Review of shock-turbulence interaction with a focus on hypersonic flow

Abstract

Hypersonic flight involves a variety of complex flow phenomena that directly impact the aerothermodynamic loading of high-speed vehicles. The turbulence encountered during a typical flight trajectory influences and interacts with the shock waves on and around the surface of a vehicle and its propulsion system, affecting both aerodynamic and power plant performance. These interactions can be studied by isolating a turbulent flow convected through a normal shock, commonly referred to as the canonical shock-turbulence interaction (STI) problem. Scale-resolving computational fluid dynamics (CFD) and linear interaction analysis (LIA) have been crucial in studying this problem and formulating scaling laws that explain the observed behavior. In this work, an extensive review of the theoretical (LIA) and numerical (CFD) work on the canonical STI is presented. The majority of the work conducted to date has focused on calorically perfect gases with constant heat capacities. However, in hypersonic flows, chemical and thermal non-equilibrium effects may alter the nature of the interaction. As a result, relevant LIA and CFD studies addressing high-enthalpy phenomena are also succinctly discussed.

Full text

Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow
Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow
Alberto Cuadra,1Mario Di Renzo,2, 3 Jimmy-John O.E. Hoste,4Christopher T. Williams,3Marcos Vera,1and
César Huete1, a)
1)Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid,
Leganés 28911, Spain
2)Department of Engineering for Innovation, University of Salento, Lecce 73100,
Italy
3)Center for Turbulence Research, Stanford University, Stanford, 94305, USA
4)Destinus SA, Payerne, Switzerland
(Dated: 17 February 2025)
Hypersonic flight involves a variety of complex flow phenomena that directly impact the aerothermodynamic
loading of high-speed vehicles. The turbulence encountered during a typical flight trajectory influences and
interacts with the shock waves on and around the surface of a vehicle and its propulsion system, affecting
both aerodynamic and power plant performance. These interactions can be studied by isolating a turbulent
flow convected through a normal shock, commonly referred to as the canonical shock-turbulence interaction
(STI) problem. Scale-resolving computational fluid dynamics (CFD) and linear interaction analysis (LIA)
have been crucial in studying this problem and formulating scaling laws that explain the observed behavior.
In this work, an extensive review of the theoretical (LIA) and numerical (CFD) work on the canonical STI is
presented. The majority of the work conducted to date has focused on calorically perfect gases with constant
heat capacities. However, in hypersonic flows, chemical and thermal non-equilibrium effects may alter the
nature of the interaction. As a result, relevant LIA and CFD studies addressing high-enthalpy phenomena
are also succinctly discussed.
NOMENCLATURE
Roman Symbols
ℓIntegral length scale of the upstream turbulence
ℓTThickness of the post-shock thermochemical
nonequilibrium region
ITurbulent intensity
KSimilarity parameter
PPressure ratio
RDensity ratio
TTemperature ratio
UPost-shock streamwise velocity perturbation
amplitude
VPost-shock spanwise velocity perturbation am-
plitude
VSGS
ij Subgrid viscous stress
WO2to N2molecular weight ratio
Da Damköhler number
LλTaylor microscale length
PProbability-density distribution
ReℓTurbulent Reynolds number
ReλTaylor microscale Reynolds
e
Sij Favre-averaged strain-rate tensor
cSpeed of sound
cpSpecific heat at constant pressure
a)Electronic mail: chuete@ing.uc3m.es
cvSpecific heat at constant volume
dDegrees of freedom
EEnergy spectrum
eInternal energy
hEnthalpy
KTurbulent kinetic energy amplification factor
kWavenumber amplitude
koCharacteristic wavenumber
PPressure
RgSpecific gas constant
Rij Reynolds’ stress tensor
TTemperature
tTime
uStreamwise/longitudinal velocity
vSpanwise/transverse velocity
WMolecular weight
XMolar fraction
x,y,zSpatial coordinates
YMass fraction
M1Shock Mach number
MtTurbulent Mach number
kWavenumber vector
vVelocity vector
Greek Symbols
βDownstream to upstream sound speed ratio
χUpstream entropic to rotational perturbation
amplitude ratio
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 2
δνThickness of the post-shock viscous dissipation
region
δvib Thickness of the post-shock vibrational relax-
ation region
ηUpstream dilatational to solenoidal kinetic en-
ergy ratio
η′Upstream dilatational to total kinetic energy ra-
tion
ηkKolmogorov length scale
ΓNormalized slope of the Hugoniot curve
γAdiabatic index, or heat capacity ratio
ˆσij Viscous stress tensor
µDynamic viscosity
νKinematic viscosity
ωVorticity
ΦEnergy spectrum tensor
ϕPhase angle
ρDensity
τDimensionless time
τSGS
ij Subgrid stress tensor
ΘCharacteristic temperature
θWave angle
εaUpstream acoustic perturbation amplitude
εeUpstream entropic perturbation amplitude
εrUpstream rotational perturbation amplitude
ξsShock displacement
ζFrequency
ωVorticity vector
Subscripts and superscripts
1Upstream condition
2Downstream condition
aAcoustic
eEntropic
rRotational
vib Vibrational
Acronyms
rms Root mean square
CEA Chemical Equilibrium with Applications
CFD Computational Fluid Dynamics
CFL Courant-Friedrichs-Lewy
CPU Central Processing Unit
CT Combustion Toolbox
DNS Direct Numerical Simulation
DSM Dynamic Smagorinsky Model
ENO Essentially-Non-Oscillatory
EoS Equation of State
FD Finite-Difference
GPU Graphics Processing Unit
HIT Homogeneous Isotropic Turbulence
HTR Hypersonic Task-based Research
ISA International Standard Atmosphere
LES Large Eddy Simulation
LIA Linear Interaction Analysis
NASA National Aeronautics and Space Administration
NASP National Aerospace Plane
RANS Reynolds-Averaged Navier-Stokes
RH Rankine-Hugoniot
SDT Spatially Developed Turbulence
SGS Subgrid-Scale
STI Shock-Turbulence Interaction
TENO Targeted Essentially Non-Oscillatory
TKE Turbulent Kinetic Energy
WENO Weighted-ENO
I. INTRODUCTION
Hypersonic flows encompass a wide range of complex
physical phenomena, including shock waves, chemical
and thermal non-equilibrium, radiation, and plasma for-
mation.1In the continuum flow regime, turbulence may
also play a major role. For instance, hypersonic vehicles
such as the National Aerospace Plane (NASP) are ex-
pected to develop turbulent boundary layers along their
flight trajectory, as illustrated in Fig. 1 of Settles and
Dodson.2Accurately capturing the effects of turbulence
and its interaction with these phenomena, particularly
shock waves, is crucial for assessing reliable predictions
of hypersonic vehicle performance throughout the entire
mission.
As pointed out by Theofilis et al.,3the least under-
stood region of the flight envelope ranges from 40 to 70
km in altitude, where thermochemical non-equilibrium
effects can be relevant. In order to further a funda-
mental understanding of these types of flows, turbulence
must be described through scale-resolving simulations,
which introduce substantial challenges for computational
fluid dynamics (CFD). In addition to the high demands
on CPU/GPU resources and the limitations of current
computational architectures, there is a critical need to
develop numerical methods and frameworks that can
effectively capture the combined effects of turbulence,
shock waves, and thermochemical processes. The need
for CFD-based studies on hypersonics is apparent, as ex-
perimentally reproduced equivalent flight conditions are
strongly dependent on the type of test facility and the
associated test time limitations.4,5
Research codes specifically tailored to a set of physics
of interest have historically been used for fundamental
studies. For example, the Hybrid code6,7 was devel-
oped with turbulence and shock waves in mind, with the
spatial discretization implemented using high-resolution
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 3
finite-difference schemes. In recent years, new such finite-
difference numerical frameworks were developed taking
advantage of the rapidly evolving GPU architectures.8–13
These developments are key in enabling the study of
more complex hypersonic flow problems by continuously
pushing the boundaries of flow conditions (Reynolds
number) and multi-physics (degrees of freedom). This
has led to an increased interest in scale-resolving studies
of turbulent shock wave/boundary layer interactions14–17
as well as hypersonic boundary layers with thermochem-
ical effects10,15,18. In these studies, it is crucial to isolate
the various thermo-physical flow phenomena involved in
order to draw meaningful conclusions, as exemplified by
Larsson et al.19 Thus, while boundary layers play a key
role in the aero-thermomechanical design of hypersonic
vehicles and associated heat management,20 the funda-
mental interaction between shock waves and turbulence
also warrants analysis through isolated studies.
The canonical shock-turbulence interaction (STI)
problem can be considered the simplest representation of
turbulence interacting with a shock wave, isolating the
physics from surrounding effects such as boundary layers,
cross-flows, or traveling pressure waves induced by com-
bustion. As depicted in Fig. 1, this setup provides a con-
trolled environment to investigate fundamental physics
relevant to supersonic and hypersonic flows. The ad-
vent of accessible computational resources, starting in
the early 1990s, has enabled the use of scale-resolving
simulations to further our understanding of this isolated
problem. Numerical studies have proven crucial, as ex-
periments of such interactions are extremely difficult to
realize because of the need to measure pre- and post-
shock turbulence states while controlling the shock front
position. Since the pioneering work of Lee et al.,21–23
the prescribed complexity of the interaction has incre-
mentally increased, driven by the increase in available
computational resources. Scale-resolving simulations of
the canonical STI have not only improved our current
fundamental understanding but have also informed mod-
eling for practical engineering problems.24–34
Before addressing the key issues that characterize STI
problems in the hypersonic regime, it is important to first
define the canonical configuration. This requires a pre-
cise characterization of the turbulence properties ahead
of the shock, as illustrated in Fig. 1. Specifically, the na-
ture of the incoming turbulence must be described by its
most critical parameters: turbulence intensity and char-
acteristic lenght scale. The compressibility level of the
turbulence, which plays a significant role in hypersonic
STI, will be discussed later, while anisotropy effects will
not be considered in this work.
In the canonical STI, where only upstream solenoidal
disturbances are considered, the levels of the incoming
turbulence fluctuations are characterized by the turbu-
lent Mach number, Mt, defined as
Mt=√Rkk
˜c=√2TKE
˜c=√3urms
˜c,(1)
where the quantities involved are the trace of the
Reynolds stress tensor (Rkk), the Favre-averaged speed
of sound (˜c), the turbulent kinetic energy (TKE), and
the root mean square velocity fluctuations (urms). The
higher the Mtfor a given shock strength, the stronger
the incoming fluctuations. In addition to Mt, another
dimensionless quantity is required to fully characterize
the turbulence. This is typically the Taylor microscale
Reynolds number, defined as
Reλα=ρurmsLλα
µ,(2)
in terms of the time-averaged density, ρ, and dynamic
viscosity, µ, and the Taylor microscale
Lλα=λα=qu′
α2
qu′
α,α2
,(3)
where u′
αdenotes velocity fluctuations in direction α. In
this expression, Einstein notation is adopted, and u′
α,α
represents the derivative of velocity fluctuation u′
αwith
respect to the spatial direction α. Each flow direction
can be associated with its own Taylor microscale and
corresponding Reynolds number. In the case of isotropic
turbulence considered here, this reduces to a single value,
which is why the following discussion will only use sin-
gular references for it.
In accordance with every Reynolds number definition,
the lower the Reλ, the more viscous the flow. It is also
possible to use an integral scale Reynolds number, ReL
(or ReT) instead, which typically relies on the dissipa-
tion length scale, Lϵ. The latter length scale also has
different definitions throughout the literature but gener-
ally relates to Rkk (or TKE) and the dissipation rate of
TKE, ϵ. The above two dimensionless numbers, Mtand
Reλα, are used to characterize the upstream turbulence
when it is composed of solenoidal disturbances only. In
addition, the shape of the turbulent kinetic energy spec-
trum, which provides information on how much kinetic
energy is contained in eddies with a given wavenumber
or length, should also be taken into account.
In terms of analysis, the three-dimensional flow prob-
lem is often reduced to one dimension by performing
spatial and temporal averaging (in that order) at each
streamwise planar location within the domain. Various
flow statistics, including TKE, are analyzed, as discussed
in Sec. III A. For TKE in particular, its evolution is best
understood by examining its budget, which is derived
from the trace of the Reynolds stress tensor transport
equations, as outlined by Lee et al.21
The definition of the Reynolds’ stress components is
given in terms of the Favre average
Rij =
]
u′′
iu′′
j=ρu′′
iu′′
j
¯ρ.(4)
The mechanisms underlying the amplification of
Reynolds stress components and TKE are extensively
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 4
FIG. 1. Illustration of the canonical shock-turbulence interaction problem.
discussed in the literature (see, for example, 21, 22, 35–
37). The energy spectrum tensor
Φij =1
(2π)2ZRij(r) e−k·rd3r(5)
is the Fourier transform of the velocity correlation tensor
Rij. The advantage of Fourier space is that it can be used
to define the energy spectrum function, which under the
isotropic assumption employed in this work, takes the
form
E(k) = 2πk2X
i
Φii(k).(6)
When evaluating the interaction of the turbulence box
with the shock plane (see Fig. 1), the relevant param-
eters include the shock strength—represented by the
shock Mach number M1=u1/c1—as well as the tur-
bulence flow properties defined above. When the Mach
number is sufficiently high, as in hypersonic flows where
M1≥5, turbulence compressibility effects can become
significant, directly impacting the behavior and struc-
ture of the turbulence ahead of the shock. For suffi-
ciently small turbulent fluctuations, compressible turbu-
lence can be approximated as a linear superposition of
three types of modes (or waves): acoustic (sound-wave),
vorticity (also known as solenoidal), and entropy modes.
