Identification of jet noise source using causality method based on large-eddy simulation of a round jet flow

Abstract

An acoustic analogy analysis based on a decomposition of the source term in Lighthill’s equation is discussed in light of a large-eddy simulation of a subsonic turbulent jet exhausting from a baseline round nozzle at Mach number M = 0.9 with Reynolds number Re = 10⁵. The decomposed sub-terms show the nonlinear reciprocal interactions of density, velocity, vorticity, and dilatation fields. To understand the aerodynamic sound generation mechanism, intrinsic links between turbulent flow and emitted acoustic signals are made and applied to the large-eddy simulation data. Cross-correlation functions are used for the links between the far-field sound signals and the sub-terms as well as major flow variables in the jet flow domain. The spatial distributions of cross-correlations are examined to identify the sound source distribution throughout the domain and observe the mutual interactions and cancellations between the decomposed sub-terms. The contributions of sub-terms are also studied in frequency domain.

Keywords

Lighthill’s equation cross-correlation

Introduction

Identifying the physical mechanism of sound production by turbulence has been a long standing quest because of its complexity. Despite many theoretical formulations and descriptions, involved experiments, and measurements, the sound generation mechanism has not been well understood and is still debated in the literature. This subject has gained more attention due to its possible applications to environmental noise reduction problems in order to satisfy strict aircraft noise regulations. An intense source of aircraft noise lies in the jet during take-off. To reduce the jet noise, substantial efforts have been made to develop better theoretical descriptions and more precise experimental measurements, improve existing noise reduction devices, and create realistic numerical simulations. The focus of this work is to develop theoretical descriptions and methodology, directly applied to the large-eddy simulation (LES) of a turbulent flow exhausting from a baseline round jet nozzle configuration to provide a better understanding.

To investigate the sound generation mechanisms, cause-and-effect methods have been studied since the late 1960s to identify the sound source. One of these methods is to use cross-correlation functions, which provide the most straightforward way to establish direct links between flow variables in the jet flow domain and sound signals at the far-field locations simultaneously. Clark and Ribner¹ first applied this method to aero-acoustics by cross-correlating fluctuating lift of an airfoil with its consequent instantaneous sound based on Curle’s equation. For the flow variables in the jet domain, velocity fluctuation and pressure fluctuation were often cross-correlated with the far-field signals in the early studies of jet problems.^2–8 In those works, the normalized correlation levels were examined with different Mach numbers and at different far-field locations. It was found that the correlation level decreases drastically as the Mach number decreases and it also drops as the inlet angle (θ) increases, where the inlet angle is defined as the angle between the jet exit centerline and the line connecting the jet exit center to the far-field location, shown in Figure 4.

In recent works, with improvement of experimental measurement techniques and numerical simulations, more in-depth research on the cross-correlation has emerged. In the early 2000s, Panda and Seasholtz⁹ studied the correlation in peak noise emission direction by using a molecular Rayleigh scattering-based technique. Panda and Seasholtz¹⁰ also studied the correlation between each directional full source term ( $ρ u \otimes u$ ) of Lighthill’s equation^11,12 and the far-field pressure fluctuation based on experiments using different nozzles, one convergent and two convergent-divergent, with Mach numbers $M = 0.95, 1.4$ , and 1.8. Bogey and Bailly investigated the noise sources in subsonic round jets with different Mach numbers and Reynolds numbers to see their influences,¹³ using Mach numbers $M = 0.6$ and 0.9, and Reynolds numbers $Re = 1700$ and $\geq 105$ in their simulations. The normalized cross-correlations were calculated with the broadband signals of jet turbulence, such as the fluctuating velocities, the Reynolds stresses, the turbulent kinetic energy and the vorticity norm, linked with the radiated sound signals at different far-field locations. They observed that maximum correlation occurs at the end of the potential core regardless of the far-field locations with different inlet angles but it decreases as the inlet angle increases. The correlation is higher at the lower Reynolds number and lower at the lower Mach number. Similar observations were made experimentally by Tam et al.¹⁴ They used four different approaches, one of which is to calculate the normalized cross-correlations based on experiments of a subsonic jet at Mach number $M = 0.9$ with a convergent nozzle of 1.6-inch exit diameter and a supersonic jet at Mach number $M = 1.67$ with a convergent–divergent nozzle of 2-inch exit diameter. The point made in their work is that there are two types of noise sources; the nearly omni-directional fine-scale turbulence structure and the highly directional large-scale turbulence structure in the downstream direction.

