2.1. Governing equations in resolvent form

The compressible N-S equations are given in compact form as

(2.1)\begin{equation} \partial_t\boldsymbol{q}=\mathcal{N}(\boldsymbol{q}), \end{equation}

where $\mathcal {N}$ is the N-S operator, $\boldsymbol {q}=[\nu, u_x, u_r, u_\theta, p]^\top$ is the state vector, $\nu$ is specific volume and $p$ is pressure, $\boldsymbol {u}=[u_x, u_r, u_\theta ]^\top$ is the velocity vector in cylindrical coordinates, and $x$, $r$ and $\theta$ refer respectively to the streamwise, radial and azimuthal directions. All variables are non-dimensionalised by the ambient speed of sound, $c_\infty$, the density, $\rho _\infty$, and the nozzle diameter, $D$. We consider a discretised system in space, for which linearisation around the mean, $\bar {\boldsymbol {q}}$, yields

(2.2)\begin{equation} \partial_t\boldsymbol{q}^{\prime}-{{\boldsymbol{\mathsf{A}}}}\boldsymbol{q}^{\prime}=\boldsymbol{f}, \end{equation}

where ${{\boldsymbol{\mathsf{A}}}}=\partial _q\mathcal {N}|_{\bar {\boldsymbol {q}}}$ is the linear operator obtained from the Jacobian of $\mathcal {N}$ and $\boldsymbol {f}$ contains all remaining nonlinear terms, which are referred to henceforth as the forcing terms. Equation (2.2) is Fourier transformed and rearranged to obtain

(2.3)\begin{equation} \hat{\boldsymbol{q}}={{\boldsymbol{\mathsf{R}}}}\hat{\boldsymbol{f}}, \end{equation}

where the hat indicates Fourier-transformed quantities and

(2.4)\begin{equation} {{\boldsymbol{\mathsf{R}}}}=(i\omega{{\boldsymbol{\mathsf{I}}}}-{{\boldsymbol{\mathsf{A}}}})^{{-}1} \end{equation}

is the resolvent operator. The resolvent operator can be modified to limit the response to prescribed measurements via a linear transformation of the state using a matrix, ${{\boldsymbol{\mathsf{C}}}}$,

(2.5)\begin{equation} \hat{\boldsymbol{y}}={{\boldsymbol{\mathsf{C}}}}\hat{\boldsymbol{q}}, \end{equation}

such that the input-output relation between forcing, $\hat {\boldsymbol {f}}$, and measurement, $\hat {\boldsymbol {y}}$, is given by

(2.6)\begin{equation} \hat{\boldsymbol{y}}={{\boldsymbol{\mathsf{C}}}}{{\boldsymbol{\mathsf{R}}}}\hat{\boldsymbol{f}}.\end{equation}

The measurement matrix ${{\boldsymbol{\mathsf{C}}}}$ can be space-, variable- and even frequency-dependent, although a frequency-dependent ${{\boldsymbol{\mathsf{C}}}}$ would imply convoluting the state in the time domain to measure $\boldsymbol {y}$. Throughout this paper, we will focus on the pressure in the acoustic field as our measured quantity; therefore, ${{\boldsymbol{\mathsf{C}}}}$ is an $N_A\times (N_V\times N)$ matrix that is one in the elements that correspond to pressure in the acoustic field and zero for the rest. Here, $N_A$ and $N$ denote the number of discrete points in the acoustic field and the full domain, respectively, and $N_V$ denotes the number of variables, e.g. $N_V=5$ for the compressible N-S equations. Note that spatial discretisation of a system with $N_V$ variables leads to a state vector with $N_V\times N$ elements.

It is also possible to impose restrictions on the forcing in (2.6) via a control matrix, ${{\boldsymbol{\mathsf{B}}}}$, as

(2.7)\begin{equation} \hat{\boldsymbol{y}}_B=\tilde{{{\boldsymbol{\mathsf{R}}}}}\hat{\boldsymbol{f}},\end{equation}

where $\tilde {{{\boldsymbol{\mathsf{R}}}}}\triangleq {{\boldsymbol{\mathsf{C}}}}{{\boldsymbol{\mathsf{R}}}}{{\boldsymbol{\mathsf{B}}}}$ denotes the modified resolvent operator. Note that $\hat {\boldsymbol {y}}_B\neq \hat {\boldsymbol {y}}$ in general. Additionally, ${{\boldsymbol{\mathsf{B}}}}$ will later be used to identify irrelevant forcing components for the jet-noise problem, i.e. those terms which, when suppressed by ${{\boldsymbol{\mathsf{B}}}}$, do not lead to changes in the acoustic field, such that $\hat {\boldsymbol {y}}_B\approx \hat {\boldsymbol {y}}$.

2.2. Resolvent-based extended spectral proper orthogonal decomposition

One of the goals of this study, and indeed one of the broader goals of jet-noise modelling, is to obtain simplified representations or models of the nonlinear interactions that underpin jet noise. One such approach is to search for a useful rank reduction. A known trait of turbulent jets and their sound is the marked difference in complexity between the turbulence and acoustic fields. This difference implies that, given a discretised turbulent jet database, it may be possible to represent the acoustic field using a compact basis with a small number of vectors, yielding a low-rank system, while a substantially larger basis will be required to define the turbulent fluctuations, and thus the forcing, in the near field, yielding a high-rank system. The fact that one can accurately calculate the acoustic field using (2.7) (or alternatively, using an acoustic analogy) assuming full access to the source implies that the resolvent operator (or the Green's function in an acoustic analogy) filters the silent structures from the high-rank turbulent source region leading to a low-rank acoustic field. A central idea underlying the approach we follow here is as follows: given the linear relation between the forcing and the target response in (2.6), the low-rank structure of the acoustic field suggests the existence of a low-rank, acoustically active forcing subspace. It is necessary to identify this subspace, because it is there that modelling work can be done.

There exist several ways to identify the forcing associated with a given response. A detailed analysis was provided by Karban et al. (Reference Karban, Martini, Cavalieri, Lesshafft and Jordan2022a), where a method referred to as RESPOD was used to achieve the abovementioned identification. RESPOD is based on the extended proper orthogonal decomposition presented by Borée (Reference Borée2003) and is related to SPOD (Lumley Reference Lumley1970; Picard & Delville Reference Picard and Delville2000; Towne et al. Reference Towne, Schmidt and Colonius2018). The aim in RESPOD is to find a forcing mode, $\boldsymbol{\chi }^{(p)}$, that is correlated with the $p$th SPOD mode, $\boldsymbol{\psi }^{(p)}$, of the measured response, $\hat {\boldsymbol {y}}$. It was first presented by Towne et al. (Reference Towne, Colonius, Jordan, Cavalieri and Brès2015) and later discussed by Karban et al. (Reference Karban, Martini, Cavalieri, Lesshafft and Jordan2022a) to identify the forcing structures that generate wall-attached eddies. Here, we briefly review the method highlighting how it can be adapted to find the low-rank forcing subspace associated with sound generation.

