2.1 Representation of coherent structures

The focus of this work is on the peak jet noise, which is widely agreed to originate from ‘large-scale’ or ‘coherent’ turbulent structures in the jet (McLaughlin, Morrison & Troutt Reference McLaughlin, Morrison and Troutt1975; Tam & Burton Reference Tam and Burton1984). The RANS flow field, of course, is devoid of any time-resolved information that one could connect to coherent structures. To bridge this gap, we look at the main contributions of the large eddies: the transport of quantities such as momentum, heat, species, etc., across the jet. Focusing on the momentum transport, in a statistical sense the effect of turbulent eddies is captured by the velocity correlation $\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }}$ , where $\overline{(\text{ })}$ denotes the ensemble average, or the associated Reynolds stress tensor $-\overline{\unicode[STIX]{x1D70C}}\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }}$ . The coherent structures induce the largest contributions to the Reynolds stress. The Reynolds stress itself controls the production of turbulence, as expressed by the evolution equation for the turbulent kinetic energy (Mathieu & Scott Reference Mathieu and Scott2000)

(2.1) $$\begin{eqnarray}\frac{\text{D}k}{\text{D}t}=-\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }}\boldsymbol{ : }\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}-\unicode[STIX]{x1D716}.\end{eqnarray}$$

Here $\text{D}/\text{D}t$ means the total derivative associated with the mean flow, $\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}$ is the mean velocity gradient and $\unicode[STIX]{x1D716}$ is the dissipation. Even though this equation is written in a simplified form for homogeneous turbulence, it nevertheless captures the essential premise of the current work: the action of the turbulent eddies is best represented by the Reynolds stress, not the turbulent kinetic energy. The turbulent kinetic energy $k$ is an integral effect of the production and dissipation terms in (2.1). It will be shown that there are significant differences in the distributions of the Reynolds stress and turbulent kinetic energy in the jet flow field, which have a direct impact on the modelling attempted here.

In summary, the Reynolds stress will be a central element of the modelling effort. It will guide the appropriate definition of a convective Mach number, and will influence the amplitude of the space–time correlation.

2.2 Suppressed communication through the jet flow

A central assumption of the model is that the sound generated by coherent structures in the direction of peak emission (shallow polar angles to the jet axis) radiates mostly outward, with minimal radiation inward (through the jet flow). For a physical explanation, consider first a single-stream jet. The convective velocity of the shear layer eddies has been measured by a number of studies to be in the range of 60–70 % of the jet exit velocity (Doty & McLaughlin Reference Doty and McLaughlin2005; Morris & Zaman Reference Morris and Zaman2010b ). As a result, the convective Mach number of the eddies relative to the ambient is larger than the convective Mach number relative to the jet flow. For exhaust conditions typical for aeroengines, the outer convective Mach number is high subsonic or supersonic, while the inner convective Mach number is low subsonic. This means high radiation efficiency (a term that will be defined in § 3.7) for outward propagation and very low radiation efficiency for inward propagation. The sound that propagates inward and emerges from the opposite side of the jet is very weak compared to the outward-propagated sound. This concept will be generalized to a multistream jet in § 3.7.

The suppression of inward radiation is supported by measurements of the azimuthal coherence of the jet pressure field. For separation angle of 180 $^{\circ }$ , and for frequencies of relevance to aircraft noise (Strouhal numbers of the order of one or higher), the azimuthal coherence is zero (Viswanathan, Underbrink & Brusniak Reference Viswanathan, Underbrink and Brusniak2011). If even a tiny fraction of the eddy-generated sound ‘leaked’ through the other side of the jet, a finite coherence would be expected. In fact, the azimuthal coherence is very weak for much smaller separation angles, indicating (i) the finite azimuthal scale of the eddies and (ii) the suppression of inward propagation. Finally, the suppression of inward propagation, and finiteness of the azimuthal scales, are evident by a wealth of data on the sound emission of jets with induced asymmetry (including data in this paper) which show azimuthal variations of up to 15 dB, a factor of 30 in pressure amplitude. Such large azimuthal changes would not be possible if inward propagation were appreciable. The experimental evidence is not limited to asymmetric jets. Jets from nozzle with inserts or lobes show distinct azimuthal variations in the far-field sound (Powers et al. Reference Powers, McLaughlin and Morris2015).

The picture becomes murkier and more complex at large polar angle to the jet axis. There, the outward radiation efficiency can be very weak, even at high convective Mach number. So, the inner and outward propagation could be of competing strengths. Indeed, experiments show that, at large polar angles, loud events on one side of the jet can increase the sound emission on the opposite side. Until a better physical understanding of sound refraction at large polar angle is developed, the arguments presented in the previous two paragraphs can only be confidently applied in or near the direction of peak emission. Consequently, the scope of the analysis that follows is confined to the peak radiated sound.

2.3 Dominance of outer shear layer

As a corollary to the notion of suppressed communication through the jet flow, we argue that the sound generated by the coherent structures of the outermost shear layer of the jet is not significantly effected by refraction effects. In past works refraction has been approached from the standpoint of localized sources embedded in a mean flow (Mani Reference Mani1976; Tam & Auriault Reference Tam and Auriault1998). This concept is questionable as far as outward radiation from large-scale coherent structures is concerned. These coherent structures are in direct contact with the irrotational ambient medium, so the sound generation involves a direct coupling between the turbulent motion and the pressure field. Mean flow–acoustic interactions are deemed negligible, except in polar directions close to the angle of growth of the jet flow. We will further argue that, in multistream jets of relevance to aircraft propulsion, the outermost shear layer is the strongest contributor to peak noise. This is because, for velocity ratios typical of turbofan engines, the convective Mach numbers of the inner shear layers are expected to be much lower than the convective Mach number of the outer shear layer (Papamoschou Reference Papamoschou2004), and thus the inner shear layers are expected to radiate sound at a reduced efficiency compared to the outer shear layer. This point will be illustrated by the data of the present study and is further supported by recent investigation of the pressure in the very near field of single- and dual-stream jets (Papamoschou & Phong Reference Papamoschou and Phong2017).

3.1 Fundamental solution

We review briefly the Lighthill acoustic analogy (Lighthill Reference Lighthill1954), emphasizing features that are salient to the present modelling effort. Referring to figure 1, the noise source region occupies a volume ${\mathcal{V}}$ , locations $\boldsymbol{y}$ and $\boldsymbol{y}^{\prime }$ refer to points inside the source region, and location $\boldsymbol{x}$ is a field point outside the source region. The distances between the field point and the source locations are $r=|\boldsymbol{x}{-}\boldsymbol{y}|$ and $r^{\prime }=|\boldsymbol{x}{-}\boldsymbol{y}^{\prime }|$ . Through a rearrangement of the Navier–Stokes equations, the pressure fluctuation $p^{\prime }$ outside the source region can be shown to satisfy the linear inhomogeneous wave equation

(3.1) $$\begin{eqnarray}\frac{1}{a_{\infty }^{2}}\frac{\unicode[STIX]{x2202}^{2}p^{\prime }}{\unicode[STIX]{x2202}t^{2}}-\frac{\unicode[STIX]{x2202}^{2}p^{\prime }}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{i}}=\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D61B}_{ij}}{\unicode[STIX]{x2202}y_{i}\unicode[STIX]{x2202}y_{j}},\end{eqnarray}$$

where $a_{\infty }$ is the speed of sound of the uniform stationary medium surrounding the source and $\unicode[STIX]{x1D61B}_{ij}$ is the Lighthill stress tensor

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D61B}_{ij}=\unicode[STIX]{x1D70C}u_{i}u_{j}+(p-a_{\infty }^{2}\unicode[STIX]{x1D70C})\unicode[STIX]{x1D6FF}_{ij}-\unicode[STIX]{x1D70F}_{ij}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D70C}$ is the density, $p$ is the pressure, $u_{i}$ is the velocity vector and $\unicode[STIX]{x1D70F}_{ij}$ denotes the viscous stress tensor. The exact solution of (3.1) in three-dimensional free space is

(3.3) $$\begin{eqnarray}p^{\prime }(\boldsymbol{x},t)=\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{j}}\int _{{\mathcal{V}}}\unicode[STIX]{x1D61B}_{ij}\left(\boldsymbol{y},t-\frac{r}{a_{\infty }}\right)\frac{1}{4\unicode[STIX]{x03C0}r}\,\text{d}^{3}\boldsymbol{y},\end{eqnarray}$$

where $1/(4\unicode[STIX]{x03C0}r)$ represents the spatial distribution of the free-space Green’s function. Applying the chain rule, and neglecting terms that decay faster than the inverse first power of the distance, the double divergence is converted to a second time derivative,

(3.4) $$\begin{eqnarray}p^{\prime }(\boldsymbol{x},t)=\frac{1}{a_{\infty }^{2}}\int _{{\mathcal{V}}}\unicode[STIX]{x1D717}_{i}\unicode[STIX]{x1D717}_{j}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D61B}_{ij}}{\unicode[STIX]{x2202}t^{2}}\left(\boldsymbol{y},t-\frac{r}{a_{\infty }}\right)\frac{1}{4\unicode[STIX]{x03C0}r}\,\text{d}^{3}\boldsymbol{y},\end{eqnarray}$$

where

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D717}_{i}=\frac{x_{i}-y_{i}}{r}\end{eqnarray}$$

is the direction cosine between observer and source. Even though the derivative transformation in (3.4) is commonly associated with a far-field approximation, it is important to note that (3.4) gives the acoustic pressure everywhere, that is, in the near field and in the far field (Lighthill Reference Lighthill1954; Harper-Bourne Reference Harper-Bourne2002, Reference Harper-Bourne2003). This is because the neglected terms in the transformation decay faster than $r^{-1}$ and thus comprise the hydrodynamic pressure.

Figure 1. Set-up of the Lighthill acoustic analogy model.

