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

STOCHASTIC MODELING OF THREE-SCALAR MIXING IN A COAXIALJET USING ONE-DIMENSIONAL TURBULENCE

Marten Klein
Christian Zenker
Lehrstuhl Numerische Strömungs- undFachgebiet Numerische Mathematik und
Gasdynamik
Wissenschaftliches Rechnen
Brandenburgische Technische Universität Brandenburgische Technische Universität
Siemens-Halske-Ring 15A
Platz der Deutschen Einheit 1
D-03046 Cottbus, Germany
D-03046 Cottbus, Germany
marten.klein@b-tu.de
christian.zenker@b-tu.de

Tommy Starick
Heiko Schmidt
Lehrstuhl Numerische Strömungs- undLehrstuhl Numerische Strömungs- und
Gasdynamik
Gasdynamik
Brandenburgische Technische Universität Brandenburgische Technische Universität
Siemens-Halske-Ring 15A
Siemens-Halske-Ring 15A
D-03046 Cottbus, Germany
D-03046 Cottbus, Germany
tommy.starick@b-tu.de
heiko.schmidt@b-tu.de

ABSTRACT

ing process. Such models are, therefore, of limited applicability for flows with complex mixing and possibly active scalars
with multiphysicale small-scale processes.

Modeling complex mixing processes is a standing challenge for a number of applications ranging from chemical tomechanical and environmental engineering. Here, the gasphase turbulent mixing in a three-stream concentric coaxial
jet is investigated as a canonical problem. Reynolds-averaged
Navier–Stokes simulations (RANS) suggest that the gas-phasemixing can be accurately modeled by air doped with passive
scalars, for which small-scale resolving numerical simulations
are performed with the one-dimensional turbulence (ODT)
model as stand-alone tool. We show that both the spatial (SODT) and temporal (T-ODT) model formulations yield qualitatively similar results exhibiting reasonable to good agreement with available reference experiments, Reynolds-averaged
and large-eddy simulations, as well as mixing models. This isdemonstrated for low-order statistics, like the scalar varianceand dissipation, but also the two-scalar joint probability density functions that can not be obtained with RANS. Our results suggest that S-ODT has better capabilities than T-ODT
to model the mixing processes in the jet which we attribute to
the account of local advective time scales.

Taking chemically-reacting jets as a relevant application,
it is well known that the location and maximum of the heatrelease in a non-premixed jet flame crucially depends on thehydrodynamic mixing of chemical species prior to combustion (e.g. Pitsch & Steiner (2000); Echekki et al. (2001);
Cabra et al. (2005); Walter et al. (2015)). In order to better understand the mixing process, a single passive scalar
has been investigated for various settings (e.g. Dowling &Dimotakis (1990); Klein et al. (2019)). Albeit the relevant
scales have been resolved by these experimental and numerical approaches, cross-correlations and differential diffusion
effects between different chemical species were not captured.
Therefore, multiscalar mixing has been investigated recently(e.g. Giddey et al. (2018); Li et al. (2021)). A feasible andcanonical application case is the three-stream mixing (e.g. Cai
et al. (2011)) or, equivalently, two-conserved-scalar mixing(e.g. Rowinski & Pope (2013)), in a coaxial round jet that is
considered here.
In this study, we address the standing numerical challenges associated with the high-fidelity representation of multiple scalar mixing in a concentric coaxial round jet (see figure 1) by utilizing the lower-order stochastic one-dimensional
turbulence (ODT) model (Kerstein (1999)) as stand-alone tool.
Here, we apply and compare the temporal and spatial model
formulations for cylindrical geometry (Lignell et al. (2018)).
We demonstrate that ODT is able to capture scalar joint probability density functions of the multiscalar mixing but also discuss its shortcomings.

