Assessment of Off-Design Low-Boom Characteristics of a Low-Boom Supersonic Transport

Assessment of Off - Design Low - Boom Characteristics

o f a Low - Boom Supersonic Transport

* † Wu Li and Karl Geiselhart NASA Langley Research Center, Hampton, Virginia 23681, USA Most conceptual d esign methods for low - boom supersonic aircraft are developed for achieving a n undertrack low - boom goal at the start of cruise. A lthough a ttempts have been made to design a supersonic aircraft achieving a low - boom goal for multiple azimuthal angles at the start of cruise , o n e unanswered question in inverse design optimization is how to define an achievable low - boom goal for multiple azimuthal angles. This paper proposes a new analysis method to assess off - design low - boom characteristics of a low - boom supersonic transport designed for an undertrack low - boom goal at the start of cruise. This method opens a potentially feasible path for conceptual design of a supersonic aircraft that has desired low - boom characteristics over the primary boom carpet for the entire cruise segment.

Nomenclature A = any equivalent area including A and A , ft e e,CFD e,r A (  ) = equivalent area for azimuthal angle  , ft e target 𝐴 (  ) = selected Pareto point as the target for A of a supersonic concept for azimuthal angle  , ft e,r e , r A = equivalent area defined using Mach angle cut method and computational fluid dynamics, ft e,CFD A (  ) = Pareto optimal equivalent area target for a pair of cruise condition and azimuthal angle  , ft e, pareto A = reversed equivalent area defined by using reverse propagation of off - body pressure , ft e,r A (  ) = reversed equivalent area for azimuthal angle  , ft e,r d = vector of shape parameters for A (  ) e dp/p = nondimensional overpressure ( p – p )/ p ∞ ∞ D , D = vector of design variables that define a super s onic aircraft l = effective length of supersonic configuration, which is the largest effective distance where Mach e angle cut plane intersects the configuration, ft p = longitudinal pressure distribution at off - body location, lb/ft p = ambient pressure, lb/ft ∞ PLdB( A ) = perceived level in decibels for ground signature of A e e t = variable for a parametric Bezier curve x = effective distance for A , ft e e є = acceptable error tolerance for inverse design of A , ft e,r  = azimuthal angle for sonic boom propagation , deg 𝑙 e 2 ‖ ‖ ‖ ‖ | ( ) |  = 2 - norm of a function f on interval [0, l ] defined by 𝑓 , which equals ( ∫ 𝑓 𝑡 𝑑𝑡 / 𝑙 ) e e I. Introduction H E ban on overland supersonic flights is a key barrier for the development of commercial supersonic aircraft. To

T

help lift th is ban, NASA has been developing low - boom technologies that will enable a commercially successful supersonic transport to fly overland with a quiet s onic thump on the ground , instead of a loud sonic boom . The NASA N+3 low - boom goal [1] is that a commercial supersonic transport will only create sonic boom ground signatures with perceived level s in decibels (PLdB ) [ 2 ] below 70, which is significantly lower than the sonic boom loudness values exceeding 100 PLdB for Concorde (see page 6 of Ref. [1]) . The X - 59, NASA’s quiet supersonic experimental aircraft, has been designed using computational fluid dynamics (CFD) based low - boom shaping technologies to demonstrate * Senior Research Engineer, Aeronautics Systems Analysis Branch † Aerospace Engineer, Aeronautics Systems Analysis Branch a sonic thump of about 75 PLdB on the ground [3,4] . Although low - boom shaping technologies have reasonably matured, a simultaneous achievement of design goals for aerodynamic efficiency and low - boom shaping of the outer mold line (OML) of a commercial supersonic transport is still extr emely challenging. Our recent work using a multiobjective multidisciplinary optimization (MDO) method called the block coordinate optimization (BCO) method demonstrated success on that front in the conceptual design phase . The BCO method dissects a multiobjective low - boom MDO problem into several weakly coupled and solvable optimization subproblems . These optimization subproblems are solve d successively and repeatedly to find a low - boom supersonic transport that also satisfies the specified missio n performance requirements [ 5,6 ] , even when an aeropropulsive CFD simulation is required for sonic boom analysis . The “ low - boom ” qualifier in this paper means the potential to attain an undertrack sonic boom ground noise level below 70 PLdB at the start of overland cruise (SOC) . It is quantified by a close match between the reversed equivalent area ( A ) [ 7 ] of the supersonic aircraft and an A target with an undertrack sonic boom ground noise level e,r e ,r below 70 PLdB. The sonic boom ground noise limit of 70 PLdB is based on the NASA N+3 low - boom goal [1], which could be replaced by another limit if needed. The specified mission performance requirements include a l ow - boom overland mission for airport pairs over the continental US and an unrestricted overwater mission for transatlantic commercial flights.