This classification stems from Kovásznay’s work,38 which
involved linearizing the Navier-Stokes equations in terms
of small perturbations (with respect to the mean). The
analysis revealed several linearly independent solutions,
which were categorized into the three aforementioned
categories. The interaction of any of these waves with
a shock wave leads to the generation of all three modes
through non-linear processes. The linear independence of
these modes (in a first-order approximation for Mt≪1)
forms the basis of Linear Interaction Analysis (LIA), a
technique first introduced by Ribner39,40 and Moore,41
and later extended by Chang42 to account for entropy
waves. Further details are provided in Sec. II.
LIA has become a useful tool in the analyses of canon-
ical STI as it provides rapid estimations against which
DNS or other numerical results can be compared. It can
also be used in a predictive manner, but due to the un-
derlying assumptions, its validity to the desired interac-
tion (turbulence type and shock strength) must be care-
fully evaluated first. For instance, for isotropic turbulent
flows, LIA finds that the TKE amplification factor does
not depend on the shape of the spectrum E(k), nor on
Reλαor Mt, the only independent parameters being the
shock intensity M1and the gas compressibility through
the specific heat ratio γ. Also noteworthy is the adop-
tion of Rapid Distortion Theory (RDT) to study the STI
problem analytically;43–45 however, its assumptions are
even more restrictive than those of LIA.37,46
Turbulence compressibility is not the only factor
that distinguishes STI in hypersonic flow conditions.
High-temperature effects, such as vibrational excitation,
molecular dissociation, and ionization, may also become
significant as the Mach number and stagnation enthalpy
exceed certain thresholds.47 However, most canonical
STI work to date has adopted constant heat capacities,
i.e., a calorically perfect gas behavior, which may not be
accurate when high-enthalpy effects, such as vibrational
excitation or chemical non-equilibrium, become relevant.
In a range of flight conditions, these effects may coexist
with turbulent flows (see, e.g., Fig. 1 of Longo et al.1).
As a result, it is also important to study, understand, and
characterize the influence of non-equilibrium gas effects
on the canonical STI.
When the turbulent Mach number is not sufficiently
small, and/or the characteristic turbulence scales are
comparable to the size of the thermochemical non-
equilibrium region, the limitations of LIA hinder its ap-
plicability, making numerical simulations necessary. The
limitations in LIA lead to a cut-off in the turbulence
spectrum E(k), where the effect of the smallest eddies
(corresponding to the largest values of k) is not consid-
ered, as occurs in LIA applied to detonation-turbulence
interaction.48–50 In other words, for LIA to remain valid,
variations in the upstream Mach number should be small
compared to the mean incoming Mach number, and the
residence time at the shock plus non-equilibrium region
should be small compared to the turbulence time scale.
In fact, detonation-turbulence interaction exhibits strong
parallels to the shock-turbulence interaction problem, es-
pecially when relaxation times are of the same order. In
such cases, DNS has proven effective in capturing the
key turbulence properties in the post-shock reacting re-
gion. These studies reveal that turbulence accelerates
the consumption of fuel and oxidizer by enhancing the
production of intermediate radicals.51–54 Similarly, the
primary motivation of this work is to investigate the ul-
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 5
timate role of turbulence in hypersonic phenomena in air
and, conversely, the impact of hypersonic effects on the
post-shock turbulence field.
This study is inspired by the efforts of the NATO-
STO AVT-352 working group on hypersonic turbulence.
The group’s contributions range from characterizing ex-
perimental hypersonic facilities to applying CFD to tur-
bulence challenges. This work focuses on the funda-
mental physics of compressible turbulence in hypersonic
conditions from a theoretical and numerical perspective.
On the theoretical side, we employ a linear interaction
analysis framework built on previous works.47,55,56 This
framework enables LIA for multi-component mixtures
by using the Combustion Toolbox57 to incorporate com-
pressible and thermochemical effects. Theoretical (LIA)
and numerical (DNS) results are presented across a wide
range of Mach numbers to characterize the influence of
upstream turbulence compressibility in the hypersonic
regime.
The paper is structured as follows. Section II discusses
the LIA developments that enable the study of canonical
STI for both calorically perfect and imperfect gases. Sec-
tion III focuses on CFD work, offering a comprehensive
overview of DNS and LES studies related to canonical
STI, along with recent advancements addressing hyper-
sonic thermochemical non-equilibrium effects. Finally,
the conclusions are presented in Sec. IV. This work aims
to provide a comprehensive review of the canonical STI
problem, serving as a foundation for further fundamental
studies and explorations.
II. LINEAR INTERACTION ANALYSIS
The section is structured as follows. First, the con-
cept of linear interaction analysis (LIA) is introduced
in Sec. II A within the context of the canonical shock-
turbulence interaction (STI) problem: an isotropic sp ec-
trum of velocity, density and pressure weak disturbances
impacting on a planar shock wave. This includes the
mathematical formulation and quantities of interest for
such studies. This is followed by Sec. II B, which exam-
ines LIA developments specifically adapted to hypersonic
regimes.
A. LIA in canonical STI
1. Mathematical Formulation
To characterize the interaction of a shock wave with
incoming turbulence, it is necessary to first consider the
nature of the turbulent disturbances approaching the
shock. Following Kovásznay’s decomposition,38 these
disturbances can be categorized into three fundamental
types: vortical, entropic, and acoustic fluctuations. Each
type of upstream fluctuation generates a distinct set of
post-shock vortical, entropic, and acoustic perturbation
eigenmodes. To examine how these disturbances inter-
act with and are affected by the shock, we employ the
LIA formalism introduced by Ribner39,40 and Moore.41
LIA decomposes the weak turbulent field ahead of the
shock into a Fourier superposition of statistically inde-
pendent, infinitesimally small, single-mode shear waves.
For the sake of clarity, this subsection focuses initially on
the vortical disturbance case, as it has been extensively
studied and provides a straightforward introduction to
the problem. The analysis closely follows the approach
outlined in Huete et al.47
Hereafter, we define a Cartesian coordinate system
where the unperturbed shock lies in the (y′,z′) plane,
and the x′-axis aligns with the mean streamwise direc-
tion of the incoming flow. If the pre-shock turbulence is
isotropic, there is no preferred direction for the wavenum-
ber vector k, so the amplitude of the vorticity modes
depends solely on k=|k|. Likewise, there is no fa-
vored orientation of the wavenumber vector relative to
the shock surface. In principle, this would require a
three-dimensional formulation to describe the interac-
tion. But by simply rotating the reference frame, the
problem can be reduced to two-dimensions. Thus, for
an incident wavenumber vector oriented arbitrarily in
space, defined by the latitude and longitude angles θand
φ, the reference frame can be rotated counterclockwise
around the x′-axis by an angle ψ, corresponding to the
longitudinal inclination of the incident wave, as shown in
Fig. 2. Use of the new reference frame (x,y,z) reduces
the interaction problem to two dimensions, eliminating
all variations with respect to z.
After the aforementioned rotation, the wavenumber-
vector components in the streamwise and transverse di-
rections are
kx=kcos θ, ky=ksin θ, (7)
respectively, with kz= 0 by construction. Similarly, in
the laboratory reference frame, the vorticity vector of the
incident wave in the pre-shock gas can be expressed as
ω1= (δω1)ei(kxx+kyy),(8)
with
δωx,1=−εk⟨c2⟩sin θcos φ, (9a)
δωy,1=εk⟨c2⟩cos θcos φ, (9b)
δωz,1=−εk⟨c2⟩sin φ, (9c)
being the vorticity amplitude in each direction. In this
formulation, ⟨c2⟩denotes the mean speed of sound in the
post-shock gas, and εis a dimensionless velocity fluctua-
tion amplitude, which is small in the linear theory, ε≪1.
The vorticity of the incident wave engenders a fluctu-
ating velocity field in the pre-shock gas given by
v1= (δv1)ei(kxx+kyy),(10)
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 6
FIG. 2. Simplification of a three-dimensional problem of a
shock interacting with an arbitrarily oriented vorticity wave
to a two-dimensional problem by rotating the reference frame
around the streamwise axis. Repro duced from C. Huete, A.
Cuadra, M. Vera, and J. Urzay. Phys. Fluids 33, 086111
(2021), with the permission of AIP Publishing.
whose amplitude is
δu1=ε⟨c2⟩sin θsin φ, (11a)
δv1=−ε⟨c2⟩cos θsin φ, (11b)
δw1=ε⟨c2⟩cos φ(11c)
in the x,y, and zdirections, respectively. Specifically,
the z-component of the fluctuation velocity vector is uni-
form along z. This component will not be considered fur-
ther in the analysis, as it passes through the shock unal-
tered due to the conservation of tangential momentum.
Additionally, note that (8) and (10) are related by the
definition of vorticity, ω1=k×v1. Moreover, the veloc-
ity field given by (10)–(11) satisfies the incompressibility
condition, k·v1= 0. Lastly, 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, 2π|v1|/(kν1), is infinitely large.
To illustrate the analysis, a particular form of the pre-
shock vorticity fluctuation corresponding to the inviscid
Taylor-Green vortex
ωz,1(x, y) = εr⟨c2⟩k2
kycos (kxx) sin (kyy)(12)
is employed in the numerical results presented below,
with ωx,1=ωy,1= 0. The corresponding streamwise
and transverse components of the velocity fluctuations
in the pre-shock gas are given by
u1(x, y) = εr⟨c2⟩cos (kxx) cos (kyy),(13a)
v1(x, y) = εr⟨c2⟩kx
kysin (kxx) sin (kyy),(13b)
respectively. In this formulation, εris the amplitude of
the pre-shock streamwise velocity fluctuations
εr=εsin θsin φ, (14)
with εr≪1in the linear theory. In this linear theory,
the vorticity and the streamwise and transverse velocity
components in the post-shock gas reference frame are
expanded to first order in εras
ω=εrky⟨c2
2⟩ˆω, u =εr⟨c2⟩ˆu, v =εr⟨c2⟩ˆv, (15)
respectively, with ˆω,ˆu, and ˆvbeing the corresponding
dimensionless fluctuations. The post-shock pressure and
density can be similarly expressed as
P=⟨P2⟩+εr⟨ρ2⟩⟨c2⟩2ˆp, ρ =⟨ρ2⟩(1 + εrˆρ),(16)
with ˆpand ˆρbeing the dimensionless fluctuations of pres-
sure and density, respectively. Angle brackets indicate
time-averaged quantities associated with perturbation-
free shock jump conditions. In this way, all fluctuations
are defined to have a zero-time average. Assuming that
the Reynolds number of the post-shock fluctuations is in-
finitely large, expansions (15)–(16) can be used to express
the linearized Euler conservation equations, which can be
combined into a single two-dimensional periodically sym-
metric wave equation for the post-shock pressure fluctu-
ations. If the transient evolution is sought, this sound-
wave equation is integrated within the spatio-temporal
domain bounded by the leading reflected sonic wave trav-
eling upstream and the shock front moving downstream.
For the long-time asymptotic evolution, the analysis can
be simplified to that of obtaining the amplitudes of the
different modes through the normal mode analysis, which
necessitates from the distorted shock boundary condi-
tion.
The boundary condition at the shock front is obtained
from the linearized Rankine-Hugoniot jump conditions
assuming that the displacement of the shock ξs=ξs(y, t)
from its average flat shape is much smaller than k−1
y. In
this limit, at any transverse coordinate, the Rayleigh-
Hugoniot jump conditions can be applied at the mean
shock front location, and can be linearized about the
mean thermochemical-equilibrium post-shock gas state,
namely R=ρ2/ρ1,P=P2/P1, and M2=u2/c2, given
by
R=(γ+ 1)M2
1+γ−1
(γ−1)M2
1+ 2 ,P=2γM2
1−γ+ 1
γ+ 1 ,(17)
and M2=M1/√RP for an ideal gas EoS in thermo-
chemical equilibrium. Additionally, the inverse of the
slope of the Hugoniot curve normalized with the slope of
the Rayleigh line
Γ = −P2−P1
1/ρ1−1/ρ2d(1/ρ2)
dP2
=γM2
1
R2∂P
∂R−1
(18)
takes the form Γ = M−2
1for an ideal gas in thermochem-
ical equilibrium. These expressions can be used to write
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the linearized RH equations as follows:
∂ˆ
ξs
∂τ =R(1 −Γ)
2M2(R − 1) ˆps−ˆu1,(19a)
ˆus=1 + Γ
2M2
ˆps+ ˆu1,(19b)
ˆvs= ˆv1− M2(R − 1) ∂ˆ
ξs
∂ˆy,(19c)
ˆρs=Γ
M2
2
ˆps,(19d)
where τis the dimensionless time coordinate and (19c)
corresponds to the conservation of tangential velocity.
