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Optimization of Viscous Waveriders Derived
from Axisymmetric Power-Law Blunt Body Flows
B. Mangin∗
Centre National de la Recherche Scientifique, 45077 Orléans, France
R. Benay†and B. Chanetz‡
ONERA DAFE, 92190 Meudon, France
and
A. Chpounx
Centre National de la Recherche Scientifique, 45077 Orléans, France
DOI: 10.2514/1.20079
A method based on a Euler code is used to study waveriders generated from the flowfield around cones or
axisymmetric power-law blunt bodies. In the cone case, the results are compared with those given by the Taylor–
Maccoll system and inviscid hypersonic small-disturbance theory. This last theory shows its limits at a Mach number
of 5, for cone angles providing the best lift-to-drag ratios. In the axisymmetric power-law blunt body flows case, the
optimization is led using a nonlinear simplex method to get the upper surface that enables the waverider to have the
best lift-to-drag ratio, the thickness-to-length ratio being fixed. Compared with the cone-derived waverider, the
proposed blunt-body-derived model allows a 20% gain in volume for near equal optimized lift-to-drag ratios.
Nomenclature
CDw = wave drag coefficient
Cf = friction coefficient
CL = lift coefficient
CmG = pitching moment coefficient at the center of volume
CP = center of pressure
CV = center of volume
D= total drag
Df= friction drag
Dw= wave drag
dS= triangular half-cell surface
H= shape factor
L= total lift
Lf= vertical component of the friction strain
Lw= waverider length
M= Mach number
Moz = pitching moment of the waverider at the nose of the
reference body
n= power of the reference body
p= pressure
q= dynamic pressure
Reu = unitary Reynolds number
Ro = radial distance at ’0
Rs = shock radius in the base plane
Sp = planform surface area (on the z–x plane)
Sw = wetted surface area
V= volume
Xt = transition location
X= freestreamwise coordinate
Y= downward vertical coordinate
Z= spanwise coordinate
= cone angle or body thickness angle
= volumetric coefficient. V2=3=Sp
= radius
1= radial distance at ’l=3
2= radial distance at ’2l=3
l = dihedral angle, defining the waverider width
’= azimuthal angle
Subscript
1= freestream
max = maximum
I. Introduction
PROJECTS about future hypersonic transport vehicles and the
recent relevance of military hypersonic drones have led to a
renewed interest in the waverider concept. This concept, providing
interesting aerodynamic coefficients for a hypersonic vehicle, will
contribute to fuel efficiency and may raise the economic viability of
civil hypersonic transport aircraft. In the 1950s, confronted with the
challenge of avoiding leakage due to shock standoff (for bodies with
blunt noses), Nonweiler [1] found a method for defining hypersonic
vehicle shapes that simultaneously solved this problem and allowed
him to take into account the 3-D geometry in a simple way.
Waveriders are generated by an inverse method: the undersurface is a
stream surface of a known inviscid flow around a reference body after
its refraction by the front shock (see Fig. 1). The leading edge is the
intersection of a freestream surface (which provides the upper
surface) and the shock created by the reference body. Leading-edge
sharpness prevents leakage and so provides high lift-to-drag ratios
(the compression zone is isolated by the shock). The undersurface
being a stream surface, the waverider cruising at supersonic or
hypersonic speeds creates the same shock wave as that due to the
reference body and seems to be riding on top of this shock (thus its
name). If the reference body is a wedge or an axisymmetric body, the
flowfield will be two-dimensional.
High lift-to-drag ratios are very difficult to obtain at hypersonic
speeds due to high wave drag and massive viscous effects.
Kuchemann [2] exhibits an empirical correlation for L=Dmax based
on experiments:
L=Dmax 4M13=M1
Received 15 September 2005; revision received 23 March 2006; accepted
for publication 10 April 2006. Copyright © 2006 by the American Institute of
Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper
may be made for personal or internal use, on condition that the copier pay the
$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood
Drive, Danvers, MA 01923; include the code $10.00 in correspondence with
the CCC.
∗Ph.D. Student, Laboratoire d’Aérothermique, 1C Avenue de la Recherche
Scientifique; manginbo@yahoo.fr.
†Assignment/Theoretical Exploitation Head, 8 rue des Vertugadins.
‡Research Director, 8 rue des Vertugadins.
xProfessor, Laboratoire d’Aérothermique, 1C Avenue de la Recherche
Scientifique.
JOURNAL OF SPACECRAFT AND ROCKETS
Vol. 43, No. 5, September–October 2006
990
(see Fig. 2). Studies lead by Corda et al. [3] and Bowcutt [4] show that
waveriders break this barrier.
Cone-derived waveriders present better volumes than the wedge-
derived ones because the concave streamlines are closer to the shock.
Thus, waveriders’streamlined shapes were originally obtained from
the flowfields around cones and wedges for their ease of calculation.
