sect_nearfield.tex

\section{The Near- and Far-field Response to Excitation}
\label{sect:nearfield}
\subsection{Previous Results}
Previous work at the GDTL by \citet{Sinha2012} studied the irrotational near-field response of a subsonic jet subjected to excitation with plasma actuators by decomposing the instantaneous fluctuating pressure field into a coherent `wave' component (which corresponds to the large-scale structure generated by the excitation) and incoherent residual fluctuations (which correspond to the natural turbulence in the jet). 
Fundamentally, this decomposition is similar to the triple decomposition used by \citet{Hussain1970}.
Sinha \etal~found that each pulse from the actuators produces a coherent large-scale structure that would grow, saturate, and decay as it advects through the jet shear layer. 
In the irrotational near-field, the signature of these large-scale structures takes the form of a compact waveform. 
At very low excitation frequencies, the characteristic period of this waveform is much less than the excitation period, and hence, the structures seeded by the excitation do not interact with one another as they evolve downstream. 
Therefore, their behavior can be thought of as representing the response of the jet to a single perturbation: the `impulse' response of the jet which is produced by the impulsive excitation by LAFPAs.
As the period of actuation approaches the characteristic period of the impulse response, the waveforms extracted by the phase-averaging technique are largely unmodified from that of the impulse response. 
Above this frequency, significant interaction between the structures is observed, with noticeable modifications to the waveform shape and amplitude. 
As the structures are growing as they advect through the shear layer, the frequency at which the structures begin to interact is dependent on the axial location.
This behavior can be observed in \fig{fig:ch3_nearfield}a.
\begin{figure}
	\centering
	\includegraphics[width=0.45\linewidth]{Figures/ch3_nearfield_phavg_v2.png}
	\includegraphics[width=0.45\linewidth]{Figures/ch3_nearfield_linear_v2.png}
	\caption{Phase-averaged waveforms along the first array position at $x/D = 3, r/D = 1.35$ (a) and a linear superposition of the phase-averaged waveform for the impulse excitation ($St_{DF} = 0.05$) compared against periodic excitation ($St_{DF} = 0.50$) (b).}
	\label{fig:ch3_nearfield}
\end{figure}

For a certain range of excitation frequencies ($St_{DF} \leq 0.50$ at $x/D = 3$, for example), the structures interact in a quasi-linear manner, insofar as their near-field pressure signatures are concerned. 
That is, the response of the jet in the irrotational near-field could be well-predicted by a linear summation of the impulse response of the jet, repeated at the periodic excitation frequency. 
This concept has been illustrated in \fig{fig:ch3_nearfield}b, where the periodic response of the jet to excitation with $St_{DF} = 0.50$ has been reproduced at $x/D = 3$. 
Additionally, a linear superposition of the impulse response for $St_{DF} = 0.05$, repeated to match the excitation frequency of $St_{DF} = 0.50$, has been overlaid. 
The linear superposition has been arbitrarily shifted in time in order to match the phase of the periodic response; this phase difference is likely due to the dependence of convection velocities on structure wavelength \citep{Veltin2011}. 
Upstream of the end of the potential core ($x/D \simeq 6$, as will be found in \sect{sect:velocity}), the quasi-linear interaction model produces close predictions of the waveform amplitude and shape, despite the significant difference in both peak amplitude and waveform shape between the impulse and periodic responses. 

This quasi-linear interaction of the jet response to excitation is not limited exclusively to the hydrodynamically-dominated regions of the jet, but in fact holds for the acoustic far-field as well, at aft angles (where the acoustic signal is strongest and is known to correlate well with large-scale structures). 
This can be observed in \fig{fig:ch3_farfield}a, where the phase-averaged response of the jet has been plotted for the far-field signal at a polar angle of $30^\circ$. 
For legibility, only a select number of excitation Strouhal numbers have been included. 
As with the irrotational near field, the acoustic far field exhibits a compact waveform for the lowest excitation Strouhal numbers. 
Though nearly a direct inverse from the waveform observed in the hydrodynamically-dominated near field, the far-field waveform is quite reminiscent of the phase-averaged waveforms observed by \citet{Kambe1983} for the acoustic radiation towards aft angles produced by the head-on collision of vortex rings. 
At higher $St_{DF}$, a continuous oscillation between sharp expansion and compression waves is again observed, though the amplitude begins to decay above moderate excitation Strouhal numbers. 
\begin{figure}
	\centering
	\includegraphics[width=0.45\linewidth]{Figures/ch3_farfield_phavg_v2.png}
	\includegraphics[width=0.45\linewidth]{Figures/ch3_farfield_linear_v2.png}
	\caption{Phase-averaged waveforms of the far-field at $30^\circ$ (a) and a linear superposition of the phase-averaged waveform for the impulse excitation ($St_{DF} = 0.05$) compared against periodic excitation ($St_{DF} = 0.25$) (b).}
	\label{fig:ch3_farfield}
\end{figure}

