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

abogus@imc.pcz.czest.pl

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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