5 Numerical analyses of wing body models 5.1 Introduction

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5 Numerical analyses of wing body models

5.1 Introduction

The most significant aerodynamic properties of an aircraft are summarized by two dimensionless quantities, the lift and drag coefficients CL and CD; they are functions only of the aerodynamic forces between fluid and aircraft, a reference surface, fluid density and velocity. In aerodynamics, the lift-to-drag ratio, or L/D ratio, is the amount of lift generated by a wing or vehicle, divided by the aerodynamic drag it creates by moving through the air. A higher or more favorable L/D ratio is typically one of the major goals in aircraft design; since a particular aircraft's required lift is set by its weight, delivering that lift with lower drag leads directly to better fuel economy in aircraft climb performance, and glide ratio. Since the aerodynamic forces are scaled by the same factor to get CL and CD, the computation of L/D is equal to CL/CD.

The cruise speed of modern airliners operates in transonic conditions, where the Mach number over a finite part of the lifting surfaces is supercritical. In this regime, a portion of the kinetic energy of the supersonic fluid flow, across the shock wave, is transformed into heat because of dissipative effects associated to the local aerodynamic field. Such phenomena leads to subsonic velocities in an infinitely thin layer with abrupt change of air properties and associated losses. This produces a large component of total drag, the Wave Drag, with a drastic reduction of L/D ratio coefficient. In addition to this the shock wave produced causes a series of unsteady phenomena dangerous for the wings.

A method to reduce these problems is the implementation of a crescent sweep angle at the wing leading edge. This work, such as the previous studies at the Department of Aerospace Engineering of Pisa University [1-5], aims to a preliminary design of models with curved planform wing, starting from a swept geometry validated on the experimental test carried out in NTF.

In the first section of this paragraph, a CFD comparison between three different geometries with curved planform wing it will be carried out in order to choose the best one among them. Finally the comparison will be made between the original geometry and the chosen curved wing model. Thanks to the replay script, and the particular geometry implementation (see chapter2), only the geometries change but the CFD grid still remains the CASE 73 validated in the previous chapter.

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5.2 Preliminary curved wing optimization

The replay script and the automated process of rearrangement of the original geometry to the new curved wings configuration allow the numerical analysis of different wing- body geometries without changing the CFD model (CASE 73) validated in the previous chapter. Hence, a preliminary curved wing optimization study is carried out before comparing the original swept wing to a curved one.

The comparison is made between the three models introduced in Section 3.3:  Case 6000 53° with ∆tip=6 m and Λtip=53°;

 Case 6400 59° with ∆tip=6.4 m and Λtip=59°;

 Case 3000 45° with ∆tip=3 m and Λtip=45°.

The simulations are performed with Mach number of 0.85, different angles of attack for Reynolds number of 3∙107 and 5.6∙107. Re=3∙107 is set up in correlation to validation process of Section 4.2.1; the second one is derived from a different input condition.

The ANSYS FLUENT setup for Re=3∙107_{is the same employed for the validation}

process. While the setup for Re=5.6∙107 exploits different input parameters in order to get the desired output. This time Mach and physical properties of fluid at a relative altitude are chosen first (it is like to impose Mach and flight altitude), then by solving the same equations system, the Gauge pressure and the Reynolds number are derived.

In order to maintain the temperature used in the first setup, the altitude chosen for the second simulation package is 1∙104 meters.

The input parameters for that altitude becomes:  M=0.85;

 T=223.25 [K];  ρ= 0,4135[kg/m3_].

Input equations derived

M μ= μ0*(T/T0)3/2 (T0+110,56)/(T+110,56) μ [Pa∙s]

T[K] ρ=Re μ/ U L Re

ρ [kg/m3_] _p=ρRT _{P [Pa]}

a=√ (γRT) a [m/s]

U=M∙a U [m/s]

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The inputs and derived parameters are listed in the following table together with material properties and reference values.

Input: ρ [kg/m^3] 0,4135 M 0,85 T [K] 223,25 R [j/kg*K] 287,05 γ 1,4 c [m] 7,00532 T0 [k] 273,11 μ0 [Pa*s] 0,00001716 Ref A [m^2] 191,8447776

cp specific heat coeff 1002,7

k thermal conductivity 0,02007 turbulent intensity 0,017422834 derived: Re 50583576,61 μ [Pa*s] 1,45798E-05 P [Pa] 26498,69782 a [m/s] 299,5287591 U [m/s] 254,5994452

Table 5-2 input, derived and reference value for 10000m

The aerodynamic comparison is focused especially on the achievement of best lift-to-drag ratio which is equivalent to CL/CD ratio.

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 Case 6000 53°, Re=3∙10 7_Mach=0.85, _angle _of _attack

α=0°,

α=1°,α=2°,α=2.5°,α=3°;

 Case 6400 59°, Re=3∙107 _Mach _=0.85, _angle _of _attack

α=0°,

α=1°,α=2°,α=2.5°,α=3°;

 Case 3000 45°, Re=3∙107 _Mach _=0.85, _angle _of _attack

α=0°,

α=1°,α=2°,α=2.5°,α=3°;

 Case 6000 53°, Re=5.6∙107 _Mach _=0.85, _angle _of _attack

α=0°,

α=1°,α=2°,α=2.5°,α=3°;

α=0°,

α=1°,α=2°,α=2.5°,α=3°;

α=0°,

α=1°,α=2°,α=2.5°,α=3°.

The optimization process between the curved models is carried out by the comparison of the resulting value of global drag coefficient CD, of global lift coefficient CL and of CL/CD ratio listed in Table 5.3-5.6 and from CL-α, CD-α, CL-CD and CL/CD-α charts plotted in Figure 5.1-5.10.

