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Controlled mixing enhancement in
turbulent rectangular jets responding
to periodically forced inflow conditions
a
bc
Artur Tyliszczak & Bernard J. Geurts
a
Department of Mechanical Engineering and Computer Science,
Czestochowa University of Technology, Czestochowa, Poland
b
Multiscale Modelling and Simulation, University of Twente,
Enschede, The Netherlands
c
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Anisotropic Turbulence, Faculty for Applied Physics, Eindhoven
University of Technology, Eindhoven, The Netherlands
Published online: 08 Apr 2015.
To cite this article: Artur Tyliszczak & Bernard J. Geurts (2015) Controlled mixing enhancement
in turbulent rectangular jets responding to periodically forced inflow conditions, Journal of
Turbulence, 16:8, 742-771, DOI: 10.1080/14685248.2015.1027345
To link to this article: http://dx.doi.org/10.1080/14685248.2015.1027345
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Journal of Turbulence, 2015
Vol. 16, No. 8, 742–771, http://dx.doi.org/10.1080/14685248.2015.1027345
Controlled mixing enhancement in turbulent rectangular jets
responding to periodically forced inflow conditions
Artur Tyliszczaka∗ and Bernard J. Geurtsb,c
a
Department of Mechanical Engineering and Computer Science, Czestochowa University of
Technology, Czestochowa, Poland; b Multiscale Modelling and Simulation, University of Twente,
Enschede, The Netherlands; c Anisotropic Turbulence, Faculty for Applied Physics, Eindhoven
University of Technology, Eindhoven, The Netherlands
Downloaded by [Universiteit Twente] at 13:20 13 April 2015
(Received 8 October 2014; accepted 5 March 2015)
We present numerical studies of active flow control applied to jet flow. We focus on
rectangular jets, which are more unstable than their circular counterparts. The higher
level of instability is expressed mainly by an increased intensity of mixing of the main
flow with its surroundings. We analyse jets with aspect ratio Ar = 1, Ar = 2 and Ar =
3 at Re = 10,000. It is shown that the application of control with a suitable excitation
(forcing) at the jet nozzle can amplify the mixing and qualitatively alter the character
of the flow. This can result in an increased spreading rate of the jet or even splitting
into nearly separate streams. The excitations studied are obtained from a superposition
of axial and flapping forcing terms. We consider the effect of varying parameters such
as the frequency of the excitations and phase shift between forcing components. The
amplitude of the forcing is 10% of the inlet centreline jet velocity and the forcing
frequencies correspond to Strouhal numbers in a range St = 0.3–0.7. It is shown that
qualitatively different flow regimes and a rich variety of possible flow behaviours can
be achieved simply by changing aspect ratio and forcing parameters. The numerical
results are obtained applying large eddy simulation in combination with a high-order
compact difference code for incompressible flows. The solutions are validated based on
experimental data from literature for non-excited jets for Ar = 1 at Re = 1.84 × 105
and Ar = 2 at Re = 1.28 × 105 . Both the mean velocities as well as their fluctuations
are predicted with good accuracy.
Keywords: active and passive controls; mixing enhancement; bifurcation phenomenon;
LES
1. Introduction
Interest in flow control techniques is driven by the potential to gain considerable improvement of performance, safety and efficiency of various technical devices with limited energy
input. Flow control may be divided into two categories: passive and active.[1,2] The former
most often relies on optimisation procedures which are based on geometric shaping or
adding fixed elements (obstacles, swirlers, etc.) to the flow domain. Active methods require
the input of energy to the flow whose type and level may be constant (predetermined control
methods) or varying in response to the instantaneous flow behaviour (interactive methods).
An evident advantage of passive flow control is its low cost and simplicity. However,
modifications to the flow domain are usually not optimal simultaneously for different flow
∗
Corresponding author. Email: [email protected]
C 2015 Taylor & Francis
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Journal of Turbulence
743
conditions. From this point of view, active flow control, while being more intricate to realise,
is much more flexible and could result in good overall response under a variety of different
flow conditions.
Fundamental research is conducted presently into both active and passive flow control
techniques. A prominent example of successful alteration of flow dynamics by flow control
is found for the class of jet flows. Research on passive control techniques concentrates on
geometrical modifications of the jet nozzle. It turns out that non-symmetric jets emanating
from rectangular or elliptical nozzles enhance mixing between the jet and the surrounding
flow. Particularly, the large-scale mixing of rectangular jets was found substantially larger
than in the case of classical circular jets. This aspect is important in many applications (e.g.,
combustion processes), and therefore, rectangular jets were extensively studied for many
years, both experimentally [3–5] and numerically using large eddy simulation (LES) [6–9]
and direct numerical simulation (DNS).[10,11] A comprehensive discussion on theoretical
issues related to rectangular jets and a list of their possible applications may be found in
review papers.[12,13]
Concerning active flow control methods applied to jet type flows, most of the research
is devoted to circular jets. The work of Crow and Champagne [14] was probably the first in
this category. It was reported that for properly chosen forcing (excitation), the jet behaviour
may change qualitatively. An enhanced mixing and the existence of two maxima in the
turbulence intensity as function of forcing frequency were found, not seen previously in
natural jets. This seminal work initiated many experimental studies which revealed the large
potential of active control techniques.[15–23]
In this paper, we combine passive and active flow control techniques to rectangular
turbulent jets, and analyse the resulting flow field. The former is obtained by using a
rectangular shape with various aspect ratios of the jet and the latter is obtained by imposing
at the jet inlet an excitation with a specific amplitude and frequency. In [18], it was shown
that for plane jets the influence of a symmetric excitation on the mean and fluctuating
velocity fields is much weaker than that in the circular jet. Recently, DNS studies [24] of
a planar jet with a steady modulation characterised in terms of a Beltrami flow have been
performed. It was shown that the dynamics and size of the mixing layers can be considerably
changed by varying the wave number of the applied modulation. The large-scale mixing
could be stimulated when the length scale of the imposed modulation had a size comparable
to the width of the jet. In the present work, unlike in [18,24], the excitation is a combination
of the symmetric and flapping forcing modes and has an unsteady character. We focus
on excited jets with aspect ratio Ar = H/W = 1, Ar = 2 and Ar = 3, where H and W
are the inlet dimensions along its major and minor axes, respectively. Emphasis is put on
determining inflow perturbation procedures that give rise to a global alteration of the flow
pattern and modification or even creation of large-scale flow structures. It is found that such
a combined excitation significantly changes the spreading rate of evolving jets, amplify the
vortices and for some forcing parameters even initiates a division of the main jet stream
into separate streams. This last phenomenon is called the bifurcation phenomenon [25]
and is a spectacular example of a modified flow pattern arising from an upstream forcing.
The existence of a bifurcation was revealed also for square [11] and rectangular jets.[9]
However, they were found to bifurcate with greater difficulty as compared to circular jets.
This was attributed to the presence of rib vortices at the corners. In the present work,
we extend these studies and quantify the effect of the frequency of the excitation on the
bifurcation phenomenon.
The paper is organised as follows. The next section gives details of the governing
equations, LES modelling and applied numerical method. In Section 3, the simulation
744
A. Tyliszczak and B.J. Geurts
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parameters and forcing method are described. The results of simulation are presented in
Sections 4 and 5. First, the validation of solutions is performed based on comparison with
literature data for non-excited jets at aspect ratio Ar = 1 and Ar = 2. Next, the simulation
results for the excited jets are presented and discussed, emphasising (1) effect of the forcing
frequencies of the axial and flapping excitations, (2) phase shift between these forcing
modes and (3) differences between the excitation effects for varying Ar . Finally, concluding
remarks are given in Section 6.
2. Governing equations and numerical algorithm
In this section, we introduce the LES model used for the simulations and summarise the main
characteristics of the numerical method. The LES approach has been developing for more
than 30 years now, and is regarded as a reliable tool in computational fluid dynamics (CFD)
simulations.[26,27] In this period, considerable effort has been put on many different issues,
including filtering techniques, commutation errors, sub-filter modelling and also mutual
interactions between numerically induced errors and modelling errors (for instance,[28–
33]) All this resulted in the current maturity of the LES approach.