INTRODUCTION
Turbulent mixing denotes the filamentation and diffusionof conserved flow variables on a range of scales in a chaotic,
notionally random flow field. Filamentation occurs due to reversible stirring motions, whereas molecular diffusion is irreversible by nature. While direct numerical simulation (DNS)
resolves all relevant scales of the flow directly, it is often
far to expensive for extrapolation to parameter regimes relevant to applications. Conventional turbulence models used
in Reynolds-averaged Navier–Stokes simulation (RANS) andlarge-eddy simulation (LES) aim to overcome this bottleneckby treating turbulent mixing as a continuous, irreversible mix-

Below, we briefly describe the configuration, the modeling approach, and the simulation set-up. After that, we discuss
the mixing by comparing ODT to RANS and available reference data. Last, we give some concluding remarks.

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
x =0
R

ODT domain
φ3 = 1r4
r1dθ

φ1 + φ2 = 1
φ1 = 1xz

φ1 + φ2 < 1

φ2 + φ3 = 1

r3 r2
r

x = 6.99 d
inflow/outflow
boundary

Ub

hux i
∆θ

y
x

pipe
Y1

annular
pipe
Y2

∆x

co-flow
Y3∆r

Figure 1. Sketch of the three-stream coaxial jet configuration investigated. A round and a concentric coaxial annular jet issue intoa slow co-flow at downstream location x/d = 0 with the sketched mean velocity profile hux i(r) due to walls at radial locations r1 –r4 .
The domain boundary is located at R  r4 . The inflow streams i = 1, 2, 3 contain different passive scalars with initially uniform massfraction Yi or mixture fraction φi , respectively. The ODT domain is a two-sided infinitesimal wedge that spans the diameter and isadvected downstream with bulk velocity Ub . See table 1 for further details.
Table 1. Details of the inflow streams i = 1, 2, 3 in figure 1.
Nominal reference values (ref) are based on Cai et al. (2011).

A THREE-STREAM COAXIAL JET
Figure 1 shows a sketch of the cylindrical three-stream
configuration. The mixing of a central round and a concentric coaxial annular jet surrounded by a slower co-flow is investigated downstream of a three-stream nozzle. The set-up iscomparable to that of Cai et al. (2011), who experimentally investigated the mixing of an inner acetone-doped jet of air withan annular ethylene jet and co-flow air. The inflow streams aregases with mass fraction Yi (i = 1, 2, 3) and mixture fraction
φi = Yi (Y1 + Y2 + Y3 ). Following Rowinski & Pope (2013)and Dowling & Dimotakis (1990) for gas-phase concentration
mixing, we simplify the configuration to three inflow streamsof air with homogeneous kinematic viscosity ν and mass density ρ. Every stream is doped with a different passive scalarof diffusivity Γi as detailed below and in table 1. Here, two
low-fidelity RANS simulation cases have been considered, one
replicating the experimental inflow streams and the other treating gas-phase concentrations as passive scalars. The results are
virtually indistinguishable (in agreement with Dowling & Dimotakis (1990)) so that we adopt the case of passive scalar for
our ODT and RANS simulations shown below.
The coaxial jet is characterized by the jet Reynolds numbers Re1 = Ub,1 r1 /ν = 7130 and Re2 = Ub,2 (r3 − r2 )/ν =3974 that are based on the bulk velocity Ub,i in each inflow stream in between the positive radial intervals [0, r1 ] and[r2 , r3 ]. The co-flow is uniform and laminar in [r4 , R]. The
gas-phase composition is characterized by the concentration
diffusivity relative to air. This is expressed for the two jets by
the Schmidt numbers Sc1 = ν/Γ1 = 1.29 and Sc2 = ν/Γ2 =0.565. Note that the scalar in the co-flow has zero diffusivity(Γ3 = 0). This scalar only labels the co-flow air, which does
not diffuse in itself. However, we keep track of its advection
since turbulent entrainment of co-flow fluid affects the mixture
fractions φi downstream of the nozzle.

property

central jet

annular jet

co-flow

i

1

2

3

rinner,i [mm]

—

3.175

4.75

router,i [mm]

