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Direct numerical simulations of temporal compressible mixing layers in a Bethe–Zel'dovich–Thompson dense gas: influence of the convective Mach number

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The present article investigates the effects of a BZT (Bethe-Zel'dovich-Thompson) dense gas (FC-70) on the development of turbulent compressible mixing layers at three different convective Mach numbers Mc = 0,1; 1,1 and 2,2. This study extends previous analysis conducted at Mc = 1,1 (Vadrot et al. 2020). Several 3D direct numerical simulation (DNS) of compressible mixing layers are performed with FC-70 using the fifth order Martin-Hou thermodynamic equation of state (EoS) and air using the perfect gas (PG) EoS. After having carefully defined self-similar periods using the temporal evolution of the integrated streamwise production term, the evolutions of the mixing layer growth rate as a function of the convective Mach number are compared between perfect gas and dense gas flows. Results show major differences for the momentum thickness growth rate at Mc = 2:2. The well-known compressibility-related decrease of the momentum thickness growth rate is reduced in the dense gas. Fluctuating thermodynamics quantities are strongly modified. In particular, temperature variations are suppressed leading to an almost isothermal evolution. The small scales dynamics is also influenced by dense gas effects, which calls for a specific sub-grid scale modelling when computing dense gas flows using large eddy simulation (LES). Additional dense gas DNS are performed at three others initial thermodynamic operating points. DNS performed outside and inside the BZT inversion region do not show major differences. BZT effects themselves therefore only have a small impact on the mixing layer growth.
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Direct Numerical Simulations of temporal
compressible mixing layers in a BZT Dense
Gas: influence of the convective Mach
number
Aur´elien Vadrot1, Alexis Giauque1and Christophe Corre1
1LMFA - Laboratoire de M´ecanique des Fluides et d’Acoustique
Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France
(Received xx; revised xx; accepted xx)
The present article investigates the effects of a BZT (Bethe-Zel’dovich-Thompson) dense
gas (FC-70) on the development of turbulent compressible mixing layers at three different
convective Mach numbers Mc= 0.1, 1.1 and 2.2. This study extends previous analysis
conducted at Mc= 1.1 (Vadrot et al. 2020). Several 3D direct numerical simulation
(DNS) of compressible mixing layers are performed with FC-70 using the fifth order
Martin-Hou thermodynamic equation of state (EoS) and air using the perfect gas (PG)
EoS. After having carefully defined self-similar periods using the temporal evolution of
the integrated streamwise production term, the evolutions of the mixing layer growth
rate as a function of the convective Mach number are compared between perfect gas
and dense gas flows. Results show major differences for the momentum thickness growth
rate at Mc= 2.2. The well-known compressibility-related decrease of the momentum
thickness growth rate is reduced in the dense gas. Fluctuating thermodynamics quantities
are strongly modified. In particular, temperature variations are suppressed leading to an
almost isothermal evolution. The small scales dynamics is also influenced by dense gas
effects, which calls for a specific sub-grid scale modelling when computing dense gas
flows using large eddy simulation (LES). Additional dense gas DNS are performed at
three others initial thermodynamic operating points. DNS performed outside and inside
the BZT inversion region do not show major differences. BZT effects themselves therefore
only have a small impact on the mixing layer growth.
Key words:
1. Introduction
Dense gases (DG) are single-phase vapours characterized by long chains of carbon
atoms and by medium to large molecular weights. They have been widely used in
Organic Rankine Cycles (ORCs) industry over the past forty years. Their large heat
capacity and their low boiling point temperature make them suitable working fluids
for low-temperature heat sources (solar, geothermal, biomass,...). The coupling with a
turbine enables power generation. Recently, because of issues caused by fossil energies,
there has been a strong research effort in developing this technology by improving ORC
turbines efficiency.
Email address for correspondence: aurelien.vadrot@ec-lyon.fr
2A. Vadrot, A. Giauque and C. Corre
Figure 1. The initial thermodynamic state and its distribution at τ= 4000 (which corresponds
to the beginning of the self-similar period) are represented in the non-dimensional pvdiagram
for BZT dense gas FC-70 at Mc= 2.2. The dense gas zone (Γ < 1) and the inversion zone
(Γ < 0) are plotted for the Martin-Hou equation of state. pcand vcare respectively the critical
pressure and the critical specific volume. The initial value of the fundamental derivative of gas
dynamics is equal to Γinitial =0.284.
Rotating elements are a main source of losses for turbines. Their use in transonic
and supersonic regimes generates shocks associated with entropy production. However,
for dense gases, entropy jumps through shocks are significantly reduced in specific
thermodynamic regions (Cinnella & Congedo 2007). This feature could enable to
increase ORC turbines efficiency, but the lack of knowledge about dense gases in these
particular thermodynamic regions close to the vicinity of the critical point restrains
ORC designers. This study seeks to widen knowledge about turbulence characteristics
of these gases by comparing their behaviour to perfect gases on a classical configuration:
the mixing layer.
A specific type of dense gas is used in these simulations: the Bethe Zel’dovich Thompson
(BZT) gases, whose name was given at first by Cramer (1991) to acknowledge pioneering
works of Bethe (1942), Zel’dovich (1946) and Thompson (1971). Unlike other dense gases,
they comprise an inversion thermodynamic region where the fundamental derivative of
gas dynamics Γbecomes negative as shown in Fig. 1. Thompson (1971) defines Γas:
Γ=v3
2c2
2p
∂v2s
=c4
2v3
2v
∂p2s
= 1 + ρ
c
∂c
∂ρ s
(1.1)
where vis the specific volume, ρthe density, c=p∂p/∂ρ|sthe speed of sound, pthe
pressure and sthe entropy. For thermally and calorically perfect gases, the fundamental
derivative is equal to (γ+ 1)/2, with γthe heat capacity ratio. In this case, its value
is always greater than one, unlike dense gas flows, where Γcan become lower than one
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc3
and even be negative for BZT dense gases. In that case, rarefaction shock-waves can
occur, which is forbidden by the Second law of thermodynamics in usual gases, where
only compression shock-waves are allowed.
Bethe (1942) expressed the entropy jump expression across shock-waves as a function
of the fundamental derivative.
∆s =s2s1=2p
∂v2s
∆v3
12T+O∆v4=c2Γ
v3
∆v3
6T+O∆v4(1.2)
with T the temperature. In the case of compression shock-waves, the specific volume
variation is negative (∆v < 0), so that the fundamental derivative must be positive
(Γ > 0) to ensure that the entropy jump remains positive (∆s > 0), thus satisfying the
Second law of thermodynamics. Only compression shock-waves are physically admissible
for classical ideal gases since Γ > 1. For BZT gases, the fundamental derivative being
negative (Γ < 0), physically admissible shock-waves in the inversion region are expansion
shock-waves such that the specific volume variation is positive (∆v > 0) to ensure the
entropy jump remains positive. Moreover, since entropy jumps are proportional to the
fundamental derivative Γ, which is of small amplitude in DG flows, intensity of shocks is
significantly reduced (Cramer & Kluwick 1984). In addition to a peculiar thermodynamic
behaviour, the sound speed is much lower in dense gases when compared to perfect gases,
which makes compressibility regimes much more easily accessible.
Up to now, although dense gas flows comprise non-classical phenomena, in absence of
a better option, perfect gas Reynolds-Averaged Navier Stokes (RANS) and Large Eddy
Simulation (LES) turbulence closure models have been used for dense gas flows (Cinnella
& Congedo 2005; Wheeler & Ong 2014; Dur´a Galiana et al. 2016)). This choice implicitly
assumes that turbulent structures are not affected by dense gas effects. This hypothesis is
not yet verified and constitutes an open-research field. There is currently no experimental
data to verify this hypothesis because maintaining the flow in the vicinity of the critical
point where physical quantities are experiencing strong variations is a very complex task.
Direct Numerical Simulation (DNS) is the tool of choice used in this study to assess
this hypothesis. DNS enables to solve every turbulent scales down to the smallest
ones corresponding to the Kolmogorov length scale without resorting to any turbulence
closure model. So far, few DNS of dense gas flows have been achieved. DNS of decaying
Homogeneous Isotropic Turbulence (HIT) performed by Giauque et al. (2017) shows
that the dynamic Smagorinsky sub-grid scale model is not able to correctly capture
the temporal decay of the turbulent kinetic energy. They extended their analysis by
performing a forced HIT highlighting significant differences in the SGS baropycnal work
and the resolved pressure-dilatation, which is reduced by a factor of 2 in the dense gas
(DG) when compared to the perfect gas (PG) (Giauque et al. 2020).
Sciacovelli et al. (2017b) performed DNS of decaying HIT and notice reduced levels
of thermodynamic fluctuations in dense gas flows due to the decoupling of thermal
and dynamic phenomena caused by the large heat capacity. The Eckert number, which
quantifies the ratio between the kinetic energy and the internal energy, is indeed much
smaller in dense gas flows. They also display a more symmetric probability density
function (PDF) of the velocity divergence in BZT DG flows, explained by the presence
of expansion shocklets and by the attenuation of compression shocklets. They show that
turbulence structures are modified by expansion regions: the occurrence of non-focal
convergent structures in DG flows diminishes the vorticity and counterbalances enstrophy
destruction. Sciacovelli et al. (2017a) analyse DG flow behaviour in a turbulent channel
flow. Initial thermodynamic state was this time chosen in a non-BZT DG region. They
observe significant differences with respect to PG flows in thermodynamic variables. Tem-
4A. Vadrot, A. Giauque and C. Corre
perature variations are negligible in DG which leads to an almost isothermal evolution.
The viscosity decreases from the wall towards the centreline unlike in PG flows. They
also notice significant differences in the shape and rates of the fluctuating density and
temperature distributions. It is also found that the structure of turbulence is not deeply
affected in DG flows. An extent of this study to the BZT DG region and to a larger
Mach number would help to conclude on BZT DG effect on turbulence development.