In (19), the function ˆ
ξs=kyξs/εrrepresents the dimen-
sionless shock displacement, whereas ˆps,ˆρs,ˆus, and ˆvs
are, respectively, the dimensionless fluctuations of pres-
sure, density, streamwise velocity and transverse velocity
immediately downstream of the shock front.
In a reference frame that moves with the shock front
immediately downstream the perturbed shock surface
(denoted by subscript 2), a corresponding set of pertur-
bations is generated. For the velocity components and
pressure, these downstream perturbations are evaluated
as
ˆu2s(τ) = δu2s
εrc2
= (Ur+Ua) cos (ωr
sτ+ϕ),(20a)
ˆv2s(τ) = δv2s
εrc2
= (Vr+Va) sin (ωr
sτ),(20b)
ˆp2s(τ) = δ P2s
εrρ2c2
2
=Psin (ωr
sτ),(20c)
with ωr
s=RM2/tan θ. The amplitudes of the post-
shock modes Ur(θ),Ua(θ),Vr(θ),Va(θ), and P(θ)are
unknown functions of θto be determined with the aid of
the shock boundary condition and the isolated-shock as-
sumption. In this work, the far-field amplitudes are com-
puted in the asymptotic limit M2τ≫1and sufficiently
far from the shock, as discussed in Ref. 47. Note that in
(20), the transverse coordinate dependence is omitted,
∼cos(kyy)for the longitudinal and ∼sin(kyy)for the
transverse velocity field, as it is not affected by the shock
passage in the linear regime where the transverse period-
icity is not broken. To distinguish between the different
contributions, the subscripts r, a refer to the nature of
the perturbation: rotational or acoustic.
2. Turbulent Kinetic Energy amplification factor
The weak isotropic turbulence in the pre-shock gas can
be modeled as a linear superposition of incident vortic-
ity waves, with amplitudes εthat vary according to the
wavenumber, consistent with an isotropic energy spec-
trum E(k) = ε2(k). The root mean square (rms) of the
velocity and vorticity fluctuations in the pre-shock gas
can be derived by applying the assumption of isotropy.
This assumption states that the probability of an inci-
dent wave having orientation angles between θand θ+dθ,
and between φand φ+dφ, is proportional to the solid
angle element sin θdθdφ/(4π). This leads to the follow-
ing expressions:
⟨u′2
1⟩
ε2⟨c2⟩2=1
3,⟨v′2
1⟩
ε2⟨c2⟩2=1
6,⟨w′2
1⟩
ε2⟨c2⟩2=1
2(21)
for the pre-shock rms velocity fluctuations. The TKE
amplification factor across the shock wave is defined as
K=⟨u′2
2⟩+⟨v′2
2⟩+⟨w′2
2⟩
⟨u′2
1⟩+⟨v′2
1⟩+⟨w′2
1⟩=⟨u′2
2⟩+⟨v′2
2⟩
ε2⟨c2⟩2+1
2(22)
=1
2"Zπ/2
0ˆu2+ ˆv2sin3θdθ+ 1#,(23)
where use of (21) has been made. Furthermore, Kcan
also be decomposed linearly into acoustic and vortical
modes as K=Ka+Kr, with
Ka=1
3Z∞
1U2
a+V2
aP(ζ)dζ=1
3Z∞
1
Π2
sP(ζ)dζ,
Kr=1
2+1
3Z∞
0U2
r+V2
rP(ζ)dζ. (24)
The entropic mode does not contain any kinetic energy,
since entropy fluctuations are decoupled from velocity
fluctuations in the inviscid linear limit. In equation (24),
P(ζ)is a probability-density distribution given by
P(ζ) = 3
2M4
2R4p1− M2
2
[M2
2R2+ζ2(1 − M2
2)]5/2,(25)
which satisfies the normalization condition R∞
0P(ζ)dζ=
1. Additionally, the velocity amplitudes Ua,Ur,Va, and
Vrare obtained using the long-time far-field asymptotic
expressions. The lower integration limit of Kais ζ= 1
because the acoustic mode decays exponentially with dis-
tance downstream of the shock in the long-wave regime
ζ < 1. However, such decaying contribution needs to be
added to Ka, and therefore to K, when evaluating the
solution in the near field.
B. LIA in hypersonic STI
1. Some prior considerations of hypersonic shocks
Hypersonic flows are influenced by the rates governing
internal energy relaxation and chemical reactions. The
relatively slow relaxation of a gas vibrational modes can
interact with the gas dynamics in several ways. Most no-
tably, the vibrational modes absorb energy and alter the
post-shock conditions when they are active. Addition-
ally, the vibrational state of the gas significantly affects
its dissociation rate.58 Even if dissociation is not crucial,
as can often be the case in the hypersonic regime, the
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relaxation time tvib of the vibrational modes is used to
define the thickness of the relaxation layer δvib ∼u2tvib.
In STI, there always exits a post-shock relaxation region
where turbulence is dissipated by viscous effects, with a
characteristic thickness δν. These two relaxation layers
must be compared to determine the characteristic vibra-
tional Damköhler number: Da = δν/δvib =δν/(u2tvib ).
For shock waves with M1≥5, thermochemical non-
equilibrium arises in the post-shock region, where pref-
erential excitation of rotational and translational en-
ergy modes gives rise to finite-rate relaxation of the
vibrational-electronic energy modes. Energy exchange
between these non-equilibrium internal degrees of free-
dom and the rotational-translational modes proves cru-
cial not only for predicting heat fluxes but also for the
evaluation of chemical production rates, with the effec-
tive vibrational excitation modulating the reaction rates
through vibration-dissociation coupling.59 For finite-rate
thermochemical relaxation that proceeds on the advec-
tive timescale, namely Da ≃1, direct numerical solution
of the relevant non-linear conservation laws generally
proves necessary for characterizing the non-equilibrium
flow physics60–62.
The canonical non-hypersonic STI, for which the up-
stream turbulence remains isotropic, can be simply for-
mulated in terms of the dominant dimensionless param-
eters alone. In particular, according to LIA, the TKE
amplification factor Kdepends on the type of gas, as
characterized by its adiabatic coefficient (γ≃1.4for
air), and shock strength, typically through the upstream
Mach number, M1. More realistic conditions can be
studied through numerical simulations, which introduce
two additional dimensionless numbers: turbulent inten-
sity (through the turbulent Mach number, Mt) and the
Taylor microscale Reynolds number, Reλ. Efforts have
been made to consolidate and simplify the parametric
dependencies. For instance, Donzis63,64 defined a novel
parameter K=Mt/[Re1/2
λ(M1−1)] based on similarity
arguments related to the instantaneous shock thickness.
This parameter represents the ratio of the Kolmogorov
length scale to the laminar shock thickness and effec-
tively collapses all available data regarding the stream-
wise component of the TKE amplification factor.37 In the
strong-shock limit, this parametric collapse relies in the
concept of classical dimensional similarity through the
so-called hypersonic similitude. Consequently, the TKE
amplification factor approaches a constant value as the
shock Mach number increases indefinitely.
At low-hypersonic flight speeds, the calorically-perfect
description, in which the gas is treated as having a con-
stant specific heat ratio, γ, can often suffice, with two
perfect gases sharing similar values of γexhibiting phys-
ical similarity. At the hypersonic frontier, the equation
of state (EoS) for an ideal gas almost universally re-
mains applicable; however, even for modest Mach num-
bers, air becomes thermally rather than calorically per-
fect, meaning that γis at least a function of temperature
γ(T). Thus, by simple dimensional analysis, we can con-
clude that the strong temperature dependence exhibited
by some of these phenomena precludes the reduction of
the problem by using parameter Kalone in the hyper-
sonic flow regime. Depending on the relevant flow-field
conditions, to include the freestream pressure, chemical
composition, and Mach number, the characteristic vi-
brational Damköhler number can vary significantly. For
large-scale turbulence, with δvib ≪δν, the separation
of scales allows the vibrational relaxation layer to be
considered infinitesimally thin. Consequently, the spe-
cific heat ratio can be evaluated using the equilibrium
post-shock state, γ(T2) = constant, before viscous dissi-
pation occurs. This situation corresponds to Da → ∞.
When δvib is of the order of δν, the specific heat ra-
tio immediately behind the shock is determined by up-
stream conditions, γ(T1), and its evolution towards γ(T2)
roughly coincides with the process of viscous dissipation,
so thermochemical non-equilibrium phenomena must be
solved along the post-shock turbulence decay, with the
two processes being coupled. In the opposite limit, where
δvib ≫δν, the ratio of specific heats remains constant at
γ=γ(T1)throughout the entire viscous dissipation pro-
cess. This scenario corresponds to Da →0. Note that
further effects associated with dissociation and ioniza-
tion are characterized by different (chemical) Damköhler
numbers, with similar limiting behavior arising in these
phenomena.
2. The Rankine-Hugoniot curve in hypersonics
The simplest method to evaluate the deviation from
the calorically perfect assumption involves consider-
ing the characteristic vibrational temperature of the
molecules. For example, for air in thermal equilibrium,
the specific internal energy behind the shock can be de-
scribed as
e2=cvT2[1 + fv(T2)] = 1
γ−1
P2
ρ2
[1 + fv(T2)] ,(26)
where the auxiliary function fvaccounts for the energy
stored in the molecules in form of vibrational modes.
In the harmonic-oscillator model, which works well suf-
ficiently far from dissociation, the auxiliary functions
reads
fv(T2) = γ−1
YO2+W(1 −YO2)YO2
ΘO2/T2
eΘO2/T2−1
+W(1 −YO2)ΘN2/T2
eΘN2/T2−1,(27)
where W=WO2/WN2= 32/28 = 8/7is the ratio
of molecular weights of molecular oxygen to molecu-
lar nitrogen, and ΘO2= 2270 K and ΘN2= 3390 K
are the corresponding vibrational temperatures, respec-
tively. In the case of air, the mass fraction of oxygen
is YO2= 0.2315, and the diatomic adiabatic index is
γ= 7/5. Note that fv→0as T2/Θv→0. Ahead of
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 9
the shock, where the gas is presumed to be cold (mean-
ing sufficiently cold to neglect vibrational effects), i.e.,
T1≪Θv, the equation for the specific internal energy
reduces to e1= (5/2)RgT1, whereas in the post-shock
gas, where T2∼ M2
1T1, the specific internal energy e2
requires consideration of translational, rotational, and
vibrational degrees of freedom.
The advantage of (26) is that it enables evaluating the
relative importance of vibrational effects by simply com-
paring the corresponding temperature with the value of
Θv. This facilitates discussions grounded in fundamen-
tal properties. However, the equation assumes a perfect
harmonic oscillator model for the molecular vibrations,
which becomes less accurate as vibrations intensify and
approach the dissociation threshold. For air, which is
predominantly composed of O2and N2, the generaliza-
tion of the analysis is straightforward, with molecular
oxygen being the first to exhibit non-negligible vibra-
tional effects.
Although this complicates the physical discussion, for
the quantitative study of these phenomena it is con-
venient to introduce the NASA polynomials to eval-
uate cp(T), and consequently e(T) = [cp(T)−Rg]T,
along with the corresponding specific heat ratio γ(T) =
cp(T)/[cp(T)−Rg], to compute the Rankine-Hugoniot
equations for thermally perfect gases. This approach
offers significant advantages: i) the range of validity is
higher, and ii) most numerical codes used in combus-
tion processes already incorporate NASA polynomials.
In particular, the results presented in this work make
use of the NASA 9-coefficient polynomials database,65
which ranges up to 20 000 K.
As previously discussed, for Mach numbers signifi-
cantly exceeding 5, dissociation and/or ionization effects
must be included in the analysis. To evaluate their rela-
tive importance in the thermochemical equilibrium flow,
an illustrative example is provided in Fig. 3(a), where the
Hugoniot curve (P=P2/P1vs. 1/R=ρ1/ρ2) is plot-
ted for normal shocks in air with pre-shock conditions:
T1= 300 K, P1= 1 atm, and volume % composition
{N2,O2}={79,21}. Full equilibrium calculations have
been carried out with the Combustion Toolbox57,66,67 as-
suming a 26-species mixture using NASA 9-coefficient
polynomials. The thermochemical equilibrium results
are compared with the analytical solution obtained for
a thermochemically frozen gas with constant composi-
tion. It is seen that, due to the endothermicity of the
dissociation processes, a significantly higher compression
ratio is achieved compared to that in the thermochem-
ically frozen case. Figure 3 (b, left y-axis) shows the
temperature jump T=T2/T1vs. upstream Mach num-
ber M1=u1/c1for the same conditions. Both pan-
els exhibit a similar behavior, with both the post-shock
specific volume and temperature jump decreasing signif-
icantly below their frozen flow values because of the fur-
ther increase in density and decrease in velocity. This
trend finishes after a clear turning point associated with
the change in the mean molecular structure. At some de-
(a)
(b)
FIG. 3. Log-log RH curve (a), log-log temperature jump (b,
left y-axis), and linear-log adiabatic index (b, right y-axis), as
a function of the downstream Mach number for normal shocks
propagating in air at T1= 300 K and P1= 1 atm; solid line:
Combustion Toolbox (CT);57,66,67 dashed line: thermochemi-
cal frozen gas approximation; circles: results by CEA code;68
Roman numerals: regions with the dominant reactions la-
beled.
gree of dissociation, endothermic effects associated with
bond-breaking are no longer dominant, and the balance
of diatomic molecules vs. monoatomic species shifts to
the latter. As the flow conditions continue to intensify,
ionization processes play an increasingly significant role,
leading to the production of ionized species and further
modifying the thermodynamic state through additional
energy absorption. If ionization is neglected, the theoret-
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 10
ical maximum compression ratio, d+ 1, decreases from
6 to 4, with dstanding for the degrees of freedom of the
species involved (d= 5 for rigid diatomic molecules and
d= 3 for single atoms). Examining the adiabatic index
γin Fig.3 (b, right y-axis), we can clearly observe the
effect of vibrational excitation, which leads to a lower
adiabatic index due to the increased degrees of freedom.
