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Hybrid Lattice Boltzmann Method for Turbulent
Non-Ideal Compressible Fluid Dynamics
Lucien Vienne, Alexis Giauque, Emmanuel Lévêque
To cite this version:
Lucien Vienne, Alexis Giauque, Emmanuel Lévêque. Hybrid Lattice Boltzmann Method for Turbulent
Non-Ideal Compressible Fluid Dynamics. Physics of Fluids, 2024, 36 (11), �10.1063/5.0234603�. �hal-
04772738�
Hybrid Lattice Boltzmann Method for Turbulent Non-Ideal
Compressible Fluid Dynamics
Lucien Vienne,a) Alexis Giauque, and Emmanuel L´evˆeque
CNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1, LMFA,
UMR5509, 69134 ´
Ecully, France
The development and application of a compressible hybrid lattice Boltzmann
method to high Mach number supercritical and dense gas flows are presented.
Dense gases, especially in Organic Rankine Cycle turbines, exhibit non-classical
phenomena that offer the possibility of enhancing turbine efficiency by reducing
friction drag and boundary layer separation. The proposed numerical framework
addresses the limitations of conventional lattice Boltzmann method in handling
highly compressible flows by integrating a finite-volume scheme for the total energy
alongside a non-ideal gas equation of state supplemented by a transport coeffi-
cient model. Validations are performed using a shock tube and a three-dimensional
Taylor-Green vortex flow. The capability to capture non-classical shock behaviors
and compressible turbulence is demonstrated. Our study gives the first analysis of
a turbulent Taylor-Green vortex flow in a dense Bethe-Zel’dovich-Thompson gas
and provides comparisons with perfect gas flow at equivalent Mach numbers. The
results highlight differences associated with dense gas effects and contribute to a
broader understanding of non-ideal fluid dynamics in engineering applications.
Keywords: lattice Boltzmann method; non-ideal compressible fluid dynamics; dense
gas; shock tube; Taylor-Green vortex
I. INTRODUCTION
Waste heat recovery or heat harvesting from geothermal reservoirs represents a promising
way of generating emissions-free power. To achieve this, one of the preferred technical
solutions has been known for the last 50 years and is named after the thermodynamic cycle
that it uses to convert heat into recoverable work (such as electricity): the Organic Rankine
Cycle (orc). Orc uses an organic fluid instead of water in a steam turbine cycle. Because
of the moderate boiling temperature of the organic fluid (such as siloxane), new heat sources
such as waste heat in industry can be harvested. A ma jor challenge in orc systems is to
enhance the overall efficiency of the cycle to get as close as possible to the Carnot limit
imposed by the laws of thermodynamics, i.e. η∞= 1 −Tcold/Thot where Tcold and Thot
denote the temperatures of the cold and warm sources, respectively. The overall efficiency
results from the combined optimization of all sub-elements that make up the cycle. One
of the most important sub-system is the turbine stage in which the fluid undergoes a very
strong expansion creating shock-waves and large turbulence levels.
The working fluids used in orc systems usually have a high molecular complexity. Such
fluids are particularly suitable because of their high heat capacity and low boiling tem-
perature. They also often exhibit strong non-ideal behaviors when their thermodynamic
state is close to the critical point1. In this region of the p−vdiagram, the so-called fun-
damental derivative of gas dynamics can become lower than unity (Γ <1), which leads to
some unusual behaviors2, such as expansion shock-waves or composite waves. The term
“fundamental” here underlines the importance of Γ in determining the non-linear behavior
of the gas. The fundamental derivative of gas dynamics was introduced by Hayes3and later
a)Corresponding author: contact@lvienne.com
2
rewritten by Thompson2as
Γ = v3
2c2
∂2p
∂v2s
=c4
2v3
∂2v
∂p2s
= 1 + ρ
c
∂c
∂ρ s
(1)
where vdenotes the specific volume, ρis the mass density, c=p∂p/∂ρ|sdefines the speed
of sound, pis the pressure and sis the entropy of the fluid. According to its definition (1), Γ
may be viewed as a measure of the rate of change of the speed of sound during an isentropic
transformation. It is also directly related to the curvature of isentropic lines in the p−v
diagram, i.e. ∂2p/∂v2|s. In general, three main regimes can be identified according to the
value of the fundamental derivative Γ (see Fig. 6 later):
•Γ>1 corresponds to a classical ideal (perfect) gas behavior. Let us mention that
for thermally and calorically perfect gases, the constant fundamental derivative is
expressed as Γ = (γ+ 1)/2 with γdenoting the heat capacity ratio.
•0<Γ<1 is specific to the classical non-ideal gas behavior. In this regime, the speed
of sound decreases in isentropic compressions, ∂c/∂ρ|s<0.
•Γ<0 is particularly interesting and indicates the presence of non-classical non-ideal
gas behavior also known as Bethe-Zel’dovich-Thompson (bzt) effect. The name bzt
was given by Cramer4to acknowledge the pioneering works of Bethe5, Zel’dovich6and
Thompson2on dense gases, since widely used in the industry. Hydrocarbons, per-
fluorocarbons, and siloxanes are examples of such bzt gases. In the p−vdiagram, it
corresponds to a limited region where the negative sign of the fundamental derivative
allows the (unusual) development of expansion shock-waves (see Fig. 6).
Several studies examined the non-classical phenomena occurring in dense bzt gases (such
as expansion shock-waves) by considering at first the fluid as inviscid7–12. Adding viscosity
effects allowed the study of boundary layers and the interaction between shocks and bound-
ary layers13–16. The benefits of using dense gases in orc turbines were demonstrated when
operating within the region 0 <Γ<1 at transonic speeds. Under these conditions, dense
gas effects minimize friction drag and boundary layer separation17. When the expansion
occurs within the inversion region Γ <0, the shock intensity decreases and entropy losses
are reduced, improving the efficiency of the turbine.
For years, the efficiency of orc systems has been addressed through in situ optimization.
Meanwhile, modern numerical simulation and design tools for non-ideal fluids have emerged
in the turbomachinery field. Therefore, techniques originally developed for ideal flows have
been effectively applied to optimize components operating under non-ideal flow conditions,
such as supersonic vanes in orc turbines, with Vitale et al.18 employing automatic differ-
entiation tools to develop adjoint Non-Ideal Compressible Fluid Dynamics (nicfd) solvers,
albeit facing challenges with external thermodynamic property libraries as noted by Rubino
et al.19. The evaluation of geometrical uncertainties was initially explored by Razaaly et
al.20, while shape optimization under uncertainty, encompassing operating conditions and
model parameters, was pioneered by Cinnella and Hercus21. A recent review by Guardone et
al.1reports a comprehensive theoretical framework embedding the fundamentals of nicfd.
However, one of the main obstacles to using simulations for the optimal design of orc
turbines is the “time to result”. To date, only high-fidelity techniques22–24 have achieved
the requested level of accuracy, but are also too time-consuming to be integrated into a
real-time shape optimization task. The “time to result” for industrial applications involv-
ing complex boundaries is primarily determined by two factors: grid generation and the
efficiency of solver algorithms. Implementing a Cartesian octree grid along with immersed
boundary conditions for wall-modeled large eddy simulation has drastically reduce grid
generation time from 1-2 months to 1-2 hours in the case of aircraft high-lift devices25.
