[American Institute of Aeronautics and Astronautics 46th AIAA Aerospace Sciences Meeting and Exhibit - Reno, Nevada ()] 46th AIAA Aerospace Sciences Meeting and Exhibit - CFD-Based

American Institute of Aeronautics and Astronautics

1

CFD-based Shape Optimization of Hypersonic Vehicles

Considering Transonic Aerodynamic Performance

Atsushi Ueno1 and Kojiro Suzuki

2

The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan

For the success of hypersonic vehicles, their shape must be optimized to achieve a high

lift-to-drag ratio (L/D) as well as a low aerodynamic heating rate in the hypersonic regime.

In addition, the transonic L/D must also be optimized to realize quick acceleration to the

hypersonic cruise speed. As the first step, we conducted a three-dimensional shape study in

which the CFD (Euler analysis)-based optimization is made only for two-dimensional airfoil

shape of the outer wing and determined the initial shape for a fully three-dimensional shape

optimization. An airfoil optimization study was done considering the hypersonic lift-to-drag

ratio (l/d), transonic l/d and the leading edge heating. The hypersonic l/d of the airfoil is

improved from 4.4 to 5.1 with the leading edge temperature kept at 1200K, while the

transonic l/d is decreased from 51 to 45. Based on the results of the two-dimensional airfoil

optimization, the initial shape for a fully three-dimensional shape optimization is obtained.

The three-dimensional Euler analysis shows that the aerodynamic performance is improved

though only the two-dimensional optimization is done for the outer wing.

Nomenclature

x, y, z = coordinate system shown in Fig. 1

LEq& = aerodynamic heating rate at leading edge

RLE = leading edge radius

TLE = wall temperature at leading edge

x = design variable vector

F, G = objective function and constraint function, respectively

c,LB = chord length and body length, respectively

Cl, Cd = 2-dimensional aerodynamic coefficients of lift and drag based on chord length, respectively

CL, CD = 3-dimensional aerodynamic coefficients of lift and drag based on wing area, respectively

Cp = pressure coefficient

XAC, XCP = positions of aerodynamic center and pressure center in the x direction, respectively

superscripts:

p, f = pressure drag and skin friction drag, respectively

I. Introduction

ESEARCH projects to develop a hypersonic transport have recently started in various aerospace communities

worldwide. In Japan, the Japan Aerospace Exploration Agency (JAXA) is investigating the hypersonic vehicle

that can fly from Tokyo to Los Angeles within two hours1. To realize a low-cost hypersonic transport with high

cruise efficiency and without using a fragile and expensive thermal protection system, such as, C/C composite

material and ceramic tiles, the optimum shape that achieves a high lift-to-drag ratio as well as a low heating rate

must be determined. In addition, a high lift-to-drag ratio is important also in the transonic regime, because the excess

thrust is relatively small in the transonic regime during acceleration to hypersonic speed. Consequently, an

appropriate compromise between the transonic and hypersonic lift-to-drag ratio is needed.

Hypersonic vehicles have been extensively studied by applying a waverider configuration2. This configuration

can produce a high lift-to-drag ratio in the hypersonic regime by riding on its own shock wave, in other words, by

using the compression lift. The leading edge of a waverider configuration should be sharp, because a rounded

1 Graduate Student, Department of Aeronautics and Astronautics, AIAA Student Member.

2 Associate Professor, Department of Advanced Energy, Senior Member AIAA.

R

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-288

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

leading edge results in a low lift-to-drag ratio for two reasons: (a) a detached shock wave is generated and it causes

large drag force, and (b) the high pressure flow behind the detached shock wave acts not only on the lower surface

but also on the upper surface of the vehicle, which decreases the lift force. However, the sharp leading edge of a

waverider configuration must endure a high aerodynamic heating rate, because the convective aerodynamic heating

rate is inversely proportional to the square root of the leading edge radius3. Furthermore, the large base area of a

waverider configuration with a truncated tail induces a large base drag in the transonic regime, thus making it

difficult to achieve a good compromise between transonic and hypersonic lift-to-drag ratio.

Our future goal is to determine the three-dimensional shape that can realize a low aerodynamic heating rate as

well as a high lift-to-drag ratio both in the hypersonic regime and in the transonic regime by a fully three-

dimensional shape optimization. As the first step prior to a fully three-dimensional shape optimization which

requires large computation time, we conducted a three-dimensional shape study in which the CFD (Euler analysis)-

based optimization is made only for two-dimensional airfoil shape of the outer wing and determined the initial shape

for the fully three-dimensional shape optimization. By this method, a well-considered initial shape is obtained,

which reduces the computation time for the fully three-dimensional shape optimization. First, the preliminary

conceptual study on the hypersonic vehicle was made as described in Section II. The body is divided into three parts,

that is, a fuselage, inner wing with a large sweep-back angle and outer wing with smaller sweep-back angle. The

airfoil shape of the outer wing was optimized from a viewpoint of an appropriate compromise between the transonic

and hypersonic lift-to-drag ratios with a constraint on the aerodynamic heating rate, as described in Section III.