In this paper, we consider incompressible flow described by the continuity and the
Navier–Stokes equations. In the framework of LES, we have for a commuting filter:
∂uj
=0
∂xj
(1)
f
∂τij
∂ui uj
∂τij
∂ui
1 ∂ p¯
+
=−
+
+
∂t
∂xj
ρ ∂xi
∂xj
∂xj
(2)
where ui are the velocity components, p the pressure and ρ the density. The overbar denotes
spatially filtered variables: f¯ = G ∗ f with G being the filter function.[26,27] The stress
f
tensor of the resolved field, τ ij , and the unresolved sub-filter stress tensor τij due to filtering
of the non-linear advection terms are
τij = 2νSij ,
f
τij = ui uj − ui uj
(3)
∂ u¯ j
∂ u¯ i
is the rate of strain tensor of the
where ν is the kinematic viscosity and Sij = 12 ∂x
+
∂x
j
i
resolved velocity field. The sub-filter tensor is modelled in this paper by an eddy-viscosity
f
f
f
model with τij = 2νt Sij + τkk δij /3. The diagonal terms τkk are added to the pressure
f
resulting in the so-called modified pressure P¯ = p¯ − ρτkk δij /3.[26] Hence, we have
∂ui uj
∂ u¯ j
∂ui
1 ∂ P¯
∂
∂ u¯ i
(ν + νt )
+
=−
+
+
∂t
∂xj
ρ ∂xi
∂xj
∂xj
∂xi
(4)
In the present implementation, P¯ is calculated with the help of the projection method. The
eddy viscosity ν t is computed using the model proposed by Vreman [34]:
νt = C
Bβ
αij αij
(5)
Journal of Turbulence
αij =
∂uj
,
∂xi
βkl = 2 αmk αml
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2
2
2
+ β11 β33 − β13
+ β22 β33 − β23
Bβ = β11 β22 − β12
745
(6)
(7)
where the constant in Equation (5) is taken as C = 2.5 × 10−2 .[34] The filter width is
computed as = (xyz)1/3 with x, y, z being the mesh spacings. This model
yields a vanishing eddy viscosity close to a solid wall, in laminar flows or in pure shear
regions. This is a very important property of a sub-filter model for jet flows where turbulence
develops in the shear layer region. Other sub-filter models, such as Smagorinsky’s model,
could be too dissipative and prevent turbulent fluctuations to develop properly in the flow.
The flow solver used in this work is an academic high-order code (SAILOR) which
is based on a projection method with time integration performed by a predictor–corrector
(Adams-Bashforth, Adams-Moulton) method. The eddy viscosity is computed every time
step at the beginning of the projection step. The spatial discretisation is based on a compact
difference method developed for half-staggered meshes,[35,36] where the pressure nodes
are shifted half a cell size from the velocity nodes. The solution algorithm requires both
interpolation procedures and derivative approximations which are performed using 10thorder and 6th-order formulas,[36] respectively.
The SAILOR code was used previously in various studies, including laminar/turbulent
transition in free jet flows,[37–39] near-wall flows,[40] multi-phase flows [41] and
flames.[42,43] Recently, the SAILOR code has been used in parametric studies of excited circular jets.[44] It was shown that it enables accurate prediction of the bifurcation
phenomenon in circular jets. Grid-refinement studies showed that the applied high-order
discretisation schemes yield grid-independent solutions already at relatively coarse meshes.
This allowed for a comprehensive analysis of various forcing parameters and identification
of the most important ones from the point of view of large-scale alteration of jet flows. This
knowledge is partially discussed later in relation to the dynamics of rectangular jets.
3. Simulation set-up and forcing procedure
In this section, we first introduce the flow domain and subsequently address the procedure
with which the flow is forced at the jet inflow.
A schematic view of the nozzle geometry is shown in Figure 1. In the computations, we
do not explicitly resolve the flow inside the nozzle. The computational domain is a simple
rectangular box with inflow conditions defined by the mean velocity components and their
fluctuations.
In Figure 1, the symbols zh and xh denote the widths in the lateral (x) and spanwise (z)
directions at which the mean velocity is half the value in the centre, Ucl . These distances are
introduced to enable comparisons with literature data that are presented in non-dimensional
form using zh and xh as the reference values.
The sizes of the computational domain are 12W in x-direction, 16W in y-direction and
12W in z-direction, with W the width of the minor side of the jet nozzle. To check the level
of influence of the side boundaries, additional test computations were performed using a
wider computational domain with 16W × 16W along the x–z directions. Effects of the
boundaries on the dynamics of the jets were found to be negligible, as will be quantified
momentarily.
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746
Figure 1.
A. Tyliszczak and B.J. Geurts
Schematic view of the rectangular nozzle.
The inlet boundary conditions are specified in terms of the mean velocity profile
superimposed with fluctuating components as
u(x, t) = umean (x) + uexcit (x, t) + uturb (x, t)
(8)
where the mean velocity is a hyperbolic-tangent profile:
Uj
1 Rx |x| Rx
1 − tanh
−
umean (x) = Uc +
4
4 δθ R x
|x|
1 Rz |z| Rz
× 1 − tanh
−
4 δθ R z
|z|
(9)
in which Uj denotes the inlet velocity at the jet axis and Uc = 0.05Uj is the velocity of a
co-flow introduced in order to mimic an inflow entrainment observed when a jet issues from
a nozzle into an open domain.[45] Without adding the co-flow, a free jet would become a jet
in an enclosure with a recirculation zone created close to the inlet. Adding the co-flowing
stream is a standard procedure in jet flow simulations.[23,46,47] At the level Uc < 0.1Uj ,
the co-flow has only a minor influence on the jet dynamics.[47] The symbols x, z in Equation
(9) denote the in-plane coordinates, Rx = W/2 and Rz = H/2 are the nozzle half-width of
the minor and major axes. The parameter δ θ is the momentum thickness of the initial shear
layer. In all simulations performed in this work, we take δ θ = 0.05Rx which is the same as
in [44,47] for natural and excited circular jets. This choice is also motivated by the fact that
for thinner shear layers, i.e., δ θ ≤ 0.05Rx , and turbulence intensity of the order of 1% the
jets exhibit very similar behaviour.[39]
The velocity component uexcit (x, t) acts as the deterministic forcing term which in
physical experiments is usually produced by a membrane (loudspeaker) located upstream
of the nozzle or by plasma actuators or by a mechanical forcing obtained by especially
designed flap actuators placed at the nozzle lip.[17,48,49] These excitations change the
direction and magnitude of the flow leaving the nozzle. In this work, we add the forcing
Journal of Turbulence
747
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Figure 2. Schematic view of the forcing terms: axial mode is shown on the left side of the figure
and the flapping mode is displayed on the right-hand side.
only to the streamwise velocity as
uexcit (x, t) = Forcing × Mask
(10)
πx Forcing = Aa sin 2π Sta Uj t/W + Af sin 2π Stf Uj t/W + sin
W
(11)
where
is the superposition of the axial forcing mode with amplitude Aa and the flapping forcing
mode with amplitude Af . The mask is defined as
Mask =
1 Rx |x| Rx
1 Rz |z| Rz
1
1 − tanh
1 − tanh
(12)
−
−
4
4 δθ R x
|x|
4 δθ R z
|z|
and provides a smooth transition with the co-flowing stream. The applied forcing locally
alters the magnitude of the inlet velocity but the direction of the flow remains unchanged.
This method was successful in previous studies [44,47,50] for the circular jets. The Strouhal
numbers of the excitations are defined as Sta = fa W/Uj and Stf = ff W/Uj , where fa , ff are the
frequencies of the axial and flapping excitations. The symbol is the phase shift between
axial and flapping forcing. In this paper, we consider the flapping excitation along the lateral
direction only, as shown schematically in Figure 2. Additional studies performed with the
excitation applied along the spanwise direction showed that for the cases with Ar ≥ 2, the
only effects of such forcing were an increase of the turbulence intensity in the flow field,
but not a global change of the flow pattern.
