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This study quantifies the mixing that results from a pulsed jet in cross-flow in the near jet region. By large eddy simulation computations, it also helps to understand the physical phenomena involved in the formation of the pulsed jet in cross-flow. The boundary conditions of the jet inlet are implemented via a Navier–Stokes characteristic boundary condition coupled with a Fourier series 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 a pulsed jet in cross-flow flow fields. Different flow configurations, pulsed and non-pulsed, are computed and compared, keeping the root mean square value of the signal constant. This comparison not only allows the characterization of the mixing but also illustrates some of the properties of the mixing characterization.
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Under consideration for publication in J. Fluid Mech. 1
Large Eddy Simulation of a Pulsed Jet in
A X E L C O U S S E M E N T12 3, O. G I C Q U E L2 3
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
Key words: Computational fluid dynamics, Jet in Crossflow, Large Eddy Simulation,
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.
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:
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 Stjf 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
(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:
A D Zτ
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 JρjUj
velocity ratio RUj/U(see figure 2):
distinct vortex rings, followed or not by a fluid column, for α < 0.20.5 and L/D < 4
(figure 3 A).
bifurcation of the jet in two streams, for α < 0.50.8 and L/D < 20 25 (figure 3
steady like jet in crossflow structure on which a turbulent puff can be seen, for
α < 0.80.9 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 7.5e3 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 UUjPulse 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
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= 7.5e3 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 ∆x100.106m.
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
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
while the sub-grid energy is obtained using the following approximation:
kSGS (x, t)0.5hτii iThν2
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:
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.
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:
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
lines). To do so a Fourier series is used:
v(t) = a0+
ancos (2πf tn) +
bnsin (2πf tn) (3.8)
In the case of a pulse wave, coefficients can easily be computed:
v(t)dt (3.9)
v(t)cos 2πtn
Tdt (3.10)
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:
= (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 W10 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
In case 7, a frequency of 55Hz results in a Womersley’s number of W37, 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
∂t =(pps) + ρ
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:
∂t =p+ρ
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
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
∇ × ~
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:
2(Ωij ij Sij Sij )i, j [1,2,3] (4.6)
where Sij is the symmetric part of the velocity tensor:
Sij =1
∂xii, j [1,2,3] (4.7)
ij its antisymmetric part:
ij =1
∂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
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
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
Z Z "hs1i − hs1i
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
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
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
This new indicator is defined as follow:
M(Y1, Y2, t) = RY2
˙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
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
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 103in
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%
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 x3/4while a classical jet in crossflow decay is x2/3but 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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... The value of k SGS , however, cannot be obtained directly from the velocity statistics, thus the expression by Coussement et al. (2012) is used: ...
... where ∆ is the filter size, a is a calibration parameter (Coussement et al., 2012) and f s is a turbulence similarity parameter, ...
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Confluences of open-channel flows commonly occur in natural and artificial water networks and in hydraulic structures. The flow in these junctions is complex, three-dimensional and is steered by the shape of the bed. This research focuses on schematized confluences with a T-shaped planform in which the bed level of the tributary channel is higher than the one in the main channel. This phenomenon is ubiquitous in natural river confluences, but also frequently occurs in human-made configurations. The effect on the flow of such a difference in bed elevations between the two channels is investigated by means of numerical simulations and laboratory experiments. Also, the effect on the mixing of suspended or dissolved substances in merging water streams is explored. The applied Large Eddy Simulations allow not only the study of the time-averaged flow features but also their time-dependent behaviour. Besides providing fundamental knowledge, this work also contributes to the optimization of artificial confluences.
... In order to obtain more accurate results and spend less computational cost, GR2 is chosen for the subsequent study. The turbulence resolution M is calculated as M = k SGS /(K + k SGS ) [30], where k SGS represents the subgrid energy, and K represents the resolved part of the energy. The turbulence resolution of all grids in GR2 was less than 0.1, that is, the percentage of turbulence kinetic energy resolved is greater than 90%. Figure 2 shows the comparison between the normalized time-mean streamwise velocity deficit of GR cases and the results of Churchfield et al. [16]. ...