If these profiles actually offered the desired good lift-to-drag ratios,
on the other hand, they did not respond to the requirements in terms
of useful volume vs. overall dimensions (the obtained profiles were
very thin). Progress in numerical methods and increasing computer
performances have allowed the field of possible generating shapes to
enlarge. Waverider generation from the aerodynamic field calculated
around blunt bodies, coupled with an optimization process, should
better take into account both aerodynamic performance objectives
and volume constraints. The goal is to improve the carrying capacity
and to facilitate motor integration.
Blunt-body-derived waveriders are more voluminous because the
streamlines are convex. Rasmussen and Brandes-Duncan [6] used
hypersonic small-disturbance theory to study power-law blunt-body-
derived waveriders. The nose profile was defined as a power of the
streamwise coordinate. It was found that inviscid lift-to-drag ratio
was maximal for a power nclose to 0.7 and about 15% higher than in
the conical case. In this study, a CFD Euler code is used to create
blunt body flows and an optimization including viscous stresses
evaluation is led. This optimization uses a nonlinear simplex method
to get the upper surface that enables the waverider to have the best
lift-to-drag ratio, the thickness-to-length ratio being fixed.
II. Creation of a Waverider
Presented next are the steps of a waverider creation and how the
strains it undergoes are evaluated.
A. Steps
A waverider is considered as the forebody of an entire
configuration and the base design is not considered. To accurately
determine the flow around a blunt reference body, including the
subsonic zone at the nose, where the method of characteristics is not
applicable, the use of an Euler code is suitable. The hypersonic
flowfields around the power-law axisymmetric body considered is
thus obtained by use of the ONERA code FLU3M [7]. When the
reference body is a cone, calculations are possible by two other
means, one by solving the Taylor–Maccoll system and the other by
use of hypersonic small-disturbance assumptions. The three methods
have consequently been compared on this last configuration.
The cylindrical upper surface is defined by its cross section in a z–y
plane (or in the corresponding –’plane when using polar
coordinates) with Yas an even function of Zor as an even function
of ’. The leading edge is the intersection of this freestream surface
and the shock created by the reference body. The undersurface is the
stream surface starting from this curve in the 2-D or axisymmetric
reference flow. Indeed, viscosity being neglected, a streamline can be
considered as a solid wall on which the neighboring streamlines slip.
As a result, the waverider will create the same shock as the reference
body (Fig. 1).
Integrating static overpressures on the undersurface provides Dw
(Xcomponent) and L(Ycomponent). The upper surface being
parallel to the freestream, its pressure is at freestream conditions so
that there is no created drag or lift.
Wall shear stresses are evaluated by semiempirical means with the
temperature reference method (see the Viscous Effects section).
Their sum over the wetted surface provides Df(Xcomponent), and a
very small component opposed to the Lf. The base drag is not
considered.
Some geometrical characteristics are calculated. In particular, the
under and upper surface areas, the planform area (on the Y0
plane), and the volume are given (see [8]). The center of volume (i.e.,
the center of gravity for homogeneous waveriders) and the center of
pressure (where aerodynamic moments are null) are computed too.
The center of gravity is computed in the same manner as the volume:
basic shapes as tetrahedron, pyramid, and regular prism are
superimposed to form the volume confined between a cell of the
wetted surface and a reference plane Y0. The shift between the
undersurface and upper surface shapes provides the waverider.
B. Flowfields
Two kinds of axisymmetric flowfields are considered following
the type of reference body (a cone or a blunt power-law axisymmetric
body).
1. Conical Flowfield
Aerodynamic flowfields on cones have been calculated using the
Euler code and two other methods: the first solves the Taylor–
Maccoll system derived from the Euler equations with conical
assumption, and the second is based on the hypersonic small-
disturbance theory.
Conical assumption is that polar angle is the only variable
(axisymmetric field and no length scale). So Euler equations can take
the following form of the Taylor–Maccoll system [9]:
u0v;v0uv22a=a2va=a2cot
a=a2v2
with uand vare polar components of the speed nondimensionalized
by a, the freestream critical speed of sound. Flow conditions
downstream of the shock are obtained, thanks to attached-shock
relations for a perfect gas. The Taylor–Maccoll system is solved by
iterations using a fourth-order Runge–Kutta algorithm. A conical
flowfield is pictured in Fig. 3.
Hypersonic small-disturbance theory assumption [10,11] enables
us to solve the Taylor–Maccoll system analytically. It requires
the assumption that the cone is slender enough and the freestream
Mach number high enough to apply small angle hypothesis. The
Fig. 1 Inverse method to create a waverider.
Fig. 2 Maximum lift-to-drag ratios (without base drag) vs. Mach
number and empirical barriers.
MANGIN ET AL. 991
orthoradial speed component is then
v=V112=2u0=V1
The shock angle-to-cone angle ratio is given by the similarity relation
= 1=21=K2
1=2
with KM1:. The undersurface is created from the mass flow
conservation between it and the cone and from the hypothesis of a
constant density downstream of the shock.
