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Design and Optimization Method for Hypersonic Quasi-Waverider
Wen Liu,∗Chen-An Zhang,†and Fa-Min Wang‡
Institute of Mechanics, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China
and
Zheng-Yin Ye§
Northwestern Polytechnical University, 710072 Xi’an, People’s Republic of China
https://doi.org/10.2514/1.J059087
The waverider has an extensive application prospect in the design of hypersonic vehicles due to its excellent
aerodynamic efficiency. However, it is known that the original cone-derived waverider is longitudinally unstable. To
solve the problem, a design method for hypersonic quasi-waverider configuration is first proposed andthen a genetic
algorithm optimization framework is constructed to obtain optimum quasi-waveriders with different constraints.
During the optimization, the aerodynamic performance is evaluatedby an efficient aerodynamic model that considers
the impacts of strong viscous interaction effects. Results from numerical simulations show that, for the optimum
quasi-waveriders without constraints, good shock wave attachment along the leading edge is achieved, and the
maximum lift-to-drag ratio (L∕D) is even higher than that of the original waverider. Optimized quasi-waveriders are
also generated based on the constraints of volumetric efficiency and stability. The L∕Dfirst increases and then
decreases with the increase in volumetric efficiency. In addition, an interesting phenomenon is found that the L∕Dis
reduced almost linearly with the increase in the degree of stability at each design condition. Furthermore, a linear
relationship is also constructed between the variation of L∕Dwith respect to degree of stability and the viscous
interaction parameter
V0.
Nomenclature
ai= coefficient of base function
bi= power of the curve equation
CD= drag coefficient
CDfric = friction drag coefficient
CDwave = wave drag coefficient
CL= lift coefficient
Cm= pitching moment coefficient
CN= normal force coefficient
ds = degree of stability
H= flight altitude
L= length of the waverider, 4 m
M∞= Mach number
p= pressure
Slower = surface area of the lower surface
Supper = surface area of the upper surface
T= temperature
V= volume of the waverider
V0= viscous interaction parameter
Xac = aerodynamic-center location
Xcp = center-of-pressure location
xle =Xcoordinate of the leading-edge point
yte = height of the waverider at the symmetry plane
α= angle of attack
β= shock wave angle
γ= ratio of the specific heats, 1.4 for perfect gas
Δamax = maximum relative variation of the design variables ai
during the optimization
δ
x= boundary-layer displacement thickness
θ= deflection angle of body surface relative to the free-
stream direction
ρ= density
I. Introduction
HYPERSONIC flight is drawing more and more attention from
researchers all over the world. Several past and ongoing pro-
grams have been executed to investigate the key technologies of
hypersonic vehicles and advance the state-of-the-art in hypersonic
aerodynamics, such as National Aerospace Plane [1], Force Applica-
tion and Launch from Continental United States [2], and Hypersonic
International Flight Research [3]. Various kinds of configurations are
designed based on different mission requirements. A common chal-
lenge encountered by different hypersonic vehicles is how to obtain
high lift-to-drag ratio (L∕D). Higher L∕Dmeans larger down and
cross range, which is usually a key driving parameter behind any
vehicle design. However, the improvement of L∕Dis especially
difficult due to the severe wave drag and friction drag at hypersonic
flight conditions. In fact, according to a well-known survey by Kuche-
mann [4], a type of “L∕Dbarrier”exists for traditional hypersonic
vehicles.
To break the L∕Dbarrier, the concept of waverider proposed by
Nonweiler [5] came to renaissance in the 1980s. Differing from the
flow physics around traditional hypersonic configurations, the shock
wave is attached to the entire leading edge of the waverider at the
design condition, thus preventing the leakage of high-pressure gas
from the lower surface onto the upper surface and achieving excellent
aerodynamic efficiency. Nevertheless, the L∕Dadvantage of waver-
ider was widely questioned at early stage by main concerns for
hypersonic viscous flow effects, aerothermodynamic effects of the
sharp leading edge, limited volumetric efficiency, and off-design
performance. Such skepticism began to be eliminated gradually
Since the concept of viscous optimized waverider was proposed by
Bowcutt et al. [6] and Corda and Anderson [7], where the viscous
effects were incorporated into the optimization process for the first
time. The viscous optimized waveriders became the first hypersonic
configurations to break the aforementioned L∕Dbarrier. Since then,
lots of researches have been carried out to help expand the appli-
cability of waveriders to realistic aerospace missions. For example,
various waverider design and optimization methods from different
generating flowfield were developed to improve the volumetric
efficiency, payload ability, and flexibility of the geometry [8–17];
Received 23 September 2019; revision received 30 November 2019;
accepted for publication 31 December 2019; published online 27 January
2020. Copyright © 2019 by the American Institute of Aeronautics and
Astronautics, Inc. All rights reserved. All requests for copying and permission
to reprint should be submitted to CCC at www.copyright.com; employ
the eISSN 1533-385X to initiate your request. See also AIAA Rights and
Permissions www.aiaa.org/randp.
*Assistant Professor, State Key Laboratory of High-Temperature
Gas Dynamics; lw@imech.ac.cn.
†Associate Professor, State Key Laboratory of High-Temperature
Gas Dynamics; zhch_a@imech.ac.cn (Corresponding Author).
‡Professor, State Key Laboratory of High-Temperature Gas Dynamics.
§Professor, School of Aeronautics.
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several leading-edge blunting methods were studied to achieve a
good balance between the aerodynamic and aerothermodynamic
concerns [18,19]; off-design performance was evaluated from vari-
ous aspects [20,21]. In addition, efforts on incorporation of the
waveriders into different practical hypersonic vehicles were also
made [22–25].
Despite the enriched researches mentioned above, one urgent topic
for waveriders to be solved is the problem of longitudinal stability.