Based on a Cartesian grid, the lattice Boltzmann method (lbm) provides a relatively sim-
ple and efficient core algorithm that is also easy to parallelize. These advantages can lead
to speed-ups of up to ten times compared to traditional Navier-Stokes solvers in industrial
3
large eddy simulations, including applications in landing gears aeroacoustics26–28, aircraft
high-lift devices29,30, automotive aerodynamics31 , and reacting flows32. While no equiv-
alent comparison with traditional Navier-Stokes solvers is available for turbo-machinery
flows, recent studies33–35 demonstrate the adoption of the lbm in turbo-machine flow simu-
lations. Exploiting the lbm to tackle compressible turbulent flows of a non-ideal fluid would
represent a first step in efficient optimal design of orc turbines.
The lbm is a fully discretized (in space, time, and velocity) kinetic approach that origi-
nates from the Lattice Gas Automata method, first introduced in 1973 by Hardy et al.36. It
may be interpreted as a minimalist statistical model, where populations of particles move
simultaneously along a regular lattice, following simple rules of streaming and collision
from which macroscopic dynamics emerges naturally. The method gained momentum in
the computational-fluid-dynamics community after the publication of the so-called lattice-
bgk model in 199237,38, in which the collision was simplified to a relaxation towards a local
equilibrium, and the consistency with the Navier–Stokes dynamics made explicit. Although
historically introduced in the context of dilute gas dynamics, the method was rapidly ex-
tended to encompass more complex dynamics39.
The lbm algorithm faces challenges when simulating highly compressible flows. Indeed,
the drastic decimation of the (microscopic) velocity space generates errors that increase with
the Mach number, which limits its application to weakly compressible flows at low speed.
This can a priori be overcome by extending the discrete set of microscopic velocities40–44,
but this entails a tremendous expansion of the spatial stencil of the method, resulting in an
increased algorithmic complexity, greater memory requirements, and difficulties in maintain-
ing numerical stability45. To enhance the applicability of the lbm to thermal, compressible,
and reactive flows, a hybrid approach combining an lbm algorithm for the fluid momentum
with a finite-difference or finite-volume algorithm for the total energy, has recently been
proposed. This approach uses the same grid and time step for both discretization schemes,
providing a unified solution that has proven efficient, as demonstrated by Feng et al.46
and Boivin et al.32. These recent advances open new opportunities for the simulation of
realistic orc systems, with an additional need to implement equations of state (eos) for
non-ideal dense gas thermodynamic conditions. This later aspect has recently been exam-
ined by Hosseini et al.47, focusing mainly on the added complexity of the van der Walls’
eos and applying their developments to phase-transition problems. Non-ideal compressible
fluid dynamics in a turbulent regime has yet to be addressed.
In this context, our main objective is twofold. Firstly, to introduce a compressible hybrid-
lbm approach encompassing a third-order eos and, secondly, to apply it to study for the
first time the dense and bzt effects in the well-known Taylor-Green Vortex (tgv) flow. The
article is organized as follows. First, a new hybrid-lbm scheme is described, which is capable
of handling turbulent dense gas flows. A focus is made on the specific aspects related to
the energy equation, the implementation of the third-order Peng-Robinson eos48 and the
transport coefficient model introduced by Chung49. Results for highly compressible flows
of dense gases are validated against the literature in the second section. The third section
further validates the method by examining the tgv flow in a supersonic perfect gas and
then extends the analysis to a dense bzt gas. The tgv configuration is usually considered
a benchmark in turbulence simulations. This is the first time it has been analyzed in the
context of dense gases. This paper compares the results with the ideal gas flow at the same
turbulent Mach number and reports the differences found. The final section summarizes
the main findings and provides a roadmap for further research.
II. NON-IDEAL GAS MODELING IN HYBRID COMPRESSIBLE LATTICE BOLTZMANN
The lbm stems from the Boltzmann equation and can be considered a kinetic method.
Using a kinetic approach to simulate continuum flows may seem unreasonable at first glance,
but it has several advantages. Firstly, the velocity space is amenable to a radical reduction
so that only a small set of microscopic (or kinetic) velocities, e.g. nine in two dimensions, is
4
sufficient to reconstruct relevant isothermal fluid dynamics at the macroscopic level (with a
third-order error in Mach number). The most important feature of the lbm is certainly that
its kinetic equations dissociate non-locality and non-linearity, thus facilitating numerical
integration. As a result, the so-called stream-and-collide algorithm is simple, accurate,
and formidably efficient in terms of computations. The applications of the lbm are now
numerous in both academic and industrial configurations, and extend to a much wider scope
than the initial domain of weakly compressible dynamics of ideal gases50–53.
Simulating compressible non-ideal gas dynamics using the lbm involves two major chal-
lenges: (i) handling strong compressibility effects associated with high density and pressure
gradients, and (ii) incorporating thermodynamic relations including a thermal and a caloric
eos for the pressure and the internal energy, i.e. p(ρ, T ) and e(ρ, T ).
(i) Significant progress has been made in addressing compressibility in the lbm. The
multi-speed approach40–44 was introduced to include third-order statistical moments (related
to heat exchanges) in the dynamics. However, this comes at the cost of an increase in
the number of microscopic velocities, which also implies more difficulties in implementing
boundary conditions and ensuring stability45. Alternatively, the off-lattice method uses
an adaptive (microscopic) velocity space that directly accounts for compressibility effects.
This approach ensures Galilean invariance and is thermodynamically consistent, but it
requires interpolations to redistribute particles onto a fixed lattice at each time step54,55,
which compromises mass conservation and increases significantly the computational costs.
In addition, it is usually combined with multi-speed strategy to mitigate the Mach number
error. Despite these drawbacks, the off-lattice method is based on solid physical grounds and
has recently proven its effectiveness56. Another method for simulating compressible flows
involves solving the density and momentum equations with the standard lbm scheme, while
the energy equation is handled separately, possibly using a different scheme. This splitting
leads to the double-distribution-function approach when the energy equation is addressed
using a second lbm scheme57,58. This method is limited to weakly compressible thermal
flows, unless it is enhanced by the multi-speed or off-lattice approach. The hybrid approach,
on the other hand, relies on a conventional finite-volume or finite-difference scheme to
solve the energy equation.59. This strategy will be continued in the following. Coupling
two distinct numerical schemes is always challenging when it comes to stability. For this
reason, the entropy was first chosen as the base variable for the energy equation. As a
characteristic variable of the Euler equations, this choice guarantees that the energy scheme
can remain linearly decoupled from the lattice Boltzmann (lb) scheme. More recently
Wissocq et al.60 proposed a robust and stable compressible hybrid lb approach based on a
total energy formulation60,61 to restore missing conservation features in the previous hybrid
entropy-based approach. Let us mention that Guo and Feng62 have recently introduced an
alternative strategy for coupling lb and finite-volume schemes.
(ii) Regarding non-ideal thermodynamics, lb solvers have introduced various strategies
to go beyond the perfect gas eos, in particular in the context of multi-phase flows. How-
ever, these methods are generally limited to weakly compressible flows, making stable and
accurate simulations of compressible flows difficult. Notable exceptions are the works of
Reyhanian et al.56,63, who used the so-called particle-on-demand scheme — an advanced
off-lattice approach — in conjunction with the van der Waals equation of state and the
assumption of a calorically perfect gas.
In the following section, an extension of the work59,60 is built up to include arbitrary
non-ideal thermal and caloric equations of state, opening up the hybrid lbm approach to
general non-ideal compressible fluid dynamics.
