Exp Fluids (2013) 54:1621
DOI 10.1007/s00348-013-1621-4
RESEARCH ARTICLE
Scalar gradient trajectory measurements using high-frequency
cinematographic planar Rayleigh scattering
Markus Gampert • Venkat Narayanaswamy
Norbert Peters
•
Received: 13 December 2012 / Revised: 29 September 2013 / Accepted: 15 October 2013 / Published online: 31 October 2013
Ó Springer-Verlag Berlin Heidelberg 2013
Abstract In this work, we perform an experimental
investigation into statistics based on scalar gradient trajectories in a turbulent jet flow, which have been suggested
as an alternative means to analyze turbulent flow fields by
Wang and Peters (J Fluid Mech 554:457–475, 2006,
608:113–138, 2008). Although there are several numerical
simulations and theoretical works that investigate the statistics along gradient trajectories, only few experiments in
this area have been reported. To this end, high-frequency
cinematographic planar Rayleigh scattering imaging is
performed at different axial locations of a turbulent propane jet issuing into a CO2 coflow at nozzle-based Reynolds numbers Re0 = 3,000–8,600. Taylor’s hypothesis is
invoked to obtain a three-dimensional reconstruction of the
scalar field in which then the corresponding scalar gradient
trajectories can be computed. These are then used to
examine the local structure of the mixture fraction with a
focus on the scalar turbulent/non-turbulent interface. The
latter is a layer that 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 Fluid Mech 626:333–365, 2009). Based
on the latter, we investigate the probability to find the
respective regions as a function of the radial distance to the
centerline, which turns out to reveal the meandering nature
M. Gampert (&) N. Peters
Institute for Combustion Technology, RWTH Aachen
Templergraben 64, 52056 Aachen, Germany
e-mail: [email protected]
V. Narayanaswamy
Department of Mechanical and Aerospace Engineering, North
Carolina State University, Raleigh, NC 27695, USA
e-mail: [email protected]
of the scalar T/NT interface layer as well as its impact on
the local structure of the turbulent scalar field.
1 Introduction
Turbulence tends to be created locally where the flow is
most unstable, which can be observed, for instance, in jet
flows, wakes and boundary layers, cf. Bisset et al. (2002)
and Philip and Marusic (2012). In these examples, turbulent regions are located adjacent to non-turbulent (NT)
ones, where no turbulence is generated. Townsend (1948,
1949) quantified this behavior in terms of an intermittency
factor c, defined as the fraction of the signal that is turbulent. Corrsin and Kistler (1955) first termed the layer
separating the turbulent from the non-turbulent (T/NT)
region as the ‘laminar superlayer’. Here, because we are
considering a scalar quantity only, we will refer to the
region where the flow changes its character from laminar
(irrotational) to turbulent (rotational) as the scalar T/NT
interface.
Detailed spatial analyses of this region have recently
been carried out experimentally (e.g., Westerweel et al.
2002, 2005, 2009; Holzner et al. 2007a, b) and numerically
(e.g., da Silva and Pereira 2008, 2011; da Silva and Taveira
2010), where the latter authors argue that in the presence of
a mean shear, the characteristic length scale d associated
with the width of the scalar T/NT interface is of the order
of the Taylor microscale k. In a previous work, cf. Gampert
et al. (2013a, e), the impact of the T/NT interface on
the mixture fraction pdf P(Z) at various axial and radial
locations was examined, where the mixture fraction is
defined as the mass fraction of fuel stream in a given fuelair mixture. To this end, the composite model by Effelsberg
and Peters (1983) was used to identify the structure of the
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Page 2 of 15
mixture fraction pdf in this free shear flow 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. In addition, statistics such
as the pdf of the location of the T/NT interface and of the
scalar profile across the latter were investigated and were
found to be in good agreement with literature data, cf.
Westerweel et al. (2009). Furthermore, the scaling of the
thickness d of the scalar T/NT interface was analyzed. It
was observed that d/L * Re-1
k , where L is an integral
length scale and Rek the local Taylor-based Reynolds
number, meaning that d is proportional to k—a finding that
is in good agreement with the dimensional scaling arguments postulated by da Silva and Taveira (2010).
The region of the T/NT interface layer was recently
further analyzed by Mellado et al. (2009), who investigated
the DNS of a temporally evolving shear layer using gradient trajectories. Based on these gradient trajectories, the
latter authors partition the scalar field into a fully turbulent
zone, a zone containing the T/NT interface layer and the
outer laminar flow. Based on the different regions, they
examine the probability of these three zones at different
locations in the shear layer and investigate the scalar
probability density function and the conditional scalar
dissipation rate in the zones in the presence of external
intermittency. This approach was adopted Gampert et al.
(2013d), where zonal statistics of the scalar pdf P(Z) as
well as the scalar difference along a scalar gradient trajectory DZ and its mean scalar value Zm were examined
based on experimentally obtained scalar fields in a jet flow.
In addition, the latter authors reconstruct the scalar probability density function P(Z) from zonal gradient trajectory statistics of the joint probability density function
PðZm ; DZÞ and observe a very good qualitative and quantitative agreement with the experimental data.
Originally, the concept of field analysis by gradient
trajectories in direct numerical simulations (DNS) has been
introduced by Wang and Peters (2006, 2008) in their theory
of dissipation elements to describe the turbulent flow field
in an unambiguous, space-filling and non-arbitrary manner.
These dissipation elements are defined as the ensemble of
points in space from which a gradient trajectory reaches the
same to two extrema as end points. Further, the concept of
dissipation elements is employed for the reconstruction of
statistical properties of the turbulent field and flow visualization purposes. Examples of the statistical analysis of
turbulent flow field using dissipation elements can be found
in Schaefer et al. (2010b, 2011) and Gampert et al. (2011,
2013b), where for instance the coefficients of the k-e-model
often employed in RANS simulations were calculated
using dissipation element theory. Taking the mixture
fraction Z as the underlying scalar field also allows the
physical interpretation of dissipation elements in the
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Exp Fluids (2013) 54:1621
context of the flamelet approach in non-premixed combustion, as shown by Peters (2009), who used dissipation
elements to study the instantaneous scalar dissipation rate
as a function of the mixture fraction.
