Elsevier

Journal of Sound and Vibration

Volume 330, Issues 18–19, 29 August–12 September 2011, Pages 4474-4492
Journal of Sound and Vibration

Jittering wave-packet models for subsonic jet noise

https://doi.org/10.1016/j.jsv.2011.04.007Get rights and content

Abstract

Three simplified wave-packet models of the coherent structures in subsonic jets are presented. The models comprise convected wave-packets with time-dependent amplitudes and spatial extents. The dependence of the radiated sound on the temporal variations of the amplitude and spatial extent of the modulations are studied separately in the first two model problems, being considered together in the third. Analytical expressions for the radiated sound pressure are obtained for the first and third models.

Results show that temporally localised changes in the wave-packet can lead to radiation patterns which are directional and which comprise high-amplitude bursts; such intermittency is observed in subsonic jets at the end of the potential core, and so the models may help explain the higher noise levels and intermittent character of the sound radiated to low emission angles for subsonic jets. By means of an efficiency metric, relating the radiated acoustic power to the fluctuation energy of the source, we show that the source becomes more powerful as its temporal localisation is increased. This result extends that of Sandham et al. (Journal of Sound and Vibration 294(1) (2006) 355–361) who found similar behaviour for an infinitely extended wavy-wall.

The pertinence of the model is assessed using two sets of data for a Mach 0.9 jet. One corresponds to a direct numerical simulation (DNS) of a Reynolds number 3600 turbulent jet and the other to a large eddy simulation (LES) of a Reynolds number 4×105 jet. Both time-averaged and time-dependent amplitudes and spatial extents are extracted from the velocity field of the numerical data. Computing the sound field generated by the wave-packet models we find for both simulations that while the wave-packet with a time-averaged envelope shows discrepancies of more than an order of magnitude with the sound field, when the wave-packet ‘jitters’ in a way similar to the intermittency displayed by the simulations, we obtain agreement to within 1.5 dB at low axial angles. This shows that the ‘jitter’ of the wave-packet is a salient source feature, and one which should be modelled explicitly.

Highlights

► Three wave-packet models to coherent structures in jets are proposed. ► Models include jitter in the amplitude and the spatial extent of the wave-packet. ► Analytical results for the first and the third models. ► Jitter increases sound radiation for subsonic flows. ► Agreement of 1.5 dB is obtained with numerical databases of Mach 0.9 jets.

Introduction

Estimation of the sound radiation from a turbulent flow, using an acoustic analogy, requires the solution of a propagation equation, assuming some given form for a corresponding source term. This form must be such that the sound producing kinematic structure of the turbulence is approximated in a physically pertinent manner. When Lighthill first provided us with a theoretical foundation from which to model, study and understand jet noise, turbulence, both generally and in the specific case of the round jet, was considered to comprise a space–time chaos, devoid of any underlying order. The standard at that time for the kinematic description of turbulence structure could be found in turbulence theories such as that of Batchelor [1]: Attempts to understand and model turbulence were based on the Reynolds averaged Navier–Stokes equations, where the only conceptual constructs invoked, aside from those expressed in the conservation equations, are those required for closure (Boussinesq's notion of eddy viscosity, for instance) on one hand, and, on the other, the flow entities supposed to participate in the physical process associated with the various terms that appear in the RANS equations: fluctuation energy is ‘produced’, ‘transported’, ‘dissipated’ by virtue of interactions between stochastic flow ‘scales’ or ‘eddies’. Source terms in acoustic analogies were constructed in accordance with this conceptual picture of turbulence. Lighthill assumed a statistical distribution of uncorrelated eddies throughout the source region; and this led to the well known U8 power law for an isothermal turbulent jet [2].

