PHYSICS OF FLUIDS 20, 066101 共2008兲
Issues with measurements of the convective velocity of large-scale
structures in the compressible shear layer of a free jet
Brian S. Thurow,1,a兲 Naibo Jiang,2 Jin-Hwa Kim,2 Walter Lempert,2 and Mo Samimy2
1
Auburn University, 211 Davis Hall, Auburn, Alabama 36849, USA
The Ohio State University, 2300 West Case Road, Columbus, Ohio 43235, USA
2
共Received 15 October 2007; accepted 8 April 2008; published online 3 June 2008兲
Experiments were conducted to measure convective velocity of large-scale structures in the shear
layer of a Mach 2.0 free jet with a convective Mach number of 0.87 and Reynolds number based on
a nozzle diameter of 2.6⫻ 106. Real-time flow visualization and planar Doppler velocimetry 共PDV兲
measurements along with space-time correlation were used. The results reveal that regardless of the
measurement technique employed, whether qualitative flow visualization or quantitative PDV, the
method of seeding can have a significant effect on the ability to accurately identify, track, and
measure convective velocity of large-scale structures. This is apparent by the relatively large spread
in convective velocity results obtained in the same jet by using various methods of seeding and
qualitative and quantitative visualization. Convective velocity measurements made under the best
seeding conditions available with PDV indicate a convective velocity of structures that is close to
the value predicted by theory up to the convective Mach number of 0.87 used in the current work.
This is in contrast to previously reported convective velocity measurements 共including those of the
present authors兲 that indicated a deviation from the theoretically predicted value but were based on
measurements made under less than ideal seeding conditions. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2926757兴
I. INTRODUCTION
Uc,i =
Compressible free shear layers are encountered in a variety of modern applications. Some examples include the
base flow behind a missile or projectile, the separated flow of
a stalled airfoil, and the noise producing high-speed exhaust
of a jet engine. Over the past two decades, the study of
compressible free shear layers has largely revolved around
the concept of the convective Mach number M c, first introduced in the numerical work of Bogdanoff1 and later in the
experimental work of Papamoschou and Roshko,2 and its use
as a compressibility measure. In these works, it was found
that this parameter is effective in characterizing the growth
rate of compressible planar shear layers with respect to their
incompressible counterparts. Physically, the convective
Mach number represents the normalized velocity of largescale turbulence structures in the shear layer with respect to
either the fast or slow-speed streams. Schematics showing an
idealized view of compressible free shear layers in laboratory
and convective frames of reference are given in Figs. 1 and
2, respectively 共modeled after Coles3 and Papamoschou and
Roshko2兲. In this context, a large-scale structure is modeled
as a coherent vortex that spans the entire shear layer. For two
pressure-matched parallel streams with equal specific heat
ratios, the convective Mach number and convective velocity
are given as
Mc =
U1 − U2 U1 − Uc,i Uc,i − U2
=
=
,
a1 + a2
a1
a2
a兲
Telephone: 334-844-6827. Electronic mail: [email protected].
1070-6631/2008/20共6兲/066101/15/$23.00
共1.1兲
a 1U 2 + a 2U 1
,
a1 + a2
共1.2兲
where U1 and U2 are the high- and low-speed free stream
velocities, a1 and a2 are the speeds of sound, and Uc,i is the
theoretical isentropic convective velocity.
The convective Mach number concept has been experimentally investigated in numerous studies over the past
20 years.4–26 In many of these works, it was observed that
the convective velocity substantially differs from that theoretically predicted by Eq. 共1.2兲, results which have led to
much debate, including, for example, speculation on the
presence of shocks 共or shocklets兲 within the shear layer 共e.g.,
Hall et al.8 and Papamoschou27兲. A clear physical understanding of these results, however, has eluded researchers as
the scatter in reported values is too wide to establish any
definitive trends, including, in some cases, data which directly contradict measurements made by other researchers.
Further complicating these findings is the wide variety of
techniques 共and corresponding differing results兲 that have
been used to measure the convective velocity. These include
the following:
• traditional two-point space-time correlations,7,12,21,24
• measurements of Mach wave angles emanating from
structures contained in the shear layer,8,18 and
• time-correlated sequences of flow visualization
images.4–6,9,11,13–17,19,20,22,25
Of these three methods, the focus of this work is on convective velocity measurements attained from time-correlated sequences of planar flow images. Measurements made using
20, 066101-1
© 2008 American Institute of Physics
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066101-2
U1
M1
p1
Phys. Fluids 20, 066101 共2008兲
Thurow et al.
U1-UC
Uc
U2
M2
p2
FIG. 1. Schematic of a planar free shear layer showing the presence of
large-scale coherent structures.
UC-U2
FIG. 2. Schematic of large-scale coherent structures in a frame of reference
moving at the structure’s convective velocity.
two-point space-time correlations, by their very nature, will
sense a local convective velocity of structures based on their
transverse location in the shear layer and are not considered
representative of a large-scale structure as a whole, except in
very low-speed and low Reynolds number cases with robust
structures spanning the entire width of the shear layer. Similarly, the formation of Mach waves by turbulence structures
is not a well understood process and it is not clear that the
entire extent of the structure contributes to the formation of
the Mach waves. Flow visualization, on the other hand, captures a more complete picture of turbulence structures and
could therefore yield a better representation of the structures.
Still, the definition of a large-scale coherent structure
remains somewhat ambiguous in the context of these works.
A commonly used definition of a coherent structure is that
given by Hussain:28 “A coherent structure is a connected
turbulent fluid mass with instantaneously phase-correlated
vorticity over its spatial extent.” Applying this definition in
the context of flow visualization images of high-speed flows,
however, is not possible and time-resolved measurements of
vorticity over a two-dimensional domain in high-speed flow
field remain a difficult and daunting proposition. Thus, given
the information that is available in a typical image of a highspeed shear layer, large-scale structures are loosely defined
as any feature of the flow that has a size/shape that roughly
spans the observable shear layer. As such, the definition of a
structure will largely depend on the flow visualization technique used to image the structures. The flow visualization
techniques that have been used in the past include:
• Schlieren or shadowgraph imaging,4,5
• flow visualization based on planar laser induced fluorescence 共PLIF兲,14,17 and
• flow visualization based on light scattering from particles formed in the flow due to condensation
processes.6,9,11,13,15,16,19,20,22,25
These methods, as implemented, produce inherently qualitative images of the shear layer. The physical mechanisms responsible for the observed signal widely vary and undoubtedly will influence the nature and characteristics of the
structures observed.
For example, in the case of Schlieren or shadowgraph
images, the observed signal corresponds to the integrated
effect of density gradients over the line of sight in the flow
field, whereas for PLIF images, the signal is proportional to
the local number density of fluorescent seed molecules excited by a laser sheet. In the compressible planar shear layer
measurements of Papamoschou and Bunyajitradulya14 and
Murakami and Papamoschou,17 structures were visualized by
seeding one of the two free streams with acetone, which
fluoresces when illuminated with the fourth harmonic of a
Nd:YAG 共yttrium aluminum garnet兲 laser 共266 nm兲. By
seeding only a single stream, the shear layer is visualized by
observing the contrast in signal between the two streams.
The transverse location within the shear layer where the signal contrast is highest, however, is influenced by numerous
factors including the local flow density and the local mixture
fraction between the two streams.