3.2 Spectral density

Using (3.4) the autocorrelation of the pressure at observer location $\boldsymbol{x}_{0}$ is

(3.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{p^{\prime }(\boldsymbol{x}_{0},t)p^{\prime }(\boldsymbol{x}_{0},t+\unicode[STIX]{x1D70F})}=\frac{1}{16\unicode[STIX]{x03C0}^{2}a_{\infty }^{4}}\int _{{\mathcal{V}}}\int _{{\mathcal{V}}}[\unicode[STIX]{x1D717}_{i}\unicode[STIX]{x1D717}_{j}\unicode[STIX]{x1D717}_{k}^{\prime }\unicode[STIX]{x1D717}_{l}^{\prime }]_{0}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\overline{\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D61B}_{ij}(\boldsymbol{y},t-r_{0}/a_{\infty })}{\unicode[STIX]{x2202}t^{2}}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D61B}_{kl}(\boldsymbol{y}^{\prime },t+\unicode[STIX]{x1D70F}-r_{0}^{\prime }/a_{\infty })}{\unicode[STIX]{x2202}t^{2}}}\frac{1}{r_{0}r_{0}^{\prime }}\,\text{d}^{3}\boldsymbol{y}^{\prime }\,\text{d}^{3}\boldsymbol{y}.\end{eqnarray}$$

Here $\overline{(\text{ })}$ denotes the expected value or ensemble average, $r_{0}=|\boldsymbol{x}_{0}-\boldsymbol{y}|$ , and $r_{0}^{\prime }=|\boldsymbol{x}_{0}-\boldsymbol{y}^{\prime }|$ . We assume stationarity in time and define accordingly the space–time correlation of the Lighthill stress tensor as

(3.7) $$\begin{eqnarray}R_{ijkl}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F})=\overline{\unicode[STIX]{x1D61B}_{ij}(\boldsymbol{y},t)\unicode[STIX]{x1D61B}_{kl}(\boldsymbol{y}^{\prime },t+\unicode[STIX]{x1D70F})}.\end{eqnarray}$$

The stationarity allows us to take the time differentiation outside the correlation of (3.6), writing it as $\unicode[STIX]{x2202}^{4}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}^{4}\overline{(\text{ })}$ (Papoulis Reference Papoulis1965). In addition, it enables the time shift $t-r_{0}/a_{\infty }\rightarrow t$ . These steps result in

(3.8) $$\begin{eqnarray}\displaystyle \overline{p^{\prime }(\boldsymbol{x}_{0},t)p^{\prime }(\boldsymbol{x}_{0},t+\unicode[STIX]{x1D70F})} & = & \displaystyle \frac{1}{16\unicode[STIX]{x03C0}^{2}a_{\infty }^{4}}\int _{{\mathcal{V}}}\int _{{\mathcal{V}}}[\unicode[STIX]{x1D717}_{i}\unicode[STIX]{x1D717}_{j}\unicode[STIX]{x1D717}_{k}^{\prime }\unicode[STIX]{x1D717}_{l}^{\prime }]_{0}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\unicode[STIX]{x2202}^{4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}^{4}}R_{ijkl}\left(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F}+\frac{r_{0}-r_{0}^{\prime }}{a_{\infty }}\right)\frac{1}{r_{0}r_{0}^{\prime }}\,\text{d}^{3}\boldsymbol{y}^{\prime }\,\text{d}^{3}\boldsymbol{y}.\end{eqnarray}$$

The spectral density is the Fourier transform of the autocorrelation,

(3.9) $$\begin{eqnarray}S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})=\int _{-\infty }^{\infty }\overline{p^{\prime }(\boldsymbol{x}_{0},t)p^{\prime }(\boldsymbol{x}_{0},t+\unicode[STIX]{x1D70F})}\,\text{e}^{-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

Inserting (3.8),

(3.10) $$\begin{eqnarray}S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}^{4}}{16\unicode[STIX]{x03C0}^{2}}\int _{{\mathcal{V}}}\int _{{\mathcal{V}}}\int _{-\infty }^{\infty }[\unicode[STIX]{x1D717}_{i}\unicode[STIX]{x1D717}_{j}\unicode[STIX]{x1D717}_{k}^{\prime }\unicode[STIX]{x1D717}_{l}^{\prime }]_{0}R_{ijkl}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F})\frac{\exp [\text{i}\unicode[STIX]{x1D6FC}(r_{0}-r_{0}^{\prime })-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}]}{r_{0}r_{0}^{\prime }}\,\text{d}\unicode[STIX]{x1D70F}\,\text{d}^{3}\boldsymbol{y}^{\prime }\,\text{d}^{3}\boldsymbol{y},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D714}/a_{\infty }$ is the acoustic wavenumber. Equation (3.10) gives the acoustic component of the spectral density everywhere. At this point the only assumption is the stationarity in time of the flow statistics.

3.3 Coordinate system

The study of azimuthal effects necessitates the use of a cylindrical polar coordinate system in the implementation of (3.10). The complexity of the problem requires the inclusion of Cartesian and spherical coordinates. The three coordinate systems used here are illustrated in figure 2: Cartesian $(X,Y,Z)$ ; cylindrical polar $(X,y,\unicode[STIX]{x1D719})$ ; and spherical $({\mathcal{R}},\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ . The Cartesian coordinate system will also be described by indices 1 ( $X$ ), 2 ( $Y$ ) and 3 ( $Z$ ), with the index $23$ referring to combined properties on the cross-stream ( $Y{-}Z$ ) plane. Index 4 will refer to time.

Selection of an appropriate jet axis, on which the definitions of radial distance $y$ and azimuthal angle $\unicode[STIX]{x1D719}$ are based, is critical for capturing the azimuthal effects on noise emission. In this regard, the nozzle axis is a poor choice because asymmetric jets have distorted mean velocity profiles and could be vectored in directions off the nozzle axis. In § 3.6 the Lighthill stress tensor will be connected to the Reynolds stress, whose dominant component scales with the magnitude of the mean velocity gradient

(3.11) $$\begin{eqnarray}G=|\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}|.\end{eqnarray}$$

The decision then is to define the centre of the jet as the point where the Reynolds stress vanishes, or $G=0$ , within the jet flow. This definition is straightforward for the region past the end of the primary potential core, where the profile of the mean flow is Gaussian-like. There, the location of $G=0$ coincides with the location of the maximum mean velocity $\overline{u}_{max}$ . For the region of the jet comprising the primary potential core, the locations of $G=0$ or $\overline{u}_{max}$ are ill defined. However, one can calculate fairly reliably the centroid of the high-speed region defined by a criterion like $\overline{u}>0.9\overline{u}_{max}$ . In fact, this criterion can be extended to the region past the end of the primary potential core where, for noisy experimental or numerical data, it provides a more reliable estimate of the location of $\overline{u}_{max}$ . Therefore, for a given axial station $X=X_{1}$ , we identify the region of high-speed flow using the criterion

(3.12) $$\begin{eqnarray}\overline{u}(X_{1},Y,Z)>0.9\overline{u}_{max}(X_{1}).\end{eqnarray}$$

Considering a flow with symmetry about the $X{-}Y$ plane, we denote $Y_{i}$ , $i=1,\ldots ,N$ , the $Y$ locations at which this criterion is satisfied. Then, the jet centroid is computed according to

(3.13) $$\begin{eqnarray}Y_{c}(X_{1})=\frac{1}{N}\mathop{\sum }_{i=1}^{N}Y_{i}.\end{eqnarray}$$

Subsequently, the $Y$ -coordinates of all the data points at this axial station are decremented by $Y_{c}$ , so that $Y=0$ becomes the centroid location. This process is applied to all the axial stations within the computational domain.

Figure 2. Coordinate systems.

3.4 Far-field approximation

The far-field version of (3.10) is now developed, using the coordinate systems depicted in figure 2. The source locations are described in cylindrical polar coordinates

(3.14a,b ) $$\begin{eqnarray}\boldsymbol{y}=(X,y,\unicode[STIX]{x1D719}),\quad \boldsymbol{y}^{\prime }=(X^{\prime },y^{\prime },\unicode[STIX]{x1D719}^{\prime }).\end{eqnarray}$$

In the spherical coordinate system, the observer is situated at

(3.15) $$\begin{eqnarray}\boldsymbol{x}_{0}=({\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0}).\end{eqnarray}$$

For ${\mathcal{R}}\gg \ell$ , where $\ell$ is a characteristic dimension of the source, $\unicode[STIX]{x1D717}_{i}\approx \unicode[STIX]{x1D717}_{i}^{\prime }\approx x_{i}/{\mathcal{R}}$ and $1/(r_{0}r_{0}^{\prime })\approx 1/{\mathcal{R}}^{2}$ . Further,

(3.16) $$\begin{eqnarray}r_{0}-r_{0}^{\prime }\approx (X^{\prime }{-}X)\cos \unicode[STIX]{x1D703}_{0}+\sin \unicode[STIX]{x1D703}_{0}[y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})].\end{eqnarray}$$

Although the axial source separation $X^{\prime }{-}X$ readily appears on the right-hand side, the radial and azimuthal separations are interconnected and cannot be separated cleanly into distinct terms. This is an important consequence of using the polar-cylindrical coordinate system to express the source location; it will prevent the formulation of the spectral density as a four-dimensional Fourier transform of the space–time correlation, a common procedure in past treatments of the acoustic analogy (Morris & Farassat Reference Morris and Farassat2002; Dowling & Hynes Reference Dowling and Hynes2004).

On defining the projection of $R_{ijkl}$ along the observer direction as

(3.17) $$\begin{eqnarray}R_{0000}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F})=[\unicode[STIX]{x1D717}_{i}\unicode[STIX]{x1D717}_{j}~\unicode[STIX]{x1D717}_{k}\unicode[STIX]{x1D717}_{l}]_{0}~R_{ijkl}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F})\end{eqnarray}$$

the spectral density for the far-field observer becomes

(3.18) $$\begin{eqnarray}\displaystyle & & \displaystyle S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}^{4}}{16\unicode[STIX]{x03C0}^{2}{\mathcal{R}}^{2}}\int _{{\mathcal{V}}}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\int _{0}^{\infty }\int _{0}^{\infty }\int _{-\infty }^{\infty }R_{0000}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\exp (\text{i}\unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D703}_{0}(X^{\prime }{-}X)-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\exp \{\text{i}\unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}_{0}[y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})]\}\,\text{d}\unicode[STIX]{x1D70F}\,\text{d}X^{\prime }y^{\prime }\,\text{d}y^{\prime }\,\text{d}\unicode[STIX]{x1D719}^{\prime }\,\text{d}^{3}\boldsymbol{y}.\qquad\end{eqnarray}$$

In (3.18) the integrals over the shifted space and time coordinates are shown explicitly, while the integration over the source volume ${\mathcal{V}}$ is displayed compactly. The spatial coordinates in the exponent arise from the free-space Green’s function in the frequency domain.

3.5 Model for the space–time correlation

The space–time correlation model used here is defined in a fixed frame of reference. It is guided by experimental measurements of space–time correlations in the flow or near acoustic field of turbulent jets (Harper-Bourne Reference Harper-Bourne2003; Doty & McLaughlin Reference Doty and McLaughlin2005; Morris & Zaman Reference Morris and Zaman2010b ; Viswanathan et al. Reference Viswanathan, Underbrink and Brusniak2011), with important simplifications and modifications. Figure 3 sketches the typical shape of the axial space–time correlation of a fluctuating quantity (velocity, velocity squared, pressure, etc.). The evolution of the timewise correlation $R_{4}$ reflects the convection of turbulence with a velocity $U_{c}$ and its decorrelation with increasing axial separation $|X^{\prime }{-}X|$ . At zero spatial separation, $R_{4}$ is the autocorrelation and decays approximately exponentially with the time separation $\unicode[STIX]{x1D70F}$ . With increasing $|X^{\prime }{-}X|$ , the timewise correlation peaks at $\unicode[STIX]{x1D70F}=(X^{\prime }{-}X)/U_{c}$ and is modulated by the axial correlation $R_{1}(X^{\prime }{-}X)$ ; in addition, the shape of $R_{4}$ broadens and becomes more Gaussian-like. Negative loops are evident throughout the evolution of $R_{4}$ . For finite axial separation, the space–time correlation is not symmetric around $\unicode[STIX]{x1D70F}=0$ , reflecting the non-stationarity of spatial statistics and the associated increase of length and time scales with downstream distance.

