11. Hypersonic Aerodynamics - the AOE home page - Virginia Tech

11.1 Introduction
11. Hypersonic Aerodynamics
Hypersonic vehicles are commonplace. There are more of them than the true supersonic
aircraft discussed in the last chapter. Applications include missiles, launch vehicles and entry
bodies. Nevertheless, there is no exact definition defining the start of the hypersonic flow
regime. Possibilities are:
a) Mach numbers at which supersonic linear theory fails
b) Where γ is no longer constant, and we must consider temperature effects on
fluid properties.
c) Mach numbers from 3 - 5, where Mach 3 might be required for blunt bodies
causing large disturbances to the flow, and Mach 5 might be the starting point
for more highly streamlined bodies.
In this section we will provide a brief outline of the key distinguishing concepts. The book by
Anderson 1 provides a starting point for further study.
Essentially there are five key points to be made:
1. Temperature and aerodynamic heating become critically important
2. In many cases pressure can be estimated fairly easily
3. Blunt shapes are commonplace
4. Control and stability issues lead to different shapes
5. Engine-airframe Integration is critical
11.2 Importance of Temperature/Aerodynamic Heating
This is easily illustrated considering the relation between stagnation temperature and static
temperature:
Or the wall temperature
T 0
T
T adiabatic
wall
γ −1 2
=1 + M
2 ∞
⎛
= ⎜ 1+ r γ −1
2 M ∞ 2
⎝
(11-1)
⎞
⎟ T
⎠ e (11-2)
where r is the recovery factor. With these relations, the limit for aluminum structure is around
Mach 2, which was the Concorde’s cruise Mach number. The SR-71 is made of titanium, and
temperature limits the speed to slightly over Mach 3. Note that some sources indicate that this
limit is actually the temperature limit on the wiring in the airplane, and that is the condition
that limited the speed.
Thus hypersonic aerodynamic configuration design means that you must always deal with
heating. In general, at sustained high speeds surfaces must be cooled, and since heating is a
critical concern, this means that viscous effects are crucial immediately. Also, unlike normal
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11-2 Configuration Aerodynamics
airplane aerodynamics, hypersonic vehicles fly at very high altitudes, and the Reynolds
number may be so low that the flow is laminar. Thus laminar flows are often of interest. This
is important because the heat transfer is much lower when the flow is laminar. In fact, being
able to estimate the transition location with certainty is a critical requirement in hypersonic
vehicle design, and is the subject of current research. 2
The X-15, shown in Figure 11-1, is the only real manned hypersonic airplane flown to
date. It was rocket powered, and started flight by being dropped from a B-52, so it was purely
a research airplane. The first flight was by Scott Crossfield in June of 1959. The X-15 reached
314,750 feet in July of 1962. An improved version reached a Mach number of 6.7 and an
altitude of 102,100 feet in October of 1967 with Pete Knight at the controls. The X-15
program flew 199 flights, with the last one being in October of 1968. * Milt Thompson’s book 3
describes the X-15 program, including the crackling sounds in the airframe as it heated up!
Figure 11-1. X-15 (NASA Dryden photo archives photo)
In case the issue of aerodynamic heating seems academic, consider the M = 6.7 flight of
the X-15. It turned out to be the last flight of that plane. A dummy ramjet installation was
* The only undergraduate from the Virginia Tech Aerospace Engineering program to earn astronaut status was
Jack McKay, who flew the X-15 as a NASA test pilot, and reached 295,000 feet in September of 1965.
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W.H. Mason Hypersonic Aerodynamics 11-3
tested by placing the ramjet mockup below the airplane. The shocks from the front of the
ramjet inlet spike impinged on the pylon supporting the scramjet. The shock interference
heating was so severe that the shocks acted as a blowtorch, cutting through the structure, and
effectively slicing off the scramjet. The internal damage to the airplane from the heating led to
the scrapping of this high-speed version of the airplane. Figure 11-2 below shows the scramjet
hanging below the airplane.