Re 3∙10

7

_{Mach 0.85}

3000 45° 6000 53° 6400 59° CL CD CL CD CL CD 0° 0.21571 0.0139 0° 0.20703 0.013614 0° 0.199887 0.013714 1° 0.34049 0.0164 1° 0.32604 0.01604 1° 0.31988 0.01613 2° 0.47218 0.02083 2° 0.44938 0.02006 2° 0.44547 0.02027 2.5° 0.54233 0.02441 2.5° 0.51448 0.02325 2.5° 0.51178 0.02364 3° 0.604621 0.02954 3° 0.578946 0.027756 3° 0.575851 0.028436

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Re 5.06∙10

7

_{Mach 0.85}

3000 45° 6000 53° 6400 59° CL CD CL CD CL CD 0° 0.218 0.01325 0° 0.212 0.01285 0° 0.2076 0.01302 1° 0.3465 0.01592 1° 0.3325 0.01516 1° 0.32824 0.01552 2° 0.47868 0.0204 2° 0.45594 0.01963 2° 0.45489 0.01976 2.5° 0.54979 0.02408 2.5° 0.52162 0.02288 2.5° 0.52183 0.02322 3° 0.61248 0.0303 3° 0.586649 0.02770 3° 0.586374 0.02815

Table 5-4 CL, CD Re 50.6M Mach 0.85 for curved wing configurations

Re 3∙10

7

_{Mach 0.85}

3000 45° 6000 53° 6400 59° CL/CD CL/CD CL/CD 0° 15.519 0° 15.207 0° 14.576 1° 20.7645 1° 20.3303 1° 19.8269 2° 22.668 2° 22.3981 2° 21.9756 2.5° 22.2174 2.5° 22.1306 2.5° 21.6528 3° 20.468 3° 20.858 3° 20.251

Table 5-5 CL/CD Re 30M Mach 0.85 for curved wing configurations

Re 5.6∙10

7

_{Mach 0.85}

3000 45° 6000 53° 6400 59° CL/CD CL/CD CL/CD 0° 16.453 0° 16.498 0° 15.943 1° 21.7599 1° 21.9327 1° 21.1541 2° 23.4647 2° 23.2303 2° 23.0219 2.5° 22.8276 2.5° 22.7987 2.5° 22,4747 3° 20.416 3° 21.179 3° 20.827

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95 Figure 5-1 CL-α Re 30 M Figure 5-2 CD-α Re 30 M 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 CL alpha

CL-α, M 0.85, Re 30 M

3000 45° 6000 53° 6400 59° 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.5 1 1.5 2 2.5 3 3.5 4 CD alpha

CD-α, M 0.85, Re 30 M

3000 45° 6000 53° 6400 59°

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96 Figure 5-3 CL-CD Re 30 M Figure 5-4 L/D-α Re 30 M 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.85, Re 30 M

3000 45° 6000 53° 6400 59° 0 2 4 6 8 10 12 14 16 18 20 22 24 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D alpha

L/D-α, M 0.85, Re 30 M

3000 45° 6000 53° 6400 59°

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97 Figure 5-5 L/D-CL Re 30 M Figure 5-6 CL-α Re 50.6M 8 10 12 14 16 18 20 22 24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL M 0,85 Re 30 M

3000 45° 6000 53° 6400 59° 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 CL alpha

CL-α, M 0.85, Re 50.6 M

3000 45° 6000 53° 6400 59°

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98 Figure 5-7 CD-α Re 50.6M Figure 5-8 CL-CD Re 50.6 M 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 CD alpha

CD-α, M 0.85, Re 50.6 M

3000 45° 6000 53° 6400 59° 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.85, Re 50.6 M

3000 45° 6000 53° 6400 59°

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99 Figure 5-9 L/D-α Re 30 M Figure 5-10 L/D-CL Re 50.6 M 8 10 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D alpha

L/D-α, M 0.85, Re 50.6 M

3000 45° 6000 53° 6400 59° 8 10 12 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M 0.85, Re 50.6 M

3000 45° 6000 53° 6400 59°

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The analysis of results underlines an higher L/D ratio of the 3000 45° model, thus, being limited to aerodynamic performances consideration, it presents the best behavior .

Taking into account that a comparison with fixed CL is not carried out for time constrains, the aerodynamic efficiency is chosen as the discriminating factor for model selection. This is the reason why 3000 45° is the configuration chosen for the comparative analysis to the original swept configuration.

5.3 CFD comparison between swept and curved

planform wing-body models

The following CFD analysis has the purpose to assess the results of previous work carried out Department of Aerospace Engineering of the University of Pisa: a better aerodynamic behavior, in transonic regime, of a curved wing body configuration in respect to a swept wing body model, consequently a wave drag reduction and a gain in terms of efficiency and aerodynamic drag. As mentioned before, the innovation and the uncertainties of this work lies on the comparison with an existing reference model designed and tested for a particular cruise condition, therefore the positive outcome is not so obvious.

The comparison is carried out between the 3000 45° curved wing configuration and the reference CRM configuration, based on a transonic transport configuration designed to fly at a cruise Mach number of M=0.85 with a nominal lift condition of CL=0.50 at an altitude of approximately 11300 meters[17].

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Figure 5-12 3000 45° curved geometry

Figure 5-13 Comparison between CRM and 3000 45°

5.3.1 CFD comparison with variable angle of attack

The simulations are first performed at CRM cruise altitude (11300 meters) with an angle sweep and a constant Mach number of 0.8, 0.85, 0.875, 0.9 and the CFD model employed is CASE 73 validates in the chapter before.