Figure 3. Illustration of the typical shape of the axial space–time correlation.

Having noted the principal features of the axial space–time correlation, we outline the simplifications and modifications implemented here. The timewise and axial correlations will be treated as symmetric functions, thus neglecting the effects of spatial non-stationarity on their distributions. The timewise correlation $R_{4}$ will have fixed shape with axial separation and will include a transverse propagation time, in addition to the axial propagation time noted above. In the transverse dimensions of the problem, we will employ a mixed correlation $R_{23}$ whose precise form will be the subject of detailed analysis. The resulting correlation has the form

(3.19) $$\begin{eqnarray}\displaystyle R_{ijkl}(\boldsymbol{y},\boldsymbol{y}^{\prime },\unicode[STIX]{x1D70F}) & = & \displaystyle A_{ijkl}(\boldsymbol{y})R_{1}\left(\frac{X^{\prime }{-}X}{L_{1}(\boldsymbol{y})}\right)R_{23}(y,y^{\prime },\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{\prime },L_{23}(\boldsymbol{y}))\nonumber\\ \displaystyle & & \displaystyle \times \,R_{4}\left({\displaystyle \frac{\unicode[STIX]{x1D70F}-\displaystyle {\displaystyle \frac{X^{\prime }{-}X}{U_{c}(\boldsymbol{y})}}-{\displaystyle \frac{d}{V_{c}(\boldsymbol{y})}}}{\unicode[STIX]{x1D70F}_{\ast }(\boldsymbol{y})}}\right).\end{eqnarray}$$

Here $A_{ijkl}$ is the amplitude of the correlation; $R_{1}$ and $R_{4}$ are the axial and timewise correlations, respectively; $R_{23}$ is a mixed radial/azimuthal correlation; $L_{1}$ and $L_{23}$ and are the correlation length scales in the axial and transverse directions, respectively; and $\unicode[STIX]{x1D70F}_{\ast }$ is the correlation time scale. The timewise correlation $R_{4}$ includes axial and transverse propagation times. The axial propagation time $(X^{\prime }{-}X)/U_{c}$ is connected to the streamwise eddy convection at velocity $U_{c}$ . The transverse propagation time $d/V_{c}$ is a special construct that will be shown to induce azimuthal directivity in the emission of the sound. It is based on a transverse distance $d$ and a transverse propagation velocity $V_{c}$ . The axial and transverse convective Mach numbers are $M_{c}=U_{c}/a_{\infty }$ and $\unicode[STIX]{x1D707}_{c}=V_{c}/a_{\infty }$ , respectively. Equation (3.19) shows explicitly the dependence of the amplitude and correlation scales on the source location $\boldsymbol{y}$ . This notation will be henceforth dropped to reduce clutter.

The notion of a transverse propagation time scale can be found in the works of Harper-Bourne (Reference Harper-Bourne2003), Raizada & Morris (Reference Raizada and Morris2006) and Miller (Reference Miller2014). In this study, the concept should not be seen as anything more than a mathematical construct to induce azimuthal influence, as will be demonstrated in the analysis of § 3.5.3. Nevertheless, it is helpful to have some insight as to the physical meaning of $V_{c}$ . Consider two points separated laterally at the same axial location. If the turbulence is highly uncorrelated spatially, so that the correlation scale is much smaller than the separation of the two points, the speed at which a disturbance propagates from the first point to the second point cannot exceed the local acoustic velocity. On the other hand, if the turbulence convects downstream in highly organized patterns whose correlation scale is much larger than the separation of the two points, then the lateral propagation speed depends on the axial convective velocity and the shape of the ‘wavefronts’. If a given wavefront arrives simultaneously at the two points, the lateral propagation speed is infinite. Experimental measurements of the second-order radial cross-correlation in subsonic jets by Morris & Zaman (Reference Morris and Zaman2010a ) suggest a very fast, yet finite, lateral convection velocity. In the uncorrelated case, a transverse convective Mach number of the order of 1 ( $\unicode[STIX]{x1D707}_{c}\sim 1$ ) represents an upper bound. In the strongly correlated case, $\unicode[STIX]{x1D707}_{c}$ can be as high as $\infty$ in which case the transverse term drops out from the argument of $R_{4}$ .

3.5.1 Generic shape for the correlations

The correlation shapes employed here fall under the class of the ‘stretched exponential’

(3.20) $$\begin{eqnarray}R_{j}(t)=\text{e}^{-|t|^{\unicode[STIX]{x1D6FD}_{j}}},\end{eqnarray}$$

also called the Kohlrausch function (Wuttke Reference Wuttke2012). The flexibility provided by this function will be used in the axial ( $j=1$ ) and timewise ( $j=4$ ) dimensions, where the range $0.7\leqslant \unicode[STIX]{x1D6FD}_{j}\leqslant 2$ will be allowed. On the transverse plane ( $j=23$ ) only the integer value $\unicode[STIX]{x1D6FD}_{23}=2$ will be considered for the sake of numerical efficiency.

Since $R_{j}$ is an even function, its Fourier transform is real and equal to twice the cosine transform:

(3.21) $$\begin{eqnarray}\widehat{R}_{j}(\unicode[STIX]{x1D702})=2\int _{0}^{\infty }R_{j}(t)\cos (\unicode[STIX]{x1D702}t)\,\text{d}t.\end{eqnarray}$$

Note that $\widehat{R}_{j}$ assumes the analytical forms

(3.22) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \widehat{R}_{j}(\unicode[STIX]{x1D702})=\frac{2}{1+\unicode[STIX]{x1D702}^{2}},\quad \unicode[STIX]{x1D6FD}_{j}=1\\ \displaystyle \widehat{R}_{j}(\unicode[STIX]{x1D702})=\sqrt{\unicode[STIX]{x03C0}}\text{e}^{-(1/4)\unicode[STIX]{x1D702}^{2}},\quad \unicode[STIX]{x1D6FD}_{j}=2.\end{array}\right\}\end{eqnarray}$$

For powers $\unicode[STIX]{x1D6FD}_{j}$ other than 1 (exponential) or 2 (Gaussian) the Fourier transform does not have an analytical expression and needs to be calculated numerically. For computational efficiency, the transform $\widehat{R}_{j}(\unicode[STIX]{x1D702})$ was computed once and was tabulated versus $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D6FD}_{j}$ ; subsequent operations used two-dimensional interpolation of the table. Great care is required in evaluating near $\unicode[STIX]{x1D6FD}_{j}=2$ where the shape of the Fourier transform is extremely sensitive on $2-\unicode[STIX]{x1D6FD}_{j}$ .

The stretched exponential will be used here with a reference scale, that is, $R_{j}(t)=\exp (-|t/\unicode[STIX]{x1D70F}|^{\unicode[STIX]{x1D6FD}_{j}})$ . Its Fourier transform is simply $\unicode[STIX]{x1D70F}\widehat{R}_{j}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D702})$ . Evaluated at $\unicode[STIX]{x1D702}=0$ , it gives the integral scale $\unicode[STIX]{x1D70F}\widehat{R}_{j}(0)$ . It can be shown that

(3.23) $$\begin{eqnarray}\widehat{R}_{j}(0)=\frac{1}{\unicode[STIX]{x1D6FD}_{j}}\unicode[STIX]{x1D6E4}\left(\frac{1}{\unicode[STIX]{x1D6FD}_{j}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}$ is the gamma function (Wuttke Reference Wuttke2012). For $0.7\leqslant \unicode[STIX]{x1D6FD}_{j}\leqslant 2$ , the corresponding range for $\widehat{R}_{j}(0)$ is $1.266\geqslant \widehat{R}_{j}(0)\geqslant 0.886$ . Thus, the integral scale is not too different from the reference scale.

Figure 4 illustrates the behaviour of the stretched exponential and its transform for $0.7\leqslant \unicode[STIX]{x1D6FD}_{j}\leqslant 2$ , the range allowed in this study. For clarity the transform is shown in decibels. The sensitivity of the transform on the power $\unicode[STIX]{x1D6FD}$ is apparent and represents a key ingredient of the optimization process employed here. For the selected range of $\unicode[STIX]{x1D6FD}_{j}$ , the Fourier transform is non-negative for all frequencies.

Figure 4. Correlation function (a) and its Fourier transformation (b) for various values of $\unicode[STIX]{x1D6FD}$ .

3.5.2 Axial and timewise Fourier transforms

The timewise integration in (3.18) amounts to a Fourier transform in the time separation $\unicode[STIX]{x1D70F}$ . Given the slow axial development of the flow, the $X^{\prime }$ integral can be approximated as an integral over the axial separation $X^{\prime }{-}X$ ranging from $-\infty$ to $\infty$ , and thus can also be treated as a Fourier transform in $X^{\prime }{-}X$ . This assumes that the scale of the axial correlation is much smaller than the distances $X$ or $X^{\prime }$ and neglects the fact that $X$ and $X^{\prime }$ have a finite origin at zero. Fourier transforms in the transverse dimensions of the problem are not feasible or appropriate. As indicated in the discussion of (3.16), the radial and azimuthal components of the Green’s function cannot be expressed in terms of separations $y^{\prime }-y$ and $\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}$ . Even if one were to overlook this fact, the concept of a Fourier transform in the radial separation $y^{\prime }-y$ breaks down because of the rapid evolution of the flow in the radial direction: the radial correlation scale cannot be considered small compared to either $y^{\prime }$ or $y$ . Similarly, the azimuthal correlation scale is not necessarily small compared to $2\unicode[STIX]{x03C0}$ to attempt a Fourier transform in $\unicode[STIX]{x1D719}^{\prime }$ .