Finally, the initial shape for the fully three-dimensional shape optimization study was defined by using the optimum

airfoil shape. Its aerodynamic performance and aerodynamic characteristics are discussed in Section IV.

II. Concept of Hypersonic Vehicle

A. Vehicle Specification & Flight

Path

In the present study, a

hypersonic vehicle was assumed as

shown in Fig. 1.

We assumed a hypersonic

transport of a similar size to the

supersonic transport Concorde. The

length was set at 60m, which is

close to length of the Concorde

(61.7m). The fuselage height (3.3m)

and fuselage width (3.0m) were the

same as the Concorde. The number

of passengers is therefore assumed

to be the same as the Concorde

(about 80 passengers). The wing

area was 600m2, which is 1.7 times larger than the Concorde. The reasons for such large wing area are: (a) it is

difficult to obtain a large lift coefficient in the hypersonic regime, and (b) the liquid hydrogen, which is one of the

candidates for the fuel, has low density and requires large fuel tank volume. The double-delta planform was adopted

to obtain a good compromise between hypersonic and transonic aerodynamic performance. The sweep-back angle of

the inner wing was 81.6deg (subsonic leading edge at Mach 5) and that of the outer wing was 30deg (supersonic

leading edge at Mach 5). The large outer wing area improves transonic aerodynamic performance. However it

60m

32.8

m

3m

6m

Fuselage

81.6deg

30deg

Inner wing

Outer

wing

Supersonic leading edge

Subsonic leading edge

Top ViewFront View

t/c=3%

3.3m

t/c=3 to 10%

x

y

z

RLE=65mm

RLE=120mm

RLE=65mm

Figure 1. Two-view drawing of hypersonic vehicle.

0 10 20 30 40 50 60-5

0

5

Fuselage

y=2.3m y=4.0m y=5.8m y=7.8m

x [m]

z [m]

Figure 2. Cross sections of fuselage and inner wing.


3

causes large wave drag in the hypersonic regime due to the supersonic leading edge. On the other hand, the large

inner wing area is required considering the large fuel tank volume. Therefore, the outer wing area was set at 142m2

and the inner wing area was set at 458m2 from a viewpoint of an appropriate compromise between transonic and

hypersonic aerodynamic performance as well as fuel tank volume. The position of the outer wing in the x direction

strongly affects the position of the aerodynamic center, because it produces large lift force in the transonic regime

due to the small sweep-back angle. Considering the longitudinal aerodynamic characteristics, the position of the

outer wing in the x direction was defined as shown in Fig. 1. The cross sections of the fuselage and inner wing are

shown in Fig. 2. The fuselage and inner wing were blended to produce large lift force in the hypersonic regime. The

parallel section at the fuselage (i.e., from x=about 10m to 45m) was defined to place a cabin. The cross sections of

the inner wing were based on the NACA 6 series airfoil which has been applied to many supersonic vehicles. The

thickness-to-chord ratio of the inner wing varied in the spanwise direction from 3 to 10%. The large thickness-to-

chord ratio is due to large fuel tank volume. To equip the nacelle under the inner wing, the front view of the lower

surface of inner wing was almost flat, though the nacelle was not considered in CFD analysis. The cross section of

the outer wing was optimized as described in Section III. The thickness of the outer wing was set at 3%, which is the

same as the Concorde, in order to reduce the wave drag. The

total weight of the vehicle was assumed to 165ton, which

includes the fuel weight (50ton). The flight path was defined

to estimate the required lift coefficient (Fig. 3). First, the

vehicle climbs to 10,000ft altitude and accelerates to Mach

0.8 and then climbs to 30,000ft altitude, which corresponds

to the cruise altitude of the existing transonic vehicles. After

that, the vehicle accelerates to Mach 1.2 with constant

altitude. At the end of acceleration, the dynamic pressure

reaches 30kPa, which is the same as that of the Concorde at

the supersonic cruise. This means that the structural design

of the present vehicle is assumed to be similar to that of the

Concorde. The vehicle continues to climb and accelerates

with constant dynamic pressure and finally reaches the cruise

point (90,000ft / Mach 5). In the present study, we consider

two representative design points, that is, 30,000ft / Mach 0.8

for the transonic flight and 90,000ft / Mach 5 for the

hypersonic flight.

B. Determination of Leading Edge Radius

The aerodynamic heating rate at the leading edge was calculated by the empirical relation for the cylinder3, as

shown in Eq. (1). The wall temperature at the leading edge was calculated from Eq. (2).

( ) [ ]2

1109550 043

50

8 cmWh

hV

R.q

aw

w.

.