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Figure 5-14 CASE 73 surface mesh for swept configuration

Figure 5-15 Figure 5-16 CASE 73 surface mesh for curved configuration

 CASE CRM, h=11300, Mach=0.8, angle of attack

α=1°, α=2°, α=2.5°, α=3°;

 CASE 3000 45°, h=11300, Mach=0.8, angle of attack

α=1°, α=2°, α=2.5°,

α=3°;

α=0°, α=1°, α=2°, α=2.5°;

 CASE3000 45°, h=11300, Mach=0.85, angle of attack

α=0°, α=1°, α=2°,

α=2.5°;

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

CASE CRM, h=11300, Mach=0.875, angle of attack

α=0°, α=1°, α=2°,

α=2.5°;

 CASE 3000 45°,h=11300,Mach=0.875, angle of attack

α=0°, α=1°,α=2°,

α=2.5°;

α=0°, α=1°, α=2°, α=2.5°;

 CASE 3000 45°, h=11300,Mach=0.9, angle of attack

α=0°, α=1°, α=2°,

α=2.5°.

The ASNSYS FLUENT setup foresees as input parameters:  Mach= 0.8, 0.85, 0.875, 0.9;

 T=216.65 [K];

 ρ= 0,347103 [kg/m3].

The input and derived parameters are listed in the following table

Input: M 0,8 0,85 0,875 0,9 ρ [kg/m^3] 0,347103 0,347103 0,347103 0,347103 T [K] 216,65 216,65 216,65 216,65 R [j/kg*K] 287,05 287,05 287,05 287,05 γ 1,4 1,4 1,4 1,4 c [m] 7,00532 7,00532 7,00532 7,00532 T0 [k] 273,11 273,11 273,11 273,11 μ0 [Pa*s] 0,00001716 0,00001716 1,716E-05 0,00001716 AREA da usare 203,2747724 203,2747724 203,27477 203,274772 Ref A [m^2] 191,8447776 191,8447776 191,84478 191,844778

cp specific heat coeff 1002,6 1002,6 1002,6 1002,6

k thermal conductivity 0,01988 0,01988 0,01988 0,01988

turbulent intensity 0,017921274 0,017785979 0,0177216 0,01765935

DERIVED:

Re 40365350,42 42888184,82 44149602 45411019,2

μ [Pa*s] 1,42197E-05 1,42197E-05 1,422E-05 1,422E-05

P [Pa] 21586,12123 21586,12123 21586,121 21586,1212

a [m/s] 295,0680184 295,0680184 295,06802 295,068018

U [m/s] 236,0544147 250,8078157 258,18452 265,561217

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The resulting value of global drag coefficient CD, of global lift coefficient CL and of CL/CD ratio are listed in Table 5.8 and Table 5.9 and the curves CL-α, CD-α, CL-CD and CL/CD-α are plotted in the Figure 5.17-5.32.

CRM M 0,8

alpha CL CD pressure CD viscous CD L/D

1,00 0,3450 0,00704 0,00895 0,01599 21,5828 2,00 0,4659 0,01091 0,00889 0,01980 23,5283 2,23 0,4945 0,01206 0,00887 0,02093 23,6247 2,50 0,5269 0,01352 0,00885 0,02237 23,5597 3,00 0,5891 0,01679 0,00877 0,02556 23,0438 3000 45° M 0,8

1,00 0,3253 0,00658 0,00896 0,01553 20,9425 2,00 0,4391 0,01007 0,00890 0,01897 23,1472 2,50 0,4961 0,01241 0,00885 0,02125 23,3425 2,54 0,5003 0,01258 0,00884 0,02143 23,3513 3,00 0,5529 0,01530 0,00877 0,02407 22,9674 CRM M 0,85

0,00 0,2343 0,00494 0,00874 0,01368 17,1216 1,00 0,3694 0,00807 0,00872 0,01679 22,0046 1,36 0,4202 0,00963 0,00869 0,01833 22,9287 1,72 0,4705 0,01140 0,00866 0,02006 23,4500 1,88 0,4975 0,01252 0,00864 0,02116 23,5147 2,00 0,5153 0,01337 0,00862 0,02200 23,4294 2,50 0,5907 0,01833 0,00852 0,02684 22,0066 3000 45° M 0,85

0,00 0,2187 0,00466 0,00877 0,01343 16,2885 1,00 0,3445 0,00737 0,00874 0,01611 21,3918 1,61 0,4229 0,00979 0,00869 0,01848 22,8836 2,00 0,4768 0,01192 0,00864 0,02056 23,1893 2,16 0,4993 0,01297 0,00862 0,02158 23,1342 2,50 0,5474 0,01567 0,00855 0,02423 22,5950

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CRM M 0,875

0,00 0,2379 0,00595 0,00857 0,01453 16,3780 1,00 0,3891 0,01014 0,00852 0,01866 20,8501 2,00 0,4993 0,01741 0,00836 0,02577 19,3731 2,01 0,5000 0,01747 0,00836 0,02583 19,3560 2,50 0,5478 0,02298 0,00828 0,03126 17,5257 3000 45° M 0,875

0,00 0,2214 0,00544 0,00861 0,01405 15,7550 1,00 0,3631 0,00875 0,00857 0,01732 20,9641 2,00 0,4676 0,01431 0,00842 0,02273 20,5703 2,29 0,5013 0,01676 0,00837 0,02512 19,9567 2,50 0,5241 0,01883 0,00832 0,02715 19,3004 CRM M 0,9