2. Power-Law Bodies Flowfields
Rasmussen and Brandes-Duncan [6] used a hypersonic small-
disturbance theory to study power-law blunt-body-derived waver-
iders. In this case, the equation of the reference body in an x–y plane
is YA:Xn. Idealized waveriders (half of the reference body as the
undersurface and two flat planes as the upper surface) were studied
with nvarying between 0.5 to 1. Viscous strains not being taken into
account, it was found that lift-to-drag ratio was maximal for nclose to
0.7 (this maximum is about 15% higher than in the conical case).
Moreover, streamlines around blunt power-law bodies enable us to
create waveriders with convex undersurfaces. Hence volume
increase appears as a supplementary advantage. The Euler code was
used to create aerodynamic flowfields around blunt power-law
bodies. The length of the body is 1.0 m and the waverider length is
0.1 m. The thickness angle being , the blunt body is defined as
Ytan:Xn
This Euler version of the ONERA FLU3M [7] code uses a Roe
scheme on structured monozone grids. No Harten correction was
needed. The two-dimensional grid is designed to obtain high point
density around the body (see Fig. 4).
Grid convergence is guaranteed by the comparison of the results
obtained with various mesh definitions. The final grid definition has
480 points in I(along the wall) and 110 in J(normal to the wall)
coordinates (spatial convergence of the calculation is obtained with
less points, especially along Icoordinates but this high definition is
needed to obtain smooth streamlines and a thin shock).
C. Leading-Edge Design
The upper surface being cylindrical, its projection in a y–z plane is
also the leading-edge projection. In this study, it is defined by a
polynom function
YfZor f’
(see Fig. 5). The first kind of parameterization has only been
considered in the case of cone-derived waveriders. In this case, a
second-degree polynom gives what is called parabolic-top
waveriders (see [10]). The leading-edge projection is the parabola
that passes through the points 0, Ro and Rs: sin l, and Rs: cos l
(of course the shock projection is a circle in this axisymmetric case).
In a more general optimization process, the polynom is the even
Lagrangian sixth-degree polynom f’that passes through the
points 0, Ro;1;’1;2;’2; and Rs,lwith ’1l=3and
’22=3l. The polar representation f’ensures a single
radial position of the leading-edge projection for a given ’.
D. Viscous Effects
Viscous stresses are evaluated using two integral methods based
on empirical formulations of the friction coefficient: the Eckert
reference temperature [12] method and the Michel–Cousteix method
[12], considering uniform and nonuniform external flowfields,
respectively. In the shock downstream area, the pressure increases
along a streamline in the case of a conical flow, whereas it decreases
in the case of a power-law blunt body flow. The two methods have
been used to evaluate these variations effects on viscous stresses.
Transition location is evaluated thanks to experimental results on
slender cones [13]. If the waverider length is lower than the transition
location, the friction coefficient is given by the corrected well-known
Blasius formula
Cfx0:664=
Rex
pf
with the local Reynolds number Rexeuex=e, and f,a
correction factor which enables us to apply incompressible Blasius
formula to supersonic flowfields, given by
f
rr=ee
p
Ter=Tre
p
Fig. 3 Streamlines of a conical flowfield (Taylor–Maccoll method).
Fig. 4 Grid around a power-law body (5 deg,n0:5).
Fig. 5 A symmetric sixth-degree upper surface and its parameters.
992 MANGIN ET AL.
where the reference temperature Tr is Monaghan’s temperature [12].
In the turbulent case (with the assumption that transition location is
quite lower than the waverider length), the friction coefficient is
given by
Cfx0:0368=Rex1=6f
where the compressible correction factor is
fr=e5=6r=e1=6Te=Tr5=6r=e1=6
The Michel–Cousteix method deals with the integration of the von
Kármán equation:
Cf=2d=dxfH2=UedUe=dx1=ede=dx
1=RdR=dxg
where is the momentum thickness, Hthe shape factor, and Rthe
distance to the symmetry axis (R1for two-dimensional cases).
Cf=2and Hare supposed to check the following rules relative to the
flat plate case:
Cf=2k:g=Rem
(with kand mcorresponding to the laminar or turbulent case, see
Table 1) and
HHi:M2
eTpTf=Te
(see the Nomenclature). The compressible correction factor is g. The
solution checks:
F:E:Rm1xF:E:Rm1xx1Zx
x1
g:F:E:Rm1
eUe=emdt
where tis the curb abscissa,
Exexp:m1Zx
x1
TpTf
Te
:dUe
Ue
and
FxUHi2
e:1
em1
Results from the two methods have been compared locally with
friction coefficient and globally with friction drag. The reference skin
friction is that of the first method. The relative shift between friction
coefficients is never greater than 5%, as for the conical case. The
resulting friction drag difference is lower than 2%, showing that local
shifts between the two methods have weak effects on it. Hence the
reference temperature method was used in the optimization process.