As the shock intensity—and consequently, the temper-
ature—rises sufficiently to dissociate molecular oxygen,
the adiabatic index increases due to the growing pres-
ence of monatomic oxygen (lower degrees of freedom).
This pattern repeats with the vibrational excitation of
molecular nitrogen, which lowers γ, followed by its sub-
sequent dissociation into monatomic species. Addition-
ally, ionization becomes a significant factor for M1≥30.
The roman numerals have been used in Fig. 3 to identify
the dominant reactions in the different Mach regimes,
starting with oxygen dissociation (region II), followed
by nitrogen dissociation (region III), whose dissociation
temperature is roughly twice that of oxygen, and finish-
ing with their first ionization (region IV). Results have
been compared with NASA’s Chemical Equilibrium with
Applications (CEA) code,68 showing always an excellent
agreement.
The resolution of the STI problem in the hypersonic
regime follows the same steps outlined in the previous
section. However, now the upstream gas is influenced
by more than just vorticity disturbances. Instead, the
shock is perturbed due to a combination of velocity (both
rotational and acoustic), density (entropic and acoustic),
and pressure (acoustic) disturbances. This leads to the
following set of linearized Rankine-Hugoniot equations56
dˆ
ξs
dτ=R(1 −Γ)
2M2(R − 1) ˆp2s−M2R21−2R−1−Γρ
2 (R − 1) ˆρ1s
+1
βˆu1s−1−Γp
2β2M2(R − 1) ˆp1s,(28a)
ˆu2s=1 + Γ
2M2
ˆp2s−M2R(1 + Γρ)
2ˆρ1s+1
βˆu1s
−1 + Γp
2β2M2Rˆp1s,(28b)
ˆv2s=1
βˆv1s− M2(R − 1) ∂ˆ
ξs
∂ˆy,(28c)
ˆρ2s=Γ
M2
2
ˆp2s− RΓρˆρ1s−Γp
β2M2
2Rˆp1s,(28d)
where the following normalized RH-slope parameters are
conveniently introduced
Γ = u2
2
∂ρ2
∂P2P1,ρ1
as in (18), and the additional factors
Γρ=Γ
u2
1
∂P2
∂ρ1P1,ρ2
,Γp= Γ∂P2
∂P1ρ1,ρ2
.(29)
For example, for an ideal gas that is calorically perfect
and is in chemical equilibrium, Γ = M−2
1, as commented
right after (18). The other two parameters reduce to
Γρ=−R−1and Γp=PM−2
1in such conditions. As
shown in Fig. 4 (a), the value of Γbecomes positive along
the upper branch of the Hugoniot curve beyond the turn-
ing point Γ = 0. Along this branch, an increase in post-
shock pressure induces a decrease in post-shock density.
The role of Γ,Γρ, and Γpin the description of the shock-
turbulence interaction problem will be discussed in next
section. These parameters can also be conveniently com-
puted using the Combustion Toolbox.57,66,67
When evaluating (28), it is readily seen that as the
shock strength increases, the ratio 1/β =c1/c2decreases
inversely with M1, diminishing the significance of up-
stream velocity and pressure perturbations relative to
the shock wave. In simpler terms, as the shock speed
and post-shock variables grow unbounded with increas-
ing Mach number, the density jump is the only one
that remains finite. Consequently, only density distur-
bances, whether acoustic or entropic in nature, can sub-
stantially affect the shock as M1→ ∞. However, since
the Mach number is finite, the relative significance of dif-
ferent types of upstream perturbations also depends on
their respective orders of magnitude. In the case of in-
terest, where pressure disturbances in the upstream flow
are expected to be very low and c2
1/c2
2≪1, they can be
safely neglected, and the focus shifts to density perturba-
tions. Thus, it is relatively easy to see that, for upstream
density disturbances to be non-negligible in STI, the fol-
lowing condition must be met
δρ1
ρ1∼c1
c2
2
M2R(1 + Γρ)
δu1
c1
.(30)
For a calorically perfect gas with γ= 1.4and M1= 5,
we find that δρ1/ρ1is non-negligible during shock pas-
sage when it is approximately one order of magnitude
smaller than δu1/c1. As the shock intensity increases,
the threshold for non-negligible density perturbations de-
creases. This conclusion is readily drawn by inspecting
Fig. 4(b), which illustrates the influence of the prefactors
Aρ, Au,and Apderived from the upstream fluctuation
terms in (28b), namely
ˆu2s=1 + Γ
2M2
ˆp2s−Aρˆρ1s+Auˆu1s−Apˆp1s,(31)
as a function of the incident Mach number. These pref-
actors quantify the respective contributions of upstream
density, velocity, and pressure fluctuations, providing in-
sights into the increasing role of density perturbations as
the Mach number grows.
Note that, although the role of upstream density dis-
turbances in STI has been addressed in the literature—
primarily to emulate the interaction of oblique shocks
with turbulent high-speed boundary layers—the key
takeaway is that in hypersonic conditions, upstream den-
sity disturbances must always be carefully evaluated.
Otherwise, the canonical prescription of STI may be in-
accurate. It should be noted that density disturbances
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ahead of the shock may also arise due to numerical tur-
bulence forcing mechanisms.69 Therefore, any compre-
hensive attempt to adequately address STI should take
these disturbances into account.
(a)
(b)
FIG. 4. Linear-log RH-slope parameters (a) and log-log influ-
ence of the upstream fluctuations on the downstream velocity
perturbation [see Eqs. (28b) and (31)](b) as a function of the
upstream Mach number for normal shocks propagating in air
at T1= 300 K and P1= 1 atm. Solid lines: calorically imper-
fect gas; Dashed lines: calorically perfect gas; Roman numer-
als: regions with the dominant reactions labeled in Fig. 3.
3. Influence of the nature of the upstream turbulence
a. Vortical fluctuations In addition to the stan-
dard assumptions of LIA applied previously in canon-
ical STI—specifically, that velocity perturbations are
much smaller than the corresponding speed of sound—
the incorporation of thermochemical effects requires that
the characteristic size of shock wrinkles be significantly
larger than the thickness of the thermochemical nonequi-
librium region behind the shock. The accuracy of this ap-
proximation in practical hypersonic systems is expected
to improve with increasing flight Mach numbers and de-
creasing flight altitudes, as will be discussed in Sec. II C.
This is because the temperature behind the shock in-
creases with the Mach number, while the upstream den-
sity increases at lower altitudes, which in turn enhances
intermolecular collisions and facilitates the rapid achieve-
ment of thermochemical equilibrium. Unlike Ref. 47,
which examined single-species symmetric diatomic gases,
the RH curve in this study is not expressed analytically in
terms of fundamental parameters like rotational, vibra-
tional, or dissociation characteristic temperatures. How-
ever, the current approach incorporates additional com-
plexities, such as recombination into multi-species gases
and ionization.
As before, the weak isotropic turbulence in the pre-
shock gas can be represented by a linear superposition
of incident vorticity waves whose amplitudes εvary with
the wavenumber in accordance with an isotropic energy
spectrum E(k) = ε2(k). The root mean square of the
velocity and vorticity fluctuations in the pre-shock gas
can be calculated by invoking the isotropy assumption.
The TKE amplification factor across the shock wave
is of utmost interest in the interaction of shock waves
with turbulence. By performing the theoretical analy-
sis described in Huete et al.,47 with the details omit-
ted here for brevity, the value of Kcan be expressed as
an integral formula—corresponding to an isentropically
weighted sum of contributions of the vorticity pertur-
bations impinging on the planar shock—that ultimately
depends on the post-shock properties: mass compression
ratio ρ2/ρ1, post-shock Mach number M2, and a non-
dimensional parameter that accounts for the RH-slope Γ,
which are again computed with the aid of the Combus-
tion Toolbox.57,66,67 In our case, we have a multi-species
mixture of gases composed mainly of O2and N2, which
have different characteristic dissociation temperatures.
The resulting curve for Kas a function of the pre-
shock Mach number M1(solid black line) is shown in
Fig. 5. The curve exhibits two distinguished peaks cor-
responding to regions (II) and (III) of Fig. 3. The non-
monotonicity of Kis dictated by the behavior of the
vorticity generation across the shock, since acoustic tur-
bulent kinetic energy is negligible in hypersonic condi-
tions. This effect, not shown explicitly in this work, was
analyzed in detail in Huete et al.47 Two main effects
are found to govern the post-shock perturbation flow:
the mass compression ratio, whose amplification via en-
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 12
FIG. 5. TKE amplification factor Kas a function of the pre-shock Mach number M1(upper solid line) for normal shocks
propagating in low turbulence intensity air with only vortical deviations (χ= 0) at T1= 300 K and P1= 1 atm. The upper
dashed line corresponds to the thermochemical frozen gas approximation. The inset represents the rate of change of the molar
fractions with the pre-shock Mach number dXj/dM1for the most relevant species in the mixture.
dothermicity increases the flow deflection and the gen-
eration of transverse kinetic energy; and the RH-slope,
which is sensitive to the different internal processes that
take place within the non-equilibrium region. The latter
is related to the rate of change of the molar fractions
with respect to the pre-shock Mach number, dXj/dM1,
for the most relevant species in the mixture, which are
also represented in Fig. 5. This is consistent with the ob-
servation by Bottin70 regarding the specific heat capacity
at constant pressure cp, where the author attributed the
local maxima in cpto the dissociation processes of O2and
N2, and their subsequent ionization. A simple method
to isolate the contribution of the dissociation and ion-
ization of species is to ad hoc freeze some of them. For
example, when the dissociation and recombination of N2
is frozen, the first peak of the K-curve still corresponds
to the peak of |dXO/dM1|∼|dXO2/dM1|. In this case,
higher temperatures are reached due to the absence of
the endothermic effects of N2dissociation. As a result,
the ionization of atomic oxygen occurs at lower Mach
numbers, exhibiting the corresponding peak in the TKE
curve.
Another effect excluded from our model, but deserving
further attention, is the second and subsequent ioniza-
tion processes of atomic oxygen and nitrogen, which are
expected to occur at temperatures exceeding 20,000 K—
the upper limit of the NASA polynomials used in this
work. To highlight this limitation, a dashed line is shown
in Fig. 5 when displaying the computations of the TKE
amplification factor above this temperature. By linearly
extrapolating the correlation between the sensitivity of
dissociation and ionization with the Mach number, we
can anticipate that further ionization phenomena, as de-
scribed in Askari71, will result in further lower-amplitude
peaks in the Kvs. M1curve.
Figure 6 depicts the amplification of the turbulent in-
tensity, namely
I2
I1
=uℓ,2/⟨u2⟩
uℓ,1/⟨u1⟩=K1/2R,(32)
and the turbulent Reynolds number
Reℓ,2
Reℓ,1
=uℓ,2ℓ2/ν2
uℓ,1ℓ1/ν1
=K1/2
T0.7r2R2+ 1
3(33)
across the shock as functions of the pre-shock Mach num-
ber M1under the same conditions. In these expressions,
uℓrepresents the velocity fluctuation at the turbulence
integral scale ℓ, and νis the kinematic viscosity of the
fluid. Consistent with Huete et al.,47 the turbulent in-
tensity ratio I2/I1exhibits a sharp increase in regions II
and III as species dissociate; see Fig. 6(a). For example,
for M1= 10, the turbulent intensity is 1.5times higher
than the value predicted for a thermochemically frozen
gas, while at M1≃26, where maximum amplification
occurs, it doubles the predicted value I2/I1≃8. This
maximum corresponds to the turning point of the Hugo-
niot curve (Γ = 0,T ≃ 42, and R ≃ 12). For M1>26,
the turbulence intensity begins to decrease; however, it is
expected to remain higher than the linear extrapolation
due to additional species dissociation, ionization, and re-
combination not accounted for in this study. A similar
trend is observed for the turbulent Reynolds number, as
illustrated in Fig. 6(b).
b. Vortical with entropic fluctuations As discussed
above, the linearized Rankine-Hugoniot equations (28)
reveal that, in hypersonic flows, upstream density distur-
bances may dominate the shock dynamics over velocity
and pressure disturbances regardless of whether they are
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 13
(a)
(b)
FIG. 6. Amplification of the turbulent intensity I2/I1(a)
and the turbulent Reynolds number Reℓ,2/Reℓ,1(b) across the
shock as a function of M1for air at T1= 300 K and P1= 1
atm. The dashed line corresponds to the thermochemical
frozen gas approximation. Roman numerals: regions with
the dominant reactions labeled in Fig. 3.
entropic42 or acoustic41 in nature, a result previously ob-
tained by Mahesh et al.72 and Quadros et al.73 Figure 7
illustrates the variation of the TKE amplification factor
Kwith the pre-shock Mach number M1for different de-
grees of vortical to entropic correlations. In the figure,
both thermochemically frozen (dashed lines) and calori-
cally imperfect (solid lines) gas models are included.
