac{\mathrm{\partial }{C}_D}{\mathrm{\partial }M}=0.1=80{(M-M_{crit})}^3\text{(3-54)}$$

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

$$M_{crit}=M_{\mathrm{DD}}-{\left(\frac{0.1}{80}\right)}^{1/3}\text{(3-55)}$$

where the drag divergence Mach number is given by the extended Korn equation, equation (3-51).

Joel Grasmeyer et al.^[66] developed a method to compute wave drag coefficient on a number of spanwise strips of a wing for use in MDO studies of a transonic strut-braced wing concept. The method employs the following relation:

$$C_{d_{\text{wave}}}=20{(M-M_{crit})}^4\frac{S_{\text{strip}}}{S_{\text{ref}}}\text{for}M>M_{crit,}\text{(3-56)}$$

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, as shown in figure 3-28. The curves represent the results of the current model using eight spanwise strips, and the discrete data points represent the Boeing 747 flight test data. 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 3-28. Based on an analysis of the Boeing 777, we used a value of 0.955 to simulate the Boeing 757-100 wave drag characteristics.

3.5.5 Supersonic Wave Drag Estimation

The Harris wave drag program^[67] is one of the most widely used methods for estimating wave drag. It is the practical implementation of the concepts described in sections 3.3.3 and 3.4.5. Figure 3-29 illustrates the procedure. At each roll angle θ, a number of x-cuts are made to use in evaluating the integral, with the x-axis running along the length of the body. Typically, fifty to 100 x-cuts are made for each θ; the total number of roll angles typically ranges from twenty-four to thirty-six. In making these calculations, the inlet capture area is removed from the area distribution. The Mach number dependence comes in through the orientation of the cutting plane at Mach angle, μ, which is defined as sin^-1(1/M).

As previously discussed in section 3.4.5, area ruling plays an important role in supersonic cruise vehicle design. We present the results of our analysis of the NASA AST3I high-speed civil transport (HSCT) concept. Figure 3-30(a) shows the general arrangement of the baseline HSCT configuration—highly blended Mach 3 with a 6,500-mile range.^[68]

Figure 3-30(b) shows the variation in drag for the initial NASA concept and our optimized design as the computed results of the integral for several “theta cuts” at Mach 3. The curve titled Optimized Design is for a geometry that also contains the results of a combined structural-aerodynamic study to improve this design using systematic advanced design methodology.^[69] Note that the drag is presented in terms of D/q. This is a traditional approach that eliminates any false impressions produced when configurations with differing reference areas are compared.

Figure 3-30(c) shows the normal cross-sectional area distribution. Here, the nacelles are seen to make a large impact on the area distribution. However, the area distribution of interest is for M = 3.0. Figures 3-30(d) and 3-30(e) present the area distributions for the θ = 0° and θ = 90° cases obtained using the Harris wave drag program. Here, the area distribution is seen to be much smoother. This is especially true for the θ = 0° case. The θ = 90° case still involves the problem of integrating the propulsion system into the configuration to obtain a smooth area distribution. Compare the area distributions presented in figures 3-30(d) and 3-30(e) with the change in drag at these two different roll angles, as this will provide some insight into the importance of shaping in producing a smooth area distribution.

Area diagrams for typical fighters are not nearly so streamlined. Figure 3-31(a) shows the original area distribution for the YF-16.^[70]

Figure 3-31(b) displays the results of refinements on the YF-16. The F-16 was not designed primarily for supersonic flight. It has a low fineness ratio and consequently a relatively high wave drag. Small aircraft are much more difficult to lay out to ensure a smooth distribution of area. Note in figure 3-31(a) that the canopy is placed to help “fill in” the area diagram. Figure 3-31(b) shows the revisions made to improve the contour forward and aft of the maximum cross-sectional area to fill in the shape and also add fuel volume. This curve has no scale. Manufacturers are sensitive about this information.

There is also a wave drag due to lift, as discussed by Ashley and Landahl.^[71] However, almost all area ruling at supersonic speeds primarily emphasizes the volumetric wave drag.

