Azimuthal Fourier Modes Research Articles

Causal analysis among azimuthal Fourier modes in linear plasma based on multivariate time series models

This study investigates the causal relationships among azimuthal Fourier modes in linear plasma turbulence using multivariate time series models. We elucidate the dynamics of mode interactions in magnetized plasmas by employing the vector autoregressive model and Granger causality analysis. Our analysis, based on data from the plasma assembly for nonlinear turbulence analysis, reveals significant variations in causality with changing pressure conditions. Modes form weakly coupled clusters at lower pressures, while higher pressures lead to stronger coupling and larger clusters. The impulse response function further provides insights into the temporal propagation and nature of influences between modes. These findings enhance the understanding of spatial pattern formation in magnetized plasmas and offer a quantitative framework for analyzing plasma turbulence dynamics.

Horizontally polarized kink oscillations supported by solar coronal loops in an asymmetric environment

Context. Kink oscillations are ubiquitously observed in solar coronal loops, and understanding them is crucial in the contexts of coronal seismology and atmospheric heating. Aims. We studied kink modes supported by a straight coronal loop embedded in an asymmetric environment using 3D magnetohydrodynamic simulations. Methods. We implemented the asymmetric effect by setting different exterior densities below and above the loop interior and initiated the simulation using a kink-like velocity perturbation perpendicular to the loop plane, mimicking the frequently measured horizontally polarized kink modes. Results. We find that the external velocity fields show fan-blade structures propagating in the azimuthal direction as a result of the successive excitation of higher azimuthal Fourier modes. Resonant absorption and phase-mixing can still occur despite an asymmetric environment, leading to the development of small-scale structures at loop boundaries. These small-scale structures nonetheless develop asymmetrically at the upper and lower boundaries due to the different gradients of the Alfvén speed. Conclusions. These findings enrich our understanding of kink modes in coronal loops embedded within an asymmetric environment, providing insights that will be helpful for future high-resolution observations.

Black Hole Polarimetry I. A Signature of Electromagnetic Energy Extraction

In 1977, Blandford and Znajek showed that the electromagnetic field surrounding a rotating black hole can harvest its spin energy and use it to power a collimated astrophysical jet, such as the one launched from the center of the elliptical galaxy M87. Today, interferometric observations with the Event Horizon Telescope (EHT) are delivering high-resolution, event-horizon-scale, polarimetric images of the supermassive black hole M87* at the jet launching point. These polarimetric images offer an unprecedented window into the electromagnetic field structure around a black hole. In this paper, we show that a simple polarimetric observable—the phase ∠β 2 of the second azimuthal Fourier mode of the linear polarization in a near-horizon image—depends on the sign of the electromagnetic energy flux and therefore provides a direct probe of black hole energy extraction. In Boyer–Lindquist coordinates, the Poynting flux for axisymmetric electromagnetic fields is proportional to the product B ϕ B r . The phase ∠β 2 likewise depends on the ratio B ϕ /B r , thereby enabling an observer to determine the direction of electromagnetic energy flow in the near-horizon environment experimentally. Data from the 2017 EHT observations of M87* are consistent with electromagnetic energy outflow. Currently envisioned multifrequency observations of M87* will achieve higher dynamic range and angular resolution, and hence deliver measurements of ∠β 2 closer to the event horizon as well as better constraints on Faraday rotation. Such observations will enable a definitive test for energy extraction from the black hole M87*.

Super-resolution of time-resolved three-dimensional density fields of the B mode in an underexpanded screeching jet

The estimation of time-resolved three-dimensional (3D) density fields of an underexpanded jet at the nozzle pressure ratio of 2.42, a so-called “spatiotemporal super-resolution” was conducted using non-time-resolved three-dimensional background-oriented schlieren (3D-BOS) and time-resolved microphone measurements. This approach aims to reconstruct three-dimensional density fields associated with the intermittent and switching behavior of the B mode of a screeching jet from the microphone data by constructing a linear regression model. An azimuthal Fourier decomposition is applied to the 3D-BOS and microphone data, and the proper orthogonal decomposition (POD) is performed for each of their azimuthal Fourier modes. The m=1 azimuthal Fourier mode is dominant in both cases, and the leading two POD modes in the m=1 azimuthal mode of the microphone data are associated with the B mode. The linear regression model is constructed from the POD modes of the m=1 azimuthal 3D-BOS data and the first two microphone POD modes of the m=1 azimuthal mode of the microphone data. The three-dimensional density fields reconstructed from each POD mode of the m=1 azimuthal mode of the microphone data have helical structures with opposite rotation directions. The amplitudes of those POD modes change with time, and the azimuthal structure associated with the B mode is determined depending on those amplitudes. The present result showed that intermittency in the flapping to helical structures and their strength can be interpreted by the temporal changes in the strengths of two rotating helical structures with opposite rotation directions.