A. Lattice Boltzmann scheme for mass and momentum
The lbm governs the evolution in time of the distribution functions (f0,· · · , fN−1) of
particles with the microscopic velocities (e0,··· ,eN−1) at each lattice node. At the macro-
scopic level, flow variables are recovered by summing the contributions from the distribution
5
functions and calculating the moments so that
ρ=
N−1
X
i=0
fiand ρuα=
N−1
X
i=0
fieiα,(2)
where ρand uαrepresent respectively the mass density and velocity components of the
fluid. The truncation of the velocity space onto a finite basis of Hermite polynomials
and resorting to Gaussian quadrature formula40,64 transform the Boltzmann equation to
the discrete-velocity Boltzmann equation. Our study truncates the velocity space to the
standard D3Q27 lattice, i.e. N= 27 distribution functions in three dimensions. This stencil
encompasses all the adjacent nodes in the uniform Cartesian grid as pictured in Fig. 1.
1
23
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0
FIG. 1: Sketch of the D3Q27 stencil. The current node is the black-filled circle at the
center. Neighboring nodes are drawn as black non-filled circles. D3Q27 distribution
functions are related to the kinetic velocities represented by all possible permutations of
eiα = 0,−1,+1 for i= 0,...,26 and α=x, y, z.
The discretization in space and time involves the integration along the characteristics and
results in the lattice Boltzmann scheme, a two-step algorithm that consists of
fcoll
i(t, x) = feq
i(t, x) + 1−∆t
τfneq
i(t, x) + ∆t
2Si(t, x)+∆tF µb
i(t, x) collision step,
(3)
fi(t+ ∆t, x) = fcoll
i(t, x−ei∆t) streaming step,
(4)
for i= 0,··· ,26. Usual notations for the lb scheme are employed. τdenotes the relaxation
time toward the equilibrium distribution function feq
i.Sirefers to the source term, and the
non-equilibrium component is defined as
fneq
i=fi−feq
i+1
2∆tSi(5)
Following Farag et al.59 but leaving the eos p(ρ, T ) undefined, the equilibrium distribution
is a third-order Hermite polynomials expansion
feq
i=wiρ1 + eiα
c2
s
uα+Hiαβ
2c4
s
uαuβ+Hiαβγ
6c6
s
uαuβuγ+di(6)
6
with
di=0 =wi
c2
sp−ρc2
sand di=0 =w0−1
c2
sp−ρc2
s,(7)
where implicit (Einstein) summation is assumed over space coordinates α, β, γ ∈ {x, y , z},
and diis a scalar that accounts for the deviation from the perfect gas assumption in the
equivalent macroscopic equations. This operator is more isotropic and is known to be more
stable than a conventional third-order Hermite polynomials expansion of the equilibrium
function65–67. Expressions of the Hermite polynomials are given in Appendix A.
The general formulation of the source term reads
Si=wi( ˙m+Hiα
c2
s
ρFα+Hiαβ
2c4
s
Corrαβ) (8)
where ˙mis a mass source term, Fαis a body force, and Corrαβ are second-order correction
terms that are detailed later. Let us mention that correction terms should a priori depend
on ˙mand Fα. In the following, we will assume that there is no mass source term ˙m= 0 nor
body force Fα= 0, and that Corrαβ corrects intrinsic errors arising from the discretization
of the velocity space.
Macroscopic quantities are derived from the statistical moments of the distribution func-
tions. Besides Eq. (2), the second and third-order moments of the distribution functions
are designated by
Παβ =X
i
eiαeiβ fi,(9)
Παβγ =X
i
eiαeiβ eiγ fi.(10)
Similarly, the moments of the equilibrium state gives
ρ=X
i
feq
i,(11)
ρuα=X
i
eiαfeq
i(12)
Πeq
αβ =X
i
eiαeiβ feq
i=ρuαuβ+δαβp, (13)
Πeq,theo
αβγ =ρuαuβuγ+ρc2
s[uδ]αβγ (14)
where [uδ]αβγ stands for a cyclic permutation [uδ]αβγ =uαδβ γ +uβδγα +uγδαβ , and δαβ is
the Kronecker delta function. Πeq,theo
αβγ represents the third-order moment of the equilibrium
function expected to recover exactly the Navier-Stokes equations. The discrete nature of
the D3Q27 lattice is responsible for this moment not being properly fulfilled. In particular,
one gets for Πeq
αβγ =Pieiαeiβeiγ feq
ithat
Πeq
ααα = 3ρc2
suα,(15)
Πeq
αβγ =ρu2
αuγ+ρc2
suγif α=β=γ(16)
leading to the lattice isotropic defect
Deq
αβγ =Zeαeβeγfeq
ide−Πeq
αβγ = Πeq,theo
αβγ −Πeq
αβγ = 0 (17)
The correction term Corrαβ addresses explicitly this discrepancy (details are provided in
Appendix B), which yields
7
Corrαβ =δαβ∂t(p−ρc2
s)
−uα∂β(p−ρc2
s)−uβ∂α(p−ρc2
s)
−∂γDeq
αβγ
+δαβ
2
3ρc2
s∂γuγ.
(18)
This expression corresponds to Eq. (52) and Eq. (54) in reference59 under the assumption
of an ideal gas law p=ρθc2
sand with κ= 0. In Eq. (18), the corrections terms will
be discretized using finite difference schemes, employing a first-order backward scheme for
the temporal derivative, a second-order central scheme for the divergence operator, and a
first-order upwind scheme for the remaining spatial derivatives.
The classical bgk collision operator given by Eq. (3) suffers from numerical instability,
particularly in high Reynolds flows. Consequently, various alternative collision operators
have been proposed in the literature68,69. Following again Farag et al.59, we adopt the third-
order recursive regularized collision operator70. This means that instead of using Eq. (5),
the non-equilibrium distribution functions are reconstructed from
¯
fneq
i=wi"Hiαβ a1
αβ
2c4
s
+Hiαβγ a1
αβγ
6c6
s#,(19)
with a1
αβ =X
i
eiαeiβ fneq
iand a1
αβγ =uαa1
βγ +uβa1
αγ +uγa1
αβ.(20)
We recall that the second-order non-equilibrium moment in Eq. (20) is related to the stress
tensor via the Chapmann-Enskog expansion, as
a1
αβ =τρc2
s∂(1)
αuβ+∂(1)
βuα−δαβ
2
3∂(1)
γuγ+O(ϵ2).(21)
To mitigate spurious pressure terms that may arise from the combination of numerical
schemes (finite difference schemes in Corrαβ and lb scheme) during the calculation of a1
αβ
using Eq. (5) and Eq. (20), we enforce its trace to be zero by redistributing the potential
discretization error as follows59:
¯a1
αα =a1
αα −1
3a1
ββ .(22)
Finally, in the collision step Eq. (3), the last term Fµb
iintroduces an had-hoc artificial
bulk viscosity to enhance stability at high Mach numbers:
Fµb
i=−0.07Ma2wi
2c4
s
Hiααρc2
s∂αuα.(23)
With the exception of leaving the pressure equation undefined and employing the D3Q27
velocity set, the present lb scheme is similar to the one proposed by Farag et al.59 . This
previous study opted for the entropy formulation for energy discretization due to its inherent
property of decoupled linear stability. We adopt the strategy proposed more recently by
Wissocq et al.60 including a total energy formulation. This conservative form of the energy
scheme ensures the recovery of the correct shock speed.