As before, a linear superposition of the impulse response can well predict the waveform shape and amplitude at the higher excitation frequencies (\fig{fig:ch3_farfield}b), though in this case only up to $St_{DF}  = 0.25$. 
From the phase-averaged waveforms alone it is not clear whether this breakdown in the linear superposition model at the higher excitation frequencies is due to nonlinear behavior or uncertainty in the phase-averaging. 
Results comparing the linear superposition of the impulse response against the measured periodic response at $St_{DF}  = 0.35$ are shown in \fig{fig:ch3_farfield_nonlinear}.
Some similarities may be found in the waveform shape and amplitude, but overall it is clear that the acoustic response of the jet to excitation at $St_{DF}  = 0.35$ is substantially modified from the response at lower frequencies.
Though this is hardly conclusive in its own right, this result does suggest either changing or competing acoustic source mechanisms are present in these excited jets.
The phase-averaged waveforms were also investigated at polar angles of $60^\circ$ and $90^\circ$; however a clear waveform was not identifiable over the statistical uncertainty inherent in the phase-averaging process (likely due to the superdirective character of the acoustic radiation \citep{Crighton1990}, which renders the amplitude at sideline angles too low to be detectable).
Additional details and analysis of the phase-averaged near- and far-field signals can be found in \citet{Crawley2015}.
\begin{figure}
	\centering
	\includegraphics[width=0.45\linewidth]{Figures/ch3_farfield_linearsuperposition_st035_v2.png}
	\caption{Linear superposition of the phase-averaged impulse response to excitation against the measured periodic response for $St_{DF} = 0.35$ at the  $30^\circ$ far-field microphone.}
	\label{fig:ch3_farfield_nonlinear}
\end{figure}

\subsection{Coherent Structures in Excited versus Natural Jets}
The effect of excitation of natural instabilities in the turbulent jet is to generate highly energetic coherent structures which persist downstream for several length-scales.
As can be seen by the strong tonal energies observed in the pressure spectra for the excited jets (as illustrated in \fig{fig:sect_nearfield_spectra_prms}a), the near-field response to these structures contains energy at the fundamental excitation frequency and numerous higher harmonics.
The presence of these highly-energetic structures can also lead to an amplification of the broadband turbulence, over a wide band of frequencies both above and below the fundamental excitation frequency.
It also produces a continuous upstream shift in the phase-averaged root-mean-squared peak amplitude, as can be seen in \fig{fig:sect_nearfield_spectra_prms}b, with the maximum being obtained near $St_{DF} \simeq 0.3$ (identified as the jet column instability frequency in our natural jet).
However, the fluctuation intensities as a function of frequency in the excited jets saturate much further upstream, $x/D \simeq 3$ for $St_{DF} \simeq 0.3$,  than in the natural jet, further downstream near the end of the potential core at $x/D \simeq 6$.
In light of this issue, an investigation of the coherent structures in the natural jet and their comparison against the excited was deemed pertinent to the analysis of the excited structure dynamics.
\begin{figure}
	\centering
	\includegraphics[width=0.44\linewidth]{Figures/sect_nearfield_spectra.png}
	\includegraphics[width=0.46\linewidth]{Figures/sect_nearfield_prms.png}
	\caption{Power spectral densities of the raw near-field pressure at $x/D = 3$ (a) and root-mean-squared pressure fluctuations of near-field pressure at (b).}
	\label{fig:sect_nearfield_spectra_prms}
\end{figure}

Obviously, it is not possible to use standard phase--averaging with the unforced case since the natural turbulent structures, although dominated by Strouhal numbers related to the instabilities in the jet, contain a broad range of energetic scales with no fixed phase relationship.
It is necessary then to use an alternative averaging method in order to extract a coherent signature of the unforced jet turbulent--structures.
Instead of phase--averaging, which cannot be applied to natural jets, a conditional--averaging method, specifically wavelet--conditioning, was applied to both unforced and forced cases to determine how closely the forced structures relate to the natural structures in the jet.
A wavelet decomposition was used as it affords an efficient methodology for analyzing intermittent events in a given signal due to the finite energy of its basis functions. 
This is in contrast to the more standard Fourier decomposition in which continuously-oscillating basis functions spread information from a single instance in time across all transform coefficients. 
The use of the wavelet decomposition has become increasingly common in aeroacoustic research; for brevity, the reader can refer to \citet{Farge1992} for an overview of the wavelet transform and its applications to turbulence and acoustics.