Various decompositions of the source term in Lighthill’s equation have been developed to identify the source of sound more precisely. Powell,¹⁵ in his study of vortex sound, decomposed the source term to two terms, (1) divergence of Lamb vector and (2) Laplacian of turbulent kinetic energy for incompressible flow. He studied the sound produced from vorticity movement in free flow and concluded that the predominant sound generator must be a lateral quadrupole, especially at low Mach numbers. M $u . .$ ller and Obermeier¹⁶ used matched asymptotic expansions, and Crighton and Lesser¹⁷ applied the asymptotic expansions to predict the dipole component from the scattered field in two-dimensional acoustical radiation problem. Freund¹⁸ adopted Reynolds decomposition of the sound source term, shown in earlier works^19,20 wherein the source term is decomposed to its average and the corresponding perturbation part. In his work, the perturbation part is again decomposed to shear noise and self-noise terms according to the linear or quadratic dependence on the fluctuating velocities, respectively. In the work by Cabana et al.,²¹ the Lighthill’s source term was decomposed to 10 sub-terms, that is one of the most detailed decompositions, and each sub-term was examined and compared with other sub-terms based on the individual magnitudes and roles. In their work, two essential events, vortex roll-up and pairing, often occurring in the aerodynamic sound production process, were considered based on the two-dimensional simulation of compressible temporal mixing-layers. The sound-radiating core is also identified by performing frequency–wave number analysis and imposing a condition of supersonic convective speed with respect to the ambient.

In the present study, we adopt the decomposition by Cabana et al.²¹ combined with cross-correlation study and directly apply it to the LES of a round jet. Two types of cross-correlation functions, normalized and un-normalized, are discussed. These cross-correlation functions are applied to build direct connections between the flow variables and the far-field sound signals, and further extended to those between fluctuating decomposed source sub-terms in Ligthill’s equation and the far-field acoustic signals, to identify the major sound sources and their distributions. The normalized cross-correlation function shows the linear dependence of the unit volume of flow variable to the unit volume of the far-field fluctuating pressure, while the un-normalized cross-correlation function shows the mutual interactions and contributions of the sub-terms to the far-field pressure fluctuations quantitatively. In this sense, the raw cross-correlation function enables us to better access the roles of the source sub-terms within the full source term. The observations are made at different far-field locations; downstream and sideline directions. The notable aspect of this work lies in examining the overall distribution of the detailed sources and their interactions throughout the entire cross-sectional flow domain based on a numerical simulation of a baseline nozzle.

This paper is organized in the following order. We describe the LES of the round jet flow in the Numerical simulation description section. In the Lighthill’s source term section, the source term in Lighthill’s equation is expanded to 10 sub-terms, the cross-correlation functions are defined, and we demonstrate how we utilize the functions in our numerical simulation of the round jet flow. We apply the methodology to the simulation and make observations in the Results and discussion section. Finally, we conclude and summarize in the last section.

Numerical simulation description

In this section, we provide a brief description of the large-eddy simulation (LES) used in our analysis. The test case is a cold jet exhausting from the SMC000 baseline round nozzle with a 5° inner contraction angle, as shown in Figure 1. The acoustic Mach number, which is defined as a ratio of the jet exit centerline velocity (U) to the ambient sound speed, is set to 0.9 for the simulation. The Reynolds number, based on the jet nozzle exit centerline velocity and the nozzle exit diameter (D), is 10⁵. Turbulent boundary layer inflow conditions are imposed.

Figure 1.

A picture of the SMC000 nozzle.

The conical-shaped nozzle configuration in Figure 1 is closely reconstructed in the numerical grid. The grid has approximately 370 million points in total to cover the nozzle interior, near-nozzle and downstream flows. The stream-wise domain size is 20D downstream from the nozzle exit plane. To control the grid density in various regions of the flow, we use an overset grid system. For the purpose of our analysis, we consider the data points only on three overlapped grids as shown in Figure 2(a). A rectangular grid (Grid 1 in the figure) for the jet core flow is adopted in order to avoid the singularity problem on the jet centerline. The other two (Grids 2 and 3) are annular cylindrical grids surrounding Grid 1. The first cylindrical grid (Grid 2) covers the first 5D downstream from the nozzle exit while the second (Grid 3) covers the rest of the further downstream domain as depicted in Figure 2(a). To capture the small-scale structure of the near-nozzle flow, Grid 2 has finer axial and azimuthal grid resolution than Grid 3. The yz-plane cross-sectional profile of the overset grid comprised of the rectangular and the cylindrical grid is shown in Figure 2(b).

Figure 2.

Overset grid system. (a) Left: side profile of overset-grid system, (b) right: front profile of overset-grid system.

