1. Introduction
The noise generated by aircraft combustors has become a significant contributor to the overall aircraft noise (Ihme Reference Ihme2017). Its significance has been recognised at low-power engine conditions during landing and approach when the jet exhaust velocity and associated jet noise is low (Ferand et al. Reference Ferand, Livebardon, Moreau and Sanjose2019). Combustion noise is typically divided into direct and indirect contributions. While direct combustion noise arises from pressure fluctuations directly caused by unsteady heat release in the combustion chamber, indirect combustion noise is generated by the acceleration of flow field perturbations when accelerated through the turbine stage or the exit nozzle. Three types of inhomogeneities are responsible for indirect noise, namely compositional noise due to species inhomogeneities (Magri, O’Brien & Ihme Reference Magri, O’Brien and Ihme2016; Ihme Reference Ihme2017), vorticity noise due to velocity fluctuations (Cumpsty Reference Cumpsty1979) and entropy noise due to temperature inhomogeneities (Marble & Candel Reference Marble and Candel1977). The remainder of the paper focuses on the latter.
The first studies on indirect noise assumed one-dimensional (1-D) low-frequency fluctuations to develop the compact nozzle (CN) theory (Marble & Candel Reference Marble and Candel1977). Extension to non-zero frequencies were subsequently performed through, for instance, piecewise linear approximations (Moase, Brear & Manzie Reference Moase, Brear and Manzie2007) and a Magnus expansion (Duran & Moreau Reference Duran and Moreau2013). Many of the more recent investigations have relaxed some of the earlier assumptions, for instance by extending analytical models to compound (Younes & Hickey Reference Younes and Hickey2019, Reference Younes and Hickey2020) and non-isentropic (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019, Reference De Domenico, Rolland, Rodrigues, Magri and Hochgreb2021) nozzles or by considering effects of wall friction (Jain & Magri Reference Jain and Magri2022) and wall heat transfer (Yeddula et al. Reference Yeddula, Guzmán-Iñigo and Morgans2022a ). Past studies have also enhanced our physical understanding of indirect combustion noise. Duran & Moreau (Reference Duran and Moreau2013) showed that both the modulus and the phase of the transfer functions of subsonic converging–diverging nozzles have a strong dependence on the frequency, leading to a significant decrease in the ratio between indirect and direct noise at high frequencies. This finding complements a model combustor study with the CN approximation from Leyko, Nicoud & Poinsot (Reference Leyko, Nicoud and Poinsot2009) who showed that indirect noise can exceed direct noise by an order of magnitude in most realistic aviation combustors. These predictions were confirmed in some combustors through experimental tests (Bake, Michel & Roehle Reference Bake, Michel and Roehle2007; Miles Reference Miles2009). Further numerical studies noted that indirect noise exceeded direct noise at low frequencies while being heavily dependent on the operating condition (Shao et al. Reference Shao, Brouzet, Rock and Ihme2022), and could be drastically reduced in next-generation aviation engines that utilise lean premixed combustion (Brouzet et al. Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024).
Despite several advancements in modelling and physical understanding, the state-of-the-art methodology for computing indirect noise from turbulent flows solely relies on planar entropy waves, as performed, for instance, in the combustion-generated noise tool CHORUS (Duran et al. Reference Duran, Leyko, Moreau, Nicoud and Poinsot2013) or by Tam (Reference Tam2021). Using this approach, Livebardon et al. (Reference Livebardon, Moreau, Gicquel, Poinsot and Bouty2016) demonstrated the importance of indirect noise generated in a helicopter engine and Tam (Reference Tam2021) showed that entropy/shock interactions were a significant noise-generating mechanism in a military jet nozzle. Yang, Guzmán-Iñigo & Morgans (Reference Yang, Guzmán-Iñigo and Morgans2022) and Yeddula et al. (Reference Yeddula, Guzmán-Iñigo, Morgans and Yang2022b ) are among the few that studied non-planar entropy waves, showing that non-planar effects cause a fall off in the entropy-to-acoustic transfer functions at moderate to high frequencies.