A chieving low - boom ground noise level goal in t he undertrack direction at SOC is only the first step in the deve lopment of a low - boom supersonic transport. In reality, a low - boom supersonic transport must achieve a low - boom ground noise level goal in all azimuthal directions for the entire mission trajectory . Ueno and Makino [ 8 ] developed a robust design method that starts with an A target for each selected off - track direction that reduces the e ,r off - track ground noise level. Then they used a conversion and blending process to generate one target for local shape modifications that might lead to a n improvement of the low - boom characteristics in five azimuthal directions. Ordaz, Wintzer, and Rallabhandi [ 9 ] used two inverse design targets for dp / p at two body lengths away from the aircraft in two azimuthal directions of 0 and 25 deg to demonstrate a robust design method . These dp / p targets were generated using an adjoint - based optimization method [1 0 ]. For inverse design optimization at multiple azimuthal angles , one unanswered question is the feasibility of simultaneously matching the specified low - boom inverse design targets in the selected azimuthal directions. If these inverse design targets cannot be theoretically achieved simultaneously by a supersonic aircraft, then only different trades of the inverse design errors are possi ble between the selected azimuthal directions, no matter what optimization method is used to minimize the inverse design errors for the selected azimuthal directions. If the specified inverse design targets in different azimuthal directions ha ve conflictin g shape modification requirements , any optimal solution w ill result in a significant inverse design error for some azimuthal direction, which might lead to a significantly higher sonic boom ground noise level than the desired goal in that azimuthal directi on.

This paper continues to push the state of the art for conceptual low - boom design and initiates a study on a n inverse design method that can potentially ensure a simultaneous achievement of a given low - boom ground noise level goal over the primary boom carpet for the entire cruise segment. A new analysis method is proposed for assessment of low - boom characteristics of a low - boom supersonic aircraft at any off - design condition. It can be considered as the first step towards developing a low - boom inverse design method that can refine a low - boom configuration to simultaneously achieve a low - boom goal for a given set of off - design conditions. For each off - design condition, the corresponding A is computed. A multiobjective optimization problem is formulated to compute an A target with e ,r e ,r the lowest sonic boom ground noise level that can be approximately matched by the A for the off - design condition.

e ,r Th e potential application of this analysis method is to predetermine a set of A targets for the specified cruise e ,r conditions and azimuthal directions that can be used for a simultaneous reduc tion of the inverse design errors.

The paper is organized as follows. Section II describes the numerical method used for sonic boom ground signature analysis. A published low - boom supersonic transport concept is reanalyzed with 46 passengers in Sec. III , which will be referred to as the Mach 1.7 46 - PAX concept . The m ethod for a ssessment of l ow - b oom c haracter istics is introduced in Sec. IV. Section V is for assessment of off - design low - boom characteristics of the Mach 1.7 4 6 - PAX concept. A potentially feasible path for robust low - boom inverse design optimization is briefly discussed in Sec. VI and the conclusi ons are given in Sec. VII.

II. Sonic Boom Analysis Method The sonic boom analysis in this paper uses the standard method documented in the Sonic Boom Prediction Workshop III [ 4,11 ] : dp/p at three body lengths (3BL) away from the aircraft in a given azimuthal direction is used as the boundary condition of an augmented Burgers equation for propagation through the atmosphere to the ground .

The Euler CFD solver Cart3D [ 1 2 ] is used for all CFD analyses in this paper. An automated meshing process [ 1 3 ] for Cart3D is used to generate a mesh to compute off - body pressure s at 3BL away from the aircraft in any azimuthal direction . The azimuthal angle of 0 deg is the undertrack direction (see the top right insert in Fig. 1 ). For off - body pressure calculation usin g a Cartesian mesh, the aircraft geometry is first rotated around the pitch axis so that the shocks below the aircraft on the symmetry plane are approximately parallel to the z - axis of the Cartesian mesh. Then the aircraft geometry is rotated around the ro ll axis so that y = 0 corresponds to the specified azimuthal direction (see the rotated aircraft geometry inserts in Fig. 1 ). With engines modeled as flow - through nacelles, t his results in a mesh of approximately 30 and 72 million cells for undertrack (  = 0 deg) and off - track (  > 0 deg) pressure calculation , respectively, at 3BL away from the aircraft . The smaller mesh size for the undertrack pressure calculation is due to the symmetry of the flow field with respect to y = 0, which means that only a mesh in y  0 is required for Cart3D analysis. Each mesh has about 275 grid points to define the pressure at 3BL away from the aircraft in the specified azimuthal direction . For visualization of the off - body pressure solution, the Cartesian mesh is rotated around the y - axis to realign the flow direction horizontally as shown in Fig. 1 , which illustrates an off - body pressure solution at 3BL away from the aircraft for azimutha l angle of 30 deg.