The correlation between vortical (εr) and entropic (εe)
fluctuations is quantified by the parameter χattached
to the curves, with positive and negative values indi-
cating different degrees of correlation or anti-correlation,
respectively. When the entropic fluctuations are of the
same order as the vortical fluctuations, εe∼εr, the de-
gree of correlation is expressed as
⟨χ⟩=⟨δρe
1δur
1⟩
⟨δur
1δur
1⟩⟨c1⟩
⟨ρ1⟩.(34)
By way of contrast, when the entropic fluctuations scale
inversely with the upstream Mach number, εe∼εrM−1
1,
the correlation parameter becomes
⟨χ⟩=1
M1
⟨δρe
1δur
1⟩
⟨δur
1δur
1⟩⟨c1⟩
⟨ρ1⟩,(35)
which emphasizes the increasing dominance of vorti-
cal structures in shock-turbulence interactions at higher
Mach numbers, particularly for M1>5. It is important
to note that the sign convention in our definition of χis
opposite to that commonly used in the literature,72,73
due to the adoption of a different reference frame in
our formulation. The bracket symbols denote mean val-
ues associated with the upstream turbulence properties,
which are considered known inputs for the linear inter-
action analysis (LIA) shown in Fig. 7.
The results shown in the figure indicate that even
relatively small levels of entropic fluctuations, such as
χ∼10−2, can significantly alter downstream turbulence
dynamics. Regarding thermochemical effects, a notice-
able change in the curve shape occurs as these effects
become active, introducing moderate bumps. However,
they play a second-order role in the modulation of the
TKE amplification ratio K. For instance, for M1= 10
and χ=−0.5M−1
1=−0.05,Kincreases by 29% with
respect to the purely solenoidal case (χ= 0), while the
endothermic effects only contribute an additional 8.6%
increase relative to the calorically perfect approximation.
c. Vortical with acoustic fluctuations As mentioned
in the discussion following equation (30), density pertur-
bations play a dominant role in the strong-shock limit,
regardless of their origin. However, due to the differing
correlation properties between steady vortical waves and
traveling acoustic perturbations, their quantitative im-
pact varies depending on the nature of the disturbances.
We define the dimensionless variable η∼ε2
a/ε2
r, which,
in terms of turbulent variables, reads as
η=⟨δu2
1a⟩+⟨δv2
1a⟩+⟨δw2
1a⟩
⟨δu2
1r⟩+⟨δv2
1r⟩+⟨δw2
1r⟩,(36)
which quantifies the relative importance of upstream
acoustic versus rotational perturbations. Due to the
propagating nature of the acoustic disturbances in the
linear regime—unlike the entropic case—the contribu-
tion of upstream compressibility to the total kinetic en-
ergy can be evaluated independently. The total contri-
bution is then the weighted sum of these components
K=⟨δu2
2⟩+⟨δv2
2⟩+⟨δw2
2⟩
⟨δu2
1⟩+⟨δv2
1⟩+⟨δw2
1⟩
=β2⟨ˆu2
2⟩+⟨ˆv2
2⟩+⟨ˆw2
2⟩
⟨ˆu2
1⟩+⟨ˆv2
1⟩+⟨ˆw2
1⟩=Kr+ηKa
1 + η.(37)
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 14
FIG. 7. TKE amplification factor Kas a function of the pre-shock Mach number M1for normal shocks propagating in low
turbulence intensity air with different correlations between vortical and entropic fluctuations χat T1= 300 K and P1= 1 atm.
The dashed lines correspond to the thermochemical frozen gas approximation.
By direct inspection in Fig. 8, where the TKE is scaled
with the Mach number squared, it becomes clear that,
unlike the effect of entropic disturbances analyzed ear-
lier, the contribution of acoustic perturbations can either
increase or decrease the downstream kinetic energy. To
understand this behavior, we need to focus on the ampli-
fication of kinetic energy due solely to upstream acoustic
energy, denoted as Ka. This is depicted in Fig. 6(b) of
Ref. 74, where the solid line crosses unity at two dis-
tinct Mach numbers, M1= 1.11 and M1= 2.35. Un-
der these conditions, there is no net change in turbulent
kinetic energy due to upstream acoustic disturbances,
regardless of the amplitude η, as indicated by the inter-
section of the different curves at these Mach numbers.
In the hypersonic regime, the inclusion of compressibil-
ity effects in upstream turbulence leads to an increase in
turbulent kinetic energy, with an asymptotic contribu-
tion that grows with M2
1. When thermochemical non-
equilibrium effects are considered, a similar trend is ob-
served, though with slightly different Mach numbers at
the intersection points. In addition, hypersonic effects
render the expected double-peak contribution associated
with high endothermicity.
From the study of the previous cases involving vortical
and vortical-entropic fluctuations (see Figs. 5 and 7), it
is evident that thermochemical effects have only a sec-
ondary effect on the amplification of K. However, when
the incoming turbulence is compressible, i.e., η > 0, the
observed amplification is significantly larger compared to
the purely vortical case, where η= 0. In this scenario,
compressibility effects dominate the leading order behav-
ior, underscoring why compressible fluctuations cannot
be neglected a priori in the hypersonic regime, even
when compressibility is less than 10%. In such condi-
tions, the interplay between compressibility and thermo-
chemical effects becomes increasingly relevant, with both
mechanisms contributing substantially to the overall am-
plification dynamics.
C. Flight altitude effects
The flight envelope of hypersonic air-breathing vehicles
presents lower and upper altitude limits beyond which
the vehicle cannot be flown. The lowest altitude limit is
constrained by structural aircraft capabilities, while the
highest altitude is imposed by the engine combustion req-
uisites that demand a minimum amount of oxygen to op-
erate. Briefly speaking, a cruise hypersonic aircraft must
fly between 15–30 km at Mach-5 and 30–45 km at Mach-
15.75 Then, since atmospheric properties are susceptible
to altitude changes of the order of 10 km, it is natural to
wonder how this effect modifies the TKE amplification
factor computed above.
Before computing the function Kvs. M1, it is con-
venient to recall an assumption underlying the LIA: the
thickness ℓTof the thermochemical nonequilibrium re-
gion behind the shock must be much smaller than the
characteristic size of the shock wrinkles, which is of the
same order as the integral length scale of the free-stream
turbulence ℓ. Thus, to properly assess the assumption
ℓ≫ℓT, we must evaluate the characteristic length of
the chemical nonequilibrium region as a function of the
flight altitude for a representative flight Mach number,
say M1= 10. To this end, we use the Hypersonic Task-
based Research (HTR) solver.8,9,76 The HTR solver is a
high-order Navier–Stokes solver targeted towards direct
numerical simulations of chemically-reactive compress-
ible turbulent flows. In particular, we utilize a recent
update of the HTR solver77 that describes thermochemi-
cal nonequilibrium via a two-temperature model.59 This
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 15
(a)
(b)
FIG. 8. Scaled TKE amplification factor KM−2
1as a func-
tion of the pre-shock Mach number M1for normal shocks
propagating in air with different vortical and acoustic devi-
ations η= [0,0.25,0.5,1], at T1= 300 K and P1= 1 atm.
Panel (a) represents the calorically perfect gas solution, while
panel (b) displays results considering a calorically imperfect
gas.
type of model uses two temperatures: T, related to the
translational and rotational energy modes of the chemi-
cal species in the gas, and Tve , related to vibrational and
electronic excitation. The dissociation rate constants are
evaluated at the geometric mean temperature √T Tve.
The chemistry of air (79% N2, 21% O2in volume) is
modeled with a 5-species mixture {N2, O2NO, O, N}
using NASA’s 9-coefficient polynomials. The interested
reader is referred to Di Renzo et al.,8Di Renzo and Piroz-
zoli,9and Di Renzo76 for further details on the numer-
ical solver itself. The calculations are carried out in a
one-dimensional computational domain discretized with
1200 grid points. Supersonic inflow conditions are im-
posed at the upstream boundary, while the downstream
boundary features a characteristic multi-component non-
reflecting outflow boundary condition.62,78 Calculations
are advanced in time with a constant time step that im-
plies a Courant–Friedrichs–Lewy number CFL ∼0.1, un-
til a steady state is reached.
The numerical results are displayed in the insets of
Fig. 9, where the upstream pressure (left subplot) is
plotted versus the thermochemical relaxation length for
M1= 10 at different flight altitudes in the International
Standard Atmosphere (ISA).79 The right subplot repre-
sents the temperature profile of the nonequilibrium zone
at an altitude of 10 km above mean sea level. The charac-
teristic relaxation length ℓT, shown as a diamond sym-
bol, is found to grow from ∼10−4m to ∼10−2m as
the flight altitude increases from 0to 30 km. As a re-
sult, the minimum characteristic length of the turbulent
eddies that can be described using the LIA increases cor-
respondingly with the flight altitude. During the climb
and acceleration segment, the value of ℓTis expected to
decrease as the Mach number increases (since the post-
shock temperature increases with the Mach number) and
to increase as the altitude increases (because of the expo-
nential pressure drop with altitude), the latter being the
expected dominant contribution. This is better analyzed
by looking at Fig. 9(b), which shows the temperature
and pressure profiles in the ISA model. First, the tem-
perature decreases linearly in the troposphere (from sea
level to 11 km), it remains roughly constant at the lower
layer of the stratosphere (from 11 km to 20 km), and
finally increases again until reaching the mesosphere at
50 km of altitude. By contrast, the pressure exhibits a
monotonous power-law decay with altitude that becomes
exponential in the constant temperature layer from 11
km to 20 km.
Returning to the TKE amplification factor, we refer
again to Fig. 9, which shows the variation of Kwith
the pre-shock Mach number M1at five flight altitudes:
0,5,10,20, and 30 km above sea level. For all cases,
the qualitative picture remains the same, with small
changes associated with the variation of the upstream
conditions with altitude. In general, endothermic effects
are found to increase the value of Kacross the whole
Mach number range. Moreover, the higher the altitude,
the stronger the peaks in the amplification of the turbu-
lent kinetic energy. It must be emphasized that Krep-
resents the TKE amplification factor, meaning that the
evaluation of the total turbulent intensity requires in-
formation on the upstream turbulent flow, which is also
expected to change with flight altitude. In this case,
lower (higher) pre-shock temperatures shift the curves
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 16
FIG. 9. TKE amplification factor Kas a function of the pre-shock Mach number M1at different flight altitudes (0, 5, 10,
20, and 30 km) above sea level in the ISA model (a) and the corresponding temperature-pressure vs. altitude profiles (b).
The curve for the lowest altitude (0 km) corresponds to conditions indicated in Fig. 5. The dashed line corresponds to the
thermochemically frozen gas approximation. The insets represent the variation of the pressure (left) and temperature (right;
translational and rotational) with the relaxation length as obtained from one-dimensional direct numerical simulations with
the HTR solver8,9,76 using a two-temperature model. A. Cuadra, M. Vera, M. Di Renzo and C. Huete, AIAA SciTech 2023
Forum, 2023–0075, 2023;55 licensed under a Creative Commons Attribution (CC BY) license.
to the right (left), which means that a higher (lower)
pre-shock velocity is required to achieve similar condi-
tions. Flight altitude also affects the integral length scale
of the upstream turbulence, making the conditions im-
posed by LIA (ℓT≪ℓ)less likely to occur. The effect
of turbulent perturbations with a characteristic length
of the order of the thermochemical relaxation length is
out of the scope of this work, but it can be investigated
using direct numerical simulations, as done by Kerkar
and Ghosh.60 Their findings suggest chemical reactions
within the nonequilibrium region enhance the produc-
tion of turbulence, primarily due to the amplification of
streamwise velocity perturbations, as opposed to the lat-
erally dominating turbulence predicted by LIA.
III. NUMERICAL SIMULATIONS
Turbulence and shock waves interact in various flow
problems relevant to supersonic and hypersonic flight.
This entails, for instance, deflected control surfaces or
shock-based lifting bodies (waveriders) in external flows,
or intakes and isolators of air-breathing propulsion flow
paths. The canonical STI setup, shown in Fig. 1, rep-
resents the simplest model of how turbulence and shock
waves interact with each other. In its simplest form,
isotropic turbulence is convected through a normal shock
wave, where a few parameters define the type of interac-
tion (see Sec. I). The isolated setup enables furthering the
fundamental understanding of the interaction’s mecha-
nism and drives the numerical framework required to
study the problem accurately. The numerical approach
requires low dissipation for the turbulence part, while
an increased dissipation is needed to capture the shock
wave, particularly at higher pre-shock Mach numbers.
This section presents a comprehensive review of the
numerical work devoted to the canonical STI problem.
