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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

Physics of Fluids ARTICLE scitation.org/journal/phf

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

Rankine–Hugoniot 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 u‘needs 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 ‘=u‘needs to be much smaller than the molecular diffu-

sion time ‘2= based on the kinematic viscosity , or equivalently, the

turbulent Reynolds number Re‘¼u‘‘= 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

Rankine–Hugoniot 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

P¼P2=P1;T¼T2=T1;R¼q2=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

P¼1þ7

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, P¼1andR¼1, 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

2ð1þaÞaR12

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

@RT¼að1aÞ

2a;(17)

aT¼T @a

@TR¼aR

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

T¼

6R

12abd2ð1aÞbv=ebv=T1

ðÞ

2ðaþ3ÞRð1þaÞ(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

a’0:7, where T’9 (corresponding to 2700 K when T1¼300 K),

M1’13, 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 Rankine–Hugoniot jump condition for a calo-

rically perfect diatomic gas

R311

T

1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þT

9T1

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 CM

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

Rbdþ4T3þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðbdþ4T3Þ2þ2T

q2T(24)

and

C7M2

1R4

ðÞ

2

5R2ð8bd23Þ;(25)

respectively, with bd>23=8 in the conditions tested here. At very

high Mach numbers M1ﬃﬃﬃﬃﬃ

bd

p,whenbd=T1, Eq. (24) simpli-

ﬁes to R4 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 xs¼xhu1itin 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 hu1ihu2iand is therefore deﬁned as xc¼xðhu1i

hu2iÞt. 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;1¼ekhc2isin hcos u;dxy;1¼ekhc2icos hcos u;

dxz;1¼ekhc2isin 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;dv1¼ehc2icos 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 Taylor–Green 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

P¼hP2iþeuhq2ihc2i2

p;q¼hq2ið1þeu

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,

xc¼s, 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 Rankine–Hugoniot 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

Rayleigh–Hugoniot 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

2M2R1

ðÞ

ps

u1;(39a)

us¼1þC

2M2

psþ

u1;(39b)

vs¼

v1M

2R1

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 ky‘T1. In these relations,

u1¼u1=ðeuhc2iÞ and

v1¼v1=ðeuhc2iÞ are 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

xc¼s. This gives a pressure ﬂuctuation

ps¼2M2=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 tðkyhc2iÞ1, 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, x¼fﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1M

2

2

qis the dimen-

sionless frequency, where fis a frequency parameter deﬁned as

f¼M2R

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1M

2

2

q

kx

ky

! (41)

with kx=ky¼1=jtan hj.Caseswithf1 correspond to sufﬁciently

small streamwise wavenumbers, kxkyﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1M

2

2

q

=ðM2RÞ,

whereas the opposite (sufﬁciently large streamwise wavenumbers)

holds for f1. The corresponding amplitudes of the pressure wave

(40) are

Pl1¼ ð1R

1Þðrbf2rcÞ

f2ð1f2Þþðrbf2rcÞ2f2RM2

2

1M

2

2

!

;(42a)

Pl2¼ð1R

1Þfﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1f2

p

f2ð1f2Þþðrbf2rcÞ2f2RM2

2

1M

2

2

!

;(42b)

Ps¼ ð1R

1Þ

fﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

f21

pþrbf2rc

f2RM2

2

1M

2

2

!

;(42c)

where r

b

and r

c

are auxiliary factors deﬁned as

rb¼1þC

2M2

;rc¼RM2

1M

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

x¼xaM

2ja. Upon substituting this relation into (45),the

expressions

ja¼M2x6ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

x21þM

2

2

q

1M

2

2

;(46a)

xa¼x6M2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

x21þM

2

2

q

1M

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<ð1M

2

2Þ1=2.

Different forms of the solution arise depending on the value of

the dimensionless frequency x. At frequencies x<ð1M

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>ð1M

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

T¼ðThT2iÞ=ðeuhT2iÞ is 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=Ps¼2ð1þaÞaR12

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

xc1Þ¼1þaþaR

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

xc1Þ¼ahai

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

Xl¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

X2¼ðR1Þð1CÞ=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

2þ1M

2

2

f2X;(57a)

Vr¼jr

1þj2

r

X¼fM2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1M

2

2

q

M2

2þ1M

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

2j¼c2ej

uaþ

urjsin hsin u;(60a)

jv0

2j¼c2ej

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

1M

2

2

q

M2

2R2þf21M

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

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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 Rankine–Hugoniot 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

1hu1ihu2i

ðÞ

@ns

@y;(67)

which represents the dimensional counterpart of the linearized

Rankine–Hugoniot 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

(a’0:7;T’9;M1’13, and R’12) with a value I2=I1’19,

as shown in Fig. 10(a). This is in contrast to the maximum value

I2=I1’8 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‘;2‘2=2

u‘;1‘1=1¼K1=2

T0:7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

(a’0:7;T’9;M1’13, and R’12) with a value of

Re‘;2=Re‘;1’5. 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 M1’20,

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

w2i2h

u2i

h

v2iþh

w2iþ2h

u2i¼12KL

KLþKT

(70)

with 1W1. The cases W¼1 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;1i¼Rþ3Wz

4(72)

with

Wz¼hx02

z;2i

hx02

z;1i¼ð1

0

X2R2M2

2PðfÞ

R2M2

2þð1M

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

1M

2

2

q

M2

2R2þf21M

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

M1ⱗ10 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 Rankine–Hugoniot 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 M1’13 (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

ha02i¼Aeð1

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.

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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.

(g) The density variance in the post-shock gas is almost exclu-

sively generated by entropic modes radiated by the shock

wrinkles and nearly doubles the value predicted for calorically

perfect gases. Similarly, the shock front generates entropic

ﬂuctuations of the concentration of atomic species that might

be relevant for applications in supersonic combustion if the

post-shock gas is going to be employed to oxidize the fuel.

14,76

The LIA predictions for the overall TKE ampliﬁcation factor in

calorically perfect gases have been previously found to be in fair agree-

ment with numerical simulations

36,38,39,49

and experiments.

33

However, the way LIA predicts the ampliﬁed TKE to be partitioned in

the streamwise and transverse directions has not been as successful. In

particular, computational and experimental studies at supersonic

Mach numbers have often reported Reynolds stress tensors with domi-

nant streamwise contributions, this being an effect typically attributed

to convective non-linearities and molecular transport.

49

Whether these

discrepancies subside or persist at hypersonic Mach numbers is an

open question of research.

The theoretical results provided here indicate ampliﬁed levels of

post-shock ﬂuctuation energies that could perhaps be unexpected at

ﬁrst, because of the high post-shock temperatures prevailing at hyper-

sonic Mach numbers. These ﬁndings would greatly beneﬁt from com-

parisons with simulations and experiments in future studies.

This theory could be extended to include additiona