Modern active control system makes it possible for the vehicles to be
acceptably unstable at hypersonic conditions [26]. However, a good
type of configuration should have the potential to flexibly change the
stability margin during the aerodynamic design according to the
overall performance requirements. Past researches have shown that
the cone-derived waverider is statically unstable because the center of
pressure is in front of the center of gravity when the latter is located at
the center of volume [27]. In fact, in actual aircraft design, the center
of gravity usually does not coincide with the center of volume and can
be moved forward by appropriate adjustment of the payload position
or using some ballast weight. Jia et al. [28] derived the variation trend
of the center of pressure with angle of attack for different streamlines
by applying the Newtonian theory on a simplified streamline model,
which is introduced in detail in the Appendix. They have found that,
for a concave streamline, the center of pressure moves forward
monotonously as the angle of attack increases, whereas the trend is
just opposite for a convex one. A sketch map of different curves and
corresponding center of pressure is given in Fig. 1. A self-trimmed
vehicle is generally preferred because no extra trim drag is produced
by the elevator. To satisfy the trim requirement, the center of gravity
should coincide with the center of pressure at the design angle of
attack. Then a concave streamline would be statically unstable in that
the center of pressure being in front of the center of gravity tends to
produce a nose-up moment as the angle of attack increases. From this
perspective, the idealized cone-derived waverider is statically unsta-
ble because the streamlines of the lower surface are concave.
Several design and optimization methods may be employed to
improve the longitudinally static stability of the cone-derived waver-
ider. First, the freestream upper surface can be disturbed to alter the
center-of-pressure location. However, minor modification may only
work at small angles of attack due to the expansion effects, whereas
major modification may lead to large loss of L∕D. Second, compared
with the cone-derived waveriders, ones derived from the axisymmetric
power-law body flows may exhibit better longitudinal stability beca use
the streamlines are convex [29]. However, no analytical solutions exist
for such base flows, and instead they are usually calculated using
computational fluid dynamics (CFD) codes [29,30]. Then the process
for choosing an optimum base flow according to different stability
constraints may be time-consuming and complex. An initial study on
this topic is given by Wang et al. [30]. Another alternative is to directly
modify the original concave lower surface of cone-derived waveriders.
In fact, affected by the strong viscous interactions at large Mach
numbers and high altitudes, the effective shape would differ from the
original shape apparently and reasonable modification of the lower
surface can even obtain a waverider with better aerodynamic perfor-
mance [31]. Accordingly, appropriate deviation and optimization from
an idealized waverider surface may be feasible.
Following the last idea mentioned above, a design and optimiza-
tion method for hypersonic quasi-waverider configuration is pro-
posed in this paper. First, the definition and generation process of the
lower surface is introduced. Then the optimization by genetic algo-
rithm (GA) is carried out, during which the volumetric efficiency,
trim, and longitudinal stability (the location of center of pressure and
aerodynamic center) can be taken into account. To improve the
optimization efficiency, the aerodynamic performances are evaluated
by an aerodynamic model that can consider the strong viscous
interaction effects by a semi-empirical method. Finally, CFD is used
to calculate the aerodynamic forces of the resulting optimum shapes
with different constraints.
II. Definition of the Quasi-Waverider
The proposed configuration is named as “quasi-waverider”in that
it can be generated from any original waverider by the follow-
ing steps:
1) The leading edge of the original waverider is kept unchanged.
2) At a different longitudinal plane, the profile of the lower surface
is determined by the same curve equation, started from the point at the
leading edge and cutoff at the base plane. The curve equation is
determined by the sum of a series of power law functions, whose
coefficients are varied during the optimization according to different
design objectives. In such ways, good uniformity of pressure distri-
bution is expected and the flexibility of the geometry generation can
also be improved greatly.
In fact, the profile curve of the lower surface can also be located
along the circumferential direction just as the streamline of the
waverider does. However, we find that no essential difference exists
between the two methods and the obtained configurations have very
close aerodynamic performances. Therefore, here the profile curve is
located along the longitudinal plane for simplicity.
3) The height of the symmetry plane is kept the same as that of the
original waverider or slight deviation from it, so that the shock wave
position and volumetric efficiency is held not changed much.
4) Finally, the freestream upper surface is employed.
Later results will show that good shock wave attachment can be
obtained for the quasi-waveriders. However, they cannot “ride”on
the shock wave along the whole leading edge as perfectly as that of
the original waverider. Then referring to the definition and difference
of the steady flow,quasi-steady flow, and unsteady flow, the proposed
configuration here is named as “quasi-waverider.”
A. Determination of the Leading Edge
In practical engineering applications, the leading edge of any kind
of waveriders can be used according to various design requirements.
Referring to the idea of Bowcutt et al. [6], here the leading edge of a
typical viscous optimized cone-derived waverider is employed. The
a) Curves
X
Y
01234
-0.4
-0.3
-0.2
-0.1
0convex
concave
α
(deg)
Xcp
0246810
0.3
0.4
0.5
0.6
0.7 convex
concave
b) Center of pressure
Fig. 1 Sketch map of different curves and corresponding center of pressure.
LIU ET AL. 2133
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detailed optimization process is introduced in Ref. [31]. The design
condition and relevant parameters are listed in Table 1.
The waverider is determined by the base curve on the base plane
shown in Fig. 2 and the generated viscous optimized waverider is
shown in Fig. 3. The base curve on the left half ( Z≥0), defined on the
shock wave circle with a nondimensionalized radius of 0.149, can be
expressed as the following third-order polynomial:
Y−0.0687–6.36Z224.12Z3(1)
B. Generation of the Lower Surface
To change the concavo-convex characteristics of the lower surface
flexibly, here a series of power law functions are taken as the base
functions and the profile curve can be determined as follows:
y
LX
11
i1
aix
Lbi
(2)
where aiis the coefficient of the base function and a1–a10 are the
design variables; the power biis determined as bi0.5
0.1i1≤i≤11. After the height of the quasi-waverider at the
symmetry plane yte and the values of control variables are given,
a11 can be calculated as
a11 −yte
L−X
10
i1
ai(3)
Here the height yte is set to be the same as that of the original cone-
derived waverider at the symmetry plane, which is equal to 0.44376 m.