0,00 0,1941 0,01050 0,00827 0,01878 10,3390

1,00 0,3386 0,01502 0,00826 0,02328 14,5452

2,00 0,4548 0,02407 0,00821 0,03228 14,0889

2,50 0,5051 0,03012 0,00818 0,03829 13,1916

3000 45° M 0,9

0,00 0,1818 0,00950 0,00833 0,01784 10,1934

1,00 0,3248 0,01278 0,00832 0,02111 15,3901

2,00 0,4411 0,02051 0,00825 0,02876 15,3354

2,62 0,5007 0,02721 0,00819 0,03540 14,1437

Table 5-9 CL and CD pressure and viscous contributions, for different angle of attack and M 0.875, 0.9

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Figure 5-17 CL-α curve for Mach number 0.8, at a flight altitude h 11300 meters

Figure 5-18 CL-α curve for Mach number 0.85, at a flight altitude h 11300 meters 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4

Cl -α, M 0.8, h 11300 m

CRM 3000 45° 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 CL α

CL-α, M 0.85, h 11300 m

CRM 3000 45°

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Figure 5-19 CL-α curve for Mach number 0.875, at a flight altitude h 11300 meters

Figure 5-20 CL-α curve for Mach number 0.9, at a flight altitude h 11300 meters 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 CL α

CL-α M 0.875, h 11300 m

CRM 3000 45° 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 CL α

CL-α, M 0.9, h 11300 m

CRM 3000 45°

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Figure 5-21 CD-α curve for Mach number 0.8, at a flight altitude h 11300 meters

Figure 5-22 CD-α curve for Mach number 0.85, at a flight altitude h 11300 meters 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 CD α

CD-α, M 0.8 h 11300 m

CRM 3000 45° 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 CD α

CD-α, M 0.85 h 11300 m

CRM 3000 45°

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Figure 5-23 CD-α curve for Mach number 0.875, at a flight altitude h 11300 meters

Figure 5-24 CD-α curve for Mach number 0.9, at a flight altitude h 11300 meters 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 CD α

CD-α, M 0.875, h 11300 m

CRM 3000 45° 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 CD α

CD-α, M 0.9, h 11300m

CRM 3000 45°

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Figure 5-25 Polar CL-CD curve for Mach number 0.8, at a flight altitude h 11300 meters

Figure 5-26 Polar CL-CD curve for Mach number 0.85, at a flight altitude h 11300 meters 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.8, h 11300 m

CRM 3000 45° 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.85, h 11300 m

CRM 3000 45°

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Figure 5-27 Polar CL-CD curve for Mach number 0.875, at a flight altitude h 11300 meters

Figure 5-28 Polar CL-CD curve for Mach number 0.9, at a flight altitude h 11300 meters 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.875, h 113000 m

CRM 3000 45° 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 CL CD

CL-CD, M 0.9, h 11300 m

CRM 3000 45°

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Figure 5-29 L/D-α curve for Mach number 0.8, at a flight altitude h 11300 meters

Figure 5-30 L/D-α curve for Mach number 0.85, at a flight altitude h 11300 meters 8 10 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D α

L/D-α, M 0.8, h 11300 m

CRM 3000 45° 8 10 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D α

L/D-α, M 0.85, h 11300 m

CRM 3000 45°

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Figure 5-31 L/D-α curve for Mach number 0.875, at a flight altitude h 11300 meters

Figure 5-32 L/D-α curve for Mach number 0.9, at a flight altitude h 11300 meters 8 10 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D α

L/D-α, M 0.875, h 11300 m

CRM 3000 45° 8 10 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D α

L/D-α, M 0.9, h 11300 m

CRM 3000 45°

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In previous tables and charts, for Mach of 0.8 and 0.85 the CRM configuration show better aerodynamic properties, transposed in an higher efficiency, in respect to the CASE 3000 45°. On the contrary for Mach of 0.875 and 0.9, the curved wing configuration reach an higher L/D ratio even if the absolute values are performed by the swept model for Mach=0.8 and Mach=0.85.

5.3.2 CFD comparison with fixed CL 0.5

The first part of simulations is focused on the study of aerodynamic properties at constant Mach with a sweep of the angle of attack. The aerodynamic drag is investigated by comparing the drag polars curves of swept and curved wing aircraft.

The second section of analysis is carried out with constant CL coefficient of 0.5 for both models and has the purpose to the investigate drag as function of Mach number. In order to perform the simulations with an equal CL is necessary to extrapolate the values of angles corresponding to CL=0.5by interpolating the CL-α curves studied in the first part of simulation. Once the angles are found the simulation can carried out at constant CL=0.5, altitude of 11300 meters for Mach= 0.8; 0.85; 0.875; 0.9.

 CASE CRM, h=11300, Mach=0.8, angle of attack α=2.23475°;  CASE 3000 45°, h=11300, Mach=0.8, angle of attack α=2.53694°;  CASE CRM, h=11300, Mach=0.85, angle of attack α=1.88384°;  CASE 3000 45°, h=11300, Mach=0.85, angle of attack α=2.16106°;  CASE CRM, h=11300, Mach=0.875, angle of attack α=2.00705°;  CASE 3000 45°, h=11300, Mach=0.875, angle of attack α=2.28933°;  CASE CRM, h=11300, Mach=0.9, angle of attack α= 2.44858875°;  CASE 3000 45°, h=11300, Mach=0.9, angle of attack α=2.61672398°.

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Figure 5-33 L/D-CL curve for Mach number 0.8, at a flight altitude h 11300 meters

Figure 5-34 L/D-CL curve for Mach number 0.85, at a flight altitude h 11300 meters 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M 0.8, h 11300 m

CRM 3000 45° 8 10 12 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M 0.85, h 11300 m

CRM 3000 45°

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Figure 5-36 L/D-CL curve for Mach number 0.9, at a flight altitude h 11300 meters 8 10 12 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M 0.875 h 11300 m

CRM 3000 45° 8 10 12 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M 0.9 h 11300 m

CRM 3000 45°

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The setup process used for these second package of simulation is the same of the first one so it would be redundant to listed again the input and the derived values.