Dynamics of tilted atmospheric vortices under asymmetric diabatic heating

Päschke et al. (J Fluid Mech, 2012) studied the nonlinear dynamics of strongly tilted vortices subject to asymmetric diabatic heating by asymptotic methods. They found, inter alia, that an azimuthal Fourier mode 1 heating pattern can intensify or attenuate such a vortex depending on the relative orientation of the tilt and the heating asymmetries. The theory originally addressed the gradient wind regime which, asymptotically speaking, corresponds to vortex Rossby numbers of order unity in the limit. Formally, this restricts the applicability of the theory to rather weak vortices. It is shown below that said theory is, in contrast, uniformly valid for vanishing Coriolis parameter and thus applicable to vortices up to low hurricane strengths. An extended discussion of the asymptotics as regards their physical interpretation and their implications for the overall vortex dynamics is also provided in this context. The paper’s second contribution is a series of three-dimensional numerical simulations examining the effect of different orientations of dipolar diabatic heating on idealized tropical cyclones. Comparisons with numerical solutions of the asymptotic equations yield evidence that supports the original theoretical predictions of Päschke et al. In addition, the influence of asymmetric diabatic heating on the time evolution of the vortex centerline is further analyzed, and a steering mechanism that depends on the orientation of the heating dipole is revealed. Finally, the steering mechanism is traced back to the correlation of dipolar perturbations of potential temperature, induced by the vortex tilt, and vertical velocity, for which diabatic heating not necessarily needs to be responsible, but which may have other origins.

Effects of targeted wall geometries on response of turbulent pipe flow at high Reynolds number

The three-dimensional turbulent pipe flow response to targeted wall conditions was numerically examined at a high Reynolds number of 1.58×105. The perturbed wall was designed based on azimuthal Fourier modes of m=3 (Case I) and m=15 (Case II), as well as their superposition at m∈3+15 (Case III). The mean flow and turbulent fields were investigated and compared, which depicted long-lasting changes in flow properties. A rapid decay of turbulence levels immediately past the perturbation was observed for all cases. However, Case I showed a faster overall recovery rate compared to the higher Fourier Mode cases. While the flow recovered by 19D in Case I, the higher Fourier Mode wall shapes (Cases II and III) showed a longer recovery length at 45D. The flow response also differed between these wall shapes. While Case I depicted a monotonic response, the other two wall shapes exhibited a non-monotonic second-order response characteristic along with a delayed recovery trend. The second-order response of turbulence was also evident in the Reynolds stress variations for cases with a higher Fourier mode wall shapes. The overshoot of Reynolds stresses, accompanied by their rapid decay rate downstream of the modified wall shape, was observed to follow known second-order turbulent response in pipe flow. Dominant flow structures were further identified in the downstream wake of each insert which provided a qualitative description of potential mechanisms behind the observed recovery trends.

An azimuthal Fourier domain formulation of the Ffowcs Williams and Hawkings equation.

The acoustic field of high-speed turbulent jets is dominated by a small number of low-wavenumber azimuthal Fourier modes. Accordingly, it is of interest to directly obtain individual azimuthal modes of the acoustic field from simulation data or models of the jet near-field. To this end, we manipulate the Ffowcs Williams and Hawkings (FW-H) equation to obtain a new formulation that lives entirely in the azimuthal Fourier domain-it delivers individual azimuthal modes of the acoustic field as a function of the same azimuthal modes of the near-field FW-H source term or, upon linearization of the source terms, as a function of the same azimuthal modes of the near-field flow variables. As an added benefit, all surface integrals are converted into line integrals in the streamwise-radial plane. After verifying and validating our formulation using a monopole problem with an exact solution and large-eddy simulation data, respectively, we show how our method can be used to efficiently and naturally compute the acoustic field associated with resolvent modes of a Mach 1.5 jet, thus avoiding the need to compute the modes on a large computational domain to capture their acoustic radiation.