%The wavelet coefficients are noted $\tilde{w} \left( s, \tau \right)$, where $s$ is the wavelet scale (inversely proportional to Fourier frequency) and $\tau$ is the time--shift as in the Short Time Fourier Transform (STFT) formulation.
%As STFT, the wavelet decomposition is a convolution of analyzed signal with a family of functions, the wavelets family.
%To get the wavelet family, the mother wavelet is stretched/compressed and shifted.
%The strength of the wavelet decomposition with regards to STFT is the variation of its time--scale, which for STFT would correspond to its window's length.
%This allows a more localized information.
%In STFT the size of the window (time--scale) is fixed at the beginning of the analysis but not the frequency, which allows to retrieve the frequency content.
%Inversely, the wavelet decomposition operate using a fixed frequency (usually noted $\omega_0$) and a window of variable length in order to retrieve the different time--scales.
%This fundamental difference between these two decomposition allows to the wavelet decomposition to be more flexible and have a better resolution at both high and low frequencies.
%For more details about the wavelet decomposition, the reader may refer to \citet{Farge1992}.

Wavelet--based conditional averaging has been used previously for structure identification in jets and other turbulent flows \citep{Camussi1997,Camussi1999,Guj1999,Camussi2002,Guj2003}.
The technique educes the signature of coherent structures based on a conditional--average of a turbulent signal using a short--time window centered around highly energetic events identified by the wavelet transform.
Identification of the energetic events is accomplished using the Local Intermittency Measure (LIM) \citep{Farge1992}, which is the ratio between a local energy of the wavelet coefficient for a specific time and scale and a time-averaged energy at that same scale.
The LIM was demonstrated to be a well--suited indicator for coherent structure identification \citep{Camussi1997}.
It provides information about the instantaneous fluctuation of energy and by choosing a proper threshold, $T$, it is possible to select a set of times $\{\tau_{i}\}$ corresponding to intermittent energetic events in a signal.

Several methods exist to select the threshold: by selecting an arbitrary threshold, an iterative process to retrieve a specific value of the flatness factor (kurtosis, fourth moment), or by evaluating the Merit index \citep{Grassucci2015}. 
In the present study, a new method inspired by the Merit index was used to select the threshold. 
In contrast to the original method, which uses a global threshold across all scales, the new method iterates at each scale to identify a proper threshold. 
At each iteration, the coefficient $R(T)$ is evaluated:
\begin{equation} \label{eqn:tEvaluation}
R(T) = -\frac{\log\left(\frac{N_e}{N_w}\right)}{\log\left(\frac{\sigma_{w_e}}{\sigma_{w_w}}\right)}.
\end{equation} 
%$N_p$ is the length of the set $\{L > T\}$, $N_m$ is the length of the set $\{L \leqslant T\}$, $\sigma_{w_p}$ the standard deviation of the set $\{w\left( \tau_p\right) | L\left( \tau_p \right) > T\}$ and $\sigma_{w_m}$ the standard deviation of the set $\{w\left( \tau_m\right) | L\left( \tau_m \right) \leqslant T\}$. 
$N_e$ is the number of energetic samples ($LIM > T$), $N_w$ is the number of incoherent, weak samples ($LIM \leqslant T$), and $\sigma_{w_e}$ and $\sigma_{w_w}$ are the standard deviations for the energetic and weak wavelet coefficients.
The threshold is selected as the value maximizing $R(T)$, in the same manner as with the original Merit index method \citep{Grassucci2015}.