3.5.6 Trim Drag

For equilibrium flight, the airplane must be trimmed. The forces must be such that the moments about the center of gravity in all axes are zero. To achieve this condition, the control surfaces are usually deflected to generate the required trimming moments.

Figure 3-32 shows a schematic of the requirement. Two typical situations are shown in figure 3-32(a). In one case, the center of gravity is ahead of the wing center of pressure, the aircraft is stable, and a download on the tail is required to balance the moment due to lift of the wing. In the other case, the center of gravity is behind the wing center of pressure, the airplane is unstable, and an upload on the tail is required to balance the moment due to lift of the wing. Other situations are possible, but these two illustrate the key idea.

Figure 3-32(b) illustrates the difference between stable and unstable configurations. For a stable airplane, the basic C_m0 is typically positive, while for an unstable aircraft, the basic C_m0 is negative. In each case, a control surface has to be deflected over a range of settings to maintain trim over a range of lift coefficients (unless the configuration is neutrally stable). On modern aircraft, the required moment could be provided by the deflection of the thrust through thrust vectoring.

Control surface deflections change the drag from the undeflected reference value. This difference in drag is known as a trim drag. There are many definitions of trim drag because it is difficult to be precise in defining the term. Some definitions consider only the drag due to the lift of the trimming surface. Some analyses allow for a negative trim drag. However, for a given flight condition, the total lift must be fixed and any change in lift on the trimming surface requires a change in lift and thus drag, on the primary surface. On a well designed aircraft, the trim drag should be small. Canard concepts are often considered advantageous because the canard and wing both supply positive lift to trim, whereas for traditional aft-tail configurations, the tail load is negative and the wing must operate at a higher lift to compensate. However, for modern aft-tail designs, the tail load is near zero, resulting in little trim drag.

Trim drag has always been an important consideration in airplane design. However, trim drag became especially important with the development of stability and control augmentation systems that allowed the designer much greater freedom in choosing a location for the center of gravity. While the static stability condition had frequently made it difficult to obtain minimum trim drag, natural static stability was no longer required. This meant that trim drag could become a key criteria for the placement of the center of gravity in a configuration; this is part of the motivation for so-called control configured vehicle (CCV) concepts.

Trim drag is especially important for several specific classes of aircraft. Supersonic aircraft demand special consideration because of the aerodynamic center shift from subsonic to supersonic flight. To control trim drag as well as stability, fuel is transferred fore and aft between subsonic and supersonic flight to achieve proper balance on supersonic cruise aircraft. Variable-sweep wing aircraft also have aerodynamic center locations that vary with wing sweep, potentially leading to high values of trim drag. Finally, maneuvering aircraft can suffer high trim drag at high lift coefficients, severely limiting sustained turn performance. This was especially true of the first generation of supersonic-capable fighters. Examples of the contribution of trim drag to the total drag are shown in figure 3-33, taken from the aircraft and airship design book by Nicolai and Carichner.^[72]

A more useful approach to the trim drag analysis is to consider the value of trimmed drag. In this approach, it is difficult to define a specific trim drag value. The best way to assess the trim penalty is to define the difference between the minimum drag attainable for the system and the minimum trimmed drag for a specified center of gravity position. This provides the designer with a measurement of the drag penalty being paid for a particular center of gravity location. This approach also demonstrates the direct connection between static margin and minimum trimmed drag. Different configuration concepts lead to different values of static margin to obtain minimum trimmed lift. In general, for aft-swept wings with aft tail configurations, the minimum trimmed drag occurs at a slightly unstable center of gravity (~5%–10%), canard configurations have minimum trim drag at slightly more unstable conditions (~15%), and forward-swept wing canard configurations must be even more unstable to achieve minimum trimmed drag (e.g., the X-29 is about 30%–35% unstable). Many studies of these fundamental properties of various configuration concepts have been made. See the study by Landfield and Rajkovic^[73] and the references contained in the study for more information.