We conclude that Fourier transformation is only possible in the timewise and axial directions; the procedure is rigorous in the timewise dimension and acceptable as an approximation in the axial dimension. Inserting the correlation form (3.19) in (3.18), and carrying out the Fourier transforms in $\unicode[STIX]{x1D70F}$ and $X^{\prime }{-}X$ , we obtain

(3.24) $$\begin{eqnarray}\displaystyle & & \displaystyle S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}^{4}}{16\unicode[STIX]{x03C0}^{2}{\mathcal{R}}^{2}}\int _{{\mathcal{V}}}A_{0000}\unicode[STIX]{x1D70F}_{\ast }L_{1}\widehat{R}_{1}\left[\unicode[STIX]{x1D6FC}L_{1}\left(\frac{1}{M_{c}}-\cos \unicode[STIX]{x1D703}_{0}\right)\right]\widehat{R}_{4}[\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{\ast }]\exp \left(-\text{i}\frac{\unicode[STIX]{x1D6FC}d}{\unicode[STIX]{x1D707}_{c}}\right)\qquad \nonumber\\ \displaystyle & & \displaystyle \quad \times \,\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\int _{0}^{\infty }R_{23}\exp \{\text{i}\unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}_{0}[y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})]\}y^{\prime }\,\text{d}y^{\prime }\,\text{d}\unicode[STIX]{x1D719}^{\prime }\,\text{d}^{3}\boldsymbol{y}.\end{eqnarray}$$

Omitting the arguments, we write this compactly as

(3.25) $$\begin{eqnarray}S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}^{4}}{16\unicode[STIX]{x03C0}^{2}{\mathcal{R}}^{2}}\int _{{\mathcal{V}}}A_{0000}\unicode[STIX]{x1D70F}_{\ast }L_{1}\unicode[STIX]{x03C0}L_{23}^{2}\widehat{R}_{1}\widehat{R}_{4}\widetilde{R}_{23}\,\text{d}^{3}\boldsymbol{y},\end{eqnarray}$$

where

(3.26) $$\begin{eqnarray}\displaystyle \widetilde{R}_{23} & = & \displaystyle \frac{1}{\unicode[STIX]{x03C0}L_{23}^{2}}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\int _{0}^{\infty }R_{23}(y,y^{\prime },\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \times \,\exp \left\{\text{i}\unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}_{0}[y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})]-\text{i}\unicode[STIX]{x1D6FC}\frac{d}{\unicode[STIX]{x1D707}_{c}}\right\}y^{\prime }\,\text{d}y^{\prime }\,\text{d}\unicode[STIX]{x1D719}^{\prime }.\qquad\end{eqnarray}$$

In (3.25) the term $\unicode[STIX]{x03C0}L_{23}^{2}$ represents a cross-stream correlation area, and the product $\unicode[STIX]{x1D70F}_{\ast }L_{1}\unicode[STIX]{x03C0}L_{23}^{2}$ can be viewed as a four-dimensional correlation ‘volume’. As discussed in § 3.5.1, the functions $\widehat{R}_{1}$ and $\widehat{R}_{4}$ are real and non-negative. The meaning and behaviour of $\widetilde{R}_{23}$ will be the topic of the discussion that follows.

3.5.3 Cross-stream correlation

As noted in § 3.5.2, the transverse correlation $R_{23}$ is not amenable to Fourier transforms. Instead, the spectral transformation of $R_{23}$ takes the form of the integral of (3.26). Evaluation of this integral, and determination of allowable forms for $R_{23}$ and the separation distance $d$ , are governed by the requirement that the power spectral density $S(\boldsymbol{x}_{0},\unicode[STIX]{x1D714})$ , given by (3.25), be real and non-negative. To satisfy this requirement for an arbitrary source distribution, $\widetilde{R}_{23}$ must be real and non-negative (recall that $\widehat{R}_{1}$ and $\widehat{R}_{4}$ are real and non-negative for the class of correlation functions selected here). A further requirement is that $R_{23}$ be periodic in the azimuthal separation $\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}$ .

Figure 5. Geometric relations on the cross-stream plane.

Figure 6. Geometric relations in shifted coordinate system on the cross-stream plane. Without loss of generality, the observer is placed at $\unicode[STIX]{x1D719}_{0}=0$ .

Equation (3.26) entails integration over the cross-stream plane. Its evaluation is facilitated by examining key geometric relations on this plane. Figure 5 depicts the projections of source elements $\boldsymbol{y}$ and $\boldsymbol{y}^{\prime }$ and the resulting geometric relations on the cross-stream plane, with the observer located at azimuthal angle $\unicode[STIX]{x1D719}_{0}$ . All the distances discussed here will be projected distances on the cross-stream plane. The distance between elements $\boldsymbol{y}$ and $\boldsymbol{y}^{\prime }$ is

(3.27) $$\begin{eqnarray}s=\sqrt{y^{\prime 2}+y^{2}-2yy^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719})}\end{eqnarray}$$

and the projection of this distance on the observer radial line is $y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})$ . This is precisely the term that appears in the exponent of (3.26). It thus becomes evident that a coordinate system centred at the source location $\boldsymbol{y}$ , rather than at the centroid, is preferred for evaluating (3.26). Accordingly, the origin is shifted from the centroid to the location of source element $\boldsymbol{y}$ , as shown in figure 6. All the azimuthal angles are now defined with respect to the observer angle $\unicode[STIX]{x1D719}_{0}$ . The observer being in the far field, the coordinate shift does not change the angular relations. In the new coordinate system, the azimuthal angle of element $\boldsymbol{y}^{\prime }$ is $\unicode[STIX]{x1D713}$ . The term $y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}_{0})-y\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})$ reduces to $s\cos \unicode[STIX]{x1D713}$ . Changing the integration variables from $(y^{\prime },\unicode[STIX]{x1D719}^{\prime })$ to $(s,\unicode[STIX]{x1D713})$ we obtain

(3.28) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \widetilde{R}_{23}=\frac{1}{\unicode[STIX]{x03C0}L_{23}^{2}}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\int _{0}^{\infty }R_{23}\text{e}^{\text{i}\unicode[STIX]{x1D6FE}s\cos \unicode[STIX]{x1D713}-\text{i}\unicode[STIX]{x1D6FF}d}s\,\text{d}s\,\text{d}\unicode[STIX]{x1D713}\\ \unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}_{0}\\ \displaystyle \unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D707}_{c}.\end{array}\right\}\end{eqnarray}$$

Although an exhaustive treatise of this integral is beyond the scope of the current work, a straightforward strategy for satisfying the requirement of real non-negativeness will be set forth by invoking the integral representation of the Bessel function of the first kind and of order zero:

(3.29) $$\begin{eqnarray}2\unicode[STIX]{x03C0}J_{0}(x)=\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\text{e}^{\text{i}x\cos \unicode[STIX]{x1D713}}\,\text{d}\unicode[STIX]{x1D713}.\end{eqnarray}$$

First, if $d$ is related to $s$ through a projection of the type $d=s\cos (\unicode[STIX]{x1D713}-\unicode[STIX]{x1D712})$ , where $\unicode[STIX]{x1D712}$ is a reference angle, integration over $\unicode[STIX]{x1D713}$ yields $2\unicode[STIX]{x03C0}J_{0}(\unicode[STIX]{x1D701}s)$ , where $\unicode[STIX]{x1D701}$ is a real positive number. Then, on selecting $R_{23}=R_{23}(s)$ , the integral over $s$ becomes the Hankel transform of $R_{23}$ . Here the natural choice for $d$ is

(3.30) $$\begin{eqnarray}d=s\cos (\unicode[STIX]{x1D713}-\unicode[STIX]{x1D719})=y^{\prime }\cos (\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719})-y,\end{eqnarray}$$

that is, $d$ is the projection of $s$ on the radial $\unicode[STIX]{x1D719}$ . Integration over $\unicode[STIX]{x1D713}$ results in a Bessel function, and (3.28) becomes

(3.31) $$\begin{eqnarray}\widetilde{R}_{23}=\frac{2}{L_{23}^{2}}\int _{0}^{\infty }R_{23}(s)J_{0}\left(s\sqrt{\unicode[STIX]{x1D6FE}^{2}+\unicode[STIX]{x1D6FF}^{2}-2\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})}\right)s\,\text{d}s\end{eqnarray}$$

yielding the Hankel transform of $R_{23}(s)$ . For computational efficiency, we seek forms for $R_{23}(s)$ that result in analytical, non-negative solutions (see appendix A for further discussion). Among several candidates, a Gaussian kernel satisfies these conditions and yields a simple analytical solution (Bateman Reference Bateman1954). Accordingly, we select

(3.32) $$\begin{eqnarray}R_{23}(s)=\exp \left[-\left(\frac{s}{L_{23}}\right)^{\unicode[STIX]{x1D6FD}_{23}}\right]\end{eqnarray}$$

and restrict $\unicode[STIX]{x1D6FD}_{23}=2$ . Evaluation of the Hankel transform gives

(3.33) $$\begin{eqnarray}\widetilde{R}_{23}=\exp \left\{-\left(\frac{\unicode[STIX]{x1D6FC}L_{23}}{2}\right)^{2}\left[\sin ^{2}\unicode[STIX]{x1D703}_{0}+\frac{1}{\unicode[STIX]{x1D707}_{c}^{2}}-2\frac{\sin \unicode[STIX]{x1D703}_{0}}{\unicode[STIX]{x1D707}_{c}}\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})\right]\right\}.\end{eqnarray}$$

Therefore, we satisfy the requirement for a real non-negative spectral density. Note that $R_{23}$ is periodic with the azimuthal separation $\unicode[STIX]{x1D719}^{\prime }-\unicode[STIX]{x1D719}$ , as is readily observed by inserting (3.27) in (3.32). Importantly, the term $\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})$ in (3.33) induces an azimuthal influence that simulates the effect of suppressed communication through the jet flow discussed in § 2.2.

3.5.4 Azimuthal influence

The azimuthal directivity of $\widetilde{R}_{23}$ arises from the term $\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})$ in (3.33) and is directly controlled by the transverse convective Mach number $\unicode[STIX]{x1D707}_{c}$ . For $\unicode[STIX]{x1D707}_{c}=\infty$ , $\widetilde{R}_{23}$ does not have an azimuthal variation. For $\unicode[STIX]{x1D707}_{c}$ finite and positive, the azimuthal influence has an extent that is controlled by $\unicode[STIX]{x1D707}_{c}$ and by the transverse non-dimensional wavenumber $\unicode[STIX]{x1D6FC}L_{23}$ . Figure 7 illustrates these dependencies for observer polar angle $\unicode[STIX]{x1D703}_{0}=30^{\circ }$ . At fixed $\unicode[STIX]{x1D6FC}L_{23}$ , the strongest azimuthal directivity is obtained for $\unicode[STIX]{x1D707}_{c}=1/\sin \unicode[STIX]{x1D703}_{0}$ ( $\unicode[STIX]{x1D707}_{c}=2$ in this example). At fixed $\unicode[STIX]{x1D707}_{c}$ , the directivity sharpens with increasing $\unicode[STIX]{x1D6FC}L_{23}$ (increasing frequency). These observations would tempt one to set $\unicode[STIX]{x1D707}_{c}=1/\sin \unicode[STIX]{x1D703}_{0}$ to maximize the azimuthal influence. Of course, this is not a legitimate step because the correlation parameters should be independent of observer location. The approach in this study is to set

(3.34) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{c}=\frac{1}{\sin \unicode[STIX]{x1D703}_{peak}},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{peak}$ is the angle of peak emission, in terms of the overall sound pressure level, for the baseline axisymmetric jet. The combined influence of $\unicode[STIX]{x1D707}_{c}$ and $L_{23}$ on the azimuthal influence places some constraints on the transverse correlation scale $L_{23}$ . Specifically, a lower constraint should be placed on the coefficient that controls $L_{23}$ such that, at given frequency, $\unicode[STIX]{x1D6FC}L_{23}$ is not too small. The present model for $\unicode[STIX]{x1D707}_{c}$ is selected for its simplicity. More sophisticated models, where $\unicode[STIX]{x1D707}_{c}$ depends on flow conditions and frequency, may provide higher levels of fidelity.