So far, however, it is unclear to what extent the multi-modal nature of turbulent entropy fields impacts indirect noise. In this study, we address this question by (i) developing an analytic framework to study indirect noise generation by non-planar and multi-modal entropy waves and (ii) performing indirect noise simulations of turbulent flows through subsonic nozzles to quantify the contribution of the non-planar entropy modes. Section 2 describes the methodology used for the simulations and the transfer function methodology for multi-modal fields. Section 3 presents the configurations considered in this study while § 4 discusses the computational results. Finally, the conclusions are summarised in § 5.
2. Methodology
2.1. Linearised Euler equations solver
The flow field evolution and acoustic propagation are described considering the linearised Euler equations (LEE). In addition, we assume an ideal, calorically perfect gas composed of a single species and an isentropic flow. The LEE system is written as three-dimensional (3-D) unsteady transport equations for the velocity fluctuations
$\boldsymbol{u}'$
, the pressure fluctuations
$p'$
and the entropy fluctuations
$s'$
\begin{align} \partial _{t}{\boldsymbol{u}^\prime } + \langle {\boldsymbol{u}} \rangle \cdot \nabla \boldsymbol{u}^\prime &= -\boldsymbol{u}^\prime \cdot \nabla \langle {\boldsymbol{u}} \rangle - \frac {1}{\langle {\rho } \rangle } \nabla p^\prime -\left ( \frac {p^\prime }{\gamma \langle {p} \rangle } - \frac {s^\prime }{c_p} \right ) \langle {\boldsymbol{u}} \rangle \cdot \nabla \langle {\boldsymbol{u}} \rangle, \end{align}
\begin{align} \partial _{t}{p^\prime }+ \langle {\boldsymbol{u}} \rangle \cdot \nabla p^\prime &= - \boldsymbol{u}^\prime \cdot \nabla \langle {p} \rangle - \gamma \left ( \langle {p} \rangle \nabla \cdot \boldsymbol{u}^\prime + p^\prime \nabla \cdot \langle {\boldsymbol{u}} \rangle \right )\!, \end{align}
\begin{align} \partial _{t}{s^\prime } + \langle {\boldsymbol{u}} \rangle \cdot \nabla s^\prime &= - \boldsymbol{u}^\prime \cdot \nabla \langle {s} \rangle. \end{align}
Here
$\rho$
is the density,
$c_p$
is the heat capacity,
$\gamma$
is the heat capacity ratio, while
$\langle {\phi } \rangle$
and
$\phi ^\prime$
represent the Reynolds average and fluctuations of
$\phi$
so that
$\phi (t,x,y,z) = \langle {\phi } \rangle (x,y,z) + \phi ^\prime (t,x,y,z)$
. The streamwise, transverse and spanwise directions are represented by
$x$
,
$y$
and
$z$
, respectively The numerical solver described in O’Brien et al. (Reference O’Brien, Kim and Ihme2015) is employed to solve (2.1)–(2.3). These equations are spatially discretised using a 4th-order dispersion-relation-preserving (DRP) finite difference scheme, while the time advancement is performed using a 4th-order Runge–Kutta method. To ensure numerical stability, an 8th-order acoustic-optimised explicit filter is utilised at every time step. Walls are treated as slip with no-penetration boundary conditions for
$\boldsymbol{u}'$
, while Von Neumann boundary conditions imposing a zero normal gradient are used for
$s'$
and
$p'$
. Characteristic boundary conditions are applied at the inlet and the outlet of the domain. A sponge region is used at the outlet to avoid any significant vorticity or entropy inhomogeneities to exit the domain. This solver has been used to study indirect noise in a realistic gas-turbine combustor (Shao et al. Reference Shao, Maeda and Ihme2021, Reference Shao, Brouzet, Rock and Ihme2022) and indirect noise due to compositional inhomogeneities (Magri et al. Reference Magri, O’Brien and Ihme2016).
