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Large Scale Fluctuations in an Axisymmetric Sudden Pipe Expansion with Large Aspect Ratio

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The aim of the present work is the investigation of the turbulent flow field downstream of an axisymmetric sudden expansion with a large aspect ratio of D/d = 12,3. For the fundamentally understanding of the flow some numerical results are presented. They were achieved by using the RANS approach and SST turbulence model. The flow field is characterized by a jet-like flow near the nozzle exit and a large toroidal recirculation zone. The x-component of the velocity u was measured using one-component laser Doppler velocimetry. Axial and radial velocity distributions as well as some velocity spectra were measured. The spectra were calculated from the velocity signal using the Sample-And-Hold method together with the refinement technique. At the axial half length of the recirculation zone at the edge of the jet flow a narrow band peak was observed in spectra, suggesting the existence of large-scale fluctuations or instability of the flow field. Further investigations reveal that this effect is locally limited and shows no sensitivity against changes of the inlet conditions, e.g. the Reynolds number and velocity profile.
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Journal of Applied Fluid Mechanics, Vol. 11, No. 4, pp. 877-883, 2018.
Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.
DOI: 10.29252/jafm.11.04.28311
Large Scale Fluctuations in an Axisymmetric Sudden Pipe
Expansion with Large Aspect Ratio
A. Ringleb1†, W. Schlüter2, O. Sommer1 and G. Wozniak1,3
1 Institute of Mechanics and Thermodynamics, Chemnitz University of Technology, Chemnitz, Saxony, 09126,
Germany
2 Faculty of Engineering Sciences, Ansbach University of Applied Sciences, Ansbach, Bavaria, 91522,
Germany
3 Suvis GmbH, Altchemnitzer Str. 11, Chemnitz, Saxony, 09120, Germany
Corresponding Author Email: ansgar.ringleb@online.de
(Received August 12, 2017; accepted January 24, 2018)
ABSTRACT
The aim of the present work is the investigation of the turbulent flow field downstream of an axisymmetric
sudden expansion with a large aspect ratio of D/d = 12,3. For the fundamentally understanding of the flow
some numerical results are presented. They were achieved by using the RANS approach and SST turbulence
model. The flow field is characterized by a jet-like flow near the nozzle exit and a large toroidal recirculation
zone. The x-component of the velocity u was measured using one-component laser Doppler velocimetry.
Axial and radial velocity distributions as well as some velocity spectra were measured. The spectra were
calculated from the velocity signal using the Sample-And-Hold method together with the refinement
technique. At the axial half length of the recirculation zone at the edge of the jet flow a narrow band peak was
observed in spectra, suggesting the existence of large-scale fluctuations or instability of the flow field. Further
investigations reveal that this effect is locally limited and shows no sensitivity against changes of the inlet
conditions, e.g. the Reynolds number and velocity profile.
Keywords: Axisymmetric sudden pipe expansion; Laser Doppler velocimetry; Sample-and-Hold;
Incompressible turbulent flow; Large scale fluctuations; Narrow-band peak spectra.
NOMENCLATURE
d nozzle diameter
D pipe diameter
f frequency
fc particle cut-off
lBW focal length
n burst rate
N number of bursts
P laser power
Red Reynolds number
S11 power spectral density
Sr Strouhal number
Srdom dominant Strouhal number
t wall thickness
T period length
UJ jet nozzle velocity
U0 velocity along the pipe axis
u' velocity fluctuations
xR reattachment point
xS length of jet-like region
κ wave number
λ wave length
μ molecular viscosity
ρ fluid density
1. INTRODUCTION
The axisymmetric sudden expansion is character-
ized by its inlet of diameter d and the circumferen-
cial wall (pipe) of diameter D (Fig. 1). The inlet
flow is affected by the design of the inlet which
could be a smooth contraction nozzle or a straight
pipe and will cause a significantly different inlet
profile. The inlet flow is characterized by the aver-
age velocity UJ and the decay of the jet by the ve-
locity along the pipe axis U0. Due to the pipe wall a
toroidal recirculation zone encloses the jet flow and
reattaches at xR. The straightened flow goes down-
stream and leads to a developed pipe flow.