The far-field acoustic noise signal is calculated by coupling the LES data with the Ffowcs Williams–Hawkings (FW–H) method which is a surface integral acoustic technique.²² The FW–H surface is placed near the outer edge of the jet and encloses the turbulent flow region. It has an open outflow and does not intercept any turbulence-containing region. This surface is sufficiently long (20D) in the axial direction so that inclusion of the additional volume integral term, which models the contribution of the region not enclosed by the surface, has negligible effect on the predictions. This numerical simulation has been validated against an experiment and documented in Uzun and Hussaini.²³ The spectra comparison provided for validation of the methodology in Figure 3 shows close agreement between the predicted far-field noise and the experimental measurement at two different inlet angles, $θ = 30 \circ$ and 90°. These two angles are used for the analysis later in this paper.

Figure 3.

Spectra comparison. Narrowband spectra at $θ = 30 \circ$ (top) and $θ = 90 \circ$ (bottom) obtained with turbulent inflow conditions and comparison with experiment.

Figure 4.

xz-plane cross-sectional profile of vorticity norm, $| ω |$ , at $y = 0$ in the overset grids 1, 2, and 3 shown in Figure 2(a). Two far-field observation points are located 50D away from the jet exit center with the inlet angles $θ = 30 \circ, 90 \circ$ , respectively.

This LES ran on 1840 processors in parallel for 40 days to produce the required data for the analysis. The simulation was performed with implicit time-stepping. The computational time step is $Δ t = 0.00125 D / U$ . For our analysis, we collect statistical data every $20 Δ t$ for 2048 discrete times. For later use, we note that the length of the potential core in this simulation is about 10.5D. The detailed description and comparison with experiments of this simulation can be found in Uzun and Hussaini.²³

Lighthill’s source term

To understand the aerodynamic sound production process, we start with the classical Lighthill’s equation derived from the mass and momentum conservation equations, describing the propagation of the acoustic waves from an inviscid flow developing in a quiescent medium. The source term of this equation can be interpreted as the driver of propagating acoustic waves and it consists of not only the sound-production mechanisms but also the propagation of acoustic waves through the turbulent flow region, and their interactions. In this section, we examine how Lighthill’s source term can be decomposed and interpreted, and how the contribution of each sub-term can be measured.

Source term decomposition

The Lighthill’s equation is obtained by combining the time derivative of the mass conservation equation and the spatial derivative of the momentum conservation equation

\frac{\partial 2 ρ}{\partial t 2} - c_{0}^{2} δ ρ = \nabla \cdot (\nabla \cdot (ρ u \otimes u)) + \nabla \cdot (\nabla (p - c_{0}^{2} ρ))

(1)

where

u, ρ

and p are velocity field, density and pressure, and

c 0

is the ambient speed of sound. For isothermal flows, the second term in the right side of equation (1) is negligible. By decomposing the first source term, we have

\frac{\partial 2 ρ}{\partial t 2} - c_{0}^{2} δ ρ = \nabla \cdot (\nabla \cdot (ρ u \otimes u)) = \nabla \cdot (ρ u (\nabla \cdot u) + ρ (\nabla u) u + (u \otimes \nabla ρ) u) = \nabla \cdot (\underset{T 1}{\underset{︸}{ρ u (\nabla \cdot u)}} + \underset{T 2}{\underset{︸}{ρ \frac{\nabla u 2}{2}}} + \underset{T 3}{\underset{︸}{ρ (\nabla \times u) \times u}} + \underset{T 4}{\underset{︸}{(u \otimes \nabla ρ) u}})

(2)

Each term, $T i$ , can be again decomposed as

\nabla \cdot T 1 = ρ u \cdot \nabla Θ + ρ Θ 2 + Θ u \cdot \nabla ρ, \nabla \cdot T 2 = ρ \nabla 2 \frac{u 2}{2} + \nabla \frac{u 2}{2} \cdot \nabla ρ, \nabla \cdot T 3 = ρ u \cdot \nabla \times ω + u \cdot \nabla ρ \times ω - ρ ω \cdot ω, \nabla \cdot T 4 = Θ u \cdot \nabla ρ + u \cdot ((\nabla \nabla ρ) u) + (u \times \nabla ρ) : (\nabla u)

(3)

where

ω

and Θ are vorticity and dilatation fields respectively, defined as

ω = \nabla \times u, Θ = \nabla \cdot u

By substituting equation (3) back to equation (2), we obtain the final decomposed form

\frac{\partial 2 ρ}{\partial t 2} - c_{0}^{2} δ ρ = \underset{S 1}{\underset{︸}{ρ u \cdot \nabla Θ}} + \underset{S 2}{\underset{︸}{ρ Θ 2}} + \underset{S 3}{\underset{︸}{2 Θ u \cdot \nabla ρ}} + \underset{S 4}{\underset{︸}{ρ \nabla 2 \frac{u 2}{2}}} + \underset{S 5}{\underset{︸}{\nabla \frac{u 2}{2} \cdot \nabla ρ}} + \underset{S 6}{\underset{︸}{ρ u \cdot \nabla \times ω}} + \underset{S 7}{\underset{︸}{u \cdot \nabla ρ \times ω}} \underset{S 8}{\underset{︸}{- ρ ω \cdot ω}} + \underset{S 9}{\underset{︸}{u \cdot ((\nabla \nabla ρ) u)}} + \underset{S 10}{\underset{︸}{(u \times \nabla ρ) : (\nabla u)}}