According to the definition of the lower surface depicted above, the
profile curve of the longitudinal cross section at any other position
can be determined as
y
LX
11
i1
aix−xle
Lbi
(4)
where xte denotes the Xcoordinate of the leading-edge point at the
corresponding cross section.
In practical applications, the number of design variables and the
power bican be assigned flexibly based on the specified optimization
problem. Generally, more control variables means more iterative
steps and less efficient optimization process, but can produce more
refined optimization results. Because the aerodynamic forces are
calculated using an efficient aerodynamic model during the optimi-
zation, the time spent by more iterative steps is acceptable. Conse-
quently, a total of 10 design variables are used for this problem.
Figure 4 describes the profile curves determined by different single
base functions. It can be inferred that the geometry of lower surface
can be changed in a large range through the combination of the
different base functions given above.
III. Aerodynamic Model
Aerodynamic m odels are widely used in various kinds of hypersonic
aerodynamic shape optimization during the preliminary design due to
the high efficiency and reasonable accuracy, such as the Newtonian
flow, tangent-cone/wedge, and shock-expansion theory. Among these
approximate methods, the tangent-cone method is especially popular
and frequently yields very reasonable results when applied to three-
dimensional hypersonic slender shapes, which is further improved by
Cruz and Sova [32] with higher accuracy. Therefore, the improved
tangent-cone method is employed here to approximate the inviscid
pressure on the windward surface and the hypersonic expansion-wave
relation is used on the leeward surface [33]:
p8
>
>
>
>
>
<
>
>
>
>
>
:
12γ
γ1M2
∞sin2β−γ−1
γ1⋅1γM∞β−θ2cos2β
1γ−1∕2M2
∞sin2β−1K≥0
1γ−1
2K2γ∕γ−1−2
γ−1<K<0
0K≤− 2
γ−1(5)
where ppw∕p∞, the hypersonic similarity parameter KM∞θ,
and θis the deflection angle of body surface relative to the freestream
direction. The shock wave angle βcanbeapproximatedas[34]
βγ1
γ−12
41
12γ3
γ12M2
∞θ2
s3
5⋅θ(6)
One phenomenon that cannot be neglected in hypersonic flow is
the strong viscous interactions, which have a significant effect on the
pressure distribution of the body surface. Such influence can be
captured by the concept of effective shape, namely, the original body
plus the boundary-layer displacement thickness. A feasible effective
shape determination method was put forward recently in [31] accord-
ing to a vorticity criterion. However, the method is based on the
numerical solutions, which prevents its application to the rapid opti-
mization problem. Therefore, an engineering approach put forward by
Bertram [35] is adopted to obtain the boundary-layer displacement
thickness rapidly, which was also employed by Anderson et al. [36] in
the waverider optimization. The expression is as follows:
Table 1 Design condition and
geometric parameters used to
define the waverider
HM
∞βL
60 km 15 8.5 deg 4 m
Fig. 2 Generation of cone-derived waverider.
Fig. 3 Viscous optimized waverider.
X(m)
Y(m)
0
-0.5
-0.4
-0.3
-0.2
-0.1
0b=0.6
b=0.8
b=1.0
b=1.2
b=1.6
1234
Fig. 4 Profile curves determined by different single base functions.
2134 LIU ET AL.
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dδ
x
dx ξ
M∞
p
p1ξ
2p⋅dp
dξ(7)
where
ξ0.425γ−1Tw
Tad 0.35
χ
χM3
∞
Rex;∞
p
Cw
p
CwT
T∞1∕2T∞110.4
T110.4 (8)
where the subscripts ∞and wrepresent the quantities based on undis-
turbed freestream conditions and wall conditions, respectively; Tad is
the adiabatic wall temperature; and Tis the reference temperature,
which can be approximated using the following expressions [37,38]:
Tad
T∞1
Pr
p⋅γ−1
2M2
∞(9)
T
T∞1.28 0.023M2
∞0.58Tw
T∞
−1.0(10)
where Pr0.72 and Tw1000 K are assumed.
The detailed steps for calculating the aerodynamic forces are
summarized as follows:
1) Calculate the pressure distribution on the original body surface
using Eq. (4).
2) Calculate the viscous interaction parameters
χand ξby
Eqs. (7–9).
3) Combine Eqs. (4) and (6) to obtain an ordinary differential
equation about dp∕dξ, which is then solved by the fourth-order
Runge–Kutta method starting at the leading edge and progressing
in the Xdirection with the initial conditions: χ0,ppw;0∕p∞,
where pw;0is the pressure calculated in step 1. More detailed solving
process is given in [35].
4) The flight altitude studied in this paper is above 40 km. There-
fore, a full laminar flow is assumed due to the low Reynolds number.
A laminar skin friction formula that can consider the strong viscous
interaction effects is employed [37]:
Cf0.664
pxCw
Rex;∞
s(11)
5) The base pressure is approximated by the freestream static
pressure.
6) Obtain the final aerodynamic performance by the integration of
forces on body surface.
IV. Genetic Algorithm Optimization Framework
GAs perform a global search from a population of individuals by
mimicking the process of evolution without depending on the gra-
dient information, which can not only overcome the defect of being
liable to obtain the local optimum for some traditional algorithms, but
also be easily incorporated into existing frameworks. Therefore, GAs
have been used extensively in the problem of aerodynamic shape
optimization [39–41]. Here a real-coded-based GA is applied to the
quasi-waverider optimization, where the fitness, chromosomes, and
genes correspond to the objective function, design candidates, and
design variables, respectively. After numerous attempts for achieving
rapid convergence and robust optimum results, the parameters listed
in Table 2 are employed for the present study.