Fig. 1 Rotated view of Cartesian mesh for off - body pressure calculation in azimuthal direction .

The sonic boom ground signature is computed by solving an augmented Burgers equation [14 - 1 8 ] to propagate the off - body pressure at 3BL away from the aircraft in an azimuthal direction to the ground through the standard atmosphere. The humidity profile for the standard atmosphere is based on ANSI S1.26 [ 1 9 ]. T he PLdB value [2] of the ground signature is used to measure the sonic boom loudness on the ground in the azimuthal direction at a given cruise condition , which is defined by the cruise Mach, cruise altitude, and angle of attack (AoA) of the aircraft.

An A target is an ideal shape of A for a supersonic configuration to achieve a low sonic boom ground noise e ,r e,r level . The calculation of a ground signature of an A target requires an additional step of converting the A target to e ,r e ,r dp/p , which is based o n a formula by Whitham that approximates a supersonic configuration by a body of revolution for sonic boom analysis ( see E qs. (14), (69), and (70) in Ref. [ 20 ]) . The approximation formula for convert ing any A e to dp/p at 50 ft away from the underlying supersonic configuration represented by A is E q. (6) in Ref. [ 2 1 ] , which e simplifies Whitham’s formula by removing the nonlinear correction term for longitudinal location . The distance of 50 ft is based on the best practic es for computing the ground signature of A . The numerical formulas for solving Eq. (6) e in Ref. [2 1 ] are Eqs. (12) and (13) in Ref. [2 2 ].

A solver of an augmented Burgers equation, sBOOM 2.8.6 [15], was updated as sBOOM 2.9.0 using a fully conservative an d dual - consistent numerical scheme [16]. Due to initial discrepancies of the two versions of sBOOM that were caused by a bug in a prerelease version of sBOOM 2.9.0 , we were concerned about the accuracies of our previous low - boom design results in Refs. [5, 6] based on sBOOM 2.8.6 . So, another solver [17] of the augmented Burgers equation in PCBoom 7.3 [ 18 ] was used to verify the accuracy of the computed ground signature. Even though the three solvers use different numerical methods to solve the same augmented Burgers equation, the computed ground signatures from these solvers are nearly identical for every test case we have computed . This provides an indirect assurance that these solvers solve the same augmented Burgers equation consistently . Figure 2 shows the dp/p at 90 ft below a supersonic configuration with body length of 90 ft, along with the ground s ignatures for dp/p computed by sBOOM and PCBoom. The cruise altitude is 50,397 ft and the cruise Mach is 1.4. For each solver , the input control parameters are selected to yield a high - resolution solution of the ground signature with negl igible differences of the analysis results if the signature resolution is further refined. The ground signatures in Fig. 2 have 40,001 and 32,768 data points from sBOOM and PCBoom, respectively. Because the three solvers generate nearly identical ground signatures with the same PLdB value , we use sBOOM 2.8.6 for sonic boom analysis in this paper.

Fig. 2 Comparison of ground signatures generated by three solvers of augmented Burgers equation .

III. Low - Boom Supersonic Concept with Low - Boom Mission Profile The Mach 1.7 4 6 - PAX concept in Fig. 3 is selected for demonstration of the proposed analysis method for assessment of off - design low - boom characteristics . The Mach 1.7 46 - PAX concept is derived from a low - boom concept with 40 pas sengers , referred to as the Mach 1.7 40 - PAX concept [6] , which was designed to approximately match an A target of 69.9 PLdB in the undertrack direction at SOC [5]. The Mach 1.7 46 - PAX concept has the same e,r OML as the Mach 1.7 40 - PAX concept . The low - boom cruise altitude and maximum takeoff gross weight of the Mach 1.7 46 - PAX concept are derived iteratively to ensure that i) its AoA at SOC is 2.3 deg (which is the AoA for the Mach 1.7 40 - PAX concept at SOC), and ii) its undertrack A matches an A target with an undertrack sonic boom ground e,r e,r noise level below 70 PLdB at SOC (see the PLdB value of the target signature in Fig. 4).

Fig. 3 Mach 1.7 46 - PAX low - boom concept.

Fig. 4 Equiv alent areas and ground signatures of Mach 1.7 46 - PAX concept in undertrack direction at SOC .

