LES PREDICTIONS OF ABSOLUTELY UNSTABLE ROUND LOW-
DENSITY JET
A. Boguslawski, A.Tyliszczak Institute of Thermal Machinery
Czestochowa University of Technology
al. Armii Krajowej 21
42-201 Częstochowa
Poland
1 Introduction
Absolute instability of round low-density jets was
studied with spatio-temporal linear stability theory by
Monkewitz and Sohn (1988) and later by Jendoubi
and Strykowski (1994). They showed, using Briggs
(1964) and Bers (1975) criterion, that in parallel axi-
symmetric low-density jet an absolutely unstable
mode growing exponentially at the location of its
generation can be triggered. Linear stability theory
shows that critical density ratio is dependent on the
shear layer thickness. For thin shear layers for which
�/� > 40 (� - jet diameter, � - momentum
thickness) the critical density ratio � = �� ��⁄ (��-jet
density, ��-ambient fluid density) is relatively
insensitive to shear layer thickness and equal
��� ≈ 0.7, while for shear layers with �/� < 40 the
critical density ratio decreases rapidly with
increasing shear layer thickness. The results of linear
stability theory are shown in Fig.1 for axisymmetric
mode (� = 0) and first and second helical modes
(� = 1 and 2 respectively).
Figure 1: Absolute-convective instability
boundary for axisymmetric (� = 0), first (� = 1)
and second (� = 2) helical modes.
It can be seen from Fig.1 that the critical density
ratio for axi-symmetric absolutely unstable mode is
much higher than the one for helical modes. The
results stemming from the linear stability theory were
confirmed in two fundamental experimental works
on heated jets by Monkewitz et al. (1990) and air-
helium jets by Kyle and Sreenivasan (1993). In both
experiments strong oscillations were observed for
low-density jets. In the case of heated jets, studied by
Monkewitz et al. (1990), the oscillations identified as
the absolutely unstable mode were observed for
density ratios lower than the critical one ��� ≈ 0.65.
This oscillations are called Mode II. In the case of
air-helium jets the critical density ratio, below which
oscillating mode emerged, established by Kyle and
Sreenivasan (1994), was slightly lower ��� ≈ 0.61.
In both experiments axi-symmetric vortical structures
undergoing vortex pairing were observed.
Characteristic frequencies of experimentally
observed oscillations agreed very well with the
results of linear stability theory. However, in both
experiments some differences were also indicated. In
the case of heated jets additional oscillations, called
Mode I, were measured for density ratio � < 0.69,
while air-helium jets revealed broadband oscillations
for very thin shear layer. The origin of these two
types of oscillations in low-density jets is not
understood till now.
LES and/or DNS could bring new insight into
understanding of low-density jets transition
mechanisms. However, there are surprisingly few
numerical studies on variable density jets available in
the literature so far. LES predictions for variable-
density jets were performed recently by Zhou et al.
(2001), Tyliszczak and Boguslawski (2006), Wang et
al. (2008), Tyliszczak et al. (2008). These LES
results did not show clear presence of absolutely
unstable mode which could be compared with the
experimental results of Monkewitz et al. (1990) and
Kyle & Sreenivasan (1993). DNS predictions of low
density jets with wide range of density ratios and
shear layer thicknesses were recently performed by
Lesshafft et al. (2007) for jet at ��� = 7500. The
frequency predictions of the DNS results were
substantially higher than those found by linear
theory. Recently LES predictions of global mode in
round low-density jet at ��� = 7000 were presented
by Foysi et al. (2010) for density ratio � = 0.14 and
shear layer thickness �/� = 27. They found
excellent agreement with experimental data as far as
oscillations frequency is concerned. They observed
also the strong vortex pairing and side jets
phenomena confirming presence of global instability
in the LES predictions.
The present paper is aimed at LES predictions,
similar to those presented by Foysi et al. (2010),
��� = 7000, but for wider range of density ratio.
The LES predictions were limited to two different
shear layer thicknesses �/� = 27 as in the work of
Foysi et al. (2010) and �/� = 40. Choosing a
relatively thick shear layer, any interaction with self-
sustained convective oscillations reported by
Boguslawski et al. (2013) that appear for shear layer
characterized by �/� > 50 is avoided. Moreover, for
parameter�/� < 40 the critical density ratio
depends strongly on the shear layer thickness (see
Fig.1). Hence, finding in LES predictions a critical
density ratio corresponding to the one predicted by
the linear stability theory confirms that the absolutely
unstable mode is correctly reproduced.