2.2. Transfer function analysis for multi-modal fields
The present section discusses the transfer function modelling framework in the context of indirect noise generation. We consider entropy fluctuations
$s^{\prime}_1$
which are accelerated through a turbine stage or nozzle, and which generate an upstream-propagating acoustic wave and a downstream propagating acoustic wave
\begin{align} \begin{array}{l} \unicode{x03C0} ^- = \left[\dfrac{ p'}{(\gamma \langle {p} \rangle )}- \dfrac{u'}{\langle {c} \rangle}\right]_{1} \\[8pt]\unicode{x03C0} ^+ = \left[ \dfrac{p'}{(\gamma \langle {p} \rangle )} + \dfrac{u'}{\langle {c} \rangle }\right]_2 \end{array}, \end{align}
where
$c$
represents the speed of sound.
Here and in the following, subscripts 1 and 2 refer to the inlet plane (
$x=x_1$
) and outlet plane (
$x=x_2$
) of the turbine stage or nozzle, respectively. In a quasi-1-D framework, we assume
$s^{\prime}_1$
and
$\unicode{x03C0} ^{\pm }$
to be a function only of
$t$
, implying that these fields are planar. Note that this approach is referred to as quasi-1-D because the flow and sound generation process are still multi-dimensional. Only
$s^{\prime}_1$
and
$\unicode{x03C0} ^{\pm }$
are assumed planar. With these assumptions, the acoustic transfer functions
$H^\pm$
which relate the entropy input to the acoustic outputs are
\begin{equation} H^{\pm }(\omega ) = \widehat {\unicode{x03C0} }^{\pm }(\omega ) / \widehat {s^{\prime}_1}(\omega ) , \end{equation}
where
$\widehat { \cdot }$
denotes the Fourier transform in frequency space, and
$\omega$
represents the angular frequency.
We now relax the assumption of a planar incoming entropy field so that
$s^{\prime}_1=s^{\prime}_1(t,y,z)$
. Note that we still assume planar acoustic waves such that
$\unicode{x03C0} ^{\pm } = \unicode{x03C0} ^{\pm }(t)$
. For ease of comprehension and to simplify the upcoming mathematical derivations, we remove the spanwise dependence such that
$s^{\prime}_1 = s^{\prime}_1(t,y)$
. In that case, rewriting (2.5) considering the
$y$
-dependence yields
\begin{equation} \widehat {\unicode{x03C0} }^{\pm }(\omega ) = \int _{-L_y/2}^{L_y/2} \widehat {s^{\prime}_1}(\omega, y) H^{\pm }(\omega, y) {\rm d}y , \end{equation}
where the transverse width of the domain at the inlet is
$L_y$
. Equation (2.6) can be interpreted as integrating the acoustic contributions of every streamline throughout the width of the domain. However, computing
$H^{\pm }(\omega, y)$
through LEE simulations appears to be a challenging task. To obtain a more useful expression for (2.6), we first perform a Fourier mode decomposition of the entropy field
$s^{\prime}_1$
in the
$y$
-direction:
\begin{align} s^{\prime}_1(t,y) = \sum _{n=0}^{\infty } A_{n}(t) \cos \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) + \sum _{n=1}^{\infty } B_{n}(t) \sin \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) , \end{align}
where the amplitudes of the
$n$
th time-dependent symmetric and anti-symmetric modes are denoted by
$A_{n}$
and
$B_{n}$
, respectively. Note that the planar mode is represented by the
$n=0$
term. A temporal Fourier transform of (2.7) yields
\begin{equation} \widehat {s^{\prime}_1}(\omega, y) = \sum _{n=0}^{\infty } \widehat {A}_{n}(\omega ) \cos \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) + \sum _{n=1}^{\infty } \widehat {B}_{n}(\omega ) \sin \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) . \end{equation}
Combining (2.6) and (2.8), we can express the acoustic outputs as
\begin{align} \widehat {\unicode{x03C0} }^{\pm }(\omega ) = &\sum _{n=0}^{\infty } \widehat {A}_{n}(\omega ) H^{\pm }_{A_n}(\omega ) + \sum _{n=1}^{\infty } \widehat {B}_{n}(\omega ) H^{\pm }_{B_n}(\omega ) , \end{align}
where
\begin{align} H^{\pm }_{A_n}(\omega ) = \int _{-L_y/2}^{L_y/2} \cos \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) H^{\pm }(\omega, y) {\rm d}y , \end{align}
\begin{align} H^{\pm }_{B_n}(\omega ) = \int _{-L_y/2}^{L_y/2} \sin \left (\frac {2 \unicode{x03C0} n y}{L_y}\right ) H^{\pm }(\omega, y) {\rm d}y . \end{align}
The transfer functions for the symmetric and the anti-symmetric entropy modes are denoted by
$H^{\pm }_{A_n}$
and
$H^{\pm }_{B_n}$
, respectively. Equation (2.9) shows that the non-planar Fourier modes contribute independently to the noise generation through their respective transfer functions.
