# Vibronic State-Specific Modelling of High-Speed Nitrogen Shocked Flows. Part
II: Shock Tube Simulations

Élio Pereira  Instituto de Plasmas e Fusão Nuclear, Instituto Superior
Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, Lisboa, 1049-001,
Portugal  Jorge Loureiro  Instituto de Plasmas e Fusão Nuclear, Instituto
Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, Lisboa,
1049-001, Portugal  Mário Lino da Silva<EMAIL_ADDRESS>Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade
de Lisboa, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal

###### Abstract

The conditions of thermochemical and radiative non-equilibrium attained in
nitrogen shocked flows were quantified using a vibronic state-specific model.
This model, being described in a companion paper, was implemented in Euler
one-dimensional simulations for shots $19$, $20$ and $40$ of the EAST’s
$62^{\text{th}}$ campaign. It was found that the peak values of the
instrumental radiative intensities were underestimated by one to two orders of
magnitude, and sensitivity tests performed on several parameters of the
simulations were not successful in getting a reasonable agreement. The shapes
of the instrumental radiative intensities’ profiles obtained in the low speed
shot were correctly predicted, unlike the ones of the medium and high speed
shots which revealed non-null plateaus proceeding or coalescing with peaks.
These plateaus were not predicted at all. It is suspected that such
discrepancies may have resulted from neglecting other shock tube related
phenomena, as pointed out by other researchers in the literature: the
absorption of radiation emitted by the driver gas and the EAST electric arc,
and/or the conduction of heat due to downstream plasma being subjected to a
stronger shock wave.

###### keywords:

Nitrogen, Plasma, State-to-State, Vibronic, Thermodynamics, Non-equilibrium,
Radiation, Shock tube

##  1 Introduction

The main objective of Spacecraft Design with respect to atmospheric entries is
to devise a vehicle that can sustain the harsh conditions of the flight,
descend in a stable and controllable way, and decelerate smoothly so that
landing can safely happen. Regarding the survival of the spacecraft against
the harsh conditions, one should focus on the strong convective and radiative
heats received by the vehicle. The distribution of the heat around the body
needs to be well predicted for a correct estimation of the thickness of the
thermal protection system (TPS). A TPS with an underestimated thickness would
compromise the integrity of the vehicle due to the removal of a considerable
part of the protective material by ablative processes. On the other hand, a
TPS with an overestimated thickness would make the spacecraft unnecessarily
heavier. And the higher the structural weight, the higher the required rate of
ejected fuel to provide the same acceleration, at launch and during manoeuvres
in outer Space. To properly design the TPS, the interaction between the flow
and the body, in a macroscopic and microscopic way needs to be understood.
Such phenomena are dictated by the history of the elements of fluid that
traverse the shock wave and arrive in the boundary layer of the body. In this
work, post-shock flows that develop inside shock tubes were studied as an
approximation to the ones that occur in atmospheric entries before the arrival
of the elements of fluid in the boundary layer. In fact, the main objective of
the work was to validate the developed vibronic-specific state-to-state model
for pure nitrogen shocked flows, which was described in a companion paper 1,
by performing simulations of benchmark shots done in a shock tube whose
results were available in the literature. Note that nitrogen is the main
component of Earth’s and Titan’s atmosphere. By firstly neglecting the other
components, one may focus on a single part of the entire model, and validate
it through shock tube tests that consider pure nitrogen as test gas. One would
then shift the focus to the other components in future works, validating the
respective models and considering models for the interactions between the
components, finally adding them to get the complete model. The shock tube
tests which were regarded in this work were the ones from the $62^{\text{th}}$
campaign of the Ames Electric Arc Shock Tube (EAST), performed in $2018$ by
Brandis and Cruden 2. The curious reader may get to know more about this shock
tube facility by reading the works of Sharma and Park 3, 4. In the
$62^{\text{th}}$ campaign, a driven gas of pure nitrogen (N2) was considered.
The tests comprised upstream (of the shock wave) flow speeds $u_{\infty}$
between approximately $6$ and $11\,\text{km}/\text{s}$, a static pressure of
value $p_{\infty}=0.2\,\text{Torr}$, and a temperature of value
$T_{\infty}=300\,\text{K}$.

The shock tube tests were here simulated using the SPARK code 5. The following
sections present a set-up of the numerical simulations, their results, and a
discussion on the discrepancies between these and experimental results.

##  2 Theory

In this section, the shock tube experiment, the variables measured by Brandis
and Cruden in the experiment, as well as the fluid flow governing equations
and their boundary conditions which were regarded in this work, will be
described.

###  2.1 The shock tube experiment

Figure 1 presents a simplistic scheme of the flow inside the Ames Electric Arc
Shock Tube during a shot, introducing the working frame of reference.

Figure 1: Upper view of a longitudinal cross-section of the shock tube,
showing the measured radiative field.

The shock tube, which is made of stain-less steel 6, has an inner diameter
$D=10.16\,\text{cm}$ 2. In each shot, the driver gas is heated by an electric
arc, and the diaphragm that separates the driver from the driven gas opens.
This produces a pressure discontinuity, resulting in a shock wave that
propagates in the driven gas. The radiation emitted by the driven gas in the
test section of the shock tube is then measured through a window by four
spectrometers, each one associated with an interval of wavelengths $\lambda$.
In the present work, the considered wavelength intervals were the ones of the
vacuum ultra-violet radiation (VUV), with $\lambda\in[145,\,195]\,\text{nm}$,
ultra-violet/visible radiation (“Blue”), with
$\lambda\in[330,\,480]\,\text{nm}$, visible/near infra-red radiation (“Red”),
with $\lambda\in[480,\,890]\,\text{nm}$, and infra-red radiation (IR), with
$\lambda\in[890,\,1450]\,\text{nm}$. The post-shock flow may be considered to
be stationary (not varying in time, but in space) when observed from a frame
of reference that moves with the shock wave if this shock wave does not
decelerate too much while travelling inside the shock tube (note, however,
that such deceleration is indeed known to occur6). The frame of reference here
mentioned is depicted in Figure 1, where its origin coincides with the point
of intersection of the shock tube axis with the shock wave plane, its $x$-axis
has the direction of the shock tube axis and points in the downstream
direction, its $z$-axis points upwards, and its $y$-axis is such that
$\vec{e}_{x}\times\vec{e}_{y}=\vec{e}_{z}$, pointing in a radial direction. It
is also worthy to refer here that the flow may be regarded as unidimensional
if the boundary layer does not grow too much.

###  2.2 Measured variables

Brandis and Cruden measured the post-shock specific radiative intensity
$I_{\lambda}$ associated with photons propagating in the radial direction of
the tube $\vec{e}_{y}$, at the farthest radial point of the plasma, $y=D/2$,
as depicted in Figure 1. One may further quantify the specific radiative
intensity through the so-called equation of radiative transfer 7 for the
propagation direction $\vec{e}_{y}$ disregarding variations in time (due to
the hypothesis of stationarity), the dependence of the emission and absorption
coefficients on $y$ and $z$ (due to hypothesis of unidimensionality), and
induced emission, as well as initially assuming the medium to be optically
thin and then introducing the concept of an escape factor 8,
$\Lambda_{s,e,v}^{e^{\prime},v^{\prime}}\in[0,\,1]$. This escape factor
corresponds to the fraction of the emitted photons due to spontaneous
emissions from the vibronic level $(e,\,v)$ to $(e^{\prime},\,v^{\prime})$
which escape the system, or in other words, that are not absorbed. One has
then

$I_{\lambda}(x,\lambda)=\frac{hcD}{\lambda}\left[\sum_{\text{se}}\Lambda_{s,e,v}^{e^{\prime},v^{\prime}}\frac{A_{s,e,v}^{e^{\prime},v^{\prime}}}{4\pi}\,\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}(x,\lambda)\,n_{s,e,v}(x)\right]\text{
,}$ (1)

where the sum is done in all spontaneous emission processes. Note that in (1),
$h$ is the Planck’s constant, $c$ is th speed of light,
$A_{s,e,v}^{e^{\prime},v^{\prime}}$ is the Einstein coefficient for
spontaneous emission, $\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}$ is the
line-shape factor and $n_{s,e,v}$ is the number density of $s$-th species
particles in the $(e,\,v)$ vibronic level.

Among the data issued by Brandis and Cruden there are the instrumentally
resolved radiative intensities $\hat{I}(x)$ and the instrumentally resolved
non-equilibrium metrics $\hat{I}_{\lambda}^{\,\text{ne}}(\lambda)$ for each of
the four intervals of wavelengths in a total of $42$ shots. The hat denotes
instrumentally resolved quantities, in contrast to the real quantities. The
former are the ones that are actually obtained in the experiment, being
related to the latter through a transformation “applied” by the instruments.
The instrumentally resolved radiative intensity associated with the $l$-th
wavelength interval, with
$l\in\\{\text{VUV},\,\text{``Blue''},\,\text{``Red''},\,\text{IR}\\}$, is
given by an integration of the instrumentally resolved specific radiative
intensity $\hat{I}_{\lambda}(x,\lambda)$ with respect to the wavelength
$\lambda$ from $\lambda=\lambda_{\text{min}}^{l}$ to
$\lambda=\lambda_{\text{max}}^{l}$:

$\hat{I}^{l}(x)=\int_{\lambda_{\text{min}}^{l}}^{\lambda_{\text{max}}^{l}}\hat{I}_{\lambda}(x,\lambda)\,d\lambda\text{
.}$ (2)

And the instrumentally resolved non-equilibrium metric associated with the
$l$-th interval of wavelengths is given by an integration of the
instrumentally resolved specific radiative intensity
$\hat{I}_{\lambda}(x,\lambda)$ with respect to the position $x$ from
$x=x_{\text{min}}^{l}$ to $x=x_{\text{max}}^{l}$ \- a region characterised by
a strong thermodynamic non-equilibrium - divided by the shock tube inner
diameter $D$:

$\hat{I}_{\lambda}^{\text{ne},l}(\lambda)=\frac{1}{D}\int_{x_{\text{min}}^{l}}^{x_{\text{max}}^{l}}\hat{I}_{\lambda}(x,\lambda)\,dx\text{
.}$ (3)

One should note here that it is not possible to know the exact position of the
shock wave through the spectra obtained in the shock tube experiments done by
Brandis and Cruden for multiple reasons: the onset of the radiative field is
associated with an increase of the number of excited species (which occurs at
some distance downstream of the shock wave), and each spectrometer captures
the spectra after some reaction time 9. The $x$ axis that Brandis and Cruden
work with may not correspond to the one depicted in Figure 1. The issued
position values are actually with respect to a particular origin for each shot
and interval of wavelengths which may not coincide with a point in the shock
wave. Therefore, they will be denoted here as $\hat{x}$ and termed “relative
positions” to distinguish them from the previously introduced ones, $x$, which
are with respect to an origin that coincides with a point in the shock wave.
The relative position of the shock wave for some shot and the $l$-th interval
of wavelengths, say $\hat{x}_{\text{sw}}^{l}$, was defined in this work as the
point where the instrumentally resolved radiative intensity starts to rise
abruptly, as also considered by Cruden in his work 9. The position $x$ is then
given by $x=\hat{x}-\hat{x}_{\text{sw}}^{l}$. The wavelength integration
limits $\lambda_{\text{min}}^{l}$ and $\lambda_{\text{max}}^{l}$, and relative
position integration limits $\hat{x}_{\text{min}}^{l}$ and
$\hat{x}_{\text{max}}^{l}$ are issued by Brandis and Cruden in their work. The
instrumentally resolved specific radiative intensity
$\hat{I}_{\lambda}^{l}(x,\lambda)$ departs from the real one,
$I_{\lambda}^{l}(x,\lambda)$, either spectrally and spatially. The non-
ideality of the instrumental apparatus is such that the measured specific
radiative intensity $\hat{I}_{\lambda}^{l}(x,\lambda)$ associated with some
particular wavelength $\lambda$ and position $x$ is in fact the result of a
distribution of the real radiative intensity on intervals of wavelengths and
positions around the reference values. Or, mathematically

$\hat{I}_{\lambda}(x,\lambda)=\int_{-\infty}^{\infty}\hat{\phi}^{\text{spa}}(x^{\prime})\left[\int_{-\infty}^{\infty}\hat{\phi}^{\text{spe}}(\lambda^{\prime})I_{\lambda}(x-x^{\prime},\lambda-\lambda^{\prime})\,d\lambda^{\prime}\right]\,dx^{\prime}\text{
,}$ (4)

with $\hat{\phi}^{\text{spe}}(\lambda^{\prime})$ and
$\hat{\phi}^{\text{spa}}(x^{\prime})$ being the so-called instrument line-
shape factor and spatial resolution function, respectively, which are issued
by Brandis and Cruden in their work.

