## Abstract

Jets are one of the most fascinating topics in fluid mechanics. For aeronautics, turbulent jet-noise modelling is particularly challenging, not only because of the poor understanding of high Reynolds number turbulence, but also because of the extremely low acoustic efficiency of high-speed jets.

Turbulent jet-noise models starting from the classical Lighthill acoustic analogy to state-of-the art models were considered. No attempt was made to present any complete overview of jet-noise theories. Instead, the aim was to emphasize the importance of sound generation and mean-flow propagation effects, as well as their interference, for the understanding and prediction of jet noise.

## 1. Introduction

For several decades, the acoustic analogy of Lighthill (1952) has been a dominant jet-noise theory. In the Lighthill theory, the Navier–Stokes equations are rearranged to a linear wave equation form for the density fluctuation,
1.1
where the source on the right-hand side includes all the nonlinear terms , and the problem of computing the far sound reduces to the integral
1.2
where the irrelevant constant pressure *p*_{0} is suppressed.

Then, by neglecting the effect of viscous dissipation, considering the turbulence eddies acoustically compact *λ*=*c*_{0}/*f*≫*l*, the Lighthill law for scaling the total acoustic power of the jet with the jet speed is obtained as *v*^{8}∼*l*^{−1}*ρv*^{3}*M*^{5}*V*_{0}, where *l* is a characteristic length scale, *λ* is the acoustic wavelength and *M*=*v*/*c*_{0}≪1 is the Mach number corresponding to the characteristic jet velocity *v*. Despite the Lighthill theory assumptions, the *v*^{8} law works surprisingly well in many jet experiments, e.g. Viswanathan (2009).

In the last two decades, most of the jet-noise reduction for civil aircraft came from increasing the size of the jet engines. This allowed engineers to reduce the jet speed for the same amount of thrust and, since jet-noise scales as a high power of the jet exit velocity, in accordance with the Lighthill law, this reduces the noise. For further jet-noise reduction, however, the detailed mechanisms of turbulent jet noise need to be quantified, and it is their modelling that remains a big challenge.

Since Lighthill, acoustic analogies were developed by Lilley (1958), Ffowcs Williams (1963), Ribner (1964), Goldstein & Rosenbaum (1973), Tester & Morfey (1976)—to name but a few. These formulations differ by the model decomposition into acoustic sources and propagation. For example, the turbulent eddies that give rise to the Reynolds stresses are convected by the jet velocity and their motion alters the radiated sound, the effect that was first correctly accounted for by Ffowcs Williams (1963). The mean jet velocity has yet another influence: it refracts the sound (Mani 1976), altering the propagation of sound from the sources to the far field. As noted in Colonius *et al.* (1997), the approach based on Lilley’s (1974) equation to describe this propagation through a specified mean-flow velocity profile, taken to be a function of radius only, can be rather restrictive. A more accurate approach to modelling linear propagation effects is based on the generalized acoustic formulation by Goldstein (2003). This formulation considers solving linearized Euler equations with a source term that includes the residual Reynolds stresses. By conducting a direct numerical simulation (DNS) of unsteady Navier–Stokes equations for a two-dimensional shear-layer problem, Samanta *et al.* (2006) showed that different acoustic analogy models have different sensitivity to the errors in the source. They found that the models that explicitly account for more propagation effects, such as the Goldstein (2003) generalized acoustic analogy, tend to be more accurate.

For high-speed jets, the jet-noise properties at small and large observation angles to the flow are very different. This difference is associated with the anisotropy of jet turbulence that is observed in numerous experimental and numerical investigations (e.g. Freund 2001; Viswanathan 2004, 2009). Figure 1*a* shows a picture of a transitional jet that is a modified Schlieren image from Van Dyke (1982). Large coherent structures developing from Kelvin–Helmholtz instability waves are shown that originate in the shear layer and grow downstream from the nozzle exit. Having reached nonlinear amplitudes, these structures grow quickly at the end of the jet potential core and remain coherent over long distances in the axial direction, until they fully mix out and dissipate into heat downstream of the jet potential core.