Equation (2.8) is an orthogonal basis decomposition of
$\widehat {s^{\prime}_1}(\omega, y)$
, which represents the input entropy field in spectral space. While any orthogonal basis decomposition of
$\widehat {s^{\prime}_1}$
could have been performed, Fourier modes possess a useful feature. Indeed, in the case of symmetric nozzles, such as those studied therein, the anti-symmetric contributions will vanish, as the integral in (2.11) will be zero. For this reason, we only analyse the acoustic contributions of the symmetric modes in the following. In § 4.2, we use this methodology to compute the noise generated by a turbulent flow field and quantify the non-planar contributions. Specifically, to compute
$ H^{\pm }_{A_n}(\omega )$
, we invert (2.9) by considering one mode
$n$
at a time.
3. Flow and nozzle configuration
We consider two types of entropy fields: whitenoise signals and a turbulent flow field from a realistic aviation combustor. As shown in figure 1(a), these fields are accelerated through an exhaust nozzle, leading to the generation of the
$\unicode{x03C0} ^{\pm }$
acoustic waves.
We refer to whitenoise signals as entropy fields that have equal amplitude over a range of frequencies. These signals result in the superposition of temporal harmonic waves with different frequencies and random phase shifts
$\varphi$
:
\begin{equation} s^{\prime}_1(t,y) = \sum _{m=0}^M C \sin \left (m\omega _0 t + \varphi _m\right ) \cos \left (2 \unicode{x03C0} n y / L_y\right ) , \end{equation}
where
$\omega _0$
and
$M\omega _0$
denote the lowest and highest angular frequencies considered, and
$C=0.01$
is a constant. This approach enables the computation of transfer functions with a single simulation instead of considering several tonal cases. Following the Fourier mode decomposition described in § 2.2, non-planar symmetric modes are also considered by transversely modulating the whitenoise signals with the cosine term in (3.1).
In addition to these whitenoise signals, we also consider a turbulent field from a single injector segment of a realistic gas-turbine combustor to assess non-planar effects in a practical configuration. These results were obtained as part of a previous study (Brouzet et al. Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024), where a large-eddy simulation (LES) of a next-generation aviation combustor was performed. This LES reproduced the experimental study of a single sector combustor segment by McCormick, Hultgren & Mendoza (2022). Therefore, potential thermo-acoustic coupling with azimuthal or helical entropy modes are not captured. The combustor features axial staging using separate pilot and main injectors. The pilot injector possesses an air-blast atomiser and creates a non-premixed flame, which anchors the main mixer flame through a reacting jet in crossflow. A take-off condition is considered with a high operating pressure and preheated air. Further details on the configuration, numerical methods, and physics of the combustor can be found in the original study (Brouzet et al. Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024). The entropy field downstream of the main flame is used as an inlet to the nozzle described in this section.
(a) Schematic of the configuration considered with (b) nozzle height and (c)
$M_x = \langle {u_x} \rangle /\langle {c} \rangle$
and
$M_y = \langle {u_y} \rangle /\langle {c} \rangle$
for the cases with
$l=$
5 mm, 25 mm and 50 mm. Here,
$M_x$
and
$M_y$
are shown for a streamline starting at
$y=0.95 h_1$
, close to the nozzle wall. The full length of the nozzle is denoted by
$L_x$
.