The approach to use trajectories to examine DNS data
has also been extended to other applications. Wang (2009)
showed that there is a linear scaling of the mean absolute
value of the velocity difference with the curvilinear distance along gradient trajectories for large elements. In
addition, it is argued that due to a conditioning of the
statistics on gradient trajectories, regions of large extensive
strain smoothing the scalar field are preferentially extracted, thereby allowing a gradient trajectory to extend over
large distances of the order of the Taylor microscale. In
addition, Wang (2010) and Schaefer et al. (2012, 2013a, b)
extended the concept of dissipation elements to so-called
streamline segments to investigate vector fields.
Previous direct numerical simulations that investigated
gradient trajectories in turbulent scalar fields revealed that
a resolution of the order of the Kolmogorov scale O(g) is
needed to obtain grid independent statistics. This resolution requirement makes the experimental study of the
latter very challenging, in particular as originally a resolution below the Kolmogorov scale was demanded.
However, recent studies showed, cf. Gampert et al.
(2011), that such a strict criterion was only necessary for a
sound theoretical derivation of dissipation element theory
that is free of any numerical artifacts. In contrast, the
statistics under investigation, such as the length distribution, and the conditional mean do not require such a high
resolution as in addition new algorithms have been
implemented that make the computation of gradient trajectories and dissipation elements much more robust, see
Sect. 2.5 for more details.
The first experimental study on dissipation elements was
performed by Schaefer et al. (2010a), who used tomographic PIV to obtain three-dimensional measurements of
the velocity field in the core region of a channel flow. The
study showed several interesting results on the length scale
of dissipation elements as, for instance, the theoretically
derived exponential tail of the probability density function
(pdf) for the length distribution of dissipation elements in
normalized form could be confirmed. However, many of
the results were limited by the resolution and rather low
signal-to-noise ratio (SNR) that is characteristic of PIV
technique. Another attempt was made by Soliman et al.
(2012), who used Rayleigh imaging to obtain the concentration distribution of two gases in a turbulent shear flow
based on which the authors then examine dissipation element statistics. However, as the images are recorded in a
planar cut through the centerline, only two-dimensional
projections of these highly corrugated three-dimensional
geometries could be analyzed.
Exp Fluids (2013) 54:1621
Page 3 of 15
The development of advanced laser optical techniques
with a high-pulse energy at a high-repetition rate has
facilitated the experimental investigation into spatially
three-dimensional conserved scalar quantities. These
techniques allow to gain phenomenological and statistical
understanding of turbulent mixing in gas- and liquid-phase
flows. While Prasad and Sreenivasan (1989), Dahm et al.
(1991), Buch and Dahm (1996) used planar laser-induced
fluorescence to study conserved scalar fields liquid flows,
Everest et al. (1996), Feikema et al. (1996), Buch and
Dahm (1998), Su and Clemens (1999, 2003), Frank and
Kaiser (2010) measured scalar fields in the gas phase based
on highly resolved Rayleigh scattering. The use of scalar
imaging by Rayleigh scattering to study dissipation elements, therefore, appears to be a more attractive option
owing to its potential to obtain relatively high SNR scalar
images, see for instance Patton et al. (2012), compared to
the velocity field from PIV. The large SNR of the scalar
images leads to a better resolution, which is crucial to study
gradient trajectories. The challenge, however, is to obtain
three-dimensional volumetric scalar field data with sufficient signal quality that allows computing the total gradient
with high precision. To this end, a scalar field, namely the
mixture fraction Z, is investigated in this study, which is
defined as the mass fraction of fuel stream in a given fuelair mixture
Z¼
mf
;
mf þ mair
ð1Þ
where the subscripts f and air refer to fuel stream and air,
respectively. According to this definition, Z varies between
Z = 0 and Z = 1.
A wide range of experimental investigations into scalar
fields can be found in the literature, see for instance Antonia
et al. (1984) and Mydlarski and Warhaft (1998). Threedimensional volumetric scalar gradient measurements,
however, are very limited and often involved multi-point or
two-dimensional measurements in combination with Taylor’s hypothesis. This approximation estimates the spatial
derivative in the streamwise x-direction from the local
instantaneous value of the time derivative from a singlepoint or planar measurement, when the required threedimensional multipoint measurements are impractical or
unavailable. In the limit of low turbulence intensities, the
motion of gradients relative to the local mean flow can be
approximated as one of pure convection. Due to the importance of two-point statistics and spatial gradient quantities in
turbulence, it is common to use Taylor’s hypothesis to
estimate spatial derivatives, see Dahm and Southerland
(1997) for a critical discussion of its accuracy. Even in
multipoint probe measurements of velocity gradients, it has
been invoked to estimate pdfs and derivatives along the
mean streamwise direction, cf. Tsinober et al. (1992) and
Kholmyansky and Tsinober (2009). Among the most widespread uses of Taylor’s hypothesis is the estimation of scalar
dissipation rates in turbulent shear flows defined as
v ¼ 2DðrZ Þ2 ;
ð2Þ
where D is a molecular diffusion coefficient. As usual not
all three spatial components of the gradient are measured,
the missing ones need to be computed from the time series
of the scalar quantity, sometimes even only using the
temporal signal when local isotropy is assumed, see for
instance Antonia and Sreenivasan (1977), Anselmet and
Antonia (1985) and Talbot et al. (2009).
In measurements, the three-dimensional information is
often found by imaging in parallel, spatially distinct twodimensional planes or via a sweeping of a single twodimensional laser sheet in sheet normal direction, see for
instance Su and Clemens (1999) for a discussion. However,
these two- and three-dimensional measurements of conserved scalar fields in the gas phase have often been limited
in temporal resolution so that the dynamic nature of turbulence is not resolved satisfactory. In later works, Ganapathisubramani et al. (2007, 2008) used a cinematographic
imaging technique for velocity measurements using PIV. In
this technique, high-speed 2-D imaging was performed in
the plane normal to the bulk flow direction; the volumetric
reconstruction of the velocity field was then performed
invoking Taylor’s hypothesis. Later works of Ganapathisubramani et al (2011a, b) have validated this technique for
obtaining the velocity gradient tensor and for the threedimensional reconstruction of the gradient properties such
as vorticity and dissipation. To this end, we use in the
following a volumetric reconstruction technique for a
scalar field inspired by Ganapathisubramani et al. (2007) to
obtain three-dimensional data at different axial locations in
the near field of a turbulent jet issuing into a coflow, see
Sect. 2 for a detailed description of the experiment. In Sect.