(4)

These sub-terms, $S i, i = 1, 2, \dots, 9, 10$ , in equation (4) consist of density, velocity, vorticity, and dilatation fields, representing their reciprocal nonlinear interactions. We note that for an incompressible flow, the density gradient-related terms, $S i, i = 3, 5, 7, 9, 10$ , disappear, and the dilatation-related terms, $S i, i = 1, 2$ , are also zero from the mass conservation equation. In other words, only terms, $S i, i = 4, 6, 8$ , remain for an incompressible case. These three terms are nothing but the Laplacian of turbulent kinetic energy, $S 4$ , and the divergence of the Lamb vector, $S 6$ and $S 8$ , where the Lamb vector (L) is defined as a cross product of velocity and vorticity. We expect that these three terms will play major roles in the jet sound generation, also mentioned in Powell¹⁵ and Golanski et al.²⁴ The details of this decomposition with index notation can be found in Cabana et al.²¹

Cross-correlation functions

The source distribution of sound generation can be identified by a causality method, directly associating turbulent flow events with emitted sound signals. We apply the causality method directly to the LES data by calculating cross-correlation functions. Two cross-correlation functions are defined here; one is a raw cross-correlation function, $RC f, g$ , and the other is a normalized cross-correlation function, $NC f, g$ . These are defined as

RC f, g (x, y, τ - t) = < f (x, t), g (y, τ) >,

(5)

NC f, g (x, y, τ - t) = < f (x, t), g (y, τ) > / (< f 2 (x, t) > 1 / 2 < g 2 (y, τ) > 1 / 2)

(6)

where

< \cdot, \cdot >

is a time average over infinite time. However, it can be considered as a finite time average or an ensemble average when the noise field is a stationary random process. For the correlation, f and g can be any variables at two different spatial points x and y and at two times t and τ, respectively. For our analysis, flow variables and fluctuations of each source sub-term in equation (4) in the jet flow domain are used for f and pressure fluctuation at each far-field location is taken for g. To calculate the fluctuating part of a variable, we use Reynolds decomposition

f = \bar{f} + f'

(7)

where

\bar{f}

is the time average of

f

and

f'

is the fluctuation part.

For the far-field positions, we choose two locations that are 50D away from the jet exit center at two different inlet angles, $θ = 30 \circ$ and $90 \circ$ , as depicted in Figure 4, where D is the jet nozzle exit diameter and θ is the inlet angle between the jet exit centerline and the line connecting between the jet exit center and the far-field location in Figure 4. These specific angles are chosen based on the far-field overall sound pressure level (OASPL) that is calculated by

OASPL = 20 \log 10 (\frac{p' rms}{2 \times 10 - 5 Pa})

(8)

where

p' rms

is the root-mean-squared pressure fluctuation obtained from

p'

in the time domain. Figure 5 shows the OASPL at observation angles of 30°, 45°, 60°, 75° and 90°. The OASPL achieves the maximum and minimum values at those two angles,

θ = 30 \circ

and 90°, respectively, among the selected five far-field locations. For each selected inlet angle, we consider four azimuthal planes to obtain a big enough sample size for the ensemble average in equation (5). One plane is shown in Figure 4 with the profile of vorticity norm, for instance, and the other three can be pictured by rotating this plane by intervals of 90° about the x-axis. These four sets of the cross-correlations computed on the planes are averaged in the analysis. After the averaging process, all the results will be plotted on the xz-plane in the next section.

Figure 5.

Far-field overall sound pressure level at five different angles.

We take the retarded time $τ 0$ as $τ - t$ in equation (5) where the retarded time is defined as the time for the sound wave to travel from the position x in the flow domain to the far-field position y. The refraction effect of the propagating waves is not taken into consideration in this study since the far-field locations are assumed to be far enough from the jet exit not to cause notable errors in the calculation. The retarded time, $τ 0 = τ - t$ , is a reasonable choice in equation (5), considering the solution of the Lighthill’s equation

p' (y, τ) = \int d t \int d x G (x, y, τ - t) S (x, t) = \int d x \frac{S (x, τ - τ 0)}{| x - y |}

(9)

where S is the source term and G is the Green’s function which is defined by

G (x, y, τ - t) = \frac{δ 0 ((τ - t) - τ 0)}{| x - y |}

(10)

where

δ 0

is the Dirac delta function. Equation (9) shows the acoustic pressure at time τ to be determined by the source field at the earlier time

τ - τ 0

since Lighthill assumed the turbulent flow to be surrounded by an ambient acoustic medium. Bogey and Bailly also showed that the peaks of the cross-correlations occur at or near this retarded time in their work.¹³ For a fixed far-field location y, the cross-correlation function in equation (5) depends only on x in the flow domain with a retarded time

τ - t

, identifying the spatial distribution of jet noise source over the entire cross-sectional flow domain.