A nozzle with a tangent hyperbolic profile is considered, where the nozzle height is defined as
$h(x) = 0.5(h_2+h_1) + 0.5(h_2-h_1) \text{tanh}(x/l)$
, with
$-L_x/2\lt x\lt L_x/2$
where
$L_x=300$
mm denotes the full length of the nozzle. The characteristic nozzle length, representative of the constriction rate, is denoted by
$l$
. A ratio
$h_1/h_2 = 5.7$
is chosen to obtain a subsonic Mach number
$M_2=0.30$
with
$M_1 = 0.05$
. Thirty equidistantly spaced numerical probes record
$p'$
and
$u'$
at the streamwise locations
$x=-0.45 L_x$
and
$x=0.45 L_x$
. The
$p'$
and
$u'$
traces are then ensemble averaged before computing
$\pi^{-}$
and
$\pi^+$
with the equations shown in the first paragraph of § 2.2. When studying non-zero frequency effects, the geometry of the nozzle affects the streamwise and transverse velocity profiles and, in turn, the generated indirect noise (Emmanuelli et al. Reference Emmanuelli, Zheng, Huet, Giauque, Le Garrec and Ducruix2020; Huet, Emmanuelli & Le Garrec Reference Huet, Emmanuelli and Le Garrec2020). We therefore consider different nozzles with
$l=$
5 mm, 25 mm and 50 mm. The profiles of the nozzles and their flow Mach number are shown in figure 1(b–c) as a function of the streamwise coordinate. As the characteristic length of the nozzle decreases, the amplitude of the transverse flow increases. To reduce computational costs and simplify the analysis, the calculations are two-dimensional (2-D), considering only the streamwise and transverse directions,
$x$
and
$y$
. Therefore, the inlet entropy fluctuations correspond to the spanwise-average of the turbulent field. For all cases, we consider inlet conditions similar to those found in the realistic aviation combustor, with
$\langle {p_1} \rangle =2.8$
MPa,
$\langle {c_1} \rangle =820$
m s–1 and
$\gamma =1.4$
. Reynolds average fields necessary to solve (2.1) to (2.3) are computed from Reynolds-averaged Navier–Stokes (RANS) simulations with the commercial software ANSYS.
4. Results
4.1. Whitenoise entropy fields
Magnitude of the transfer functions
$H^\pm _{A_n}$
for the symmetric modes (
$n \leqslant 5$
) in the (a,b)
$l=50$
mm and (d,e)
$l=5$
mm nozzles. The phases of
$H^{-}_{A_n}$
are shown for (c)
$l=50$
mm and (f)
$l=5$
mm. The horizontal dashed lines represent the CN solution and the vertical dot-dashed lines show the threshold frequency
$f_{thresh}$
at which the difference between the planar mode and the loudest non-planar mode is less than 3 dB.
We have performed LEE computations for whitenoise entropy signals with planar and non-planar symmetric modes up to
$n = 5$
. The magnitude of the transfer functions
$H^\pm _{A_n}$
for the cases with
$l=50$
mm and
$l=5$
mm are shown in figure 2(a,b) and figure 2(d,e), respectively, as a function of the convective Helmholtz number
$He = fh_1/u_1$
. The transfer functions presented in this study have been filtered over a
$1/12$
-octave band for improved convergence at high frequencies. The results from the CN theory are also displayed, showing good agreement with
$H^{\pm }_{A_0}$
at low frequencies. In the low-frequency range, the non-planar modes are at least an order of magnitude smaller in amplitude, which corresponds to a relative difference of 20 dB or more. The vertical dot-dashed lines represent the frequency
$f_{thresh}$
at which the difference between the loudest non-planar mode and the planar mode is less than 3 dB. This frequency therefore indicates when the non-planar modes become acoustically significant. As the characteristic length of the nozzle decreases,
$f_{thresh}$
decreases and the amplitude of the non-planar modes approaches the CN solution. Specifically, the
$n=1$
mode in figure 2(e) reaches an amplitude corresponding to 35 % of the CN solution at 370 Hz (or
$He=0.68$
), showing the relevance of non-planar modes for indirect noise at intermediate frequencies. Note that minor acoustic reflections are expected at the inlet due to the injection of non-planar waves. As discussed in the supplementary material (available at https://doi.org/10.1017/jfm.2025.358), however, these reflections are not expected to significantly affect the results presented in this paper.
