Maximum Lift Prediction -- Specific Conceptual Design Method

When the distribution of lift is not computed, it is still possible to make a rough estimate of maximum lift capability. This section describes a simple method appropriate for early design of conventional aircraft.

Outer Panel Section C_lmax

One starts by estimating the section C_lmax of the outer wing panels. If the airfoil is known, this value may be based on experimental data or computations. A typical variation of section C_lmax with thickness for peaky-type transport aircraft airfoils is shown in figure 1. Note that outer panel airfoil thickness ratio is generally less than the average value. Assuming that the outer panel has a t/c about 90% of the average value is reasonable. The increase in C_lmax with thickness up to about 12%-15% reflects the larger nose radius of the thicker airfoils. Increased nose radius reduces the leading edge suction peak, the associated adverse pressure gradient, and the tendency to stall. Since supercritical airfoils have large nose radii, their C_lmax is about 0.1 greater than the conventional sections shown here.

Figure 1. Section C_lmax for Various Families of Airfoils.

The section C_lmax is also affected by Reynolds number. Some data on this effect is shown in figure 2. The effect of Reynolds number is sometimes very difficult to predict as it changes the location of laminar transition and boundary layer thickness. Thin airfoils are less Reynolds number sensitive, thick sections are more sensitive and show effects up to 15 million.

Figure 2. Effect of Reynolds Number

Recent experiments have suggested that, especially for slotted flap systems, significant variations with Reynolds number may occur even above Reynolds numbers of 6 to 9 million. But for initial design purposes, the variation of C_lmax with Reynolds number may be approximated by:
C_lmax = C_{lmax_ref} * (Re / Re_ref)^0.1

Relating Wing C_Lmax to Outer Panel C_lmax

The plot in figure 3 shows the ratio of wing C_Lmax to the section C_lmax of the outer wing panel as a function of wing sweep angle and taper ratio. This plot was constructed by computing the span load distribution of wings with typical taper ratios and twist distributions. The results include a reduction in C_Lmax due to tail download of about 0.05, a value typical of conventional aircraft; they also include a suitable margin against outer panel stall. (This margin is typically about 0.2 in C_l.)
When estimating the C_lmax of the wing outer panel, one should use the chord of the outer panel (typ. at about 75% semi-span) to compute the Reynolds number effect on that section.

Figure 3. Effect of Taper and Sweep on Wing / Outer Panel C_lmax

Additional corrections to wing C_Lmax

FAR Stall Speed
The formula for stalling speed given earlier in this section refers to the speed at which the airplane stalls in unaccelerated (1-g) flight. However, for the purposes of certificating a transport aircraft, the Federal Aviation Agency defines the stalling speed as the minimum airspeed flyable at a rate of approach to the stall of one knot per second. Slower speeds than that corresponding to 1-g maximum lift may be demonstrated since no account is taken of the normal acceleration. The maximum lift coefficient calculated from the FAA stall speed is referred to as the minimum speed C_Lmax or C_{Lmax_Vmin}. The increment above the 1-g C_Lmax is a function of the shape of the lift, drag, and moment curves beyond the stall. These data are not usually available for a new design but examination of available flight test data indicate that C_{Lmax_Vmin}averages about 11% above the 1-g value (based on models DC-7C, DC-8, and KC-135). A typical time history of the dynamic stall maneuver is shown in figure 4.

Figure 4. Typical Record of Dynamic Stall Maneuver Power-off Stall, Thrust Effect Negligible, Trim Speed 1.3 to 1.4 V_s, Wings Held Level, Speed Controlled by Elevator

FAR Stall C_L is value of C_Ls when DV/Dt = 1kt/sec and: C_Ls = 2W / S r V_s²

Figure 5. Flight Data showing FAA C_Lmax vs. C_Lmax based on 1-g flight.

Wing-Mounted Engines
The presence of engine pylons on the wings reduces C_Lmax. On the original DC-8 design, the reduction associated with pylons was 0.2. When the pylons are "cut-back" so they do not extend over the top of the leading edge, the reduction can be kept to within about 0.1 with respect to the best clean-wing value.