The previous CFD analysis with fixed CL=0.5 and altitude of 11300m simulate the cruise condition for M=0.85 [16]. Assuming it could be the cruise condition also for M=0.8, M=0.875 and M=0.9, the Lift (the same for all the simulation because of the fixed CL coefficient) equals the weight of the aircraft and the thrust T necessary to model flying is equal to aerodynamic drag D. Hence a drag reduction corresponds to a fuel saving and to a better aerodynamic efficiency, this is the reason why it is interesting to investigate the

8 10 12 14 16 18 20 22 24 26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, h 11300 m

CRM, M=0.9 3000 45°, M=0.9 CRM, M=0.875 3000 45°, M=0.875 CRM, M=0.85 3000 45°, M=0.85 CRM, M=0.8 3000 45°, M=0.8

----

Mach 0,8

----

Mach 0,875

----

Mach 0,85

----

Mach 0,9

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CD coefficient (so the aerodynamic drag) in order to find the better configuration for set Mach. CRM 3000 45° Mach CD CD 0.8 0.02093361 0.02141425 0.85 0.02116056 0.02158839 0.875 0.02584607 0.02511949 0.9 0.03829384 0.0354091 Mach ΔCD% 0.8 -2.295993248 0.85 -2.021824122 0.875 2.811171349 0.9 _7.533161795

Table 5-10 CD percentage variation for CL 0.5 and M0.8, 0.85, 0.875, 0.9.

Figure 5-38 CD-M curve for varying Mach numbers, at a flight altitude h 11300 meters 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 CD Mach

CD-Mach h 11300 m

CRM 3000 45° MDD curved=0,8591 MDD swept=0,8541

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The Table 5.10 points out a better behavior of the swept wing model for Mach=0.8 and Mach=0.85 while the curved wing configuration shows a lower CD for Mach=0.875 and Mach=0.9 with a significant reduction for higher Mach. These results are confirmed and explained in the CD-Mach chart. The curves are very close in the first section, with the swept one slightly under the curved one. In correspondence of the elbow the trend is inverted. Where the curves present the elbow usually is placed the point of Drag Divergence Mach (MDD) that for definition is reached when d(CD)/dM=0.10, these configurations are not an exception and shows the MDD for M=0.8454(swept) and for M=0.8591(curved). From the MDD onwards the inclination is different and it is exactly the different gradient which implies a lower increment of CD with the Mach rise, as regarding the curved wing model in respect to the swept one. This is the reason why the curved planform wing aircraft results better than the swept one from the MDD point onward.

Under transonic condition a good portion of the total aerodynamic drag is generated by the so called “Wave Drag”. Across the shock wave, developed over the wings, there’s an energy dissipation and at the same time an entropy rising. Because of this loss, a pressure increment suddenly occurs and the wave drag is produced.

The pressure is strictly proportional to pressure coefficient, so a Cp comparison over the wing profiles at a corresponding wing span, it is necessary.

The analysis is made for:

 M=0.8 for η=0.15, η=0.3, η=0.45, η=0.6, η=0.7, η=0.8, η=0.9;  M=0.85 for η=0.15, η=0.3, η=0.45, η=0.6, η=0.7, η=0.8, η=0.9;  M=0.875 for η=0.15, η=0.3, η=0.45, η=0.6, η=0.7, η=0.8, η=0.9;  M=0.9 for η=0.15, η=0.3, η=0.45, η=0.6, η=0.7, η=0.8, η=0.9.

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Figure 5-39 cp profile for M 0.8 15%b

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Analyzing the charts for M=0.8 it can be said the Cp curves over the profile are more or less the same for curved and swept configuration along the entire wingspan. For M=0.85, 0.875, 0.9 the Cp profiles start to have a different behavior especially from the 70% of wingspan onwards. The shock wave, for curved configuration, is anticipated and the intensity of Cp drop decreases, therefore the wave drag.

It is interesting to note also the inclination of Cp curves, in correspondence of shock wave generation, result lower in curved model than in the swept one. This is a favorable effect in terms of flow separation, indeed for low pressure gradients the separation process is more difficult.

In the following figures the supersonic zone is depicted for CL 0.5 and Mach=0.8 (Fig.5.67-5.60), M=0.85(Fig 5.71-5.74), M=0.875(Fig 5.75-5.78), M=0.9(Fig 5.79-5.82) for swept and curved planform wing geometries. The supersonic region for curved wing is reduced respect to swept wing, except for M=0.8 where they show almost the same status, and present a twisted shape near the tip maybe due to the twisted angle distribution not optimized for that configuration. The reduction in supersonic region correspond to reduction in dissipated energy and in wave drag. Therefore the positive trend for M=0.875 and M=0.9 are confirmed by this analysis.

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Figure 5-68 sonic region, curved wing M 0.8 x view

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Figure 5-70 sonic region, curved wing M 0.8 3D view

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A final consideration on the results achieved in this paragraph can only remarks the better behavior of curved planform wing configuration at higher Mach values but, at the same time, the improvements respect to the swept configuration don’t seem to be good enough. Indeed in previous studies conducted in Pisa [1-5] the curved wing shows a higher aerodynamic efficiency already for M=0.85, moreover that configuration reaches the maximum L/D ratio. On the contrary in this thesis the maximum value is obtained for swept configuration (Fig 5.32). In addition to this, there are other discrepancies from previous works, such as the double sonic region in the tip region, underlined by the Cp charts (Fig 5.39-5.66) and the picture of sonic regions (Fig 5.73, 5.78, 5.82).