Figure 7. Azimuthal distribution of $\widetilde{R}_{23}$ for observer polar angle $\unicode[STIX]{x1D703}_{0}=30^{\circ }$ . (a) Fixed $\unicode[STIX]{x1D6FC}L_{23}$ and varying $\unicode[STIX]{x1D707}_{c}$ ; (b) fixed $\unicode[STIX]{x1D707}_{c}$ and varying $\unicode[STIX]{x1D6FC}L_{23}$ .

3.6 Amplitude of the correlation

The amplitude $A_{ijkl}$ in (3.19) represents the correlation

(3.35) $$\begin{eqnarray}R_{ijkl}(\boldsymbol{y},\boldsymbol{y},0)=\overline{\unicode[STIX]{x1D61B}_{ij}(\boldsymbol{y},t)\unicode[STIX]{x1D61B}_{kl}(\boldsymbol{y},t)}.\end{eqnarray}$$

It is important to recall, however, that the source term in (3.8) is not $R_{ijkl}$ itself but $\unicode[STIX]{x2202}^{4}R_{ijkl}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}^{4}$ . This means that only terms that depend on $\unicode[STIX]{x1D70F}$ can contribute to $A_{ijkl}$ . It is thus convenient to express $A_{ijkl}$ as

(3.36) $$\begin{eqnarray}A_{ijkl}(\boldsymbol{y})=\overline{[\unicode[STIX]{x1D61B}_{ij}]_{a}[\unicode[STIX]{x1D61B}_{kl}]_{b}},\quad a\rightarrow b,\end{eqnarray}$$

where $a$ and $b$ represent different times. Only correlations that involve both $a$ and $b$ are to be retained.

We assume that the principal component of the Lighthill stress tensor is $\unicode[STIX]{x1D61B}_{ij}=\overline{\unicode[STIX]{x1D70C}}u_{i}u_{j}$ and write the velocity components as

(3.37) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle u_{1}=\overline{u}+u^{\prime }\\ \displaystyle u_{2}=v^{\prime }\\ u_{3}=w^{\prime },\end{array}\right\}\end{eqnarray}$$

where $u^{\prime },v^{\prime },w^{\prime }$ are the fluctuating velocity components in Cartesian coordinates. The distinct Lighthill tensor components are

(3.38) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D61B}_{11}=\overline{\unicode[STIX]{x1D70C}}(\overline{u}^{2}+2\overline{u}u^{\prime }+u^{\prime }u^{\prime })\\ \displaystyle \unicode[STIX]{x1D61B}_{12}=\overline{\unicode[STIX]{x1D70C}}(\overline{u}v^{\prime }+u^{\prime }v^{\prime })\\ \displaystyle \unicode[STIX]{x1D61B}_{13}=\overline{\unicode[STIX]{x1D70C}}(\overline{u}w^{\prime }+u^{\prime }w^{\prime })\\ \displaystyle \unicode[STIX]{x1D61B}_{22}=\overline{\unicode[STIX]{x1D70C}}v^{\prime }v^{\prime }\\ \displaystyle \unicode[STIX]{x1D61B}_{23}=\overline{\unicode[STIX]{x1D70C}}v^{\prime }w^{\prime }\\ \displaystyle \unicode[STIX]{x1D61B}_{33}=\overline{\unicode[STIX]{x1D70C}}w^{\prime }w^{\prime }.\end{array}\right\}\end{eqnarray}$$

Following the rule accompanying (3.36), only cross-terms like $\overline{u_{a}^{\prime }v_{b}^{\prime }}$ will be retained; terms like $\overline{u_{a}^{\prime }v_{a}^{\prime }}$ do not contribute to the source. Under the assumption of isotropic turbulence, the volume integral of the third-order correlations vanishes (Ribner Reference Ribner1969). Although the validity of this assumption needs to be evaluated thoroughly, here we will neglect third-order correlations like $\overline{u_{a}^{\prime }v_{b}^{\prime 2}}$ . The resulting evaluation of $A_{ijkl}$ leads to terms containing second-order correlations $\overline{u_{i}^{\prime }u_{j}^{\prime }}$ , usually referred to as ‘shear noise’; and terms containing fourth-order correlations $\overline{u_{i}^{\prime }u_{j}^{\prime }u_{k}^{\prime }u_{l}^{\prime }}$ , typically described as ‘self-noise’. A preliminary evaluation of the self-noise terms, using the quasi-normal hypothesis (Morris & Zaman Reference Morris and Zaman2010b ) and the approximations that follow, indicates that their contribution to peak noise is at least 10 dB below the contribution of the shear-noise terms. Therefore, the fourth-order correlations are deemed irrelevant to the prediction of peak noise.

Figure 8. Azimuthal relationships of source and observer.

Thus, the problem boils down to modelling the second-order correlations, that is, the components of the Reynolds stress tensor. To this end, we use the constitutive relation that forms the foundation of turbulence modelling (Mathieu & Scott Reference Mathieu and Scott2000)

(3.39) $$\begin{eqnarray}\overline{u_{i}^{\prime }u_{j}^{\prime }}={\textstyle \frac{2}{3}}k\unicode[STIX]{x1D6FF}_{ij}-\unicode[STIX]{x1D708}_{T}S_{ij},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}_{T}$ is the turbulent viscosity, $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta and

(3.40) $$\begin{eqnarray}S_{ij}=\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}\overline{u}_{j}}{\unicode[STIX]{x2202}x_{i}}.\end{eqnarray}$$

Given that the jet flow is slowly diverging, the dominant component of $S_{ij}$ is the transverse gradient of the mean axial velocity. The approximate magnitude of this gradient is

(3.41) $$\begin{eqnarray}G=\sqrt{\left[\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}Y}\right]^{2}+\left[\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}Z}\right]^{2}}\end{eqnarray}$$

and its azimuthal direction is $\unicode[STIX]{x1D719}_{g}$ . Figure 8 describes the azimuthal relations between source and observer. The convention here is to assign an outward azimuthal direction for an inward gradient, and vice versa. For an axisymmetric jet with monotonically declining radial velocity profile (that is, without a wake component), $\unicode[STIX]{x1D719}_{g}$ coincides with the geometric azimuthal angle $\unicode[STIX]{x1D719}$ . For a jet whose velocity isocontours are not circular, or that has a wake defect, $\unicode[STIX]{x1D719}_{g}$ and $\unicode[STIX]{x1D719}$ will generally be different.

It is now argued that the principal turbulent transport is in the direction of the mean flow gradient with an associated turbulent velocity fluctuation $q^{\prime }$ . The corresponding velocity correlation

(3.42) $$\begin{eqnarray}g=\langle u^{\prime }q^{\prime }\rangle =\unicode[STIX]{x1D708}_{T}G\end{eqnarray}$$

is deemed the dominant contributor to the momentum transport and hence to the Reynolds stress. The transport normal to the direction of the mean flow gradient is considered negligible. The correlation $g$ will henceforth be loosely referred to as the ‘Reynolds stress’ and will be treated as non-negative. The direction of the mean flow gradient, and its impact on the individual terms of the Reynolds stress tensor, will be accounted for by the angle $\unicode[STIX]{x1D719}_{g}$ . Returning to the constitutive relation (3.39), using $v^{\prime }=-q^{\prime }\cos \unicode[STIX]{x1D719}_{g}$ and $w^{\prime }=q^{\prime }\sin \unicode[STIX]{x1D719}_{g}$ , we are now able to make the following approximations:

(3.43) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \overline{u^{\prime }u^{\prime }}\approx {\textstyle \frac{2}{3}}k\\ \displaystyle \overline{v^{\prime }v^{\prime }}\approx {\textstyle \frac{2}{3}}k\\ \displaystyle \overline{w^{\prime }w^{\prime }}\approx {\textstyle \frac{2}{3}}k\\ \overline{u^{\prime }v^{\prime }}=-\langle u^{\prime }q^{\prime }\rangle \displaystyle \cos \unicode[STIX]{x1D719}_{g}\approx -g\cos \unicode[STIX]{x1D719}_{g}\\ \overline{u^{\prime }w^{\prime }}=\langle u^{\prime }q^{\prime }\rangle \displaystyle \sin \unicode[STIX]{x1D719}_{g}\approx g\sin \unicode[STIX]{x1D719}_{g}\\ \overline{v^{\prime }w^{\prime }}\approx 0.\end{array}\right\}\end{eqnarray}$$

It is recognized that in the actual jet the axial velocity fluctuations are stronger than the transverse fluctuations, as measured by a variety of experiments (for example, Morris & Zaman Reference Morris and Zaman2010b ). However, here it is preferred to stay faithful to the constitutive relation (3.39).

Based on the convention of figure 2, the direction cosines for the far-field observer are

(3.44) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D717}_{1}=\cos \unicode[STIX]{x1D703}_{0}\\ \displaystyle \unicode[STIX]{x1D717}_{2}=-\!\sin \unicode[STIX]{x1D703}_{0}\cos \unicode[STIX]{x1D719}_{0}\\ \displaystyle \unicode[STIX]{x1D717}_{3}=\sin \unicode[STIX]{x1D703}_{0}\sin \unicode[STIX]{x1D719}_{0}.\end{array}\right\}\end{eqnarray}$$

Due to the symmetry of $\unicode[STIX]{x1D61B}_{ij}$ and the resulting pairwise symmetry of $A_{ijkl}$ (that is, $A_{ijkl}=A_{klij}$ ), the 81 elements of $A_{ijkl}$ comprise single or multiple occurrences of 21 distinct terms. Of those, only 6 terms have the potential to contribute to shear noise; these are the terms where the index $1$ appears at least once in $ij$ and at least once in $kl$ . Table 1 lists the shear-noise terms, their multipliers (frequencies), their expressions according to (3.38), their approximations according to (3.43) and their directivities according to (3.44).

Table 1. Distinct shear-noise terms of $A_{ijkl}$ and associated directivities.

The total contribution in the direction of the far-field observer is

(3.45) $$\begin{eqnarray}\frac{A_{0000}}{\overline{\unicode[STIX]{x1D70C}}^{2}\overline{u}^{2}}=\frac{8}{3}k\cos ^{2}\unicode[STIX]{x1D703}_{0}+8g\cos ^{3}\unicode[STIX]{x1D703}_{0}\sin \unicode[STIX]{x1D703}_{0}\cos (\unicode[STIX]{x1D719}_{g}-\unicode[STIX]{x1D719}_{0}).\end{eqnarray}$$

The second term on the right-hand side arises from the 1112 and 1113 components of the Lighthill stress tensor. In past works on axisymmetric jets, these components were neglected because they were thought to integrate to zero when inserted in the formula for the spectral density (Ribner Reference Ribner1969). This is not the case if we accept that the space–time correlation induces an azimuthal directivity along the lines of (3.33). Then, the second term of (3.45) does not integrate to zero and makes a finite, positive contribution to the spectral density. Of course, the components $A_{1112}$ and $A_{1113}$ must be retained for asymmetric jets regardless of the chosen form of the space–time correlation.

