• Laminar boundary layer,

• Turbulent boundary layer,

• Location of laminar-to-turbulent transition,

• Flow separation regions, and

• Compressibility.

(1) Basic pressure drag is caused by the pressure differential formed by the airplane that acts parallel to the tangent to the flight path.

(2) Skin friction drag is caused by the “rubbing” of molecules along the surfaces of the airplane.

(3) Lift-induced drag is caused by the circulation around the wing, which tilts the lift vector backwards, creating a force component that adds to the total drag.

(4) Wave drag is caused by the rise in pressure around a body due to the formation of a normal shock wave on the airplane. This effect begins at high subsonic Mach numbers.

(5) Miscellaneous drag is caused by a number of “small” contributions that are often easily overlooked, such as small inlets and outlets, access panels, fuel caps, to name a few.

15.1.1 The Content of this Chapter

α = angle-of-attack

β = angle-of-yaw

ρ = air density

V_∞ = far-field airspeed

R_e = Reynolds number

M = Mach number

C_D = total drag coefficient

S_ref = reference area (typically wing area)

V_∞ = far-field airspeed

ρ = air density

15.2.1 Basic Drag Modeling

Basic drag modeling is the mathematical combination of all sources of drag for a vehicle, such that the effect of changing its orientation with respect to its path of motion and fluid velocity is realistically replicated. This modeling culminates in the determination of the total drag coefficient, C_D.

As stated in the introduction, the total drag coefficient comprises the effect of basic pressure drag, skin friction drag, lift-induced drag, wave drag, and contributions from other sources, commonly referred to as miscellaneous drag. Typically, basic pressure drag, skin friction drag, and miscellaneous drag are lumped together and are represented using a single number, called the minimum drag, as is done in the so-called quadratic drag modeling. This is accomplished using a special method called the component drag build-up method, presented later in the chapter.

In short, the method estimates the total drag by determining the skin friction drag of a particular component (e.g. the wing or the HT). Then the basic pressure drag is accounted for using a booster factor called a form factor, presented in detail later in this chapter. Thus, the designer estimates the skin friction coefficient for the surface and then multiplies it by the form factor, yielding a number that, as stated earlier, combines the effect of friction and pressure. Additionally, it is necessary to account for the numerous protrusions and discontinuities in the generally smooth surfaces of the aircraft, as it results in greater than expected pressure drag. This drag is called miscellaneous drag.

As the AOA or AOY changes, the contribution of the pressure drag terms increases, adding non-linearity to the formulation that is usually not accounted for in the modeling. It is acceptable to omit this effect for low values of AOA and AOY. However, once a certain orientation is reached, the pressure drag grows to a magnitude so large that it can no longer be ignored. Then, standard drag models are no longer acceptable unless a means to capture this effect is included. A method to account for this high AOA effect is presented in Section 15.2.3 of this chapter.

15.2.2 Quadratic Drag Modeling

A standard way to present the drag coefficient is to relate it to the lift coefficient using a quadratic polynomial in accordance with the derivation presented in Section 15.3.4, The lift-induced drag coefficient: C_Di. The method can provide an accurate prediction over a range of lift coefficients, although the accuracy drops rapidly at the extremes of the drag polar. A standard “simplified” quadratic presentation for the drag coefficient is:

(15-5)

where

A far more “realistic” presentation for the drag coefficient is the adjusted drag model, represented graphically in Figure 15-2.

(15-6)

where C_{L minD} = the lift coefficient where drag becomes a minimum.

Two important properties of the adjusted drag polar are shown in Figure 15-2. Changing the C_LminD will only shift the polar sideways. If C_LminD < 0 then the curve will be shifted left, but if C_LminD > 0 it will be shifted right. Similarly, changing the value of the C_Dmin will shift the polar up or down. Of course C_Dmin is always larger than zero, whereas C_LminD can range from a negative to a positive number depending on the camber of the airfoil, or, in the case of a complete aircraft, the effective camber of the entire geometry.

15.2.3 Approximating the Drag Coefficient at High Lift Coefficients

Figure 15-3 reveals a critical problem in all drag modeling; at higher C_Ls, the drag model deviates drastically from the actual drag and underpredicts it severely. The deviation is caused by a rapid growth of flow separation with AOA and the second-order approximation cannot keep up with the resulting increase in drag. This is a serious problem when estimating aircraft performance at low airspeeds. Therefore, important airspeeds, such as best angle of climb, minimum rate of descent, and others, are shifted to lower airspeeds than observed in practice. This section addresses this shortcoming and develops a method to approximate the rapid rise in drag at higher lift coefficients.