The special role of linear instability waves for jet noise was first noted by Michalke & Fuchs (1975), Laufer *et al.* (1976), Michalke (1977), Moore (1977) and, later, by Tam & Auriault (1999), Tam & Burton (1984), Tam & Chen (1994) and Tam *et al.* (2008). After an extensive analysis of experimental data, Tam *et al.* (1996) empirically constructed two seemingly universal ‘similarity sound spectra’ that are able to give the best fits to all jet-noise data available to them for small and large observer angles to the jet. The fitting functions for high observer angles are referred to as the fine-scale spectrum (FSS) and the large-scale spectrum (LSS) for the small observer angles (Viswanathan 2004). Figure 1*b* shows the comparison of two similarity spectra plotted as functions of *f*/*f*_{0}, where *f* is the frequency and *f*_{0} is the peak frequency, with the experimental data at 30^{°} and 90^{°} observer angles. The experimental data correspond to a *M*=0.75 isothermal axisymmetric jet from the recent European jet-noise experiment (Power *et al.* 2004). It can be seen that the empirical ‘similarity sound spectra’, indeed, approximate the measurements well.

The differences in noise spectra at large and small observation angles appears to suggest the existence of two distinct physical mechanisms of jet noise. The first mechanism (FSS; Tam & Auriault 1999), by analogy with the kinetic-gas theory, is attributed to the ‘effective turbulence pressure’ that is exerted by fine-scale turbulence in the shear layer. As shown by Morris & Farassat (2002), the FSS model is equivalent to the classical Lighthill acoustic analogy with the isotropic turbulent source model. The FSS model is valid for high observer angles. For jet noise at small observer angles, the LSS model is suggested. This mechanism is attributed to large turbulence structures that are spatially coherent in the axial jet direction and can be viewed as the growth and decay of linear instability waves (Tam & Burton 1984; Tam & Chen 1994; Tam *et al.* 2008). The semi-empirical two-source FSS–LSS model leads to good predictions for a wide range of jet conditions (e.g. Tam *et al.* 2008; Morris 2009). In addition to the ability to collapse the experimental jet-noise spectra to the canonical *G* and *F* shapes, the FSS–LSS model explains (i) the directivity of the Mach number exponent as the correction to the Lighthill *v*^{8} law, (ii) the enhanced sound radiation close to the Mach wave radiation angle, (iii) the dominance of the axisymmetric *m*=0 component of jet mixing noise for angles close to the jet axis, (iv) that jet-noise spectra scale on the Helmholtz number rather than the Strouhal number for small angles to the jet, and (v) the change of spectral sound density shapes from small to large observer angles to the jet axis.

## 2. The role of interference within the distributed noise sourcein acoustic analogy

By including the interference effects within the distributed sound source, Michalke & Fuchs (1975) and Michel (2007) showed that many important jet-noise characteristics, i.e. (i)–(iv), can be explained within the classical Lighthill acoustic analogy model.

By rearranging and analysing the components of the acoustic integral, the importance of the inner integral of the power spectral density
2.1
over the coherent source volume, where **Δ**=(Δ_{1},Δ_{2},Δ_{3}) (coordinate 1 is in the jet direction) is the spatial separation between two points within the source location, **x** is the observer coordinates, *ψ*_{q} is due to the source convection in the flow, *ψ*_{r} describes the sound propagation to the far field by the influence of the emission distance on the retarded time due to the distributed source and *γ*_{q}(**Δ**) is the coherence function that decays with the spatial separation, was demonstrated. In the geometrical far field and assuming a simple line source convected in the flow, this integral can be evaluated analytically for a suitable coherence function *γ*_{q}(Δ_{1}).

In addition to the coherence function, the key parameters of the Michel model include: (i) non-dimensional length scale, *fL*_{x}/*v*_{p}, and (ii) non-dimensional phase speed, *v*_{p}/*c*_{0}, where and *v*_{p} are the dimensional frequency, the integral correlation length scale and the phase speed of acoustic disturbances, respectively.

The best agreement with the experiments was obtained by taking *γ*_{q}(Δ_{1}) in the form of a linear combination of two exponentials,
2.2
and assuming that *fL*_{x}/*v*_{p}=1 and *v*_{p}/*c*_{0}=1.2.