For any Euler solver and fixed OML with flow - throw nacelles , an aircraft’s undertrack dp/p at 3BL below the aircraft is invariant with respect to altitude change (see Fig. 5 a ) as long as the AoA (or lift coefficient) and cruise Mach number are fixed. So, if the undertrack A of a supersonic aircraft is an acceptable inverse design of an A e,r e,r target with groun d noise level of 69.9 PLdB, then this aircraft has the same undertrack A at any higher altitude with e,r the same AoA . Moreover, the same A target at the higher cruise altitude also has a ground noise level below 69.9 e,r PLdB. For example, the Mach 1.7 46 - PAX concept with AoA of 2.3 deg is also an acceptable inverse design for the same A target in Fig. 4 with undertrack sonic boom ground nois e level of 67.4 PLdB at the cruise ceiling of 60,000 e,r ft (see Fig. 5 b ).

Fig. 5 dp/p for various altitudes and undertrack signatures at cruise ceiling when AoA = 2.3 deg .

Fig. 6 Low - boom mission profile .

Table 1 Mach 1.7 4 6 - PAX concept Variable Value Variable Value Seat pitch, in 38 Low - boom angle of attack, deg 2.30 Max takeoff gross weight , lb 144, 760 Altitude at start of overland cruise, ft 5 1 , 7 40 Zero fuel weight, lb 7 5 ,6 83 PLdB of low - boom inverse design target 69.9 Overwater cruise Mach 1.8 Weight at start of cruise, lb 13 6 , 6 4 3 Overland cruise Mach 1.7 Block fuel for maximum range, lb 64,655 Takeoff field length, ft 8 504 Overwater range, nm 3 671 Approach velocity, kts 1 2 6 Overland range, nm 3 7 2 5 So, the overland mission profile with a fixed low - boom cruise altitude (see Fig. 5 in Ref. [6]) is replaced by the low - boom mission profile in Fig. 6 for assessment of off - design low - boom characteristics and analysis of the mission performance. This low - boom mission profile allows a low - boom concept to retain the designed low - boom characteristics at SOC as much as possible during the cruise. It also improves the overland cruise performance of a low - boom concept due to the inc reasing cruise altitude. The weight and performance data of the Mach 1.7 46 - PAX concept using the low - boom mission profile in Fig. 6 are listed in Table 1. If the mission profile for overland flight used a fixed cruise altitude of 51,740 ft to analyze the overland performance, then its overland range would be 3314 nm instead of 3725 nm using fixed AoA.

IV. Method for Assessment of Low - Boom Characteristics In this paper, the low - boom characteristics of a supersonic aircraft is defined by the sonic boom ground noise level of a n A target that can be approximately matched by the aircraft’s A in an azimuthal direction at a given cruise e,r e,r condition of the low - boom mission profile in Fig. 6 . The A of the aircraft for each azimuthal direction is obtained by e,r reversely propagating the off - body pressure at 3BL to 50 ft away from the aircraft (see Fig. 1 ) using an augmented Burgers equation in the azimuthal direc tion, converting the reversely propagated pressure to Whitham’s F - function, and calculating the corresponding equivalent area of Whitham’s F - function. The details for calculating the A e,r in the undertrack direction can be found in Ref. [ 2 3 ]. The same meth od works for A in any azimuthal direction after rotating e,r the aircraft geometry as shown in Fig. 1 . A supersonic aircraft typically has different A shapes for different azimuthal e,r directions .

The assessment of low - boom characteristics of a supersonic aircraft in an azimuthal direction of  deg is formulated as the following multiobjective optimization problem for simultaneous minimization of the ground noise level of an A target, denoted by A (  ) , and the difference between A (  ) and the aircraft’s A , denoted by ‖ 𝐴 (  ) − 𝐴 (  ) ‖ .

e,r e e e,r e e , r min { PLdB [ 𝐴 (  ) ] , ‖ 𝐴 (  ) − 𝐴 (  ) ‖ } ( 1 ) e e e , r 𝒅 The notation PLdB[ A (  )] represents the PLdB value of the ground signature of A (  ) in the azimuthal direction of  e e deg. The A (  ) in Eq. (1) is parameterized by the control points of a Bezier curve . For 0 ≤ t ≤ 1, the corresponding e point ( x , A ( x ,  )) on a Bezier curve is defined as follows.

e 9 !

9 𝑗 9 − 𝑗 ( ) 𝑥 𝑡 = 𝑙  𝑡 + ∑ 𝑥   𝑡  ( 1 − 𝑡 ) 𝑒 𝑐 , 𝑗 ( ) 𝑗 ! 9 − 𝑗 !

𝑗 = 1 ( 2 ) 9 !

9 𝑗 9 − 𝑗 ( ( ) ) 𝐴 𝑥 𝑡 ,  = 𝑦  𝑡 + ∑ 𝑦   𝑡  ( 1 − 𝑡 ) e 𝑐 , 8 𝑐 , 𝑗 𝑗 ! ( 9 − 𝑗 ) !