Finally, the turbulent fluctuations uturb (x, t) are computed by applying a digital filtering
method to random inflow perturbations as proposed by Klein et al. [51]. This method
guarantees spatially correlated velocity fields which can be tuned to reflect real turbulent
flow conditions.[51–55]
At the side boundaries, the streamwise velocity is assumed equal to Uc and the remaining
velocity components are equal to zero. Hence, there is no flow through the side boundaries.
The pressure is computed from the Neumann condition n · ∇p = 0 with n the outward
normal vector. At the outlet, all velocity components are computed from a convective
boundary condition ∂ui /∂t + VC ∂ui /∂n = 0. The convection velocity VC is computed every
time step as the mean velocity in the outlet plane, limited such that VC = max (VC , 0).
748
A. Tyliszczak and B.J. Geurts
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Table 1. Parameters of the computational cases for non-excited jets, specified by aspect ratio Ar ,
Reynolds number Re and the computational grids. The cell sizes are expressed as x∗ = x/W,
y∗ = y/W, z∗ = z/W with x, y and z taken from the non-uniform grid at x = y = z = 0.
Case
Ar
A1-1
A1-2
A1-3
A2-1
A2-2
A2-3
A3-1
1
1
1
2
2
2
3
Nx × Ny × Nz
Re
1
1.84
1.84
1
1.28
1.28
1
×
×
×
×
×
×
×
104
105
105
104
105
105
104
192
192
320
192
192
320
192
×
×
×
×
×
×
×
256
256
384
256
256
384
256
×
×
×
×
×
×
×
192
192
320
192
192
320
192
(x∗ , y∗ , z∗ )
(0.034, 0.042, 0.034)
(0.034, 0.042, 0.034)
(0.020, 0.028, 0.020)
(0.034, 0.042, 0.054)
(0.034, 0.042, 0.054)
(0.020, 0.028, 0.032)
(0.034, 0.042, 0.058)
The term ∂ui /∂n is discretised with a second-order tri-points upwind scheme.[56] The
pressure at the outflow is assumed constant and equal to zero. This formulation of the
outflow boundary conditions is stable and allow to pass turbulent flow structures without
any visible distortion.[42–44]
4. Natural, non-excited jets – validation test cases
The first set of simulations is performed for non-excited jets. These solutions will be used
as point of reference for comparison with the excited jets analysed in the next section.
Additionally, the results for the jets with aspect ratios Ar = 1 and Ar = 2 illustrate the
accuracy of the applied numerical method and assess the dependence of the solution on
inlet parameters and on the density of the computational grid.
We refer to experimental data [3,4] obtained for Reynolds numbers Re = 1.84 × 105
for Ar = 1 and Re = 1.28 × 105 for Ar =√2. The Reynolds number for rectangular jets
is defined as Re = Uj De /ν, where De = 2 W H /π is the so-called equivalent diameter.
In order to check the dependence of the solutions on Re and also to enable comparison
between the solutions for different Ar , we additionally consider flow at Re = 1 × 104 .
The computational domain is 12W × 16W × 12W. The numerical grids are compacted
towards the jet centre and along the axial direction. In the x- and z-directions, the mesh
nodes are distributed using a tangent hyperbolic function with parameters chosen such that
almost uniform cell sizes are generated across the jet inlet. In the axial direction, the grid
nodes are distributed using an exponential function. Two computational meshes are used.
The first one consists of 192 × 256 × 192 nodes and will be further referred to as the basic
mesh. We also include a refined mesh with 320 × 384 × 320 nodes to assess the level of
dependence of the results on the mesh density. Table 1 reports the analysed test cases and
gives details of the applied numerical grids. The cell sizes x, z and y correspond to the
location x = z = y = 0. The symbols ‘A1’, ‘A2’ and ‘A3’ indicate the aspect ratio and the
different flow cases are numbered 1, 2 and 3. Ratios of the maximum (near the boundaries)
to minimum (in the inlet jet region) cell sizes are (1) 6.41 and 6.47 in the x-direction, (2)
2.081 and 2.083 in the y-direction, (3) 2.06 and 2.07 in the z-direction for Ar = 2 and 1.56
for Ar = 3, where the first number refers to the coarse mesh and the second one to the ratio
on the dense mesh.
LES computations for the reference cases analysed in [3,4] were carried out previously
in [7] for three different velocity profiles. Zero and a non-zero spanwise and lateral velocity
components were analysed and compared with mean values taken from experiments. It was
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Journal of Turbulence
749
found that the best agreement with the reference data was obtained when the streamwise
velocity profile was assumed as a sharp rectangular profile and the remaining velocity
components were set to zero. Here, we also adopt this approach and take the mean spanwise
and lateral velocities zero, whereas the streamwise velocity profile is approximated by
Equation (9). This profile closely resembles both the experimental profile and the profile
used in [7]. The mean profile (9) with δ θ = 0.05Rx yields a shear layer thickness δ 99 =
0.48Rx which is defined here as the region where Uc < umean (x) ≤ 0.99Uj . In the lateral
direction, the number of grid nodes located across δ 99 is equal to 7 and 12 on the basic and
dense mesh, respectively. In the spanwise direction for Ar = 2, there are, respectively, five
and eight nodes depending on the mesh density, and for Ar = 3, there are four nodes on the
basic mesh.
Concerning the distribution of turbulence intensities and turbulent length/time scales,
no detailed data were documented in the experiments. Only the minimum value of the
turbulence intensity on the jet axis was reported equal to Ti = 0.005Uj . In LES reported in
[7], the turbulent fluctuations were neglected and the inflow was treated as laminar arguing
that a precise reproduction of the time-dependent inflow conditions is very difficult. In the
present simulations, we generate approximate inflow perturbations such that the fluctuating
velocity components are calculated from an assumed magnitude of Ti and assumed turbulent
length and time scales ll , lt needed to generate the turbulent inflow conditions using the
digital filtering method. Such fluctuations are then added to the mean velocity profile
computed from Equation (9). For simplicity, we assume a uniform turbulence intensity
across the nozzle at Ti = 0.01Uj and will investigate the influence of variations in ll , lt
later.
In the simulations, initially, the velocity field is set to Uc everywhere in the computational
domain except the inflow plane where the velocity is computed from Equation (8). During
an initial transient phase of about 100T0 (T0 = W/Uj ), the jet develops in the domain.
Subsequently, the solution is averaged over 800T0 which was found to be sufficient to obtain
steady statistics. The convergence of the time averaging of the solutions was monitored by
comparing the averaged velocity profiles between 600T0 and 800T0 , observing less than
3% maximal deviation.
4.1. Influence of the turbulent scales ll , lt
In several simulations, we analyse the influence of ll , lt on results obtained for the cases A1-2
and A2-2. The computations are performed with ll = 0.2W, 0.1W, 0.05W and lt = 0.5W/Uj ,
W/Uj , 2W/Uj , 5W/Uj . The differences between the results obtained are readily apparent.
Figure 3 shows the profiles of the time-averaged streamwise velocity and its fluctuations
along the centreline. The comparison is presented for four cases with ll = 0.1W, ll = 0.2W
and lt = W/Uj , lt = 2W/Uj . Results with fluctuations generated using random noise are also
included in Figure 3, denoted as ll = lt = 0. Typical for the randomly generated perturbations
is their rapid dissipation immediately downstream of the inlet (for instance, see Figure 1 in
[52]). Indeed, in the zooms showing the profiles of fluctuations close to the inlet, it is seen
that the fluctuations corresponding to ll = lt = 0 first decrease to zero and recover only
after some distance. Contrary, results obtained with the turbulence generated using digital
filtering show that the initial level of Ti is approximately preserved. Further downstream,
the results are dependent on ll , lt and the best agreement with the experimental data seems
to be obtained for the case with ll = 0.2W and lt = 2W/Uj . We adopt these parameters in
the sequel of this paper.
