^{1}, Lin Zhou

^{2}, Hai-Hua Yang

^{1}and De-Jun Sun

^{1,a)}

### Abstract

Large eddy simulation is performed for investigating the local and far-field behaviors of free and swirling jets at moderate Reynolds number. By solving compressible boundary layer equations, the inflow profiles with different swirl number are calculated, and then their stability characteristics are analyzed based on linear stability theory. The amplification rates of swirling jets are higher than the free one, particularly for higher or negative azimuthal wavenumber modes. Multiple unstable modes are superimposed to construct inflow forcing. The quantities of flow and acoustic are presented and compared against the results of existed experiments and other computations, besides, the comparisons are also made among themselves. For swirling jets, the spreadings of jet half-width and vorticity thickness at the initial and transition stage are enhanced, but they are surpassed by the free jet at turbulent mixing stage. In all cases, the development of mixing layer initially is greatly influenced by frequencies f 0 and f 0/2 associated with upstream forcing. As the swirl intensity is increased, the growth rates of fluctuation quantities on the centerline and the nozzle lip line are raised, but the peak levels on the centerline are reduced substantially. In swirling jets, the strength of vortex pairing is decreased, and the pairing noise is weakened correspondingly. The overall sound pressure levels are lower than that of the free jet at all observation angles (ϕ), and about 3 decibels (dB) is reduced at ϕ ≃ 30° in the strong swirling case at a distance of 60 radii. The Fourier analyses of pressure and acoustic sources show that the modes are varied greatly, which suggests that the noise reduction should be corresponding to the change of instability waves.

This work was supported by National Natural Science Foundation of China (Grant Nos. 11072238 and 11232011) and 111 project of China (Grant No. B07033). The authors especially thank the reviewer's many helpful suggestions to improve this manuscript. The calculations have been done with the help of Supercomputing Center of USTC.

I. INTRODUCTION

II. SIMULATION METHOD AND PARAMETERS

A. Near- and far-field approach

B. Initial flows and parameters

III. RESULTS AND DISCUSSION

A. Visualization of initial flow development

B. Mean flow properties

C. Fluctuation properties

D. Acoustic far-field

E. Mode analysis of pressure field

F. Noise sources

IV. CONCLUSION

##### B01F3/00

## Figures

The schematic figure of the computational domain, in which the physical domain is set to [0, ^{40} r 0 in axial direction and [0.0175,18]r 0 in radial direction (r 0 is the radius of the jet). The shadow areas are buffer zones, and the inflow forcing is applied in inflow buffer zone.

The schematic figure of the computational domain, in which the physical domain is set to [0, ^{40} r 0 in axial direction and [0.0175,18]r 0 in radial direction (r 0 is the radius of the jet). The shadow areas are buffer zones, and the inflow forcing is applied in inflow buffer zone.

Inflow profiles of (a) density; (b) axial velocity (u z ); (c) radial velocity (u r ); and (d) azimuthal velocity (u θ). —– SJ0; −∇ − SJ1; and −□ − SJ2.

Inflow profiles of (a) density; (b) axial velocity (u z ); (c) radial velocity (u r ); and (d) azimuthal velocity (u θ). —– SJ0; −∇ − SJ1; and −□ − SJ2.

The amplification rates for different azimuthal wavenumber: (a) SJ0; (b) SJ1; and (c) SJ2.

The amplification rates for different azimuthal wavenumber: (a) SJ0; (b) SJ1; and (c) SJ2.

The instantaneous vortical structures shown by |Ω| and dilatation field in r − z plane. The gray contour levels from light to dark are scaled by (0.001,10) for vorticity and (−0.001,0.001) for dilatation. (a) SJ0; (b) SJ1; and (c) SJ2.

The instantaneous vortical structures shown by |Ω| and dilatation field in r − z plane. The gray contour levels from light to dark are scaled by (0.001,10) for vorticity and (−0.001,0.001) for dilatation. (a) SJ0; (b) SJ1; and (c) SJ2.

The instantaneous three-dimensional vortical structures shown by Q-criterion with Q = 25(U j /D)^{2}: (a) SJ0; (b) SJ1; and (c) SJ2.

The instantaneous three-dimensional vortical structures shown by Q-criterion with Q = 25(U j /D)^{2}: (a) SJ0; (b) SJ1; and (c) SJ2.

The decay of mean centerline velocity in axial direction. The axial coordinate is normalized by Witze correlation, ^{41} where z c is the end position of the potential core. ▶ DNS of Freund, ^{40} • Experiment of Tanna et al., ^{42} ■ Experiment of Bridges and Wernet, ^{43} and present computation: —— SJ0, −− − SJ1, and −· · − SJ2.

The decay of mean centerline velocity in axial direction. The axial coordinate is normalized by Witze correlation, ^{41} where z c is the end position of the potential core. ▶ DNS of Freund, ^{40} • Experiment of Tanna et al., ^{42} ■ Experiment of Bridges and Wernet, ^{43} and present computation: —— SJ0, −− − SJ1, and −· · − SJ2.

