an author's
https://oatao.univ-toulouse.fr/27038
https://doi.org/10.2514/6.2016-2984
Castelain, Thomas and Gojon, Romain and Mercier, Bertrand and Bogey, Christophe Estimation of convection speed
in underexpanded jets from schlieren pictures. (2016) In: 22nd AIAA/CEAS Aeroacoustics Conference, 30 May 2016 -
1 June 2016 (Lyon, France).
Estimation of convection speed in underexpanded jets
from schlieren pictures
Thomas Castelain
1,2,∗
, Romain Gojon
1, †
, Bertrand Mercier
1, ‡
& Christophe Bogey
1, §
1- Univ Lyon, Laboratoire de M´ecanique des Fluides et d’Acoustique, UMR 5509,
´
Ecole Centrale de Lyon, 36 av. Guy de Collongue, F-69134 Ecully C´edex, France
2- Univ Lyon, Universit´e Lyon 1, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne C´edex, France
In this paper, the quality of the estimation of the convection velocity in jet shear layers
using schlieren pictures is investigated. The aim is to discuss whether the convection ve-
locity is likely to be biased if determined from schlieren images obtained at a hi gh frame
rate, as in previous experiments using the phase shift method. For this, a numerical pro-
cedure is developed in order to generate schlieren-like images on the basis of simulation
data, and applied to the results provided by the large-eddy simulation (LES) of an under-
expanded round jet at an ideally expanded Mach number of 1.56. The results obtained
from the schlieren pictures are compared with those obtained directly from the LES den-
sity fields. It is notably found that the location of the maximum of gray level fluctuations
in the s chlieren pictures corresponds well to that of the maximum of density fluctuations,
and that the convection velocity estimated for low frequencies using schlieren pictures is
underestimated for small separation distances between the two points used for the phase
shift cal culation.
I. Introduction
In aeroac oustics models for supersonic jet noise,
1, 2
one input parameter related to the flow itself is the
convection velocity of the turbulence in the jet shear layer. In recent years, different experimental methods
3–8
were applied to measure the convection velocity in high-speed jets, and when possible, its de pendency on
the frequency. In s everal studies, a schlieren system is set up together with imaging optics so that the
light intensity in the image plane of the jet is measured by use of two combined photodiodes
6, 9, 10
or of a
high speed camera.
7, 8
The data analysis seeks for instance at determining the phase difference between two
signals from a couple of sens ors. This phase s hift is related (at least partly) to c onvec tion and hence, an
estimate of the convection velocity is obtained as soon as the distance b etween the two sensors is known.
Conventional schlieren imaging is known to provide pictures whose c ontrast results from the integration of
all the light beam perturba tions from the light source to the sensor. Sharp-focusing systems
11, 12
can provide
two-dimensional slices of a flow, with a focus depth depending on the exper imental apparatus chosen. In
practical,
11
the focus depth is of the order of the diameter of the Lab-scale round supersonic jets.
This paper is an attempt to determine if the characteristics of conventional schlieren imaging (space
integration over the light path, sensitivity to gr adients of density rather than the density itself) can affect
the convection velocity estimation.
In this purpose, experimental s chlieren images of an underexpanded jet at M
j
= 1.50 are first exploited.
To estimate the convection velocity, the phase shift method is applied by using two time signals extra cted
from the time-series of schlieren pictures a cquired at a high frame ra te. The distance between the two points
is varied and its influence on the convection velocity e stimation is highlighted. To go further, a study base d
on synthetic schlieren pictures processed from 3-D data o btained by Large Eddy Simulation (les) of a round
underexpanded M
j
= 1.56 jet is presented. These pictures, obtained by using a specific numerical proce dure,
∗
Assistant Professor, Universit´e Lyon 1, France
†
PhD, currently Post-doctoral student at KTH Royal Institute of Technology, Sweden
‡
PhD student, Univ Lyon,
´
Ecole Centrale de Lyon
§
CNRS Research Scientist, AIAA Senior Member and Associate Fel low
22nd AIAA/CEAS Aeroacoustics Conference
30 May - 1 June, 2016, Lyon, France
Aeroacoustics Conferences
•• ••
~z
~y
Figure 1. Single-frame schlieren image for M
j
= 1.50. The w hite dashed area indicates the field of view
recorded at a high frame rate; the symbols indicate the typical locations where the data are extracted to appl y
the convecti on velocity estimation procedure by two-poi nt phase difference, the ax ial distance between the two
points ranging from • 2δ
z
= 8 px (ie 2δ
z
/D = 0 .055) to • 2δ
z
= 6 4 px (ie 2δ
z
/D = 0.44)
are validated aga inst reference quasi-analytic solutions. The convection velocity is es tima ted again using the
phase shift method, as for the experiments. The results are compared with those derived from the same
metho dology applied to the density field - the raw data provided by the numerical s imulations. The analysis
aims to show that the spacing between the two points affects the convection velocity estimation from the
schlieren pictures. Before presenting these results, the parameters and methods used in the experiments and
in the simulation a re first given.