The azimuthal dependencies contained in the cosine terms of (3.33) and (3.45) complicate the evaluation of the spectral density (3.25) when the computational domain is restricted to one of the symmetric halves of the jet flow. In addition, for axisymmetric jets, it is desirable to cut down the expense of computing the spectral density by considering only an azimuthal slice of the domain, a task also complicated by the cosine terms. These issues are addressed in appendix A.

3.7 Outer surface of peak stress (OSPS)

In the expression for the spectral density (3.25), the effect of the axial convection of the turbulent eddies is captured by the term

(3.46) $$\begin{eqnarray}\widehat{R}_{1}\left[\unicode[STIX]{x1D6FC}L_{1}\left(\frac{1}{M_{c}}-\cos \unicode[STIX]{x1D703}_{0}\right)\right].\end{eqnarray}$$

We will call this term ‘radiation efficiency’, realizing that this term has been used in the past under varying contexts. Here it means the efficiency with which a four-dimensional correlation volume $\unicode[STIX]{x1D70F}_{\ast }L_{1}\unicode[STIX]{x03C0}L_{23}^{2}$ radiates sound to the far field at fixed amplitude, frequency and correlation functions. The radiation efficiency is controlled by the convective Mach number $M_{c}=U_{c}/a_{\infty }$ . We gain insight into the underlying physics by considering special values of $M_{c}$ . For a very low-speed jet where $M_{c}\rightarrow 0$ , the argument of $\widehat{R}_{1}$ becomes very large and thus $\widehat{R}_{1}\rightarrow 0$ . This is the limit of zero radiation efficiency. The limit $M_{c}=\infty$ signifies disturbances being transmitted instantaneously throughout the length of the object, like in the case of an oscillating solid cylinder. Then the radiation efficiency becomes $\widehat{R}_{1}(-\unicode[STIX]{x1D6FC}L_{1}\cos \unicode[STIX]{x1D703}_{0})$ and the peak radiation occurs at $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ . In general, the radiation efficiency peaks at $\cos \unicode[STIX]{x1D703}_{0}=1/M_{c}$ , where the argument of $R_{1}$ is zero. For $M_{c}\geqslant 1$ , this represents the well-known Mach wave emission in high-speed jets that occurs near $\unicode[STIX]{x1D703}_{0}=\arccos (1/M_{c})$ . For $M_{c}<1$ the radiation efficiency does not reach its peak value and increases monotonically towards $\unicode[STIX]{x1D703}_{0}=0$ . A physical constraint in applying the above arguments is the spreading rate of the jet flow, which is around 10 $^{\circ }$ . Sound emission at observer angles close to the spreading angle is bound to be influenced by flow–acoustic interactions.

The importance of the radiation efficiency term, coupled with the need to connect it to sound generated by coherent structures, make the selection of $M_{c}$ one of the most critical decisions in the modelling. Here we argue that radiation efficiency is governed by the eddies of the outermost shear layer of the jet. This follows the observation that just outside the jet, in the near pressure field, the pressure distribution reflects the ‘footprint’ of the most energetic eddies, as confirmed by several studies of single-stream jets (Ho Reference Ho1985; Zaman Reference Zaman1986). For multistream jets, we pose an additional condition that these eddies be in direct contact with the ambient fluid, a position supported by recent measurements of the convective velocity in the near field of dual-stream jets (Papamoschou & Phong Reference Papamoschou and Phong2017). The noise radiated by the outer eddies involves a direct coupling between the turbulent motion and the sound field; this does not involve mean flow/acoustic interaction or propagation effects as long as the observer polar angle is not too close to the jet spreading angle. In terms of the mean flow, and following the arguments presented in § 2.1, the action of these eddies is represented by the peak Reynolds stress of the outermost shear layer. Accordingly, we define the outer surface of peak stress (OSPS) as the locus of the first peak of the Reynolds stress as one approaches the jet from the ambient towards the jet axis. Denoting the radial location of the OSPS as $y_{OSPS}(X,\unicode[STIX]{x1D719})$ , and letting $\boldsymbol{y}=(X,y,\unicode[STIX]{x1D719})$ represent the location of a volume element in polar coordinates (figure 2), the convective Mach number of that element is defined as

(3.47) $$\begin{eqnarray}M_{c}(X,y,\unicode[STIX]{x1D719})=\frac{\overline{u}(X,y_{OSPS}(X,\unicode[STIX]{x1D719}),\unicode[STIX]{x1D719})}{a_{\infty }}.\end{eqnarray}$$

This means that all the volume elements at a particular $X$ and $\unicode[STIX]{x1D719}$ are assigned the same value of $M_{c}$ as determined by (3.47).

Accurate detection of the OSPS requires very good resolution of the thin layers near the nozzle exit. The near-field region affects the mid to high frequencies and is thus of paramount importance to aircraft noise. The detection scheme is illustrated in figure 9. The RANS flow field is divided into axial slices of very fine spacing near the nozzle exit and coarser spacing downstream. Each axial slice is divided into fine azimuthal segments, typically in $2.5^{\circ }$ increments. Within each azimuthal segment, the data (velocity, Reynolds stress) are sorted in order of the decreasing radius $y$ . The search process for the first (outermost) peak of the Reynolds stress starts at the radial location where the mean axial velocity is one third of the tertiary exit velocity, a position that is well outside the dividing streamline of the outermost shear layer but still within the jet flow. Starting the search within the jet flow prevents spurious detection of peaks that may occur if one started the search further out where the velocity is very low and the data can be noisy. Denoting $g_{j}$ the discrete values of the Reynolds stress, the operation $h_{j}=\max (g_{j},g_{j+1})$ is carried out as we move inward towards the jet axis. We seek the first occurrence where $h_{j}$ remains invariant for $J$ consecutive points. This indicates that the first peak of the Reynolds stress occurred at point $j-J$ . The proper value of $J$ will depend on the resolution of the RANS data (population of each axial/azimuthal segment) and needs to be determined carefully by the user. Examples of the OSPS will be shown figures 16–18.

Figure 9. Detection scheme for the location of the outer peak of the Reynolds stress (triangular marker).

3.8 RANS-based scales

The correlation length and time scales follow the traditional definitions, based on the RANS flow field, used in past acoustic analogy models (Morris & Farassat Reference Morris and Farassat2002). They are constructed from the turbulent kinetic energy $k$ and the dissipation $\unicode[STIX]{x1D716}$ . The specific dissipation is defined as $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D716}/k$ . The equation that follows describes the axial and transverse length scales, and the time scale.

(3.48) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle L_{1}=C_{1}\frac{k^{3/2}}{\unicode[STIX]{x1D716}}=C_{1}\frac{k^{1/2}}{\unicode[STIX]{x1D6FA}}\\ \displaystyle L_{23}=C_{23}\frac{k^{3/2}}{\unicode[STIX]{x1D716}}=C_{23}\frac{k^{1/2}}{\unicode[STIX]{x1D6FA}}\\ \displaystyle \unicode[STIX]{x1D70F}_{\ast }=C_{4}\frac{k}{\unicode[STIX]{x1D716}}=C_{4}\frac{1}{\unicode[STIX]{x1D6FA}}.\end{array}\right\}\end{eqnarray}$$

The turbulent viscosity $\unicode[STIX]{x1D708}_{T}$ in (3.39) is obtained from the usual dimensional construct

(3.49) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{T}=c_{\unicode[STIX]{x1D707}}\frac{k}{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

The value $c_{\unicode[STIX]{x1D707}}=0.09$ was used here.

4.1 Source parameters

The prediction of the far-field spectral density is dependent on a parameter vector $\boldsymbol{V}=(V_{1},\ldots ,V_{K})$ that defines the correlation functions used in formulating the space–time correlation in (3.19). Here the parameter vector comprises the scale coefficients $C_{1},C_{23},C_{4}$ in (3.48) and the exponent powers $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{4}$ in (3.20). Recall that the exponent power for the cross-stream correlation is fixed at $\unicode[STIX]{x1D6FD}_{23}=2$ .

We denote the parameter vector

(4.1) $$\begin{eqnarray}\boldsymbol{V}=[C_{1},C_{23},C_{4},\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{4}].\end{eqnarray}$$

The far-field power spectral density can then be expressed as

(4.2) $$\begin{eqnarray}S(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714}).\end{eqnarray}$$

It is convenient to work with the sound pressure level (SPL) spectrum, in units of decibels. The modelled SPL spectrum is

(4.3) $$\begin{eqnarray}\text{SPL}_{mod}(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})=10\log _{10}\left[\frac{S(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})}{S_{norm}}\right],\end{eqnarray}$$

where $S_{norm}=4\times 10^{-10}~\text{Pa}^{2}$ is the commonly used normalization value. The experimental SPL spectrum is $\text{SPL}_{exp}({\mathcal{R}}_{exp},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})$ where ${\mathcal{R}}_{exp}$ is the microphone distance or the distance to which the experimental spectrum is referenced.

4.2 Determination of parameter vector

Determination of the parameter vector is based on knowledge of the spectral density of the axisymmetric reference jet. Specifically, we seek a parameter vector that minimizes the difference between the modelled and experimental SPL spectra for the reference jet: $\text{SPL}_{mod}^{ref}(\boldsymbol{V},R,\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714})$ and $\text{SPL}_{exp}^{ref}({\mathcal{R}}_{exp},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714})$ , respectively. We facilitate the optimization by normalizing the experimental and modelled spectral densities by their respective maximum values versus frequency. Equivalently, in units of decibels we subtract the maximum values. The normalization removes the effect of the distances ${\mathcal{R}}$ and ${\mathcal{R}}_{exp}$ , so the normalized spectra depend only on the parameter vector (for the modelled spectrum), the observer polar angle and the frequency. The normalized modelled and experimental SPL spectra for the reference jet are:

(4.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \overline{\text{SPL}}_{mod}^{ref}(\boldsymbol{V},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714})=\text{SPL}_{mod}^{ref}(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714})-\text{SPL}_{mod,max}^{ref}(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0})\\ \displaystyle \overline{\text{SPL}}_{exp}^{ref}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D714})=\text{SPL}_{exp}^{ref}({\mathcal{R}}_{exp},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714})-\text{SPL}_{exp,max}^{ref}({\mathcal{R}}_{exp},\unicode[STIX]{x1D703}_{0}).\end{array}\right\}\end{eqnarray}$$

This normalization removes the amplitude as a variable, so we are concerned only with matching the shape of the spectra.