2 Numerical method
The flow solver used in this work is an academic
high-order code based on the low Mach number
approximation. This code (SAILOR) may be used for
solving a wide range of flows under various
conditions, varying from isothermal and constant
density to situations with considerable density and
temperature variations. For research purposes the
SAILOR code includes a variety of sub-grid models
used when the code is operated in Large Eddy
simulation (LES) mode (Geurts (1997), Sagaut
(2001)). In the present work we incorporate the sub-
grid model as proposed by Vreman (2004). In this
model the subgrid viscosity vanishes in laminar flows
or pure shear regions. This is an important aspect in
jet flows with low turbulent intensity at the inlet
conditions. An excess of dissipation coming from the
subgrid part would hinder the transition and
developed turbulence regimes - this is not the case
with the selected model.
The SAILOR code was used previously in
various studies including laminar/turbulent transition
in near-wall flows, free jet flows, multi-phase flows
and flames. The solution algorithm is based on a
projection method with time integration performed
by a predictor-corrector (Adams-Bashforth/Adams-
Moulton) method. The spatial discretization is based
on 6th order compact differencing developed for half-
staggered meshes (Laizet & Lamballais (2009)).
Unlike in the fully staggered approach the velocity
nodes are common for all three velocity components
whereas the pressure nodes are moved half a grid size
from the velocity nodes. This greatly facilitates
implementation of the code and is computationally
efficient as there is only a small amount of
interpolation between the nodes. As shown in
Laizet&Lamballais (2009) the staggering of the
pressure nodes is sufficient to ensure a strong
velocity-pressure coupling which eliminates the well
known pressure oscillations occurring on collocated
meshes.
3 LES predictions of absolutely unstable
jet
Fig.2 presents the non-dimensional frequency
based on the jet diameter and maximum velocity at
the nozzle exit (�� = ��/����) of the global mode
predicted by LES compared to the results of linear
spatio-temporal stability of the inlet velocity and
density profiles and LES results of Foysi et al.
(2010) and DNS predictions of Lesshafft et al.
(2007). The present LES results for thicker shear
layer (�/� = 27) coincide very well with the DNS
of Lesshafft et al. (2007) for �/� = 30 . In the case
of thinner shear layer characterized by the
parameter �/� = 40 current LES results indicate
higher frequencies than DNS predictions of
Lesshafft et al. (2007). The global frequency
predicted with LES is substantially higher than the
absolute mode frequency obtained from linear
stability theory. The discrepancies between LES
predictions and stability calculations are increasing
for lower density ratios. Present LES prediction of
the global mode frequency for the density ratio
� = 0.14 differs also from the results of Foysi et al.
(2010).
Figure 2: Global frequency predicted with LES
and DNS compared with absolute frequency of the
inlet profile .
Fig.3 presents sample spectra of the axial velocity
fluctuations registered at the jet axis and distance
/� = 3 from the nozzle exit for the density ratio
varying in the range� = 0.2 ÷ 0.7 and shear layer
thickness �/� = 27. For the density ratio � = 0.7
there are no visible periodic oscillations in the
velocity field fluctuating component, while strong
peak is emerging for density ratios S=0.6. Hence, the
critical density ratio for the global mode predicted by
LES is close to the result of linear stability theory for
the shear layer thickness �/� = 27 (see Fig.1).
Amplitude of the velocity oscillations shown in
Fig.3 is normalised by the maximum velocity at the
nozzle exit. The amplitude for the density ratio
� = 0.6 is smaller than for lower ones as this density
ratio is very close to the critical value. Decreasing
further the density ratio the amplitudes are insensitive
to the density ratio suggesting that its value is limited
by non-linear interactions which is characteristic
feature of the global mode. The fundamental peak is
present in spectra at the constant level in the whole
distance /� � 1 � 6, however further downstream
for /� � 5 � 6 is gradually merging in the
background turbulence. These oscillations are
generated by the vortex structures formed as a result
of absolute/global instability. Similarly to the DNS
results of Lesshafft et al. (2007), for the shear layer
thickness �/� � 27, vortex pairing process is not observed as in the spectra shown in Fig. 3, there is no
subharmonic mode present. It means that vortex
pairing process observed experimentally by
Monkewitz et al. (1990) and air-helium jets by Kyle
and Sreenivasan (1993) in absolutely unstable low-
density jets, can be observed only for thin shear
layers for which the vortex structures are triggered
sufficiently close each other to interact and initiate
vortex pairing process.
Figure 3: Evolution of spectral distribution of
axial velocity fluctuations at the jet axis and distance
x/D=1÷3 from the nozzle exit, D/θ=27.
Figure 4: Mean axial velocity profile for the
shear layer characterized by D/θ=27.
Fig.4 shows the mean axial velocity profiles for
the case �/� � 27 and density ratio range � � 0.2 �
1. For the density ratios corresponding to convective
instability (� � 0.7 � 1) influence of the density
ratio on mean velocity profile along the jet axis is
relatively weak. Starting from the density ratio
� � 0.6 significant influence of the density ratio on mean velocity decay is observed. The rapid mean
velocity decay is associated with strong velocity
fluctuations shown in Fig. 5. Velocity fluctuations
resulting from the global mode are as high as
turbulence level in fully developed region of
convectively unstable jet.