The overview adopts a grouping based on numerical
frameworks and STI subtopics of interest (Sec. III A),
and continues by compiling key results from the litera-
ture (Sec. III B) alongside direct comparisons with rele-
vant advancements in LIA, discussed earlier in Sec. II A.
These comparisons highlight the specific developments
related to hypersonic flows, which illustrate deviations
from the assumptions of a calorically perfect gas.
A. Literature Review
The following review summarizes key developments
in the numerical simulation of shock-turbulence inter-
action, focusing on both direct numerical and large-eddy
simulation approaches. To start, numerical frameworks
for solving the compressible Navier-Stokes equations are
considered, followed by a review of direct numerical sim-
ulation for the canonical shock turbulence interaction.
Finally, key conclusions pertaining to subgrid-scale mod-
eling and large-eddy simulation are presented from the
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 17
shock-turbulence literature.
1. Numerical frameworks
The main challenge in performing a numerical inves-
tigation of STI comes from the conflicting requirements
of stability and accuracy.80 To capture the correct dy-
namics of turbulence one needs to select a numerical
scheme that adds as little dissipation as possible to pre-
serve the Reynolds number of the flow. On the other
hand, the presence of discontinuities can easily induce
instability of the calculations via Gibbs artifact if not
treated correctly. Several solutions have been devised to
address these competing effects, which may be classified
into three categories.
The first category seeks to discretize the shock wave
assuming its thickness is similar to the laminar value and
calculations performed with this strategy are referred to
as “shock-resolving”. Moin and Mahesh 81 demonstrated
that the ratio between the Kolmogorov length scale of the
incoming turbulence (ηk) and the shock wave thickness
(δ) scales as
ηk
δ= 0.13pReλM1−1
Mt
.(38)
If this ratio is order unity, one could aim at using
a single discretization scheme for the Navier-Stokes
equations inside and outside the shock wave. It is clear
however that this strategy becomes rapidly unfeasible
as soon as the upstream Mach number exceeds 1.5.
This class of numerical framework, which has been
utilized in studies of weak interactions,21–23 is therefore
unsuitable for evaluation of strong shocks and will not
be further discussed in this review about hypersonic
shock-turbulence interactions.
The second type of framework is referred to as “shock-
capturing” and it regularizes the presence of disconti-
nuities by adding numerical dissipation to stabilize the
solution. As this numerical dissipation can strongly in-
fluence the accuracy of turbulence prediction, it is crit-
ical to carefully select the regions where stabilization is
required. For this reason, several algorithms have been
developed and utilized over the years to predict canoni-
cal STI. Numerical dissipation is either added by locally
changing the numerical scheme to a dissipative formu-
lation or artificially introducing additional viscosity in
the flow. The former approach has been mainly applied
by considering non-linear shock-capturing schemes, such
as Essentially-Non-Oscillatory (ENO),82 Weighted-ENO
(WENO),83 Targetted-ENO (TENO),84 and their vari-
ants, combined with flux-splitting methods. As these
non-linear schemes might still prove significantly dissipa-
tive, a common approach is to mix them with centered
schemes creating what is termed a hybrid discretization.
Specifically, a selection process based on a sensor is uti-
lized to distinguish smooth regions of the flow from sten-
cils that go across shocks. In this way, energy-preserving
formulations can be deployed in smooth regions of the
flow, while shock-capturing is limited to the vicinity of
shock waves. Examples of works that utilized this tech-
nique for STI studies are due to Lee et al.,85 and Lars-
son and Lele.46 It is noteworthy that the overall nu-
merical dissipation strongly depends on the specificity
of the considered shock sensor, and, for this reason, re-
cent works are focusing on deriving more advanced and
precise sensors.61,86 The idea of including additional dis-
sipation by means of an artificial viscosity was first pro-
posed by von Neumann and Richtmyer,87 and Jameson
et al.88 and then developed in several different forms by
others, particularly in the context of Large-Eddy Simula-
tions (LES).89,90 This approach, which is less invasive in
the overall numerical algorithm of a code, provides sim-
ilar stabilization properties as the ENO-type approach,
though some extra dissipation is sometimes observed. In
both shock-capturing approaches, a computational grid
size large enough to capture all the scales of the shock
wave wrinkling is necessary. A thorough review of the
shock-capturing framework performance in the context of
STI direct numerical simulation is provided by Johnsen
et al.6
The third class of numerical frameworks is the “shock-
fitting", whereby the shock-wave is treated as a pure dis-
continuity, and Rankine-Hugoniot jump conditions are
considered across the nodes of the computational grid91.
This methodology is inherently more costly from a com-
putational point of view, though its ability to consider
strong shock waves without introducing additional dissi-
pation, might be particularly beneficial in computing hy-
personic STI. For this reason, experiments of very intense
STI calculations have been performed over the years92–95
though the Reynolds number of the interactions has al-
ways remained low to avoid excessive computational cost.
Most studies involving canonical STI have been per-
formed using one of the aforementioned shock treatments
in conjunction with a finite-difference (FD) discretiza-
tion of the Navier-Stokes equations. In fact, the sim-
ple geometrical configuration of these interactions has
favored the use of FD schemes, which are usually easy
to implement using high-order discretization necessary to
achieve an accurate representation of turbulence at mod-
erate computational cost. Typical FD schemes utilized
in STI for the smooth region of the flow are either skew-
symmetric96,97 or compact.98 However, the need to ex-
tend the study of STI to more complicated configurations
has motivated the development of finite volume35,99–101
and discontinuous Galerkin102–104 formulations with sat-
isfactory results.
2. Direct numerical simulation
The direct numerical simulations of Lee21,22,85 com-
prised the first fully nonlinear analysis of shock-
turbulence interaction, enabling a detailed characteriza-
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 18
tion of the evolution of turbulence statistics throughout
the interaction. These simulations confirmed that the
interaction of vortical isotropic turbulence with a shock
wave leads to an amplification of turbulence kinetic en-
ergy, together with a corresponding reduction in the
characteristic turbulence length scales due to preferen-
tial amplification of small-scale (i.e., high-wavenumber)
turbulence motion.22 Substantial agreement between the
fully nonlinear numerical simulations with LIA implies
this augmentation of turbulence kinetic energy arises
primarily from linear mechanisms. In addition to re-
producing TKE amplification, linear interaction analysis
was likewise shown to correctly predict a reduction in
the Taylor length scale across the interaction, as well as
the Mach-number dependence for the post-shock varia-
tion in the Kolmogorov scale.85 In particular, LIA and
DNS both corroborated that at modest bulk Mach num-
bers, i.e., M1<1.65, the post-shock dissipation scale
is increased relative to the pre-shock turbulence, while
further increasing the Mach number gives rise to a re-
duction in the Kolmogorov scale across the interaction.
As such, the resolution requirements of direct numeri-
cal simulation were confirmed to be heightened through
the interaction for high supersonic and hypersonic Mach
numbers, necessitating post-shock refinement to main-
tain a constant grid spacing relative to the local Kol-
mogorov scale.46,85 This aspect has so far limited the ex-
ecution of direct numerical simulations in the hypersonic
regimes at high Reynolds numbers. The highest inves-
tigated pre-shock Mach numbers remain in the neigh-
borhood of M1∼5and have been considered only in
conjunction with moderate Reynolds numbers based on
the Taylor microscale (Reλ∼40).46,56,105,106 In contrast
to LIA’s success in predicting turbulence kinetic energy
and length-scale evolution across canonical shock turbu-
lence interaction, the linear theory was proven to be un-
able to capture the evolution of post-shock anisotropy,
which arises primarily from further augmentation of the
streamwise normal component of the Reynolds stress ten-
sor via pressure-transport effects following the primary
interaction.22
Relative to the purely vortical isotropic turbulence
considered by Lee,21,22,85 the introduction of entropic
modes, associated with correlated velocity/temperature
fluctuations upstream of the shock, was subsequently
shown utilizing direct numerical simulation to further
augment TKE amplification across the interaction for
negative correlations.35,72 The presence of positively
correlated velocity/temperature fluctuations correspond-
ingly diminished the TKE amplification. For both pos-
itively and negatively correlated temperature/velocity
fluctuations, Mahesh72 demonstrated that the modula-
tion of post-shock TKE amplification by entropic modes
arises from the baroclinic torque induced by upstream
density fluctuations. Furthermore, the presence of up-
stream dilatational energy, associated with intrinsic com-
pressibility in the pre-shock turbulence, likewise serves to
further augment the TKE amplification, particularly at
elevated bulk Mach numbers, as evidenced by the numer-
ical simulations of Grube and Martin105 and Cuadra et
al.56
The study of non-linear interactions between turbu-
lence and shock waves has also led to the identifica-
tion of three primary regimes of the shock deforma-
tions.46,64,107–111 When the upstream perturbations are
weak, the shock wave remains coherent, and its posi-
tion is only locally displaced upstream and downstream
with respect to its mean location. If the turbulence p er-
turbations that impinge upon the shock are sufficiently
strong, they prove able to locally alter the intensity of
the discontinuity and even generate holes within its front.
These two regimes were identified at first by Larsson46
and are termed “wrinkled” and “broken”, respectively.
Much later, Chen and Donzis110 demonstrated that a
third regime arises when pre-shock turbulence fluctua-
tions become very intense. This new state, the so-called
“vanished” regime, corresponds to conditions where the
shock is not able to produce any amplification of the
Reynolds stresses, while the mean-field jump conditions
remain satisfied. The transition from the wrinkled to
the broken shock regime appears to be well described by
the parameter K=Mt/(M1−1). For K>0.6the
shock will be broken, whereas otherwise, it will be sub-
ject to wrinkling.46,64 The appearance of the vanished
state seems instead correlated with the scale-separation
parameter δ/ηk∼7.69. It is noteworthy that as the pre-
shock Mach number increases, both δ/ηkand Kdecrease,
making the appearance of broken or vanished shocks un-
likely in hypersonic flows. However, the confinement of
the shock response to the wrinkled regime favors the
agreement between LIA and DNS results for highly hy-
personic conditions. In fact, several studies have shown
that, in supersonic conditions, having a low δ/ηkemerges
as a determining factor for achieving convergence be-
tween LIA and DNS.63,110,112–114
3. Large-eddy simulation
Reducing the overall degrees of freedom, and therefore
computational cost, relative to direct numerical simu-
lation, large-eddy simulation (LES) instead provides a
coarse-grained representation of shock-turbulence inter-
action. As LES entails directly resolving only the largest
turbulence length scales, modeling the effect of physical
processes transpiring beneath the grid scale, i.e. subgrid
scale (SGS), is required. Formally, this closure problem
in the low-pass-filtered momentum equation comprises
the right-hand side of
∂
∂t (ρeui) + ∂
∂xjρeuieuj+P δij −ˆσij
=∂
∂xjτSGS
ij +VSGS
ij (39)
where ρis the filtered density, eui=ρui/ρ is the Favre-
filtered velocity, and ˆσij is the viscous stress tensor as
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 19
evaluated with the resolved primitive variables. The un-
closed terms arising from the nonlinearity in the Navier-
Stokes equations, i.e., those which cannot be trivially
expressed in terms of the filtered variables, consist of the
subgrid stress tensor τSGS
ij =ρ(euieuj−guiuj)and the sub-
grid viscous stress VSGS
ij = ˆσij −σij, with σij denoting
the filtered viscous stress tensor. Within the LES formal-
ism, while the closure of the continuity equation proves
trivial when expressed in terms of the filtered density
and Favre-averaged velocity, the evolution of the tem-
perature field typically entails numerical solution of a
filtered energy equation115,116 or total-energy equation
as constructed based on the filtered primitive variables
themselves.23,117,118
Whereas closure of the subgrid viscous stress, as well
as the analogous closure terms involving molecular dif-
fusivities, is generally neglected on the basis of a priori
analysis117 and scale-separation arguments, LES predic-
tions of shock-turbulence interaction have been shown to
depend sensitively on the closure of the (inviscid) subgrid
stress tensor, typically modeled on the basis of the eddy-
viscosity approximation, namely τSGS
ij = 2µte
Sij where
µtis the effective eddy "viscosity" and e
Sij is the Favre-
averaged strain-rate tensor.99 In particular, the early
study of Garnier et al. 119 demonstrated that inclusion of
the explicitly-modeled subgrid stress from the Dynamic
Smagorinsky model120,121 provided significant improve-
ment in the prediction of the filtered turbulent kinetic en-
ergy and Reynolds stresses for well-resolved calculations,
relative to both coarsened DNS and the static Smagorin-
sky model, which proved under- and over-dissipative,
respectively. At coarser resolutions, namely when the
shock corrugations become subgrid, Garnier et al. 119
likewise demonstrated that LES-based predictions of the
canonical shock-turbulence interaction deteriorate signif-
icantly, not necessarily due to breakdown of the post-
shock subgrid model performance but instead due to
insufficient resolution of the shock wave itself. Subse-
quent coarse-grid LES calculations by Bermejo-Moreno
et al.,122 for which the nominal spanwise grid spacing
exceeded that of the DNS by a factor of 16, similarly
suggest that even the state-of-the-art LES approaches
naturally yield inaccurate predictions of the Reynolds-
stress evolution through the canonical shock-turbulence
interaction at such coarse resolutions.
