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Thermochemical effects on hypersonic shock waves interacting with weak turbulence

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Abstract

The interaction between a weakly turbulent free stream and a hypersonic shock wave is investigated theoretically by using linear interaction analysis (LIA). The formulation is developed in the limit in which the thickness of the thermochemical nonequilibrium region downstream of the shock, where relaxation toward vibrational and chemical equilibrium occurs, is assumed to be much smaller than the characteristic size of the shock wrinkles caused by turbulence. Modified Rankine–Hugoniot jump conditions that account for dissociation and vibrational excitation are derived and employed in a Fourier analysis of a shock interacting with three-dimensional isotropic vortical disturbances. This provides the modal structure of the post-shock gas arising from the interaction, along with integral formulas for the amplification of enstrophy, concentration variance, turbulent kinetic energy (TKE), and turbulence intensity across the shock. In addition to confirming known endothermic effects of dissociation and vibrational excitation in decreasing the mean post-shock temperature and velocity, these LIA results indicate that the enstrophy, anisotropy, intensity, and TKE of the fluctuations are much more amplified through the shock than in the thermochemically frozen case. In addition, the turbulent Reynolds number is amplified across the shock at hypersonic Mach numbers in the presence of dissociation and vibrational excitation, as opposed to the attenuation observed in the thermochemically frozen case. These results suggest that turbulence may persist and get augmented across hypersonic shock waves despite the high post-shock temperatures.
Thermochemical effects on hypersonic shock
waves interacting with weak turbulence
Cite as: Phys. Fluids 33, 086111 (2021); doi: 10.1063/5.0059948
Submitted: 14 June 2021 .Accepted: 28 July 2021 .
Published Online: 20 August 2021
C. Huete,
1,a)
A. Cuadra,
1
M. Vera,
1
and J. Urzay
2
AFFILIATIONS
1
Grupo de Mec
anica de Fluidos, Universidad Carlos III, Legan
es, Madrid 28911, Spain
2
Center for Turbulence Research, Stanford University, Stanford, California 94305, USA
a)
Author to whom correspondence should be addressed: chuete@ing.uc3m.es
ABSTRACT
The interaction between a weakly turbulent free stream and a hypersonic shock wave is investigated theoretically by using linear interaction
analysis (LIA). The formulation is developed in the limit in which the thickness of the thermochemical nonequilibrium region downstream
of the shock, where relaxation toward vibrational and chemical equilibrium occurs, is assumed to be much smaller than the characteristic
size of the shock wrinkles caused by turbulence. Modified Rankine–Hugoniot jump conditions that account for dissociation and vibrational
excitation are derived and employed in a Fourier analysis of a shock interacting with three-dimensional isotropic vortical disturbances. This
provides the modal structure of the post-shock gas arising from the interaction, along with integral formulas for the amplification of enstro-
phy, concentration variance, turbulent kinetic energy (TKE), and turbulence intensity across the shock. In addition to confirming known
endothermic effects of dissociation and vibrational excitation in decreasing the mean post-shock temperature and velocity, these LIA results
indicate that the enstrophy, anisotropy, intensity, and TKE of the fluctuations are much more amplified through the shock than in the ther-
mochemically frozen case. In addition, the turbulent Reynolds number is amplified across the shock at hypersonic Mach numbers in the
presence of dissociation and vibrational excitation, as opposed to the attenuation observed in the thermochemically frozen case. These results
suggest that turbulence may persist and get augmented across hypersonic shock waves despite the high post-shock temperatures.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0059948
I. INTRODUCTION
Strong shock waves participate in a number of problems in phys-
ics, including the dynamics of high-energy interstellar medium,
1–4
the explosions of giant stars,
5–8
the fusion of matter in inertial-
confinement devices,
9–11
and the ignition of combustible mixtures by
lasers.
12,13
In addition to those, an important contemporary problem
of relevance for aeronautical and astronautical engineering is the aero-
thermodynamics of hypersonic flight.
14,15
In hypersonics, similarly to
the aforementioned problems, the intense compression of the gas
through the shock waves generated by the fuselage leads to high tem-
peratures that can activate complex thermochemical phenomena.
16
In
particular, at high Mach numbers of up to approximately 25 in the ter-
restrial atmosphere, corresponding to sub-ionizing, sub-orbital stagna-
tion enthalpies of up to approximately 15–30 MJ/kg depending on
altitude, vibrational excitation, and air dissociation are the dominant
thermochemical phenomena typically observed in the gas downstream
of shock waves around hypersonic flight systems.
Turbulence can also play an important role at the high Mach
numbers mentioned above, particularly in low-altitude hypersonic
flight because of the correspondingly larger Reynolds numbers of the
airflow around the fuselage.
17–19
However, the way in which turbu-
lence influences the thermomechanical loads and the thermochemistry
around hypersonic flight systems remains largely unknown. To com-
pound this problem, experiments in the area of hypersonic turbulence
are curtailed by the exceedingly large flow powers required to move
gases at sufficiently high Mach and Reynolds numbers in order to
observe shock waves simultaneously with turbulence and thermo-
chemistry. In addition, the airflow in most ground facilities is poisoned
with weak free-stream turbulence that interacts with the shock waves
enveloping the test article. The fluctuations in the post-shock
gases induced by this interaction oftentimes lead to artificial transition
to turbulence in hypersonic boundary layers in wind tunnel
experiments.
20
Most early work on the interaction of shock waves with turbu-
lence has been limited to calorically perfect gases in boundary
layers
21–30
and isotropic free streams.
31–35
Large-scale numerical simu-
lations, including Direct Numerical Simulations (DNS),
36–50
Large
Eddy Simulations (LES),
51–53
and Reynolds-Averaged Navier-Stokes
Phys. Fluids 33, 086111 (2021); doi: 10.1063/5.0059948 33, 086111-1
Published under an exclusive license by AIP Publishing
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Simulations (RANS),
54,55
have been the pacing item for those investi-
gations. Nonetheless, the rapid progress in large-scale numerical simu-
lations during the last decades has not abated the fundamental role
that theoretical analyses have played in understanding shock/turbu-
lence interactions by providing closed-form solutions. In problems
dealing with shock waves propagating in turbulent free streams, as in
the problem treated in the present study, the most successful theoreti-
cal approach has been the linear interaction analysis (LIA) pioneered
by Ribner.
56–58
Under the assumption that turbulence is comprised of small lin-
ear fluctuations that can be separated using Kovaznay’s decomposition
into vortical, entropic, and acoustic modes,
59
LIA describes their two-
way coupled interaction with the shock by using linearized
RankineHugoniot jump conditions coupled with the linearized Euler
equations in the post-shock gas. The resulting formalism describes the
wrinkles induced by turbulence on the shock and the corresponding
Kovaznay’s compressible turbulence modes radiated by the interaction
toward the downstream gas.
Despite its simplicity and limitations, LIA has not only provided a
valuable insight into the underlying physical processes of shock/turbu-
lence interactions, but has also worked sufficiently well for predicting
the amplification of the turbulent kinetic energy (TKE), that is, com-
monly used for bench-marking numerical simulations.
38–40
However,
there exist known discrepancies between LIA and numerical simulations
in the way that TKE is distributed among the diagonal components of
the Reynolds stress tensor. For instance, LIA yields a smaller (larger)
amplification of TKE associated with streamwise (transverse) velocity
fluctuations relative to that observed in numerical simulations. These
discrepancies are typically attributed to the fact that LIA treats the shock
as a discontinuity, in that DNS results are observed to converge to those
obtained by LIA when the ratio of the numerical shock thickness to the
Kolmogorov length scale becomes sufficiently small.
41,43,45
In this study, an extension that incorporates thermochemical
effects of vibrational excitation and gas dissociation is made to the
standard LIA previously applied to calorically perfect gases.
56–58,60
As
in the standard LIA, the following conditions must be satisfied: (a) the
root mean square (rms) of the velocity fluctuations uneeds to be
much smaller than the speed of sound in both pre-shock and post-
shock gases; (b) the amplitude of the streamwise displacement of the
distorted shock from its mean position n
s
needs to be much smaller
than the upstream integral size of the turbulence ; and (c) the eddy
turnover time ‘=uneeds to be much smaller than the molecular diffu-
sion time 2= based on the kinematic viscosity , or equivalently, the
turbulent Reynolds number Re¼u‘= needs to be large.
In addition to the conditions [(a)–(c)] stated above, the incorpo-
ration of thermochemical effects requires that the characteristic size of
theshockwrinkles,whichisofthesameorderas,needstobemuch
larger than the thickness Tof the thermochemical nonequilibrium
region behind the shock, as depicted in Fig. 1. For instance, the value
of Tbehind a Mach-14 normal shock at a pressure equivalent to
45 km of altitude is approximately 1 cm (see page 503 in Ref. 61). In
this thermochemical nonequilibrium region, the gas relaxes toward
vibrational and chemical equilibrium in an intertwined manner, in
that the vibrational energy of the molecules and their dissociation
probability are coupled.
16,62
The value of Tis approximately given by
the mean post-shock velocity multiplied by the sum of the characteris-
tic time scales of dissociation and vibrational relaxation. Since both of
these characteristic time scales depend inversely on pressure and expo-
nentially on the inverse of the temperature, the veracity of the approxi-
mation T=‘ 1 in practical hypersonic systems is expected to
improve as the flight Mach number increases and the altitude
decreases.
The LIA results provided in this study yield integral formulas for
the amplification of the enstrophy, composition variance, and TKE as
a function of the post-shock Mach number, the density ratio, and the
normalized inverse of the slope of the Hugoniot curve. The latter
undergoes a change in sign at high Mach numbers due to the thermo-
chemical effects. As a result, at Mach numbers larger than approxi-
mately 13 in the conditions tested here, a local decrement (increment)
in post-shock pressure—due, for instance, to shock wrinkling—
engenders an increment (decrement) in post-shock density. This pecu-
liar structure of the Hugoniot curve at hypersonic Mach numbers is
found to strongly amplify turbulence in the post-shock gas, where
most of the TKE is observed to be contained in transverse velocity
fluctuations of the vortical mode. For instance, the present LIA results
in a maximum TKE amplification factor of approximately 2.9, whereas
this value drops to 1.7 when the gas is assumed to be thermochemi-
cally frozen (i.e., diatomic calorically perfect).
The remainder of this paper is structured as follows. The
RankineHugoniot jump conditions across the shock are derived in
Sec. II accounting for dissociation and vibrational excitation in the
post-shock gas. A linearized formulation of the problem is presented
in Sec. III for the interaction of a normal shock with monochromatic
vorticity disturbances. A Fourier analysis is carried out in Sec. IV to
address the interaction of a normal shock with weak isotropic turbu-
lence composed of multiple and linearly superposed vorticity modes.
Finally, conclusions are given in Sec. V.
II. RANKINE–HUGONIOT JUMP CONDITIONS WITH
VIBRATIONAL EXCITATION AND GAS DISSOCIATION
We consider first the problem of an undisturbed, normal shock
wave in a cold, inviscid, irrotational, single-component gas consisting
FIG. 1. Sketch of the model problem: a normal shock wave interacts with a hyper-
sonic free stream of weak isotropic turbulence (velocities are shown in the shock
reference frame).
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 086111 (2021); doi: 10.1063/5.0059948 33, 086111-2
Published under an exclusive license by AIP Publishing
of symmetric diatomic molecules. The pre-shock density, pressure,
temperature, specific internal energy, and flow velocity in the reference
frame of the shock are denoted, respectively, as q
1
,P
1
,T
1
,e
1,
and u
1
.
The corresponding flow variables in the post-shock gas are denoted as
q
2
,P
2
,T
2
,e
2,
and u
2
.
A. Conservation equations across the shock
In the reference frame attached to the shock front, the conserva-
tion equations of mass, momentum, and enthalpy across the shock are
q1u1¼q2u2;(1a)
P1þq1u2
1¼P2þq2u2
2;(1b)
e1þP1=q1þu2
1=2¼e2þP2=q2þu2
2=2þqd;(1c)
respectively. In this formulation, the symbol q
d
denotes a positive
quantity that represents the net change of specific chemical enthalpy
caused by the gas dissociation reaction
A2AþA (2)
with A2being a generic molecular species and A its dissociated atomic
counterpart. In particular, q
d
can be expressed as
qd¼aRg;A2Hd;(3)
where Rg;A2is the gas constant based on the molecular weight of A2,
and H
d
is the characteristic dissociation temperature. In addition, the
variable ais the degree of dissociation defined as the ratio of the mass
of dissociated A atoms to the total mass of the gas, or, equivalently, the
mass fraction of A atoms.
Equations (1a)–(1c) are supplemented with the ideal-gas equa-
tions of state in the pre-shock gas
P1=q1¼Rg;A2T1(4)
and in the post-shock gas
P2=q2¼ð1þaÞRg;A2T2:(5)
In addition, the specific internal energy in the pre-shock gas e
1
is given
by the translational and rotational components
e1¼ð5=2ÞRg;A2T1;(6)
whereas in the post-shock gas e
2
requires consideration of transla-
tional, rotational, and vibrational degrees of freedom along with mix-
ing between molecular and atomic species, which gives
e2¼Rg;A2T23aþð1aÞ5
2þHv=T2
eHv=T21

