ÍDICE DE SIGLAS Y ARCHIVO DE PAPELES DE TRABAJO

Planform geometry primarily affects the aircraft’s stability characteristics, passenger-payload capacity, and vortex drag. With the latter dominated by span, which is already fixed, atten-tion is restricted to the first two aspects.

Centrebody Width

The cabin/centrebody width is constrained by passenger comfort requirements. Consider the solution to a pure-rolling motion for a step aileron input (Nelson (1998))

dφ

dt =p_ss

1−e^−t/τR

. (6.1)

Equation 6.1 describes how bank angle varies with time for a given roll-mode time constant, where p_ss is the steady-state roll rate. On integrating Eq. 6.1, with the initial boundary condition thatφ= 0 at t= 0,

p_ss= φ_tg

t_tg−τ_R 1−e^−ttg/τR. (6.2) Here we see thatpss depends partly on the time-constant (τR), and partly on the bank angle (φtg) and time (ttg) within which a manoeuvre should be executed — MIL-F-8785C specifies 30^◦ and 5 s, respectively.

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

The roll-induced accelerationais

a = rd²φ dt²

= rp_ss 1 τR

e^−t/τR (6.3)

where r is the cabin half-width. The maximum roll acceleration (at t = 0), amax, and outboard distance, r, from the roll axis are thus related via

r= a_max d²φ

dt²

t=0

=a_maxτ_R

p_ss. (6.4)

Figure 6.1 shows the variation of cabin half-width with the time constantτRfor a range of maximum roll accelerations. Half-widths in the range of 5 – 30 m are expected over the range of vertical accelerations detailed. Furthermore, we see that, as τ_R becomes very large, the cabin half-width tends to a constant value. This is confirmed by considering the asymptotic solution to Eq. 6.2 and combining it with Eq. 6.4, showing that r→a_max2φ^t2tg

tg.

0 2 4 6 8 10 12 14 16 18 20

0 5 10 15 20 25 30 35

τR

r (m)

0.15g

0.05g 0.10g

Figure 6.1: Variation of cabin half-width with roll-mode time constant for a range of vertical accelerations.

For a cabin half-width of 10 m, roll-mode time constants in excess of 8 s are necessary to reduce the maximum vertical acceleration to a level recommended by Pratt (2000) of

0.05-g, at the cost of compromised flying qualities. Assuming a ‘flying plank’ geometry with the allowable weight uniformly distributed gives a roll moment of inertia (Ixx) of 59.8×10⁶ kg.m², which, combined withClp =−0.481 rad⁻¹(see Sec. 6.1.2.3), provides an approximate estimate for τ_R of 0.62 s through Eq. 6.5 (provided by Nelson (1998)).

lp = Clp

b 2U∞

QS_refb Ixx

(6.5) τ_R = −1

l_p

where Q = 1/2ρ∞U_∞² . This leads to a roll acceleration of 0.20-g. Therefore, the roll-mode time constant must be artificially increased. As a compromise, a slightly higher than recom-mended vertical acceleration of 0.06-g is accepted, with τ_R= 4.5 s.

To achieve an acceptable payload capacity the centrebody region is given a straight trailing edge. With the leading-edge sweep and cabin width now set, and chord length at the spanwise extremes specified by t/c= 0.20, the centrebody planform geometry is fully defined.

Stability Characteristics

In Chap. 2 it was revealed that, provided the C.G. lies ahead of the neutral point (NP), satisfactory longitudinal stability characteristics can be achieved; furthermore, despite large differences in the lateral stability derivatives relative to conventional aircraft, flying wings still exhibit satisfactory lateral-directional stability characteristics. Therefore, work here is restricted to investigating how the planform parameters affect static stability. Zero wing-twist and symmetrical aerofoil sections are assumed.

The C.G. position is not yet known. Therefore, given the low sweepback and hence danger of the NP being ahead of the C.G., a neutral point that is as far aft as possible is targeted.

For directional static stability, the lateral NP must also lie aft of the C.G. Furthermore, for lateral-directional stability, C_lβ <0 is a necessary condition.

To avoid adverse boundary-layer stability effects, the leading edge-sweep is fixed at 25^◦. With the centrebody region defined, two geometric parameters remain: outboard wing taper ratio and wingtip-fin height (assuming the fins are untapered and aligned perpendicular to the flow direction). Outboard wing taper ratios between 0.1 to 1, and wingtip-fin heights of 2 m to 5 m are considered. Figure 6.2 provides an illustration of the planform geometry for extreme combinations of these two parameters — notice that for higher outboard wing taper ratios the centrebody becomes more distinguishable.

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

(a)

(b)

Figure 6.2: Schematic diagram of aircraft planform geometry: (a) TR = 0.1 andHw = 5 m and (b)T_R= 1 and H_w = 2 m.

Longitudinal Neutral Point

Figure 6.3 shows the variation in longitudinal NP position over a range of outboard wing taper ratios and wingtip-fin heights. The neutral point has a strong dependence on taper ratio, moving further aft asTRincreases. For low taper ratios, wingtip-fin height has a weak effect, but becomes increasingly influential at higher taper ratios.

T_R Hw (m)

x_NP (m)

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

10 10.5 11 11.5

Figure 6.3: Variation in longitudinal NP position over a range of outboard wing taper ratios and wingtip-fin heights

For the geometric constraints imposed here, there is an increase in quarter-chord sweep angle associated with a high taper ratio, and hence the neutral point is further aft. At low taper ratios the effect of fin height is weak due to the reduced moment arm, despite the increase in tip loading with fin-tip height. To summarise, an aft longitudinal NP is favoured by a high taper ratio (higher quarter-chord sweepback) and large wingtip-fin height (increased tip loading).

Lateral Neutral Point

The lateral NP is defined as the location at which the sideforce due to sideslip effectively acts.

Equation 6.6 shows that it depends on the weathercock and sideforce stability derivatives,

6.1. LAMINAR-FLYING-WING DESIGN CONCEPT

and wing span

x_{lat,N P} −x_ref =−bC_nβ

C_Yβ, (6.6)

where x_ref is the point about which the stability derivatives are evaluated.

The effects of taper ratio and wingtip-fin height on the position of the lateral NP are shown in Fig. 6.4. The lateral NP moves further aft with increasing taper ratio due to the associated increase in sweepback; interestingly, this effect is strongest at low wingtip-fin heights. As for the longitudinal NP, wingtip-fin height effects become more apparent at higher taper ratios; however, somewhat counter-intuitively, the lateral NP moves further aft as wingtip-fin height decreases. For mid-range values of T_R, X_{lat,N P} travels forward then aftward with increasing Hw. Further examination of the dependence of the weathercock and sideforce stability derivatives on these parameters is required to understand these observations.

TR Hw (m)

xlat,NP (m)

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

18 18.2 18.4 18.6 18.8 19 19.2 19.4 19.6 19.8 20

Figure 6.4: Variation in lateral NP position over a range of outboard wing taper ratios and wingtip-fin heights

Figure 6.5(a) shows how Cn_β varies with TR and Hw. The yawing moment coefficient increases with wingtip-fin height due to the increased pressure force associated with a larger surface area. For a constant wingtip-fin height,C_nβ initially increases with taper ratio due to the increased sweepback and chord length (and hence area) combination; however, this ap-proaches a maximum for mid-range values ofTRand subsequently, though rather mildly, drops with further increases in T_R — this effect becomes weaker as H_w is reduced. Figure 6.5(b)

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