By the use of the π-theorem of Buckingham and
under consideration of the molecular viscosity μ
and the fluid density ρ a set of two similarity pa-
rameters can be found - the Reynolds number (1)
A. Ringleb et al. / JAFM, Vol. 11, No. 4 pp. 877-883, 2018.
878
and the aspect ratio (2).
Fig. 1. Schematic of sudden pipe expansion flow
1
Red
J
dU

 (1)
DD
d
(2)
Furthermore, for unsteady flows another parameter
for the description of its characteristics is necessary,
the periodic time T or its correspondent frequency
f = 1/T. Therefore, an additional similarity
parameter can be found - the Strouhal number Sr
(3).
fJ
d
f
Sr
U
 (3)
For small aspect ratios (e.g. D/d < 3) and for a wide
range of Reynolds numbers (101 < Red < 105) this
type of flow is very well investigated. Gould et al.
(1990) presented a data set of two-dimensional laser
Doppler velocimeter (LDV) measurements for a
diameter ratio of D/d = 2 and a Reynolds number of
Red = 56,000. In this study numerous radial profiles
of the axial velocity u and radial velocity v, it’s
normal stresses, Reynolds stresses and turbulent
kinetic energy are given up to a distance of x/d = 14
downstream the nozzle. It makes the study suitable
for validation of the present experimental apparatus
also because of a detailed description of the used
flow system.
In Furuichi et al. (2002) the ultrasound Doppler
velocity profiling (UVP) method was used for
measurements of a flow with D/d = 1.8 and a
moderate Reynolds number of Red = 3,820. In the
recirculation zone complex large-scale structures
were observed that interact with the separated shear
layer upstream and grows further downstream.
For similar diameter ratios two numerical studies
deal with flows at much higher Reynolds numbers
of Red 50,000. Sagar et al. (2011) applied the
renormalized group (RNG) k-ε turbulence model
and found good agreement of the skin friction factor
in comparison to experimental results. Bae et al.
(2013) compared several turbulence models and
pointed out that the standard k-ε model and
Reynolds stress model (RSM) provide better
agreement with experimental data than the SST
model.
Hammad et al. (1999) used the particle image
velocimetry (PIV) technique for the investigation of
the same diameter ratio of D/d = 2, but at a much
lower Reynolds number of 20 < Red < 211, which
caused laminar flow. The paper contained vector
plots as well as streamline plots that will help to
gain an understanding of the two-dimensional
velocity field. A major issue is the knowledge that
the reattachment length of the recirculation zone
increases linearly with Red. The same flow was
used by Ray et al. (2012) and Carrillo et al. (2014)
for the validation of their numerical studies. Both
showed grid studies and confirm good agreement
between their numerical results and the
experimental results of Hammad.
However, a comparable data base for larger aspect
ratios (e.g. D/d > 6) does not exist. This was firstly
stated in a study by Rinaldi (2003), where a jet ends
in a rectangular space with large length ratios
(approx. 10 times the nozzle diameter). He
presented LDV measurements that proof a self-
similarity of the flow in terms of free jet flow.
A study of a confined jet flow with large diameter
ratios of D/d = {15.7; 24; 40.8} is given by Khoo et
al. (1992). Using the PIV method, they investigated
the turbulent spatial length scales and pointed out
the existence of isotropic turbulence. But the
differences in the experimental setup should be
noted. In Khoo a grid at the nozzle wall at x = 0 was
used, which allows a backward flow. Moreover, a
linear decay of the root mean square (rms) velocity
up to x/D = 5 was observed.
The effect of the inlet design on the flow field for
the jet flow was studied by Mi et al. (2001). Due to
the nozzle contraction a block velocity profile result
in the nozzle exit with a laminar shear layer that
become instable downstream the nozzle and result
in a vortex roll-up of the shear layer. This effect can
be detected as a narrow-band peak in the velocity
spectra near the nozzle exit.
The aim of the present work is the investigation of
the turbulent flow of an axisymmetric sudden pipe
expansion with a large diameter ratio of D/d = 12.3
and a nozzle Reynolds number of Red = 104. At the
axial half-length of the recirculation zone at
x = xR/2 a narrow band peak in the velocity spectra
can be observed, which will be discussed in detail.
