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Universit`
a degli Studi di Roma “La Sapienza”
Dipartimento di Meccanica e Aeronautica
Dottorato di Ricerca in Meccanica Teorica e Applicata
XX ciclo
Ph.D. Thesis
Modeling of ablation phenomena in
space applications
Daniele Bianchi
Supervisor: Prof. M. Onofri
2006/2007
Contents
Introduction 1
Hypersonic aerodynamic heating . . . . . . . . . . . . . . . . . . . . . 1
Early Reentry Vehicles: Blunt Bodies and Ablatives . . . . . . . . . . . 2
Ablative materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CFD methods for ablating systems . . . . . . . . . . . . . . . . . . . . 10
Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
I One-dimensional transient ablation 15
1 Physical approach to the ablation problem 17
1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Conservation of energy in moving coordinate system . . . . . . . 19
1.4 Conservation equations for the chemically reacting boundary-layer 23
1.5 Transfer-coefficient correlation equations . . . . . . . . . . . . . 26
1.5.1 Transfer-coefficient approaches . . . . . . . . . . . . . . 26
1.5.2 Element conservation equation . . . . . . . . . . . . . . . 27
1.5.3 Surface mass balance . . . . . . . . . . . . . . . . . . . . 28
1.5.4 Energy equation . . . . . . . . . . . . . . . . . . . . . . 29
1.5.5 Surface energy balance . . . . . . . . . . . . . . . . . . . 31
1.5.6 Blowing correction of heat-transfer coefficient . . . . . . 32
1.6 Ablation thermochemistry . . . . . . . . . . . . . . . . . . . . . 33
1.6.1 ablation rate . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.7 Boundary-layer and material response coupling . . . . . . . . . . 37
2 Numerical approach to the ablation problem 39
2.1 Finite-difference method for the in-depth solution . . . . . . . . . 39
2.2 Nodal coordinate layout . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Crank-Nicholson algorithm . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 Interior nodes . . . . . . . . . . . . . . . . . . . . . . . . 41
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CONTENTS
2.3.2 The surface node . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 The last node . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.4 Tri-diagonal matrix form . . . . . . . . . . . . . . . . . . 43
2.4 Computational strategy for the coupled solution . . . . . . . . . . 45
2.4.1 Reduction of the Tri-diagonal matrix . . . . . . . . . . . . 45
2.4.2 Coupling in-depth responseto SEB . . . . . . . . . . . . 47
2.4.3 Completing the in-depth solution . . . . . . . . . . . . . 49
2.4.4 Solution without energy balance . . . . . . . . . . . . . . 49
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.1 Solution check-out . . . . . . . . . . . . . . . . . . . . . 50
2.5.2 Blunt body analysis . . . . . . . . . . . . . . . . . . . . . 56
2.5.3 SRM nozzle throat analysis . . . . . . . . . . . . . . . . 63
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
II CFD method for ablating surfaces 73
3 Thermodinamic model 75
3.1 High-temperature gas dynamics . . . . . . . . . . . . . . . . . . 76
3.2 Internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4 Frozen speed of sound . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Thermodynamic data . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.1 Data for individual species . . . . . . . . . . . . . . . . . 81
3.5.2 Mixture properties . . . . . . . . . . . . . . . . . . . . . 81
4 Mathematical model and numerical method 83
4.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Numerical technique . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Inflow and outflow conditions . . . . . . . . . . . . . . . 86
4.3.2 Wall conditions . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.3 Error accumulation on the boundaries . . . . . . . . . . . 93
4.3.4 Multi-block technique . . . . . . . . . . . . . . . . . . . 93
5 Ablation model and boundary conditions 95
5.1 Surface mass and energy balance . . . . . . . . . . . . . . . . . . 96
5.1.1 Steady-state surface energy balance . . . . . . . . . . . . 98
5.1.2 Surface equilibrium assumption . . . . . . . . . . . . . . 99
5.2 Thermochemical ablation model . . . . . . . . . . . . . . . . . . 100
5.2.1 Thermochemical table model . . . . . . . . . . . . . . . . 100
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5.2.2 Fully-coupled ablation model . . . . . . . . . . . . . . . 102
5.2.3 Evaluation of wall chemical composition . . . . . . . . . 103
5.3 Implementing the ablative boundary conditions . . . . . . . . . . 106
5.4 Inviscid conditions . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.5 Viscous conditions . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5.1 Isothermal ablation . . . . . . . . . . . . . . . . . . . . . 109
5.5.2 Steady-state ablation . . . . . . . . . . . . . . . . . . . . 111
5.6 Computational Requirements . . . . . . . . . . . . . . . . . . . . 112
6 2-D planar results 113
6.1 Existing approaches . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Isothermal ablation results . . . . . . . . . . . . . . . . . . . . . 114