For a given ensemble of realisations $\hat {{{\boldsymbol{\mathsf{Y}}}}}=[\hat {\boldsymbol {y}}_1\cdots \hat {\boldsymbol {y}}_P]$ of an $N$ dimensional discretised system, where $P$ is the number of Fourier realisations, SPOD involves eigendecomposition of the CSD matrix $\hat {{{\boldsymbol{\mathsf{S}}}}}\triangleq \hat {{{\boldsymbol{\mathsf{Y}}}}}\hat {{{\boldsymbol{\mathsf{Y}}}}}^H$,

(2.8)\begin{equation} \hat{{{\boldsymbol{\mathsf{S}}}}} = \hat{\boldsymbol{\varPsi}}\hat{\boldsymbol{\varLambda}}\hat{\boldsymbol{\varPsi}}^H, \end{equation}

where the eigenvectors, $\hat {\boldsymbol{\varPsi }}$, and eigenvalues, $\hat {\boldsymbol{\varLambda }}$, of $\hat {{{\boldsymbol{\mathsf{S}}}}}$ are the SPOD modes and gains, respectively. An alternative way to obtain the SPOD modes, as shown by Towne et al. (Reference Towne, Schmidt and Colonius2018), is to perform the eigendecomposition

(2.9)\begin{equation} \hat{{{\boldsymbol{\mathsf{Y}}}}}^H{{\boldsymbol{\mathsf{W}}}}\hat{{{\boldsymbol{\mathsf{Y}}}}}=\hat{\boldsymbol{\varTheta}}\hat{\boldsymbol{\varLambda}} \hat{\boldsymbol{\varTheta}}^H, \end{equation}

where ${{\boldsymbol{\mathsf{W}}}}$ is a positive-definite weight matrix and $\hat {\boldsymbol{\varTheta }}$ is a matrix containing the eigenmodes of $\hat {{{\boldsymbol{\mathsf{Y}}}}}^H{{\boldsymbol{\mathsf{W}}}}\hat {{{\boldsymbol{\mathsf{Y}}}}}$. The eigenmodes, $\hat {\boldsymbol{\varPsi }}$ and $\hat {\boldsymbol{\varTheta }}$, are related as

(2.10)\begin{equation} \hat{\boldsymbol{\varPsi}}=\hat{{{\boldsymbol{\mathsf{Y}}}}}\hat{\boldsymbol{\varTheta}}\hat{\boldsymbol{\varLambda}}^{{-}1/2}, \end{equation}

or alternatively as

(2.11)\begin{equation} \hat{\boldsymbol{\varTheta}}=\hat{{{\boldsymbol{\mathsf{Y}}}}}^H{{\boldsymbol{\mathsf{W}}}}\hat{\boldsymbol{\varPsi}}\hat{\boldsymbol{\varLambda}}^{{-}1/2}. \end{equation}

Equation (2.10) indicates that it is possible to obtain the SPOD modes as a linear combination of the realisations. Writing (2.7) with ${{\boldsymbol{\mathsf{B}}}} = {{\boldsymbol{\mathsf{I}}}}$ for the ensemble of realisations as

(2.12)\begin{equation} \hat{{{\boldsymbol{\mathsf{Y}}}}}=\tilde{{{\boldsymbol{\mathsf{R}}}}}\hat{{{\boldsymbol{\mathsf{F}}}}}, \end{equation}

where $\hat {{{\boldsymbol{\mathsf{F}}}}}\triangleq [\hat {\boldsymbol {f}}_1\cdots \hat {\boldsymbol {f}}_P]$ is the matrix of the forcing realisations, and multiplying (2.12) by $\hat {\boldsymbol{\varTheta }}\hat {\boldsymbol{\varLambda }}^{-1/2}$ yields

(2.13)\begin{equation} \hat{\boldsymbol{\varPsi}}=\tilde{{{\boldsymbol{\mathsf{R}}}}}\hat{{{\boldsymbol{\mathsf{F}}}}}\hat{\boldsymbol{\varTheta}} \hat{\boldsymbol{\varLambda}}^{{-}1/2}. \end{equation}

Equation (2.13) can be written for the $p$th SPOD mode by extracting the corresponding columns in the matrices, $\hat {\boldsymbol{\varPsi }}$, $\hat {\boldsymbol{\varTheta }}$ and $\hat {\boldsymbol{\varLambda }}$,

(2.14)\begin{equation} \hat{\boldsymbol{\psi}}^{(p)}=\tilde{{{\boldsymbol{\mathsf{R}}}}}\hat{{{\boldsymbol{\mathsf{F}}}}}\hat{\boldsymbol{\theta}}^{(p)}{\lambda^{(p)}}^{{-}1/2}, \end{equation}

where $\hat {\boldsymbol{\theta }}^{(p)}$ denotes the $p$th column in $\hat {\boldsymbol{\varTheta }}$ and $\lambda ^{(p)}$ denotes the $p$th diagonal element in $\hat {\boldsymbol{\varLambda }}$. We then define the RESPOD mode of the forcing, $\hat {\boldsymbol{\chi }}^{(p)}$, as

(2.15)\begin{equation} \hat{\boldsymbol{\chi}}^{(p)}\triangleq\hat{{{\boldsymbol{\mathsf{F}}}}}\hat{\boldsymbol{\theta}}^{(p)}{\lambda^{(p)}}^{{-}1/2}. \end{equation}

Following Borée (Reference Borée2003), it can be shown that the RESPOD mode, $\boldsymbol{\chi }^{(p)}$, contains all the forcing components correlated with the SPOD mode, $\boldsymbol{\psi }^{(p)}$. Furthermore, (2.14) indicates that the two modes are connected via the resolvent operator as

(2.16)\begin{equation} \hat{\boldsymbol{\psi}}^{(p)}=\tilde{{{\boldsymbol{\mathsf{R}}}}}\hat{\boldsymbol{\chi}}^{(p)}. \end{equation}

We explain in § 3.2 how to calculate (2.16). The ability to identify a RESPOD mode of forcing with each SPOD mode of the response implies, for the jet-noise problem, that one can use this approach to identify the low-rank forcing subspace that is correlated with the low-rank acoustic field, and which, furthermore, generates the low-rank acoustic field when applied to the resolvent operator. Note that the above analysis could be applied when using an acoustic analogy as well. A Green's function that is used to solve the acoustic propagation problem in an acoustic analogy establishes a linear relation between the source and the acoustic field, and hence, can be described by a matrix left-multiplying the source for the discretised system, which reduces it to an input-output form as shown above (Abreu et al. Reference Abreu, Nogueira, Nilton and Cavalieri2019).

We will use this approach to obtain a low-rank representation of the forcing that is responsible for most of the acoustic energy radiated by a turbulent jet. The advantage of identifying a low-rank forcing is twofold: (i) one needs to model only that piece in the forcing; (ii) not dealing with the entire CSD matrix of the forcing is convenient in terms of computing the response, which would otherwise require multiplication of the resolvent operator by an $N\times N$ matrix, where $N$ is generally very large. Given the number of realisations, $P$, and the degree of freedom, $N$, which usually satisfy $P\ll N$, this approach provides a computationally inexpensive means of obtaining a low-rank system.

3.1. Large-eddy simulation database

The numerical database used in this study to develop an empirical forcing model consists of LES of four subsonic jets, one at jet Mach number, $M_j\triangleq U_j/c_j=0.4$, with no flight effect and others at $M_j=0.9$ with or without flight effect. We use three other LES databases at $M_j=0.7$ with or without flight effect and at $M_j=0.8$ without flight effect to test the model. The LES was conducted using the unstructured flow solver ‘Charles’ (Brès et al. Reference Brès, Ham, Nichols and Lele2017). In all cases, the jets are isothermal and ideally expanded. Two of the cases at $M_j=0.9$ contain a flight stream at $M_\infty =0.15$ and 0.3, respectively. All of the jets are turbulent thanks to a synthetic forcing applied inside the nozzle, generating a fully turbulent boundary layer at the jet exit (Brès et al. Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018; Maia et al. Reference Maia, Brès, Lesshafft and Jordan2022). Other parameters related to each simulation are tabulated in table 1, where $Re=\rho _j U_jD/\mu _j$ denotes the Reynolds number, $\mu$ denotes the dynamic viscosity, $D$ is nozzle diameter, $U$, $P$ and $T$ denote the mean streamwise velocity, pressure and temperature, respectively, c.v. stands for control volume, ${\rm d} t=\widetilde {{\rm d} t}c_\infty /D$ and $t_{{sim}}=\tilde {t}_{{sim}}c_\infty /D$ are time step and total time of the simulation in acoustic units, where $\tilde {t}$ is the physical time, and ${\rm \Delta} t$ is the time step in acoustic units used for data storage. The subscripts, $j$, $\infty$ and $0$ denote jet exit, free stream and stagnation conditions, respectively. Throughout this paper, velocities are non-dimensionalised with the ambient speed of sound, $c_\infty$, lengths with nozzle diameter, $D$, pressure with $\rho _\infty c_\infty ^2/2$ and time with $c_\infty /D$. Frequencies are reported in Strouhal number, $St=\tilde {f}D/U_j$, where $f$ is the dimensional frequency.