LE

LE

−

= − ρ

& (1)

4

LELE Tq σε=& (2)

In Eq. (1), ρ is the free stream density, V is the flight velocity, haw is the adiabatic wall enthalpy, and hw is the wall

enthalpy at the wall temperature. In Eq. (2), σ is the Stefan-Boltzmann constant and ε is the emissivity which was set

at 0.8 in this study. The leading edge radius of the fuselage was set at 65mm so that the wall temperature at the

leading edge becomes 1,200K at the hypersonic cruise condition. The leading edge radius of the outer wing at the

wing-tip section was also set at 65mm. Because the outer wing was defined by the same airfoil as that of the wing-

tip section, the leading edge radius of the outer wing which depends on the chord length varied in the spanwise

direction. The leading edge radius at the joint section of the inner and outer wings was 120mm (Fig. 1), because the

chord length at that section is about 1.8 times larger than that at the wing-tip section. The aerodynamic heating rate

at the leading edge of the inner wing was not considered as the design constraints because no shock wave is

generated at the leading edge at any flight condition due to the subsonic leading edge.

Mach number

Altitude

[ft]

Flight path

Hypersonic flight condition

Transonic flight condition

dynamic pressure=30kPa

0 1 2 3 4 5 6

20000

40000

60000

80000

100000

Figure 3. Flight path of hypersonic vehicle.


4

III. Airfoil Optimization Study

A fully three-dimensional shape optimization requires large computation time. However, the computation time

for the two-dimensional airfoil optimization is small. Therefore, CFD (Euler analysis)-based optimization for two-

dimensional airfoil was conducted to obtain better initial guess for the fully three-dimensional shape optimization.

The flow over the outer wing is expected to show two-dimensional nature compared to that over the inner wing

because the sweep-back angle of the outer wing is small. Design of the two-dimensional airfoil of the outer wing is

important to realize high aerodynamic performance. The outer wing causes a large wave drag at Mach 5 due to the

supersonic leading edge, while it produces a large lift force at Mach 0.8 due to its small sweep-back angle.

Therefore, the airfoil of the outer wing should be optimized from a viewpoint of an appropriate compromise

between transonic and hypersonic aerodynamic performance. On the other hand, the flow over the inner wing will

be highly three-dimensional because of the large sweep-back angle. Therefore the airfoil shape of the inner wing is

less effective to improve the overall aerodynamic performance of the vehicle than that of the outer wing.

Furthermore, the inner wing has a subsonic leading edge and no shock wave is generated at the leading edge even at

Mach 5. Consequently, a large lift-to-drag ratio is expected at hypersonic speeds even when the airfoil shape itself is

not optimized for hypersonic flight. Therefore, the airfoil of the inner wing was not optimized and the inner wing

was defined by the airfoil based on the NACA 6 series airfoil that has been applied to many supersonic vehicles.

The optimization of the airfoil that is applied to the outer wing at the wing-tip section is discussed in the

following subsections. The outer wing was defined by the same airfoil as that of the wing-tip section.

A. Method of Airfoil Optimization at wing-tip section

The airfoil optimization was conducted by applying a two-

dimensional Euler flow solver and the Sequential Quadratic

Programming (SQP) method4, 5

as an optimizer.

In the flow analysis, the symmetric TVD scheme6 was used

to discretize the convective term and the LU-SGS7 method was

applied for the implicit time integration. The lift coefficient and

the pressure drag coefficient were obtained from the Euler

analysis. The skin friction drag coefficient was calculated by

using the empirical relation based on the turbulent skin friction

coefficient over a flat plate considering the effect of

compressibility8. The boundary layer flow is assumed to be

fully turbulent, because the Reynolds numbers based on the

chord length (6m at the wing-tip section) were 1.7x107 at Mach

5 and 4.5x107 at Mach 0.8. The drag coefficient is obtained as

the sum of the pressure drag coefficient and the skin friction drag coefficient. All the aerodynamic coefficients used

in the airfoil optimization study were normalized by the chord length. The grid topology was C-type in both

transonic and hypersonic analyses. The number of grid points was determined as 221 (parallel to the surface) by 60

(normal to the surface) in transonic analyses and 121 by 50 in hypersonic analyses considering the grid convergence

on the lift-to-drag ratio that was defined as the objective function in our optimization study. Figure 4 shows the grid

for the initial airfoil. The grid for transonic analysis is shown only in the near field of the airfoil, and the far field

boundary is located at a distance 25 times larger than the chord length from the airfoil.

A constrained optimization problem in which the hypersonic lift-to-drag ratio is maximized while the transonic

lift-to-drag ratio and the aerodynamic heating rate (i.e., the leading edge radius) are constrained was solved by the

SQP method. In this method, the Quadratic Programming (QP) problem is solved and the design variable vector is

updated by the solution of the QP problem. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) approximation9 was

applied to the second derivative of the objective function in the QP problem. To solve the QP problem, the

augmented Lagrange multiplier method10

was applied. This method is convenient because it yields not only the

solution of the QP problem but also the corresponding Lagrange multiplier required in the BFGS approximation.