However, predictions based on Lighthill's analogy, using such kinematical models for the turbulence, do not explain all of the features of subsonic jet noise: at low emission angles (with respect to the downstream jet axis) the U8 power law does not hold for example, and the narrower spectral shape is generally not well predicted. This of course does not necessarily imply that there is anything wrong with the acoustic analogy as a theoretical framework for jet noise, as is sometimes suggested in the literature. It simply means that the models used for the source and propagator have not been adequately appropriated to the physical mechanisms which are at work in a turbulent jet.

An acoustic analogy constitutes an exact rearrangement of the fluid mechanics equations. If a flow solution is known exactly, an accurate calculation of the sound field can be obtained, regardless of the source–propagator split. This has been demonstrated by means of direct numerical simulation (hereafter DNS). Colonius et al. [3], for instance, calculated sound generation by a mixing layer using Lilley's analogy, Freund [4] used Lighthill's analogy to determine the acoustic field of a Mach 0.9 jet. More recently both Samanta et al. [5] and Sinayoko et al. [6] have used Goldstein's [7] generalisation of the acoustic analogy to explore other source–propagator splits. In all cases the correct sound field can be retrieved.

However, despite the possibility of accurate calculation of the acoustic fields, the source terms obtained from DNS have not, to date, clearly revealed which features of a turbulent jet underpin the production of sound. It is therefore not clear how the turbulence of a jet should be manipulated and modified in order to reduce noise. Some broad guidelines can of course be provided. The solution of an acoustic analogy will place the far-field sound energy in relation to certain turbulence quantities, such as the mean velocity field, the integral space- and time-scales, convection velocities, etc. And we thereby have an indication of how changes in these turbulence quantities will impact the radiated sound power. But we do not currently have tools capable of clarifying what kind of actuation, or design modification, will lead to a desired change in such time-averaged quantities; the space- and time-local physical processes which underpin these statistical measures are not clearly understood, and so we are generally obliged to operate on a trial and error basis. The evolution of fluid mechanics, and aeroacoustics, will see the implementation of real-time, closed-loop control; and the aforesaid time-averaged quantities will be of little use in this context. It will be necessary to directly model, both kinematically and dynamically, the space- and time-local characteristics of flow events which generate sound. This paper, and others by the first two authors, are motivated by this necessity.

Efforts to understand and model the space- and time-local behaviour of turbulence would be in vain were it not for the underlying order which is now known to exist in most turbulent flows: soon after the first attempts by Lighthill and his successors to predict the sound radiated by turbulent jets a change occurred in the way turbulence is perceived. Turbulent flows were observed to be more ordered than had previously been believed, and a new conceptual flow entity was born, sometimes referred to as a ‘coherent structure’, or, alternatively, a ‘wave-packet’. Mollo-Christensen was one of the first to report such order in the case of the round jet: ‘…although the velocity signal is random, one should expect to see intermittently a rather regular spatial structure in the shear layer’ [8]. A series of papers followed, confirming these observations and postulating on the nature of this order ([9], [10], [11] to cite just a few). Stability theory was evoked as a possible theoretical framework for the modelling of such flow behaviour, and kinematical and/or dynamical models for coherent structures, based on ideas derived from stability theory, can be found, for instance, in the works of Michalke [12], [13], Crow [14], Crigthon and Gaster [15], Tam and Morris [16], Mankbadi and Liu [17] and Crighton and Huerre [18]: coherent turbulent structures are modelled by means of a hydrodynamic stability analysis of the mean flow; for slowly spreading mean flows solutions comprise waves which amplify spatially and then decay. From the point of view of sound production, the salient features of such flow organisation, as identified by the said modelling efforts, are the process of amplification and decay, and the high level of axial coherence, which makes of these structures a non-compact, ‘semi-infinite antenna for sound’ [8]. This spatial modulation can help to explain, for instance, the results of Laufer and Yen [19], whose excited low Mach number jet showed superdirective radiation.