2.77

4.19

75

Ub,i [m/s]

34.5

32.5

0.4

ρref,i [kg/m3 ]

1.26

1.14

1.17

[mm2 /s]

13.4

8.3

15.9

ρi = ρ [kg/m3 ]

1.168

1.168

1.168

νi = ν [mm2 /s]

15.9

15.9

15.9

Γi [mm2 /s]

10.39

14.69

0

νref,iaveraged Navier–Stokes and scalar conservation equations are
solved by modeling the mean effects of unresolved fluctuations
with the aid of the Boussinesq closure assumptions. Here, 3-D
RANS equations have been solved numerically using Ansys®
Academic Research CFD, Release 2021 R2. The computational domain is 100d long and contains an inflow section oflength 20d such that fully-developed pipe and annular pipe
flows exit the nozzle at x = 0. No-slip zero-scalar-flux boundary conditions are prescribed at the radial walls, except forr = R, which is treated as free-slip wall. Inflow and outflow
boundary conditions are prescribed in axial direction. The gridcontains ≈ 5 × 107 cells with local refinement at the nozzle
walls and across the conical influence region of the jet. A conventional k–ω shear-stress transport (SST) closure model withdefault model parameters is used that can handle wall-attached
and free-stream turbulence. The main purpose of the present
RANS analysis is to assess the simplification to passive scalar
mixing and to provide additional low-fidelity reference data.

MODEL OVERVIEW AND SIMULATION SET-UP
Modeling The Mean State Using RANS
Computational fluid dynamics has become a powerfultool in analyzing engineering problems over the last decades.
An established approach is RANS in which the Reynolds-

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
fluxes since the continuity equation is not part of the model.
This deficiency is partly compensated by the domain displacement x(t) mentioned above, but, strictly, T-ODT is applicable
only to flows that are statistically invariant in downstream andspanwise direction such as channel or pipe flows.
In S-ODT, a local time-to-space transformation is used
(Kerstein (1999)). In the adaptive formulation (Lignell et al.
(2018)), grid cells are displaced with their local axial velocityux (r, x). Hence, for a fixed spatial increment δx , molecular diffusion has more time to take effect where ux is low. The ratioof the local advective (δx /ux (r)) to the diffusive (δx2 /ν, δx2 /Γi )time scales is more accurately represented in S-ODT than TODT. S-ODT avoids the ambiguity with respect to downstream
advection but it requires ux (r) > 0. The S-ODT equations specialized to round jets readThe differences between gas-phase and passive scalar mixing
are marginal so that we limit our attention to the latter case.

Stochastic Modeling Using ODTThe key idea of the one-dimensional turbulence (ODT)
model (Kerstein (1999)) is to resolve the flow on all relevant
scales, but only for a notional line-of-sight. Deterministic processes (like molecular diffusion) are directly resolved along
the lower-order computational domain, whereas turbulence
is modeled by a stochastically-sampled sequenceof discretep
eddy events with momentary rate τ −1 = C 2E/l 2 , where C
is the eddy rate parameter, l the eddy size, and E the specific available eddy energy. E is obtained based on the momentary flow state as E = Ekin − ZEvp , where Ekin = u2x,K and
Evp = ν 2 /l 2 denote the kinetic and viscous penalty contributions, respectively. ux,K is a kernel-weighted eddy scale velocity and Z the small-scale suppression parameter.
Eddy events punctuate the deterministic flow evolution by
application of a conservative (measure-preserving) map that
instantaneously modifies flow profiles by a permutation of
fluid elements along the 1-D domain. Eddy events are characterized by three random variables: the eddy size, radial location, and time (or axial location) of occurrence. Energeticallyplausible (E > 0) candidate events are considered for probabilistic acceptance if an eddy turnover fits into the elapsed
simulation time, that is, τ ≤ βLS t for T-ODT or with t by inversion of x(t) for S-ODT, where βLS is the large-scale suppression parameter (Echekki et al. (2001)).
For the round jet sketched in figure 1, the ODT domain
is an infinitesimal double-wedge that spans the diameter andhas open boundaries in the co-flow region. The two subdomains for positive and negative radial (r) locations are coupled
by a symmetric center cell in order to minimize flux imbalances due the the coordinate singularity at the axis within thestand-alone model (Lignell et al. (2018)). The ODT domain isadvected downstream with bulk velocity Ub during a simulation run, which is done a differently in the temporal (T-ODT)
and spatial (S-ODT) model formulations (Kerstein (1999)).
In T-ODT, the ODT domain is interpreted as a closed system and lower-order conservation equations are formulated forthe physically conserved quantities. Synthetic flow profiles areresolved along the radius r and evolve in time t during a simulation run while the entire ODTdomain is uniformly displacedRin axial direction as x(t) = 0t Ub (t 0 ) dt 0 , where Ub (t) is radially
uniform. Following Echekki et al. (2001); Klein et al. (2019),
but taking into account the cylindrical geometry(Starick et al.
.R
RRR(2019)), Ub = U3 + −R
ρ(u − U3 )2 r dr −R
ρ(u − U3 )r dr.
The elliptic flow problem is, hence, approximated by a parabolized set of conservation equations. Here, we assume a minimal model and resolve only the axial velocity component ux
together with the scalar mass fractions Yi (i = 1, 2, 3) such that