Results and discussion

In this section, we apply the cross-correlation functions introduced in the Lighthill’s source term section to the LES data described in the Numerical simulation description section. The observation is made on the xz-plane, for the sake of clarity, to see the spatial distribution over the entire cross-sectional domain. This plane covers the jet turbulence region of our interest, as shown in Figure 4. We first calculate the cross-correlations between the flow variables and the far-field pressure fluctuations. The cross-correlation functions are then applied between each or combined decomposed source sub-term fluctuations in the flow domain and the pressure fluctuations at a couple of far-field locations.

Flow variables

The normalized cross-correlations between the major flow variables in the jet domain and the pressure fluctuations at far-field locations have been examined in the recent literature. In particular, Bailly and Bogey¹³ showed the profiles of the normalized cross-correlations between fluctuating directional velocities, Reynolds stresses, the turbulent kinetic energy and vorticity norm on different axial lines, and the fluctuating pressures at different far-field points with different time delays. It was shown that the peak values of the normalized cross-correlations occur near the end of the potential core with the time delay at or near the retarded time, $τ 0$ . In the present study, the vorticity norm ( $| ω |$ ), Lamb vector (L), kinetic energy (K), and density (ρ) in the flow domain and the pressure fluctuations at the far-field locations at the inlet angles $θ = 30 \circ$ and $90 \circ$ are correlated, with the retarded time $τ 0$ . The results are shown in Figure 6. The normalized cross-correlations at $θ = 30 \circ$ have large values mainly over two regions, the end of the potential core (x = $10.5 D$ ) and the outer shear layer region near the interface between the shear layer and the ambient flow. We also notice that, near the end of the potential core, $NC | ω |, p'$ and $NC L, p'$ have positive correlations while $NC K, p'$ and $NC ρ, p'$ have negative correlations at $θ = 30 \circ$ . The sideline directional correlations at $θ = 90 \circ$ show the similar patterns but have overall smaller correlation values throughout the domain, compared with those in the downstream direction at $θ = 30 \circ$ . The result indicates that the propagation of acoustic waves is less intense in the sideline direction. However, the major difference between the profiles of these two directional correlations is that, in the sideline direction, the normalized correlations near the end of the potential core show nearly no significance unlike those in the downstream direction.

Figure 6.

Normalized cross-correlations between flow variables ( $| ω |, L, K$ and ρ) and pressure fluctuations ( $p'$ ). (a) $NC | ω |, p'$ . (b) $NC L, p'$ . (c) $NC K, p'$ . (d) $NC ρ, p'$ . The range of grey color scale (black, white)=(−0.3,0.3) is used in common for all the figures. Left: $θ = 30 \circ$ , right: $θ = 90 \circ$ .

We also examine the raw cross-correlations, $RC | ω |, p'$ and $RC L, p'$ , at the two different inlet angles, $θ = 30 \circ$ and $90 \circ$ , in Figure 7. The profiles for other variables ( $RC K, p'$ and $RC ρ, p'$ ) are omitted because of their negligibly small magnitudes $(< O (10 - 7))$ . In this figure, it is interesting to see the difference of the profiles between the two inlet angles, downstream and sideline directions. These raw cross-correlations between the vorticity-related variables ( $| ω|$ and L) and the far-field pressure fluctuation ( $p'$ ) show that the relatively large-scale turbulent structures are more pronounced in the downstream direction, while the fine-scale structures appear in both directions but more noticeable for the sideline direction, supporting the claims by Tam et al.¹⁴ The vorticity norm $| ω |$ has strong negative correlation values within the initial jet shear layer from the nozzle exit extending to near the end of the potential core in the downstream direction while it has strong positive values in the sideline direction, as shown in Figure 7(a). However, the Lamb vector, L, does not have noticeable patterns of cross-correlation near the initial shear layer but shows strong positive correlation at the end of the potential core at $θ = 30 \circ$ as shown in Figure 7(b).

Figure 7.

Raw cross-correlations between flow variables ( $| ω |$ and L) and pressure fluctuations ( $p'$ ). (a) $RC | ω |, p'$ . (b) $RC L, p'$ . The ranges of grey color scale are (black, white) = $(- 10 - 5, 5 \times 105)$ for the top raw ( $| ω |$ ) and $(- 10 - 5, 10 - 5)$ for the bottom raw (L). Left: $θ = 30 \circ$ , right: $θ = 90 \circ$ .