The GA optimization framework of the quasi-waverider is shown
in Fig. 5, where the constraints of volumetric efficiency, trim, and
longitudinal static stability are included. Note that, if no constraint is
assigned, the program will directly pass the corresponding step to the
next one.
In addition, the fitness is assigned according to the weight of L∕D:
fitnessiL∕Di
P30
i1L∕Di
(12)
The volumetric efficiency is defined as
Veff V2∕3
Supper Slower
(13)
V. Computational Fluid Dynamics Solver
A. Numerical Method
A cell-centered finite volume method is employed to solve
the three-dimensional compressible Euler or Navier–Stokes (N-S)
equations. The AUSMspatial discretization scheme is adopted,
with an implicit lower–upper symmetric Gauss–Seidel scheme for
the temporal integration to accelerate convergence. More details
about the CFD solver and its validation at hypersonic conditions
can be found in [31,42].
B. Grid Independence Validation
A half-model grid with approximately 600,000 cells is used,
shown in Fig. 6. The grid independence study is conducted using a
grid with approximately 1,200,000 cells, refined along the stream-
wise direction on the body. The corresponding force coefficients are
listed in Table 3, including the lift coefficient, wave drag coefficient,
friction drag coefficient, and the center of pressure. The reference
area of the half-model is 1.828 m2, being equal to the projected area
toward the X–Zplane. The laminar flow model is employed. We can
see from Table 3 that the results from two grids are very close.
Therefore, the coarser grid is used herein to save the computa-
tional costs.
VI. Results and Analysis
The current study is focused on the design condition of Mach 15,
including the Euler results and N-S results at three typical flight
altitudes of 40, 50, and 60 km. Then the optimum quasi-waverider
configurations are obtained through the GA optimization framework
by incorporating different constraints: 1) no constraint; 2) constraint
of volumetric efficiency; 3) constraint of trim and stability. Note that,
for the former two problems, the maximum L∕Dat small angles of
attack is taken as the design objective; whereas for the third problem,
the maximum L∕Dfor given lift coefficient is pursued due to the trim
requirement. In addition, the calculation condition is kept identical to
the corresponding design condition of the optimization program.
Before detailed analysis of the results, the aerodynamic model
used for this study is first validated in comparison with CFD. Taking
the optimum quasi-waverider obtained at H60 km without con-
straint (named “QW2,”to be introduced later) as an example, Fig. 7
compares the aerodynamic forces calculated by CFD and the
Table 3 Comparison of force
coefficients at the condition M∞15,
H60 km,α0 deg
Performance Coarser grid Finer grid
CL×10−22.910 2.914
CDwave ×10−33.771 3.774
CDfric ×10−37.557 7.565
Xcp 0.6353 0.6351
Table 2 Parameters used in GA
Population Generation pc pm ai(1≤i≤10)Δamax
30 200 0.8 0.1 −yte;y
te10%
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aerodynamic model (AeroModel) at angles of attack from 0 to 6 deg.
Obviously, the results from the efficient aerodynamic model match
the CFD data reasonably well at the strong viscous interaction con-
dition, with the maximum relative difference of the lift coefficient,
drag coefficient, L∕D, and center of pressure being only 3.52, 3.01,
3.71, and 1.06%, respectively.
A. Optimum Quasi-Waveriders Without Constraint
1. Inviscid Design Condition
When the viscous effects are not taken into account during the
optimization, the maximum L∕Dat angles of attack from 0 to 4 deg is
taken as the design objective. Several comparisons between the
generated optimum quasi-waverider (named “QW1”) and the origi-
nal viscous optimized waverider (CW) are shown in Fig. 8, including
the profile, L∕D, and pressure contour of the flowfield. In Fig. 8a, it
can be observed through the curves at the symmetry plane that the
quasi-waverider is slightly thinner than the waverider. According to
the result in Fig. 8b, the L∕Dof the quasi-waverider is even slightly
higher than that of the original waverider, with the maximum value
being improved from 9.28 to 9.38. The reason can be explained by
Fig. 8c, where the pressure contour of the flowfield is compared.
It shows that good shock wave attachment along the leading edge of
the quasi-waverider is maintained with only a little spillage at cross
sections near the base, which proves the rationality of the quasi-
waverider design method. In addition, we can also find that the
thinner nose of the quasi-waverider leads to a weaker shock wave,
making the drag wave lower and L∕Dhigher than those of the
original waverider.
2. Design Condition at Different Altitudes
When the viscous effects at different flight altitudes are considered,
the maximum L∕Dat angles of attack from 0 to 6 deg is taken as the
design objective during the optimization. The profile curves at the
symmetry plane of optimum quasi-waveriders at different conditions
are shown in Fig. 9, where “Inviscid”corresponds to the profile curve
of QW1 mentioned above. It shows that different profile curves are
close near the end due to the same height constraint and the difference
mainly exists near the nose: the deflection angle turns smaller as the
flight altitude becomes higher. Such results can be explained by the
strong viscous interactions: as the flight altitude increases, the viscous
interaction effects become stronger and the boundary-layer displace-
ment thickness becomes thicker, making the shock wave stronger for
the same shape. Therefore, in order to weaken the shock wave and
correspondingly reduce the wave drag, the deflection angle of the
optimum quasi-waveridernear the nose tip is smaller at higher altitude.
Comparison of the maximum L∕Dat different conditions between
the original waverider and different optimum quasi-waveriders is
shown in Fig. 10. It shows that at each design condition, the optimum
quasi-waverider has higher maximum L∕Dthan that of the original
waverider. Taking the quasi-waverider optimized at H60 km
(named “QW2”) as an example, the comparison of L∕Dand wave
drag coefficient with the original waverider is given in Fig. 11,
Fig. 5 GA optimization framework of the quasi-waverider.
Fig. 6 Half-model grid used in numerical simulations (approximately
600,000 cells).