3, we examine the scalar field by gradient trajectories based
on the method introduced by Mellado et al (2009) with a
focus on the T/NT interface layer. Finally, the paper is
concluded in Sect. 4 with a brief summary.
2 Experimental investigation
In the course of this section, we will present the measurement techniques, the experimental arrangement as well
as the data processing procedure that have been employed
in this study. Furthermore, some data validation in terms of
scalar spectra, spatial resolution, axial decay of the scalar
and the velocity along the centerline and radial self-similarity of the latter two quantities is given. Finally, the
volumetric reconstruction and the gradient trajectory
search algorithm are described.
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Page 4 of 15
Exp Fluids (2013) 54:1621
The experiments were performed in a coflowing turbulent jet facility, which consists of a center steel tube with an
inner nozzle diameter d = 12 mm. The surrounding coflow
tube had a diameter of 150 mm, which was large enough to
reduce the experimental setup to a two-stream-mixing
problem. Research grade propane (99.95 % pure) was fed
through the center tube using a mass flow controller
(OMEGA FMA-2600A) at various flow rates to achieve the
desired jet exit Reynolds number. The coflow gas was
chosen as carbon dioxide owing to its larger Rayleigh cross
section compared to air, which was necessary to obtain
highly resolved mixture fraction fields, from which an
accurate investigation also of the outer region of the jet
flow is possible. Naturally, the smaller difference in Reyleigh cross section of Co2 and propane increases the error
associated with the computation of gradients. However, as
discussed in the context of scalar spectra, the combination
of CO2 and propane seems to be a good trade-off between
SNR in the outer region of the jet and the gradient-to-noise
ratio in the jet’s inner part. The mean velocity of the CO2
coflow was 0.05 m/s, as determined from laser Doppler
anemometry (LDA) measurements. The different experimental cases that have been investigated are shown in
Table 1. At the measurement locations between x/d = 10
and x/d = 30, the jet exit velocity was always chosen in
such a way that the out-of-plane resolution (see Table 1 for
further details) is always good enough for a proper calculation of gradient trajectories. This results in a variation of
the mean jet exit velocity U0 between 1.15 and 3.30 m/s
resulting in jet Reynolds numbers Re0(=U0 d/m) between
3,000 and 8,610. Furthermore, Rek (=urms ku/ mCl) shown in
Table 1 is the local Taylor-based Reynolds number on the
center line. For the calculation of this quantity, urms has
been measured using laser Doppler anemometry (LDA),
and the kinematic viscosity on the centerline mCl has been
determined using the local concentration of the two gases,
which varies due to the mixing of propane and CO2 in
downstream direction (note that the Schmidt number
Sc (= m/D) remains close to unity due to the same molecular weights of propane and carbon dioxide). The Taylor
Table 1 Experimental parameters
x/d
10
15
20
30
Jet exit velocity U0 (m/s)
1.15
1.76
1.82
3.30
Mean centerline velocity UCl (m/s)
0.57
0.61
0.50
0.62
Mean centerline mixture fraction ZCl
0.38
0.24
0.18
0.13
Kolmogorov scale g(mm)
0.18
0.20
0.26
0.24
2.30
3.32
4.26
4.61
Centerline viscosity mCl ðmm =sÞ
6.50
6.95
7.40
7.50
Nozzle-based Reynolds number Re0
3,000
4,500
4,750
8,610
Taylor-based Reynolds number Rek
61
72
71
96
Taylor Scale k(mm)
2
123
microscale k ð¼ ð15mCl u2rms =eÞ1=2 ) has been computed using
an approximation formula to estimate the mean energy
dissipation e taken from Friehe et al. (1971), which has
also been applied for the calculation of the Kolmogorov
scale gð¼ ðm3 =eÞ1=4 Þ.
2.1 Measurement technique
In a first step, LDA was performed to obtain the radial and
axial profiles of the flow velocity at different flow conditions. This technique is non-intrusive and commonly used
for single-point measurements of the velocity using tracer
particles. Furthermore, it is characterized by a high accuracy in combination with a high spatial and temporal resolution. The technique makes use of the Doppler effect—
first when incident laser light impinges on a tracer particle
that is moving with the flow, second when the laser light is
scattered by this particle and received by a detector, thus
recording the Doppler shift of the incident light wave frequency. The latter is directly proportional to the difference
of the normal vectors, which appear when the propagation
direction of the incident and the scattered light differ as
well as to the velocity of the particle—for a detailed discussion of the LDA technique please refer to Tropea et al.
(2007).
An LDA system, a Coherent Innova 300 laser coupled to
a one-component fiber optic of Aerometrics Inc. was used
for the experiments. In the present case, small glass spheres
with a diameter between 3.5 and 7 lm were used as tracer
particles. Two beams, with a wavelength of 488 and
514 nm, were passed through a Bragg cell, which resulted
in two more beams whose frequency was offset by 40 kHz
from the input wavelengths. These beams were focused to a
spot, approximately 100 lm diameter, using a biconvex
lens (f.l. = 300 mm). The Doppler signals were focused by
another biconvex lens (f.l. = 300 mm) to the detector. The
two velocity components were computed using the data
processing software Real-Time Signal Analyzer provided
by Aerometrics Inc.
Afterward, high-speed (kHz-rate) two-dimensional
Rayleigh scattering imaging was performed in the y-zplane, normal to the bulk flow direction x. Laser Rayleigh
scattering has been used and documented in many previous
studies, see for instance Talbot et al. (2009), Dowling and
Dimotakis (1990) and Su and Clemens (2003), and is
therefore like LDA only briefly described here. The technique makes use of the fact that gas molecules elastically
scatter photons and that different molecules have different
Rayleigh scattering cross sections. In the present study, for
instance, the cross section of propane is roughly six times
higher than the one of the surrounding CO2. The Rayleigh
scattering light intensity in the perpendicular direction to
Exp Fluids (2013) 54:1621
the light source from a binary gas mixture is related in a
linear manner to the concentration of the gas exiting from
the jet. Hence, the two end points of pure propane and pure
CO2, respectively, of this linear relation are recorded for
calibration purposes, before the conversion from signal to
concentration is simply accomplished by linear interpolation, see Eckbreth (1996) and Tropea et al. (2007) for
further details.
