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Under consideration for publication in J. Fluid Mech. 1

Large Eddy Simulation of a Pulsed Jet in

Crossﬂow

A X E L C O U S S E M E N T12 3†, O. G I C Q U E L2 3

AND G. D E G R E Z1

1Aero-Thermo-Mechanics Departement, Universit´e Libre de Bruxelles, Avenue F.D. Roosevelt

51, CP 165/41. 1050 Bruxelles. Belgium

2Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry. France

3CNRS, UPR 288 ”Laboratoire d’´

Energ´etique molculaire et macroscopique, combustion”,

Grande Voie des Vignes, 92295 Chatenay-Malabry. France.

(Received ?; revised ?; accepted ?. - To be entered by editorial oﬃce)

This study quantiﬁes the mixing resulting of a pulsed jet in crossﬂow in the near jet

region. By large eddy simulation computations, it also helps to understand the physi-

cal phenomena involved in the formation of the pulsed jet in crossﬂow. The boundary

conditions of the jet inlet are implemented via an NSCBC coupled with a Fourier se-

ries development. The signals used to pulse the jet inlet are a square or a sine wave. A

new way of characterizing the mixing is introduced with the goal of easily interpreting

and quantifying the complicated mixing process involved in pulsed jet in crossﬂow ﬂow

ﬁelds. Diﬀerent ﬂow conﬁgurations, pulsed and non-pulsed, are computed and compared,

keeping the RMS value of the signal constant. This comparison not only allows the char-

acterization of the mixing but also illustrates out some of the properties of the mixing

characterization.

Key words: Computational ﬂuid dynamics, Jet in Crossﬂow, Large Eddy Simulation,

Mixing

1. Introduction

Steady jets in uniform crossﬂow have been investigated by analytical, experimental

and numerical approaches over the past 50 years. In addition to being interesting from a

theoretical point of view, the jet in crossﬂow is commonly used in various applications,

including V/STOL aircraft in transition ﬂight, turbine ﬁlm cooling, chimneys, roll control

in rockets or fuel mixing in combustion chambers (Priere et al. 2005; Williams et al. 1966;

Gritsch et al. 1998; He et al. 1999; Xu & Zhang 1997; Thole et al. 1997; Ferrante et al.

2011).

For example, in a combustion chamber, the mixing of the two ﬂows can have a critical

inﬂuence on the size and eﬃciency of the chamber. This is even more important in high

speed combustion, such as in scramjets where ultrafast mixing is the key to the design

of an eﬃcient combustion chamber. This mixing eﬃciency is also required for pollutant

reduction such as NOxwhich requires the reduction of the duration of high-temperature

exposure (Priere et al. 2005).

It has long been recognized that jets in crossﬂow are more eﬃcient mixers than a

free jet or a mixing layer (Kamotani & Greber 1972; Broadwell & Breidenthal 1984).

†Email address for correspondence: axcousse@ulb.ac.be

2A. Coussement, O. Gicquel and G. Degrez

Figure 1. Jet in crossﬂow vortices.

This enhanced mixing is attributed to the counter-rotating vortex pair whose role in the

mixing process is explained by Cortelezzi & Karagozian (2001). The mixing properties of

the jet in crossﬂow come from complex three dimensional interactions, most of which were

identiﬁed in the seventies (Fearn & Weston 1974; Kamotani & Greber 1972). Nowadays,

lots of investigations have been undertaken to understand the formation and the behavior

of the jet in crossﬂow ﬂow structures. One can divide the vortical structure of the jet in

crossﬂow into four principal vortices (see ﬁgure 1):

•The counter-rotating vortex pair (CVP) is the dominant structure downstream of

the jet injection point. This structure is generated by the deﬂection of the jet and is

convected by the transverse ﬂow. Experimental studies (Kelso et al. 1996) suggested

that the CVP ﬁnds its origin in a process of roll up, tilting and folding of the shear layer

vortices. This was conﬁrmed latter on by the work of Cortelezzi & Karagozian (2001).

•The horseshoe vortex which is due to the adverse pressure gradient just in front of

the injection hole. This structure and the vortex-shedding frequency are quite similar to

those observed around a solid cylinder (Krothapalli et al. 1990).

•Shear-layer ring vortices are generated in the boundary layer of the jet oriﬁce and

evolve in the ﬂow to generate the CVP (Kelso et al. 1996; Cortelezzi & Karagozian 2001).

The process involved in the generation of these vortices is a Kelvin-Helmoltz instability

in the shear layer when the two streams meet.

•Wake vortices are the least understood. The experiments of Fric & Roshko (1994)

suggested that these vortices originate from the wall boundary layer.

Experiments of M’Closkey et al. (2002); Johari (2006); Johari et al. (1999); Hermanson

et al. (1998) and Eroglu & Breidenthal (2001) have shown that one can further improve

the mixing by pulsating the jet speed Uj. These experiments have been undertaken both

in liquid and gaseous phase. For example, Johari et al. (1999) who study the penetration

of a fully modulated liquid jet in crossﬂow, show that the maximum penetration occurs at

a forcing frequency, f, which corresponds to a jet Strouhal number Stj≡f D/Uj= 0.004

where Dis the inner diameter of the jet oriﬁce and Ujthe mean jet speed of the steady

jet in crossﬂow (i.e. the mean of the jet velocity minus the pulsation). Moreover this

maximum penetration occurs at a duty cycle α= 0.20, the duty cycle being the ratio

between the temporal pulse width and the period of the pulsation.Note that this quantity

was originally deﬁned for a rectangular signal in time.

To better understand the physics behind the pulsed jet in crossﬂow, Johari (2006)

compiled all the experiments performed in gaseous or liquid phase (Wu et al. 1988; Eroglu

& Breidenthal 1991; Hermanson et al. 1998; Johari et al. 1999; Eroglu & Breidenthal 2001;

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 3

Figure 2. Classiﬁcation of the pulsed jet in crossﬂow ﬂow regimes by Johari (2006). Symbols

represent diﬀerent experimental conﬁgurations of pulsed jet in crossﬂow (see Johari (2006) for

details.

(A) (B) (C)

Figure 3. Smoke visualization of pulsed jet in crossﬂow, from the work of M’Closkey et al.

(2002) with Uj= 3.1m/s,U∞= 1.2m/s, an RMS value of the pulsation of 1.7 m/s and (A)

α= 0.31 f= 110Hz (B) α= 0.15 f= 55H z (C) α= 0.62 f= 220Hz

M’Closkey et al. 2002), and introduced a stroke ratio:

L

D=1

A D Zτ

0ZA

UjdA dt (1.1)

where Ujis the instantaneous jet velocity at the nozzle exit, Ais the nozzle area and L is

the penetration length. He then uses this stroke ratio, along with the duty cycle of pulsa-

tions, α, to analyse experimental data and ﬁnd a relationship between the ﬂow structure,

αand L/D, independent of the jet-to-crossﬂow momentum ratio J≡ρjUj

2/ρ∞U∞

2and

velocity ratio R≡Uj/U∞(see ﬁgure 2):

•distinct vortex rings, followed or not by a ﬂuid column, for α < 0.2−0.5 and L/D < 4

(ﬁgure 3 A).

•bifurcation of the jet in two streams, for α < 0.5−0.8 and L/D < 20 −25 (ﬁgure 3

B).

•steady like jet in crossﬂow structure on which a turbulent puﬀ can be seen, for

α < 0.8−0.9 and L/D < 75.

•steady like jet in crossﬂow structure, in other cases (ﬁgure 3 C).