750
A. Tyliszczak and B.J. Geurts
1.2
ll=0W; lt=0W/U
j
0.7
ll=0.1W; lt=1W/Uj
ll=0.1W; lt=2W/Uj
ll=0.2W; lt=1W/Uj
0.6j
ll=0.2W; lt=2W/U
Quinn & Militzer(1988)
1
Ucl/Umax
0.4
0.6
0.02
0.3
0.01
0.4
0
0.5
0.2
1
0.2
0
0.1
0
2
4
y/De
6
8
0
10
(a) Case: A1-2; Ar = 1, Re=1.84 × 105
1.2
0.7
ll=0W; lt=0W/Uj
ll=0.1W; lt=1W/Uj
ll=0.1W; lt=2W/Uj
ll=0.2W; lt=1W/Uj
ll=0.2W; lt=2W/Uj
Quinn (1992)
1
0.6
0.5
0.4
0.6
0.02
0.3
0.01
0.4
0
0
0.5
0.2
1
0.2
0
(uu)1/2/Ucl
0.8
Ucl/Umax
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0
(uu)1/2/Ucl
0.5
0.8
0.1
0
2
4
y/De
6
8
0
10
(b) Case: A2-2; Ar = 2, Re=1.28 × 105
Figure 3. Mean stremwise velocity profile and its fluctuating component for the cases A1-2 and
A2-2 for different turbulent length and time scales used to generate turbulent inflow conditions based
on the digital filtering method.[51] The mean centreline velocity is normalised using the maximum
velocity in the flow domain, Umax . The fluctuations are normalised by the centreline velocity, Ucl .
Journal of Turbulence
751
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Figure 4. Solution for the non-excited jet with aspect ratio Ar = 1 for the case A1-1. Left figure:
isosurface of the vorticity magnitude (|| = 3(Uj /W)). Right figure: contours of the time-averaged
axial velocity in the x–y plane.
4.2. Jet with Ar = 1
The instantaneous and the time-averaged results obtained for the non-excited jet with Ar =
1 for the case A1-1 are presented in Figure 4. The instantaneous solution is visualised by
the isosurface of the vorticity magnitude equal to || = 3(Uj /W). A highly turbulent flow
behaviour emerges at the end of the potential core that extends to ≈4De (De = 1.128W for
Ar = 1). Here, the end of the potential core is taken as the distance from the inlet, where
the time-averaged axial velocity at the centreline drops to 98% of Uj . Figure 4 shows also
the contours of the time-averaged axial velocity in the lateral cross-section plane ‘x–y’.
Note that because of the symmetry of the flow for Ar = 1, the results in the streamwise
plane ‘z–y’ are the same which confirms that the averaging time is sufficiently long. The
profiles of the time-averaged axial velocity obtained from the simulation A1-1, A1-2 and
A1-3 are shown in Figures 5 and 6. These figures present the mean values and fluctuating
component along the lateral direction. The solutions are compared with experimental data
[3] at the axial locations y/De = 0.28, 2.658, 4.484, 7.088, where the measurements were
performed. The results are normalised by the centreline velocity Ucl and the half-velocity
width xh computed at the respective y/De locations.
In Figure 5(a), it is seen that the experimental data exhibit behaviour typical for jets
issuing from sharp-edged slots or from short converging nozzles, as the mean axial velocity
at the shear layer is higher than the centreline value. As shown in [3], the overshoot at the
nozzle edges was about 20%. This is not taken into account in the present simulations,
and therefore in the flow region adjacent to the inflow plane, some discrepancies may
be expected. Indeed, concerning the mean velocity profiles, small differences between the
experimental and numerical results are observed for the solutions at y/De = 0.28 and y/De =
2.658 as shown in Figure 5(a) and 5(b). Further downstream at y/De = 4.484 and y/De =
7.088, the numerical results match the experimental data very accurately. It is apparent
that the influence of the Reynolds number is very small and the profiles for different Re
coincide to a high degree.
In Figure 6, the velocity fluctuations are seen to be very close to each other regardless of
Re. Additionally, it is worth noting that the velocity fluctuations of solution A1-3 obtained
on the dense mesh are virtually the same as the fluctuations of the solution found on the basic
mesh. This means that the mesh density 192 × 256 × 192 is sufficient to provide highfidelity results for mean and fluctuating quantities. The profiles of the velocity fluctuations
agree reasonably well with the experimental data at y/De = 0.28 (Figure 6(a)) and y/De =
752
A. Tyliszczak and B.J. Geurts
1.2
1.0
1.0
0.8
0.8
U/Ucl
U/Ucl
1.2
0.6
0.4
0.4
0.2
0.2
0.0
-3
-2
-1
0
1
2
3
0.0
-3
-2
-1
(a) y/De = 0.28
1
2
3
(b) y/De = 2.658
1.2
1.2
1.0
1.0
0.8
0.8
Quinn (’88)
A1-1
A1-2
A1-3
U/Ucl
0.6
0.6
0.4
0.4
0.2
0.2
0.0
-3
0
x/xh
x/xh
U/Ucl
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0.6
-2
-1
0
1
x/xh
(c) y/De = 4.484
2
3
0.0
-3
-2
-1
0
1
2
3
x/xh
(d) y/De = 7.088
Figure 5. Profiles of the mean axial velocity along the lateral direction for the jet with Ar = 1 for
the cases A1-1, A1-2 and A1-3.
7.088 (Figure 6(d)), whereas in the transient region where turbulent flow develops, i.e.,
at y/De = 2.658 and y/De = 4.484 (Figure 6(b) and 6(c)), differences are clearly visible.
At these locations, the shapes of the profiles reflect the experimental data correctly, but
the maxima of the fluctuations significantly overpredict the measured values, particularly
at x/xh = ±1. This may be caused by the fact that in the simulation the level of Ti and
the turbulent scales were taken uniform across the jet radius which may lead to a larger
perturbation growth downstream.
4.3. Jet with Ar = 2
An isosurface of instantaneous vorticity magnitude and the contours of the time-averaged
axial velocity obtained for the non-excited jet with Ar = 2 for the case A2-1 are presented in
Figure 7. The results are presented both in the lateral and the spanwise planes. The potential
Journal of Turbulence
0.30
0.30
Quinn (’88)
A1-1
A1-2
A1-3
0.25
1/2
(uu) /Ucl
(uu)1/2/Ucl
0.20
0.15
0.15
0.10
0.10
0.05
0.05
-2
-1
0
1
2
3
0.00
-3
-2
-1
(a) y/De = 0.28
1
2
3
(b) y/De = 2.658
0.30
Quinn (’88)
A1-1
A1-2
A1-3
0.25
0.20
0.30
Quinn (’88)
A1-1
A1-2
A1-3
0.25
0.20
(uu)1/2/Ucl
0.15
0.15
0.10
0.10
0.05
0.05
0.00
-3
0
x/xh
x/xh
(uu)1/2/Ucl
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Quinn (’88)
A1-1
A1-2
A1-3
0.25
0.20
0.00
-3
753
-2
-1
0
1
x/xh
(c) y/De = 4.484
2
3
0.00
-3
-2
-1
0
1
2
3
x/xh
(d) y/De = 7.088
Figure 6. Profiles of the axial velocity fluctuations along the lateral direction for the jet with Ar =
1 for the cases A1-1, A1-2 and A1-3.
core is shorter than for the jet with Ar = 1 (see Figure 4) and extends only to 2.2De (De =
1.595W for Ar = 2 ), which in terms of De is about 40% less than found at Ar = 1. Analysis
of the time-averaged results allows to find a location of the so-called crossover point, i.e.,
the distance from the inlet where the jet dimensions along the lateral and spanwise axes
become equal. In the present case, this is found at y/De ≈ 4.25 which fits within the range of
crossover points, i.e., 3 < y/De < 7 as put forward in [12]. The profiles of the axial velocity
obtained for the cases A2-1, A2-2 and A2-3 are compared with the experimental data from
[4] in Figure 8. The solutions are presented along the lateral and spanwise directions at the
axial distances y/De = 2, y/De = 5 and y/De = 10 as selected in [4].