B. Finite-volume scheme for total energy
By adapting Riemann solvers to the lb formalism, Wissocq et al.60 introduced a con-
servative scheme that is linearly equivalent to its non-conservative entropy formulation. In
8
particular, this scheme preserves the low dissipation characteristic of the lbm for isentropic
phenomena such as acoustic propagation and vorticity. The conservative formulation for
the total energy is expressed as
∂t(ρE) + ∂αFρE
α= 0,(24)
where FρE
αdenotes the total energy flux, encompassing both advective and diffusive com-
ponents. Detailed expressions of these fluxes are provided in the study by Wissocq et al.60.
These expressions depend not only on the macroscopic quantities (ρ, uα, E) and the flux
discretization scheme used, but also on the distribution functions fi. Additional terms in
FρE
αensure consistency between the mass and momentum, solved by lbm, and the total
energy solved by the finite-volume method. The same computational grid as the lbm is
used. We tested four discretization schemes for the fluxes: a first-order upwind scheme,
two muscl-Hancock schemes as suggested in references59,60 (mhm1d) or in references61,71
(mhm2d) and a tvd Heun scheme as used in Wissocq et al.61.
C. Thermal and caloric equations of states
The governing equations require a model to establish the relationships between the ther-
modynamic variables p, ρ, and e. We need to provide the thermal and caloric eos, respec-
tively:
p=p(ρ, T ) and e= (ρ, T ).(25)
Tdenotes temperature, and erepresents the density of internal energy (per unit mass),
which is related to total energy through E=e+1
2uαuα. Both thermal and caloric eos are
linked via the relation
e=eref +ZT
Tref
cv∞(T′)dT ′+Zρ
ρref "T∂p
∂T ρ−p#dρ′
ρ′2,(26)
where the ref subscript refers to an arbitrary reference state, cv∞denotes the specific heat
at constant volume in the ideal gas limit. The last term in Eq. (26) represents the deviation
from the assumption of calorically perfect gas and is contingent upon the eos. Following
Guardone and Argrow72, the ideal isochoric heat capacity can be approximated by a power
law of the form
cv∞(T) = cv∞(Tc)T
Tcn
,(27)
with Tcdenoting the critical temperature, and cv∞(Tc) and nbeing dependent on the
simulated gas.
In this study, we consider three thermal equations. The simplest relation, known as the
ideal gas assumption, is valid in the dilute gas region:
p(ρ, T ) = ρRgT, (28)
where Rgdenotes the specific gas constant. The van der Waals eos refines the ideal gas
assumption by considering the finite size of gas molecules and the attractive forces between
them:
p(ρ, T ) = ρRgT
1−bρ −aρ2,(29)
where the aand bparameters depends on the critical pressure pcand temperature Tcas
follows:
a=27R2
gT2
c
64pc
and b=RgTc
8pc
.(30)
9
More complex yet more accurate eos are necessary to realistically explore the non-classical
gas dynamic phenomena. Third-order thermal eos have been found to offer a good com-
promise in terms of complexity and accuracy. They have been validated in the thermody-
namic region where non-classical gas phenomena are expected to be observed for heavier
substances73. One the available third-order eos is the Peng-Robinson eos48. It is expressed
as
p(ρ, T ) = ρRgT
1−bρ −a(T)ρ2
1+2bρ −b2ρ2,(31)
where
a= 0.45724R2
gT2
c
pc
α(T) and b= 0.07780 RTc
pc
,with α(T) = 1 + κ(ω)(1 −pT/Tc)2,
(32)
where arelates to the attractive effect of van der Waals forces and bto the effective molar
volume. The κcoefficient varies according to the accentric factor ω, which measures the
non-sphericity of molecules of the selected gas.
κ(ω)=0.37464 + 1.54226ω−0.26992ω2,if ω≤0.491 (33)
κ(ω)=0.379642 + 1.487503ω−0.164423ω2+ 0.016666ω3,if ω > 0.491 (34)
Obtaining the temperature as a function of internal energy and density from Eq. (26) for
the Peng-Robinson eos is not straightforward. We resort to an iterative Newton-Raphson
method to compute T=T(ρ, e).
Finally, the gas viscosity and thermal conductivity must be described according to the
thermodynamic variables. When the perfect gas assumption is involved, the Sutherland law
for air and a constant Prandtl number equal to 0.71 is used. For dense gas simulations,
we use the generalized laws for viscosity and thermal conductivity developed by Chung et
al.49.
D. Summary of the hybrid compressible lattice Boltzmann algorithm for non-ideal gases
The hybrid compressible lattice Boltzmann algorithm for non-ideal gases is depicted in
the flowchart presented in Figure 2.
Finally, we would like to highlight that this algorithm has been ported to a multi-GPU
architecture, offering advantageous execution times. For instance, the perfect and dense
gas tgv simulations presented later, at a resolution of N= 768 with the mhm2d flux-
reconstruction scheme, were completed in approximately 2.25 and 6 hours, respectively,
using two AMD MI250x cards in double precision.
III. VALIDATION
In this section, the hybrid lattice Boltzmann method for non-ideal fluids is validated across
two distinct scenarios. Firstly, we showcase its capability to capture the non-classical be-
havior of a bzt gas within a shock tube. Secondly, we assess its effectiveness in simulating
compressible turbulence by considering the Taylor-Green vortex flow. Each case demon-
strates its correctness and accuracy in capturing complex fluid dynamics.
A. Non-classical van der Waals shock tube
Shock tube simulations are a standard one-dimensional test problem for evaluating nu-
merical schemes. In the context of bzt fluids, they offer the opportunity to showcase a
10
Correction term Corrαβ from Eq. (18)
Artificial bulk viscosity Fµb
ifrom Eq. (23)
Viscosity µfrom transport coefficient model49
Relaxation time τfrom Eq. (B20)
Equilibrium distributions from Eq. (6)
Non-equilibrium distributions from
Eq. (19) with ¯a1
αα from Eq. (22)
lb collision from Eq. (3)
feq
i(t) ; fneq
i(t)
Energy fluxes from Eq. (24)
fcoll
i(t)
Energy update from Eq. (24)
FρE
α(t)
lb streaming from Eq. (4)
fcoll
i(t)
Macroscopic variables from Eq. (2)
fi(t+ ∆t)
ρE(t+ ∆t)
ρ(t+ ∆t) ; uα(t+ ∆t) ; E(t+ ∆t) ; T(t+ ∆t) ; p(t+ ∆t)
t=t+ ∆t
FIG. 2: Temporal flowchart of the proposed algorithm, with spatial dependencies between
steps omitted for clarity.
non-classical shock behavior. The fluid is assumed to be calorically perfect and modeled
by the van der Waals eos, with the thermodynamic properties summarized in Table I.
Viscosity is set to zero. A 1000-point grid represents a one-meter length tube, L= 1m.
Additionally, we define a dimensionless time as t∗=t/L ×ppc/ρcand choose a constant
time step of ∆t∗= 9 ×10−4resulting in a Courant–Friedrichs–Lewy number close to 0.92.
The first-order upwind scheme is used for the energy fluxes. Table II lists the selected ini-
tial state on both sides of the diaphragm. Both states are located in the bzt region where
Γ<0.
M[kg.mol−1]Tc[K] pc[atm] cv∞(Tc)/Rg[−]
0.574 632.15 15.98 1/0.0125
TABLE I: Thermodynamic properties of the gas used in the shock tube case. Van der
Waals eos is considered.