Consider the hypothetical wind tunnel test data in Figure 15-5. Simply stated, the data presented cannot be approximated satisfactorily using a quadratic polynomial. Neither the simplified nor the adjusted drag models will provide acceptable drag prediction at high (or low) lift coefficients. For this reason, only the data points at lower lift coefficients are used to create a curve-fit (or a trend curve). For the wind tunnel data shown in Figure 15-5, assume that a standard adjusted drag model has been determined and, as shown, it agrees well with the measured drag at lower lift coefficients. This model is given by Equation (15-6), reproduced below for convenience:

(15-6)

As is very evident from the graph, the test data begins to deviate sharply from the curve starting at C_L = 1.15 (and C_L = −0.45). Of course we are primarily interested in the positive lift coefficient, as this is needed for low-speed performance predictions. In order to work around this predicament, the author has used the following methodology in the past with very satisfactory results. In this method, once the C_L exceeds a certain value, which here will be called C_Lm, a quadratic (or cubic) spline is created to simply replace the values of the adjusted drag model.

The method presented here effectively defines a new quadratic polynomial and splices it to the adjusted drag model at C_Lm. Other splines are certainly possible; however, the advantage of the quadratic spline is the simplicity of its determination and the acceptable accuracy it provides. The method allows the spline to blend smoothly with the underlying adjusted drag model. The first step is to define a modified drag coefficient to be used for C_L > C_Lm. It can be represented by:

(15-11)

Ultimately, the task is to define the constants A, B, and C so C_Dmod can be evaluated. To do this, two conditions have to be satisfied at C_Lm:

A third condition is needed to finalize the determination of the coefficients; it is that the value of C_Dmod at C_Lmax must match that of the wind tunnel data. That aside, the function for C_Dmod has three constants (A, B, C) so three equations are required to determine them. One of the equations requires the derivative of both C_D and C_Dmod to be determined. These are presented below:

Now the three equations that allow the constants A, B, and C to be determined can be written as follows:

Rearranging this in a matrix form yields the following expression that allows A, B, and C to be determined using any matrix method, Cramer’s rule or matrix inversion methods:

(15-12)

Then, once A, B, and C have been determined the drag model is further refined as follows:

This model has been implemented in Figure 15-6. The improvements in the capability of the drag model are demonstrated by how the quadratic spline, represented by C_Dmod, smoothly follows the wind tunnel data starting at C_Lm. Note that although wind tunnel data is assumed here, the same approach can also be used for analytical estimation of the drag coefficient. For such work, it is reasonable to select C_Lm as the value ½(C_LminD + C_Lmax) as a first guess.

15.2.4 Non-Quadratic Drag Modeling

Figure 15-3 shows that the quadratic model may indeed work well for a specific range of lift coefficients provided a proper value of span efficiency, e, is selected. However, this is not always the case. There are two other instances where the quadratic modeling is simply inadequate and must be abandoned. The first one involves very high or very low lift coefficients, when large areas of separated flow have formed on the airplane. This was treated in the previous section. The second one pertains to aerodynamically clean aircraft, such as sailplanes, for which a well-defined drag bucket exists and is not masked by the drag of other sources that cannot be described by the quadratic formulation. These problematic regions are easily visible in Figure 15-8. In this case, the performance formulation in Chapters 13 through 18 must be revised to account for those cases.

Generally, the presence of the drag bucket will prevent the use of polynomials, as these are not capable of following the sharp change in curvature of the drag polar. Even polynomials of very high order (16+) will not be adequate, as they tend to oscillate inside the prediction region. There are primarily two other methods that can be used to represent such drag polars; a spline (e.g. B-spline) or a lookup table.

15.2.5 Lift-Induced Drag Correction Factors

The estimation of drag due to lift of a three-dimensional wing requires a correction of the two-dimensional airfoil data. This is expressed in Equations (15-5) and (15-6) using the factor e with the induced drag components. There are generally two kinds of such correction factors (although some use them interchangeably):

15.2.6 Graphical Determination of L/D_max

In the absence of knowing the exact numbers of a particular drag polar, it is possible to determine the maximum lift-to-drag ratio, as well as the lift coefficient at which it occurs, graphically. This is shown in Figure 15-9 for a “conventional” drag polar and one for a NLF airfoil (or aircraft) featuring a drag bucket. This is done by fairing a line from the origin so it becomes a tangent to the polar. The reason why this yields L/D_max can be visualized by recognizing that the tangent aligns with the smallest value of C_D at the greatest value of C_L.

15.2.7 Comparing the Accuracy of the Simplified and Adjusted Drag Models

The accuracy of the simplified and adjusted drag models can best be illustrated by comparing them to wind tunnel data. One such comparison is shown in Figure 15-10, which compares experimental wind tunnel test results to the simplified and adjusted drag models using Equations (15-5) and (15-6), respectively. Of course it should be emphasized that ordinarily one would not use these models with the drag polar of a two-dimensional airfoil, but a three-dimensional wing. However, this particular illustration can be justified based on Equation (15-9) as the purpose is to show the importance of the term C_lminD, here using the lower-case subscript for forces.