E. Stresses
Each generating streamline has the same number of points (100
points for the 100 streamlines to ensure stresses convergence) giving
a100 100 grid that structures the undersurface (defined by indices
Iand J). The wave drag and lift of each cell are the components of the
average pressure strain multiplied by the cell area. For each cell, the
average component of the strain due to the Cf parallel to the local
inviscid speed multiplied by the cell area gives its contribution to the
drag. By summing on all the cells, the lift-to-drag ratio is given by
Eq. (1):
L
DLLf
DwDf
RRSWCpp1dS:ey0q1RRSW Cf:dStey0
RRSWCpp1dS:ex0q1RRSW Cf:dStex0
(1)
with e
y0e
y. In the small-disturbance-based method, pressure forces
are evaluated thanks to analytic formulas [10,11] derived from a
momentum flux balance.
III. Optimization
The aim of this work is to find waveriders with the highest lift-to-
drag ratios able to perform useful tasks (e.g., a sufficient volume). A
first study is made to understand the parameters influence on
aerodynamic and geometric characteristics in the particular case of
the cone-derived waveriders with parabolic tops. Then a more
general optimization process based on a nonlinear simplex method is
used to find the leading edge providing the best lift-to-drag ratio for
waveriders created from power-law body generated flowfields.
A. Parameters Influences
The first type of parameters defining a waverider are those relative
to the reference body. In the conical case, it is only the cone angle .
In the case of a power-law blunt body, these parameters are the
thickness-to-length ratio tanand the power nin the reference body
defining equation ytanxn. The second type of parameters are
the waverider length Lw and the previously defined Ro,1,2, and
l. When the last four parameters increase, the undersurface
streamlines go away from the reference body, and so are less
deflected, hence the waverider is thinner (pressure strains decrease).
But when Ro and lincrease, the width and the wetted area also
increase and so does the viscous strain. There is an equilibrium to find
between the thickest waverider, which is the half-reference-body
(Ro !0), and the flat plate (Ro,1,1, and 2!infinity). Indeed,
for thick waveriders, the friction drag can be negligible in front of the
wave drag and for thin waveriders the contrary can happen. Hence,
with Ro,1, and 2being fixed, there is a particular reference body
thickness-to-length ratio that provides the highest lift-to-drag ratio
(see the optimal cone angle in Fig. 6).
B. Nonlinear Simplex Optimization Process
Most optimization schemes are based on variational methods that
require analytical descriptions. However, a nonlinear simplex
method for function minimization by Nelder and Mead [14] enables
us to converge toward the best configuration just by evaluating the
function to minimize in a pool of configurations. Here it is considered
for minimizing the inverse of the lift-to-drag ratio. With n
parameters, the simplex is made of n1points, a point or
configuration representing a waverider. At each step, the worst
configuration (with the highest value) is eliminated and replaced by a
Table 1 Friction evaluation coefficients
Laminar Turbulent
k0.2205 0.0086
m11=5
Hi 2.6 1.4
0.667 0.4
2.9 1.22
Fig. 6 Lift-to-drag ratios vs. cone angle (Taylor–Maccoll and
Rasmussen methods).
MANGIN ET AL. 993
new one: it is the image of the worst one by the reflection through the
center (or average configuration) of the others. If the function at this
point has the lowest value, an expansion occurs: this point is replaced
by the image of the last center by the reflection through it. If the
function at this point has the highest value, a contraction occurs: this
point is replaced by the center of itself and the last center. Finally the
process converges to the best configuration vicinity.
In this study, the simplex method was not applied directly with all
the parameters. The reference body parameters and ncannot take all
the values because each couple corresponds to a mesh. Indeed,
automation of the entire process (mesh creation, calculus, and
optimization with the remaining parameters) has not been under-
taken because of the complexity of the chain and the subsequent
enormous time required by a step. The method resembles the
optimization of cone-derived waveriders lead by Bowcutt et al. [4],
where the simplex method was used for a fixed shock angle (hence
fixed cone angle), the final best waverider being the best of the bests
obtained for each of the studied shock angles.
Two different freestream conditions (see Table 2) have been
studied. Case 1 (ONERA Meudon R2-Ch wind tunnel) has been
chosen for comparing the different methods in the parabolic-top case
and case 2 (CNRS Orléans SH2 wind tunnel) corresponds to the
optimization part. In the second case, the waverider has a totally
laminar boundary layer. The first and second conditions,
respectively, were applied on waveriders whose length was 0.25
and 0.1 m, respectively. Experiments led on slender cones furnished
a correlation between unitary Reynolds number and transition
Reynolds number (see [13] and Table 2). Hence it appears that the
waverider tested in the second case would have a laminar boundary
layer on its first half and a turbulent one on its second. Meanwhile, it
is considered that the boundary layer is totally turbulent. (Friction
strains have been overevaluated rather than considering a nonprecise
transition location.)