The conditional--averaging (\eq{eqn:ensembleAverage}) over the set of times $\{\tau_{i}\}$ is then performed in order to get the coherent structures signature as for the phase--averaging.
At each time location corresponding to a peak of energy, a window $W$ of fixed time--length $l_{W}$ is selected from the original signal $p \left( t \right)$. The conditional--average, $\tilde{p}$ is evaluated from this set of windows:
\begin{equation} \label{eqn:ensembleAverage}
	\tilde{p}^n\left( W \right) = \frac{1}{N_e} \sum^{N_e}_{i = 1} p^l \left(\xi_{i}\right),
\end{equation}
where the superscripts $n$ and $l$ correspond to the position of the reference signal and of any other signal of the array, respectively.
Auto--conditioning occurs when the conditional--average is performed on a signal $p^{n}$ by using its own set of times ${\tilde{\tau}^n}$, making $n = l$. 
The subscript $s$ corresponds to the scale, $N_e$ is the number of detected energetic events, $\tilde{\tau}^n_{s}$ is the set of corresponding times for a specific scale $s$ at which these events are occurring and $\{\xi_{i}\}$ is the interval surrounding each peak, $\xi_{i} \in \left[ \tilde{t}_{i} - \frac{x}{2}, \tilde{t}_{i} + \frac{x}{2} \right]$, $\tilde{t}_{i} \in \tilde{\tau}^n_{s}$.
Based on the phase-averaged results, which indicated that a single coherent event could include both positive and negative fluctuations in the pressure domain, only positive pressure peaks were used for identifying the energetic events in the conditional-average.
%A first analysis using \eq{eqn:ensembleAverage} was performed on the microphones line--array at $r/D = 1.20$.
%A second analysis was then performed by doing the conditional--average only with negative/positive valued peaks in the real domain (pressure).
%
%\begin{figure}
%	\centering
%	\includegraphics[width=0.45\linewidth]{Figures/negativePeak.eps}%
%	\includegraphics[width=0.45\linewidth]{Figures/positivePeak.eps}\\	
%	\includegraphics[width=0.45\linewidth]{Figures/compPeaks}
%	\caption{Auto--conditioning signatures of the signal at $x/D = 3$, (a) with only negative peaks, (b) with only positive peaks, (c) comparison between both} \label{fig:compPeaks}
%\end{figure}
%\fig{fig:compPeaks} presents the two subfigures for the different cases (negative/positive peaks) and a third subfigure to compare them.
%\begin{itemize}
%  \item $(a)$ conditional--average (\eq{eqn:ensembleAverage}) using only the negative peaks
%  \item $(b)$ only the positive peaks
%  \item a comparison between the two previous one ($(a)$ was inverted in order to compare it to $(b)$)
%\end{itemize}
%\fig{fig:compPeaks}c presents the resemblance between the signature of subfigures (a) and (b) on which the conditional--average was performed using only negative or positive peaks (it is important to understand that the differentiation is done by evaluating the value of each peak in the real domain and not in the LIM/wavelet domain). %At this point, it is believed to be the result of the passage of two successive large scale turbulent--structures, in the zone where their interaction take place.
%The formulation of the conditional--average was modified to take this observation into account and reformulated as follows:
%\begin{equation} \label{eqn:condAvgSign}
%\tilde{p}_{m}^n\left( W \right) = \left< p_{m} | P_{k} \right>_{\tilde{\tau}^n_{s}} = \frac{1}{N^n} \sum^{N^n}_{i = 1} sign\left( p_{n} \left( \tilde{t}_{i} \right) \right) p_{m} \left( \xi_{i} \right),
%\end{equation}
%$sign\left( p_{n} \left( \tilde{t}_{i} \right) \right)$ is introduced to take into account the sign of the peak in the real domain. This operation might be unnecessary for an experimental database (with a long time--series) and the conditional--average be done with one or the other peak' sign.
%For a numerical database which is strongly limited in its time duration, it improves the results obtained with the wavelet--conditioning.
%The conditional--average using \eq{eqn:ensembleAverage} with all the peaks (not presented here) is not relevant as additional proof for the separation between negative/positive peaks.
%When the conditional--average is performed on a signal $p_{n}$ by using its own set of times ${\tilde{\tau}^n_{s}}$, making $n = m$; it becomes the auto--conditioning. If the conditional--average of a signal $p_{m}$ is done by using the set of times of a signal $p_{n}$, and so $n \neq m$; it is then called the cross--conditioning. 
%The auto/cross--conditioning are presented in \fig{fig:condNcond}: the top is the auto--conditioning where the selection of $W$ is done in the reference signal $n$; and the bottom is the cross--conditioning of another signal $m$ where $W$ is centered around the times obtained with signal $n$.
%The cross--conditioning is only presented in theory but no results with valuable meaning were obtained for this specific study.
%\begin{figure}
%	\centering
%	\includegraphics[width=1\textwidth]{Figures/condNcond.eps}
%	\caption{\textit{Auto--conditioning of a signal $p_n(t)$ and cross--conditioning of a signal $p_n(t)$}}
%	\label{fig:condNcond}
%\end{figure}

For sake of brevity, the results for only four cases, baseline and $St_{DF} = 0.05, 0.25,$ and $0.35$, are presented here in \fig{fig:WaveAutoCond}, which illustrates the auto--conditioning of the closest microphones array at $r/D = 1.2$.
For the natural jet, the near-field signature of the coherent structures educed by the wavelet auto-conditioning take the form of a strong compression region surrounded by a weaker expansion region on both the leading and trailing edge. 
As the structure convects downstream, the local time--scale of the fluctuation grows rapidly.
This variation in time-scale with axial position is expected, given prior knowledge of the simple auto--correlation of the near-field pressure at each axial position.
The amplitude of the compression region is relatively strong over most of the domain; up to $x/D=10$ the amplitude of the event is greater than twice the local standard deviation before slowly decaying further downstream. 
\begin{figure}
	\centering
	\includegraphics[width=0.85\textwidth]{Figures/wavelet_conditioning.png}
	\caption{Wavelet auto--conditioned pressure field along the microphone array at $r/D = 1.20$ for the natural jet (a), $St_{DF} = 0.05$ (b), $St_{DF} = 0.25$ (c), and $St_{DF} = 0.35$ (d). The results have been non-dimensionalized by the local standard deviation, and the color-scale is consistent across all plots.}
	\label{fig:WaveAutoCond}
\end{figure}