2136 LIU ET AL.
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a) CLb) CD
c) L/D d) Xc
p
α
(deg)
CL
0123456
0.02
0.04
0.06
0.08
0.1 CFD
AeroModel
α
(deg)
CD
0123456
0.01
0.015
0.02
0.025
0.03
CFD
AeroModel
α
(deg)
L/D
0123456
2.4
2.7
3
3.3
3.6
CFD
AeroModel
α
(deg)
Xcp
0123456
0.6
0.62
0.64
0.66
0.68
0.7
CFD
AeroModel
Fig. 7 Comparison of results calculated by CFD and aerodynamic model.
b) L/D c) Pressure contour of the flowfield at =0 deg
α
α
(deg)
L/D
01234
5
6
7
8
9
10
CW
QW1
a) Profile
Fig. 8 Comparison of the original waverider and quasi-waverider optimized at the inviscid condition.
LIU ET AL. 2137
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where the abscissa is the lift coefficient. Note that only the wave drag
is compared because the friction drag of the two configurations is
very close. It is found that, for the same lift coefficient, the L∕Dof
the quasi-waverider is higher mainly due to the lower wave drag. The
maximum L∕Dis improved by 2.07% (from 3.38 to 3.45) and the
improvement is larger as the angle of attack increases.
Furthermore, the pressure of the two configurations is compared in
Fig. 12, including the pressure contour of the flowfield and the lower
surface,and pressure distribution at different cross sections. In Fig. 12a,
an apparently weaker shock wave exists around the lower surface of
the quasi-waverider, which is similar to what is shown in Fig. 8c. The
weaker shock wave results in the lower pressure distribution near the
leading edge shown in Figs. 12b and 12c. Therefore, the wave drag of
the quasi-waverider is lower than that of the original waverider.
Based on the above results, the optimum quasi-waveriders opti-
mized at different design conditions all have higher maximum L∕D
compared with the original waverider, which mainly results from the
lower wave drag.
B. Optimum Quasi-Waveriders with Constraint of Volumetric
Efficiency
The volumetric efficiency is an indication of the volume relative to
the surface area and is often incorporated into waverider optimization
program to maintain a balance between usability (high volume) and
aerodynamic performance (high L∕D) [43]. Although the quasi-
waveriders obtained above have higher L∕D, their volumetric effi-
ciency is lower than that of the original waverider. For example, the
volume efficiency of the configuration QW2 and the waverider is
0.1039 and 0.1165, respectively.
Therefore, the constraint of volumetric efficiency is incorporated
into the quasi-waverider optimization. Optimum quasi-waveriders
are obtained at four typical design conditions. The maximum L∕Dof
different quasi-waveriders and the original waverider is plotted in
Fig. 13. It shows that, as the volumetric efficiency increases, the
maximum L∕Dfirst increases and then decreases.
The following analysis is focused on the performance of quasi-
waveriders with the same volumetric efficiency as the original waver-
ider. The corresponding data are rearranged in Table 5. At the inviscid
design condition, the maximum L∕Dof the quasi-waverider is slightly
lower than that of the original waverider; however, when the viscous
effects are considered, the maximum L∕Dof the quasi-waverider is
slightly higher, and the advantage is enlarged at higher altitudes. Also
taking the quasi-waverider at H60 km (named “QW3”)asan
example, Fig. 14 compares the profile and L∕Dbetween the two
configurations. In fact, only very minor difference exists between
the two shapes: the nose region is slightly thinner and the end region
is slightly thicker for the quasi-waverider. The geometric feature
Fig. 9 Comparison of curves at the symmetry plane of optimum quasi-waveriders at different conditions.
H(km)
L/D
340 50 60
3.5
4
4.5
5
5.5
6
CW
QWs
Fig. 10 Comparison of the maximum L∕Dbetween the original waver-
ider and different optimum quasi-waverider at different altitudes.
a) L/D b) CD_wave
C
L
L/D
0.02 0.04 0.06 0.08 0.1
2.4
2.6
2.8
3
3.2
3.4
3.6
CW
QW2
CL
CD_wave
0.02 0.04 0.06 0.08 0.1
0
0.005
0.01
0.015
0.02
0.025
CW
QW2
Fig. 11 Comparison of aerodynamic performance between the quasi-waverider QW2 and the original waverider at H60 km.
2138 LIU ET AL.
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a) Pressure contour of the flowfield b) Pressure contour of the lower surface
c) Pressure distribution at different cross sections
Z(m)
P/P
-0.500.5
2
4
6
8
10
12 Slc: X=1m
Slc: X=2m
Slc: X=3m
Slc: X=4m
QW2
CW
Fig. 12 Comparison of pressure between QW2 and the original waverider at H60 km,α0 deg.
a) Inviscid b) H=40km
c) H=50km d) H=60km
Veff
L/D
0.1 0.11 0.12 0.13
8
8.5
9
9.5
CW
QWs
Veff
L/D
0.1 0.110.120.13
5.2
5.4
5.6
5.8
6
CW
QWs
Veff
L/D
0.1 0.11 0.12 0.13
4.1
4.2
4.3
4.4
4.5
4.6
CW
QWs
Veff
L/D
0.1 0.11 0.12 0.13
3.2
3.3
3.4
3.5 CW
QWs
Fig. 13 Maximum L∕Dof the quasi-waveriders with different volumetric efficiency constraints and the original waverider at different conditions.
LIU ET AL. 2139
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of the thinner nose leads to the lower wave drag and slightly
higher L∕D.
According to the above results, because the viscous effects (includ-
ing the impacts of strong viscous interactions and friction drag) are
taken into account during the optimization process, the optimum
quasi-waverider has minor advantage of L∕Dover the original
waverider even with the same volumetric efficiency at flight con-
ditions of strong viscous interactions.