IV. Results
A. Comparison of Taylor–Maccoll/Small Disturbances
It is known that the hypersonic small-disturbance theory (which
will be called the Rasmussen method) works for small cone angles
(less than 5 deg) and for high Mach numbers (greater than 10).
Meanwhile results for a Mach number of 3 can be seen in the
literature. To appreciate how it performs at Mach 5 with the
considered cone angles, its results are compared with those given by
Taylor–Maccoll calculations in the particular case of cone-derived
waveriders with flat upper surfaces.
Each method allows its own determination of the undersurface
geometry. In each case, strains are evaluated by the previously
defined method. Ro being fixed (Ro=Lw 0:2), in Fig. 6 we first
compare lift-to-drag ratio evolutions vs. the cone angle given by the
two methods. Lift-to-drag ratios are paradoxically different for lower
cone angles. This shift, which can be seen when the viscosity is
neglected, is amplified in the laminar and turbulent cases. This means
that waverider geometries do not coincide. Indeed, for 5 deg,
there is a non-negligible difference between the stream surfaces
given by the two methods, although these are very close for
12:95 deg. (In Fig. 7 the undersurfaces are represented in
reversed position relative to the ground by their projection in various
transverse cutting planes.)
The geometry shift is due to the fact that Kmust be high enough:
according to Anderson [15], Kmust be higher than 0.5 and 1.5,
respectively, for equal to 5 and 20 deg, respectively, corresponding
to a Mach number higher than 5.73 and 4.30, respectively. In the
present study, the Mach number is too low for less than 10 deg.
Moreover, the similarity relation provides shock angles whose
relative shifts compared with Taylor–Maccoll results reach 4%
(0.5 deg) for 5 deg. This explains why the hypersonic small-
disturbances theory has not been used to study Mach 5 power-law
body derived waveriders.
These results look like those of the flat plate under incidence
(curves passing from pure skin friction at zero incidence to a pure
wave drag at high incidence, crossing a maximum between the two
states). All the curves are the same if the cone angle is greater than
13 deg, because friction strains are negligible compared with
pressure strains, and shock angles are close.
B. Comparison of Taylor–Maccoll vs. Euler Calculations
The ONERA code FLU3M has been used to obtain the flowfields
around power-law blunt bodies. Specific subroutines such as a shock
localizer and a streamlines extractor have been implemented to create
waveriders. The results given by the use of Taylor–Maccoll
equations are compared with those of the Euler-code-based method.
A difference between these two methods can emerge when
looking at the shock localization. Indeed, the Taylor–Maccoll
method fits an infinitely thin shock whereas the Euler-code-based
method creates a shock that smears over several grid points. Lobbia
and Suzuki [16] encountered this problem with a drag shift of 13%
and a lift shift of 22% between a 3-D Euler-code-based method and
Taylor–Maccoll for a 10 deg cone. The use of a 3-D code prevented
them from excessively increasing the number of points to thin the
shock.
In our study, shock capturing consists of localizing the maximal
pressure first derivative (between two neighboring grid points
labeled as Nand N1) of each gridline (in the Ior Jdirection). The
shock is considered as the curve passing through the middle of the
intervals joining calculation points between which the pressure
variation is maximal. Then the “ladder”curve is smoothed.
For a cone angle equal to 12.95 deg and Ro=Lw equal to 0.2,
waveriders with a flat upper surface are obtained by using either of
the two methods. Strains are very close, relative differences being
lower than 4% (see Table 3). The aerodynamic coefficients are
related to the wetted surface area. Geometries are very close too.
These results validate the Euler-code-based method to create
waveriders. It will be used to create power-law blunt-body-derived
waveriders.
C. Cone-Derived Waveriders with Parabolic Tops
The influence of parameters on aerodynamic and geometric
characteristics will be pictured in the particular case of cone-derived
Table 2 Flow conditions
Case 1 Case 2
Mach number 5 4.96
Total pressure (Pa) 32e5 8.5e5
Total temperature (K) 500 453
Reu (m-1) 40e6 12.66e6
Rext 4.78e6 2.75E6
Xt (m) 0.12 0.21
Length (m) 0.25 0.1
Fig. 7 Reversed undersurfaces in transverse cutting planes Y-Z.
994 MANGIN ET AL.
waveriders with parabolic tops. The flowfield conditions are those of
case 1. The definition of the leading-edge/upper-surface projection in
a transverse y–z plane is described by YA:Z Ro, with Achosen
to enable the curve to pass through the point (Rs sin l,Rs: cos l).
The three parameters defining these waveriders are the cone angle ,
Ro, and l.