The comparison was implemented using wind tunnel data for the NACA 2412 airfoil obtained from NACA R-824. This airfoil is used on a number of Cessna aircraft designs, for instance the Model 150, 152, 172, and 182, to name a few. First, the drag polar was digitized and the smallest experimental value of the C_d selected for use as the C_dmin for each drag model. Second, the corresponding C_lminD to use with the adjusted model was determined by trial and error. Third, the induced drag contribution was calculated using the expressions for the simplified model and for the adjusted model and added to the C_dmin. Then, k was varied to get the best fit of each model (note both models used the same k). The value of k that best fit the experiment was found to be k ≈ 62.³ The graphs show that the adjusted drag model provides a substantially better fit than the simplified one, in particular for the positive values of the C_l. Another lesson to be learned is the dire consequences of estimating range for an aircraft using the simplified model. Since range depends so explicitly on the C_L/C_D (see Section 20.2, Range Analysis) a prediction using the simplified drag model would yield a range much less than the adjusted (or experiment) indicates. The opposite can also happen, i.e. the C_L/C_D using the simplified model is sometimes higher than experiment. The important lesson is that the adjusted model better matches experiment; the simplified does not and should be avoided.

15.3.1 Basic Drag Coefficient: C_D0

The basic drag is a pressure drag force caused by resultant pressure distribution over the surface of body. It can be thought of as the component of the pressure force parallel to the tangent to the flight path. For instance, consider a cylinder in a moving fluid. Its drag consists of the friction between its surface and the moving fluid, and the difference in pressure along its surface. The latter will be the focus in this section. The force is the product of the pressure acting on a cross-sectional area of the body, normal to the flight path.

(15-13)

where

A simple interpretation of the basic drag is shown in Figure 15-11, which shows a sphere moving through a fluid forming a high-pressure region in front of it (left) and a low-pressure region behind it as the turbulent wake. The pressure differential across its cross-sectional area yields a drag force. Equation (15-13) can be rewritten for simple geometry and uniform pressure distributions as follows:

(15-14)

For other geometries, this force increases at high (and low) AOAs as the separation regions grow. If the basic drag force is known, the basic drag coefficient it is defined as follows:

(15-15)

where

The basic drag coefficient is typically a function of:

(15-16)

where

Ultimately, the basic drag force can be thought of as the increase in the skin friction forces due to the applied form factor (FF), i.e. the difference (FF − 1). However, and as will be seen later, this distinction is better suited for explanation than utility, as it is far more practical to combine the two (i.e. the skin friction and pressure drag contributions). This is due to the complex interaction between the two and our desire to maintain a level of simplicity in our analyses.

15.3.2 The Skin Friction Drag Coefficient: C_Df

The skin friction drag coefficient is defined as follows:

(15-17)

where

Note the difference between the two coefficients C_f and C_Df.

Skin friction is caused by a fluid’s viscosity as it flows over a surface. Its magnitude depends on the viscosity of the fluid and the wetted (or total) surface area in contact with it, as well as the surface roughness. The analysis of skin friction drag is complicated by the process of transition, when the laminar boundary layer becomes turbulent. Consequently, this is called a mixed boundary layer. A realistic drag analysis always assumes a mixed boundary layer.

An airfoil that can sustain laminar flow as far aft as 50% of the chord, naturally and on its own merit, is referred to as a natural laminar flow airfoil (NLF). Such airfoils generate substantially less drag than airfoils not capable of this.

Figure 15-12 shows an airfoil immersed in airflow, on which the laminar boundary layer extends from the leading edge to a point on the upper surface denoted by X_{tr_upper} and X_{tr_lower} on the lower surface. At those points, enough instability has developed in the boundary layer to change the laminar profile into a thicker turbulent one. This is the consequence of dynamic flow forces becoming larger than the viscous forces (see Figure 15-13) and tends to occur once the Reynolds number (R_e), as measured from the leading edge (LE) along the surface, approaches and exceeds 1 million (on a flat and smooth plate) [3, pp. 2–8]. As the speed of the fluid increases (increasing the R_e) the transition points will move farther and farther forward, although they will never fully get to the LE. Therefore, if smooth, the LE will always develop some laminar BL.

The stability of the boundary layer also depends on the geometry and quality of the surface over which the fluid flows. The presence of rivet-heads, uneven plate joints, insects, and even paint chips can destabilize the laminar boundary layer and initiate an early transition (see Section 8.3.13, The effect of leading edge roughness and surface smoothness). Figure 15-14 depicts how the boundary layer changes with AOA. At a low AOA, the transition location on the upper surface is close to the trailing edge, stabilized by the high pressure along the upper surface. At a high AOA, the transition point moves forward, leaving less area covered by the laminar boundary layer. However, and this is important to keep in mind, the laminar boundary layer is more stable on the lower surface because of a net reduction in airspeed.