Figure 11-2. X-15 with dummy ramjet (Picture from the NASA Dryden Photo Web Site)
11.3 Surface pressure estimation
In many cases surface pressures are relatively easy to estimate at hypersonic speeds. At
supersonic speed we have a local relation for two-dimensional flows relating surface slope
and pressure:
C p =
2θ
M 2 −1
(11-3)
However, this relation is not particularly useful in most cases on actual aircraft
configurations. In comparison, hypersonic rules are useful. The most famous relation is based
on the concepts of Newton. Although Newton was wrong for low-speed flow, his idea does
apply at hypersonic speeds. The idea is that the oncoming flow can be though of as a stream
of particles, which lose all their momentum normal to a surface when they “hit” the surface.
This leads to the relation:
C p = 2sin 2 θ (11-4)
where θ is the angle between the flow vector and the surface. Thus you only need to know the
geometry of the body locally to estimate the local surface pressure. Also, particles impact only
the portion of the body facing the flow, as shown in Figure 11-3. The rest of the body is in a
“shadow”, and the Cp is zero. See Anderson 1 for the derivation of this and other pressureslope
rules.
Two key observations come from the Newtonian rule. First, the Mach number does not
appear! Second, the pressure is related to the square of the inclination angle and not linearly as
it is in the supersonic formula. This illustrates how the situation in hypersonic flow is
significantly different than the linearized flow models at lower speed.
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11-4 Configuration Aerodynamics
oncoming
flow
Shadow
Region,
Cp = 0.0
Figure 11-3. Shadow sketch, showing region where Cp is zero
This figure also shows how the situation changes from low speed to high speed. Look at the
“ultimate” lift plot in Figure 11-4. Here the lower surface pressure is taken equal to the
stagnation value, and the upper surface is taken equal to the vacuum value. We can see that at
low speeds the lift is generated on the upper surface, while at high speed the lift is almost
completely generated on the lower surface.
CL
ultimate
14.0
12.0
10.0
8.0
"Ultimate" CL
Note the switch from upper surface
to lower surface dominated, and the
relatively low values from Mach 0.9
to Mach 1.5.
6.0
Total Lift
4.0
Upper
Surface
2.0
Lower
Surface
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Mach
Figure 11-4. The “Ultimate CL” plot (showing the dominance of the lower
surface at hypersonic speeds)
The Newtonian flow model can be refined to improve agreement with data. This form is
known as the Modified Newtonian flow formula,
where the stagnation Cp is a function of Mach and γ,
C p = C pmax sin 2 θ (11-5)
C pmax = p 0 2
− p ∞
1 2
ρ ∞ V ∞
2 (11-6)
and P 02 is the stagnation or total pressure behind a normal shock. This expression gets both the
Mach number and ratio of specific heats back into the problem. The classical Newtonian
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W.H. Mason Hypersonic Aerodynamics 11-5
theory is actually the limit as M → ∞, and γ → 1. As briefly noted above, there are lots of
other local rules, and Anderson’s book 1 should be consulted for a complete discussion.
11.4 Round LE shapes to reduce heat transfer
Since round shapes have lots of drag, why do we see such highly rounded shapes The answer
is the heat transfer. The maximum value of the heat transfer is related to the leading edge
radius as shown in this relation:
q ˙ max ≈
laminar
1
R LE
(11-7)
where R LE is the leading edge radius at the stagnation point.
Thus, we need a nose or leading edge radius large enough for the tip not to melt. H. Julian
Allen and A.J. Eggers, Jr., at NASA Ames did the analysis that showed that blunt bodies were
required to survive entry from orbit. 4 Thus, the solution of hypersonic flows about the nose of
an entering body became an important analysis problem. In fact, this was the first great
challenge problem for CFD (this was in the 1960s, and the computational solution of the
flowfield wasn’t called CFD until the early 1970s). It is known as “the Blunt Body problem”.
It was particularly difficult because the flow is subsonic behind a normal, or nearly normal,
shock wave. The flow then accelerates quickly to supersonic/hypersonic speed. This is the
reverse problem of transonic flow, where the freestream is supersonic/hypersonic rather than
subsonic. In particular, we want to know the shock standoff distance, the shock shape, and the
flow properties at the nose, where the aerodynamic heating is largest. The shape of the shock
determines the distribution of flow properties such as entropy, which vary as the shock shape
changes.