In the present work, as the physical phenomenon to be described is highly unsteady,

Reynolds Averaged Navier Stokes (RANS) modeling is not suitable and due to the di-

mension of the computational domain Direct Numerical Simulation (DNS) remains too

expensive. Moreover Unsteady RANS (URANS) is known to be insuﬃcient when com-

plex interaction between vortical structures and boundary layers can are present, as it

is the case for a jet in crossﬂow (see Menter & Egorov (2005)). Therefore Large Eddy

Simulation is retained as it gives the best results for a reasonable computational cost..

4A. Coussement, O. Gicquel and G. Degrez

LES computations are based on the case of M’Closkey et al. (2002) which is a gaseous

phase jet in crossﬂow having a stroke ratio L/D ≈7.5 (see ﬁgure 3 B).

Note that in their work Sau & Mahesh (2010) have also simulated a pulsed jet in

crossﬂow based on their experiments (see Sau & Mahesh (2008)) for various ﬂow regimes

and largely completed the work of Johari (2006). However, the goal of the present work

is somewhat diﬀerent: here the physical phenomena behind the initial vortex ring forma-

tion and propagation along with the mixing characterization will be deeply investigated.

Moreover focus will only be put on the near ﬁeld eﬀects of the jet pulsation.

First, the computational cases considered will be detailed and the soundness of the

computation veriﬁed using Pope’s criterion (Pope 2004), and the steady jet in crossﬂow

solution will be compared with experimental work. The boundary conditions used will

then be described and results presented. Finally, an analysis of the physical phenomenon

controlling the behavior of the pulsed jet in crossﬂow will be given and the question of

the diﬀerent ﬂow regimes assessed. This analysis will ﬁrst focus on the generation of a

puﬀ by the pulsation, then the interaction of the puﬀ with the crossﬂow and ﬁnally a

new mixing criterion will be introduced. An analysis of the mixing enhancement of the

pulsed jet in crossﬂow with respect to the steady jet in crossﬂow will be completed on

that basis.

2. Flow parameters

The present work focuses on reproducing one of the experimental cases of M’Closkey

et al. (2002), speciﬁcaly the 55 Hz case. Therefore, the ﬂow parameters are derived from

that work, mostly because the computation can be compared, at least visually, to some

experimental results. The main parameters of the steady jet in crossﬂow case are:

•A jet inlet speed of 3.1 m/s ( Uj)

•A crossﬂow inlet speed of 1.2 m/s (U∞)

•A nozzle exit diameter of 7.5e−3 m

•Crossﬂow and jet are composed of nitrogen

From this conﬁguration, several pulsed or non pulsed jet cases were derived and their

ﬂow variables are detailed in table 1, where case 7 is the 55Hz pulsed jet in crossﬂow

of M’Closkey et al. (2002). The pulsed cases 6, 7 and 8 were chosen so that they have

the same RMS value of the pulsed signal (≈1.7 m/s). In this way the same energy is

injected in the jet and this RMS value is derived from the work of M’Closkey et al. (2002).

Moreover, case 2 is chosen because it has the same mean velocity over a pulsation period

as cases 6 and 8. Case 1 is the nominal case in the sense that it has the same velocity as

the base velocity of the pulsed ﬂow, i.e. the velocity minus the pulsation. Case 3, 4 and 5

are computed to compare mixing properties of various steady jets in crossﬂow, to see if

some classical scaling law could be applied to the mixing characterization proposed here.

Case 6, 7 and 8 are comparable in terms of energy injected in the jet ﬂow while cases

6, 7 and 2 are comparable in terms of mass ﬂow. Table 1 also indicates the maximum

Reynolds number in the pipe Remax pipe.

In all computations, the jet and the crossﬂow are assumed to be composed of two

diﬀerent species. For all cases except case 5 both species thermodynamic properties are

the same, namely those of nitrogen. The molecular diﬀusion between the two ”species”

is modeled using Fick’s law.

Table 1 also gives the velocities and momentum ratios, Rand Jrespectively, computed

with the pulsations.

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 5

Table 1. Computational cases. The * denotes the cases presented in the work of M’Closkey

et al. (2002).

Case U∞UjPulse Pulse Frequency Jet J R Remax pipe

Type Amplitude Material

1* 1.2 m/s 3.1 m/s no pulse n.a. n.a. nitrogen 6.67 2.58 1310

2 1.2 m/s 3.79 m/s no pulse n.a. n.a. nitrogen 9.975 3.15 1610

3 6 m/s 15.5 m/s no pulse n.a. n.a. nitrogen 6.67 2.58 6570

4 1.2 m/s 9.3 m/s no pulse n.a. n.a. nitrogen 60 7.75 3940

5 1.2 m/s 11.55 m/s no pulse n.a n.a. hydrogen 6.67 9.625 4890

6 1.2 m/s 3.79 m/s sine wave 2 m/s 55Hz. nitrogen 9.975 3.15 2450

7* 1.2 m/s 3.1 m/s α= 0.15 4.6 m/s 55H z. nitrogen 9.975 3.15 3260

8 1.2 m/s 3.1 m/s α= 0.60 2.25 m/s 55Hz. nitrogen 13.75 3.7 2270

3. Numerical Simulation LES

The fully compressible unstructured grid ﬂow solver AVBP is used (Moureau et al.

2005). The numerical implementation is based on a Taylor- Galerkin (TTGC) ﬁnite ele-

ment discretisation and an explicit fourth order fourth stage Runge-Kutta time stepping

scheme.

The dynamic Smagorinsky model (Sagaut 2006) is retained for the LES sub-grid scale

viscosity model:

νt= (Cs∆)2q2e

Sij e

Sij (3.1)

where ∆ is the ﬁlter characteristic length, i.e. the cubic root of the cell volume, and

Csis the Smagorinsky constant which is dynamically computed to adapt to local ﬂow

conditions. The turbulent heat ﬂux is modelled using a turbulent Prandtl number and

the turbulent diﬀusion by a turbulent Schmidt number.

3.1. Mesh and domain

Figures 4 A and 4 B show the computational domain. The jet diameter is D= 7.5e−3 m.

The jet oriﬁce center is located at a distance A= 4Dfrom the inlet of the computational

domain, and at a distance B= 26.6Dfrom the outlet. The width, W= 10.6D, is

chosen to avoid the jet impact on side boundaries. As stated in the introduction, this

work focuses on near ﬁeld eﬀects of the pulsation on the jet in crossﬂow so that even

if the spatial extents of the domains do not represent the full M’Closkey et al. (2002)

experiment, it allows an accurate representation of the near ﬁeld eﬀects (i.e: within 10

D downstream of the jet).

The computational grid contains 253,495 nodes and 1,436,506 cells. Thanks to the low

speed of both ﬂows, such a number of cells allows the computation of a resolved LES,

as will be explained further. The mesh is reﬁned in the inlet pipe and near the mixing

zone as shown on ﬁgures 5 A and B. This was necessary to correctly capture the physics

of the pulsations and the interaction between the crossﬂow and the pulsation. Indeed, it

is recognized that a good description of the ﬂow in that particular region is crucial to

obtain accurate results (Cortelezzi & Karagozian 2001). The order of magnitude of the

mesh size, and so the ﬁlter size, in that particular region is ∆x≈100.10−6m.

