Journal of Turbulence

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Journal of Turbulence
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Gradient trajectory analysis in a
Jet flow for turbulent combustion
modelling
M. Gampert
a
, P. Schaefer
a
, V. Narayanaswamy
a
& N. Peters
a
a
Institut für Technische Verbrennung, RWTH-Aachen University,
Templergraben 64, Aachen, Germany
Version of record first published: 14 Mar 2013.
To cite this article: M. Gampert , P. Schaefer , V. Narayanaswamy & N. Peters (2013): Gradient
trajectory analysis in a Jet flow for turbulent combustion modelling, Journal of Turbulence, 14:1,
147-164
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Journal of Turbulence,2013
Vol.14,No.1,147–164,http://dx.doi.org/14685248.2012.747688
Gradient trajectory analysis in a Jet flow for turbulent combustion
modelling
M.Gampert

,P.Schaefer,V.Narayanaswamy and N.Peters
Institut f
¨
ur Technische Verbrennung,RWTH-Aachen University,Templergraben 64,
Aachen,Germany
(Received 29 August 2012;final version received 2 November 2012)
Based on planar high-speed Rayleigh scattering measurements of the mixture fraction
Z of propane discharging from a turbulent round jet into co-flowing carbon dioxide at
nozzle-based Reynolds numbers Re
0
= 3000–8600,we use scalar gradient trajectories
to investigate the local structure of the turbulent scalar field with a focus on the scalar
turbulent/non-turbulent interface.The latter is located between the fully turbulent part
of the jet and the outer flow.Using scalar gradient trajectories,we partition the turbulent
scalar field into these three regions according to an approach developed by Mellado et al.
(J.P.Mellado,L.Wang,and N.Peters,Gradient trajectory analysis of a scalar field with
external intermittency,J.Fluid Mech.626 (2009),pp.333–365.).Based on these differ-
ent regions,we investigate in a next step zonal statistics of the scalar probability density
function (pdf) P(Z) as well as the scalar difference along the trajectory Zand its mean
scalar value Z
m
,where the latter two quantities are used to parameterize the scalar profile
along gradient trajectories.We showthat the scalar pdf P(Z) can be reconstructed from
zonal gradient trajectory statistics of the joint pdf P(Z
m
,Z).Furthermore,on the one
hand we relate our results for the scalar turbulent/non-turbulent interface to the findings
made in other experimental and numerical studies of the turbulent/non-turbulent inter-
face,and on the other hand discuss themin the context of the flamelet approach and the
modelling of pdfs in turbulent non-premixed combustion.Finally,we compare the zonal
statistics for P(Z) with the composite model of Effelsberg and Peters (E.Effelsberg and
N.Peters,A composite model for the conserved scalar pdf,Combust.Flame 50 (1983),
pp.351–360) and observe a very good qualitative and quantitative agreement.
Keywords:jet flow;gradient trajectory;turbulent/non-turbulent interface;scalar pdf;
flamelet theory
1.Introduction
Turbulent mixing is a subject of immense interest owing to its occurrence in numerous
engineering applications,which involve the mixing of a scalar in a turbulent flow field.In
a two-feed system,the state of mixing can be uniquely defined by a parameter called the
mixture fraction Z,which is defined as the mass fraction of fuel streamin a given fuel–air
mixture,
Z =
m
f
m
f
+m
air
,(1)

Corresponding author.Email:m.gampert@itv.rwth-aachen.de
C

2013 Taylor &Francis
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148 M.Gampert et al.
where the subscripts f and “air” refer to fuel streamand air,respectively.According to this
definition,Z varies between Z = 0 and Z = 1.The mixture fraction and the associated
scalar dissipation rate χ,which is defined as
χ = 2D
￿
∂Z
∂x
i
￿
2
,(2)
where Ddenotes the molecular diffusivity and repeated indices i imply summation over the
three spatial dimensions,are very important parameters in non-premixed combustion;for
instance,these describe the turbulent flame structure on the basis of the laminar flamelet
theory [1,2].Combustion occurs when the fuel mass fraction in a fuel/oxidizer mixture
reaches a particular value,called the stoichiometric mixture fraction Z
st
.The value of Z
st
is
about 0.06 for different hydrocarbon/air mixtures;for instance,for a propane–air mixture,
Z
st
= 0.06095.Owing to the low values of stoichiometric mixture fraction,combustion
occurs at the outer boundary of a turbulent fuel jet,which is characterized by turbulent
regions adjacent to non-turbulent ones.When experimental investigations are performed,
for instance,at the edge of a turbulent jet flow,the signal varies abruptly between a turbulent
and a non-turbulent character for measurements of scalar quantities.Corrsin and Kistler [3]
first termed the layer at the outer edge of this turbulent/non-turbulent (T/NT) interface
the laminar superlayer.We are considering a scalar quantity only so that we will refer
to the region in which the scalar signal changes from a turbulent to a laminar character in
the following as the scalar T/NT interface to point out that all analyses are conducted in a
scalar field.At the outer edge of this scalar T/NT interface a layer is present that will in the
following be termed diffusive scalar sublayer in analogy to [3].
Detailed spatial analyses of this region have been carried out experimentally (e.g.[4–7])
and numerically (e.g.[8–10]),see [11] for a review of recent investigations,giving deeper
insight into the vorticity dynamics close to the T/NT interface.In addition,Westerweel
et al.[12] examined the temperature field of a non-isothermal jet and observed a good
agreement of statistics with the ones obtained from the investigation of concentration and
axial momentum[5,7,8,13,14] so that the findings presented in the following are considered
to also apply to the fields of velocity and vorticity.
In a previous work,cf.[15],the contribution of the T/NT interface to the mixture
fraction probability density function (pdf) P(Z) at various axial and radial locations has
been examined,and the composite model proposed in [16] for the mixture fraction pdf in
non-premixed combustion has been used.Thereby,the structure of the scalar T/NTinterface
in this free shear flowhas been identified and it was concluded that the T/NT interface and
its contributions to the mixture fraction pdf are of major importance,particularly in the
early part of the jet.Then statistics such as the pdf of the location of the T/NT interface and
the scalar profile across the latter were investigated and found to be in good agreement with
literature data,cf.[7].In addition,the scaling of the thickness δ of the scalar T/NT interface
was analyzed at Reynolds numbers,Re
λ
= 60–140,where Re
λ
denotes the local Reynolds
number based on the Taylor scale λ,using the mixture fraction profile in interface normal
direction.It was observed that δ/L ∝ Re
−1
λ
,where L is an integral length scale,meaning
that δ ∝ λ – a finding that is in good agreement with dimensional scaling arguments
postulated by da Silva and Taveira [9].
The region of the T/NT interface was recently further analyzed by Mellado et al.[17].
In this work,the latter authors investigate the Direct Numerical Simulation (DNS) of a
temporally evolving shear layer using gradient trajectories.Mellado et al.[17] applied
this analysis to partition the scalar field into a fully turbulent zone,a zone containing
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Journal of Turbulence 149
the T/NT interface and the outer laminar flow.Based on different regions,they examined
the probability of these three zones at different locations in the shear layer and investigated
the scalar pdf and the conditional scalar dissipation rate in the zones in the presence of
external intermittency.
To this end,scalar gradient trajectories are calculated fromeach grid point in ascending
and descending directions until a local extremum is reached,at which the scalar gradi-
ent vanishes and the Hessian is either positive-definite (minimum) or negative-definite
(maximum).We define the local direction n of a gradient trajectory as
n =
∇Z
| ∇Z |
,(3)
whichis followeduntil the trajectoryhits anextreme point.The projectedvelocitydifference
u
n
,with
u
n
= u
+
· n
+
−u