Table 1. Details of the LES database. The first four and the last three cases are used to tune and test the empirical model, respectively.

The axisymmetric nature of jets renders possible decomposing the flow into azimuthal Fourier modes and analysing them separately. To facilitate azimuthal Fourier decomposition, for each case, the LES data are interpolated onto a cylindrical grid with mesh size $(N_x,N_r,N_\theta )=(656,138,128)$, where $N_x$, $N_r$ and $N_\theta$ are the number of grid points in streamwise, radial and azimuthal directions. The cylindrical grid extends in $x,r,\theta \in [0,30]\times [0,6]\times [0,2{\rm \pi} ]$.

The linearised N-S equations in the time domain for the $m$th azimuthal Fourier mode are given as

(3.1)\begin{equation} \partial_t{\boldsymbol{q}^{\prime}}^{(m)}-{{\boldsymbol{\mathsf{A}}}}^{(m)}{\boldsymbol{q}^{\prime}}^{(m)}=\boldsymbol{f}^{(m)}, \end{equation}

where the superscript $(m)$ denotes the azimuthal mode number. We limit our analysis to the acoustic field in the first azimuthal mode, $m=0$, only. In the interest of developing modelling approaches, in a step-by-step methodology, it is common in the literature to focus first on this component, as it is clear that it corresponds to coherent structures (Sandham & Salgado Reference Sandham and Salgado2008; Freund & Colonius Reference Freund and Colonius2009; Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012, Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013; Jordan & Colonius Reference Jordan and Colonius2013). In addition to this strategical reason, the mode $m=0$ is the dominant mode for the downstream noise where jet noise has its peak as stated by Cavalieri et al. (Reference Cavalieri, Jordan, Colonius and Gervais2012) and many other studies. These reasons makes the axisymmetric mode the natural choice to start modelling jet noise.

Algorithm 1 Computing the forcing

In figure 1, snapshots of pressure and temperature for the first azimuthal Fourier mode, $m=0$, are shown for the four cases used for tuning of the empirical model. Temperature is shown as an indicator of turbulent fluctuations in the shear layer. Pressure is saturated to show the acoustic waves propagating from the jets. It is seen that the case M09Mc00 (see table 1 for case abbreviations) has the strongest pressure gradients, and thus, highest noise level. Existence of the flight stream suppresses both the turbulent fluctuations and the noise generated by the jet, at a level increasing with the flight velocity. The flow fields for the remaining three cases are not shown for brevity. All seven cases are validated against experimental data. A detailed validation has been reported by Brès et al. (Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018) for the cases M04Mc00 and M09Mc00 and by Maia et al. (Reference Maia, Brès, Lesshafft and Jordan2022) for the remaining cases. The M09Mc00 case is publicly available as part of a database for reduced-complexity modelling of fluid flows (Towne et al. Reference Towne2022).

Figure 1. Snapshots of the first azimuthal Fourier mode of pressure (grey) and temperature (colour) for the cases (a) M04Mc00, (b) M09Mc00, (c) M09Mc15, (d) M09Mc30. Colour-scale for pressure linearly varies in the range of $[-6\times 10^{-3},6\times 10^{-3}]$ for all cases. Colour-scale for temperature is given as $[1,1.01]$, $[1,1.03]$, $[1,1.02]$, $[1,1.02]$ for the abovementioned four cases, respectively.

The case M04Mc00 contains both the state and the forcing data while others contain only the state data. The forcing data, once the state data are stored, are obtained via the procedure devised by Towne (Reference Towne2016) and summarised in . The extraction of the forcing data is restricted to the M04Mc00 case, as aliasing effect in the database (see the Appendix and Karban et al. Reference Karban, Martini, Jordan, Brès and Towne2022b for a detailed discussion) increase with Mach number, rendering resolvent-based prediction using forcing data unreliable for the remaining flow cases. Both the state and forcing data are Fourier transformed using blocks containing 512 snapshots in time with an overlap ratio of 75 %. To minimise spectral leakage, we use an exponential windowing function (Martini et al. Reference Martini, Cavalieri, Jordan and Lesshafft2019),

(3.2)\begin{equation} W(t) = \exp\left({n\left(4-\frac{T}{t(T-t)}\right)}\right), \end{equation}

with $n=1$ and window size $T=512{\rm \Delta} t$. The correction discussed by Martini et al. (Reference Martini, Cavalieri, Jordan and Lesshafft2019), which is necessary to satisfy (2.3) when a windowing function is applied during the temporal Fourier transform (FT), is implemented while computing the forcing terms in the frequency domain. The correction is shown by Nogueira et al. (Reference Nogueira, Morra, Martini, Cavalieri and Henningson2021) to significantly improve the convergence of resolvent-based prediction of the response via (2.3).

3.2. Resolvent analysis

Resolvent-based prediction of the response using the forcing data is achieved via a custom resolvent analysis code (Bugeat et al. Reference Bugeat, Chassaing, Robinet and Sagaut2019). The code uses the finite-volume method to solve the linearised N-S equations decomposed into azimuthal Fourier modes. The input-output relation in (2.3) is written for a given azimuthal mode, $m$, as

(3.3)\begin{equation} \hat{\boldsymbol{q}}^{(m)}={{\boldsymbol{\mathsf{R}}}}^{(m)}\hat{\boldsymbol{f}}^{(m)}, \end{equation}

where ${{\boldsymbol{\mathsf{R}}}}^{(m)}\triangleq ({\rm i}\omega {{\boldsymbol{\mathsf{I}}}}-{{\boldsymbol{\mathsf{A}}}}^{(m)})^{-1}$. In practice, the response, $\hat {\boldsymbol {q}}^{(m)}$, to a given forcing, $\hat {\boldsymbol {f}}^{(m)}$, is computed by solving the linear system

(3.4)\begin{equation} {{\boldsymbol{\mathsf{L}}}}^{(m)}\hat{\boldsymbol{q}}^{(m)}=\hat{\boldsymbol{f}}^{(m)}, \end{equation}

where ${{\boldsymbol{\mathsf{L}}}}^{(m)}={{{\boldsymbol{\mathsf{R}}}}^{(m)}}^{-1}=({\rm i}\omega {{\boldsymbol{\mathsf{I}}}}-{{\boldsymbol{\mathsf{A}}}}^{(m)})$ is a sparse linear operator. The resolvent code solves (3.4) via lower–upper (LU) decomposition using the PETSc library (Balay et al. Reference Balay, Gropp, McInnes and Smith1997). Once the LU decomposition is performed, (3.4) can be solved efficiently for numerous forcing vectors. This allowed us to calculate at once the responses to different Fourier realisations of the forcing, which then are used to compute the power spectral density (PSD) of the resulting acoustic field via Welch averaging. When calculating the response of the modified resolvent operator to a RESPOD forcing mode as in (2.16), we simply replaced the vector on the right-hand side of (3.4) with ${{\boldsymbol{\mathsf{B}}}}\hat {\boldsymbol{\chi }}^{(p)}$.