Upstream, the flow shows characteristics of the free
jet flow, where the flow is parallel to the pipe axis.
Downstream, a strong velocity decay due to the end
of the recirculation zone can be observed.
2. EXPERIMENTAL INVESTIGATION
2.1 Apparatus and Instrumentation
A one-component laser Doppler velocimeter (LDV)
of typ fp50Shift (ILA Intelligent Laser Applications
GmbH, Juelich, Germany) was used to measure the
axial component of the flow velocity u. The LDV
probe was arranged in backward scattering mode
(Fig. 2) and uses a laser of typ Nd:YAG with a
maximum power of P = 75 mW, a wave length of
λ = 532 nm and a focal length of lBW = 140 mm. A
pipe of silica glass with a wall thickness of
t = 1.5 mm was used as optical access. The
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879
accuracy of the test facility was evaluated for
adjusting the flow rate or rather the nozzle Reynolds
number Red. Therefore, the measurement error can
be determined to be about ΔRed = 4%, which leads
to an overall measurement error for the Doppler
frequency fD of the LDV system of ΔfD = 7%.
Fig. 2. Schematic of the test facility and LDA
probe
For the present study two types of nozzles were
used. First, a smooth contraction nozzle (illustrated
in Fig. 2) with an inlet diameter d (nozzle or pipe)
of d = 0.006 m. A contraction ratio of the diameter
of the stagnation chamber to the nozzle diameter of
23 results. Therefore, a block-like velocity profile
can be expected that is similar to Fig. 3. Second, a
straight pipe expansion of the nozzle with a length
to diameter ratio of l/d = 13.2 was applied. This will
cause a non-fully developed turbulent pipe flow and
a sufficiently developed turbulent boundary layer
can be expected. The exact jet exit conditions were
not measured. While Mi et al (2001) used a
contraction ratio of 5.7 and a length to diameter
ratio of l/d = 72, similar jet exit conditions can be
assumed.
The aspect ratio of the pipe and nozzle diameter
was chosen to be D/d = 12.3 to establish a typical
large aspect ratio flow field. For air flow a
Newtonian viscid behavior can be assumed, the
viscosity amounts to ν = 1,6·10-5 m2/s. An
appropriate flow velocity UJ was chosen to achieve
turbulent flows with Reynolds numbers of
Red = {5·103; 1·104; 2·104}. While a maximum
Reynolds number of Red = 2·104 lead to a velocity
of UJ 53 m/s incompressible flow (Ma < 0.3) can
be assumed.
2.2 Experimental Procedure
The power spectral density S11 was calculated by
using the software „kern.exe” that is provided by
Nobach (2015). Hence, the Sample-And-Hold
(S+H) reconstruction method in combination with
the refinement technique was applied. The S+H is
one of the fundamental strategies for estimating a
power spectral density (PSD) based on a non-
equidistant sampled velocity-time signal like it is
caused by LDV measurements, wherein the velocity
at each Doppler burst is sampled and held. A
detailed investigation of the S+H is given for
instance in (Adrian and Yao 1987). The PSD
estimation using the S+H method is limited due to
the mean particle rate, the so-called particle cut-off
frequency fc. The so-called refinement technique
overcomes this limitation (Nobach et al. 1998).
Thus, the measurement system noise can be
estimated that leads to a valid PSD estimation
above the particle cut-off frequency. The velocity
signal was low pass filtered with a cut-off frequency
of fLP = 10·fc. The particle cut-off fc=n/2π is caused
by the burst rate n and amounts for all
measurements approximately fc = 80 Hz. The
number of bursts was N = 105.
2.3 Data Validation
The performance of the experimental apparatus and
the applied LDV technique were validated for the
mean and variance of the velocity component u at
the sudden pipe expansion with experimental data
provided by Gould et al. (1990). Therefore, the
nozzle diameter amounts to d = 13.05 mm, the
diameter ratio to D/d = 1.9 and the nozzle
contraction ratio to 10.6. A block profile of the
mean axial velocity (Fig. 3) that causes a thin layer
profile of its axial fluctuations (Fig. 4) can be
observed at the nozzle exit, like it can be expected
from a smooth nozzle contraction. These inlet
conditions are very close to that one of Gould.
