.
1
Jet-edge interaction tones
P. Jordan
1
, V. Jaunet
1
, A. Towne
2
, A.V.G. Cavalieri
3
,
T. Colonius
4
, O. Schmidt
4
, A. Agarwal
5
1
D ́epartement Fluides, Thermique, Combustion, Institut PPRIME,
CNRS—Universit ́e de Poitiers—ENSMA, Poitiers, France
2
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
3
Divis ̃ao de Engenharia Aeron ́autica, Instituto Tecnol ́ogico de Aeron ́autica,
S ̃ao Jos ́e dos Campos, SP, Brazil
4
Division of Engineering and Applied Science, California Institute of Technology,
Pasadena, CA 91125, USA
5
Department of Engineering, University of Cambridge, UK
Abstract.
Motivated by the problem of jet-flap interaction noise, we study the tonal
dynamics that occur when a sharp edge is placed in the hydrodynamic nearfield of an
isothermal turbulent jet. We perform hydrodynamic and acoustic pressure measurements
in order to characterise the tones as a function of Mach number and streamwise edge po-
sition. The distribution of spectral peaks observed, as a function of Mach number, cannot
be explained using the usual edge-tone scenario, in which resonance is underpinned by
coupling between downstream-travelling Kelvin-Helmholtz wavepackets and upstream-
travelling sound waves. We show, rather, that the strongest tones are due to coupling
between the former and upstream-travelling jet modes recently studied by Towne
et al.
(2017) and Schmidt
et al.
(2017). We also study the band-limited nature of the reso-
nance, showing a high-frequency cut-off to be due to the frequency dependence of the
upstream-travelling waves. At high Mach number these become evanescent above a cer-
tain frequency, whereas at low Mach number they become progressively trapped with
increasing frequency, a consequence of which is their not being reflected in the nozzle
plane. Additionally, a weaker, low-frequency, forced-resonance regime is identified that
involves the same upstream travelling jet modes but that couple, in this instance, with
downstream-travelling sound waves. It is suggested that the existence of two resonance
regimes may be due to the non-modal nature of wavepacket dynamics at low-frequency.
1. Introduction
Resonant phenomena are widely encountered in fluid systems, where they are un-
derpinned by diverse flow physics. They may be exploited to produce pleasant, desired
effects, as is the case with musical wind-instruments (Howe 1975; Coltman 1976; Fabre
et al.
2012). Or they may constitute an undesired behaviour that complicates the design
of engineering systems. This can occur for flow in the presence of sharp edges (Richardson
1931; Powell 1953
a
; Curle 1953) or cavities (Rossiter 1964; Rowley
et al.
2002; Kegerise
et al.
2004); it is the case for imperfectly expanded supersonic jets, that ‘screech’ (Powell
1953
b
; Alkislar
et al.
2003; Edgington-Mitchell
et al.
2014), impinging jets (Powell 1988;
arXiv:1710.07578v1 [physics.flu-dyn] 20 Oct 2017
2
Jordan et al..
Krothapalli
et al.
1999; Henderson
et al.
2005) and globally unstable flows more generally
(Huerre & Monkewitz 1990; Monkewitz
et al.
1993).
Some kind of feedback is usually at work when a fluid system undergoes resonance.
Typically this involves a disturbance, initiated at some point in the flow, and that travels
in a given direction, triggering, at a distant point, a second disturbance that travels
back toward the inception point of the first. Synchronisation of the two can occur if
their phases are appropriately matched at the inception and reflection points. In this
‘long-range’ feedback scenario, the inception and reflection points may correspond to
physical boundaries, as is the case in cavity flows, or they may arise due to other flow
phenomena, such as shocks in underexpanded supersonic jets, or turning points in slowly
spreading mean-flows (Rienstra 2003; Towne
et al.
2017). In certain globally unstable
flows, resonance may occur in the absence of solid boundaries, between disturbances
of opposite generalised group velocity when their frequencies and wavenumbers become
matched; this is the case for the saddle-point ringing that underpins absolute instability
in wake flows or low-density jets for instance (Huerre & Monkewitz 1990).