The reduced density and pressure and the fundamental derivative of gas are shown at
different times in Fig. 3. This case is representative of the non-classical behavior of a bzt
fluid, where an expansion shock moves to the left and a compression fan moves to the right,
with a contact discontinuity connecting both waves. The results are in close agreement
with the simulation of Guardone and Vigevano, who used a Roe linearization method for
11
p/pcρ/ρc
left 1.090 0.879
right 0.885 0.562
TABLE II: Initial conditions for the shock tube case, “D2” case in Argrow74 or “DG2”
case in Guardone and Vigevano75.
the van der Waals gas75, as well as with the results obtained by Reyhanian et al.63 with an
off-lattice lb scheme. However, in contrast to this latter simulation, no overshoot at the
expansion shock is visible in our case. The current simulation confirms the capacity of the
hybrid lbm to accurately capture the non-classical behavior of gases in the bzt region.
0.0 0.2 0.4 0.6 0.8 1.0
x
[m]
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
r
reduced density [-]
ref Guardone et al.
ref Reyhanian et al.
0.0 0.2 0.4 0.6 0.8 1.0
x
[m]
0.90
0.95
1.00
1.05
1.10
pr
reduced pressure [-]
t
*
t
*
0.0 0.2 0.4 0.6 0.8 1.0
x
[m]
1.2
1.0
0.8
0.6
0.4
0.2
0.0
fondamental derivative [-]
FIG. 3: Numerical solution of the shock tube problem. Solution at the dimensionless time
t∗= 0,0.09,0.18,0.27,0.36,0.45. Lines with color gradient indicating the time evolution:
present simulation, markers at t∗= 0.45; Orange marker +from Guardone and
Vigevano75 and green marker xfrom Reyhanian et al.63 . Close agreement is obtained.
The shock tube test case has served as a valuable benchmark for evaluating the ability
of our scheme to handle steep gradients, or discontinuities, without introducing numerical
artifacts such as spurious oscillations (dispersion errors), which is an essential prerequisite
for compressible flow simulations. The Taylor-Green vortex flow test case is now consid-
ered by introducing turbulence associated with the complex entanglement of compressible
vortex structures. Together, these two test cases are expected to provide a more compre-
hensive assessment of the robustness and accuracy of our scheme in handling a wide range
of compressible flow situations.
B. Compressible Taylor-Green vortex flow with a perfect gas
The Taylor-Green Vortex (tgv) flow is a standard test case that exhibits a transition
from laminar to turbulent flow in a decaying regime. This flow is a popular benchmark to
evaluate the accuracy and efficiency of numerical schemes. However, it is usually examined
under the condition of incompressibility or at low Mach number (Ma ≲0.1). The effect of
compressibility on the tgv flow has only recently been examined. In that situation, the
interaction of multiple shock waves makes this configuration particularly challenging. Peng
and Yang76 described the evolution of the vortex-surface fields in (perfect gas) compressible
tgv flows at Reynolds number Re0= 800 and Mach number up to Ma0= 2. Lusher and
Sandham77 compared different shock-capturing schemes up to Mach number Ma0= 1.25 at
Re0= 1600. Wilde et al.78 simulated this later case to validate a semi-Lagrangian lattice
Boltzmann method combining off-lattice and multispeed strategies. Similarly, we validate
our approach with the perfect gas compressible tgv flow and extend this flow configuration
to dense gas.
12
The fluid domain of the tgv flow is a three-dimensional periodic box of size 0 ≤x, y, z ≤
2πL. The initial solution includes counter-rotating vortices and is given by
ux(x, y, z, t = 0) = U0sin( x
L) cos( y
L) cos( z
L),(35)
uy(x, y, z, t = 0) = −U0cos( x
L) sin( y
L) cos( z
L),(36)
uz(x, y, z, t = 0) = 0,(37)
p(x, y, z, t = 0) = p0+ρ0U2
0
16 cos(2x
L) + cos(2y
L)cos(2z
L)+2,(38)
T(x, y, z, t = 0) = T0,(39)
The initial density field is obtained from the thermal eos.
In the ideal gas configuration, the operating point is located within the dilute region with
T0= 293.15, ρ0= 1.204, Rg= 287.05, p0=ρ0RgT0, and a heat capacity ratio γ= 1.4. The
reference velocity U0is determined by the initial Mach number, while the length Lis defined
by the initial Reynolds number. We set Ma0=U0/c0= 1, resulting in an initial turbulent
Mach number of 0.5, and Re0=Lρ0U0/µ0= 1600 for all tgv simulations presented in this
article. Simulations are advanced until a non-dimensional time of t∗=t/tconv = 20, where
the convective time is defined as tconv =L/U0. A fixed time step is set for all simulations,
given by ∆t∗= 0.012 ×128/N , where Ndenotes the grid resolution. This yields an initial
Courant–Friedrichs–Lewy number of 0.49 for all grids.
Fig. 4 addresses the grid resolution sensitivity for N= 128,256,384,512 and 768 with the
muscl-Hancock scheme based on the total energy (mhm2d61,71). We plot the dimension-
less mean kinetic energy, Ek, and enstrophy or solenoidal viscous dissipation, En, defined
respectively as
Ek=⟨1
2ρuiui⟩1
ρ0U2
0
,(40)
En=⟨1
2µ|ω|2⟩L
ρ0U3
0
,(41)
where ⟨·⟩ indicates averaging over the whole grid, and ω=∇ × urepresents the vorticity
of the fluid and µits dynamic viscosity. Our results are compared with those of Lusher and
Sandham77, who applied a sixth-order targeted essentially non-oscillatory (teno) scheme
on a 5123grid.
The kinetic energy shows little sensitivity to changes in grid size, unlike the enstrophy.
For N= 128 and 256, the enstrophy is significantly underestimated. A closer agreement is
observed between N= 384, N = 512 and N= 768. As Nincreases, vanishing discrepancies
are only observed near the plateau around t∗= 9, at the moment of the maximum enstrophy
(t∗≈11.2), and during the decrease that follows immediately (11.2< t∗<13). The
enstrophy perfectly matches the reference for N= 768. The maximum Mach number over
the entire domain acts as a sharp indicator of grid convergence. At grid resolutions of
N= 512 and N= 768, the differences are marginal, demonstrating that the simulations
accurately capture discontinuities with minimal numerical dissipation. This consistency
suggests that the N= 768 grid size is sufficient to effectively resolve the critical flow
features.
Fig. 5 compares, at a fixed resolution N= 512, different flux-reconstruction schemes
used in the finite-volume integration of the total energy equation. All schemes yield similar
results. The upwind scheme better captures the enstrophy plateau, whereas the muscl-
Hancock methods (mhm1d and mhm2d) are less dissipative near the maximum of enstrophy.
The mhm2d scheme offers the best accuracy, however, the improvement over the mhm1d
scheme is only marginal in this test case.
Following the grid-sensitivity analysis and flux-reconstruction-scheme comparisons, the
next tgv flow simulations will be performed by using the mhm2d scheme with a N= 768
grid size.
13
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.02
0.04
0.06
0.08
0.10
0.12
Ek
total kinetic energy [-]
128^3
256^3
384^3
512^3
768^3
ref Lusher & Sandham
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.000
0.002
0.004
0.006
0.008
0.010
En
total enstrophy [-]
10 12
0.0095
0.0100
0.0105
0.0110
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.6
0.8
1.0
1.2
1.4
1.6
Maximum Mach number [-]
FIG. 4: Influence of the grid resolution on the mean kinetic energy (left) and enstrophy
(middle), and the maximum Mach number (right) for the perfect gas compressible tgv
flow at Ma0= 1, using the mhm2d scheme for the total energy. The dotted black lines of
Lusher and Sandham77 are perfectly superimposed on the plots of the N= 512 and
N= 768 simulations for the energy and enstrophy, demonstrating grid convergence.