Fig. 3. Mean axial velocity profile at x/d=0.5
Fig. 4. Normalized axial fluctuations at x/d=0.5
The overall deviation was smaller than 5%. In
regions of large gradients (u/r or u'²/r, e.g. at
the edge of the jet flow) the maximum difference of
the mean value is less than 20% and of the variance
less than 30%. Especially in the region of the
recirculation zone, what is important for the spectral
results that presented later on, the present
measurement results are in very good agreement
with the data of Gould (Figs. 5 and 6). There, the
maximum difference of the mean value is less than
6% and of the variance less than 25%.
The estimation of the power spectral density was
validated on the free jet with data provided by Mi et
al. (2001). Therefore, measurements were
performed at Red = 16,000 (nozzle) in the shear
layer of the laminar core at x/d = 3 and r/d = 0.25.
The signal length was N = 105 and the particle cut-
off was approximately fc = 200 Hz. In comparison
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880
Fig. 5. Mean axial velocity profile at x/d=2.5
Fig. 6. Normalized axial fluctuations at x/d=2.5
to the experimental data provided by Mi a similar
spectrum could be measured (Fig. 7). The observed
large-scale structures which were caused by a
nozzle formed inlet lead to a spectrum with a local
maximum and a dominant frequency. The present
apparatus and procedure shows a dominant
frequency at Sr = 0.37 which corresponds to the
results of Mi where Sr = 0.4 was found.
Fig. 7. Spectra in the laminar shear layer of a jet
3. NUMERICAL SIMULATION
The numerical simulations were carried out by using
the software STAR-CCM+® version 8.02. Therefore,
the RANS approach and the standard SST turbulence
model were applied. The flow was described as a 2D
region due to its axisymmetric behavior. At the inlet a
velocity boundary condition was set. According to the
experimental data (see Fig. 3) an ideal block profile of
the velocity was assumed. Neglecting the fluctuations
profile a uniform turbulence intensity of 1% was set.
At the outlet a pressure boundary condition with a
reference pressure of zero was adopted. A detailed
description of the numerical procedure e.g. mesh data
and validation on test cases is given in Ringleb et al.
(2016).
4. RESULTS AND DISCUSSION
For the investigated diameter ratio of D/d=12.3 a
flow field with a large recirculation zone can be
expected (Fig. 8), which extends downstream up to
a length of 36 times the diameter (x/d = 30). In the
upstream corner a smaller recirculation zone is
simulated. Both are arranged circumferentially
around the jet flow at the pipe axis, which leads to
the description of a toroidal recirculation zone.
The jet flow exits the nozzle at x = 0 and forms a
jet-like flow up to an axial distance of x/d = 20.
Further downstream the jet spreading increases due
to a stronger streamline curvature that is caused by
the end of the recirculation zone, which is marked
by the point of reattachment at x = xR. For x > xR
pipe flow develops.
Fig. 8. Numerical prediction of streamline
distribution and normalized mean axial velocity
In the proximity to the nozzle exit the laminar core
region extends up to x/d = 5. At its edge the shear
layer expands downstream, where the axial velocity
fluctuations have their maximum (Fig. 9). The
fluctuations preserve much far downstream beyond
the reattachment point. Caused by the strong
curvature of the streamlines the jet flow spreads and
must slow down because of mass continuity. The
turbulent fluctuations cannot slow down in the same
way because of their mass inertia. This leads to a
high amount of fluctuations, whereas the flow
velocity is quite small, which causes a high amount
of normalized fluctuations.
Fig. 9. Numerical prediction of the normalized
axial fluctuations
The static pressure upstream the reattachment point
is strongly subjected to the jet flow (Fig. 10). In the
core region a significant pressure gradient (p/r) is
caused by the jet shear-layer. A much more constant
radial pressure distribution arises for x/d 15,
where the center of the recirculation zone is located,
that will become more smooth further downstream.
The main pressure drop occurs at the end of the
recirculation zone between 20 < x/d < 30, where a
maximum streamline curvature exists and the flow
velocity slows down.