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Due to abundant wind resources and land saving, offshore wind farms have been vigorously developed worldwide. The wake of wind turbines is an important topic of offshore wind farms, in which the wake expansion is a key issue for the wake model and the layout optimization of a wind farm. The large eddy simulation (LES) is utilized to investigate various offshore wind farms under different operating conditions. The numerical results indicate that it is more accurate to calculate the wake growth rate using the streamwise turbulence intensity or the total turbulence intensity in the environment. By fitting the results of the LES, two formulae are proposed to calculate the wake growth rate of the upstream wind turbine. The wake expansion of the downstream wind turbine is analyzed, and the method of calculating the wake growth rate is introduced. The simulation indicates that the wake expansion of the further downstream wind turbines is significantly reduced. The smaller lateral distance of wind turbines in the offshore wind farm has the less wake expansion of the wind turbines. The wake expansion under different inflow wind speeds is also analyzed, while the wake expansion of wind turbines under more complex conditions needs to be further studied.
... JICFs have a wide variety of practical applications including plume dispersion, film cooling for combustors and turbines, and discharges of sewage and industrial effluents into channels and natural rivers [22,27,30]. JICFs have also gained considerable attention in many experimental and numerical studies in fluid mechanics due to their capability to provide higher mixing in comparison with free jets [22,31]. This is mostly attributed to the existence of various complicated vortical structures resulting from the interaction of the jet and cross-flow as the jet expands downstream [28,30,32]. ...
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Significant environmental effects from the use of marine outfall discharges have led to increased efforts by both regulatory bodies and research groups to minimize the negative impacts of discharges on the receiving water bodies. Understanding the characteristics of discharges under conditions representative of marine environments can enhance the management of discharges and mitigate the adverse impacts to marine biota. Thus, special attention should be given to ambient cross-flow effects on the mixing behaviors of jet discharges. A buoyant jet in cross-flow has different practical applications such as film cooling and dilution, and provide a higher mixing capability in comparison with free jets or discharges into stationary environments. The main reason for this is believed to be the existence of various complicated vortical structures including a counter-rotating vortex pair as the jet expands downstream. Although tremendous research efforts have been devoted to buoyant jets issuing into cross-flows over the past five decades, the mixing process of an effluent at the discharge point is not yet well understood because of the highly complex fluid interactions and dispersion patterns involved. Therefore, there is a need for a deeper understanding of buoyant jets in cross-flows in order to obtain better predictive methods and more accurate design guidelines. The main aims of this study were (i) to establish the background behind the subject of buoyant jets in cross-flows including the flow structures resulting from the interaction of jets and cross-flows and the impacts of current on mixing and transport behavior; (ii) to present a summary of relevant experimental and numerical research efforts; and finally, (iii) to identify and discuss research gaps and future research directions.
... The specific vorticity distribution associated with such unsteady jets is governed by the applied velocity program. While jet characteristics can be altered effectively through sinusoidal forcing as reported by Paschereit et al. (1995), Muldoon & Acharya (2010) or Coussement et al. (2012), the velocity signal is often characterised by a rapid acceleration of finite amounts of fluid that are at rest initially. The transient motion of these starting, or fully pulsed, jets leads to the generation of large-scale primary vortex rings that are mainly responsible for the desired effects mentioned above. ...