In all computations, a sponge zone is placed within $x/D=[20,30]$ and $r/D=[6,12]$ at the downstream and outer ends of the domain, respectively, to avoid spurious reflections. The domain is extruded to $12D$ in the radial direction to accommodate the top sponge region. A single damping function of the form

(3.5)\begin{equation} 1-\frac{1-\exp\left({\kappa\dfrac{(\eta-\eta_s)^2}{(\eta_s-\eta_{max})^2}}\right)}{1- \textrm{exp}(\kappa)} \end{equation}

is used to damp fluctuations in both $x$ and $r$ directions, where $\kappa =4$, $\eta$ denotes either the $x$ or $r$ coordinate, $\eta _s$ denotes the beginning of the sponge zone and $\eta _{max}$ denotes the domain end position. Further details can be found from Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019).

The original code was written based on conservative variables, while the LES forcing database was generated using the primitive-like variable set, $\boldsymbol {q}=[\nu,\boldsymbol {u},p]$, as discussed in § 2.1, yielding a compatibility issue. To overcome this issue, a correction derived by Karban et al. (Reference Karban, Bugeat, Martini, Towne, Cavalieri, Lesshafft, Agarwal, Jordan and Colonius2020) is implemented.

4.1. Masking the forcing vector

Before performing a dedicated analysis to obtain a low-rank forcing model, we first reduce the number of the forcing terms to model. This is achieved by applying spatial and component-wise masking in the forcing data to observe the effect of the masked regions/components on noise generation. This masking involves zeroing certain parts of the forcing vector using the matrix ${{\boldsymbol{\mathsf{B}}}}$. Figure 2 shows the result of different masks in terms of the PSD of the acoustic pressure at $St=0.6$. Masking the forcing beyond $r>2D$ or $r>4D$ yields nearly identical results in the entire flow domain. Masking beyond $r>1.5D$ also yields nearly identical noise fields in the downstream region, $x>6D$, while a slight discrepancy is observed in the region $x/D\in [3,6]$. The turbulent regions of different jet configurations shown in figure 1 suggest that the jet spreading rate, which determines the forcing region, is similar in the cases M04Mc00 and M09Mc00 while it is reduced for the coflow cases. Given these observations, the $r<2D$ limit defined for the M04Mc00 case is considered applicable for the remaining cases as well. We therefore limit the forcing to the region $r<2D$ for the rest of the analysis.

Figure 2. PSD of pressure predicted using resolvent analysis with masking applied (a) in space and (b) in variables in comparison to the LES data at $r=5D$ for the case M04Mc00 at $St=0.6$.

Component-wise masking of the forcing shows that the components $f_\nu$ and $f_p$, which correspond to the mass and energy equations, respectively, have negligible contribution to the acoustic field. Masking the component $f_{u_r}$, which is the forcing associated with the radial momentum equation, causes significant reduction in the sideline noise while not affecting the acoustic field in the downstream region. However, masking the component $f_{u_x}$, which is the forcing associated with the streamwise momentum, causes significant reduction in the downstream noise while having limited effect on the sideline noise. These results indicate that the forcing term $f_{u_x}$ in the region $r<2D$ is solely responsible for downstream noise generation, consistent with the observation of Freund (Reference Freund2001) using Lighthill's analogy. In what follows, focusing on the downstream noise generation only, we aim to identify the acoustically active subspace associated with this single forcing component.

4.2. Applying RESPOD to obtain low-rank forcing

We now aim to obtain a low-rank representation of the subspace of the forcing associated with the most-energetic components of the acoustic field. To achieve this, we first compute the SPOD modes of the acoustic field, and we then use RESPOD to extract the associated forcing modes, as described in § 2.2. In figure 3, the SPOD eigenvalues of the pressure in the downstream acoustic field, defined as $x/D,r/D\in [6,30]\times [4,6]$, and those of the forcing term $f_{u_x}$ in the turbulent region, defined as $x/D,r/D\in [0,30]\times [0,2]$, are shown for $St=0.6$. For the acoustic field, the leading SPOD eigenvalue corresponds to more than 75 % of the total acoustic energy. The sum of the first five SPOD eigenvalues corresponds to 99 % of the total acoustic energy, indicating a low-rank organisation in the acoustic field. For the forcing in the near field, however, the leading SPOD mode contains less than 6 % of the total energy in $f_{u_x}$. Approximately one hundred modes are required to capture 90 % of the total forcing energy, indicating an extremely high-rank structure. As discussed earlier, this difference between the near-field turbulence forcing and the acoustic field is the crux of the jet-noise problem, which we aim to overcome by educing the small portion of the forcing responsible for the acoustic field.

Figure 3. (a) SPOD eigenvalues of the pressure in the acoustic field and (b) streamwise forcing, $f_{u_x}$, in the near field for the case M04Mc00 at $St=0.6$.

Using RESPOD, we extract from this high-rank forcing data a low-rank subspace that is correlated with the low-rank pressure structures observed in the acoustic field. In figure 4, we show the leading SPOD mode of pressure in the acoustic field and the associated RESPOD mode of the forcing in the near field, together with the energy distribution of the first twenty RESPOD forcing modes. The leading SPOD mode takes the form of an acoustic wave propagating at some angle, while the associated RESPOD forcing mode contains a disorganised structure, which may imply underconvergence in the forcing mode, despite the very long time-series used for the analysis. The first RESPOD mode contains less than 0.8 % of the total forcing energy, but it is associated with the leading SPOD mode of the acoustic field, corresponding to 75 % of the total noise in the downstream region. This result shows the importance of applying such an identification prior to any modelling effort. Without this extraction of the low-rank forcing structure, a fitting function optimised using the forcing data will be affected by the existence of energetic structures that do not significantly contribute to sound generation.

Figure 4. (a) Optimal SPOD mode of acoustic pressure and the associated RESPOD mode of the forcing together with (b) the energy distribution in the first twenty RESPOD modes of the forcing for the case M04Mc00 at $St=0.6$. The acoustic and forcing fields in the top plot are denoted by the black and green dashed boxes, respectively.

No smooth trend is observed in the energy of the first twenty RESPOD forcing modes, in contrast to the SPOD modes in the acoustic field. As discussed by Karban et al. (Reference Karban, Martini, Cavalieri, Lesshafft and Jordan2022a), RESPOD does not impose a strict filtering on the forcing to extract the active structures that actually drive the acoustic field, but finds the correlated part which includes silent-but-correlated structures. Lack of a smooth trend in the energy of the RESPOD forcing modes implies underconvergence of these modes. Given that the active part in the RESPOD forcing modes are linearly related to the SPOD modes of the acoustic field, they should experience the same convergence rate. The underconvergence in the RESPOD forcing modes can then be attributed to the contribution of the silent-but-correlated structures, causing also the mode shape to be significantly less organised compared with the associated SPOD mode. This underconvergence observed in the forcing modes does not pose a problem in the following analysis. The SPOD modes of the response and the RESPOD modes of the forcing are computed using Fourier realisations of response and forcing that exactly correspond to the same time window. In the ideal case of an error-free database, (2.3) is therefore satisfied for each pair of response-forcing realisations. So, no matter how underconverged the forcing data are, the structures generating the converged acoustic field are, by construction, ensured to be contained in the forcing mode seen in figure 4.