Figure 2 shows that for the same choice of parameters, the Michel model captures both the deviation of the Mach number exponent from the Lighthill *v*^{8} law (figure 2*a*) and the strong sound amplification (figure 2*b*) for small observation angles to the jet flow quite well (Viswanathan 2004). The over all sound pressure level (OASPL) shown in figure 2*b* is computed in the standard dB units: , where *p*′ is acoustic fluctuation at the observer location and *p*_{ref} is a reference pressure. The experimental data correspond to a *M*=0.9 jet (Viswanathan 2004). It can be seen that by playing with the parameter *A*, it is possible to improve the agreement of the Michel model with the experiment even further (figure 2*b*).

It is worth noting that the acoustic analogy model of Michalke & Fuchs (1975) and Michel (2009) shares the simplicity of the two-source model of Tam and co-workers and, yet, it does not need to make any assumptions about the existence of two separate mechanisms of jet noise to predict many integral characteristics of jet noise. Moreover, the Michel results indicate that the best agreement with the experiment corresponds to *fL*_{x}∼*v*_{p}∼*c*_{0}. This makes physically short scales play the role of large structures for high frequencies, when judged by the influence on the sound integral, and suggests that the distinction between the small-scale and large-scale effects on jet noise can be very subtle.

The problem with using the Michel model for a detailed prediction of jet noise (e.g. for predicting the noise spectra) is the need for the functional dependencies: *γ*_{q}, *ψ*_{q} and *ψ*_{r}. These include a mixture of sound generation and propagation effects and cannot be easily obtained either from the experimental data or computational modelling. A more simple and generic modelling of these effects is possible within the generalized acoustic analogy (Goldstein 2003) that is discussed in the next section.

## 3. Acoustic analogy: informed by unsteady jet simulations

For a detailed investigation of acoustic jet properties, a more comprehensive acoustic analogy model can be used that does not depend on empirical parameters (Karabasov *et al.* 2008). This methodology may be summarized as follows:

— Acoustic propagation is handled using the Goldstein (2003) decomposition of the Navier–Stokes equations 3.1

The flow fluctuations are defined from the decomposition of density, pressure, velocity and enthalpy into a steady mean and unsteady perturbation of the form where represents a time average, a Favre average and single and double primes represent the corresponding variation about the mean.

In this formulation, a momentum pertubation variable is defined (with zero time average) as ; and the Favre-averaged stagnation enthalpy, and its perturbation, take the special definitions and . The source terms are , and for isothermal jets, 3.2 where D/D

_{τ}is the convective derivative, .The linearized Euler equations (LEEs) are solved numerically using an efficient adjoint method in the frequency domain (Karabasov & Hynes 2006), and the power spectral density integral is obtained as a convolution of the source term with the propagator function, 3.3 where is the Fourier transform of the temporal–spatial (two-time, two-point) cross correlation of the turbulent sources 3.4 where

**x**is observer coordinates, and the components of the second-rank wave-propagation tensor, , are defined by 3.5 where is the adjoint vector Green’s function for the linearized Euler equations.— The input needed for solution of the LEE system is extracted from a mix of computational fluid dynamics (CFD) models, balancing computational affordability and ability to predict different aspects of the flow, and turbulence structure.

— The time-averaged flow field is taken from a Reynolds-averaged Navier–Stokes (RANS) CFD solution using a

*k*–*ε*turbulence model.— The acoustic source terms are scaled by a large-eddy simulation (LES) CFD solution, specifically.

— The non-dimensional shapes of the fourth-order 2-point 2-time correlations of the turbulent Reynolds stresses

*R*_{ijkl}are determined by fitting 3.6 to LES predicted correlations.— The correlation coefficients obtained from the LES solution compare well with recent measurements (Harper-Bourne 2003).

— The time and length scales used in the non-dimensional

*R*_{ijkl}are at any point in the flow field and are extracted from RANS*k*–*ε*predictions, but with proportionality constants fitted to time/length scales extracted from LES predictions.— The relative amplitudes of individual

*R*_{ij,kl}(*y*,0,0)/*R*_{11,11}(*y*,0,0) components are extracted from LES predictions.