The boundary layer develops on the surfaces of a “streamlined” three-dimensional shape in a similar fashion as it does on a flat plate and is affected by all the same shortcomings. The velocity distribution in a laminar or turbulent boundary is also similar between the two, albeit being affected by the pressure distribution of the three-dimensional geometry [3, pp. 2–6].

Figure 15-15 shows an important property of flow around an airfoil that ensures the maintenance of the laminar boundary layer (assuming the surface quality is smooth). This property is an extensive region of a favorable pressure gradient. The figure shows the pressure distribution along the chord of two airfoils of 12% thickness-to-chord ratio; a NACA 64-012A and NACA 0012. Both airfoils are at an AOA of 0°. The pressure distribution of the former is represented by the solid line and the latter by the dashed line. The figure also shows that the pressure along the surfaces of the NACA 64-012A airfoil is dropping, starting behind the leading edge until 40% of the chord. The pressure on the NACA 0012 airfoil drops sharply until 12% of the chord. The laminar flow of the 64-012A extends to 55% of the chord, versus 50% of the 0012, and, consequently, it generates slightly less drag. In practice, the transition point of the NACA 0012 would be expected to be farther forward than that.

One way to achieve such an extensive laminar boundary layer is to place the maximum thickness of the airfoil as far aft as possible. The goal is to allow the fluid to accelerate as far aft as possible and then decelerate without separation. Ultimately, how far aft the maximum thickness can be placed depends on the distance along which the fluid is allowed to decelerate. Too short a distance and the fluid will separate near the trailing edge and increase the drag. Figure 15-15 shows that the negative (“favorable”) pressure gradient of the former extends to 40% of the chord, but merely 12% of the latter. A favorable pressure gradient means that fluid molecules are accelerating along the surface and the acceleration stabilizes the boundary layer, allowing the laminar boundary layer to extend farther toward the trailing edge. A direct computational analysis, using the airfoil analysis software Xfoil, predicts the transition point of the NACA 64-012A airfoil to be at 55% of the chord and 50% on the NACA 0012 airfoil at a Reynolds number of 3 million and Mach number of 0.10. Bringing the transition point beyond this will require the maximum thickness to be placed even farther aft. This requires a careful attention to detail and depth of understanding of the aerodynamics of the particular geometry. Some NLF airfoils sustain laminar BL as far aft as 70% of the chord.

So far we have seen how the laminar boundary layer depends on the geometry of the surface over which it flows. The greatest drawback of such flow is its sensitivity to geometry and even uncontrollable factors like the quality of the airflow. Figure 15-16 shows how the flat plate drag coefficient changes with Reynolds number. It is evident how laminar flow reduces the magnitude of the drag coefficient. As an example, the laminar boundary layer drag coefficient at R_e = 1 × 10⁶ is approximately 0.00135, while it is approximately 0.00445 for a turbulent boundary layer; the turbulent skin friction is about 3.3 times the laminar one.

Figure 15-16 also shows how susceptible to transitioning the laminar boundary layer is as it flows over a flat plate, operating within Reynolds numbers ranging from 4 × 10⁵ to 1 × 10⁷. A method to estimate this transition is presented below.

15.3.3 Step-by-step: Calculating the Skin Friction Drag Coefficient

15.3.4 The Lift-Induced Drag Coefficient: C_Di

Consider a finite wing featuring a single airfoil mounted in space in airflow at some specific AOA and airspeed. Contrast that with a straight wing featuring exactly the same airfoil, extending from wall to wall in a wind tunnel at the same AOA and airspeed (ignoring typical wind-tunnel interferences). In this situation, the finite wing will generate less lift per unit span than the one in the wind tunnel. This is caused by the fact that the airflow makes its path around the wingtips of the finite wing through the formation of vortices, one at each wingtip. The short consequence of this flow is that in order for the wing to generate a specific amount of lift, it must always be at a higher AOA than the corresponding two-dimensional airfoil in the wind tunnel. This leads to the formation of drag that increases the wing drag beyond that of the airfoil only. This drag is called a lift-induced, or induced, or vortex drag and is denoted by D_i. As usual the standard formulation of this drag involves the determination of a lift-induced drag coefficient. A number of methods to determine this drag coefficient are presented below. The lift-induced drag is a pressure drag force.