In many of the examples evoked above, the downstream-travelling disturbance is a
convectively unstable Kelvin-Helmholtz wave, and the upstream-travelling disturbance a
sound wave. But in the case of round jets there are other kinds of wave available for both
upstream and downstream transport of fluctuation energy. Tam & Hu (1989) discuss one
such upstream-travelling wave, originally observed by Michalke (1970), who disregarded
it as an artefact of the locally parallel framework of his analysis. It has been suggested by
Tam & Ahuja (1990) that this wave may be important in explaining the tonal behaviour
of impinging subsonic jets, and a recent numerical study by Bogey & Gojon (2017) shows
that this may also be the case for resonance in impinging supersonic jets. Towne
et al.
(2017) and Schmidt
et al.
(2017) recently discovered that the round jet can support a
number of additional waves. Resonance possibilities in jets are therefore more numerous
than had previously been thought.
The present study was motivated by the new generation of Ultra-High-Bypass-Ratio
turbofan engines and the potential problems posed by the closely coupled jet-flap con-
figurations that such systems involve. With this in mind, we consider the problem of
subsonic jets grazing a sharp edge. Such a configuration was studied by Lawrence &
Self (2015), who observed a tonal behaviour that could not be explained in terms of
upstream-travelling, free-stream sound waves. We perform similar experiments, involv-
ing a flat rectanguler plate whose edge is positioned in the hydrodynamic nearfield of a
round turbulent jet. The nozzle is the same used in recent work (Cavalieri
et al.
2013;
Br`es
et al.
2015; Jordan
et al.
2017; Jaunet
et al.
2017; Towne
et al.
2017; Schmidt
et al.
2017) and among the exit conditions considered are those of the cited studies. Our ob-
jective is to establish if the strong hydrodynamic and acoustic tones that are observed
in this closely coupled jet-edge configuration can be understood in the framework of the
waves considered by Towne
et al.
(2017) and Schmidt
et al.
(2017).
The remainder of the paper is organised as follows. We present the experimental setup
in section
§
2. This is followed, in section
§
3, by an overview of the hydrodynamic and
acoustic pressure fields that result from interaction of the turbulent jet, whose Mach
number is varied, with the plate edge, whose streamwise position is varied. In section
§
4
we recall briefly the theoretical framework established by Towne
et al.
(2017) and then
use this to explain the observed tones. Some concluding remarks are provided in section
§
5.
Jet-edge interaction tones
3
(a)
(b)
Figure 1.
Experimental setup: (a) front view showing nozzle, plate, nearfield microphone and
azimuthal array of 18 microphones in the acoustic field; (b) close-up view of the nearfield mi-
crophone.
2. Experimental setup
The experiments were performed at the
Bruit et Vent
jet-noise facility of the PPRIME
Institute, Poitiers, France. The experimental setup, shown in figure 1, involved a rectan-
guler plate situated in the nearfield of a round, isothermal jet of diameter,
D
= 0
.
05
m
.
The boundary layer inside the nozzle is turbulent due to tripping by a carborundum strip
situated 2.7
D
upstream of the exit plane (Cavalieri
et al.
2013).
In the nearfield, microphone and plate positions are described using a cylindrical co-
ordinate system, (
x,r,φ
), where
x
,
r
and
φ
refer, respectively, to streamwise, radial
and azimuthal positions. In the acoustic field, a spherical coordinate system, centered
on the jet centerline in the nozzle exit plane, is used, (
θ,φ,R
), where
θ
is measured
from the downstream jet axis, and
φ
is measured in the clockwise direction with the
jet viewed from the downstream,
φ
= 0 being vertically above the jet. All distances are
non-dimensionalised using the jet diameter,
D
.
The plate was inclined to form an angle of 45
◦
with the jet axis; its edge was situated
at
r/D
= 0
.
5 and the streamwise position varied from
x/D
= 2 to
x/D
= 4 in steps of
1
D
. For each plate position, the Mach number of the jet was varied from
M
= 0
.
6 to
M
= 1 in increments of ∆
M
= 0
.