Finally, the greatest number of reported convective velocity measurements is based on the flow visualization of
scattered light from small particles formed via condensation
of a suitable vapor, such as water, acetone, or ethanol. This
process can take one of two forms. In the first, commonly
referred to as a “passive scalar” technique, the vapor is
seeded into a high-speed stream and condensation occurs
upon expansion of the flow to low temperatures within the
nozzle 共i.e., supersonic velocities兲. This is the case for Refs.
11, 13, 16, 19, and 25. It should be pointed out that although
termed passive scalar, it is not a “conserved” passive scalar
as commonly seen in many incompressible flow experiments. In the second form, commonly referred to as product
formation, condensation occurs when vapor contained in a
relatively warm stream 共i.e., low speed兲 is entrained into the
shear layer where mixing with a cold stream 共i.e., high
speed兲 causes the temperature to drop. This is the case for
Refs. 6, 9, 15, 20, and 22. The exact nature of both of these
condensation processes is quite complex, as the formation
and growth of condensed particles are dependent on many
factors including the local temperature, the extent of mixing,
and the kinetic time scales for nucleation and agglomeration.
These methods tend to produce images with relatively high
contrast between the shear layer and the surrounding fluid,
but little contrast within the shear layer.
To date, the only convective velocity measurements
based on quantitative data, other than those presented here,
are the supersonic and high subsonic axisymmetric jet studies of Adelgren et al.23 and Kim et al.29 In these works, the
convective velocity of large-scale structures in forced jets
were measured using particle image velocimetry 共PIV兲. The
forcing condition allowed for phase locking of the measurements to the flow control actuator, from which an average
convective velocity could be determined. The very nature of
these experiments precludes their use to investigate naturally
occurring structures.
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066101-3
Phys. Fluids 20, 066101 共2008兲
Issues with measurements of the convective velocity
In this work, we investigate the influence that flow seeding and imaging techniques can have on convective velocity
measurements made in a Mach 2.0 共M c = 0.87兲 axisymmetric
jet. In particular, we focus on the influence that particle seeding methods have on these measurements. In addition, we
compare and contrast these results with measurements obtained from time-correlated sequences of velocity data obtained in the same flow field. For velocity measurements, we
utilize the planar doppler velocimetry technique 共PDV兲,
which determines fluid velocity based on the Doppler shift in
frequency of light scattered by particles contained in the flow
field. A distinct advantage of PDV over other techniques,
such as PIV, is that condensation processes can be used to
seed the flow field in a manner virtually identical to the flow
visualization techniques highlighted above. In this manner,
we are able to obtain both qualitative flow visualization images and quantitative velocity data from the same flow field
共and in some cases the same exact set of data兲 without having to modify the flow or experimental arrangement. This
allows us to make a direct comparison between convective
velocity measurements based on inherently qualitative flow
visualization images and convective velocity measurements
based on quantitative velocity data.
II. EXPERIMENTAL ARRANGEMENT
A. Experimental facility
All experiments were conducted in the free jet facility
located at The Ohio State University’s Gas Dynamics and
Turbulence Laboratory 共GDTL兲. The facility consists of a jet
stand and stagnation chamber to which a variety of nozzles
may be attached. Air is supplied from three five-stage compressors; it is filtered, dried, and stored in two cylindrical
tanks with a total capacity of 42.5 m3 at 16.5 MPa 共1600 ft3
at 2500 psi兲. The stagnation chamber contains a perforated
plate and two screens of varying porosity to condition the
flow to be as uniform as possible prior to entering the nozzle.
More details concerning the facility can be found in Hileman
and Samimy30 and Kerechanin et al.31
Pressure in the stagnation chamber is controlled through
the actuation of one of two Fisher control valves, arranged in
parallel, one configured for low and the other for high mass
flow rate conditions. Actuation is automatic through the use
of a Fisher–Rosemount PID-based process controller 共Model
DPR 960兲, which measures the stagnation pressure via an
attached pressure transducer and accordingly adjusts the
valve. Constant pressure can be maintained with an accuracy
of approximately ⫾0.5%. The nozzle used in this study had
a design Mach number of 2.0 and an exit diameter of
25.4 mm 共1 in.兲. The flow is unheated with a stagnation temperature of approximately 280 K. The diverging portion of
the nozzle was designed using the method of characteristics
for uniform flow at the nozzle exit. The Mach number was
experimentally determined using a pitot probe to be 2.06,
with an associated Reynolds numbers based on nozzle diameter of 2.6⫻ 106.
B. Megahertz rate PDV and flow visualization
PDV is a powerful optical diagnostic technique that can
be used to measure all three components of instantaneous
velocity over a two-dimensional plane with high spatial
resolution. PDV accomplishes this task by measuring the
Doppler shift in frequency of light scattered by moving particles in the flow field. The Doppler shift ⌬f d is related to the
fluid velocity by the expression
⌬f d =
共s៝ − o៝ 兲 ៝
· V,
␭
共2.1兲
where s៝ is the unit vector in the direction of the scattered
light, o៝ is the unit vector in the direction of the incident laser
៝ is the velocity
light, ␭ is the wavelength of the light, and V
vector of the flow. The relatively small frequency shift 共on
the order of hundreds of megahertz兲 is discriminated using
an atomic or molecular vapor filter.
The method, first described by Komine and Brosnan32
and Meyers and Komine33 共and termed Doppler global velocimetry兲, is conceptually similar to what is now known as
filtered Rayleigh scattering.34 Since these initial works, the
technique has been further developed by numerous research
groups.35–43 In the course of development, many researchers
began using the term PDV to describe the technique. This
term will subsequently be used in this work.
A typical one-component PDV system utilizes a pulsed
injection-seeded Nd:YAG laser, one or two scientific grade
charge-coupled device 共CCD兲 cameras, and a molecular iodine filter. The laser is used to illuminate a plane of the flow
with narrow spectral linewidth light. The Doppler shifted
scattered light is then split into two paths, a signal path and a
reference path, using a beam splitter and imaged onto the
camera共s兲. In this manner, the absolute absorption of scattered light, as it passes through an iodine cell placed in one
of the beam paths, is measured at every spatial location
within the object plane. For scattering by relatively large 共as
compared to molecular dimension兲 particles, this absorption
is a function of particle velocity only. Accurate calibration
and image mapping algorithms have been developed with the
result that velocity accuracies of ⬃1 – 2 m / s are now achievable. More details concerning the history of PDV, the art of
its application, and recent advances can be found in comprehensive review articles by Elliott and Beutner44 and Samimy
and Wernet.45
Typical PDV systems are limited by commercially available lasers and cameras to repetition rates on the order of
10– 100 Hz. Megahertz rate PDV extends the PDV technique
to high-repetition rates through the use of a custom built
pulse burst laser system, two ultrahigh framing rate cameras,
and a molecular vapor filter. Details of the laser can be found
in Wu et al.46 and Thurow et al.47 In this work, the megahertz rate PDV system was used to acquire both flow visualization and velocity images, eliminating the need for separate experimental arrangements. For the sake of brevity, only
a brief description of the technique and equipment are provided here with the reader referred to Thurow et al.48 for
more details.
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066101-4
Phys. Fluids 20, 066101 共2008兲
Thurow et al.