The schematic of the experimental setup is shown in
Fig. 1 and a blow-up of the nozzle region in Fig. 2. Two
frequency-doubled beams (k = 527 nm) from a high-frequency dual-head Nd:YLF laser (Litron Lasers
LDY303HE-PIV) were made coincident, both spatially and
temporally, to deliver a total energy of about 32 mJ/pulse
at 1 kHz (32 W). To this end, the perpendicularly polarized
beams are split; then, the polarization of one beam is
flipped using a waveplate before the two beams are
recombined by a prism. 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 of the beams was normal to the jet axis
to maximize the Rayleigh scattering signals in the imaging
plane. The beams were transformed into a horizontal collimated sheet using a combination of spherical and cylindrical lenses. The width and the thickness (FWHM) of the
resultant sheet were approximately 10 and 0.3 mm,
respectively.
Images were acquired at 1 kHz using a high-speed
CMOS camera (LaVision HighSpeedStar 6, 1,024 9 1,024
pixels) fitted with a camera lens (Nikon f.l. = 85 mm)
stopped at f/1.4, which is mounted via a Scheimpflug
adapter. This is needed to make the tilted focal plane
coincide in its angle with that of the detector as the camera
is installed in a 20° angle to avoid any interaction with the
jet flow. To correct for the image distortion, we have
recorded the same area with tilted and perpendicular view
Page 5 of 15
and then used commercial software (DaVis) to map the two
images. This correction is in the following applied automatically when the data are recorded. This effect has also
been considered for the calculation of the resolution.
However, compared to the filtering effects that are
described in the following, the impact of the image distortion is small. An extension ring was placed between the
camera and the lens to minimize the working distance; the
resulting field of view was about 60 mm 9 60 mm. The
SNR calculated from the raw images was above 20 in the
pure CO2 region and that in the pure propane region was
above 40. The time interval between the successive images
was 1 ms.
2.2 Data processing procedure
The Rayleigh scattering images were corrected for background scattering, camera dark signal and laser sheet
inhomogeneities on an ensemble average basis. The
resulting signal is related to the number density and the
scattering cross section by the following expression Eckbreth (1996)
IRay ¼ CI0 nrmix ;
ð3Þ
where C is a constant that describes the collection volume
and the efficiency of the optical setup, I0 is the incident
laser intensity, n is the number density, and r is the
mixture-averaged differential cross section. For a flow that
occurs under isothermal and isobaric conditions, as is the
case in the present experiments, the Rayleigh signal, IRay, is
only a function of rmix. For a two-stream mixing process
(propane issuing into CO2), the above simplifies to
IRay ¼ CI0 nðX1 r1 þ ð1 X1 Þr2 Þ;
ð4Þ
where Xi and ri are the mole fraction and the differential
cross section of species i respectively. Using Eq. 4, and
Fig. 1 Experimental setup for the high-speed Rayleigh scattering measurements
123
Page 6 of 15
Exp Fluids (2013) 54:1621
Fig. 2 Geometry of the jet, the coflow and the measuring plane
using the Rayleigh scattering signals of pure propane and
pure CO2, the mole fraction of propane in the propane/CO2
mixture is given as
XC3 H8 ¼
IRay ICO2
:
IC3 H8 ICO2
ð5Þ
For calibration purposes, measurements of pure propane
(Z = 1) and CO2 (Z = 0), respectively, have been
performed. Based on these, the mixture fraction was
determined from the mole fraction according to Eq. 6 using
linear interpolation. The mixture fraction, which in the
present case equals the propane mass fraction, was then
computed from the mole fraction using the relation
Z ¼ Y C3 H 8 ¼
XC3 H8 WC3 H8
:
XC3 H8 ðWC3 H8 WCO2 Þ þ WCO2
ð6Þ
In Eq. 6, XC3 H8 is the propane mole fraction and WC3 H8 and
WCO2 are the molecular weights of propane and carbon
dioxide, respectively.
In a next step, one-dimensional energy spectra of the
individual mixture fraction fields were computed to assess
the in-plane resolution of the images in y- and z-direction,
see Fig. 3a. As the spectra that are obtained at x/d = 20 are
almost identical in y- and z-direction due to the radial
symmetry of the jet flow only the spectrum in y-direction
Eyy is shown for a better visibility. It has been computed
from all data columns in y-direction that are within a
square of 128 9 128 pixels which is centered around the
centerline. The spectrum is relatively flat at the low spatial
frequencies, which decay close to exponentially with
increasing spatial frequency due to the absence of an
inertial subrange as the Reynolds number is relatively low
and finally end with a flat region that corresponds to the
123
noise floor. The noise floor in the scalar spectrum for both
Eyy and Ezz begins at approximately D y ¼ D z 0:9 mm as
jg & 0.3, which demonstrates that the measurements have
a spatial resolution of at least 3g.
The mixture fraction fields were then filtered with a
finite impulse response like filter, see for instance Gamba
and Clemens (2011) for a similar treatment. This procedure
minimizes the influence of the experimental noise on the
statistics and other derived quantities related to gradient
trajectories that are discussed in this work. This type of
filter is preferred as it allows better control of the effect that
the filter has on the energy content of the measurements.
Any filtering scheme applied to the data such as, for
instance, a Gaussian filter might result in a modification of
the energy and dissipation frequency content which could
eventually mask the correct dissipation roll off. Note that in
the coflow region, the mixture fraction value fluctuates
between Z = 0 and 0.03, which is caused by the residual
noise that is left after data processing. Figure 4 shows the
comparison of sample scalar fields from x/d = 15 and x/
d = 20 before and after filtering, illustrating the high
quality of the data.
The out-of-plane resolution was studied in streamwise
direction along the centerline using several sets of 100
successive mixture fraction fields, see Fig. 3b. Here, Taylor’s hypothesis was invoked to convert the temporal frequency of imaging to the corresponding spatial frequency.