· n

,(4)
along such a trajectory,where the superscripts denote the location at the maximum and
minimum points,respectively,as well as between its bounding extreme points has been
investigated for instance by Wang [18] and Gampert et al.[19,20],where a linear scaling
u
n
| s ∝ s · a

has been found to be universally valid,where s denotes the separation
arc length along a gradient trajectory and a

is the asymptotic value of the conditional
mean strain rate acting on long gradient trajectories.
Peters [21] related such a gradient trajectory formulation in the mixture fraction field
to the flamelet theory [1] in non-premixed combustion.The latter is a detailed multiscale
approach in combustion modelling,which employs a non-equilibriumformulation in a thin
layer in the vicinity of stoichiometric mixture Z
st
as a microscale model.This microscale
model,called flamelet equations,introduces the mixture fraction Z as an independent
coordinate.Such flamelets often exist in the scalar T/NT interface,as the stoichiometric
mixture is also frequently located here.Though the mixture fraction is a conserved scalar
that represents the element mass fraction,Peters [21] shows that it may also be interpreted
as a scaled coordinate along gradient trajectories,
∂Z
∂n
=
￿
χ
Z
2D
￿
1/2
,(5)
where ∂/∂n = n · ∇ and χ
Z
is the instantaneous,fluctuating scalar dissipation rate that is
evaluated at a mixture fraction isoline.Based on this ansatz,Peters and Wang [22] derived
a balance equation for the instantaneous scalar dissipation rate as a function of the mixture
fraction.
The present study continues the investigation of the scalar T/NT interface in a turbulent
jet flow and analyses the scalar pdf conditioned on different regions of the flow field using
gradient trajectories.To this end,we perform high-frequency planar Rayleigh scattering
measurements of pure propane C
3
H
8
discharging froma free round jet into co-flowing pure
CO
2
.In this case,the local mass fraction Y
C
3
H
8
of propane is equal to the mixture fraction Z,
Z = Y
C
3
H
8
=
X
C
3
H
8
W
C
3
H
8
X
C
3
H
8
(W
C
3
H
8
−W
CO
2
) +W
CO
2
.(6)
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150 M.Gampert et al.
In Equation (6),X
C
3
H
8
is the propane mole fraction and W
C
3
H
8
and W
CO
2
are the molecular
weights of propane and CO
2
,respectively.In Section 2,we describe the experimental
arrangement together with the calculation of scalar gradient trajectories,present the results
for the zonal statistics in Section 3 and those for the composite model in Section 4 and
finally conclude our paper in Section 5.
2.Experiment
The measurement technique and the post-processing data have been described and validated
in companion papers,cf.[15,23] so that only a brief description is given in the following.
The experiments were performed in a co-flowing turbulent jet facility.Figure 1 shows
the schematic of experimental setup.The facility consists of a center tube made of steel
with an inner diameter of 12 mm.The surrounding co-flow tube had a diameter of 150
mm and a length of 450 mm,which was large enough to reduce the experimental setup
to a two-stream problem.In the latter tube a honeycomb is installed in the lower third to
guarantee a uniformvelocity profile.
Research grade propane (99.95% pure) was fed through the center tube using a flow
controller (OMEGA FMA-2600A) at various flow rates to achieve the desired Reynolds
number.The co-flow gas was chosen as CO
2
owing to its larger Rayleigh cross-section,
which was necessary to obtain an accurate determination of scalar gradient trajectories.
For different experimental runs,the mean velocity of CO
2
was 0.05 m/s.Table 1 shows the
flowconfigurations of all experimental runs made in this work.The Reynolds number Re
D
based on jet exit conditions was varied between 3000 and 8610 and the corresponding jet
exit velocity was between 1.15 m/s and 3.3 m/s.Furthermore,Re
λ
(= u
rms
λ
u