The reason of these differences can be due to:

 the necessity to compute a FSI simulation in order to find undeformed configuration for swept model, then perform again the FSI simulation over the undeformed curved wing geometry, obtained by shearing the aforementioned swept aircraft;

 The necessity to optimized wing profile and twist angle distribution for curved wing;

 The wrong angle of incidence between curved wing and fuselage.

In the following paragraph this last problem is faced in order to investigate the possible improvements deriving from a hypothetical angle of attack optimization.

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5.4 Wing-only case: Drag penalty caused by off-design

angle of incidence

From the detailed description of the CRM geometry (see Table 1.2), it could be observed the marked angle between fuselage and the wing root chord, known as angle of incidence or mounting angle, of 6.7166°.

For an entire wing-body model more than one design constraint should be considered in the final choice of the angle of incidence, such as pilot visibility during take-off and landing phases or high lift requirements during taxiing, but again the most relevant is the aerodynamic efficiency optimization in the cruise phase. Being the fuselage an elongated low lifting body, its angle of attack for minimum drag is usually close to zero. The wing design angle of attack for the cruise phase dictates the optimum value for the mounting angle in order to maximize global lift to drag ratio of the aircraft.

The results from section 5.3 for the two wing-body models give rise to some concern about how much the angle of incidence could affect the aerodynamic performances, particularly by an increased fuselage drag, thus lower the lift to drag ratio, in the curved wing if there is proof that the fuselage is in off-design conditions.

The CRM with curved wing geometry has a higher sweep angle outward from the kink station, this implying lower lift generation than the CRM with swept wing at the same angle. Indeed the design lift coefficient CL=0.5 for the original CRM corresponds to an angle of attack α=1.888°, while for the curved wing the required for global CL=0.5 is an angle α=2.162°, giving a difference Δα=0.274°. Remembering also that in literature [1-6] the curved wing showed a better behaviour without the presence of the fuselage, we need a way to estimate the various contributions to lift and drag of fuselage and wing respectively.

The entire wing-body model is split in different surfaces in Ansys ICEM CFD, before the mesh is imported in Ansys FLUENT, where it is possible to compute the force acting on the fuselage and the wing separately. Lift and drag components are computed multiplying by the appropriate directional cosines.

To obtain the wing-only nondimensional lift coefficient we divide, as usual, the lift component by the reference dynamic pressure and the wing-a planform area. This differ from the wing-body CL dimensionless formula where the aircraft reference Wimpress area is used, thus taking into account the fuselage contribution to total lift.

The optimized wing-body junction in the CRM geometry has been developed in order to reduce interference drag between the wing and the fuselage; it smooths out the pressure

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differences, thus avoiding major boundary layer separation. The presence of the fairing surface limits the implementation of a different angle of incidence for the curved wing-body model, because the surface should be properly remodelled. Filling the gaps generated by a rotation of the wing around the spanwise axis would produce new entities numbering in the CAD geometry making impossible the automation procedure described in Chapter 3.4 to work. In addition, this problem would represent a great effort to investigate the effects on the flow near the wing root with more CFD analysis computational demand.

Respect to previous works [1-5], in which only a swept and a curved wing were compared without any fuselage interference affecting the flow in proximity of the root, this method represents as an improvement for a more realistic correlation.

Wing-only data extrapolated in this way are presented in Figures from 5.83 to 5.102 together with wing-body data. For each Mach number the following graphs are shown:

 Lift coefficient versus the angle of attack α (Fig. 5.83-5.86);  Drag coefficient versus the angle of attack α (Fig. 5.87-5.90);

 Polar curves of lift coefficient versus drag coefficient (Fig. 5.91-5.94);  Aerodynamic efficiency or lift to drag ratio versus α (Fig. 5.95-5.98);

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CRM wing M 0,8

1,00 0,3492 0,00555 0,00472 0,01027 33,9973 2,00 0,4708 0,00862 0,00465 0,01327 35,4681 2,23 0,4995 0,00953 0,00463 0,01416 35,2781 2,50 0,5320 0,01072 0,00460 0,01531 34,7457 3,00 0,5943 0,01341 0,00451 0,01793 33,1490 3000 45° wing M 0,8

1,00 0,3279 0,00499 0,00474 0,00973 33,6848 2,00 0,4419 0,00767 0,00467 0,01234 35,8070 2,50 0,4988 0,00948 0,00461 0,01409 35,4071 2,54 0,5030 0,00961 0,00460 0,01421 35,3927 3,00 0,5554 0,01179 0,00452 0,01631 34,0483 CRM wing M 0,85

0,00 0,2380 0,00390 0,00459 0,00849 28,0168 1,00 0,3743 0,00661 0,00456 0,01117 33,5117 1,36 0,4253 0,00793 0,00453 0,01246 34,1308 1,72 0,4759 0,00943 0,00450 0,01393 34,1522 1,88 0,5032 0,01041 0,00448 0,01489 33,7990 2,00 0,5211 0,01118 0,00446 0,01564 33,3188 2,50 0,5969 0,01584 0,00435 0,02018 29,5709 3000 45° wing M 0,85

0,00 0,2211 0,00351 0,00463 0,00814 27,1461 1,00 0,3474 0,00576 0,00460 0,01036 33,5240 1,61 0,4258 0,00775 0,00455 0,01230 34,6305 2,00 0,4797 0,00954 0,00450 0,01403 34,1845 2,16 0,5023 0,01044 0,00447 0,01491 33,6799 2,50 0,5502 0,01287 0,00440 0,01727 31,8560