Figure 2(c,f) show the phases of the
$H_{A_n}^-$
transfer functions for the two nozzles considered. The results for
$H_{A_n}^+$
are similar and not shown for the sake of brevity. With
$l=50$
mm, the phases are linear for all modes up to
$n=5$
in the frequency range of interest. This means that the finite spatial extent of the nozzle causes a constant time delay in the acoustic wave response regardless of the transverse nature of the entropy field. However, the phases of the
$n \geqslant 2$
modes have a nonlinear behaviour in the
$l=5$
mm nozzle, showing that the indirect noise generated by non-planar modes can be drastically affected by the transverse flow.
4.2. Turbulent entropy field
(a) Turbulent entropy field from Brouzet et al. (Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024) with (b) planar mode
$A_0$
and (c) non-planar mode
$A_1$
. The corresponding spectra are shown in (d).
In § 2.2, we have established that the indirect noise generated by a multi-modal entropy field can be computed by considering its different spatial Fourier modes independently. This strategy is now applied to the turbulent field described in § 3. Figure 3(a) shows the entropy fluctuations
$s^{\prime}_1$
obtained at a probe located in the downstream section of the aviation combustor studied by Brouzet et al. (Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024). The corresponding planar mode
$A_0$
and first non-planar mode
$A_1$
are displayed in figures 3(b) and (c), respectively. The amplitude of
$A_0$
is noticeably smaller than
$s^{\prime}_1$
due to the cancellation between positive and negative fluctuations when performing spatial averaging in the transverse direction. Furthermore,
$A_1$
appears to have a greater magnitude than
$A_0$
, indicating stratification of the flow field. Such stratification is commonly found in aviation combustors, especially when dilution and/or effusion cooling is employed (Blomeyer et al. Reference Blomeyer, Krautkremer, Hennecke and Doerr1999; Shao et al. Reference Shao, Brouzet, Rock and Ihme2022).
For a quantitative analysis, figure 3(d) shows the spectra of
$s^{\prime}_1$
(dot-dashed line) and all modes
$A_n$
for
$n \leqslant 5$
(solid lines) as a function of
$He$
. The spectra have been filtered over a
$1/3$
-octave band, as oscillations were present when using narrower band filtering, which introduced variations in the uncertainties but did not change the results quantitatively. While the spectra for
$A_0$
and
$A_1$
are similar, the modes become less significant as
$n$
increases. The
$s^{\prime}_1$
spectrum features turbulence characteristics: monotonous decay with increasing frequency, power-law dependence in the inertial range, and exponential decay in the dissipation range. The
$A_0$
mode spectrum has a smaller amplitude at all frequencies, consistent with figure 3(a,b), and does not possess the same power-law decay in the inertial range. This discrepancy highlights the necessity of using the relevant turbulent entropy spectrum associated with the modes
$A_n$
when computing indirect noise through a transfer function approach. With the turbulent flow considered in figure 3, an over-prediction of 10 dB would be observed at
$He=1$
if
$s^{\prime}_1$
was used instead of the planar mode fluctuations
$A_0$
. This result has direct implications for the experimental investigation of indirect noise since a single probe measurement is not sufficient to compute a planar or non-planar mode spectrum. Computational approaches could provide additional information and could be used to quantify combustor exit mode shapes.