The pulse burst laser system, discussed extensively in
Thurow et al.,47 is Nd:YAG based and is capable of producing a burst of 1–99 short duration 共10 ns兲 and high energy
共order of 10 mJ/pulse兲 pulses over a timespan of approximately 100 ␮s, with repetition rates as high as 1 MHz. Images were obtained with a pair of high-speed cameras manufactured by Princeton Scientific Instruments, which have the
ability to capture sequences of 28 images at frame rates as
high as 1 ⫻ 106 frames/ s. High frame rates are achieved by
shifting charge produced on the active area of the chip to an
array of individual memory modules contained next to each
pixel location. When incorporated into a common PDV configuration, the pulse burst laser and high-speed cameras allow for the acquisition of planar velocity data at megahertz
rates with accuracies on the order of 10– 20 m / s. The development of this technique is the subject of Thurow et al.48
For the experiments described herein, the laser sheet
propagates in the upstream direction at an 18° angle relative
to the jet axis. This results in a one-component PDV system
with sensitivity to velocities in the 0.67i − 0.22j + 0.71k direction, where i is the unit vector in the streamwise direction, j
in the transverse direction, and k is the out-of-plane unit
vector. This configuration was chosen to maximize the sensitivity of the system to the streamwise component of velocity, which is an order of magnitude higher than the other two
components, but cannot be measured directly with a single
component PDV system 关see Eq. 共2.1兲兴. Experiments conducted with the laser sheet propagating at a shallower angle
produced severe aero-optic aberrations within the laser sheet
and could not produce reliable measurements. To alleviate
concerns over the abnormal velocity component being measured and used in this study, a two component megahertz rate
PDV experiment was conducted that allowed for a measurement of the streamwise component of velocity directly. Results obtained using this configuration yielded similar results
to those presented here. Two component megahertz rate PDV
was not used for the current experiments, however, due to its
decreased field of view, increased complexity, and decreased
accuracy.
In addition to velocity data, flow visualization data were
also acquired. Flow visualization images are inherently acquired in the PDV acquisition process. The signal of the
reference image, which does not pass through a molecular
filter, is unaffected by the velocity of light scattering particles
in the flow and, for all intensive purposes, is therefore identical to an image acquired using a standard flow visualization
arrangement. This unique aspect of PDV allows for the simultaneous acquisition of flow visualization and velocity images without the need for separate experiments.
For the experiments described herein, the laser was operated at 250 kHz and illuminated a 127⫻ 63.5⫻ 0.1 mm3
共laser sheet thickness兲 volume. This area was imaged onto a
160⫻ 80 pixel CCD camera, corresponding to a resolution of
0.8 mm per pixel. Both velocity and flow visualization images were postprocessed with a 3 ⫻ 3 pixel low-pass filter to
reduce noise sources such as speckle. Thus, the spatial resolution of the measurement system is reduced to effectively
⬃2.4 mm over 53⫻ 26 points. In addition, flow visualization
images were normalized on an image-to-image basis to ac-
count for fluctuations in laser intensity from shot-to-shot as
well as intensity variations across the laser sheet. This was
accomplished by fitting a smooth curve to the peak intensity
of an image profile taken transverse to the propagation direction of the laser light sheet. The implementation is similar to
that employed by Dimotakis et al.49 and effectively removes
any variations in image intensity due to the distribution of
energy across the laser sheet.
C. Particle seeding considerations
As will become clear in the next section, consideration
of the details of particle seeding is an important aspect of this
work. In contrast to PIV, resolving individual particles is not
necessary for accurate measurements with PDV. Rather, the
signal at each pixel corresponds to light scattered from large
number of small 共on the order of 50 nm兲 particles. In order to
make full field measurements in the current jet experiment,
three different regions of the flow field, the mixing layer, the
jet core, and the ambient air/coflow, must be uniformly
seeded with such small particles. In general, each of these
regions was seeded in a different manner, the details of
which are given below.
1. Mixing layer
The mixing layer is the turbulent region between the
high-speed jet core flow and low-speed ambient flow 共stagnant ambient or a very low-speed coflow兲. In this work, the
mixing layer is seeded via product formation from naturally
occurring water vapor condensation, when moist ambient air
mixes with the cold dry air in the jet core. Particles, therefore, will only form in regions of the mixing layer where
sufficient entrainment of ambient air or coflow and mixing
have occurred to lower the local temperature of the mixture
to the condensation point.
2. Jet core
The jet core is seeded by the injection of a small amount
of acetone 共⬃0.4% by mass兲 approximately 10 m upstream
of the stagnation chamber, where it evaporates and mixes
with the jet air before reaching the nozzle. Upon expansion
to supersonic flow within the nozzle, the acetone condenses
into small particles 共on the order of 50 nm兲 similar to those
produced via product formation in the mixing layer. This
technique also promotes the product formation process
within the mixing layer, as the entrained acetone particles
serve as nuclei for water condensation. Thus, even with seeding of only the jet core, the scattering signal levels within the
mixing layer are often enhanced.
3. Ambient/coflow
Product formation and acetone condensation mark the
majority of the flow field but are unable to mark the relatively warm low-speed periphery of the jet. The extent of this
unseeded region depends on the product formation processes, which depend heavily on ambient conditions 共humidity and temperature兲 as well as the jet Mach number or temperature. For these experiments, a smoke generator 共Corona
Vicount Compact 1300兲 was used to ensure adequate seeding
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066101-5
Phys. Fluids 20, 066101 共2008兲
Issues with measurements of the convective velocity
throughout the periphery of the jet. Smoke particles
共⬃0.3 ␮m兲 are seeded into a low-speed coflow of a few m/s,
with particle number density controlled to yield scattering
intensity which is approximately equal to that in the mixing
layer.
D. Two-dimensional space-time correlations
N
1 tot
兺 Fk共x,y,tk兲,
Ntot k=1
共2.2兲
共2.3兲
The correlation C between two instances of the signal is
m
Ci,j =
n
1
兺 兺 F ⬘共x,y,ti兲F j⬘共x,y,t j兲,
mn x=1 y=1 i
共2.4兲
where higher values of C represent a better degree of correlation. The correlation coefficient R is a normalized representation of the correlation and defined as
Ri,j =
Ci,j
冑Ci,iC j,j ,
Uc =
where R takes a value of 1.0 for perfect correlation 共i = j兲 and
−1.0 for perfect anticorrelated 共Fi⬘ = −F j⬘兲. For the data considered herein, we are interested in the movement of a portion of the signal 共defined as the structure兲 with time. Thus,
the overall domain 共M ⫻ N兲 is larger than the feature of interest, F 共m ⫻ n兲. One must account for the movement of the
feature in time and space by including a ⌬x and a ⌬y term,
冑Ci,i共0,0兲C j,j共⌬x,⌬y兲 .
⌬x
.
⌬t
共2.7兲
共2.8兲
As there are up to 28 共⌬x , ⌬t兲 pairs for each image sequence,
a least squares fit is used to calculate the convective velocity
for the sequence.
Space-time correlation coefficients are calculated both
on an instantaneous basis and an ensemble average basis.