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.02
0.04
0.06
0.08
0.10
0.12
Ek
total kinetic energy [-]
Upwind
TVD Heun
MHM1D
MHM2D
ref Lusher & Sandham
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.000
0.002
0.004
0.006
0.008
0.010
En
total enstrophy [-]
10 12
0.0095
0.0100
0.0105
0.0110
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.6
0.8
1.0
1.2
1.4
1.6
Maximum Mach number [-]
FIG. 5: Influence of the finite-volume scheme on the mean kinetic energy (left) and
enstrophy (middle), and the maximum Mach number (right) for the perfect gas
compressible tgv flow at resolution N= 512. Dotted black line from Lusher and
Sandham77. Marginal improvements are observed according to the finite-volume schemes
used for the total energy.
IV. TAYLOR-GREEN VORTEX FLOW WITH A DENSE GAS
Sciacovelli et al.79 were the first to investigate the impact of dense gas effects on decaying
compressible homogeneous isotropic turbulence. They employed the van der Walls thermal
eos, assuming a calorically perfect gas, and neglected viscous effects. Later, the same
authors22 focused on viscous effects occurring in the small-scale dynamics of dense gas
decaying compressible homogeneous isotropic turbulence. They used the Martin-Hou eos
and the Chung formulation for transport coefficients to represent the dense gas behavior
more accurately. Subsequently, Giauque et al.80 continued the study by considering isotropic
homogeneous forced turbulence to extend the time span and analyze turbulence in a quasi-
stationary context. In this section, the impact of dense gas effects on the transition from
laminar to turbulent flow in a decaying vortex is investigated. The well-known Taylor-
Green vortex (tgv) flow has never been studied in the context of a bzt gas. We consider the
dense gas named perfluorotripentylamine (FC-70, C15F33 N), modeled by the Peng-Robinson
equation of state and the Chung formula for transport coefficients. The thermodynamic
properties of the dense gas are provided in Table III.
The operating point for the dense gas simulation is taken within the inversion region, i.e.
14
M[kg.mol−1]Tc[K] pc[atm] cv∞(Tc)/Rg[−]Zc[−]n[−]ω[−]
0.821 608.2 10.2 118.7 0.270 0.4930 0.7584
TABLE III: Thermodynamic properties of the dense gas FC-70. Peng-Robinson eos is
used.
Γ<0, with initial conditions p0= 0.96pcand ρ0=ρc/1.65 resulting in a compressibility
factor Z0=p0/(ρ0RgT0)≈0.43[−] as indicated in Fig. 6. Contrary to section III B in
Eq (38), the initial pressure field is constant p(x, y, z, t = 0) = p0. It was observed that this
modification, which allows the system to adapt the pressure field to the velocity field by
itself, was necessary to avoid crossing the saturation line and the presence of a liquid phase
during the simulation. The perfect gas simulation follows a similar setup and operating
point to the previous section III B, but with an initial constant pressure field. Fig. 7 shows
the influence of the grid resolution on the total kinetic energy, enstrophy and maximum
Mach number for the dense gas simulation. The minor differences between N= 512 and
N= 768 indicate that a grid size of N= 768 allows for an accurate resolution of all scales
of motion, much like the perfect gas simulation with a non-constant initial pressure field.
0.5 1.0 1.5 2.0 2.5 3.0
vr
[-]
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
pr
[-]
= 0
= 1
liquid
supercritical
phase change gas
T
=
Tc
saturation curve
DG initial condition
0.0
0.2
0.4
0.6
0.8
1.0
Z
=
p
/(
RgT
)
FIG. 6: pr−vrdiagram of the FC-70 gas with the Peng-Robinson eos. The red cross
marks the initial condition for the dense gas simulation.
The tgv flows for perfect and dense gases at an initial Mach number of one are now
compared. Fig. 8 displays the temporal evolution of the dimensionless kinetic energy and
dimensionless enstrophy (solenoidal viscous dissipation) integrated over the entire domain
and the evolution of the maximum local Mach number.The initial drop in kinetic energy
during the first convective (or acoustic) time (0 < t∗<2) indicates a transfer from ki-
netic to internal energy, attributed to the “imbalanced” (constant) initial pressure field.
Conversely, the kinetic energy increases during the early stage (2 < t∗<4) of the flow,
consistent with the simulation with a non-constant initial pressure field as well as with
previous observations76,77. Two small peaks appear later, indicating subsequent internal to
kinetic energy conversions. The evolution of kinetic energy and enstrophy in both perfect
and dense gas simulations follow a similar trend, except for an enstrophy plateau in the
perfect gas case compared to a peak in the dense case. In contrast to the previous plots,
the evolution of the maximum Mach number differs significantly between the perfect and
dense gas simulations until the enstrophy reaches its maximum value, i.e. approximately
t∗≈9.
In Fig. 9, the (isotropic) kinetic energy spectra ∥
\
√ρu′
α∥2(k) normalized by ρ0U2
0are
plotted as a function of the wavenumber k=∥k∥at different times for the perfect gas
15
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.02
0.04
0.06
0.08
0.10
0.12
Ek
total kinetic energy [-]
128^3
256^3
384^3
512^3
768^3
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.000
0.002
0.004
0.006
0.008
0.010
En
total enstrophy [-]
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.6
0.8
1.0
1.2
1.4
Maximum Mach number [-]
FIG. 7: Influence of the grid resolution on the total kinetic energy, total enstrophy, and
maximum Mach number for the dense gas compressible tgv flow at Ma0= 1. The minor
differences between N= 512 and N= 768 indicate that a grid size of N= 768 allows for
an accurate resolution of all scales of motion.
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.02
0.04
0.06
0.08
0.10
0.12
Ek
total kinetic energy [-]
Perfect gas
Dense gas
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.000
0.002
0.004
0.006
0.008
0.010
En
total enstrophy [-]
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
t
* [-]
0.6
0.8
1.0
1.2
1.4
1.6
Maximum Mach number [-]
FIG. 8: Total kinetic energy, total enstrophy and maximum Mach number for perfect gas
and dense gas compressible tgv flow at Ma0= 1. A similar trend is observed in
integrated quantities like total energy and total enstrophy; however, local values, such as
the maximum Mach number, differ significantly.
and dense gas tgv simulations, both initialized with a constant pressure field. Given the
definition of the tgv flow, the spectra are initially fed at k= 2k0= 4π/L. An intense
energy transfer then operates from low to high wavenumbers driven by the quadratic non-
linearity of gas dynamics. For the perfect gas simulation, more energy is contained in the
(very) small scales (k∗>50) compared to the dense gas simulation up to t∗<6. For t∗>6,
the turbulent kinetic energy spectra become similar for the perfect and dense gas cases and
exhibit a classical k−5/3Kolmogorov power-law scaling. To examine the influence of the
initial pressure field, a simulation initialized with a non-constant pressure (cf III B) is also
considered for the perfect gas. The growth of the small-scale structures is slower in this
case. However, by t∗= 6, the kinetic energy spectrum reaches and maintains a similar level
to that observed with a constant initial pressure field. This suggests that the initial pressure
field has a minor influence (after a short transient period) on the subsequent distribution
of kinetic energy across the different scales of motion.