The validation tests for the new jet-noise model include the JEAN experimental data (figure 1*b*) that correspond to an isothermal axisymmetric jet at *M*=0.75 and *Re*=10^{6} (Power *et al.* 2004). Figure 3*a* shows a surface of instantaneous vorticity downstream from the nozzle obtained from the LES solution and figure 3*b* shows the computational grid layout (McMullan *et al.* 2008).

The three-dimensional computational grid used in the LES calculations has a total of 17×10^{6} cells. The jet-nozzle wall geometry is included in the domain. The grid was uniformly refined near the nozzle exit to produce the height of the first off-wall grid cell *y*^{+}=*u***y*/*v*∼30, where *v* is the kinematic viscosity, *u** is the characteristic slip velocity near the boundary and *y* is the dimensional cell size. The LES solution was run over 100 000 time steps until it reached a statistically converged state and then sampled for the acoustic post-processing over the next 100 000 time steps.

Figure 4 shows the prediction results based on the new acoustic analogy method informed by LES. For the small and large observer angle, the model predictions are within 2 dB of the measurements for a wide range of frequencies.

The same methodology was applied for acoustic modelling of an isothermal chevron jet (Karabasov *et al.* in press) at *M*=0.75 and Reynolds number 10^{6} from the North Atlantic Space Agency (NASA) Small Hot Acoustic Rig (SHJAR) experiment (Bridges & Brown 2004). Chevron nozzles have been proposed as noise-reduction devices that may potentially significantly reduce the jet peak noise with very little effect on the thrust (Saiyed *et al.* 2000). The case considered corresponds to the chevron nozzle SMC006 that corresponds to six chevrons with a large penetration angle and leads to strong jet mixing in comparison to the baseline axisymmetric flow. Figure 5 shows unsteady jet solution details obtained from the LES calculation (Xia *et al.* 2009). The total mesh used in the LES calculation has 57 blocks and 12.5×10^{6} cells. The size of the first off-wall grid cell in the radial, axial and polar directions in wall units corresponds to Δ*r*^{+}∼2.5, Δ*x*^{+}∼300 and Δ*rθ*^{+}∼150, respectively.

Figure 6 shows the comparison of sound pressure level (SPL) predictions of the new acoustic analogy model with the narrowband NASA measurements. The figure also includes results of the previous acoustic predictions based on the standard integral Ffowcs Williams–Hawkings (1969) (FW–H) method with an open cylindrical control integration surface where the sources have been specified from the same LES solution sampled over the same number of time steps (approx. 250 000) after the solution reached a statistically converged state.

The prediction based on the integral FW–H method leads to more uneven spectra results, which might be caused by the averaging interval being not long enough for this method. Since the experimental spectra correspond to the high-resolution narrow-band data (Bridges & Brown 2004), no attempt was made to artificially smooth the FW–H prediction. It can also be noted that the FW–H solution exhibits a rapid spectra drop off at high frequencies (*St*>2), which is caused by the mesh resolution limits at the location of the FW–H control surface. A further improvement of the FW–H method accuracy at high frequencies is possible by refining the LES computational grid. However, this is an expensive solution: a conservative estimate shows that extending the accurate spectra predictions to *St*∼4 with the present FW–H method would increase the amount of computing time required by a factor of 16.

For the new acoustic model, the agreement with the experiment is very encouraging over a wide range of Strouhal numbers *St*=*fU*_{j}/*D*_{i}, both for small and large observer angles. The high-frequency limit of spectra resolution within the 2–2.5 dB error band is *St*∼8 and *St*∼6 for the observer locations at 90^{°} and 30^{°} to the jet flow, respectively.

Despite the encouraging sound-prediction results of the chevron jet-noise model, further work remains to be done to increase its resolution since the maximum noise reduction achieved by applying chevrons in comparison with the baseline axisymmetric jet in experiments (Bridges & Brown 2004) is within 2–3 dB. However, the fact that the new jet-noise model is able to capture high frequencies for the same amount of LES cost appears very encouraging for predicting the experimentally measured effect of chevron noise increase at high frequencies. The lesser sensitivity of the new LES-based acoustic analogy method to the LES grid resolution in comparison with the classical acoustic analogy model is also consistent with the study of Samanta *et al.* (2006), which was mentioned in §1.