Gloerfelt et al. (2020) performed the DNS of a dense gas compressible boundary layer
at Mach numbers ranging from 0.5 to 6. They especially confirm the decoupling between
dynamical and thermal effects, which leads to a suppression of friction heating. The most
remarkable consequence is that the boundary layer thickness remains equal to its value
in the incompressible regime as the Mach number increases.
Recently, Vadrot et al. (2020) performed DNS of temporal compressible mixing layers
for BZT DG flow and PG flow at a convective Mach number Mc= 1.1, which is defined
as:
Mc= (u1u2)/(c1+c2) (1.3)
where uiand cidenotes the flow speed and the sound speed of stream i(upper or lower)
of the mixing layer.
They show that the mixing layer is significantly affected by dense gas effects during the
initial unstable growth phase, revealing a much faster unstable growth in the DG flow.
However, only slight differences are observed during the self-similar period, which is the
regime of interest when studying mixing layers. Self-similarity is thoroughly described
in section 3.1. Results from this initial study at Mc= 1.1 also show that the turbulent
Mach number (Equation 1.4) is in the low-limit to get shocklets.
Mt=qu0
iu0
i
c(1.4)
The authors expect that shocklets, which exhibit very different properties in DG flows
when compared to PG flows, would have an impact on the mixing layer growth. In order
to account for these additional effects, an extent of the study to larger convective Mach
numbers is hereby considered.
Since it is known that there are major differences between BZT DG flow and PG flow
in shocklet generation, a study in a higher compressible regime would help to answer the
following question: is the mixing layer growth rate modified in BZT dense gas flows ?
Since the past thirty years, many DNS of mixing layers have been achieved. The
first ones were performed by Sandham & Reynolds (1990); Luo & Sandham (1994);
Vreman et al. (1996). These DNS use the perfect gas hypothesis. A common feature
of compressible mixing layers, shown by experiments at first and DNS afterwards,
is the reduction of the mixing layer growth rate with the increase of the convective
Mach number. However, detailed mechanisms responsible for this trend are still under
investigations.
At first, additional terms in the turbulent kinetic energy equation due to compress-
ibility effects: compressible dissipation dand pressure-dilatation Πii were suspected
to be responsible for the growth rate reduction. Zeman (1990) and Sarkar et al. (1991)
especially proposed models for the dilatation dissipation. However, it was shown by Sarkar
(1995) that the growth rate diminution is primarily due to the reduction of turbulent
production and not to dilatation terms. Vreman et al. (1996) confirmed that dilatation
terms play a minor role in mixing layer growth and extended previous analysis, showing
that pressure-strain terms Πij diminution is responsible for the turbulent production
decrease. They also noticed thanks to DNS that this diminution is mainly due to the
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc5
decrease of pressure fluctuations normalised by the dynamic pressure (prms/(1
2ρ0∆u2)).
Pantano & Sarkar (2002) later demonstrated analytically the aforementioned observation.
Hamba (1999) performed the DNS of an homogeneous shear flow varying Mtfrom 0.1
to 0.3. The author identifies a dissipative term, responsible for the normalised pressure
fluctuations diminution, in the transport equation for p02called pressure-variance dissi-
pation and which depends on the thermal conductivity. Several turbulence models were
next proposed, based on the normalised pressure fluctuations reduction (Fujiwara et al.
2000; Park & Park 2005; Huang & Fu 2008).
However, few experiments and DNS have been achieved at high Mc. Rossmann et al.
(2001) have experimentally studied higher compressibility regimes until Mc= 2.25
and Matsuno & Lele (2020) recently performed DNS of temporal mixing layers up to
Mc= 2.0, but none of them is performed for real gas let alone for DG flows.
In the present article, several 3D DNS of compressible dense gas mixing layers are
performed for the first time at Mc= 2.2. A comparison is made between PG and DG
flows. Evolution of the mixing layer growth rate as a function of the convective Mach
number is compared between perfect gas and dense gas flows. This study extends previous
analysis conducted at Mc= 1.1 (Vadrot et al. 2020).
An unusual behaviour is noticed, as the decrease of the mixing layer growth rate with
the convective Mach number does not follow the same evolution between DG and PG
flows. Discrepancy is not significant at lower Mach number Mc= 1.1 (Vadrot et al. 2020)
but when the convective Mach number increases, DG mixing layer growth is influenced
by modified thermodynamics behaviour. Differences are first analysed in the context of
the peculiar shocklets properties in BZT DG flows. Finally, thermodynamics behaviour
of DG flows is also investigated.
The first section is devoted to the problem description exposing the main physical
and numerical parameters. Results are validated for the perfect gas flow in the second
section with a comparison to available results in the literature. Comparison is made
between dense gas and perfect gas in section 4. Finally, a physical analysis of discrepancies
between DG and PG flows is conducted thanks to additional DNS performed at different
thermodynamic operating points (Section 5). The aim of this analysis is to highlight and
explain differences between BZT DG and PG flows at large convective Mach number.
2. Problem formulation
2.1. Initialisation
The problem consists in extending the analysis conducted at Mc= 1.1 in Vadrot et al.
(2020) by performing a DNS of a 3D mixing layer at a convective Mach number Mc= 2.2
for air considered as a perfect gas and for a BZT dense gas: the perfluorotripentylamine
(FC-70, C15F33N). Physical parameters associated to FC-70 and used in these DNS are
given in table 1.
The initial thermodynamic state is chosen inside the inversion region in order to
favour the occurrence of expansion shocklets, physically allowed in BZT dense gases.
Figure 1 shows the initial state in the pvdiagram and its distribution during the
beginning of the self-similar regime at τ= 4000 for DG flow. The initial value of the
fundamental derivative is Γinitial =0.284 which makes possible the appearance of
expansion shocklets. The distribution spreads inside and slightly outside the inversion
region. One can also note that the distribution does not perfectly follow the initial
adiabatic curve. Mechanical dissipation and shocklets entropy losses are responsible for
this discrepancy because their effect cannot be neglected at Mc= 2.2.
6A. Vadrot, A. Giauque and C. Corre
Tc(K) pc(atm) ZcTb(K) m(= cv(Tc)/R)n
FC-70 608.2 10.2 0.270 488.2 118.7 0.493
Table 1. Physical parameters of FC-70 (Cramer, 1989). The critical pressure pc, the critical
temperature Tc, the boiling temperature Tband the compressibility factor Zc=pcvc/(RTc) are
the input data for the Martin-Hou equation. The critical specific volume vcis deduced from the
aforementioned parameters. The acentric factor nand the cv(Tc)/R ratio are used to compute
the heat capacity cv(T) (R=R/M being the specific gas constant computed from the universal
gas constant Rand M, the gas molar mass).
Mcρ12Reδθ,0Lx×Ly×LzNx×Ny×Nz∆u (m.s1)δθ,0(nm)L0
Air 0.1 1.0 160 344 ×344 ×86 1024 ×1024 ×256 34.11 135.8Lx/48
Air 1.1 1.0 160 344 ×172 ×86 1024 ×512 ×256 375.2 12.35 Lx/48
Air 2.2 1.0 160 688 ×688 ×172 1024 ×1024 ×256 753.0 6.153 Lx/8
FC-70 0.1 1.0 160 344 ×344 ×86 1024 ×1024 ×256 5.665 2070 Lx/48
FC-70 1.1 1.0 160 344 ×172 ×86 1024 ×512 ×256 62.32 188.2Lx/48
FC-70 2.2 1.0 160 688 ×344 ×172 1024 ×512 ×256 125.1 93.77 Lx/8
Table 2. Simulation parameters. Lx,Lyand Lzdenote computational domain lengths
measured in terms of initial momentum thickness. Nx,Nyand Nzdenote the number of grid
points. L0denotes the size of initial turbulent structures (k0= 2π/L0) measured in terms of
initial momentum thickness. All grids are uniform.
For air, the same values of reduced specific volume and reduced pressure are selected
for the initial thermodynamic state. Critical values used for air are the critical pressure
pc= 3.7663 ×106P a and the specific volume vc= 3.13 ×103m3.kg1(Stephan &
Laesecke 1985).
Key non-dimensional parameters are the convective Mach number (Equation 1.3) and
the Reynold number based on the initial momentum thickness δθ,0:
Reδθ,0=∆uδθ,0(2.1)
where νdenotes the kinematic viscosity and the momentum thickness at time tis defined
as:
δθ(t) = 1
ρ0∆u2Z+
−∞
ρ∆u2
4˜u2
xdy (2.2)
with ρ0= (ρ1+ρ2)/2 the averaged density and ˜uxthe Favre averaged streamwise velocity
defined in Eq. 2.9.
The initial momentum thickness Reynolds number is set equal to 160 for all the DNS
following Pantano & Sarkar (2002). Table 2 summarizes the computational parameters
of simulations performed for different Mc(domain size, number of grid elements, di-
mensional values of velocity, initial momentum thickness and initial turbulent structures
sizes). Additional DG simulations given in Appendix A have been performed for other
domain sizes and resolutions to validate the current DNS. The impact on the selection
of the self-similar period is also analysed in Appendix A.
The temporal mixing layer consists in two streams flowing in opposite directions. The
velocity in the upper part of the domain U1is set equal to ∆u/2, whereas U2is set
to ∆u/2. A representation of the computational domain is provided in figure 2. Periodic
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc7
Figure 2. Temporal mixing layer configuration.
boundary conditions are imposed in the xand zdirections and non-reflective conditions
are set in the ydirections using the NSCBC model proposed by Poinsot & Lele (1992).
The streamwise velocity field is initialised using an hyperbolic tangent profile:
¯ux(y) = ∆u
2tanh y
2δθ,0(2.3)
The complete streamwise velocity field is obtained by adding fluctuations to the average
velocity. For the yand zcomponents, the average velocity is set equal to zero. A Passot-
Pouquet spectrum is imposed for initial velocity fluctuations:
E(k) = (k/k0)4exp(2(k/k0)2) (2.4)
where kdenotes the wavenumber. The peak wavenumber k0controls the size of the initial
turbulent structures. Its influence on the mixing layer growth is investigated in Appendix
A. Its value only influences the initial unstable growth regime. It has been noted that
a larger value of k0accelerates the transition to the unstable growth. Its value for each
DNS is given in table 2. The velocity field is then filtered to initialize turbulence only
inside the initial momentum thickness.