𝑗 = 2 In this paper, a Bezier curve with 10 control points is used as a parametric form of A (  ). In Refs. [5,6,2 3 ] , t he e parametric form of an A target has only 8 control points. However, the George - Seebass - Darden low - boom inverse e,r design theory [ 2 1 ,2 4 ] implicitly depends on the validity of the inverse relation between an equivalent area and an F - function (see Theorems 1 and 2 in Ref. [ 2 2 ]) . This inverse relation is valid if and only if the slop e of the equivalent area is zero at x = 0 and x = l (see Remark 2 after Theorem 1 in Ref. [ 2 2 ]). So, theoretically, the slope of an A e e e e,r target must be zero at x = 0 and x = l , even though minor deviation s of the slope at x = 0 and x = l from zero could e e e e e e be considered as acceptable numerical errors for low - boom inverse design of A . To be consistent with the theory in e,r Ref. [2 2 ] , two additional control points are added to enforce the zero - slope constraint s at x = 0 and x = l . The e e e param etric form of A (  ) in Eq. (2) always satisfies the zero - slope constraint at x = 0 and x = l . To keep the number e e e e of design variables for an A target the same as before, x and x are fixed with x = 0.00 1  l and x = 0.99 9  l in e,r c, 1 c,8 c,1 e c,8 e this paper. So, the design vector d in Eq. (1) has x (2  j  7) and y (2  j  8) as the components with a total of 13 c,j c,j design variables . The choices for the resulting parametric form of an A target are empirical . But this parametric form e,r is adequate for the applications in this paper . That is, every A (  ) in this paper can be fitted by an A (  ) with e ,r e ‖ 𝐴 (  ) − 𝐴 (  ) ‖  є . If this parametric form were not flexible enough for A targets , x and x w ould also be used e,r c,1 c,8 e e , r as design variables in Eq. (1). If needed, more control points could be used to define an A target in Eq. (1) . But more e,r design variables will require more time to solve Eq. (1).

The selected optimal solution of Eq. (1) as the A target of A (  ) is the Pareto optimal solution with the minimum e,r e ,r value of PLdB[ A (  )] while the inverse design error of A (  ) for the selected A target is within the error tolerance , e e ,r e,r target i.e., ‖ 𝐴 (  ) − 𝐴 (  ) ‖  є . For easy reference, this Pareto optimal solution will be denoted by 𝐴 (  ). In this paper, e e , r e , r the acceptable error tolerance є is 0.007  A ( l ,  ) , which was used in Refs. [5,6]. If A (  ) is an acceptable inverse e ,r e e ,r target design for 𝐴 (  ), then the underlying configuration has the potential to achieve a ground noise level of e , r target PLdB[ 𝐴 (  )] at the specified cruise condition and azimuthal direction. Otherwise, the underlying configur ation e , r without any shape modification is not consider ed to hav e any quantifiable low - boom characteristics at the specified cruise condition and azimuthal angle. In this case, a more flexible parametric model of A (  ) is required for assessment e of off - design low - boom characteristics of the underlying supersonic configuration for the required inverse design error constraint .

target To gain an intuitive understanding of ‖ 𝐴 (  ) − 𝐴 (  ) ‖ in terms of shape modifications for a low - boom e , r e , r inverse design, we need the following approximation relation between A and A .

e,r e,CFD 𝐴 ( 𝑥 , 𝑫 ) − 𝐴 ( 𝑥 , 𝑫 )  𝐴 ( 𝑥 , 𝑫 ) − 𝐴 ( 𝑥 , 𝑫 ) for 0 ≤ x ≤ l ( 3 ) e e e , r e e , r e 0 e , CFD e e , CFD e 0 Equation (3) was first formulated and verified in Ref. [ 2 5 ] for the undertrack direction. But it is also true for an off - track direction, which is the undertrack direction for a rotated aircraft geometry (see Fig. 1). For simplicity, we omit the azimuthal angle  in Eq. (3). Note that A [ 20 ,2 6 ] is the sum of the equivalent area due to lift and the equivalent e,CFD area due to volume for the rotated aircraft geometry. So, the values of A around a location of x can be e,CFD e increased/decreased by increasing/decreasing the lift and volume distributions of the rotated aircraft geometry around the same location of x . By Eq. (3), this corresponds to increasing/decreasing the A values around the same location e e,r target of x . Now, assume that D is indeed the inverse design whose A ( D ) matches 𝐴 (  ) for A (  ) = A ( D ) , the e e,r e,r e,r e , r 0 reversed equivalent area of a baseline design D . Then Eq. (3) implies the following approximation formula , which target ( ) ( ) indicates that the inverse design error ‖ 𝐴  − 𝐴 𝑫 ‖ for the baseline design D does approximately represent e , r e , r 0 0 the amount of the required lift and volume modifications of D to match the A target .