As discussed in § 2.2, the first RESPOD forcing mode contains all structures correlated with the leading SPOD mode. This indicates that the remaining forcing that amounts to 99 % of the total forcing energy can generate only 25 % of the total acoustic energy. In figure 5, we show a comparison of the true acoustic field and the acoustic fields obtained using rank-5 and rank-1 forcing truncations obtained using RESPOD for a number of frequencies, $St\in [0.4,1]$. The rank-5 forcing, which contains the RESPOD forcing modes corresponding to the first five SPOD modes of the acoustic field, recovers nearly the entire acoustic field in the downstream region. The rank-1 forcing, i.e. the RESPOD forcing mode corresponding to the optimal SPOD mode of the acoustic field, also recovers a significant portion of the downstream acoustic field. Although the acoustic field predicted by the rank-1 forcing should correspond to 75 % of the total acoustic energy at $St=0.6$, as the response should be identical to the optimal SPOD mode of the acoustic field, the actual prediction amounts to approximately 60 %. A similar discrepancy is observed for the rank-5 prediction, which recovers 80 % of the acoustic energy while it should generate 99 %. This is due to the errors contained in the LES database, causing a loss in the correlation information between the response and the forcing. Despite all the limitations of the existing database as discussed in the Appendix, we see that it is still possible to define a rank-1 forcing that can generate most of the downstream noise.

Figure 5. PSD of acoustic pressure generated using rank-5 and rank-1 forcing obtained by RESPOD, in comparison to the acoustic field obtained from LES data (corresponding to full-rank forcing in the ideal case) at different frequencies ranging from $St=0.4$ to 1 (from a–d).

In what follows, we further decompose the rank-1 forcing obtained by RESPOD to extract the acoustically active forcing components which drive the leading SPOD mode of the acoustic pressure seen in figure 4.

4.3. Isolating the radiating component of the low-rank forcing

In figure 6, the acoustic fields generated by the first two RESPOD modes are shown for the case M04Mc00 at a number of frequencies, $St\in [0.4,1.0]$. The first modes at all frequencies contain a single wave propagating at some angle with no jump in the phase. However, there exists a phase shift in the second modes that moves upstream with increasing frequency. The phase shift appears to satisfy orthogonality between the first and the second modes, which is expected as RESPOD finds the forcing modes that generate the SPOD modes, which comprise an orthogonal basis. Note however that no such orthogonality is ensured for the forcing modes.

Figure 6. Real part of the pressure generated using (a,c,e,g) first and (b,d, f,h) second RESPOD mode of the forcing at different frequencies ranging from $St=0.4$ to 1 (from top to bottom). Colour scale is in the range of [$-1\times 10^{-6},1\times 10^{-6}$].

Looking at the acoustic field of the first RESPOD mode, it is apparent that the propagation angle is nearly constant, around $30^\circ$ when measured from the downstream end, for all frequencies, reminiscent of a Mach-wave-like mechanism (cf. Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008). To explore this trend, we consider a wave in the streamwise direction defined by $\exp (-{\rm i}k_xx)$, where $k_x$ is the streamwise wavenumber associated with a phase speed

(4.1)\begin{equation} c_x=\omega/k_x, \end{equation}

where $\omega =2{\rm \pi} St$ is the angular frequency. For a Mach-wave-like mechanism, the phase speed is greater than the speed of sound, $c_\infty$, and the propagation angle is given by $\cos ^{-1}(c_\infty /c_x)$ (Ffowcs Williams Reference Ffowcs Williams1963; Crighton Reference Crighton1975). We project the first and the second RESPOD forcing modes onto this wave, varying in the phase speed over the range $[c_\infty, 2c_\infty ]$, yielding

(4.2)\begin{equation} a^{(p)}(k_x,St)=\langle \boldsymbol{\chi}^{(p)}(x,r,St), \exp({-{\rm i}k_xx})\rangle\triangleq\int_S\boldsymbol{\chi}^{(p)}(x,r,St)\exp({-{\rm i}k_xx})\,{\rm d} S, \end{equation}

where $p$ is the RESPOD mode number and $S$ is the two-dimensional (2-D) domain spanning the $x$ and $r$ directions. The results are shown in figure 7. It is seen that at all frequencies, the projection coefficient, $a^{(1)}$, peaks around 1.1–1.2$c_\infty$, which corresponds to an angle of ${\sim }30^\circ$, consistent with the propagation angle observed in the acoustic response field. The coefficient, $a^{(2)}$, however, has a dip around the same value at all frequencies, reminiscent of the orthogonality observed in the response modes of figure 6.

Figure 7. Projection of first and second RESPOD modes of the forcing onto streamwise harmonic waves with supersonic phase speeds. Different frequencies ranging from $St=0.4$ to 1 are shown in panels (a–d).

These results suggest that projection of the forcing onto supersonic waves is the relevant mechanism for generation of downstream noise, consistent with previous hypotheses and models (Freund Reference Freund2001; Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012; Jordan & Colonius Reference Jordan and Colonius2013; Cavalieri et al. Reference Cavalieri, Jordan and Lesshafft2019). To test this hypothesis, we define the following FT in the streamwise direction,

(4.3)\begin{equation} \mathcal{F}_x({{\boldsymbol{\mathsf{a}}}})=\int_0^L{{\boldsymbol{\mathsf{a}}}}\exp({-{\rm i}k_xx})\,{{\rm d}x}, \end{equation}

where $L=30D$ is the domain length. Using this FT, we decompose the first RESPOD mode of the forcing, $\boldsymbol{\chi }^{(1)}$, into two parts, $\boldsymbol{\chi }^{(1-)}$ and $\boldsymbol{\chi }^{(1+)}$, containing subsonic and supersonic components, respectively (Sinayoko et al. Reference Sinayoko, Agarwal and Hu2011). The resulting forcing fields are depicted in comparison to the original forcing mode in figure 8. It is seen that most of the forcing energy is contained in the subsonic part of the mode, $\boldsymbol{\chi }^{(1-)}$, making it indistinguishable from $\boldsymbol{\chi }^{(1)}$. The supersonic component, $\boldsymbol{\chi }^{(1+)}$, takes the form of a compilation of radially thin wavepackets with a disorganised radial phase structure.

Figure 8. (a) Real part of the first RESPOD mode of the forcing compared with its (b) subsonic and (c) supersonic parts. Different frequencies ranging from $St=0.4$ to 1 are shown from top to bottom.

The acoustic response generated by these subsonic and supersonic modes, $\boldsymbol{\chi }^{(1-)}$ and $\boldsymbol{\chi }^{(1+)}$, respectively, are compared with the response of the entire first RESPOD mode of the forcing, $\boldsymbol{\chi }^{(1)}$, in figure 9 for a range of frequencies. It is clear that the supersonic modes underpin noise generation at all frequencies. Removing the supersonic components leads to more than an order-of-magnitude reduction in sound generation. The energy contained in the supersonic part of the RESPOD modes of the forcing is shown in figure 10 for different mode numbers and frequencies. The following definition is used to calculate the energy of a given mode, $\boldsymbol{\varphi }$, defined in a domain $\varOmega$,

(4.4)\begin{equation} \int_\varOmega \boldsymbol{\varphi}^\top\boldsymbol{\varphi}\,{\rm d}\varOmega. \end{equation}

Note that an energy norm is not required in the above definition since all the modes we investigate contain a single variable. For all the modes and frequencies, the supersonic components contain less than 5 % of the mode energy. The first RESPOD mode of the forcing already contains less than $1\,\%$ of the total forcing energy, which means that the energy fraction of the supersonic part of the first RESPOD mode of the forcing, $\boldsymbol{\chi }^{(1+)}$, with respect to the total forcing energy at the same frequency, is $\sim$0.04 %, while it generates $\sim 75\,\%$ of the total acoustic energy in the downstream region for a frequency range $St=[0.4,1.0]$ at $M_j=0.4$.