###  2.3 Fluid flow governing equations

As referred before, in this work it was assumed that the post-shock flow was
unidimensional and stationary. Moreover, it was assumed that the transport
phenomena (mass diffusion, viscosity and heat conduction) were negligible (the
flow is then said to be of the Euler type) as well as the external body
forces, and that the translational and rotational energy modes of the heavy
particles were in equilibrium with each other (which is normally true
immediately downstream of the shock wave 10). The equations which were dealt
with were then the ones of balance of the species’ mass in their vibronic
levels (due to the regarded vibronic-specific state-to-state model), momentum,
total energy, and free electrons energy subjected to the above-mentioned
assumptions:

$\displaystyle\frac{dc_{s,e,v}}{dx}=\frac{\dot{\omega}_{s,e,v}}{\rho
u},\,\,\,\,\,\,\,\,\,\forall s,\,e\,\text{and }v\text{ ,}$ (5a)
$\displaystyle\left(\frac{\rho
u^{2}}{p}-1\right)\frac{du}{dx}+\frac{u}{p}\left[\left(\sum_{s\in\\{\text{h}\\}}\frac{p_{s}}{T_{\text{tr}_{\text{h}}}}\frac{dT_{\text{tr}_{\text{h}}}}{dx}\right)+\frac{p_{\text{e}}}{T_{\text{tr}_{\text{e}}}}\frac{dT_{\text{tr}_{\text{e}}}}{dx}\right]=-\frac{1}{p\rho}\left[\left(\sum_{s\in\\{\text{h}\\}}\frac{\dot{\omega}_{s}p_{s}}{c_{s}}\right)+\frac{\dot{\omega}_{\text{e}}p_{\text{e}}}{c_{\text{e}}}\right]\text{
,}$ (5b) (5c)
$\displaystyle\frac{dT_{\text{tr}_{\text{e}}}}{dx}+\frac{u}{C_{p,\text{e}}}\frac{du}{dx}=\frac{\left(\sum_{s}\dot{\Omega}_{s,\text{e}}^{\text{int}}\right)-\dot{\omega}_{\text{e}}\left(h_{\text{e}}+\frac{1}{2}u^{2}\right)}{\rho
uc_{\text{e}}C_{p,\text{e}}}\text{ .}$ (5d)

In the equation for the balance of the species’ mass in their vibronic levels
(5a), $c_{s,e,v}$ is the mass fraction of $s$-th species particles in the
$(e,\,v)$ vibronic level, $\dot{\omega}_{s,e,v}$ is the respective volumetric
mass source term, $\rho$ is the mixture’s mass density, and $u$ is the
$x$-component of the mixture’s flow velocity vector. In the equation for the
balance of the mixture’s momentum (5b), $\\{h\\}$ denotes the set of heavy
species, $p$ is the mixture’s pressure, $p_{s}$ is the partial pressure
associated with the $s$-th species particles, $T_{\text{tr}_{\text{h}}}$ is
the heavy particle translational temperature, $T_{\text{tr}_{\text{e}}}$ is
the free electron translational temperature, and the quantities
$\dot{\omega}_{\text{e}}$, $p_{\text{e}}$ and $c_{\text{s}}$ are
$\dot{\omega}_{\text{s}}$, $p_{\text{s}}$ and $c_{\text{s}}$ in which the
$s$-th species corresponds to the free electron. In the equation for the
balance of the total energy (5c), $C_{p,s,\text{tr-rot}}$ is the
translational-rotational specific heat at constant pressure of the $s$-th
species particles, $\dot{\Omega}_{\text{rad}}$ is the volumetric radiative
energy source term, $\sum_{s}\dot{\Omega}_{s,\text{e}}^{\text{int}}$ is the
energy transferred per unit of time and volume from the inner particles to the
inner free electrons of the element of fluid, $h_{s}$ is the specific enthalpy
of the $s$-th species particles, $\epsilon_{s,\text{el-vib},e,v}$ is the
sensible vibronic energy of a $s$-th species particle in its $(e,\,v)$
vibronic level, and $m_{s}$ is its mass. Finally, in the equation for the
balance of the free electrons energy (5d), $C_{p,\text{e}}$ is the free
electrons specific heat at constant pressure.

####  2.3.1 Initial conditions

Initial values need to be assigned to the unknowns of the problem. These are
associated with the conditions of the post-shock flow immediately downstream
of the shock wave - which will be labelled here by “2”. It can be assumed that
no chemical processes neither electronic or vibrational excitation occur
throughout the shock wave 10. Hence, $c_{s,v,e,2}=c_{s,v,e,\infty}$, where the
symbol “$\infty$” denotes the upstream conditions. Particular attention should
be given to the free electron translational temperature
$T_{\text{tr}_{\text{e}}}$: since the free electrons are much lighter than the
heavy particles, the excitation of their translation energy modes occurs in a
different way, and it is not certain that an equilibration between them occurs
immediately downstream of the shock wave. Also, the mole fraction of free
electrons should be negligible throughout the shock wave, which may let us to
regard the problem as a purely numerical one. Since no discussion about the
topic was found in the literature, it was decided to regard the choice of
other researchers, such as Kadochnikov and Arsentiev 11, and consider the
equality $T_{\text{tr}_{\text{e}},2}=T_{\text{tr}_{\text{h}},2}$. This
equality implies that the immediately freed electrons have the same
translational temperature as the heavy particles to which they were initially
bonded.

The low $T_{\infty}$ value (of $300\,\text{K}$) allows one to disregard the
contribution of the vibrational and electronic energy modes to the particles
energy 12 throughout the shock wave. This means that both mixture specific
heat at constant volume $C_{V}$ and mixture specific heat at constant pressure
$C_{p}$, from the upstream to the immediately downstream conditions, only
depend on the translational and rotational energy modes of the particles. It
is then possible to obtain the well-known Rankine-Hugoniot jump conditions,
which allows one to compute the conditions immediately downstream of the shock
wave from the ones upstream of it 7.

###  2.4 The line-shape factor

In this work, the line-shape factor
$\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}$ regarded in (1) was considered
to be the result of four contributions: Doppler, collisional, Stark, and
resonance broadening. One may determine a line-shape factor for each isolated
contribution, and then compute the global line-shape factor
$\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}$ through a triple convolution
13:

$\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}(\vec{r},t,\lambda)=\int_{-\infty}^{\infty}\left(\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{D}}(\vec{r},t,\lambda_{s,e,v}^{e^{\prime},v^{\prime}}+\lambda^{\prime})\left\\{\int_{-\infty}^{\infty}\left(\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{col}}(\vec{r},t,\lambda_{s,e,v}^{e^{\prime},v^{\prime}}+\lambda^{\prime\prime})\left[\vphantom{\int_{-\infty}^{\infty}}\right.\right.\\\
\left.\left.\int_{-\infty}^{\infty}\left(\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{S}}(\vec{r},t,\lambda_{s,e,v}^{e^{\prime},v^{\prime}}+\lambda^{\prime\prime\prime})\cdot\left(\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{res}}(\vec{r},t,\lambda-\lambda^{\prime}-\lambda^{\prime\prime}-\lambda^{\prime\prime\prime})\,d\lambda^{\prime\prime\prime}\right]\,d\lambda^{\prime\prime}\right\\}\,d\lambda^{\prime}\text{
.}$ (6)

The line-shape factor for isolated Doppler broadening
$\left(\phi_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{D}}$
associated with radiative transitions between the vibronic levels $(e,\,v)$
and $(e^{\prime},\,v^{\prime})$ can be shown to be approximately given by a
Gaussian function 13 $G(\lambda,w_{G},\lambda_{0,G})$ of half-width at half-
maximum 14
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{D}}=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}\sqrt{[2\ln
2\,k_{B}T_{\text{tr}_{\text{h}}}]/(m_{s}c^{2})}=:w_{G}$, centred at
$\lambda=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}=:\lambda_{0,G}$, with $k_{B}$
being the Boltzmann constant, and $\lambda_{s,e,v}^{e^{\prime},v^{\prime}}$
the wavelength of the photon in the absence of line broadening and shift.

For the case of collisional broadening, the formula based on the electron
theory of Lorentz and referred by Penner 14 was considered for the respective
line-shape factor. This is approximately given by a Lorentzian function 13
$L(\lambda,w_{L},\lambda_{0,L})$ of half-width at half-maximum
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{col}}=(\lambda_{s,e,v}^{e^{\prime},v^{\prime}})^{2}\cdot(Z_{\text{opt},s,e,v}+Z_{\text{opt},s,e^{\prime},v^{\prime}})/(2\pi
c)=:w_{L}$, centred at
$\lambda=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}=:\lambda_{0,L}$, where
$Z_{\text{opt},s,e,v}$ is the optical collisional frequency per $s$-th species
particle in the $(e,\,v)$ vibronic level with the other particles, and
$Z_{\text{opt},s,e^{\prime},v^{\prime}}$ is the one with respect to the
$(e^{\prime},\,v^{\prime})$ vibronic level. The former quantity is given by

$Z_{\text{opt},s,e,v}=\left(\sum_{q\in\\{\text{h}\\},e^{\prime\prime},v^{\prime\prime}}\frac{n_{q,e^{\prime\prime},v^{\prime\prime}}\cdot\sigma_{\text{opt},s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}}{1+\delta_{s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}}\sqrt{\frac{8k_{B}T_{\text{tr}_{\text{h}}}}{\pi\,\mu_{s,q}}}\right)+n_{\text{e}}\cdot\sigma_{\text{opt},s,e,v}^{\text{e}}\sqrt{\frac{8k_{B}T_{\text{tr}_{\text{e}}}}{\pi\,m_{\text{e}}}}\text{
.}$ (7)

The sum in (7) is done in all the heavy species, their electronic levels and
vibrational levels. The quantity
$\delta_{s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}$ is a Kronecker delta
giving $1$ if $(s,e,v)=(q,e^{\prime\prime},v^{\prime\prime})$, and $0$ if not.
The quantity $\mu_{s,q}=m_{s}\cdot m_{q}/(m_{s}+m_{q})$ is the reduced mass of
the collision partners. The term
$\sigma_{\text{opt},s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}$ corresponds
to the optical collisional cross section for a collision between a particle of
the $s$-th species in $(e,\,v)$ with a particle of the $q$-th species in
$(e^{\prime\prime},\,v^{\prime\prime})$. And
$\sigma_{\text{opt},s,e,v}^{\text{e}}$ corresponds to
$\sigma_{\text{opt},s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}$ for the case
of the $q$-th species being a free electron. Penner 14 refers that, according
to the available data, such quantities have the same order of magnitude as the
respective collisional cross sections
$\sigma_{s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}$, and therefore, it was
decided to consider
$\sigma_{\text{opt},s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}\approx\sigma_{s,e,v}^{q,e^{\prime\prime},v^{\prime\prime}}$.
No data were found for the case of collisions with free electrons, and,
therefore, its contribution was disregarded.