Lighthill’s source sub-terms

The cross-correlations between the Lighthill’s source sub-terms and the far-field sound signals are examined in this section. To understand the role of each sub-term, interacting with other sub-terms within the full source term, the raw cross-correlation is more instructive than the normalized cross-correlation. It is because the normalized cross-correlation is a standardized quantity that gives the comparable values for the sub-terms so that it is hard to show which sub-term in quantity has the largest contribution to the sound generation within the entire source. In other words, we are more interested in the relative contributions of sub-terms compared to the entire source so that we can compare their contributions quantitatively within the source.

In the Lighthill’s source term decomposition shown in equation (4), the sub-terms, $S i, i = 4, 5, 6, 7, 8$ , are related to kinetic energy and the Lamb vector, named here as ‘sound production’ terms. In the original work,²¹ these terms are also labeled as ‘production’ terms and the rest, $S i, i = 1, 2, 3, 9, 10$ , are ‘acoustic’ terms. In the case of incompressible flow, only three sub-terms, $S 4, S 6$ , and $S 8$ , are left since the density gradient and the dilatation-related terms become negligible, so that one might expect these terms would play the major roles, which is also mentioned in Golanski et al.²⁴

Figure 8 shows that the raw cross-correlations between the fluctuations of the three major sub-terms, $S' 4, S' 6$ , and $S' 8$ , and the far-field pressure fluctuations. The correlations of other sub-terms are not presented here since those have extremely small correlation values with $O (10 - 7)$ or less, which we will discuss later again this section. In Figure 8, $RC S' 4, p'$ shows smaller oscillatory structures all over the flow domain in both directions while $RC S' 6, p'$ and $RC S' 8, p'$ show relatively larger structures near the end of potential core and within the jet shear layer especially in the downstream direction. We also notice that these two raw cross-correlations, $RC S' 6, p'$ and $RC S' 8, p'$ , have the opposite signs over the regions near the end of potential core and the outer shear layer at $θ = 30 \circ$ , indicating the major structural cancellation. In the sideline direction, the overall correlation values decrease and the emphases near the end of potential core and in the outer shear layer weakens as expected, showing the fine-scale turbulent structures.

Figure 8.

Raw cross-correlations between fluctuations of the Lighthill’s terms, $S' i$ ( $i = 4$ , 6 and 8), and pressure fluctuations ( $p'$ ). (a) $RC S 4', p'$ . (b) $RC S 6', p'$ . (c) $RC S 8', p'$ . The range of grey color scale, (black, white) = (−0.0001,0.0001), is used in common for all the figures. Left: $θ = 30 \circ$ , right: $θ = 90 \circ$ .

The notable reciprocal cancellation of the larger-scale turbulent structures near the end of the potential core as well as within the shear layer occurs by the interaction between $S 6$ and $S 8$ , examined in Figure 9(a). The cross-correlation $RC S' 4, p'$ , showing smaller structures, cancels out with the sum of $RC S' 6, p'$ and $RC S' 8, p'$ in Figure 9(b). We note that some minor oscillations with different signs (black for the minimum and white for the maximum of the grey scale) in Figures 8 and 9 are due to the grey scale cut-off, which is necessary to compare the cross-correlations of these sub-terms and see their significant patterns in the same grey scale but do not have much impact on the results.

Figure 9.

Raw cross-correlations between fluctuations of sub-sums and pressure fluctuations ( $p'$ ). (a) $RC (S' 6 + S' 8)', p'$ . (b) $RC (S' 4 + S' 6 + S' 8)', p'$ . (c) $RC (S' 1 + S' 2 + \dots + S' 9 + S' 10)', p'$ . The range of grey color scale, (black, white) = (−0.0001,0.0001), is used in common for all the figures. Left: $θ = 30 \circ$ , right: $θ = 90 \circ$ .