C. Optimum Quasi-Waveriders with Constraint of Trim and Stability
Trim and static stability are two key factors that must be considered
during the design of any practical vehicle. In order that no extra trim
drag is produced by the elevator at the design condition, the constraint
of trim is incorporated into the optimization of the quasi-waverider,
which includes two requirements: first, the lift coefficient is specified
to balance the weight; second, the center of pressure is coincident
with the center of gravity to make the pitching moment be zero. The
static stability is evaluated by calculating the difference between the
aerodynamic center and the center of gravity (also the center of
pressure here). For example, when the degree of stability is asked
to be over 1% and the center-of-pressure location is evaluated to be
0.6Lfor the given lift coefficient, the aerodynamic-center location
needs to be over 0.61L. In a word, this is a lift-constrained L∕D
maximization problem, where the locations of both the center of
pressure and the aerodynamic center are constrained. The two param-
eters are calculated as follows:
Xcp Cm;nose
CN
;X
ac ∂Cm;nose
∂CL
(14)
where Cm;nose denotes the pitching moment coefficient relative to the
nose of the configuration and CNdenotes the normal force coeffi-
cient. Note that, referring to the standard convention, a positive
pitching moment corresponding to a nose-up moment is ruled here.
First, the characteristic of trim and static stability of the original
waverider is analyzed at different design conditions. The variation of
the center of pressure with angle of attack at different conditions is
shown in Fig. 15, including the lower surface (not integrating the
forces of the upper surface) and the total shape. It shows that, for the
lower surface, the center of pressure moves forward as the altitude
increases at the same angle of attack. This is because the viscous
interaction effects are stronger at higher altitudes, which lead to a
larger pressure increase near the leading edge, as shown in Fig. 16.
However, when the angle of attack increases at the same altitude, the
compression effects of the flow around the lower surface turn
stronger, tending to weaken the viscous interaction effects. There-
fore, at larger angles of attack, the difference of the center of pressure
among different conditions is reduced.
The combination of the above two results changes the variation
trend of the center of pressure with angle of attack. For the inviscid
condition, the center of pressure moves forward as the angle of
attack increases, but the trend is just opposite at H60 km. Such
Table 5 Trim and stability of the waverider at different
conditions, α0 deg
Xcp Xac
Degree of
stability, %
Condition Lower Total Lower Total Lower Total
Inviscid 0.6223 0.6223 0.6104 0.6107 −0.88 −0.96
H40 km 0.6157 0.6201 0.6113 0.6107 −0.20 −0.86
H50 km 0.6102 0.6195 0.6123 0.6108 0.40 −0.77
H60 km 0.6016 0.6157 0.6142 0.6115 1.48 −0.39
Table 4 Maximum L∕Dof the waverider and quasi-
waveriders with the same volumetric efficiency
Design/calculate condition L∕DCW L∕DQW ΔL∕D,%
Inviscid 9.28 9.23 −0.54
H40 km 5.75 5.77 0.35
H50 km 4.43 4.45 0.45
H60 km 3.38 3.41 0.89
a) Profile b) L/D
CL
L/D
0.02 0.04 0.06 0.08 0.1
2.6
2.8
3
3.2
3.4
CW
QW3
Fig. 14 Comparison of profile between the original waverider and QW3 with the same volumetric efficiency at H60 km.
a) Lower surface
α(deg)
Xcp
02468
0.6
0.61
0.62
0.63 Inviscid
40km
50km
60km
α(deg)
Xcp
02468
0.6
0.61
0.62
0.63 Inviscid
40km
50km
60km
α(deg)
ΔXcp
02468
-0.005
0
0.005
0.01
0.015 Inviscid
40km
50km
60km
b) Total waverider c) Difference
Fig. 15 Variation of the center of pressure with angle of attack at different conditions.
2140 LIU ET AL.
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phenomenon can also be explained directly from the geometric
perspective by the variation of the effective shape (ES), which is
shown in Fig. 17. Note that the effective shapes here are obtained
based on a vorticity criterion and the detailed introduction was given
in Ref. [31]. Obviously, as the viscous interaction effects turn
stronger, the effective shape changes gradually from a concave sur-
face to a convex one. Then the result is consistent with the conclusion
from Ref. [28], which is mentioned earlier in the Sec. I.
However, the variation trend of the center of pressure with angle of
attack is different when the upper surface is taken into account, as
shown in Fig. 15b. The difference of the center of pressure (ΔXcp)
between the total surface and the lower surface is also plotted in
Fig. 15c. It can be found that the influence of the aerodynamic force
on the upper surface is more and more evident as the altitude
increases, which is almost negligible at the inviscid condition. This
result is also attributed to the impacts of strong viscous interactions:
the pressure near the leading edge of the upper surface is increased,
shown in Fig. 16, and thus the center of pressure of the total waverider
moves backward.
Furthermore, the center of pressure, the aerodynamic center, and
the degree of stability at α0 deg is listed in Table 5, including the
lower surface and the total waverider. Note again that the degree of
stability is evaluated by calculating the difference between the aero-
dynamic center and the center of pressure. It is shown that, although
the center of pressure moves forward as the altitude increases, the
aerodynamic center moves backward slightly. However, only the
lower surfaces at the conditions of H50 km and H60 km
are statically stable, which is caused by the variation trend of the
center of pressure moving backward with angle of attack. In fact, the
aerodynamic center varies little with angle of attack. Then taking any
angle of attack plotted in Fig. 15 as the design condition, the lower
surface is statically stable because the aerodynamic center is located
behind the center of pressure. Nevertheless, the aerodynamic forces
on the upper surface reduce the degree of stability, making the total
waverider statically unstable.
From the above analysis, we can infer that in order to make the total
configuration satisfy the requirement of trim and static stability with
the loss of L∕Dbeing as low as possible, an optimized lower surface
with more convex geometric feature is necessary, which can be
realized by the quasi-waverider optimization method.
Here the lift coefficient that corresponds to the maximum L∕Dof
the original waverider is specified at each design condition, which is
listed in Table 6. In addition, assume that the center of gravity can be
varied along the Xaxis and the variation range is also given in Table 6.