e,r target ‖ 𝐴 (  ) − 𝐴 ( 𝑫 ) ‖  ‖ 𝐴 ( 𝑫 ) − 𝐴 ( 𝑫 ) ‖ (4) e , r 0 e , CFD e , CFD 0 e , r Equation (1) is solved using the non - dominated sorting genetic algorithm II (NSGA II) in ModelCenter [ 2 7 ].

target NSGA II require s finite lower and upper bounds for the design variables. The target 𝐴 (  ) c ou ld be obtained by e , r solving Eq. (1) with a set of bound constraints on d : 0  x  l and 0  y  1. 5  A ( l ,  ). In practice, more restrictive c,j e c,j e,r e target bounds of d are used empirically to reduce the size of the design space in search of 𝐴 (  ).

e , r The computing equipment for solving Eq. (1) i s two Windows servers with 40 cores of Intel Xeon CPUs at 2.4 GHz , with ModelCenter running on one server and the analysis codes running on another . The optimization run uses ModelCenter ’s parallel computing capability to perform 16 design analyses simultaneously. I t takes about 12 hours to generate approximately 5 , 0 00 design s . While the wall - clock time for solving Eq. (1) depends on A (  ) and the loads e,r target on the Windows servers , as well as the accuracy requirement for the Pareto frontier, 𝐴 (  ) is typically obtained e , r within 5 days for all the analysis cases in this paper , using multiple optimization runs with various bounds of d .

Fig. 7 Selected A targets from Pareto frontier for Eq. (1) with appropriate bounds of d .

e,r target The analysis method of computing 𝐴 (  ) is applied to the Mach 1.7 46 - PAX concept at SOC with an azimuthal e , r 2 2 angle of 30 deg as an example . In this case, A ( l ,  ) of the aircraft is 223.2 ft , l = 258.1 ft, and є = 1.56 ft . Here, l e,r e e e is determined by the undertrack Mach angle cut plane for the rotated aircr aft geometry in Fig. 1. The Pareto frontier in Fig. 7 is generated using a design space of d defined by a range of 0.2  l for x and a range of 0.4  A ( l ,  ) for y .

e c,j e,r e c,j The design space is centered around a design vector satisfying ‖ 𝐴 (  ) − 𝐴 (  ) ‖  є. The reason to find a solution of e e , r ‖ 𝐴 (  ) − 𝐴 (  ) ‖  є first is to ensure that the parametric form of A (  ) is flexible enough to fit A (  ) accurately .

e e,r e e , r The Pareto frontier in Fig. 7 is derived from about 18,000 designs generated by NSGA II within a wall - clock time of 46 hours. Three targets are plotted in Fig. 7 to show trades between the PLdB value and inverse design error. These A targets have appr oximately identical shapes but noticeably different PLdB values. This shows a high sensitivity e,r of the PLdB value of the ground signature with respect to the shape of A target.

e,r target The Mach 1.7 46 - PAX concept is an acceptable inverse design of 𝐴 (  ) that has a sonic boom ground noise e , r level of 7 1 .1 PLdB in azimuthal direction of 30 deg. Such a n analysis method provides a quantitative assessment of low - boom characteristics of a supersonic configuration at different cruise conditions and azimuthal directions. Note that, once the Mach 1.7 46 - PAX concept is refined for minimization of the sonic boom ground noise level in a preliminary design phase, an off - design assessment of low - boom characteristics could be easily determined by the PLdB value of the ground signature of the refined configuration at the given off - design condition. So, this assessment method of low - boom characteristics is specifically de veloped for conceptual design of low - boom supersonic aircraft.

V. Assessment of Low - Boom Characteristics of Mach 1.7 4 6 - PAX Concept The inverse design optimization for the Mach 1.7 46 - PAX concept at SOC starts with a n A target with undertrack e,r sonic boom gr ound noise level below 70 PLdB and matches the undertrack A of the aircraft with the A target within e,r e,r the tolerance є (see the inverse design error curve Fig. 4). The analysis method in Sec. IV will be applied to assess off - design low - boom characteris tics of the Mach 1.7 46 - PAX concept. An off - design condition for the Mach 1.7 46 - PAX concept is defined by either a cruise condition not at SOC or a nonzero azimuthal direction for sonic boom analysis. A representative set of off - design conditions will be used to give an overall assessment on whether the Mach 1.7 46 - PAX concept has any significant deficiency in low - boom characteristics over the primary boom carpet for the entire cruise segment. Low - boom characteristics at a non - cruise condition is also important but they are not discussed in this paper because different sonic boom analysis methods are required for non - cruise conditions.