15.3.5 Total Drag Coefficient: C_D

Once the constituent drag contributions have been estimated, the total drag coefficient is simply determined by addition using the following expression:

(15-52)

Note that the combination C_Do + C_Df is highly internally dependent and, therefore, in this book, they are combined and called the minimum drag, C_Dmin. However, in light of the above definition, inserting the explicit forms of the other coefficients yields:

(15-53)

For internal consistency it is better to write:

(15-54)

15.3.6 Various Means to Reduce Drag

As has already been stated, drag is generally a detrimental force in aircraft design, in particular for operational missions that require fuel efficiency. In addition to “obvious” means to reduce drag, such as retractable landing gear and smooth NLF surfaces, people have been creative in their attempts to reduce drag. Below are a selected number of ideas conceived with this intent, presented here to inspire the reader.

15.4.1 The Effect of Aspect Ratio on a Three-Dimensional Wing

In 1923, Prandtl [20] presented the results of well-known experiment which depict well the effect aspect ratio has on the generation of lift and drag of a wing. A copy of of his actual results is shown in Figure 15-26. The graph to the right shows what has already been discussed in Chapter 9, The anatomy of the wing, the impact of AR on lift. The graph to the left shows the effect of AR on drag. It shows that two wings (call them Wing 1 and Wing 2) whose geometry differs only in the AR (wing area and airfoil, and thus the minimum drag of the airfoil, C_dmin, are assumed the same) will generate drag coefficient at the same C_L, which can be expressed as follows:

(15-55)

From this it is evident if the drag of one wing, say Wing 1, is known, the drag of the other one can be estimated from the difference of the two drag polars:

(15-56)

Of course, the expression holds as long as the difference in the ARs is not too great. This is because the chord length of the airfoil will affect the minimum drag of the airfoil, C_dmin, and the AR will affect the Oswald span efficiency factor, e.

Figure 15-27 shows the effect of changing the AR of a three-dimensional wing on the drag polar using modern notation of coefficients. It shows that as the magnitude of the aspect ratio does not change the minimum drag (not accurate if the change is large), but rather the width of the drag polar is reduced. This is a consequence of a reduction in the magnitude of the C_Lmax and the lift curve slope, C_Lα. The reduction is easily estimated through the use of Equations (15-38), (15-45), and (15-47).

15.4.2 The Effect of Mach Number

Generally, as the airplane approaches the speed of sound, the drag coefficient will begin to rise sharply. The Mach number at which the rise begins depends on the geometry of the airplane. An airplane with thick airfoils will experience it perhaps as early as M = 0.6, whereas a sleek high performance jetliner begins to experience it at M ≈ 0.85. This rise is handled in the drag estimation by the addition of a compressible drag coefficient, ΔC_DC, which is most accurately estimated in wind tunnel testing or by sophisticated CFD methods. The Mach number at which this happens is called the drag-divergence Mach number, denoted by M_dd, and it occurs slightly above the critical Mach number (see Section 8.3.7, The critical Mach number, M_crit).

The drag-divergence Mach number is defined either when ΔC_DC = 0.002 or when dC_DC/dM = 0.10.

Figure 15-28 shows a hypothetical scenario in which an airplane flying at a constant C_L experiences a sharp rise in the compressible drag coefficient near M = 0.7. As stated earlier, this is the critical Mach number, denoted by M_CRIT. Its value largely depends on the geometry of the aircraft and details such as the thickness of the wing airfoils and the geometry of the wing/fuselage juncture, to name few. In fact, any part of the airplane which features interferences due to the joining of disparate components is suspect. Such interferences typically result in increased local airspeeds, which promote the formation of normal shocks (and therefore higher drag) in the juncture.

A method to estimate this contribution to the total drag is presented in the Section 15.5.12, Drag due to compressibility effects.

15.4.3 The Effect of Yaw Angle β

As discussed in Section 15.3.4, The Lift-Induced Drag Coefficient: C_Di, drag varies greatly with AOA. This effect extends to yaw as well, as shown in Figure 15-29; it too increases the drag and is usually minimum when β = 0°. This fact has been used for a long time by pilots when landing. Pilots “coming in too high” for a landing will yaw the airplane, decreasing its L/D ratio, allowing the airplane to temporarily lose altitude more rapidly. This addition depends entirely on the geometry of the aircraft. Generally, it is an acceptable approximation to assume the flow to stay attached on the aircraft for a β up to ±10°. However, beyond that, flow separation is certain and the associated drag increase must be accounted for.

15.4.4 The Effect of Control Surface Deflection – Trim Drag

Deflecting control surfaces usually increases the drag of the aircraft. For instance, deflecting the elevator shifts the drag polar vertically, increasing the C_Dmin. This change is referred to as the trim drag. If a conventional tail-aft airplane is loaded with its CG far forward, a higher deflection of the elevator will be required to trim it. This implies additional drag, in addition to the extra lift the wing must generate that increases lift-induced drag. This is why it is important to size the tail so the elevator is close to neutral in cruise (usually referred to as an elevator “in trail”).