Representing rectangular jet dynamics through azimuthal Fourier modes

Rectangular propulsion nozzles have advantages over circular nozzles, including easier thrust-vectoring and air-frame-integration. Jet noise is easier to study with circular jets (CJ), however, due to azimuthal homogeneity, which, together with low-rank acoustic dynamics enables simpler acoustic models. Using Large Eddy Simulations of rectangular jets (RJ) of various aspect ratios we show that acoustic fluctuation components exhibit comparably rapid convergence in azimuthal Fourier space even for high aspect ratios. A reduced-order model for RJ that retains near-field acoustic asymmetry can be constructed using only three leading azimuthal modes, but with two additional terms relative to CJ.

Response of turbulent pipe flow to targeted wall shapes at a range of Reynolds numbers

The response and recovery of turbulent pipe flow to three-dimensional perturbed wall changes were examined numerically in a wide range of Reynolds numbers between Re=5×103 and 1.58×105. The perturbations were based on distinct azimuthal Fourier modes corresponding to m = 3, 15, and 3 + 15. The long-lasting response of the flow was examined by characterizing both the mean and turbulent field in the wake of pipe inserts for each Re. The variation of the recovery with increasing Reynolds number revealed an asymptotic behavior for Re≥7.5×104, which scaled with Re4 for both mean velocity and turbulence kinetic energy. Two peaks were observed for the mean velocity along the wake centerline, where the location of peaks followed a power-law trend in the form of Lp/D∝Re4/3, where D is the pipe diameter. A fast decay of turbulence past the wall change further suggested that maximum Reynolds shear stress in the downstream wake decays as (x/D)−1/3 for all Re. The flow also exhibited long-lasting responses that obstructed its relaxation at 20D downstream of the perturbation, even for low Re of 5×103. Overall, the recovery exhibited a second-order response.

Azimuthal decomposition of the radiated noise from supersonic shock-containing jets.

Acoustic measurements of unheated supersonic underexpanded jets with ideally expanded Mach numbers of 1.14, 1.38, and 1.50 are presented. Of the three components of supersonic jet noise, the focus is on the broadband shock-associated noise (BBSAN) component. Motivated by the modelling of BBSAN using the wavepacket framework, a traversable microphone ring is used to decompose the acoustic pressure into azimuthal Fourier modes. Unlike noise radiated downstream, BBSAN is dominated by azimuthal modes 1-3, which are approximately 3-4 dB/St stronger than the axisymmetric component. Crucially, the relative contribution of successive modes to BBSAN is sensitive to the observer angle and jet operating condition. Four azimuthal modes are necessary to reconstruct the total BBSAN signal to within 1 dB/St accuracy for the conditions presented here. The analysis suggests, however, that the number of modes required to maintain this accuracy increases as the peak frequency shifts upward. The results demonstrate the need to carefully consider the azimuthal content of BBSAN when comparing acoustic measurements to predictions made by jet noise models built on instability theory.

Azimuthal capillary waves on a hollow filament – the discrete and the continuous spectrum

We study the temporal spectrum of linearised, azimuthal, interfacial perturbations imposed on a cylindrical gaseous filament surrounded by immiscible, viscous, quiescent fluid in radially unbounded geometry. Linear stability analysis shows that the base state is stable to azimuthal perturbations of standing wave form. Normal mode analysis leads to a viscous dispersion relation and shows that in addition to the discrete spectrum, the problem also admits a continuous spectrum. For a given azimuthal Fourier mode and Laplace number, the discrete spectrum yields two eigenfunctions which decay exponentially to zero at large radii and thus cannot represent far field perturbations. In addition to these discrete modes, we find an uncountably infinite set of eigenmodes which decay algebraically to zero. The completeness theorem for perturbation vorticity may be expressed as a sum over the discrete modes and an integral over the continuous ones. We validate our normal mode results by solving the linearised, initial value problem (IVP). The initial perturbation is taken to be an interfacial, azimuthal Fourier mode with zero perturbation vorticity. It is shown that the expression for the time dependent amplitude of a capillary standing wave (in the Laplace domain, ) has poles and branch points on the complex plane. We show that the residue at the poles yields the discrete spectrum, while the contribution from either side of the branch cut provides the continuous spectrum contribution. The particular initial condition treated here in the IVP, has projections on the discrete as well as the continuous spectrum eigenmodes and thus both sets are excited initially. Consequently the time evolution of the standing wave amplitude and the perturbation vorticity field have the form of a sum over discrete exponential contributions and an integral over a continuous range of exponential terms. The solution to the IVP leads to explicit analytical expressions for the standing wave amplitude and the vorticity field in the fluid outside the filament. Linearised analytical results are validated using direct numerical simulations (DNS) conducted using a code developed in-house for solving the incompressible, Navier–Stokes equations with an interface. For small perturbation amplitude, analytical predictions show excellent agreement with DNS. Our analysis complements and extends earlier results on the discrete and the continuous spectrum for interfacial viscous, capillary waves on unbounded domain.