2.2. Governing equations
In order to describe the temporally evolving mixing layer, the unsteady, three-
dimensional, compressible Navier-Stokes equations are solved:
∂ρ
∂t +(ρui)
∂xi
= 0 (2.5)
(ρui)
∂t +(ρuiuj)
∂xj
=∂p
∂xi
+∂τij
∂xj
(2.6)
(ρE)
∂t +[(ρE +p)uj]
∂xj
=(τij uiqj)
∂xj
(2.7)
8A. Vadrot, A. Giauque and C. Corre
where τij =µ(∂ui
∂xj+uj
∂xi2
3
∂uk
∂xkδij ) denotes the viscous stress tensor (µthe dynamic
viscosity), E=e+1
2uiui, the specific total energy (e, the specific internal energy),
qj=λ∂T
∂xj, the heat flux given by Fourier’s law (λthe thermal conductivity).
Part of this study is conducted thanks to the analysis of the turbulent kinetic energy
(TKE) equation terms. It requires to decompose density, pressure and velocity into mean
and fluctuating components as follows:
ρ= ¯ρ+ρ0
p= ¯p+p0
ui= ˜ui+u00
i
(2.8)
where ¯
φdenotes the Reynolds average for a flow variable φwhile the Favre average ˜
φis
defined as :
˜
φ=ρφ
ρ(2.9)
Reynolds fluctuations are noted φ0while Favre fluctuations are noted φ00. Reynolds
averaging is equivalent to plane averaging along xand zdirections because of the use
of periodic boundary conditions. The TKE equation is obtained from the Navier-Stokes
equation by applying the averaging process:
¯ρ˜
k
∂t +¯ρ˜
k˜uj
∂xj
=ρu00
iu00
j
˜ui
∂xj
| {z }
Production
τ0
ij
∂u00
i
∂xj
| {z }
Dissipation
1
2
∂ρu00
iu00
iu00
j
∂xj
| {z }
Turbulent transport
∂p0u00
i
∂xi
| {z }
Pressure transport
+∂u00
iτ0
ij
∂xj
| {z }
Viscous transport
+p0∂u00
i
∂xi
| {z }
Pressure dilatation
u00
i¯p
∂xi¯τij
∂xj
| {z }
Mass-flux term
(2.10)
where ˜
k=1
2
]
u00
iu00
idenotes the specific turbulent kinetic energy. The main terms of
equation 2.10 are production, dissipation and transport terms. Pressure dilatation
and mass-flux term (the later comprises the baropycnal work) are equal to zero in
the incompressible case. The dissipation term can be decomposed into a solenoidal, a
low-Reynolds number and a dilatational component. The latter is associated to losses
occurring in eddy shocklets. Lee et al. (1991) expressed the dilatational dissipation also
called the compressible dissipation as:
d=4
3ν∂u00
k
∂xk2
2u00
k
∂ν0
∂xk
∂u00
k
∂xk
(2.11)
This expression comprises the effect of viscosity variations unlike Sarkar & Lakshmanan
(1991) and Zeman (1990) who expressed it as d=4
3¯ν u00
k
∂xk2
, neglecting viscosity
variations. For decaying compressible turbulence, Lee et al. (1991) found that Sarkar
& Lakshmanan (1991) and Zeman (1990)’s expression overestimates by about 15% the
compressible dissipation.
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc9
In addition to equations 2.5, 2.6 and 2.7, thermal and calorific perfect gas equations
of state (EoS) are used for air:
p=ρRT
e=eref +ZT
Tref
cv(T0)dT 0
(2.12)
where Ris the specific gas constant, cvthe specific heat capacity, pthe pressure, Tthe
temperature, ρthe density.
For FC-70, the Martin-Hou EoS (referred as MH) will be retained to provide an
accurate representation of BZT dense gas thermodynamic behaviour (Guardone et al.
2004).
p=RT
vb+
5
X
i=2
Ai+BiT+CiekT /Tc
(vb)i
e=eref +ZT
Tref
cv(T0)dT 0+
5
X
i=2
Ai+Ci(1 + kT /Tc)ekT /Tc
(i1)(vb)i1
(2.13)
where (.)ref denotes a reference state, b=vc(1 (31,883Zc+ 20.533)/15), k= 5.475
and the coefficients Ai,Biand Ciare numerical constants determined by Martin & Hou
(1955) and Martin et al. (1959) from physical parameters summarized in table 1.
To complete the thermodynamic description of the BZT dense gas, Chung’s model is
used to compute dynamic viscosity and thermal conductivity (Chung et al. 1988). FC-70
is assumed to behave as a nonpolar gas, its dipole moment is neglected (Shuely 1996).
For PG transport coefficients, the Sutherland’s model is used associated to a constant
Prandtl number set equal to 0.71. The selected constants for Sutherland’s law are the
ones given by White (1998).
2.3. Numerical Setup
DNS are performed using the explicit and unstructured numerical solver AVBP. It
solves the 3D unsteady compressible Navier-Stokes equations coupled with the perfect
gas EoS (Equation 2.12) for Air and the MH EoS for FC-70 (Equation 2.13) using a
two-step time-explicit Taylor Galerkin scheme (TTGC) for the hyperbolic terms based
on a cell vertex formulation (Colin & Rudgyard 2000). The scheme provides high spectral
resolution and low numerical dissipation ensuring a third-order accuracy in space and in
time. AVBP is designed for massively parallel computation and can be used to perform
LES as well as DNS simulations (Desoutter et al. 2009; Cadieux et al. 2012). The scheme
is completed with a shock capturing method. In regions where strong gradients exist, an
additional dissipation term is added following the approach of Cook & Cabot (2004). Its
impact on the resolution of the smallest scales has been analysed in a previous article
(Giauque et al. 2020).
10 A. Vadrot, A. Giauque and C. Corre
Figure 3. Temporal evolution of the mixing layer momentum thickness for Mc= 0.1/1.1/2.2
using air with PG EoS. Slopes are non-dimensional and standard deviations computed over the
self-similar period are indicated on the plot.
3. DNS of Perfect gas mixing layer: verification and validation
This section is devoted to the selection of self-similar periods and the assessment of the
quality of perfect gas DNS performed for air at three different convective Mach numbers
(Mc= 0.1/1.1/2.2).
3.1. Temporal evolution and self-similarity
Figure 3 shows the temporal evolution of the momentum thickness normalized by its
initial value. This key quantity characterizes the development of mixing layers. Time is
non-dimensional (τ=t∆u/δθ,0). The evolution is plotted for three different convective
Mach numbers (Mc= 0.1/1.1/2.2). Results at Mc= 1.1 are extracted from Vadrot
et al. (2020). The same Reynolds number (Reδθ,0= 160) based on the initial momentum
thickness is used for the three different DNS. Simulation parameters are given in table 2.
At Mc= 2.2, the size of initial turbulent structures has been enlarged in order to speed
up the development of the mixing layer.
One can identify three main phases: an initial delay caused by a transition of modes
from the modes in which turbulent kinetic energy is initially injected to the most unstable
ones; an unstable over-linear growth; and the self-similar period, during which the mixing
layer evolves linearly with time. The procedure used to select the self-similar period is
detailed in subsequent paragraphs.
At Mc= 2.2, one can notice that the mixing layer takes a much longer time to develop.
This is consistent with observations of Pantano & Sarkar (2002) who noticed that the
time necessarily to reach self-similar regime increases with compressibility. Self-similarity
is reached around τ11500 after a long unstable growth phase. As a comparison,
at Mc= 0.1 and Mc= 1.1, self-similarity is reached respectively at τ= 700 and
τ= 1700. Moreover, the self-similar period is also stretched as the convective Mach
number increases.
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc11
A long time delay is observed at the beginning of the simulation. That delay is
associated to the transition of modes. Turbulent kinetic energy is initially injected at
a given integral length set equal to Lx/8. Afterwards, energy is distributed over the
whole spectrum and some unstable modes are amplified leading to the unstable growth
phase. In order to reduce this time delay, initial turbulent structures have been chosen
larger in proportion to the initial momentum thickness at Mc= 2.2 (Table 2). This
modification of initial turbulent structures size does not impact the growth rate over the
self-similar regime. This has been carefully verified for DG flows in Appendix A.
In addition, domain lengths are doubled in xand zdirections and multiplied by four
in the ydirection when compared to DNS at Mc= 1.1 relatively to initial momentum
thicknesses. This enables the mixing layer to develop until larger values of δθ(t)θ,0and
to obtain a long enough self-similar period without reaching domain boundaries. Other
simulations performed with smaller domains did not allow the flow to reach self-similarity.
Slopes and standard deviations mentioned in figure 3 are computed over the self-similar
period. One can observe that the growth rate is divided by a factor of about two between
DNS at Mc= 2.2 and at Mc= 1.1. Indeed, compressibility effects tend to reduce mixing
layer development as the convective Mach number increases.
DNS performed at Mc= 0.1 constitutes our reference incompressible case used to plot
˙
δθ/˙
δθ,inc =f(Mc). The computed growth rate is about 0.0131 which is relatively close
to the empirical value of 0.016 given by Pantano & Sarkar (2002). One can notice a very
short unstable growth phase when compared to larger convective Mach numbers cases.
Self-similarity is a major characteristic of mixing layers: during the self-similar period,
flow development can be described using single length and velocity scales. The momentum
thickness linearly evolves with time. This particular state in the development of mixing
layers is widely used to extract key features of mixing layers. The well known chart
giving the evolution of the mixing layer growth rate as a function of the convective Mach
number (Papamoschou & Roshko 1988) is plotted during the self-similar regime. This
period is also used to investigate the balance of the TKE equation, because temporal
solutions can be averaged during self-similarity since the flow is in a statistically stable
state.