With the transfer functions
$H^{\pm }_{A_n}$
(figure 2) and the entropy spectra
$\widehat {A}_n$
(figure 3), we can now compute the acoustic waves
$\unicode{x03C0} ^{\pm }$
by simplifying (2.9) and truncating the summation as
\begin{equation} \widehat {\unicode{x03C0} }^{\pm }_N = \sum _{n=0}^N H^{\pm }_{A_n} \widehat {A}_n . \end{equation}
Specifically, we compute
$\widehat {\unicode{x03C0} }^{\pm }_0$
, i.e. using only the planar mode, and
$\widehat {\unicode{x03C0} }^{\pm }_5$
, i.e. adding the contributions of the first five non-planar symmetric modes. Figure 4 compares
$\widehat {\unicode{x03C0} }^{\pm }_0$
and
$\widehat {\unicode{x03C0} }^{\pm }_5$
with the prediction from the LEE simulations (
$\widehat {\unicode{x03C0} }^{\pm }_{LEE}$
) for the
$l=50$
mm (panels (a–c)) and
$l=5$
mm (panels (d –f)) nozzles. First, the planar mode prediction
$\unicode{x03C0} ^{\pm }_0$
fails at mid to high frequencies, when the contribution from the non-planar modes becomes significant. The vertical lines represent the frequency
$f_{thresh}$
at which the difference between
$\widehat {\unicode{x03C0} }^{\pm }_{LEE}$
and
$\widehat {\unicode{x03C0} }^{\pm }_{0}$
exceeds 3 dB. As for the transfer functions,
$f_{thresh}$
decreases when the nozzle becomes more compact, confirming that non-planar modes can be acoustically significant at frequencies below 1 kHz. The
$\widehat {\unicode{x03C0} }^{\pm }_5$
spectra show a better agreement with the LEE results in this mid- to high-frequency range. To better quantify the performance of the two spectra computed via the transfer function approach, we report the differences
$\Delta ^{\pm }_{0} = \unicode{x03C0} ^{\pm }_0 - \unicode{x03C0} ^{\pm }_{LEE}$
and
$\Delta ^{\pm }_{5} = \unicode{x03C0} ^{\pm }_5 - \unicode{x03C0} ^{\pm }_{LEE}$
as a function of
$He$
in figure 4(c,f). While the model considering non-planar modes performs significantly better only at high frequencies (
${\gt } 3$
kHz) for the
$l=50$
mm nozzle, smaller
$\Delta ^{\pm }_5$
values are observed around 1 kHz with
$l=5$
mm. Overall, considering non-planar modes in the transfer function formulation significantly improves the prediction of the noise spectra generated by turbulent flow fields. Further improvements in the acoustic wave prediction at high frequencies could be obtained by considering modes
$A_n$
for
$n \gt 5$
.
Spectra of
$\unicode{x03C0} ^{\pm }_{LEE}$
(solid lines),
$\unicode{x03C0} ^{\pm }_0$
(dashed lines) and
$\unicode{x03C0} ^{\pm }_5$
(dot-dashed lines) for the (a,d)
$\widehat {\unicode{x03C0} }^-$
and (b,e)
$\widehat {\unicode{x03C0} }^+$
acoustic waves. Panels (c,f) show
$\varDelta ^{\pm }_{0}$
and
$\varDelta ^{\pm }_{5}$
. Results for the
$l=50$
mm (a–c) and
$l=5$
mm (d–f) nozzles are shown.