3.2. LES Resolution

The LES quality can be expressed in terms of energy: the simulation precision improves

when kSGS , which is the sub-grid energy, decreases. The limit case where kSGS equals

6A. Coussement, O. Gicquel and G. Degrez

(A) (B)

Figure 4. (A) Computational domain side view (B) Computational domain up view

(A)

(B)

Figure 5. (A) Mesh side view (B) Mesh top view

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 7

zero corresponds to a DNS. If Kis deﬁned as the resolved part of the energy, the turbulent

resolution M(x, t) reads:

M(x, t) = kS GS (x, t)

K(x, t) + kSGS (x, t)(3.2)

Starting from this, Pope (Pope 2004) introduced the concept of adaptative LES: the

turbulent resolution must be less than 0.2, which corresponds to a resolution of 80%

of the kinetic energy. The parts of the domain in which this criterion is not respected

are judged insuﬃciently resolved. Knowing this, it is possible to judge the quality of

the LES simulation by computing this criterion at every point. Unfortunately evaluating

exactly the turbulent resolution is nearly impossible, so approximations are made. The

approximative resolved part of the kinetic energy reads:

K(x, t)≈0.5heu2iT− heui2

T(3.3)

while the sub-grid energy is obtained using the following approximation:

kSGS (x, t)≈0.5hτii iT≈hν2

tiT

(Cm∆)2(3.4)

where hiTdenotes the temporal average on a time Tand euis the resolved part of the

speed. ∆ is the ﬁlter size and Cmis an auto-similar function of the turbulence:

Cm=r2

3

A

πK

3

2

o

(3.5)

where Koand Aare equal respectively to 1.4 and 0.44 and νtis the sub-grid scale

viscosity. This formulation is known to overestimate kSGS (Sagaut 2006), so that the

criterion is more restrictive.

In the present case, the temporal average is done over one pulsation cycle i.e T= 1/55 s.

Only one pulsation was used because the increase in νtis mainly due to the pulsation,

so that the information contained in a one cycle average is complete enough.

Figure 6 shows the turbulent resolution Mfor the most critical conﬁguration, case 7

in which the pulsation amplitude is the strongest. This induces high velocity gradients

and thus higher sub-grid viscosity νt. Most of the cells are in the zone where M60.2;

those for which M > 0.2 correspond to the bigger cells located at the domain exit, in

which kSGS is higher due to the higher mesh size.

Finally the y+(equation 3.6) was also computed for the various cases presented here,

and its value ranges between y+≈0.6 and y+≈4.4 in the pipe and near the pipe

exit, which, according to Sagaut (2006) is below the critical value of 5, meaning that

the viscous sublayer is resolved in the zone where vortical structures interact with the

boundary layer.

The LES is thus judged to be well resolved.

y+=y

νrτwall

ρ(3.6)

3.3. Steady Jet Solution

To further verify the behavior of the computations, averaged results for the steady jet

case 1 will be benchmarked against steady jet experimental results. To compare with

experimental data, the scaling law (Broadwell & Breidenthal 1984), which establishes a

relation between the position of the locus of jet material maximum concentration, Ycl,

8A. Coussement, O. Gicquel and G. Degrez

Figure 6. Number of cell for diﬀerent values of M, computation case 7

and the distance downstream of the jet nozzle, is used. All the coordinates are normalized

by the product of the velocity ratio and the jet diameter: RD.

To verify the scaling law, a converged average ﬁeld is used (ﬁg. 7 A). From this average

ﬁeld the normalized penetration:

Ycl

Rd =Ax

Rd n(3.7)

is computed and the values of nand Adeduced (see ﬁgure 7 B).

Both best ﬁt and one third power law give A= 1.191, nfor the best ﬁt is 0.325 and

the adjusted R-square is 99%. The value of nis similar that of the analytical scaling law,

which is why the best ﬁt and one third power law are superposed, but the Acoeﬃcient

is ten percent lower than reported by Niederhaus et al. (1997); Priere (2005) and Johari

(2006). Most of the laws from the literature are established for a jet velocity ratio ten

to ﬁfty percent larger than the ratio of case 1, which could cause diﬀerence in mixing.

Indeed, in the work of Johari et al. (1999) a slight diﬀerence is shown between the A

coeﬃcient of two jets in crossﬂow with diﬀerent velocity ratios, which seems to indicate

that the Acoeﬃcient depends on the velocity ratio. This could be due to diﬀerences

in the boundary layer proﬁles or Reynolds number eﬀects, which are known to have an

inﬂuence on the behavior of a jet in crossﬂow (Cortelezzi & Karagozian 2001)

The roll-up frequency was also computed from the steady jet numerical solution. A

value of 226Hz was found which is similar to the frequency reported by M’Closkey et al.

(2002) (i.e 220 Hz) and fully validates the numerical solution.

Since the ﬂow solver AVBP is known to correctly simulate a jet in crossﬂow (Jouhaud

et al. 2007) no further validations are undertaken.

3.4. Boundary Conditions

To generate a pulsed jet in crossﬂow based on the experiments of M’Closkey et al. (2002),

a velocity pulsation must be imposed at the jet inlet. This pulsation will be imposed us-

ing the so-called Navier-Stokes Characteristic Boundary Condition (NSCBC) formalism

proposed by Baum et al. (1995). Because of the relaxation used in this method, imposing

a pure pulsation is impossible. However since it is impossible experimentally to impose

a real pulse and as the goal of the simulation is to reproduce experimental results, there

is no need to impose a pure pulsation. To reproduce the experimental results one has to

mimic the signal imposed by M’Closkey et al. (2002) at the nozzle exit (see 8 B dashed

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 9

(A) (B)

Figure 7. (A) Average jet mass fraction in section z= 0, computation case 1 (B) Normalized

penetration of steady jet; solid and dashed curves represent the one-third scaling and the best-ﬁt

respectively

lines). To do so a Fourier series is used:

v(t) = a0+

∞

X

n=1

ancos (2πf tn) +

∞

X

n=1

bnsin (2πf tn) (3.8)

In the case of a pulse wave, coeﬃcients can easily be computed:

a0=1

TZT/2

−T/2

v(t)dt (3.9)

an=2

TZT/2

−T/2

v(t)cos 2πtn

Tdt (3.10)

bn=−2

TZT/2

−T/2

v(t)sin 2πtn

Tdt (3.11)

where v(t) is the signal to reproduce, Ais the pulse amplitude and αis the duty cycle.

The signal measured by M’Closkey et al. (2002) at the exit of the pipe was digitized

to compute the Fourier series coeﬃcients. From there the series was limited to its ﬁrst

10 coeﬃcients, so that n= 1,2. . . 10. The use of only 10 coeﬃcients allows the repro-

duction of the signal showed in ﬁgure 8 A, which accounts for 90 % of the energy present

in M’Closkey et al. (2002) signal. To compensate for the missing 10 %, the pulsation

amplitude was increased accordingly.