We seek to minimize the difference between the modelled and experimental spectra at observer polar angle $\unicode[STIX]{x1D703}_{0}$ and at a set of frequencies $\unicode[STIX]{x1D714}_{n},n=1,\ldots ,N$ . We construct the cost function

(4.5) $$\begin{eqnarray}F(\boldsymbol{V})=\sqrt{\frac{1}{N}\mathop{\sum }_{n=1}^{N}[\overline{\text{SPL}}_{mod}^{ref}(\boldsymbol{V},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714}_{n})-\overline{\text{SPL}}_{exp}^{ref}(\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D714}_{n})]^{2}}+\mathop{\sum }_{k=1}^{K}P_{k}(V_{k}).\end{eqnarray}$$

The square root term represents the ‘error’ between model and experiment in units of decibels; $P_{k}$ are appropriately defined penalty functions that constrain the parameters within reasonable ranges. The parameter vector $\boldsymbol{V}$ is determined by minimizing the cost function. The minimization process of (4.5) utilized the restarted conjugate-gradient method of Shanno & Phua (Reference Shanno and Phua1976) (ACM TOM Algorithm 500). The minimization typically used $N=10$ frequencies spaced at equal logarithmic intervals, covering the entire relevant part of the spectrum. The scheme converged after approximately 30 function calls to an error around 1.0 dB and zero penalty function.

4.3 Application to non-reference jets

Upon a satisfactory match of the reference modelled and experimental spectra, the parameter vector $\boldsymbol{V}$ becomes determined. This parameter vector is now applied to the non-reference (typically asymmetric) jet, yielding $\text{SPL}_{mod}(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})$ . Direct comparison with the SPL spectrum of the experimental non-reference jet is enabled by the amplitude adjustment

(4.6) $$\begin{eqnarray}\text{SPL}_{mod}(\boldsymbol{V},{\mathcal{R}}_{exp},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})=\text{SPL}_{mod}(\boldsymbol{V},{\mathcal{R}},\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D714})+\text{SPL}_{exp,max}^{ref}-\text{SPL}_{mod,max}^{ref}.\end{eqnarray}$$

5.1 Experimental

5.1.1 Experimental set-up

The experiments utilized three-stream nozzles as part of UCI’s recent effort in characterizing and suppressing noise from three-stream jets representative of the exhausts of future supersonic aircraft. The nozzles comprised axisymmetric (reference) configurations as well as asymmetric configurations that involved reshaping of the secondary and tertiary ducts that surround the primary duct. The intent of the asymmetric nozzles was to reduce noise directed downward, that is, towards airport communities. This report covers two sets of nozzles, nominal plug and enlarged plug. The enlarged-plug nozzles were motivated by sonic-boom compatibility, as explained in Phong & Papamoschou (Reference Phong and Papamoschou2017). Nozzle naming is consistent with that used in past reports for ease of reference. The subscripts $p$ , $s$ and $t$ are used to denote the primary, secondary and tertiary streams, respectively. Key features of the nozzles are presented in figures 10 and 11 for the nominal- and enlarged-plug sets, respectively, including the azimuthal distributions of the widths of the annuli at the terminations of the secondary and tertiary ducts. All the nozzles share the same exit areas. The effective (area-based) primary exit diameter is $D_{p,eff}=13.33~\text{mm}$ and the area ratios are $A_{s}/A_{p}=1.44$ and $A_{t}/A_{p}=1.06$ . The plug diameter and length (the latter measured from the exit plane of the primary duct) are 6.08 and 18.26 mm, respectively, for the nominal-plug set; and 11.90 and 38.40 mm, respectively, for the enlarged-plug set. Denoting the width of the tertiary annulus $W_{t}$ , and noting that $D_{p,eff}$ provides a scale for the lateral extent of the strongest noise sources, we use the ratio $W_{t}/D_{p,eff}$ to describe the relative size of the tertiary stream. Similarly, for the secondary stream we use the ratio $W_{s}/D_{p,eff}$ , where $W_{s}$ is the width of the secondary annulus. The azimuthal angle $\unicode[STIX]{x1D719}$ is defined relative to the downward vertical direction.

Considering first the set of nominal-plug designs (figure 10), variation of the nozzle shape involved changing the exit shape of the tertiary duct only. Nozzle AXI03U is a coaxial design, and is used as the reference nozzle for this set. The tertiary annulus thickness is uniform with $W_{t}/D_{p,eff}=0.119$ . Nozzle ECC06U features a shaped offset tertiary duct wherein the tertiary annulus becomes thicker over the azimuthal range $-110^{\circ }\leqslant \unicode[STIX]{x1D719}\leqslant 110^{\circ }$ and thinner outside this range. The ratio $W_{t}/D_{p,eff}$ is constant at 0.155 over $-60^{\circ }\leqslant \unicode[STIX]{x1D719}\leqslant 60^{\circ }$ and thins gradually to 0.05 near the top of the nozzle. The tertiary outer wall is recessed at the top of the nozzle to prevent formation of a long thin duct. Nozzle ECC08U retains the same features of ECC06U but adds a wedge deflector at the top of the tertiary duct. The deflector dimensions are $\ell /D_{p,eff}=1.50$ and $\unicode[STIX]{x1D6FF}=25^{\circ }$ , where $\ell$ is the deflector length and $\unicode[STIX]{x1D6FF}$ is the wedge half-angle. The deflector blocks an azimuthal extent of 40 $^{\circ }$ at the top of the nozzle, which allows thickening of the tertiary annulus on the underside of the nozzle while preserving the cross-sectional area. The ratio $W_{t}/D_{p,eff}$ increases to 0.165 over $-60^{\circ }\leqslant \unicode[STIX]{x1D719}\leqslant 60^{\circ }$ . The tertiary exit diameters are $D_{t}=31.15$ , 32.09 and 32.19 mm for AXI03U, ECC06U and ECC08U, respectively. The slight variation in outer diameter is due to the reshaping of the tertiary duct while maintaining constant area.

The investigation of the enlarged-plug nozzles (figure 11) included variations of the shapes of the secondary and tertiary ducts. Nozzle AXI04U is a coaxial reference design, with uniform distributions of the secondary and tertiary annuli at $W_{s}/D_{p,eff}=0.219$ and $W_{t}/D_{p,eff}=0.127$ , respectively. Nozzle ECC12U features asymmetric distributions of both the secondary and tertiary annuli. The overall design of the tertiary duct is similar to that of ECC08U, with a wedge deflector having $\ell /D_{p,eff}=1.20$ and $\unicode[STIX]{x1D6FF}=18^{\circ }$ . The tertiary exit diameters are $D_{t}=38.10$ and 40.60 mm for AXI04U and ECC12U, respectively.

Figure 10. Three-stream nozzles with nominal plug. Left to right: perspective view, cross-sectional view and azimuthal distribution of the tertiary annulus width. Azimuthal angle $\unicode[STIX]{x1D719}$ is defined relative to the downward vertical direction.

Figure 11. Three-stream nozzles with enlarged plug. Left to right: perspective view, cross-sectional view and azimuthal distributions of the widths of the secondary and tertiary annuli. Azimuthal angle $\unicode[STIX]{x1D719}$ is defined relative to the downward vertical direction.

The nozzles were tested at cycle conditions that were representative of three-stream turbofan engines operating at take-off power. Table 2 lists the main flow conditions at the nozzle exit. The Reynolds number of the primary stream, based on $D_{p,eff}$ , was 280 000. The exit velocities ( $U$ ) and Mach numbers ( $M$ ) were matched exactly using helium–air mixture jets (Papamoschou Reference Papamoschou2006).

Table 2. Cycle point for three-stream jets.

Noise measurements were conducted inside an anechoic chamber equipped with twenty-four 1/8-in. condenser microphones (Bruel & Kjaer, Model 4138) with frequency response up to 120 kHz. Twelve microphones were mounted on a downward arm (azimuth angle $\unicode[STIX]{x1D719}=0^{\circ }$ ) and twelve were installed on a sideline arm ( $\unicode[STIX]{x1D719}=60^{\circ }$ ). On each arm, the polar angle $\unicode[STIX]{x1D703}$ ranged approximately from $20$ to $120^{\circ }$ relative to the downstream jet axis, and the distance to the nozzle exit ${\mathcal{R}}_{exp}$ ranges from 0.92 to 1.23 m. This arrangement enabled simultaneous measurement of the downward and sideline noise at all the polar angles of interest. In selected cases, rotation of the nozzle enabled a richer coverage of azimuthal angles. The microphones were connected, in groups of four, to six conditioning amplifiers (Bruel & Kjaer, Model 2690-A-0S4). The 24 outputs of the amplifiers were sampled simultaneously, at 250 kHz per channel, by three 8-channel multi-function data acquisition boards (National Instruments PCI-6143) installed in a Dell Precision T7400 computer with a Xeon quad-core processor. National Instruments LabView software is used to acquire the signals. The temperature and humidity inside the anechoic chamber are recorded to enable computation of the atmospheric absorption. The microphone signals were conditioned with a high-pass filter set at 300 Hz. Narrowband spectra were computed using a 4096-point fast Fourier transform, yielding a frequency resolution of 61 Hz. The spectra were corrected for microphone actuator response, microphone free field response and atmospheric absorption, thus resulting in lossless spectra.

5.1.2 Acoustic results

Figure 12 plots narrowband SPL spectra in the downward direction ( $\unicode[STIX]{x1D719}_{0}=0^{\circ }$ ) for jets ECC06U and ECC08U, with comparison to AXI03U; and for jet ECC12U, with comparison to AXI04U. The spectra of the two reference coaxial jets, AXI03U (nominal plug) and AXI04U (enlarged plug), are very similar, with AXI04U having a 1–2 dB benefit at medium to high frequencies. This is partly attributed to source–observer shielding by the enlarged plug (Bauer, Kibens & Wlezien Reference Bauer, Kibens and Wlezien1982; Chase, Garzón & Papamoschou Reference Chase, Garzón and Papamoschou2013). Considering the nominal-plug asymmetric jets, ECC06U offers reductions of the order of 10 dB at polar angles near the angle of peak emission and in the medium to high frequency range. Addition of the wedge deflector in nozzle ECC08U increases these reductions to ${\sim}$ 15 dB. For the enlarged-plug asymmetric jet ECC12U, the combined asymmetry of the secondary and tertiary streams increases the noise reduction to ${\sim}17~\text{dB}$ . Change in noise emission in the broadside direction ( $\unicode[STIX]{x1D703}\approx 90^{\circ }$ ) is dependent on the aggressiveness of the asymmetry, with jets ECC06U and ECC08U showing a slight increase, and jet ECC12U causing a moderate increase. Spectra at different azimuthal angles will be presented later in the discussion of the model predictions. Considering the sideline direction $\unicode[STIX]{x1D719}_{0}=60^{\circ }$ , jets ECC06U and ECC08U do not produce a significant benefit, while jet ECC12U provides a distinct reduction at low frequency.