Several key papers examining trim drag from a nearfield point of view have been written, including works by McKinney and Dollyhigh,^[74] Lutze,^[75] and Sachs.^[76] In the nearfield, extreme care must be taken to include the downwash incidences and induced angles of attack correctly. Alternately, an analysis can be made in the Trefftz plane. Lamar^[77] developed a code for finding the minimum trimmed induced drag for two surfaces, which was extended by Mason^[78] to include (approximately) the effects of profile drag. Note that a farfield analysis combining the minimization of induced drag and wave drag due to lift has been presented by Tulinius and Margason.^[79] A more general approach to treating multiple surfaces was given by Kuhlman.^[80] More recently, three-surface configuration has been introduced, and the three-surface minimum trim drag problem has been solved by Goodrich, Sliwa, and Lallman^[81] using a nearfield approach.

An example of the possible dramatic effects of CG location on trimmed drag is presented in figure 3-34.^[82] All the results contained in the figure are for the minimum trimmed drag at different values of a specified CG location. These results were obtained during early forward-swept wing configuration studies, illustrating why an aft CG position and resulting highly unstable configuration are required to obtain the full benefits of a forward-swept wing configuration similar to the X-29.* The very high drag values reflect a transonic maneuver condition. Trim drag should be much smaller for the cruise condition (certainly less than 2%–4%). As shown here, modern technology should allow the aircraft to fly with no trim drag. The difference between the minimum trimmed drag at Δx_cg = -40% and any other CG could be considered the trim drag. The figure contains both induced and profile drag contributions to the total trimmed drag. As the CG moves forward (x positive in this nomenclature; x = 0 corresponds to neutral stability), the additional load on the canard leads to a rapidly increasing value of the minimum trimmed drag. Because of the increasing load on the canard, the canard airfoil section becomes important. Near the CG for minimum drag, the canard airfoil is not important because the canard is lightly loaded. This figure shows why canard configurations are most efficient when used with unstable configuration concepts. Stable canard configurations are not necessarily the most efficient aerodynamically. Because of the high loads on a stable canard configuration, the canard airfoil section is very carefully selected. Specifically, it is usually highly cambered to achieve the high lift coefficients and has a small leading-edge radius so that it will stall before the main wing.

3.6 Accurate Drag Prediction: State of the Art

During the 1990s, significant advances were made in the numerical simulation of viscous flows on complex aircraft configurations using proprietary, commercial, and open source CFD codes based on the Reynolds-averaged Navier–Stokes (RANS) equations with a variety of turbulence models. It goes without saying that RANS codes could compute absolute drag—a quantity of great interest for aerodynamic design of aircraft configurations. Tinoco^[83] went so far as to state, “The application of Computational Fluid Dynamics (CFD) to the design of commercial transport aircraft has revolutionized the process of aerodynamic design. Today, CFD stands alongside the wind tunnel in terms of importance.” His assertion was, of course, reflective of his extensive experience at Boeing and his awareness of published work of other organizations.

One of the challenges facing the aerodynamics community worldwide was the lack of an impartial assessment of the accuracy of CFD predictions of aerodynamic characteristics in general and of drag in particular. To address this challenge, the AIAA Applied Aerodynamics Technical Committee formed a technical working group (TWG) which recommended conducting an international workshop with the primary goal of assessing state-of-the-art CFD methods as practical aerodynamic tools for the prediction of forces and moments on industry-relevant geometries, with a focus on absolute drag. The motivation was to identify areas needing additional research and development. Several members of the TWG formed the AIAA Drag Prediction Workshop (DPW) organizing committee that hosted a workshop in 2001—the first in a series of workshops discussed next in section 3.6.1.