;(7)
where H
v
is the characteristic vibrational temperature. The first term
inside the square brackets in (7), proportional to the dissociation
degree a, corresponds to the translational contribution of the mon-
atomic species. The second term, proportional to the factor 1 a,
includes the translational, rotational, and vibrational contributions of
the molecular species, where it has been assumed that the rotational
degrees of freedom are fully activated and the molecules vibrate as har-
monic oscillators.
The formulation is closed with the chemical-equilibrium condi-
tion downstream of the shock, namely,
63
a2
1a¼GmHr
pmkB
h2

3=2ffiffiffiffiffi
T2
p
q2
eHd
T21eHv
T2

;(8)
where H
r
is the characteristic rotational temperature, mis the atomic
mass of A, k
B
is the Boltzmann’s constant, his the reduced Planck’s
constant, and G¼ðQa
elÞ2=Qaa
el is a ratio of electronic partition func-
tions of A atoms (Qa
el) and A2molecules (Qaa
el ). Upon neglecting the
variations of the specific internal energy with temperature due to elec-
tronic excitation, the electronic partition functions in Gcan be approx-
imated as the ground-state degeneracy factors. Typical values of H
r
,
H
v
,H
d
,G, and mare provided in Table I for a wide range of molecular
gases.
B. Dimensionless formulation
A dimensionless formulation of the problem can be written by
introducing the dimensionless parameters
B¼GmHrT1=2
1
q1
pmkB
h2

3=2
;bd¼Hd
T1
;bv¼Hv
T1
(9)
along with the pressure, temperature, and density jumps
P2=P1;T2=T1;q2=q1(10)
across the shock. In the expressions below, the solution for a vibration-
ally and chemically frozen gas (i.e., a calorically perfect diatomic gas) is
recovered by taking the limits bv!1and bd!1(or a!0).
Using these definitions, the dimensionless Rayleigh line
1þ7
5M2
111
R

;(11)
which relates Pand R, is obtained by combining the mass and
momentum conservation equations (1a) and (1b).In(11), the symbol
M1denotes the pre-shock Mach number defined as
M1¼u1=c1;(12)
where c1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð7=5ÞRg;A2T1
pis the speed of sound of the pre-shock
gas. Regardless of the value of M1, the Rayleigh line always emanates
from the pre-shock state, 1and1, as a straight line with
negative slope in the fR1;Pgplane.
In contrast, since the post-shock gas is calorically imperfect, its
Mach number
M2¼u2
c2¼M1
R
c1
c2
(13)
requires a more elaborate calculation of the speed of sound
TABLE I. Rotational (H
r
), vibrational (H
v
), and dissociation (H
d
) characteristic tem-
peratures, along with the factor Gand the atomic mass mof relevant molecular
gases.
H
2
O
2
N
2
F
2
I
2
Cl
2
H
r
(K) 87.53 2.08 2.87 1.27 0.0538 0.0346
H
v
(K) 6338 2270 3390 1320 308 805
H
d
(K) 51 973 59 500 113 000 18 633 17 897 28 770
G2
2
/1 5
2
/3 4
2
/1 4
2
/1 4
2
/1 4
2
/1
m(kg) 1026 0.167 35 2.6567 2.3259 3.1548 21.072 5.8871
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 086111 (2021); doi: 10.1063/5.0059948 33, 086111-3
Published under an exclusive license by AIP Publishing
c2
2¼P2
q2
2
@P2
@T2q2
@ðe2þqdÞ
@T2q2
@P2
@T2q2
@ðe2þqdÞ
@q2T2
@ðe2þqdÞ
@T2q2
þ@P2
@q2T2
:(14)
Upon substituting (5) and (7) into (14),theexpression
c2
2
c2
1¼5T
71þaþaRþ1þaþaT
ðÞ
2ð1þaÞaR12
evib þ2bd=T
ðÞ
5þaþ2ð1aÞ
e2
vibebv=TþaT12
evib þ2bd=T
ðÞ
(15)
is obtained, where
evib ¼bv=T
ebv=T1(16)
is the dimensionless component of the specific internal energy corre-
sponding to vibrational excitation in equilibrium. In addition, the coef-
ficients a
R
and a
T
in (15) are given by
aR¼R@a
@RT¼að1aÞ
2a;(17)
aT¼T @a
@TR¼aR
1
2þbd
T
11þbv
bd

