Lesson 3 · 40 min

Estimating L/D from the Sketch

The Breguet equations need a lift-to-drag ratio, but there is no drag analysis yet: only a sketch. Fortunately, the sketch already holds the two numbers that matter most, the span and the wetted area. Together with a skin-friction coefficient typical of the class, they give an \(\LDmax\) good enough to size the aircraft, long before a full drag build-up is possible.

Learning objectives

Estimate the wetted area of wings, tails and fuselage from a sketch.
Estimate \(\CDz\) with the equivalent skin-friction method, \(\CDz = C_{fe}\Swet/\Sref\).
Compute the wetted aspect ratio and \(\LDmax = \tfrac12\sqrt{\pi e A_{\text{wet}}/C_{fe}}\).
Choose the \(L/D\) for cruise and loiter for jet and propeller aircraft.

Wetted area from the sketch

Zero-lift drag at subsonic speeds is mostly skin friction, so it grows with the area the air touches: the wetted area \(\Swet\). Quick estimates come from the views of a sketch:

Wetted area (quick approximations)

\[ \text{Wings and tails:}\ \ \Swet \approx S_{\text{exposed}}\left(1.977 + 0.52\,\tfrac{t}{c}\right) \] \[ \text{Fuselage:}\ \ \Swet \approx 3.4\,\frac{A_{\text{top}} + A_{\text{side}}}{2} \]

\(S_{\text{exposed}}\) is the planform area outside the fuselage; a little more than twice this area is wetted (top and bottom, plus a thickness correction). \(A_{\text{top}}\) and \(A_{\text{side}}\) are the projected areas of the fuselage in the top and side views. Add nacelles, pods and external stores the same way.

(a) Exposed planform

Area outside the fuselage, in the top view

(b) Wing or tail section

Both sides wetted, plus a little for thickness

Projected areas of the top and side views

Figure 3.1 What the approximations measure. (a) Only the planform outside the fuselage, \(S_{\text{exposed}}\), is wetted; the reference area \(\Sref\) also counts the hatched part inside. (b) The air wets both surfaces, each a little longer than the chord, so \(\Swet\) is just over twice \(S_{\text{exposed}}\): the factor is \(1.977 + 0.52\,t/c\), or 2.04 at \(t/c = 0.12\). (c) The fuselage is wrapped around its top and side views. For a circular cylinder of diameter \(d\), the wetted area \(\pi d L\) is exactly \(\pi\) times the average projected area \(dL\); the 3.4 is an empirical value that fits real fuselages.

Example 3.1 — The business jet's wetted area

The sketch shows a wing of \(\Sref = 28\ \text{m}^2\) and aspect ratio 8, of which 85% is exposed, with \(t/c = 0.12\); exposed tails of \(6.0\ \text{m}^2\) (horizontal) and \(4.5\ \text{m}^2\) (vertical) with \(t/c = 0.10\); a fuselage \(14\ \text{m}\) long and \(1.8\ \text{m}\) in diameter whose top and side views each cover 85% of the bounding rectangle; and two nacelles of \(6\ \text{m}^2\) wetted area each. Find the total wetted area and \(\Swet/\Sref\).

Top view

Side view

Wing: \(\Sref = 28\ \text{m}^2\), \(A = 8\), 85% exposed, \(t/c = 0.12\)
Tails: \(6.0 + 4.5\ \text{m}^2\) exposed, \(t/c = 0.10\)
Fuselage: \(14 \times 1.8\ \text{m}\), each view 85% of the rectangle
Nacelles: two, \(6\ \text{m}^2\) wetted each

Figure 3.2 The business jet of Example 3.1, drawn to scale from the data in the example (the wing taper and sweep are illustrative). Each colour is one line of the wetted-area build-up.

Show solution

\[ \text{Wing: } 0.85(28)(1.977 + 0.52(0.12)) = 23.8(2.039) = 48.5\ \text{m}^2 \] \[ \text{Tails: } (6.0 + 4.5)(1.977 + 0.052) = 21.3\ \text{m}^2 \] \[ A_{\text{top}} = A_{\text{side}} = 0.85(14)(1.8) = 21.4\ \text{m}^2, \qquad \text{fuselage: } 3.4(21.4) = 72.8\ \text{m}^2 \]

Wetted area build-up
Component	\(\Swet\) (m²)	Share
Wing	48.5	31%
Tails	21.3	14%
Fuselage	72.8	47%
Nacelles	12.0	8%
Total	154.7	100%

Wing 31%14%Fuselage 47%8%

\[ \frac{\Swet}{\Sref} = \frac{154.7}{28} = 5.52 \approx 5.5 \]

Almost half the wetted area of a business jet is fuselage: a small aircraft must still have a cabin people can sit in. That is one reason small jets have lower \(L/D\) than airliners.