The problem was solved by computing the unsteady flowfield, which is always
mathematically hyperbolic. If the solution is actually steady, the computation will converge to
the steady state result, while overcoming the difficulty of the mixed elliptic-hyperbolic
equation type that describes the steady state problem. The successful method invented to
obtain a practical blunt body calculation method is generally attributed to Moretti. 5
11.5. Aerodynamic control
We described the benefits of blunt shapes above. Many hypersonic vehicles display another
unusual geometric feature. It turns out that thick bases are often used on hypersonic vehicles,
and in this section we illustrate why they are desirable. Our example relates to the thick
trailing edge on the vertical tail of the X-15.
The example of the difference in the flow characteristics between supersonic and
hypersonic speed can be illustrated with the hypersonic directional stability problem. To start,
we consider the yawing moment, which is
where the standard definition of C N is:
N VT = q VT S VT l VT C YVT (11-8)
C n =
N
q ref S ref b ref
(11-9)
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11-6 Configuration Aerodynamics
We can write the yawing moment due to the vertical tail as:
C nVT =
l VT S VT
b ref S
ref
vertical tail
volume
coefficient, V VT
q VT
⋅ C YVT (11-10)
q
ref
ratio of
dynamic pressures
assume ≈ 1
The nomenclature associated with the problem and the equations is illustrated in Figure 11-5.
⋅
β
V
∞
N
l VT
Y VT
US
LS
θ
Figure 11-5. Sketch defining the hypersonic directional stability problem
Now, for a high-speed flow, we will assume that the vertical tail is a two-dimensional
surface with a constant pressure on each side, so that C YVT = C PLS − C PUS . We will consider
two cases, one supersonic, the other hypersonic. We compare the results for directional
stability at high Mach number using the two-dimensional rule for linearized supersonic flow
and Newtonian theory for hypersonic flow, as shown below.
θ
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W.H. Mason Hypersonic Aerodynamics 11-7
Linear theory
2θ
C p =
M 2 −1
Newtonian theory
C p = 2sin 2 θ (11-11)
Case 1: Linearized supersonic theory
( )
( )
C YVT = ΔC p = 2 θ + β
M 2 −1 − 2 θ − β
M 2 −1 =
showing that the θ’s cancel. We use this expression to get C nβ :
C nβ
VT = V VT
∂C YVT
∂β
= V VT
4
M 2 −1
4β
M 2 −1
This expression shows that C nβ is positive, but vanishes for hypersonic Mach numbers.
(11-12)
(11-13)
Case 2: Hypersonic flow theory, Newtonian theory
This time the expression for the side force is:
Using trig functions:
Which reduces to:
and
C Yβ
VT
C YVT = ΔC p = 2sin 2 ( θ + β) − 2sin 2 ( θ − β) (11-14)
⎡
= 2 sin2 θ cos 2 β + 2cosθ sinβ sinθ cos β + cos 2 θ sin 2 β
⎤
⎢
⎥
⎢
⎣ −sin 2 θ cos 2 β + 2 cosθ sin βsinθ cos β − cos 2 θ sin 2 ⎥
β⎦
C YβVT
= 8cosθ sin β
sinθ cosβ
≈1 ≈β ≈θ ≈1
≅ 8θβ
(11-15)
(11-16)
C nβ
VT = 8V VTθ (11-17)
If θ is zero, so is C nβ ! But opening up the wedge angle rapidly increases C nβ . In addition,
there is no Mach number dependence. The Case 2 results were verified experimentally, and
the wedge vertical tail concept literally saved the X-15 program. 6 This effect is also the reason
for flared “skirts” seen on some launch vehicles.
The application of this effect is clearly shown in the first picture of the X-15 above and the
front view of the X-15 shown in Figure 11-6 below.
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11-8 Configuration Aerodynamics
Figure 11-6. Picture of the X-15 showing the wedge used for the vertical tail “airfoil”
section. (from the US Air Force Museum Web Site)
11.6. Engine-airframe Integration
Any air-breathing hypersonic vehicle will have a highly integrated engine and airframe. It will
be hard to tell exactly what counts as the vehicle, and what counts as the propulsion system.