ebv=T
1ebv=T
2
43
5:(18)
Equation (15), along with definitions (16)–(18), determines the post-
shock Mach number (13).
The equations of state (4) and (5) can be combined into a single
equation as
P¼ð1þaÞRT :(19)
Upon substituting (4)–(7) into the conservation equations (1a)–(1c)
and using the normalizations (9) and (10),therelation
6R
12abd2ð1aÞbv=ebv=T1
ðÞ
2ðaþ3ÞRð1þaÞ(20)
is obtained between a,R,andT. Finally, the problem is closed by
rewriting the chemical-equilibrium condition (8) in dimensionless
form using (9) and (10) as
a2
1a¼Bebd=Tffiffiffiffi
T
p
R1ebv=T
ðÞ
;(21)
which provides an additional relation between a,R,andT. In particu-
lar, given the dimensionless parameters b
v
,b
d,
and B,thecombination
of (19)–(21) provides the Hugoniot curve P¼PðR
1Þ, which in the
present case is a laborious implicit function, that is, evaluated numeri-
cally and is shown in Fig. 2. As a result, given a pre-shock Mach num-
ber M1, the post-shock state is completely determined by the
intersection of the Hugoniot curve and the Rayleigh line (11).
C. The turning point in the Hugoniot curve at
hypersonic Mach numbers
It is worth discussing some peculiarities of the Hugoniot curve,
that is, obtained by including dissociation and vibrational excitation
in the post-shock gas, since they are of some relevance for the shock/
turbulence interaction problem studied in Secs. III and IV.
The main panel in Fig. 2 shows Hugoniot curves in light colors
for H
2
,O
2
,N
2
,andF
2
using the simple theory provided above particu-
larized for the parameters B,b
v
,andb
d
listed in Table II.Asshownin
Fig. 2,thecurvesforO
2
and N
2
compare well with the more complex
numerical calculations obtained with NASA’s chemical equilibrium
with applications (CEA) code.
64
The latter incorporates variations of
the specific heat with temperature due to both vibrational and elec-
tronic excitation through the NASA polynomials.
65
To narrow down the exposition, the main panel in Fig. 2 also
shows a Hugoniot curve colored by the degree of dissociation and
obtained using the representative values B¼106;bv¼10, and
bd¼100. This is a particular choice of values that nonetheless
approximately captures the order of magnitude of these parameters
observed among the different gases listed in Table II (with exception
FIG. 2. Hugoniot curves for different molecular gases at pre-shock temperature
T1¼300 K and pressure P1¼1 atm [gray lines: present formulation; symbols:
numerical results obtained with NASA’s Chemical Equilibrium with Applications
(CEA) code
64
excluding ionization], along with the Hugoniot curve of a gas with
B¼106;bv¼10, and bd¼100 (line colored by the degree of dissociation). The
latter is compared in the inset with the Hugoniot curves of a calorically perfect mon-
atomic gas (gray line corresponding to c¼5=3) and a calorically perfect diatomic
gas (gray line corresponding to c¼7=5).
TABLE II. Dimensionless parameters B,b
v
, and b
d
for relevant molecular gases at
pre-shock temperature T1¼300 K and pressure P1¼1 atm.
H
2
O
2
N
2
F
2
I
2
Cl
2
B1062.0668 6.472 14.0452 9.818 7.1796 0.6818
bv1012.1127 0.7567 1.13 0.44 0.1027 0.2683
bd1021.7324 1.9833 3.7667 0.6211 0.5966 0.959
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 086111 (2021); doi: 10.1063/5.0059948 33, 086111-4
Published under an exclusive license by AIP Publishing
of the much larger value of Bobserved for N
2
, which translates into
much higher dimensionless post-shock temperatures being required to
attain significant dissociation of N
2
).
The inset in Fig. 2 shows that the Hugoniot curve starts departing
significantly from that of a calorically perfect diatomic gas [corre-
sponding to an adiabatic coefficient c¼7=5 and a maximum density
ratio R¼ðcþ1Þ=ðc1Þ¼6] at a rather modest degree of dissocia-
tion a1% attained at M15. Despite the smallness of this cross-
over value of a, large changes in chemical enthalpy occur because of
the large bond-dissociation specific energy of most relevant species
(e.g., approximately 15 MJ/kg for O
2
). As a result, a1% renders
abd¼Oð1Þin (20), which represents a balance between the heat
absorbed by dissociation q
d
and the pre-shock internal energy e
1
in the
conservation equation (1c).Asais further increased, q
d
becomes of
thesameorderase
2
, and the departure from calorically perfect behav-
ior becomes increasingly more pronounced.
As abecomes increasingly closer to unity, which requires the
kinetic energy of the pre-shock gas to be increasingly larger than q
d
(or
equivalently, it requires the pre-shock Mach number M1to be
increasingly larger than ffiffiffiffiffi
bd
p), the slope of the Hugoniot curve under-
goes a change in sign and turns inward toward larger specific volumes.
For the parameters investigated in Fig. 2, the turning point occurs at
a0:7, where T’9 (corresponding to 2700 K when T1¼300 K),
M113, and R’12, the latter being almost double (triple) the
density ratio of a calorically perfect diatomic (mono-atomic) gas.
There, the inverse of the slope of the Hugoniot curve normalized with
the slope of the Rayleigh line
C¼ P2P1
1=q11=q2

dð1=q2Þ
dP2¼7
5M2
1
R2
@P
@R

1
(22)
attains a zero value. The role of Cin the description of the shock/tur-
bulence interaction problem will be addressed in Secs. III and IV.
As shown in Fig. 3,thevalueofCbecomes negative along the
upper branch of the Hugoniot curve beyond the turning point C¼0.
Along that branch, an increment (decrement) in post-shock pressure
induces a decrement (increment) in post-shock density. For the
parameters tested here, the value Cin the upper branch of the
Hugoniot curve is always larger than the critical values for the onset of
(a) shock instabilities associated with multi-wave
66,67
and multi-
valued
68
solutions, and (b) D’yakov–Kontorovich pseudo-instabilities
associated with the spontaneous emission of sound.
8,69
Similar charac-
teristics of the Hugoniot curve have been observed elsewhere for
shocks subjected to endothermicity.
70–73
D. Limit behavior in the post-shock gas
Typical distributions of the density ratio R, the post-shock Mach
number M2, and the pre-shock Mach number M1are provided in
Fig. 4 as a function of the temperature ratio T. The curves also show
the limit behavior for a!0andbv!1(corresponding to a calori-
cally perfect diatomic gas at low temperatures), and for a!1(corre-
sponding to a fully dissociated gas at high temperatures). Some insight
into these limits is provided below.
In Fig. 4(a), the low-temperature limit of the density ratio corre-
sponds to the standard RankineHugoniot jump condition for a calo-
rically perfect diatomic gas
R311
T