Zero-lift drag: the equivalent skin friction

Real aircraft have more drag than flat-plate skin friction alone: form drag, interference, gaps, antennas, leakage. These are all lumped into an equivalent skin-friction coefficient \(C_{fe}\), measured on existing aircraft of each type:

Equivalent skin-friction method

\[ \CDz = C_{fe}\,\frac{\Swet}{\Sref} \]

Equivalent skin-friction coefficients for subsonic flight (source: Raymer, Table 12.3)
Type of aircraft	\(C_{fe}\)

Light aircraft have higher \(C_{fe}\) than airliners: fixed landing gear, exposed engines, less careful finish and, above all, a lower Reynolds number, at which skin friction is higher.

The wetted aspect ratio

Substitute \(\CDz = C_{fe}\Swet/\Sref\) into Module 1's \(\LDmax = \tfrac12\sqrt{\pi A e/\CDz}\), with \(A = b^2/\Sref\). The reference area cancels:

Maximum \(L/D\) from the sketch

\[ A_{\text{wet}} = \frac{b^2}{\Swet} = \frac{A}{\Swet/\Sref}, \qquad \LDmax = \frac12\sqrt{\frac{\pi e\,\colL{A_{\text{wet}}}}{C_{fe}}} \]

Span gives lift efficiently (induced drag falls with \(b^2\)); wetted area costs friction drag. Their ratio is what matters. Take \(e\) about 0.75 to 0.85 for a first estimate (Module 9 estimates it properly).

Plotted against \(A_{\text{wet}}\), the \(\LDmax\) of real aircraft falls in a band for each class, and a first estimate can be read off the band. The formula above is the physics behind those bands; it is a little optimistic because it ignores trim, compressibility and the drag of things the sketch does not show, so many designers take a few percent off.

Example 3.2 — The business jet's \(L/D\)

With \(A = 8\), \(\Swet/\Sref = 5.5\), \(C_{fe} = 0.0030\) (civil transport) and \(e = 0.75\), find \(A_{\text{wet}}\), \(\CDz\), \(\LDmax\), and the \(L/D\) to use in cruise.

Show solution

\[ A_{\text{wet}} = \frac{8}{5.5} = 1.455, \qquad \CDz = 0.0030(5.5) = 0.0165 \] \[ \LDmax = \frac12\sqrt{\frac{\pi(0.75)(1.455)}{0.0030}} = \frac12\sqrt{1142} = 16.9 \]

A jet gets its best range at \(L/D = 0.866\,\LDmax\) (Module 1, Lesson 6), so cruise uses \(0.866(16.9) = 14.6\). The 45-minute loiter reserve uses \(\LDmax = 16.9\).

Figure 3.3 \(\LDmax\) against wetted aspect ratio, \(\tfrac12\sqrt{\pi e A_{\text{wet}}/C_{fe}}\), for each type of aircraft in the \(C_{fe}\) table (gray), with the chosen type in blue. Set the aspect ratio and the wetted-area ratio of a sketch to place it. A longer span helps; a bigger fuselage for the same wing hurts.

Which \(L/D\) for cruise and loiter

An aircraft does not fly every segment at \(\LDmax\). What it maximizes depends on how its fuel consumption varies with speed:

\(L/D\) for the Breguet segments
Engine	Cruise (best range)	Loiter (best endurance)
Jet	\(0.866\,\LDmax\)	\(\LDmax\)
Propeller	\(\LDmax\)	\(0.866\,\LDmax\)

A jet burns fuel in proportion to thrust, so it loiters where drag is least (\(\LDmax\)) and cruises a little faster, where speed times \(L/D\) is greatest. A propeller engine burns fuel in proportion to power, drag times speed, so it cruises at \(\LDmax\) and loiters slower, where power is least, at \(0.866\,\LDmax\). Lesson 4 shows why from the fuel consumption.

Check your understanding

Key takeaways

Wings and tails: \(\Swet \approx S_{\text{exposed}}(1.977 + 0.52\,t/c)\); fuselage: \(\Swet \approx 3.4(A_{\text{top}} + A_{\text{side}})/2\).
\(\CDz = C_{fe}\Swet/\Sref\), with \(C_{fe}\) about 0.0030 for transports and 0.0055 for light singles.
\(A_{\text{wet}} = b^2/\Swet\) and \(\LDmax = \tfrac12\sqrt{\pi e A_{\text{wet}}/C_{fe}}\).
Jets cruise at \(0.866\,\LDmax\) and loiter at \(\LDmax\); propeller aircraft the other way round.
Next, Lesson 4 estimates the propulsive ingredient, \(C\).