Trim drag is most accurately estimated using precise wind tunnel tests, although it can certainly be estimated using analytical methods too. A graph similar to the one in Figure 15-30 is usually obtained, with several curves showing the drag polar for an elevator deflection of, say, 0°, 2°, 4°, 6°, 10°, and so on. These allow the performance engineer to better estimate the capability of the airplane at different loading and flight conditions. In the absence of wind tunnel testing, it is imperative to be able to assess it with some level of accuracy, in particular if the airplane features a high (or low) thrustline.

A method to estimate trim drag based on CG location and thrust setting is presented in Section 15.5.2, Trim drag.

15.4.5 The Rapid Drag Estimation Method

The rapid drag estimation method is without question the fastest way to estimate the drag of an airplane. It is based on the assumption that there is a correlation between the averaged skin friction coefficient of an airplane, its total wetted area, and its minimum drag coefficient. Naturally, with this speed comes inaccuracy, so the method should only be used to figure out a “ballpark” value for the drag to compare to other methods. For instance, the method does not account for any peculiarities in the airplane design, such as extent of laminar flow, specific geometric features, and others. It is possible to account for the rudimentary effects of flaps and landing gear, but again, the designer should not use this method for anything but to get an idea of the magnitude of the drag coefficient, and not be surprised if it is vastly different from the value obtained by more accurate methods. The method calculates the minimum drag coefficient using the concept of EFPA (see Section 15.2.2, Quadratic drag modeling) and Equation (15-8):

(15-57)

where

The method involves reading the value of f for a “clean” aircraft from Figure 15-31, using the approach outlined in the figure. Ideally, the estimate of the aircraft’s wetted area should be as precise as possible. Also note that the equivalent skin friction coefficient means a normalized value representative for the entire aircraft. It is inevitable, considering the variety in the surface quality of a normal airplane, that some engineering judgment will be required to assess the value.

The drag increments due to flaps, landing gear, speed brakes, etc., are accounted for using the following relation and then added to the C_Dmin:

(15-58)

where

The equivalent parasite areas and coefficients are obtained from the Table 15-6, where A_C stands for maximum cross-sectional area, S_ref is the wing reference area and S_HT is the HT planform area.

15.4.6 The Component Drag Build-Up Method

The component drag build-up method (CDBM) is used to estimate the drag of the complete aircraft. The method is primarily based on the estimation of flat plate skin friction over the surfaces of the airplane that are exposed to the airflow (wetted area). The method accounts for geometric differences between components and changes in drag resulting from bringing one component into the neighborhood of another (interferences).

The CDBM bases the parasitic drag (the combination of C_D0 + C_Df) using flat plate skin friction coefficients that are modified using two special factors called a component form factor (FF) and interference factor (IF). The FF is a measure of the pressure drag due to viscous separation. It accounts for the fact that the drag force generated by a sphere and a box of equal wetted area is different from one another. Interference effects between aircraft parts (wing and fuselage, engine and wing, HT and VT, etc.) are accounted for using a special factor called component interference factor (IF). This factor is based on the fact that as two bodies are brought together in fluid flow, the drag of the combination is greater than the drag of the individual bodies on their own. The IF is denoted by Q in Raymer, but by IF by most other authors.

A flow chart showing the procedure is shown in Figure 15-32. In the flow chart, the “+” sign means that contributions are added, whereas the “×” sign means multiplication.

The method requires the skin friction coefficients to be calculated for all components in direct contact with air (or wetted by air) . The components are parts like the wings, horizontal tail, vertical tail, and so on. Even though the wing, HT, VT, and fuselage are shown, other components, such as engine nacelles, pylons, winglets, external fuel tanks, dorsal, or ventral fins, should be added to the list if present. Naturally, they should be excluded only if other means to estimate their drag is selected (e.g. see Section 15.5, Miscellaneous or additive drag) and then be included as miscellaneous drag, C_Dmisc. These are adjusted with FF through a multiplication operation, but this ensures dissimilar components don’t contribute equally to the overall drag even if both have identical wetted areas. Then, IFs are applied , also through a multiplication operation, to account for the increase in drag when individual components are brought into close proximity.

Once the skin friction drag (with its associated pressure and interference drag boosting) has been determined, the next step is to determine all remaining sources of drag. This is simply called miscellaneous or additive drag . This is drag attributed to miscellaneous sources, such as antennas, fuel caps, air flow through small gaps (such as those between control surfaces and the lifting surface to which they are mounted), as well as control system components (bellcranks, hinges, etc.), inlets, outlets, antennas, and so on. These are then summed up and multiplied by a crud-factor , which accounts for contributions that are practically impossible to account for otherwise, such as surface panel misalignments, dents, small vents and outlets, and so on. The crud factor is typically 25%, which means that the sum of the above contributions is multiplied by 1.25. This operation returns the minimum drag coefficient . We add to it the lift-induced drag coefficient at the flight condition to obtain the total drag coefficient .