Increment in C_Lmax Due to Slats
When leading edge slats are deployed, the leading edge pressure peak is suppressed. The introduction of a gap between the leading edge device and the wing leading edge increases the energy of boundary layer above what it would have been without a gap. For this reason, the section lift coefficient is increased dramatically. The specific amount depends on the detailed design of the slat, its deflection, and the gap size. For the purposes of our preliminary design work, the value is estimated based on Douglas designs shown in figure 6. The effect of sweep reduces the lift increment due to slats by the factor shown in figure 7. A better method would include the observation that when leading edge devices are employed, the favorable effect of nose radius (and increased t/c) would not be realized. Although this data applies for 5 deg of flap deflection, this slat increment can be used for preliminary estimates at all flap angles.

Figure 6. Effect of slat deflection on C_lmax increment due to slats. Prediction based on maximum Mach number constraint. This data is for a 17% slat.

Figure 7. Effect of wing sweep on slat maximum lift increment.

Increment in C_Lmax Due to Flaps
A simple method for estimating the C_Lmax increment for flaps is described by the following expression. It is highly approximate and empirical, but the next level of sophistication is very complex, and sometimes not much more accurate.

DC_{Lmax_flaps}= S_wf / S_ref DC_{Lmax _flaps}K(sweep)

where:
S_wf = wing area affected by flaps (including chord extension, but not area buried in fuselage)
S_ref = reference wing area
DC_{lmax_flaps} = increase in two-dimensional C_lmax due to flaps
K = an empirical sweepback correction

The wing area affected by flaps is estimated from a plan view drawing. Typical flaps extend over 65% to 80% of the exposed semi-span, with the outboard sections reserved for ailerons. The resultant flapped area ratios are generally in the range of 55% to 70% of the reference area. (See table at the end of this section.)

DC_{lmax_flaps} is determined empirically and is a function of flap type, airfoil thickness, flap angle, flap chord, and sweepback. It may be estimated from the expression:

DC_{lmax_flaps} = K1 K2 DC_{lmax_ref}
DC_{lmax_ref} is the two-dimensional increment in C_lmax for 25% chord flaps at the 50 deg landing flap angle and is read from the experimentally-determined curve below at the mean thickness ratio of the wing.

Figure 8. Section C_lmax increment due to flaps. The results are for double slotted flaps. For single slotted flap multiply this value by 0.93. For triple slotted flaps, multiply by 1.08.

K1 is a flap chord correction factor. It includes differences between the flap chord to wing chord ratio of the actual design to that of the reference wing with 25% chord flaps.

Figure 9. Effect of Flap Chord.

K2 accounts for the effect of flap angles other than 50 deg.

Figure 10. Flap Motion Correction Factor

K(sweep) is an empirically-derived sweep-correction factor. It may be estimated from:
K = (1-0.08*cos²(Sweep)) cos^3/4(Sweep)

Effect of Mach Number
The formation of shocks produces significant changes in the airfoil pressure distribution and limits the maximum lift coefficient. In fact, a strong correlation exists between the C_lmax of a slat and the C_l at which flow near the slat becomes supersonic. In general, as the freestream Mach number is increased, the aircraft C_Lmax is reduced. The figure below shows this effect for the DC-9-30.

Figure 11. Effect of Mach number on maximum lift.

As a first approximation this data can be used to estimate the effect for another aircraft as follows:
C_Lmax(M) = C_{Lmax_l.s.} * C_{Lmax_ref} (M') / C_{Lmax_l.s.ref}

Where:
C_{Lmax_l.s.} is the C_Lmax at low speed (Mach number < 0.3)
and M' = Modified Mach number based on equivalent normal Mach = M*cos(sweep) / cos(DC-9sweep),
where the DC-9, which provides the reference data here, has a sweep of 24.5 deg.

The final figures show the approximate C_Lmax values for a number of aircraft.

Figure 12. C_Lmax Values for a variety of transport aircraft.

Airplane	Swf / Sref	Flap Type	Flap Chord Ratio	Sweep (deg)
DC-3S	0.575	Split	0.174	10
DC-4	0.560	Single Slot	0.257	0
DC-6	0.589	Double Slot	0.266	0
DC-7C	0.630	Double Slot	0.266	0
DC-8	0.587	Double Slot	0.288	30.5
DC-9-30	0.590	Double Slot	0.360	24.5
DC-10-10	0.542	Double Slot	0.320	35

Figure 13. Effect of Flap and Slat Deflections on C_Lmax for several Douglas airplanes. The results are based on the FAA measured stall speeds and reflect the 1 kt/sec deceleration.