Cl
) is the
local Taylor-based Reynolds number on the center line.For the calculation of this quantity,
u
rms
has been measured using Laser Doppler Anemometry (LDA),the kinematic viscosity
on the centerline ν
Cl
has been determined using the local concentration of two gases,while
the Taylor length λ
u
(= (15 u
rms
ν
Cl
/ε)
1/2
) of the velocity field has been computed using an
approximation formula to estimate ε taken from [24].The latter has also been employed
for the calculation of the Kolmogorov scale,η (=(ν
3
/ε)
1/4
).Although these length scales
are consequently not calculated directly from the measurements,and therefore have to be
treated with caution,previous results suggest a quite accurate prediction,cf.[15,23] so that
these will be used in the following to interpret the data.
The objective of the experiments was to obtain temporally and spatially highly re-
solvedscalar fieldimages,whichwere acquiredusinghigh-speedtwo-dimensional Rayleigh
Figure 1.Experimental setup for high-speed Rayleigh scattering measurements.
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Journal of Turbulence 151
Table 1.Experimental parameters.
x/d 10 15 20 30
Jet exit velocity,U
0
(m/s) 1.15 1.76 1.82 3.30
Mean centerline velocity,U
Cl
(m/s) 0.57 0.61 0.50 0.62
Mean centerline mixture fraction,

