1. Noise source-identification and propagation in a supersonic jet
1 Introduction
Jet noise is a major nuisance, especially near airports and on aircraft carriers. Sustained exposure
to high noise levels constitute health hazards to crew and results in environmental impact and dis-
ruption of the community near airports. Despite decades of research, the fundamental mechanism
by which the turbulent energy is filtered into acoustic energy through non-linear processes remains
unknown. This is a physical ramification of the non-linearity in the governing laws of fluid flows -
the Navier–Stokes equations (NSE). A satisfactory theory of sound genesis will provide scientific
insight and aid exploration of control techniques.
In order to better understand these non-linear interactions and identify sound sources, we an-
alyze a jet by decomposing its flowfield into the constituent modes – acoustic, hydrodynamic and
vortical, the results of which are discussed in Section 2. For this, we use the Momentum Potential
Theory (MPT), put forward by Doak (1989). Further, to understand the transport of acoustic en-
ergy into the farfield of the jet, we characterize the evolution of small perturbations in this turbulent
flow using a novel technique called Synchronized Large-Eddy Simulation (SLES) which solves the
full 3D, compressible, unsteady Navier-stokes equations (Section 3). Results to date have analyzed
a supersonic cold air-jet with a nozzle-exit Mach number of 1.3.
2 Mode decomposition and sound-sources of the jet – MPT
The flowfield of the turbulent jet is highly complex, with a broad range of spatio-temporal scales,
as can be seen in Fig. 1(a). The governing NSE do not split the different forms of energy, which
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2. Figure 1: (a) Complex flowfield of the jet, (b) Hydrodynamic mode, (c) Acoustic mode with
wavepacket-nature, (d) Radiation from the acoustic wavepacket.
inhibits clarity in understanding the acoustic dynamics. To resolve this, we have applied for the
first time (Unnikrishnan and Gaitonde (2016a)), the decomposition proposed by Doak (1989). The
momentum density of the jet ρu, (where ρ is the fluid-density and u is the particle velocity vector)
is split as; ρu = B+B − ψA − ψT . B is the mean hydrodynamic, B is the fluctuating hydro-
dynamic component and, − ψAand − ψT are the fluctuating acoustic and thermal components
respectively. The hydrodynamic component is defined as the divergence-free part of ρu and the
acoustic and thermal components are irrotational.
The magnitude of the hydrodynamic mode (||B ||) in (b) highlights the shear-layer growth of
the jet and the roll-up of vortices. The acoustic-mode component (−∂ψA/∂x) in (c) captures (for
the first time in published literature) a well-defined axial wavepacket in the highly turbulent core
of the jet. This coherent form of the acoustic mode is significant, since the acoustic dynamics of
the jet are better represented here and this wavepacket can be reproduced to model sound radiation.
The highly-directional downstream sound radiation induced by this wavepacket is shown in (d).
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3. Figure 2: (a) Schematic of sound-source mechanism, (b) Time-averaged behavior of sound-source.
Upon further analyzing Doak’s theory, the interaction of the fluctuating hydrodynamic mode
B with fluctuating Coriolis acceleration (ω × u) , (where ω = × u is the vorticity, defined
as the curl of velocity vector) was found to be the most significant acoustic source mechanism
of the jet. The schematic in Fig. 2(a) summarizes this source mechanism; whenever rolled-up
vortices from the shear-layer of the jet intrude into its high-speed core, the local total-enthalpy H
(H = cpT + u.u/2, where cp is the specific heat at constant pressure and T is the temperature)
surges, and these fluctuations in total enthalpy H , are carried away by the momentum fluctuations
(ρu) , resulting in the acoustic-energy-flux from the jet. The time-averaged behavior of this source
mechanism, −B .(ω×u) in (b) identifies the prominent sound-source region to be within the
lipline of the jet (marked by the dotted line). The peak source-values are observed along the
center-region of the inner shear-layer, between two to seven jet-diameters in the axial direction.