The numerical results for the lateral direction agree very well with the experimental
data, as seen in Figure 8. The solutions are practically independent of the Reynolds number
and the mesh density. The same applies to the results in the spanwise direction. Small
discrepancies are found only at y/De = 2 in the shear layer region, where the numerical
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754
A. Tyliszczak and B.J. Geurts
Figure 7. Solution for the non-excited jet with aspect ratio Ar = 2 for the case A2-1. Left figures:
isosurface of the vorticity magnitude (|| = 3(Uj /W)) in the x–y (upper figures) and z–y plane (bottom
figures). Right figures: contours of the time-averaged axial velocity in the x–y and z–y plane.
results show a slightly slower velocity decay. We note that a similar effect was observed
in lattice Boltzmann LES reported in [7], where the same experimental data were used
as point of reference. Hence, the faster velocity decay in the shear layer region in the
spanwise direction may be characteristic for the experimental set-up used in [4]. Finally,
the comparison of the turbulent kinetic energy (TKE) defined as q2 = 0.5( < u2 > + < v2 >
+ < w2 > ) is shown in Figure 9. The shape of the TKE distribution is captured correctly,
although the maxima of the TKE differ from the experiment by 5%–30%, depending on the
simulation (A2-1,2,3) and the y/De location. At the distance y/De = 10, not only the level
of TKE is underestimated, but also the profiles are flattened in the central part of the jet
(−1 ≤ x/xh ≤ 1; −1 ≤ z/zh ≤ 1). This may well be related to the proximity of the outflow
boundary. Simulations with a slightly longer domain of 12.5De also displayed the tendency
towards flattened profiles.
Figure 10 shows amplitude spectra of the axial velocity for the case A2-1 calculated
at the location of the shear layer x/W = 0.5. The Strouhal number used on the horizontal
axis in Figure 10 is defined as StW = fW/Uj , where f is the frequency of the harmonics
of the velocity time signal. The presented spectra were calculated from time series taken
from mesh points located at the axial distances y/De = 0.0, 0.5, 1.0, 3.0. From Figure 10,
it can be seen that there is no distinct dominant frequency which could be interpreted
as the preferred mode frequency, instead a broad region of more intense fluctuations is
observed. The results for the jets with Ar = 1 and Ar = 3 also do not show the existence of
a well-defined peak at a specific isolated frequency, as usually seen in circular jets.[57,58]
Journal of Turbulence
1.2
1.2
y/De=2
1.2
y/De=5
1.0
0.8
0.8
0.8
U/Ucl
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
-3
-2
-1
0
1
2
0.0
-3
3
-2
-1
x/xh
1.2
0
1
2
0.0
-3
3
1.2
1.2
y/De=5
0.8
0.8
0.8
U/Ucl
0.6
0.4
0.4
0.4
0.2
0.2
0.2
1
2
0.0
-3
3
1
2
3
Quinn (’92)
A2-1
A2-2
A2-3
y/De=10
0.6
0.6
0
0
U/Ucl
1.0
U/Ucl
1.0
-1
-1
x/xh
1.0
-2
-2
x/xh
y/De=2
0.0
-3
Quinn (’92)
A2-1
A2-2
A2-3
y/De=10
U/Ucl
1.0
U/Ucl
1.0
0.6
-2
-1
z/zh
0
1
2
0.0
-3
3
-2
-1
0
1
2
3
z/zh
z/zh
Figure 8. Profiles of the mean axial velocity along the lateral direction (upper figures) and the
spanwise direction at y/De = 2, 5, 10 for the jet with Ar = 2, cases A2-1, A2-2 and A2-3.
0.06
0.08
y/De=2
Quinn (’92)
A2-1
A2-2
A2-3
0.10
y/De=5
Quinn (’92)
A2-1
A2-2
A2-3
0.06
y/De=10
Quinn (’92)
A2-1
A2-2
A2-3
0.08
q2/Ucl2
0.06
q2/Ucl2
q2/Ucl2
0.04
0.04
0.04
0.02
0.02
0.00
-3
-2
-1
0
1
2
3
0.00
-3
0.02
-2
-1
x/xh
0.06
0
1
2
3
0.00
-3
-2
-1
Quinn (’92)
A2-1
A2-2
A2-3
y/De=2
0.08
0
1
2
3
x/xh
x/xh
Quinn (’92)
A2-1
A2-2
A2-3
y/De=5
0.06
0.10
Quinn (’92)
A2-1
A2-2
A2-3
y/De=10
0.08
0.04
q2/Ucl2
0.06
q2/Ucl2
q2/Ucl2
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755
0.04
0.04
0.02
0.02
0.00
-3
-2
-1
0
z/zh
1
2
3
0.00
-3
0.02
-2
-1
0
z/zh
1
2
3
0.00
-3
-2
-1
0
1
2
3
z/zh
Figure 9. Profiles of the turbulent kinetic energy q2 along the lateral direction (upper figures) and
the spanwise direction at y/De = 2, 5, 10 for the jet with Ar = 2, cases A2-1, A2-2 and A2-3.
756
A. Tyliszczak and B.J. Geurts
0.15
0.15
y/De=0.5
y/De=0.0
0.10
Amplitude
Amplitude
0.10
0.05
0.05
0.00
0.00
10-2
10-1
StW
100
101
0.15
10
-2
0
10
0
10
StW
10
StW
10
1
y/De=3.0
0.10
Amplitude
Amplitude
0.10
0.05
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-1
0.15
y/De=1.0
0.05
0.00
10
10
0.00
-2
10
-1
StW
10
0
10
1
10
-2
10
-1
1
Figure 10. Axial velocity spectrum for the non-excited jet for the case A2-1 at the location of the
shear layer x/W = 0.5 at four distances from the inlet y/De = 0.0, 0.5, 1.0, 3.0.
In laboratory experiments [59,60], the preferred mode frequency for a jet with Ar = 2
was found in the range StW, pref = 0.25–0.35. In the present results, the peak at StW = 0.3
is also visible; however, it is not distinctly larger than the fluctuation level in the broad
region corresponding to the range of frequencies StW = 0.3–0.7. Hence, from the current
simulations it cannot be said univocally that the peak at StW = 0.3 defines the preferred
frequency. The discrepancies may result from different characteristics of the inlet conditions
in the experiments compared to the present studies. It would be very meaningful if a more
complete identification of the experimental inlet conditions could be determined to allow
further scrutiny of the remaining differences.
4.4. Jet with Ar = 3.
The results obtained for the non-excited jet with Ar = 3 in case A3-1 are presented in
Figure 11. As before, the vorticity modulus and the time-averaged axial velocity in two
cross-section planes are presented. Compared to the results obtained for the jet with Ar =
2, larger vortical structures are formed closer to the inlet, at y/De ≈ 1.75 (De = 1.954W
for Ar = 3). The length of the potential core becomes shorter y/De = 1.5, and, on the other
hand, the crossover location is longer and shifts downstream to y/De ≈ 7.
5. Excited jets
The simulations for the non-excited jets established the level of accuracy of the applied
LES and correctness of the numerical settings (grid density, spatial disretisation, time
integration). It was demonstrated that the inlet boundary conditions defined by Equation
(9) allow for a physical representation of the dominant features of jets issuing from square
and rectangular nozzles without actual inclusion of the flow upstream of the nozzles in the
computational domain. The mean velocity profiles agreed very well with the measurements
both for Ar = 1 and Ar = 2. Also, turbulent quantities, e.g., the axial velocity fluctuations
and TKE, were found to be in acceptable agreement with the experimental data. This is
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Journal of Turbulence
757
Figure 11. Solution for the non-excited jet with the aspect ratio Ar = 3 for the case A3-1. Left
figures: isosurface of the vorticity magnitude (|| = 3(Uj /W)) in the x − y (upper figures) and z − y
plane (bottom figures). Right figures: contours of the time averaged axial velocity in the x − y and
z − y plane.
expected to carry over to the simulation results for the excited jets presented in this section,
since the dynamic range of scales is not essentially larger compared to the unforced jets.