3.6.1 The Drag Prediction Workshops

In the first AIAA Drag Prediction Workshop in 2001, eighteen international participants using fourteen different codes submitted CFD results for a DLR-F4 wing-body configuration. An AIAA paper by Levy et al.^[84] highlights the results, including those of a statistical analysis conducted by Hemsch.^[85] Analysis of the results showed a code-to-code scatter of the force and moment predictions that was an order of magnitude larger than anticipated. The CFD drag predictions exhibited a 2-sigma confidence interval of roughly +/- 40 counts (1 count represents a 0.0001 change in drag coefficient). In contrast, the experimental data used for assessing the accuracy of computational predictions exhibited a 2-sigma confidence interval of roughly +/- 8 drag counts.

From these unexpected results, it was easy to conclude that numerical simulations of viscous flows using the then state-of-the-art CFD methods were not accurate enough to earn the trust and confidence of most aerodynamicists. However, the workshop was successful in demonstrating intense interest in the community by bringing together many CFD developers and practitioners, which greatly facilitated the sharing of diverse ideas and best practices. The DPW organizing committee also identified several deficiencies in the quality of the grids supplied to the participants and in the assumption of fully turbulent flow made by many participants. Therefore, the DPW committee decided to organize and host a second workshop, DPW-2, in 2003 with DLR-F6 as the test case. Laflin et al.^[86] provide highlights of the important findings.

The DPW-2 established the need and framework for conducting a series of workshops over the subsequent years. Each successive workshop demonstrated improvements in our capabilities for drag prediction while also pointing to the shortcomings to be addressed in the future workshops.

• DPW-3 (2006)—Summary of data and findings compiled in a paper in 2007 by Vassberg et al.^[87]
• DPW-4 (2009)—Summary of data and findings compiled in a paper in 2010 by Vassberg et al.^[88]
• DPW-5 (2012)—Summary of data and findings compiled in a paper in 2013 by Levy et al.^[89]
• DPW-6 (2016)—Summary of data and findings compiled in papers by Roy and Tinoco,^[90] Keye and Mavriplis,^[91] Tinoco et al.,^[92] and Derlaga and Morrison.^[93]

Clearly a lot of progress was made in the fifteen years since the first workshop. We encourage interested readers to review the papers cited above to learn about the progress. However, we share three key DPW-6 findings from the paper by Tinoco et al.^[94]

1. Computational solutions for a NASA Common Research Model (CRM) wing-body configuration exhibited “tighter” convergence of total drag with a spread of less than ten counts.
2. Results for a NASA CRM wing-body-pylon-nacelle configuration showed that predicted drag increment was within the uncertainty of the test data—a finding of significant importance to aerodynamicists in the industry.
3. Results for NASA CRM wing-body configuration that incorporated static aeroelastic effects showed that lift was overpredicted, whereas the pitching moment was more negative (i.e., nose down) at a given lift coefficient than observed in test data.

The seventh workshop, DPW-7, was held in 2022, with a summary of data and many interesting findings presented in a paper by Tinoco.^[95] One of the test cases involved significant shock-induced separation and an assessment of more than thirty different solutions prompted the authors to conclude, “It is obvious from this and prior workshops that there is an interaction between solver, grid, and turbulence model that becomes most prevalent when there is significant shock-induced separation that we don’t understand.”

In the more than two decades of effort since the first workshop, we have clearly come a long way in terms of developing a comprehensive understanding and assessment of capabilities and shortcomings of the state-of-the-art CFD codes for accurate drag prediction. But we have a long way to go—especially when the flow is complex due to boundary layer separation or dominated by free vortices—to be able to credibly predict drag. We consider a prediction to be credible if it replicates reality. The state-of-the-art viscous flow simulation methods that are largely based on the RANS equations continue to face serious challenges in providing credible predictions.

3.7 Drag Reduction Possibilities

So far, we have discussed various aspects of aircraft drag prediction because the fuel consumption (or in a broader sense, energy consumption) is heavily dependent on the value of drag. Clearly, the lower the value of drag, the higher the energy efficiency of the aircraft. To explore drag reduction possibilities, it is instructive to assess the relative magnitudes of the drag components. For a typical transport aircraft in the cruise segment of flight that usually dominates its missions, major drag components are skin friction drag and lift-induced drag. The former is about 50%, and the latter about 40%. The remaining 10% is contributed by form drag, interference drag, and wave drag. The challenge for aerodynamicists is quite clear; their main focus ought to be on reducing skin friction and lift-induced drags. We discuss some of the related aspects in this section. Bushnell^[96] provides an excellent summary of state-of-the-art drag reduction across the speed range, as well as describing drag reduction approaches.