The interaction between starting jets and a steady cross-flow with a zero-pressure-gradient, turbulent boundary layer is studied experimentally. A device typically used in flow control applications is employed as jets of compressed air are injected into the cross-flow through a rectangular high-aspect-ratio outlet. Investigating different velocity ratios between starting jets and cross-flow within the interval r = ujet/Uinf = 2.4, . . . , 11, we identify two regimes of different flow structure appearance, transferring the classification map applicable to parallel circular starting jets in cross-flow established by Sau & Mahesh (J. Fluid Mech., vol. 604, 2008, pp. 389–409). At r < 4, the vorticity associated with the upstream part of the starting jet is cancelled by the cross-flow boundary layer. Hairpin vortices are observed. At r > 4, the starting jets penetrate through the boundary layer, and vortex rings are generated. They are asymmetric in shape as the windward vortex ring core is thinner due to the interaction with the cross-flow. As the limiting case of zero cross-flow (r → ∞) is approached, the asymmetry decreases and the formation time corresponding to maximum vortex ring circulation converges to the characteristic value of t∗ ≈ 12 recently determined for this type of non-parallel planar starting jets when emitted into quiescent surroundings (Steinfurth & Weiss, J. Fluid Mech., vol. 903, 2020, A16). The findings presented in the current article can promote the sophisticated selection of actuation parameters in active mixing and separation control.
... The above-mentioned studies led to a better understanding of the complex flow structure of vertical jets in crossflows, and showed that these discharges generate four main types of vortices: counterrotating vortex pairs (CVPs), horseshoe vortices, jet shear layer vortices, and wake vortices ( Fig. 1). Jets discharged in crossflows provide a higher mixing capability compared to free jets or mixing layers [28], and one of the reasons for this is the formation of CVP vortices as the jet expands downstream into the crossflow [22]. The wake vortices and shear-layer vortices are naturally unsteady and are eliminated from the velocity measurements by averaging over time. ...
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The dispersion of surface jets in crossflows such as rivers or channels can cause critical environmental problems in the form of chemical or thermal pollution of these water bodies. The turbulent flow structures occurring in such crossflows play an important role in the mixing of surface jets with the surrounding water bodies. In this study, experimental measurements of the time history of the 3-D velocity field were conducted to better understand the flow structure of surface jets in crossflow conditions. Stereoscopic Particle Image Velocimetry was used to measure the instantaneous spatial and temporal velocity distribution downstream of the jet’s discharge point. In addition to the mean velocity distribution, turbulent flow characteristics such as the turbulent kinetic energy (\(k\)), turbulent kinetic energy dissipation rate (\(\epsilon\)), and turbulent eddy viscosity (\(\nu_{t}\)) were calculated. The formation and evolution of a vortex in the surface jet’s flow structure was detected over the measurement zone. The vortex in the surface jets in crossflow resembled to half of the vortices in a counter-rotating vortex pair (CVP) of submerged jets in crossflows. It can be inferred that the water surface performed like a plane of symmetry.
... Muldoon et al. [28] presented the mechanism for the development of the wake vortices, and the obtained results indicated the potential of using external forcing as a potential control strategy for controlling the jet penetration and mixing with the crossflow. Coussement et al. [29] proved that the more the velocity increases, the more the vortex emitted would be coherent independently of the duty cycle by large eddy simulation with a square or a sine pulse wave. The effects of pulsing of high-speed subsonic jets on mixing and jet trajectory in turbulent subsonic crossflows were investigated by Srinivasan et al. [30] through large-eddy simulation, and they discovered that the classical momentum-ratio-based scaling could still be used for the high Reynolds and jet Mach number cases. ...
The mixing process plays an important role in the combustion realization of the scramjet engine. In the current study, the steady jet, as well as pulsed jets with different wave shapes namely sine, square and triangle waves, is investigated in order to achieve adequate fuel/air mixing. Flow field properties are studied numerically based on grid independency analysis and code validation. The vortex structures, as well as the flow field parameters such as mixing efficiency, total pressure recovery coefficient, fuel penetration depth and mixing length, are deeply analyzed for different jet-to-crossflow pressure ratios. The obtained results predicted by the three-dimensional Reynolds-average Navier-Stokes (RANS) equations coupled with the two equation SST κ-ω turbulence model show that the grid scale makes only a slight difference to wall pressure profiles for all cases studied in this article. Compared with the steady jet, the pulsed jets with different wave shapes are beneficial for the mixing efficiency improving of the transverse jet, and the pulsed jets have special advantages on reducing the total pressure loss and mixing length but not for improving the fuel penetration depth. When the jet-to-crossflow pressure ratio is high, the performance of pulsed jets is better, and different wave shapes in the pulsed jet result in different vortex structures. This should be studied and discussed deeply in the near future.
