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Annual Review of Fluid Mechanics

Rate Effects in Hypersonic

Flows
Graham V. Candler
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Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis,

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Minnesota 55455, USA; email: candler@umn.edu

Annu. Rev. Fluid Mech. 2019. 51:379–402 Keywords

The Annual Review of Fluid Mechanics is online at hypersonic aerodynamics, aerothermodynamics, high-temperature gas
fluid.annualreviews.org
dynamics, nonequilibrium flows, finite-rate processes
https://doi.org/10.1146/annurev-fluid-010518-
040258 Abstract
Copyright  c 2019 by Annual Reviews. Hypersonic flows are energetic and result in regions of high temperature,
All rights reserved
causing internal energy excitation, chemical reactions, ionization, and gas-
surface interactions. At typical flight conditions, the rates of these processes
are often similar to the rate of fluid motion. Thus, the gas state is out of
local thermodynamic equilibrium and must be described by conservation
equations for the internal energy and chemical state. Examples illustrate
how competition between rates in hypersonic flows can affect aerodynamic
performance, convective heating, boundary layer transition, and ablation.
The conservation equations are outlined, and the most widely used models
for internal energy relaxation, reaction rates, and transport properties are
reviewed. Gas-surface boundary conditions are described, including finite-
rate catalysis and slip effects. Recent progress in the use of first-principles
calculations to understand and quantify critical gas-phase reactions is dis-
cussed. An advanced finite-rate carbon ablation model is introduced and is
used to illustrate the role of rate processes at hypersonic conditions.

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1. INTRODUCTION: THE HYPERSONIC FLOW ENVIRONMENT

A flow is typically considered hypersonic if its Mach number is greater than five. The Mach number
is the ratio of the flow speed to the sound speed, and the square of the Mach number is proportional
to the ratio of the kinetic energy to the internal energy of the flowing gas. When a hypersonic
flow is brought to rest or passes through a shock wave, much of its kinetic energy is converted
to internal energy and the temperature increases. Thus, hypersonic flows are energetic and are
characterized by high temperatures. For example, when the Space Shuttle orbiter reentered the
Earth’s atmosphere, it was traveling at 7.5 km/s or about Mach 25, and the kinetic energy of the
flow was approximately 124 times greater than the thermal energy of the free-stream gas.
In addition to planetary-entry flows, there is interest in the development of vehicles capable
of sustained hypersonic flight in the atmosphere, which is driving the development of hypersonic
gliders and air-breathing scramjet-powered systems. Many of the current areas of research in
hypersonic aerodynamics and aerothermodynamics are discussed in a recent article (Leyva 2017).
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Another recent article reviews key fluid dynamics issues associated with the operation of scramjets
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(Urzay 2018). An older review of planetary entry gas dynamics is also relevant (Gnoffo 1999).
Figure 1 illustrates several hypersonic flows: the 1960s-era X-15 Mach 6.7 rocket plane, a
generic lifting hypersonic vehicle, and the Mars Science Laboratory capsule flying at 6 km/s
in the Mars atmosphere. The latter two images are from computational fluid dynamics (CFD)
simulations of the flow field, and both visualize the temperature on the flow symmetry plane. The
hypersonic speeds result in high temperatures in the shock layer that envelops the vehicle and in
the boundary layer where there are extreme levels of shear.
The elevated temperatures in hypersonic flows give rise to many processes, such as vibra-
tional and electronic energy excitation, chemical reactions, ionization, and gas-surface interac-
tions. When these processes occur, the perfect gas shock relations are no longer valid and the
equations of state become nontrivial. For example, a Mach-6 normal shock wave produces con-
ditions such that the vibrational energy is excited to about 10% of the total internal energy. At
higher Mach numbers, additional processes become important, and therefore hypersonic flows are

a c

V∞

Figure 1
Visualizations of three hypersonic flows: (a) the NASA X-15, (b) a generic lifting hypersonic body, and (c) the Mars Science Laboratory
capsule at Mars entry conditions. Panel a courtesy of NASA.

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usually characterized by imperfect gas effects. The rates of vibrational excitation and chemical re-
actions depend on the local thermodynamic state, and these rates increase with increasing density
and temperature. At typical hypersonic conditions, these rates are often similar to the advection
rates.
In this review, we discuss the effects of the internal energy relaxation and chemical reaction
rate processes on hypersonic flows. We see that it is seldom possible to assume that the gas is in
a state of local thermodynamic equilibrium; rather, the thermodynamic state changes as the gas
responds to rapid compressions and expansions. Several examples are provided to illustrate what
happens in a typical hypersonic flow and to motivate the development of the governing equations
and the scaling properties of these flows. Specific examples of interactions between finite-rate gas-
phase and gas-surface reactions are discussed. These include interaction of acoustic disturbances
and vibrational energy relaxation, chemical freezing in high-enthalpy wind tunnel nozzles, and
control flap effectiveness at hypersonic flight conditions.
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1.1. Gas-Phase Processes

Let us consider a typical hypersonic environment to illustrate the types of gas-phase rate processes
that occur in these flows. Figure 2 shows CFD simulations (see Candler et al. 2014, 2015) of
5-km/s airflow over a 0.1-m radius sphere-cone geometry at three altitudes (40 km, 60 km, and
80 km). The primary difference between each case is the free-stream density, which decreases by
a factor of 13.4 between altitudes of 40 and 60 km and by a factor of 18.3 between 60 and 80 km.
These differences in density and the corresponding differences in pressure have significant effects
on the rate processes and the resulting flow field. The free-stream conditions are given in Table 1.
First, consider the 40-km altitude case. Note that there is a strong shock wave that separates
the free-stream flow from the high-temperature gas that envelops the sphere-cone geometry. This
shock wave is commonly called a bow shock, due to a vague similarity to the bow wave in front
of a boat. On the centerline the bow shock is normal, and its strength decreases with increasing
distance from the centerline. The shock heats the gas to a temperature of about 6,000 K in this
region; this is a significantly lower level than given by perfect-gas normal shock theory, which
predicts a temperature of 12,300 K. This difference in temperature is due to internal energy
excitation, chemical reactions, and ionization, which absorb energy and reduce the postshock
temperature. All of these processes take place at rates that depend on the local flow state. We see
that in hypersonic flows, these rates are often similar to the advection rate. Thus, a reaction may
be initiated at one point in a flow field, but not reach completion before the gas has moved to a
location with a different thermodynamic state.
Now consider the evolution of the flow along the streamline highlighted in the figure. The
postshock pressure is about 0.86 bar, and the temperature jumps to about 10,000 K before rapidly
dropping to 5,500 K. Two temperatures are plotted: the translational–rotational temperature,
T, and the vibrational–electronic temperature, Tve . In the present model, it is assumed that the
rotational energy modes of the gas molecules are in equilibrium with the translational or thermal
modes. However, the internal energy in the vibrational and electronic modes is allowed to evolve
at its own rate, and its energy is characterized by Tve . This is a two-temperature representation
of the gas energy state. In the present case, the two temperatures are close to being equilibrated
with one another.
After passing through the shock, the gas rapidly expands as it accelerates around the spherical
nose, which causes the pressure and temperature to drop before reaching approximately constant
values on the conical section of the body. Note that the gas has significant levels of reaction, with
O2 almost fully dissociated and N2 7.3% dissociated. Nitric oxide (NO) is present at about the