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CRM wing M 0,875

0,00 0,2418 0,00502 0,00446 0,00948 25,4978 1,00 0,3939 0,00892 0,00441 0,01333 29,5602 2,00 0,5091 0,01594 0,00426 0,02020 25,2065 2,01 0,5099 0,01599 0,00426 0,02025 25,1767 2,50 0,5577 0,02138 0,00417 0,02555 21,8290 3000 45° wing M 0,875

0,00 0,2238 0,00436 0,00452 0,00888 25,2083 1,00 0,3658 0,00728 0,00448 0,01176 31,0933 2,00 0,4756 0,01243 0,00435 0,01678 28,3396 2,29 0,5100 0,01471 0,00429 0,01900 26,8381 2,50 0,5331 0,01669 0,00424 0,02093 25,4665 CRM wing M 0,9

0,00 0,1987 0,00995 0,00423 0,01418 14,0110 1,00 0,3460 0,01433 0,00420 0,01853 18,6726 2,00 0,4620 0,02327 0,00414 0,02741 16,8574 2,50 0,5115 0,02926 0,00409 0,03335 15,3362 3000 45° wing M 0,9

0,00 0,1858 0,00874 0,00432 0,01306 14,2296

1,00 0,3318 0,01174 0,00429 0,01603 20,6926

2,00 0,4480 0,01919 0,00420 0,02339 19,1550

2,50 0,4954 0,00000 0,00000 0,02854 17,3589

2,62 0,5066 0,02572 0,00413 0,02985 16,9727

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Figure 5-83 CL-α curves, wing-only and wing-body models for swept and curved planforms, M=0.8

Figure 5-84 CL-α, wing-only and wing-body models for swept and curved planforms, M=0.85 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.5 1 1.5 2 2.5 3 3.5 CL Axis Title

CL-α, M=0.8

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3 3.5 CL α

CL-α, M=0.85

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body

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Figure 5-85 CL-α, wing-only and wing-body models for swept and curved planforms, M=0.875

Figure 5-86 CL-α, wing-only and wing-body models for swept and curved planforms, M=0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3 3.5 CL Axis Title

CL-α, M=0.875

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3 3.5 CL Axis Title

CL-α, M=0.9

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Figure 5-87 CD-α curves, wing-only and wing-body models for swept and curved planforms, M=0.8

Figure 5-88 CD-α, wing-only and wing-body models for swept and curved planforms, M=0.85 0.005 0.01 0.015 0.02 0.025 0.03 0.5 1 1.5 2 2.5 3 3.5 CD Axis Title

CD-α, M=0.8

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.5 1 1.5 2 2.5 3 3.5 CD Axis Title

CD-α, M=0.85

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Figure 5-89 CD-α, wing-only and wing-body models for swept and curved planforms, M=0.875

Figure 5-90 CD-α, wing-only and wing-body models for swept and curved planforms, M=0.9 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.5 1 1.5 2 2.5 3 3.5 CD Axis Title

CD-α, M=0.875

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.5 1 1.5 2 2.5 3 3.5 CD Axis Title

CD-α, M=0.9

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Figure 5-91 Polar CL-CD curves, wing-only and wing-body models for swept and curved planforms, M=0.8

Figure 5-92 Polar CL-CD, wing-only and wing-body models for swept and curved planforms, M=0.85 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.008 0.011 0.014 0.017 0.02 0.023 0.026 0.029 CL CD

CL-CD, M=0.8

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 CL CD

CL-CD, M=0.85

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Figure 5-93 Polar CL-CD, wing-only and wing-body models for swept and curved planforms, M=0.875

Figure 5-94 Polar CL-CD, wing-only and wing-body models for swept and curved planforms, M=0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 CL CD

CL-CD, M=0.875

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 CL CD

CL-CD, M=0.9

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Figure 5-95 L/D-α, wing-only and wing-body models for swept and curved planforms, M=0.8

Figure 5-96 L/D-α, wing-only and wing-body models for swept and curved planforms, M=0.85 18 20 22 24 26 28 30 32 34 36 38 0.5 1 1.5 2 2.5 3 3.5 4 L/D Axis Title

L/D-α, M=0.8

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D Axis Title

L/D-α, M=0.85

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Figure 5-97 L/D-α, wing-only and wing-body models for swept and curved planforms, M=0.875

Figure 5-98 L/D-α, wing-only and wing-body models for swept and curved planforms, M=0.9 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D Axis Title

L/D-α, M=0.875

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 L/D Axis Title

L/D-α, M=0.9

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Figure 5-99 L/D -CL, wing-only and wing-body models for swept and curved planforms, M=0.8

Figure 5-100 L/D-CL, wing-only and wing-body models for swept and curved planforms, M=0.85 18 20 22 24 26 28 30 32 34 36 38 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 L/D CL

L/D-CL, M=0.8

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M=0.85

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Figure 5-101 L/D-CL, Swept and curved planforms cases for wing-alone and wing-body models, M=0.875

Figure 5-102 L/D-CL, Swept and curved planforms cases for wing-alone and wing-body models, M=0.9 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M=0.875

CRM wing CRM wing-body 3000 45° wing 3000 45° wing-body 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L/D CL

L/D-CL, M=0.9

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A comparative analysis between the swept (CRM case) and the curved (3000 45° case) wing planform is realized in this section using wing-only data; this is possible by separating the lift and drag forces acting on fuselage and wing surfaces. Because of the discrepancy in the angle of attack at which the curved and the swept wing-body models reach the design CL=0.5 value, also the fuselage is at a slightly different angle respect to the undisturbed flow. This implies a higher fuselage drag component for the curved planform at the same lift coefficient.