Large-scale streaky structures in turbulent jets

Streaks have been found to be an important part of wall-turbulence dynamics. In this paper, we extend the analysis for unbounded shear flows, in particular a Mach 0.4 round jet, using measurements taken using dual-plane, time-resolved, stereoscopic particle image velocimetry (PIV) taken at pairs of jet cross-sections, allowing the evaluation of the cross-spectral density of streamwise velocity fluctuations resolved into azimuthal Fourier modes. From the streamwise velocity results, two analyses are performed: the evaluation of wavenumber spectra (assuming Taylor’s hypothesis for the streamwise coordinate) and a spectral proper orthogonal decomposition (SPOD) of the velocity field using PIV planes in several axial stations. The methods complement each other, leading to the conclusion that large-scale streaky structures are also present in turbulent jets where they experience large growth in the streamwise direction, energetic structures extending up to eight diameters from the nozzle exit. Leading SPOD modes highlight the large-scale, streaky shape of the structures, whose aspect ratio (streamwise over azimuthal length) is approximately 15. The data were further analysed using SPOD, resolvent and transient growth analyses, good agreement being observed between the models and the leading SPOD mode for the wavenumbers considered. The models also indicate that the lift-up mechanism is active in turbulent jets, with streamwise vortices leading to streaks. The results show that large-scale streaks are a relevant part of the jet dynamics.

Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets

To investigate the effects of the nozzle-exit conditions on jet flow and sound fields, large-eddy simulations of an isothermal Mach 0.9 jet issued from a convergent-straight nozzle are performed at a diameter-based Reynolds number of $1\times 10^{6}$. The simulations feature near-wall adaptive mesh refinement, synthetic turbulence and wall modelling inside the nozzle. This leads to fully turbulent nozzle-exit boundary layers and results in significant improvements for the flow field and sound predictions compared with those obtained from the typical approach based on laminar flow in the nozzle. The far-field pressure spectra for the turbulent jet match companion experimental measurements, which use a boundary-layer trip to ensure a turbulent nozzle-exit boundary layer to within 0.5 dB for all relevant angles and frequencies. By contrast, the initially laminar jet results in greater high-frequency noise. For both initially laminar and turbulent jets, decomposition of the radiated noise into azimuthal Fourier modes is performed, and the results show similar azimuthal characteristics for the two jets. The axisymmetric mode is the dominant source of sound at the peak radiation angles and frequencies. The first three azimuthal modes recover more than 97 % of the total acoustic energy at these angles and more than 65 % (i.e. error less than 2 dB) for all angles. For the main azimuthal modes, linear stability analysis of the near-nozzle mean-velocity profiles is conducted in both jets. The analysis suggests that the differences in radiated noise between the initially laminar and turbulent jets are related to the differences in growth rate of the Kelvin–Helmholtz mode in the near-nozzle region.

Mach-number scaling of individual azimuthal modes of subsonic co-flowing jets

The Mach-number scaling of the individual azimuthal modes of jet mixing noise was studied for jets in flight conditions, i.e. with co-flow. The data were obtained via a series of direct numerical simulations (DNS), performed of fully turbulent jets with a target Reynolds number, based on nozzle diameter, of $Re_{jet}=8000$. The DNS included a pipe 25 diameters in length in order to ensure that the flow developed to a fully turbulent state before exiting into a laminar co-flow, and to account for all possible noise generation mechanisms. To allow for a detailed study of the jet mixing noise component of the combined pipe–jet configuration, acoustic liner boundary conditions on the inside of the pipe and a modification to the synthetic turbulent inlet boundary condition of the pipe were applied to minimize internal noise in the pipe. Despite these measures, the use of a phased-array source breakdown technique was essential in order to isolate the sources associated with jet noise mechanisms from additional noise sources that could be attributed to internal noise or unsteady flow past the nozzle lip, in particular for the axisymmetric mode. Decomposing the sound radiation from the pipe–jet configuration into its azimuthal Fourier modes, and accounting for the co-flow effects, it was found that at $90^{\circ }$ the individual azimuthal Fourier modes of far-field pressure for the jet mixing noise component exhibit the same $M^{8}$ scaling with the centreline jet Mach number as that experimentally documented for the overall noise field. Applying the phased-array source breakdown code to the DNS data at smaller angles to the jet axis, an increase of the velocity exponent of the jet noise source was found, approaching 10 at $30^{\circ }$. At this smaller angle the higher azimuthal modes again showed the same behaviour as the axisymmetric mode.