A further feature of the unsteadiness associated with the orderly part of a turbulent jet is its intermittency. The above citation from Mollo-Christensen recognises this. Crow and Champagne observed, by means of flow visualisation, the appearance of a train of coherent ‘puffs’ of turbulence. These were characterised by an average Strouhal number of 0.3, but the authors noted how ‘three or four puffs form and induct themselves downstream, an interval of confused flow ensues, several more puffs form, and so on’. Further experimental evidence of this kind of intermittency can be found in the work of Broze and Hussain [20], who observed that even when a forcing at St=0.4 is applied at the nozzle exit, significant jitter remains in the flow velocity signature.

An illustration of the space–time flow structure that we here loosely qualify as a jittering wave-packet is provided in Figs. 1(a) and (b). We see in Fig. 1(a) a space–time plot of the azimuthal mean of the streamwise component of the velocity, taken from the Mach 0.9 jet DNS of Freund [21]. A pattern of convected waves is observed from xD to x6D. These are characterised by some average frequency, but they undergo a modulation which is both spatial and temporal: the maximum amplitude of the wave changes in time, as does the position where it breaks down.

It is more difficult to access such information experimentally, as PIV systems do not currently provide sufficient temporal resolution. However, the near-field pressure contains the footprint of evanescent waves generated by convected hydrodynamic coherent structures, and nearfield microphone arrays (such as used by Picard and Delville [23], Suzuki and Colonius [24], Tinney and Jordan [22], Reba et al. [25]) can be used to measure this footprint. Fig. 1(b) shows the near pressure field of a coaxial jet measured by Tinney and Jordan [22]. A similar space–time structure to that observed in Fig. 1(a) DNS is observed.2

The analysis and/or calculation of sound radiation by such spatio-temporal structure can be performed in the frequency domain. The flow quantities are Fourier transformed in time, and the calculations are performed separately for each frequency component, which is periodic by definition and therefore cannot represent temporal jitter; the signature of jittering events is in this case spread across a range of frequencies. This approach is chosen, for instance, by Morris [26] and Reba et al. [25] and is perfectly adequate for the evaluation of source models and for sound prediction.

If, on the other hand, we are interested in determining the instantaneous features of a jet that generate sound (which closed-loop controllers will be required to manipulate in order to reduce noise), with a view to understanding flow mechanisms and thence constructing more sophisticated models, an analysis in the time domain can be advantageous, for it is no longer necessary to project the flow data onto infinitely extended basis functions. Such an approach proved useful in the analysis by Cavalieri et al. [27] of the noise-controlled mixing layers of Wei and Freund [28]. The noise reduction in the controlled flow was seen to be caused by a temporally localised action of the control; this prevented a space–time localised flow event and an associated space–time localised energy burst in the acoustic field. A number of other studies can be cited where time-domain analysis allowed a provision of insight which would not have been possible using spectral analysis: experimental studies include those of Juvé et al. [29], Guj et al. [30], Hileman et al. [31] and Kœnig et al. [32] for example; numerical examples can be found in the work of Kastner et al. [33] and Bogey and Bailly [34]. Fig. 2 shows a scalogram taken from Kœnig et al., where time-local high-amplitude energy bursts in the acoustic field at 30° can be seen at t0.323, 0.338, and 0.392 s.

In addition, therefore, to the spatial modulation evoked earlier, the ordered part of a turbulent jet is seen to comprise temporal modulation, or ‘jitter’. The only works, to the best of our knowledge, where such behaviour has been explicitly modelled, are those of Ffowcs Williams and Kempton [35] and Sandham et al. [36]. In the present work we wish to evaluate how space–time modulation such as that shown in Figs. 1(a) and (b) impacts the radiated sound field (Fig. 2). We follow the ideas of Ffowcs Williams and Kempton [35], who introduced some jitter in the phase speed of a convected wave, but maintained a fixed envelope function. Our contribution is in the evaluation of the effect of temporal variations of this envelope function, which are observable in Figs. 1(a) and (b). We propose three source models whose basic form is that of a convected instability wave, but one which undergoes modulation in both space and time; in this sense, our model includes both the characteristics of Crow's [14] and Sandham et al.'s [36] models.