So far, the differences between tgv simulations of perfect and dense gas have been rela-
tively small. However, as the shock tube results reported in section III A indicate, a distinct
feature of dense gases is related to fluid expansions and compressions. In Fig. 10, we plot the
dilatation field θ(x, t) = ∂αuα(x, t) normalized by its standard deviation at time t∗= 9.5 in
a slice at x=L/5 for both gases. At this instant, the maximum Mach number (in the whole
domain) is high, above 1.4. Four symmetric quadrants reminiscent of the initial condition
are visible in the slice for both gases, but the dilatation field shows distinct differences
16
100101102
108
1010
1012
1014
1016
t
*= 0
Perfect gas
Dense gas
Perfect gas
p
(
t
= 0)
p
0
k
* 5/3
100101102
108
1010
1012
1014
1016
t
*= 2
100101102
108
1010
1012
1014
1016
t
*= 4
100101102
108
1010
1012
1014
1016
t
*= 6
100101102
108
1010
1012
1014
1016
normalized kinetic energy spectra
t
*= 8
100101102
108
1010
1012
1014
1016
t
*= 9.5
100101102
108
1010
1012
1014
1016
t
*= 10
100101102
108
1010
1012
1014
1016
t
*= 12
100101102
k
*= ||
k
||/
k
0 [-]
108
1010
1012
1014
1016
t
*= 14
100101102
k
*= ||
k
||/
k
0 [-]
108
1010
1012
1014
1016
t
*= 16
100101102
k
*= ||
k
||/
k
0 [-]
108
1010
1012
1014
1016
t
*= 18
100101102
k
*= ||
k
||/
k
0 [-]
108
1010
1012
1014
1016
t
*= 20
FIG. 9: Turbulent kinetic energy spectra at different times for the compressible tgv flow;
k0=2π
L. The light blue dotted line corresponds to a perfect gas with an initial
non-constant pressure field given by Eq. (38). The blue and orange continuous lines refer
to simulations with an initial constant pressure field p(x,0) = p0for the perfect and dense
gas, respectively. At early times, the perfect gas simulation contains more kinetic energy
in the small scales than the dense gas simulation.
between the perfect and dense gas. In particular, thin layers of moderate compression are
visible in the perfect gas but not in the dense gas. The supersonic flow, contained in the area
delimited by the green isocontour Ma = 1, occurs in small regions at the same locations for
both gases. However, as already observed in homogeneous isotropic turbulence79,80 , large
expansions (red spots where θ/θstd >5) are only present in the dense gas simulation, inside
these supersonic regions.
Previous results suggest that integral quantities behave similarly for perfect and dense
gas in the compressible tgv flow at initial Ma0= 1. Nevertheless, the maximum Mach
number along the simulation differs between the two gases as the flow decays, as shown
in Fig. 8. To highlight how spatial fluctuations evolve during the simulation, we plot the
standard deviation normalized by the averaged value over the entire domain at different
times for the pressure, density, temperature, dynamic viscosity (µ), thermal conductivity
(λ) and speed of sound (c)in Fig. 11. Mass densities show the same level of fluctuations.
However, for the pressure and temperature variations, the level of fluctuations is up to five
times lower in the dense gas than in the perfect gas. The almost isothermal behavior of
the dense gas can be attributed to its larger heat capacity. This is also reflected in the
Eckert number of the flow, defined as the ratio between kinetic energy and enthalpy, i.e.
Ec=U2
0/cp0T0. For the dense gas, the Eckert number at the initial operating point is
significantly lower with EcDG ≈6 10−4compared to EcPG = 0.4 for the perfect gas. On
the other hand, viscosity and speed of sound exhibit greater spatial variation in the dense
gas, primarily due to fluctuations in density (or pressure).
17
FIG. 10: Normalized dilatation field θ/θstd at time t∗= 9.5 in the slice x=L/5. The
green curves represent the isocontours Ma = 1. The color map is saturated at ±5 to
highlight large compression and expansion events. Left: perfect gas; Right: dense gas.
Large expansions occur only in the dense gas simulation, specifically within the supersonic
regions.
0 2 4 6 8 10 12 14 16 18 20
t* [-]
0
5
10
15
20
STD
/ [%]
Perfect gas
p
Perfect gas
Perfect gas
T
Dense gas
p
Dense gas
Dense gas
T
0 2 4 6 8 10 12 14 16 18 20
t* [-]
0
2
4
6
8
10
12
STD
/ [%]
Perfect gas
c
Perfect gas
Perfect gas
Dense gas
c
Dense gas
Dense gas
FIG. 11: Relative standard deviation (in space) along time for the perfect and dense
gases. Left: state variables p,ρ,T; Right: transport coefficients µ(dynamic viscosity), λ
(thermal conductivity) and speed of sound c. Because of the constant Prandtl assumption,
µand λremain proportional in the perfect gas. The state variables, transports
coefficients, and speed of sound evolve differently for the dense gas and perfect gas.
V. CONCLUSION AND PERSPECTIVES
To answer the needs of the energy transition, designing efficient orc turbines require new
numerical methods capable of handling complex flow structures (such as expansion shock-
waves and compressible turbulence) arising from the non-ideal thermodynamic behavior
of the fluids at play. This paper presents a novel hybrid lattice-compressible Boltzmann
method suitable for simulating turbulent bzt dense gas flows. This method integrates a
third-order Peng-Robinson eos and Chung’s transport coefficient model, thereby enhanc-
ing the capacity of the conventional lbm to manage non-ideal fluids. The proposed hybrid
scheme addresses the challenges of simulating highly compressible flows by combining the
strengths of the lbm and the finite-volume method for handling the evolution of the total
energy. The validation was conducted through two distinct test cases: the non-classical
behavior of a bzt gas within a shock tube and the compressible tgv flow. The results
18
from the shock tube simulations aligned closely with existing literature, validating the
method’s accuracy in capturing non-classical shock behaviors, such as rarefaction shock
waves. For the tgv flow, an in-depth analysis of the sensitivity to the grid resolution and
flux-reconstruction scheme (for the total energy) has been carried out, confirming the effec-
tiveness of the method in simulating compressible turbulence. Simulations demonstrated
the algorithm’s ability to capture strong compressibility effects, using high grid resolution
and advanced flux reconstruction schemes. A significant contribution of this work is the first
application of hybrid lbm to investigate dense and bzt effects in a tgv flow configuration.
The simulations revealed significant differences between dense and perfect gas flows at the
same initial Mach number, underlining the importance of taking into account the effects of
non-ideal gases in the analysis of compressible turbulence.
Future work includes the study and development of new numerical flow schemes to en-
sure numerical stability while maintaining accessible grid resolution, at even higher Mach
numbers, more representative of industrial orc applications. The treatment of bound-
ary conditions is another important project in perspective. When simulating flows in orc
turbines, handling boundary conditions is particularly challenging due to the interaction
of shock waves with solid structures. These interactions can lead to complex phenomena
such as shock reflections, boundary layer separation, and pressure wave distortions, which
require precise treatments to be properly captured. The lattice Boltzmann scheme, being
constrained by its fixed Cartesian grid, further complicates the situation and requires the use
of immersed boundary techniques to account for arbitrarily curved, possibly moving, solid
boundaries. This requires careful interpolation and boundary force modeling to accurately
represent the dynamics of fluid-structure interaction, especially in the presence of highly-
compressible and high-speed flows typical of orc turbines. To the best of our knowledge,
this remains an open area of research, pushing the limits of current lattice Boltzmann-based
simulation capabilities.
