The AltSeq class

shows that the magnitude of the side-force stability derivative increases with tip-fin height, whilst it increases to a maximum and then drops withT_R for a givenH_w. Similar arguments for the effects on weathercock stability can also be applied to the side force. Therefore, it is because the sideforce stability derivative decreases at a faster rate than the weathercock stability derivative that, the lateral neutral point moves further aft with decreasing tip-fin height.

Planform Summary

With values for the longitudinal NP and lateral NP between 10 – 11.5 m and 18 – 20 m, respectively, the longitudinal NP position is more critical. Furthermore, C_lβ <0 is satisfied for all combinations of taper ratio and fin height. Values for taper ratio and wingtip-fin height below the maximum considered are selected for structural efficiency, therefore: T_R= 0.9 and Hw= 3.5 m.

The final planform design is summarised in Fig. 6.8. The reference area and span are 1087.9 m² and 80 m, respectively. The aspect ratio is 5.9. The centrebody has a half-span of 10 m and a quarter-chord sweep of 18.9^◦. The outboard quarter-chord sweep angle is 24.5^◦. The cruise and climb-out static margins, and longitudinal neutral points are detailed in Tab. 6.1. The aircraft exhibits close to neutral static stability characterisitcs over all flight phases of interest. Furthermore, the condition for spiral mode stability isCl_βCnr/ClrCnβ >1:

the LFW has values in the range 0.44 – 0.86. A flight control system is therefore required to provide neutral spiral stability characteristics.

Table 6.1: Static stability — C.G. locations (see Sec. 6.1.5.5): 11.38 m (start of cruise and climb out) and 11.44 m (end of cruise).

Parameter Cruise (with suction) Cruise (no suction) Climb out

M∞ 0.67 0.39 0.21

XN P 11.38 11.41 11.43

Static margin (%) 0/-0.5 0.2/-0.2 0.4/-0.1

ClβCnr/ClrCnβ 0.86 0.60 0.44

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

T_R Hw (m)

Cn β

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 2.5 3 3.5 4 4.5 5

0.01 0.015 0.02 0.025 0.03

(a)

TR Hw (m)

C_Y

β

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 2.5 3 3.5 4 4.5 5

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

(b)

Figure 6.5: Change in (a) yawing moment coefficient and (b) side force with sideslip angle over a range of outboard wing taper ratios and wingtip-fin heights.

The need for an aerofoil section exhibiting minimal nose-down pitching moment at zero-lift was used, as an aerodynamic balance condition, to justify the selection of a NACA 4-digit section in Chap. 5. These are low-speed aerofoils, with a maximum thickness close to the leading edge (Abbott and Von Doenhoff (1959)). A more sophisticated approach, that maximises internal volume and improves transonic performance, is required here.

Methodology

Figure 6.6 shows a bespoke section generator (developed by Eastwood (2008)), which requires six variables to define an aerofoil section geometry. The section generator first takes the suction surface of the leading edge of a RAE2822 supercritical section, whose extent is marked by the first control point CP0, to ensure good performance in this aerodynamically critical region (Drela and Giles (1987)); the remaining surface is fitted by a cubic B´ezier curve to the trailing edge, set by the control point CP3 whose vertical location lies alongx/c= 1 (y/c >0 leads to a blunt trailing-edge section). To ensure continuity between the adjoining sections, the second control point CP1 is constrained to move along ‘Track 1’, which is tangential to the surface at the join. By defining a trailing-edge wedge angle θte, the third control point CP2 is constrained to move along ‘Track 2’, which is tangential to the surface at the trailing edge. The curve is then mirrored to form a symmetric section, and expanded to the required thickness.

Figure 6.6: Aerofoil section generator.

To simplify the structural design task: a multi-bubble geometry comprising intersecting transverse cylinders is chosen for the pressurised cabin (see Sec. 6.1.5). The bubble overlap is chosen to: a) give a minimum height of 1.9 m, based on the statistic, in Torenbeek (1976), that 99% of passenger heights are below this value, and b) have a seat pitch of 80 cm to

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

90 cm, which Torenbeek (1976) cites as typical of conventional-type passenger carriers. A bubble diameter of 2.14 m is chosen, leading to a bubble pitch of 1 m. Each seat row is assumed to lie on a bubble axis; consequently, the number of bubbles sets the number of rows. The number of seats abreast is determined by assuming one aisle every three seats;

Torenbeek (1976) quotes values for seat and aisle widths of 42.5 cm and 50.8 cm, respectively, which give an effective seat width of 59.4 cm. The aerofoil section and number of bubbles are determined via trial and error:

a) multi-bubble arrangement arbitrarily defined between, and ahead, of the front and rear vertical spar positions (see Sec. 6.1.5);

b) aerofoil cross-section fitted to meet the various geometric constraints;

c) inviscid calculation using VGK in the perpendicular plane to ensure the flow is subcrit-ical, and viscous calculation to ensure attached flow in cruise and takeoff; if not, iterate from b).

Figure 6.7 shows a cross-section of a representative multi-bubble cabin embedded within a centrebody wing section. Markers are placed at the front and rear bubble locations to denote minimum clearance requirements for the placement of suction hardware components and structural elements. At the spar locations, the bubbles intersect at the same height to prevent localised bending moments. To reduce surface curvature over the rear, a blunt trailing edge geometry is utilised. In contrast, sharp trailing-edge wing sections are employed outboard of the cabin.

Figure 6.7: Schematic diagram of the multi-bubble cabin embedded within an aerofoil section

— vertical spar locations are represented by the dash-dot lines, whilst the minimum spacings between the cabin and wing surface are indicated by the ‘x’ markers.

Specification

Figure 6.8 details the final aerofoil and multi-bubble section geometries, and associated Cp

distributions calculated at the local section C_l values (see Sec. 6.1.2). Specific details of the aerofoil sections are provided in Tab. 6.2.

Table 6.2: LFW section geometry variables specification.

Parameter y = 0.84 m y = 4.20 m y = 9.18 m y = 10 - 40 m z = 0 - 3.5 m

(t/c)max 0.149 0.164 0.194 0.20 0.10

CP0X 0.03 0.03 0.08 0.10 0.10

CP1X 0.15 0.15 0.26 0.40 0.40

CP2X 0.91 0.90 0.80 0.85 0.85

CP3Y 0.012 0.008 0.002 0 0

θte (deg) 28 26 17 5 5

Along the central axis, the reduced maximum thickness-to-chord ratio permits the place-ment of bubbles ahead of the front spar, whilst a blunt trailing edge is required to alleviate flow conditions over the rear. Moving out across the centrebody, the (non-dimensional) rear spar location effectively moves forwards as the chord drops. This permits a lower trailing-edge wtrailing-edge angle and thickness. With the leading trailing-edge swept, the distance between the front spar and nose drops, therefore the number of bubbles ahead of the front spar decreases. To allow for a smooth transition to the outboard region, the outer centrebody section is scaled to the required t/c, and a sharp trailing-edge is enforced. The fin sections are scaled down versions of the outer sections witht/c= 0.10.

The total cabin floor area is around 138 m², but approximately 7 m² is required for items such as: wardrobes, toilets, etc., for a 200-passenger aircraft (Torenbeek (1976)). Therefore, the passenger specification for the LFW is220 persons.

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

Figure6.8:Threedimensionalviewofwinghalf-span,spanwisevariationsofaerofoilsectiongeometry,multi-bubblecabin arrangement,andCpdistributionatthecruise(withsuction)condition:centrebodyM⊥=0.63,outboardM⊥=0.61, wingtip-finM⊥=0.67;seeFig.6.9forCl⊥values.