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