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þT
9T1
ðÞ
2
s
2
43
5;(23)
which can be derived by taking the limits a!0andbv!1in (20).
In this low-temperature limit, the normalized slope of the Hugoniot
curve becomes CM
2
1,asindicatedinFig. 3.
In the opposite limit, when the post-shock gas is hot and almost
fully dissociated, a!1, the density jump and the normalized slope of
the Hugoniot curve become
Rbdþ4T3þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðbdþ4T3Þ2þ2T
q2T(24)
and
C7M2
1R4
ðÞ
2
5R2ð8bd23Þ;(25)
respectively, with bd>23=8 in the conditions tested here. At very
high Mach numbers M1ffiffiffiffi
bd
p,whenbd=T1, Eq. (24) simpli-
fies to R4 in the first approximation, whereas (25) yields very small
and negative values of C. Remarkably, unlike R;M1,andM2,the
normalized inverse of the slope Cis not bounded by its asymptotic
limits at low and high Mach numbers. The relevance of this property
for the problem of shock/turbulence interaction will be discussed in
Secs. III and IV.
The results mentioned above for a!1 indicate that the post-
shock gas increasingly resembles a monatomic calorically perfect gas
FIG. 3. Normalized inverse of the slope of the Hugoniot curve Cas a function of
the temperature jump across the shock Tfor B¼106;bv¼10, and bd¼100
(line colored by the degree of dissociation). Dashed lines represent asymptotic
limits for a calorically perfect diatomic gas (bv!1and a!0), and for a highly
dissociated gas (a!1).
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(corresponding to an adiabatic coefficient c¼5=3) at infinite Mach
numbers, an effect that can also be visualized in Fig. 2 as the Hugoniot
curve asymptotes the abscissa R11=4. However, this limit is of lit-
tle practical relevance because it would require such exceedingly high
temperatures that additional effects like electronic excitation, radia-
tion, and ionization would have to be included in the formulation,
thereby invalidating these considerations.
III. THE INTERACTION OF A HYPERSONIC SHOCK
WAVE WITH AN INCIDENT MONOCHROMATIC
VORTICITY WAVE
For small-amplitude velocity fluctuations and vanishing turbu-
lent Mach numbers, the free-stream turbulence in the pre-shock gas
can be represented as a linear superposition of Kovaznay’s three-
dimensional vorticity modes, which are solutions of the incompress-
ible Euler equations.
59,74
This section addresses the interaction of the
shock with a single one of those vorticity modes.
A. Laboratory, shock, and post-shock reference frames
Three reference frames are used in the analysis. Whereas the
spanwise and transverse axes of all the frames coincide, the streamwise
axis differs depending on whether the frames are attached to the labo-
ratory (x), the mean shock front (x
s
), or the mean absolute post-shock
gas motion (x
c
).
In the laboratory reference frame, the streamwise coordinate is
denoted by xand is attached to the bulk of the pre-shock gas, which is
at rest on average. In contrast, in the shock reference frame, which cor-
responds to the one visualized in Fig. 1, the streamwise coordinate x
s
moves at the mean shock velocity hu1iand is therefore defined by the
relation xs¼xhu1itin terms of the time coordinate t. The integral
formulation of the conservation equations across the shock can be
readily written in the shock reference frame, as done in Sec. II.
Whereas the incident vorticity wave remains stationary in space in the
laboratory frame, it becomes a wave traveling at velocity hu1itoward
theshockintheshockreferenceframe.
In the reference frame moving with the post-shock gas, the
streamwise coordinate x
c
moves with the post-shock mean absolute
velocity hu1ihu2iand is therefore defined as xc¼xðhu1i
hu2t. In this frame, the vorticity and entropy fluctuations in the
post-shock gas are stationary in space, which facilitates the description
of the problem, as shown below.
B. Orientation and form of the incident vorticity wave
Anticipating that the pre-shock turbulence is isotropic, there is
no privileged direction of the wavenumber vector k, and therefore, the
amplitude of the vorticity modes depends exclusively on k¼jkj.
Similarly, because of this isotropy, there is no preferred wavenumber-
vector orientation relative to the shock surface. In principle, this would
require the formulation of a three-dimensional problem to describe
the interaction. However, a simple rotation of the reference frame can
transform the problem into a two-dimensional one, as described below
(see also Refs. 36,60,and75).
For an incident wavenumber vector arbitrary oriented in space at
latitude and longitude angles hand u, respectively, the reference
frames described in Sec. III A can be rotated counterclockwise around
xby an angle equal to the longitudinal inclination of the incident wave
w, as indicated in Fig. 5. In this way, the interaction problem becomes
two-dimensional, in that all variations with respect to zare zero.
Using the aforementioned rotation, the wavenumber-vector
components in the streamwise and transverse directions are
kx¼kcos h;ky¼ksin h;(26)
respectively, with k
z
¼0 by construction. Similarly, in the laboratory
reference frame, the vorticity vector of the incident wave in the pre-
shock gas can be expressed as
FIG. 4. Distributions of (a) density jump R, (b) post-shock Mach number M2, and (c)
pre-shock Mach number M1as a function of the temperature jump Tfor
B¼106;bv¼10, and bd¼100 (lines colored by the degree of dissociation; refer to
Fig. 3 for a colorbar). Dashed lines represent asymptotic limits for a calorically perfect
diatomic gas (bv!1and a!0), and for a highly dissociated gas (a!1).
FIG. 5. Simplification of a three-dimensional problem of a shock interacting with an
arbitrary-oriented vorticity wave to a two-dimensional problem by rotating the refer-
ence frame around the streamwise axis.
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x1¼ðdx1ÞeiðkxxþkyyÞ(27)
with
dxx;1¼ekhc2isin hcos u;dxy;1¼ekhc2icos hcos u;
dxz;1¼ekhc2isin u(28)
being the vorticity amplitude in each direction. In this formulation,
hc2idenotes the mean speed of sound in the post-shock gas, and eis a
dimensionless velocity fluctuation amplitude, which is small in the lin-
ear theory, e1. The vorticity of the incident wave engenders a fluc-
tuationvelocityfieldinthepre-shockgasgivenby
v1¼ðdv1ÞeiðkxxþkyyÞ(29)
whose amplitude is
du1¼ehc2isin hsin u;dv1¼ehc2icos hsin u;
dw1¼ehc2icos u(30)
in the x,y,andzdirections, respectively. Specifically, the z-component
of the fluctuation velocity vector is uniform along z.Thiscomponent
will not be carried any further in the analysis, since it is transmitted
unaltered through the shock because of the conservation of tangential
momentum. Note also that (27) and (29) are related by the definition
of the vorticity x1¼kv1. Furthermore, the velocity field (29)–(30)
is one that satisfies the incompressibility relation kv1¼0. Finally,
implicit in the definitions given above is that the incident vorticity
wave is inviscid, or equivalently, that the pre-shock Reynolds number
of the fluctuation, 2pjv1j=ðk1Þ, is infinitely large.
To illustrate the analysis, a particular form of the pre-shock vor-
ticity fluctuation corresponding to the inviscid TaylorGreen vortex
xz;1x;y
ðÞ
¼euhc2ik2
ky
!
cos kxx
ðÞ
sin kyy
 (31)
is employed in the numerical results highlighted below, with
xx;1¼xy;1¼0. The corresponding streamwise and transverse com-
ponents of the velocity fluctuations in the pre-shock gas are given by
u1x;y
ðÞ
¼euhc2icos kxx
ðÞ
cos kyy

