12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

A VOLUME FORCING METHOD BASED ON A RECONSTRUCTION-LIKE
PROCEDURE FOR HYBRID RANS/LES

Jérémie Janin
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)Saint Paul Lez Durance, 13115 France
Aix-Marseille Univ., CNRS,
Centrale Marseille, M2P2 Marseille, France
jeremie.janin@irsn.fr

Fabien Duval
Christophe Friess
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
Aix-Marseille Univ., CNRS,Saint Paul Lez Durance, 13115 FranceCentrale Marseille, M2P2 Marseille, France

Pierre Sagaut
Aix-Marseille Univ., CNRS,
Centrale Marseille, M2P2 Marseille, France

ABSTRACTwith long transients. In this frame, hybrid RANS/LES methods are a very attractive alternative but pose the difficulty ofgenerating turbulent fluctuations in transition zones. In thiswork, this difficulty is adressed by means of a volume forcing
method. Numerical calculations using the open-source software CALIF3 S-P2 REMICS (2021) are carried out on a planar
turbulent jet at Re = 10000 to assess and illustrate the interestof the proposed approach.

Hybrid RANS/LES methods usually require a forcing
method to trigger turbulence in LES regions, especially when
there is no hydrodynamic instabilities to generate turbulence.
A forcing method is proposed based on a reconstruction approach which consists in enriching the resolved velocity signal in the momentum equations. Only the extra terms that areconsidered to be prevalent for the turbulent production are retained. The proposed approach is assessed on the configurationof the planar jet case.

Forcing methods consist usually in adding a body force to
the Navier-Stokes equations. Most of the time, target statisticalquantities such as mean velocity and turbulent kinetic energy
need to be defined as inputs for the forcing term. These statistical quantities could come from either a precursor simulation
as in Pamier et al. (2009) or from the RANS upstream domain
as in de Laage de Meux et al. (2015). For the latter, the forcing
is applied to a channel flow which has an homogeneous energy
distribution in the streamwise direction. Therefore, the use ofvalues from the RANS upstream domain as targets, appears
relevant. In non-homogeneous flows, the turbulent kinetic energy can significantly vary in the streamwise direction. For
this reason, we propose here another approach which consistsof deriving the target quantities from a prescribed kinetic energy ratio between resolved and subgrid scales.

INTRODUCTION
Explosion risks, especially those related to hydrogen releases, are encountered in a large number of situations. These
include for instance hydrogen leaks from pipes or storages inventilated rooms (Taveau (2011)) or, during a severe accident
in nuclear reactor containment vessels, hydrogen production
due to the oxidation of zirconium (Bentaib et al. (2015)). In
evaluating explosion hazards, the first stage consists in predicting the release and the turbulent mixing of flammablespecies. The cost and reliability of numerical simulations
depend on the approach adopted for the turbulence modeling. One usually distinguishes two modeling approaches:
the Reynolds-Averaged Navier-Stokes (RANS) statistical approach, that consists in modeling all turbulent scales, and theLarge Eddy Simulation (LES) approach that consists in a direct calculation of the large scales and a modeling of the small
ones. The RANS approach has the advantage of presenting
reasonable computational costs but the reliability of the simulations may depend strongly on the level of sophisticationof the turbulence model. On the other hand, the LES approach is much more predictive but requires larger computing
times that can be out of reach when simulating large facilities

The forcing method is tested upon the planar jet test casefor which there exists various Direct Numerical Simulation(DNS) data in the literature such as Stanley (2002), Klein(2003), Le Ribault et al. (1999) and Engelmann et al. (2021).
Stanley (2002) and Le Ribault et al. (1999) performed respectively DNS and LES simulations at Reynolds number of 3000.
Klein (2003) provided DNS results for Reynolds number inthe range of 1000 to 6000 but only results at Re = 4000 arereported here. The simulations presented here are performed
with Reynolds number Re = 10000 as in Engelmann et al.
(2021). In the present work, the first step was to compare a1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
The length scale LE−DES is defined as a function of a target
turbulent kinetic energy ratio (Friess et al., 2015) as