Boundary conditions that are used for the simulations are summarized in table 2. Note

that, in the NSCBC method not only is the velocity imposed at the inlet but also the

temperature and the mass fractions along with relaxation coeﬃcients. Since the goal is

to simulate a pulsed signal at the inlet, the relaxation coeﬃcient of the velocity is high

(see Table 2). In this way, the velocity in the pipe will be as close as possible to the

velocity imposed at the inlet. The other relaxation coeﬃcients are chosen low enough so

that they do not perturb the velocity (see Table 2). Moreover the inlets do not involve

any kind of boundary layer model nor turbulence injection. This last fact is justiﬁed by

the relatively low maximum Reynolds number at the pipe inlet Remax = 3260 during the

pulsations. At the outlet only the pressure is imposed, with a low relaxation coeﬃcient,

10 A. Coussement, O. Gicquel and G. Degrez

(A) (B)

Figure 8. (A) Pulse signal, α= 0.15 and A= 4.6m/s with n= 10 (B) Actuator signal in volts

(dash-dotted) line, velocity signal at the jet nozzle with compensation (i.e. mean substracted,

dashed line) and velocity signal at the jet nozzle without compensation (solid line) at 73.5Hz

and α= 0.22%

Table 2. Boundary condition with associated target values and relax coeﬃcients

Boundary name Fixed quantities Relaxation coeﬃcient

Inlet crossﬂow Unormal, Utransversal , T , Y σUnormal = 10000 σUtransv ersal ,T ,Y = 1000

Inlet Jet Unormal, Utransv ersal, T , Y σUnormal = 1000 σUtransver sal ,T ,Y = 100

Outlet P σP= 100

Jet wall qwall = 0, powerl aw N/A

Bottom wall qw all = 0, powerlaw N/A

Side walls qwall = 0 N/A

Top wall qwall = 0 N/A

as indicated in Table 2, to evacuate the acoustic wave as easily as possible. The pipe walls

and the plate wall use a no-slip wall boundary condition. A slip adiabatic wall boundary

condition is enforced on the top and side boundaries. Since the computational domain is

big enough, there is no need to impose another type of boundary condition.

4. Simulation Results

Simulation results and subsequent analysis will now be given. A ﬁrst qualitative analy-

sis can be made on the basis of ﬁgure 9 which shows the instantaneous ﬁeld of jet material

mass fraction in a slice located at z= 0 for cases 1, 6, 7 and 8.

First, three of the four ﬂow regime identiﬁed by Johari (2006) are present. Case 7

presents a bifurcation of the jet in two streams, case 6 a steady like jet in crossﬂow

structure with some turbulent puﬀs, and ﬁnally case 8 presents only a steady like jet in

crossﬂow structure. Second, the instantaneous, YJ, ﬁeld can be compared to the work

of M’Closkey et al. (2002) but since no mixing, nor velocity measurement were made

during the experiments, the comparison will only be qualitative. Figure 10 shows the

experimental ﬂow of M’Closkey et al. (2002) corresponding to case 7, where the zone

corresponding to ﬁgure 9 C has been highlighted. Comparison between the ﬁgures shows

a good agreement between the experimental and computational work.

The pulsed jet simulations results are therefore consistent against experimental data,

and further analysis will now be performed to fully understand the pulsed jet physics:

•First the physics of the ﬂow in the inlet pipe will be investigated on the basis of the

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 11

(A : Case 1 ) (B : Case 6 )

(C : Case 7 ) (D : Case 8)

Figure 9. Instantaneous mass fraction of jet material YJfor four computation case

inviscid ﬂow theory and the link between the pulsation and the vortex ring strength will

be explained.

•Then, from the conclusions of the previous point, the vortex ring dynamics in the

crossﬂow will be explained which will lead to an explanation of the diﬀerent ﬂow regimes.

•Finally, a mixing criterion will be presented and an analysis of the mixing enhance-

ment of the pulsed jet in crossﬂow will be performed based on this criterion.

Note that the mass fraction of jet material is noted as YJand the mass fraction of

the crossﬂow material as YC. Finally, for representing time variations in the ﬂow, a non-

dimentional number is used: Tf, which is deﬁned as the ratio of the physical time and

the period of the pulsation.

4.1. Vortex ring intensity

This ﬁrst part will focus on the ﬂow behavior in the pipe during the pulsation and the

strength of the vortex rings that will interact with the crossﬂow. First, the Womersley

number must be introduced:

W=D

2ω

ν1/2

= (2πReSt)1/2(4.1)

ω= 2 π f where fis the pulsation frequency. This number can also be expressed in

terms of the Strouhal and Reynolds numbers. When Wis small, i.e <1, it means that the

12 A. Coussement, O. Gicquel and G. Degrez

Figure 10. Experimental pulsed jet in crossﬂow at 55Hz , duty cycle 0.15, RMS 1.7 from the

work of M’Closkey et al. (2002). The area corresponding to ﬁgure 9 C has been highlighted.

Figure 11. Velocity proﬁle at the jet exit in the symmetry plane for case 7 at Tf= 0.76

pulsation frequency is suﬃciently low to permit the velocity proﬁle to develop during each

cycle. The velocity will be in phase with the pressure gradient, resulting in a Poiseuille

law for the velocity proﬁle. On the contrary, when W≈10 or more the velocity proﬁle

lags the pressure gradient, resulting in a plug-like velocity proﬁle. In that case the ﬂow

can be considered as quasi-inviscid; the pulsation eﬀects are higher than the viscous

eﬀects.

In case 7, a frequency of 55Hz results in a Womersley’s number of W≈37, so that

the velocity proﬁle lags the pressure gradient. Viscous eﬀects can be neglected, and a

ﬂat velocity proﬁle, as predicted by the inviscid theory, must therefore be found. This

assumption is demonstrated by ﬁgure 11 where the velocity proﬁle looks almost uniform.

Knowing that the ﬂow in the pipe is considered inviscid easily allows the computation

of the vortex intensity. First, one must consider Bernoulli’s equation for the velocity

potential:

ρ∂Φ

∂t =−(p−ps) + ρ

2(v2−v2

s)(4.2)

in which the contribution of gravity is neglected. The subscript sdenotes the steady

conﬁguration, pis the pressure and vthe velocity, that is Ujhere. If one considers only

the increase of ρ∆Φ during a pulsation of length δt with respect to the steady case,

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 13

Bernoulli’s equation for potential becomes:

ρ∂Φ

∂t =−p+ρ

2v2(4.3)

To compute the increase in ρΦ one must then perform an integration over δt. The increase

in potential, which will be called the pulsation amplitude in what follows, can therefore

be easily computed from the ﬂow variables during the pulsation.

The pulsation amplitude, ρ∆Φ, can be related to the intensity of the vortex that will

be created by the pulsation at the exit of the nozzle. For example, if a pulsation is applied

in a pipe exiting in a suﬃciently large cavity, a vortex ring will be created. This is what

happens at the start of a jet (see Krueger et al. (2006),Gharib et al. (1998) ). In that

case, the vortex ring intensity, Γ, can be computed by:

ρ∆Φ = ρΓ = −Zt+δt

t

p+ρ

2v2dt (4.4)

To verify the linear relationship between the increase in potential due to the pulsation

and the vortex intensity, the vortex intensity must also be computed from the ﬂow vari-

able. To do so, vorticity is computed from an instantaneous ﬂow ﬁeld using the following

deﬁnition:

~

∇ × ~

V=Γ

Vl

(4.5)

where Vlis the volume of the vortex ring. Vorticity is so extracted from the three dimen-

sional ﬂow ﬁeld and integrated over the main vortex, using the Q-criterion to distinguish

the vortex ring vorticity from the ﬂow vorticity. The Q-criterion is deﬁned as:

Q=1

2(Ωij Ωij −Sij Sij )i, j ∈[1,2,3] (4.6)

where Sij is the symmetric part of the velocity tensor:

Sij =1

2∂ui

∂xj

+∂uj

∂xii, j ∈[1,2,3] (4.7)

Ωij its antisymmetric part:

Ωij =1

2∂ui

∂xj

−∂uj

∂xii, j ∈[1,2,3] (4.8)

The instant where the vortex intensity, Γ, is computed corresponds to the moment

when the jet nozzle velocity is maximum, therefore all the vorticity generated during the

speed increase is contained within the ﬂow ﬁeld.