Owing to differences in the largest resolvable frequency,
this spectrum does not extend to the largest spatial wavelengths of the cross-stream spectra Eyy and Ezz. We note
the absence of a noise floor at the largest frequency, concluding that the accuracy of gradients computed in the outof-plane direction is not limited by the noise but by the
resolution of the experiment, which is Dx 5g as
jg & 0.18. With respect to the out-of-plane resolution of
Taylor’s hypothesis as well as the computation of dissipation elements, we conclude that a satisfactory resolution
is obtained. As the latter geometries are of the order O ðkÞ;
this corresponds to O(12-19g) in the present study.
Though Wang (2008) showed that in particular long dissipation elements are oriented perpendicular to the
streamwise x-direction, i.e., favorable with respect to the
resolution in the present measurement approach, a conservative estimate yields that a dissipation element on
average only spans over three consecutive images in outof-plane direction.
Finally, the major source of systematic uncertainty in
the determination of the mixture fraction is the departure
from linearity of the camera response, which is within 4 %
as quoted by the manufacturer. The combined uncertainty
arising from all the sources, where in particular mode
fluctuations of the laser are of importance, is estimated to
be below 5 %.
Exp Fluids (2013) 54:1621
Page 7 of 15
10
10
(a)
0
−3
E
filtered
E
unfiltered
yy
10
−6
yy
10
−1
(a)
0
0.8
Z
z/d
0.2
0
-0.2
-2
-1
0
y/d
(b)
0
Z
0.8
0.2
z/d
0
-0.2
-2
E(κ) [arbitrary scale]
E(κ) [arbitrary scale]
Fig. 3 Scalar spectra a in-plane
spectrum Eyy(j) before and after
filtering and b unfiltered out-ofplane spectrum Exx(j) computed
from the measurements
obtained at x/d = 20
-1
0
y/d
κη
10
0
(b)
0
10
−1
10
−2
10
E
xx
unfiltered
−2
10
−1
κη
10
over the respective values U0(Z0) at the nozzle are plotted
along the jet axis at the downstream locations presented in
Table 1 and at a fixed jet exit velocity of U0 = 3.3 m/s,
which corresponds to a jet exit Reynolds number of 8,660.
Note that here and in the following the subscript c denotes
a quantity on the center line, hi indicates an ensemble
average within time at a fixed radial location, U(Z) is the
mean and u(Z’) the fluctuating component of the velocity
(mixture fraction).
Following the results discussed in the literature, the
mean axial velocity and mixture fraction follow a hyperbolic decreasing law, cf. Pope (2000) (p. 96 ff). In order to
determine the slope of the curves, we have fitted the data to
the following expressions
Uc
d
¼ ku
ð7Þ
x x0
U0
and
(c)
0
0.8
Z
z/d
0.2
0
-0.2
-2
-1
0
y/d
(d)
0
0.8
Z
z/d
0.2
0
-0.2
-2
-1
0
y/d
Fig. 4 Exemplary a raw and b post-processed images obtained at
x/d = 15 and c raw and d post-processed images obtained at x/d = 20
2.3 Downstream evolution of the mean velocity
and the mean mixture fraction along the jet axis
In Fig. 5, the inverse of the mean centerline velocity Uc
and the inverse of the mean centerline mixture fraction Zc
Zc
d
¼ kZ
x x0
Z0
ð8Þ
in which x0 denotes the virtual origin of the jet. In the
present study, it is found to be x0/d = -3.2 for the
velocity, which is close to the values reported by Talbot
et al. (2009) (x0/d = -2.52) and Amielh et al. (1996)
(x0/d = -2.9) and x0/d = -1.75 for the scalar, which is
also well in the range of values reported by different
authors, cf. Talbot et al. (2009), Dowling and Dimotakis
(1990), Dibble et al. (1987) and Lubbers et al. (2001).
Furthermore, we find ku = 6.08 which is very close to the
values of Talbot et al. (2009) (ku = 6.2) and Amielh et al
(1996) (ku = 6.1) and kZ = 4.85, which is slightly below
the values of kZ = 5.3 and kZ = 5.5 found by Talbot et al.
(2009) and Lubbers et al. (2001), respectively.
Figure 6 depicts the evolution of kZ as a function of the
distance to the nozzle orifice taken from Lubbers et al.
(2001) via
Zc x x0 kZ ¼
:
ð9Þ
Z0
d
In addition to the DNS results of Lubbers et al. (2001), the
experimental values of Talbot et al. (2009), Dowling and
123
Page 8 of 15
6
10
(b)
(a)
5
8
Z0 /Ζ c
4
U0 /Uc
Fig. 5 Variation with the axial
distance of the mean velocity
(a) and the mean mixture
fraction (b) along the centerline
Exp Fluids (2013) 54:1621
3
2
4
2
1
0
6
0
5
10
15
20
25
30
35
0
0
5
10
15
20
25
30
35
(x−x 0 )/d
(x−x 0 )/d
addition, the half-width radius r1/2 of the mean velocity
corresponds to r~ ¼ 0:08. Radial r.m.s. profiles of the
velocity versus r~ are presented in Fig. 7b. We observe a
good collapse of the profiles at the different measurement
locations with a normalized peak magnitude of
approximately hu2 i1=2 =Uc ¼ 0:255 at r~ ¼ 0:045, which is
close to the values reported by Talbot et al. (2009)
(hu2 i1=2 =Uc ¼ 0:26 at r~ ¼ 0:05). The solid line indicates
a fit, which in this case represents a fourth-order
polynomial
Fig. 6 kZ as a function of the distance to the nozzle orifice (Figure
based on Lubbers et al. 2001)
Dimotakis (1990), Becker et al. (1967), Birch et al. (1978)
and Lockwood and Moneib (1980) are shown. We find a
variation of kZ between 4.2 and 5.1, which is in very good
agreement with the data obtained by the other authors
though these values lie at the lower end close to the ones of
Dowling and Dimotakis (1990) and Lockwood and Moneib
(1980).
2.4 Radial profiles of mean and root-mean-square
values of velocity and mixture fraction
In a next step, we examine the radial profiles of the mean
U, Z and root-mean-square (r.m.s.) values hu2 i1=2 ; hZ 02 i1=2
of velocity and mixture fraction normalized with the
respective mean value on the jet axis as a function of the
non-dimensional similarity coordinate r~ ¼ r=ðx x0 Þ.