6.2.1 Test case 1: Helium environment . . . . . . . . . . . . . . 115
6.2.2 Test case 2: Nitrogen environment . . . . . . . . . . . . . 119
6.2.3 Test case 1 and Case 2: comparison with blowing correc-
tion equation . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.4 Effect of surface temperature . . . . . . . . . . . . . . . . 126
6.2.5 Effect of boundary-layer finite-rate chemistry . . . . . . . 131
6.2.6 Comparison with thermochemical table approaches . . . . 134
6.3 Steady-state ablation results . . . . . . . . . . . . . . . . . . . . 139
6.3.1 Test case 3: Air environment . . . . . . . . . . . . . . . . 139
6.3.2 Effect of Mach number . . . . . . . . . . . . . . . . . . . 142
6.3.3 Effect of boundary-layer finite-rate chemistry . . . . . . . 149
6.3.4 Comparison with thermochemical table approaches . . . . 152
7 Rocket nozzle applications 159
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3 Chamber equilibrium calculations . . . . . . . . . . . . . . . . . 162
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 163
8 Conclusions 177
A Governing equations 181
A.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A.3 Equations in terms of a/δ,v,s,yi. . . . . . . . . . . . . . . . . 186
A.4 Nondimensionalform of the Navier-Stokes equations . . . . . . . 188
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B Lambda scheme and solving technique 191
B.1 The λ-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
B.2 Equations in the computational plane . . . . . . . . . . . . . . . . 196
B.3 Two-Dimensional axisymmetric problems . . . . . . . . . . . . . 202
B.4 Time-marching finite difference method . . . . . . . . . . . . . . 205
C Difference form of the in-depth energy equation 209
C.1 Interior nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.2 The surface node . . . . . . . . . . . . . . . . . . . . . . . . . . 210
C.3 The last node . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C.4 Equations for coefficients in energy equation array . . . . . . . . 212
Introduction
”...re-entry...is perhaps one of the most difficult problems one can imagine...It is
certainly a problem that constitutes a challenge to the best brains working in these
domains of modern aerophysics...possible means [include] mass transfer cooling,
consisting of a coating that sublimates or chemically dissociates...
-Theodore von Karman
Hypersonic aerodynamic heating
The matter of aerodynamic heating is an extremely important aspect of hypersonic
vehicle design and the understanding and accurate prediction of surface heat flux
is a vital part of the study and design of a hypersonic vehicle. The kinetic energy
of a high-speed, hypersonic flow is dissipated by friction inside the boundary-
layer. The viscous dissipation that occurs within hypersonic boundary-layers can
produce very high heat-transfer rates to the surface. The surface itself must be
designed in order to sustain the heat flux without collapsing and to prevent the
heat load from damaging the underlying structure. Therefore surface heat transfer
is usually one of the dominant aspect that drives the design of hypersonic vehicles
and also of rocket nozzles. To understand why the aerodynamic heating becomes
so large at hypersonic speeds, we can derive some useful relations from the flat
plate theory.
The local heat-transfer coefficient can be expressed by any one of the several
defined parameters, such as the Stanton number Chdefined as follows:
Ch=qw
ρeue(haw hw)(1)
where qwis the heat transfer (energy per second per unit area) into the wall, haw
and hware the adiabatic wall enthalpy and the wall enthalpy, respectively, and the
subscript edenotes local properties at the outer edge of the boundary layer. If
we consider the case of a flat plate parallel to the flow, these local properties are
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simply freestream values, namely ρand u. The wall heat flux is therefore:
qw=ρuCh(haw hw)(2)
Assuming for simplicity a recovery factor of unity, the adiabatic wall enthalpy has
the following expression:
haw =h0=h+u2
2(3)
where h0is the total enthalpy of the flow. Since at hypersonic speeds u2/2is
much larger than h, from (3) we obtain:
haw u2
2(4)
Moreover, even if the surface temperature can be high in this kind of application,
it is still limited by the material itself, i.e. it cannot exceed the melting or failing
temperature of the protection material. Hence, the surface enthalpy h0is usually
much smaller than h0at hypersonic speeds. That is, using also (4):
(haw hw)haw u2
2(5)
Substituting Eq. (5) into (2) we obtain the approximate relation:
qw1
2ρu3
Ch(6)
The main purpose of Eq. (6) is to demonstrate that aerodynamic heating increases
with the cube of flight velocity and hence increases very rapidly in the hyper-
sonic flight regime, such is the case of an atmospheric reentry. By comparison,
aerodynamic drag is given by:
D=1
2ρu2
SCD(7)
Hence, at hypersonic speeds, aerodynamic heating increasing much more rapidly
with velocity than drag, and this is the primary reason why aerodynamic heating
is a dominant aspect of hypersonic vehicle design.