II. Jets parameters and methods used
A. Experimental set-up
The experiments were done in the 10 m×8 m×8 m anechoic r oom of the Centre Acoustique, Laborato ire
de M´eca nique des Fluides et d’Acoustique at
´
Ecole Centrale de Lyon. A contoured convergent nozzle of
diameter D = 38 mm, is continuously supplied by a centrifugal compressor in unheated dry air. The wall
static pressure is measured 15 nozzle diameters upstream of the exit. Stagnation pressure is then retrieved
from the wall static pressure value thr ough the estimate of the local Mach number in the measurement
section. The superso nic jet s tudied here is associated to an ideally expanded Mach number M
j
= 1.50 and a
total temperature around 30
◦
C. The associa ted Reynolds number based on the fully expanded velocity and
the nozzle diameter D is Re
D
= 2×10
6
. A representative view of the choked jet is provided in Figure 1 using
a conventional Z-type schlieren system. The imaging system used in the following consists of a continuous
Cree XHP LED light source, a knife edge set perpendicular to the jet axis , and two f /8, 203.2-mm-diam
parabolic mirro rs arranged so that the off-axis setting is limited to 10 deg. The schlieren images are rec orded
by a high-speed P hantom V12 CMOS camera whose fra me rate is set to 430 769 Hz, with an exposur e time o f
1.87 µs. The total length of one recording is 1.21 s which corres ponds to 521472 successive images. To ensure
the high acquisition frame rate, the recorded image area is limited to a 640 px×16 px region centered on the
upper jet shear layer. Using appropriate collimating optics, the ima ge resolution is set to 0.261 mm/px.
With these settings, the field of vie w corres ponds to 4.4D in the longitudinal direction and to 0.11D in the
radial direction, as can be observed in the typical image depicted in Figure 1. In this Figure the location of
the couple points used for the estimations of co nvection velocity, arbitrary placed on the nozzle lipline and
in the middle of the third shock cell, is also represented. This position is noted x
0
in the following. Similar
results as thos e presented in section III were also obtained for other locations at various distances to the
nozzle exit along the lipline.
B. Numerical simulation of a round underexpanded M
j
=1.56 jet
A supersonic round jet has been computed by s olving the unstea dy compressible Navier-Stokes equations
using low-dispersion and low-dissipation schemes
13–15
. Further details on the simulation are provided in a
previous paper.
16
The jet is underexpanded, and is characterized by a Nozzle Pres sur
e Ratio of NPR =
P
r
/P
amb
= 4.03, where P
r
is the stagnation press ure and P
amb
is the ambient pressure. The fully expanded
Mach number is M
j
= 1.56, the exit Mach number M
e
= 1, and the Reynolds number is Re
D
= 5 × 10
4
.
The mesh contains 400 million points, with mesh spacings allowing acoustic waves with Strouhal numbers
up to St
D
= 5.6 to be well propagated. The 3-D flow density of the simulation is recorded at a sampling
frequency of St
D
= 6.4, over a volume defined in cylindrical coordinates by 200 points in the radial direction
with a maximum radial position of r = 1.5 D, 512 points in the azimuthal direction covering 2π and 491
points along the jet axis ranging up to z = 5.15D. In the following, a light beam along the Cartesian ~x axis
is supposed to go through the underexpanded jet, as represented in the figur e 2. The equations governing
refractions of the light beam due to the variations in the flow density, derived from Fermat’s principle, are
provided in the next section.
~x
~y
~z
Figure 2. Time-averaged density field o f an underexpanded, M
j
= 1.56 and Re
D
= 5 × 10
4
, jet, obtained by
simulation.
16
Isovalues of density f rom 1 to 2.6 kg.m
−3
, by step 0.2 kg.m
−3
, are presented.
C. Two-point method for convection velocity estimation
The convection velocity may be estimated from the phase lag between two signals represe ntative of the flow,
recorded at two different points in the jet shear layer. These could be density signals,
3
velocity signals
4, 5
or
optical signals depe nding on density gradients.