The rationale of the method presented above is to estimate the aerodynamic performance penalties, which could be corrected by a variation in the wing mounting angle design at the root chord respect to the body longitudinal line in the 3000 45° case.

The off-design condition of the forces acting on the fuselage affects the examined parameters: as shown in Figures 5.83-5.86, the lift coefficient variation between wing-only and wing-body data is almost negligible, due to the low contribution of the fuselage to the total lift; instead, the gap between the two wing planforms increases significantly in the drag curves (Fig. 5.87-5.90).

Consequently, the lift to drag ratio varies significantly and Figures 5.96-5.102 highlight the abovementioned trend: for each Mach number the highest value of the aerodynamic efficiency is associated to the curved wing planform for the wing alone data. On the contrary, at Mach numbers M=0.8 and M=0.85 for wing-body configuration the original CRM shows the maximum lift to drag ratio.

A comparison of the aerodynamic efficiencies, taking into account the original design conditions (M=0.85, CL=0.5), is performed in the following tables for equal CL=0.5 ± 0.001. CRM 3000 45° M L/D L/D Δ% 0.8 35.29 35.39 0.325 % 0.85 33.80 33.68 -0.353 % 0.875 25.18 26.84 6.60 % 0.9 15.34 16.97 10.67 %

Table 5-13 Wing-only case, L/D for CL=0.5±0.001 for swept and curved planform shapes with percentage difference

At the aircraft design point the curved and the swept wing have very similar trend, but with a relative difference of -0.353 % the more favourable is the original CRM planform shape. This is probably due to the fact that the curved wing should be optimized in terms of

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geometric characteristics in the spanwise direction (i.e. twist, thickness and camber distributions) for the operative point. Future works should consider a new optimized design for the curved wing planform in terms of pressure distributions. For example, considering the higher sweep angle in the outboard sections, a more triangular lift distribution is obtained respect to the original CRM nearly elliptical distribution [16], as confirmed by the pressure coefficients at the outboard sections analysed in 5.3.2. Furthermore an optimization technique could produce a more uniform isobar and shock wave pattern along the wing. Lacking proper tools and computational resources, that kind of iterative design, involving geometric parametrization and CFD interaction to compute the proper pressure distribution in transonic flow, is very challenging and it is not the purpose of this thesis.

Alternatively a different design point could be considered both in terms of lift coefficient or Mach number for the curved planform wing, as can be seen by the following considerations.

Table 5.11 summarize the maximum L/D and its matching CL: the maximum value is computed by interpolation of the polynomial curve among the simulated points and its derivation, for different Mach number, with and without the fuselage contribution. While for the wing-body case the curved planform shape shows worst performances compared to the constant sweep planform, the opposite happens considering only the forces acting on the wing surfaces.

Wing-only Wing-body

CRM 3000 45° CRM 3000 45°

Mach L/Dmax CL L/Dmax CL L/Dmax CL L/Dmax CL

0.8 35.52 0.454 35.81 0.449 23.62 0.499 23.36 0.487

0.85 34.34 0.448 34.66 0.428 23.51 0.486 23.19 0.474

0.875 29.73 0.368 31.09 0.363 20.88 0.402 21.17 0.401

0.9 18.67 0.347 20.77 0.351 14.70 0.375 15.78 0.379

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L/Dmax % difference between swept and curved wing

Mach Wing-only Wing-body

0.8 0.82 % -1.10 %

0.85 0.93 % -1.36 %

0.875 4.57 % 1.39 %

0.9 11.3 % 7.35 %

Table 5-15 Maximum L/D percentage increase from swept to curved wing

Recalling that for the curved planform wing-body model the fuselage is at a different angle of attack when at the same CL of the original CRM, its effect is evaluated by removing the contribution to lift and drag. This way, for a Mach equal to 0.85, a difference of about 2.3% in the maximum L/D value is obtained relatively to the wing-body. Hence the curved wing could save fuel in the order of 0.93% (see Tab. 5.15), but this is true only for a new design CL of 0.448. In the case of a lower bending moment at the root of the wing, structural mass would be saved, together with some fuel mass, reducing in the end the cruise lift coefficient needed for a level flight respect to the swept wing design CL=0.5.

As described in Chapter 1.5, because the CRM design point does not coincide with the point of highest L/D, additional logistic constraints should be examined. For example, independently of propulsion and structural characteristics, the range of an aircraft is linearly dependent on the aerodynamic efficiency and the cruise speed (i.e Mach number for a given altitude). If a slightly larger Mach number, just below the drag rise, maintains an equal L/D, then the design point could be shifted with a range gain.

At Mach numbers greater than 0.85 the improvement of the curved wing planform results in a significantly increased L/D, over 10% at M=0.9, about 6% at M=0.875; here the wave drag reduction from kink to tip sections becomes considerable thanks to the crescent sweep. However at such speeds the wave drag is already dominant and the drop in efficiency is too large to be operative in a cruise phase, roughly 20% lower at M=0.875. The useful characteristic is the delayed drag divergence at higher values of MDD, which is

usually very close to the design point of a modern airliner.

From Figure 5.38 the drag divergence Mach number is estimated to be 0.8591 instead of 0.8541. Additional numerical simulations could be implemented for values between 0.855 to 0.86 with the aim to demonstrate the possibility to cruise at a slightly higher velocity without major increases of wave drag thus with a longer range capability; this could been accomplished in future works, as well as the study of a larger number of curved

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leading edge shapes, without changes to the automated process developed described in Chapter 3.