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T (K) 1.0 12 100

p
40 km altitude 11,500 T
2.0 Tve 10 10–1
10,500 0.8

Pressure, p (bar)
9,500

T, Tve × 10 3 (K)

Mass fraction
8,500 8 10–2 YN
1.5
0.6 YO
7,500
y/R 6 10–3 YNO
6,500 YNO+
1.0 4,500 0.4
5,500 4 10–4
3,500
0.5 0.2
2,500 2 10–5
1,500
0 500 0 0 10–6 0
0.08 12 100
60 km altitude
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2.0 10 10–1
0.06
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Pressure, p (bar)

T, Tve × 10 3 (K)

Mass fraction
8 10–2
1.5

y/R 0.04 6 10–3

1.0
4 10–4
0.02
0.5
2 10–5

0 0 0 10–6
4 12 100

2.0 80 km altitude 10 10–1

Pressure, p × 10 –3 (bar)

3
T, Tve × 10 3 (K)

Mass fraction
8 10–2
1.5

y/R 2 6 10–3
1.0
4 10–4
1
0.5
2 10–5

0 0 0 10–6
0 0.5 1.0 1.5 2.0 2.5 0 1 2 0 1 2
x/R Distance on streamline, s/R Distance on streamline, s/R

Figure 2
Temperature contours in the flow field of a sphere-cone at three altitudes and a speed of 5 km/s (left), along with flow properties
extracted on the streamline (center and right). The symbols at s/R = 2 in the mass fraction plots (right) denote the equilibrium state of
air at the local p, T. Variables: p, pressure; R, nose radius (10 cm); s, distance along the streamline; T, translational–rotational
temperature; Tve , vibrational–electronic temperature; Y , mass fraction.

Table 1 Free-stream conditions for example sphere-cone cases

Altitude (km) Density (kg/m3 ) Temperature (K) Pressure (Pa) Mach number Mean free path (µm)
40 3.85 × 10−3 251 278 15.7 19.7
60 2.88 × 10−4 245 20.3 15.9 261
80 1.57 × 10−5 197 0.886 19.9 4430

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1% level, and its ion, NO+ , at a much lower level. This is the most prevalent ionized species at
the present conditions.
In the high-pressure region immediately behind the shock, the gas is in local thermodynamic
equilibrium. However, at s/R = 2, the gas state is far from equilibrium, with four orders of
magnitude more N atoms present than predicted by equilibrium. This out-of-equilibrium, or
nonequilibrium, state is a result of the rapid expansion of the gas around the body. The drop in
pressure and temperature results in a sudden reduction in the chemical recombination rate, and the
gas is approximately frozen near its postshock thermochemical state. Thus, there is a competition
between the reaction and advection rates; this is one of the main features of hypersonic flows.
Now let us compare the 40-km altitude flow field with the higher-altitude cases. At 60 km, the
flow field is generally similar, but the elevated temperature region behind the shock wave is more
pronounced, and the postshock temperature is higher, reaching about 11,400 K. The vibrational–
electronic modes are close to equilibrium until the gas expands, then Tve lags the change in T.
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Also note that at s/R = 2, there are three orders of magnitude more N atoms than predicted by
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local thermodynamic equilibrium. These differences are due to the lower pressure and density at
this altitude, which results in reduced rates of internal energy relaxation and chemical reaction.
At 80 km, the flow field is significantly different, with a more diffuse bow shock wave and
a large region of elevated temperature. The shock standoff distance is also notably larger. The
vibrational–electronic temperature lags the translational–rotational temperature at all locations
on the extracted streamline. The level of dissociation is greatly reduced, and at s/R = 2, there
are fewer N and O atoms than predicted by equilibrium. This is caused by the low-density free-
stream conditions, which suppress the reaction rates, resulting in subequilibrium levels of reaction
throughout the flow field. Thus, this case is significantly different than the previous two.

1.2. Gas-Surface Processes

We have seen that the gas-phase processes of internal energy excitation, chemical reactions, and
ionization affect hypersonic flows. There is another important class of rate process in these flows—
the interaction of the high-temperature gas with surfaces. Depending on the surface material
properties and the relative reaction rates, the heat transfer rate to the surface can be significantly
affected. Several examples are discussed.
Consider the heat transfer rate from the high-temperature air to the sphere-cone geometry
discussed above. We have performed two sets of simulations: one with an inert, noncatalytic
surface, and a second with a catalytic surface that promotes recombination of those N and O
atoms that impinge on the surface. We use an isothermal surface temperature of 1,000 K for both
cases. The details of the catalytic gas-surface interaction boundary condition are discussed below.
Figure 3 quantifies the heat transfer rate to the surface as a function of the distance from the
stagnation point for the three altitudes considered previously. At 40 km, the catalytic boundary
condition increases the stagnation-point heat flux by 64% relative to the noncatalytic surface. This
occurs because when the atoms recombine on the surface, the chemical bond energy is deposited
on the surface. Figure 3 also shows how the temperatures and mass fractions vary with distance
from the surface along the stagnation streamline (on the symmetry axis). Note that the 40-km
noncatalytic case has significant levels of O atoms and several percent of N atoms at the surface,
while in the catalytic case these species have close to zero mass fraction on the surface.
Figure 3 shows that at 60 km, the effect of surface catalysis is somewhat larger than at
40 km, with a 69% increase relative to the noncatalytic heat transfer rate. At 80 km, there is no
detectable difference between the catalytic and noncatalytic cases because there are very low levels
of dissociation, as shown in the lower right panel.

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300 12 0.25
Catalytic T catalytic
Noncatalytic T noncatalytic
250 10 Tve catalytic
Heat transfer rate (W/cm 2 )

0.20
Tve noncatalytic
40 k m a l titu d e

200 8

T, Tve × 10 3 (K)

Mass fraction
0.15
150 6

0.10
100 4

0.05
50 2
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0
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150 12 0.25

125 10
Heat transfer rate (W/cm 2 )

0.20
60 k m altitud e

100 8
T, Tve × 10 3 (K)

Mass fraction
0.15
75 6

0.10
50 4

0.05
25 2

0 0 0
0 1 2 3 0 0.02 0.04 0.06 0.08 0.10 0 0.02 0.04 0.06 0.08 0.10
40 12 10 0

10 10 –1
Heat transfer rate (W/cm 2 )

O2 catalytic
30 O2 noncatalytic
80 km altitud e

8 10 –2 N catalytic
T, Tve × 10 3 (K)

Mass fraction

N noncatalytic
O catalytic
20 6 10 –3 O noncatalytic

4 10 –4
10
2 10 –5

0 0 10 –6
0 1 2 3 0 0.1 0.2 0.3 0 0.1 0.2 0.3
Distance along surface, ξ/R Distance from surface, n/R Distance from surface, n/R

Figure 3
Heat transfer rate to the sphere-cone surface (left), the temperature distribution along the stagnation streamline (center), and the species
mass fraction distribution along the stagnation streamline (right) for three altitudes: 40 km, 60 km, and 80 km. Two surface models are
used: catalytic, such that all atomic species impacting the surface recombine, and noncatalytic, in which the surface does not promote
recombination. Variables: ξ , distance along the surface from the stagnation point; n, the surface-normal distance at the stagnation point.