;(32a)
v1x;y
ðÞ
¼euhc2ikx
ky
!
sin kxx
ðÞ
sin kyy

;(32b)
respectively. In this formulation, euis the amplitude of the pre-shock
streamwise velocity fluctuations
eu¼esin hsin u(33)
with u1 in the linear theory.
C. Linearized formulation of the interaction problem
In this linear theory, the vorticity and the streamwise and trans-
verse velocity components in the post-shock gas reference frame are
expanded to first order in euas
x¼eukyhc2
2i
x;u¼euhc2i
u;v¼euhc2i
v;(34)
respectively, with
x;
u,and
vbeing the corresponding dimensionless
fluctuations. The post-shock pressure and density can be similarly
expressed as
P¼hP2euhq2ihc2i2
p;q¼hq21þeu
qÞ(35)
with
pand
qbeing the dimensionless fluctuations of pressure and
density, respectively. The brackets indicate time-averaged quantities,
which are given by the solution obtained in Sec. II.Inthisway,allfluc-
tuations are defined to have a zero time average.
Assuming that the Reynolds number of the post-shock fluctua-
tions is infinitely large, the expansions (34) and (35) can be employed
in writing the linearized Euler conservation equations of mass, stream-
wise momentum, transverse momentum, and energy as
@
q
@sþ@
u
@
xcþ@
v
@
y¼0;(36a)
@
u
@sþ@
p
@
xc¼0;(36b)
@
v
@sþ@
p
@
y¼0;(36c)
@
p
@s¼@
q
@s;(36d)
in the reference frame moving with the post-shock gas. In this nota-
tion, the space and time coordinates have been non-dimensionalized
as
xc¼kyxc;
y¼kyy;s¼kyhc2it:(37)
The linearized Euler equations (36) can be combined into a
single, two-dimensional periodically symmetric wave equation
@2
p
@s2¼@2
p
@
x2
cþ@2
p
@
y2(38)
for the post-shock pressure fluctuations. Equation (38) is integrated
for s0 within the spatiotemporal domain bounded by the leading
reflected sonic wave traveling upstream,
xc¼s, and the shock front
moving downstream
xc¼M
2s,withM2¼hu2i=hc2i.
In the integration of (38), the boundary condition far down-
stream of the shock is provided by the isolated-shock assumption,
whereby the effect of the acoustic waves reaching the shock front from
behind is neglected. The boundary condition at the shock front is
obtained from the linearized RankineHugoniot jump conditions
assuming that (a) the thickness of the thermochemical non-
equilibrium region Tis much smaller than the inverse of the trans-
verse wavenumber k1
y; and (b) the displacement of the shock
ns¼nsðy;tÞfrom its mean, flat shape (see Fig. 1) is much smaller
than k1
y. In these limits, at any transverse coordinate
y,the
RayleighHugoniot jump conditions can be applied at the mean shock
front location
xc¼M
2sand can be linearized about the mean
thermochemical-equilibrium post-shock gas state P;R;T;M2,and
acalculated in Sec. II, thereby yielding
@
ns
@s¼R1C
ðÞ
2M2R1
ðÞ
ps
u1;(39a)
us¼1þC
2M2
psþ
u1;(39b)
vs¼
v1M
2R1
ðÞ
@
ns
@
y;(39c)
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qs¼C
M2
2
ps:(39d)
In (39),
ns¼kyns=euis the dimensionless shock displacement,
whereas
ps;
qs;
us,and
vsare, respectively, the dimensionless fluctua-
tions of pressure, density, streamwise velocity, and transverse velocity
immediately downstream of the shock front, where thermochemical
equilibrium is reached in the limit kyT1. In these relations,
u1¼u1=ðeuhc2and
v1¼v1=ðeuhc2are the normalized compo-
nents of the pre-shock velocity field (29) engendered by the incident
wave described in Sec. III B. Note that, at the turning point of the
Hugoniot curve (C¼0), the compression of the gas exerted by the
shock is isochoric in the near field and therefore leads to vanishing
density fluctuations immediately downstream of the shock, as pre-
scribed by the linearized jump condition (39d).
The flow is periodic in the transverse direction
y. As a result, the
terms involving partial derivatives with respect to
yin (36a),(36c),
(38),and(39c) can be easily calculated from the transverse functional
form of the post-shock flow variables given the incident vorticity wave
(31). In particular, it can be shown that the fluctuations
p;
u,and
ns
are proportional to cos ð
yÞ,whereas
vis proportional to sin ð
yÞ.These
prefactors are henceforth omitted in the analysis, but should be
brought back when reconstructing the full solution from the dimen-
sionless fluctuations.
The initial conditions required to solve (38) assume that the
shock is initially flat,
ns¼
vs¼0ats¼0. Correspondingly, the initial
values of the fluctuations of pressure and streamwise velocity immedi-
ately downstream of the shock must satisfy the relation
usþ
ps¼0at
s¼0, as prescribed by the first acoustic wave traveling upstream
xc¼s. This gives a pressure fluctuation
ps¼2M2=ð1þC
þ2M2Þimmediately downstream of the shock front at s¼0.
The linearized problem (38), along with its boundary and ini-
tial conditions provided above, describe the fluctuations in the post-
shock gas in the LIA framework. Remarkably, this problem can be
integrated using the mean post-shock flow obtained from the ana-
lytical formulation provided in Sec. II, as done in the remainder of
this paper, or by considering a mean post-shock flow obtained
numerically with more sophisticated thermochemistry. For instance,
instead of the formulation presented in Sec. II, a one-dimensional
chemical equilibrium code like CEA (see Fig. 2 and Sec. II C) could
be used to calculate numerically the mean post-shock conditions
incorporating (a) different models for the variations of the specific
heats such as the NASA polynomials,
65
which include both vibra-
tional and electronic excitation, and (b) additional chemical effects
such as ionization. This can be understood by noticing that (38),
along with its boundary and initial conditions, depend only on the
following dimensionless parameters: the mean density jump R, the
mean post-shock Mach number M2, and the inverse of the slope of
the Hugoniot curve C, all of which can be computed numerically
solving a one-dimensional shock wave subject to arbitrary
thermochemistry.
D. Far-field and long-time asymptotic analysis
At long times tðkyhc21, the solution to the wave equation
(38), subject to the boundary conditions described in Sec. III C, yields
the pressure fluctuations
ps¼Pl1cos ðxsÞþPl2sin ðxsÞif f1;
Pscos ðxsÞif f1
((40)
behind the shock. In this formulation, x¼fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
qis the dimen-
sionless frequency, where fis a frequency parameter defined as
f¼M2R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
q
kx
ky
! (41)
with kx=ky¼1=jtan hj.Caseswithf1 correspond to sufficiently
small streamwise wavenumbers, kxkyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
q

=ðM2,
whereas the opposite (sufficiently large streamwise wavenumbers)
holds for f1. The corresponding amplitudes of the pressure wave
(40) are
Pl1¼ ð1R
1Þðrbf2rcÞ
f2ð1f2Þþðrbf2rcÞ2f2RM2
2
1M
2
2
!
;(42a)
Pl2¼ð1R
1Þfffiffiffiffiffiffiffiffiffiffiffiffi
1f2
p
f2ð1f2Þþðrbf2rcÞ2f2RM2
2
1M
2
2
!
;(42b)
Ps¼ ð1R
1Þ
fffiffiffiffiffiffiffiffiffiffiffiffi
f21
pþrbf2rc
f2RM2
2
1M
2
2
!
;(42c)
where r
b
and r
c
are auxiliary factors defined as
rb¼1þC
2M2
;rc¼RM2
1M
2
2
1C
2