4.3. Scaling relation for
$f_{thresh}$
We use scaling analysis to estimate dependencies of
$f_{thresh}$
and identify the frequencies at which non-planar waves become significant for a given nozzle geometry. To provide a phenomenological understanding and scaling, we show in figure 5(a) a schematic of the nozzle with two streamlines, one at the centreline and one close to the nozzle’s wall. Positive and negative entropy contributions are depicted by
$+$
and
$-$
circles. At the inlet of the nozzle, the positive and negative entropy contributions cancel each other when a transverse integration is performed. As the entropy fluctuations are convected by the mean flow, the shear will distort the original shape of the wave, leading to a different acoustic response compared with a quasi-1-D flow. Specifically, shear will induce a difference in resident times
$\tau$
between streamlines. An associated phase difference
$\Delta \phi$
can be computed as
\begin{equation} \Delta \phi = 2\unicode{x03C0} f \tau . \end{equation}
If
$\Delta \phi$
is of the order of
$\unicode{x03C0}$
or larger, as depicted in figure 5, cancellation effects will not occur when a transverse integration is performed, or at least not completely. As a result, the indirect noise generated by non-planar waves will become larger. By rearranging (4.2), we can therefore identify the threshold frequency at which we expect non-planar modes to be acoustically significant:
\begin{equation} f_{thresh} = \frac {1}{2 \tau _{max}} , \end{equation}
where
$\tau _{max}$
represents the maximum resident time difference between two streamlines. Practically speaking, 50 streamlines equidistantly spaced across the nozzle height are first computed based on
$\langle {u} \rangle$
. Their residence time from the inlet to the outlet of the nozzle is then computed by integrating
$\langle {u} \rangle$
along the streamline paths. Here,
$\tau _{max}$
is defined as the difference between the smallest and largest residence times.
Figure 5(b) shows the result of the scaling relation (4.3), compared with the results from the LEE. In addition to the nozzles presented in § 3, we consider two additional cases, i.e. a converging–diverging (CD) nozzle and the linear-velocity (LV) nozzle, both presented in the supplementary material. The quantitatively good predictions of the scaling suggest that 2-D shear effects are responsible for significant acoustic contributions of non-planar entropy modes. Furthermore, it shows that a simple streamline computation can provide an estimation of the frequencies at which non-planar modes are expected to be significant.
(a) Schematic explaining the effect of shear on the non-planar entropy waves. (b) Comparison between
$f_{thresh}$
obtained with the LEE results and the scaling model (4.3) for a converging–diverging (CD) nozzle, a linear-velocity (LV) nozzle and the tangent-hyperbolic (Tanh) nozzle introduced in § 3. Open and closed symbols represent
$\unicode{x03C0} ^-$
and
$\unicode{x03C0} ^+$
results, respectively.
5. Conclusions
We studied the indirect noise generated by multi-modal, non-planar entropy waves when accelerated through a nozzle by performing LEE computations. The simulations showed that, while the planar mode was acoustically dominant at low frequencies, non-planar modes become significant above a threshold frequency
$f_{thresh}$
, found in the mid- to high-frequency range. These contributions from the non-planar modes were explained by 2-D shear effects from the mean flow, which distort the entropy waves. A nozzle with a higher constriction rate will therefore result in non-planar modes contributing to the indirect noise at lower frequencies. A scaling relation that uses the residence time along streamlines was developed for estimating
$f_{thresh}$
. Reasonable agreement with LEE computations for several nozzle configurations was obtained.
To compute the indirect noise generated by multi-modal entropy fields, a transfer function methodology that accounts for the contribution of non-planar Fourier modes was developed. The application of this methodology to a turbulent flow field showed a significant improvement in the indirect noise spectra prediction compared with a conventional method that relies only on the planar mode. Furthermore, this study shows that the indirect noise from non-planar entropy modes found in aviation combustors can be significant at frequencies of interest. Using the data of a next-generation aviation combustor (Brouzet et al. Reference Brouzet, Krisna, McCormick, Reimann, Mendoza and Ihme2024), the present study shows significant non-planar acoustic contributions at frequencies below 1 kHz, which can be relevant in situations of thermo-acoustic instabilities coupled to indirect noise. In many industrial combustors, flows can be stratified to even a higher degree, for instance in the presence of dilution and effusion cooling, or can feature strong azimuthal/helical modes. In these cases, the energy content of the non-planar modes might be significantly larger than that of the planar mode, which could lead to even larger acoustic contribution of the non-planar entropy modes.
Supplementary material.
Supplementary material is available at https://doi.org/10.1017/jfm.2025.358.
Funding.
This work was partially supported by the AeroAcoustics Research Consortium (AARC) under award OAI-AARCS-24125. Information about AARC can be found at https://oai.org/aeroacoustics/.
Declaration of interests.
The authors report no conflict of interest.
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