ρ∆Φ is then computed using 4.4, increasing integration limits corresponding to the

time interval during which the velocity is rising, which correspond to the leading edge of

the pulsation. If the previous assumption is true, a linear relation must be found between

ρ∆Φ and Γ : Γ ∝ρ∆Φ for all the diﬀerent pulsed cases. To complete the set of pulsed

cases two more are added:

•Case 7 bis: it has the same properties as case 7 but with a pulse amplitude of 5.1 m/s

and duty cycle of 0.2, which yields an RMS value of the pulse signal of 2.2 m/s against

1.7m/s for both case 7 and 8.

•Case 7 ter : it also shares the characteristics of case 7 but with a pulse amplitude of

2.4 m/s yielding an RMS value of 0.9m/s.

An instantaneous slice in the symmetry plane (z= 0) of the YJﬁeld is showed on ﬁgure

12 for those two ﬂows.

14 A. Coussement, O. Gicquel and G. Degrez

(A) (B)

Figure 12. Instantaneous mass fraction of jet material YJfor: A, a pulse signal of 55Hz am-

plitude 5.1 m/s and duty cycle 0.2; B: a pulse signal of 55Hz amplitude 2.4 m/s and duty cycle

0.15

Γ is plotted as a function of −ρ∆Φ for all cases in ﬁgure 13 and the linear law predicted

by the theory is recovered. The vortex intensity is linked to the pulsation no matter what

the duty cycle is, since this curve is computed using the same integration time for the 4

non-sine cases, independently of the duty cycle. Indeed the increase in speed has the same

duration for the four cases. Only the sine case does not display the linear relationship,

simply because the increase in velocity lasts longer: trying to compute Γ emitted during

the velocity increase leads to a lower value than expected, because the vortex has already

started to dissipate. Figure 13 emphasizes a lower than expected value of Γ due to the

vortex dissipation. Nevertheless, this result justiﬁes that a puﬀ-like ﬂow is also obtained

with a sine wave.

As a result the vortex intensity is only proportional to the pulsation amplitude so that

the higher the velocity increase is, the stronger the vortex emitted is. This analysis, and

the values found, also seem to conﬁrm the analysis of Johari (2006) using the stroke

ratio deﬁned in equation 1.1. In the next section the interaction of this vortex ring with

the crossﬂow will be investigated. The source of the vortex generated is therefore the

pulsation and not a Kelvin Helmholtz like instability, as it is the case for the shear layer

vortices. Note that, in their work Gharib et al. (1998), have reached the same conclusion:

the initial vortex is directly linked to the pulsation. Moreover the linear relationship

found here is conﬁrmed by the results of Gharib et al. (1998). Indeed, Gharib et al.

(1998) found a linear relationship between the stroke ratio, computed by integrating the

increase in velocity as for the pulsation, and the vortex circulation which is proportional

to the vortex intensity. However the limit of the linear relationship found by Gharib et al.

(1998) was not reached here.

4.2. Crossﬂow - vortex ring interactions

For the three pulsed cases presented here: case 6, case 7 and case 8 the vortex ring evolu-

tion during the pulsation will be detailed. Figures 14, 15 and 16 show three dimensional

iso-contour of the Q-criterion for, respectively, case 6, 7 and 8.

First, for case 7, ﬁgure 17 shows the Q-criterion and ﬁgure 18 the velocity proﬁle at the

jet exit at diﬀerent Tffor a slice in the symmetry plane (z= 0). At Tf= 0.42, Q-criterion

shows the vortex shedding process at the nozzle exit. It has been demonstrated in the

work of Cortelezzi & Karagozian (2001) that this vortex shedding process is related to

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 15

Figure 13. Γ as a function of −ρ∆Φ

Figure 14. Three dimensional iso-contour of Q-criterion (Q = 7e4) for case 6 at Tf= 0.98

the CVP. Vortices are shed from the border of the nozzle outlet and convected by the jet

ﬂow to the far ﬁeld. Later these vortices are rolled up to become the CVP, those are the

shear layer vortices. The velocity proﬁle is not symmetric, mainly due to the inﬂuence

of the crossﬂow on the jet ﬂow at the nozzle exit (Majander & Siikonen 2006), which is

consistent with the work of Andreopoulos & Rodi (2006). The complete vortex shedding

process can also be seen in ﬁgure 15, in the steady jet part of the ﬂow.

At Tf= 0.67 a new vortex is generated as the velocity starts to rise. The velocity proﬁle

is almost uniform over a large part of the nozzle width, which is a typical behavior of

velocity proﬁles in ﬂows at high Womersley numbers, a phenomenon even clearer at

Tf= 0.72.

16 A. Coussement, O. Gicquel and G. Degrez

Figure 15. Three dimensional iso-contour of Q-criterion (Q = 7e4) for case 7 at Tf= 0.20

Figure 16. Three dimensional iso-contour of Q-criterion (Q = 7e4) for case 8 at Tf= 0.50

The plug shape is less visible on the left part of the velocity proﬁle, which is mainly due

to a strong interaction between the two ﬂows in that region. Moreover, the shed vortices

forming the CVPs described above are stronger and more coherent in the front of the jet

(Cortelezzi & Karagozian 2001), and have a priori a greater inﬂuence in the ﬂow ﬁeld.

Hence, the zone in front of the jet has more inertia and resists velocity perturbations.

As a result, the left part of the velocity proﬁle lags the right part of it. This diﬀerence

of inertia is also pointed out by the velocity proﬁle at Tf= 0.86 in which the left part

is somewhat higher than the right part. Comparing the velocity between Tf= 0.42,

Tf= 0.67, Tf= 0.86 and Tf= 0.99, clearly illustrates the phenomenon.

At Tf= 0.72, one circular vortex is developing at the nozzle exit, This highly coherent

and strong vortex is generated only by the leading edge of the pulsation.

At Tf= 0.76 the pulse related vortex begins to be convected in ﬂow because the jet

exit velocity stays high, which tends to re-enforce the vortex. Another small vortex also

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 17

Tf= 0.42 Tf= 0.67 Tf= 0.72

Tf= 0.76 Tf= 0.86 Tf= 0.99

Figure 17. Slice in the Q-criterion, in the symmetry plane z= 0 for case 7 at diﬀerent Tf

Tf= 0.42 Tf= 0.67 Tf= 0.72

Tf= 0.76 Tf= 0.86 Tf= 0.99

Figure 18. Velocity proﬁle at the jet exit in the symmetry plane for case 7 at diﬀerent Tf

appears at the exit of the jet pipe, in agreement with the work of Cortelezzi & Karagozian

(2001) and tends to indicate that the jet in crossﬂow changes its behavior to adapt to

the new speed. This last phenomenon is clearly visible on ﬁgure 15; the strong vortex is

followed by a small vortex, located at the lips of the pipe which is shedded by the strong

vortex located above.

During the velocity decrease, if one applies the reasoning presented in the previous

18 A. Coussement, O. Gicquel and G. Degrez

Tf= 0.10 Tf= 0.30 Tf= 0.37

Tf= 0.42 Tf= 0.67 Tf= 1

Figure 19. Slice in the Q-criterion, in the symmetry plane z= 0 for case 8 at diﬀerent Tf

section, another vortex ring with the same intensity should be generated but rotating in

the opposite direction. Around Tf= 0.86 another vortex is therefore generated but it is

partially compensated by the vortex seen at Tf= 0.76 at the nozzle exit, and no strong

coherent structure can be visualized after. During the last two Tf, the main vortex is

convected in the crossﬂow.