In conjunction with the finding that high-fidelity LES
of canonical shock-turbulence interaction requires direct
resolution of shock corrugations by the grid, LES predic-
tions have likewise been shown to improve significantly
by setting the subgrid fluxes to zero in cells for which
a shock-capturing scheme is active, limiting the acti-
vation of the explicit SGS models to cells for which a
central numerical scheme is utilized to evaluate the re-
solved inviscid fluxes.122 In this way, deactivation of the
SGS models in the vicinity of the shock precludes SGS
dissipation from further augmenting the numerical dis-
sipation introduced by the shock-capturing scheme. Re-
cently, by leveraging adaptive mesh refinement to en-
sure resolution of shock corrugations and limited acti-
vation of subgrid fluxes away from detected discontinu-
ities, Braun et al. 123 recently developed an explicit large-
eddy simulation framework based on a hybrid stretched-
vortex model,124 producing resolved Reynolds-stress pro-
files largely independent of the post-shock resolution for
high-Reλupstream turbulence.
To date, this joint development and assessment of
large-eddy simulation models for shock-turbulence inter-
action have focused almost entirely on supersonic flows.
However, as the free stream Mach number increases, sev-
eral key physical processes emerge, which will demand
novel subgrid-scale modeling. In particular, as the tur-
bulent Mach number through the interaction increases,
giving rise to increasingly significant dilatational mo-
tion, treatment of the isotropic component of the subgrid
stress tensor becomes increasingly important.121,125,126
Representing the subgrid kinetic energy, τSGS
kk must be
modeled to close not only the filtered momentum equa-
tions but the energy-transport equation as well.127 As
the bulk Mach number further increases and viscous dis-
sipation gives rise to high-temperature post-shock condi-
tions, finite-rate thermochemical effects emerge, includ-
ing vibrational excitation, chemical dissociation, and ion-
ization. Filtering of the relevant vibration-energy con-
servation and species’ continuity equations gives rise not
only to additional subgrid advective closure problems for
the transport of vibrational energy and partial densities
but subgrid contributions to the reactive and thermal
relaxation processes as well. Analysis of explicit sub-
grid modeling approaches for thermochemical relaxation
in the context of hypersonic large-eddy simulation has
been the subject of only limited investigation in the lit-
erature,128,129 and therefore represents a key area for fu-
ture development to enable coarse-grained calculations
of shock-turbulence interaction at high Mach numbers.
B. Compilation of numerical results
Following the CFD overview in Sec. III A, some of
the DNS results on STI are presented herein. Table I
summarizes the upstream conditions adopted by various
authors in canonical STI studies for calorically perfect
gases,22,37,46,72,85,105,110–112,130–132 and more recently, for
thermally perfect gases with compressibility effects at
Mach 5.56 Over the past two decades, significant progress
has been made in understanding the complex physics
governing STI. As shown in Fig. 10, a substantial amount
of DNS data has been gathered over a wide range of up-
stream Mach numbers, M1∈[1.1,6], turbulent Mach
numbers Mt∈[0.02,0.691], and Taylor Reynolds num-
bers, Reλ∈[9,74]. The upstream conditions are pre-
sented in a linear-log fashion as a function of the factor
M1−1for better visualization. The data covers different
types of shock regime as defined based on the similar-
ity parameter provided by Larsson et al.37 Despite these
advances, the hypersonic regime, defined by M1≥5
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 20
(highlighted in grey), remains largely unexplored due
to the significant computational demands imposed by
strong shocks, which exacerbate the inherent complex-
ities of thermochemical non-equilibrium flows. Further-
more, achieving the ultimate goal of simulating fully de-
veloped turbulence in the STI problem, which requires
Reλ>100,113,133 remains a formidable challenge, as will
be discussed in Sec. IV.
Since Fig. 10 only depicts the input parametric space
explored in STI through DNS, they do not provide infor-
mation on the turbulence amplification ratio. This ratio
is presented and compared with far-field predictions us-
ing the LIA framework in Figs. 11 and 12. The LIA
results, based on the analysis presented int Sec. II B,
consider both calorically perfect and calorically imper-
fect gases. It must be noted that accurately determining
the precise value from the DNS data for direct compari-
son with the LIA results can be challenging because the
shock-capturing method typically involves numerical dis-
sipation factors that have an impact on the turbulence
evolution. To define the far-field solution in the DNS,
several works (see e.g., Chen and Donzis 110,111) take
advantage of the non-monotonic behavior of the stream-
wise Reynolds stress (R11). In particular, the position
at which the streamwise Reynolds stress peaks, typi-
cally normalized with its corresponding upstream value
(R11/R11,1), is the position at which the comparison is
carried out. Alternatively, one can extrapolate the tur-
bulent statistics back to the average shock position, as
suggested by Larsson and Lele.46 This approach aims
to minimize the influence of viscosity effectively. Addi-
tional discrepancies may also arise because LIA results
are derived from Reynolds-averaged statistics, while DNS
of compressible flows commonly relies on Favre-averaged
statistics. The latter choice is proved convenient when
dealing with the convective term of the Navier-Stokes
equations.
We first focus on the longitudinal and transverse com-
ponents of the turbulent kinetic energy, quantified by
the streamwise Reynolds stress R11 and the transverse
Reynolds stress R22. Both LIA (lines) and DNS data
(markers) are presented in Fig. 11. In particular, solid
lines correspond to a calorically imperfect gas in ther-
mochemical equilibrium, while dashed lines assume the
calorically perfect approximation with a constant adia-
batic index γ= 1.4. The LIA predictions are color-coded
from light-to-dark blue, reflecting increasing upstream
compressibility levels, denoted by η= [0.001,0.05,0.1].
The remaining input parameters for the simulations at
M1= 5, provided in Table I and corresponding to
Cuadra et al.,56 were selected with the aim of enabling a
direct comparison with LIA. Similar DNS setups, which
disregard non-linearities and viscous effects not captured
by LIA, are represented using the same color scheme for
consistency and presented in a log-log scale as a func-
tion of the factor (M1−1) for better visualization. In
particular, light-to-dark blue circles and squares denote
their results for γ= 1.4and γ=γ(T), respectively. The
(a)
(b)
FIG. 10. Compilation of upstream conditions considered in
direct numerical simulations for solving the shock-turbulence
interaction problem, displaying (a) turbulent Mach number
and (b) Taylor Reynolds number as functions of the Mach
number. The dashed line denotes the limit from broken to
wrinkle regime proposed by Donzis.63,64 The grey area de-
notes the hypersonic regime M1≥5. The symbols repre-
sent direct numerical simulations from other studies: (▷),22,85
(△),72 (▽),37,46 (∗),112 ( ), 130 ( ),131 ( , ,□,■,⋄),110,111
(),132 ( ),105 and ( , ).56
data collected from other numerical simulations, not cor-
responding to the hypersonic regime, are denoted by dif-
ferent symbols (see Table I for reference), maintaining
the notation of Chen and Donzis110,111 for the results
presented there. For example, the hollow symbols rep-
resent results obtained using HIT, while the black filled
symbols correspond with spatially developed turbulence
(SDT), which allows the interaction between the fluctu-
ations and the mean flow. This procedure resembles a
larger streamwise component of velocity variance com-
pared to the transverse components (slightly anisotropic
flow), similar to the conditions found in wind tunnel
tests.110,134
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 21
TABLE I. Summary of several computational works in canonical shock-turbulence interactions.
Source Year M1MtReλγSymbol
Lee et al.22,85 1993, 1997 1.2, 2, 3 0.102-0.11 14.9-19.7 1.4 ▷
Mahesh et al.72 1997 1.29 0.14 19.1 1.4 △
Larsson et al.37,46 2009, 2013 1.28-6 0.05-0.56 39-74 1.4 ▽
Ryu and Livescu112 2014 1.1-2.2 0.02-0.27 10-45 1.4 ∗
Tian et al.130 2017 2 0.09-0.38 9-45 1.4
Boukharfane et al.131 2018 1.7, 2, 2.3 0.17 21 1.4
Chen and Donzis110,111 2019, 2022 1.1, 1.2, 1.4 0.05-0.54 10-65 1.4 , ,□,■,⋄
Gao et al.106 2020 1.28-5 0.1-0.4 37-74 1.4
Grube and Pino105 2023 2.98-4.69 0.224-0.691 18-48 1.4
Cuadra et al.56 2024 5 0.2, 0.4 40 1.4, γ(T),
By direct inspection of Fig. 11, it is observed that the
two components of the Reynolds stresses are amplified
in most of the region tested. For the purely solenoidal
case (η= 0), at pre-shock Mach numbers up to M1∼2,
most of the TKE produced across the shock belongs to
rotational modes of the streamwise velocity fluctuations
(the acoustic contribution is negligible in the far-field
for η= 0). However, for M1≳2,R11 plunges below
R22 and the transverse component dominates, reaching
its first peak around M1∼10. This value is approxi-
mately 15% higher than the obtained with the thermo-
chemically frozen approximation. In contrast, when di-
latational modes are present (η > 0), as in the case of
η= 0.1, the acoustic contribution of the TKE ampli-
fication accounts for approximately ∼3% for a Mach-
2 and 5% at Mach-5. More significantly, even with a
modest upstream compressibility of 5%, both R11 and
R22 show notable amplification as M1increases. For
instance, at M1= 5, LIA predicts an amplification of
around 42% for R11 and 16% for R22 with η= 0.05, while
with η= 0.1, these values rise to approximately 80% and
31%, respectively, compared to the purely solenoidal case
with γ= 1.4. The Reynolds stresses components are fur-
ther modulated when accounting for high-temperature
endothermic effects. For example, with η= 0.1, LIA
predicts a 7% decrease in R11 and an increase of 5% in
R22. Note that at Mach-5 vibrational excitation domi-
nates over the dissociation process. However, it is read-
ily seen how the caloric imperfect solution deviate more
from the γ= 1.4case as M1exceeds 5.
Therefore, we can conclude that, although discrepan-
cies between the final values of R11 and R22 in LIA and
DNS are observed, as expected under the conditions ex-
plored by Cuadra et al.,56 the relative impact of com-
pressibility effects at Mach 5 are accurately captured. In
particular, the light-to-dark blue symbols indicate vary-
ing degrees of compressibility, with circles and squares
representing the caloric perfect and caloric imperfect ap-
proximations, respectively. In all the cases tested, the
Reynolds stresses R11 (panel a) and R22 (panel b) in-
crease as compressibility levels rise, as predicted by LIA.
On the other hand, with regard to high-temperature en-
dothermic effects, LIA predicts a decrease in R11 and an
increase in R22, whereas DNS results show amplification
in both components. This translates into an different
prediction is what concerns the turbulence anisotropy:
the one predicted by LIA (R22 > R11)is totally opposite
to the DNS (R11 > R22). Although this effect tends to
balance out when considering the total kinetic energy, as
discussed later, further studies are needed to fully un-
derstand the discrepancies. One plausible explanation47
is that LIA overpredicts the transverse velocities due to
its idealization of transverse velocity conservation across
the shock, particularly in non-equilibrium regions. The
endothermicity and the higher compression ratio result
in a greater local deflection than that computed with a
finite non-equilibrium zone. Consequently, the overpre-
diction of lateral velocity leads to an underprediction of
the streamwise component, based on the conservation of
energy. Further explanations pertain to the energy trans-
fer occurring from the acoustic field in the longitudinal
direction. This phenomenon leads to DNS predictions
for R11 being higher than those provided by LIA when
M1≳2.5.
With use made of the Reynolds stresses, R11 and R22,
it is straight forward to construct the total TKE am-
plification factor according to K= 1/3(R11 + 2R22), as
shown in Fig. 12. Here, the DNS data from Larsson
and Lele46 and Larsson et al.37 are also included (red
symbols). These DNS results correspond to a slightly
different setup, in which the authors artificially elimi-
nated viscous dissipation by extrapolating to an infinite
Reynolds number [see Eq. (3.4) from Ref. 37]. It was
found that, in contrast to the values obtained for the in-
dependent Reynolds stresses R11 and R22, the total TKE
amplification factor produced by DNS is much closer to
the LIA prediction. As expected, the DNS results ap-
proximate LIA prediction when Mt≪1and Reλ→ ∞.
Specifically, similar values are obtained when there is
sufficient separation of scales between the characteristic
size of the smallest eddies and the shock thickness. Pre-
vious studies also confirm the convergence between LIA
and DNS when the turbulent Mach number approaches
zero.110,112 Regarding the effects of upstream compress-
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 22
(a)
(b)
FIG. 11. Streamwise R11 (a) and transverse R22 (b) compo-
nents of the TKE amplification factor as a function of M1−1.
The dashed lines correspond to the values of R11 and R22 cal-
culated assuming that the post-shock gas is thermochemically
frozen. The symbols represent direct numerical simulations
from other studies: (▷),22,85 (△), 72 (▽),37,46 (∗),112 ( ),130
(),131 ( , ,□,■,⋄),110,111 ( ),105 and ( , , light-to-dark
blue for η= [0.001,0.05,0.1]).56
ibility and temperature-dependent heat capacity, which
are found to be non-negligible in the hypersonic regime,
both LIA and DNS provide similar estimates of the rel-
ative influences of these factors.56
IV. CONCLUSIONS
This work investigates the fundamental physics of
compressible turbulence interacting with planar shocks
in hypersonic conditions through both theoretical and
numerical approaches. The theoretical framework ex-
tends LIA.47,55,56 to multi-component mixtures by in-
corporating the Combustion Toolbox57, accounting for
compressible and thermochemical effects. New results
across a broad range of Mach numbers characterize the
impact of upstream turbulence compressibility in the hy-
personic regime, utilizing both LIA and DNS.