Figure 9. PSD of the acoustic pressure generated by the first RESPOD mode of the forcing (solid) compared with its subsonic (dashed) and supersonic (dash–dotted) parts. Different frequencies ranging from $St=0.4$ to 1 are shown in panels (a–d).

Figure 10. Energy ratio of the supersonic part of the RESPOD mode of the forcing.

The analysis above identifies the forcing subspace that actively contributes to noise generation in the M04Mc00 case. As discussed earlier, this data-driven approach is only applicable for this case, since the errors contained in the LES database prevent us from extracting reliable forcing data at higher Mach numbers. In the following section, we will discuss how to develop an empirical model for these forcing structures and extend it to other flow cases.

5.1. Empirical source modelling for $M_j=0.4$

As shown in figure 8, the supersonic mode, $\boldsymbol{\chi }^{(1+)}$, roughly follows the jet spreading and has the form of thin wavepackets elongated in the streamwise direction spanning most of the flow domain. Given the radial randomness of these thin wavepackets, a model for the $x$–$r$ structure is not feasible. We therefore make use of the characteristics of the modified resolvent operator, $\tilde {{{\boldsymbol{\mathsf{R}}}}}$, which yields the acoustic pressure as the response thanks to the measurement matrix ${{\boldsymbol{\mathsf{C}}}}$.

In resolvent analysis, a singular value decomposition (SVD) of the resolvent operator is used to identify mechanisms by which the output (acoustic pressure in our case) is driven by the input (forcing). The SVD is given as

(5.1)\begin{equation} \tilde{{{\boldsymbol{\mathsf{R}}}}}={{\boldsymbol{\mathsf{U}}}}\boldsymbol{\varSigma}{{\boldsymbol{\mathsf{V}}}}^H, \end{equation}

where ${{\boldsymbol{\mathsf{U}}}}$ and ${{\boldsymbol{\mathsf{V}}}}$ are the response and forcing modes, respectively, and $\boldsymbol{\varSigma }$ denotes the resolvent gain. The RESPOD forcing mode is projected onto the forcing modes of the modified resolvent operator and then amplified by the resolvent gains to generate the acoustic response whose spatial organisation is defined by the response modes. The projection between the forcing and RESPOD modes can be considered as a two-dimensional discrete integration in the streamwise and radial directions. It was reported by Jeun et al. (Reference Jeun, Nichols and Jovanović2016) and later by Bugeat et al. (Reference Bugeat, Karban, Agarwal, Lesshafft and Jordan2022) that the forcing modes of the acoustic resolvent operator, $\tilde {{{\boldsymbol{\mathsf{R}}}}}$, take the shape of streamwise supersonic waves with almost constant radial support within the region where forcing is active. When projecting the RESPOD modes onto the forcing modes, the radially thin supersonic wavepackets observed in figure 8 are integrated in the radial direction without any modulation by the optimal forcing mode. This implies that the forcing modes of figure 8 can be replaced by a line source obtained by radial integration, justifying the use of line-source models in the literature (Michalke Reference Michalke1970; Michel Reference Michel2009; Lesshafft, Huerre & Sagaut Reference Lesshafft, Huerre and Sagaut2010; Cavalieri et al. Reference Cavalieri, Jordan, Agarwal and Gervais2011; Cavalieri & Agarwal Reference Cavalieri and Agarwal2014; Maia et al. Reference Maia, Jordan, Cavalieri and Jaunet2019; da Silva et al. Reference da Silva, Jordan and Cavalieri2019). The resulting line sources at different frequencies are shown in figure 11 for the case M04Mc00. We observe wavepackets spanning $x/D=[0,20]$ with a dominant wavenumber at all frequencies. Note that the forcing data do not decay to zero at the end of the domain, which is a side effect of using a streamwise Fourier transform, which assumes periodicity in the streamwise direction. These spurious oscillations are filtered when applying these modes to the resolvent operator thanks to the sponge zone beyond $x/D=20$.

Figure 11. Real (blue) and imaginary (orange) parts of the supersonic part of the first RESPOD mode of the forcing integrated in the radial direction. Different frequencies ranging from $St=0.4$ to 1 are shown in panels (a–d).

To characterise these wavepackets, we perform an FT in the streamwise direction. The amplitude and phase of the Fourier coefficients are shown in figure 12 for a range of frequencies. The wavenumber has been scaled by $c_\infty /\omega$ so that the abscissa in the figure is the inverse phase speed, with the range $[-1,1]$ corresponding to the supersonic phase speeds. We see the same peak corresponding to a phase speed, $c_x=1.1722c_\infty$ at all frequencies, with energy contained in the immediate neighbouring wavenumbers as well. The ordinate of the figure is scaled with $St$, and the peak has a nearly constant amplitude under this scaling. We ignore the rest of the spectrum and investigate the phase relation between this peak and the neighbouring wavenumbers only, as shown in figure 12. The exact value of the phase affects only the global phase of the response, and thus is random. The shape of the forcing modes is determined by the change in phase with respect to wavenumber. There exists a phase difference of ${\sim }0.5{\rm \pi}$ between the central and leftmost wavenumbers. The phase difference between the central and rightmost wavenumbers is either approximately $-1.7{\rm \pi}$ or approximately $0.3{\rm \pi}$, which correspond to the same phase shift as $0.3{\rm \pi} =\textrm {mod}(-1.7{\rm \pi},2{\rm \pi} )$.

Figure 12. (a,c,e,g) Amplitude and (b,d, f,h) phase of the streamwise FT of the integrated line source. Red dashed line shows the isolated spectrum used for modelling. Vertical dashed black lines indicate the phase speed, $c_x=1.1722c_\infty$, and dashed blue lines mark the neighbouring wavenumbers to that. Different frequencies ranging from $St=0.4$ to 1 are shown from top to bottom.

Given these observations, we need to find a model consisting of three wavenumbers, the wavenumber corresponding to the constant phase speed observed and the neighbouring ones, with some empirical phase and amplitude relations in between and a global amplitude which scales with $St$. We propose the following empirical model,

(5.2)\begin{equation} \mathcal{F}_x(\xi)=\frac{A}{St}\left(\exp({{\rm i}{\rm \pi} k_x^p}) + B\left(\exp({\phi_1{\rm i}{\rm \pi} k_x^{p-}}) + \exp({\phi_2{\rm i}{\rm \pi} k_x^{p+}})\right) \right), \end{equation}

where $A=4.6\times 10^{-7}$, $B=0.8$, $\phi _1=0.5$, $\phi _2=-0.7$, $k_x^p=St/(\beta c_\infty )$ is the wavenumber corresponding to the peak observed in figure 12 with $\beta =1.1722$ and $k_x^{p\pm }$ denotes the neighbouring wavenumbers with ${\rm \Delta} k_x=1/30$ where 30 is the domain length. The corresponding forcing model, $\xi$, can then be obtained by taking the inverse FT of (5.2) in $x$. Note that the neighbouring wavenumbers provide the wavepacket amplitude envelope without having to define a Gaussian-like form. Choosing a different domain length would change ${\rm \Delta} k_x$ and therefore the resulting forcing model. However, we anticipate that the results are not sensitive to this parameter, which will be justified later when showing the results for models at higher Mach numbers using the same value for $L$. Figure 13 shows a comparison of the forcing model obtained from (5.2) and the line source obtained from the LES data. The model has a similar spatial support and the same dominant wavelength corresponding to the line source. Similar to the line source data, the empirical model is also assumed to be periodic in the streamwise direction, causing the spurious oscillations at the end of the domain, as mentioned above, to appear in the model as well. Once again, these artefacts are damped by the sponge zone within the resolvent operator.

Figure 13. Real (blue) and imaginary (orange) parts of (a,c,e,g) the line source compared with (b,d, f,h) the line-source model given by (5.2). Different frequencies ranging from $St=0.4$ to 1 are shown from top to bottom.