B. Definition of Airfoil Optimization at wing-tip section

As the initial airfoil, whose leading edge radius was 65mm to satisfy the aerodynamic heating requirement

discussed in Section II, an airfoil which approximates the NACA 64A-203 (Fig. 5) was chosen. The upper and lower

surfaces of the airfoil were defined separately by the Bezier curve11

with seven control points as shown in Fig. 5.

The points P0U and P0L were identical and the points P6U and P6L were identical. If a control point moves, then

the airfoil shape also changes. The shape optimization was done by defining the position vectors of the control

(a) Mach 5 (b) Mach 0.8

Figure 4. Grid for airfoil optimization.


5

points as design variables. The z-coordinates of points P2 to P5 were defined as design variables, while points P0

and P6 were fixed. The point P1 was automatically determined to set the leading edge radius to be 65mm.

Table 1 summarizes the parameters used for the airfoil optimization.

1) Design variable vector (x):

The design variable vector was composed of the z-coordinates of the control points (P2 to P5) and the angle of

attack, which is automatically determined as the angle between the uniform flow and the line segment P0 to P6.

2) Objective function (F):

The hypersonic lift-to-drag ratio was the objective function and was maximized.

3) Constraint functions (G1 to G10):

The leading edge radius was constrained to 65mm (G1). The maximum transonic lift-to-drag ratio of the initial

airfoil is about 64 and the corresponding lift coefficient is about 0.51. The transonic lift-to-drag ratio was

constrained to 45 (G2) and the transonic lift coefficient was constrained to 0.36 (G3), because we allowed for a

decrease in transonic aerodynamic performance (30% decrease) rather than in aerodynamic performance at

hypersonic cruise. In the transonic regime, it is desirable to enlarge the supersonic region on the upper surface in

order to minimize the effect of the shock-induced boundary layer separation. To prevent formation of a shock wave

ahead of x=15%c, the pressure gradient along the x-axis (dCp/d(x/c)) was constrained to less than 3.75 in the

transonic regime (G4). This constraint value corresponds to the pressure gradient of the shock wave formed on the

upper surface of the initial airfoil. The lift coefficient at hypersonic cruise is 0.07 to 0.08 considering the concept of

the vehicle discussed in Section II. The hypersonic lift coefficient was therefore constrained to 0.07 (G5). The

hypersonic angle of attack was constrained to less than 5deg considering the pitch attitude angle at hypersonic cruise

(G6). We set constraints for the airfoil thickness (G7 to G10), because the airfoil thickness is expected to become

thin to reduce the hypersonic wave drag. Finally, for all constraint functions, tolerance up to ±1% of the constraint

value was allowed.

C. Result of Airfoil Optimization at wing-tip section

Table 2 shows the result of the airfoil optimization. Figures 6(a) and (b) show the pressure distribution along the

surface at Mach 0.8 and 5, respectively. Figure 6(c) shows the optimum airfoil and the initial airfoil. In this figure,

the z-axis is expanded to emphasize the difference between these airfoils. Figures 6(d) and (e) show the Cp contour

plot of the optimum airfoil at Mach 0.8 and 5, respectively.

The lift coefficient should be increased and the drag coefficient should be decreased to increase the hypersonic

lift-to-drag ratio. The airfoil shape should be changed to increase the lift coefficient, because the hypersonic angle of

attack was constrained to less than 5deg (G6). The location of the crest on the upper surface of the optimum airfoil is

lower than that of the initial airfoil. The flow over the upper surface is expanded more strongly due to the convex

upper surface, resulting in smaller pressure coefficient on the upper surface (Fig. 6(b)). The pressure coefficient on

the lower surface of the optimum airfoil is larger than that of the initial airfoil (Fig. 6(b)) due to the compression

caused by larger front projection area (Fig. 6(c)). As a result of these changes in the airfoil shape, the lift coefficient

was increased from 0.065 to 0.070 (Table 2), which satisfies the constraint (G5). The drag reduction was realized

also by a shape change. The thickness of the optimum airfoil near the leading edge is smaller than that of the initial

Table 1. Parameters for airfoil optimization.

x

z-coordinates of control points (P2 to P5)

Transonic AoA (initially 1.5deg)

Hypersonic AoA (initially 5deg)

F Hypersonic Cl/Cd Maximize

G1 Leading edge radius =65mm

G2 Transonic Cl/Cd =45

G3 Transonic Cl =0.36

G4 dCp/d(x/c), x<15%c <3.75

G5 Hypersonic Cl =0.07

G6 Hypersonic AoA <5deg

G7 Thickness, x=5%c >0.5%c




P0U

Control points

(upper surface)

Control points

(lower surface)

x/c

z/c

P0L

P1L

P2L

P3L

P4LP5L

P6L

P6U

P5U

P4U

P3U

P2U

P1U Initial Airfoil

NACA 64A-203

0.0 0.2 0.4 0.6 0.8 1.0-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Figure 5. Definition of initial airfoil using Bezier curve.