First , consider the low - boom characteristics in the undertrack direction for the entire cruis e segment. Note that dp/p at 3BL away from the aircraft with a fixed AoA is identical at any cruise condition in a fixed azimuthal direction.

As a result, the Mach 1.7 46 - PAX concept is an acceptable low - boom inverse design for the same undertrack A e,r tar get at any cruise condition before it changes the AoA after reaching the cruise ceiling in the low - boom mission profile. For the same dp/p , the corresponding ground signature has a lower noise level if the cruise altitude is higher, which is based on our n umerical verifications of a variety of dp/p shapes. An intuitive interpretation is that the loudness of sonic boom generated by a fixed waveform decreases as the propagation distance increases. So, the ground noise level of the same A target decreases w hen the cruise altitude increases (see Figs. 4 and 5).

e,r After reaching the cruise ceiling of 60,000 ft, the Mach 1.7 46 - PAX concept must reduce the AoA to match the reduced cruise weight as the fuel is consumed during the cruise. At the end of cruise, the undertrack A of the Mach e,r 1.7 46 - PAX concept is very different from the A target at SOC as expected. However, the analysis method in Sec.

e,r IV shows that the Mach 1.7 46 - PAX concept is also an acceptable inverse design for an undertrack A target with e,r sonic boom ground noise level of 66.0 PLdB (see Fig. 8). This shows that the Mach 1.7 46 - PAX concept has low - boom A shapes in the undertrack direction with the potential of achieving sonic boom ground noise levels below 70 e,r PLdB for the entire cruise se gment.

Fig. 8 A distributions and signatures for aircraft cruising at 60,000 ft in undertrack direction .

e For low - boom characteristics in off - track directions, four off - track angles are used to show the off - track low - boom target ( ) characteristics of the Mach 1.7 46 - PAX concept at SOC. The analysis method in Sec. IV is used to compute 𝐴  e , r for A (  ) of the Ma ch 1.7 46 - PAX concept at SOC with  = 25, 30, 35, and 40 deg. Each A (  ) is an acceptable e,r e,r target target inverse design for the A target of 𝐴 (  ) with inverse design error ‖ 𝐴 (  ) − 𝐴 (  ) ‖  є. The A (  ) shapes e,r e,r e , r e , r e , r are approximately identical for x e < 1 2 0 ft (see the A e shapes in Fig. 9), so t he configuration does behave like a body of revolution at SOC for x < 1 2 0 ft. A small portion of the wing is within the range of x < 1 2 0 ft (see Fig. 3) and the e e fuselage shape is not symmetric with respect to the roll axis of the aircraft at SOC. This indirectly confirms that Whitham - Walkden ’s body - of - revolution approximation [20 ,2 8 ] of a supersonic configuration for sonic boom analysis is accurate for a wing - body configuration if the lift force is w eak, which is consistent with the assumption o f a wing at a small incidence to the stream in Refs. [20 ,2 8 ]. Note that A and A in the undertrack direction at SOC are e, CFD e,r approximately identical for x e < 1 2 0 ft (see the A e shapes in Fig. 4), which indicates that A e,r is an aft correction of A for sonic boom analysis of any supersonic aircraft using a body - of - revolution approximation. The nearly e, CFD identical ground signatures for dp/p at 3BL and A (see the ground signatures in Fig . 4) show that A represents an e,r e,r accurate body - of - revolution approximation of the supersonic aircraft for sonic boom analysis.

Fig. 9 E quivalent areas and ground signatures in different azimuthal directions at SOC .

A s  increases from 0 to 25 deg, the aft shape of A (  ) gradually bends downward proportionally starting at about e,r target ( ) x = 150 ft (see the A shapes for  = 0 and 25 deg in Fig. 9) . This allows the aft shape of 𝐴  to bend in a similar e e e , r target way to match A (  ) while maintaining the ground noise level below 70 PLdB for 𝐴 (  ) . For  = 30 deg , the aft e,r e , r shape of A (  ) still bends downward proportionally starting at about x = 150 ft, but the best aft fit of A (  ) requires e,r e e,r a slightly positive slope near the end of effective range, which conflicts with the zero slope constraint at x = l . Even e e target target though 𝐴 (  ) has an analytical zero slope at x = l , the forced change of 𝐴 (  ) from a slightly positive slope e e e , r e , r target to the zero slope near x = l i s manifested by a small kink in the ground signature of 𝐴 (  ) around 180 ms (see e e e , r target the target signature for  = 30 deg in Fig. 9) . The ground noise level of 𝐴 (  ) is 7 1 .1 PLdB , slightly higher than e , r target those for azimuthal directions of 0 and 25 deg . For  = 35 and 40 deg , 𝐴 (  ) requires a n approximately linear fit e , r with a pos itive slope for x  210 ft. The corresponding A targets have ground noise levels of 79.7 and 8 7 . 0 PLdB , e e,r respectively . The shapes of the target and actual ground signatures are quite similar for each azimuthal direction, which is another indirect confirmation of the validity of inverse design optimization using A . Th e PLdB value of e,r target 𝐴 (  ) provides a quantitative assessment on degradation of the undertrack low - boom A shape in the azimuthal e,r e , r direction in conceptual low - boom design phase .