For the case of Stark broadening, it was decided to consider in this work the
approach of Johnston 15, which solely accounts for the contribution of free
electrons. The associated wavelength-specific line-shape factor
$\left(\phi_{\lambda,s,e,v}^{e^{\prime}v^{\prime}}\right)_{\text{S}}$ is given
by a Lorentzian function of half-width at half-maximum
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{S}}$, centred
at $\lambda=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}$. The half-width at half-
maximum regarded by Johnston has a form identical to the one of Park 16:

$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{S}}=\left(w_{\lambda,s,e,v,\text{ref}}^{e^{\prime},v^{\prime}}\right)_{\text{S}}\left(\frac{T_{\text{tr}_{\text{e}}}}{T_{\text{tr}_{\text{e}},\text{ref}}}\right)^{n}\cdot\left(\frac{n_{\text{e}}}{n_{\text{e},\text{ref}}}\right)\text{
,}$ (8)

where
$\left(w_{\lambda,s,e,v,\text{ref}}^{e^{\prime},v^{\prime}}\right)_{\text{S}}$
is the half-width at half-maximum at a free electron translational temperature
$T_{\text{tr}_{\text{e}}}=10,000\,\text{K}:=T_{\text{tr}_{\text{e}},\text{ref}}$
and at a free electron number density
$n_{\text{e}}=10^{16}\,\text{cm}^{-3}:=n_{\text{e},\text{ref}}$. Park 16
solely treated the case of argon, obtaining the exponent $n=0.33$ by fitting
the curve (8) to the values of half-width at half-maximum
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{S}}$ at
$n_{\text{e}}=n_{\text{e},\text{ref}}$ for different
$T_{\text{tr}_{\text{e}}}$, issued by Griem 17, 18, 19. Johnston considers
this exponent value to be also acceptable for the case of the nitrogen and
oxygen atoms. In the present work, it was decided to take such assumption one
step further by regarding this same value for all the considered heavy species
(N, N+, N2 and N2+). Johnston then considers a model for the reference half-
width at half-maximum of the form

$\left(w_{\lambda,s,e,v,\text{ref}}^{e^{\prime},v^{\prime}}\right)_{\text{S}}=\frac{C\cdot\left(\lambda_{s,e,v}^{e^{\prime},v^{\prime}}\right)^{2}}{\left(\epsilon_{s}^{+}-\epsilon_{s,\text{el-
vib},e,v}\right)^{n}}\text{ ,}$ (9)

with $\epsilon_{s}^{+}$ being the ionisation energy of the a $s$-th species
particle from its ground level, and $C$ and $n$ some constants. The values for
the ionisation energies of all the heavy species considered in this work were
taken from the literature, being listed in Table 1.

Table 1: Ionisation energies from the ground level of N, N+, N2 and N2+. $s$ |  $\epsilon_{s}^{+}$ [eV] |  Reference
---|---|---
N |  14.534 |  Biémont et al. 20
N+ |  29.601 |  Biémont et al. 20
N2 |  15.581 |  Trickl et al. 21
N2+ |  27.9 |  Bahati et al. 22

Model (9) is a variant of the one considered by Cowley in its theoretical work
23. Johnston obtained $C=1.69\times
10^{10}\,[(\text{cm}^{-1})^{2.623}/\text{cm}^{2}]\cdot\text{cm}$ (here the
ambiguous unit of energy $\text{cm}^{-1}$ was used which actually means
$\text{cm}^{-1}\cdot h\cdot c$) and $n=2.623$ by fitting the curve (9) to the
values of
$\left(w_{\lambda,s,e,v,\text{ref}}^{e^{\prime},v^{\prime}}\right)_{\text{S}}$
for the nitrogen and oxygen atoms, issued by Griem 19 and by Wilson and
Nicolet 24.

For the case of resonance broadening, the theory developed by Griem 18 was
considered. The wavelength-specific line-shape factor corresponds to a
Lorentzian function centred at
$\lambda=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}$ and of half-width at half-
maximum

$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{res}}=\frac{3}{32}\sqrt{\frac{g_{s,e,v}}{g_{s,e^{\prime},v^{\prime}}}}\frac{\left(\lambda_{s,e,v}^{e^{\prime},v^{\prime}}\right)^{5}\,A_{s,e,v}^{e^{\prime},v^{\prime}}}{\pi^{3}c}n_{s,e^{\prime},v^{\prime}}\text{
,}$ (10)

with $g_{s,e,v}$ being the degree of degeneracy of the $(e,\,v)$ vibronic
level of the $s$-th species.

By noting that the convolution of two Lorentzian functions is also a
Lorentzian function whose half-width at half-maximum and centre is the sum of
them, one may show that the global line-shape factor defined by (6)
corresponds to a Voigt function 13 $V(\lambda,w_{G},w_{L},\lambda_{0V})$ of
Gaussian and Lorentzian half-widths at half-maxima
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{D}}=:w_{G}$ and
$\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{col}}+\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{S}}+\left(w_{\lambda,s,e,v}^{e^{\prime},v^{\prime}}\right)_{\text{res}}=:w_{L}$,
centred at $\lambda=\lambda_{s,e,v}^{e^{\prime},v^{\prime}}=:\lambda_{0,V}$.
The definition of the Voigt function involves an improper integral which is
not analytically solvable, and therefore, one should consider a numerical
method such as the trapezoidal integral rule, or an empirical approximation
such as the formula proposed by Whiting 25, to account for it. Although the
former may give better results, it requires much more computational resources
than the latter. The empirical formula of Whiting was then chosen, which
according to his words, matches the exact function within 5 per cent at worst.
The formula of Olivero and Longbothum 26 for the half-width at half-maximum of
the Voigt function $w_{V}$ was considered. It has an accuracy of about 0.01
per cent.

##  3 Results

###  3.1 The test matrix

Brandis and Cruden 2 issue data for a total of 42 shock tube shots from which
17 are of benchmark quality (in some or all wavelength intervals) and may be
used for validation of the developed numerical model. For the sake of
compactness, it was decided to regard solely 3 of these 17 shots, more
precisely, shots that spawned conditions of low, medium, and high speed
hypersonic flows, and of benchmark quality in the four wavelength intervals
(VUV, “Blue”, “Red”, and IR). These would allow a study about the dependence
of the results on the upstream flow speed $u_{\infty}$ in the whole range of
wavelength values. Under such criteria, the shots 40 (with
$u_{\infty}=6.88\text{ m}/\text{s}$), 19 (with $u_{\infty}=10.32\text{
m}/\text{s}$), and 20 (with $u_{\infty}=11.16\text{ m}/\text{s}$) were taken.

###  3.2 The analysis methodology

In the following sections, the numerical and experimental results are
compared, and possible causes for their discrepancies are enunciated. The
dependence of the results on different adjustable parameters of the
simulations is reported. To properly guide the reader over the extensive set
of obtained results, it was decided to treat the three shots simultaneously
for each of the four wavelength intervals and to show side-by-side the
respective graphs for the two quantities that were measured in the
experiments: the instrumentally resolved radiative intensities $\hat{I}^{l}$
and non-equilibrium metrics $\hat{I}_{\lambda}^{\text{ne},l}$, with
$l\in\\{\text{VUV},\text{``Blue''},\text{``Red''},\text{IR}\\}$. Further
comments on the behaviour of important physical quantities such as the
translational temperatures and the species’ mole fractions, as well as on the
evolution of the system to equilibrium are provided.

###  3.3 Simulations of post-shock flows generated by a shock tube

####  3.3.1 The case of the VUV radiation

The instrumentally resolved radiative intensities $\hat{I}^{\text{VUV}}(x)$
obtained from the numerical simulations are depicted in Figures 2 and 3, and
the non-equilibrium metrics
$\hat{I}_{\lambda}^{\text{ne},\text{VUV}}(\lambda)$ are depicted in Figure 4.
The three graphs appearing in each figure are ordered by increasing free
stream speed $u_{\infty}$ from top to bottom - the top one is with respect to
shot $40$, the middle one to $19$, and the bottom one to $20$. The solid
coloured lines represent the numerically obtained contributions of the
different systems of spontaneous emission processes to $\hat{I}^{\text{VUV}}$
and $\hat{I}_{\lambda}^{\text{ne},\text{VUV}}$. The solid black lines
represent the numerically obtained overall quantities (i.e. the sums of the
contributions). The dotted black lines represent the experimentally obtained
overall quantities. In Figures 3 and 4 the numerical values are quantified in
the left $y$-axis and the experimental ones in the right $y$-axis. These
$y$-axes are scaled differently to make the two sets of values visually
comparable. In the case of $\hat{I}^{\text{VUV}}$, the scales are such that
the heights of the peaks match each other. And in the case of
$\hat{I}_{\lambda}^{\text{ne},\text{VUV}}$, the scales are such that there is
a coincidence between the peaks associated with spontaneous emission of N at
$\lambda=149\,\text{nm}$. The necessity of using different scales unveils
immediately a significant discrepancy: the numerically obtained values are
much lower than the experimental ones. In fact, the heights of the
experimental $\hat{I}^{\text{VUV}}$ and
$\hat{I}_{\lambda}^{\text{ne},\text{VUV}}$ peaks are one to two orders of
magnitude higher than the numerical ones (see Figure 2 for the case of the
instumental radiative intensities).

One should then analyse relative discrepancies, i.e. the discrepancies between
the numerical and experimental results as if they were scaled to the same
order of magnitude. Figure 3 reveals that the shape of the profile of
$\hat{I}^{\text{VUV}}$ was well predicted for the case of the low speed shot.
It also shows that spontaneous emissions of the type
$\ch{N2+}(\text{C}-\text{X})$ were the main contribution to the numerical
$\hat{I}^{\text{VUV}}$ values. However, the same cannot be said for the
experimental counterpart: Figure 4 shows many
$\hat{I}_{\lambda}^{\text{ne},\text{VUV}}$ peaks of considerable heigh
assigned to $\ch{N2+}(\text{C}-\text{X})$ which were not actually observed in
the experiment. Although the two peaks that appear at $\lambda=156\,\text{nm}$
and $\lambda=166\,\text{nm}$ seem to overlap others of
$\ch{N2+}(\text{C}-\text{X})$, this should be mere coincidence. The two peaks
may be predicted if one considers atomic carbon C - a contaminant species - in
the database, as Cruden and Brandis 27 did. Furthermore, according to Cruden
and Brandis 27, the experimental peak at $\lambda=193\,\text{nm}$ should also
be due to C. One finds the peak at $\lambda=149\,\text{nm}$ to be higher than
the peak at $\lambda=174\,\text{nm}$ in the numerical spectra, but lower in
the experimental one. These two peaks are due to spontaneous emission of N.
Such discrepancy may be due to the absorption of radiation associated with the
first peak being greater than the one associated with the second peak in the
experiment. Meanwhile, in the simulation, absorption was completely
disregarded. The discrepancy may also be due to an improper modelling of the
rate coefficients which dictate the population of the nitrogen atoms that
spontaneously emit radiation at wavelengths $\lambda=149\,\text{nm}$ and
$\lambda=174\,\text{nm}$. One way of making the ratio between the heights of
the two peaks more coherent with the experimental result is by decreasing the
rate of excitation of atomic nitrogen to the upper level of the system
associated with the first peak and increasing the one associated with the
second. However, in the present case, the upper levels of the system
associated with the two peaks are the same, corresponding to the fifth level,
say111The electronic levels of the atomic particles will be labelled here by
integer numbers $e=0,\,1,\,...$ $\ch{N}(4)$. The first peak is a result of the
transitions $\ch{N}(4-1)$, and the second is a result of the transitions
$\ch{N}(4-2)$. Therefore, changing the values of the rate coefficients for
excitation would not make any difference to the ratio between the heights of
the peaks.