## 4. The importance of mean-flow effects for identification of jet-noise sources

One advantage of the new acoustic analogy, informed by LES, is that equation (3.5) can be used to investigate the effect of mean flow on source coherence separately from the detailed modelling of sound generation produced by the fluctuating turbulent Reynolds stresses (3.4). Let us consider the following example. A ring of point sources is introduced in a jet mean flow in the shear-layer location. The mean-flow field corresponds to the jet from the JEAN experiment (Power *et al.* 2004). Assume that each element of the ring source is a linear combination of several point quadrupole sources of different directivity. Subsequently, each point quadrupole can be understood as a combination of two point dipole sources or four point monopole sources whose volume strength is zero when the distance between each two neighbouring elementary sources tends to zero (Howe 2003). Suppose that the resulting directivity of the ring source is such that the correlation function (3.4) satisfies
4.1
where 1 is the axial coordinate and 3 is the radial of the cylindrical-polar coordinate basis (*y*_{1}, *y*_{2}, *y*_{3}) and (*η*_{1}, *η*_{2}, *η*_{3}), *δ* is the Dirac delta function and *α*_{ijkl} are constant directivity parameters. The irrelevant constant proportionality coefficient in equation (4.1) to satisfy the normalization condition is suppressed.

For the point-source correlation function (4.1), the sound integral (3.3) reduces to an integral of the linear propagation operator (3.4) convoluted with *α*_{ijkl} over the circumferential angle. The non-dimensional coefficients *α*_{ijkl} are chosen such that they are only non-zero for the six main components of the tensor and components equal to them from the symmetry (table 1) and correspond to the typical directivity of the most significant correlation amplitudes of the LES source *α*_{ijkl}≈*R*_{ij,kl}(**y**,0,0)/*R*_{11,11}(**y**,0,0) in the vicinity of the end of the jet potential core (Karabasov *et al.* 2008).

For several axial locations of the ring source and a range of the source frequencies, the acoustic power spectra (3.3) are then calculated at two observer locations at 100 jet diameters from the nozzle exit and for 90^{°} and 30^{°} to the jet axis. Figure 7 shows the numerical experiment layout. For illustration purposes, the contours of pressure fluctuations, which correspond to the refraction of acoustic waves through the jet that were emitted by a point source in the jet shear layer, are also shown.

The spectra predictions are first obtained by using the full linearized Euler propagator model. The solutions obtained are then used as the reference ‘true’ data for solving a series of inverse problems for identifying the apparent acoustic source location from the computed sound spectra with an approximation introduced in the mean-flow propagation model. The approximation corresponds to ignoring most of the jet spreading effects and does not include the nozzle. For this model, the jet is regarded as a locally parallel shear flow.

Comparison of the spectra obtained based on the approximate propagation model with the reference spectra shows that the best agreement with the reference spectra may not be necessarily obtained for the same source location in the jet. Indeed, figure 8 shows that while at 90^{°} observer angle, the spectrum of the approximate jet model virtually coincides with the reference spectrum (figure 8*a*), and for 30^{°} angle, the jet mean-flow effects change the apparent source location in the jet dramatically (figure 8*b*). Indeed, for a range of frequencies of the ring source located at the end of the jet potential core the under-prediction of the apparent source location that occurs from using the approximate mean-flow model at 30^{°} angle to the jet is as large as four jet diameters. Figure 8*c* shows the measured variation of the axial location of the observed peak sound source in a *M*=0.9 jet at different angles to the jet from a recent elliptic mirror microphone experiment (Tam *et al.* 2008). The acoustic mirror has two foci: the far focus is placed in the region of interest in the jet and the near focus is placed at a far-field location where the measurements are taken for different observer angles. In the experiment, all mean-flow effects on sound propagation are neglected and the sound source that is largely contributing to the far-field signal is always assumed to be located along the focal axis. Whereas for large angles to the jet, the mean-flow propagation effects are small and the acoustic mirror prediction of the effective source location are expected to be accurate, for 30^{°} angle, this is not the case and a significant (approx. 5 *D*) shift of the apparent jet-noise source from the end of the jet potential core is observed. The fact that the upstream shift of the apparent source location from the end of the jet potential core in the experiment (Tam *et al.* 2008) is similar to the one calculated from solving the idealized inverse problems (approx. 4 *D*) may be coincidental, but it illustrates the importance of mean-flow propagation effects for the identification of effective noise sources at small angles to the jet that correspond to the peak sound directivity.