Early Reentry Vehicles: Blunt Bodies and Ablatives
Although various people, including Wernher von Braun and other experts, had
studied spaceflight during the 1940’s nobody began thinking about how a vehicle
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would actually return from space until the early 1950s. The few who did, like
von Braun, realized that probably the best way to do it was to build a very big
vehicle and circulate a fluid through its skin to soak up the heat of reentry. Clearly
the problem of reentry to Earth’s atmosphere was a significant challenge for the
early spaceflight researchers, as they considered how best to overcome the heat
generated by friction. However, not all reentry vehicles were spacecraft such as
the atomic warheads launched atop ballistic missiles. They would fly up in a
cannonball arc above most of the atmosphere and then come back through it at
around 20 times the speed of sound, heating up tremendously.
Early research on missile reentry vehicles during the 1950’s focused upon
long, needle-like designs. When tested in wind tunnels, so much heat was trans-
ferred to these vehicles that they burned up. Scientist H. Julian Allen at the Ames
Aeronautical Laboratory made a rather counter-intuitive discovery in 1952: he
found that by increasing the drag of the vehicle, he could reduce the heat it gen-
erated. Much of the heat of reentry was actually deflected away from the vehicle.
The best designs were what Allen and another scientist, Alfred J. Eggers, called
”blunt-body” designs. Instead of needle-noses, they had blunt noses that formed
a thick shock wave ahead of the vehicle that both deflected the heat and slowed it
more quickly, thereby protecting the vehicle.
Figure 1: Prototype version of the Mk-2 reentry vehicle (RV).
Based upon this research, in 1955 General Electric (GE) engineers began work
on the Mark 2 reentry vehicle (see Figure 1) for the Thor, Jupiter and Atlas mis-
siles. The Mark 2 was a blunt body design. Much of the heat was deflected
away from the vehicle via the shock wave. But some heat still reached the surface
through the superheated air that formed in front of the vehicle. Getting rid of this
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CONTENTS
excess heat was a problem. GE decided to use the heat-sink concept, whereby the
heat of reentry was conducted from the surface of the vehicle to a mass of mate-
rial that could soak it up quickly. The key was to conduct the heat away from the
surface fast enough so that the surface material itself did not melt. GE’s engineers
tested several materials as heat-sinks, including beryllium, cast iron, and steel.
But the best proved to be copper. By putting a big mass of copper just below the
outer shell of the vehicle they could prevent the craft from burning up. Figure 2
shows the copper heat sink of the intercontinental ballistic missile. A 1000 pound,
copper-clad 316 stainless steel shell was manufactured by electroforming, pos-
sessing an outer skin of nickel and a reflective platinum final surface. The design
was the precursor for the manned Mercury flights with beryllium heat sinks and
the subsequent Gemini and Apollo flights, which had head shields instead of heat
sinks, but the same blunt shapes.
Figure 2: The copper heat sink of the intercontinental ballistic missile (ICBM).
The Mark 2 had what was called a low ballistic coefficient, or beta. The ballis-
tic coefficient was a calculation of weight, drag and cross-section. Vehicles with a
high beta, usually slender and smoother and with less drag, travelled through the
upper atmosphere without decelerating much and did most of their slowing down
in the thick-lower atmosphere. They took longer to slow down and generated less
heat, but experienced this heat over a longer period of time. GE’s Mark 2 had
a low beta. It was a flattened cone on its leading edge. It spent a lot of time in
the upper atmosphere, trailing a stream of ionized gas that showed up on radar,
which was not good for a warhead. Although this design was adequate, it was not
ideal. What ballistic missile designers wanted was a vehicle that travelled as fast
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as possible through the atmosphere so that it could not be intercepted. A high-
beta vehicle was the best choice. GE engineers doubted that heat-sink technology
would work for a high-beta vehicle. In addition, the heat-sink concept was heavy
and the copper took up valuable payload weight.