Figure 7a depicts the radial evolution of U/Uc and we find
a collapse of all the mean quantities from x/d = 10 to 30
on a unique curve, which is fitted to
U
¼ expðKu r~2 Þ;
Uc
ð10Þ
yielding a value of Ku = 77.5. This agrees excellently with
the one of Talbot et al. (2009) (Ku = 77.4) and is close to
the ones obtained by Lubbers et al. (2001) (Ku = 76.1) and
Panchapakesan and Lumley (1993) (Ku = 75.2). In
123
hu2 i1=2
¼ p0 þ p1 r~ þ p2 r~2 þ p3 r~3 þ p4 r~4 ;
Uc
ð11Þ
with p0 = 0.26, p1 = 1.03, p2 = -23.59, p3 = 74.16,
p4 = -46.87. With respect to the afore discussed accuracy of Taylor’s hypothesis, we note that in regions with
large r.m.s. values one may question the three-dimensional
reconstructions. However, the r.m.s. fluctuations are of the
order of the integral time and length scale. The imposed
gradient is therefore proportional to the inverse of the
integral time and consequently much smaller than the local
velocity gradient, which is proportional to the inverse of
the Kolmogorov time scale. Figuratively speaking, one
eddy is moved across our measurement volume within a
velocity fluctuation, while our recording is performed with
a frequency proportional to the inverse of Kolmogorov
time scale. Therefore, the time interval between two
recordings is sufficient to regard the inner structure of an
eddy also in this region of the flow in our case as being
frozen.
Figure 8a depicts the radial evolution of Z and we find a
collapse of all the mean quantities from x/d = 10 to 30 on
a unique curve, which in this case is fitted to
Z
¼ expðKZ r~2 Þ:
Zc
ð12Þ
This fit yields a value of KZ = 59.2, which again agrees
excellently with the one of Lubbers et al. (2001)
(KZ = 59.1) and is close to the one obtained by Talbot
et al. (2009) (KZ = 58.2). In addition, the half-width radius
bc of the mean mixture fraction corresponds to r~ ¼ 0:11.
Radial r.m.s. profiles of the mixture fraction versus r~ are
Exp Fluids (2013) 54:1621
Page 9 of 15
(b)
1
x /d = 10
x /d = 15
x /d = 20
x /d = 30
U / Uc
0.8
0.6
0.4
0.2
0
x /d = 10
x /d = 15
x /d = 20
x /d = 30
0.25
<u2>1 / 2 /Uc
(a)
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0
0.25
0
0.05
0.1
0.15
0.2
0.25
r / (x-x0)
r /(x-x0)
Fig. 7 Radial profiles for mean value U/Uc (a) and for r.m.s. value hu2 i1=2 =Uc (b) at x/d = 10, 15, 20 and 30
(b)
1
Fit
x/d=10
x/d=15
x/d=20
x/d=30
0.8
0.6
0.4
0.2
0
Fit
x/d=10
x/d=15
x/d=20
x/d=30
0.25
<Ζ’2>1/2/Z c
(a)
Z/Ζ c
Fig. 8 Radial profiles for mean
value Z/Zc (a) and for r.m.s.
value hZ 02 i1=2 =Zc (b) at x/
d = 10, 15, 20 and 30
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
r / (x-x0 )
presented in Fig. 8b. The solid line again indicates a
fourth-order polynomial fit
hZ 02 i1=2
¼ p0 þ p1 r~ þ p2 r~2 þ p3 r~3 þ p4 r~4 ;
Zc
ð13Þ
with p0 = 0.21, p1 = 0.42, p2 = 8.78, p3 = -118.19,
p4 = 276.61. These values are close to the ones reported by
Talbot et al. (2009) and Richards and Pitts (1993). Furthermore, we observe the normalized peak magnitude of
approximately hZ 2 i1=2 =Zc ¼ 0:25 at r~ ¼ 0:09, which is in
the range of values reported by Talbot et al. (2009),
hZ 2 i1=2 =Zc ¼ 0:25 at r~ ¼ 0:11, and Richards and Pitts
2 1=2
(1993), hZ i =Zc ¼ 0:26 at r~ ¼ 0:10 as well as by
Schefer and Dibble (1986).
2.5 Volumetric reconstruction and gradient trajectories
The volumetric reconstruction of the two-dimensional
mixture fraction field was performed using computer codes
developed in-house. The measured mixture fraction field
was low-pass filtered using the procedure described in Sect.
2.2 To employ Taylor’s hypothesis on the two-dimensional
mixture fraction fields for the volumetric reconstruction,
the radial profile of the mean axial velocity U(r), cf. Sect.