The ensemble average is given by
N
具R1,j典 =
共2.5兲
Ci,j共⌬x,⌬y兲
In the current experiments, F represents either the image
intensity 共in the case of the flow visualization images兲 or the
measured component of the velocity 共in the case of the PDV
data兲. Within the first frame of each sequence, a structure is
defined as the signal contained within a window of fixed size
and location. This becomes the template with which each
additional frame in the sequence is correlated. The window
width is 3␦, where ␦ is the local shear layer thickness as
determined from flow visualization images. In addition, results not presented here show that the measurements are
largely insensitive to the window width.51 The window
height is large enough to encompass the entire transverse
extent of the mixing layer on one side of the jet core. For the
majority of measurements reported here, the center location
of the template is fixed at ⬃5.0 x / D, whereas the potential
core is ⬃9 jet diameters in length. At this location, large
structures are expected to be present within the shear layer,
yet small enough such that they may evolve without interference from the opposite side of the jet.
By definition,R1,1 is the autocorrelation coefficient with
the maximum of 1.0 for ⌬x = ⌬y = 0. The maximum correlation coefficient in R1,2 will be lower than 1.0 and its location
共⌬x , ⌬y兲 represents the new location of the structure at time,
t = t0 + ⌬t, where ⌬t is the time separation between consecutive frames in a sequence 共4 ␮s in this study兲. In a likewise
fashion, R1,3 is used to find the location of the structure at
time, t = t0 + 2⌬t and so on through all 28 frames. The convective velocity is calculated as
and the fluctuating signal is
Fk⬘共x,y,tk兲 = Fk共x,y,tk兲 − 具F共x,y兲典.
n
1
兺 兺 F ⬘共x,y,ti兲F j⬘共x + ⌬x,y + ⌬y,t j兲,
mn x=1 y=1 i
共2.6兲
Ri,j共⌬x,⌬y兲 =
A two-dimensional space-time correlation is used to
track structures in the flow as they evolve and move downstream. A structure is defined in the first frame and tracked
across the remaining frames. This allows for the calculation
of large-scale structures’ convective velocities as well as
their coherent lifetime. The procedure used here is similar to
that used by Fourgette et al.,6 Mahadevan et al.,11 Poggie and
Smits,13 Papamoschou and Bunyajitradulya,14 Smith and
Dutton,16 Murakami and Papamoschou,17 Thurow et al.,20,22
and Samimy et al.50 This procedure has been used in the past
to analyze time-correlated flow visualization data but has not
been used on quantitative data of a flow variable such as
velocity until recently.29 Conceptually, the space-time correlation used here is similar to conventional space-time correlations where data are acquired over time at two points 共i.e.,
two hot wires or pressure transducers兲; the distance between
the probes can then be varied to produce multiple separations. Data in this study, however, are acquired across a plane
and separations can be achieved by looking at different regions of each image.
Consider a two-dimensional time varying signal,
F共x , y , t兲, with dimension m ⫻ n. The signal is defined as being within an M ⫻ N domain and has Ntot realizations, or
measurements, in time. The ensemble average of the temporally fluctuating signal is
具F共x,y兲典 =
m
Ci,j共⌬x,⌬y兲 =
1 tot
兺 R1,j关x,y,tk + 共j − 1兲⌬t兴,
Ntot k=1
共2.9兲
where k represents a specific image sequence and j represents the time step within the k image sequence. The ensemble average convective velocity can then be determined
by fitting a curve through the location of peak correlation
coefficients in the ensemble-averaged data.
For the analysis of instantaneous data, the tracking of the
correlation coefficient peak can become unreliable after long
periods of time where the correlation coefficient level can
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066101-6
Phys. Fluids 20, 066101 共2008兲
Thurow et al.
t = 0 µsec
TABLE I. Summary of convective velocity measurements for a Mach 2.0
axisymmetric jet using various measurement techniques and seeding conditions.
Template
location
共x / D兲
Data type and
seeding
Mixing layer
Histogram
average Uc
共m/s兲
Histogram
rms Uc
共m/s兲
175
375
101
Flow visualization
8.0
Core/mixing layer
5.0
340
63
Mixing layer/coflow
5.0
371
62
t = 32 µsec
Velocity
Core/mixing layer/coflow
Mixing layer
5.0
8.0
295
456
64
23
Core/mixing layer
共high humidity兲
5.0
321
63
Core/mixing layer
共high humidity兲
8.0
315
62
Core/mixing layer
共low humidity兲
8.0
251
50
Mixing layer/coflow
5.0
386
64
Theoretical value 关Eq. 共1.2兲兴
287
drop to low values 共below 0.5兲. To avoid problems associated
with this, we imposed a criterion such that images were only
included in the analysis so long as a linear fit was maintained
in the x-t diagram to within 10%. In most instances, we
found that we could track structures through approximately
the first 16 frames while maintaining a ⫾10% linear fit in the
x-t diagram of the structure location. In addition, a convective velocity calculation was only deemed valid if at least
four points were used in the curve fit. Thus, instantaneous
convective velocity measurements are determined by tracking the structure through as many frames as possible. The
measurement is also subject to a bias error due to errors in
determining the scale of the image. The scale is based on the
imaging of a ruler placed at the location of the laser sheet.
This error is estimated to be ⬃5 ⫻ 10−6 m / pixel assuming
single pixel accuracy in reading the ruler image, which results in a possible bias on the order of 5 m / s for the data
reported here. This is small compared to the uncertainty in
the curve fit. Thus, we conclude that the accuracy of the
instantaneous velocity measurements is ⬃10% of the measurement.
III. EXPERIMENTAL RESULTS
A. Overview
Time-correlated flow visualization and velocity image
sequences were obtained from a Mach 2.0 axisymmetric jet,
for a wide variety of seeding conditions. For flow visualization images, three distinct seeding conditions were used:
• Mixing layer only
• Mixing layer and jet core
• Mixing layer and co-flow 共smoke兲
t = 64 µsec
t = 96 µsec
FIG. 3. 共Color online兲 Typical flow visualization image sequence obtained
using natural product formation method of seeding where only the mixing
layer is visualized. The sequence shows four images out of a series of 28
images acquired at 250 kHz.
For velocity images, four distinct seeding conditions
were used:
•
•
•
•
Mixing layer only
Mixing layer and jet core
Mixing layer and co-flow
Jet core, mixing layer and co-flow
Data were acquired over a period spanning two years. Initially, the work was focused on developing an understanding
of the physical mechanisms that lead to deviations in convective velocity from Eqs. 共1.1兲 and 共1.2兲. As the work
evolved, however, the importance of seeding conditions became clearer and the scope of the project was expanded to
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066101-7
Issues with measurements of the convective velocity
Phys. Fluids 20, 066101 共2008兲
FIG. 4. 共Color online兲 Flow visualization images acquired with 共a兲 the mixing layer and jet core seeded and 共b兲 the mixing layer and coflow seeded.
include more cases. For the earliest sets of data, the imaging
region spanned from x / D = 6.0 to x / D = 12.0 with the center
of the structure template located at x / D = 8.0. For the most
recent set of experiments, however, the imaging region was
moved upstream to visualize from x / D = 3.5 to x / D = 8.5
with the template located at x / D = 5.0. This change was
implemented to ensure that the structures being tracked
were not influenced by interactions between the upper and
lower mixing layers of the jet, which begin to coalesce at
x / D = ⬃ 9. The template location for each set of data is noted
in Table I, which gives a summary of all of the results.