6, 7, 9, 10
For one physical position x within the flow, let g(x,t)
be the signal measured by the sensor used at the time t. Calling G(x,f) the Fourier transform of g(x,t) and
taking the x location as a r eference point, the cross-spec trum G
δz
(x, f) between the signal coming fro m x +
δz~z and the one coming from x - δz~z is computed using :
G
δz
(x, f) = G(x + δz~z, f) × G
⋆
(x − δz~z, f) (1)
where the
⋆
denotes complex conjugate.
The estimation o f the convection velocity U
c
is derived from the phase Φ
δz
(x, f) of the cross- spectrum
G
δx
(x, f) using :
U
c
(x, f) = 2πf
2δz
Φ
δz
(x, f)
=
4πfδz
Φ
δz
(x, f)
(2)
0
5
10
15
20
0 5 10 15
St
D
(a)
Φ
δz
(x
0
) [rad]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 5 10 15
St
D
(b)
U
c
/U
j
Figure 3. (a ) Phase lag Φ
δz
(x
0
for different couples of points centered on x
0
: δ
z
= 4 px; δ
z
= 8 px;
δ
z
= 12 px; δ
z
= 16 px; δ
z
= 32 px, (b) Estimation o f the non-dimensio ned convection velocity U
c
/U
j
for
the same couples of points centered on x
0
III. Experimental results for an underexpanded M
j
=1.50 jet
High-speed schlieren cinematography has previous ly been proposed using the Cranz-Schardin principle,
17
which a llowed for up to 4 c onsecutive schlieren images with a minimum time separ ation of 1 µs. The idea
proposed here is to use the tr emendous capac ities of high-speed digital cameras to construct time signals
extracted from a sequence of several hundreds of thousands consecutive images. Grayscale time signals
from the schlieren images would then be used as the input signals for cross- spectrum computation as in (1).
Because of the knife-edge orientation, orthogonal to the jet axis in this experimental study, the signal will
be deno ted as g
z
, w he re the subscript z is used to indicate that the contrast in these experimental schlieren
images is due to density gradients along the ~z axis.
In this study, the e xperimental signa ls were segmented into 9997 blocks of 104 p oints, obtained using a
50% overlap, to compute the mean cross-spectrum of w hich the phase is extracted. C onsidering in Figure
3(a) the evolution of the phase Φ
δz
(x
0
) as a function of the Strouhal number St
D
= f D/U
j
based o n the
nozzle diameter and the perfectly expanded jet velocity U
j
, one notices that the phase delay mo notonically
increases with St
D
for the tes ted values of δz. For the two largest values of δz, the phase shift is limited
to approximately 20 radians because noise limits the e stimation accuracy for larger values. The convection
velocity, estimated using (2) and given in Figur e 3(b), follows the behavior obtained in previo us studies,
by globally incr easing monotonically with St
D
and reaching values around 0.7U
j
or 0.8U
j
. Nevertheless, as
pointed out earlier,
3, 6, 7
the distance δz be tween the probes a ffects the estimation of the convection velocity
over the whole range of St
D
.
This influence must be taken into acc ount also becaus e the differences in U
c
are noticeable (around 10%)
for the smallest separations 2δz= 8 px and 16 px (respectively 2δz/D= 0.055 a nd 0.110, which corresponds
to 2δz= 2.1 mm and 4.2 mm), tha t are of the order of the local shear layer momentum thickness (δ
θ
≈3.3
mm for x
0
=(3.8D,0.5D) as mentioned in Fig 3.42 of a previous study
7
) and thus small with respect to the
jet diameter D. Panda
3
reminds us that the distor tion of the eddies that occurs over the s eparation distance
also influences in the phase, thus there should be at least one part of the error in the estimation of the
convection velocity that comes from the phase lag method itself. This effect is noticeable in Panda
3
for the
case of a round M
j
= 0.95 jet; for a s eparation between the two probes of around one jet diameter D at
St
D
= 1.6, the lack of coherence b etween the two experimental signals induces errors in the phase estimate.
For smaller pro bes separations, the linear dependency of the cros s-spectral phase with the probes se paration
is remarkable, which indicates that the separation between the probes induces only a small, if any, bias in
the convection velocity estimate.
To summarize, probes separation in high-speed conventional schlieren imaging affects the convection
velocity estimation, as it is also the case in previous studies
3, 6
where it is proposed to calculate this qua ntity
by averaging over multiple separation points. Considering the large differe nc e be tween these estimations