The experimental instrumentally resolved radiative intensity
$\hat{I}^{\text{VUV}}$ obtained for the case of the medium speed shot has a
peak proceeded by a non-zero plateau. This plateau was not predicted at all. A
transition to zero occurs instead. And for the case of the high speed shot,
the experiment did not produce a sole peak, but a coalescence between a peak
and a plateau surpassing it, which was also not predicted.

Figure 2: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved radiative intensities $\hat{I}^{\text{VUV}}(x)$
obtained for the low, medium and high speed shots.

Figure 3: Same as Figure 2, but with two differently scaled $y$-axes: one for
the numerical values (at left), and another for the experimental values (at
right).

Figure 4: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{VUV}}(\lambda)$ obtained for the low,
medium and high speed shots.

####  3.3.2 The case of the “Blue” radiation

The obtained instrumentally resolved radiative intensities
$\hat{I}^{\text{Blue}}(x)$ and non-equilibrium metrics
$\hat{I}_{\lambda}^{\text{ne},\text{Blue}}(\lambda)$ are depicted in Figure 5
and Figure 6, respectively. The main contributors to the overall quantities
are $\ch{N2+}(\text{B}-\text{X})$, $\ch{N2}(\text{C}-\text{B})$ and
$\ch{N2+}(\text{A}-\text{X})$, and the residual contributors correspond to N
and $\ch{N2}(\text{A}-\text{X})$. Meanwhile, Cruden and Brandis 27 showed that
$\ch{N2}(\text{C}-\text{B})$, $\ch{N2+}(\text{B}-\text{X})$, N, and the
contaminant species CN (cyanogen radical) are enough to describe the
experimentally obtained spectrum for the case of the medium speed shot.

As for the case of the VUV radiation, the instrumentally resolved radiative
intensities and non-equilibrium metrics were underestimated by one to two
orders of magnitude. The scales of the $y$-axes of the non-equilibrium metrics
graphs were chosen such that there was a coincidence between the numerical and
experimental peaks associated with spontaneous emission of
$\ch{N2+}(\text{B}-\text{X})$ at $\lambda=391\,\text{nm}$. The profile of the
numerical radiative intensity $\hat{I}^{\text{Blue}}$ obtained in the low
speed shot corresponds to a peak in which the transition from the maximum to
zero is faster than in the experimental counterpart. For the case of the
medium and high speed shots, the shapes of the peaks were remarkably well
predicted, contrary to the plateaus that proceed them: the numerical radiative
intensities transit to zero instead of converging to these plateaus. Such
behaviour was also obtained in the case of the VUV radiation.

Sets of peaks appearing in the experimental non-equilibrium metrics
$\hat{I}_{\lambda}^{\text{ne},\text{Blue}}$ may be discerned. These are in
some way predicted by the numerical model. However, the heights of the
individual peaks for each set do not agree particularly well with the
experimental ones, namely, the ones associated with spontaneous emissions of
the types $\ch{N2}(\text{C},0-\text{B},0)$ and
$\ch{N2+}(\text{B},0-\text{X},0)$, at $\lambda=337\,\text{nm}$ and
$\lambda=391\,\text{nm}$, respectively. These were undoubtedly overestimated,
prevailing over the others. The discrepancies between the numerical and
experimental non-equilibrium metrics may in some part be due to the regarded
spontaneous emissions being vibronic-specific instead of rovibronic. If
rovibronic-specific spontaneous emissions were considered, the peaks
associated with the vibronic levels would be divided into several others (each
one associated with a rotational level) whose centres may be more or less far
from each other, depending on the energy of the rotational levels. The
importance of the rotational levels to the spectra should be accessed in the
future.

Figure 5: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved radiative intensities $\hat{I}^{\text{Blue}}(x)$
obtained for the low, medium and high speed shots.

Figure 6: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Blue}}(\lambda)$ obtained for the low,
medium and high speed shots.

####  3.3.3 The case of the “Red” radiation

Figures 7 and 8 depict the obtained instrumentally resolved radiative
intensities $\hat{I}^{\text{Red}}(x)$ and non-equilibrium metrics
$\hat{I}_{\lambda}^{\text{ne},\text{Red}}(\lambda)$, respectively. The scales
of the $\hat{I}_{\lambda}^{\text{ne},\text{Red}}$ graphs are such that the
experimental and numerical peaks at $\lambda=869\,\text{nm}$ match each other.
The heights of the experimental $\hat{I}^{\text{Red}}$ and
$\hat{I}_{\lambda}^{\text{ne},\text{Red}}$ peaks were underestimated by one to
two orders of magnitude. As happened for the case of the “Blue” radiation, in
the low speed shot, the transition of the numerical radiative intensity
$\hat{I}_{\lambda}^{\text{ne},\text{Red}}$ from the peak maximum value to zero
was slighter faster than the experimental one. For the medium and high speed
shots, plateaus proceeding peaks were observed experimentally. These were not
predicted by the numerical model.

There is strong evidence of a relative underestimation of spontaneous
emissions of N, since the most prominent peaks (such as the ones at
$\lambda=649$, $745$, $821$, $862$ and $869\,\text{nm}$) from the experimental
spectra, which are due to N, are exceeded in the numerical spectra by the ones
associated with $\ch{N2}(\text{B}-\text{A})$. Also, spontaneous emissions of
the type $\ch{N2+}(\text{A}-\text{X})$ should not be as relevant as they were
found to be in the obtained numerical spectra. For the case of the medium
speed shot, Cruden and Brandis 27 solely considered spontaneous emissions from
N (the dominant ones), $\ch{N2}(\text{B}-\text{A})$ and H(as a contaminant
species) in this wavelength interval.

Figure 7: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved radiative intensities $\hat{I}^{\text{Red}}(x)$
obtained for the low, medium and high speed shots.

Figure 8: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Red}}(\lambda)$ obtained for the low,
medium and high speed shots.

####  3.3.4 The case of the IR radiation

Figures 9 and 10 depict the obtained instrumentally resolved radiative
intensities $\hat{I}^{\text{IR}}(x)$ and non-equilibrium metrics
$\hat{I}_{\lambda}^{\text{ne},\text{IR}}(\lambda)$, respectively. The scales
of the $\hat{I}_{\lambda}^{\text{ne},\text{IR}}$ graphs are such that the
experimental and numerical peaks at $\lambda=940\,\text{nm}$ match each other.
As for the other wavelength intervals, it was found that the numerical model
underestimated the radiative intensities and non-equilibrium metrics by one to
two orders of magnitude. In the case of the low speed shot, the shape of the
profile of $\hat{I}^{\text{IR}}$ was well predicted. For the cases of medium
and high speed shots, individual peaks cannot be discerned from the experiment
but coalescences between peaks and plateaus occurring above them. Again, such
phenomena was not predicted by the numerical model.

The obtained non-equilibrium metrics show that spontaneous emissions of the
type $\ch{N2}(\text{B}-\text{A})$ seem to be relatively overestimated,
specially the ones at $\lambda=1047\,\text{nm}$ \- due to transitions from
$v=0$ to $v^{\prime}=0$ \- and $\lambda=1232\,\text{nm}$ \- due to transitions
from $v=0$ to $v^{\prime}=1$ \- which prevail over the others. Conversely, the
peaks associated with spontaneous emission of N (such as the ones at
$\lambda=905\,\text{nm}$, $\lambda=940\,\text{nm}$, $\lambda=1012\,\text{nm}$
and $\lambda=1053\,\text{nm}$) seem to be relatively underestimated, since
these predominate in the experimental spectra but not in the numerical
spectra.

Figure 9: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved radiative intensities $\hat{I}^{\text{IR}}(x)$
obtained for the low, medium and high speed shots.

Figure 10: Numerical (solid lines) and experimental (dotted lines)
instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{IR}}(\lambda)$ obtained for the low,
medium and high speed shots.

###  3.4 Mole fractions, temperatures, and evolution to equilibrium

Figure 11 depicts the predicted heavy particle and free electron translational
temperatures, $T_{\text{tr}_{\text{h}}}(x)$ and $T_{\text{tr}_{\text{e}}}(x)$.
According to the Rankine-Hugoniot condition, the values of the translational
temperatures immediately downstream of the shock wave ($x=0$) correspond to
$T_{\text{tr}_{\text{h}},2}=T_{\text{tr}_{\text{e}},2}=22,434\,\text{K}$,
$50,122\,\text{K}$ and $58,565\,\text{K}$ for the low, medium and high speed
shots, respectively. As expected, these decrease with $x$, due to the
endothermic processes, to somewhat identical values, and with a rate which is
as high as the immediately downstream temperature.

Figure 11: Heavy particle (solid lines) and free electron (dashed lines)
translational temperatures $T_{\text{tr}_{\text{h}}}(x)$ and
$T_{\text{tr}_{\text{e}}}(x)$, obtained for the low (blue), medium (red) and
high (green) speed shots.

Figure 12 presents the evolutions of the mole fractions $x_{s}(x)$ of the five
species considered in the simulations - N2, N, N2+, N+ and e- - for the low,
medium and high speed shots. One finds that an higher upstream speed implies
an earlier dissociation and ionisation of the particles. In the medium and
high speed shots, the dissociation is such that the mole fraction of atomic
nitrogen N surpasses the mole fraction of molecular nitrogen N2. There is an
increase and posterior decrease of the mole fractions of atomic nitrogen ions
N+, molecular nitrogen ions N2+ and free electrons e-. The former is
associated with ionisation and the latter to recombination. The higher the
upstream speed, the more abrupt the increases and decreases of the mole
fractions of the charged particles are.

Figure 12: Species-specific mole fractions $x_{s}(x)$ obtained for the low,
medium and high speed shots.