In Figure 10(a), the ranges of the 10 raw cross-correlations between the sub-terms and the pressure fluctuation in the observed xz-plane are compared. In both far-field directions, the cross-correlation values of the three sub-terms, $RC S' 4, p', RC S' 6, p'$ and $RC S' 8, p'$ are dominant. The ranges of the raw cross-correlations for the major sums are also given in Figure 10(b). As expected, the strong cancellation patterns are examined. Figure 10 shows that the sub-terms, $S 4$ and $S 6 + S 8$ , majorly interact with each other in quantity while the other terms have relatively a lot smaller interactions. It is interesting to observe that the two sub-terms in the divergence of the Lamb vector, $S 6$ and $S 8$ , mutually interact and cancel the prominent patterns of the larger turbulent structures near the end of the potential core and the outer shear layer in Figure 9(a), but the major quantitative cancellation occurs between the sub-term $S 4$ , the Laplacian of turbulent kinetic energy, and the sub-term $S 6$ in the divergence of the Lamb vector in Figures 9(b) and 10. The sub-terms other than these three major sound production sub-terms do not have any significant role in the noise radiation to the far-field locations based on the raw cross-correlation values even though they are in the similar range of normalized cross-correlation values, in the range of (−0.25, 0.2), which are calculated but not shown in this paper. As we discussed earlier, it is difficult to compare the contributions of the sub-terms within the entire source term based on the normalized cross-correlation function because of the normalization process.

Figure 10.

Range comparison of $RC S' i, p'$ for $i = 1, 2, \dots, 9, 10$ and their summations throughout the flow domain. (a) Left: individual sub-terms, (b) right: major sums.

The distribution of point-wise dominant raw cross-correlation terms is also depicted in Figure 11. At the inlet angle $θ = 30 \circ$ , we observe that the Lamb vector-related terms (in green and yellow) are relatively dominant near the end of the potential core and the outer shear layer while the patterns are weaker at $θ = 90 \circ$ . The sub-term $RC S' 4, p'$ (in dark brown) is more uniformly distributed except near the end of potential core at $θ = 30 \circ$ . Inside the potential core, very little source is found at both $θ = 30 \circ$ and $θ = 90 \circ$ , mostly $RC S' 1, p'$ (in pink) or none (in light grey). Here, one might consider the mean square pressure fluctuation which is a quantity of major interests in the noise generation problem which, from equation (9), can be written as

E [p' 2 (y, τ)] = \int d x \frac{E [S (x, τ - τ 0) p' (y, τ)]}{| x - y |}

(11)

where

E [\cdot]

is the expectation operator. This equation shows that the mean square pressure fluctuation at a far-field location depends upon the integral of the cross-correlation over the entire source field. From the earlier observations in this paper as well as in Figure 11, the cross-correlation values of the three major sub-terms are dominant throughout the entire flow domain.

Figure 11.

Distribution of maximum raw cross-correlation among $RC S' i, p'$ . The color bar shows sub-term numbers, $i$ , for $S i$ . The light grey area has almost no energy which is smaller than $O (10 - 7)$ . Left: $θ = 30 \circ$ , right: $θ = 90 \circ$ .

The contributions of the major sub-terms based on the cross-correlation is also studied in frequency domain by using the cross-power spectral density, $P f, g$ , and the coherence, $C f, g$ . $P f, g$ is defined as the Fourier transformation of the raw cross-correlation function and $C f, g$ as the normalized cross-power spectral density

P f, g (ω) = \int_{- \infty}^{\infty} RC f, g (τ) e - i ω τ d τ

(12)

C f, g (ω) = | P f, g | 2 / (P f, f P g, g)

(13)

where

| \cdot |

is the magnitude of the complex quantity, i.e.

| P f, g | 2

= Re

(P f, g) 2

+ Im

(P f, g) 2

. The auto power spectral density

P f, f

shows the distribution of power per unit frequency. The coherence

C f, g

is the normalized value of the cross-power spectral density based on the auto power spectral densities of the individual variables so that it satisfies

0 \leq C f, g \leq 1

. It also can be interpreted as a measure of the linear relationship between the two variables in frequency domain, and the closer to unity the coherence is, the stronger is the linear relationship.

The magnitude of the cross-power spectral density and the corresponding coherence are calculated at x = 10D near the end of the potential core and depicted in Figures 12 and 13. In these figures, the three major terms are compared at the two different far-field positions, $θ = 30 \circ$ and $90 \circ$ , that were used earlier. As expected, the magnitude of the cross-power spectral density is larger at low frequencies in the downstream direction than in the sideline direction, indicating the large-scale turbulent structures, shown in Figure 12. In the sideline direction, we have slightly larger magnitudes of the cross-power spectral density at high frequencies than in the downstream direction. This is a consistent result with what we observed earlier based on the raw cross-correlation functions. The sub-terms, $S 4$ and $S 6$ , that showed the major quantitative reciprocal cancellation in their raw cross-correlations shown in Figure 10, also have very similar cross-power spectral densities in frequency domain.

Figure 12.

Cross-power spectral density, $P S' i, p'$ for $i = 4, 6, 8$ at x = 10D near the end of the potential core. (a) Left: $θ = 30 \circ$ , (b) right: $θ = 90 \circ$ .

Figure 13.

Coherence, $C S' i, p'$ for $i = 4, 6, 8$ near the end of the potential core. (a) Left: $θ = 30 \circ$ , (b) right: $θ = 90 \circ$ .