This section provides a final perspective on the shock-
turbulence interaction problem, tracing its development
from early theoretical approaches, such as LIA, to the
latest advancements, including applications in hyper-
sonic regimes. We summarize the main conclusions and
key challenges, focusing on issues related to numerical
discretization, the limitations of DNS, and the poten-
tial of LES. We also compare these methods with LIA,
highlighting its most significant assumptions. Finally, we
provide an outlook on future research directions in the
field.
A. Concluding remarks
Regarding the canonical STI, various historical trends,
summarized below, can be found in the literature since
the pioneering work of Lee et al.21–23 An important aim
in the analysis of the generated data is to devise simi-
larity scaling laws (e.g., see Refs. 63 and 112) to explain
the observed amplification behavior as well as to identify
shock regimes (e.g., see Refs. 37 and 64). Moreover, from
an application perspective, fundamental studies can be
used to devise or improve predictive models, as exem-
plified by several authors.24–34,135 One common feature
of the CFD studies discussed in Sec. III A, excluding
the reacting STI, is the assumption of a constant spe-
cific heat capacity. In order to extend the relevance to
hypersonic flows, temperature dependence of this prop-
erty must be considered in either a thermally perfect
gas (for lower enthalpy and low Knudsen numbers) or
a thermally non-perfect gas, where chemical- and ther-
mal non-equilibrium effects become significant. In this
work, the latter topic has been discussed in the context
of the canonical STI.
1. Challenges of numerical discretization
Owing to the multi-scale character of turbulent flows,
minimal numerical dissipation proves necessary for an
accurate and computationally efficient representation of
high-wavenumber dynamics in shock-turbulence interac-
tion. While the seminal works on the topic did not con-
sider specific shock treatment in their studies which was
possible due to the low shock Mach numbers (M1<1.5),
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 23
FIG. 12. TKE amplification factor Kas a function of M1−1. The dashed lines correspond to the value of Kassuming that
the post-shock gas is thermochemically frozen. The symbols represent direct numerical simulations from other studies: (▷),22,85
(△), 72 (▽),37,46 (∗),112 ( ), 130 ( , ,□,■,⋄),110,111 and ( , , light-to-dark blue for η= [0.001,0.05,0.1]).56 The red-hollow
triangles (▽) represent the values from Larsson et al.37 with artificially removed viscous dissipation.
increasing M1requires alternative methods which are
either shock capturing or shock-fitting. Shock captur-
ing methods have received wider attention in the liter-
ature. They rely on the local addition of extra numer-
ical dissipation to the solution either by upwinding the
numerical scheme or by considering artificial flow prop-
erties. As the introduction of numerical dissipation has
the side effect of damping turbulence fluctuations, con-
fining its application to cells containing discontinuities
is paramount to retaining accuracy in the flow descrip-
tion. This problem has motivated the development of
improved shock sensors able to discern when the solu-
tion becomes discontinuous. In this context, a common
solution is to deploy hybrid schemes, which use skew-
symmetric or low-dissipation schemes in smooth regions
of the flow and shock-capturing schemes for stencils that
intersect discontinuities.
Shock-fitting techniques eliminate the need for shock
sensors and have been used in the literature to investigate
stronger shocks. The rationale behind their application
is that the method removes the need to have very fine
cells in the shock region which can become very stringent
when the shock front corrugation is high. However, these
methods are often associated with very high computa-
tional costs that is usually untenable on computational
grids that are fine enough to support turbulence.
The simple geometrical features of the STI problem
have favored the use of high-order numerics, which en-
sure high-computational intensity per grid point, for
the discussed calculations, while relatively fewer studies
have addressed the setup with low-order numerics, which
are usually better suited for complex geometries and
unstructured grids.35,99,101 Additional work on finite-
volume and low-order schemes would be valuable for ex-
tending the discussed findings to more complex configu-
rations.
2. Direct numerical simulations and their limits
Direct numerical simulations have been the primary
tool used to gain fundamental understanding of the
canonical STI. Experimental methods are limited due
to the difficulty of measuring pre- and post-shock states
while controlling the shock wave position. In contrast,
DNS allows precise control over the flow and several vari-
ations of the setup have been considered over the years.
Key parameters investigated in these simulations, be-
sides the Reynolds number and convective Mach number
of the incoming flow, include the properties of the in-
coming turbulence, namely the intensity of the vortical
fluctuations, entropy, and acoustic fluctuations.
The main limitation of the DNS methodology stands
in its high computational cost. In fact, the computa-
tional grid that is utilized to discretize the Navier–Stokes
equations must be fine enough to correctly describe all
the scales of the flow (except for the shock itself) and
large enough to accommodate a sufficient number of in-
tegral length scales. Moreover, the presence of the shock
wave modifies the small scales of the flow by reducing
their size and increasing their anisotropy. The parame-
ter that mainly determines the computational cost of a
configuration is the Taylor microscale Reynolds, Reλ, of
the incoming turbulence. This dependency is readily jus-
tified by the broadening of the turbulence spectrum at
higher Reynolds numbers. Note that current boundaries
of this phase space explored with DNS pertain Reλbe-
tween approximately 40 and 70. These values are quite
low if we consider that a proper inertial rage of turbu-
lence is usually observed for Reλ∼100.
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 24
The cost of a DNS of STI also depends on the Mach
number of the incoming flow. In particular, the ratio
of Kolmogorov length scales (ηk,2/ηk,1) between the up-
stream and downstream turbulence scales as
ηk,2
ηk,1∼ PT 11/8,(40)
where Pand Trepresent the pressure and temperature
ratios across the shock.46 It can easily be inferred that
the worst-case scenario happens at the border of the hy-
personic regime around Mach 5.
The turbulence Mach number, and in particular its ra-
tio with the convective Mach number, also contribute to
the computational cost. This parameter linearly affects
the integration time window necessary to capture the
interaction. A small value of Mt/M1, typical of hyper-
sonic STI, results in a strong separation of scales between
the integral time scales of turbulence and the convective
time scales. Considering that the size of the computa-
tional domain is usually of the order of the integral scale
of the turbulence, a large number of flow-through times
is required to describe one eddy turnover time of the
incoming flow. Conversely, a high Mt/M1limits the
computational cost of the calculations by reducing the
separation of time scales between mean advection and
turbulence.
Moving toward the hypersonic regime and consid-
ering stronger shock waves, the inclusion of high-
temperature phenomena, such as chemical and thermal
non-equilibrium, will also increase the computational
cost of the calculations by augmenting the set of trans-
port equations that need to be solved. To have a sense
of the impact of the inclusion of these phenomena in
the calculations, one could consider that the typical ra-
tio of computational costs of a standard shock-capturing
scheme scales with the number of unknowns to the power
two or three. However, the specifics of the numerical
methods deployed during the numerical simulations must
be considered to identify a proper scaling.
3. Large eddy simulations
Improving dramatically the computational tractabil-
ity of scale-resolving simulations, large-eddy simulation
(LES) requires significantly fewer degrees of freedom
than direct numerical simulation. As the LES paradigm
involves resolving only the largest scales of turbulent mo-
tion, physical modeling of the effect of small-scale pro-
cesses on the resolved quantities proves necessary. To
enable predictive LES in the context of supersonic canon-
ical STI, in particular, three crucial elements have been
identified. Firstly, dynamic modeling was shown to pro-
vide significant improvement in the prediction of filtered
Reynolds stresses as compared to both coarsened DNS
(i.e., without an explicit SGS model) and static LES
modeling. Secondly, ensuring that the computational
grid is sufficiently fine in the vicinity of the shock to
directly resolve corrugations was shown to be crucial
for predictive simulations of second-order statistics. Fi-
nally, the inclusion of the SGS fluxes only in smooth
regions of the flow, i.e., those cells in which the hybrid
scheme reverts to a central discretization, prevents artifi-
cial damping of turbulent fluctuations. However, the de-
velopment and evaluation of LES capabilities for shock-
turbulence interaction in the hypersonic regime remain
essentially absent from the literature, even for calorically
perfect gases. In light of the canonical STI’s relevance to
propulsion-system design and broader aerospace appli-
cations, more sophisticated LES models will be required
not only for subgrid chemistry but also advective trans-
port of both scalars (species, temperature, etc.) and mo-
mentum, particularly for high-Mach conditions.
4. CFD VS. LIA
LIA has been used alongside DNS as it provides a the-
oretical framework for comparing and understanding be-
havioral trends in STI. Historically, challenges in achiev-
ing a fair comparison between LIA and DNS have led
to very different results between both methods, particu-
larly regarding the amplification of the Reynolds stresses.
These differences were attributed mainly to variations
in the definition of the post-shock (or post-interaction)
location and the impact of viscous effects. To address
this, viscous decay was compensated by extrapolating
the DNS solution behind the shock region (defined based
on the bounds of shock oscillations) back to the mean
shock location.37,46,85 This resulted in better agreement
between the two frameworks. An important milestone in
canonical STI studies is the reconciliation between DNS
and LIA, which was achieved by considering the ratio
of laminar shock thickness to the pre-shock Kolmogorov
length scale (the scale-separation parameter) as a refer-
ence scaling parameter. When this parameter gets closer
to zero, DNS predictions of Reynolds stresses and vortic-
ity variance amplifications converge towards LIA predic-
tions. Building on this, LIA has enabled to increase the
range of Reλaccessible in post-shock turbulence stud-
ies by its use in generating a shock-processed state113,114
(also referred to as shock-LIA). Furthermore, LIA can
be used in a predictive sense over a wide range of condi-
tions which in turn would allow the improvement of, for
instance, RANS models (see, e.g., Refs. 25 and 29).
In what concerns the hypersonic regime, we have ex-
plored the potential of LIA under the following con-
siderations: i) the upstream turbulence may consist of
small-amplitude perturbations of various types, such as
rotational, entropic, and acoustic; ii) the thermochemical
non-equilibrium region may involve processes like vibra-
tional excitation, molecular dissociation/recombination,
and ionization. The latter assumes that the character-
istic size of the turbulence is much larger than the non-
equilibrium length, a condition strongly influenced by
the nature of the turbulence (modal composition), the
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Review of Shock-Turbulence Interaction with a Focus on Hypersonic Flow 25
shock Mach number, and the upstream flow conditions,
as evidenced by the flight-altitude effects commented on
before. This is due to the strong temperature dependence
of relaxation processes and chemical kinetics.47 From a
practical perspective, the LIA model presented in this
work assumes that the high-frequency part of the turbu-
lence spectrum can be neglected. It should be noted that
the opposite limit, which is not covered in this section,
involves separating the interaction between turbulence
and an inert shock from the study of post-shock tur-
bulence evolution in a still-compressing non-equilibrium
flow. This approach, as explored in Huete et al.136 for
detonation-turbulence interaction, is left for future inves-
tigation.
B. Outlook
Based on the discussion presented in this review, sev-
eral insights can be drawn regarding the specific numer-
ical methods, including DNS and LES:
•Numerical methods for STI. The transition to
the hypersonic regime introduces even greater chal-
lenges for numerical methods, as increasingly in-
tense shocks demand higher precision in their iden-
tification to preserve the stability of the calcula-
tion. At the same time, the appearance of eddy
shocklets will blur the boundaries of the flow con-
ditions, complicating when numerical dissipation
must be added. The hybrid schemes appear for
now a valuable solution to the problem, though ad-
ditional work in shock sensors is necessary to min-
imize numerical dissipation without compromising
stability.
•Direct numerical simulations. Future calcu-
lations must be performed well into the hyper-
sonic regime, incorporating high-Reynolds-number
incoming turbulence with a wide inertial subrange.
High-temperature effects must also be included to
shed light on the interaction between thermal and
chemical relaxation and the highly anisotropic tur-
bulence downstream of the shock wave. Exascale
supercomputers must be leveraged to achieve this
goal.
•Large eddy simulations. New sub-grid-scale
models must be developed to enable prediction
post-shock turbulence with numerical solution of
the the filtered Navier–Stokes equations, particu-
larly for high-enthalpy hypersonic flows. In this
context, new DNS databases at high Reynolds and
Mach numbers can facilitate their development and
evaluation via a priori analysis.
ACKNOWLEDGMENTS
CH and ACL work has been partially supported
by projects PID2022-139082NB-C51 and TED2021-
129446B-C41 funded by MCIN/AEI. MDR acknowledges
the CINECA award under the ISCRA initiative, for the
availability of high-performance computing resources and
support. C.W. acknowledges support by the National
Science Foundation Graduate Research Fellowship Pro-
gram under Grant No. DGE2146755. JH work was sup-
ported by the “DLR DAAD Research Fellowship Pro-
gramme".
DATA AVAILABLITY
The data that support the findings of this study are
available from the corresponding author upon reasonable
request.
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