:(43)
To describe the far-field post-shock gas, it is convenient to split the
fluctuations of velocity, pressure, and density into their acoustic (a),
vortical (r), and entropic (e) components as
uðxc;sÞ¼
uað
xc;sÞþ
urð
xcÞ;(44a)
vð
xc;sÞ¼
vað
xc;sÞþ
vrð
xcÞ;(44b)
pð
xc;sÞ¼
pað
xc;sÞ;(44c)
qð
xc;sÞ¼
qað
xc;sÞþ
qeð
xcÞ:(44d)
The acoustic pressure wave emerging from (38) is of the form
pae6iðxasja
x
yÞ,wherex
a
and j
a
are the dimensionless acoustic
frequency and longitudinal wavenumber reduced with c2kyand k
y
,
respectively, which are related as
x2
a¼j2
aþ1:(45)
In the shock reference frame
x¼M
2s, the oscillation frequency at
shock front, x, is related to the post-shock Mach number as
x¼xaM
2ja. Upon substituting this relation into (45),the
expressions
ja¼M2x6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x21þM
2
2
q
1M
2
2
;(46a)
xa¼x6M2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x21þM
2
2
q
1M
2
2
(46b)
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are obtained. In (46), the solution corresponding to the positive sign in
front of the square root must be excluded since it represents nonphysi-
cal acoustic waves whose amplitude increases exponentially with dis-
tance downstream of the shock when x<ð1M
2
2Þ1=2.
Different forms of the solution arise depending on the value of
the dimensionless frequency x. At frequencies x<ð1M
2
2Þ1=2,or
equivalently f<1, the amplitude of the acoustic pressure decreases
exponentially with distance downstream of the shock. On the other
hand, for x>ð1M
2
2Þ1=2,orf>1,theacousticpressurebecomes
a constant-amplitude wave
pð
xc;sÞ¼Pscos xasja
xc
ðÞ
;(47)
which corresponds to a downstream-traveling sound wave for ja<0
(or x<1), and to an upstream-traveling sound wave for ja>0
(x>1), both cases being referenced to the post-shock gas reference
frame. In this case, the acoustic modes of the density, temperature,
and velocities are
qað
xc;sÞ¼Pscos xasja
xc
ðÞ
;(48a)
Tað
xc;sÞ¼Hacos xasja
xc
ðÞ
;(48b)
uað
xc;sÞ¼Uacos xasja
xc
ðÞ
;(48c)
vað
xc;sÞ¼Vasin xasja
xc
ðÞ
;(48d)
respectively, where
T¼ðThT2=ðeuhT2is the dimensionless
post-shock temperature fluctuation.
The amplitudes of the acoustic modes of the streamwise and
transverse velocity fluctuations in (48) are proportional to the ampli-
tude of the acoustic pressure, Ua=Ps¼ja=xaand Va=Ps¼1=xa,
as prescribed by second and third equations in (36).Similarly,the
amplitude of the acoustic mode of the post-shock temperature fluctua-
tions can be expressed relative to P
s
as
Ha=Ps¼2ð1þaÞaR12
evib þ2bd=T
ðÞ
5þaþ2ð1aÞ
e2
vibebv=TþaT12
evib þ2bd=T
ðÞ
(49)
with a
R
,a
T,
and
evib being defined in (17) and (18),and(16),respec-
tively. Note that (49) simplifies to Ha=Psc1 in both the calori-
cally perfect diatomic gas limit (a!0andbv!1,forwhich
c!7=5) and in the fully dissociated gas limit (a!1, for which
c!5=3).
The entropic mode of the density fluctuations is determined by
the linearized Rankine–Hugoniot jump condition (39d) after subtract-
ing the acoustic mode
qeð
xcÞ¼ C
M2
2
psðs¼
xc=M2Þ
qað
xc;s¼
xc=M2Þ(50)
to give
qeð
xc1Þ¼ Dl1cos ðje
xcÞþDl2sin ðje
xcÞif f1;
Dscos ðje
xcÞif f1
((51)
in the asymptotic far field. In (51),je¼Rkx=kyis a dimensionless
wavenumber, and Dj¼ðCM2
21ÞPjis a fluctuation amplitude
that depends on fthrough the pressure amplitudes Pl1;Pl2,andP
s
defined in (42). Since the pre-shock gas contains only vortical velocity
fluctuations, all entropic modes are generated at the shock. The entro-
pic density fluctuations
qeare related to the entropic temperature
fluctuations
Teð
xc1Þ¼1þaþaR
1þaþaT
qeð
xcÞ;(52)
and both
qeand
Teinduce entropic fluctuations in the degree of
dissociation, as shown in (8). As a result, the thermochemical equi-
librium state in the post-shock gas fluctuates depending on the
local shock curvature. Specifically, there exist fluctuations of the
concentrations of the chemical species A and A2in the post-shock
gas that are in phase with the entropic modes of density and tem-
perature fluctuations. The normalized fluctuation of the degree of
dissociation is
að
xc1Þ¼ahai
eu¼aR
qeð
xcÞþaT
Teð
xcÞ
¼ðaRaTÞð1þaÞ
1þaþaT
qeð
xcÞ:(53)
In a similar manner, the vorticity fluctuations
xdefined in (34)
can be expressed in terms of fas
xð
xc1Þ¼ Xlcos ðjr
xcþ/rÞif f1;
Xscos ðjr
xcÞif f1;
((54)
where, as found in the entropic perturbation field, the dimensionless
rotational wavenumber is simply given by the compressed upstream
wavenumber ratio jr¼je¼Rkx=ky. The amplitudes are
Xl¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðX1þX2Pl1Þ2þðX2Pl2Þ2
q;(55a)
Xs¼X1þX2Ps;(55b)
where X1¼Rð1þk2
x=k2
yÞquantifies the amplification of the pre-
shock vorticity as a direct result of the shock compression, and
X2¼ðR1Þð1CÞ=ð2M2Þmeasures the vorticity production by
the discontinuity front rippling. The corresponding phase for f<1is
given by tan /r¼X2Pl2=ðX1þX2Pl1Þ, which is different to that
associated with entropic fluctuations tan /e¼Pl2=Pl1.
Figure 6 shows the value of jXj2as a function of the shock
strength M1for six arbitrary values of the frequency parameter f.
FIG. 6. Square of the vorticity amplitude jXj2as a function of the pre-shock Mach
number M1for B¼106;bv¼10;bd¼100 and six different values of the
frequency parameter: f¼0:6, 0.7, 0.8, 1.1, 1.5, and 2.
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Three of them pertain to the long-wavelength regime f<1(X¼Xl)
and the other three to the short-wavelength regime f>1(X¼Xs). It
is found that the shape of the curve qualitatively changes depending
on the wavelength regime. For instance, when compared to interac-
tions with frequency f<1, cases for f>1rendercurveswithwider
peaks and whose location corresponds to lower Mach numbers.
The streamwise and transverse components of the vortical mode
of the velocity read
urð
xc1Þ¼Urcos ðjr
xcþ/rÞ;(56a)
vrð
xc1Þ¼Vrsin ðjr
xcþ/rÞ;(56b)
where the phase angle is /r¼0forf>1. The amplitudes are propor-
tional to the vorticity fluctuations as
Ur¼1
1þj2
r
X¼M2
2
M2
2þ1M
2
2

f2X;(57a)
Vr¼jr
1þj2
r
X¼fM2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
q
M2
2þ1M
2
2

f2X;(57b)
where Xdepends on frequency, as shown in (55) and Fig. 6.
IV. THE INTERACTION OF A HYPERSONIC SHOCK
WAVE WITH WEAK ISOTROPIC TURBULENCE
The weak isotropic turbulence in the pre-shock gas can be repre-
sented by a linear superposition of incident vorticity waves whose
amplitudes evary with the wavenumber in accord with an isotropic
energy spectrum EðkÞ¼e2ðkÞ. The root mean square (rms) of the
velocity and vorticity fluctuations in the pre-shock gas can be calcu-
lated by invoking the isotropy assumption, which states that the prob-
ability the incident wave has of having orientation angles ranging from
hto hþdh,andfromuto uþdu, is proportional to the solid angle
sin hdhdu=ð4pÞ. This assumption provides the expressions
hu02
1i
e2hc2i2¼1
3;hv02
1i
e2hc2i2¼1
6;hw02
1i
e2hc2i2¼1
2(58)
for the pre-shock rms velocity fluctuations, and
hx02
x;1i
e2k2hc2i2¼1
3;hx02
y;1i
e2k2hc2i2¼1
6;hx02
z;1i
e2k2hc2i2¼1
2(59)
for the pre-shock vorticity fluctuations. In this section, a linear analysis
is performed to calculate the variations of the rms of the velocity and
vorticity fluctuations across the shock.
A. Amplifications of turbulent kinetic energy,
turbulence intensity, and turbulent Reynolds
number across the shock
The analysis begins by expressing pre-shock components of the
velocity fluctuation modulus as
ju0
2c2ej
uaþ
urjsin hsin u;(60a)
jv0
2c2ej
vaþ
vrjsin hsin u;(60b)
jw0
2j¼jw0
1j;(60c)
where the acoustic and vortical modes of the dimensionless velocity fluc-
tuations in the far field are given in (48) and (56).Therelationsbetween
the modes of the streamwise and transverse velocity fluctuations are pro-
vided by the irrotationality condition
va¼ja
uafor the acoustic mode,
and by the solenoidal condition ky
vr¼Rkx
urfor the vortical mode.
The TKE amplification factor across the shock wave is defined as
K¼hu02
2iþhv02
2iþhw02
2i
hu02
1iþhv02
1iþhw02
1i¼hu02
2iþhv02
2i
e2hc2i2þ1
2
¼1
2ðp=2
0
u2þ
v2
ðÞ
sin3hdhþ1
"#
;(61)
where the use of (58) has been made. Furthermore, Kcan also be
decomposed linearly into acoustic and vortical modes as
K¼KaþKr,with
Ka¼1
3ð1
1U2
aþV2
a

PðfÞdf¼1
3ð1
1
P2
sPðfÞdf;
Kr¼1
2þ1
3ð1
0U2
rþV2
r

PðfÞdf:
(62)
The entropic mode does not contain any kinetic energy, since entropy
fluctuations are decoupled from velocity fluctuations in the inviscid
linear limit.
In Eq. (62),PðfÞis a probability-density distribution given by
PðfÞ¼3
2M4
2R4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
q
M2
2R2þf21M
2
2