Cl
0.38 0.24 0.18 0.13
Kolmogorov scale,η (mm) 0.18 0.20 0.26 0.24
Velocity Taylor scale,λ
u
(mm) 2.30 3.32 4.26 4.61
Centerline viscosity,ν
Cl
(mm
2
/s) 6.50 6.95 7.40 7.50
Nozzle-based Reynolds number,Re
0
3000 4500 4750 8610
Taylor-based Reynolds number,Re
λ
61 72 71 96
scattering imaging.Laser-Rayleigh scattering is used to determine the instantaneous mass
fraction of the binary mixture of jet and reservoir gas in a small focal plane within the
turbulent core of the jet around the center line.
For the Rayleighscatteringimaging,twofrequency-doubledbeams (λ
f
= 527nm) from
a high-frequency dual-head Nd:YLF laser (Litron Lasers LDY303HE-PIV) were combined
to deliver the energy of about 32 mJ/p at 1 kHz (32 W).To account for energy fluctuations,
the signal is corrected on a shot-by-shot basis by a 12 bit energy monitor (LaVision
Online Energy Monitor).The polarization of both beams was normal to the jet axis which
maximized the Rayleigh scattering signals fromthe radial-azimuthal plane.The beams were
transformed into a horizontal collimated sheet using a combination of a Galilean telescope
(expansion ratio of 1.5) and a cylindrical lens.The width and the thickness (FWHM) of
the resultant sheet were approximately 10 mm and 0.3 mm,respectively.Images were
acquired at 1 kHz using a high-speed CMOS camera (LaVision HighSpeedStar 6) fitted
with a camera lens (Nikon f.l.= 85 mm) stopped at f/1.4.An extension ring of 20 mm
length was placed between the camera and the lens to minimize the working distance;the
resulting field of view is about 60 ×60 mm.The signal-to noise ratio (SNR) in the pure
propane region of rawimages was over 40.The time interval between the successive images
was 1 ms,while the in-plane resolution was 100 microns.This resolution is of the order
of the Kolmogorov scale η,thus allowing a detailed investigation of gradient trajectories.
The images were recorded using commercial software (LaVision 7.2) and were further
processed using computer codes written in-house.
The major sources of systematic uncertainties are the departure from linearity of the
camera response and the presence of noise in both propane and CO
2
streams.The departure
from the linearity of the camera response is within 4%,as quoted by the manufacturer.
The image noise was minimized using an optimal filter designed for the propane streamso
that the scalar spectrumfollows Pope’s model [25].However,there is residual noise left in
the CO
2
after applying the optimal filter,which induces uncertainty in the calculation of
scalar gradients.The combined uncertainty arising from these sources is estimated to be
below 5%.Based on these corrected images,the signals corresponding to pure CO
2
and
pure propane,respectively,are calibrated and used to convert the recorded photon counts to
propane mass fraction.The impact of post-processing is illustrated in Figure 2,where (a) a
sample raw image recorded at x/d = 20 and (b) the corresponding post-processed image
are shown.
In the next step,the recorded time series of the plane at a fixed downstream location
x/d is transformed into a spatial signal in streamwise direction with x = U · t based
on Taylor’s hypothesis,see [26].Hence,we obtain a frozen three-dimensional mixture
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152 M.Gampert et al.
Figure 2.(a) Examplary raw and (b) post-processed images obtained at x/d = 20.
fraction field.Due to the importance of two-point statistics and spatial gradient quantities
inturbulence,it is commontouse Taylor’s hypothesis toestimate spatial derivatives [27–29].
The camera resolves a plane of 1024 ×1024 pixels at a frequency of 1 kHz.This
bounds the jet exit velocity U
0
to a value,at which the resolution in the x-direction remains
at the order of the Kolmogorov scale at the various downstreampositions x/d.Finally,the
mean concentration
on the center line is given.Note that the analysis of the mixture
fraction volume has been restricted in radial direction to
˜
r(= r/(x −x
0
)) < 0.1,where
x
0
denotes the virtual origin of the jet and has been found to be x
0
/d = −1.75,cf.[23],
and is performed at each axial location using three statistically independent sets of 5400
consecutive images.
3.Zonal statistics based on gradient trajectories
As described in Section 1,we will examine in the following the mixture fraction fields
based on the procedure developed by Mellado et al.[17] using gradient trajectories.To this
end,the flowis partitioned into three different regions using the statistics of scalar gradient
trajectories – namely a fully turbulent zone,an outer flow region and embedded within
these two the scalar T/NT interface.
In the following,gradient trajectories together with scalar minimum and maximum
points are used to detect the different regions of the scalar field,see Figure 3 for an illustra-
tion:If a gradient trajectory associated with one specific grid point connects one minimum
and one maximum point,this point is considered to be inside the fully turbulent zone.
Figure 3.Flow partitioning based on gradient trajectories:A:Trajectory from minimum to maxi-
mum,fully turbulent zone;B:from upper stream to maximum,upper turbulent interface;C:from
minimumto lower stream,lower turbulent interface (Band Cwill be called the scalar T/NTinterface);
D:fromupper streamto lower stream,quasi-laminar diffusion layers (figure taken from[17]).
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Journal of Turbulence 153
On the contrary,if the trajectory connects a maximum with the outer stream,where the
mixture fraction Z = 0,then that point belongs to the scalar T/NT interface.In addition,
the trajectory might theoretically proceed through the studied flowregion without any inter-
mediate extreme point,thus defining the so-called quasi-laminar diffusion layer.However,
this effect is not observed in the present study.Finally,all points whose trajectories do not
reach an extreme point are considered to be in the outer flow.The approach has already
been successfully applied in [23],where it has been used to study the profiles of the zonal
probability of three different regions over the non-dimensional radial coordinate
˜
r.It has
been found that for
˜
r < 0.08 the scalar field is dominated by the fully turbulent region
with an increasing contribution of the scalar T/NT interface,which is already present on
the centerline with a zonal probability of approximately 0.05.For
˜
r > 0.18 in contrast,
it is most probable to find the outer flow,whose contribution is not negligible starting at
˜
r < 0.06.In-between,however,the structure of the scalar field is mainly dominated by
the scalar T/NT interface.In the present study,we will investigate in particular the zonal
statistics of the scalar pdf P(Z) in different regions in more detail.
Let us note that this partitioning is based on non-local information,as the grid points at
a given radial distance fromthe centerline with a scalar value between the free-streamand
the centerline value might belong to the scalar T/NT interface and the distinction is only
possible by following the corresponding trajectory.Mellado et al.[17] showed that this
non-local approach allows detecting engulfed regions,which is not possible if the interface
definition is based on a single-valued envelope surface.However,an outer limit to the
interface is also set by a threshold in the magnitude of the scalar gradient,belowwhich the
scalar is approximately a homogenous field with the outer flow value Z = 0.This second
criterion defines the conventional intermittency function and separates the non-turbulent
zones fromthe scalar T/NT interface.
As an example,in Figure 4 trajectories within the fully turbulent zone are shown to
share a minimumpoint and reach eight different maximumpoints.Obviously,the resulting
gradient trajectories are of strongly varying shape and intertwisted nature.
This differentiation between the outer non-turbulent zones and the T/NT interface has
been introduced for several reasons.First,it is needed from the numerical point of view
because the gradient approaches zero as one moves toward the outer homogeneous region
so that below a threshold there is only noise,and the gradient direction is numerically
undetermined.Second,this distinction is the conventional one used to define the intermit-
tency factor and can be used to compare with traditional results using only this quantity.
Finally,it is also useful to simplify possible models,since the pdf of the scalar field in these
non-turbulent regions is just a delta function at the outer flow value Z = 0 and the scalar
dissipation can be approximated by zero.In summary,a point at a given distance r from
the centerline can be a part of the non-turbulent outer flow,belonging to the scalar T/NT
interface or be located within a turbulent region.
In a first step,we calculate the extreme points in the experimentally obtained three-