To understand the micro-vortex structure of Microjet Vortex Generators, a plate with pitched and skewed jet holes was established. Simulations using SST-DDES method based on WENO scheme were conducted to analyse the effect of jet pressure, hole interval and incoming boundary layer thickness on the flow structure. The pitch angle and skew angle are 55 degrees and 45 degrees respectively. The total pressure of microjets ranges from 2.5 to 3.5 times the mainstream pressure. The interval of microjets ranges from 6.75 to 13.5 diameters of the hole, and the thickness of incoming boundary layer ranges from 2 to 4 diameters of the hole. The flow field is dominated by three vortices, including a main vortex and two secondary vortices which are significantly influenced by jet pressure and interval. The interaction between secondary vortices and main vortex plays an important role in the process of boundary layer mixing.
Purpose This study aims to numerically investigate the flow features and mixing/combustion efficiencies in a turbulent reacting jet in cross-flow by a hybrid Eulerian-Lagrangian methodology. Design/methodology/approach A high-order hybrid solver is employed where, the velocity field is obtained by solving the Eulerian filtered compressible transport equations while the species are simulated by using the filtered mass density function (FMDF) method. Findings The main features of a reacting JICF flame are reproduced by the large-eddy simulation (LES)/FMDF method. The computed mean and root-mean-square values of velocity and mean temperature field are in good agreement with experimental data. Reacting JICF’s with different momentum ratios are considered. The jet penetrates deeper for higher momentum ratios. Mixing and combustion efficiency are improved by increasing the momentum ratio. Originality/value The authors investigate the flow and combustion characteristics in subsonic reacting JICFs for which very limited studies are reported in the literature.
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Radiation plays an important role in a broad range of thermal engineering applications comprising turbulent flows. The growing need for accurate and reliable numerical simulations to support the design stages of such applications is the main motivation of this thesis.Of special interest in this work are the free-shear flows and the fundamental understanding of how radiation can modify their fluid dynamics and heat trans- port as well as how their turbulence fluctuations can alter radiative transfer. The goal of this thesis is to provide high-fidelity data of turbulent free jets coupled with thermal radiation in order to develop and validate free-shear turbulent models accounting for coupling interactions. To this end, turbulent free jets are described by direct numerical simulations (DNS) coupled to a reciprocal Monte- Carlo method to solve the radiative transfer equation. The spectral dependency of the radiative properties is accounted for with an accurate Correlated-k (ck) method. The numerical study is carried out with state-of-the-art fidelity to be as representative as possible of an actual jet in a participating medium. The simulation is optimized in terms of processing time taking advantage of an acceleration method called Acoustic Speed Reduction and by injecting artificial turbulence to enhance inlet boundaries.Two direct simulations of heated jets coupled with thermal radiation are carried out. On the one hand, a heated jet with moderate radiation is simulated. The analysis of its high-fidelity coupled DNS data has allow to derive a new scaling law for the decay of the temperature profile. This scaling accounts for the effects of modified density due to moderate radiation. Moreover, it allows for distinguishing whether thermal radiation modifies the nature of heat transfer mechanisms in the jet developed region or not. On the other hand, a strongly heated free jet is computed in order to quantify the effects of radiation on mean temperature and velocity fields as well as on second order moments.Besides the coupled DNS data, a RANS solver for variable-density flows coupled with thermal radiation has been implemented during the course of this thesis. The goal is to directly quantify the accuracy of the existing turbulent models, and to identify key parameters for further modeling of coupling interactions.