It is noted that for all the cases reported here, the mixing
layer was naturally seeded 共by condensation of water vapor
contained in the entrained air by product formation兲. Also
note that previous convective velocity measurements reported by the authors were all based on flow visualization,
with only the mixing layer seeded, corresponding to the first
case listed above.20,22 For completeness, these measurements
were repeated for this work and are presented here.
B. Flow visualization and velocity image sequences
Figure 3 shows a typical flow visualization image sequence for the Mach 2.0 axisymmetric jet with only the mixing layer seeded. The images span a streamwise location
from x / D = 6.8 to x / D = 11.5. The image sequence 共acquired
at 250 kHz兲 shows a set of 4 out of 28 images, each separated by 32 ␮s. Flow is from left to right with bright regions
corresponding to the portion of the mixing layer where water
vapor has condensed into small particles. The jet core and
ambient air remain unseeded and thus do not produce any
signal. This image sequence is typical of the data presented
in previous works and illustrates the type of information obtainable with high-repetition rate imaging and to form a basis
for comparison.
Figures 4共a兲 and 4共b兲 show single flow visualization images 共each taken from a sequence of 28 images兲 acquired
with 共a兲 the mixing layer and jet core seeded and 共b兲 the
mixing layer and coflow seeded. Seeding of the jet core produces a fairly uniform signal within the jet core, with a lower
intensity than the signal in the mixing layer, thus resulting in
a fairly distinct boundary between the jet core and the mixing layer. However, the boundary between the mixing layer
and the ambient is very distinct. Seeding of the coflow with
a smoke generator 关Fig. 4共b兲兴 affords a greater level of control, such that the signal levels can be better matched between the coflow and mixing layer. Thus, the boundary between coflow and mixing layer is less distinct when the
coflow is seeded. However, the boundary between the jet
core and the mixing region is very distinct.
Figures 5共a兲–5共c兲 show typical velocity images for the
Mach 2.0 jet acquired for 共a兲 the jet core, mixing layer, and
coflow seeded; 共b兲 the jet core and mixing layer seeded on a
hot, humid day; and 共c兲 the mixing layer and coflow seeded.
As can be seen in Fig. 5共a兲, under full seeding conditions,
FIG. 5. 共Color online兲 Typical velocity images for the Mach 2.0 jet acquired for 共a兲 the jet core, mixing layer, and coflow seeded; 共b兲 the jet core and mixing
layer seeded on a hot, humid day; and 共c兲 the mixing layer and coflow seeded.
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066101-8
Thurow et al.
Phys. Fluids 20, 066101 共2008兲
FIG. 6. 共Color online兲 Mach 2.0 jet flow visualization images: 共a兲 average intensity and 共b兲 rms intensity fluctuations images.
velocity measurements can be made throughout the entire
image. In Fig. 5共b兲, no seeding is provided in the coflow, but
the high-humidity levels allow for measurements to be made
fairly far into the coflow region. Still, some regions exist at
the periphery of the jet where measurements could not be
made and the velocity is therefore arbitrarily set to zero, the
ramifications of which will be discussed in Sec. IV. Figure
5共c兲 illustrates a velocity measurement where no signal is
present in the jet core and the velocity is also arbitrarily set
to zero. Clearly, this case is not physical; however, it is included here to clearly illustrate the role that seeding can play
in convective velocity measurements.
Figure 6 shows average and fluctuating intensity values
of the flow visualization images with only the mixing layer
seeded. 共The short horizontal dotted line in the upper half of
the mixing layer corresponds to a band of bad camera pixels,
which are excluded from the convective velocity measurements.兲 Statistics were calculated based on over 250 image
sequences of 28 images each. The fluctuating intensity values show a bimodal distribution within the mixing layer with
the highest fluctuation levels at the edges due to intermittent
passage of large-scale structures. In the central portion of the
mixing layer, mixed fluid is nearly always present and, thus,
the fluctuations are much lower. It is noted that fluctuations
are slightly higher near the outer edge of the mixing layer,
the reason for which is not entirely clear at this time.
Figure 7 shows average and fluctuating values of velocity for the fully seeded flow field. Over 500 image sequences
were obtained, resulting in a total of over 10 000 individual
velocity images that were used for statistical calculations.
The images span a region from x / D = 3.5 to x / D = 8.5. Unlike
the flow visualization images, the fluctuating values exhibit a
single maximum, located near the center of the mixing layer.
Figure 8 shows average jet centerline velocity as a function
of downstream location 共x / D兲, obtained from both PIV and
PDV measurements. The average centerline velocity measured by PDV at x / D = 4.0 is ⬃340 m / s, which corresponds
to a streamwise velocity component of ⬃507 m / s, in excellent agreement with the isentropic jet exit velocity of
514 m / s at Mach 2.06, which is the measured Mach number
for the jet. The wavy variation of the PIV measured average
velocity along the jet’s centerline is due to a weak shock/
expansion wave train, which is not clearly captured by PDV
due to the lower measurement precision 共⫾10 m / s兲. Within
the jet core, velocity fluctuations at a fixed spatial location
were on the order of 18 m / s, which is ⬃5% of the jet exit
velocity and in general agreement with earlier PDV and
FIG. 7. 共Color online兲 Mach 2.0 jet velocity images: 共a兲 average and 共b兲 fluctuating velocity values obtained using PDV with full flow field seeding.
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066101-9
Phys. Fluids 20, 066101 共2008兲
Issues with measurements of the convective velocity
sented here are for the same flow field, under nearly the same
conditions. Thus, any reported changes in convective velocity are due to the measurement, and/or seeding technique, as
opposed to a change in the flow itself.
1. Flow visualization images:
Seeding of mixing layer only
FIG. 8. 共Color online兲 Jet centerline Mach number vs streamwise location as
calculated from both PIV and PDV data on the same jet.
LDV studies on Mach 2.0 jets.41 Fluctuations within the mixing layer peak around 60 m / s, which is 17%–18% of the jet
exit velocity.
C. Space-time correlation results
Convective velocity results obtained using space-time
correlation, summarized in Table I, are presented as follows.
First, we present results for flow visualization images with
only the mixing layer seeded. These results best represent the
nature of measurements previously made by the authors, and
others, and form the motivation for the measurements that
followed. Next, we present results for the fully seeded velocity images, which represent the best data available to date.
Finally, we consider the remaining seeding conditions, focusing on how seeding influences the measured convective velocity. The reader is reminded that all measurements pre-
Figure 9 summarizes the space-time correlation data for
flow visualization image sets where only the mixing layer is
seeded. Due to the rich nature of the data, the figure is divided into four parts. Parts 共a兲–共c兲 give ensemble-averaged
results for the entire data set, while part 共d兲 is a histogram of
the distribution of convective velocities obtained from each
individual image sequence. Graph 共a兲 gives streamwise slices
of the full two-dimensional ensemble-averaged correlation
coefficient level as a function of downstream displacement
关R共⌬x , ⌬y = 0兲 versus ⌬x兴 at several time separations, ⌬t.