Table 2 makes a synopsis on the temperatures $T_{\text{tr}_{\text{h}}}$ and
$T_{\text{tr}_{\text{e}}}$, and mole fractions $x_{s}$, obtained from the
simulations of the three shots, at a downstream point far from the shock wave,
$x=5\,\text{cm}$ \- where the radiation intensities were found to be
stagnated. The table also shows the values issued by Cruden and Brandis 27 for
the medium speed shot. These were obtained by fitting the conditions of a
hypothetical system such that the computed “Blue” and “Red” spectra matched
the experimental ones, using the NEQAIR tool 8, 28, at a $x$ position for
which the plateaus of radiative intensities occurred. With that objective, a
two-temperature model - of temperatures $T_{\text{tr}_{\text{h}}-\text{rot}}$
and $T_{\text{vib}-\text{el}-\text{tr}_{\text{e}}}$ \- was assumed and the
populations of some particular energy levels were adjusted. Regarding the
values of the temperatures at $x=5\,\text{cm}$ obtained in the present work,
one can say that the heavy particle translational temperature departed from
the free electron translational temperature by $-18\,\text{K}$ (or
$-0.23\,\%$), $-3\,\text{K}$ (or $-0.04\,\%$) and $-1\,\text{K}$ (or
$-0.03\,\%$) for the cases of the low, medium and high speed shots. These
values are sufficiently small for one to assume that an equilibrium between
the respective energy modes was attained. The values of the temperatures
obtained by Cruden and Brandis are significantly higher than the ones of this
work. The analysis of the “Blue” spectra resulted in a
$T_{\text{tr}_{\text{h}}}$ value which is greater by $\left(1,986\pm
170\right)\,\text{K}$ (or $\left(28\pm 2\right)\,\%$) and a
$T_{\text{tr}_{\text{e}}}$ value which is greater by $\left(3,213\pm
300\right)\,\text{K}$ (or $\left(45\pm 4\right)\,\%$). The analysis of the
“Red” spectra resulted in a $T_{\text{tr}_{\text{h}}}$ value which is greater
by $\left(2,606\pm 2,490\right)\,\text{K}$ (or $\left(36\pm 35\right)\,\%$)
and a $T_{\text{tr}_{\text{e}}}$ value which is greater by $\left(2,013\pm
130\right)\,\text{K}$ (or $\left(28\pm 2\right)\,\%$). The mole fraction of N
obtained from the simulation of the medium speed shot in the present work, at
$x=5\,\text{cm}$, is almost the double of the one obtained from the simulation
of the low speed shot, showcasing a very meaningful increase on the
dissociation of N2. The mole fraction of N2+ is lower (by $38\,\%$) - due to a
much faster recombination - and the one of N+ is almost the double. The
differences between the values of the mole fractions obtained from the
simulation of the high speed shot and the ones obtained from the simulation of
the medium speed shot are not significant enough, as also are not the
differences between the upstream speeds. Regarding the results of the work of
Cruden and Brandis 27 for the medium speed shot, one may say that not only the
differences between the obtained chemical compositions in that work and the
present one are staggering but also the differences between the chemical
compositions resulting from the analysis of the “Blue” and “Red” spectra in
the former. The results of the analysis of the “Blue” spectra showed a greater
importance of the molecular particles N2 and N2+ and free electrons e- in the
composition whereas the results of the analysis of the “Red” spectra showed a
greater importance of atomic nitrogen N. In fact, the mole fractions of N2,
N2+ and e- obtained in the the former case are $22$, $133$ and $30$ times
greater than the ones obtained in the latter case, and the mole fraction of N
is $57\,\%$ lower. This incoherence evidences that the models regarded in
NEQAIR may not represent sufficiently well the reality. The results of Cruden
and Brandis also evidence a much stronger dissociation of N2 and ionisation of
N than the ones of the present work: the predicted mole fraction of N2 is at
least $9$ times lower, and the mole fraction of N+ is at least $20$ times
greater. Moreover, this greater dissociation and ionisation are accomplished
with a much lower cost of heavy particle and free electron translational
temperatures. Such result means that the post-shock elements of fluid
simulated in the present work may have lost more energy than they should, or
did not gain as much. Since it was found that the radiation variables were
underpredicted by several orders of magnitude, the former hypothesis should be
disregarded. It is then suspected that some source of energy which is external
to the regarded system occurs in the experiment. Also, the unaccounted
absorption of radiation and conduction of heat within the system may have some
impact in the redistribution of the energy.

Table 2: Temperatures $T_{\text{tr}_{\text{h}}}$ and
$T_{\text{tr}_{\text{e}}}$, and mole fractions $x_{s}$, at $x=5\,\text{cm}$,
obtained from the simulations of the low, medium and high speed shots, as well
as the ones obtained by Cruden and Brandis 27.

$u_{\infty}[\text{m}/\text{s}]$ Ref. $T_{\text{tr}_{\text{h}}}[\text{K}]$
$T_{\text{tr}_{\text{e}}}[\text{K}]$ $x_{\ch{N2}}$ $x_{\ch{N}}$ $x_{\ch{N2+}}$
$x_{\ch{N+}}$ $x_{\ch{e-}}$ $x_{\ch{CN}}$ $x_{\ch{H}}$ $6.88$ This work
$7,793$ $7,811$ $0.52$ $0.48$ $6.5(-5)$ $2.6(-4)$ $3.2(-4)$ — — $10.32$ This
work $7,184$ $7,187$ $0.27$ $0.73$ $4.0(-5)$ $4.5(-4)$ $4.9(-4)$ — — NEQAIR,
“Blue” 27 $9,170\pm 170$ $10,400\pm 300$ $2.9(-2)$ $0.42$ $1.6(-2)$ $0.27$
$0.27$ $7.3(-4)$ — NEQAIR, “Red” 27 $9,790\pm 2,490$ $9,200\pm 130$ $1.3(-3)$
$0.98$ $1.2(-4)$ $8.9(-3)$ $8.9(-3)$ — $1.0(-3)$ $11.16$ This work $7,098$
$7,100$ $0.26$ $0.74$ $3.7(-5)$ $4.4(-4)$ $4.7(-4)$ — —

  * •

The number between parenthesis in the mole fractions cells correspond to the
orders of magnitude of these quantities.

To ascertain the impact of energy loss by radiation on the developed
conditions of the post-shock flow, it was decided to perform simulations
disregarding spontaneous emission processes and comparing the respective
results with the ones including these processes. Table 3 makes a synopsis on
the resultant temperatures $T_{\text{tr}_{\text{h}}}$ and
$T_{\text{tr}_{\text{e}}}$, and mole fractions $x_{s}$, at $x=5\,\text{cm}$.
All of the temperatures are higher than the respective ones obtained when
regarding spontaneous emission processes, and increasing with the upstream
speed. While the differences are negligible for the case of the low speed shot
- $T_{\text{tr}_{\text{h}}}$ differs by just $266\,\text{K}$ (or $3.4\,\%$) -
they are meaningful for the case of the medium and high speed shots -
$T_{\text{tr}_{\text{h}}}$ differs by $2080\,\text{K}$ (or $29\,\%$) and
$3,016\,\text{K}$ (or $42\,\%$). Meanwhile, a greater dissociation and
ionisation have occurred, as evidenced by the lower value for the mole
fraction of N2 and the higher values for the mole fractions of N and N+. These
results show that the energy lost by spontaneous emission in the highly
excited systems has a much higher proportion than in the less excited ones.
The more excited the particles are, the higher the number of spontaneous
emissions and the higher the energy that is lost through radiation.
Surprisingly, the conditions for the medium speed shot disregarding
spontaneous emission processes are now coherent with the ones obtained by
Cruden and Brandis 27, with the exception of the mole fraction of N2+ which is
substantially lower. This result sheds light on the previously enunciated
hypothesis of the simulated elements of fluid not receiving the energy that
their counterparts seem to receive in the experiment.

Table 3: Temperatures $T_{\text{tr}_{\text{h}}}$ and $T_{\text{tr}_{\text{e}}}$, and mole fractions $x_{s}$, obtained from the simulations of the low, medium and high speed shots, at $x=5\,\text{cm}$, disregarding spontaneous emission processes. $u_{\infty}[\text{m}/\text{s}]$ | $T_{\text{tr}_{\text{h}}}[\text{K}]$ | $T_{\text{tr}_{\text{e}}}[\text{K}]$ | $x_{\ch{N2}}$ | $x_{\ch{N}}$ | $x_{\ch{N2+}}$ | $x_{\ch{N+}}$ | $x_{\ch{e-}}$
---|---|---|---|---|---|---|---
$6.88$ | $8,059$ | $8,076$ | $0.46$ | $0.54$ | $1.0\times 10^{-4}$ | $4.6\times 10^{-4}$ | $5.7\times 10^{-4}$
$10.32$ | $9,264$ | $9,265$ | $0.022$ | $0.93$ | $3.1\times 10^{-5}$ | $0.022$ | $0.022$
$11.16$ | $10,114$ | $10,114$ | $0.0041$ | $0.89$ | $2.4\times 10^{-5}$ | $0.052$ | $0.052$

Another important result that may be ascertained from the numerical
simulations is the systems departure from thermal equilibrium. For that
purpose, representative temperatures were computed. The representative
vibrational temperature associated with the $e$-th electronic level of the
$s$-th species, $T_{s,\text{vib},e}$, was obtained by fitting the curve

$\ln\left(\frac{x_{s,e,v}}{g_{s,\text{el},e}\cdot
g_{s,\text{vib},e,v}}\right)=-\frac{1}{k_{B}T_{s,\text{vib},e}}\epsilon_{s,\text{vib},e,v}+\ln\left(\frac{x_{s,e}}{g_{s,\text{el},e}\cdot
Q_{s,\text{vib},e}(T_{s,\text{vib},e})}\right)\text{ ,}$ (11)

where $Q_{s,\text{vib},e}$ is the vibrational partition function associated
with the $e$-th electronic level of the $s$-th species, to the points of
abscissae $\epsilon_{s,\text{vib},e,v}$ and ordinates
$\ln(x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v}))$. Equation (11)
tells that under vibrational self-equilibrium, the natural logarithm of the
mole fraction of $s$-th species particles in a state of the $e$-th electronic
level and $v$-th vibrational level, i.e.
$\ln(x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v}))$, decreases
linearly with their vibrational energy $\epsilon_{s,\text{vib},e,v}$. The
representative electronic temperature associated with the $s$-th species,
$T_{s,\text{el}}$, may be obtained by fitting the curve

$\ln\left(\frac{x_{s,e}}{g_{s,\text{el},e}}\right)=-\frac{1}{k_{B}T_{s,\text{el}}}\epsilon_{s,\text{el},e}+\ln\left(\frac{x_{s}\cdot
Q_{s,\text{vib},e}(T_{s,\text{vib},e})}{Q_{s,\text{el-
vib}}(\\{T_{s,\text{vib},e}\\},T_{s,\text{el}})}\right)\text{ ,}$ (12)

where $Q_{s,\text{el-vib}}$ is the vibronic partition function associated with
the $s$-th species, to the points of abscissae $\epsilon_{s,\text{el},e}$ and
ordinates $\ln(x_{s,e}/g_{s,\text{el},e})$. In thermal equilibrium conditions,
one has
$T_{\text{tr}_{\text{h}}}=T_{\text{tr}_{\text{e}}}=T_{s,\text{vib},e}=T_{s,\text{el}}=:T$,
$\forall$ $s$ and $e$. By inserting (12) into (11) one gets in such conditions

$\ln\left(\frac{x_{s,e,v}}{g_{s,\text{el},e}\cdot
g_{s,\text{vib},e,v}}\right)=-\frac{1}{k_{B}T}\left(\epsilon_{s,\text{el},e}+\epsilon_{s,\text{vib},e,v}\right)+\ln\left(\frac{x_{s}}{Q_{s,\text{el-
vib}}(T)}\right)\text{ .}$ (13)

Therefore, at thermal equilibrium, the points of abscissae
$\epsilon_{s,\text{el},e}+\epsilon_{s,\text{vib},e,v}=:\epsilon_{s,\text{el-
vib},e,v}$ and ordinates $\ln(x_{s,e,v}/(g_{s,\text{el},e}\cdot
g_{s,\text{vib},e,v}))$, with $s$ fixed, lay in the same curve, this being of
slope $-1/(k_{B}T)$ and of $y$-intercept $\ln(x_{s}/Q_{s,\text{el-vib}}(T))$.
If the vibrational partition function $Q_{s,\text{vib},e}$ does not vary too
much with the electronic level $e$ such that $Q_{s,\text{vib},e}\approx
Q_{s,\text{vib}}$, one may also show from relation (12) that in thermal
equilibrium the points of abscissae $\epsilon_{s,\text{el},e}$ and ordinates
$\ln(x_{s,e}/g_{s,\text{el},e})$, with $s$ fixed, lay in the same curve, this
being of slope $-1/(k_{B}T)$ and of $y$-intercept
$\ln(x_{s}/Q_{s,\text{el}}(T))$.

The representative vibrational and electronic temperatures may in their turn
be used to compute Boltzmann representative vibronic mole fractions
$x_{s,e,v}^{B}:=x_{s,e,v}(T_{s,\text{vib},e})$ \- given by (11) - and
Boltzmann representative electronic mole fractions
$x_{s,e}^{B}:=x_{s,e}(T_{s,\text{vib},e},T_{s,\text{el}})$ \- given by (12).