∂ ux
1 ∂
∂ ux
1 dP
+ eddies =
νr
−
,
∂t
r ∂r
∂r
ρ dx


∂Yi
1 ∂
∂Yi
+ eddies =
Γi r
,
∂t
r ∂r
∂r



∂ ux ux 1 ∂ rur ux
1 ∂
∂ ux
+
+ eddies =
νr
,
∂x
r ∂r
r ∂r
∂r


∂ uxYi 1 ∂ rurYi
1 ∂
∂Yi
+
+ eddies =
Γi r
.
∂x
r ∂r
r ∂r
∂r

(3)
(4)

It is worth noting that the radial velocity ur is not resolved directly but induced by the application of physical conservation
laws whenever needed. This is done separately for the deterministic advancement and eddy implementations. In the algorithm, ur 6= 0 manifests itself by radial displacements of the
Lagrangian grid cell interfaces such that the axial mass andmomentum flux is conserved for each Lagrangian cell.
For further technical details, the reader is deferred toLignell et al. (2018). Following their notation, we utilize thetriplet map TMA with eddy-rate evaluation for the radial coordinate. A symmetric center cell of size ∆raxis = 0.15 mm
is used which is by a factor 10 larger (factor 2 smaller) than
the smallest (largest) allowed grid cell size to ensure coupling of the subdomains with r ≷ 0. Grid adaption is performed upon triplet map implementation and enforced after
two diffusive advancement steps independent of eddy event
occurrences. The ODT model parameters have been estimatedby pre-simulations with a small ensemble yielding C = 5,
Z = 400, and βLS = 3.5 for S-ODT (all similar to Lignell et al.
(2018)), whereas βLS = 0.4 for T-ODT (as in Echekki et al.
(2001); Klein et al. (2019)). These parameters are kept constant for the jet simulations shown below.

Generation Of Inflow Conditions
Figure 2 shows low-order statistics and an instantaneous
ODT radial profile of the axial velocity at the nozzle exit. For
RANS, jet inflow conditions are obtained by extending thesimulated flow domain with nozzle walls to negative x. There,
initially uniform flow profiles are prescribed that develop towards the statistically steady-state and exit the nozzle at x/d =0. For ODT, we follow the self-consistent procedure describedin Klein et al. (2019). Mean profiles are consistent with our but
differ from the reference RANS. T-ODT simulations of pipe
(C = 5, Z = 350, maximum eddy size lmax = 2r1 /3) and annular pipe (C = 10, Z = 600, lmax = 2(r3 −r2 )/3) flows are therefore conducted as pre-simulations. A large number of flow
profiles is sampled from the statistically steady state of bothinternal flows in order to generate an N = 50,000 ensembleof perturbed inflow profiles from each of which an independent ODT jet simulation was started after adding the co-flow
and the scalar to the inflow streams. For S-ODT, the inflow
condition has been regularized by truncation of ux (r, 0) withux,min = U3 /2 across the wall regions [r1 , r2 ] and [r3 , r4 ].