5=2;(63)
which satisfies the normalization Ð1
0PðfÞdf¼1. In addition, the
velocity amplitudes Ua;Ur;Va,andVrare obtained using the long-
time far-field asymptotic expressions (48) and (57).Thelowerintegra-
tion limit of K
a
is f¼1sincetheacousticmodedecaysexponentially
with distance downstream of the shock in the long-wave regime f<1.
However, the integral 1=3Ð1
0ðP2
l1þP2
l1ÞPðfÞdfneeds to be added to
K
a
when evaluating the solution in the near field
xs
xc1.
Figure 7 shows the TKE amplification factor K,givenbythesum
of the acoustic and vortical contributions in (62), as a function of the
pre-shock Mach number M1. Similarly to the results observed in
Sec. II, the onset of vibrational excitation at M13 begins to produce
small departures of Kfrom the thermochemically frozen result corre-
sponding to a diatomic calorically perfect gas. These departures are
exacerbated as the degree of dissociation increases and become signifi-
cant even at small values of aof order 1% at M15, where Ksignifi-
cantly departs from the curve predicted in the thermochemically frozen
limit corresponding to a diatomic calorically perfect gas. The latter was
shown to plateau at K¼1:78 for M11inearlywork,
37,60
whereas
the present study indicates that such plateau does not exist when ther-
mochemical effects at hypersonic Mach numbers are accounted for.
The resulting curve of Kin Fig. 7 is non-monotonic and contains
two peaks in the hypersonic range of Mach numbers. This behavior
cannot be guessed by a simple inspection of the post-shock density
and Mach number shown in Fig. 4. Instead, the non-monotonicity of
Kis related to the strong dependence of the enstrophy amplification
on the wavenumber. Specifically, the vortical mode of the velocity fluc-
tuation, which is shown below to be the most energetic, is proportional
to the post-shock vorticity amplitude Xgiven in (55),whichpeaksat
Physics of Fluids ARTICLE scitation.org/journal/phf
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different pre-shock Mach numbers depending on the frequency
parameter f,asshowninFig. 6.
The first peak of Kreaches a value of 2.1 and occurs at M16,
where a5%. In contrast, the second peak at K2:9 nearly doubles
the value predicted in the thermochemically frozen limit, and occurs at
amuchhigherMachnumberM119 where dissociation is almost
complete. At very large Mach numbers M1>40, in the fully dissoci-
ated regime, Kasymptotes to the value K1:69 predicted for mon-
atomic calorically perfect gases. However, as discussed in Sec. II D,this
limit has to be interpreted with caution because additional thermochem-
ical effects not included here, such as ionization and electronic excita-
tion, play an important role at those extreme Mach numbers.
Most of the TKE produced across the shock belongs to transverse
velocity fluctuations of the vortical mode. To see this, we consider the
decomposition of the TKE amplification factor into longitudinal (KL)
and transverse (KT) components as
K¼1
3ðKLþ2KTÞ(64)
with
KL¼ð1
1U2
aPðfÞdf
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Ka
L
kinetic energy of the
longitudinal acoustic mode
þð1
0U2
rPðfÞdf
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
Kr
L
kinetic energy of the
longitudinal vortical mode
;(65a)
KT¼1
2ð1
1V2
aPðfÞdf
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
Ka
T
kinetic energy of the
transverse acoustic mode
þ3
4þ1
2ð1
0V2
rPðfÞdf
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Kr
T
kinetic energy of the
transverse vortical mode
:(65b)
The contribution of the acoustic mode to KLand KTyields negligible
TKE over the entire range of Mach numbers, as shown in Fig. 8(a).In
contrast, the contribution of the vortical mode is significant. Whereas
the longitudinal TKE of the vortical mode Kr
Ldominates over the
transverse one Kr
Tat supersonic Mach numbers, it plunges below Kr
Tat
hypersonic Mach numbers around the turning point of the Hugoniot
curve. The value of Kr
Tpeaks at M119 with Kr
T3:8, as observed
in Fig. 8(b).ThispeakisresponsibleforthepeakinKobserved Fig. 7
at the same Mach number, thereby indicating that most the TKE there
is stored in vortical gas motion in the transverse direction.
The mechanism whereby high-temperature thermochemistry
augments the TKE across the shock in this LIA framework is
explained by the linearized RankineHugoniot jump condition (39c)
and is schematically shown in Fig. 9. In particular, the conservation of
the tangential velocity across the wrinkled shock requires
vt¼hu1icos #þv0
1sin #
¼hu2icos #þv0
2sin #; (66)
where #¼p=2þarctanð@ns=@yÞis a local shock incidence angle
whose departures from p=2 are of order
u
,sincekyns¼OðuÞin this
linear theory. The streamwise velocity fluctuations u0
1and u0
2have
been neglected in writing (66), since their multiplication by cos bis
smaller by a factor of order
u
relative to the other terms. Equation
(66) yields the transverse post-shock velocity fluctuation
v0
2¼v0
1hu1ihu2i
ðÞ
@ns
@y;(67)
which represents the dimensional counterpart of the linearized
RankineHugoniot jump condition (39c).InEq.(67),@ns=@y<0
in both configurations sketched in Fig. 9.Notethat(67) holds
FIG. 7. TKE amplification factor Kas a function of the pre-shock Mach number
M1for B¼106;bv¼10, and bd¼100 (line colored by the degree of dissocia-
tion). Dashed lines correspond to limit behavior of Kcalculated using the asymptotic
expressions (23) and (24) for small and high Mach numbers, respectively.
FIG. 8. (a) Acoustic and (b) vortical modes of the streamwise (KL) and transverse
(KT) components of the TKE amplification factor as a function of the pre-shock
Mach number M1for B¼106;bv¼10, and bd¼100 (lines colored by the
degree of dissociation; refer to Fig. 7 for a colorbar). Dashed lines correspond to
limit behavior of KLand KTcalculated using the asymptotic expressions (23) and
(24) for small and high Mach numbers, respectively.
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independently of whether the gas is thermochemically frozen or equili-
brated. However, the thermochemistry influences (67) by flattening
the shock front (i.e., by decreasing @ns=@y) while strongly decreasing
the mean post-shock velocity hu2i¼hu1i=R, with the latter effect
prevailing over the former. As a result, v0
2and its associated kinetic
energy KTare larger relative to those observed in a diatomic calorically
perfect gas.
The TKE amplification, along with the aforementioned decrease
in the mean post-shock velocity hu2icaused by the thermochemical
effects, also leads to a strong amplification of the turbulence intensity
across the shock. Specifically, the ratio of post- to pre-shock turbulence
intensities
I2
I1¼u‘;2=hu2i
u‘;1=hu1i¼K1=2R(68)
is found to peak at the turning point of the Hugoniot curve
(a0:7;T’9;M113, and R’12) with a value I2=I119,
as shown in Fig. 10(a). This is in contrast to the maximum value
I2=I18 predicted by the theory of calorically perfect gases.
Although the theory presented above is formulated in the inviscid
limit, the ratio of post- to pre-shock turbulent Reynolds numbers
Re‘;2
Re‘;1¼u‘;22=2
u‘;11=1¼K1=2
T0:7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2R2þ1
3
s(69)
is a finite quantity that can be calculated. In the last term of (69),the
use has been made of the fact that the only wavenumber, that is, dis-
torted through the shock is the longitudinal one, which changes from
k
x
in the pre-shock fluctuations to kxRin the post-shock ones. In
addition, the molecular viscosity is assumed to vary with temperature
raised to the power of 0.7.
Remarkably, the vortical post-shock fluctuations downstream of
the hypersonic shock are not only much more intense than those
upstream, but they also have a higher turbulent Reynolds number
Re‘;2>Re‘;1,asshowninFig. 10(b). Similarly to the turbulence inten-
sities, the maximum ratio of turbulent Reynolds numbers across the
shock is reached at the turning point of the Hugoniot curve
(a0:7;T’9;M113, and R’12) with a value of
Re‘;2=Re‘;15. In contrast, the theory of calorically perfect gases pre-
dicts an attenuation of the turbulent Reynolds number at those condi-
tions. When thermochemical effects are accounted for, the
amplification of the turbulent Reynolds number lasts until M120,
beyond which the increase in post-shock temperature and the decrease
in post-shock density make Re‘;2=Re‘;1to plummet below unity.
In summary, the increase in transverse velocity fluctuations of the
vortical mode across the shock is responsible for the TKE amplification
FIG. 9. Schematics of the mechanism of TKE amplification for (a) thermochemically
frozen (i.e., diatomic calorically perfect) post-shock gas, and (b) thermochemically
equilibrated post-shock gas, both panels simulating the same pre-shock conditions.
The flow is from right to left. The magnitude of the shock displacement and velocity
perturbations has been exaggerated for illustration purposes.
FIG. 10. Amplification of (a) turbulence intensity and (b) turbulent Reynolds number
across the shock as a function of the pre-shock Mach number M1for
B¼106;bv¼10, and bd¼100 (lines colored by the degree of dissociation;
refer to Fig. 7 for a colorbar). The dashed lines correspond to the values of I2=I1
and Re‘;2=Re‘;1calculated assuming that the post-shock gas is thermochemically
frozen (i.e., diatomic calorically perfect).
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in this linear theory. In addition, the results indicate that the TKE is
more amplified when dissociation and vibrational excitation are
accounted for at high Mach numbers. In the conditions tested here, the
post-shock fluctuations resulting at hypersonic Mach numbers can be—
at most—19 times more intense and can have—at most—a five times
larger turbulent Reynolds number than the pre-shock fluctuations.
B. Amplifications of anisotropy, enstrophy, and
variances of density and degree of dissociation across
the shock
The weak isotropic turbulence in the pre-shock gas becomes
anisotropic as it traverses the shock wave. An anisotropy factor that
quantifies this change can be defined as
60
W¼h
v2iþh
w2i2h
u2i
h
v2iþh
w22h
u2i¼12KL
KLþKT
(70)
with 1W1. The cases W¼1 and 1 represent anisotropic
turbulent flows dominated by longitudinal and transverse velocity
fluctuations, respectively. In contrast, W¼0 corresponds to an isotro-
pic turbulent flow, KT¼KL¼K.Figure 11 shows that dissociation
and vibrational excitation dissociation lead to larger anisotropy factors
in the post-shock gas compared to the thermochemically frozen
(diatomic calorically perfect) case in the relevant range of hypersonic
Mach numbers up to the fully dissociated gas limit.
The vortical motion downstream of the shock is quantified by
the enstrophy amplification factor
W¼hx02
x;2iþhx02
y;2iþhx02
z;2i
hx02
x;1iþhx02
y;1iþhx02
z;1i¼1
3þ2
3W?;(71)
wheretheuseof(59) and of the invariance of the normal vorticity
across the shock has been made. In (71),W?is the enstrophy amplifi-
cation factor in the transverse direction
W?¼1
3þ2
3hx02
y;2iþhx02
z;2i
hx02
y;1iþhx02
z;1i¼3Wz
4(72)
with
Wz¼hx02
z;2i
hx02
z;1i¼ð1
0
X2R2M2
2PðfÞ
R2M2
2þð1M
2
2Þf2df(73)
being the amplification factor of the rms of the zcomponent of the
vorticity. Equation (73) includes the asymptotic amplitudes defined in
(55) and the relation
sin2hðfÞPðfÞ¼3
2M6
2R6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1M
2
2
q
M2
2R2þf21M
2
2