6

airfoil, while the leading edge radius is the

same (i.e., 65mm). The thin airfoil weakens

the bow shock wave and the pressure

coefficient near the leading edge becomes

smaller (Fig. 6(b)), resulting in the smaller

pressure drag coefficient. However, the

constraint function G7, which constrained the

thickness at x=5%c, is not active for the

optimum airfoil (Table 2). To decrease the

hypersonic drag coefficient, the optimum

airfoil should be thin such that the function

G7 is active. Such a scenario is acceptable,

when aerodynamic performance only at

hypersonic speeds is concerned. However,

transonic aerodynamic performance must be

considered at the same time. When the

constraint function G7 is active, the surface

has a large curvature around the leading edge,

as shown schematically in Fig. 7. The

transonic flow expands rapidly along this

large curvature and becomes supersonic on

the upper surface immediately after the

leading edge. But flow can no longer expand

because the upper surface is concave with

respect to the uniform flow in order to satisfy

the constraint function G8. Then a shock

wave is formed and the flow decelerates to subsonic, which means violation of the constraint function G4.

Furthermore, the drag force becomes large due to the high pressure flow behind the shock wave, resulting in the

violation of the constraint function G2. Therefore, considering an appropriate compromise between hypersonic and

transonic aerodynamic performance, the thickness near the leading edge becomes thin as long as the constraint

function regarding the shock wave and the lift-to-drag ratio in the transonic regime can be satisfied. As a result, the

hypersonic drag coefficient was decreased from 0.0146 to 0.0137 (Table 2). The hypersonic lift-to-drag ratio was

increased by 16% from 4.4 to 5.1 (Table 2) due to the increase in the lift coefficients as well as the decrease in the

drag coefficient, while the transonic lift coefficient and lift-to-drag ratio were decreased by 30%.

Table 2. Result of airfoil optimization‡.

Initial Optimum

F 4.4 5.1

G1 65mm 65mm

G2 64 45

G3 0.514 0.362

G4 1.38 3.79

G5 0.065 0.070

G6 5deg 5deg

G7 1.5%c 0.7%c

G8 3.0%c 3.0%c

G9 1.3%c 1.0%c

G10 0.4%c 0.5%c

AoA (Mach 0.8) 1.5deg 1.8deg

Cd p (Mach 5) 0.0125 0.0116

Cd f (Mach 5) 0.0021 0.0021

‡Shaded cells indicate that the inequality

constraint function is active.

x/c

Cp Cp

x/c

Optimum

Initial

Optimum

Initial

0.0 0.5 1.0

-1.2

-0.8

-0.4

0.0

0.4

0.80.0 0.5 1.0

-0.1

0.0

0.1

0.2

0.3

0.4

(a) Cp distribution, Mach 0.8 (b) Cp distribution, Mach 5

x/c

z/c

Optimum Initial

0.0 0.5 1.0-0.04

0.00

0.04

(c) Airfoil

(d) Cp contour plot, Mach 0.8 (e) Cp contour plot, Mach 5

Figure 6. Optimization results.

Leading edge radius is fixed

Large curvature to thin the airfoil

Expand

Supersonic

region

Subsonic

region

Shockwave

Flow

Drag

Figure 7. Schematic of the leading edge.


7

IV. Three-dimensional Shape Study

By using the two-dimensionally optimized airfoil, the initial shape for a fully three-dimensional shape

optimization (referred to as Type I configuration) was defined. In order to evaluate its aerodynamic performance, the

three-dimensional Euler analysis has been carried out. To confirm the improvement in hypersonic aerodynamic

performance due to the airfoil optimization, the configuration whose outer wing was defined by the initial airfoil in

Section III (referred to as Type II configuration) was also analyzed. The wing planform and the cross section of the

fuselage and inner wing were the same between the two configurations (Figs. 1 and 2). Figure 8 shows the three-

dimensional view of the Type I configuration.