The auto--conditioning of the impulse excitation case, $St_{DF} = 0.05$, is presented in \fig{fig:WaveAutoCond}b.
General agreement is observed between the impulsive structures and the natural jet turbulence; they take the form of a strong compression region bookended by expansion regions, the time-scales of which increase with axial position.
However, noticeable differences are also apparent; the excited structures exhibit a much stronger trailing expansion region than the natural structures, and the decay in amplitude of the expansion and dominant compression regions is much more rapid with axial position.
As a result, in the downstream region the energetic event of the impulse excitation case becomes similar in shape to the unforced one, consisting of a central peak and surrounded by two lobes of similar amplitude expansion regions.
Results for the periodic excitation cases ($St_{DF} = 0.25$ and 0.35) are presented next in \fig{fig:WaveAutoCond}c and \fig{fig:WaveAutoCond}d, respectively.
As with the phae-averaged near-field pressure results for these excitation cases, a repetition of energetic events with the same time--scale is observed in the upstream region; rather than one strong compression region and two expansion regions, a single compression and expansion event occurs per excitation cycle.
Further downstream however the continuous oscillations abruptly end however, at $x/D = 6$ for $St_{DF} =0.25$ and $x/D = 4$ for $St_{DF} =0.35$, and the shape of the coherent event begins to revert to that of the natural jet.

\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{Figures/wavelet_cross_conditioning.png}
	\caption{Wavelet cross--conditioned pressure field along the microphone array at $r/D = 1.20$ for $St_{DF} = 0.05$ (a,b,c), $St_{DF} = 0.25$ (d,e,f), and $St_{DF} = 0.35$ (g,h,i). Reference microphones were set to $x/D = 1$ (a,d,g), $x/D = 6$ (b,e,h), and $x/D = 9$ (c,f,i).}
	\label{fig:WaveCrossCond}
\end{figure}
The abrupt shift in the behavior of the wavelet auto-conditioned pressure fields for the periodically-excited jets was surprising, particularly in light of the phase-averaged results which show a continuous evolution of the near-field signature of the coherent structures. 
The nature of this abrupt shift can be better understood by comparing the wavelet cross-correlations for the three excitation cases, as has been done in \fig{fig:WaveCrossCond} using reference microphones of $x/D = 1$, 6, and 9.
For the impulsive structures (\fig{fig:WaveCrossCond}a, b, and c), the wavelet cross-conditioning is identifying a single energetic event which is coherent over the entire domain shown as it convects downstream.
Identical trends were found for the baseline jet results, which are not shown here for brevity.
In contrast, the results for $St_{DF} = 0.25$ indicate two distinctly separate coherent phenomena, one in the upstream region ($x/D \leq 6$) and another in the downstream region.
Increasing the excitation frequency further to $St_{DF} = 0.35$ results in three distinct energetic events, one upstream ($x/D \leq 4$), one downstream ($x/D > 7$), and one near the end of the potential core.
The exact nature of the flow phenomena which produce these distinct energetic signatures in the near-field pressure will be further elucidated in \sect{sect:velocity}.
%The characteristic energetic--event of the forced cases, a positive valued peak followed by a negative valued one of the same amplitude, are not similar in shape to the unforced case, a central peak surrounded by two smaller of the opposite sign; at least for the few lengthscales directly after the nozzle exit.
%The forced structures are more energetic and organized than the natural ones, especially close to the nozzle exit, but will have a similar behavior further downstream, principally the lowest excitation case at $S_t = 0.05$.
%For the highest frequency excitation, due to the distance between successive structures, which is shorter (\fig{fig:autoCond}c and \fig{fig:autoCond}d), the jet does not have sufficient length (from the nozzle exit till the end of the potential core) to completely recover the natural jet behavior as for $S_t = 0.05$.
%Because of the short distance between successive events, the coherent structures break down which lead to the energy cascade between large scales and fine scales is not as efficient as for $S_t = 0.05$ and the unforced cases.
%It can be noted that the distance from the nozzle for which the excitation takes effect is becoming shorter as the frequency of the excitation is higher: on \fig{fig:autoCond}c the excitation effect is maintained up to $x/D=6$ while on the following \fig{fig:autoCond}d, it is up to $x/D=4$, shortened by around $2D$.
%Forcing the jet with plasma actuators drive it to have more regular and ordered structures, which are more energetic than the ones observed for an unforced jet. The effect on the acoustic field is weaker as the shape of the spectra is the same \fig{fig:sect_nearfield_spectra_prms} with extra peaks due to the excitation and and a small elevation of the amplitude for the strongest excitation $S_t = 0.35$.

