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This draft was prepared using the LaTeX style ﬁle belonging to the Journal of Fluid Mechanics 1

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 p−vdiagram

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 =s2−s1=−∂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= (u1−u2)/(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 p−vdiagram 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).

Mcρ1/ρ2Reδθ,0Lx×Ly×LzNx×Ny×Nz∆u (m.s−1)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 ×10−3m3.kg−1(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/ν (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 ui−qj)

∂xj

(2.7)

8A. Vadrot, A. Giauque and C. Corre

where τij =µ(∂ui

∂xj+∂uj

∂xi−2

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

v−b+

5

X

i=2

Ai+BiT+Cie−kT /Tc

(v−b)i

e=eref +ZT

Tref

cv(T0)dT 0+

5

X

i=2

Ai+Ci(1 + kT /Tc)e−kT /Tc

(i−1)(v−b)i−1

(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 τ≈11500 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)/δθ,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=dδθ

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 τ≈2500.

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∆u3/δθ(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 y−turbulent 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∆u3/δθ(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 p−vdiagram 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 τ≈1100, 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 ppc/ρc. 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.

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