The profile curves of the optimum quasi-waveriders with different
degree-of-stability (ds) constraints are shown in Fig. 18. It is evident
that, as the degree of stability increases, the profile curve becomes
more and more convex, and consequently the shock wave is detached
from the leading edge gradually, which is shown in Fig. 19 (inviscid
condition) and Fig. 20 (H60 km). Note that the pressure increase
around the upper surface in Fig. 20 is partly caused by the influence of
strong viscous interaction effects.
The aerodynamic performance of the original waverider and opti-
mum quasi-waveriders at the specified lift coefficient of each con-
dition is listed in Table 7. The results from both the aerodynamic
model and CFD are listed for further comparison. It can be found that,
for some quasi-waveriders, the degree of stability based on CFD
results is less than the given constraint of the optimization program,
such as the configurations obtained at H40 km. Apparently, this
is caused by the error of the aerodynamic model. However, the
difference is generally small, which is especially acceptable in the
preliminary aerodynamic design.
Furthermore, based on the CFD results, the L∕Dvariation with
degree of stability for the quasi-waveriders is plotted in Fig. 21.
Interestingly, the L∕Dis reduced almost linearly with the increase
in degree of stability at each condition, and the corresponding
expressions through the least square fitting are also given in Fig. 21.
The first-order coefficient of the expression denotes the loss ratio of
L∕Dwith degree of stability, and the constant term represents the
L∕Dfor the quasi-waverider being critical statically stable. Accord-
ing to the expressions, the L∕Dof the optimum quasi-waverider with
different degree of stability can be estimated easily and rapidly.
Another point deserved to be noted in Fig. 21 is that the slope of the
curve, namely, the first-order coefficient of the linear expression,
becomes smaller at higher altitude. As mentioned earlier, the higher-
altitude condition means the stronger viscous interaction effects,
which can be quantified by a widely used viscous interaction param-
eter
V0, defined as [33]:
V0M∞
Cw
p
Re∞;L
p(15)
According the Ref. [33], both similarity parameters
χand
V0
govern the laminar viscous interactions. The main difference is that
the former parameter governs the induced pressure increment and the
latter one governs the pressure coefficient and force coefficients by
the viscous interactions. Therefore, the parameter
V0is used here to
discuss the influence of viscous interactions on L∕D.
X(m)
P/P
001234
5
10
15
20 Inviscid
40km
50km
60km
Lower Surface
Upper Surface
Fig. 16 Pressure distribution along the symmetry.
X(m)
Y(m)
01234
-0.6
-0.4
-0.2
0
0.2
0.4
0.6 Origin
ES-40km
ES-50km
ES-60km
Lower Surface
Upper Surface
Fig. 17 Original curve and effective shapes along the plane at
different conditions, α0 deg. Symmetry plane at different conditions,
α0 deg.
Table 6 Lift coefficient and
center-of-gravity range specified
during the optimization
Condition CLΔXcgΔXcp
Inviscid 0.031 [0.55, 0.65]
H40 km 0.041 [0.55, 0.65]
H50 km 0.051 [0.55, 0.65]
H60 km 0.073 [0.55, 0.65]
LIU ET AL. 2141
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a) Inviscid b) H=40km
c) H=50km d) H=60km
X(m)
Y(m)
01234
-0.5
-0.4
-0.3
-0.2
-0.1
0ds>0%
ds>1%
ds>2%
ds>3%
X(m)
Y(m)
01234
-0.5
-0.4
-0.3
-0.2
-0.1
0ds>0%
ds>1%
ds>2%
ds>3%
X(m)
Y(m)
01234
-0.5
-0.4
-0.3
-0.2
-0.1
0ds>0%
ds>1%
ds>2%
ds>3%
X(m)
Y(m)
01234
-0.5
-0.4
-0.3
-0.2
-0.1
0ds>0%
ds>1%
ds>2%
ds>3%
Fig. 18 Comparison of the profile curve with different degree-of-stability constraints.
Fig. 19 Comparison of flowfield among quasi-waveriders with different degree of stability at the inviscid design condition.
2142 LIU ET AL.
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The values of
V0at different altitudes are listed in Table 8 and the
variation of first-order coefficient, namely, ∂L∕D∕∂ds, with
V0is
plotted in Fig. 22. Again an almost linear relationship exists between
the two variables. Then the relationship between ∂L∕D∕∂dsand
V0can be constructed directly:
∂L∕D
∂ds524.5
V0−21.82 (16)
According to the above equation, the loss rate of L∕Dwith degree
of stability can be easily calculated at different flight conditions,
which is useful for the evaluation of the overall performance require-
ments in the preliminary design. In addition, we should note that
Eq. (15) is valid for the degree of stability typically ranging from −0.5
to 3.5%. Larger degree of stability is beyond the optimization target
of the current paper in that such requirement may be unnecessary for
practical hypersonic vehicles.
Fig. 20 Comparison of flowfield among quasi-waveriders with different degree of stability at H60 km,α4 deg.