The total component drag build-up is expressed as follows:

(15-59)

where

15.4.7 Component Interference Factors

As previously stated, interference factors (IF) are used to account for the proximity of one component to another. For instance consider the juncture between the wing and fuselage. The presence of both bodies constrains the airflow compared to that of the individual components, increasing the local airspeeds greatly, which increases the drag. However, this does not account for additional drag that may arise due to early separation due to a poorly designed wing/fuselage juncture.

Table 15-7 lists typical IFs that are partially derived from Refs [3] and [25] and partially using factors that have worked well in drag analyses performed by the authors. No claim is made about their accuracy beyond that. Note that when using the factors for multiple objects, for instance, a triple-tail, the presented IF must be applied to all surfaces. Thus, when summing up the four surfaces of such a tail (one HT and three VTs), their corresponding C_Df (= C_{f i}·(S_{wet i}/S_ref)) must be multiplied each time. The following formulation illustrates how its contribution would be accounted for in the CDBM:

15.4.8 Form Factors for Wing, HT, VT, Struts, Pylons

As stated earlier, a form factor (FF) reflects the geometric shape of components and, therefore, methods to estimate this value varies greatly with classes of components. Thus there is a set of FFs that are only used with geometry capable of generating lift (“wing-like” surfaces). Others are only used with geometry that serves as fuselages, and so on. The following expressions are used to estimate FF for lift for wing-like surfaces. Such surfaces include wings, horizontal tail, vertical tail, struts, and pylons, but can also be extended to wing-shaped antennas and landing gear pant fairings. These form factors are typically derived by semi-empirical methods that emphasize the thickness-to-chord ratio of the structure. The following form factors are found in the literature.

The inevitable question that comes up is “which one do I pick?” Unless a specific application is cited (like the ones by Hoerner) the answer is often based on engineering judgment. In that case the unsure designer can take the average of two or three methods.

Hoerner [3, p. 6-6] suggests the following form factors for lifting surfaces featuring airfoils whose (x/c)_max = 30%:

(15-60)

Hoerner [3] suggests the following form factors for lifting surfaces featuring airfoils whose (x/c)_max = 40% to 50%, such as NACA 64 and 65 series airfoils:

(15-61)

Neither of the above models account for wing sweep or compressibility effects. Torenbeek [26] suggests the following form factors for lifting surfaces featuring airfoils whose t/c ≤ 21%:

(15-62)

Shevell [27, p. 178] suggests the following form factors for lifting surfaces and introduces compressibility and sweep effects:

(15-63)

Nicolai [28] and Raymer [25] suggest the following form factor for lifting surfaces that also corrects for compressibility but considers the sweep of the maximum thickness line rather than that of the quarter chord. The equation, as shown, is only valid as long as M > 0.2 because the compressibility correction (the bracket on the right-hand side) becomes less than 1 at a lower value. For this reason the compressibility correction term should be set to 1 for airspeeds below M = 0.2.

(15-64)

Jenkinson [29] suggests two kinds of form factors: one for the wing and another for tail surfaces. The form factor for the wing is given by:

Furthermore, the reference recommends the interference factor to use with the expression is IF = 1.0 for well filleted low or mid wings, and 1.1–1.4 for small or no fillet. The form factor for tail surfaces is given by:

The reference recommends an IF = 1.2 for tail surfaces.

In the above equations:

Note that both Equations (15-63) and (15-64) approach Torenbeek‘s form, shown in Equation (15-62), for wings whose quarter-chord (Shevell) or maximum thickness (Nicolai, Raymer) sweep angle is 0°, when M = 0 (Shevell) or M = 0.2 (Nicolai, Raymer).

15.4.9 Form Factors for a Fuselage and a Smooth Canopy

This section discusses form factors intended for use with geometry that represent fuselages, so-called streamlined bodies, and smooth canopies. These form factors are typically represented in terms of the fineness ratio, defined as length (l) divided by the average diameter (d), as shown in Figure 15-33 and Figure 15-34. The figure indicates the proper way to evaluate the required parameters. When it comes to fuselages, it is of importance to focus on the body itself. The upper part of the figure shows a fuselage with the tail and engine pods. The lower part shows the fuselage stripped of these, representing the geometry used in this analysis method. Sometimes the distinction is not so clear, leaving no option but to depend on engineering judgment.

A streamlined body usually refers to a body-of-revolution, whose cross-section is similar to what is shown in Figure 15-34. A streamlined body is used to represent many types of fuselages, but also the hulls of airships. They can also be used to represent engine nacelles in drag estimation. A canopy refers to the external shape of an airplane’s glass helmet and is sized geometrically as shown in Figure 15-34. For more realistic canopies, refer to Section 15.5.10, Drag of canopies.