Figure 13 shows the electronic state-specific mole fractions
$x_{s,e}/g_{s,\text{el},e}$ (markers) and respective Boltzmann representatives
$x_{s,e}^{\text{B}}/g_{s,\text{el},e}$ (lines) obtained from the simulations
of the low, medium and high speed shots, at $x=5\,\text{cm}$. The markers in
the figure sparsely agree with the respective lines. This result shows that
self-equilibrium of the electronic energy modes of the different species was
not attained at $x=5\,\text{cm}$.

Figure 13: Electronic state-specific mole fractions
$x_{s,e}/g_{s,\text{el},e}$ (markers) and respective Boltzmann representatives
$x_{s,e}^{\text{B}}/g_{s,\text{el},e}$ (lines) as functions of the electronic
energies $\epsilon_{s,\text{el},e}$, obtained from the simulations of the low,
medium and high speed shots, at $x=5\,\text{cm}$.

The obtained representative electronic temperatures are presented by Table 4.
It shows that these temperatures depart from each other and from the obtained
heavy particle and free electron translational ones in the order of the
thousands of kelvins. In fact, the variables $T_{s,\text{el}}$ are all lower
than $T_{\text{tr}_{\text{h}}}$, by amounts ranging from $3,359\,\text{K}$ (or
$43.10\,\%$) to $1,407\,\text{K}$ (or $18.05\,\%$), $2,811\,\text{K}$ (or
$39.13\,\%$) to $968\,\text{K}$ (or $13.47\,\%$), and $2740\,\text{K}$ (or
$38.60\,\%$) to $932\,\text{K}$ (or $13.13\,\%$), for the cases of the low,
medium and high speed shots, respectively. Therefore, one cannot state that
the electronic and the translational energy modes attained an equilibrium with
each other at $x=5\,\text{cm}$.

Table 4: Heavy particle and free electron translational temperatures $T_{\text{tr}_{\text{h}}}$ and $T_{\text{tr}_{\text{e}}}$, and representative electronic temperatures $T_{s,\text{el}}$ obtained from the simulations of the low, medium and high speed shots, at $x=5\,\text{cm}$. $u_{\infty}[\text{m}/\text{s}]$ | $T_{\text{tr}_{\text{h}}}[\text{K}]$ | $T_{\text{tr}_{\text{e}}}[\text{K}]$ | $T_{\ch{N2},\text{el}}[\text{K}]$ | $T_{\ch{N},\text{el}}[\text{K}]$ | $T_{\ch{N2+},\text{el}}[\text{K}]$ | $T_{\ch{N+},\text{el}}[\text{K}]$
---|---|---|---|---|---|---
$6.88$ | $7,793$ | $7,811$ | $4,434$ | $5,551$ | $6,386$ | $6,093$
$10.32$ | $7,184$ | $7,187$ | $4,373$ | $5,682$ | $6,216$ | $5.762$
$11.16$ | $7,098$ | $7,100$ | $4,358$ | $5,699$ | $6,166$ | $5,719$

Figure 14 shows the obtained vibronic state-specific mole fractions
$x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ and respective
Boltzmann representatives $x_{s,e,v}^{\text{B}}/(g_{s,\text{el},e}\cdot
g_{s,\text{vib},e,v})$ at $x=5\,\text{cm}$. In this figure, there are markers
that agree remarkably well with the respective lines (vibrational self-
equilibrium was achieved in those cases), and others (the ones associated with
the seemingly horizontal lines) which do not agree at all. The deviations of
the latter are due to the non-modelling of the spontaneous emission processes
of the higher vibronic levels, implying these to not be depopulated through
these processes in contrast to the lower ones. When fitting the respective
vibronic state-specific mole fractions, a nearly horizontal line is obtained,
which is associated with a very high representative vibrational temperature.
This result is clearly unphysical. To avoid it in the future, one should
consider a redistribution procedure on the Einstein coefficients for
spontaneous emission to model the vibronic levels whose data are not
available.

Figure 14: Vibronic state-specific mole fractions
$x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ (markers) and
respective Boltzmann representatives
$x_{s,e,v}^{\text{B}}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ (lines)
as functions of the vibronic energies $\epsilon_{s,\text{el-vib},e,v}$,
obtained from the simulations of the low, medium and high speed shots, at
$x=5\,\text{cm}$.

The non-absurd values of $T_{s,\text{vib},n}$ deviate from
$T_{\text{tr}_{\text{h}}}$ by amounts from $-843\,\text{K}$ (or $-10.82\,\%$)
to $3,730\,\text{K}$ (or $47.86\,\%$), $-383\,\text{K}$ (or $-5.33\,\%$) to
$1,112\,\text{K}$ (or $15.48\,\%$), and $-311\,\text{K}$ (or $-4.38\,\%$) to
$1,188\,\text{K}$ (or $16.74\,\%$), for the cases of the low, medium and high
speed shots, respectively. These values are too large for one to assume that
equilibrium between the vibrational and translational energy modes was
attained for all the considered electronic levels. It is important to note
that markers associated with the same species do not lay on the same line,
showing that the respective electronic energy mode is not at self-equilibirum
(as also evidenced by the fact that the markers in Figure 13 do not coincide
with their Boltzmann representatives).

Since absorption and induced emission processes were not considered in this
work, spontaneous emission occurred without any counterbalance. This may have
led to the observed self-non-equilibrium of the electronic energy modes.
Figure 15 shows the electronic state-specific mole fractions
$x_{s,e}/g_{s,\text{el},e}$ and respective Boltzmann representatives
$x_{s,e}^{\text{B}}/g_{s,\text{el},e}$ obtained from the simulations of the
low, medium and high speed shots, at $x=5\,\text{cm}$, disregarding
spontaneous emission. The markers in this figure agree much better with the
respective lines than in Figure 13. Self-equilibrium of the electronic energy
modes seems to be attained by all species with the exception of N2 and N in
the case of the low speed shot, as the respective markers slightly depart from
the lines. Table 5 presents the obtained representative electronic
temperatures. The variables $T_{s,\text{el}}$ depart from
$T_{\text{tr}_{\text{h}}}$, by amounts ranging from $-948\,\text{K}$ (or
$-11,76\,\%$) to $432\,\text{K}$ (or $5.36\,\%$), $-83\,\text{K}$ (or
$-0.90\,\%$) to $39\,\text{K}$ (or $0.42\,\%$), and $-81\,\text{K}$ (or
$-0.80\,\%$) to $11\,\text{K}$ (or $0.11\,\%$), for the cases of the low,
medium and high speed shots, respectively. All these values are negligibly
small with the exception of the ones from the former case, the lower limit
being due to N and the higher limit due to N2, which, as mentioned above, were
also found to not be at electronic self-equilibrium. One may state that the
electronic and the translational energy modes of the particles attained an
equilibrium between themselves at $x=5\,\text{cm}$, with the exception of N
and N2 in the case of the low speed shot.

Figure 15: Electronic state-specific mole fractions $x_{s,e}/g_{s,\text{el},e}$ (markers) and respective Boltzmann representatives $x_{s,e}^{\text{B}}/g_{s,\text{el},e}$ (lines) as functions of the electronic energies $\epsilon_{s,\text{el},e}$, obtained from the simulations of the low, medium and high speed shots, at $x=5\,\text{cm}$, disregarding spontaneous emission processes. Table 5: Heavy particle and free electron translational temperatures $T_{\text{tr}_{\text{h}}}$ and $T_{\text{tr}_{\text{e}}}$, and representative electronic temperatures $T_{s,\text{el}}$ obtained from the simulations of the low, medium and high speed shots, at $x=5\,\text{cm}$, disregarding spontaneous emission. $u_{\infty}[\text{m}/\text{s}]$ | $T_{\text{tr}_{\text{h}}}[\text{K}]$ | $T_{\text{tr}_{\text{e}}}[\text{K}]$ | $T_{\ch{N2},\text{el}}[\text{K}]$ | $T_{\ch{N},\text{el}}[\text{K}]$ | $T_{\ch{N2+},\text{el}}[\text{K}]$ | $T_{\ch{N+},\text{el}}[\text{K}]$
---|---|---|---|---|---|---
$6.88$ | $8,059$ | $8,078$ | $8,491$ | $7,111$ | $8,100$ | $8.063$
$10.32$ | $9,264$ | $9,271$ | $9,181$ | $9.216$ | $9.303$ | $9.271$
$11.16$ | $10,114$ | $10,116$ | $10,033$ | $10,113$ | $10,125$ | $10,116$

Figure 16 shows the vibronic state-specific mole fractions
$x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ and respective
Boltzmann representatives $x_{s,e,v}^{\text{B}}/(g_{s,\text{el},e}\cdot
g_{s,\text{vib},e,v})$, obtained from the simulations of the low, medium and
high speed shots, at $x=5\,\text{cm}$, disregarding spontaneous emission. All
markers in Figure 16 lay in their respective lines, evidencing that
vibrational self-equilibrium occurs. However, the lines associated with N2 for
the case of the low speed shot depart slightly from each other. This is a
result of an electronic self-non-equilibrium of N2, which was already
mentioned. The values of $T_{s,\text{vib},n}$ deviate from
$T_{\text{tr}_{\text{h}}}$ by amounts from $-1,557\,\text{K}$ (or
$-19.32\,\%$) to $46\,\text{K}$ (or $0.57\,\%$), $-1,052\,\text{K}$ (or
$-11.36\,\%$) to $42\,\text{K}$ (or $0.45\,\%$), and $-729\,\text{K}$ (or
$-7.21\,\%$) to $13\,\text{K}$ (or $0.13\,\%$), for the cases of the low,
medium and high speed shots, respectively. The lower limits of the deviations
are significant and in the same order of magnitude as the ones obtained when
considering spontaneous emission. The upper limits are, however, much smaller.
The vibrational and translational energy modes should therefore be closer to
an equilibrium between each other when disregarding spontaneous emission than
when regarding.

Figure 16: Vibronic state-specific mole fractions
$x_{s,e,v}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ (markers) and
respective Boltzmann representatives
$x_{s,e,v}^{\text{B}}/(g_{s,\text{el},e}\cdot g_{s,\text{vib},e,v})$ (lines)
as functions of the vibronic energies $\epsilon_{s,\text{el-vib},e,v}$,
obtained from the simulations of the low, medium and high speed shots, at
$x=5\,\text{cm}$, disregarding spontaneous emission processes.

All of the above-mentioned results allow the authors to conclude that the
spontaneous emission processes strongly contribute to the observed thermal
non-equilibrium conditions of the systems at $x=5\,\text{cm}$. It should be
ascertained in the future if the inclusion of absorption and induced emission
processes may reduce this thermal non-equilibrium. The results also showed
that collisional processes, alone, do not ensure equilibrium between the
electronic and translational energy modes of N2 and N in the case of the low
speed shot.