computation with turbulent inlet boundary conditions and acomputation with laminar inlet boundary condition in addition with the forcing method. For these two simulations, it
was observed that the results were very similar. This is dueto the location of the forcing area which is situated at the entrance of the domain, just after the laminar velocity inlet. In
this case the forcing acts ultimately as a turbulent boundary
condition. In order to move away from this configuration, we
proposed here a configuration inspired by RANS/LES zonal
methods by imposing the turbulent kinetic energy ratio on thefirst diameters of the jet to enforce a RANS zone, and thenswitching to a non-zonal hybrid method with a self-adaptive
reconstruction method. We focus in this work on the capability of the method to reproduce the expected self-similar profiles downstream from the RANS domain. We focus further
on the center-line resolved and subgrid scale turbulent kinetic
energy profiles with respect to the prescribed kinetic energy
ratio.
In the next section, the reconstruction approach is presented as well as the employed turbulent model. In addition,
the synthetic velocity used in the reconstruction procedure isbriefly recalled. Finally, the advantages and drawbacks of themethod on the studied case of the planar jet are discussed.

3/2

LE−DES =where rk = km /k is the energy ratio, km the turbulent kinetic
energy related to the unresolved scales, and k the total turbulent kinetic energy.
Here, we take benefit of the prescribed kinetic energy
ratio to drive the synthetic velocity as suggested in a previous work (Janin et al., 2021). The synthetic velocity involved in the reconstruction approach is derived from the Random Fourier Modes (RFM) method developed by Kraichnan(1970), and extended later by Fung et al. (1992), which isbriefly recalled below:Nusi (x j ,t) = 2 ∑ ûn cos(κln xl + ψn + ωn t)σin

where ûn , ψn , ωn and σin correspond respectively to the amplitude, the phase, the time frequency and the direction of thenth Fourier mode related to the wave vector κin . The amplitudepis written as ûn = E(κn )δ κn , in which E(κ) is a prescribedenergy spectrum. The stochastic frequency ωn = λUrms κn corresponds to the sweeping hypothesis where λ follows a normal distribution. A turbulent kinetic energy k and an integral
length scale Lt are required to set up the given energy spectrum. The monitoring of the resolved turbulent kinetic energy
is performed as in the work of Janin et al. (2021) and the target
resolved kinetic energy is here defined askr† = k(1 − rk )(4)

In the present paper, an energy spectrum model suggested by
Chaouat & Schiestel (2009) has been chosen as it is easy tointegrate and thus leading to a straightforward expression for
the kinetic energy ratio rk .
hi−2/9
rk = 1 + β 2/9 (αL Lt κcut )3∂ τi j ∂ usi
∂ ũi
∂ ũi
1 ∂ p̃
∂ 2 ũi∂ ũi+ ũ j
=−
+ν
−
−
− usj
∂t
∂xj
ρ ∂ xi∂ x j∂ x j
∂xj
∂t
∂xj

(5)

in which β = (2/3Ck )2/9 , αL = 0.85 and κcut is the cutoff
wave number.
It mus be pointed out that the formulation of the target
resolved kinetic energy is based on an assumption of conservation of the turbulent kinetic energy. Indeed, the proposed
method aims to balance the resolved and subgrid turbulent kinetic energy budgets. Additional terms from the reconstruction procedure in Eq. 1 lead to a new production term in the
resolved turbulent kinetic budget, as intented by the method.
Note that the production term is not clipped, meaning it can
exhibits negative values. This production term is added as asink term to the subgrid kinetic energy transport equation. This
allows to have a transfer of energy between the subgrid andthe resolved part while formally keeping the total turbulent kinetic energy budget unchanged. This energy transfer must takeplace in under-resolved regions.
These regions are encountered when the observed turbulent kinetic energy ratio rko = ksgs /k is greater than the kinetic

In the above equation ũi , p̃, ν are respectively the resolved
velocity, the resolved pressure, the kinematic viscosity and τi j
refers to the subgrid stresses which are modeled here using theBoussinesq’s closure in combination with a subgrid scale eddy
viscosity. The subgrid scale eddy viscosity is estimated in theframe of the Equivalent-Detached-Eddy Simulation (E-DES)
approach proposed by Friess et al. (2015) using a two-equation
ksgs -εsgs subgrid scale model. In this approach, the controlof the energy partition between resolved and subgrid scales
is performed in a similar way as in the usual DES method byintroducing the following limiter that multiplies the dissipation
term in the ksgs -equation:
3/2