A. Method of Analysis

The three-dimensional Euler equations were used for

the governing equations in the flow analysis. The

symmetric TVD scheme was used for the convective term

and the Matrix Free Gauss-Seidel method12

was applied for

the implicit time integration. The lift coefficient and the

pressure drag coefficient were obtained from the Euler

analysis. The skin friction drag coefficient was estimated

by the same method described in Section III. The Reynolds

numbers based on the length of the mean aerodynamic

chord (32.7m) were 9.0x107 at Mach 5 and 2.4x10

8 at

Mach 0.8. Therefore the boundary layer flow is assumed to

be fully turbulent. All the aerodynamic coefficients used in

the three-dimensional shape study were based on the wing

area (600m2) and the length of the mean aerodynamic

chord. The grid topology was C-H type (i.e., C-type in the

chordwise direction and H-type in the spanwise direction) in both

transonic and hypersonic analyses. The numbers of grid points were

281 (in the chordwise direction), 71 (in the spanwise direction), and

60 (in the normal direction to the surface) in transonic analyses and

201, 61, and 46 in hypersonic analyses. Figure 9 shows the grid for

hypersonic analyses (Type I configuration). The grid for transonic

analyses was similar to that for hypersonic analyses except for the

location of the far field boundary which is located at a distance 15

times larger than the body length from the vehicle.

B. Result of Three-dimensional Shape Study

1) Aerodynamic performance

The pressure distributions along the airfoil are shown in Fig. 10

and the surface density contour plot (normalized by the free stream

density) with the plot of the surface streamlines is shown in Fig. 11

for the Type I configuration at Mach 5. It should be noted that the

pressure distribution at y=14.9mm (Fig. 10(c)) is almost the same as

that of the optimum airfoil in Section III (Fig. 6(b)). This fact implies

that our method of the two-dimensional airfoil optimization is

reasonable. The characteristic phenomenon of this optimum airfoil is the compression near the leading edge on the

lower surface and this phenomenon can be seen over the entire outer wing (Fig. 11).

The pressure coefficient at the leading edge of the fuselage and outer wing is large due to the shock wave (Figs.

10(a) and 10(c)). The inner wing has a subsonic leading edge, which results in a small pressure coefficient at the

leading edge (Fig. 10(b)). Therefore, the wave drag is small at the inner wing, even though the thickness-to-chord

ratio is large (i.e., 10%c at the thickest section) to have a large amount of fuel tank volume in it. The local surface

inclination, which is positive when the surface is windward, is negative on the upper surface of the fuselage and the

inner wing (e.g., behind x=75%c at y=0m in Fig. 10(a)), which makes the flow expand. As a result of this expansion,

the suction force acts on the surface where the local surface inclination is negative, resulting in the increase in the

drag force. Therefore, the dominant source of the drag force is different in each section: (a) the drag force at the

outer wing is mainly due to the wave drag, (b) at the inner wing, the drag force due to the suction force is large, and

Figure 8. Three-dimensional view (Type I).

Figure 9. Grid for three-dimensional

hypersonic analyses.


8

(c) the drag force at the fuselage is

mainly caused by both the wave drag

and the suction force. Table 3 shows

the breakdown of the lift and drag

coefficients. The largest contribution

to the drag force is made by the inner

wing. This is mainly due to the large

wing area (458m2), because the two-

dimensional drag coefficient of the

airfoil is small due to the small wave

drag. The largest contribution to the

lift force is made by the inner wing.

The suction force due to the negative

pressure coefficient on the upper

surface (Fig. 10(b)) produces not only

drag force but also lift force. As a

result, the lift-to-drag ratio of the inner

wing alone is comparable to that of the outer wing created by using the optimum airfoil (Table 3). On the other hand,

the fuselage and the outer wing produce a lift force mainly by the compression on the lower surface (Figs. 10(a) and

10(c)). The total lift coefficient is about 0.07 which satisfies the required lift coefficient at the hypersonic cruise

(0.07-0.08). The lift-to-drag ratio is 4.87 which is larger than that of the Type II configuration (4.72) by 3%. This

improvement in the lift-to-drag ratio is attributed to the optimization of the airfoil at the outer wing for the following

reasons: (a) the surface stream lines over the outer wing are almost parallel to the x-axis (Fig. 11), which means that

the flow is two-dimensional and the design of the airfoil is important to realize high aerodynamic performance, and

(b) the lift-to-drag ratio of the outer wing for the Type I configuration (5.73) is larger than that for the Type II

configuration (5.06), while those of the fuselage and inner wing are almost the same (Table 3). Consequently, it is

useful to apply the two-dimensionally-optimized airfoil to the wing when the sweep-back angle is small.

The pressure distributions along the airfoil are shown in Fig. 12 and the surface density contour plot (normalized

by the free stream density) with the plot of the surface streamlines is shown in Fig. 13 for the Type I configuration at

Mach 0.8.

Unlike the result of the hypersonic analysis, the pressure distribution at y=14.9mm (Fig. 12(c)) is not the same as

that of the optimum airfoil in Section III (Fig. 5(c)), mainly because the angle of attack is higher than that of the

optimum airfoil (Table 2) to realize the lift coefficient that is required for the transonic flight (about 0.18). However,

the flow over the outer wing is almost two-dimensional (Fig. 13) due to the small sweep-back angle and is

-0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0-0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0

x/c

Cp

x/c x/c

Cp Cp

-0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0

Cl=0.0475

Cdp=0.0121

Cl=0.0539

Cdp=0.0095

Cl=0.0766

Cdp=0.0114

(a) y=0m (Fuselage) (b) y=5.8m (Inner wing) (c) y=14.9m (Outer wing)

Figure 10. Pressure distributions along airfoil (Type I, Mach 5, AoA=5deg).