The selection of the self-similar period is thus a key point in the study of turbulent
mixing layers, but this choice is difficult especially at high compressible regimes which
require lengthy simulations. One can note that in our case the time required to achieve
self-similarity is multiplied by a factor of about five when the convective Mach number
increases from Mc= 1.1 to Mc= 2.2.
Lots of authors evoke difficulties in reaching self-similarity (Pantano & Sarkar 2002;
Pirozzoli et al. 2015) particularly because of computational domain lengths. Moreover,
criteria to define self-similarity are not standardised. Superposition of the mean velocity
profiles, linear evolution of the momentum thickness, collapse of the Reynolds stress
profiles are three different ways to define the self-similar period.
The same methodology used in Vadrot et al. (2020) is applied here to select the self-
similar period: it relies on the stabilisation of the streamwise production term integrated
over the whole domain. The underlying reason for using this criterion comes from Vreman
et al. (1996) who demonstrated the following relation between the mixing layer growth
rate and the production power (¯ρPxx =ρu00
xu00
y˜ux
∂y ):
δ0
θ=θ
dt =2
ρ0∆u2Z¯ρPxx dy (3.1)
Figure 4 shows the temporal evolution of the non-dimensional streamwise production
12 A. Vadrot, A. Giauque and C. Corre
a) b)
c)
Figure 4. Temporal evolution of the non-dimensional streamwise turbulent production term
integrated over the whole domain P
int = (1/(ρ0∆u3)) RLy¯ρPxxdy (with ¯ρPxx (y) = ρu00
xu00
y˜ux
∂y )
at Mc= 0.1 (a), Mc= 1.1 (b) and Mc= 2.2 (c). Results are shown for the air using PG EoS.
Selections of self-similar period are indicated on each plot.
integrated over the whole domain for the three DNS at Mcranging from 0.1 to 2.2
performed for air using the PG EoS. A constant integrated production is directly related
to a self-similar regime according to Eq. 3.1. Selected self-similar periods are indicated
on each plot. As the convective Mach number increases, the maximum peak of integrated
turbulent production decreases which is consistent with the decrease of the momentum
thickness growth rate. Time required to achieve self-similarity lengthens but self-similar
periods last longer.
Difficulties can be encountered to get a fully stable plateau with an almost constant
integrated turbulent production. Domain lengths have a major influence on self-similarity.
The evolution of the turbulent production follows a piecewise decrease, reaching several
plateaus. It is observed that these piecewise plateaus are directly related to integral
lengths scales. When some turbulent structures grow and become too large for the
computational domain, the integrated turbulent production decreases and reaches
another plateau lower than the previous one. The mixing layer therefore adapts its
growth to domain lengths when the computational box is not large enough. Since the
integrated turbulent production is related to the mixing layer growth rate, a lower
plateau leads to a smaller mixing layer growth rate. Great care therefore needs to be
taken selecting the size of the computational domain as well as a good stabilization of
the integrated turbulent production in order to precisely select the self-similar period.
Influence of the domain size on self-similarity is thoroughly investigated in Appendix A
for dense gas flows and correlations with integral length scales are analysed.
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc13
Figure 5. Evolution of the mixing layer growth rate with respect to the convective Mach
number for air using PG EoS. Comparison is made with available DNS results in literature and
experimental results by Rossmann et al. (2001). Standard deviations are indicated on the plot.
3.2. Validation over the self-similar period
Since self-similar periods are now well defined for each DNS, it is possible to plot the
evolution of the mixing layer growth rate with respect to the convective Mach number.
Figure 5 shows a comparison between current PG results and available numerical (Freund
et al. 2000; Pantano & Sarkar 2002; Kourta & Sauvage 2002; Fu & Li 2006; Zhou et al.
2012; Mart´ınez Ferrer et al. 2017; Matsuno & Lele 2020) and experimental results
(Rossmann et al. 2001) from the literature. Current DNS follow the tendency observed
and described in the literature: the well-known compressibility-related reduction of the
momentum thickness growth rate as Mcincreases. From the incompressible case to
Mc= 2.2, the mixing layer growth rate is divided by a factor of about five. Standard
deviations have also been computed and are reported on the plot. It represents about
5% of the computed growth rates. It is rather difficult to reduce this uncertainty because
of difficulties encountered in reaching perfect self-similarity. This is also illustrated by
the scattering of literature results, which might be a consequence of this phenomenon.
Moreover, the lack of numerical results at highly compressible regimes makes the
validation process more complex.
Yet, numerical parameters given in table 3 confirm the validation of the current DNS.
The integral lengths lxand lzare computed using the streamwise velocity field:
lx=1
2u2
xZLx/2
Lx/2
ux(x)ux(x+rex)dr (3.2)
lz=1
2u2
xZLz/2
Lz/2
ux(x)ux(x+rez)dr (3.3)
Integral length scales show that the domain is chosen sufficiently large. The largest
14 A. Vadrot, A. Giauque and C. Corre
McReδθReλxr=Lη/∆x lx/Lxlz/Lz
Air (τ= 700) 0.1 1879 209 0.63 0.10 0.04
Air (τ= 1450) 0.1 3444 194 0.81 0.11 0.13
FC-70 (τ= 550) 0.1 1448 135 0.58 0.04 0.05
FC-70 (τ= 900) 0.1 2176 201 0.7 0.07 0.06
Air (τ= 1700) 1.1 1874 143 0.97 0.07 0.06
Air (τ= 2550) 1.1 2413 156 1.09 0.12 0.08
FC-70 (τ= 1700) 1.1 2469 176 0.80 0.09 0.05
FC-70 (τ= 2550) 1.1 3304 241 0.87 0.20 0.05
Air (τ= 11500) 2.2 3487 146 1.44 0.12 0.07
Air (τ= 14100) 2.2 3700 191 1.64 0.11 0.10
FC-70 (τ= 4000) 2.2 4663 263 0.52 0.10 0.06
FC-70 (τ= 6000) 2.2 6259 390 0.57 0.16 0.05
Table 3. Non-dimensional parameters computed at the beginning and at the end of the
self-similar period for Mc= 2.2 simulations. Reλxdenotes the Reynolds number based on the
longitudinal Taylor microscale λx=q2u02
x/(∂u0
x/∂x)2computed at the centreline. Lηdenotes
the Kolmogorov length scale computed at the centreline.
value 0.20 is obtained at the end of the self-similar period for DG flow at Mc= 1.1.
Otherwise, values do not exceed 0.16 in the streamwise direction and 0.13 in the z
direction. As a comparison, Pantano & Sarkar (2002)’s integral length scale reaches
0.178 in the streamwise direction for a configuration with Mc= 0.7 and a density ratio
of 4. Appendix A also confirms that domain lengths have been properly chosen for DG
mixing layer at Mc= 2.2.
The ratio r=Lη/∆x characterizes the resolution of simulations. Larger is the ratio,
better is the resolution. Minimum value is about 0.52 computed for DNS at Mc= 2.2.
For other simulations, values are larger than 0.6 and the maximum value is 1.64 for PG
at Mc= 2.2 because of small dissipation in high compressible regimes. As a comparison
Pantano & Sarkar (2002)’s ratio is about 0.38 for the most resolved simulation and
recently Matsuno & Lele (2020) performed a DNS at Mc= 2.0 with a Lη/dx ratio
equal to 0.41. One can thus consider that turbulent scales are adequately resolved for
all simulations presented in this paper since in addition the turbulent kinetic energy is
very low close to the Kolmogorov scale (Moin & Mahesh 1998).
4. Dense gas effect on mixing layer growth
4.1. Temporal evolution
As previously done for the perfect gas mixing layer, it is required to precisely define
the self-similar range for the dense gas flow. This is done through both figures 6 and
7. Figure 6 enables the comparison of normalised DG momentum thickness over time
at three different convective Mach numbers : Mc= 0.1/1.1/2.2. The three DNS are
performed at the same initial Reynolds number Reδθ,0= 160. Additional simulation
parameters are given in table 2. At Mc= 0.1, similarly to PG mixing layer, the domain
length is doubled in the ydirection to get a long enough self-similar period. At Mc= 2.2,
the domain length is divided by two in the ydirection when compared to PG flow. The
domain is therefore large enough to reach a self-similar period which lasts 4000τ. Initial
turbulent structures are chosen six times larger at Mc= 2.2 when compared to other Mc
to be consistent with PG simulation. It is nevertheless shown in Appendix A that the size
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc15
Figure 6. Temporal evolution of the mixing layer momentum thickness for DG at
Mc= 0.1/1.1/2.2.
of initial turbulent structures does not influence the growth rate during self-similarity.
This choice was motivated by the will to shorten the simulation. Enlarging the size of
initial turbulent structures accelerates the unstable growth phase. As a consequence, in
figure 6, Mc= 1.1 and Mc= 2.2 curves overlap after τ2500.
Slopes and standard deviation computed over the self-similar range are given in figure
6. At Mc= 0.1, because of the suppression of compressibility effects, growth rate is very
close to that of PG flow: the difference is about 1.5% and is below the standard deviation
range. Like for PG, the DNS at Mc= 0.1 is considered as the reference incompressible
case and is used to plot the dependence of the normalised momentum thickness growth
rate with respect to Mc. At Mc= 1.1, comparison between DG and PG flows is detailed
in Vadrot et al. (2020) during unstable growth and self-similar phases.
Figure 6 shows that the momentum thickness growth rates are very close between
Mc= 2.2 and Mc= 1.1 unlike the perfect gas case. The well-known decrease of
the growth rate with the convective Mach number is modified by dense gas effects.
Despite being a highly compressible fluid, compressibility effects decrease in F C 70.
Explanations for this effect are given in section 5.
Slopes provided in figure 6 are determined using the same methodology used for PG in
section 3.1. For each convective Mach number, the non-dimensional integrated turbulent
production term P
int is plotted over time. The three main phases described for the PG
flow can also be identified for DG. One can notice that, at Mc= 2.2, the initial phase
corresponding to an energy transfer to the most unstable modes is much shorter for DG
flow, likely because unstable modes are different between the two types of gas. After this
phase, turbulent production reaches a maximum which decreases as Mcincreases. Finally,
self-similar periods are defined selecting the range during which turbulent production is
almost constant. As observed for PG flow, the self-similar period extends as Mcincreases.