(3)

n=1

In this study we suggest to use a reconstruction-like procedure as in Janin et al. (2021) to trigger turbulence in transition zones. The reconstruction procedure consists in adding
a fluctuating part, that corresponds here to a synthetic velocity, to the resolved velocity. This operation introduces additional terms into the filtered Navier-Stokes equations. Usually,
only the unsteady term is retained (Schmidt & Breuer (2017))
requiring, at least formally, that turbulence is statistically homogeneous. However, the flow of interest involves shear flow
turbulence and this calls to retain a convective term related to
the reconstruction procedure. We follow the motivations of
Lundgren (2003) regarding the physical meaning of his linear
forcing and we also retain here the term proportional to theresolved velocity gradient. Similar developments have been
followed using either an unsteady forcing term (Schmidt &Breuer (2017)) or a forcing term proportional to the velocity
gradient (?,Zhang (2021)). Here, both contributions are retained with the proposed reconstruction approach and the resulting filtered Navier-Stokes equations read:ksgs
FE−DES = max 1,
εsgs LE−DES(2)

1

METHODOLOGY

"

rk
k3/2
h
i
cε1 /cε2 εsgs
cε2 −cε1
1 + cε1 − rk#(1)

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
ferent inlet parameters are summarized in Table 1.

energy ratio defined by Eq 5. Thanks to the control of theresolved turbulent kinetic energy, the forcing is only active inthese regions. This makes the method self-adapting which prevents forcing in areas not of interest as coflow regions. This
ensures that the solution does not deteriorate and saves computing times.
An attractive feature of the method is that the control of
the resolved turbulent kinetic energy is done by a selective
forcing. This selective forcing could either operates at low
or high wavenumbers. For homogeneous isotropic turbulence,
forcing at low wavenumbers is preferred as forcing at highwavenumbers alters the inertial zone of energy spectrum. In
the case of turbulent shear flows the question arises. In theirwork, Haering (2022) argued for a forcing at high wavenumbers. That way large scales that may be present or the mean
velocity flows are not disturbed by the forcing. Although we
support this approach, preliminary results are not satisfactory.
Therefore, the selective forcing at low wavenumbers is used inthis work.

Table 1: Summary table of inlet parameters for the zonal

simulation with forcing and the simulation with turbulent Boundary Condition (BC)

Inlet parameters

Forcing/RANS

Turbulent BC

Re

104

104

Length scale

0.4d j0.4d jTurbulent intensity

10%

10%

Synthetic method

hyperbolic
tangent
peaks in theshear layer

RFM
hyperbolic
tangent
peaks in theshear layer

1

0.3

Mean profiles
Turbulent kinetic
energyTurbulent kinetic
energy ratio

NUMERICAL SET UP
The proposed method is applied to a planar jet at a Re =10000 which is based on the nozzle width d j = 0.05m. Simulations are performed with the E-DES model and a mesh
containing 7.5105 grid points. The domain size is [0; Lx ] ×[0; Ly] × [0; Lz ] with Lx = 40d j , Ly = 40d j and Lz = 6.4d j ,doubling the streamwise and normalwise dimensions compared to Engelmann et al. (2021). The fluid considered is
air with constant physical properties, ρ = 1.2kg.m−3 and µ =1.810−5 kg.m−1 .s−1 . Periodic conditions are used in the zspanwise direction. An outlet-like condition that allows a control of the kinetic energy is imposed at the outlet that corresponds to the right, top and bottom boundaries as illustrated on
Fig. 1. At the inlet, an hyperbolic tangent profile is specified
for the mean axial velocity U j as in Le Ribault et al. (1999),
with a coflow Uco /U j = 9%. No fluctuations are superimposedon the mean velocity profile in boundary conditions. In order
to assess the overall effect of the reconstruction procedure, theRANS mode is enforced over a 6d j long area (i.e. rk = 1)
downstream the nozzle while this ratio keeps its original formulation Eq. 5 downstream the black dashed line reported inFig. 1.
Simulation results are obtained using the in-house
CALIF3 S-P2 REMICS software. Time discretization is carriedout by using a fractional step algorithm that consists in a pressure correction method. Space discretization is performed byusing a staggered finite volume scheme for which scalar unknowns are located at cell centers while the velocity is locatedat cell faces. The numerical scheme is discretely kinetic energy
conserving (Boyer et al. (2014)) and corresponds to a centeredsecond-order spatial discretization of both convective and diffusive fluxes together with the semi-implicit Crank-Nicolson
time scheme. The time step is fixed according to the CFL
based on the inlet mean velocity and the mesh size to be 0.5.
The final time of the simulation is 40 flow time units and a
restart is performed at 10 flow time units. The flow time units
is defined as in Stanley (2002): Tt = 2Lx / U j +Uco .
Results are compared with a simulation with turbulent inlet boundary condition, refereed here as turbulent BC. Turbulent fluctuations are generated with the RFM method andadded to the mean velocity profile at the inlet. A RANS
computation is also performed with a classical k-ε turbulencemodel. For both above mentioned computations, the same numerical parameters and physical properties are used. The dif-