3.7.1 Skin Friction Drag Reduction

The skin friction drag mainly depends on two factors: (i) wetted area, which depends on the size of the configuration, and (ii) Reynolds number, which is a major determinant as to whether the boundary layer will be laminar or turbulent. It is well known that the boundary layer on a simple flat plate is laminar near the leading edge and transitions to turbulent when the Reynolds number exceeds a critical value. Considerable research has been done over the years on drag reduction technologies based on management of boundary layers. Many researchers have focused their efforts on reducing the skin friction of turbulent boundary layers, while others focus on extending the extent of laminar flow region.

3.7.1.1 Wetted Area Reduction

Since skin friction drag is directly proportional to wetted area, it stands to reason that reducing the wetted area is a straightforward way of reducing the skin friction drag. However, the feasibility of wetted area reduction is rather limited for conventional wing-body-tail configurations (e.g., B-777). Achieving smaller wetted areas requires dramatically different aircraft configuration, as discussed by Torenbeek.^[97] Examples of dramatically different aircraft include blended-wing body (figure 1-1) and strut-braced wing (figure 1-2).

3.7.1.2 Turbulent Skin Friction Reduction

Since the boundary layer on nearly the entire surface may be considered to be turbulent in most conventional designs at cruise conditions, many researchers have focused their efforts on devising means of reducing the associated friction drag. Riblets is one of the relatively mature technologies for achieving this. Viswanathan^[98] reviews the progress made in the last two decades of the twentieth century. Riblets are small longitudinal striations on the surface that impose spanwise viscous forces to reduce skin friction by 8% to 10%. Riblets can be applied to the aircraft surface in the form of a self-adhesive film. This technology has been flight tested on the Airbus A320 fuselage. About 75% of the surface was covered with 3M riblet film, which resulted in approximately 1.5% reduction in aircraft drag.

3.7.1.3 Extended Laminar Flow Region

It is well known that skin friction drag of a laminar boundary layer is roughly one order of magnitude less than that of a turbulent boundary layer. Considering the significant impact of the lift-to-drag (L/D) ratio on energy efficiency, let us examine flow on a wing as an example. On most wings, the boundary layer starts out as laminar near the leading edge but transitions to turbulent a short distance downstream. The main contributing factors are (i) Tollmien–Schlichting instability, which amplifies streamwise disturbances to promote transition; (ii) crossflow instabilities that dominate transition for wings with high sweep at high Reynolds number; and (iii) attachment-line instability that is due to disturbances originating in the fuselage boundary layer and propagating along the leading edge. Extensive research has been and is being conducted to develop a comprehensive understanding of these factors and to devise techniques for managing them to increase the extent of the laminar regions. Both passive and active techniques have been devised.

3.7.1.4 Natural Laminar Flow

Natural laminar flow (NLF) is a prime example of a passive technique which relies on shaping the airfoil geometry to extend the laminar-flow region and does not require any extra energy during flight operations. The airfoil geometry is modified to move the maximum thickness point as far aft as possible. This produces favorable pressure gradient on the forward part of the airfoil, which delays transition. Campbell and Lynde^[99] and Lynde and Campbell^[100] are excellent sources on the advances in and promise of the NLF technique.