A parallelized multidomain compact solver for incompressible turbulent flows in cylindrical geometries

We present an efficient parallelized multidomain algorithm for solving the 3D Navier–Stokes equations in cylindrical geometries. The numerical method is based on fourth-order compact schemes in the two non-homogeneous directions and Fourier series expansion in the azimuthal direction. The temporal scheme is a second-order semi-implicit projection scheme leading to the solution of five Helmholtz/Poisson equations. To handle the singularity appearing at the axis in cylindrical coordinates, while being able to have a thinner or conversely a coarser mesh in this zone, parity conditions are imposed at r=0 for each flow variable and azimuthal Fourier mode. To simulate flows in irregularly shaped cylindrical geometries and benefit from a hybrid OpenMP/MPI parallelization, an accurate perfectly free-divergence multidomain method based on the influence matrix technique is proposed. First, the accuracy of the present solver is checked by comparison with analytical solutions and the scalability is then evaluated. Simulations using the present code are then compared to reliable experimental and numerical results of the literature showing good quantitative agreements in the cases of the axisymmetric and 3D unsteady vortex breakdowns in a cylinder and turbulent pipe flow. Finally to show the capability of the algorithm to deal with more complex flows relevant of turbomachineries, the turbulent flow inside a simplified stage of High-Pressure compressor is considered.

Acoustic scattering from a water-filled cylindrical shell: measurements, modeling, and interpretation.

Understanding the physics governing the interaction of sound with targets in an underwater environment is essential to improving existing target detection and classification algorithms. To illustrate techniques for identifying the key physics, an examination is made of the acoustic scattering from a water-filled cylindrical shell. Experiments were conducted that measured the acoustic scattering from a water-filled cylindrical shell in the free field, as well as proud on a sand-water interface. Two modeling techniques are employed to examine these acoustic scattering measurements. The first is a hybrid 2-D/3-D finite element (FE) model, whereby the scattering in close proximity to the target is handled via a 2-D axisymmetric FE model, and the subsequent 3-D propagation to the far field is determined via a Helmholtz integral. This model is characterized by the decomposition of the fluid pressure and its derivative in a series of azimuthal Fourier modes. The second is an analytical solution for an infinitely long cylindrical shell, coupled with a simple approximation that converts the results to an analogous finite length form function. Examining these model results on a mode-by-mode basis offers easy visualization of the mode dynamics and helps distinguish the different physics driving the target response.

Wavepackets in the velocity field of turbulent jets

AbstractWe study the velocity fields of unforced, high Reynolds number, subsonic jets, issuing from round nozzles with turbulent boundary layers. The objective of the study is to educe wavepackets in such flows and to explore their relationship with the radiated sound. The velocity field is measured using a hot-wire anemometer and a stereoscopic, time-resolved PIV system. The field can be decomposed into frequency and azimuthal Fourier modes. The low-angle sound radiation is measured synchronously with a microphone ring array. Consistent with previous observations, the azimuthal wavenumber spectra of the velocity and acoustic pressure fields are distinct. The velocity spectrum of the initial mixing layer exhibits a peak at azimuthal wavenumbers $m$ ranging from 4 to 11, and the peak is found to scale with the local momentum thickness of the mixing layer. The acoustic pressure field is, on the other hand, predominantly axisymmetric, suggesting an increased relative acoustic efficiency of the axisymmetric mode of the velocity field, a characteristic that can be shown theoretically to be caused by the radial compactness of the sound source. This is confirmed by significant correlations, as high as 10 %, between the axisymmetric modes of the velocity and acoustic pressure fields, these values being significantly higher than those reported for two-point flow–acoustic correlations in subsonic jets. The axisymmetric and first helical modes of the velocity field are then compared with solutions of linear parabolized stability equations (PSE) to ascertain if these modes correspond to linear wavepackets. For all but the lowest frequencies close agreement is obtained for the spatial amplification, up to the end of the potential core. The radial shapes of the linear PSE solutions also agree with the experimental results over the same region. The results suggests that, despite the broadband character of the turbulence, the evolution of Strouhal numbers $0. 3\leq St\leq 0. 9$ and azimuthal modes 0 and 1 can be modelled as linear wavepackets, and these are associated with the sound radiated to low polar angles.