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104 1.0

a b

Heat transfer rate divided by ρ∞0.5 u3∞

0.8
Heat transfer rate (W/cm 2 )

8 km/s
7
6
103 5
0.6 4
4 km/s

0.4 5
102
6
7
0.2 8
Catalytic Catalytic
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Noncatalytic Noncatalytic
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101 0
10 – 5 10 – 4 10 – 3 10 – 2 10 – 1 10 0 10 – 5 10 – 4 10 – 3 10 – 2 10 – 1 10 0
Free-stream density (kg/m 3) Free-stream density (kg/m 3)

Figure 4
(a) Heat transfer rate to the stagnation point of a 10-cm sphere in air at 5 km/s as a function of free-stream density. (b) Stagnation-point
√
heat transfer rate normalized by the square root of the free-stream density and free-stream speed cubed, ρ∞ u3∞ , for five free-stream
speeds (in meter–kilogram–second units).

Figure 4a shows the variation of the stagnation-point heat transfer rate as a function of the
free-stream density for the 0.1-m radius sphere at 5 km/s and a surface temperature of 1,000 K.
The noncatalytic boundary condition has a large effect on the heat transfer rate at moderate
densities, but it has a minimal effect at high and low densities. At low densities, there is little to
no dissociation, as shown in the 80 km case discussed above. At high densities, the atomic species
recombine in the low-temperature boundary layer, reducing the effect of surface catalysis.
Figure 4b summarizes a series of CFD simulations comparing catalytic and noncatalytic bound-
ary conditions at different free-stream conditions. The same sphere-cone is run at speeds ranging
from 4 km/s to 8 km/s across a wide range of free-stream densities. The ratio of the assumed wall
temperature to the free-stream total temperature is fixed at Tw /To = 0.079 for all cases. The figure
√
plots the stagnation-point heat transfer rate scaled by ρ∞ u3∞ , which approximately collapses the
data (see Tauber et al. 1987). Note that with this scaling, the catalytic surface stagnation-point
heat flux is approximately independent of density, while the noncatalytic surface condition shows
strong variations over a range of free-stream densities. The largest effect is for the highest speed
condition, but even at 4 km/s there is a substantial reduction in the stagnation-point heat flux for
densities of about ρ∞  3 × 10−4 kg/m3 . As in Figure 4a, the reduction in heat flux diminishes
when the flow is close to chemically frozen (low ρ∞ ) and when the near-surface boundary layer is
close to chemical equilibrium (high ρ∞ ).
These computational results are consistent with the reacting boundary layer theory results of
Fay & Riddell (1958), as reproduced in Figure 5. The heat transfer coefficient on a flat plate is
plotted as a function of the gas-phase recombination rate and the catalytic activity of the surface.
The heat flux is shown to be independent of the recombination rate for a catalytic surface and
falls off significantly if the surface is noncatalytic and the recombination is slow. This effect was
known at the time of the design of the Space Shuttle orbiter, but it was not possible to test the
reaction-cured glass tiles at flight conditions, and therefore the conservative fully catalytic heat
flux was used to design the thermal protection system. During the early reentries of the orbiter,

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0.5

Total heat transfer, catalytic wall

0.4

Heat transfer coefficient

wall
lytic ll
cata c wa
0.3 No
n yti
atal
, c
fer
ns
t tra
0.2 ea
fh
no
ort
io L = 1.4
ep
Conductiv α = 0.71
0.1 β = 0.5
Tw = 300 K

0
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10 – 6 10 – 5 10 – 4 10 – 3 10 – 2 10 – 1 10 0 10 1 10 2 10 3 10 4
Recombination rate parameter
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Figure 5
Nondimensional stagnation-point heat transfer rate for a reacting boundary layer with finite recombination
rates and limiting values of surface catalytic activity. Adapted with permission from Fay & Riddell (1958).

experiments were performed to assess the catalytic surface effects. Several tiles were coated with
a highly catalytic material, and in-flight measurements showed a large heat flux increase on those
tiles due to catalytic effects (Stewart et al. 1983, Curry 1993), reflecting the effects shown in
Figures 4 and 5.

1.3. Effects of Rate Processes on Hypersonic Flight Systems

The examples discussed above illustrate that finite-rate processes can significantly affect hypersonic
flows and may result in complex out-of-equilibrium behavior of the gas. In this section, we discuss
the effects of rate processes on several flight and ground-test systems.
One of the best known hypersonic rate effects is the Space Shuttle orbiter pitch anomaly. During
the first reentry of the orbiter at Mach numbers above 16, the body flap had to be deflected by 16◦
instead of the predicted 7◦ in order to trim the vehicle at a 40◦ angle of attack (Maus et al. 1984, Iliff
& Shafer 1993). This discrepancy is typically attributed to so-called real gas effects (Weilmuenster
et al. 1994), which refers to imperfect gas or chemical reaction effects. However, this conclusion
is not universally accepted in the literature and has also been attributed to a viscous–inviscid
flow interaction (Koppenwallner 1987). This occurs because at hypersonic conditions, boundary
layers tend to be thicker than their equivalent–Reynolds number low-speed counterparts. The
near-surface shear raises the boundary layer temperature, lowering the density and increasing
the viscosity. The resulting thick, highly viscous boundary layer can reduce the effectiveness of
control surfaces. The high boundary layer temperatures can also produce finite-rate processes
in the boundary layer and on the surface. It is very difficult, if not impossible, to replicate the
combination of these effects in ground-test facilities.
The level of chemical reaction has a direct effect on the bow shock standoff distance in hy-
personic flows; this can change the pressure distribution and aerodynamic performance of blunt
capsules. For example, it was shown that the Mars Pathfinder capsule’s aerodynamic stability
changes significantly with Mach number and the amount of chemical reaction (Gnoffo et al.
1996). This effect occurs because increasing reaction increases the density rise across the bow

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shock, which causes the shock standoff distance to decrease. This in turn changes the location of
the sonic line and the resulting pressure distribution on the capsule forebody.
Experiments, theory, and numerical simulations show that finite-rate processes (vibrational
relaxation in particular) can affect laminar–turbulent transition in hypersonic boundary layers.
For example, Adam & Hornung (1997) and Germain & Hornung (1997) correlated experimental
data on a sharp cone as a function of free-stream total enthalpy for air, nitrogen, and carbon
dioxide. They showed that reactive CO2 has a larger transition Reynolds number relative to air,
and particularly in contrast to low-reactivity nitrogen. Subsequent theoretical analyses showed
that the relaxation of the CO2 vibrational modes damps instability growth in these flows, resulting
in transition delay. This interaction is discussed in more detail in Section 4.2.
Many hypersonic flight vehicles require the use of ablative thermal protection systems, in
which the surface material interacts with the external high-temperature flow and undergoes finite-
rate gas-surface reactions. These processes include oxidation and nitridation reactions, catalytic
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recombination, sublimation, and erosion. Ablation results in surface mass loss and the injection of
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surface reaction products into the flow. The mass injection pushes the boundary layer away from
the surface and shields the body from the high-temperature shock-layer gas. The ablation species
undergo gas-phase reactions in the boundary layer and wake. Such a thermal protection system is
not reusable because the ablation process consumes a portion of the surface material. An example
of graphite ablation is given in Section 6.
The operation of many high-enthalpy test facilities is affected by finite-rate processes. For
example, shock tunnels are used to generate a short-duration flow to simulate hypersonic flight in
the atmosphere. To do so, a shock wave is driven into a quiescent test gas; the shock wave reflects
off of the shock tube end wall, producing a slug of high-temperature and -pressure gas. This gas
then expands through a converging-diverging nozzle to hypersonic conditions. However, when
the test gas is compressed, it undergoes chemical reactions, and it is this reacted gas that expands
through the nozzle. Just as the stagnation region gas chemical state is frozen in the sphere-cone
flow, the test gas state may freeze during the expansion through the nozzle, resulting in a partially
reacted test gas. This gas is at low temperature, but with potentially large mass fractions of atoms
and other reaction products. In some cases, particularly N2 flows, the vibrational energy can also
be frozen near the nozzle throat temperature.