This is exemplified in the concepts developed for recent aerospace planes. In these concepts
the propulsion is at least in part provided by a scramjet engine, which obtains thrust with a
combustion chamber in which the flow is supersonic. Figure 11-7, from a presentation by
Walt Engelund of NASA Langley, shows this concept. The entire forebody of the vehicle
underside is used as an external inlet to provide flow at just the right conditions to the engine.
The entire underside afterbody is the exhaust nozzle.
The first US vehicle to use a scramjet engine was the Hyper-X, now known as the X-43.
The initial flight attempt was made in June of 2001, but failed before having a chance to
demonstrate the operation of the scramjet. After a long investigation and a revised procedure,
it flew successfully on March 27, 2004. 7 The X-43A is shown in the Figures 11-8 and 11-9.
The November-December 2001 issue of the Journal of Spacecraft and Rockets had a
special section devoted to the Hyper-X.
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W.H. Mason Hypersonic Aerodynamics 11-9
Figure 11-7. Example of engine-airframe integration for a hypersonic aircraft
(from Walt Engelund, NASA Langley, May 2001).
Figure 11-8. Artist conception of the X-43
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11-10 Configuration Aerodynamics
Figure 11-9. The X-43 at NASA Dryden in flight preparation early in 2001.
11.7 Airbreathing Hypersonic Vehicle Design
The successful design of an airbreathing hypersonic vehicle is an excellent example of the
need for multidisciplinary optimization methods. This is a very hard problem. The recent
success of the scramjet test flight means that we can expect to see designs proposed using this
technology. Key surveys are available based on numerous studies conducted at NASA
Langley. 8,9
Finally, we note that one concept not discussed so far deserves mention. Waveriders
potentially provide efficient hypersonic flight. These designs can be thought of as placing a
shape in the streamline of body generated flowfield so that the bottom surface is resting on the
pressure field generated by the shock wave of this flowfield. The arrangement is designed to
obtain lift with very low drag. 10
11.10 Exercises
1. Derive expressions for the lift curve slope of a flat plate and a wedge using linearized
supersonic theory and Newtonian theory. Comment on the differences.
11.9 References
1 John Anderson, Hypersonic and High Temperature Gas Dynamics, McGraw-Hill, New
York, 1989. (reissued as a second edition by the AIAA)
2 John J. Bertin and Russell M. Cummings, “Fifty years of hypersonics: where we’ve been,
where we’re going”, Progress in Aerospace Sciences, Vol. 39, pp. 511-536, 2003 (Note:
ignore the error in the first line of the abstract, that has the wrong date and pilot for the Mach
6.7 flight)
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W.H. Mason Hypersonic Aerodynamics 11-11
3 Milton O. Thompson, At the Edge of Space: The X-15 Flight Program, Smithsonian
Institution Press, Washington, 1992.
4 H. Julian Allen and A.J. Eggers, Jr., “A Study of the Motion and Aerodynamic Heating of
Ballistic Missiles Entering the Earth’s Atmosphere at High Supersonic Speeds,” NACA
Report 1381, 1958.
5 Gino Moretti and Mike Abbett, “A Time Dependent Computational Method for Blunt Body
Flows,” AIAA Journal, Vol. 4, No. 12, 1966, pp. 2136-2141.
6 Charles McLellan, “A Method for Increasing the Effectiveness of Stabilizing Surfaces at
High Supersonic Mach Numbers,” NACA RM L54F21, Aug. 1954, Declassified: 1958.
7 Michael A. Dornheim, “A Breath of Fast Air,” Aviation Week & Space Technology, April 5,
2004.
8 Mary K. O’Neill and Mark J. Lewis, “Design Tradeoffs on Engine-Integrated Hypersonic
Vehicles,” AIAA Paper 92-1205, Feb. 1992.
9
Mary K. Lockwood, Dennis H. Petley, John G. Martin and James L. Hunt, “Airbreathing
hypersonic vehicle design and analysis methods and interactions,” Progress in Aerospace
Sciences, Vol. 35, pp. 1-32, 1999.
10 Charles E. Cockrell, Jr., Lawrence D. Huebner and Dennis B. Finley, “Aerodynamic
Characteristics of Two Waverider-Derived Hypersonic Cruise Configurations,” NASA TP
3559, July 1996.
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