(1)
(2)

where the mean pressure gradient dP/dx is only used for thegeneration of turbulent inflow conditions as described below.
An eddy event is implemented by an instantaneous mappingthat moves fluid fromlocation f (r) to mapped location r suchthat ux (r) → ux f (r) and Yi (r) → Yi f (r) . A shortcomingof T-ODT is that it can not capture induced radial momentum3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Joint Probability Density Function
Figure 6 shows two-scalar joint probability density functions (JPDFs) for T-ODT (top row) and S-ODT (bottom row)in terms of the mixture fractions φ1 and φ2 for radial locations
r/d = 0.387, 0.635, 0.992, but fixed axial location x/d = 6.99.
Detailed S-ODT statistics are well within the 99 percentile of
the reference experiments from Cai et al. (2011) satisfactorily
reproducing the shape and mean state at all points across theshear region of the three-stream annular jet. The JPDFs obtained with T-ODT are degraded which is reminiscent of multiscalar mixing in isotropic turbulence (Giddey et al. (2018)).
The T-ODT mean state is consistent with RANS but the agreement of the JPDF is reasonable only for r/d = 0.635. The bulk
velocity Ub is 20.8 m/s (18.0 m/s) for S-ODT (T-ODT) andreached at r/d = 0.660 (0.520). The ensemble-averaged axial
velocity profile hux i(r) (not shown here) is monotonically decreasing with r > 0 for x > 4d, so local advection time scales
at r/d = 0.387 (0.992) are smaller (larger) than that associatedwith the uniform Ub . This explains why S-ODT captures thecomplex mixing better than T-ODT.

Figure 2. Radial profiles of the axial velocity at x/d = 0
across 0 ≤ r/d ≤ 1. Mean (m.) and instantaneous (one realization, o. r.) profiles from ODT, RANS, and reference RANS(RANS-PDF, Rowinski & Pope (2013)) are shown.

Dissipation Of Scalar Fluctuations
Figure 7 shows the dissipation rate of the scalar fluctuation variance hχi i = ∑3j=1 2Γi h(∂ φi /∂ x j )2 i for i = 1, 2 which
is a relevant measure for micromixing and an important quantity for the evaluation of mixing-model fidelity as it is related
to mixing time scales. In ODT, the dissipation rate is approximated as hχi i ≈ 2Γi h(∂ φi /∂ r)2 i. Present results indicate that
only S-ODT is able to yield dissipation rates hχ1 i and hχ2 ithat are close to the reference experiments and a RANS-PDF(IEM) method. T-ODT simulations, by contrast, fail to capturehχ2 i. Diffusive closure models as those used in the reference
LES (or stand-alone RANS) likewise tend to overestimate thedissipation of scalar fluctuations.

RESULTS
Flow Visualization
Figure 3 shows axial-radial snapshots of the jet colored by
the scalar mixture fraction φ2 from the annular region for our
RANS and an S-ODT realization demonstrating that ODT iscapable of capturing the mean state. A T-ODT realization (notshown) gives qualitatively similar results. ODT thus provides
a synthetic flow solution that aims to be statistically representative in terms of mean but contains additional information on
the large and small scale fluctuations.