Fig. 10. Numerical prediction of normalized static
pressure field
For sufficiently large aspect ratios a spectrum with a
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881
narrow band peak can be observed at the edge of the
jet flow at the axial half length of the recirculation
zone (Fig. 11). Its maximum is located at
approximately Sr = 5·10-4 and corresponds for
Red = 104 to a frequency of approximately 2 Hz.
This seems to be a locally restricted effect because
it could only be observed at the edge of the bounded
jet at the end of the jet-like region. The jet-like
region extends ap-proximately to xS/d xR/2 and
shows a similar decay of the centerline velocity U0
as well as a similar progress of the turbulence
degree at u'/U0 0,3 to the free jet. After the jet-
like region a transition from jet to pipe flow
follows, which includes the reattachment of the
recirculation zone at xR/d 50. In contrast to the
jet edge spectra (Fig. 11), the spectra in the
proximity to the inlet and to the reattachment point
do not show any peak. Especially the latter shows
turbulent like spectra with its typical -5/3 decay of
the cascade process (Fig. 12).
Fig. 11. Spectra at the edge of the jet flow at
r/d=2 and x/d=20
The results described above refer to an inlet formed
as a smooth contraction (nozzle) which results in a
block velocity profile and a sharp shear layer at the
inlet. The use of a straight pipe upwards the inlet
leads to a turbulent velocity profile with a typical
boundary layer and an appropriate turbulence
profile. For the free jet problem, the change from
nozzle to pipe inlet leads to different flow
structures. Caused by the nozzle a vortex roll-up in
the laminar shear layer of the core region can be
observed (Mi et al. 2001) which leads to spectra
with a narrow band peak. The vortex roll-up results
in large-scale turbulent structures and causes a
faster decay of the centerline velocity than the pipe
inlet flow. The latter one causes sufficiently smaller
turbulent structures and a softer decay of the
centerline velocity. This phenomenon is known as
“jet inlet anomaly” (Boersma et al. 1998). Such a
behavior cannot be determined in any spectra of the
present flow field. Both spectra for pipe and nozzle
inlet look similar to each other and show no
indication to different turbulent structures (Figs. 11
and 12).
Moreover, there is a significant sensitivity to the
inlet velocity or rather the Reynolds number Red
which is one of the similarity parameters (1). The
spectra for Red = {5·103; 1·104; 2·104} are shown in
Fig. 13. Both - the power spectral density and the
Fig. 12. Spectra at the end of the recirculation
zone at x/d=40 and r/d=4
frequency - are normalized to the large scales of the
flow problem (the inlet velocity UJ and inlet
diameter d). The normalization of the frequency
results in the Strouhal number Sr (3). The spectra
look quite similar and the characteristic narrow
band peak is located at the same range of Strouhal
number 10-4 < Sr < 10-3. The resulting constant
dominant Strouhal number at Srdom = 5·10-4 refers to
a doubling of the dominant frequency fdom by
doubling the inlet velocity (Table 1). It can be
assumed that this peak is caused by large scale
structures. However, a more profound
understanding about its underlying mechanism has
to await further investigations.
Fig. 13. Spectra at x/d=20 and r/d=2 for
different Reynolds numbers
For a better understanding of the underlying flow
structures a few further considerations will be
discussed.
Firstly, in comparison to the present flow the
unsteady flow around a cylinder, also known as
Karman vortex shedding, leads to a Strouhal
number of Sr = 0.2. The vortex roll-up of the jet
flow results in Strouhal numbers of Sr = 0.4 which
corresponds to Karman vortex shedding.
The Strouhal number can also be comprehended as
a length scale by the use of Eq. (4). For Karman
vortex shedding wave numbers of κ = 2/5π·d-
1 1.3/d result and amount approximately the
reciprocal cylinder diameter. The same
characteristic can be determined for the vortex roll-
up of a jet. The present flow results in Strouhal
numbers with two orders of magnitude less than the
cylinder or jet flow. Hence, the resulting wave
number is κ = 0,5 m-1 and its reciprocal corresponds
to 100 times the inlet diameter or 10 times the pipe
A. Ringleb et al. / JAFM, Vol. 11, No. 4 pp. 877-883, 2018.
882
Fig. 14. Spectra at x/d=20 and r=0 for different
Reynolds numbers
diameter. Therefore, turbulent structures as a reason
for the spectrum peak are hardly conceivable.