The procedure is similar in concept to a conventional spacetime correlation coefficient except that the roles of space and
time are reversed. Graph 共b兲 is the ensemble-averaged peak
correlation coefficient level versus time and provides an indication of the coherence of large-scale structures. Graph 共c兲
gives the location of maximum correlation coefficient 共⌬x兲
versus time 共⌬t兲 and is used to calculate an ensembleaveraged convective velocity 共Uc = ⌬x / ⌬t兲. For both 共b兲 and
共c兲, the peak correlation coefficient and location are determined from the analysis of the ensemble-averaged spacetime correlation, as represented by Eq. 共2.9兲. Lastly, graph
共d兲 is a histogram of the convective velocities calculated for
each individual image sequence. The uncertainty of each realization is approximately 10% of the measured value 共see
Sec. II D兲. Figure 9 constitutes new data, similar to that pre-
FIG. 9. 共Color online兲 Twodimensional space-time correlation results for Mach 2.0 flow visualization
data with only the mixing layer
seeded: 共a兲 streamwise slice at t = 0,
16, 32, 64, and 108 ␮s; 共b兲 peak correlation coefficient vs time; 共c兲 position of peak correlation coefficient vs
time; and 共d兲 histogram of instantaneous convective velocity.
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066101-10
Phys. Fluids 20, 066101 共2008兲
Thurow et al.
a)
Correlation
0 s
1
R (x)
b)
1
20 s
40 s
60 s
0.5
0
0
20
40
x
(mm)
c)
Normalized Occurrences
Position (mm)
dx/dt = 276 m/s
15
10
5
0
0
20
40
Time (s)
0
60
20
0.5
60
1
0
20
40
Time (s)
d)
60
mean = 295 m/s std = 64 m/s
FIG. 10. 共Color online兲 Twodimensional space-time correlation results for Mach 2.0 velocity data for a
fully seeded flow field: 共a兲 streamwise
slice at t = 0, 20, 40, and 60 ␮s; 共b兲
peak correlation coefficient vs time;
共c兲 position of peak correlation coefficient vs time; and 共d兲 histogram of instantaneous convective velocity.
0.5
0
100
200
viously reported in Thurow et al.22 The most notable features
in Fig. 9 are the bimodal ensemble-averaged correlation coefficient peak in Fig. 9共a兲 and the corresponding bimodal
distribution exhibited by the histogram of Fig. 9共d兲. Further
analysis shows that the two peaks in the histogram correspond to convective velocities centered at approximately 375
and 175 m / s. The ensemble-averaged convective velocity
obtained using the first 共and largest兲 peaks from each slice in
Fig. 9共a兲 is 182 m / s, as shown in the x-t plot of Fig. 9共c兲.
These values are in stark contrast to the theoretically expected value of 287 m / s 关see Eq. 共1.2兲兴. Figure 9共b兲 indicates a structure “lifetime” of approximately 60 ␮s, based on
a 1 / e value of the correlation coefficient. These results compare very well to previously reported results22 on the same
jet but using a different camera system and imaging location.
2. Velocity images: Full flow field seeding
Figure 10 is analogous to Fig. 9, but where velocity
rather than gray scale intensity forms the basis for correlation
and where the jet core, mixing layer, and coflow are all
seeded. This condition represents the best available data set
and is based on the analysis of 500 full image sequences of
28 images each. Similar to Fig. 9共a兲, the correlation coefficient maxima in Fig. 10共a兲 are seen to decrease in magnitude
and shift downstream with increasing time. However, it is
clear that there is only a single maximum and, from Fig.
10共c兲, that this ensemble-averaged correlation coefficient
maximum translates downstream at a velocity of approximately 276 m / s. Similarly, the histogram shows a Gaussianlike distribution of convective velocities with a mean value
of 295 m / s and standard deviation of 64 m / s. Both the
ensemble-averaged and histogram-averaged convective ve-
300
400
dx/dt (m/s)
500
locity are close to the theoretically expected value of
287 m / s, as indicated by Eq. 共1.2兲. This is a striking result as
it is in direct contrast with the flow visualization results and
other measurements reported in the literature. A large-scale
structure lifetime can be estimated by looking at the time
required for the correlation value to drop to 1 / e. Extrapolating, this value is ⬃80 ␮s, which is longer than that based on
flow visualization images.
3. Other seeding conditions
The remaining cases considered in this study will now
be briefly discussed. While data similar to that presented in
Figs. 9 and 10 were produced and analyzed for each case, the
full set of graphs will not be presented here for the sake of
brevity. Table I presents the average value of convective velocity and the standard deviation determined from the instantaneous velocity realizations. For the data presented in Fig.
9, we report values for each peak of the histogram to illustrate the bimodal shape of this data set. For all other cases,
we report the average value of all realizations. The accuracy
of the mean calculation can be assessed by using the central
limit theorem, which states that
␴x̄ =
␴
冑N ,
where ␴x̄ and ␴ are the standard deviation of the mean and
the standard deviation of a set of measurements of size N.
For the fully seeded velocity data, the set consists of 500
image sequences giving a statistical accuracy of ⬃3 m / s.
The smallest data set consisted of 50 image sequences giving
an uncertainty in the mean of ⬃9 m / s. Combined with uncertainties in image scaling, we conclude that the accuracy of
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066101-11
Issues with measurements of the convective velocity
FIG. 11. 共Color online兲 Histogram of convective velocities for flow visualization image sequences with 共a兲 jet core and 共b兲 coflow seeded in addition
to mixing layer.
the average measurement for all cases is approximately
10 m / s. In addition, it should be noted that the mean value
does not correspond to the most frequently occurring value
except for a normal distribution.
In discussing these results, we begin with the additional
two sets of flow visualization images, which were obtained
under different seeding conditions. Figure 11 gives the histogram of convective velocities for flow visualization image
sequences with 共a兲 the jet core and 共b兲 the coflow seeded, in
addition to naturally occurring seeding of the mixing layer.
In both cases, the histogram indicates a relatively high value
of convective velocity with respect to theory.
There are a few issues that must be considered to properly interpret these results. One, as can be seen from the
images in Fig. 4共a兲, seeding of the jet core does not completely eliminate the boundary between the jet core and the
mixing layer, due presumably to differences in the average
seed density and/or the average size of the condensed particles. On the other hand, seeding of the coflow 关see Fig.
4共b兲兴 is effective at reducing or eliminating the distinct
boundary between coflow and mixing layer. Thus, in both
seeding cases, correlations in the images are dominated by
the interface between the high-speed jet core and the mixing
layer. Second, seeding of the jet core promotes the product
formation process within the mixing layer, as it provides
small particles that serve as a nucleus for the condensation
process to occur. It must also be noted that these data were
acquired on a hot summer day when ambient humidity levels
were fairly high. This promotes the product formation process as more water is available in the ambient air and the
condensation point is more easily reached. The net effect is
that the observed signal in the mixing layer extends further
out under these conditions with the boundary between mixing layer fluid and the coflow becoming less distinct.
Thus, for both seeding cases, the location of highest contrast in the images is located at the inner edge of the mixing
layer, at its boundary with the high-speed core. It follows
then that the measured convective velocity would be higher
for these cases, as this inner interface dominates the correlation.
We next consider the influence of seeding on the velocity
results. In the PDV technique, a velocity is computed at every pixel location where sufficient signal has been measured.
Unlike the flow visualization images, the velocity images
effectively remove any information about the underlying
Phys. Fluids 20, 066101 共2008兲
seeding levels 共i.e., size and number of scattering particles兲
and only the local flow velocity is reported. However, at
locations with weak or no signal, the velocity is arbitrarily
set to zero. For the case where only the mixing layer is
seeded, measurements are not possible in the jet core or in
the outer periphery of the jet. From a space-time correlation
perspective, the contrast between the jet core, where velocity
is wrongly set to zero, and the mixing layer is much greater
than between the mixing layer and the coflow, where the
velocity is relatively low and zero is nearly a valid assumption. Thus, the most defining flow features will be at the
high-speed interface between the mixing layer and core fluid.