We concentrate on identifying forcing conditions causing an increased spreading rate of
the jet or even a splitting of the jet into two separate branches (bifurcation phenomenon). In
round jets, it has been observed that this phenomenon occurs in a wide range of Reynolds
numbers. Experiments as well as DNS and LES [25,47,50] studies of round jets confirmed
this for Reynolds numbers 1.5 × 103 < Re < 105 . Moreover, it was found that this
phenomenon is only weakly dependent on Re. Necessary conditions leading to a bifurcation
were formulated in terms of forcing frequencies yielding a Strouhal criterion Sta /Stf = 2
with 0.35 < Sta < 0.7,[25] with Sta , Stf as defined in Equation (11). In parametric studies of
excited circular jets [44], it was shown that the strength of the bifurcation directly depends
on Sta . The effect of the forcing was observed to be largest when Sta is close to the preferred
mode frequency.
As shown in the previous subsection in the rectangular jets, a specific preferred mode
frequency could not be clearly identified, but rather a broader region of more prominent
frequencies was observed. For that reason, the simulations are performed for a wide set
of frequencies chosen to cover the particular broad region observed in Figure 10, i.e., we
consider Sta = 0.3, 0.4, 0.5, 0.6, 0.7, and use a fixed ratio Sta /Stf = 2. In all cases, the
Reynolds number is Re = 1 × 104 for which the non-excited jets closely resemble high-Re
findings in the near -field (y/De ≤ 10). The influence of Re on properties of turbulent square
758
A. Tyliszczak and B.J. Geurts
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Table 2. Parameters of the computational cases for the excited jets simulations. The superscript
‘fine’ denotes simulations on denser mesh and ‘ext’ simulations in a wider computational domain.
The symbols ‘St0.3 -’, ‘St0.4 -’ etc. denote the axial forcing frequency and ‘0 ’ and ‘ π4 ’ denote the
phase shift.
Case
Ar
Sta = fa W/Uj
A1-St0.3 − 0.7 -0, π4
A2-St0.3 − 0.7 -0, π4
A2fine -St0.3 − 0.7 -0
A2exp -St0.3 − 0.7 -0
A3-St0.3 − 0.7 -0, π4
1
2
2
2
3
0.3, 0.4, 0.5, 0.6, 0.7
0.3, 0.4, 0.5, 0.6, 0.7
0.3, 0.5, 0.7
0.3, 0.5, 0.7
0.3, 0.4, 0.5, 0.6, 0.7
0, π4
0, π4
0
0
0, π4
jets in a range 8 × 103 ≤ Re ≤ 5 × 104 was analysed in recent experimental work by Xu
et al.[5] It was shown that the near-field region (y/De ≤ 10) is not significantly affected by
Re. Results in the far field were found quite independent of Re only beyond Recrit ≥ 3 ×
104 . In the current simulations, the analysis is limited to the near-field region, where the
excitation plays the most important role.
Table 2 gives details of the analysed test cases. The notation in Table 2 is constructed
as follows: ‘A1-’, ‘A2-’, ‘A3-’ denote the aspect ratio; ‘St0.3 -’, ‘St0.4 -’ etc. denote the axial
forcing frequency which determines the flapping mode forcing frequency from Sta /Stf = 2;
‘0 ’ and ‘ π4 ’ denote the phase shift (see Equation (11)). The computations for the phase
shift = π4 were motivated by the results obtained in [44] for circular jets. There, the phase
shift between the axial and flapping forcing caused qualitative differences. In all cases, the
turbulence intensity is equal to Ti = 0.01Uj and the excitation amplitudes are taken as Aa =
Af = 0.1Uj . Such a relatively high level of excitation is chosen following [44], where it was
shown that the excitation should have an amplitude larger or at least comparable to the Ti
level.
Most of the computations are carried out using the basic mesh. The test computations
aiming to asses the influence of the mesh density are performed for the jet with Ar = 2
using the mesh with 320 × 384 × 320 nodes. In Table 2, these computations are denoted
by the superscript ‘fine’. Moreover, to check to which extent the side boundaries influence
the inner part of the flow domain, an additional set of simulations is carried out using an
expanded domain 16W × 16W (in the lateral and spanwise directions). In this case, denoted
by the superscript ‘exp’, the dense mesh is used with stretching parameters chosen such that
the cell sizes in the vicinity of the jet are smaller than in the basic domain. They are equal
to x∗ = 0.017 and z∗ = 0.036 (cf. Table 1). Hence, this case allows to check further
the influence of the mesh density in the ‘x–z’ plane. Sample results obtained from these
validation tests are shown in Figure 12 presenting the radial profiles of the time-averaged
axial velocity (mean and fluctuations) at two locations downstream of the inlet. It is seen
that the results obtained on the basic mesh and using the basic computational domain
(A2-St0.5 -0 ) are only slightly different from those obtained on the denser meshes on the
basic computational domain (A2fine -St0.5 -0 ) and on the extended domain (A2exp -St0.5 0 ). Further analyses (not presented here) show that small differences are observed only
for turbulent quantities (u, v, uv, etc.) and only in the region of high turbulence intensity
y/De = 2.0–4.0. Similar behaviour is found in simulations with Sta = 0.3 and Sta = 0.7.
Influence of the side boundaries of the basic domain on dynamics of the flow in the central
part is also very small. Later, in contour plots, it will be shown that for a few cases the
forcing leads to very large spreading rate, which causes that, in the region far from the
Journal of Turbulence
759
1.2
0.5
A2-St0.5- 0
A2fine-St0.5A2exp-St0.5-
1
0
0
0.4
1/2
U/Uj
0.3
(uu) /Uj
y/De = 3.0
0.8
0.6
0.2
0.4
0.1
0.2
1
x/De
0
3
2
1.2
0.5
A2-St0.5- 0
A2fine-St0.5A2exp-St0.5-
1
0
0
0.4
1/2
0.3
(uu) /Uj
y/De = 7.0
0.8
U/Uj
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0
0
0.6
0.2
0.4
0.1
0.2
0
0
1
x/De
2
0
3
Figure 12. Comparison of the axial velocity profiles at axial distances y/De = 3 and y/De = 7 in the
lateral cross-section plane at z = 0. Cases for A2-St0.5 -0 , A2fine -St0.5 -0 and A2exp -St0.5 -0 .
inlet (y/De ≈ 10), the jets approach the boundaries of the computational domain. In these
situations, the influence of the boundaries is expected to be significant; however, the farfield region is not of primary importance for the present studies. Hence, one may conclude
that the basic mesh and the basic domain are sufficient to obtain accurate numerical results.
The dynamics of the excited jets are illustrated in Figures 13 and 14 showing the
isosurfaces of instantaneous vorticity modulus and the contours of the time-averaged axial
velocity for the cases A1-St0.3 -0 and A2-St0.3 -0 . It can be seen that almost in the whole
flow domain, the excited jets in the lateral planes are significantly wider compared to the
non-excited jets (cf. Figures 4 and 7). In the non-excited jet for Ar = 2, the crossover point
was found at y/De ≈ 4.25, while in the present case, the crossover point does not exist in the
entire computational domain, and the jet in the lateral plane is everywhere wider than in the
spanwise plane. The isosurfaces of the vorticity clearly show that the potential core region
of the excited jets is very short and strong vortical structures are seen already at y/De ≈ 1,
resembling deformed toroidal rings due to the axial forcing. In fact, careful inspection of
the zoomed part of Figures 13 and 14 allows identifying the so-called rib vortices (pointed
by the arrows) with elongated shapes originating from the corners of the nozzle. These
structures are also present in the non-excited jets but much shorter and less pronounced.
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A. Tyliszczak and B.J. Geurts
Figure 13. Solution for the excited jet with aspect ratio Ar = 1, the case A1-St0.3 -0 . Left figure:
isosurface of the vorticity magnitude (|| = 3(Uj /W)) in the x–y plane. Right figure: contours of the
time-averaged axial velocity in the x–y plane.