7=2:(74)
The enstrophy amplification factor Wis provided in Fig. 12 as a
function of the pre-shock Mach number. Similarly to Fig. 7 for K,the
curve of Wdisplays two maxima, but the differences with respect to
the thermochemically frozen case are much larger for W.Thefirst
peak of Wis dominated by the increase in short-wavelength vorticity,
as shown in Fig. 6, and it represents an amplification of nearly four
times the enstrophy predicted by the theory of calorically perfect gases.
Whereas the pre-shock density is uniform because of the vortical
character of the incident modes, the density in the post-shock gas fluc-
tuates due to both acoustic and entropic modes generated by the shock
wrinkles. To investigate these fluctuations, we consider the normalized
density variance
FIG. 11. Anisotropy factor Was a function of the pre-shock Mach number M1for
B¼106;bv¼10, and bd¼100 (line colored by the degree of dissociation).
Dashed lines correspond to limit behavior of Wcalculated using the asymptotic
expressions (23) and (24) for small and high Mach numbers, respectively.
FIG. 12. Enstrophy Was a function of the pre-shock Mach number M1for
B¼106;bv¼10, and bd¼100 (line colored by the degree of dissociation).
Dashed lines correspond to limit behavior of Wcalculated using the asymptotic
expressions (23) and (24) for small and high Mach numbers, respectively.
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hq02
2i
hq2
2i¼ðGaþGeÞð1
0
EðkÞk2dk;(75)
which depends on the integral of the energy spectrum Eover the entire
wavenumber space. The prefactors G
a
and G
e
represent density-
variance components induced by acoustic and entropic modes, respec-
tively, and are given by
Ga¼Ka;(76a)
Ge¼CM2
21

2ð1
0
P2
l1þP2
l2

PðfÞdf
þCM2
21

2ð1
1
P2
sPðfÞdf;(76b)
wheretheuseof(51) has been made. Figure 13(a) shows that, while
the vortical fluctuations across the shock are increased by dissociation,
the density variance induced by the entropic mode is small for
M110 but increases sharply thereafter up to M119, where it
achieves a maximum value. As observed by comparing Figs. 8(b) and
13(a), the acoustic prefactor G
a
is found to be negligible compared to
the entropic one G
e
.
Whereas the RankineHugoniot jump condition (39d) evalu-
ated at the turning point of the Hugoniot curve C¼0 indicates that
the density fluctuations immediately downstream of the shock are
zero, the entropic prefactor in Fig. 13(a) at M113 (where Cvan-
ishes) leads to a non-zero density variance. The two results can be
reconciled by noticing that the formulation in (76) and the approxi-
mation GeGaare applicable only to the far-field downstream of
the shock. In contrast, the acoustic mode needs to be retained near
the shock. Specifically, the post-shock density fluctuations in the
near field vanish as C!0 because of a destructive interference
between the acoustic and entropic modes. In contrast, the entropic
mode dominates in the far field and leads to non-zero post-shock
density fluctuations.
The entropic component of the density variance engenders a var-
iance of the degree of dissociation given by
ha02Aeð1
0
EðkÞk2dk;(77)
where
Ae¼ðaRaTÞ2ð1þaÞ2
ð1þaþaTÞ2Ge(78)
is the corresponding prefactor. Figure 13(b) shows that A
e
attains a
maximum value at M115, and becomes negligible both in the
absence of dissociation and when dissociation is complete.
V. CONCLUSIONS
The interaction between a hypersonic shock wave and weak iso-
tropic turbulence has been addressed in this work using LIA. Contrary
to previous studies of shock/turbulence interactions focused on calori-
cally perfect gases, the results provided here account for endothermic
thermochemical effects of vibrational excitation and gas dissociation
enabled by the high post-shock temperatures. Important approxima-
tions used in this theory are that the thickness of the thermochemical
non-equilibrium region trailing the shock front is small compared to
the characteristic size of the shock wrinkles, and that all fluctuations in
the flow are small relative to the mean.
The results presented here indicate that the thermochemical
effects act markedly on the solution in a number of important ways
with respect to the results predicted by the theory of calorically perfect
gases:
(a) Significant departures from calorically perfect-gas behavior
can be observed in the solution even at modest degrees of dis-
sociation of 1%, corresponding to Mach 5 and therefore to
the beginning of the hypersonic range. This is because the
associated bond-dissociation energies of typical molecules are
large. As a result, the chemical enthalpy invested in dissocia-
tion in the post-shock gas can easily surpass the pre-shock
thermal energy and become of the same order as the pre-
shock kinetic energy.
(b) A turning point in the Hugoniot curve is observed at approxi-
mately Mach 13 and 70% degree of dissociation that leads to
a significant increase of the mean post-shock density of
approximately 12 times its pre-shock value, which represents
nearly twice the maximum density jump predicted by the the-
ory of calorically perfect gases.
(c) The aerothermodynamic behavior of the post-shock gas
changes fundamentally around the turning point in the
Hugoniot curve. As the Mach number increases above 13,
positive fluctuations of streamwise velocity engender positive
pressure fluctuations in the post-shock gas that are accompa-
nied by negative density fluctuations. In this way, the local
post-shock density and pressure are anticorrelated, although
the shock remains stable to corrugations in all operating con-
ditions tested here.
FIG. 13. Entropic prefactors of (a) the post-shock density variance and (b) the post-
shock degree of dissociation as a function of the pre-shock Mach number M1for
B¼106;bv¼10, and bd¼100 (lines colored by the degree of dissociation;
refer to Fig. 12 for a colorbar). Dashed lines correspond to limit behavior of G
e
calculated using the asymptotic expressions (23) and (24) for small and high Mach
numbers, respectively.
Physics of Fluids ARTICLE scitation.org/journal/phf
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(d) The amplification of TKE is larger than that observed in calo-
rically perfect gases. Whereas the streamwise velocity fluctua-
tions across the shock are decreased, the transverse ones are
greatly increased (i.e., much more than in a diatomic calori-
cally perfect gas). This phenomenon can be explained in the
linear theory by using the conservation of tangential momen-
tum, which elicits larger transverse velocity fluctuations as a
result of the increase in post-shock density that occurs due to
dissociation and vibrational excitation. This effect also leads
to a much more significant increase of anisotropy and enstro-
phy across the shock than that observed in a diatomic calori-
cally perfect gas.
(e) Most of the amplified content of TKE is stored in vortical
velocity fluctuation modes in the post-shock gas. The trend of
the TKE amplification factor with the pre-shock Mach num-
ber is non-monotonic and involves two maximum values,
one equal to 2.1 at Mach 6 (corresponding to a degree of dis-
sociation of 5%), and a second one equal to 2.9 at Mach 19
(corresponding to a degree of dissociation larger than 99%).
(f) The turbulence intensity and turbulent Reynolds number
increase across the shock and reach maximum amplification
factors of 19 and 5, respectively, both occurring at the turning
point of the Hugoniot curve (Mach 13 and degree of dissocia-
tion of 70%). The maximum amplification factor of the tur-
bulence intensity is more than twice the one attainable in a
diatomic calorically perfect gas. The amplification of the tur-
bulent Reynolds number observed here is in contrast with the
attenuation predicted by the theory of calorically perfect gases
at hypersonic Mach numbers.