We now compare the acoustic response generated by this model to the response of the first RESPOD mode of the forcing, $\boldsymbol{\chi }^{(1)}$, in figure 14 for a range of frequencies. The model accurately predicts downstream noise generation for high frequencies while it yields an underprediction at lower frequencies. The sound directivity is seen to be well captured at all frequencies, which implies that the underprediction at low frequencies can be fixed by adding a tuning parameter to the model. In the following section, we apply corrections to the model using acoustic data to improve predictions and to include Mach-number and flight effects.

Figure 14. PSD of the acoustic pressure generated by the first RESPOD mode of the forcing (solid) compared with that of the line-source model (dashed) for the case M04Mc00. Different frequencies ranging from $St=0.4$ to 1 are shown in panels (a–d).

5.2. Tuning the model using acoustic data from the LES

We discussed earlier that the errors in the LES database cause the correlation between the forcing and the acoustic field to be partially contaminated, yielding the response generated by the first RESPOD forcing mode, $\boldsymbol{\chi }^{(1)}$, to globally underpredict the optimal SPOD mode of the acoustic field. As the forcing model in (5.2) is constructed based on the supersonic part of this forcing mode, the effect of the errors in the database is inherited in the model, $\xi$. To minimise this effect, we tune the model using the acoustic data obtained directly from the LES. The tuning process is summarised as follows: we first correct the mode amplitude using the acoustic field from the M04Mc00 case to better capture the overall sound level and its change with respect to frequency. The same frequency scaling will then be used for all other flow cases. To tune the model for Mach-number effects, we will use the acoustic field from the M09Mc00 case. Finally, to tune the model for flight-stream effects, we will use the acoustic fields from the M09Mc15 and M09Mc30 cases.

We start tuning the model, $\xi$, by adding a scalar correction to better match the optimal SPOD mode of the acoustic field. In figure 15, the energy ratio of the response generated by the model, $\xi$, is compared with that of the optimal SPOD mode of the acoustic pressure as a function of frequency. Normalisation is done using the total acoustic energy in the downstream region. It is seen that the underprediction of the model increases as the frequency decreases. A correction in the amplitude and changing the $St^{-1}$ scaling to $St^{-3/2}$ yields the corrected trend seen in figure 15.

Figure 15. Energy ratio of the response generated by the line-source model (dashed) compared with that in the optimal SPOD mode of the acoustic pressure (solid). The energy ratio obtained using the corrected model is also shown (dash–dotted). Normalisations are done using the acoustic energy in the downstream region at each frequency.

We now test the ansatz (5.2) to see if it can capture the Mach-number effect. The phase difference between $k_x^p$ and $k_x^{p-}$ mainly determines the shape of the envelope of the wavepacket while the phase difference between $k_x^p$ and $k_x^{p+}$ determines its streamwise position. It was also found that the resulting acoustic field does not strongly depend on the value of $B$. Given these observations, we set $\beta$, $A$, $\phi _2$ as free parameters to tune, and we keep $\phi _1$ fixed to keep the wavepacket shape unchanged.

Assuming that the parameter $\beta$ varies linearly with the jet Mach number $M_j$, and matching the observed value at $M_j = 0.4$ and the observed phase velocity at $M_j = 0.9$ yields the expression

(5.3)\begin{equation} \beta=0.7722 + M_j. \end{equation}

For $M_j=0.9$, this results in a propagation angle of $53.3^\circ$, very close to the value observed by Bugeat et al. (Reference Bugeat, Karban, Agarwal, Lesshafft and Jordan2022).

To set the amplitude, $A$, and the phase constant, $\phi _2$, we performed tests to find the parameters that best match the acoustic field in the case M09Mc00. We finally obtained the empirical relations

(5.4)$$\begin{gather} A =3.22\times10^{{-}6}M_j^{7/2}, \end{gather}$$

(5.5)$$\begin{gather}\phi_2=0.1-St. \end{gather}$$

The resulting model equation with these corrections reads

(5.6)\begin{align} \mathcal{F}_x(\xi)=3.15\times10^{{-}6}\sqrt{\frac{M_j^7}{St^3}} (\exp({{\rm i}{\rm \pi} k_x^p}) + B(\exp({0.5{\rm i}{\rm \pi} k_x^{p-}}) + \exp({(0.1-St){\rm i}{\rm \pi} k_x^{p+}}))), \end{align}

where

(5.7)\begin{equation} k_x^p=\frac{St}{(0.7722+M_j)c_\infty}, \end{equation}

and $k_x^{p\pm }$ denotes once again the neighbouring wavenumbers with ${\rm \Delta} k_x=1/30$. The resulting acoustic field is shown in figure 16 for M04Mc00 and M09Mc00, and compared with the corresponding LES data. It is seen that the model given in (5.6) yields a prediction that well matches the LES data in the downstream region at all frequencies. The acoustic response does not noticeably differ for the case M04Mc00 whether one uses a constant or linearly varying value for the phase, $\phi _2$.

Figure 16. PSD of the acoustic pressure obtained using line-source model (5.6) to that extracted from the LES for the cases (a,c,e,g) M04Mc00 and (b,d, f,h) M09Mc00. Different frequencies ranging from $St=0.4$ to 1 are shown from top to bottom.

Finally, we extend the empirical model given in (5.6) to take into account the flight effect. It is known that the effect of flight is to suppress noise, largely due to the suppression of turbulence in the shear layer (Maia et al. Reference Maia, Brès, Lesshafft and Jordan2022). We compare the noise generated in the cases M09Mc00, M09Mc15 and M09Mc30, respectively, in figure 17 with two different scalings. Defining

(5.8)\begin{equation} k_I(M_j,M_\infty)=\int_SK(M_j,M_\infty)\,{\rm d} S, \end{equation}

where $K(M_j,M_\infty )$ denotes the turbulent kinetic energy as a function of the jet Mach number, $M_j$, and the flight Mach number, $M_\infty$, and using the scaling $k_I^2$ causes the sidestream noise in the three cases to collapse on top of each other. The dominant terms in the forcing have the form $\boldsymbol {u}\boldsymbol{\cdot }\boldsymbol {\nabla }\boldsymbol {u}$, which has the same dimension with turbulent kinetic energy differentiated in space. Inspired by this, we defined a scaling factor using $\partial _xK$ as

(5.9)\begin{equation} k_{I,x}(M_j,M_\infty)=\int_S\partial_xK(M_j,M_\infty)\,{\rm d} S. \end{equation}

The scaling $k_{I,x}^2$ yields a slightly improved match in the peak noise level of the three cases. To determine if this scaling should be directly adopted in the forcing model, one needs to understand the effect of the free stream on the efficiency of the noise generation mechanisms embedded in the resolvent operator. For this, we performed a resolvent-based noise prediction using the same forcing in the three cases at $M_j=0.9$. The resulting acoustic fields are shown in figure 18, where it is seen that the change in the mean flow does not strongly affect the noise level, but causes a change in the directivity in a similar fashion as observed in the LES data shown in figure 17. This suggests that one may use the same mathematical form for the source model for jets with or without flight effect, applying an amplitude correction. We adopt $k_{I,x}^2$ scaling for the empirical model as the peak noise, which occurs in the downstream region, is more relevant for the present study. In addition to the amplitude correction, comparison of the model prediction with the LES data with flight-stream effect yielded that the phase constant, $\phi _2$, is to be updated as

(5.10)\begin{equation} \phi_2=0.1 - St\frac{M_j-M_\infty}{M_j}, \end{equation}

resulting in the final equation for the jet-noise source model,

(5.11) \begin{align} \mathcal{F}_x(\xi) &= 3.15\times10^{{-}6}\varGamma\sqrt{\frac{M_j^7}{St^3}} (\exp({{\rm i}{\rm \pi} k_x^p}) + B(\exp({0.5{\rm i}{\rm \pi} k_x^{p-}}) \nonumber\\ &\quad +\exp({(0.1-St(M_j-M_\infty)/M_j){\rm i}{\rm \pi} k_x^{p+}}))), \end{align}

where

(5.12)\begin{equation} \varGamma\triangleq\frac{k_{I,x}(M_j,M_\infty)}{k_{I,x}(M_j,0)}. \end{equation}

The resulting acoustic response of the model forcing obtained by taking the inverse FT of (5.11) is compared with the acoustic fields coming from the LES data in the cases M09Mc15 and M09Mc30, respectively, in figure 19. The error at all the frequencies remains within 2 dB for the downstream region.