02. Pressure measurements were performed for each
configuration using a microphone located at (
x/D,r/D
) = (0
.
08
,
0
.
55), shown in figure
1(b), in order to record the hydrodynamic nearfield pressure signature (all references to
‘nearfield’ data correspond to measurements provided by this microphone), and by means
of an azimuthal array of 18 microphones situated in the acoustic field, at
R/D
= 14
.
6
and
θ
= 80
◦
, as shown in figure 1. Twenty seconds of data were recorded at a sample
rate of 200kHz and power spectral densities were estimated from this data using Welch’s
method. Frequency is expressed in non-dimensional form in terms of the Strouhal number,
St
=
fD/U
j
, where
f
is frequency in
Hz
, and
U
j
the jet exit velocity in
ms
−
1
.
3. Acoustic and hydrodynamic tones
The high sample rate and finely resolved Mach-number variation allow us to obtain
high-resolution power-spectral-density (PSD) maps (shown in Figures 2 and 3 for the
hydrodynamic and acoustic regions, respectively) that comprise a rich ensemble of spec-
tral tones. Peak levels in the acoustic field are of order 170
dB/St
, while levels in the
hydrodynamic field are, naturally, considerably higher.
The strongest tones have a similar
St
−
M
pattern in both acoustic and hydrodynamic
4
Jordan et al..
Figure 2.
Power-spectral-density maps estimated from hydrodynamic, nearfield pressure
recordings for three streamwise edge positions, from left to right:
L/D
= 2, 3 & 4.
Figure 3.
Power-spectral-density maps estimated from acoustic pressure recordings on the
‘shielded’ side of the plate, (
θ,φ,R
) = (80
◦
,
70
◦
,
14
.
6), and for three the same streamwise edge
positions as figure 2.
regions, which does not correspond to the usual edge-tone scenario in which resonance oc-
curs between downstream-travelling Kelvin-Helmholtz wavepackets and upstream trav-
elling sound waves. Following Towne
et al.
(2017), we use the terms ‘upstream-’ and
‘downstream-travelling’ to refer to the sign of the generalised group velocity (Briggs
1964) of a wave, rather than its phase velocity.
The pattern of the strongest tones is characterised, for all plate-edge streamwise posi-
tions, by a ‘squeezing-together’ of the peaks with increasing frequency and Mach number.
A second kind of pattern is more clearly visible in the low-Strouhal region (approxi-
mately
St <
0
.
6) of the sound-field PSD maps. This comprises spectral peaks that are
both weaker and broader than those at higher frequencies, and in which the ‘frequency-
squeezing’ is not so clearly apparent. Our goal is to understand and model the flow
dynamics responsible for these two behaviours.
4. Understanding and predicting the tones
Exploration of the mechanisms responsible for the tonal behaviour discussed above
requires consideration of the different kinds of wave that are supported by a turbulent
Jet-edge interaction tones
5
jet. This has been done in a pair of recent papers, in locally parallel and weakly non-
parallel frameworks by Towne
et al.
(2017), and in a fully global framework by Schmidt
et al.
(2017). The studies show that the turbulent jet in isolation can support diverse,
weak, forced-resonance mechanisms, due to a rich variety of waves that is briefly discussed
in what follows. For a more involved discussion the reader should refer to the said papers.
4.1.
Overview of wave behaviour supported by turbulent jets
The resonance mechanisms studied by Towne
et al.
(2017) and Schmidt
et al.
(2017) in an
isolated, isothermal, Mach 0.9 turbulent jet involve two kinds of downstream-travelling
(
k
+
) wave: (1) the well-known Kelvin-Helmholtz instability, that we denote
k
+
KH
(blue
in figures 4 and 5); and, (2) a wave discovered by Towne
et al.
(2017), denoted
k
+
T
(green
in figures 4 and 5), that only exists in the Mach number range, 0
.
82
< M
≤
1, over a
restricted range of frequencies, and whose physics vary within that range: at the low-
frequency end the waves are largely trapped within and guided by the jet, behaving in
the manner of acoustic waves propagating in a porous-walled cylindrical duct; at the
high-frequency end, on the other hand, the waves have support in the shear-layer and it
is more appropriate to think of them as shear-layer modes.