One can also notice that integrated production terms in DG flows are consistent with
16 A. Vadrot, A. Giauque and C. Corre
a) b)
c)
Figure 7. Temporal evolution of the non-dimensional streamwise turbulent production term
integrated over the whole domain P
int = (1/(ρ0∆u3)) RLy¯ρPxxdV (with ¯ρPxx (y) = ρu00
xu00
y˜ux
∂y )
at Mc= 0.1 (a), Mc= 1.1 (b) and Mc= 2.2 (c). Results are shown for the FC-70. Self-similar
periods are indicated on each plot.
momentum thickness growth rates: the values of P
int are very close between Mc= 2.2
and Mc= 1.1 and the value of P
int at Mc= 0.1 is twice larger than the one at Mc= 1.1.
This observation confirms the relevance of Vreman et al. (1996) relationship given in
equation 3.1. Beginning and ending times for each DNS self-similar periods are provided
in table 3.
4.2. Comparison with perfect gas over the self-similar period
Self-similar periods have been selected for both types of gas. It is thus possible to
plot the evolution of self-similar growth rates as a function of the convective Mach
number. Slopes are usually normalised using an incompressible reference case at very
low convective Mach number for which compressibility effects can be neglected. DNS at
Mc= 0.1 is considered here as the reference incompressible case. For example, Pantano
& Sarkar (2002) use a simulation at Mc= 0.3 as a reference case. There is no consensus
on this choice, which can partly explain the spreading of PG results observed in figure
8 - where the same literature results used in figure 5 are reported. DG mixing layer
results are added with their corresponding error bars coloured in black, which length is
equal to twice the standard deviation computed over the self-similar range. Unlike PG
mixing layer which shows a fairly abrupt decrease as Mcincreases, DG mixing layer
seems to be much less influenced by compressibility effects as Mcbecomes larger than
1.1. Differences between DG and PG mixing layers are large enough when compared to
standard deviations to reveal that turbulence development is actually modified by dense
gas effects in mixing layer flows.
In order to analyse the impact of compressibility effects, Pantano & Sarkar (2002)
study the TKE equation and particularly the importance of the turbulent production
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc17
Figure 8. Evolution of the mixing layer growth rate over the convective Mach number for air
and for FC-70. Comparison is made with available DNS results in literature and experimental
results in Rossmann et al. (2001).
Figure 9. Distribution of the volumetric normalized powers over the non-dimensional
cross-stream direction y/δθ(t) at Mc= 2.2. P: Production, D: Dissipation and T: Transport
are normalized by ρ0∆u3θ(t). Distributions have been averaged between the upper and the
lower stream to get perfectly symmetrical distributions.
18 A. Vadrot, A. Giauque and C. Corre
term. They find that this term is decreasing in consistent proportion with the growth
rate as the convective Mach number increases. The computation of TKE equation terms
requires to statistically average the terms. This can only be done during the self-similar
period during which both mixing layers are in a statistically stable state. Figure 9 shows
the comparison between DG and PG mixing layers of the normalized main terms of
the TKE equation over the non-dimensional cross-stream direction y/δθ(t). Production,
dissipation and transport terms are averaged during corresponding self-similar ranges.
The production term (denoted P) is always positive and is responsible for the growth of
the mixing layer. Viscous dissipation (denoted D) is always negative and counterbalances
the production term. The transport term (denoted T) enables the propagation of TKE
from the center to the edges of the mixing layer. It is thus negative at the center and
positive at the edges. Consistently with the comparison of slopes between DG and PG
flows, all main terms and particularly the production term are two to three times larger
for DG.
Another noticeable feature which was highlighted in the previous analysis at Mc= 1.1
(Vadrot et al. 2020) is confirmed here: curves are wider for the PG mixing layer, when
compared to the DG mixing layer. For the DG mixing layer, TKE is more localised at
the center. This is directly linked to the thermodynamic profiles, which are wider for
PG mixing layer (see Figure 19 in section 5.3).
Other terms of the TKE equation, namely the compressible dissipation, the mass-flux
coupling term, the convective derivative of the TKE and even the pressure dilatation are
negligible for both types of gas. The pressure dilatation term which is directly linked to
shocklets effects is carefully analysed in section 5.1 to quantify shocklets effects on the
mixing layer growth.
As mentioned in the introduction, Pantano & Sarkar (2002) demonstrate that the
compressibility-related reduction of the momentum thickness growth rate is induced
by the reduction of pressure-strain terms Πij, which causes a reduction of turbulent
production. In the TKE equation, which is obtained from the sum Rii, the pressure-
strain terms do not appear. Their sum Πii, which constitutes the pressure-dilatation
term, appears in the TKE equation but is negligible. In order to study pressure-strain
terms, one needs to plot turbulent stress tensor equations terms. Figure 10 shows x
and yturbulent stress tensor equations main terms. In the streamwise direction, the
pressure-strain term counterbalances the streamwise production, whereas in the cross-
stream directions, pressure-strain term is positive and is balanced by viscous dissipation.
In the cross-stream direction, turbulent production term can be neglected unlike in the
streamwise direction for which it is maximal.
One can notice that pressure-strain terms are significantly reduced for PG flows when
compared to DG flows at Mc= 2.2: streamwise pressure strain term is twice larger
for DG when compared to PG. This is consistent with the comparison of momentum
thickness growth rates. For both types of gas, growth rates are identically linked to their
pressure-strain terms. Compressibility effects impacts the same terms for both DG and
PG.
It remains to verify the last step in Pantano & Sarkar (2002)’s explanation, which
is that the reduction of pressure-strain terms is caused by a reduction of normalised
pressure fluctuations. Figure 11 shows the cross-stream evolution of the root-mean
squared value of pressure normalised by the dynamical pressure 1
2ρ0∆u2. Comparison is
made between DG and PG flows at Mc= 1.1 and Mc= 2.2.
At Mc= 1.1, DG and PG distributions are very close as are their corresponding
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc19
Figure 10. Distribution of the main non-dimensional volumetric power terms of the x- (top)
and y- (bottom) turbulent stress tensor (Rxx and Ryy ) equations over the non-dimensional
cross-stream direction y/δθ(t). Pxx and Pyy : Streamwise and cross-stream production, Πxx
and Πyy : Streamwise and cross-stream pressure-strain and Dxx and Dyy : Streamwise and
cross-stream dissipation terms are normalized by ρ0∆u3θ(t). Results are computed at
Mc= 2.2. Distributions have been averaged between the upper and the lower stream to get
perfectly symmetrical distributions.
momentum thickness growth rates. As the convective Mach number increases, DG
non-dimensional pressure fluctuations experience a 20% decrease also consistent with
the observed decrease in the growth rate. This decrease is yet much smaller than that
of the PG mixing layer, in which normalised pressure fluctuations are approximately
divided by a factor of two. To sum up, although the same mechanism is responsible
for the growth rate decrease in both types of gas (i.e. the reduction of non-dimensional
pressure fluctuations), its effect is significantly different between the two types of gas. For
DG flows, the well-known compressibility-related reduction of the momentum thickness
growth rate is almost suppressed by dense gas effects at convective Mach numbers above
Mc= 1.1.
Figure 12 shows the comparison between PG and DG streamwise specific turbulent
kinetic energy spectra computed over the centreline. Spectra are normalised by ∆u2δθ(t)
20 A. Vadrot, A. Giauque and C. Corre
Figure 11. Distributions of the root mean square value of pressure averaged over the self-similar
period, plotted along the ydirection and compared between FC-70 and Air at Mc= 1.1 and
Mc= 2.2. Distributions have been averaged between the upper and the lower stream to get
perfectly symmetrical distributions.
Figure 12. Streamwise specific TKE spectra computed at the centreline.
in the same way as Pirozzoli et al. (2015) and averaged over the self-similar period.
The longitudinal Taylor microscale λxis also indicated for each gas in figure 12. Its
value is much larger for DG flow consistently with Reynold numbers computed from
Taylor microscales given in table 3. The inertial phase is thus significantly reduced for
the PG flow. Dissipation occurs at much larger scales making the comparison between
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc21
Figure 13. Temporal evolution of the turbulent Mach number Mt.
the two inertial phase slopes difficult. Spectra yet confirm previous results observed at
Mc= 1.1 (Vadrot et al. 2020): dense gas effects tend to increase small scales energy. The
dissipation term, which is the main term at these scales, is significantly reduced. These
results suggest a need for a specific sub-grid scale modelling of dense gas flows whose
dynamic over scales is significantly modified with respect to perfect gas flows.
5. Analysis of discrepancies between DG and PG
5.1. First hypothesis: Shocklets effect
Previous analysis conducted at Mc= 1.1 shows that the growth rate is not influenced
by dense gas effect during the self-similar period (Vadrot et al. 2020). However, significant
differences are observed during the unstable growth phase. At Mc= 1.1, the evolution of
the turbulent Mach number shows that shocklets might be detected during the unstable
growth phase but not during the self-similar range, during which Mtdecreases well
below the range of values for which shocklets are expected. It is known that shocklet
generation is different between BZT DG flow and PG flow (Giauque et al. 2020), yet
can shocklets alone explain discrepancies between DG and PG flows ?
In the current analysis, we increase the convective Mach number in order to reach
larger turbulent Mach numbers during the self-similar period and to analyse the influence
of shocklets. Figure 13 shows the temporal evolution of the turbulent Mach number
Mt(see equation 1.4). Turbulent Mach numbers increase during the initial phase up to
1.1 and 0.9 respectively for DG and PG flows. Then Mtdecreases and reaches a rather
stable plateau corresponding to the self-similar period. During this phase, average values
of turbulent Mach numbers are respectively equal to 0.67 and 0.49 for DG and PG flows.
Shocklets can thus be observed during both DG and PG self-similar periods.