## 5. Conclusion

Jet-noise modelling remains very challenging because of the coupling between high Reynolds number turbulence phenomena and sound-propagation effects. Starting from the classical Lighthill theory, a number of jet-noise models have been proposed in the literature to explain these effects. The models range from semi-empirical scaling laws to more comprehensive models that are free from empirical input parameters and applicable for variable nozzle geometries.

One of the jet-noise source models discussed (FSS–LSS) is based on assuming two special jet-noise mechanisms of a distinct far-field sound directivity. Examples of two recent acoustic analogy models, which do not need this assumption, are also provided. These are the recent Michel model and the model based on the Goldstein acoustic analogy—informed by LESs. Together they appear to be able to explain a number of key jet-noise effects: (i) the directivity of the Mach number exponent as the correction to the Lighthill *v*^{8} law, (ii) the enhanced sound radiation close to the Mach wave radiation angle, (iii) the dominance of the axisymmetric *m*=0 component of jet mixing noise for angles close to the jet axis, (iv) that jet-noise spectra scale on the Helmholtz number rather than Strouhal number for small angles to the jet, and (v) the change of the spectral shapes from small to large observer angles to the jet. Despite different acoustic analogy formulations used, the two models have a number of important elements in common

— a unified statistical description of the source model; the Michel model uses the source coherence function

*γ*_{q}(**Δ**) (2.2) and the model of line source convected in the flow*ψ*_{q}(**Δ**) (2.1), and the LES-based acoustic analogy model uses the fourth-order velocity correlation coefficients*R*_{ijkl}(**y**,**Δ**,*τ*) (4.1) that include both the source coherence and convection effects and— a model of acoustic interference within the distributed source that depends on the observer location; the Michel model computes the retarded time change

*ψ*_{r}(**Δ**,**x**) equation (2.1) based on the free-space propagation operator, and the LES-based acoustic analogy model accounts for a more complex linear acoustic interaction with the non-uniform jet flow based on the linearized Euler propagator equation (3.5).

In addition to the challenges of modelling the turbulent-source statistics, the accuracy of identifying apparent sound sources in turbulent jets may be very sensitive to the detailed modelling of mean-flow sound propagation/interference effects. Therefore, despite recent progress in jet-noise modelling, quantitatively accurate, empiricism-free modelling of effective jet-noise mechanisms, which leads to effective mitigation of turbulent jet noise, remains an open challenge for the future.

## Author Profile

**Sergey Karabasov**

Sergey Karabasov has a Master’s Degree in Physics and Applied Mathematics from the Moscow Institute of Physics and Technology and a PhD in Mathematical Modelling that he received from the Department of Computational Mathematics and Cybernetics of Moscow State University in 1999. Since 2000, he has been working in Cambridge University Engineering Department on mathematical modelling of transonic helicopter noise and jet engine noise. In 2003, a paper that he co-authored won the best paper award at the American Helicopter Society International Forum. Since 2005, he has held a Royal Society University Research Fellowship devoted to research into new hybrid numerical methods for aeroacoustics. In addition to aeroacoustics, his research interests include computational fluid dynamics, geophysical fluid-dynamics modelling and numerical methods for convection/advection-dominated problems in fluid mechanics. In 2008, he received a Mary Sears Visitor Award from the Woods Hole Oceanographical Intuition for the implementation of novel high-resolution numerical methods for ocean modelling. In 2010, he received a Royal Society Award for organizing an International Scientific Seminar ‘Frontiers of numerical jet modelling:from engineering to environmental flows’ at the new Royal Society Kavli Centre. His recreational interests include swimming and Tai Chi.

## Footnotes

One contribution of 18 to a Triennial Issue ‘Visions of the future for the Royal Society’s 350th anniversary year’.

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