In order to validate our models we have chosen to use Lighthill's acoustic analogy [2]. The simplicity of this analogy, compared for instance with those of Lilley [37] or Goldstein [7], means that we are able to obtain most of the results analytically, and this permits a comprehensive exploration of the parameter space of the jittering wave-packet models. Furthermore, the availability of an analytical time domain Green's function for Lighthill's analogy means that the effect of intermittent temporal variations can be obtained without the application of a Fourier transform, which might mask the intermittent bursts, as it did in the case of Wei and Freund's optimal controlled mixing-layer, by spreading their signature over a large frequency range [28], [27].

The work we pursue here is, largely, an exercise in system reduction. We are interested in the simplest possible representation of the jet, as a source of sound, but which provides an acceptably accurate prediction of the radiated sound field. This raises the question of optimal source description and robustness of the formulation used to describe the sound generation process; these two issues are closely related.

The question of the robustness of acoustic analogies is an interesting one, that has been addressed by Samanta et al. [5] in an ad hoc manner. The solutions of different acoustic analogies are calculated for the same direct numerical simulation of a two-dimensional mixing layer. The sound fields computed by all analogies show good agreement with the DNS. The flow fields are then artificially modified so as to introduce errors in the source q; this is done through a manipulation of the coefficients of POD modes of the full solution. Errors result in the calculation of the radiated sound; for most cases similar errors are obtained for the different formulations, but for one of the cases (division of the first POD mode coefficient by 2) Lighthill's analogy presents greater errors than the other formulations for the sound radiation in the upstream direction.

The problem can be thought of as follows. Consider the integral solution of an acoustic analogy, written in the general form Lq=p. The parameter space of the source q is expressed in terms of an orthonormal basis and possesses an inner product; such is the case, for instance, for the POD basis of Samanta et al. If we now consider the eventual impact of the introduction of a small disturbance (modification) to the source, δq (as done by Samanta et al. for a number of different analogies), we are interested in the impact that this will have on the acoustic field, i.e. δp. The problem comes down to the following situation: if δqL then the sound field will be sensitive to small perturbations in the source, δq. δq is in this case aligned with the direction of maximum sensitivity of the operator L in the parameter space considered. If, on the other hand, δqL, then changes in q will have no impact on the sound field, p.

This shows that the arbitrary introduction of disturbances to, and subsequent comparison of, two different analogies cannot provide an unambiguous assessment, in an absolute sense, of the relative robustness of the two formulations. For, if the gradients L1 and L2 have different directions in the parameter space, one will always be able to find a perturbation which causes one operator to appear less robust than the other. This shows that, while the study of Samanta et al. is of considerable interest as a first step in addressing this question of the robustness of acoustic analogies, it does not allow a conclusive evaluation in this regard; the said authors recognise that the conclusions of their work are particular to the studied flow and to the proposed disturbances.

The approach that we follow in the present paper amounts to an ad hoc, physics-based, filtering (or system reduction) of flow data such that directions in the source parameter space that are irrelevant for the peak sound radiation at low emission angles are removed. The system reduction is achieved in a number of ways. For example, by means of reasoning based on the source compactness we choose to suppress the radial dimension of the jet—this choice is associated with the Lighthill formulation of the problem; the choice of a convected wave ansatz, on the other hand, is based on observation (by us and past researchers), as is the wave-packet ‘jitter’. The success of the jittering wave-packet ansatz in reproducing the low angle radiation indicates that we have identified a robust source representation for the operator that we consider.

The paper is organised as follows. In Section 2 we obtain an analytical solution for the sound radiation of a wave-packet whose spatial and temporal modulations are given by Gaussian functions; Section 3 presents numerical results for a source model based on the reduction of the spatial extent of modulation during a certain period of time. Finally, we obtain in Section 4 an analytical expression for a wave-packet whose amplitude and spatial extent both change slowly with time, and a comparison of the model results with DNS (Freund [21]) and LES data (Daviller [38]; see also Cavalieri et al. [39]) is presented in Section 5.