This is clearly evident in the convective velocity measurements, which have an average value of 456 m / s, much
greater than expected. Moreover, the standard deviation is
only 23 m / s, indicating that this is a very dominant feature.
Conversely, consider the case where the jet core is
seeded, but the coflow is not. Two instances are reported
here: one attained with low-humidity ambient conditions and
the other at high-humidity ambient conditions. We consider
the low-humidity case first. In this instance, as shown in
Table I, the convective velocity is ⬃250 m / s, which is lower
than expected. Under low-humidity conditions, there is no
sufficient signal to allow velocity measurements to extend to
the low-speed coflow surrounding the jet. In fact, analysis of
individual images indicates that the lowest measurable velocity is ⬃100 m / s. Arbitrarily setting the velocity to zero, for
lack of better information, in these regions introduces an
artificial interface between the mixing layer and coflow that
is represented in the template used for structure tracking. It
then follows that a lower convective velocity results.
Under high-humidity conditions, velocity measurements
can be made further out in the mixing layer. Inspection of
images for high-humidity conditions shows that velocities as
low as 30 m / s could be measured. This still produces an
artificial gradient between the mixing layer and coflow; however, the magnitude of the gradient is not as strong as the
low-humidity case. This is consistent with the higher convective velocity measurement of ⬃320 m / s measured in this
case, as shown in Table I, which is higher than the theoretically expected value. Actually, in this case, the naturally occurring velocity gradients within the structure are now on par
with or greater than the gradients occurring at the low-speed
interface between seeded and unseeded fluids. These gradients, however, do not extend into the low-speed portion of
the jet. One possible consequence is that the correlation measurement may be biased toward higher speeds as a large
portion of the low-speed section of the structures is missing.
To test this hypothesis, an algorithm was devised where
the correlation was only calculated using portions of the image where a velocity measurement was obtained. Constructing the algorithm is complicated by the time-evolving nature
of the flow and the constantly changing regions where measurements are available, but the general results support the
ideas given above. For example, for the low-humidity case,
the revised algorithm computed a convective velocity measurement of 347 m / s. This shows that the artificial interface
between the mixing layer and coflow was removed, but that
now the measurement is biased by the fact that a significant
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066101-12
Thurow et al.
portion of the low-speed side of the structure is missing in
the data. More details on this analysis can be found in
Thurow.51
Lastly, velocity measurements with core seeding and
high-humidity ambient conditions were made on two
separate occasions at two different downstream locations,
x / D = 5.0 and x / D = 8.0. The convective velocities for these
two cases are 321 and 315 m / s. This indicates two things:
one, the experimental results are repeatable, and two, the
effect of downstream location of the structure template is not
significant for the cases reported here. This is also supported
by additional data reported in Thurow.51
IV. DISCUSSION OF RESULTS
In this work, a large combination of experimental techniques and/or seeding conditions was used to investigate the
convective velocity of large-scale structures in the shear
layer of a jet. However, rather than producing identical results, as might be expected, the experimentally determined
convective velocities ranged from 175 to 450 m / s, an approximately ⫾50% variation from the predicted mean value
of 287 m / s. We consider this to be a significant finding for
many reasons. First, these experiments demonstrate the limitations of inherently qualitative techniques, such as the flow
visualization technique employed here. Second, even when
using quantitative diagnostics, it is clear that the importance
of adequate and uniform seeding cannot be overlooked. Finally, when properly implemented, these results represent a
new contribution to our understanding of the convective velocity of large-scale structures in compressible free shear layers. We will now look at these points in more detail.
A. Convective velocity measurements based
on flow visualization images
Flow visualization is a significant but an inherently
qualitative tool for the investigation of turbulent flow fields.
A focus of this study, and many earlier studies, has been on
the determination of convective velocity from time sequences 共often pairs兲 of such images. The results presented
here illustrate that consideration of the details of the flow
visualization method is essential for proper interpretation of
the results. This is particularly evident when comparing the
data presented in Fig. 9 共traditional flow visualization兲 and
Fig. 10 共fully seed PDV data兲. Both sets of data were acquired from the same flow field but give completely different
pictures of the dynamics of large-scale turbulence structures.
To better understand how these drastically different pictures
can arise, we take a closer look at the correlation procedure
used to track structures in these image sequences.
In this study, structures are defined by the signal 共either
intensity or velocity兲 contained within a predefined region of
each image. For the calculation of convective velocity, one is
most concerned with determining the location of peak correlation in space, ⌬x and ⌬y, at different instances in time. In
this context, what are the defining features of a structure that
will influence this determination? Consider a signal with
strong spatial gradients 共i.e., high contrast features such as
Phys. Fluids 20, 066101 共2008兲
sharp edges兲. In this case, the correlation level will be quite
sensitive to the values of ⌬x and ⌬y as even a slight shift
will cause a large mismatch between the two instances in
time being compared. Conversely, consider a signal whose
spatial gradients are more spread out and moderate. Increasing or decreasing ⌬x and ⌬y will have a much smaller effect
on the correlation level as the signal gradually changes over
these distances. It follows that the most distinguishing features of a large-scale structure will correspond to the sharpest
spatial gradients.
Flow visualization images acquired using product formation for particle seeding are primarily characterized by the
boundary between mixed and unmixed fluids. Within regions
of intense mixing 共i.e., the center of a structure兲, the intensity
is fairly constant. In regions with little to no mixing, condensation does not occur and the intensity level is approximately
zero. The largest spatial gradients thus occur at the interface
between mixed 共seeded兲 and unmixed 共unseeded兲 fluids. This
is supported by the mean and fluctuating intensity images
given in Fig. 6, where the largest fluctuations are found near
the edges of the mixing layer and only small fluctuations are
found at the center. It follows that space-time correlations of
the flow visualization data will exhibit two convecting peaks,
one associated with each boundary/edge. When a seeded region protrudes into the high- or low-speed flow, the observed
convection velocity will be, respectively, faster or slower
than the predicted value. Thus, the measurement of convection velocity based on product formation flow visualization
tends to be biased either high or low, as observed in the
Mach 2.0 jet data of Fig. 9.
Should this imply that structures within the Mach 2.0 jet
are moving at two different speeds? No, rather, it is simply
stating that the boundaries between mixed and unmixed fluids 共at least as defined via product formation兲 is moving at
these two speeds. As a structure moves, it will be subjected
to shear forces such that the local velocity will be higher near
the high-speed stream and lower near the low-speed stream.
As these high contrast boundaries are located in these regions, it is reasonable to expect that they would move at, or
near, the local velocity. With this interpretation, these results
are consistent with the one-dimensional correlation results of
Samimy et al.7 and Elliott et al.,10 which showed significant
variation of convective velocity in the lateral direction 共direction of shear兲.
These results should not be taken as a blanket statement
that flow visualization images can never be used for quantitative measurements. Rather, they place additional emphasis
on understanding the physics of the visualization process and
the mathematical procedure for extracting the desired quantities. Properly understood, flow visualization still remains a
powerful tool for the investigation of turbulence structures.