Significance of the method: By focusing on the decomposed fields, this analysis has identified
the prominent features of the acoustic mode, and how it derives its energy to radiate sound.
3 Perturbation evolution in the turbulent jet – SLES
As mentioned in the introduction, the identification of sound-source mechanism was followed by
development of a novel technique to study small-perturbation evolution in the jet. The idea is
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4. Figure 3: (a) Forcing location in the twin simulation, Instantaneous visualization of (b) Basline,
(c) Twin and (d) Perturbation-field of the jet.
to understand how the turbulence filters and modulates signals from the core of the jet to yield
its observed acoustic signature. Traditional methods study propagation of perturbations in a time-
invariant mean base flow by linearizing the NSE. Here, we use two synchronous full Navier-Stokes
simulations - a baseline and a twin, to propagate the perturbations in the time-dependent evolving
turbulent flow (Unnikrishnan and Gaitonde (2016b)). The twin simulation is forced at a select
location using the signal from the baseline at the corresponding location and the difference between
the two simulations is obtained. This difference-field (denoted perturbation-field) shows the effect
of the forcing on the instantaneous flow, yielding insight into jet-noise directivity and intermittency.
Several insights were obtained by studying perturbations evolving from the core of the jet.
Figure 3(a) schematically shows the forcing location of the twin, point A. The background contours
of time-averaged streamwise velocity show the high-speed jet-core and the developing shear-layer
surrounding it. An instantaneous snapshot of the baseline and twin simulations are shown in (b)
and (c) respectively. Although they appear similar at this level, their difference in (d) indicates how
the forcing at point A evolved into the perturbation-field in the turbulent flow.
Looking at more details, the perturbation-field in Fig. 4(a) shows the effects of forcing at
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5. Figure 4: (a) Perturbation field of the jet, Time-frequency analysis of the perturbation field at P1
(b) and P2 (c).
point A, leading to intermittent wavefronts in the nearfield of the jet, and secondary downstream
amplification (between x = 15 and x = 20). An analysis of the time-frequency properties of the
perturbation-field at two points P1 (sideline) in (b) and P2 (downstream) in (c) indicates well-
defined intermittent events. (St, Strouhal is the non-dimensional frequency). For e.g., the sideline
signals are dominated by high-frequency content as marked by the dotted curve in (b). Conversely,
high-energy shifts to lower frequencies in the downstream direction which shows peak acoustic
radiation in jets. These results are consistent with experimental observations (Tam, 1995; Tam et
al., 2008). Significance of the method: The current technique provides information on sound
directivity and farfield signature without simplifications associated with mean-flow analyses and
localizes the effect of specific regions in the core on the acoustic signature of the jet.
Future work will combine the two methods described above– MPT and SLES to study evolution
of each mode from specific forcing locations in the jet, by applying Doak’s decomposition to
the perturbation-field obtained from SLES. The sound-source and nearfield characteristics of the
modes have to be then analyzed using statistical tools. Finally, these techniques would be applied
to subsonic, heated and coaxial jets, which mimic real-life aircraft-engine-exhaust configurations.
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6. References
Doak, P. E. (1989). Momentum potential theory of energy flux carried by momentum fluctuations.
Journal of sound and vibration, 131(1), 67–90.
Tam, C. K. W. (1995). Supersonic jet noise. Annual Review of Fluid Mechanics, 27(1), 17–43.
Tam, C. K. W., Viswanathan, K., Ahuja, K. K., & Panda, J. (2008). The sources of jet noise:
experimental evidence. Journal of Fluid Mechanics, 615, 253–292.
Unnikrishnan, S., & Gaitonde, D. V. (2016a). Acoustic, hydrodynamic and thermal modes in a
supersonic cold jet. Journal of Fluid Mechanics, 800, 387–432.
Unnikrishnan, S., & Gaitonde, D. V. (2016b). A high-fidelity method to analyze perturbation
evolution in turbulent flows. Journal of Computational Physics.
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