Figure 11. Surface density contour plot & surface streamline

(Type I, Mach 5, AoA=5deg).

Table 3. Breakdown of CL & CD (Mach 5, AoA=5deg).

Type I Type II

CL CD p

CD f

CL/CD CL CD p

CD f

CL/CD

Fuselage 0.0138 0.0033 0.0005 3.69 0.0138 0.0033 0.0005 3.69

Inner wing 0.0379 0.0061 0.0014 5.08 0.0378 0.0061 0.0014 5.07

Outer wing 0.0191 0.0030 0.0004 5.73 0.0178 0.0031 0.0004 5.06

Total 0.0707 0.0123 0.0022 4.87 0.0694 0.0125 0.0022 4.72


9

accelerated to supersonic speeds. As a

result, the outer wing produces as

large lift force as the inner wing,

despite its small wing area, as shown

in Table 4. The outer wing also

realizes a large lift-to-drag ratio.

Therefore, the airfoil of the outer wing

is important to obtain high transonic

aerodynamic performance.

However, the airfoil of the outer

wing was optimized to improve

hypersonic aerodynamic performance

with the compromise on transonic

aerodynamic performance. The lift-to-

drag ratio of the Type I configuration

is therefore smaller than that of the

Type II configuration whose outer

wing is based on the NACA 6

series airfoil which is suitable for

transonic vehicles (Table 4).

Figure 14 shows the relation

between the lift-to-drag ratio and

the lift coefficient. The

hypersonic lift-to-drag ratio of the

Type I configuration is larger than

that of the Type II configuration

at the lift coefficient for the

hypersonic cruise (0.07-0.08). The

maximum hypersonic lift-to-drag

ratio of the Type I configuration is

about 5, which is comparable to

that of the caret-wing waverider (about 5 to 6)13

. On the other hand, the transonic lift-to-drag ratio of the Type I

configuration is smaller than that of the Type II configuration at the lift coefficient for the transonic flight (about

Table 4. Breakdown of CL & CD (Mach 0.8, AoA=3deg).

Type I Type II

CL CD p

CD f

CL/CD CL CD p

CD f

CL/CD

Fuselage 0.0201 0.0022 0.0011 6.14 0.0218 0.0023 0.0011 6.49

Inner wing 0.0854 0.0050 0.0031 10.52 0.0941 0.0058 0.0031 10.61

Outer wing 0.0749 0.0016 0.0008 31.28 0.0920 0.0020 0.0008 32.41

Total 0.1804 0.0088 0.0050 13.09 0.2079 0.0101 0.0050 13.80

Type I

Type II

CL

CL/C

D

CL/C

D

CL

Type I

Type II

0.00 0.02 0.04 0.06 0.08 0.101

2

3

4

5

6

0.05 0.10 0.15 0.20 0.2510

11

12

13

14

15

(a) Mach 5 (b) Mach 0.8

Figure 14. Lift-to-drag ratio (Type I & II).

-2

-1

0

1

0.0 0.5 1.0-2

-1

0

1

0.0 0.5 1.0

x/c

Cp

x/c x/c

Cp Cp

-2

-1

0

1

0.0 0.5 1.0

Cl=0.0671

Cdp=0.0083

Cl=0.1464

Cdp=0.0091

Cl=0.2827

Cdp=0.0032

(a) y=0m (Fuselage) (b) y=5.8m (Inner wing) (c) y=14.9m (Outer wing)

Figure 12. Pressure distributions along airfoil (Type I, Mach 0.8, AoA=3deg).

Figure 13. Surface density contour plot & surface streamline

(Type I, Mach 0.8, AoA=3deg).


10

0.18). However, the maximum transonic lift-to-drag ratio of the

Type I configuration is about 13, which is acceptable considering

transonic aerodynamic performance of the Concorde (i.e.,

L/D=11.5 at Mach 0.95)14

. To realize higher aerodynamic

performance, the fully three-dimensional shape optimization

should be performed. The three-dimensional shape in our study

shows good aerodynamic performance both in the transonic and

hypersonic regimes. Therefore, it can be used for a good initial

guess for the fully three-dimensional shape optimization.

2) Aerodynamic center and pressure center

The positions of the aerodynamic center (XAC) and pressure

center (XCP) are shown in Fig. 15 for the Type I configuration.

The pressure center is evaluated along the path shown in Fig. 2.

Because the outer wing is located at the rear of the vehicle and

produces large lift force at Mach 0.8 (Table 4), the aerodynamic

center at Mach 0.8 is located relatively backward (i.e., 68.5%LB).