Finally, analysis of the excited structure evolution vis-a-vis that of the coherent structures in the natural jet was performed.
By comparison of the results obtained between the phase-averaging and wavelet-conditioning for the impulsively-excited jet ($St_{DF} = 0.05$) and the natural jet (\fig{fig:phase-wavelet_comparison}), several conclusions may be reached.
First, while there are small discrepancies between the results obtained from the phase-averaging and wavelet-conditioning methods for the same signal, overall very similar waveforms are produced for the coherent structures.
Clearly, the wavelet-conditioning is successful in identifying the excited structures over the natural turbulence of the jet, even without being given explicit phase information.
In the upstream region of the jet, there are discrepancies between the waveform of the natural coherent structures educed by the wavelet-conditioning and the ones educed for the impulsively-excited jet.
The expansion region that precedes the primary compression region of the natural structures is nearly non-existent for the excited structures, but the trailing expansion region is more pronounced.
However, as the structures convect downstream near the end of the potential core, the coherent waveforms for the natural and excited exhibit clear similarity.
This indicates that, though the excited structures are far more energetic than those normally found in the jet, the downstream evolution of the natural turbulent structures mirrors (in a statistical sense) that of the intermittent structures generated by the impulse-excitation.
Therefore, it is reasonable to expect that any acoustic source mechanisms identified downstream for the excited structures are valid for the natural turbulent structures as well.
\begin{figure}
	\centering
	\includegraphics[width=0.45\textwidth]{Figures/conditioning/Phase-wavelet_comparison_x3D_pa.png} %
	\includegraphics[width=0.45\textwidth]{Figures/conditioning/Phase-wavelet_comparison_x6D_pa.png} 
	\caption{Comparison of results for natural and excited ($St_{DF} = 0.05$) jet at $x/D = 3$ (a) and $6$ (b).}
	\label{fig:phase-wavelet_comparison}
\end{figure}

\subsection{Identifying the Acoustic Source Region}
\label{sect:near_field_source_region}
Much of the difficulty in identifying the aeroacoustic source terms revolves around the dissimilar range of scales and fluctuation intensities of the turbulent eddies in the shear layer and the resulting radiated noise. 
Outside the jet shear layer, in the irrotational near-field of the jet, strong hydrodynamic pressure fluctuations associated directly with the passage of coherent structures in the shear layer and their resultant weak acoustic radiation coexist \citep{Arndt1997}. 
Beyond this, in the acoustic far-field, the hydrodynamic signature of the coherent structures is nonexistent owing to their strong exponential decay with radial distance.
As a result, identification of pure acoustic waves and their corresponding source events is problematic.
A decomposition of the pressure field into its constitutive hydrodynamic and acoustic components is therefore required. 
By identification and prediction of coherence nulls in the near field, \citet{Coiffet2006} showed that the full irrotational near-field consistent primarily as a linear superposition of its hydrodynamic and acoustic components, which lead subsequent researchers to propose linear filters to extract the individual components from the near-field pressure, with varying degrees of success. 

As discussed by \citet{Tinney2008}, in a transonic jet in which the large-scale structures are convecting subsonically with respect to the ambient speed of sound, a demarcation of the hydrodynamic and acoustic energy fields can be observed with phase velocity.
This is because the hydrodynamic pressure fluctuations will be aligned with the jet axis, and traveling subsonically. 
Acoustic pressure fluctuations will impinge on the linear microphone array at oblique angles, and therefore will appear as having either sonic or supersonic phase velocity, based on the source location. 
Therefore, a demarcaction between the hydrodynamic and acoustic energy components should be readily identifiable about the sonic wavenumber, $k_a = \omega / a_\infty$.
Decomposition of the irrotational near-field pressure is therefore straightforward in Fourier space.

However, there is also a great drawback associated with Fourier analysis: while it analyzes a given signal at a distinct frequency, local information for a given event is spread over all spectral coefficients. 
This is due to the fact that the basis functions used by the Fourier transform oscillate indefinitely. 
For a signal composed of completely random fluctuations this is not an issue, however it has become increasingly clear that the jet noise phenomenon is not a random process \citep{Kearney-Fischer2013}.
Transient events, such as intermittency or the spatial and temporal modulation of a wavepacket, have been shown to be important in the noise generation process. 
It was for this reason that a continuous spatio-temporal wavelet filter, based off of the work of \cite{Antoine2004} and \cite{Kikuchi2010}, was instead used in the current work to decompose the acoustic and hydrodynamic near-fields based on phase-velocity.
It has been shown that by using a temporally/spatially localized fluctuation as a basis, the wavelet transform compresses the information in a turbulent field much more efficiently (and accurately) than the Fourier transform \citep{Farge1992}.
A much more detailed explanation of the spatio-temporal wavelet transform, the justification for its use in decomposing the near-field pressure, and validation of the methodology using the current database can be found in \citet{Crawley2016}. 