###  3.5 Dependence on the escape factor

It is well known that the emitted VUV radiation is strongly absorbed by the
particles in the medium 27 and, therefore, the assumption of it being
optically thin may not be valid. By considering an escape factor
$\Lambda^{\text{VUV}}<1$ in the simulations, it is possible to take into
account, in a crude way, the effect of auto-absorption (i.e. absorption by the
same emitting source) of the VUV radiation on the resultant radiation
variables. Part of the energy which was once lost due to emitted VUV radiation
escaping from the system may now be used in other processes. More energy
becomes available for excitation of the particles, and as a consequence, more
energy is radiated in the other wavelength intervals. There is even the
possibility of more radiative energy in the VUV wavelength interval escaping
from the system, since though a significant fraction of the photons do not
escape, the number of emitted photons becomes as high as the number excited
particles. Values of $\Lambda^{\text{VUV}}=0.1$ and $0.01$ were tried, and the
respective overall radiation variables are shown in Figures 20, 21, 22, 23,
24, 25, 26 and 27 in A. Figure 20 shows that by decreasing the escape factor,
the peak values of the instrumentally resolved radiative intensities in the
VUV wavelength region decreased as much as a half in the three shots. On the
other hand, the contributions of N increased significantly becoming now much
more dominant than the others (i.e. the ones of $\ch{N2+}(\text{C}-\text{X})$,
$\ch{N2}(\text{a}-\text{X})$ and $\ch{N2}(\text{A}-\text{X})$), as shown in
Figure 21). The obtained spectra show better agreement with the experimental
results. The radiative intensity profiles show steeper rises and falls, and,
in the case of the low speed shot, get closer to the experimental ones. Still,
no plateaus were predicted in the cases of the medium and high speed shots.
For the case of the “Blue” wavelength region, the decrease in the escape
factor made the peak values of the instrumentally resolved radiative
intensities to increase with some significance: as much as 10% in the low
speed shot, and 40% in the medium and high speed shots. The change on the
peaks shapes is not noticeable (see Figure 22). Regarding the shape of the
instrumentally resolved non-equilibrium metric, Figure 23 shows that only the
peak associated with $\ch{N2}(\text{C}-\text{B})$ at $\lambda=337\,\text{nm}$
experienced a meaningful change, increasing its height with the decrease in
the escape factor value. For the case of the “Red” and IR wavelength regions,
the peak values of the instrumentally resolved radiative intensities also
increased: as much as 10% in the low speed shot, and 60% (for the case of the
former) and 80% (for the case of the latter) in the medium and high speed
shots. The shapes of the peaks did not change significantly. The contributions
from N to the non-equilibrium metrics were preferentially augmented (in
particular in the medium and high speed shots), agreeing better with the
experimental results.

These results show strong evidence for the medium being optically thick in the
VUV wavelength region.

###  3.6 Dependence on the dissociation rates of N2

The evolution of the post-shock conditions is highly dependent on the
dissociation rates of N2. Greater dissociation rates would imply a production
of a greater amount of N atoms. And, due to dissociation being an endothermic
process, lesser energy would be available for excitation. However, it is hard
to say how an increase of the dissociation rates could affect the excitation
of the particles to a given energy level since a lesser population of N2 would
mean a lower number of N2 molecules to be excited and the converse for the
case of N. As such, one may not say, in a straightforward manner if the number
of particles in some excited energy level should be lower or higher. To test
the dependence of the numerical results on the dissociation rates of N2, we
decided to consider dissociation rates of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (the ground electronic level of
molecular nitrogen222The ground electronic level of N2 is usually more
populated than the other ones, being, therefore, more important in what
concerns dissociation.) scaled by $0.1$ and $10$, keeping
$\Lambda^{\text{VUV}}=0.01$. The obtained results are shown in Figures 28, 29,
30, 31, 32, 33, 34 and 35 in A. These show that the peak values of the
instrumentally resolved radiative intensities increased with a decrease of the
rates of dissociation with this being more relevant for the case of all
wavelength intervals for the low speed shot and the wavelength intervals
“Blue”, “Red” and IR for the medium and high speed shots. The increase went
from insignificant to as high as $100\,\%$. For the case of an increase of the
dissociation rates, the decrease of the peak values went from as low as
$20\,\%$ to as high as $70\,\%$. The figures for the non-equilibrium metrics
show that both contributions of N and N2 to the radiation variables increased
with the decrease of the rates of dissociation, with the effect on the latter
contribution being much more significant. The energy that was once spent in
dissociation was instead spent on the excitation of N and N2, and since more
N2 and less N particles were obtained, the effect on N2 had a greater
importance. Note, however, that to get a better agreement with the
experimental spectra, the contributions of N should be the ones to get
enhanced and not the contributions of N2. Also, one should point out that in
the case of the low speed shot, the radiative intensity peaks widened with the
decrease of the rates of dissociation, agreeing better with the experimental
profiles for all wavelength intervals except for VUV. For the other shots, the
change on the peaks shape was not significant. The increase of the
dissociation rates made the peaks to narrow in the case of the low speed shot,
agreeing worse with experimental counterparts. Conversely, the increase
widened the peaks in the case of the “Blue” wavelength interval of the medium
and high speed shots, getting closer to the experimental results.

In short, solely decreasing or increasing the dissociation rates of
$\ch{N2}(\text{X})$ do not unequivocally lead to better or worse results.

###  3.7 Dependence on the excitation rates of N

It was mentioned in the section that precedes the above one that by
considering an escape factor $\Lambda^{\text{VUV}}<1$ the contributions of N
to the radiation variables were still underestimated when compared with the
molecular contributions in the “Red” and IR wavelength intervals. It was then
decided to try scaling the excitation rate coefficients of N, keeping
$\Lambda^{\text{VUV}}=0.01$, to enhance them. The crude formulae of Annaloro
et al. 29 and Panesi et al. 30 regarded in the database333 See companion paper
CITE COMPANION PAPER. as models for excitation of N by heavy particle and
electron impact, respectively, may indeed accommodate some degree of
uncertainty. Their multiplication by $10$ and $100$ was tried, and numerical
simulations solely converged for the case of low speed shot. This is an
indication of the rate coefficients being actually high enough in the
conditions of the medium and high speed shots, wherein increasing their values
may lead to a physical incoherence. The results are shown in Figures 36 and 37
in A. With the increase in the rate coefficients for excitation of N, the
peaks values of the instrumentally resolved radiative intensities rose as much
as to the quintuple in the case of the VUV radiation, the double in the case
of the “Blue” and “Red” radiation, and the quadruple in the case of the IR
radiation. However, the values are still one to two orders of magnitude lower
than the experimental ones. Regarding the shape of the profiles, it was found
that the rising and falling parts got steeper, deviating significantly from
the experimental profiles. This result may be justified by a greater rate
excitation rate of N producing a greater rate of radiative emission, and,
therefore, the radiative intensity rises faster and higher. As a higher
excitation requires more energy, which is then lost in the form of radiation,
the system suddenly gets incapable of continually exciting the particles, and
the radiative intensity falls faster. Figure 37 shows that the contribution of
N to the instrumentally resolved non-equilibrium metrics increased in all
wavelength regions, agreeing better with the experimental spectra except in
the “Blue” wavelength region, for which the contributions of the molecular
particles should prevail over the ones of the atomic particles.

In short, the increase of the rate coefficients for excitation of N cannot be
said to unequivocally lead to better results.

###  3.8 A synopsis about the dependence of the results on the different
parameters

Figure 17 shows the peak values of the instrumentally resolved radiative
intensities $\hat{I}^{l}_{\text{peak}}$ (with
$l\in\\{\text{VUV}$,“Blue”,“Red”,$\text{IR}\\}$) obtained with the different
models (i.e. the default model and its transformations) and in the experiment
for the low, medium and high speed shots. And Figures 18 and 19 show the
temperatures $T_{\text{tr}_{\text{h}}}$ and $T_{\text{tr}_{\text{e}}}$ and
mole fractions $x_{s}$ attained at $x=5\,\text{cm}$, respectively, which were
obtained with the different models of this work and of the work of Cruden and
Brandis 27.

By considering escape factors $\Lambda^{\text{VUV}}=0.1$ and
$\Lambda^{\text{VUV}}=0.01$ for the VUV wavelength interval, the peak values
of the instrumentally resolved radiative intensities increase for all
wavelength intervals except VUV, for which it decreases. The attained
temperatures rise just slightly, but the mole fractions of the species do not
change appreciably.

Decreasing the rates of dissociation of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$, keeping
$\Lambda^{\text{VUV}}=0.01$, makes the peak values of the instrumentally
resolved radiative intensities to increase, and increasing the rates of
dissociations makes them to decrease. The impact is, however, not so relevant
for the case of the VUV wavelength interval of the medium and high speed
shots. Figure 18 shows that by decreasing the rates of dissociation, less
energy is spent on endothermic processes (since higher translational
temperatures are attained). Curiously, increasing the rates of dissociation
does not decrease the attained translational temperatures in the cases of the
medium and high speed shots (the translational energy is then redistributed in
a different way). It is important to mention here that the values of all of
the referred temperatures are still much lower (by several thousands of
kelvins) than the ones inferred by Cruden and Brandis. Figure 19 shows a
tendency for an increase of the attained mole fractions of N and N+ and a
decrease for the ones of N2 and N2+ with an increase of the rates of
dissociation. The increase on the mole fraction of N+ and the decrease on the
one of N2+ may be justified by the fact that a stronger dissociation implies
that, in contrast to N2, more N particles become available to be ionised. The
values for the mole fractions obtained by increasing the rate coefficients are
actually the ones of all the tried models (regarding spontaneous emission)
which better agree with the ones of Cruden and Brandis. Still, their results
indicate that the system should have endured stronger dissociation and
ionisation.

By keeping $\Lambda^{\text{VUV}}=0.01$, and increasing the excitation rates of
N (which was only successful for the case of the low speed shot), a meaningful
rise of the peak radiation values occurs, in particular the ones for the IR
and VUV wavelength intervals. Still, these are one to two orders of magnitude
lower than the ones obtained in the experiment (labelled by “Exp.”). The
increase in the radiation variables comes at a cost of the translational
temperatures ($T_{\text{tr}_{\text{h}}}$ decreases as much as $698\,\text{K}$)
and a weaker dissociation and ionisation of N2. The mole fraction of N+ at
$x=5\,\text{cm}$ gets lower due to a faster recombination. The peak value of
the mole fraction of N+ actually increases as the highly excited N particles
are more easily ionised. It can be concluded that more energy is used in
endothermic processes, with excitation of N being the mainly one.

As shown by Figures 18 and 19, the translational temperatures and the mole
fractions of all species except N2+ attained in the numerical simulations
disregarding spontaneous emission (labelled by “No s. emission”) agree well
with the ones derived by Cruden and Brandis. This endorses the hypothesis of
the simulated elements of fluid not receiving the amount of energy that they
should, as only by disregarding spontaneous emission (therefore, retaining a
lot more energy) the thermodynamic conditions get reasonably closer to the
inferred ones.

Figure 17: Peak values of the instrumentally resolved radiative intensities
$\hat{I}^{l}_{\text{peak}}$ obtained with the different models and in the
experiment for the low, medium and high speed shots.

Figure 18: Temperatures $T_{\text{tr}_{\text{h}}}$ and
$T_{\text{tr}_{\text{e}}}$ at $x=5\,\text{cm}$ obtained with the different
models for the low, medium and high speed shots.

Figure 19: Mole fractions $x_{s}$ at $x=5\,\text{cm}$ obtained with the
different models for the low, medium and high speed shots.

###  3.9 Possible causes of the significant underestimation of the
experimental results by the numerical ones

Several studies in the literature reported the implementation of analytical
and numerical models which significantly underestimated the radiation
variables obtained trough shock tube experiments. In fact, these discrepancies
had been so recurrently observed that Cruden et al. 31 even used the term
“long-standing” to describe the problem regarding results obtained from the
Japan Aerospace Exploration Agency’s High-Enthalpy Shock Tunnel (JAXA-HIEST).
Cruden et al. 31 studied the impact that contamination species have on the
numerical results, fitting their concentrations to match some particular JAXA-
HIEST experimental results. They concluded that it may well be possible for
contaminants such as Fe (atomic iron) and CN (cyanogen radical) to cause the
observed discrepancies. In the case of the experiments that produced the
benchmark data analysed in the present work - the EAST test 62 - Cruden and
Brandis 27 found evidence for the presence of the contaminants C (atomic
carbon), H (atomic hydrogen) and CN(cyanogen radical). However, it seems
improbable that this could explain the deviations of several orders of
magnitude for the obtained radiation variables, since the observed spectrum is
almost completely dominated by non-contaminant species (namely N, N2 and N2+).
Furthermore, Brandis et al. 32 refer that in the latest EAST campaigns many
upgrades to the system have been made to reduce the level of contamination.