2. GOVERNING EQUATIONS
The governing equations and associated boundary conditions for hypersonic flows are discussed in
this section. The examples described above show that these flows are not in local thermodynamic
equilibrium and that there can be significant effects of rate processes on important flow features.
Thus, it is necessary to track the evolution of the thermochemical state of the gas with appropriate
conservation equations. The overall form of the governing equations is described; more details
may be found in Gnoffo et al. (1989) and Lee (1985), for example.

2.1. Conservation Equations

The evolution of the mass of chemical species s is given by the conservation equation

∂ρs ∂
+ (ρs u j + ρs vs j ) = ωs , 1.
∂t ∂xj

where ρs is the species-s density, u j is the mass-averaged velocity in the x j direction, vs j is the
species-s mass diffusion velocity, and ωs is the species-s chemical source term.

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The momentum conservation equation is

∂ρui ∂
+ (ρui u j + pδij − σij ) = 0, 2.
∂t ∂xj
where p is the pressure and σij is the stress tensor. As discussed above, the internal energy may be
out of equilibrium with the translational (thermal) energy in a hypersonic flow. In that case, its
evolution is governed by an additional internal energy conservation equation of the form
 
∂ρe int ∂ 
+ ρe int u j + ρs e int,s vs j + q int, j = Q int , 3.
∂t ∂xj s

where e int is the mass-averaged internal energy (in the previous examples, e int is the sum of the
vibrational and electronic energies), and q int, j is the j -direction energy flux due to gradients of
internal energy. The source term, Q int , represents the rate of internal energy relaxation due to
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collisional processes in the gas mixture. The internal energy, e int can be written as

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ρe int = ρs e int,s , 4.
s

where e int,s is the species-s internal energy. The specific form of e int,s used to represent a particular
flow depends on the flow conditions, gas properties, and quantities of interest being simulated. In
some cases, multiple internal energy equations must be solved. For example, it may be necessary
to track several vibrational energies or to solve a separate electronic energy equation.
Finally, the total energy conservation equation is given by
∂E ∂   
+ (E + p)u j − σij ui + ρs h s vs j + q j = 0, 5.
∂t ∂xj s

where E is the total energy per unit volume,

 1
E= ρs e s + ρuk uk . 6.
s
2

Here, e s is the species-s total specific energy, including all energy modes (translational, rotational,
vibrational, electronic, and chemical):
e s = e tr,s + e int,s + h ◦s , 7.
where e tr,s is the translational–rotational energy and h ◦s is the heat of formation. In the above
conservation equation, h s is the species-s specific enthalpy, h s = e s + p s /ρs . The pressure is the
sum of the partial pressures of the gas mixture.
Unless all scales are resolved, which is currently prohibitive at vehicle scales, these equa-
tions must be augmented for turbulent flows, either by adding transport equations for Reynolds-
averaged Navier–Stokes turbulence model variables or by including large-eddy simulation subgrid-
scale models. The present state of turbulence modeling for hypersonic flows is deficient, and
turbulence models developed for low-speed flows are typically used.

2.2. Thermodynamics
It is critical to correctly represent the thermodynamics of high-temperature hypersonic flows.
For moderate–Mach number flows, there is little to no electronic excitation, and the vibrational
energy modes can be adequately represented with a simple harmonic oscillator. Above about
Mach 15 in air, it is important to include electronic energy excitation. This can be done with
algebraic expressions for the electronic energy states of each species, utilizing data for energy

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levels and degeneracies. However, this can be cumbersome and it is usually preferable to use
curve fits for thermodynamic quantities such as those of McBride et al. (2002) and Scoggins &
Magin (2014). With a two-temperature model for the translational–rotational and vibrational–
electronic energies, care must be taken to correctly evaluate the different energies at the appropriate
temperatures.

2.3. Source Terms

Let us consider nitrogen dissociation, written as
N2 + M  N + N + M, 8.
where M represents a collision partner. The law of mass action for the source term, ωN2 , is
 
ρN2  ρs ρN 2  ρs
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ωN2 = −MN2 kf,s − kb,s . 9.

MN2 s Ms MN M
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s s

Ms is the species-s molecular weight, and kf,s and kb,s are the forward (dissociation) and backward
(recombination) reaction rates, respectively. Usually, the dissociation rate is computed from a
temperature-dependent Arrhenius reaction rate, and the recombination rate is set so that the
source term is zero in thermodynamic equilibrium.
The internal energy relaxation term, Q int , is complicated for ionizing gases and is not discussed
in full here; readers are referred to Gnoffo et al. (1989) and Lee (1985) for more details. Rather, we
focus on the term that is responsible for vibrational energy relaxation through collisional processes.
The standard approach is to use a Landau–Teller model,
 ρs e v,s
∗
(T) − ρs e v,s 
Qv = + ωs ẽ v,s , 10.
s
τv,s s

where τv,s is the species-s translational–vibrational relaxation time that is appropriately averaged
to account for different relaxation times for each collision pair. Here, ẽ v,s is the average vibrational
energy removed or added to the vibrational energy pool due to dissociation and recombination
reactions, respectively. As discussed below, vibrationally excited molecules dissociate more readily,
and therefore this value should be larger than the average vibrational energy.

2.4. Transport Properties

Gas transport properties are critical for the accurate representation of high-temperature rate-
dependent flows. There are several approaches to computing the transport properties, ranging
from the approximate Wilke mixing rule (Wilke 1950) to more accurate representations of the
results from kinetic theory (Hirschfelder et al. 1954, Gupta et al. 1990). Seldom is the full multi-
component mass diffusion model used because of its cost; rather, the self-consistent effective binary
diffusion model (Ramshaw & Chang 1993) is used because of its sufficient accuracy and relatively
low cost. For ionized flows, it is critical to accurately compute the Coulombic interactions to obtain
the correct overall transport properties. The individual species transport properties are often taken
from tabulations of Blottner et al. (1971), Gupta et al. (1990), and Wright et al. (2005), for example.