The heat-sink’s drawbacks became even more apparent when it was consid-
ered for a space vehicle. First, a space vehicle would reenter at a faster velocity
than a ballistic missile and so it would get much hotter that the missile, requiring
more copper, and perhaps other means of transferring the heat away from the sur-
face. In addition, all that weight was prohibitive. Even worse, the extra-weight
had cascading effects. Not only would all that copper require more fuel to get it
into orbit, but it would require more fuel to get it out of orbit. An even bigger
problem was the high-temperature reached by the heat-sink itself, which could be
sustained by a warhead but not by a human being. A reentry at much higher ve-
locities was needed for a lunar mission, but the metal composite heat-sink would
vaporize like a meteor. The aerodynamicist’s answer from experiments with gas
guns and theoretical calculations was the concept of ablation by vaporizing a pro-
tection material as the thermal barrier. This was the heat shield concept in oppo-
sition to the heat-sink concept.
By the mid-1950’s, GE engineers were designing lightweight, medium-beta,
reentry vehicles for missile warheads. GE engineers evaluated several different
concepts. One was transpirational cooling, which essentially boiled off a liq-
uid, using the change from liquid to gas to take away the heat. Another was
re-radiation, whereby the heat would be radiated away from the vehicle. Another
proposal was liquid metal cooling, whereby a liquid metal, such as mercury, was
circulated through the heat shield and conducted the heat away very efficiently.
But the most promising proposal was a technique called ablation. By 1956, some
researchers were noting that reinforced plastics had proven more resistant to heat-
ing than most other materials. They proposed using these plastics in the inlets of
supersonic cruise missiles. GE engineers realized that they could use this same
technique for reentry. They could coat the vehicle with a material that absorbed
heat, charred, and either flaked off or vaporized. As it did so, it took away the
absorbed heat.
The ablation technique worked for both spacecraft and ballistic missile re-
quirements, for low and high-beta reentry vehicles. Ablation reduced tempera-
tures. A bluntbody, low-beta reentry vehicle returning from space could keep the
external temperature relatively low. Ablative material on the vehicle would lower
this temperature even further. A streamlined, high-beta missile warhead, however,
would experience much higher heating for shorter periods of time. But ablation
could also reduce this temperature as well, so that a missile warhead could reen-
ter very fast and minimize its chance of interception, keeping cool by burning off
layers a special plastic. The key was selecting the right material. Ultimately, they
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CONTENTS
decided upon a phenolic resin plastic. They decided to use a nylon cloth impreg-
nated with the phenolic resin and molded into the needed shape. Eventually, this
and similar materials were used to coat the surfaces of nuclear missiles warheads.
Figure 3: Mercury spacecraft ablative heat shield after recovery.
Figure 4: Charred ablative heat shield from the first KH-4 Corona mission.
The first Mercury spacecraft used a blunt body design and a heat sink, but later
versions used the blunt body design and an ablative surface (see Figure 3). GE
built an ablative semi-blunt (slightly rounded) reentry vehicle for Air Force and
CIA Discoverer/CORONA spacecraft (see Figure 4), which returned film from
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spy satellites. Other companies, such as AVCO, also developed ablative reentry
vehicles for missiles. Blunt body designs and improved ablative materials were
also used on the Gemini and Apollo spacecraft, advancing rapidly during 1960s.
By the end of the decade, other technologies and techniques for surviving the
tremendous heating of atmospheric reentry were developed.
Figure 5: SRM nozzle structure.
Figure 6: Ablative materials in SRM nozzles.
Ablation is affected by the freestream conditions, the geometry of the reentry
body, and the surface material. Ablation occurs during the reentry of planetary
expeditions or of ballistic projectiles and it occurs inside the nozzles of solid pro-
pellant rocket motors. Reentry vehicles range from blunt configurations, such as
the Apollo spacecraft, to slender sphere-cone projectiles. For low heating levels,
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CONTENTS
low-temperature ablators such as teflon are used and for more demanding reentry
conditions, graphites and carbon-based materials are often used.
Graphitic materials have received much attention in the last decades for ap-
plication to both planetary entry probe heat shields and ballistic missile nose tip
and heat shields. Ablation of graphite on atmospheric reentry continues to be ac-
tively studied, both to achieve greater fidelity of simulation and to support new
concepts. Moreover carbon/carbon composites and other graphitic materials have
found increasing use in the manufacture of nozzles for solid-propellant rocket mo-
tors (see Figures 5 and 6) because of their high-temperature resistance, exce