RESULTSIn this section, we investigate the influence of the reconstruction method by focusing on several quantities of interest
such as the mean and root mean square values of the velocity
in the self-similar region as well as the axial evolution of the
turbulent kinetic energy. The slope coefficients of the jet halfwidth evolution and the decrease of the axial mean velocity excess are also compared to the results from the literature. Since
we are emulating a zonal hybrid RANS/LES model, there is atransition area in between the two domains. Hence, it is interesting to assess these slope coefficients both in the transitionarea and further downstream.
This transition zone is highlighted on the snapshot of theinstantaneous velocity displayed in Fig. 1. The black dashedline delimits the upstream RANS domain (rk = 1) and the zone
where the E-DES model behaves self-adapting. As expected,
no velocity fluctuations are observed upstream of the black
dashed line while the forcing enables the generation of fluctuations downstream. The transition area takes place just after
the RANS domain. In this transition region, it appears that thejet-half width evolution is reduced over a distance of at least 7
diameters.
In order to discuss quantitative results, the effect of the
proposed approach on statistical quantities needs to be assessed. All statistical quantities presented after are averaged
over 30Tt . First of all, the results in the self similar region
are presented in Fig. 3 and are compared to results from theliterature. Results are also compared to the simulation using
turbulent boundary condition.
Radial profiles are presented for seven different axial locations between x = 11d j and x = 28d j . Fig. 2a shows thenormalwise evolution of the mean axial velocity for different
axial positions. Results are in good agreement with resultsfrom the literature and the turbulent BC simulation. Mean velocity profiles collapse to a self-similar state except the profile
at x = 11d j meaning that the self-similarity region is not yet
reached at this location.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 1: Instantaneous snapshot of the velocity magnitude. The black dashed line delimits the upstream RANS domainfrom the downstream E-DES domain at the axial position x = 6d j .
1

0.6
x = 11dj
x = 15dj
x = 20dj
x = 24dj
x = 26dj
x = 28dj
Turbulent BC
Le Ribault et al.
Stanley et al.

∆U/∆Uc

0.6
0.4

0.5
0.4
Urms /∆Uc

0.8

0.3

0.2

0.2

0

0.1

−0.2

0

0.5

1

1.5
y/dj

2

2.5

x = 11dj
x = 15dj
x = 20dj
x = 24dj
x = 26dj
x = 28dj
Turbulent BC
Le Ribault et al.
Stanley et al.
Klein et al.

0

3

0

0.5

1

(a)

2.5

3

0.6
x = 11dj
x = 15dj
x = 20dj
x = 24dj
x = 26dj
x = 28dj
Turbulent BC
Stanley et al.
Klein et al.

0.4
0.3

0.4
0.3

0.2

0.2

0.1

0.1

0

0.5

1

1.5
y/δ0.52

2.5

x = 11dj
x = 15dj
x = 20dj
x = 24dj
x = 26dj
x = 28dj
Turbulent BC
Stanley et al.
Klein et al.

0.5

Wrms /∆Uc

0.5

Vrms /∆Uc

2

(b)

0.6

0

1.5
y/δ0.5

0

3

(c)

0

0.5

1

1.5
y/δ0.5

2

2.5

3

(d)

Figure 2: Radial profiles in the self-similarity region. (a) Mean streamwise velocity, (b) Streamwise velocity fluctuations
Urms , (c) Normalwise velocity fluctuations Vrms and (d) Spanwise velocity fluctuations Wrms .

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.15

0.15
Forcing

kr - Forcing

Turbulent BC

ksgs - Forcing

RANS

kr - Turbulent BC

Stanley et al.

ksgs - Turbulent BC

0.10

RANS

k/∆Uc2

k/∆Uc2

0.10

0.05

0.00

0.05

0

5

10

15
x/dj

20

25

0.00

30

0

5

10

(a)

15
x/dj

20

25

30

(b)

Figure 3: (a) Axial evolution of the turbulent kinetic energy. Comparison between the forcing method, the turbulent BC

simulation and the DNS results from Stanley (2002), (b) Axial evolution of the resolved and subgrid turbulent kinetic
energy. Comparison between the forcing method, the turbulent BC simulation and the RANS turbulent kinetic energy.