2.5. Relaxation and Reaction Rates

Hypersonic flows depend on the rates that govern the internal energy relaxation and chemical
reaction processes. Until recently, the rates used were primarily based on shock tube data from

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the 1960s and 1970s. Standard approaches use Millikan & White’s (1963) correlation for the
vibrational relaxation times, with some corrections for particular relaxation pairs (Park 1993). A
standard set of air reaction rates was tabulated by Park (1993) and, for gas-phase reactions of
ablation products, by Martin et al. (2015).
As is discussed in more detail in Section 5, computational chemistry is now providing accurate
potential energy surfaces (PESs) that can be used to simulate the reaction dynamics of air species.
These are being used to compute more accurate vibrational relaxation times and reaction rates,
which are starting to become available in the literature.

2.6. Gas-Surface Boundary Conditions

The surface boundary conditions used for high-enthalpy hypersonic flows are significantly more
complicated than the usual no-slip boundary condition used for conventional flows. Here, we
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provide two examples and refer to appropriate references for more complex boundary conditions.
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2.6.1. Catalytic surface boundary condition. The basic statement of mass balance for species
s at a reactive surface is given by (MacLean et al. 2011, Marschall & MacLean 2011, Marschall
et al. 2015):
∂ ys
− (ρ Ds )w + (ρu)w ys w = ωs . 11.
∂n w
We have assumed Fickian diffusion; Ds is the mass diffusivity, ys is the species-s mass fraction, and
u is the rate of motion of the surface due to reactive mass loss by the gas-surface reactions with
rate ωs . The w subscript indicates the wall condition. These reactions produce a flux of reacted
species to or from the surface, rather than a volumetric rate of mass production, as in the gas-phase
reactions.
Let us consider the example of a catalytic surface discussed above. The catalytic efficiency, α,
is the fraction of atoms impinging on a surface that undergo reaction (recombine to molecules)
on the surface. The surface mediates the recombination process by one of several possible surface
reaction processes. From kinetic theory, the one-way flux of species-s particles across a surface is
1
Js = ρs C̄s , 12.
4
where C̄s is the thermal speed of the species-s particles. Of those particles that impact the surface,
a fraction α of them recombine and 1−α bounce off and do not react. Thus, the rate of production
of species-s mass is ωs = 14 αρs C̄s evaluated at the wall state. Within the continuum description,
the flux of reacted particles must be equal to the diffusive flux of reactive gas to the surface. This
is represented by the above mass balance expression, and therefore we have a boundary condition
of the form
∂ ys 1
− (ρ Ds )w = αρs ,w C̄s ,w , 13.
∂n w 4
since uw = 0 (no mass change of the surface) for this catalytic surface reaction. With the assumption
that the normal pressure gradient is zero in the boundary layer, we can then solve for the surface
or wall state.
Implicit in this formulation is a rate of reaction on the surface. A noncatalytic surface with
α = 0 has surface reactions that are negligibly slow relative to the diffusion timescales. Likewise,
a large catalytic efficiency (α → 1) represents a rapid surface reaction, and the rate of catalytic
recombination is limited by the rate of diffusion to the surface. In more complex gas-surface
interaction models, there can be competing surface reactions and reactions that depend on surface

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coverage (the fraction of surface material bond sites occupied by gas-phase species). Then the
gas-surface reaction rates are explicitly temperature- and pressure-dependent, and the law of mass
action is used to develop the rate of chemical species formation on the surface (Marschall et al.
2015).

2.6.2. Velocity and temperature slip boundary conditions. The extended form of the Navier–
Stokes equations provided above can be derived using a Chapman–Enskog solution of the Boltz-
mann equation. Conceptually, a first-order perturbation to a Maxwellian velocity distribution is
substituted into the Boltzmann equation and appropriate moments are taken. This results in the
Navier–Stokes equations with assumptions for linear relationships between stress and strain and
between temperature gradients and heat flux. The surface boundary conditions that are consistent
with this kinetic theory–based derivation are the so-called slip boundary conditions. It is only in
the limit of small mean free path that the conventional no-slip boundary conditions are recovered.
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The dependence of the surface state on the mean free path is easy to visualize. For example,
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consider a rarefied gas with a mean free path, λ. The particles than impinge on the surface originate
about a distance λ from the surface, interact with and accommodate to the surface state, and then
collide again about λ from the surface. The influence of the surface is then transmitted to the rest
of the flow field through subsequent collisions. This region of adjustment to the surface state thus
scales with λ.
Maxwell (1879) derived an expression for the velocity slip in the surface-tangent direction as
(see also Gupta et al. 1985),
2 − σ ∂u
uslip = λ , 14.
σ ∂n w

where σ is the accommodation coefficient (the fraction of particles that are diffusely reflected),
and n is a surface-normal coordinate. Note that the usual no-slip boundary condition is obtained
as λ → 0.
A similar temperature jump or slip boundary condition for a perfect gas was derived by Kennard
(1938) as
2 − αT 2γ 1 ∂T
Tslip − Tw = λ , 15.
αT (γ + 1) Pr ∂n w

where Tw is the wall or surface temperature, αT is the thermal accommodation coefficient, γ is

the ratio of specific heats, and Pr is the Prandtl number.
Under low-density rarefied conditions, these slip effects may be important, and it has been
shown that greatly improved comparisons between particle-based flow simulations [e.g., direct
simulation Monte Carlo (Boyd & Schwartzentruber 2017)] and the Navier–Stokes equations are
obtained when they are included (Gupta et al. 1985, Gokcen 1989, Lofthouse et al. 2008, Bhide
et al. 2018).
An interesting rate-driven example of surface slip effects involves CUBRC shock tunnel exper-
iments, in which moderate-enthalpy N2 is shock heated and expanded to hypersonic conditions
(Nompelis et al. 2003). Due to the rapid expansion of the gas, the vibrational energy freezes close
to the nozzle throat conditions. A typical hypersonic test condition has a free-stream translational–
rotational temperature of 98 K, while the vibrational temperature is frozen close to the nozzle
throat conditions at 2,560 K. To obtain good agreement between numerical simulations and heat
transfer measurements in this nonequilibrium test gas, it is necessary to include a vibrational
energy slip model with the experimentally measured accommodation coefficient of 10−3 .

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2.7. Nondimensionalization and Scaling

The chemical source term can be nondimensionalized using the free-stream density, velocity
magnitude, and a relevant length scale, L. This results in a nondimensional source term for the
N2 dissociation reaction (Equation 9),