Studied parameters lie in the range 4–10 deg for and in the range
10–80 deg for l. The minimum value of Ro needed to ensure the
leading-edge projection to be ’-single valued depends on the first
two parameters:
Ro > Rs: cosl=2
If the obtained lift-to-drag ratios are pictured in the 3-D parameter
space, it appears that lift-to-drag ratios are the highest in a curved
tube, the axis of which is parallel to the Ro axis. This tube is
symmetrically cut by the plane 9:2deg pictured in Fig. 8 (it
corresponds to the previously evidenced optimal cone angle). For a
ratio Ro=Lw greater than 0.04 (Ro higher than 1 cm for
Lw 0:25 m), the best lift-to-drag ratios lie in the zone included
between the two curves corresponding, respectively, to waveriders as
long as wide (the lower one, solid line) and waveriders with flat upper
surfaces (the highest one, dotted line). This means that the best
waveriders are slightly wider than long and have a convex upper
surface. The best waverider with Ro=Lw 2:68 (Ro 0:67 m)is
obtained for l12:5deg, giving a lift-to-drag ratio equal to 6.53
and a width-to-length ratio equal to 1.28. This waverider is not very
useful because it is too thin to carry loads (Ro is high, see Fig. 8,
bottom right).
More useful and thicker waveriders are obtained with lower Ro.A
waverider whose thickness-to-length ratio is equal to 0.1
corresponds to a ratio Ro=Lw equal to 0.8 (Ro 0:2m). In this
case, the highest lift-to-drag ratio is equal to 6.43 and is obtained with
l29:4 deg (see Fig. 9, bottom left). The asymptotic shape of a
cone appears in the undersurface profile.
The volume distribution is pictured in Fig. 10 and shows that the
most voluminous waverider is the one shown in Fig. 9, top right. The
volume and the lift-to-drag ratio of this body are, respectively, equal
to 4:89e-3m
3and 5.95. Its volume is due to its very high aspect ratio.
It is obvious that such a body is out of aircraft applications. Practical
reasons lead, therefore, to consider the body represented at the
bottom left of Fig. 9, with a volume of 3:94e-4m
3and a lift-to-drag
ratio of 6.43, as optimal. The optimization just performed of a cone-
derived waverider, with a parabolic definition of the upper surface,
was able to be led simply by a close inspection of the numerous
performed calculations. In the case of less simple definitions of the
upper surfaces, the optimization will require the use of a more
improved technique.
D. Optimized Power-Law Blunt-Body-Derived Waveriders
The only waveriders considered in the optimization process are
those with a thickness-to-length ratio equal to 0.1. Ro is chosen to
ensure this condition and it is checked that it is high enough to be
supersonic after the shock. This last condition is necessary for the
shock wave to be attached to the leading edge: if it is not the case, the
inverse method cannot be applied. The length of the reference bodies
is 1 m.
The optimization process consists of searching the best shape for
the upper surface of the body. The function to minimize is the inverse
of the lift-to-drag ratio. The simplex process should have been
applied to 24 undersurface geometries. These surfaces are generated
by the streamlines of the Euler flowfields around reference bodies
defined by n0:5, 0.6, 0.7, 0.8, 0.9, and 1.0 and 3,5,7,and
9 deg. We recall that the cases n1are cones and the length of the
reference body is 1 m. The particular reference bodies corresponding
to 3 deg;n0:8, 0.9, and 1.0 and 5 deg;n1:0were too
slender to create waveriders as thick as a tenth of their length.
Consequently, the simplex process was applied to the 20 remaining
Table 3 Waveriders characteristics (Taylor–Maccoll/FLU3M)
Lw 0:25 m,Ro=Lw 0:2,
12:95 deg Taylor–Maccoll FLU3M "(%)
CDw 1:031E-02 0:991E-02 3.88
CL 5:334E-02 5:176E-02 2.96
Wetted surface area (m2) 7:634E-02 7:625E-02 0.12
Lift-to-drag ratio 5.175 5.231 1.18
Fig. 8 Lift-to-drag ratio of conical-derived waveriders with parabolic tops (turbulent case, Lw 0:25 m,9:2 deg).
Fig. 9 Volume of conical-derived waveriders with parabolic tops
(turbulent case, 9:2 deg).
MANGIN ET AL. 995
flowfields. The upstream conditions are those of case 2, assuming
laminar boundary layers. For each case, Ro being fixed, the
parameters are the width of dihedral angle l, and the distances of 1
and 2. One hundred simplex steps were generally necessary to
accurately approach the minimum.
1. Influence of the Reference Body Shape on Optimal Characteristics
Fig. 10 shows the bases of the optimal waveriders obtained for all
the studied reference bodies. In every case 1,2, and Ro are close,
and lift-to-drag ratios are higher than the empirical barrier (6.4).
When Ro increases, the width of the optimal waveriders increases,
but ldecreases to limit the wetted surface area. The distance 2
increases less than 1so that waveriders become more and more like
a reversed W, which we called a two-bump configuration. The lift-to-
drag distribution of these optimal waveriders is pictured in Fig. 11.
The highest lift-to-drag ratio is obtained for the thickest reference
body (9 deg) with the lowest power (n0:5) and just reaches 7.