Figure 14. Solution for the excited jet with aspect ratio Ar = 2, the case A2-St0.3 -0 . Left figures:
isosurface of the vorticity magnitude (|| = 3(Uj /W)) in the x–y (upper figures) and z–y planes (bottom
figures). Right figures: contours of the time-averaged axial velocity in the x–y and z–y planes.
For Ar = 1, hairpin vortices are created in the corners, whereas for the jet with Ar ≥ 2, only
single vortical tubes are observed, which is consistent with [12].
The velocity spectra for the excited jets for the cases A2-St0.3 -0 , A2-St0.5 -0 and
A2-St0.7 -0 are shown in Figure 15. At y/De = 0, two peaks corresponding to the axial and
flapping excitation are very pronounced. Further downstream, a series of higher harmonics
Journal of Turbulence
100
10
Sta=0.3
Stf=Sta/2
100
-2
10
10-4
10-2
Amplitude
10
100
10-6
101
Sta=0.5
Stf=Sta/2
-6
10-2
10-1
100
0
10-2
10-4
10-2
10-1
StW
100
Stf=Sta/2
10-2
100
-4
10-6
10-6
101
10
10-2
10-2
10-4
10-4
10-1
100
Stf=Sta/2
10
10
-6
10-2
10
10-1
100
0
101
10-6
10-4
10-4
-6
10-1
StW
100
(c) y/De = 1.0
100
StW
101
101
Sta=0.3
10-1
Stf=Sta/2
100
101
100
101
Sta=0.5
10-4
10-2
10-2
10-1
101
Sta=0.7
10-2
10-2
10
10-2
10-2
100
Sta=0.7
Stf=Sta/2
100
Stf=Sta/2
100
10-2
-4
10-6
101
Sta=0.5
Amplitude
10-2
100
Amplitude
10
0
10-2
10-6
101
(b) y/De = 0.5
Sta=0.3
Stf=Sta/2
10-1
Stf=Sta/2
(a) y/De = 0.0
0
100
Sta=0.5
10-4
10-2
10
10-1
10-2
10-6
101
Sta=0.7
Stf=Sta/2
10-2
100
10-4
10
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10-1
0
10-2
10
Sta=0.3
Stf=Sta/2
-2
10-4
-6
Amplitude
10
761
10-6
10-1
Sta=0.7
Stf=Sta/2
10-2
10-1
StW
100
101
(d) y/De = 3.0
Figure 15. Axial velocity spectrum for the excited jet with Ar = 2 at the location of the shear
layer x/W = 0.5 at four distances from the inlet y/De = 0.0, 0.5, 1.0, 3.0 for the cases A2-St0.3 -0 ,
A2-St0.5 -0 and A2-St0.7 -0 corresponding to different Sta explicitly indicated in the figures.
appear with amplitudes that are quite comparable with the forcing amplitudes. The presence of higher frequency harmonics reveals itself already at y/De ≈ 0.1 and indicates the
development of smaller flow scales which, at the end of the potential core region, intensify
and trigger a sudden transition to turbulent flow, as could also be seen in Figure 14. The
spectra obtained for the jets with Ar = 1 and Ar = 3 show qualitatively the same behaviour.
The forcing frequencies are characteristic of the flow even quite far downstream. Higher
harmonics are visible mainly in 0.1 < y/De < 3.0.
5.1. Time-averaged solutions
The contours of the time-averaged axial velocity in the lateral and spanwise planes obtained
from the simulations detailed in Table 2 are presented in Figure 16 for Ar = 1, Figure 17
for Ar = 2 and Figure 18 for Ar = 3. The effects of changing the excitation frequency are
readily noticed for all three aspect ratios. Also, the effect of the phase shift between the
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A. Tyliszczak and B.J. Geurts
Figure 16. Axial velocity contours in the excited jet with aspect ratio Ar = 1 for the cases A1St0.3 − 0.7 -0 and A1-St0.3 − 0.7 - π4 . The results are shown in the lateral plane (first and second rows
of figures) and in the spanwise plane (third and fourth rows).
axial and flapping excitation is pronounced. In the case of the jet with Ar = 1, the results
obtained with = 0 (upper row of Figure 16) show that for some forcing frequencies, the
maxima of the velocity are located in the shear layer regions, while for = π4 , they are
always in the centreline, as can be seen in the second row of figures. This effect is even more
pronounced in the case with Ar = 2 for the forcing frequency Sta = 0.5 (see Figure 17). In
the simulation with = 0, the jet almost splits into separate branches, while a phase shift
of = π /4 completely prevents this phenomenon. For the jets with Ar = 2, the phase shift
also influences the spreading rate. For instance, in simulations with Sta = 0.7 and = π4 ,
the spread of the jet is smaller than in case = 0. On the other hand, in the simulations
with Sta = 0.3, the opposite trend is observed. The results obtained for the jets Ar = 3,
shown in Figure 18, are only slightly dependent on the phase shift and the solutions for
= 0 and = π4 differ only quantitatively. It is surprising that the tendency to splitting
clearly observed for Ar = 2 is practically not seen at all for Ar = 3.
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Journal of Turbulence
763
Figure 17. Axial velocity contours in the excited jet with aspect ratio Ar = 2 for the cases A2St0.3 − 0.7 -0 and A2-St0.3 − 0.7 - π4 . The results are shown in the lateral plane (first and second rows
of figures) and in the spanwise plane (third and fourth rows).
The simulations show that for all values of Sta the jets are wider in the lateral plane
in which the flapping excitation is applied compared to the spanwise plane. This effect
increases with increasing Ar . In case Ar = 1, although the flow structures of the jets in the
lateral and spanwise planes differ, their sizes are quite similar, whereas in the simulations
with Ar = 2 and Ar = 3, the jets in the lateral plane are significantly wider. In many cases,
the contours of the velocity even reach the side boundaries, for instance, in the simulation
A2-St0.3 - π4 shown in Figure 17 in the second row of the first column. In these situations,
the results far from the inlet are somewhat biased by the influence of the nearby boundaries.
Such a large spread of the jet in the lateral plane implies a corresponding reduction of the jet
spreading in the spanwise direction. It is seen that in some cases, for instance, A2-St0.5 -0
or A3-St0.4 -0 , in the region y/De < 2, the jets first expand and then suddenly narrow to
a size smaller even than its inlet dimension. Analysis of the vector fields (not presented
here) shows that in this region the velocity vectors are oriented towards the centreline. This
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764
A. Tyliszczak and B.J. Geurts
Figure 18. Axial velocity contours in the excited jet with aspect ratio Ar = 3 for the cases A3St0.3 − 0.7 -0 and A3-St0.3 − 0.7 - π4 . The results are shown in the lateral plane (1st and 2nd row of
figures) and in the spanwise plane (3rd and 4th row).
illustrates the remarkable level of control that can be achieved on the far-field development
of the flow.
The axial velocity profiles are shown in Figures 19–21. These profiles correspond to
the solutions at distances y/De = 1, y/De = 3 and y/De = 7 along the lateral direction
(sub-figures (a), (c) and (e)) and along the spanwise direction ((b), (d) and (f)). At y/De
= 1, the effect of excitation is small and the results are very similar regardless of Sta .
This means that the differences between the solutions observed further downstream are
not directly related to the mechanical forcing, but rather indirectly, resulting mainly from
interactions between natural instability modes and the flow structures created by the forcing
at different Sta . A clear illustration of the effect of varying Sta appears beyond y/De ≈ 3,
where the profiles exhibit quite complicated patterns with a flattening in the lateral plane
around x/De = 1. This effect is most pronounced in the jets with Ar = 1.