Finally, we would like to mention that, although our simulations do not reach the Mach
regimes encountered in the targeted orc turbine simulations, they nevertheless serve as a
significant benchmark. They ensure that our hybrid lbm scheme can effectively address
the complexities of compressible flows, such as capturing shock waves and handling high
density and pressure gradients, which represent the basis of the phenomena observed at
higher speeds. In this regard, we believe it represents a valuable step towards modeling
real-world compressible flows encompassing non-ideal gas.
Appendix A: Hermite polynomials
The Hi’s are discrete tensor Hermite polynomials in the velocity eand serve as a basis
for the expansion of the discrete distribution functions. Zero, first, second, and third-order
Hermite polynomials are
Hi0= 1,(A1)
Hiα =eiα,(A2)
Hiαβ =eiαeiβ −c2
sδαβ,(A3)
Hiαβγ =eiαeiβeiγ −c2
s(eiαδβγ +eiβδαγ +eiγ δαβ),(A4)
with 0 ≤i < 26 for the D3Q27 velocity set. Here, α, β , γ ∈ {x, y, z }and cs= 1/√3.
Appendix B: Chapman-Enskog expansion
Macroscopic fluid dynamics follows from seeking solutions to the hierarchy of statistical
equations that vary on a much slower timescale than the collisional timescale τ. This
is usually done by using a multiple-timescales Chapman-Enskog expansion in the small
19
parameter ϵ, with
fi=f(0)
i+ϵf(1)
i+ +ϵ2f(2)
i+···
∂t=ϵ∂(1)
t+ϵ2∂(2)
t+···
∂α=ϵ∂(1)
α(B1)
The parameter ϵmay be identified physically with the Knudsen number.
The Taylor expansion up to the second order of fi(t+ ∆t, x+ei∆t) gives
fi(t+ ∆t, x+ei∆t)−fi(t, x)=∆t(∂t+eiα ∂α)fi+(∆t)2
2(∂t+eiα∂α)2fi(B2)
∆t(∂t+eiα∂α)fi+(∆t)2
2(∂t+eiα∂α)2fi=−∆t
τ(fi−feq
i)+∆t(1 −∆t
2τ)Si(B3)
At different orders, one obtains
O(ϵ0) : f(0)
i=feq
i(B4)
O(ϵ1) : ∂(1)
t+eiα∂(1)
αf(0)
i=−1
τf(1)
i+ (1 −∆t
2τ)Si(B5)
O(ϵ2) : ∂(2)
tf(0)
α+∂(1)
t+eiα∂(1)
αf(1)
i+∆t
2∂(1)
t+eiα∂(1)
α2f(0)
α=−1
τf(2)
i(B6)
Simplifying the second-order equation using the first-order yields
O(ϵ2) : ∂(2)
tf(0)
α+∂(1)
t+eiα∂(1)
α(1 −∆t
2τ)(f(1)
i+∆t
2S(1)
i) = −1
τf(2)
i(B7)
Taking the zeroth, first, and second moments of Eq. (B5) yields
∂(1)
tρ+∂(1)
α(ρuα) = 0,(B8)
∂(1)
t(ρuα) + ∂(1)
βΠeq
αβ = 0,(B9)
∂(1)
tΠeq
αβ +∂(1)
γΠeq
αβγ =−1
τΠ(1)
αβ + (1 −∆t
2τ)Corr(1)
αβ (B10)
Similarly, computing the zeroth, first, and second moments of Eq. (B7) gives
∂(2)
tρ= 0 (B11)
∂(2)
t(ρuα) + ∂(1)
β{(1 −∆t
2τ)[Π(1)
αβ +∆t
2Corr(1)
αβ ]}= 0.(B12)
Π(1)
αβ is still unknown but can be expressed using Eq. (B10) at order O(ϵ). Recalling that
∂α(abc) = a∂α(bc) + b∂α(ac)−ab∂α(c) and using Eqs. (13, B8, B9), one gets
∂(1)
tΠeq
αβ =−∂(1)
γ(ρuαuβuγ)−(uα∂(1)
βp+uβ∂(1)
αp) + δαβ∂(1)
tp, (B13)
and with the definition Eq. (14) and Eq. (B8), one obtains
∂(1)
γΠeq
αβγ =∂(1)
γ(Πeq,theo
αβγ −Deq
αβγ ) (B14)
=∂(1)
γ(ρuαuβuγ)−∂(1)
γDeq
αβγ +c2
s∂(1)
β(ρuα) + c2
s∂(1)
α(ρuβ) + δαβc2
s∂(1)
γ(ρuγ)
(B15)
=∂(1)
γ(ρuαuβuγ)−∂(1)
γDeq
αβγ +ρc2
s∂(1)
β(uα) + ρc2
s∂(1)
α(uβ)
+uα∂(1)
β(ρc2
s) + uβ∂(1)
α(ρc2
s)−δαβ∂(1)
t(ρc2
s).(B16)
20
Finally,
∂(2)
t(ρuα)−∂(1)
β(1 −∆t
2τ)τ[ρc2
s(∂(1)
αuβ+∂(1)
βuα)
+δαβ∂(1)
t(p−ρc2
s)−uα∂(1)
β(p−ρc2
s)−uβ∂(1)
α(p−ρc2
s)
−∂(1)
γDeq
αβγ −Corr(1)
αβ ]o= 0.
(B17)
To reduce this term to the divergence of the viscous stress, one sets Corr(1)
αβ =δαβ∂(1)
t(p−
ρc2
s)−uα∂(1)
β(p−ρc2
s)−uβ∂(1)
α(p−ρc2
s)−∂(1)
γDeq
αβγ . By gathering the equations at Oϵ1
and Oϵ2and multiplying each order by ϵand ϵ2respectively, one eventually obtains
∂tρ+∂α(ρuα) = 0 (B18)
∂t(ρuα) + ∂β(ρuαuβ+pδαβ) = ∂β(τ−∆t
2)ρc2
s(∂αuβ+∂βuα)(B19)
These are the Navier-Stokes equations where the dynamic viscosity reads
µ= (τ−∆t
2)ρc2
s.(B20)
To satisfy the Stokes hypothesis, i.e. a null bulk viscosity or, equivalently, a deviatoric
(traceless) viscous stress, another term is finally added in the correction term so that
Corrαβ =δαβ∂t(p−ρc2
s)
−uα∂β(p−ρc2
s)−uβ∂α(p−ρc2
s)
−∂γDeq
αβγ
+δαβ
2
Dρc2
s∂γuγ
(B21)
where Dis the space dimension, and by assuming ∂t≈ϵ∂(1)
t+O(ϵ2).
ACKNOWLEDGMENTS
This work was granted access to the HPC resources of PMCS2I (Pˆole de Mod´elisation et
de Calcul en Sciences de l’Ing´enieur de l’Information) at ´
Ecole Centrale de Lyon, ´
Ecully,
France. This project benefited from computer and storage resources by GENCI at CINES
thanks to the grant 2023-AD012A14929 on the MI250x partition of the supercomputer
Adastra.
For the purpose of Open Access, a CC-BY public copyright licence has been applied by
the authors to the present document and will be applied to all subsequent versions up to
the Author Accepted Manuscript arising from this submission.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding
author upon reasonable request.
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