B. Convective velocity measurements based on
velocity data
Convective velocity data obtained using PDV are not
subject to the same seed bias as that obtained from flow
visualization since utilizing the ratio between the signal and
reference images effectively removes any artificial seed-
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066101-13
a)
Correlation
R ( x)
b)
1
0 s
1
20 s
40 s
60 s
0.5
0
20
x (mm)
c)
Position (mm)
20
15
dx/dt = 183 m/s
10
5
0
20
40
Time (s)
60
0.5
0
40
Normalized Occurrences
0
0
Phys. Fluids 20, 066101 共2008兲
Issues with measurements of the convective velocity
1
0
20
40
Time (s)
d)
60
mean = 204 m/s std = 54 m/s
FIG. 12. 共Color online兲 Two-dimensional space-time
correlation results for Mach 1.3 velocity data: 共a兲
streamwise slice at t = 0, 20, 40, and 60 ␮s; 共b兲 peak
correlation coefficient vs time; 共c兲 position of peak correlation coefficient vs time; and 共d兲 histogram of instantaneous convective velocity.
0.5
0
100
200
dx/dt (m/s)
induced gradients. Thus, the velocity field, even only a single
component, represents an inherently more tenable description of the shear layer flow and the turbulence structures
within.
Inherent to this description is the assumption that adequate seeding is provided throughout the measurement region to obtain a velocity measurement at all points in the
flow. Inadequate seeding in PDV presents problems similar
to those encountered in the application of PIV to a flow field.
Specifically, a measurement can only be made where particles are present. In cases where complete seeding was not
provided, two effects take place that can compromise calculation of the convective velocity. First, arbitrarily setting the
velocity to zero 共or any value兲 where there is no particle to
provide a measurement creates an artificial boundary between the seeded and unseeded fluids that will bias the
space-time correlation procedure. Second, even when the unseeded region is completely excluded from the correlation
algorithm, the incomplete measurement of the distribution of
structures can also result in a bias. Thus, if the low-speed
side is inadequately seeded, the representation of the structure will be weighted toward its high-speed side. These ideas
are reflected in the results presented in Table I.
C. On the convective velocity of large-scale structures
in compressible free shear layers
Considering the arguments posed above, only one experiment met the conditions necessary to produce reliable
convective velocity measurements. Specifically, we focus on
the PDV data acquired in a fully seeded flow field, presented
in Fig. 10. The most striking result is that the convective
velocity determined from these images was very close to the
theoretically predicted value given by Eq. 共1.2兲. The convective velocity calculated from the ensemble-averaged spacetime cross correlation was 276 m / s, the average convective
velocity from the histograms was 295 m / s, and the theoreti-
300
cally expected value is 287 m / s. Thus, the experimentally
measured value agreed to within 4% of the expected value.
In light of these results, additional PDV measurements
were made on a Mach 1.3 共M c = 0.59兲 axisymmetric jet. The
jet was operated in the same facility as the Mach 2.0 jet and
both the jet core 共methyl alcohol兲 and coflow 共smoke generator兲 were seeded in addition to the natural seeding present
in the mixing layer. The field of view of the imaging system
was x / D = 1.5 to x / D = 6.0 and the template was centered at
x / D = 2.7. The results of the space-time correlation analysis
are given in Fig. 12. The theoretically expected convective
velocity was 205 m / s, while the experimentally measured
values are 183 and 204 m / s for the ensemble-averaged and
histogram-averaged determined results. The reason for the
discrepancy between ensemble-averaged and histogramdetermined values is not exactly clear; however, both values
are close to the theoretically expected value. If only the first
ten points, which have a higher correlation level, are used for
the x-t least squares curve fit in Fig. 12共c兲, the ensemble
average value is much closer to the theoretical value.
These are truly striking results considering that previous
measurements, based on flow visualization images, indicated
the potential for a higher or lower than theoretical convective
velocity to dominate for convective Mach numbers above
0.5. The present results indicate that this is not the case, but
rather that the original description presented by Bogdanoff1
and Papamoschou and Roshko2 may not be affected by compressibility.
We must temper this discussion somewhat by understanding the limitations of the current experiments. For one,
only a single component of velocity was measured. Second,
the definition of a large-scale structure in the context of the
current work is somewhat untenable. A more robust definition, such as the one based on vorticity 共e.g., Hussain28兲, is
desired, but still beyond our present experimental capabilities. It is our opinion that the issues presented in this work
Downloaded 19 Sep 2008 to 164.107.168.44. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
066101-14
Phys. Fluids 20, 066101 共2008兲
Thurow et al.
will not be completely resolved until more detailed 共i.e., resolution in space and time兲 vorticity measurements are made
in a similar flow field. Even then, the three dimensionality of
the structures must be considered and may play a role in the
measurements. In light of the present results, the recent work
by Kastengren et al.52 has examined various structure eduction methods for flow visualization and/or velocity images
that may lead to an improved definition of large-scale structures and allow for more detailed analysis of the current set
of data; this is left for a future exercise. Third, the current
measurements are confined to two convective Mach numbers, M c = 0.59 and M c = 0.87, thus preventing the extrapolation of these results to higher convective Mach numbers.
Numerical analyses suggest the presence of additional instability modes 共including fast and slow modes兲 for M c ⬎ 1.0,
an issue that cannot be addressed in this work.53–56 Finally,
the measurements presented here were performed on an axisymmetric mixing layer and thus do not account for the influence that flow geometry 共e.g., axisymmetric versus planar兲
may have. In spite of these shortcomings, the data presented
here represent the most complete set of data available.
V. CONCLUSIONS
Multiple experiments were conducted to measure the
convective velocity of large-scale structures in a Mach 2.0
axisymmetric free jet. The experiments consisted of both
qualitative flow visualization and quantitative PDV measurements under a variety of seeding conditions. Across all of the
techniques employed, the convective velocity measurements
significantly varied. For the case under the most ideal conditions 共e.g., fully seeded PDV兲, the convective velocity was
found to be close to the theoretically predicted value, a result
that has eluded measurement using other techniques. It is
quite clear from the results presented here that, at the very
least, the effects of compressibility on the convective velocity of large-scale structures need to be re-examined. As discussed in the Introduction, a large number of researchers
共including the present authors兲 have reported convective velocity measurements that significantly deviate from the theoretically expected value with increasing compressibility levels. The common trait among all of these works is that these
measurements were based on flow visualization images. It is
clear from the present work, however, that convective velocity measurements based on flow visualization images or inadequately seeded PDV data can be misleading. A comprehensive analysis of each of these techniques is beyond the
scope of this article. In general, we can comment that an
implicit assumption made in all of the previous works is that
the flow visualization method employed adequately captured
the full extent of the large-scale structures in the flow. In
reality, however, the techniques used by different researchers
vary widely and each possesses its own unique set of characteristics that lead to different representations of large-scale
structures for each case. Thus, previously reported convective velocity measurements must be carefully considered before drawing any conclusions on the dynamics of large-scale
structures in compressible free shear layers. The present results indicate that the convective velocity of large-scale
structures is close the theoretically expected value; however,
further work is necessary to validate this and to further our
understanding of compressibility effects on turbulent free
shear layers.
ACKNOWLEDGMENTS
This research was supported in part through grants provided by the National Science Foundation and the Air Force
Office of Scientific Research.
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