By decomposing the irrotational near-field pressure, the relationship between the near- and far-field can be more easily elucidated. 
Two-point correlations were computed using both the full near-field and the acoustic near-field component, between each microphone in the near field and the far-field microphone at $30^\circ$; results can be found in \fig{fig:ch3_full_vs_partial_xcorr}. 
Examination in the spatio-temporal domain shows distinct regions of positive and negative correlation spanning several jet diameters and flow time scales.
The time lag, $\tau$, in the figures have been non-dimensionalized by the ambient speed of sound, $a_\infty$, and $R$, the distance from each near-field microphone to the far-field microphone (note that this results in an ordinate that is scaled separately along the abscissa, due to the dependence of the axial position on $R$).
\begin{figure}
	\centering
		\includegraphics[width=0.475\linewidth]{Figures/sect_nearfield_fullxcorr.png}
		\includegraphics[width=0.401\linewidth]{Figures/sect_nearfield_acousticxcorr.png}
	\caption{Normalized two-point correlations for the natural jet between the near field and the far field at $30^\circ$ for the full near-field pressure (a) and the acoustic component only (b).}
	\label{fig:ch3_full_vs_partial_xcorr}
\end{figure}

Near the jet shear layer (\fig{fig:ch3_full_vs_partial_xcorr}a), four distinct correlation regions can be observed: two positive, two negative; one strong and one weak for each. 
The first correlation regions, the strong-negative and weak-positive, are noticeable beginning at the most upstream microphone and reach their peak values around $5 < x/D < 10$, decaying significantly beyond that.
The slopes of these regions indicate propagation velocities noticeably below the sonic velocity; in the upstream region, they roughly match with the measured convective velocity of the large-scale structures ($U_c \simeq 0.7 U_j$ as measured by two-point correlations between subsequent near-field microphones, see \citet{Crawley2015} for additional details) in the upstream region of the jet, and slowly decelerate downstream.
Similar behavior was observed by \citet{Bogey2007}, who noted that two-point correlations between the flow-field and acoustic near-field in a simulated jet produced strong positive correlation regions which peaked at the end of the potential core and which followed the convection of the large-scale structures.
Conversely, the strong-positive and weak-negative correlation regions exhibit propagation velocities that match well with the ambient speed of sound. 
The distinctly different propagation velocities and of the two pairs of correlation regions indicate that these correspond to different physical phenomena.
The strong-negative and weak-positive correlation regions observed near the jet shear layer are associated with the large-scale structures themselves, rather than acoustic phenomena. 
This relationship becomes even more clear when just the acoustic component of the near-field is considered, rather than the full irrotational near-field. 
Gone entirely now are the correlation regions with subsonic propagation velocities (\fig{fig:ch3_full_vs_partial_xcorr}b), and instead, a single positive correlation region corresponding to sonically-propagating waves exists over the entire domain. 

The correlations of the decomposed near-field can be used to identify the acoustic source region, at least in a rough sense, by comparing the time lag at which the greatest correlation is achieved against expected times-of-arrival for different propagation paths.
A schematic of these propagation paths is provided in \fig{fig:ch3_ToA}.
The first expected time of arrival, $\tau_a$, corresponds to the expected time lag for an acoustic wave traveling directly from the noise source to the near-field microphone and on to the far-field microphone and hence, the noise source region lies along the axis created by the near-field and far-field microphones.
Another expected time-of-arrival can be constructed by assuming the source region is stationary in space; from simple geometric considerations of the distance from the assumed source region to the near-field and far-field microphones, the time lag, $\tau_s$, between the arrival of an acoustic wave at both microphones can be computed.
The stationary source region is of course not known \textit{a priori}, but is set by the author subsequent to the computation of the two-point correlations.

For simplicity, density and convection effects on the acoustic wave as it travels through the jet shear layer have been neglected in this analysis. 
By necessity, it has been assumed that the acoustic radiation in the jet is dominated by $m = 0$ azimuthal Fourier mode (the near-field and far-field microphone arrays are not at the same azimuthal angle with respect to the nozzle). 
This assumption is easily justified in the excited jets, where the actuators have been fired in phase. 
While the near-field pressure and acoustic radiation towards aft polar angles in a natural, high Reynolds number jet is a combination of numerous azimuthal Fourier modes, previous researchers have found these fields to be dominated by the axisymmetric mode \citep{Arndt1997,Hall2006,Koenig2013,Juve1979}.
\begin{figure}
	\centering
	\includegraphics[width=3in]{Figures/ToA_tau.png}
	\caption{Expected times of arrival for on-axis acoustic propagation, $\tau_a$, and off-axis acoustic propagation, $\tau_s$ from a stationary source region centered at $x_s$.}
	\label{fig:ch3_ToA}
\end{figure}