VI. Potential Application to Robust Low - Boom Inverse Design The analysis method in Sec. III can be considered as the sonic boom analysis of low - boom characteristics of a supersonic concept over the primary boom carpet for the entire cruise segment. However, this analysis method can be slightly modified for finding a compatible set of A target for a given set of cruise conditions and azimuthal directions.

e,r Here a set of compatible A tar gets means that the inverse design errors with respect to this set of A targets can be e,r e,r simultaneously reduced within a given tolerance using modifications of the aircraft ’s OML .

Now, let A (  ) be the reversed equivalent area of a supersonic configura tion corresponding to a pair of cruise e ,r condition and azimuthal direction of  deg . Assume that there exists an appropriate Pareto optimal solution A (  ) e ,pareto of Eq. (1) such that all the inverse design error functions, A (  ) − A (  ), have the positive sign over a specified e ,pareto e ,r range of x . By Eq. ( 3 ), if the physical volume of this supersonic configuration is increased in the specified range of e x , then the values of A (  ) over the specified range of x will increase, which will lead to a simultaneous reduction e e ,r e of the inverse design errors for the specified pairs of cruise conditions and azimuthal angles. Equation ( 1 ) provides a tradeoff between an easy - to - match target and the ground noise level of the target for low - boom inverse design optimization for any pair of cruise condition and azimuthal direction . Moreover , Eq . (1) generates a variety of A e ,r targets for each pair of cruise condition and azimuthal direction (see Fig. 7) . In this paper, one of those targets is used to assess the low - boom characteristics of the underlying supersonic concept at an off - design condition. Future research will be focused on h ow to use these targets to determine a set of compatible A targets for robust inverse design e ,r optimization over the primary boom carpet for the entire cruise segment.

VII. Conclusions An analysis method for assessment of low - boom characteristics of a supersonic concept is proposed in this paper.

It aims to provide a quantitative method to assess the low - boom characteristics of a given supersonic concept at any cruise condition and azimu thal direction. This analysis method find s an A target that has the lowest sonic boom e,r ground noise level and can be approximately matched by the A of the supersonic concept for the specified cruise e,r condition and azimuthal direction. A low - boom missio n profile is defined for any supersonic concept designed for low boom at SOC. This low - boom mission not only allows a supersonic concept to retain the designed low - boom characteristics at SOC as much as possible for the entire cruise segment, but also impr oves the overland cruise efficiency of the supersonic concept.

The analysis method is applied to assess the off - design low - boom characteristics of the Mach 1.7 46 - PAX concept, which was designed for a low - boom A shape at SOC in the undertrack direction. It shows that the Mach 1.7 46 - PAX e,r concept is an acceptable low - boom inverse design for an A target with ground noise level below 70 PLdB in the e,r undertrack direction at any cruise condition. T he Mach 1.7 46 - PAX concept also has a low - boom A shape at SOC e,r for any off - track direction with azimuthal angle no more than 25 deg , while the low - boom A shape matches an A e,r e,r target with sonic boom ground noise level less than 70 PLdB . However, as the azimuthal angle increases from 25 to 40 deg, the M ach 1.7 46 - PAX concept starts to lose the low - boom characteristics designed for the undertrack direction at SOC. In particular, the A of the Mach 1.7 46 - PAX concept in the off - track angle of 40 deg can only be considered e,r as an acceptable inverse design of an A target with sonic boom ground noise level of 83 .1 PLdB.

e,r This paper is the first step of finding a low - boom inverse design method to reduce the sonic boom ground noise level over the primary boom carpet for the entire cruise segment. The ultimate goal is to generate a set of compatible A targets for inverse design optimization at specified pairs of cruise conditions and azimuthal directions .

e,r Acknowledgments This work is funded by the NASA Commercial Supersonic Technology Project. The authors ar e grateful to Joel Lonzaga for providing the template files and instructions to run PCBoom 7 for propagation of dp/p to the ground.

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