The same set of data is plotted for case 8 in ﬁgures 19 and 20 and the same analysis

framework is applied. At Tf= 0.37 the velocity starts to increase and at Tf= 0.42 a

strong coherent vortex is generated by the leading edge of the pulsation. Compared to

case 7, however the velocity increase is smaller, resulting in a weaker vortex, which is

why it dissipates quicker. Indeed the strong vortex cannot be seen on ﬁgure 16. Further

at Tf= 0.67 and Tf= 1 the ﬂow tends to stabilize itself in a new conﬁguration, so that

the process of vortex shedding and rolling-up is starting. As for case 7 the vortex at the

nozzle exit at Tf= 1 is cancelled by the vortex created by the decrease of velocity. The

biggest diﬀerence between case 7 and case 8 is a much more complicated ﬂow structure

in case 8. This could be interpreted as the combination of two initial transient ﬂows, one

with a jet speed of 3.1 m/s and one with a jet speed of 5.35 m/s because the jet stays

60 % of the period at 5.35 m/s and 40 % at 3.1 m/s. Those initial transient ﬂows, which

involve the vortex roll-up are perturbing each other resulting in complicated vortical

structures, as ﬁgure 16 indicates. It could also be seen on this ﬁgure that, as explained

above, a vortex shedding process and roll-up is present. Still the resulting ﬂow displays

a steady jet in crossﬂow like behavior, which is due to the convection of those structures

by the crossﬂow.

Finally results for case 6 are presented in ﬁgures 21 and 22 for Tf’s multiples of

the quarter of the pulse frequency, and Tf= 0.43 which allows the visualization of the

penetration of the vortex. Figure 9 B shows that some kind of turbulent puﬀs are emitted,

which is quite astonishing because case 6 and 8 have more or less the same maximum

velocity: 5.79 m/s for case 6 and 5.35 m/s for case 8. Here, the duration of the pulse

is a key parameter. For a sine wave the time taken by the speed to increase from its

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 19

Tf= 0.10 Tf= 0.30 Tf= 0.37

Tf= 0.42 Tf= 0.67 Tf= 1

Figure 20. Velocity proﬁle at the jet exit in the symmetry plane for case 8 at diﬀerent Tf

minimum value to its maximum value is much longer than for a pure pulse signal. This

means that the pulse amplitude is higher, and so is the vortex generated which explains

the puﬀs observed on the instantaneous ﬁeld. Moreover the vortex is better convected by

the jet since the jet speed increases for a longer period of time, during which the vortex

is pushed by the jet and gets stronger. At Tf= 0.43 other vortices are observed. This

could be, as for case 8, linked to the transient establishment of the new crossﬂow with

a jet speed of ≈5.79 m/s. At Tf= 0.75, no more vortices are observed which conﬁrms

the fact that during a decrease in speed, vortices with an opposite rotation direction are

generated and annihilate the vortices normally emitted by the crossﬂow. Again, this is

conﬁrmed by ﬁgure 14 where the strong initial vortex can be seen, followed by small

trailing vortices. Those structures are generated by the velocity increase, as in Gharib

et al. (1998). Indeed, the vortical structure near the pipe exit is the following strong

vortex being generated. Conﬁrming that only the vortices generated during the increase

of velocity are present in the crossﬂow. Also, the fact that velocity is becoming negative

is consistent with observations done for low velocity ratio jet in crossﬂow by Gopalan

et al. (2004), and is due to a recirculation zone at the jet nozzle exit.

From this section and the previous, the following facts have been pointed out: ﬁrst,

the strength of the vortex emitted, and therefore its coherence, is proportional to the

pulsation amplitude. Then, if during the velocity increase, there a vortex is emitted

there is also an opposite vortex of the same strength that is emitted during the velocity

decrease, which destroys the small vortices that roll-up to create the CVP. These two facts

can explain the diﬀerence between the ﬂow regimes: for case 7 a strong vortex is emitted

as the velocity increases, but since the duty cycle is small, the opposite vortex generated

shortly afterwards by velocity decrease destroys the small vortices, which prevents the

vortex roll-up process and leaves the strong coherent vortex free to be convected by

the ﬂow. Thereafter, the vortex roll-up process restarts. This creates the two observed

20 A. Coussement, O. Gicquel and G. Degrez

.

Tf= 0 Tf= 0.25 Tf= 0.43

Tf= 0.5Tf= 0.75 Tf= 1

Figure 21. Slice in the Q-criterion, in the symmetry plane z= 0 for case 6 at diﬀerent Tf

Tf= 0 Tf= 0.25 Tf= 0.43

Tf= 0.5Tf= 0.75 Tf= 1

Figure 22. Velocity proﬁle at the jet exit in the symmetry plane for case 6 at diﬀerent Tf

branches, one due to the strong vortex freely convected, the other one due to classical

crossﬂow structure which tries to establish when the velocity is low.

On the contrary, for case 6 a vortex of more or less the same strength is emitted (see

ﬁgure 13), but since the velocity increase is smoother and the velocity stays longer at

its high value, a vortex roll-up process can start at the nozzle exit, which prevents the

main vortex from being freely convected. When the velocity decreases, the vortex roll-up

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 21

is destroyed. This again explains the ﬂow structure, since the velocity stays high a long

enough time to start the roll-up process, the ﬂow behaves like a collection of transient

crossﬂows, as can be seen on ﬁgure 9 and 14: one main vortex, followed by two small

ones. The main vortex is strong enough to stays coherent during its convection, but since

there is a vortex roll-up process it cannot be convected far away in the crossﬂow. This

explains why strong pulsations with high duty cycles give steady crossﬂow like structures

with some highly coherent vortices.

Finally, for case 8, where the pulsation is low and the duty cycle is high, the same kind

of ﬂow as for case 6 is observed, except that no highly coherent vortex can be observed.

This is because the velocity increase is smaller, and this so is the vortex intensity, yielding

more interactions during the roll-up phase.

As Johari (2006) already pointed out, the pulsation amplitude and the duty cycle

determine the ﬂow conﬁguration. It is demonstrated here that the pulsation amplitude

is linked to the vortex strength and thus to its appearance in crossﬂow. Whether this

vortex will be ejected far away from the nozzle or not is also controlled by the duty cycle:

the smaller it is the more the main vortex will leave the steady like part of the jet in

crossﬂow . However, if the velocity increase is not large enough during the pulsation, the

main vortex will stay close to the steady jet like part of the ﬂow even for a small duty

cycle.

4.3. Mixing

It has long been argued that the pulsed jet in crossﬂow can improve the mixing with

respect to a non-pulsed jet in crossﬂow. To quantify this mixing eﬃciency, Denev et al.

(2009) propose, based on the work by Priere et al. (2004) to use four indices to assess the

mixing eﬃciency. The ﬁrst two are the spatial mixing deﬁciency (SMD) and the temporal

mixing deﬁciency (TMD).

SM D =1

DyDz

Z Z "hs1i − hs1i

hs1i#2

dydz

1/2

(4.9)

where

hs1i=1

DyDzZ Zhs1idydz (4.10)

is the mean of hs1iover a jet cross section and hs1iis the local temporal mean of s1,

which in the present case would be the mass fraction of the jet material YJ. Finally Dy

and Dxare the spatial extents of the integration plane.

T M D =1

DyDzZ Z phs0

1s0

1i

hs1idydz for hs1i>0.01 (4.11)

The authors also present in Denev et al. (2009) two other indices that quantify the spatial

unmixedness Us(see Priere (2005) for details) and the average temporal ﬂuctuations

(ATF).