A more feasible contribution to the discrepancies between the numerical and
experimental results is the non-modelling of the so-called precursor
phenomena: some of the VUV radiation emitted by highly excited particles in
the shock layer is absorbed by others upstream of the shock wave, inducing
their excitation, photoionisation and photodissociation. This changes the
conditions upstream of the shock wave and, inevitably, the conditions
downstream of it: Nomura et al. 33 refer that the shock layer thickness and
the non-equilibrium temperatures are increased, yielding an excess of
radiation. Yamada et al. 34 compared experimental results with numerical ones
obtained through a model that does not account for precursor phenomena, and
found that the measured radiation intensities for N2, N2+ and N started to
increase upstream of the shock wave. The radiation profiles in the shock layer
differed significantly, showing that the precursor phenomena had a great
influence on the thermochemical processes that occurred downstream of the
shock wave.

Another feasible contribution to the discrepancies corresponds to heating of
the driven gas due to downstream plasma subjected to a stronger shock wave,
and radiative energy transfer from the driver gas and the EAST electric arc.
Bogdanoff and Park 6 performed several shock tube experiments, finding the
temperature downstream of the shock wave in the observation point to be three
to four times the one obtained through the Rankine-Hugoniot relations. The
electric arc of the shock tube increases the temperature of the driver gas to
very high values (several tens of thousands of kelvins) which causes it to
radiate a lot of energy, some of it to the driven gas upstream and downstream
of the shock wave. Also, it is known that the shock wave decelerates through
the shock tube, heating more the part of the driven gas near the diaphragm
than the part of the driven gas near the observation point444In a
correspondence, Dr. Brett Cruden stated that this contribution was suspected
to be the most important one.. A transfer of energy from the former to the
later may then also occur.

##  4 Conclusions

The database of kinetic processes described in  CITE COMPANION PAPER was
employed in Euler one-dimensional simulations of the shots $19$, $20$ and $40$
of the EAST’s $62^{\text{nd}}$ campaign. It was found that the model
underpredicted the experimental radiation variables by one to two orders of
magnitude. Sensitivity tests performed on different parameters of the
simulations were unsuccessful in getting a reasonable agreement. The shape of
the radiative intensities profiles of the low speed shot was correctly
predicted, but not the ones of the medium and high speed shots which revealed
non-null plateaus proceeding or coalescing with peaks. These plateaus were not
predicted at all. The analysis of Cruden and Brandis on the spectra obtained
in the shot $19$ of the EAST’s $62^{\text{nd}}$ campaign showed that the high
experimental values of the radiation variables were attained with a lower cost
of translation temperature. There is a strong evidence for the underestimation
of the radiation variables observed in this work to result from several
contributions already pointed out by other researchers in the literature: the
non-modelling of the precursor phenomena and/or the non-modelling of shock
tube-related phenomena such as heat transfer by radiation between the driver
gas (as well as the driver arc) and the test gas, and by conduction due to
downstream plasma subjected to a stronger shock wave.

One of the next steps that should be taken in the future is to test other
kinetic databases reported in the literature and to compare their results with
the ones of this work, such that particular qualities which were actually
improved or, on the contrary, worsened may be identified. To quantify the
possible divergences between the herein implemented vibronic-specific state-
to-state model and the simpler models such as the multi-temperature ones,
these latter should also be tried.

To ascertain the effect of heat transfer within the test gas subjected to
strong shock waves generated by shock tubes, higher fidelity one-dimensional
simulations should be performed. The crude concept of an escape factor should
be disregarded, and an equation of radiative transfer should be solved
instead. Furthermore, the transport phenomena should be introduced in the
balance equations.

The database of radiative processes should be extended, accommodating
absorption, induced emission, photodissociation, photoassociation,
photoionisation, photorecombination and bremsstrahlung, beyond spontaneous
emission processes. All these processes should be treated as rovibronic
instead of simply vibronic since the numerical spectra associated with the
molecular contributions obtained in this work showed pointier profiles when
compared to the ones obtained in the experiments.

An extrapolation of the Einstein coefficients for spontaneous emission to the
energy levels whose data are not available should be performed to avoid
getting unphysical levels’ population distributions.

Finally, as more ambitious goals, the impact on the radiation variables of the
precursor phenomena, the absorption of radiation emitted by the driver gas and
the EAST electric arc, and the conduction of heat due to downstream plasma
being subjected to a stronger shock wave, should be studied.

The authors would like to acknowledge Bruno Lopez from the University of
Illinois Urbana–Champaign who provided technical support for the SPARK code,
Daniel Potter from CSIRO in Australia for insights on line-broadening
mechanisms, and Brett A. Cruden and Aaron Brandis from the NASA Ames Research
Center, who shared their obtained data regarding the EAST’s $62^{\text{nd}}$
campaign.

This work has been partially supported by the Portuguese Science Foundation
FCT, under Projects UIDB/50010/2020 and UIDP/50010/2020.

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##  Appendix A Appendix A: Sensitivity tests on different parameters of the
simulations

This appendix presents the numerical instrumentally resolved radiative
intensities $\hat{I}^{l}(x)$ and non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},l}(\lambda)$, with
$l\in\\{\text{VUV},\text{``Blue''},\text{``Red''},\text{IR}\\}$, obtained from
the numerical simulations of the shots 40, 19, and 20 of the $62^{\text{nd}}$
EAST campaign 2, considering changes of scale of the VUV escape factor
$\Lambda^{\text{VUV}}$, the dissociation rates of $\ch{N2}(\text{X})$, and the
excitation rates of N.

###  A.1 Dependence on the escape factor

Figure 20: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{VUV}}(x)$, obtained with $\Lambda^{\text{VUV}}=1$ (solid black
lines), $\Lambda^{\text{VUV}}=0.1$ (solid blue lines), and
$\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well as the respective
experimental instrumentally resolved radiative intensities
$\hat{I}^{\text{VUV},\text{exp}}(x)$ (dotted black lines).

Figure 21: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{VUV}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=1$ (solid black lines), $\Lambda^{\text{VUV}}=0.1$
(solid blue lines), and $\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well
as the respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{VUV},\text{exp}}(\lambda)$ (dotted black
lines).

Figure 22: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{Blue}}(x)$, obtained with $\Lambda^{\text{VUV}}=1$ (solid
black lines), $\Lambda^{\text{VUV}}=0.1$ (solid blue lines), and
$\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well as the respective
experimental instrumentally resolved radiative intensities
$\hat{I}^{\text{Blue},\text{exp}}(x)$ (dotted black lines).

Figure 23: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Blue}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=1$ (solid black lines), $\Lambda^{\text{VUV}}=0.1$
(solid blue lines), and $\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well
as the respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Blue},\text{exp}}(\lambda)$ (dotted
black lines).

Figure 24: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{Red}}(x)$, obtained with $\Lambda^{\text{VUV}}=1$ (solid black
lines), $\Lambda^{\text{VUV}}=0.1$ (solid blue lines), and
$\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well as the respective
experimental instrumentally resolved radiative intensities
$\hat{I}^{\text{Red},\text{exp}}(x)$ (dotted black lines).

Figure 25: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Red}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=1$ (solid black lines), $\Lambda^{\text{VUV}}=0.1$
(solid blue lines), and $\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well
as the respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Red},\text{exp}}(\lambda)$ (dotted black
lines).

Figure 26: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{IR}}(x)$, obtained with $\Lambda^{\text{VUV}}=1$ (solid black
lines), $\Lambda^{\text{VUV}}=0.1$ (solid blue lines), and
$\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well as the respective
experimental instrumentally resolved radiative intensities
$\hat{I}^{\text{IR},\text{exp}}(x)$ (dotted black lines).

Figure 27: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{IR}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=1$ (solid black lines), $\Lambda^{\text{VUV}}=0.1$
(solid blue lines), and $\Lambda^{\text{VUV}}=0.01$ (solid red lines), as well
as the respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{IR},\text{exp}}(\lambda)$ (dotted black
lines).

###  A.2 Dependence on the dissociation rates of N2

Figure 28: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{VUV}}(x)$, obtained with $\Lambda^{\text{VUV}}=0.01$, and
unscaled dissociation of $\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid
black lines), and scaled by $0.1$ (solid green lines), and by $10$ (solid blue
lines), as well as the respective experimental instrumentally resolved
radiative intensities $\hat{I}^{\text{VUV},\text{exp}}(x)$ (dotted black
lines).

Figure 29: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{VUV}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=0.01$, and unscaled dissociation of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid black lines), and scaled
by $0.1$ (solid green lines), and by $10$ (solid blue lines), as well as the
respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{VUV},\text{exp}}(\lambda)$ (dotted black
lines).

Figure 30: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{Blue}}(x)$, obtained with $\Lambda^{\text{VUV}}=0.01$, and
unscaled dissociation of $\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid
black lines), and scaled by $0.1$ (solid green lines), and by $10$ (solid blue
lines), as well as the respective experimental instrumentally resolved
radiative intensities $\hat{I}^{\text{Blue},\text{exp}}(x)$ (dotted black
lines).

Figure 31: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Blue}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=0.01$, and unscaled dissociation of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid black lines), and scaled
by $0.1$ (solid green lines), and by $10$ (solid blue lines), as well as the
respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Blue},\text{exp}}(\lambda)$ (dotted
black lines).

Figure 32: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{Red}}(x)$, obtained with $\Lambda^{\text{VUV}}=0.01$, and
unscaled dissociation of $\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid
black lines), and scaled by $0.1$ (solid green lines), and by $10$ (solid blue
lines), as well as the respective experimental instrumentally resolved
radiative intensities $\hat{I}^{\text{Red},\text{exp}}(x)$ (dotted black
lines).

Figure 33: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Red}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=0.01$, and unscaled dissociation of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid black lines), and scaled
by $0.1$ (solid green lines), and by $10$ (solid blue lines), as well as the
respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{Red},\text{exp}}(\lambda)$ (dotted black
lines).

Figure 34: Numerical instrumentally resolved radiative intensities
$\hat{I}^{\text{IR}}(x)$, obtained with $\Lambda^{\text{VUV}}=0.01$, and
unscaled dissociation of $\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid
black lines), and scaled by $0.1$ (solid green lines), and by $10$ (solid blue
lines), as well as the respective experimental instrumentally resolved
radiative intensities $\hat{I}^{\text{IR},\text{exp}}(x)$ (dotted black
lines).

Figure 35: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{IR}}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=0.01$, and unscaled dissociation of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ (solid black lines), and scaled
by $0.1$ (solid green lines), and by $10$ (solid blue lines), as well as the
respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},\text{IR},\text{exp}}(\lambda)$ (dotted black
lines).

###  A.3 Dependence on the excitation rates of N

Figure 36: Numerical instrumentally resolved radiative intensities
$\hat{I}^{l}(x)$ obtained with $\Lambda^{\text{VUV}}=0.01$, and unscaled
excitation of N (solid black lines), and scaled by $10$ (solid blue lines),
and by $100$ (solid red lines), as well as the respective experimental
instrumentally resolved radiative intensities $\hat{I}^{l,\text{exp}}(x)$
(dotted black lines), for the case of the low speed shot.

(a)

(b)

(c)

(d)

Figure 37: Numerical instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},l}(\lambda)$, obtained with
$\Lambda^{\text{VUV}}=0.01$, and unscaled excitation of N (solid black lines),
and scaled by $10$ (solid blue lines), and by $100$ (solid red lines), as well
as the respective experimental instrumentally resolved non-equilibrium metrics
$\hat{I}_{\lambda}^{\,\text{ne},l,\text{exp}}(\lambda)$ (dotted black lines),
for the case of the low speed shot.