L ρ∞ L ρ̄N2  ρ̄s ρ 2 L ρ̄N 2  ρ̄s
ω̄N2 = ωN2 = −MN2 kf,s − ∞ kb,s , 16.
ρ∞ u∞ u∞ MN2 s Ms u∞ MN s
M s

where the bar variables are nondimensional quantities. Note that the relevant scaling parameters
are:
ρ∞ L
forward (dissociation) rate : , 17.
u∞
ρ∞2
L
backward (recombination) rate : . 18.
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u∞
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The dissociation source term scales with the density, while recombination scales with the density
squared. Therefore, the relative importance of dissociation and recombination can change as
the free-stream density changes. It should also be noted that the flow-field temperature scales
with u2∞ , so that the temperature-dependent rates, kf and kb , scale with the free-stream velocity
magnitude squared.
Dissociation is a two-body binary collision process, so the ρ∞ L scaling is termed binary scal-
ing (Hornung 1972b). Other two-body processes such as exchange reactions (e.g., N2 + O 
NO + N) also have binary scaling. Similarly, the vibrational relaxation process is a two-body pro-
cess and is characterized by binary scaling (τv in Equation 10 scales inversely with density). Recom-
bination is a three-body or ternary process, accounting for its dependence on the density squared.
The flows discussed above clearly exhibit these scaling properties. The rate of dissociation
and vibrational relaxation behind the bow shock decreases with decreasing free-stream density, as
expected. However, the recombination process exhibits a stronger dependence on density. When
the gas suddenly expands around the nose and onto the cone, the density decreases and the rate of
recombination decreases quadratically with the density. This is why there is little recombination
during the flow expansion and why the chemical freezing is stronger at 60 km than at 40 km. In
contrast, because vibrational relaxation is a binary collisional process, it more closely follows the
changes in the translational temperature.
A more subtle scaling effect occurs in the near-surface boundary layer. At 40 km, the density
is high enough that significant recombination occurs in the thermal boundary layer. However, at
60 km, the density is lower and recombination is reduced due its quadratic dependence on density.
This causes the dependence of the noncatalytic heat transfer rate on density shown in Figure 4.
The scaling of the gas-surface reactions is not as clear as the gas-phase reactions. For example,
consider the catalytic recombination of oxygen according to O + O(s) → O2 ; here, O(s) is an
oxygen atom that is bonded to an open surface site. The rate of this process is not simple since it is
proportional to both the gas-phase O atom concentration and the number of O atoms bonded to
the surface. Typically, the surface coverage depends on the surface temperature and gas pressure.
These rates can be nondimensionalized by a characteristic advection rate, resulting in
Damköhler numbers for each process. In the limit of large Damköhler number, a process will
be close to equilibrium, while small values indicate that it is frozen.

3. TRANSLATIONAL AND ROTATIONAL PROCESSES

We do not usually think of translational relaxation or thermalization of a gas as a rate process.
However, in hypersonic flows, the change of the gas state can be so rapid that the translational

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modes are relatively slow to adjust. When this occurs, the fundamental assumptions of the kinetic
theory derivation of the Navier–Stokes equations are violated.
An obvious example of translational nonequilibrium occurs within a strong shock wave. The
preshock velocity distribution function has a small thermal component and a large component
in the free-stream direction. Postshock, the opposite is true, with much of the upstream directed
motion converted to thermal motion. Within the shock wave itself, the velocity distribution
function is much more complicated, with features of the pre- and postshock distribution functions.
Such a bimodal velocity distribution function cannot be represented as a perturbed Maxwellian
distribution, as assumed by the Chapman–Enskog derivation of the Navier–Stokes equations.
Thus, the Navier–Stokes equations cannot correctly represent the flow within a shock wave.
Likewise, higher-order Chapman–Enskog solutions of the Boltzmann equation (e.g., the Burnett
equations) cannot represent this effect.
When translational energy relaxation is important, a kinetic theory–based approach such as the
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direct simulation Monte Carlo method must be used (Bird 1994, Boyd & Schwartzentruber 2017).
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Typically, this occurs under rarefied conditions, which can be identified using a gradient-length
local Knudsen number (Boyd et al. 1995):
λ
KnGLL = ∇ρ. 19.
ρ
If this quantity exceeds a value of about 0.05, the Navier–Stokes equations are likely to be invalid.
Typically, rotational energy relaxation does not need to be considered separately since it is
only important for flows where the Navier–Stokes equations are already suspect or invalid. This
is because the rotational modes relax very rapidly (in about five to ten collisions), and that rate is
similar to the rate at which the translational modes relax to equilibrium. This is not the case for
the vibrational modes.

4. VIBRATIONAL PROCESSES
The relatively slow relaxation of the vibrational modes of a gas can interact with the gas dynamics
in several ways. Most obviously, the vibrational modes absorb energy and change the postshock
conditions when they are active. The vibrational state of the gas also has a strong effect on its
dissociation rate, as discussed in the next section. Vibrational freezing and thermal nonequilibrium
can be important in rapidly expanding flows such as in the wake of a planetary entry capsule or
in a hypersonic wind tunnel nozzle, as shown above. In some flows, the vibrational relaxation rate
may be tuned to match acoustic disturbances, which can lead to the dissipation and dispersion
of sound in high-temperature gases. Acoustic damping by this mechanism can counter boundary
layer instabilities and delay transition to turbulence if tuned to the dominant instabilities. In this
section, we summarize the interaction of vibrational relaxation with acoustic processes.

4.1. Vibrational Relaxation and the Bulk Viscosity

The derivation of the stress tensor for a linearly viscous fluid yields

∂ui ∂u j 2
σij = − pδij + μB δij ∇ · u + μ + − δij ∇ · u , 20.
∂xj ∂ xi 3
where μB is the bulk viscosity (e.g., Thompson 1988); the same result can be obtained from kinetic
theory (Hirschfelder et al. 1954). Deviations of the average normal stress from the pressure are
commonly ascribed to nonzero values of μB (Emanuel 1992); nonzero μB also has an effect on

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0.14 10 – 2
a b
T/θ v
0.12 10 – 3
0.1

Relaxation time, pτ (atm•s)

Damping per wavelength

0.2
0.5 16,000 K 4,600 K 750K
0.10 10 – 4
1 2,000 K 1,000K
2
0.08 5 10 – 5
N2 –N2 vibrational
N2 –N2 rotational
0.06 10 – 6
CO2 –N2 vibrational

0.04 10 – 7

0.02 10 – 8
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0 10 – 9
10 – 3 10 – 2 10 – 1 10 0 10 1 10 2 10 3 0.04 0.06 0.08 0.10 0.12
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ωτv T –1/3 (K –1/3)

Figure 6
(a) Vibrational damping rate as a function of the product of the angular frequency and relaxation time, ωτv , from the results of Meador
et al. (1996), and (b) typical vibrational and rotational relaxation times. θv is the molecular characteristic temperature of vibration.

the speed of sound and its attenuation. However, Meador et al. (1996) showed that μB is zero and
that its interpretation as an internal energy relaxation parameter is incorrect. Rather, absorption
and dispersion of sound are the result of internal energy relaxation, but these effects are not
embodied in μB . Instead, the internal energy should be represented with a conservation equation,
as discussed in Section 2.1. Meador also derived closed-form expressions for the speed of sound
and its dissipation rate and dispersion as functions of the internal energy relaxation time.
Figure 6a shows the damping rate of acoustic waves as a function of the acoustic angular
frequency times the relaxation time, ωτv , and the damping of the acoustic wave per wavelength
(Meador et al. 1996). Clearly, ωτv  1 produces significant damping, provided that the temperature
is large enough so that the vibrational modes are excited. Here, θv is the characteristic temperature
of vibration, which is 3,395 K for N2 and 2,239 K for O2 . Figure 6b shows that the relaxation
times for vibrational and rotational processes vary widely. Therefore, to obtain effective damping,
the relaxation process must be resonant with the acoustic frequency of interest. The next section
discusses how this effect may be used to damp instabilities in a hypersonic boundary layer.