In order to study their effect on the growth rate, one can analyse the compressible
22 A. Vadrot, A. Giauque and C. Corre
Figure 14. Distributions of the ratio between the compressible dissipation (d) and the total
dissipation () (see details in Equations 2.10 and 2.11). Results are averaged over the self-similar
period. Comparison is made between FC-70 and Air at Mc= 1.1 and Mc= 2.2. Distributions
have been averaged between the upper and the lower stream to get perfectly symmetrical
distributions.
component of the dissipation given in Equation 2.11. Zeman (1990) and Sarkar et al.
(1991) show that the dilatational part of the dissipation increases with the turbulent
Mach number because of the occurrence of eddy shocklets in the compressible regime.
Wang et al. (2020) perform a compressible isotropic turbulence and observe that
shocklets act as kinetic energy sinks which absorb large-scale kinetic energy. Shocklets
are thus an additional source of dissipation. The dilatational dissipation is computed over
the self-similar period. Figure 14 shows the ratio between the compressible dissipation
and the total dissipation over the cross-stream direction. Around y/δθ(t)3.5, one can
note an increase of the ratio. It corresponds to the borders of the mixing layer, outside
of which the dissipation drops to zero (see Figure 9). Except for these regions, at
Mc= 1.1, the compressible dissipation represents less than 0.5% of the total dissipation
for both DG and PG flows. At Mc= 2.2, the ratio increases consistently with the increase
of turbulent Mach numbers. The ratio is thus larger for DG flow compared to PG flow.
However, the rate of dilatational dissipation with respect to the total dissipation remains
below 4% for DG and below 1% for PG. Compressible dissipation can therefore be
neglected with respect to the total dissipation. Shocklets have a limited influence on the
TKE equation. Since the TKE equation governs the mixing layer dynamics, one cannot
explain discrepancies observed between DG and PG flows with shocklets effect.
5.2. Additional simulations varying the initial thermodynamic operating point
In order to explain discrepancies observed between DG and PG flows, we perform
additional DNS varying the initial thermodynamic operating point. Figure 15 shows the
four selected operating points. DGA corresponds to the reference simulation analysed in
section 4. DGA’s initial operating point is located inside the inversion zone also called
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc23
Figure 15. Four different initial thermodynamic states used to perform additional DNS are
represented in the non-dimensional pvdiagram for BZT dense gas FC-70 at Mc= 2.2. The
dense gas zone (Γ < 1) and the inversion zone (Γ < 0) are plotted for the Martin-Hou equation
of state. pcand vcare respectively the critical pressure and the critical specific volume.
McLx×Ly×LzNx×Ny×NzL0r=Lη/∆x lx/Lxlz/Lz
DGA 2.2 688 ×344 ×172 1024 ×512 ×256 Lx/8 0.52 0.57 0.10 0.16 0.06 0.05
DGB 2.2 688 ×344 ×172 1024 ×512 ×256 Lx/8 0.51 0.55 0.11 0.12 0.06 0.04
DGC 2.2 688 ×688 ×172 1024 ×1024 ×256 Lx/8 0.50 0.55 0.11 0.166 0.06 0.04
DGD 2.2 688 ×688 ×172 1024 ×1024 ×256 Lx/8 0.50 0.54 0.09 0.14 0.07 0.07
Table 4. Simulation parameters for additional FC-70 simulations varying the initial operating
point. r,lx/Lxand lz/Lzare given at beginning and ending times of self-similar periods.
BZT region. The operating point of the second simulation DGB is chosen outside the
inversion region and inside the dense gas zone. This enables us to investigate the impact
of BZT effects on the mixing layer growth. Finally, for DGC and DGD, initial operating
points are chosen on the same adiabatic curves as respectively DGB and DGA but
outside the dense gas zone. The diversity of targeted thermodynamic regions aims at
providing a proper insight into the effects of dense gas on the shear layer growth.
At first, one needs to validate DNS named DGB, DGC and DGD. Table 4 gives
simulations parameters including r,lx, and lzfor the four different simulations.
Achieved values are very close to DGA and since DGA has been validated previously
(see section 4 and Appendix A), one can consider that DGB, DGC and DGD are
adequately resolved. The size of computational domains have been enlarged for DGC
and DGD in the ydirection in order to provide the mixing layer with more space in
order to reach self-similarity.
Self-similar periods are defined for each DNS using the same methodology previously
24 A. Vadrot, A. Giauque and C. Corre
Figure 16. Temporal evolution of the non-dimensional streamwise turbulent production terms
integrated over the whole domain P
int = (1/(ρ0∆u3)) RLy¯ρPxxdV (with ¯ρPxx (y) = ρu00
xu00
y˜ux
∂y )
at Mc= 2.2. Results are shown for the FC-70 for four different DNS: DGA, DGB, DGC
and DGD. Self-similar periods are indicated on each plot: DGA (τ[4000/6000]); DGB
(τ[4000/6400]); DGC (τ[3800/6000]) and DGD (τ[3800/6000]).
presented in section 3.1. Plateaus showing constant integrated turbulent production
correspond to self-similar periods. They are identified with vertical lines in figure 16. In
addition, beginning and ending times are given in the caption for each case. Although
all the DNS are performed at the same convective Mach number Mc= 2.2, results are
quite different. The initial evolution is similar, but after τ1100, discrepancies appear
especially for DGD. Maximum values and self-similar regimes are influenced by the
initial thermodynamic operating point.
The comparison of mixing layer momentum thickness evolutions is done in figure 17.
Slopes with standard deviations computed during self-similar regimes are indicated on
the plot. From these results, one can deduce that BZT region does not have a major
influence on the mixing layer growth. DGC’s growth rate is indeed very close to DGA’s
although initial thermodynamic operating points are located respectively outside and
inside DG and BZT regions. The relation between the mixing layer growth and the
initial thermodynamic operating point is not obvious: operating points located on the
same adiabatic curve (respectively DGA, DGD and DGB, DGC) are far away in terms
of growth rate. Looking at the growth rate, simulations can be classified by pairs: DGA
goes with DGC and DGB goes with DGD. One can observe that slopes are all below the
Mc= 1.1 growth rate. It means that the well-known compressibility-related reduction
of the momentum thickness growth rate is still verified. Yet there is an additional effect
due to the initial thermodynamic operating point.
At the end of section 4, the physical explanation provided by Pantano & Sarkar (2002)
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc25
Figure 17. Temporal evolution of the mixing layer momentum thickness for DG at Mc= 2.2.
Results are shown for the FC-70 for four different DNS: DGA, DGB, DGC and DGD.
Figure 18. Evolution of the non-dimensional mixing layer growth rate over the center root-mean
squared value of pressure normalised by 1
2ρ0∆u2. Results are given for DG and PG at Mc= 1.1
and Mc= 2.2.
was assessed on DGA: the reduction of the momentum thickness is due to a reduction of
normalised pressure fluctuations. It remains to check whether this reduction of normalised
pressure fluctuations is also observed for DGB, DGC and DGD. Figure 18 shows the
normalised growth rate as a function of the normalised pressure fluctuations computed
26 A. Vadrot, A. Giauque and C. Corre
at the center of the mixing layer. For PG flow, the reduction is significant. Between
Mc= 1.1 and Mc= 2.2, growth rate and normalised pressure fluctuations are divided
by a factor of two. For DG, the decrease of the normalised growth rate is also correlated
with a decrease of pressure fluctuations. Among cases at Mc= 2.2, the ranking purely
based on the level of pressure fluctuations is not entirely satisfactory but this could
be explained by standard deviations caused by variations of the plateaus of integrated
turbulent production. Moreover, other effects must also be taken into account for dense
gases: this is the topic of the next section.
5.3. Analysis of discrepancies between DG and PG flows
There is a significant effect of dense gas on the well-known compressibility-related
reduction of the momentum thickness growth rate. Dense gas effects modify the decrease
at convective Mach numbers larger than Mc= 1.1. Between Mc= 1.1 and Mc= 2.2, the
growth rate slope does not vary much for DG. Several factors can be identified, which
contribute to explain the observed discrepancies between DG and PG mixing layers. The
first main difference between DG and PG flows is the ratio between internal and kinetic
energies. It is associated to the Eckert number, which is defined for the mixing layer as:
Ec =∆u2
cp0T0
(5.1)
where cp0denotes the initial specific heat capacity at constant pressure and T0, the
initial temperature. Initial Eckert numbers are computed for each DNS and results are
gathered in table 5. For DG flows, values are about two orders of magnitude lower
than PG flows. Two features of DG mixing layers are responsible for these significant
differences: the large heat capacity of FC-70 and the small differential speed ∆u. The
differential speed is defined in order to get the same initial convective Mach number
between DG and PG mixing layers. Since the sound speed is much lower in dense gases,
a much lower differential speed is obtained for a given value of the convective Mach
number, which mechanically reduces the Eckert number. With small Eckert numbers
kinetic energy becomes negligible when compared to internal energy. It is the case for all
DG flows in this study even though the convective Mach number is large. As shown by
the present results, kinetic energy also decouples from thermodynamics compressibility
effects and the growth rate of the momentum thickness is allowed to reach larger values.
It can be observed the close values of the momentum thickness growth rates for DGA /
DGC on one hand and DGB / DGD on the other hand are well correlated with the values
of the initial Eckert number reported in table 5. The lower Eckert numbers for DGA
/ DGC correspond to higher growth rates for these shear layer configurations, induced
by an even stronger decoupling between internal and kinetic energy for DGA/DGC
with respect to DGB/DGD. However, the Eckert number can not be the only factor
explaining dense gas effect on the growth rate since DGC displays a slightly lower growth
rate with respect to DGA, with a slightly lower value of the initial Eckert number.