The solutions at y/De = 7 clearly confirm that the excitation leads to a significant
widening of the jet in the lateral direction and narrowing in the spanwise direction, compared
Journal of Turbulence
1.2
y/De=1.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.4
0.2
0.2
1
x/De
2
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
0
0
y/De=1.0
1
U/Uj
1.2
765
0
0
3
1
1.2
y/De=3.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.2
0.2
x/De
2
0
0
3
1
(c)
1.2
y/De=7.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
0.8
0.6
0.2
(e)
3
2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.6
0.2
x/De
y/De=7.0
0.8
0.4
1
2
1
0.4
0
0
z/De
(d)
U/Uj
1.2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
1
y/De=3.0
1
0.4
0
0
2
(b)
U/Uj
1.2
U/Uj
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(a)
z/De
0
0
1
z/De
2
3
(f)
Figure 19. Mean axial velocity profiles for the jet with Ar = 1 for the non-excited case and for the
cases A1-St0.3–0.7 -0 . Solutions at axial distances y/De = 1, y/De = 3 and y/De = 7 in the lateral
cross-section plane at z = 0 (figures in the column on the left side) and in the spanwise cross-section
plane at x = 0.
766
A. Tyliszczak and B.J. Geurts
1.2
y/De=1.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.4
0.2
0.2
1
x/De
2
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
0
0
y/De=1.0
1
U/Uj
1.2
0
0
3
1
1.2
y/De=3.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.2
0.2
x/De
2
0
0
3
1
(c)
1.2
y/De=7.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
0.8
0.6
0.2
(e)
3
2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.6
0.2
x/De
y/De=7.0
0.8
0.4
1
2
1
0.4
0
0
z/De
(d)
U/Uj
1.2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
1
y/De=3.0
1
0.4
0
0
2
(b)
U/Uj
1.2
U/Uj
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(a)
z/De
0
0
1
z/De
2
3
(f)
Figure 20. Mean axial velocity profiles for the jet with Ar = 2 for the non-excited case and for the
cases A2-St0.3–0.7 -0 . Solutions at axial distances y/De = 1, y/De = 3 and y/De = 7 in the lateral
cross-section plane at z = 0 (figures in the column on the left side) and in the spanwise cross-section
plane at x = 0.
Journal of Turbulence
1.2
y/De=1.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.4
0.2
0.2
1
x/De
2
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
0
0
y/De=1.0
1
U/Uj
1.2
767
0
0
3
1
1.2
y/De=3.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
U/Uj
0.8
0.6
0.6
0.2
0.2
x/De
2
0
0
3
1
(c)
1.2
y/De=7.0
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
1
0.8
0.6
0.2
(e)
3
2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.6
0.2
x/De
y/De=7.0
0.8
0.4
1
2
1
0.4
0
0
z/De
(d)
U/Uj
1.2
3
no forcing
Sta=0.3
Sta=0.4
Sta=0.5
Sta=0.6
Sta=0.7
0.8
0.4
1
y/De=3.0
1
0.4
0
0
2
(b)
U/Uj
1.2
U/Uj
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(a)
z/De
0
0
1
z/De
2
3
(f)
Figure 21. Mean axial velocity profiles for the jet with aspect ratio Ar = 3 for the non-excited case
and for the cases A3-St0.3–0.7 -0 . Solutions at axial distances y/De = 1, y/De = 3 and y/De = 7 in
the lateral cross-section plane at z = 0 (figures in the column on the left side) and in the spanwise
cross-section plane at x = 0.
768
A. Tyliszczak and B.J. Geurts
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to the non-excited case. For instance, in case Ar = 3, the profiles presented in Figure 21(e)
show that at the lateral distance x/De ≈ 1.7, the velocity of the unexcited jet approaches
the co-flow velocity, while in all excited cases (A3-St0.3–0.7 -0 ), it is comparable with the
centreline velocity. On the other hand, in the spanwise direction, the jets are significantly
thinner, compared to the non-excited case. The velocities of excited jets reaches zero at
z/De ≈ 1.5, where the velocity of non-excited jet still has substantial value of about 20% of
Uj .
5.1.1. Bifurcation phenomenon
The results presented in Figures 16–18 show that the bifurcation phenomenon as observed
for round jets, strictly speaking, does not appear in any of the presented cases. In the round
jets [25,44,47,50] and also in the square jets [9,11] at low Reynolds numbers, a characteristic
effect of the bifurcation phenomenon was a clear separation of the jets into two distinct
branches and almost total vanishing of the flow near the axis. In those simulations, the
jets exhibited a ‘Y’ shape when seen as a cross section of time-averaged velocity. In the
present simulations, there are only some indications of this behaviour, as may be seen, for
instance, in the results for A2-St0.5 -0 in Figures 17 and 20(c), but a distinct splitting of
the main stream does not occur. The main effect appears a strongly anisotropic spreading
rate at some forcing conditions, e.g., a much stronger spreading rate in the lateral compared
to the spanwise direction. This illustrates the qualitative modification of the flow that is
achievable via the inflow forcing.
6. Conclusions
The paper presented the results of LES of non-excited and excited square and rectangular
jets with aspect ratios Ar = 1, Ar = 2 and Ar = 3. The simulations performed for nonexcited jets validated the applied numerical approach (subgrid modelling, mesh densities,
boundary conditions) and assessed the dependency of the solutions on the parameters, i.e.,
Reynolds number and inlet turbulence time/length scales. Importance of the inlet turbulence
characteristics was demonstrated comparing the axial velocity along the jet centrelines.
This allowed to chose the time and length scales to best match experimental data in the
region downstream of the inlet. The simulation results were in good agreement with the
experimental data taken from the literature. In the downstream region y/De < 10, the profiles
of the mean axial velocity almost exactly matched the measurements. Discrepancies were
seen for the fluctuating components, which points toward differences in small-scale features
of the inflow conditions between the experiments and the simulations. It turned out that the
Reynolds number in the range 1 × 104 to 1.8 × 105 has only a minor influence on the
near-field results.
The computations showed that active and passive flow control methods (excitation and
nozzle shaping) may be successfully combined and used to increase the mixing of the jets
or to alter their behaviour even quite far downstream of the inflow. It was shown that a rich
variety of mean flow patterns can be induced simply by changing aspect ratio and forcing
parameters. Modifications of the forcing frequencies allowed alteration of the large-scale
flow in the region downstream of the inlet boundary. Analysis of the solutions in the spectral
space showed that at a distance up to y/De ≤ 3, the induced axial and flapping forcing result
in smaller scales which were identified by high-frequency harmonics. In this region, the
Journal of Turbulence
769
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contour plots and also one-dimensional plots revealed a very complex flow pattern. Further
downstream, the observed effects of excitation were as follows:
• Increased spreading of the jet in the lateral plane—significantly larger than in the
case of the unexcited jet; the widening of the jet in the lateral direction caused its
narrowing in the spanwise plane and decreasing of the velocity on the centreline.
• For some values of the excitation frequency, the jets exhibited a tendency to the
separation of the main flow direction; the occurrence of this phenomenon for the
jets with Ar = 1 and Ar = 2 was conditioned by the phase shift between the axial
and flapping forcing; when the forcing terms were in phase, the maxima of the axial
velocity could be found away from the centreline of the jet which is characteristic
for the bifurcation phenomenon; when forcing terms were shifted by π /4, this effect
disappeared. For the jet with Ar = 3, the phase shift was found to lead to only
quantitative changes in the flow; large qualitative alterations in the flow structure
were not observed.
• The full splitting of the jet, in the way as for the round jets, was not observed in the
present simulations; earlier works [9,11] showed that the jets splitted into completely
separate branches; however, these results were reported for low Reynolds number
flows, Re=1000 and Re=3200, and appear not to be robust enough to survive higher
Reynolds numbers.
In simulations of round jets [44], the excitation amplitude had to be large compared to the
level of turbulence intensity in order to induce a strong bifurcation. Taking into account
that the rectangular jets are more unstable than the round jets, it is likely that the amplitude
of excitation will play a key role on the bifurcation in two or more jets at high Reynolds
numbers. Further research in this direction is planned.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work has been partially supported by the Polish National Science Center [grant number DEC2011/03/B/ST8/06401]. Computations have been carried out at SARA Computing Centre (Amsterdam) [grant number SH-061]; Cyfronet Computing Centre (Krakow) within the PL-Grid infrastructure.
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