All of those deﬁnitions are based on a spatial integration of a local temporal mean of

YJ. In many cases, the information that is useful to the designer is the mass ﬂow-rate

of scalar J, the jet material that crosses a section with a mass fraction YJincluded in

a certain range. For example, if the targeted design is dedicated to dilute the jet in the

crossﬂow the most important indicator will be to quantify the percentage of YJthat

crosses the section with a local mass fraction lower than a limit value. For a combustion

chamber the relevant indicator will be the fuel mass ﬂow rate crossing a section with a

22 A. Coussement, O. Gicquel and G. Degrez

local mass fraction included within the ﬂammability range. As this analysis can not be

accurately performed with the previous indicators (see section 4.4) a new indicator is

proposed.

This new indicator is deﬁned as follow:

M(Y1, Y2, t) = RY2

Y1˙mYdY

h˙mtoti(4.12)

where

˙mY∗=Z Z(ρY Vn|Y=Y∗)dS (4.13)

is the instantaneous mass ﬂow rate crossing section Swith a scalar mass fraction Y=Y∗.

Vnis the component of the velocity normal to the cross-section and h˙mtotiis the mean

mass ﬂow rate of Jinjected in the crossﬂow. Physically, the instantaneous value of

M(Y1, Y2, t) represents the net mass ﬂow that is passing through a control section, with

a mass fraction included between Y1and Y2, normalized by h˙mtoti. An average of the

instantaneous values of Min time, can then indicate the mean net mass ﬂow passing

through a control section with a mass fraction of jet material between Y1and Y2. This

time average of M, is the mixing criterion that will be used here and will be called hMi-

mixing criteria in what will follow. The control section will be in our case a crossﬂow

cross-streamwise slice.

With this mixing criterion, it is possible to easily compare diﬀerent jet in cross-ﬂow

conﬁgurations depending on what one is looking for. Using the two previous examples,

this indicator should be used as follows:

•In the case designed to dilute the jet in the main stream, the relevant indicator

will be hM(0, Ymax)iwhere Ymax is the maximum mass fraction allowed in the crossﬂow

material.

•In the case of a combustion chamber, the relevant indicator is hM(Ylean, Yrich)i

where Ylean and Yrich are the lean and rich ﬂammability limits.

An example of the mixing criterion hM(0, YJ)iis presented on ﬁgure 23 for case 1. Using

this curve it is possible to see that 60% if the jet material is passing through the section

with a mass fraction lower than 0.3 and almost 100% with a mass fraction lower that

0.65. Before presenting the results it should be noted that, as for the SMD or the TMD

criteria, the hMimixing criterion could be determined experimentally using a high speed

Planar Laser Induced Fluorescence (PLIF) technique.

For all the computational cases presented in table 1, results will be presented for slices

at x= 3Dand x= 6D, additional locations not providing any more information. The

conditional averaging process is done over 1 300 time samples, equivalent to 10 pulse

periods.

First, a comparison of the steady case is done: results for case 1, 3 and 4 are plotted

in ﬁgure 24 A and B. Case 1 and 3 are very close, which is consistent with the work of

Smith & Mungal (1998) and points out that the steady jet in crossﬂow scales with the

jet velocity ratio. This observation, also shows that the hMi-mixing criteria is consistent

with this major characteristic of jets in crossﬂows. The comparison of cases 1, 3 and 4

also conﬁrms that velocity ratio is a key parameter for the mixing. Nevertheless, the little

diﬀerence in the curves could be attributed to a Reynolds number eﬀect. The comparison

between x= 3Dand 6Dshows that the general form of the curve stays the same, cases

3 and 1 curves cross each other at more or less the same point, the only diﬀerence is

that the curve is moved closer to the y-axis and in fact some kind of stretching is applied

between the curve at x= 3Dand 6D.

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 23

Figure 23. hM(0, YJ)ifor case 1 in a cross stream-wise slice at x= 3D

(A) (B)

Figure 24. A : Comparison of hM(0, YJ)iat x= 3D, B : Comparison of hM(0, YJ)iat x= 6D

In ﬁgure 25, the same curves are plotted for steady cases 1, 2 and 5. Case 5 has the same

momentum ratio, R, as case 1, so that normally the two curves should be superposed,

but as hydrogen diﬀusivity is very high, the mixing must be increased, which is conﬁrmed

by the hMi-criteria. At x= 6Dthe hydrogen jet is already nearly completely mixed. In

the case 5 conﬁguration, a complete mixing corresponds to a mean YJof 0.95 10−3in

the crossﬂow, that is why the hMi-criteria curve is superposed to the y-axis at x= 6D

for hMivalue below 50%. Note that the same kind of stretching can also be observed

between x= 3Dand x= 6D.

To characterize this stretching the YJvalue corresponding to hMi= 90% and hMi=

50% were extracted for all the cases at x= 3Dand x= 6D(table 3). The ratios of YJat

hMi= 50% and hMi= 90% were computed and are presented in table 3. Those ratios

24 A. Coussement, O. Gicquel and G. Degrez

(A) (B)

Figure 25. A : Comparison of hM(0, YJ)iat x= 3D, B : Comparison of hM(0, YJ)iat x= 6D

Table 3. Values of YJfor hM(0, YJ)i= 99% and hM(0, YJ)i= 50% at x=3D and x=6D for all

the computational cases

Case YhMi=50% YhMi=90% YhMi=50% YhMi=90%

YhMi=50%

YhMi=90%

YhMi=50%

YhMi=90%

x= 3D x = 3D x = 6D x = 6D x = 3D x = 6D

1 0.286 0.631 0.153 0.336 2.205 2.192

2 0.241 0.451 0.116 0.223 1.871 1.922

3 0.298 0.546 0.153 0.276 1.828 1.797

4 0.163 0.306 0.112 0.203 1.869 1.813

5 0.191 0.637 0.262 0.787 3 2.998

6 0.181 0.316 0.108 0.193 1.745 1.782

7 0.171 0.444 0.101 0.271 2.596 2.678

8 0.218 0.416 0.136 0.248 1.902 1.824

are independent of the location (x= 3Dor x= 6D) but strongly depend on the case

which means that it could be related to jet characteristics, such as Ror J. The ratios at

x= 3Dare plotted as a function of the velocity ratio, R, or the momentum ratio, J, for

cases 1 to 8 and no correlation was found, as it can be seen at ﬁgure 26. As a preliminary

conclusion, it seems that this hMi-curve depends of the physics of the ﬂow, but not as a

classical power law of J or R, which are common in jet in crossﬂow and that the shape of

the curve is conversed along the crossﬂow. However those conclusions should be veriﬁed

by further studies.

The comparison of the hMi-criterion of the pulsed jet in crossﬂow (cases 6 and 7) and

a nominal jet in crossﬂow (case 2) is shown in ﬁgure 27. The ﬂows are compared to case

2 because the three ﬂows conﬁgurations have the same mean speed, and cases 6 and 7

have the same RMS value of the pulsation.

For case 7 at x= 3D, an enhancement in mixing at low YJis noticed. This could

be linked with the conclusion of Johari et al. (1999): the turbulent puﬀ must mix more

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 25

(A) (B)

Figure 26. YhMi=50%

YhMi=90% at X= 3Dand X= 6Dhas a function of : (A) the velocity ratio and

(B) the momentum ratio

(A) (B)

Figure 27. A : Comparison of hM(0, YJ)iat x= 3D, B : Comparison of hM(0, YJ)iat x= 6D

eﬃciently than a steady jet in crossﬂow. A turbulent puﬀ has a concentration that decays

with a power law ∝x−3/4while a classical jet in crossﬂow decay is ∝x−2/3but even with

this enhancement, the YJat which hMi= 1 is the same for case 2 and case 7. Further

down stream at x= 6Dthe mass fraction at which hMi= 1 is smaller for case 2 than

for case 7. The mixing is, thus, less eﬃcient for a pulsed jet in crossﬂow, which is due

to the perturbation of the steady jet in crossﬂow structure by the pulsation. Indeed, the

destruction of the vortex roll-up process has been pointed out in the previous section.