Table 7 Aerodynamic performance of the optimum quasi-waveriders with constraints of stability
and the original waverider
QW
ds > 0% ds > 1% ds > 2% ds > 3%
Condition Performance CW Aero model CFD Aero model CFD Aero model CFD Aero model CFD
Inviscid
L∕D9.28 8.94 8.99 8.81 8.71 8.58 8.29 8.33 7.86
Xcp 0.6223 0.6126 0.6056 0.5986 0.5905 0.5813 0.5735 0.565 0.5589
Xac 0.6127 0.6128 0.6063 0.6086 0.6007 0.6014 0.5955 0.595 0.593
ds,% −0.96 0.02 0.07 1.00 1.02 2.01 2.20 3.00 3.41
H40 km
L∕D5.78 5.6 5.68 5.5 5.51 5.33 5.28 5.16 5.02
Xcp 0.618 0.6158 0.6064 0.5983 0.5878 0.5791 0.5696 0.5596 0.5488
Xac 0.611 0.617 0.6014 0.6083 0.5954 0.5992 0.5895 0.5896 0.5781
ds,% −0.70 0.12 −0.50 1.00 0.76 2.01 1.99 3.00 2.93
H50 km
L∕D4.43 4.31 4.4 4.24 4.29 4.11 4.12 3.98 3.92
Xcp 0.616 0.6146 0.6071 0.598 0.5882 0.5774 0.5681 0.5563 0.5452
Xac 0.611 0.6165 0.606 0.608 0.5948 0.5974 0.5862 0.5863 0.5725
ds,% −0.50 0.19 −0.11 1.00 0.66 2.00 1.81 3.00 2.73
H60 km
L∕D3.38 3.31 3.37 3.28 3.31 3.17 3.16 3.03 2.99
Xcp 0.613 0.6122 0.6058 0.5994 0.5894 0.5766 0.5678 0.5548 0.5513
Xac 0.61 0.6132 0.6071 0.6094 0.5956 0.5967 0.5894 0.5848 0.5829
ds,% −0.30 0.10 0.13 1.00 0.62 2.01 2.16 3.00 3.16
LIU ET AL. 2143
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VII. Conclusions
To improve the longitudinally static stability of the original waver-
iders with the loss of L∕Dbeing as small as possible, a design and
optimization method for hypersonic quasi-waverider configuration is
presented in this paper. An efficient and accurate aerodynamic model
that can consider the impacts of strong viscous interaction effect is
employed during the GA optimization process.
When no constraint of volumetric efficiency or stability is given, it
is found that the resulting optimum quasi-waveriders have slightly
higher L∕Dthan the original waverider due to the lower wave drag.
Then optimum quasi-waveriders are obtained based on the constraint
of volumetric efficiency. For the optimized quasi-waveriders with the
same volumetric efficiency as the original waverider, a minor advan-
tage of L∕Dalso exists at the strong viscous interaction conditions.
Finally, constrained by the degree of stability, optimum quasi-waver-
iders are generated at different design conditions. It is discovered that
the L∕Dis reduced almost linearly with the increase in degree of
stability. Furthermore, a linear relationship is also constructed
between the variation of L∕Dwith respect to degree of stability
and the viscous interaction parameter
V0.
In addition, the results obtained in this paper may also help us to
gain more understanding about the connection between the geo-
metric feature and the aerodynamic performance, such as what
profile curve determining a configuration with higher L∕Dat the
strong viscous interaction conditions and what profile curve generat-
ing a configuration that is statically stable. Such understanding may
also be expanded to the design of any other kind of waveriders and
even the conventional lifting-body configuration.
Current work is focused on the quasi-waveriders with sharp lead-
ing edge. To obtain a more practical hypersonic configuration, future
work will try to take into account the influence of blunting effects,
payload, and lateral-directional stability during the optimization.
Appendix: Derivation for the Variation Trend of Xcp
A two-segment broken line can be used as a simplified model of a
streamline, shown in Fig. A1. For a convex streamline, we have θ1>
θ2and δθ >0. For a concave streamline, the features are just oppo-
site. The length, force, and moment are nondimensionalized by dl,
1∕2ρ∞V2
∞dl, and 1∕2ρ∞V2
∞dl2, respectively.
The dimensionless resultant force parallel to Yaxis is
F
yCp1cos θ1Cp2cos θ2(A1)
where the pressure coefficient is
Cpp−p∞
1∕2ρ∞V2
∞
(A2)
The moment produced by F
yis
M
Oy 0.5Cp2⋅cos2θ2−0.5Cp1⋅cos2θ1(A3)
Thus, the dimensionless location of pressure center is
Xcp M
Oy
F
y0.5Cp2⋅cos2θ2−0.5Cp1⋅cos2θ1
Cp1cos θ1Cp2cos θ2
(A4)
The Newtonian theory is adopted here to calculate the pressure
coefficient at hypersonic conditions:
Table 8 Values of the
viscous interaction parameter
V0at different conditions
M∞H,km L,m
V0×10−2
15 40 4 0.514
15 50 4 1.023
15 60 4 1.845
Fig. 22 Variation of ∂L∕D∕∂dswith the viscous interaction
parameter
V0.
Fig. A1 Simplified model of a streamline.
ds
L/D
-0.01 0 0.01 0.02 0.03 0.04
0
3
6
9
12 Inviscid
40km
50km
60km
y=-34.09x+9.03
y=-19.00x+5.62
y=-16.66x+4.39
y=-12.07x+3.39
Fig. 21 L∕Dvariation with degree of stability for quasi-waveriders at
different conditions.
2144 LIU ET AL.
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Cp2sin2θ(A5)
Also note
θ1θ2δθ (A6)
Substituting Eqs. (A5) and (A6) into Eq. (A4), we obtain
Xcp sin2θ2⋅cos2θ2−sin2θ2δθ⋅cos2θ2δθ
2sin2θ2δθcosθ2δθ2sin2θ2cos θ2
(A7)
In Fig. A1, if the angle of attack is defined as the angle between the
second segment of the broken line and the Xaxis, we have αθ2.
Hence, for a given δθ, Eq. (A7) depicts the variation of Xcp as a
function of α. The results are shown in Fig. A2 for different values of
δθ. We can find that, for a positive value of δθ,Xcp increases
monotonically as angle of attack increases, whereas for a negative
value of δθ,Xcp decreases monotonically as angle of attack increases.
Therefore, from the above analysis we can conclude that, as the
angle of attack increases, the center of pressure moves backward for a
convex streamline and moves forward for a concave one.
Acknowledgments
This work was supported by the Strategic Priority Research
Program (A) of Chinese Academy of Science (XDA17030000).
The computational work was carried out at the National Supercom-
puter Center in Tianjin, and the calculations were performed on
TianHe-1(A).
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