Radial profiles of the streamwise velocity fluctuations
shown in Fig. 2b exhibit the same trends, namely that the jet
becomes self-similar at x = 11d j . Profiles are not in agreement
near the center line before merging from y/δ0.5 = 1. Nevertheless, there seems to be a convergence between profiles at position x = 26d j and x = 28d j . The results are underestimated, incomparison with the turbulent BC simulation but are in good
agreement with DNS and LES data from the literature.
Fig. 2c and Fig. 2d represent respectively radial profilesof the normalwise and spanwise velocity fluctuations. Fromthese two figures, it appears that the profiles collapse to a selfsimilar behavior from x = 20d j . Profiles are again underestimated compared to the turbulent BC simulation and the DNS
results. For the Vrms radial profiles of the simulation with turbulent boundary condition, the self-similarity region is reachedat 20d j (not shown here). The same value seems to be obtained
for the zonal simulation.
The underestimation of the fluctuating velocity profiles
is reflected in Fig. 3a which represents the axial evolution of
the turbulent kinetic energy. Regarding the RANS computation, the peak of turbulent kinetic energy is clearly underestimated compare to the DNS results. The turbulent kinetic
energy reaches the same level as the simulation with turbulent boundary condition near x = 20d j . Thanks to the forcing
the turbulent kinetic energy does not decrease downstream the
RANS domain and agrees well with the RANS results.
Fig. 3b shows the axial evolution of the energy distribution between the resolved and subgrid part also compared to
the RANS turbulent kinetic energy. As expected, the increaseof the resolved part starts from x = 6d j for the zonal configuration. At the same location, the subgrid kinetic energy decays
due to the sharp decrease of the kinetic energy ratio and the
production term which is subtracted from the right hand sideof the subgrid turbulent kinetic transport equation.
It must be noticed that the reconstruction method has noinfluence on the ability of mean and fluctuating velocity profiles to converge towards self-similarity. The turbulent kinetic
energy at the jet center line, is underestimated. However thisis mainly due to the fact that the upstream RANS statistics are
not reliable.
In order to estimate the impact of the proposed approachon the jet half width evolution and the mean excess velocity
decay, the coefficients of the linear relationship that holds inthe self-similar region are compared (Stanley (2002)):

δ0.5
= K1
dj



∆U j
∆Uc



2

x
+ K2
dj


= C1



x
+C2
dj

(6)


(7)

in which, ∆U j and ∆Uc represent respectively the inlet velocity
excess and the local center line velocity excess. These coefficients are calculated over a distance of 6d j , from 10d j to 16d j .
For the forcing
are additionally computed for case, coefficientsthe region 20d j ; 26d j since the first region is closed to thetransition area. The resulting values are reported in Table 2. In
the first region, the slope of the jet half width evolution is low
compared to simulation with turbulent boundary condition. In
the second region, the slope approaches the RANS results but
is still underestimated compared to the DNS results. The sameconclusions are made for the slope of the axial evolution of thevelocity excess. This decay observed on the results is still under investigation and is part of the limitations of the method.
The quality of the synthetic fluctuations introduced in the domain might play an important role in recovering more accurate
slopes.

Table 2: Summary of the resulting coefficients K1, K2,

C1 and C2 for the region [6d j ; 12d j ].

5

Simulations

K1

K2

C1

C2

Forcing [10d j ; 16d j ]

0.062

3.44

0.138

1.19

Forcing [20d j ; 26d j ]

0.069

1.83

0.155

-0.61

Turbulent BC

0.096

1.47

0.196

0.066

RANS

0.082

2.16

0.168

0.273

Stanley (2002)