The pressure center moves backward further when the Mach

number is about 1, because the fuselage and inner wing produce

large lift force near the trailing edge (Fig. 16) due to the

expansion of supersonic flow on the upper surface where the local surface

inclination is negative. Therefore, the aerodynamic center travels backward

as the Mach number approaches to 1. The contribution of the outer wing to

the overall lift force becomes small as the Mach number becomes large

(Tables 3 & 4). Therefore the pressure center as well as the aerodynamic

center travels forward as the Mach number approaches to 5. The center of

gravity should be located ahead of 62%LB to obtain the longitudinal static

stability. This requirement is reasonable, because the centroid of the

planform area is 64%LB. The canard-wing may be needed to cancel the

pitch-down moment in the transonic and low supersonic regimes.

3) Aerodynamic heating rate

In our study, the aerodynamic heating rate is evaluated by the empirical

equation. However, the shock cone generated at the nose impinges onto the

leading edge of the outer wing (Fig. 17). The aerodynamic heating may be

significantly augmented at this impinging point. In the fully three-

dimensional shape optimization to be performed, the wing planform shape

should be determined considering such a phenomenon.

V. Conclusion

A three-dimensional shape study was performed based on the airfoil optimization considering aerodynamic

performance both in the transonic and hypersonic regimes as well as the aerodynamic heating rate. Results show

that: a) The two-dimensional hypersonic lift-to-drag ratio can be improved by the airfoil optimization, but the airfoil

is constrained to be thick near the leading edge due to the requirement for transonic aerodynamic performance,

which restricts the improvement in the hypersonic lift-to-drag ratio, b) the aerodynamic performance is improved

Centroid

of planform area

Mac

h n

um

ber

x [%LB]

XAC

XCP

0 20 40 60 80 100

1

2

3

4

5

Figure 15. Aerodynamic center & pressure

center (Type I).

Cp

x/c

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.0 0.5 1.0

y=5.8m

(inner wing)

Figure 16. Pressure distribution

(Type I, Mach 0.95, AoA=1deg).

0.013.0

Figure 17. Density contour plot normalized by free stream density (Type I, Mach 5, AoA=5deg).


11

though only the two-dimensional optimization is done for the outer wing, and c) the canard-wing will be needed to

cancel the pitch-down moment in the transonic and low supersonic regimes.

To realize higher aerodynamic performance, the fully three-dimensional shape optimization should be performed.

The three-dimensional shape based on the two-dimensionally optimized airfoil will be suitable for the initial guess

of the fully three-dimensional shape optimization.

Acknowledgments

This work is supported in part by Grant-in-Aid for Scientific Research 17360408 of the Japan Society for the

Promotion of Science.

References 1JAXA vision, URL: http://www.jaxa.jp/about/2025/index_e.html 2Lobbia, M. and Suzuki, K., “Design and Analysis of Payload-Optimized Waveriders,” AIAA Paper 2001-1849, May 2001. 3Tauber, M. E., J. V. Bowles, and Lily Yang, “Use of Atmospheric Braking During Mars Missions,” Journal of Spacecraft

and Rockets, Vol. 27, No. 5, 1990, pp. 514-521. 4Fletcher, R., Practical Methods of Optimization, John Wiley and Sons, 1987 5Powell, M.J.D., Variable Metric Methods for Constrained Optimization, Mathematical Programming: The State of the Art,

(A. Bachem, M. Grotschel and B. Korte, eds.) Springer Verlag, 1983, pp. 288-311. 6Yee, H. C., “A class of high-resolution explicit and implicit shock-capturing methods,” NASA TM 101088, 1989 7Yoon, S. and Jameson, A., “An LU-SSOR Scheme for the Euler and Navier-Stokes Equations,” AIAA Paper 87-0600, 1987 8Raymer, D. P., Aircraft Design: A Conceptual Approach, 4th ed., AIAA Education Series, AIAA, 2006 9Fletcher, R., “A New Approach to Variable Metric Algorithms,” Computer Journal, Vol. 13, 1970, pp. 317-322. 10Andrzej Ruszczynski, Nonlinear Optimization, Princeton University Press, 2006 11Samuel R. Buss, 3-D Computer Graphics: A Mathematical Introduction with OpenGL, Cambridge University Press, 2003 12Shima, E., “A Simple Scheme for Structured/Unstructured CFD,” Proceedings of the 29th Fluid Dynamic Conference, 1997,

pp. 325-328. (in Japanese) 13”Investigation report of development trend of next generation airplane/space plane, Vol. 2, investigation on development

trend of SST/HST,” The Society of Japanese Aerospace Companies, 1988 (in Japanese) 14Rech, J. and Leyman, C. S., “A Case Study by Aerospatiale and British Aerospace on the Concorde,” AIAA Professional

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