15.5.1 Cumulative Result of Undesirable Drag (CRUD)

Of course the above title is just a play on words. The word crud means dirt, filth, or refuse. In aircraft design it stands for the undesirable drag caused by things like exhaust stacks, misaligned sheet-metal panels, antennas, small inlets and outlets, sanded walkways, and so on. These parts are easily overlooked when performing drag analysis, primarily because the evaluation of their contribution is next to impossible. For instance, consider misaligned sheet-metal panels for aluminum aircraft. The problem is not that there isn’t a method to estimate the drag of a misaligned panel, because there is, but rather that it is impossible to estimate how poorly or well the panels align until the actual aircraft is built. It is imperative the aspiring designer is aware of this drag and accounts for it appropriately. This section introduces how this is typically done.

In 1940 NACA released the wartime report WR-L-489 in which 11 military aircraft were investigated to determine why they didn’t meet predicted performance [30]. The report detailed wind tunnel testing of the aircraft in the NACA Full-Scale Wind Tunnel in Langley, Virginia. The aircraft were stripped to a clean configuration by gradually removing components and the drag coefficient was measured after each removal step. The results give a timeless insight into the cumulative impact of small and easily ignored design details on the total drag.

One of the aircraft investigated was the Seversky P-35 that had been designed for the US Army Air Corps in the early 1930s (see Figure 15-41). At the time, it was an innovative single-engine fighter that offered a number of firsts. It was the Air Corp’s first all-metal fighter, the first to feature a retractable landing gear, and the first with an enclosed cockpit. The P-35 had a predicted maximum airspeed in excess of 260 KTAS. However, in practice, it fell way short of that goal as the maximum airspeed was found to be some 20 KTAS less than that. While a part of the problem was that its engine could not develop the advertized power, this did not fully explain the large difference between practice and prediction. This simply implied its drag was higher than predicted – something common to other aircraft of the period as well. After all, that is why 11 aircraft, rather than a single one, were studied in the report. This highlights a scenario that anyone estimating the drag of a new airplane may find himself in – to assume the cleanliness of Airplane 1 in Figure 15-41, when in reality it will be closer to Airplane 18.

Such under-predictions are not limited to aircraft of the yesteryear. They still happen. A recently developed small twin-engine business jet was unable to meet the predicted performance advertised on the company website. This led to a costly redevelopment effort and eventually the airplane was shown to meet the promised capabilities. A small single-engine propeller aircraft developed by a prominent car manufacturer also didn’t meet the predicted performance, allegedly due to drag higher than predicted. Both aircraft were designed by people who cannot be accused of not knowing what they were doing. It shows that no matter one’s proficiency, under-prediction of drag is a likely scenario and the subject must be handled with utmost caution.

Going back to the NACA WR-L-489, the source presents a detailed listing of the changes made to the airplane presented in a graphic and tabulates their effect on the total drag. The graphic is reproduced as Figure 15-41 and the table is recreated as Table 15-8. It should be of great interest to the aircraft designer to inspect the table and consider how the contributions of seemingly insignificant modifications, such as a sanded walkway or an opened cowl flap, affect the overall drag of the airplane. The problem is not the value of individual component, but rather their cumulative effect.

15.5.2 Trim Drag

Trim drag is the penalty paid for providing static stability. Technically, trim drag is the combination of two sources.

(1) The difference in lift-induced drag of the airplane with and without balancing forces and (2) the increase in drag due to the deflection of the elevator (see Section 15.4.4, The Effect of Control Surface Deflection –Trim Drag). This is denoted using the variable . Its value typically ranges from 0.0000 to 0.0005 per degree of elevator deflection. It must be multiplied by the elevator deflection to trim, δ_e, to determine the additive drag coefficient.

If a longitudinally stable airplane could sustain steady level flight without having to generate a balancing force there would be no trim drag. The balancing force of a statically stable conventional tail-aft aircraft is added to the weight. This implies the total lift generated by the wing must be greater than the weight alone. This requires it to operate at a higher AOA and, thus, at a higher lift-induced drag than otherwise. It does not matter whether this stability is generated using a horizontal tail, such as that of the statically stable tail-aft configuration, or using airfoils and elevons, such as that of a flying wing; the total drag of the airplane is increased. This section presents a method to estimate the trim drag.

Since trim drag is a consequence of the longitudinal stability, it is easy to derive a complicated and unwieldy formulation that is of limited practical use. Ordinarily, trim drag constitutes a small fraction of the total drag of the airplane; it should range from 1% to 2%. During the design process, it is more practical to estimate the trim drag using a simple formulation and this is what will be demonstrated here. The same methodology can be revised to account for more complex situations. Note that the method assumes the airplane is operating at a low AOA, allowing for simplification.