Weights
The wing weight estimate considers multiple load cases (Takeoff Gross Weight (TOGW), Zero Fuel Weight (ZFW), Taxi). Spars are sized according to bending loads, while non-structural weight is estimated empirically. Empirical estimates are used for control system weights, landing gear and instrumentation.

Drag
Viscous drag estimated by equivalent flat-plate skin-friction, and induced drag estimated using conventional advanced design methods. Compressible drag is computed based on expressions from R.T. Jones and JHB Smith for a yawed ellipse.

Engines
A rubberized, generic low-bypass turbofan engine model was used for estimating fuel burn and thrust at the given mission segments. Performance is on a par with that feasible by current levels of technology.

Trim
No trim drag penalties were modeled. It was assumed that the aircraft CG was located at the root section quarter-chord point at all times, presumably using fuel pumping, or some other balancing mechanism.

Geometry
A simplified parameterization was used, and the wing was assumed to be symmetric when the oblique sweep was set to 0. Parameters included:

• Reference area (S_ref)
• Aspect Ratio (AR)
• Symmetric, quarter-chord sweep (Λ_c/4)
• Trapezoidal wing taper (λ)
• Break location (η)
• Leading edge extension (LEX)
• Trailing edge extension (TEX)
• Airfoil t/c
• Oblique sweep

Other
Takeoff field length estimates are based on an empirical fit related to available thrust, weight, and aircraft stall speed. The stall speed estimate is in turn based on a user-specified value of wing CLmax. Fuel burn used for climb and acceleration to initial cruise conditions is estimated using energy methods.

Figure 6.1 Planform view of study configuration

Design variable	Minimum bound	Maximum bound
Oblique sweep (deg)	0.0	90.0
Sref (ft2)	10.0	1000.0
TOGW (lb)	500.0	10000.0
Sea Level Static (SLS) Thrust (lb)	10.0	3000.0
Initial cruise altitude (ft)	30000	65000
Final cruise altitude (ft)	30000	65000
Airfoil t/c	0.05	0.25

Table 6.2 Optimization design variables and associated bounds on values

Constraint	Minimum bound	Maximum bound
Initial cruise climb gradient	0.005	-
Final cruise climb gradient	0.005	-
Fuel weight < fuel volume	True	-
Cruise range (nmi)	1000.0	-
TOFL (ft)	-	5000

Table 6.3 Optimization constraints and associated bounds on values

The optimization objective in each case was to minimize the TOGW, unless otherwise noted.

The study consisted of four related configuration optimization cases:

1. TOGW was minimized while airfoil t/c was held constant; the TOFL constraint was disabled for this case.
2. TOGW was minimized while airfoil t/c was varied; the TOFL constraint was disabled for this case.
3. TOGW was minimized while airfoil t/c was held constant; the TOFL constraint was active for this case.
4. Range was maximized while TOGW and t/c were held constant; the TOFL constraint was active for this case.

In each case, the planform parameters (listed in Table 6.1) were held constant. For each study, the configuration was optimized at each of the following Mach numbers: 0.75, 0.85, 0.95, 1.2, 1.4, 1.6 and 1.8. Results computed are presented in the following section.

6.2 Results and Discussion

For all cases, the expected trend of decreasing L/D with increasing Mach number was observed (see Figure 6.2). When considering the closely related cases 1 and 3 (t/c fixed cases with TOGW as objective, with and without a TOFL constraint), differences in supersonic performance are negligible in the Mach 1.2-1.6 range, although there is a discrepancy at Mach 1.8 which is attributed to a poorly converged solution for the case with unconstrained field length (case 1).

For each of the supersonic study cases where a range requirement was set, fuel volume was invariably critical. In the cases where t/c was fixed, the optimizer responded by increasing the wing reference area to accommodate the mission fuel (see Figure 6.3). In the cases where t/c was allowed to vary, it was increased dramatically (see Figure 6.4), while wing area was decreased. Case 4, where the TOGW is fixed and no requirement is placed on range, supersonic performance is somewhat better in the Mach 1.4-1.8 range, no doubt the result of not having the wing size driven by the constraint on fuel volume. This latter case is probably best representative of the L/D values that could ideally be achievable, although as this study illustrates, real-world considerations may make them infeasible. The general increasing trend in required fuel is driven by the interaction between the progressively more powerful engines required to overcome the increase in wave drag with Mach number, and the larger aircraft volume required to fuel these engines.

For cases optimized for sub- and trans-sonic flight, those with a constrained TOFL have lower L/D, which can be attributed to the wing reference areas for these configurations no longer being optimally sized for efficient cruise. Instead, wing reference area is forced to increase in order to decrease the stall speed and thus the TOFL. On the other hand, the unconstrained cases are free to size the wing for optimal wing loading, and in turn have very poor field length performance (see Figure 6.5). It should be noted that weight savings due to changes in the wing structure were largely negligible due to the very small fraction of the TOGW that the wing weight represented.

For the cases with fixed t/c optimized for supersonic flight (cases 1 and 3), the TOFL was never critical. This was likely due to the cumulative effect of having engines sized for supersonic flight (providing ample thrust at the takeoff condition), coupled with the large wing area that resulted due to the fuel volume constraint. For the case with t/c as a design variable (case 2), the benefit of increased wing area was lost, since the optimizer in these cases favored the increase of t/c over an increase in S_ref. - engine thrust effect can be seen in case 4, Sref vs. M - "kink" in t/c fixed, w/TOFL vs M curve - fortuitous TOFL constraint met for case 4? - Why is the aircraft with super-thick t/c so good? Small wing area.

Figure 6.2 Plot of L/D vs. Mach number

Figure 6.3 Plot of wing reference area vs. Mach number

Figure 6.4 Plot of t/c vs. Mach number

Figure 6.5 Plot of TOFL vs. Mach number

Figure 6.6 Plot of TOGW vs. Mach number

(a)
(b)
(c)
Figure 6.7 Plots of C_Dc (a), C_Di (b) and C_Dp (c) vs. Mach number

Figure 6.8 Plot of oblique sweep vs. Mach number

Figure 6.9 Plot of normal Mach vs. Mach number

Figure 6.10 Plot of normal C_L vs. Mach number

Figure 6.11 Plot of range vs. Mach number