2Sr
d
(4)
Secondly, there are no streamwise convection
effects of flow structures detectable as it can be
seen from Figs. 11 and 12. Therefore, only diffusion
or dissipation effects are possible reasons for
transferring energy of the spectrum peak. Due to
undetectable convection effects, a longitudinal wave
structure through the pipe can be also eliminated,
because the reciprocal wave length corresponds to
six times the length of the recirculation zone or one
time the pipe length.
Table 1 Influence of the Reynolds number Red
on characteristic frequencies
Red f
0 [Hz] fdom [Hz]
5.000 2,8 0,8
10.000 11,3 1,7
20.000 - 3,4
Thirdly, the observed peak is locally limited to the
edge of the jet stream at the half length of the
recirculation zone. That is why it cannot be caused
by effects further upstream like vortex roll-up but
rather by a local effect. Therefore, the recirculation
zone itself could be responsible. One possibility is a
time-dependent variation of the length of the
recirculation zone that could result in a time
dependent position of the reattachment point
xR = f(t). This in turn may result in such large-scale
fluctuations at the half length of the recirculation
zone where streamlines are nearly parallel to the
axis and the turbulence intensity is low, whereas, at
the end of the recirculation zone the flow is much
more turbulent and axial fluctuations could be
surpassed by turbulent fluctuations.
In both spectra at the half length of the recirculation
zone (at the centerline and the edge of the jet) a
slight buckling of the spectrum can be identified
which begins at the centerline for Sr > 0.002 (Fig.
14) and in the shear-layer for Sr > 0.003 (Fig. 13).
Its behavior looks like the influence of added noise
reported by Nobach et al. (1998). However, it can
only be observed for Red = {5·103; 1·104}. Its non-
appearance for the flow at Red = 2·104 leads to the
presumption that the buckling is triggered by a
laminar to turbulent transition in the flow field.
Eventually, it should be noted, that the discussed
large-scale fluctuations exist at low frequencies for
a high power level that corresponds to high energy
rates contained by the underlying flow structures.
That is why it can be assumed that these velocity
fluctuations also result in pressure fluctuations
within the flow field.
5. CONCLUSION
A sudden pipe expansion flow with an underlying
expansion ratio of D/d = 12.3 has been investigated.
Large scale fluctuations have been observed at the
edge of the jet stream at the axial half length of the
recirculation zone. They show no sensitivity against
the inlet design but a proportional sensitivity against
the flow velocity. A doubling of the dominant
frequency results in an increase of the Reynolds
number by factor of 2. Dimensionless
considerations lead to a longitudinal wave structure
or an axial commutation of the recirculation zone as
a possible reason. Transversal or vortex like
structures are hardly to imagine because of the
resulting large wave numbers and the local limited
occurrence.
Further investigations on a wide range of Reynolds
numbers Red and different diameter ratios D/d are in
progress. The aim is to carry out much more
detailed information about the sensitivity of this
peak. Based on the present work it can be presumed
that this peak shows a linear behavior against the
Reynolds number. A similar observation was made
by Anderson et al. (2003) for a parallel plane jet
flow. Furthermore, it is necessary to get more
knowledge about the underlying flow structure.
Therefore, the ongoing investigations on Red and
D/d will be helpful.
Moreover, the observed spectrum peak can be used
for the development of a method for indirect flow
rate measurements in an axisymmetric sudden pipe
expansion flow with LDV. This technique will be
very suitable for applications in rough industry
environments such as flow rate sensors for chemical
or process engineering. Because LDV is a so called
non-invasive technique, no mechanical parts e.g.
surfaces can be damaged due to corrosion or high
temperatures. A disadvantage is the necessity of an
optical access.
ACKNOWLEDGEMENTS
Financial support of the European Regional
Development Fund (EFRE) through grant CIPP is
gratefully acknowledged.
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