3.7.1.5 Laminar Flow Control

Laminar flow control (LFC) encompasses active techniques which rely on either suction to remove small amounts of the boundary layer closest to the surface or blowing to energize the boundary layer. This results in stabilization of the boundary layer that delays transition. Both of these are active techniques because additional energy is required to operate the suction or blowing systems. Torenbeek^[101] provides concise descriptions of suction-based LFC techniques in Chapter 4. Researchers have demonstrated that laminar flow can be obtained using LFC on modern airfoil surfaces to midchord and beyond. As Bushnell^[102] points out, the major concern is not whether laminar flow can be obtained but rather whether it can be maintained, reliably, in an economic fashion. Some of the real-world challenges hampering the widespread adoption of LFC include surface roughness/waviness, joints and steps (including aeroelastic deformations thereof), flight through ice clouds (visibility under fifty miles can affect LFC system performance), acoustic fields caused by both engine and airframe, and suction-related issues such as plenum acoustics and localized vorticity generation, including clogging effects.

3.7.2 Lift-Induced Drag Reduction

The lift-induced drag is typically about 40% of the total drag in the cruise flight of an aircraft’s mission. Therefore, it is highly desirable to reduce it. Increasing the span is the most straightforward approach of reducing it because lift-induced drag is inversely proportional to the square of the span. Reducing lift-induced drag can be quite beneficial for takeoff performance as well. At low speeds, lift-induced drag can account for up to 90% of the total drag at a critical takeoff condition with a one-engine inoperative. Even a small reduction of lift-induced drag allows the aircraft to take off if an engine fails. However, the designer cannot arbitrarily increase the wingspan without paying a penalty. A larger span increases the structural weight due to the increased bending moment. Aircraft statistical weight equations (that you can find in most aircraft design books) show a strong dependence between wing structural weight and span for all classes of aircraft. Careful tradeoffs must be conducted to determine an optimum span, and this may require the use of modern multidisciplinary design optimization (MDO) methodologies. In addition, practical operational considerations, such as meeting airport gate categories, may limit the span of large transport aircraft.

The strut-braced wing (SBW) is one such configuration which employs a high-aspect-ratio wing and uses a strut to minimize the wing weight penalty that a cantilever long-span wing will pay. The origins of this concept in the 1950s are presented in section 1.2.2. For many years in the 1990s, Virginia Tech conducted extensive concept and technology maturation research.^[103],[104] You can find more references on the SBW concepts on my websites.^[105]

Designers have explored configurations using nonplanar lifting systems (e.g., box wing, joined wing, C-wing, and twin fuselage) that reduce the lift-induced drag. However, design trade studies have not proven significant improvements over the conventional monoplane. For example, wind tunnel tests of a box wing by Miranda and Dougherty^[106] verified the predicted low induced drag at high speed. However, flutter analysis revealed both symmetric and antisymmetric instabilities to occur well below the required flutter speed that could not adequately be suppressed. No significant reductions in wing and tail weight were realized for the boxplane relative to a monoplane with the same mission capability. These findings led to the abandonment of boxplane research by Lockheed. However research studies continue, and Salem et al.^[107] provide a critical review of design approaches and applications for the box-wing configuration.

3.8 Some Issues for Drag Calculation Using Computational Aerodynamics

Because of the quest for reduced drag and the difficulty in computing and measuring small changes in drag, numerous disputes have arisen in aerodynamics. Confusion introduced by nonstandard drag nomenclature also contributes to these spirited debates. One recent issue was the sheared wingtip drag reduction controversy. Here, it was speculated that wingtip shaping could lead to spans (e’s) greater than one for planar planforms.^[108] This conclusion arose based on both computations and wind tunnel tests. Refined computational investigations^[109] illustrating the need to study computational solution convergence carefully have resulted in the conclusion that e’s greater than one were not actually computed. However, it is clear that wingtip planform-shaping can lead to improved aerodynamic efficiency.

Another area is the search for more fundamental understanding of drag. These theories differ significantly from the accepted approach to drag. One key example is found in the work of Yates.^[110]

In addition to the efforts to reduce drag due to lift by tip-shaping, use of winglets, tip sails, and canard configurations, among other methods, significant efforts are being made to reduce skin friction drag. They include efforts to obtain laminar flow through passive means (NLF), suction, or a combination known as hybrid natural laminar flow control. Turbulent friction-reduction techniques are also being developed. Riblets are perhaps the most well known means. An AIAA book reviews this area.^[111]

Figure References