Here, the acoustic source region was assumed to be located at $x_s /D = 4$, which is just upstream of the end of the potential core in the unforced jet. 
(Please note that this analysis is not meant to imply that the source region is located at a specific, fixed point – it is merely a convenient way of understanding the propagation paths.)
Observed behavior of the excited jets shown in \fig{fig:ch3_xcorrOA} is similar to the natural jet behavior; note that due to numerical discrepancies at the domain boundaries (see \citet{Torrence1998} for a discussion of the `cone of influence' of wavelet coefficients and the effect thereof), the correlation values have been truncated at the most upstream and downstream microphones.
For the impulsively-excited jet, nearly identical correlation regions are observed between the excited and natural jet; in the periodically-excited jet continuous oscillations occur throughout time due to the similarity of continuously and periodically generated large-scale structures and resultant acoustic radiation.
In the upstream region of the jet, the peaks of the positive correlation region match $\tau_a$ nearly exactly. 
In the downstream region, $\tau_a$ begins to increasingly over-predict the time lag for the maximum correlation. 
On the other hand, $\tau_s$ tracks the time lags for the peak correlation consistently over the downstream region, but not the upstream region. 
The results found here appear to indicate that the dominant acoustic radiation reaching the far-field aft angles is being generated over an extended region of the jet mixing layer, roughly $x/D \leq 4$, which is just upstream of the time-averaged end of the potential core in the natural jet.
This is not too dissimilar from the findings of other researchers, who have suggested that the acoustic source region lies just \textit{downstream} of the end of the potential core \citep{Hileman2005}.
It should be clarified here though, that the interpretation of these results is not meant to suggest that only trivial levels of noise are generated outside of this apparent noise source region, just that the dominant radiation is produced in this region in a time-averaged sense.
\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{Figures/sect_nearfield_ffxcorr.png}
	\caption{Normalized two-point correlations between the acoustic component of the near field and the far field at $30^\circ$ for microphone array position starting at $x/D = 1, r/D = 1.2$ for the natural jet (a), $St_{DF} = 0.05$ (b), $St_{DF} = 0.25$ (c) and $St_{DF} = 0.35$ (d).}
	\label{fig:ch3_xcorrOA}
\end{figure}

However, these results should not be interpreted as indicating that the source mechanisms are necessarily consistent for all excitation frequencies. 
For the lower-frequency periodic excitation ($St_{DF} \leq 0.25$), the consistency in the far-field response (\fig{fig:ch3_farfield}b) coupled with the consistency in the apparent source region is suggestive of a consistent dominant source mechanism.
In contrast, the inconsistency in the far-field response for the higher-frequency periodic excitation ($St_{DF} \geq 0.35$, \fig{fig:ch3_farfield_nonlinear}) is suggestive of a change in the dominant source mechanism, just one that is associated with the vortex dynamics in the jet shear layer upstream of the end of the potential core.
It should also be noted that the peak correlation values between the acoustic near-field and the far-field are significantly lower (though certainly non-negligible) in the periodic excitation cases, suggesting a decaying coherence in the source mechanisms at these frequencies.
From the current results alone, the significance of this estimated source region is not entirely clear.
In \sect{sect:velocity} the time-resolved velocity field will be explored in detail to better elucidate the structure dynamics, with a particular focus on the region just upstream of the end of the potential core.

A special note on the differences concerning the results presented in \fig{fig:ch3_xcorrOA} and those presented by \citet{Crawley2015} is warranted here.
In that paper, the more simple Fourier filter was used to decompose the irrotational near-field; processing artifacts were noted and a parametric study was attempted to minimize their impact.
In the resulting two-point correlations of the decomposed acoustic field, a shift in the apparent source region was noted to coincide with the shift of the peak pressure fluctuations measured just outside the shear layer (higher frequency excitation cases saturating further upstream near the nozzle exit).
Because this behavior was observed across the entire range of filter parameters used, it was assumed to be representative of the true physical behavior and not a numerical artifact.
Of course, this assumption precludes the possibility that \textit{the entire parameter space produced similar numerical artifacts}. 
As discussed more thoroughly in \citet{Crawley2016}, the Fourier filter has a tendency to allow energy leakage from the hydrodynamic field into the acoustic, particularly at low frequencies. 
Since it has already been observed that the hydrodynamic signature of the large-scale structures can linearly correlate to the acoustic emission, a potential consequence of this leakage is correlation regions which instead point to the region of high hydrodynamic energy - i.e. the saturation point of the near-field pressure fluctuations.
The analysis found of \citet{Crawley2016} demonstrated that wavelet filter is far more robust to energy leakage and numerical artifacts, and as such the authors is inclined to lend more credence to the results presented herein.