dimensional mixture fraction field as well as the corresponding gradient trajectories using
the same numerical procedures already applied,for instance,in [19,20,30] so that we
can afterwards define different regions in the scalar field.Figure 5 shows representative
mixture fraction fields measured at x/D = 10.One clearly observes the following three
different regions in the figure:A:the fully turbulent part of the flow,B:the scalar T/NT
interface,where the value of the mixture fraction drops from the turbulent to the outer
flow value (Z = 0) and C:the co-flow,which,by definition,corresponds to Z = 0 (note
that in the co-flow region (C),the measured mixture fraction value fluctuates between
Z = 0 and Z = 0.03,whichis causedbythe residual noise that is left after data-processing).
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154 M.Gampert et al.
Figure 4.Example of gradient trajectories in the turbulent zone based on the mixture fraction field
Z obtained at x/d = 20 and Re
0
= 4750.All trajectories share the same minimumpoint and connect
it with eight different maximum points.The scalar value increases from minimum point (blue) to
maximumpoint (red).
In some of the instantaneous realizations,we observe the presence of multiple T/NT
interfaces due to the separation of turbulent eddies from the main flow as illustrated in
Figure 5(b).These occasional patches are disconnected from the main jet body due to the
breaking away of vortical eddies – a physical process that is called detrainment.However,
for free shear flows detrainment is a small effect,as such patches are usually re-entrained
within a few eddy time scales,cf.[31].
The parametrization of the scalar field along a gradient trajectory is necessary for
statistical investigation.To this end,Peters and Trouillet [32] propose to use the arithmetic
Figure 5.Sample instantaneous mixture fraction fields Z obtained at x/d = 10 and Re
0
= 3000
illustrating different regions of the flow field – A:inner turbulent flow,B:turbulent/non-turbulent
interface and C:outer co-flow:(a) Instantaneous mixture fraction field with a single interface,and
(b) instantaneous mixture fraction field with multiple turbulent/non-turbulent interfaces.
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Journal of Turbulence 155
Figure 6.Isocontour lines of jpdf P(Z
m
,Z) for the whole domain within
˜
r < 0.1 obtained at
x/d = 15.
mean Z
m
of the minimumand maximumvalues of extremal points that bound the gradient
trajectory,
Z
m
=
Z
max
+Z
min
2
(7)
as well as the scalar difference Z,where
Z = Z
max
−Z
min
.(8)
Based on these quantities,the latter authors study the role of quasi-one-dimensional dis-
sipation layers (Q1DLs) in turbulent scalar mixing.To this end,they replace the scalar
Z by Z
m
and Z as independent variables based on which they,on the one hand,try to
reconstruct the profiles of a scalar and its dissipation rate along gradient trajectories,and
on the other hand formulate an equation for the joint probability distribution function (jpdf)
P(Z
m
,Z) following the findings of O’Brien [33],who considered the pdf of a reactive–
diffusive scalar.Computing for each grid point in the experimentally obtained mixture
fraction field within
˜
r < 0.1 the corresponding gradient trajectories,we can in a first step
calculate P(Z
m
,Z) for the overall domain.As shown in Figure 6,we obtain a triangular
shaped jpdf,which has a distinct maximum at around Z
m
= 0.30 and Z = 0.15.The
theoretical boundaries of this jpdf are given by Z
m
= Z/2 and Z
m
+Z/2 = 0.6,as
Z = 0.6 is the maximummixture fraction value.
Based on the method described above,we distinguish between this jpdf the fully turbu-
lent P
t
(Z
m
,Z) and the interface P
s
(Z
m
,Z) regions.The resulting jpdfs are shown in
Figure 7(a) for the scalar T/NT interface and in Figure 7(b) for the turbulent region.We ob-
serve in Figure 7(a) that all values of this jpdf are concentrated around Z
m
= Z/2,as the
criterion for the interface detection is that the trajectory connects the outer flow (Z = 0)
with a maximumpoint so that Z
m
= (Z
max
+Z
min
)/2 = Z
max
/2 and Z = Z
max
−Z
min
=
Z
max
.However,thoughthe jpdf inthe T/NTinterface shouldonlybe representedbya line de-
fined by Z
m
= Z/2,it has a thin distinct width,which is limited by the above-mentioned
residual noise level after post-processing.Furthermore,it has a maximal probability at
Z
m
= 0.16 and Z = 0.32 fromwhich we find a decreasing probability toward the origin
and the longest trajectories for which Z
m
= 0.29 and Z = 0.58.FromFigure 7(b),which
shows the jpdf P
t
(Z
m
,Z) in the fully turbulent region,it is obvious that the rest of the
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156 M.Gampert et al.
Figure 7.Isocontour lines of the jpdfs (a) P
s
(Z
m
,Z) for the the T/NT interface region,and (b)
P
t
(Z
m
,Z) for the fully turbulent part of the domain within
˜
r < 0.1 obtained at x/d = 15.
total jpdf (cf.Figure 6) corresponds to the turbulent part.As a total jpdf,it has a triangular
shape with a maximumat around Z
m
= 0.1 −0.2 and Z = 0.30 with a cropped off lower
edge due to missing T/NT interface contributions.Naturally,the jpdf in the outer flow has
a peak at Z
m
= Z = 0.
In Figure 8,the marginal pdfs for (a) Z
m
and (b) Z are shown.For P
t
(Z
m
) in the
turbulent zone,we find an almost Gaussian bell-shaped pdf with a mean of approximately
Z
m
= 0.25,where also the maximum of the pdf is located.Furthermore,P
t
(Z
m
) ranges
fromZ
m
= 0 to 0.5 in contrast to P
s
(Z
m
) in the scalar T/NT interface.The latter marginal
pdf is zero for mixture fraction values larger than Z
m
= 0.28,a finding that is expected,as
in this region of the field the mixture fraction decreases fromits turbulent to the outer flow
value so that even theoretically only a maximal value of Z
m
= 0.5 is possible.
We observe in Figure 8(b) that while the turbulent pdf P
t
(Z) starts at Z
t
= 0 and
ranges up to Z
t
= 0.31,the pdf conditioned on the T/NT interface has its lowest value at
Z
s
= 0.06 and ranges up to Z
s
= 0.56.This finding translates directly to the observed
jump of the scalar value across the T/NT interface,cf.[7,15].In addition,it is evident from
Figure 8(b) that the region of stoichiometric mixture is frequently located within the scalar
T/NT interface – a finding that is in good qualitative agreement with the results of Mellado
et al.[17];quantitative differences of the actual values of P(Z) and the range of Z-values
originate in the different flow setup that is considered by the latter authors.
As the T/NT interface starts in the outer flowwith Z = 0 and ranges up to Z = Z
s
=
0.56,the value of the stoichiometric mixture for propane and air Z
st
= 0.06095 is always
located along such a gradient trajectory.Furthermore,we notice that the shape of P
s
(Z)
in the interface qualitatively strongly resembles P
s
(Z
m
),which is due to the definition of
Figure 8.Marginal pdfs (a) P(Z
m
),and (b) P(Z) for the fully turbulent part and the T/NTinterface
obtained at x/d = 15.
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Journal of Turbulence 157
the T/NT interface region based on trajectories.As the latter starts in the outer flow at
Z = 0,we obtain Z = Z
max
and Z
m
= Z
max
/2 so that we find the same shape for the
pdfs of Z and Z
m
,which for Z
m
is just shifted to smaller values and rescaled to an area of
unity.In contrast,the scalar difference between the extreme points of gradient trajectories
in the fully turbulent zone is much smaller with a mean of Z = 0.1 so that this region is
characterized by many small fluctuations in the scalar field.
Based on the jpdf of these two parameters (cf.Figures 6 and 7),we will in the following
investigate the scalar pdf P(Z) in the different regions of the field and show that it can be
reproduced from P(Z
m
,Z) obtained by the gradient trajectory statistics.In a first step,
we therefore write the overall pdf in terms of its different contributions [15–17],
P(Z) = γ[(1 −s)P
t
(Z) +sP
s
(Z)] +(1 −γ)P
o
(Z),(9)
where γ is the intermittency factor defined as the fraction of the signal that is not due to the
outer flow and s is the interface contribution,given by the fraction of the T/NT interface
within the remaining part.Furthermore,P
t
(Z) is the scalar pdf in the fully turbulent zone,
P
s
(Z) denotes the pdf stemming from the T/NT interface and P
o
(Z) is the scalar pdf in
the outer flow,which by definition is a delta peak at zero.Each of these pdfs as well as
the overall one can be calculated purely fromthe jpdf of the introduced gradient trajectory
parameters by Peters and Trouillet [32],
P
i
(Z) =
￿
1
0
￿
Z
max
0
P
l
(Z;Z
m
,Z)P
i
(Z
m
,Z)dZdZ
m
.(10)
In Equation (10),P
i
(Z
m
,Z) is the zonal jpdf of Z
m
and Z,while P
l
(Z) is the local
distribution function of Z within the gradient trajectory of length l.However,as the latter
is unknown,we follow the approximation of Peters and Trouillet [32],who assume a sine
0
0.2
0.4
0.6
0.8
1
−0.5
0
0.5
˜s
=
s/l
(Z−Z
m
)/ΔZ