4.2. Effect of Vibrational Processes on Boundary Layer Stability

During the late 1990s, experiments on a sharp 5◦ cone in the California Institute of Technology T5
free-piston shock tunnel showed that boundary layer transition depends on the free-stream total
enthalpy and the gas properties (Adam 1997, Adam & Hornung 1997, Germain & Hornung 1997).
As shown in Figure 7a, the transition Reynolds number based on the boundary layer reference
temperature increases with increasing total enthalpy, and thereby with increasing internal energy
excitation and chemical reaction. In addition, the transition Reynolds number is an order of
magnitude larger in CO2 than it is in air or N2 . At these conditions, boundary layer transition occurs
via the Mack second mode (Mack 1975), which is an acoustic disturbance in the boundary layer.
Linear stability theory at the T5 conditions shows that under certain conditions, vibrational
energy relaxation and chemical reactions result in reduced instability growth rates. Furthermore,
numerical experiments show that exothermic (heat-releasing) finite-rate processes increase the

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101 100
a 90 b Exothermic disturbances
Endothermic disturbances
80 Nonreacting disturbances
x=
70 Increasing
Incre
e x 0.106 m
60
Re*tr (10 – 6 )

x=

–α i (1/m)
50 x= 0.316 m
10 0 0.943 m
40
30

N2 Cold tunnels 20
CO2 Air cold 10
Air CO2 cold Unstable
N2 cold 0
10 –1 -10
0 2 4 6 8 10 12 14 16 0 500 1,000 1,500 2,000 2,500 3,000
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ho (MJ/kg) Frequency, f (kHz)

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Figure 7
(a) The transition Reynolds number based on reference conditions, Re∗tr , on a 5◦ cone as a function of free-stream total enthalpy, h o , for
three gases. (b) The acoustic disturbance growth rate, −αi , at various axial locations on a sharp cone in CO2 for disturbances with exother-
mic reactions, endothermic reactions, and no reactions. Panels adapted with permission from (a) Adam (1997) and (b) Johnson (2000).

growth rate, while endothermic processes decrease it. This demonstrates that there is an interac-
tion between rate processes and the growth of second-mode disturbances in a hypersonic boundary
layer. Figure 7b summarizes the linear stability theory results. The exothermic reactions are sig-
nificantly more unstable, while the endothermic reactions are damped relative to the nonreacting
flow.
Fujii & Hornung (2001) showed that at the experimental conditions, there is an overlap between
the most unstable acoustic frequencies and the vibrational relaxation rate of CO2 . Their work
shows that if boundary layer instabilities are tuned to CO2 relaxation, their amplitude may be
damped. Based on this work, Leyva et al. (2009) and Jewell et al. (2013) studied the injection of
CO2 into the boundary layer of a hypersonic cone in an attempt to delay transition to turbulence.
With careful injection, it was shown that it is possible to stabilize the boundary layer.
Wagnild & Candler (2014) used direct numerical simulations to study the tuning of acoustic
disturbances to hypersonic boundary layers. The results of this work are consistent with the
theoretical results and illustrate the effects of vibrational relaxation tuning on acoustic disturbances.
There have been many other studies of hypersonic boundary layers including rate effects; however,
there are very few that focus on vibrational rate processes and their interaction with the acoustic
modes (e.g., Hudson et al. 1997, Knisely & Zhong 2018). Many other authors have studied the
effects of equilibrium gas models and chemical nonequilibrium processes. However, chemical rate
processes are much slower than the relevant disturbances in hypersonic boundary layers and are
not tuned to relevant disturbance frequencies.
This work shows that there are three regimes for vibrational damping of acoustic waves:
(a) equilibrium, in which the vibrational state rapidly adjusts to the acoustic disturbance (this
occurs at low frequencies); (b) nonequilibrium, when the vibrational mode responds to the passage
of the acoustic disturbance with a phase lag, which absorbs energy from the acoustic mode; and
(c) frozen, in which the acoustic timescale is much smaller than the vibrational relaxation timescale
and does not respond to the passage of the wave, so that there is negligible acoustic damping.
Figure 8 illustrates these regimes for a 100-kHz acoustic wave passing through CO2 at a range
of initial temperatures. At low temperature (300 K), the acoustic modes are only weakly excited,
and the relaxation time is much slower than the characteristic time of the acoustic wave. Thus,

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T = 300 K T = 1, 000 K T = 2, 500 K

2.0
1.0
a b c
temperature × 10 –2 (K)

1.5
0.5 1.0
Disturbance

0.5
0 0
– 0.5
– 0.5 T‘ – 1.0
T ‘ve – 1.5
– 1.0
– 2.0
2 3 4 5 2 3 4 5 2 3 4 5
x × 10 –2 (m) x × 10 –2 (m) x × 10 –2 (m)

Figure 8
Damping of a 100-kHz acoustic disturbance in CO2 as a function of temperature: (a) 300 K, (b) 1,000 K, and (c) 2,500 K. Adapted from
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Wagnild & Candler (2014) with permission from the authors.

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the vibrational modes do not respond and are frozen. This is the case in room-temperature air,
for example. At 1,000 K, the bending modes of CO2 are excited (θv = 960 K) and their relaxation
times are close to the optimal frequency for acoustic damping. In this case, the vibrational energy is
excited and its response is out of phase with the translational–rotational temperature; this results in
strong acoustic damping. At higher temperatures, the vibrational energy approaches equilibrium
and rapidly responds to the temperature perturbations caused by the acoustic wave.

5. CHEMICAL REACTIONS
As illustrated in the introduction, dissociation is an important process in hypersonic flows. Disso-
ciation is complicated because its rate depends on the level of vibrational excitation; vibrationally
excited molecules have a lower energy barrier to dissociation and more readily dissociate. Fur-
thermore, when vibrationally excited molecules dissociate, they are removed from the vibrational
energy pool, suppressing the overall vibrational energy of the gas mixture. The coupling between
vibration and dissociation has been known since the 1960s, and models were postulated to repre-
sent the process (e.g., Marrone & Treanor 1963). These models make the effective dissociation
rate a function of T and Tv .
In the late 1980s, Park (1986, 1987) recognized that a new approach was needed. The dissoci-
ation rates used in flow field modeling were obtained from shock tube experiments, and models
were required to infer the reaction rates from the raw data. Park reinterpreted the shock tube
data to calibrate a model in a form that would be consistent with the data. This was an important
advance over previous models that took the reaction rates as given and then developed models for
dissociation. Such an approach is inconsistent because the  inferred reaction rates depend on the
model used to infer them. Park’s analysis resulted in the TT v model,  in which reaction rates in
Arrhenius form are evaluated at an effective temperature equal to TT v . The model suppresses
the dissociation rate when Tv is low, consistent with the understanding of the dissociation process
(e.g., Hornung 1972a). A component of this model is the average energy removed from the total
vibrational energy due to dissociation (ẽ v,s in Equation 10). Park assumed ẽ v,s = 0.3De , where De
is the dissociation energy of the molecule. This value is incorrect and causes numerical problems,
and in practice, ẽ v,s is typically approximated as the average vibrational energy. The Park model is
now widely used, not because it has proven to be accurate, but because it is more straightforward to
implement than many other models. Subsequently, other models have been proposed (e.g., Knab
et al. 1995, Luo et al. 2018, Singh & Schwartzentruber 2018).