Figure 12. Narrowband far-field spectra at various polar angles $\unicode[STIX]{x1D703}$ for jets (a) ECC06U and (b) ECC08U, with comparison to reference jet AXI03U (red); and (c) jet ECC12U with comparison to reference jet AXI04U (red). Azimuthal direction $\unicode[STIX]{x1D719}=0^{\circ }$ (downward).

5.2 Computational

5.2.2 Relevant statistics

For conciseness, the presentation of the RANS solutions will focus on the nominal-plug jets AXI03U, ECC06U and ECC08U. The trends delineated will apply broadly to the enlarged-plug jets AXI04U and ECC12U, and distinctions will be pointed out in § 5.2.3. The discussion starts with contour maps of the mean axial velocity on the symmetry ( $X{-}Y$ ) plane, shown in figure 13. $X=0$ signifies the location of the plug tip. For the axisymmetric jet AXI03U, the high-speed region extends to approximately seven tertiary diameters. The asymmetric delivery of the tertiary stream shortens moderately the high-speed region to $x/D_{t}\approx 6.5$ for jet ECC06U and $x/D_{t}\approx 6.0$ for jet ECC08U. For the asymmetric jets, the thickening of the low-speed flow on the underside of the jet is evident. Some distortion of the high-speed region is also noticeable. The plug wake is evident for all the jets.

Figure 13. Distribution of normalized mean axial velocity $\overline{u}/U_{p}$ on the symmetry plane of jets: (a) AXI03U; (b) ECC06U; and (c) ECC08U.

Figure 14. Distribution of normalized turbulent kinetic energy $k/U_{p}^{2}$ on the symmetry plane of jets: (a) AXI03U; (b) ECC06U; and (c) ECC08U.

Figure 14 plots the corresponding distributions of the normalized turbulent kinetic energy $k/U_{p}^{2}$ . For jet AXI03U, in the vicinity of the nozzle exit we can distinguish clearly the outer shear layer (between the tertiary stream and the ambient) and the weaker inner shear layer (between the primary and secondary streams). The middle shear layer (between the secondary and tertiary streams) is not apparent. We gain an understanding of the relative strengths of the shear layers by considering that the turbulent kinetic energy (or any component of the Reynolds stress) scales approximately as $(\unicode[STIX]{x0394}U)^{2}$ , where $\unicode[STIX]{x0394}U$ is the velocity difference across a particular shear layer. It is insightful to assess the strengths in terms of the dimensionless ratio

(5.1) $$\begin{eqnarray}r_{k}=\frac{(\unicode[STIX]{x0394}U)^{2}}{(U_{max}-U_{\infty })^{2}},\end{eqnarray}$$

where the denominator signifies the largest possible strength, with $U_{max}$ the maximum fully expanded velocity (in this case, the primary exit velocity) and $U_{\infty }$ the ambient velocity (in this case, zero). From the values of table 2 we have $r_{k}=0.139$ , 0.022 and 0.228 for the inner, middle and outer shear layer, respectively. This dimensional argument helps explain the dominance of the outer shear layer and the near invisibility of the middle shear layer. At approximately $x/D_{t}=2.5$ the secondary and tertiary streams are completely merged with primary shear layer. Now the primary eddies are in direct contact with the ambient, resulting in a rapid increase in the turbulent kinetic energy which reaches its maximum value near $x/D_{t}=5.5$ . Here the ratio $r_{k}$ is close to unity. The location of peak turbulent kinetic energy is very close to the end of the primary potential core, as delineated by the cone-like region of very low turbulent intensity. Further downstream, the turbulent kinetic energy declines as the jet mean velocity drops. It is notable that at large downstream distances the profile for the turbulent kinetic energy becomes Gaussian-like.

Figure 15. Distribution of normalized Reynolds stress $g/U_{p}^{2}$ on the symmetry plane of jets: (a) AXI03U; (b) ECC06U; and (c) ECC08U. Compare with the distribution of the turbulent kinetic energy in figure 14, particularly past $x/D_{t}=5$ .

For the asymmetric jets ECC06U and ECC08U the secondary and tertiary streams extend much longer on the underside of the jet, reaching $x/D_{t}=4$ for ECC06U and $x/D_{t}=5$ for ECC08U. Their elongation prevents the contact of the primary eddies with the ambient until past the end of the primary core. As a result, the turbulent kinetic energy on the underside of the jet is significantly reduced; this effect is most pronounced for jet ECC08U. The primary potential core is moderately reduced to $x/D_{t}=5.2$ for ECC06U and $x/D_{t}=4.8$ for ECC08U. All these trends bode well for noise reduction in the downward direction. However, looking at the distributions of figure 14(b,c) one has difficulty drawing a line distinguishing unambiguously the upper and lower sides of the jet.

We turn our attention to the distribution of the normalized ‘Reynolds stress’ $g/U_{p}^{2}$ , depicted in figure 15. The distribution of $g$ near the nozzle exit is very similar to that of $k$ , so the same arguments apply there. However, further downstream we note a significant difference between the $g$ and $k$ fields. The $g$ field has a clear minimum in the interior of the jet, which is practically zero once the wake region dissipates. The location of the minimum coincides with the jet centroid defined by (3.13). Downstream of the end of the potential core, the distribution of $g$ has two lobes representing the turbulence production near the edge of the jet. Compare with the Gaussian-like profile for $k$ in figure 14. We can clearly delineate the upper and lower sides of the jet, as demarcated by the location of the minimum $g$ . The lower side experiences dramatic reductions in $g$ , particularly in jet ECC08U. These results reinforce the view discussed in § 2.1 that the Reynolds stress is the appropriate statistical quantity to represent the action of the turbulent eddies.

Figure 16. Location of the outer surface of peak stress (OSPS) on the symmetry plane of jets (a) AXI03U and (b) ECC08U. Contour map shows the Reynolds stress field $g/U_{p}^{2}$ . White dots indicate the location of the outermost peak of $g$ . Solid white line is the jet centroid as defined by (3.13).

Figure 17. Outer surface of peak stress (OSPS) for jets (a) AXI03U, (b) ECC06U and (c) ECC08U. Contour levels show distribution of $M_{c}$ on the OSPS.

5.2.3 OSPS

We now examine the geometry of the outer surface of peak stress. Figure 16 plots the location of the OSPS on the symmetry plane of jets AXI03U and ECC08U. The plots of the OSPS are overlaid on contour maps of the Reynolds stress $g$ . For the axisymmetric jet, the OSPS experiences a sudden convergence where the outer streams become totally mixed with the primary shear later, near $x/D_{t}=2.5$ . This is followed by a gradual convergence near the end of the primary potential core, downstream of which the OSPS diverges slowly. For the asymmetric jet, the OSPS on the top side of the jet is similar to that for the axisymmetric jet. On the bottom side, the OSPS slightly diverges in the initial region of the jet, then suddenly shifts inward near $x/D_{t}=5$ , the location where the outer streams are totally mixed with the primary stream. For the first five diameters or so the OSPS on the bottom side is in a low velocity region of the jet, meaning that the convective Mach number there is low subsonic. This has a large impact on the prediction of noise emission from this region. Figure 16 also depicts the location of the centroid, which is seen to coincide with the locus of minimum Reynolds stress.

We gain further insight by examining three-dimensional views of the OSPS, plotted in figure 17. The distribution of the convective Mach number $M_{c}$ is shown as contour levels on the surfaces. The views cover the axial range $-0.5\leqslant x/D_{t}\leqslant 10$ . For jet AXI03U, the transition of the OSPS from the tertiary to the secondary to the primary shear layer is evident. The convective Mach number reaches the peak value of 1.20 shortly downstream of the transition to the primary shear layer. For jets ECC06U and ECC08U we observe a progressively stronger reshaping of the OSPS, starting at the lateral sides and progressing to the lower side. For jet ECC08U, on the underside of the OSPS, the convective Mach number is as low as 0.35. This corresponds to of the order of a 100-fold reduction in radiation efficiency in that region, as will be explained in § 7. On the upper half of the jet, the shape of the OSPS and the $M_{c}$ distributions are similar to those of the axisymmetric case.

For completeness we also examine the OSPS of the enlarged-plug jets AXI04U and ECC12U in figure 18. For jet AXI04U, the shape of the OSPS and $M_{c}$ distribution on it are similar to those of jet AXI03U. However, jet ECC12U shows a more distorted OSPS relative to ECC06U or ECC08U, a result of the combined asymmetry of the secondary and tertiary ducts. The secondary and tertiary flows are more evenly distributed on the underside of the jet, which suggests that this jet may have better sideline noise reduction than ECC06U or ECC08U.

Figure 18. Outer surface of peak stress (OSPS) for jets (a) AXI04U and (b) ECC12U. Contour levels show distribution of $M_{c}$ on the OSPS.

It is also instructive to examine the distribution of $M_{c}$ versus axial and azimuthal directions, as is done in figure 19 for the nominal-plug jets. The largest reduction in $M_{c}$ occurs in the downward direction $\unicode[STIX]{x1D719}=0$ . There the peak value of $M_{c}$ reduces from 1.2 for AXI03U to 1.0 for ECC06U to 0.8 for ECC08U. It is also noted that the distribution of the peak becomes confined to only about one diameter for ECC08U, versus three diameters for AXI03U. For ECC08U, the low- $M_{c}$ region persists up to about $\unicode[STIX]{x1D719}=50^{\circ }$ , after which the distribution becomes similar to that of the axisymmetric jet. The significant improvement in suppression of convective Mach number in jet ECC08U, versus jet ECC06U, came from a rather subtle reshaping of the tertiary duct, as illustrated in figure 10. This underscores the sensitivity of the offset-stream method to the fine details of the duct reshaping. The convective Mach number distribution for jet ECC12U, plotted in figure 20, shows a large suppression of $M_{c}$ extending up to $\unicode[STIX]{x1D719}=40^{\circ }$ . It is evident that the combined asymmetry of the secondary and tertiary streams results in a wider azimuthal sector with low $M_{c}$ .

The information presented in figures 17–20 constitutes important feedback one obtains from the RANS solution before proceeding to the acoustic analogy step. Examinations of the OSPS and the associated $M_{c}$ distribution provide strong clues whether a particular nozzle arrangement will produce a quiet jet and in what azimuthal directions. In the particular examples shown here, one can expect jet ECC08U to bring significant noise benefit in the downward direction but probably little or no benefit in the sideline direction, near $\unicode[STIX]{x1D719}_{0}=60^{\circ }$ . On the other hand, jet ECC12U may be expected to provide a strong downward reduction combined with a distinct, but not as large, sideline reduction.

Figure 19. Axial–azimuthal distribution of convective Mach number $M_{c}$ on the OSPS of jets AXI03U (a), ECC06U (b) and ECC08U (c).

Figure 20. Axial–azimuthal distribution of convective Mach number $M_{c}$ on the OSPS of jets AXI04U (a) and ECC12U (b).