2.4, measured using LDA, was extended to two dimensions
based on radial symmetry. This symmetry has been
investigated in the course of the LDA measurements for
various jet exit velocities, and a good agreement was
found. The instantaneous mixture fraction field at the given
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
r /( x-x 0 )
instance, t0 þ D t; was displaced from the previous realization at t = t0, using the local mean velocity. The mean
axial velocity naturally varies over the y-z-plane so that the
axial coordinates are different for different regions of the
jet. Consequently, the axial coordinates near the jet center
are stretched, while the coordinates at larger radial locations are compressed. The following equation relates the
out-of-plane displacement of the scalar field at time
t ¼ t0 þ D t; with respect to the field at t = t0:
Zðx; y; z jt¼t0 þDt Þ ¼ ZðxðtÞ þ Uðy; zÞDt; y; zÞ:
ð14Þ
Based on such a three-dimensional scalar field, the
scalar dissipation rate can be computed. However, for the
calculation of v, a mixed spatiotemporal scalar dissipation
approximation has to be formed by combining the available
spatial derivatives with the time derivative, yielding
"
2 2 #
1 oZ 2
oZ
oZ
v ¼ 2D
þ
þ
;
ð15Þ
U ot
oy
oz
as in the present case time series measurements within a
plane are performed, so that a direct evaluation is only
possible for the cross-stream spatial derivative components
in y- and z-direction. Based on Eq. 15, the scalar dissipation rate is computed based on the experimental mass
fraction field using a central differences scheme with a
five-point stencil to ensure an accurate calculation of the
gradients, cf. Hearst et al. (2012)—see Fig. 9 for an illustration of the filtered mixture fraction field in combination
with the in-plane derivatives and the out-of-plane
123
Page 10 of 15
Exp Fluids (2013) 54:1621
z/d
(a) 0.3
0.5
0
-0.3
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
y/d
z/d
(b) 0.3
20
0
-0.3
-2.5
-20
-2
-1.5
-1
-0.5
0
0.5
y/d
z/d
(c) 0.3
20
0
-0.3
-2.5
-20
-2
-1.5
-1
-0.5
0
0.5
y/d
z/d
(d) 0.3
20
0
-0.3
-2.5
-20
-2
-1.5
-1
-0.5
0
0.5
y/d
z/d
(e) 0.3
10
0
-0.3
-2.5
0
-2
-1.5
-1
-0.5
0
0.5
y/d
z/d
(f ) 0.3
1
0
-0.3
-2.5
-8
-2
-1.5
-1
-0.5
0
0.5
y/d
Fig. 9 Exemplary illustration of a fully post-processed image (a) the
mixture fraction field Z, (b) the non-dimensional spatial derivative in
streamwise direction qZ/qx/qZ/qxrms, (c) the non-dimensional spatial
derivative in y-direction qZ/qy/qZ/qyrms, (d) the non-dimensional
spatial derivative in z-direction qZ/qz/qZ/qzrms, (e) the scalar dissipation rate v in linear scale and (f) the scalar dissipation rate v in
logarithmic scale obtained at x/d = 20
derivative that has been calculated using Taylor’s hypothesis. Finally, a planar cut through the field of the scalar
dissipation rate is depicted in linear and logarithmic scale
that has been calculated according to Eq. 15. From the
latter two images, it is evident that scalar dissipation is
mainly organized in sheets, an observation that is consistent with the previous high-repetition results of Patton et al.
(2012) and the findings gained in the low-repetition studies
of Buch and Dahm (1996, 1998), Su and Clemens (1999,
2003), Frank and Kaiser (2010).
The numerical algorithm traces gradient trajectories
until the extrema are reached, see Schaefer et al. (2010b)
and Gampert et al. (2011, 2013b, c) for examples of this
method and its application to experimental and numerical
123
data. Therefore, the algorithm first reads the scalar field
discretized on a uniform grid into the main memory. It
loops over all grid points and starts for each a single gradient trajectory tracing the path through the computational
domain. Following the derivatives of the scalar field in
descending and ascending direction, the local extremal
points for each gradient trajectory are found and assigned
to the associated grid point.
The algorithm traces the trajectory path by taking small
numerical steps iteratively in direction of the interpolated
derivative. The step size depends on the local gradient of
the scalar field and has a maximum size of two percent of
the grid spacing. The interpolation scheme has to be
sophisticated for high accuracy and insensible for
numerical errors. Here, a linear interpolation of first-order
derivatives on a staggered grid is used instead of interpolating the scalar value directly. It satisfies the requirement that it works well in 2D and 3D space, the
interpolated values at grid points are consistent with the
known values, and the derivatives are continuous and
ensure smooth advancing of trajectories through the whole
field. The original and staggered grids are located at
alternating equally spaced points. Derivatives at staggered
points are calculated from the difference of the scalar
values at two adjacent original grid points. This results in
first-order accuracy for derivatives, but a second-order
accuracy for scalar values. By numerical approximation of
the scalar fields curvature at points with zero scalar gradient, a local minimum, maximum and saddle point are
distinguished.
Note, that the analysis of the mixture fraction volume is
performed at each axial location using three statistically
independent sets of 5,400 consecutive images.
3 Investigation into the mixture fraction field using
gradient trajectories
In the following, we will examine the measured mixture
fraction fields in terms of gradient trajectories based on a
procedure developed by Mellado et al. (2009). Gradient
trajectories are calculated from each grid point in ascending and descending gradient direction where
n¼
rZ
j rZ j
ð16Þ
is the normalized unit vector in gradient direction. The
trajectories are pursued until a local extreme point is
reached at which the scalar gradient vanishes. Based on
this analysis, the flow is partitioned into three different
regions—namely a fully turbulent zone, an outer flow
region and embedded within these two the scalar T/NT
interface layer.
Exp Fluids (2013) 54:1621
Gradient trajectories and scalar extreme points are used
to detect the three regions of the scalar field using the
following criteria: 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, see trajectory A in Fig. 10. On the
contrary, if the trajectory connects a maximum with the
outer stream, where the mixture fraction is Z = 0, that
point belongs to the scalar T/NT interface, see trajectory B
in Fig. 10. Note that the outer flow is assumed once the
mixture fraction value is less than the residual noise, i.e.,
Z \ 0.03. In addition, the trajectory might theoretically
proceed through the studied flow region without any
intermediate extreme point, thus defining a so-called quasilaminar diffusion layer. However, as we will see in the
following, such layers are 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.
Mellado et al. (2009) used this analysis to examine the
zonal probability of the different regions as well as the
scalar probability density function and the conditional
scalar dissipation rate in the zones in the presence of
external intermittency.
We will use this approach to determine the probability
of finding the respective regions in radial direction r at a
fixed downstream location x/d. To this end, we calculate
the extreme points in the experimentally obtained mixture
fraction fields as well as the corresponding gradient trajectories using the numerical procedures described in Sect.
2.5. As an example, trajectories in the fully turbulent zone
are shown in Fig. 11. In this case, they share a common
minimum point and reach seven different maximum points.
We observe that the resulting gradient trajectories are
strongly varying in shape and are intertwisted.
Fig. 10 Flow partition based on
gradient trajectories:
a trajectory from minimum to
maximum, fully turbulent zone;
b from outer flow to maximum,
T/NT interface
Page 11 of 15
Let us further note that this method to partition the
scalar field uses non-local information, as a gradient trajectory extends either between two extremal points (in the
case of the fully turbulent zone) or from one extremal point
to the outer stream. Mellado et al. (2009) showed that this
non-local approach allows to detect engulfed regions,
which is not possible if the interface definition is based on a
single-valued envelope surface as widely done in the literature. However, an outer limit to the T/NT interface is
also set by a threshold in the magnitude of the scalar gradient, below which the scalar is approximately a homogeneous field. This second criterion defines the
conventional intermittency function and separates the NT
zones from the scalar T/NT interface and the turbulent
region.