The downstream-travelling waves can undergo resonance with three kinds of upstream-
travelling (
k
−
) wave. One of the upstream-travelling waves is that previously discussed
by Tam & Hu (1989), that we denote
k
−
TH
(cyan in figures 4 and 5). As shown by Towne
et al.
(2017), this wave exists over the Mach-number range, 0
< M
≤
0
.
82, and its
physics depend, in a manner similar to that of the
k
+
T
wave, on the frequency considered:
at sufficiently high frequency it is trapped within and guided by the jet, behaving in
the manner of a porous-walled acoustic duct mode; at lower frequencies, on the other
hand, the mode has support in the shear layer and again must be thought of as a shear-
layer mode. The second
k
−
wave, also discovered by Towne
et al.
(2017), exists over
the Mach number range, 0
.
82
< M <
1, and has the same soft-duct-like character as
the high-frequency
k
−
TH
waves; it is therefore denoted,
k
−
d
(cyan in figures 4 and 5). We
distinguish it from the
k
−
TH
waves because, unlike these, it becomes evanescent below a
well-defined frequency (at the transition from cyan to green in figure 5). We note that
for Mach numbers close to
M
= 0
.
82, in the close vicinity of the cut-off frequency, it
also ceases to be duct-like, but the Strouhal- and Mach-number ranges over which it is
duct-like is sufficiently large to justify the denomination. The third upstream-travelling
wave behaves in a manner similar to the low-frequency end of the
k
−
TH
branch, i.e. it is
a shear-layer mode, and is distinguished from the
k
−
TH
wave by the fact that it becomes
evanescent
above
a well-defined frequency (at the transition from red to green in figure
5). This wave is denoted
k
−
p
(red in figures 4 and 5). We can add to this catalogue of
waves, upstream- and downstream-travelling freestream sound waves (black in figures 4
and 5), which are also potential candidates for the enabling of resonance.
The
k
−
TH
,
k
+
T
,
k
−
d
and
k
−
p
waves are members of hierarchical families parameterised by
two integers, (
m,j
), corresponding to the azimuthal and radial orders of the waves. In
what follows, we restrict attention to axisymmetric waves,
m
= 0, of radial order,
j
= 1.
4.2.
Dispersion relations and preliminary tone-frequency prediction
Our objective is to see if the tone patterns can be understood in terms of the waves
described above. The linearised Euler equations provide the modelling framework. These
are considered in a locally parallel setting, with normal-mode
Ansatz
,
q
(
x,r,t
) = ˆ
q
(
r
)e
i
(
kx
−
ωt
)
.
(4.1)
6
Jordan et al..
∞
∞
M
Figure 4.
Schematic depiction of waves supported by cylindrical vortex sheet; colours
correspond to those of figure 5.
Figure 5.
Vortex-sheet dispersion relations in the range 0
.
6
≤
M
≤
0
.
97. Blue:
k
+
KH
Kelv-
in-Helmholtz modes; cyan in range 0
.
6
≤
M
≤
0
.
82:
k
−
T H
modes (Tam & Hu 1989); cyan in
range 0
.
82
< M
≤
0
.
97:
k
−
d
modes (Towne
et al.
2017); red & green, respectively:
k
−
p
and
k
+
T
acoustic jet modes (Towne
et al.
2017); black:
k
+
and
k
−
freestream sound waves (solid:
M
= 0
.
6; dash-dot:
M
= 0
.
97).
Here,
k
is the streamwise wavenumber, non-dimensionalised by
D
, and,
ω
= 2
πStM
, is
the non-dimensional frequency. Three dispersion relations, corresponding to increasingly
realistic conditions, are obtained from these: (DR1) that which is obtained by considering
the jet to behave as a porous-walled cylindrical duct; (DR2) that which describes waves
supported by a cylindrical vortex sheet (Lessen
et al.
1965; Michalke 1970); and, (DR3)
that which is obtained if the shear-layer is considered to have finite thickness. These three
models have been thoroughly discussed by Towne
et al.