CONCLUSION
Small-scale resolving numerical simulations of the turbulent mixing of passive scalars in a three-stream concentriccoaxial jet using RANS and the one-dimensional turbulence(ODT) model as stand-alone tool. Both the temporal (T-ODT)
and spatial (S-ODT) model formulations for cylindrical geometry give qualitatively similar results that are consistent withRANS for the mean state and capture details of available reference data. For the spatially developing jet, S-ODT is preferredas it more accurately captures details of the complex mixing
such as the joint probability density function and the dissipation of scalar fluctuations. This property can be attributed
to the model formulation, since S-ODT aims to capture theratio of local advective and diffusive time scales. Based on
the present study, ODT can be more confidentially applied tomultiphysical mixing in free shear flows like reactive jets or
wakes that require small-scale fidelity. Forthcoming research
will build upon the detailed representation of scalar and momentum mixing by utilizing ODT as autonomous microscalemodel within a bottom-up LES approach (Kerstein (2022)).

Low-Order Scalar Statistics At The Axis
Figure 4 shows downstream profiles of the ensemble
mean hφi i and variance hφi02 i, where φi0 = φi − hφi i, for thescalar mixture fractions φ1 and φ2 , respectively, for T-ODT,
S-ODT, our RANS, and reference experiments. RANS is unable to fully capture the axial mean profiles in contrast to
ODT, which agrees with experiments capturing the potential core (x/d . 5), the local fluctuation maximum (within
7 < x/d < 9), and the decay. S-ODT predicts a two times
larger variance maximum. This behavior is likely singular due
to strongly fluctuating axial velocities which may be different
for finite radial distances addressed below.

Radial Profiles Of Low-Order Scalar Statistics
Figure 5 shows radial profiles of the mean hφi i and fluctuation variance hφi02 i of the scalar mixture fraction φi , i = 1, 2,
for T-ODT, S-ODT, our RANS, and reference experiments
form Cai et al. (2011) for x/d = 6.99. ODT results are consistent with RANS for the mean but all three models do not
reproduce low-order statistical moments in the outer region(r/d > 1.2). In the vicinity of the axis (r/d < 0.4), T-ODT
exhibits physically more reasonable fluctuation variances than
S-ODT as it does not take into account local advection timescales. Correspondingly, only S-ODT accurately captures fluctuations across the shear region (0.4 < r/d < 1.2) in which thecomplex mixing takes place.

REFERENCES
Cabra, R., Chen, J.-Y., Dibble, R. W., Karpetis, A. N. & Barlow, R. S. 2005 Lifted methane–air jet flames in a vitiatedcoflow. Combust. Flame 143, 491–506.
Cai, J., Dinger, M. J., Li, W., Carter, C. D., Ryan, M. D. &Tong, C. 2011 Experimental study of three-scalar mixing ina turbulent coaxial jet. J. Fluid Mech. 685, 495–531.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3. Radially and axially truncated axial-radial section of the jet showing the pseudo-colored mixture fraction φ2 for our
RANS (a) and one S-ODT realization (b). Black vertical lines with caps denote discrete eddy events. Gray regions and dotted lines areused for detailed comparison with available reference data.

Figure 4. Downstream profiles of low-order scalar statistics at the jet axis (r/d = 0) for our RANS, ODT, and reference experiments(EXP) from Cai et al. (2011) showing the mean (a, c) and fluctuation variance (b, d) of the scalar mixture fractions φ1 and φ2 .

Figure 5. Radial profiles of low-order scalar statistics at x/d = 6.99 for our RANS, ODT, and reference experiments (EXP). Labels
and line styles are the same as in figure 4. In addition, vertical lines give the locations r1 –r4 of the nozzle walls for orientation.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
1.0