It can be seen in Fig. 10 that it corresponds to a waverider wider than
long (reminder that Lw 0:1m), which is an unpractical leading
edge kind of waverider (the widest). Optimal waveriders that are
longer than wide or slightly as long as wide are the ones in the
outlined square in Fig. 10 and above the solid line in Fig. 11. If the
waveriders that are wider than long are not to be considered, the best
practical waverider is the one of the case 7 deg,n0:7. Note
that the friction drag varies from 30% (slender reference bodies) to
60% (thicker reference bodies) of the total drag. Indeed, the base area
increases with increasing reference body thickness, and the inviscid
lift-to-drag ratio consequently decreases. For the case 7 deg,
n0:7, the friction drag is 35% of the total drag (i.e., the wave drag
is twice the friction drag). The volume-to-Lw [10] ratio distribution
is pictured in Fig. 12. The highest volume is obtained for the case
9 deg,n0:5. As for the parabolic-top cone-derived wave-
riders, the volume is due to the width of the waverider (high Ro
implies high shock radius). All the volumetric coefficients are close
to 0.26 except for the case 3 deg;n0:6, where it is about 0.72.
It can be seen in Figs. 10 and 13 that close optimal upper surfaces
correspond to close values of Ro. Two optimal waveriders, the upper
surfaces of which are close, are compared in the following section to
consider the effect of the reference body on the lift-to-drag ratio and
the volume.
Fig. 10 Base view of optimal waveriders (Lw 0:1m).
Fig. 11 Lift-to-drag ratio of the optimal waveriders.
Fig. 12 Volume of the optimal waveriders.
Fig. 13 Ro of the optimal waveriders.
996 MANGIN ET AL.
2. Toward an Optimal Shape
By considering not only lift-to-drag ratio but also the carrying ca-
pacity of the body (its volume), the two most interesting configu-
rations, 7 deg,n0:7and 9 deg,n1:0, are shown in
Table 4. A synthetic representation of the geometries is given in
Fig. 14. The configurations 7 deg,n0:7and 9 deg,n
1:0are close, but the first one, generated with a blunt reference body,
presents a slightly higher lift-to-drag ratio of 6.90 instead of 6.87.
The volume increase from 9 deg,n1:0to the close blunt-
body-derived configuration reaches 21%. The cone-derived case
7 deg,n1:0, whose shape is more classical than the two
others, is added in Table 4 and in Fig. 14 for comparison. It illustrates
the two kinds of geometries obtained in our optimization process. As
a main result, the two-bump configuration has constantly appeared as
the most interesting one. It is to be noticed that the three
configurations overpass the empirical barrier of lift-to-drag ratios
equal to 6.4.
A higher volume being obtained in the case of the two-bump
configurations, we see clearly that, with respect to our criterion of
maximum volume maximum lift-to-drag ratio, the blunt body
generated, 7 deg,n0:7, must be retained at the end of the
process.
2. Aerodynamic Stability
Waveriders are known to be statically unstable [10] that is, their
center of pressure (where the aerodynamic resultant is applied) is
upward of the center of gravity. The pitching moments of the
optimized waveriders have been computed to evaluate their degree of
instability. Only the lift is taken into account because of the small
vertical distances making the drag work: there is a shift of two orders
of magnitude between moments due to the lift and that due to the
drag. The center of pressure is the point where the pitching moment is
null. The center of pressure location (measured from zero, the body
nose that is the origin of the frame) Xcp is equal to Moz=L (the
moment at the point zero, Moz, is negative because of the z-axis
orientation).
The center of volume is the center of gravity of the body
considered as homogeneous. Its streamwise location, measured from
zero, is Xcv.X, the distance between the two centers, is equal to a
certain percentage of the waverider length. It characterizes the degree
of instability of the considered body. The waverider length relative to
zero coincides with the base location, Xbase. Table 5 gives all these
characteristics for the cases 7 deg,n0:7and 9 deg,
n1:0. The optimized waveriders are therefore unstable. The
optimized 9 deg,n1:0waverider is slightly less unstable than
the 7 deg,n0:7one (Xequal to, respectively, 11 and 14% of
the waverider length). This can be explained by looking at the
undersurface pressure distributions in Fig. 16. In the cone case,
higher pressures are located just upstream of the base, whereas, with
a convex power-law body, higher pressures are located close to the
nose (see Fig. 15). Stability can be improved by designing upper
surfaces that create an expansion just upstream of the base.
V. Conclusion
Power-law blunt body-derived waveriders have been studied in an
attempt to improve both lift-to-drag ratio and volume. Small-
disturbance theory showed its limits at Mach 5, which explains the
use of a Euler code to create the flowfields. This last method has been
well validated for cone-derived waveriders by a comparison with
Taylor–Maccoll results. Nonuniformity of the external flowfield has
negligible effect on the friction stresses. Influences of the parameters
have been pictured in the particular case of parabolic-top cone-
derived waveriders. In the case of more complex upper surfaces, a
simplex-based method enabled us to optimize the viscous lift-to-drag
ratio of waveriders having a thickness-to-length ratio equal to 0.1. It
appears that the choice of a blunt body as a reference body rather than
that of a simple cone generates a slight increase of the lift-to-drag
ratio and a 20% rise for the volume of the waverider.