Figure 5: Fluctuating axial velocity profile for
the shear layer characterized by D/θ=27.
Fig. 5 shows also that the maximum amplitude of
velocity fluctuations is independent of the density
ratio while decreasing density ratio shifts the location
of the maximum closer to the nozzle exit.
Independence of the oscillations amplitude of the
density ratio was already observed in the velocity
spectra.
Figure 6:Iso-contours of temperature at the
distances x/D=1.5-3, S=0.2, D/θ=27.
Figure 7:Iso-surfaces of temperature (left figure)
and axial velocity (right figure) in two time instants,
S=0.2, D/θ=27.
Fig. 6 illustrates the side-jets phenomenon by iso-
contours of temperature in the jet cross-sections
located in the near field. The side jets were observed
in heated jets by Monkewitz et al. (1990) and in air-
helium jets by Kyle and Sreenivasan (1993) and
Halberg and Strykowski (2006). However, the
mechanism of side jets formation is not fully
understood so far and needs further studies, it is a
characteristic phenomenon observed when strong
vortex structures are generated in the flow field.
Instantaneous iso-surfaces of temperature and
axial velocity at two time instants are shown in Fig.
7. A development of vortex structures is visible in
temperature and velocity fields. These structures
break-up further downstream into a developed
turbulent flow.
Fig. 8 shows evolution of the spectral distribution
of axial velocity fluctuations along the jet axis for the
case of thinner shear layer characterised by�/� =
40. In this case strong periodic oscillations are seen
in the velocity spectra for density ratio � < 0.7. By
contrast to the results presented in Fig. 3 in this case
clear vortex pairing process is observed. A
subharmonic mode is visible even at the distance
/� = 1 marking the beginning of the vortex pairing
process. Then this subharmonic mode is growing
attaining its maximum at the distance /� = 3 ÷ 4.
Further downstream this peak is decreasing as the
process of vortex breakup into fully developed
turbulent flow undergoes.
As in the previous case, changing the density
ratio from � = 0.7 to � = 0.6 leads to a drastic
change of the mean velocity profile, as shown in Fig.
9. For the density ratio � < 0.7 the decay of the mean
velocity profile starts nearly at the nozzle exit and
there is no potential core predicted.
Fluctuating velocity profiles along the jet axis for
density ratios in the range � = 0.2 ÷ 1, for the case
of shear layer thickness �/� = 40, are shown in
Fig.10. As it was shown in Fig. 8, for the density
ratio � = 0.6, amplitude of the oscillations is at the
same level as for the lower density ratios in the near
field /� < 2. However, further downstream for the
density ratio lower than 0.6 the fluctuations grow
attaining the level over 30% at the distance /� =
2 ÷ 3. As it was mentioned above at this distance the
vortex pairing process is completed. Consequently,
velocity oscillations for thinner shear layer, for which
vortex pairing process is present, are significantly
stronger than in the case presented in Fig. 5.
Figure 8: Evolution of spectral distribution of
axial velocity fluctuations at the jet axis and distance
x/D=1÷3 from the nozzle exit, D/θ=40.
Figure 9: Mean axial velocity profile for the
shear layer characterized by D/θ=40.
Figure 10: Fluctuating axial velocity profile for
the shear layer characterized by D/θ=40.
4 Conclusions
The paper presents preliminary results of LES
predictions of global mode in low-density round free
jet. The results suggest that the global oscillations
were reproduced for wide range of the density ratio
below the critical value. The critical density ratio was
predicted with reasonable agreement with the results
of spatio-temporal linear stability theory. However,
characteristic frequencies of the global mode are
substantially overpredicted compared to the stability
calculation results. The present LES results predict
higher frequency of the gobal mode than the DNS
reported by Lesshafft et al. (2007). Some
discrepancies were also observed with the LES
predictions of Foysi et al. (2010). The LES
calculations were performed for two different shear
layer thicknesses characterized by the parameter
�/� � 27 and 40. In the case of thicker shear layer,
due to low frequency of vortex generation and as a
consequence large distance between consecutive
vortices, the vortex pairing process is not observed.
By contrast, in the case of thinner shear layer clear
vortex pairing process is visible in the evolution of
velocity fluctuations spectra. It was shown that
vortex pairing process leads to significantly higher
oscillations in globally unstable jet.
Acknowledgements
The research project was supported by Polish
National Science Centre, project no. DEC-
2011/03/B/ST8/06401
This research was supported in part by PL-Grid
Infrastructure.
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