For DG flows, the amount of internal energy is much larger when compared to kinetic
energy. Internal and kinetic energies are decoupled in that case. In the equation of
energy conservation (Equation 2.7), all the terms can be neglected with respect to the
temporal and convective internal energy terms. Since the Eckert number quantifies the
friction heating, it is significantly reduced in DG flows as previously shown by Gloerfelt
et al. (2020). Figure 19 shows the distribution of the Reynolds averaged temperature,
density and the root mean square value of density fluctuations over the cross-stream
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc27
McEc ˙
δθ/˙
δθ,inc
DG 1.1 0.0040 0.484
DGA 2.2 0.0162 0.395
DGB 2.2 0.0226 0.352
DGC 2.2 0.0147 0.389
DGD 2.2 0.0203 0.342
PG 1.1 1.94 0.450
PG 2.2 7.74 0.188
Table 5. Eckert numbers and normalized momentum thickness growth rates are given for
each simulation.
direction of the shear layer. Results are averaged over the self-similar period. It can
be observed in figure 19 that temperature variations are almost suppressed for DG.
Sciacovelli et al. (2017a) confirm this remark in supersonic turbulent channel flows and
state that dense gas flow are less subject to friction losses associated with Mach number
effects. For the mixing layer, above Mc= 1.1, compressibility effects associated with the
increase of convective Mach number have less influence on DG flows in part because of
the reduction of friction heating.
The evolution of the average density confirms this reduction. The PG air density
suffers a 40% decrease at the center between Mc= 1.1 and Mc= 2.2. In the PG,
friction heating is important and leads to an increase of the temperature, which induces
a decrease of the density. The mechanism is significantly reduced in dense gas flows.
For DG, the temperature is almost constant and averaged density displays very limited
variations. At Mc= 2.2, the averaged density decrease at the center of the mixing layer
represents about 8% of the initial density compared to 45% for air. Equation 3.1 shows
that this effect influences the mixing layer growth rate which depends on the density. As
the mixing layer develops in PG, strong friction occurs at the center, which decreases
the density. The momentum thickness growth rate is thus significantly reduced for PG
when compared to DG.
Figure 19 (right) displays the root mean square value of density fluctuations. Between
PG and DG flows, the distribution across the mixing layer changes shape. For PG, it
consists in two symmetric peaks with respect to the center of the mixing layer. Peaks are
located at the borders of the mixing layer, where the cross-stream gradient of averaged
density is maximal. In this region, the mixing layer flow experiences strong dynamic and
thermal variations with an important coupling between internal and kinetic energy. For
DG, the distribution is composed of a single peak located at the center of the mixing
layer. The distribution is much less affected by the variation of the averaged density. For
DG, thermal quantities are less influenced by flow dynamics because of the decoupling
of internal energy and kinetic energy. The root mean square value of density fluctuations
diffuses from the center of the mixing layer.
The amplitudes of the distributions are also quite different between DG and PG flows.
For DG, the maximum root mean square value of density fluctuations is multiplied by a
factor of three from Mc= 1.1 to Mc= 2.2. In the PG case, it is multiplied by a factor of
about two. Compressible flows are more subject to root mean square density fluctuations
which increase as the Mach number grows. An explanation can be found in the definition
of the isentropic compressibility coefficient, which is large for DG flows:
28 A. Vadrot, A. Giauque and C. Corre
a) b)
c)
Figure 19. The non-dimensional Reynolds averaged temperature (a) and density (b); and root
mean squared value of the density (c) are averaged over the self-similar regime and plotted along
the y direction. Comparison is made between FC-70 and Air at Mc= 1.1 and Mc= 2.2.
χs=1
ρ
∂ρ
∂p s
(5.2)
For flows with large values of χs, small variations of pressure lead to large variations
of density. The sound speed is directly linked to the isothermal compressibility since:
c=1
ρχs
(5.3)
For DG flows, the large isentropic compressibility factor strongly diminishes the sound
speed. As a result, the initial sound speed in the computed DG flows is about six times
smaller when compared to its initial value for the PG shear layers. Figure 20 shows the
normalised momentum growth rate at Mc= 2.2 as a function of the normalised sound
speed. A rather clear correlation appears between the momentum thickness growth rate
and the initial sound speed: the growth rate decreases with increasing sound speed.
The main conclusion that can be drawn from these observations is that the smaller
Eckert number in DG flows causes a decoupling between internal and kinetic energy and
induces less friction heating. Both phenomena influence the mean and fluctuating thermal
physical quantities, which consequently limits the compressibility-related reduction of the
momentum thickness growth rate.
6. Concluding remarks
The present work extends the previous analysis of a temporal compressible shear
layer conducted at Mc= 1.1 (Vadrot et al. 2020) to a larger convective Mach number
Mc= 2.2 for air described as a perfect gas and FC-70 (BZT gas) described using
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc29
Figure 20. Evolution of the non-dimensional mixing layer growth rate as a function of the
sound speed normalised with ppcc. Results are given for DG and PG at Mc= 2.2.
Martin Hou EoS. A reference incompressible DNS is also performed at Mc= 0.1 to
provide the incompressible growth rate ˙
δθ,inc used to normalize the growth rate ˙
δθ. The
computed evolution of the mixing layer growth rate with respect to the convective Mach
number is compared with available results from the literature for perfect gas. The PG
results are found consistent with the literature and establish the accuracy of the present
simulations.
The choice of the domain size is paramount in this study. The domain is enlarged
at Mc= 2.2 for both DG and PG DNS when compared to DNS at Mc= 1.1 in order
to ensure mixing layers reach self-similarity. An analysis presented in Appendix A is
performed to thoroughly investigate the sensitivity of the DG mixing layer to domain
extent and to the size of initial turbulent structures. Results establish the relevance of
the choices of domain extent and initial structures size made in the present study.
The selection of the self-similar period is a key point in the study of mixing layers: this
choice is complex and the diversity of criteria used for the selection process contributes to
the scattering of ˙
δθ/˙
δθ,inc =f(Mc) plots reported in the literature. In the present work,
self-similar periods are selected using the integrated streamwise production over time,
which is proportional to the momentum thickness growth rate under certain conditions
(Vreman et al. 1996).
The comparison between perfect and dense gases shows major differences for the
momentum thickness growth rates at Mc= 2.2. The dense gas flow limits the well-
known compressibility-related reduction of the momentum thickness growth rate. At
Mc= 2.2, the growth rate is twice as large for dense gas when compared to perfect
gas. Pantano & Sarkar (2002) demonstrate that for perfect gas flows the growth rate
reduction is due to the reduction of pressure fluctuations leading to the reduction
of pressure-strain terms. We show that growth rate is also correlated with pressure
fluctuations in dense gas flows. Yet, the small scales dynamics is very different. A much
larger dissipation is also observed for perfect gas mixing layer. These results call for a
30 A. Vadrot, A. Giauque and C. Corre
McLx×Ly×LzNx×Ny×NzL0
DG0 1.1 344 ×172 ×86 1024 ×512 ×256 Lx/48
DG1 2.2 344 ×172 ×86 1024 ×512 ×256 Lx/48
DG2 2.2 344 ×344 ×86 1024 ×1024 ×256 Lx/4 = 86
DG3 2.2 648 ×344 ×172 1024 ×512 ×256 Lx/8 = 86
PG0 2.2 688 ×688 ×172 1024 ×1024 ×256 Lx/4
Table 6. Simulation parameters for temporal shear layer DNS (Reδθ,0= 160) with varying
domain extent, resolution and size of initial structures. Lx,Lyand Lzdenote computational
domain lengths measured in terms of initial momentum thickness. Nx,Nyand Nzdenote
the corresponding numbers of grid points. L0denotes the size of initial turbulent structures
(k0= 2π/L0) measured in terms of initial momentum thickness. All grids are uniform.
specific sub-grid scale modelling for dense gas flows when simulated using Large Eddy
Simulation.
Additional dense gas DNS have been performed at three others initial thermodynamic
operating points. Results show that BZT effects have only a small impact on the mixing
layer growth. Shocklets indeed produce only a limited effect on mixing layer growth.
The compressible dissipation is negligible when compared with the total dissipation.
For dense gas mixing layers, several physical factors tend to reduce compressibility
effects: the decoupling of kinetic and internal energy reduces the effect of increasing Mc;
reduced friction losses in dense gas flows modify the distribution of the averaged density,
which therefore favours the momentum thickness growth rate. Finally, it is found that
increasing the initial isothermal compressibility also increases the momentum thickness
growth rate in dense gas flows. Initial sound speed could therefore be an appropriate
indicator when forecasting the mixing layer growth rate in real gas flows.
Acknowledgements - This work is supported by the JCJC ANR EDGES project,
grant #ANR-17-CE06-0014-01 of the French Agence Nationale de la Recherche. Sim-
ulation have been carried out using HPC resources at CINES under the project grant
#A0062A07564.
Appendix A. DG mixing layer: Influence of domain size, resolution
and initial turbulent structures size
Additional simulations have been performed for DG mixing layer with Reδθ,0= 160
and Mc= 2.2 in order to confirm proper resolution and domain size. The computational
parameters corresponding to these simulations are summarized in Table 6 along with
the parameters used in the previous study at Mc= 1.1.
Figure 21 shows temporal evolutions of momentum thickness for the simulations listed
in Table 6. DG1 is performed with the same domain lengths and size of initial turbulent
structures (relatively to the initial momentum thickness) as in the previous Mc= 1.1
study DG0. At τ= 4000, self-similarity is not yet achieved but flow field visualisations
indicate that the yboundaries of the domain are reached. DG2 is then conducted with
a domain size doubled in the ydirection and with smaller initial turbulent structures
corresponding to Lx/4 = 86δθ,0, in order to speed up the mixing layer development.
Simulations show that the modification of initial structures size only modifies the time
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc31
Figure 21. Temporal evolution of the mixing layer momentum thickness.
necessarily to reach the unstable growth phase but not the growth rate itself.
Yet, a large decrease of the growth rate is observed for DG2 around τ= 4000; self-
similarity cannot be reached. Figure 22 displays the time evolution of the integral length
scale in the zdirection lzfor DG2 and DG3 simulations. Around τ= 4000, the integral
length scales lz/Lzsuddenly decreases for DG2 after having reached a value of 0.2. The
domain is thus not large enough to account for spanwise turbulent structures, which
causes a growth rate decrease and prevents the transition to self-similarity.