As already said, case 7, displays a bifurcation of the jet into two streams (see ﬁgure

9 B) : the turbulent puﬀ part produces the low YJmixing enhancement, but the rest of

the curve is given by the steady like part. The disturbance of the steady-like part by the

pulsation is the cause of the reduced mixing eﬃciency at high hMivalues.

Case 6, which uses a sine wave, seems to enhance the mixing, this could be linked

to the fact that some kind of turbulent puﬀs are emitted (ﬁgure 9 C). Between those

puﬀs, crossﬂow material penetrates the jet, which enhance the mixing. As for case 7, this

26 A. Coussement, O. Gicquel and G. Degrez

(A) (B)

Figure 28. A : Comparison of hM(0, YJ)iat x= 3D, B : Comparison of hM(0, YJ)iat x= 6D

enhancement vanishes at x= 6D. This is linked to the perturbation of the steady-like

ﬂow structure by the pulsation. The perturbation reduces the mixing induced by the

CVP and so reduces the quantity of crossﬂow material which is pumped into the jet.

This hypothesis still needs to be conﬁrmed by further studies but is consistent with the

work of Cortelezzi & Karagozian (2001). Indeed, even though the vortex roll-up process

is starting during the pulsation, when the velocity is high, this does not lead to the

formation of the CVP, mainly due to limited amount of time of the vortex roll-up.

Finally, ﬁgure 28 compares the three pulsed cases, which have the same RMS value

for the pulsation but not the same mean jet mass ﬂow. The sine pulsation gives the best

results in term of total mixing, i.e. value at which hMi= 1. Case 7 is worst on that

point but enhances the mixing at low hMivalue. Case 8 is somewhere between the two,

which is, in fact, consistent with the form of the signal: a duty cycle of 0.6 is quite close

to a sine wave, but the rise and fall between the high speed part and the low speed part

of the signal are similar to case 7. Yet, no detached turbulent puﬀs are observed, which

is indicated on the mixing curve by a lack of mixing in the low hMivalue. But, the

pulsation still allows the introduction of crossﬂow material in the jet, which, as for the

sine wave, gives an overall enhancement in mixing.

In conclusion a sine signal seems to enhance the mixing in the most eﬀective way,

nevertheless if an enhancement in mixing at low hMivalues must be obtained, a pulse

signal with low duty cycle seems to be more recommended. More generally, because the

vortex roll-up starts for high duty cycle, the mixing is better, which means that, for

the mixing, large duty cycle should be used in the light of the present mixing criterion.

This emphasizes the importance of the vortex roll-up process in the mixing (Cortelezzi

& Karagozian 2001).

4.4. Comparison to the SMD criteria

The SMD criteria deﬁned in equation 4.9 has been computed, at x= 3Dand x=

6D, for case 2, 6, 7 and 8 for comparison with the mixing criteria introduced here. In

those computations, YJwas used as s1. Results are presented in Table 4 and should be

compared with ﬁgures 27 A and B and 28 A and B. Here the use of temporal averaging

in the SMD criterion leads to erroneous conclusion.

Large Eddy Simulation of a Pulsed Jet in Crossﬂow 27

Table 4. SMD criteria at x= 3Dand x= 6Dfor cases 2, 6, 7 and 8.

Case SMD at x= 3DSMD at x= 6D

2 34.52 24.17

6 29.36 23.55

7 29.73 24.01

8 32.12 25.15

Indeed while the SMD criteria indicates that the best mixer is the sine wave (case

6), it also indicates that case 7 is close in terms of mixing eﬃciency. Moreover case 7 is

considered a better mixer than case 2. In the SMD criteria temporal averaging is done

on each point individually, which in the case of pulsated ﬂows causes the criteria to

be too low, as is the case for case 7. Indeed, in the zone where the vortex ring crosses

the observation plane, YJis null between the passage of two vortex rings. This yields a

very low value of hYJiin this zone, and so when the integration is performed over the

observation plane, this low value of hYJivirtually enhance the mixing.

Moreover, the SMD by reducing the mixing characterization to a single value is less

able of representing the particularities of the ﬂow structures. For case 7 the hMicurve

shows, as explained above, two parts: the ﬁrst one, for low YJvalues, which represents

the high mixing eﬃciency of the puﬀs and the second part, which is less eﬃcient due to

the perturbed jet in crossﬂow.

5. Conclusions

This study allowed the linking of the intensity of the vortex emitted with the velocity

increase of the pulsed signal, which proves that the more the velocity increases, the

more the vortex emitted will be coherent independently of the duty cycle. Moreover the

higher the pulsation amplitude is, the more it will prevent the vortex roll-up process

when the velocity will decrease, and so the more the main vortex will be convected easily

in the crossﬂow. The duty cycle controls the vortex roll-up process during and after

the pulsation; the higher the duty cycle is, the more likely a vortex roll-up process can

be observed when the velocity is high. Combined together, these two facts explain the

physics behind the various ﬂow regimes of the pulsed jet in crossﬂow.

The quantiﬁcation of the mixing presented here allows the coherent analysis of the

mixing of strongly unsteady ﬂow by using conditional averaging. It has also been demon-

strated that this criterion behaves better than criterion deﬁned by temporal averaging

processes. This criterion here is able to condense a lot of information in a single curve

and can also be applied for any kind of ﬂow. While not designed for the analysis of the

forcing conditions, the criterion is also able to detect the presence of strong coherent

structures, as has been shown for case 7. In the present case it allowed not only to ﬁnd

out that each jet in crossﬂow seems to have the same hMicriterion shape at diﬀerent

stream-wise positions but also to show that pulsed jet in crossﬂow, using a pure pulse

signal does not seem to be the best solution for an overall enhancement of mixing. The

best mixing enhancement at low YJis achieved with a pure pulsed signal and a low duty

cycle, but the best overall mixing is realized with a sine wave, for the case presented in

this work, at a constant RMS value corresponding to a ﬂuctuation of 52% in terms of jet

velocity or 133% in terms of momentum ratio. This result is linked to the vortex roll-up

process controlled by the duty cycle.

28 A. Coussement, O. Gicquel and G. Degrez

Finally, the next logical step in the study of pulsed jets in crossﬂow is to apply the

conditional averaging process presented here on an experimental case in the near ﬁeld

and the far ﬁeld, along with numerical investigation of the mixing in the far ﬁeld. Com-

putations of a pulsed jet in crossﬂow with the same pulse amplitude but with diﬀerent

duty cycle values should also been done to numerically study the inﬂuence of the duty

cycle on the mixing, since only the inﬂuence of pulse amplitude was studied here.

6. Acknowledgement

The authors wish to thanks the CERFACS CFD Team for developing AVBP and

providing it for the study. The ﬁrst author was supported by a fellowship from the Fonds

National de la Recherche Scientiﬁque, FRS-FNRS (Communaut´e Francaise de Belgique).

Computing resources were provided by the IDRIS under the allocation 2009-i2009020164

made by GENCI (Grand Equipement National de Calcul Intensif).

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