Acoustic scattering from a water-filled cylindrical shell: Mode identification and interpretation via finite element and analytical models

Understanding the physics governing the interaction of sound with targets in an underwater environment is essential to improving upon existing target detection and classification algorithms. Simple models are viable tools for meaningful interpretation of scattering results. To illustrate this, two modeling techniques are employed to study the acoustic scattering from a water-filled cylindrical shell. The first model is a hybrid 2-D/3-D finite element (FE) model, whereby the scattering in close proximity to the target is handled via a 2-D axisymmetric FE model, and the subsequent 3-D propagation to the farfield is determined via a Helmholtz integral. This model is characterized by the decomposition of the fluid pressure and its derivative in a series of azimuthal Fourier modes, a technique that has previously facilitated mode identification [Espana et al., J. Acoust. Soc. Am. 130, 2332 (2011)]. The second is an analytical solution for an infinitely long cylindrical shell, coupled with a simple approximation that converts the results to an analogous finite length form function. These two model results, when examined together on a mode-by-mode basis, offer visualization of the mode dynamics and the ability to distinguish the different physics driving the target response (i.e., structural modes versus water-waveguide modes). [Work supported by ONR.]

Linear stability analysis of cylindrical Rayleigh–Bénard convection

AbstractThe instabilities and transitions of flow in a vertical cylindrical cavity with heated bottom, cooled top and insulated sidewall are investigated by linear stability analysis. The stability boundaries for the axisymmetric flow are derived for Prandtl numbers from 0.02 to 1, for aspect ratio $A$ ($A= H/ R= \mathrm{height} / \mathrm{radius} $) equal to 1, 0.9, 0.8, 0.7, respectively. We found that there still exists stable non-trivial axisymmetric flow beyond the second bifurcation in certain ranges of Prandtl number for $A= 1$, $0. 9$ and 0.8, excluding the $A= 0. 7$ case. The finding for $A= 0. 7$ is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed, where five kinds of steady modes $m= 1, 2, 8, 9, 10$ and three kinds of oscillatory modes $m= 3, 4, 6$ are presented. These multiple modes indicate different flow structures triggered at the transitions. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that the radial or axial shear of base flow combined with buoyancy mechanism leads to the instability results.

Axisymmetric superdirectivity in subsonic jets

AbstractWe present experimental results for the acoustic field of jets with Mach numbers between 0.35 and 0.6. An azimuthal ring array of six microphones, whose polar angle, $\theta $, was progressively varied, allows the decomposition of the acoustic pressure into azimuthal Fourier modes. In agreement with past observations, the sound field for low polar angles (measured with respect to the jet axis) is found to be dominated by the axisymmetric mode, particularly at the peak Strouhal number. The axisymmetric mode of the acoustic field can be clearly associated with an axially non-compact source, in the form of a wavepacket: the sound pressure level for peak frequencies is found be superdirective for all Mach numbers considered, with exponential decay as a function of $ \mathop{ (1\ensuremath{-} {M}_{c} \cos \theta )}\nolimits ^{2} $, where ${M}_{c} $ is the Mach number based on the phase velocity ${U}_{c} $ of the convected wave. While the mode $m= 1$ spectrum scales with Strouhal number, suggesting that its energy content is associated with turbulence scales, the axisymmetric mode scales with Helmholtz number – the ratio between source length scale and acoustic wavelength. The axisymmetric radiation has a stronger velocity dependence than the higher-order azimuthal modes, again in agreement with predictions of wavepacket models. We estimate the axial extent of the source of the axisymmetric component of the sound field to be of the order of six to eight jet diameters. This estimate is obtained in two different ways, using, respectively, the directivity shape and the velocity exponent of the sound radiation. The analysis furthermore shows that compressibility plays a significant role in the wavepacket dynamics, even at this low Mach number. Velocity fluctuations on the jet centreline are reduced as the Mach number is increased, an effect that must be accounted for in order to obtain a correct estimation of the velocity dependence of sound radiation. Finally, the higher-order azimuthal modes of the sound field are considered, and a model for the low-angle sound radiation by helical wavepackets is developed. The measured sound for azimuthal modes 1 and 2 at low Strouhal numbers is seen to correspond closely to the predicted directivity shapes.