The differentiation between the outer NT zones and the
scalar T/NT interface has been introduced for several reasons. First, it is needed from a numerical point of view as
the gradient approaches zero the further one moves toward
the outer homogeneous region so that below a threshold,
there is only numerical noise, and the gradient direction is
undetermined. Second, this distinction is the conventional
one used to define the intermittency factor and can be used
to compare with traditional results. 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
corresponding outer value and the scalar dissipation can be
approximated by zero.
In summary, a point at a given distance r from the
centerline can be part of the non-turbulent outer flow,
belong to the scalar T/NT interface or be located within the
turbulent region. A probability to be part of each of these
zones can be calculated by the area fraction that each zone
covers in the measured mixture fraction fields. These
Fully turbulent
core
Centerline
123
Page 12 of 15
Exp Fluids (2013) 54:1621
Fig. 11 Example of gradient
trajectories in the turbulent zone
based on the mixture fraction
field Z obtained at x/d = 30. All
trajectories share the same
minimum point and connect it
with seven different maximum
points. The scalar value
increases from minimum point
(blue) to maximum (red)
3 mm
Z min
0.21
Z max
0.13
Zonal probability
1
Outer flow
T/NT interface
Fully turbulent zone
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
r /( x-x0 )
Fig. 12 Profiles of the zonal probability of the three different regions
shown over a non-dimensional radial coordinate r~ ¼ r=ðx x0 Þ
obtained at x/d = 15 and Re0 = 4,500
probabilities depend on the radial distance to the centerline
and are depicted in Fig. 12 in terms of the self-similar
variable r~ ¼ r=ðx x0 Þ.
The behavior of the outer flow regions is as expected,
increasing from zero to a probability of one as we move
further outside in radial direction r~—it starts to be present
in the scalar field at approximately r~ ¼ 0:10 and is the
dominating part of the field after r~ ¼ 0:17. The scalar T/NT
interface peaks at about r~ ¼ 0:13 and drops asymmetrically
to zero as the outer flow is approached. This may be
compared to the statistics of the location of the T/NT
interface, cf. Westerweel et al. (2009), where typically a
123
close to Gaussian distribution around a mean value of
approximately two half-width radii of the velocity field is
observed. Here, we note a slight difference for the T/NT
interface location from gradient trajectory statistics. As
mentioned afore, the pdf of the location of the T/NT
interface has a maximum at around r~ ¼ 0:13 which corresponds to r = 1.5r1/2, where r1/2 is the velocity halfwidth radius as discussed in the previous section.
This slight deviation has several reasons: First, in the
present study, we investigate the mean location of the
scalar T/NT interface. Instead of only looking at the onedimensional T/NT interface, we thus investigate a structure
with a spatial extension. Second, we note that the method
to detect the envelope, cf. Westerweel et al. (2009), as
suggested by Prasad and Sreenivasan (1989) is in contrast
to the gradient trajectory approach only based on a
threshold value, though the latter is rather invariant due to
the large jump of the scalar profile across the T/NT interface, when shown over a properly normalized coordinate.
Finally, we conclude based on the above analysis that the
envelope is not embedded in the center of the scalar T/NT
interface but rather in the outer part, resulting in a mean
location at a larger radial distance to the jet’s centerline.
Figure 13 shows the results of the above analysis as
obtained at x/d = 30. In contrast to the case of x/d = 15,
this data can only be evaluated up to r~ ¼ 0:15 due to the
spreading of the jet flow and the fixed field of view. Only
small differences are found at the origin, where the
Exp Fluids (2013) 54:1621
Page 13 of 15
1
Outer flow
T/NT interface
Fully turbulent zone
Zonal probability
0.8
T/NT interface. Contributions of the latter are present on
the centerline and can be observed in radial direction up to
r~ ¼ 0:24, while the fully turbulent zone is negligible
beyond r~ ¼ 0:19.
0.6
Acknowledgments This work was funded by the NRW-Research
School BrenaRo and the Cluster of Excellence Tailor-Made Fuels
from Biomass, which is funded by the Excellence Initiative of the
German federal state governments to promote science and research at
German universities.
0.4
0.2
0
0
0.05
r /( x-x0)
0.1
0.15
Fig. 13 Profiles of the zonal probability of the three different regions
shown over a non-dimensional radial coordinate r~ ¼ r=ðx x0 Þ
obtained at x/d = 30 and Re0 = 8,610
interface is found less frequently and around the maximum
probability of the T/NT interface, which at x/d = 30 is
rather a plateau with a zonal probability of approximately
0.6. However, in general, we observe the same tendencies
as described afore for x/d = 15.
4 Conclusion
We have presented high-frequency planar Rayleigh scattering measurements of the mixture fraction Z of propane
discharging from a turbulent round jet into coflowing carbon dioxide at nozzle-based Reynolds numbers
Re0 = 3,000–8,600. Applying Taylor’s hypothesis, we
have obtained three-dimensional scalar data, based on
which we have investigated the local structure of the turbulent scalar field with a focus on the T/NT interface layer
using scalar gradient trajectories.
Therefore, a method to obtain three-dimensional data of
the mixture fraction field has been introduced in a first step.
High-speed cinematographic Rayleigh scattering imaging
is performed at different axial locations of a turbulent
propane jet issuing into CO2 coflow. Taylor’s hypothesis is
invoked to obtain a three-dimensional reconstruction of the
scalar field and the corresponding scalar gradient. As
experimental noise is present that induces artificial gradients, we apply a finite impulse response filter that allows an
accurate computation of the direction of the scalar gradient.
Based on the post-processed data, gradient trajectories
were calculated for every grid point. Examining the latter
in different regions of the scalar field allows to investigate
its local structure. To this end, the scalar field is partitioned
into a fully turbulent zone, an outer flow region and a scalar
T/NT interface. Analyzing the probability to find the
respective regions as a function of the radial distance to the
centerline revealed the meandering nature of the scalar
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