13a) T-ODT
x/d=6.99
r/d=0.387

0.8

9

φ2

0.6
0.4
0.2
0.0
1.0
0.8

φ2
0.2
0.0
0.0

5

5

5

3

3

3

1

1

1

13

e) S-ODT
x/d=6.99
r/d=0.635

0.4

φ1

EXP contour

0.6

0.8

13

f) S-ODT
x/d=6.99
r/d=0.992

11
9

11
9

7

7

7

5

5

5

3

3

3

1
0.2

9
7

9

0.4

9

11

7

11

0.6

13

c) T-ODT
x/d=6.99
r/d=0.992

11

7

13

d) S-ODT
x/d=6.99
r/d=0.387

13b) T-ODT
x/d=6.99
r/d=0.635

11

1.0

1
0.0

0.2

EXP mean

0.4

φ1

0.6

0.8

RANS mean

1.0

1
0.0

0.2

S-ODT mean

0.4

φ1

0.6

0.8

1.0

T-ODT mean

Figure 6. Scalar joint probability density functions (JPDFs) for mixture fractions φ1 and φ2 at downstream location x/d = 6.99 andradial locations r/d = 0.387 (a, d), r/d = 0.635 (b, e), and r/d = 0.992 (c, f) from N = 50,000 ODT realizations. Different color
scales visually distinguish T-ODT (a–c) from S-ODT (d–f). The dashed contour gives the 99 percentile of the experimentally measured
JPDF (Cai et al. (2011)). Open symbols denote the mean states for ODT, experiments, and our RANS. The area above the diagonal isinaccessible for passive scalars in constant-density flow.

Giddey, V., Meyer, D. W. & Jenny, P. 2018 Modeling threedimensional scalar mixing with forced one-dimensional turbulence. Phys. Fluids 30, 125103.
Kerstein, A. R. 1999 One-dimensional turbulence: Model formulation and application to homogeneous turbulence, shear
flows, and buoyant stratified flows. J. Fluid Mech. 392, 277–
334.
Kerstein, A. R. 2022 Reduced numerical modeling of turbulent
flow with fully resolved time advancement. Part 1. Theory
and physical interpretation. Fluids 7 (2), 76.
Klein, M., Zenker, C. & Schmidt, H. 2019 Small-scale resolving simulations of turbulent mixing in confined jets using
one-dimensional turbulence. Chem. Eng. Sci. 204, 186–202.
Li, W., Yuan, M., Carter, C. D. & Tong, C. 2021 Investigation of three-scalar subgrid-scale mixing in turbulent coaxial jets. J. Fluid Mech. 924, A40.
Lignell, D. O., Lansinger, V., Medina, J., Klein, M., Kerstein,
A. R., Schmidt, H., Fistler, M. & Oevermann, M. 2018 Onedimensional turbulence modeling for cylindrical and spherical flows: Model formulation and application. Theor. Comp.
Fluid Dyn. 32 (4), 495–520.
Pitsch, H. & Steiner, H. 2000 Large-eddy simulation of a turbulent piloted methane/air diffusion flame (Sandia flame D).
Phys. Fluids 12 (10), 2541.
Rowinski, D. H. & Pope, S. B. 2013 An investigation of mixing in a three-stream turbulent jet. Phys. Fluids 25 (10),105105.
Starick, T., Schmidt, H. & Lignell, D. O. 2019 Onedimensional turbulence modelling of a lifted methane/air jet
flame in a vitiated coflow. Proc. 11th Int. Symp. Turb. Shear
Flow Phen. (TSFP11) 1, 8C, ID 278.
Walter, M., Kornev, N. & Hassel, E. 2015 Large eddy
simulation of turbulent reactive mixing at high Schmidt
and Reynolds numbers. Chem. Eng. Technol. 38 (9),1608–1616.

Figure 7. Radial profiles of the dissipation rates hχ1 i (a) andhχ2 i (b) of the scalar fluctuation variance at x/d = 6.99. Reference experiments are from Cai et al. (2011). RANS-PDF andLES reference results are from Rowinski & Pope (2013).

Dowling, D. R. & Dimotakis, P. E. 1990 Similarity of the concentration field of gas-phase turbulent jets. J. Fluid Mech.
218, 109–141.
Echekki, T., Kerstein, A. R., Dreeben, T. D. & Chen, J.-Y.
2001 ‘One-dimensional turbulence’ simulation of turbulent
jet diffusion flames: Model formulation and illustrative applications. Combust. Flame 125, 1083–1105.

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