(2017). We make a preliminary
tone-prediction using DR2; fine-tuning requires consideration of DR1 and DR3.
Figure 5 shows vortex-sheet dispersion relations, DR2, in the Mach-number range
0
.
6
≤
M
≤
0
.
97. With the exception of the Kelvin-Helmholtz mode, which has non-zero
imaginary part, the lines are locii of eigenvalues with zero imaginary part, i.e. neutrally
Jet-edge interaction tones
7
Figure 6.
(a) ∆
k
between
k
+
KH
Kelvin-Helmholtz mode and all
k
−
jet modes in range
0
.
6
≤
M
≤
0
.
97; (b) Illustration of resonance-frequency identification (showing ‘frequency
squeezing’) for
L/D
= 3,
M
= 0
.
6 and
hard-hard
end conditions: horizonal lines show val-
ues of ∆
k
hh
(equation 4.2) for L/D=3.
stable, propagating waves (Towne
et al.
2017). The waves discussed above have been
colour-coded. The downstream-travelling waves are shown in blue and green: respec-
tively, the Kelvin-Helmholtz mode,
k
+
KH
, and the
k
+
T
mode. Upstream-travelling waves
are shown in cyan and red: respectively, [
k
−
TH
(0
.
6
≤
M
≤
0
.
82);
k
−
d
(0
.
82
< M <
1)]
and
k
−
p
(0
.
82
< M <
1). The black lines show dispersion relations for upstream- and
downstream-travelling free-stream sound waves at
M
= 0
.
6 (solid) and,
M
= 0
.
97 (dash-
dot).
Resonance can potentially occur between any
k
+
/k
−
mode pair; there are therefore
five different possibilities if we exclude resonance between upstream- and downstream-
travelling sound waves. Given that the Kelvin-Helmholtz mode is the only unstable wave,
all others being in reality either neutral or slightly damped, the most likely scenario is
that in which
k
+
KH
is coupled, via end conditions provided by the nozzle exit plane and
the plate edge, to a
k
−
mode. A further argument for excluding the
k
+
T
wave is the
continuity of the tones across the
M
= 0
.
82 threshold, on the lower side of which these
modes are evanescent.
We consider two possible pairs of end conditions: a
hard-hard
condition, ˆ
u
(
x
= 0
,ω
) =
ˆ
u
(
x
=
L,ω
) = 0, in which upstream-travelling and downstream-travelling waves are
reflected by solid walls (a simplified model for scattering by the nozzle lip and the plate
edge); and a
soft-hard
condition, ˆ
p
(
x
= 0
,ω
) = ˆ
u
(
x
=
L,ω
) = 0, in which the upstream-
travelling waves are reflected by pressure-release in the nozzle exit plane, downstream-
travelling waves being reflected by the plate edge, again modelled as a solid wall.
It is straightforward to show that these scenarios lead to the following resonance cri-
8
Jordan et al..
Figure 7.
Tone-frequency predictions using vortex-sheet dispersion relations and assuming res-
onance between
k
+
KH
and
k
−
jet modes, coupled by
hard-hard
(top) and
soft-hard
(bottom) end
conditions. From left to right:
L/D
= 2, 3 & 4. Cyan: resonance between
k
+
KH
and
k
−
T H
; red:
resonance between
k
+
KH
and
k
−
p
.
teria,
∆
k
hh
=
2
nπ
L
,
(4.2)
∆
k
sh
=
(2
n
+ 1)
π
L
,
(4.3)
for
hard-hard
(note that
soft-soft
end conditions would lead to the same criterion) and
soft-hard
end conditions, respectively, where ∆
k
is the difference in wavenumber between
the upstream- and downstream-travelling waves that participate in the resonance.
The difference in wavenumber, ∆
k
, can be easily computed for any
k
+
/k
−
pair as a
function of Mach number and frequency, using the dispersion relations of the two waves.