0.092

2.63

0.201

1.23

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Conclusion

Lundgren, T.S. 2003 Linearly forced isotropic turbulence. Annual Research Brief (CTR, Standford) pp. 461–473.
de Laage de Meux, B., Audebert, B., Manceau, R. & Perrin,
R. 2015 Anisotropic linear forcing for synthetic turbulence
generation in large eddy simulation and hybrid RANS/LES
modeling. Phys. Fluids 27, 035115.
P2 REMICS
2021
Collaborative
websitehttps://gforge.irsn.fr/gf/project/p2remics/.
Pamier, M, Weiss, P. E., Garnier, E, Deck, S & Sagaut, P 2009
Generation of synthetic turbulent inflow data for large eddy
simulation of spatially evolving wall-bounded flows. Phys.
Fluids 21, 045103.
Schmidt, S. & Breuer, M. 2017 Source term based synthetic
turbulence inflow generator for eddy-resolving predictionsof an airfoil flow including a laminar speration bubble.
Comput. Fluids 146, 1–22.
Stanley, S. A., Sarkar S. Mellado J.P 2002 A study of the flowfield evolution and mixing in a planar turbulent jet using
direct numerical simulation. J. Fluid Mech 450, 377–407.
Taveau, J. 2011 Explosion hazards related to hydrogen releases
in nuclear facilities. Journal of loss prevention in the process industries 24 (1), 8–18.
Zhang, J. 2021 Significance of generating synthetic turbulence
for zonal detached eddy simulation of shallow water flows.
Ocean Engineering 237, 109418.

The reconstruction procedure proposed in Janin et al.
(2021) is applied to the case of a planar jet at Re = 10000.
The approach is extended to take into account the nonhomogeneity of the flow. This results in an extra forcing term
related to the reconstruction of the convective term in the filtered Navier-Stokes equations. The overall approach is assessed on a given configuration which roughly consists in azonal hybrid RANS/LES method. By design, the forcing isonly active downstream the RANS domain and target properties such as the resolved turbulent kinetic energy are provided
by a prescribed turbulent kinetic energy ratio.
The reconstruction method allows to recover the selfsimilarity region. However, a transition zone appears in whichmean quantities such as slopes of both the jet half width evolution and the mean velocity excess decay are significantly decreased. An improvement in the quality of the synthetic fluctuations might help to fine-tune the previously computed slopes.
Typically, poor estimation of turbulent quantities in the upstream RANS domain contributes here to a large transition towards a fully developed self-similar behavior.
The expected benefits of the proposed approach should be
at first a reduction of the computational cost by using a CFL
constraint built far from the slot where the characteristic mean
velocity has decreased drastically. A substantial benefit is also
expected for buoyant jets as LES usually exhibits superior predictive capabilities far from the nozzle in the dominated buoyant region while RANS predictions remain accurate near thenozzle in the dominated inertia region.

REFERENCES
Bentaib, A., Meynet, N. & Bleyer, A. 2015 Overview on hydrogen risk research and development activities: methodology and open issues. Nuclear Engineering and Technology
47 (1), 26–32.
Boyer, F., Dardalhon, F., Lapuerta, C. & Latché, J.-C. 2014
Stability of a crank-nicolson pressure correction scheme
based on staggered discretizations. International Journal
for Numerical Methods in Fluids 74, 34–58.
Chaouat, B. & Schiestel, R. 2009 Progress in subgridscale transport modelling for continuous hybrid non-zonalrans/les simulations. Int. J. Heat Fluid Flow 30, 602–616.
Engelmann, L., Klein, M. & Kempf, A.M. 2021 A-posteriori
les assessment of subgrid-scale closures for bounded passive scalars. Comput. Fluids 218, 104840.
Friess, C., Manceau, R. & Gatski, T.B. 2015 Toward an equivalence criterion for hybrid rans/les methods. Comput. Fluids
122, 233–246.
Fung, J.C.H., Hunt, J.C.R., Malik, N.A. & Perkins, R.J. 1992
Kinematic simulation of homogeneous turbulence by unsteady random fourier modes. J. Fluid Mech 236, 281–318.
Haering, S. W., Oliver T. A. Moser R. D. 2022 Active model
split hybrid rans/les. Physical review fluids 7, 014603.
Janin, J., Duval, F., Friess, C. & Sagaut, P. 2021 A new linear forcing method for isotropic turbulence with controlled
integral length scale. Phys. Fluids 33, 045127.
Klein, M., Sadiki A. Janicka J. 2003 Investigation of the influence of the reynolds number on a plane jet using direct
numerical simulation. Heat and Fluid Flow 24, 785–794.
Kraichnan, R.H. 1970 Diffusion by a random velocity field.
Phys. Fluids 13, 22.
Le Ribault, C., Sarkar, S. & Stanley, S.A. 1999 A-posteriori lesassessment of subgrid-scale closures for bounded passive
scalars. Phys. Fluids 11, 3069.

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