The formation of vortex rings generated through impulsively started jets is studied experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity and vorticity fields of vortex rings are obtained using digital particle image velocimetry (DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate that the flow field generated by large L/D consists of a leading vortex ring followed by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected from that of the trailing jet. On the other hand, flow fields generated by small stroke ratios show only a single vortex ring. The transition between these two distinct states is observed to occur at a stroke ratio of approximately 4, which, in this paper, is referred to as the ‘formation number’. In all cases, the maximum circulation that a vortex ring can attain during its formation is reached at this non-dimensional time or formation number. The universality of this number was tested by generating vortex rings with different jet exit diameters and boundaries, as well as with various non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided for the existence of the formation number based on the Kelvin–Benjamin variational principle for steady axis-touching vortex rings. It is shown that based on the measured impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin principle correctly predicts the range of observed formation numbers.
Fully modulated, incompressible, turbulent transverse jets were studied experimentally in a water tunnel over a range of pulsing frequencies and duty cycles and at two jet-to-crossflow velocity ratios. The jet flow was completely modulated by operating an in-line solenoid valve resulting in the shutoff of jet supply during a portion of the cycle. The planar laser-induced fluorescence technique was used to determine the penetration, dilution, and structural features of the pulsed jets. The molecular mixing rate was quantified using a chemical reaction between the jet and crossflow fluids. Short injection times resulted in creation of vortex ring structures, whereas long injection times produced axially elongated turbulent puffs, similar to a segment of a steady jet. The latter case resulted in only modest enhancement of the jet penetration depth and dilution. Pulsed jets dominated by vortex rings had penetration depths significantly greater than a steady jet with the same velocity ratio, up to a factor of 5 at 50 jet diameters downstream of the exit. For short injection times, duty cycle had a significant effect on the behavior of pulsed jets. Increasing the duty cycle for a fixed injection time reduced the jet penetration. The dilution and mixing of pulsed jets with short injection time was also improved over the steady jet for duty cycles as high as 0.5. The greatest increase in the mixing rate was approximately 50% for well-separated pulses with short injection times.
This paper presents detailed measurements of the film-cooling effectiveness for three single, scaled-up film-cooling hole geometries. The hole geometries investigated include a cylindrical hole and two holes with a diffuser-shaped exit portion (i.e., a fan-shaped and a laid-back fan-shaped hole). The flow conditions considered are the crossflow Mach number at the hole entrance side (up to 0.6), the crossflow Mach number at the hole exit side (up to 1.2), and the blowing ratio (up to 2). The coolant-to-mainflow temperature ratio is kept constant at 0.54. The measurements are performed by means of an infrared camera system, which provides a two-dimensional distribution of the film-cooling effectiveness in the near field of the cooling hole down to x/D = 10. As compared to the cylindrical hole, both expanded holes show significantly improved thermal protection of the surface downstream of the ejection location, particularly at high blowing ratios. The laidback fan-shaped hole provides a better lateral spreading of the ejected coolant than the fan-shaped hole, which leads to higher laterally averaged film-cooling effectiveness. Coolant passage cross-flow Mach number and orientation strongly affect the flowfield of the jet being ejected from the hole and, therefore, have an important impact on film-cooling performance.
The scalar transport in a swirling jet in a crossflow has been investigated in water tunnel experiments. The jet to freestream velocity ratio was varied from 4.9 to 11.1, and the jet swirl numbers ranged from 0 to 0.17, The jet exit Reynolds number was kept at 1.3 x 10(4) during the experiments. Planar laser-induced fluorescence was utilized to measure planar cross sections of the mean concentration field of the jet up to 68 jet diameters downstream of the exit. The jet penetration depth, half-value radius, and maximum concentration were determined from these concentration fields. For jets without swirl, measured cross-sectional mean concentration distributions have symmetric double-lobed kidney shapes that are consistent with the counter-rotating vortex pair that is known to exist in the far field of the jet. The addition of swirl causes the far-field distributions to become nonsymmetric, with one of the lobes increasing in size and the other decreasing, resulting in a comma shape. Swirl is also observed to decrease jet penetration but not to significantly affect the decay of maximum mean concentration for the range of swirl numbers investigated.