Exp (x/d=15)
Model
Figure 9.Comparison of the model ansatz from Equation (11) for the mean scalar profile along a
gradient trajectory together with the experimental data obtained at x/d = 15.
Downloaded by [Stanford University] at 13:22 14 March 2013

158 M.Gampert et al.
function for the monotonic profile of Z,yielding
Z = Z
m
+
Z
2
sin(π
˜
s −π/2),(11)
where
˜
s = s/l is a normalized coordinate along the trajectory,which increases linearly from
zero to one.This ansatz for the scalar profile (solid line) together with the experimental data
averaged over all trajectories (diamond markers) are exemplary shown for x/d = 15 over
the normalized coordinate
˜
s in Figure 9.In this figure,we observe a very good agreement
of the model curve with the experimental data with only marginal deviations around the
inflection point.
Using such a scalar profile,we obtain [34]
P
l
(Z;Z
m
,Z) =
P(s)
| ∂Z/∂s |
(12)
that can be computed for each combination (Z
m
,Z) following [32]
P
l
(Z;Z
m
,Z) =
π
−1
| (Z −Z
m
+Z/2)
1/2
(Z
m
+Z/2 −Z)
1/2
|
.(13)
Introducing Equation (12) in Equation (10) together with the jpdfs shown in Figures 6
and 7 allows us to reconstruct the scalar pdfs within the different zones of the scalar
field;see Figure 10 calculated from the data obtained at x/d = 15.We observe for the
experimentally obtained pdf (open circles) a bimodal shape with a maximumat Z = 0.23,
a non-zero value at the origin of P(Z = 0) close to unity and and intermittency factor
γ = 0.995.In addition,the scalar pdfs computed separately for the turbulent and the
interface parts are shown weighted by its respective prefactors according to Equation (9)
together with the reconstructed overall pdf,which is also calculated according to Equation
(9) (note that the pdf of the outer flow region P
o
is not shown in Figure 10 as it is only
a delta peak at the origin).Nevertheless,it is included in the total pdf to which it only
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.5
1
1.5
2
2.5
3
3.5
4
Z
P (Z)


Exp (x/d=15)
Total pdf
Turbulent zone
T/NT Interface
Figure 10.Comparison of the measured pdf P(Z) obtained at x/d = 15 with the one calculated
fromEquation (9).In addition,the weighted zonal pdfs of the fully turbulent γ[(1 −s)P
t
(Z)] and the
scalar T/NT interface γ[sP
s
(Z)] are shown.
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Journal of Turbulence 159
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
1
2
3
4
5
Z
P (Z)