This is shown in figure 6(a) for the pairs
k
+
KH
/k
−
TH
and
k
+
KH
/k
−
d
, both shown in cyan,
and for
k
+
KH
/k
−
p
, shown in red. Having calculated ∆
k
, the resonance criteria of equations
4.2 and 4.3 can be superposed, as in figure 6(b), and the resonance frequencies provided
by the intersection of these with the lines ∆
k
(
M,St
). The example shown in figure 6(b)
is for
M
= 0
.
6,
L/D
= 3 and
hard-hard
end conditions, and it illustrates an interesting
characteristic of this kind of resonance: a frequency ‘squeezing’, due to the dispersive
Jet-edge interaction tones
9
nature of the
k
−
waves. As the Mach number is increased this ‘squeezing’ becomes more
pronounced, due to the stronger variation of phase speed with frequency.
Tone-frequency predictions are made for the Mach-number range considered, using
both
hard-hard
and
soft-hard
end conditions, and these are compared with the observed
behaviour in figure 7. The general trend is found to be satisfactorily captured, in par-
ticular the aforesaid frequency ‘squeezing’. But there remain three discrepancies: (1)
the resonance models predict a continuation of the tones to infinitely high frequency,
whereas the data clearly shows a frequency cut-off; (2) at low frequencies, the model
predicts peaks that are not observed in the data; (3) the best match is for some cases
provided by the
hard-hard
end conditions, whereas for others the
soft-hard
conditions do
better.
The third of these discrepancies may be an indication that the flow is selecting between
different end conditions, and indeed an observation of mode switching—clearly audible
in the experiments—provides support for this idea. The discrepancy may, alternatively,
be due to the actual end conditions being more subtle than ˆ
u
= 0 and ˆ
p
= 0, comprising
flow physics only amenable by a Wiener-Hopf analysis involving boundary conditions
that change at
x
= 0 between a hard-walled nozzle and a vortex sheet, or by detailed
numerical analysis. Both are beyond the scope of the present work, and we will therefore
not explore this point further, accepting that the end-condition modelling, whilst not
perfect, is sufficient to support the main conclusion of the work, which is that the
k
−
jet
modes of Towne
et al.
(2017) and Schmidt
et al.
(2017), together with Kelvin-Helmholtz
wavepackets and free-stream sound waves, underpin the observed edge-tone behaviour.
Where the two other discrepancies are concerned, on the other hand, further explanation
is possible. The high-frequeny cut-off is considered in what follows.
4.3.
High-frequency tone cut-off
In the Mach-number range 0
.
6
≤
M
≤
0
.
82 we explore the high-frequency cut off using
DR3 and DR1. The first illustrates a progressive trapping of the
k
−
TH
wave by the jet with
increasing frequency ; once entirely trapped (duct-like) it is prevented from interacting
with the nozzle lip. The second is used to show that for frequencies at which the
k
−
TH
wave is truly duct-like (trapped), it cannot be reflected by pressure release in the nozzle
exit plane and is entirely transmitted into the nozzle. In the Mach-number range, 0
.
82
≤
M <
1, on the other hand, the resonance cut-off condition is due simply to a cutting-off
of the
k
−
p
waves: as discussed earlier, they are evanescent above a well-defined frequency.
This can be seen in figure 5 by looking at the red lines, which only appear up to a given
Strouhal number; beyond this value the waves become evanescent, which corresponds to
the saddle-point
S
2 discussed in Towne
et al.
(2017).
4.3.1.
Trapped waves cannot touch the nozzle
To understand the high-frequency cut-off in the range 0
.
6
≤
M
≤
0
.
82 we first consider
the dispersion relation associated with a finite-thickness round jet,
DR
3. Eigensolutions
of the Rayleigh equation are obtained for an axisymmetric mixing layer whose radial
velocity profile has the form of a hyperbolic-tangent velocity profile with displacement
thickness similar to that of the
M
= 0
.
6 free jet at
x/D
= 1.
We are here interested in the frequency dependence of the pressure eigenfunctions
associated with the discrete mode equivalent to the
k
−
TH
mode in the vortex-sheet model.
This is shown in figure 8. At very low frequencies the waves behave much like plane free-
stream sound waves; the scale separation between their wavelength and the width of the
jet is such that they are essentially unaffected by the flow. With increasing frequency they