Figure 17. PSD of the acoustic pressure for the cases M09Mc00 (solid), M09Mc15 (dashed) and M09Mc30 (dash–dotted) with (a) no scaling, (b) $k_I^2$ scaling and (c) $k_{I,x}^2$ scaling at $St=0.6$.

Figure 18. PSD of the acoustic pressure obtained using the line-source model given in (5.6) for the cases M09Mc00 (solid), M09Mc15 (dashed) and M09Mc30 (dash–dotted) at $St=0.6$.

Figure 19. PSD of the acoustic pressure obtained using the line-source model given in (5.11) (dashed) compared with the LES data (solid) for the cases (a,c,e,g) M09Mc15 and (b,d, f,h) M09Mc30. Different frequencies ranging from $St=0.4$ to 1 are shown in panels (a–h).

5.3. Blind testing the model under different operating conditions

To test the validity of the model given in (5.11), we use three LES cases that were not used in its development: M07Mc00, M07Mc015 and M08Mc00 (see table 1 for details). The predictions are compared against the LES data in figure 20. The acoustic response generated by the forcing model predicts the downstream acoustic field with a 1 dB accuracy within the region $x/D=[10,20]$ for the static jets at $M_j=0.7$ and 0.8, except for a sharper decay observed at $St=1$ beyond $x=19D$ for both cases. For the case with flight effect, the peak noise level is predicted accurately at all the frequencies. The accuracy of the prediction is within 2 dB for the region $x/D=[10,20]$ at $St=0.4$ and 0.6 and for the region $x/D=[12,20]$ at $St=0.8$. A sharper decay is observed for $St=1$ at $x=14D$. These results show that the proposed forcing model is capable of predicting jet noise within the Mach number and Strouhal number ranges of $M_j=[0.4,0.9]$ and $St=[0.4,1]$, respectively. The suppression of jet noise due to flight effect is also well captured for the regime $M_\infty /M_j<0.33$. Beyond these limits, the validity of the model remains to be tested.

Figure 20. PSD of the acoustic pressure obtained using the line-source model given in (5.11) (dashed) compared with the LES data (solid) for the cases (a,d,g,j) M07Mc00, (b,e,h,k) M07Mc15 and (c, f,i,l) M08Mc00. Different frequencies ranging from $St=0.4$ to 1 are shown from top to bottom.

We also present a comparison of the model prediction and the LES data as a function of frequency in figure 21 for all the cases investigated in this study at two different propagation angles, $\theta =15^\circ$ and $25^\circ$ measured from the downstream end, which correspond to $x/D=18.7$ and 10.7, respectively, in previous figures showing PSD data. The region $\theta <25^\circ$ roughly determines the acoustic field dominated by the first RESPOD forcing mode and thus, the region of validity of the model. Additionally, $\theta =15^\circ$ is near the sponge zone limit used in the resolvent computations. The frequency range in this comparison is extended to $St=[0.1,1.5]$ to show the model performance beyond the range for which it has been tuned. We limit the analysis to this range since, below $St=0.1$, the hydrodynamic fluctuations reach the acoustic field as reported by Nekkanti & Schmidt (Reference Nekkanti and Schmidt2021), who used the same database for the cases M09Mc00 and M07Mc00; and at $St=1.5$, the acoustic level is already 20 dB less than the peak in all the cases.

Figure 21. PSD of the acoustic pressure obtained from LES (solid) and predicted by the line-source model (dashed) for (a,b) static jet cases and (c,d) cases with flight stream at two different propagation angles, (a,c) $\theta =15^\circ$ and (b,d) $25^\circ$.

For the range $St=[0.4,1]$, the model accurately predicts the acoustic field for all the cases, despite some underestimation for the cases M04Mc00, M07Mc15 and M09Mc30 at $\theta =25^\circ$. However, at slightly lower propagation angles, the model starts to yield better predictions for these three cases as well, as can be seen in figures 16, 19 and 20. At ${St=1.5}$, the model yields accurate predictions in all the cases except for the cases M07Mc00 and M08Mc00 at $\theta =15^\circ$. At frequencies below $St=0.4$, the model starts to overpredict the acoustic level which becomes evident at $St=0.1$ in all the cases. We expect the model to be valid only above a certain frequency. The forcing amplitude is scaled by $1/St^{3/2}$ in (5.11), which tends to infinity as $St\to 0$. The overprediction remains within 3 dB at $St=0.2$ for all the cases except M07Mc00, M07Mc15 and M09Mc30, where it reaches up to 5 dB. Another reason that may explain the poor performance at low frequencies is that the identity of the leading resolvent mode switches from Kelvin–Helmholtz to Orr mechanism at approximately $St = 0.3$ (Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020), and the very different physics of Orr modes would require a different forcing model.

Finally, we test our model against experimental data available in the literature. We choose the cold jet experiments conducted in NASA SHJAR (Khavaran & Bridges Reference Khavaran and Bridges2009) denoted with set point 7, for which an LES database was generated by Markesteijn & Karabasov (Reference Markesteijn and Karabasov2017) and the mean flow and acoustic field data were provided by Gryazev, Markesteijn & Karabasov (Reference Gryazev, Markesteijn and Karabasov2023). The jet has a Mach number $M=0.9$ with a static-to-total temperature ratio $T_j/T_0=0.835$ and nozzle pressure ratio $NPR=2.861$. The acoustic data are provided at three angles by Gryazev et al. (Reference Gryazev, Markesteijn and Karabasov2023): $\theta =30^\circ$, $60^\circ$ and $90^\circ$, out of which, only $\theta =30^\circ$ is relevant for our model considering that our study is limited to the axisymmetric mode and low-propagation angles. Figure 22 shows the comparison of the resolvent-based prediction against the experimental data. We see the model slightly underpredicts the acoustic field at low $St$ while the discrepancy increases up to 5 dB at $St=1.5$. The reason for this discrepancy is that the model predicts only the acoustic field of the axisymmetric mode while the experimental data are obtained for the total acoustic field. It is known that the axisymmetric mode dominates the acoustic field at low angles and low $St$ while the contribution of the higher-order azimuthal modes increases with frequency (Brès et al. Reference Brès, Jaunet, Rallic, Jordan, Towne, Schmidt, Colonius, Cavalieri and Lele2016; Faranosov et al. Reference Faranosov, Belyaev, Kopiev, Zaytsev, Aleksentsev, Bersenev, Chursin and Viskova2017). The trend observed here is consistent with the literature (see figure 9 of Brès et al. Reference Brès, Jaunet, Rallic, Jordan, Towne, Schmidt, Colonius, Cavalieri and Lele2016).

Figure 22. PSD of the acoustic pressure at $m=0$ predicted by the line-source model (orange) for the NASA-SHJAR-SP7 jet compared against the experimental data for the total acoustic field at $\theta =30^\circ$.