In Figure 13, the coherence near the end of the potential core between the source sub-term $S' 8$ and $p'$ over the low-frequency range is relatively high in both directions and the other two coherences for the sub-terms $S' 4$ and $S' 6$ are noticeably small in the sideline direction. The fluctuation at the high frequencies in Figure 13 can be ignored since it is mainly from the numerical error of the extremely small denominators in the definition of $C f, g$ .

Summary and conclusion

The well-known Lighthill’s equation is employed and its source term is decomposed into the ten sub-terms comprising density, velocity, vorticity, and dilatation fields. The decomposed Lighthill’s source sub-terms are calculated based on the LES of a baseline round cold jet nozzle (SMC000) at Mach number $M = 0.9$ with Reynolds number $Re = 10 5$ . To measure the contribution of each decomposed sub-term to the jet noise radiation to the far-field locations, we consider two cross-correlation functions, the raw and the normalized cross-correlation functions. The normalized cross-correlation has been conventionally used to associate the basic flow variables in the flow domain such as density, vorticity, and turbulent kinetic energy with the far-field acoustic signals. However, this normalized cross-correlation is useful only to compare the contributions of independent variables to the far-field sound generation but it is limited when the contributions of the decomposed sub-terms need to be compared within the full source term. For this reason, we calculate both cross-correlation functions for the flow variables, and calculate the raw cross-correlation functions for the decomposed Lighthill’s source sub-terms to examine the interactions between the sub-terms. The fluctuation part of each source sub-term, the major sums as well as the basic flow variables in the flow domain are cross-correlated with the pressure fluctuations at two far-field locations.

These cross-correlations are applied to flow variables first, such as vorticity norm, Lamb vector, kinetic energy, and density. They show strong normalized cross-correlation values at the end of the potential core and the outer shear layer region near the interface between the shear layer and the ambient flow. One interesting observation is that the cross-correlations applied to the vorticity norm and the Lamb vector have a positive sign at the potential core end while those applied to the kinetic energy and the density have the opposite sign, potentially indicating that these terms are canceled out when they interact with each other within the entire source. The raw cross-correlations of the vorticity norm and the Lamb vector display the most distinct directivity patterns, supporting the claim that two different noise sources, the fine-scale and the large-scale turbulent structures, exist and the high frequency noise generated by the fine scale turbulence propagates in nearly all directions, while the lower frequency noise generated by the large-scale turbulence propagates mostly in the downstream direction.

The procedure is repeated with the fluctuating decomposed Lighthill’s source sub-terms. To observe the contributions of the individual source sub-terms to the far-field sound generation within the entire Lighthill’s source term, and their mutual interactions and cancellations, the raw cross-correlation is more suitable than the normalized cross-correlation since the normalized cross-correlation function is useful mostly for the comparison between independent variables. With the raw cross-correlation, we found that the divergence of Lamb vector and the Laplacian of turbulent kinetic energy terms are of the major contributions within the entire source term. In the downstream direction, these three major sub-terms have strong patterns such that the Lamb vector-related terms show larger coherent turbulent structures while the kinetic energy-related term shows smaller structures, but a large portion of these terms is canceled out through mutual interactions. In the sideline direction, the overall cross-correlation values are smaller than those in the downstream direction throughout the entire jet flow domain. Moreover, at the end of the potential core and within the shear layer, the cross-correlations in the sideline direction are not as prominent as in the downstream direction. An interesting observation is that the major turbulent structural cancellation occurs within the divergence of the Lamb vector terms while the major quantitative cancellation occurs between the turbulent kinetic energy term and the Lamb vector related terms. The three major sub-terms are also studied in the frequency domain by calculating the cross-power spectral density and coherence at the end of the potential core and a consistent conclusion as in the earlier cross-correlation study was achieved. The importance of the divergence of Lamb vector and the Laplacian of turbulent kinetic energy that we examined throughout this study also supports the previous works such as Powell¹⁵ and Howe.²⁵

The present study provides valuable insights on how the Lighthill’s source sub-terms interact with each other and how differently the major sub-terms contribute and play roles to the far-field sound generation, by visualizing the spatial distributions of the cross-correlations over the entire cross-sectional jet flow domain. This work can be extended to different jet nozzle configurations, such as different types of chevron nozzles, that are being investigated for their jet noise reduction benefits.²⁶ A comparison of the cross-correlation profiles between the round and the chevron nozzles will possibly help us characterize the distinct structures generating the far-field sound. This is expected to lead to better understanding and development of noise control devices.

Footnotes

Acknowledgement

The authors gratefully acknowledge Dr Jay Hardin and Dr Jonghoon Bin for their helpful discussions and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the NASA Glenn Research Center for its financial support.

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