Exp (x/d=30)
Total pdf
Turbulent zone
T/NT interface
Figure 11.Comparison of the measured pdf P(Z) obtained at x/d = 30 with the one calculated
fromEquation (9).In addition,the weighted zonal pdfs of the fully turbulent γ[(1 −s)P
t
(Z)] and the
scalar T/NT interface γ[sP
s
(Z)] are shown.
has a very small contribution (1 −γ = 0.005).We observe a very good qualitative and
quantitative agreement of the reconstructed pdf with the measured one,further validating
that the ansatz (Equation (11)) is in close agreement with the real scalar profile along the
tajectories.Furthermore,we can attribute specific parts of the measured pdf to different
flowregions.In the fully turbulent part,the pdf is close to a Gaussian shape with a value of
P(Z = 0) = 0 at the origin and a maximum at Z = 0.23,the location of which coincides
with the one of the total pdf.In contrast,the pdf of the T/NTinterface recovers the non-zero
value of the measured pdf at the origin,which is not due to contributions fromthe outer flow
only.In addition,it has a non-zero value over the whole range of mixture fraction values
with a small maximumat approximately Z = 0.33.This may be explained by the fact that
the T/NT interface contains not only the T/NT interface itself but also the adjacent regions
up to the first maximumpoint.Finally,its contribution s calculated fromthe fraction of the
interface region within the non-outer flow part is s = 0.107.
The same trends described for Figure 10 can also be observed in Figure 11,where the
scalar pdfs within the different zones of the scalar field are shown,which are computed
from the data obtained at x/d = 30.For the experimental pdf again a bimodal shape is
found that seems to stem from the interface contributions to the total pdf (see dashed
line) which has a maximum at Z = 0.17,a non-zero value at the origin of P(Z = 0) and
extends up to Z = 0.46.Further,we find γ = 0.984 and s = 0.140 and observe again a
very good agreement between reconstructed and measured pdfs.In addition,the pdf of the
fully turbulent part has the shape of a Gaussian bell-curve as in Figure 10 with a value of
P(Z = 0) = 0 at the origin and a maximum,whose location again coincides with the one
of the total pdf.
4.Evaluation of a composite model for the mixture fraction pdf
The previous findings are not only of importance for the physical understanding of tur-
bulent mixing but also for the well-established presumed pdf approach to model scalar
pdfs in turbulent non-premixed combustion.One of the most widely used approaches un-
dertaken for predictions in intermittent flow regions is to presume a β-pdf,bi-variate or
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160 M.Gampert et al.
multi-variate β-distributions (see [35] and for instance [36] for a discussion and comparison
of various scalar pdf models in non-premixed combustion).It should be emphasized that,
similar to the β-pdf,the bi-variate and multi-variate distributions were also developed to
approximate the experimentally determined pdfs P(Z).Furthermore,it is vital to note that
the β-pdf can assume only the values of zero or infinity at Z = 0;this is a clear deficiency
with respect to the pdf shown in Figure 10.For highly intermittent cases,one may argue
that the β-pdf’s flexibility of generating a singularity at Z = 0 is favorable.Such an argu-
ment,however,ignores the fact that the physics of the fully turbulent region and the T/NT
interface are quite different.Whereas the physics of the fully turbulent region is determined
by the stochastic fluctuations,the interface is strongly influenced by molecular transport.
Effelsberg and Peters [16] showed that the mixture fraction pdf in intermittent regions
can be physically explained by considering separate contributions from the fully turbulent
flow,the scalar T/NT interface and the outer co-flow.As this composite model has recently
been successfully validated for the mixture fraction pdf at various radial and axial locations
of a jet flow,cf.[15],we compute the composite model and its zonal contributions according
to [16] for the experimentally obtained pdf at x/d = 15 so that the results can be compared
with the gradient trajectory statistics discussed in the previous chapter.To this end,we
first calculate the intermittency factor (γ) for the measured pdf P(Z) in the propane-CO
2
jet.Based on γ and P(Z),we then calculate the composite model pdf P
c
(Z) following
Equation (9).
To obtain the pdfs for the fully turbulent part P
t
(Z) and the interface part P
s
(Z),we
compute the model parameters (s,k,α
t
and γ
t
) fromthe first four moments of the measured
pdf of Z using the relations given in [16].
We use the parameters α
t

t
and γ
t
to construct a beta function that is hypothesized to
model the pdf contribution fromthe fully turbulent part
P
t
(Z
t
) =
(γ
t
)
(α
t
)(β
t
)
Z
α
t
−1
t
(1 −Z
t
)
β
t
−1
.(14)
Note that α
t