We see that the intermittent wave-packet can produce directive sound radiation, such as is observed in jets. We furthermore show how intermittency increases the radiated power for a given fixed source fluctuation level. We then use velocity data extracted from the DNS and the LES to compare the sound radiation of two wave-packets: one whose amplitude and spatial extent is time-averaged and a second where the same quantities ‘jitter’ in the same way as velocity data taken from the simulations. The time-averaged wave-packet shows a discrepancy of more than an order of magnitude when compared with the DNS and the LES, while the ‘jittering’ wave-packet agrees to within 1.5 dB for downstream radiation. This illustrates how the ‘jittering’ of the ‘coherent structures’ of the flow is a salient source feature, and it suggests that, where simplified modelling strategies are concerned, an effort should be made to model this source parameter explicitly.

Finally, the role played by such wave-packet jitter in a supersonic scenario is assessed in Appendix C. We find that, while it is a salient feature for convectively subsonic flows, when the convective Mach number is greater than one the jitter ceases to be so important, as would be expected from an analysis in the frequency–wavenumber space.

Section snippets

Temporally localised amplitude change of a wave-packet

In Lighthill's analogy, the acoustic field is given, for a three-dimensional flow, asp(x,t)=14π1|xy|2Tijyiyj(y,τ)τ=t|xy|/cdy,where Tij is Lighthill's stress tensor. We wish to write down as simple an expression as possible to model the source term, but one which is capable of reproducing some of the features of jet noise at shallow angles to the jet axis. Thus we model Tij only with the T11 term since it can be shown [40] that the radiation in the far field comes only from the component

Temporally localised envelope truncation

In order to provide temporal changes in the spatial extent of the envelope function, we model T11 asT11(y,τ)=2ρ0Uu˜πD24δ(y2)δ(y3)ei(ωτky1)ey12/L2(τ).

With this expression the peak amplitude of the convected wave is kept constant, but the characteristic length of the envelope, L, changes with time. We model the changes in L asL(τ)=L0κe(ττ0)2/τL2,where L0 is an initial envelope width and κ is the maximum envelope reduction, which happens at τ=τ0. This reduction of the envelope happens during

Wave-packet with slowly varying amplitude and spatial extent

In order to account for both temporal changes in the wave-packet amplitude and axial extension, and to provide a framework where it is possible to fit the wave-packet parameters with numerical data, we now model the T11 component Lighthill's stress tensor asT11(y,τ)=2ρ0Uu˜πD24δ(y2)δ(y3)A(τ)ei(ωτky1)ey12/L2(τ),where we allow temporal variations of the amplitude A, as in Section 2, and also temporal changes in L, as in Section 3.

Using this expression in Eq. (1), with the far-field assumption,

Evaluation of the model using simulation data

In order to now assess the degree to which these simplified models can reproduce the main characteristics of the downstream radiation from subsonic jets we use numerical data of Mach 0.9 jets, taken from a direct numerical simulation of an unheated jet at Re=3600 [21] and from a large eddy simulation of a Mach 0.9 isothermal jet at Re=4×105 ([38]; see also [39]), to furnish the third model with instantaneous values for A and L. These are obtained from the azimuthal mean (the axisymmetric

Conclusion

An analytical expression for the radiated far-field pressure by a temporally localised wave-packet is presented, and results are found to agree well with those obtained by a numerical calculation. The effect of temporal modulation is seen to comprise an enhancement of sound radiation at all angles. A directive radiation pattern is obtained for moderate temporal modulations. This result extends that of Sandham et al. [36].

Numerical results for a wave-packet whose axial extent changes in time

Acknowledgements

The present work is supported by CNPq, National Council of Scientific and Technological Development—Brazil. Numerical data: DNS and LES kindly provided, respectively, by Jonathan B. Freund and Guillaume Daviller.

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    Current address: Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK.

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