Fig. 14 Base view of three optimal waveriders.
Table 4 Optimized waveriders characteristics
Thickness-to-length ratio 0:17 deg,n1:07 deg,n0:79 deg,n1:0
Ro (m) 6.50e-3 0.060 0.054
1(m) 7.00e-3 0.059 0.054
2(m) 9.47e-3 0.062 0.057
l () 60.72 34.98 35.69
Dw=D (%) 50.0 65.1 65.6
Lift-to-drag ratio 6.45 6.90 6.87
Volumetric coefficient 0.264 0.266 0.261
Table 5 Stability analysis of the cases 7 deg,n0:7
and 9 deg,n1:0
Xcp=X max Xcv=X max X=Lw (%) CmG
7 deg,n0:70.828 0.881 14.0 8:45E-4
9 deg,n1:00.864 0.901 11.0 8:94E-4
Fig. 15 Pressure distribution at the undersurface (7 deg,n0:7)
and (9 deg,n1:0).
MANGIN ET AL. 997
The method exposed here is therefore well adapted, for instance,
for the optimization of an hypersonic glider that must carry a huge
charge. The most simple examples of that are the bomb bay or
weapon magazine of a military aircraft; but civil applications, like the
first given in the Introduction, are also possible. The main result is
that the conception of an easy-to-control, loaded hypersonic vehicle
flying on large distances should be facilitated.
References
[1] Nonweiler, T. R. F., “Delta Wings of Shape Amenable to Exact Shock
Wave Theory,”Journal of the Royal Aeronautical Society, Vol. 67,
No. 1, 1963, p. 39.
[2] Kuchemann, D., The Aerodynamic Design of Aircraft, Pergamon,
Oxford, 1978, pp. 448–510.
[3] Corda, S., and Anderson, J. D., “Viscous Optimized Hypersonic
Waveriders Designed from Axisymmetric Flowfields,”AIAA Paper 88-
0369, Jan. 1988.
[4] Bowcutt, K. G., Anderson, J. D., and Capriotti, D., “Viscous Optimized
Hypersonic Waveriders,”AIAA Paper 87-0272, 1987.
[5] Jacquotte, O. P., Montigny-Rannou, F., and Goussement, G.,
“Generation, Optimization and Adaptation of Multi-Block Structured
Grids for Complex Configurations,”Survey on Mathematics for
Industry, Vol. 4, 1995, pp 267–277.
[6] Rasmussen, M. L., and Brandes-Duncan, B., “Hypersonic Waveriders
Generated from Power-Law Shocks,”AIAA Paper 95-6160, 1995.
[7] Péchier, M., “Prévisions Numériques de l’effet Magnus pour des
Configurations de Munition,”Ph.D. Dissertation, Department of
Mechanical Engineering, Univ. of Poitiers, France, Sept. 1999.
[8] Mangin, B., Benay, R., and Chanetz, B., “Augmentation de la Finesse
d’un Engin Hypersonique par Application du Concept Waveriders,”
DAFE ONERA-RT 5/08424, 2003.
[9] Zucrow, M. J., and Hoffman, J. D., Gas Dynamics, Vols. 1 and 2, Wiley,
New York, 1976.
[10] Rasmussen, M. L., “Analysis of Cone-Derived Waveriders by
Hypersonic Small-Disturbance Theory,”Proceedings of the 1st
International Hypersonic Waverider Symposium, NASA/Univ. of
Maryland, Dept. of Aerospace Engineering, 1990.
[11] Kim, B. S., Rasmussen, M. L., and Jischke, M. C., “Optimization of
Waverider Configurations Generated from Axisymmetric Conical
Flows,”Journal of Spacecraft and Rockets, Vol. 20, No. 5, 1983,
pp. 461–469.
[12] Cousteix, J., Turbulence et Couche Limite, Cépadues-Éditions,
Toulouse, France, 1989.
[13] Stainback, P. C., Wagner, R. D., Owen, F. K., and Horstman, C. C.,
“Experimental Studies of Hypersonic Boundary-Layer Transition and
Effects of Wind-Tunnel Disturbances,”NASA TND 7453, 1974.
[14] Nelder, J. A., and Mead, R., “A Simplex Method for Function
Minimization,”Computer Journal (UK), Vol. 7, Jan. 1965, pp. 308–
313.
[15] Anderson, J. D., Jr., Hypersonic and High Temperature Gas Dynamics,
Series in Aeronautical and Aerospace Engineering, McGraw–Hill, New
York, 1989.
[16] Lobbia, M., and Suzuki, K., “Design and Analysis of Payload-
Optimized Waveriders,”AIAA Paper 2001-1849, 2001.
M. Miller
Associate Editor
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