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Recently, it has become possible to better understand the dissociation process through the use
of ab initio computational chemistry methods. Electronic-structure calculations are carried out
for many possible atomic configurations to build a PES for molecular interactions (e.g., Paukku
et al. 2013). Collisions between air species are simulated using the relevant PES to obtain statistical
data for relaxation and reaction rates at specified conditions (e.g., Bender et al. 2015, Valentini
et al. 2016). Such an analysis reveals the complete physics of dissociation and its dependence on

the vibrational energy state. Soon this work will supplant Park’s TT v model with physics-based
models of known fidelity. There are many recent papers related to this subject, including those
of Kim & Boyd (2013), Panesi et al. (2014), Andrienko & Boyd (2016), Schwartzentruber et al.
(2017), and Macdonald et al. (2018).

6. SURFACE PROCESSES
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heat transfer rates to hypersonic vehicles. Ablation is significantly more complicated than
surface-catalyzed recombination. The simplest ablative surface is carbon in one form or another,
such as graphite or a carbon–carbon matrix. The basic reactions that can take place are oxidation
and nitridation, surface-catalyzed recombination, and at high temperatures, sublimation. A
representative gas-surface kinetics model is (Zhluktov & Abe 1999):
O + (s) ↔ O(s), O(s) + C(b) ↔ CO + (s),
N + (s) ↔ N(s), O + O(s) + C(b) ↔ CO2 + (s),
2O(s) ↔ O2 + 2(s), O(s) + C(b) ↔ CO2 + 2(s),
O2 + (s) ↔ O + O(s), C + (s) ↔ (s) + C(b),
N2 + (s) ↔ N + N(s), C2 + 2(s) ↔ 2(s) + 2C(b),
CO2 + (s) ↔ CO + O(s), C3 + 3(s) ↔ 3(s) + 3C(b).
Here, (s) indicates an open bond site, O(s) denotes an O atom occupying a bond site, and C(b) indi-
cates a bulk or solid carbon atom. Rates for each of these processes are required to close the model.
Until recently, such a finite-rate model was not used for analysis, and a gas-surface equilibrium
approximation was made. In this case, the equilibrium composition of a mixture of air and the
surface material is computed at the surface pressure and temperature; this provides the surface
state for the gas-phase boundary condition. This approach is commonly termed a B method (the
nondimensional mass blowing rate) and is valid at high pressures where equilibrium is approached,
but is unlikely to be valid at low to moderate pressures.
Figure 9 shows CFD simulations of ablation using three models: the equilibrium B ap-
proach, Zhluktov & Abe’s (1999) finite-rate model, and a recently developed finite-rate model
(Poovathingal et al. 2017). The main difference between the finite-rate models is that the more
recent model favors the formation of CO relative to CO2 ; this is consistent with oxygen molec-
ular beam experiments (Murray et al. 2015). Here, the properties of graphite are used, and the
gas-surface mass and energy balances are solved at the interface between the gas and the solid.
The thermal response of the material is included, and the body changes shape due to ablative
mass loss. Clearly, there are large differences between the models, with the equilibrium B model
producing much larger recession relative to the finite-rate models. The Zhluktov & Abe (1999)
model produces the lowest level of recession. Experiments (Alba et al. 2016) indicate that this
model is deficient, and the third model is more consistent with data. In any case, it is clear that
the finite-rate models produce significantly reduced levels of surface recession at these conditions.
Furthermore, the boundary layer composition predicted by the models is completely different,

www.annualreviews.org • Rate Effects in Hypersonic Flows 397

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 FL51CH15_Candler ARI 12 September 2018 12:50

B' model Zhluktov & Abe model Poovathingal et al. model T (K)
2,500
6 a b c
2,300
y (cm)

4 2,100

1,900
2
1,700

0 1,500
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
x (cm) x (cm) x (cm)

Figure 9
Predicted recession and solid temperature for a graphite sphere-cone (6.35 cm in radius) at conditions corresponding to an altitude of
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30 km and a flight speed of 3.5 km/s after 830 s of exposure: (a) B model, (b) Zhluktov & Abe (1999) model, and (c) Poovathingal et al.
(2017) model. The gray region indicates the initial preablated geometry.
Annu. Rev. Fluid Mech. 2019.51. Downloaded from www.annualreviews.org

and as shown in Section 4.2, the boundary layer internal energy dynamics can lead to important
effects. Additional experimental data are required to fully validate the models.

SUMMARY POINTS
1. Hypersonic flows involve interactions of gases and thermal protection system materials
at extreme conditions.
2. In hypersonic flows, internal energy relaxation, gas-phase reactions, and gas-surface in-
teractions typically occur at rates that are similar to the gas motion rates, resulting in an
out-of-equilibrium thermochemical state.
3. These rate processes affect aerodynamic performance, heat transfer rates, instability
growth leading to boundary layer transition, catalytic gas-surface interactions, and
ablation.
4. Hypersonic flows are sensitive to the scaling of the chemical rate processes, with disso-
ciation governed by binary scaling and recombination by ternary scaling.
5. Advanced multiphysics simulations and computational chemistry methods are being used
to understand and accurately model key rate processes.

FUTURE ISSUES
1. Validation of hypersonic flow simulations and the underlying models remains a critical
issue. It is difficult to reproduce all relevant rate processes in ground-based wind tunnels,
and making nonintrusive measurements in these flows is a challenge.
2. Advanced simulations should be used to design experiments that target modeling uncer-
tainties and that can be performed in existing wind tunnels.
3. Progress has been made in the development of finite-rate models for graphite ablation,
but much work is needed to represent the interaction of more complex thermal protection
materials with air reaction products.

398 Candler
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still occur before final publication.)
 FL51CH15_Candler ARI 12 September 2018 12:50

4. Little effort has been directed toward understanding turbulent motion at the extreme
conditions of hypersonic flight.
5. The prediction of hypersonic rate-dependent flows relies on computational fluid dynam-
ics methods, which have been adapted from methods designed for lower-speed flows.
Improved numerical methods need to be developed that are less sensitive to grid imper-
fections, have low levels of dissipation and yet can capture strong discontinuities, and are
efficient for problems with many chemical species and stiff source terms.
6. The high-fidelity simulation of the hypersonic flow environment is far from complete; a
hypersonic flight system changes during flight over a large range of time scales. Relevant
processes include thermal expansion, material degradation and shape change, surface
roughness changes, and fluid–structure interactions and vibrations.
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DISCLOSURE STATEMENT
The author is not aware of any biases that might be perceived as affecting the objectivity of this
review.

ACKNOWLEDGMENTS
This work was sponsored by the Air Force Office of Scientific Research (AFOSR) under grants
FA9550-16-1-0161, FA9550-17-1-0057, and FA9550-17-1-0250. The views and conclusions con-
tained herein are those of the author and should not be interpreted as necessarily representing the
official policies or endorsements, either expressed or implied, of the AFOSR or the US Govern-
ment. The author would like thank Joseph Brock, Ross Chaudhry, Ioannis Nompelis, and Thomas
Schwartzentruber for helpful comments.

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