Because of the aforementioned observations, domain sizes have been doubled in xand
zdirections when compared to DG1. This corresponds to the DG3 simulation, which is
the reference DNS used in Section 4 to compare results between DG and PG. For DG3,
the momentum thickness evolution reaches a perfectly linear stage and self-similarity is
well achieved as confirmed by Figures 7 (right) and 21.
REFERENCES
Bethe, H. A. 1942 The theory of shock waves for an arbitrary equation of state. technical paper
545, office sci. Res. & Dev .
Cadieux, F., Domaradzki, J. A., Sayadi, T., Bose, T. & Duchaine, F. 2012 DNS and
LES of separated flows at moderate reynolds numbers. Proceedings of the 2012 Summer
Program, Center for Turbulence Research, NASA Ames/Stanford University, Stanford,
CA, June pp. 77–86.
Chung, T. H., Ajlan, M., Lee, L. L. & Starling, K. E. 1988 Generalized multiparameter
correlation for nonpolar and polar fluid transport properties. Industrial & engineering
chemistry research 27 (4), 671–679.
Cinnella, Paola & Congedo, Pietro M 2005 Numerical solver for dense gas flows. AIAA
journal 43 (11), 2458–2461.
Cinnella, P. & Congedo, P. M. 2007 Inviscid and viscous aerodynamics of dense gases.
Journal of Fluid Mechanics 580, 179–217.
32 A. Vadrot, A. Giauque and C. Corre
Figure 22. Temporal evolution of the integral length scale lz.
Colin, O. & Rudgyard, M. 2000 Development of high-order taylor–galerkin schemes for LES.
Journal of Computational Physics 162 (2), 338–371.
Cook, Andrew W & Cabot, William H 2004 A high-wavenumber viscosity for high-
resolution numerical methods. Journal of Computational Physics 195 (2), 594–601.
Cramer, M. S. 1991 Nonclassical dynamics of classical gases. In Nonlinear waves in real fluids,
pp. 91–145. Springer.
Cramer, M. S. & Kluwick, A. 1984 On the propagation of waves exhibiting both positive
and negative nonlinearity. Journal of Fluid Mechanics 142, 9–37.
Desoutter, G., Habchi, C., Cuenot, B. & Poinsot, T. 2009 DNS and modeling of the
turbulent boundary layer over an evaporating liquid film. International Journal of Heat
and Mass Transfer 52 (25-26), 6028–6041.
Dur´
a Galiana, Francisco J, Wheeler, Andrew PS & Ong, Jonathan 2016 A study of
trailing-edge losses in organic rankine cycle turbines. Journal of Turbomachinery 138 (12).
Freund, J. B., Lele, S. K. & Moin, P. 2000 Compressibility effects in a turbulent annular
mixing layer. Part 1. Turbulence and growth rate. Journal of Fluid Mechanics 421, 229–
267.
Fu, S. & Li, Q 2006 Numerical simulation of compressible mixing layers. International journal
of heat and fluid flow 27 (5), 895–901.
Fujiwara, Hitoshi, Matsuo, Yuichi & Arakawa, Chuichi 2000 A turbulence model for
the pressure–strain correlation term accounting for compressibility effects. International
Journal of Heat and Fluid Flow 21 (3), 354–358.
Giauque, A., Corre, C. & Menghetti, M. 2017 Direct numerical simulations of homogeneous
isotropic turbulence in a dense gas. Journal of Physics: Conference Series 821 (1), 012017.
Giauque, A, Corre, C & Vadrot, A 2020 Direct numerical simulations of forced homogeneous
isotropic turbulence in a dense gas. Journal of Turbulence 21 (3), 186–208.
Gloerfelt, Xavier, Robinet, Jean Christophe, Sciacovelli, Luca, Cinnella, Paola
& Grasso, Francesco 2020 Dense-gas effects on compressible boundary-layer stability.
Journal of Fluid Mechanics 893.
Guardone, ALBERTO, Vigevano, Luigi & Argrow, BM 2004 Assessment of
thermodynamic models for dense gas dynamics. Physics of Fluids 16 (11), 3878–3887.
DNS of temporal compressible mixing layers in a BZT DG: influence of Mc33
Hamba, Fujihiro 1999 Effects of pressure fluctuations on turbulence growth in compressible
homogeneous shear flow. Physics of Fluids 11 (6), 1623–1635.
Huang, Siyuan & Fu, Song 2008 Modelling of pressure–strain correlation in compressible
turbulent flow. Acta Mechanica Sinica 24 (1), 37–43.
Kourta, Azeddine & Sauvage, R 2002 Computation of supersonic mixing layers. Physics of
Fluids 14 (11), 3790–3797.
Lee, Sangsan, Lele, Sanjiva K & Moin, Parviz 1991 Eddy shocklets in decaying
compressible turbulence. Physics of Fluids A: Fluid Dynamics 3(4), 657–664.
Luo, K. H. & Sandham, N. D. 1994 On the formation of small scales in a compressible mixing
layer. In Direct and Large-Eddy Simulation I, pp. 335–346. Springer.
Martin, J. J. & Hou, Y. 1955 Development of an Equation of State for Gases. AIChE journal
2(4), 142–151.
Martin, J. J., Kapoor, R. M. & De Nevers, N. 1959 An improved equation of state for
gases. AIChE Journal 5(2), 159–160.
Mart
´
ınez Ferrer, P. J., Lehnasch, G. & Mura, A. 2017 Compressibility and heat release
effects in high-speed reactive mixing layers I.: Growth rates and turbulence characteristics.
Combustion and Flame 180 (M), 284–303.
Matsuno, Kristen & Lele, Sanjiva K 2020 Compressibility effects in high speed turbulent
shear layers–revisited. In AIAA Scitech 2020 Forum, p. 0573.
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research.
Annual review of fluid mechanics 30 (1), 539–578.
Pantano, C. & Sarkar, S. 2002 A study of compressibility effects in the high-speed turbulent
shear layer using direct simulation. Journal of Fluid Mechanics 451, 329–371.
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimental
study. Journal of Fluid Mechanics 197, 453–477.
Park, CH & Park, Seung O 2005 A compressible turbulence model for the pressure–strain
correlation. Journal of Turbulence (6), N2.
Pirozzoli, S., Bernardini, M., Mari´
e, S. & Grasso, F. 2015 Early evolution of the
compressible mixing layer issued from two turbulent streams. Journal of Fluid Mechanics
777, 196–218.
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible
viscous flows. Journal of computational physics 101 (1), 104–129.
Rossmann, Tobias, Mungal, M Godfrey & Hanson, Ronald K 2001 Evolution and growth
of large scale structures in high compressibility mixing layers. In TSFP Digital Library
Online. Begel House Inc.
Sandham, N. D. & Reynolds, W. C. 1990 Compressible mixing layer - Linear theory and
direct simulation. AIAA Journal 28 (4), 618–624.
Sarkar, S. 1995 The stabilizing effect of compressibility in turbulent shear flow. Journal of
Fluid Mechanics 282, 163–186.
Sarkar, S., Erlebacher, G., Hussaini, M. Y. & Kreiss, H. O. 1991 The analysis and
modelling of dilatational terms in compressible turbulence. Journal of Fluid Mechanics
227, 473–493.
Sarkar, Sutanu & Lakshmanan, B 1991 Application of a reynolds stress turbulence model
to the compressible shear layer. AIAA journal 29 (5), 743–749.
Sciacovelli, L., Cinnella, P. & Gloerfelt, X. 2017aDirect numerical simulations of
supersonic turbulent channel flows of dense gases. Journal of Fluid Mechanics 821, 153–
199.
Sciacovelli, L., Cinnella, P. & Grasso, F. 2017bSmall-scale dynamics of dense gas
compressible homogeneous isotropic turbulence. Journal of Fluid Mechanics 825, 515–
549.
Shuely, Wendel J 1996 Model liquid selection based on extreme values of liquid state
properties in a factor analysis. Tech. Rep.. Edgewood Research Development and
Engineering Center Aderbeen Proving Ground MD.
Stephan, Karl & Laesecke, A 1985 The thermal conductivity of fluid air. Journal of physical
and chemical reference data 14 (1), 227–234.
Thompson, P. A. 1971 A fundamental derivative in gasdynamics. Physics of Fluids 14 (9),
1843–1849.
34 A. Vadrot, A. Giauque and C. Corre
Vadrot, Aur´
elien, Giauque, Alexis & Corre, Christophe 2020 Analysis of turbulence
characteristics in a temporal dense gas compressible mixing layer using direct numerical
simulation. Journal of Fluid Mechanics 893.
Vreman, A. W., Sandham, N. D. & Luo, K. H. 1996 Compressible mixing layer growth rate
and turbulence characteristics. Journal of Fluid Mechanics 320, 235–258.
Wang, Jianchun, Wan, Minping, Chen, Song, Xie, Chenyue, Zheng, Qinmin, Wang,
Lian-Ping & Chen, Shiyi 2020 Effect of flow topology on the kinetic energy flux in
compressible isotropic turbulence. Journal of Fluid Mechanics 883.
Wheeler, Andrew PS & Ong, Jonathan 2014 A study of the three-dimensional unsteady
real-gas flows within a transonic ORC turbine. In ASME Turbo Expo 2014: Turbine
Technical Conference and Exposition. American Society of Mechanical Engineers Digital
Collection.
White, F. M. 1998 Fluid Mechanics, McGraw-Hill Series in Mechanical Engineering.
Zel’dovich, J. 1946 On the possibility of rarefaction shock waves. Zhurnal Eksperimentalnoi i
Teoreticheskoi Fiziki 16 (4), 363–364.
Zeman, Otto 1990 Dilatation dissipation: the concept and application in modeling compressible
mixing layers. Physics of Fluids A: Fluid Dynamics 2(2), 178–188.
Zhou, Q., He, F. & Shen, M. Y. 2012 Direct numerical simulation of a spatially developing
compressible plane mixing layer: flow structures and mean flow properties. Journal of
Fluid Mechanics 711, 1–32.
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