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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, RWTHAachen 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,
147164
To link to this article: http://dx.doi.org/10.1080/14685248.2012.747688
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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,RWTHAachen University,Templergraben 64,
Aachen,Germany
(Received 29 August 2012;ﬁnal version received 2 November 2012)
Based on planar highspeed Rayleigh scattering measurements of the mixture fraction
Z of propane discharging from a turbulent round jet into coﬂowing carbon dioxide at
nozzlebased 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/nonturbulent 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/nonturbulent interface to the ﬁndings
made in other experimental and numerical studies of the turbulent/nonturbulent inter
face,and on the other hand discuss themin the context of the ﬂamelet approach and the
modelling of pdfs in turbulent nonpremixed 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/nonturbulent 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 twofeed 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.rwthaachen.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 nonpremixed 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 nonturbulent 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 nonturbulent character for measurements of scalar quantities.Corrsin and Kistler [3]
ﬁrst termed the layer at the outer edge of this turbulent/nonturbulent (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 nonisothermal 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
nonpremixed 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 positivedeﬁnite (minimum) or negativedeﬁnite
(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 nonpremixed combustion.The latter is a detailed multiscale
approach in combustion modelling,which employs a nonequilibriumformulation 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 highfrequency planar Rayleigh scattering
measurements of pure propane C
3
H
8
discharging froma free round jet into coﬂowing 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 postprocessing 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 coﬂowing 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 coﬂow tube had a diameter of 150
mm and a length of 450 mm,which was large enough to reduce the experimental setup
to a twostream 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 FMA2600A) at various ﬂow rates to achieve the desired Reynolds
number.The coﬂow gas was chosen as CO
2
owing to its larger Rayleigh crosssection,
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
/ν
Cl
) is the
local Taylorbased 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 acquiredusinghighspeedtwodimensional Rayleigh
Figure 1.Experimental setup for highspeed 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
Nozzlebased Reynolds number,Re
0
3000 4500 4750 8610
Taylorbased Reynolds number,Re
λ
61 72 71 96
scattering imaging.LaserRayleigh 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,twofrequencydoubledbeams (λ
f
= 527nm) from
a highfrequency dualhead Nd:YLF laser (Litron Lasers LDY303HEPIV) 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 shotbyshot 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 radialazimuthal 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 highspeed 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 signalto 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 inplane 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 inhouse.
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 postprocessing is illustrated in Figure 2,where (a) a
sample raw image recorded at x/d = 20 and (b) the corresponding postprocessed 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 threedimensional mixture
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152 M.Gampert et al.
Figure 2.(a) Examplary raw and (b) postprocessed images obtained at x/d = 20.
fraction ﬁeld.Due to the importance of twopoint 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 xdirection 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,quasilaminar 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 socalled quasilaminar 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 nondimensional 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.Inbetween,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 nonlocal information,as the grid points at
a given radial distance fromthe centerline with a scalar value between the freestreamand
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
nonlocal approach allows detecting engulfed regions,which is not possible if the interface
deﬁnition is based on a singlevalued 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 nonturbulent
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 nonturbulent 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
nonturbulent 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 nonturbulent 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 coﬂow,which,by deﬁnition,corresponds to Z = 0 (note
that in the coﬂow region (C),the measured mixture fraction value ﬂuctuates between
Z = 0 and Z = 0.03,whichis causedbythe residual noise that is left after dataprocessing).
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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 reentrained
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/nonturbulent
interface and C:outer coﬂow:(a) Instantaneous mixture fraction ﬁeld with a single interface,and
(b) instantaneous mixture fraction ﬁeld with multiple turbulent/nonturbulent 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 quasionedimensional 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 abovementioned
residual noise level after postprocessing.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 bellshaped 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 Zvalues
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 nonzero 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 nonzero
value of the measured pdf at the origin,which is not due to contributions fromthe outer ﬂow
only.In addition,it has a nonzero 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 nonouter 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 nonzero 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 bellcurve 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 wellestablished presumed pdf approach to model scalar
pdfs in turbulent nonpremixed combustion.One of the most widely used approaches un
dertaken for predictions in intermittent ﬂow regions is to presume a βpdf,bivariate or
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160 M.Gampert et al.
multivariate βdistributions (see [35] and for instance [36] for a discussion and comparison
of various scalar pdf models in nonpremixed combustion).It should be emphasized that,
similar to the βpdf,the bivariate and multivariate distributions were also developed to
approximate the experimentally determined pdfs P(Z).Furthermore,it is vital to note that
the βpdf 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 βpdf’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 coﬂow.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 propaneCO
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
,β
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
,β
t
and γ
t
in the above equation are related as γ
t
= α
t
+β
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 βpdf 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 xaxis 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 Zvalues 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
Downloaded by [Stanford University] at 13:22 14 March 2013
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 nonpremixed combustion.
5.Conclusion
We have presented planar highspeed Rayleigh scattering measurements of the mixture
fraction Z of propane discharging froma turbulent round jet into coﬂowing 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 Gaussianbell
curve,while P
s
(Z) in the T/NT interface has a nonzero 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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