t
and γ
t
in the above equation are related as γ
t
= α
t

t
.To obtain the pdf
contribution from the T/NT interface,we assume the mixture fraction profile across the
scalar T/NT interface as follows:
Z = Z
t
￿
y
δ
￿
1/(1−k)
,(15)
where Z
t
is a random variable whose distribution is given by Equation (14),which corre-
sponds to the mixture fraction at the edge of the turbulent region and k is one of the model
parameters.Using these parameters,the pdf contribution fromthe scalar T/NT interface is
calculated as follows:
P
s
(Z) =
1 −k
Z
k
￿
1
Z
Z
k−1
t
P
t
(Z
t
)dZ
t
.(16)
For a detailed description of the derivation,refer to [16].The resulting composite pdf
together with the weighted zonal pdfs of the fully turbulent and the T/NT interface regions
are shown in Figure 12.In addition,a β-pdf is included for comparison and illustration of the
above deficiencies that has been calculated using the first two moments of the experimental
pdf.
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Journal of Turbulence 161
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.5
1
1.5
2
2.5
3
3.5
4
Z
P (Z)
Composite pdf
Turbulent part
T/NT Interface
Beta pdf
Exp (x/d=15)
Figure 12.Comparison of the measured pdf P(Z) obtained at x/d = 15 with the model pdf for
conserved scalar quantities of Effelsberg and Peters [16].Furthermore,the weighted zonal pdfs of
the fully turbulent γ[(1 −s)P
t
(Z)] and the T/NT interface regions γ[sP
s
(Z)] as calculated fromthe
composite model are shown.
Overall,we observe a very good agreement for the composite model as compared
with the measured pdf.For the calculation of the model pdf,we have solved the equations
relating the first four moments of the measured pdf to the model parameters for which we
have obtained s = 0.088,k = 0.009,α
t
= 5.243 and β
t
= 14.033.Both location and value
of the maximum lie almost on top of each other.The same is valid for the tail of the pdf,
where the composite model only slightly overpredicts the values for P(Z).The composite
model also reproduces the bimodal shape of the pdf,although it underpredicts the region
close to the origin,where it does not recover the quite steep rise that is present in the pdf
that is computed fromthe experiment.
Regarding the contributions to the composite pdf that are modelled to be stemming
from the fully turbulent part,we observe a very similar shape as for the zonal pdf shown
in Figure 10.The location and the value of the maximumare in good agreement;however,
behind the latter the model pdf decreases slower to zero than the zonal pdf obtained from
gradient trajectory statistics.Nevertheless,the intersection with the x-axis is for both pdfs
at around Z = 0.62.Another difference between model and experiment is present at the
origin.While the measured pdf starts at Z = 0 and already contributed significantly to the
overall pdf in the region ranging up to Z = 0.10,the modelled pdf only starts at Z = 0.03
and is smaller than the T/NT interface contributions up to Z = 0.08.
The same findings as for the fully turbulent zone are valid for the pdf of the scalar
T/NT interface.Qualitatively,we observe a good agreement of the model pdf with the one
calculated fromgradient trajectories.However,while the latter starts at almost at the same
value as the modelled pdf,it decreases faster to small values of P(Z) but contributes to
the overall pdf in the whole range of Z-values up to Z = 0.62.In contrast,the model pdf
for the T/NT interface has a significant contribution to the composite pdf up to Z = 0.30
but is already zero for values larger than Z = 0.41.In addition,it decreases monotonically
to zero.It is evident that the way the T/NT interface pdf is modelled by the composite pdf
model,this enables the reproduction of the bimodal shape of the overall pdf.In addition,we
note that the model predicts a value of s = 0.088,which is slightly smaller than the actual
value of s = 0.107.This difference can be explained by comparing the different shapes of
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162 M.Gampert et al.
the mixture fraction along a gradient trajectory and in the composite model,cf.Equations
(11) and (15).While along a gradient trajectory the mixture fraction is assumed to followa
sine profile,it is modelled linearly in the limit of k approaching zero in the composite pdf.
Thus,the volume fraction occupied by the T/NT interface can be expected to be larger for
the analysis based on gradient trajectories,which will result in larger values for s.
In summary,we find a very good overall agreement of the composite model with that
of the zonal analysis and the pdf that is computed fromthe measurements.In addition,the
zonal mixture fraction pdf calculated from gradient trajectory statistics and the model are
qualitatively close to each other,although slight quantitative differences are present.These
findings further highlight,on the one hand,the impact of the scalar T/NT interface on the
mixture fraction pdf in the early part and particularly at the edge of a turbulent jet flow,and
on the other hand give indications on howto develop and optimize existing models that are
currently employed in the computation of turbulent non-premixed combustion.
5.Conclusion
We have presented planar high-speed Rayleigh scattering measurements of the mixture
fraction Z of propane discharging froma turbulent round jet into co-flowing CO
2
at nozzle-
based Reynolds numbers Re
0
= 3000–8600,based on which we have investigated the
local structure of the turbulent scalar field as well as the scalar pdf using scalar gradient
trajectories.
The latter have been calculated for every grid point,and scalar profiles along the latter
are parametrized by the arithmetic mean Z
m
of minimumand maximumvalues of extremal
points that bound the gradient trajectory and the scalar difference Z between them.
Using these parameters,we have partitioned the turbulent scalar field into three regions – a
fully turbulent one,where each trajectory connects a minimum and a maximum point,the
outer flow with Z = 0 and a meandering scalar T/NT interface,where a maximum point
is connected with the outer flowvia a gradient trajectory – the latter structure is embedded
within the former two.In the next step,we have investigated the jpdf P(Z
m
,Z) as well as
the marginal pdfs P(Z
m
) and P(Z) in different zones and observe distinct characteristics
for each of them:many small fluctuations together with a large mean scalar value are typical
for the fully turbulent region,while the regularly observed large jump of scalar value is
caught by the gradient trajectory statistics in the scalar T/NTinterface.As the latter starts in
the outer flowat Z = 0,it also directly follows fromthis large jump that the T/NT interface
trajectories frequently contain the comparatively small value of stoichiometric mixture Z
st
.
Then we have presented a method to reconstruct the overall scalar pdf P(Z) based
on gradient trajectory statistics using the jpdf P(Z
m
,Z) in the different zones of the
scalar field.We observe a good agreement between the experimentally obtained pdf with
the reconstructed one,where P
t
(Z) in the turbulent part has the shape of a Gaussian-bell
curve,while P
s
(Z) in the T/NT interface has a non-zero value at the origin from which it
decreases to zero.
Finally,we discuss the impact of the former results on modelling approaches for pdfs
of conserved scalar quantities.To this end,we compare the measured pdf with the one
obtained fromthe composite model of Effelsberg and Peters [16].We observe a very good
agreement of the composite model with the experimentally obtained pdf.The fully turbulent
pdfs agree in shape and value for P(Z),while the zonal mixture fraction pdfs calculated
from gradient trajectory statistics and the model are qualitatively close to each other but
exhibit slight quantitative deviations.
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Journal of Turbulence 163
Acknowledgements
This work was funded by the Research School “BrenaRo” and the Cluster of Excellence “Tailor-
Made Fuels fromBiomass,"which is funded by the Excellence Initiative of the German Federal state
governments to promote science and research at German universities.
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