(a) The half-width of jet r 0.5 and (b) the vorticity thickness along z. —— SJ0, −− − SJ1, and −· · − SJ2, in which error bars are shown.

(a) The half-width of jet r 0.5 and (b) the vorticity thickness along z. —— SJ0, −− − SJ1, and −· · − SJ2, in which error bars are shown.

The azimuthal velocity profiles of (a) SJ1; (b) SJ2; and (c) the maximum azimuthal velocity at different z location. −△ − SJ1; −□ − SJ2.

The azimuthal velocity profiles of (a) SJ1; (b) SJ2; and (c) the maximum azimuthal velocity at different z location. −△ − SJ1; −□ − SJ2.

The power spectrum density (PSD) of pressure fluctuation on the nozzle lip line at (a) z = 4.9r 0 and (b) z = 7.9r 0 in the plane of θ = 0.

The power spectrum density (PSD) of pressure fluctuation on the nozzle lip line at (a) z = 4.9r 0 and (b) z = 7.9r 0 in the plane of θ = 0.

The fluctuations of (a) axial velocity and (b) radial velocity along the the centerline line. Experimental results of □ Ahuja et al., ^{45} △ Crow and Champagne, ^{46} ○ Zaman ^{47} (the experimental data have been shifted by −3.5r 0), and present simulation: —— SJ0, −− − SJ1, and −· · − SJ2.

The fluctuations of (a) axial velocity and (b) radial velocity along the the centerline line. Experimental results of □ Ahuja et al., ^{45} △ Crow and Champagne, ^{46} ○ Zaman ^{47} (the experimental data have been shifted by −3.5r 0), and present simulation: —— SJ0, −− − SJ1, and −· · − SJ2.

The fluctuations of (a) , (b) , (c) , and (d) along the nozzle lip line (r = r 0). —— SJ0, −− − SJ1, and −· · − SJ2.

The fluctuations of (a) , (b) , (c) , and (d) along the nozzle lip line (r = r 0). —— SJ0, −− − SJ1, and −· · − SJ2.

Two-point axial correlations (u z ) and one-dimensional axial energy spectra (u z ) at r = r 0. (a) and (b) SJ0; (c) and (d) SJ1; and (e) and (f) SJ2. “—” z/r 0 = 5; “−− −” z/r 0 = 10; “−· −” z/r 0 = 15; “· · ·” z/r 0 = 20; and “−· · −” z/r 0 = 25. The straight line has −5/3 slope.

Two-point axial correlations (u z ) and one-dimensional axial energy spectra (u z ) at r = r 0. (a) and (b) SJ0; (c) and (d) SJ1; and (e) and (f) SJ2. “—” z/r 0 = 5; “−− −” z/r 0 = 10; “−· −” z/r 0 = 15; “· · ·” z/r 0 = 20; and “−· · −” z/r 0 = 25. The straight line has −5/3 slope.

(a) The comparison of OASPL between the free jet and experimental data. Experimental data of ■ Mollon-Chirstensen et al., ^{48} ▲ Lush, ^{49} and ▼ Stromberg; ^{11} ○ Computation of Bogey et al., ^{7} —— SJ0. (b) The OASPL of free and swirling jets: —— SJ0, −− − SJ1, and −· · − SJ2.

The noise spectra at a far-field distance of R = 60r 0 at different observation angles (ϕ). (a) 30° and (b) 90°. Experimental spectra of ◀ Tanna, ^{50} ○ Bogey et al.; ^{51} spectra of present simulation:—— SJ0, −− − SJ1, and −· · − SJ2.

The azimuthal correlations of far-field sound at different observation angle ϕ at a distance of R = 60r 0 to the nozzle exit. ▲ Experimental data of Maestrello; ^{52} present simulation: —— SJ0, −− − SJ1, and −· · − SJ2.

The azimuthal correlations of far-field sound at different observation angle ϕ at a distance of R = 60r 0 to the nozzle exit. ▲ Experimental data of Maestrello; ^{52} present simulation: —— SJ0, −− − SJ1, and −· · − SJ2.

The pressure profiles of varied pairs of azimuthal wavenumber and frequency on the cylindrical shell of r = r 0. (a) SJ0, St = 0.66; (b) SJ1, St = 0.66; (c) SJ2, St = 0.64; (d) SJ0, St = 0.33; (e) SJ1, St = 0.33; and (f) SJ2, St = 0.32.

The pressure profiles of varied pairs of azimuthal wavenumber and frequency on the cylindrical shell of r = r 0. (a) SJ0, St = 0.66; (b) SJ1, St = 0.66; (c) SJ2, St = 0.64; (d) SJ0, St = 0.33; (e) SJ1, St = 0.33; and (f) SJ2, St = 0.32.

The pressure profiles of varied pairs of azimuthal wavenumber and frequency on the cylindrical surface of r = 10r 0. (a) SJ0, St = 0.66; (b) SJ1, St = 0.66; (c) SJ2, St = 0.64; (d) SJ0, St = 0.33; (e) SJ1, St = 0.33; (f) SJ2, and St = 0.32. —– n = 0; −− − n = −1; −· − n = 1; · · · n = −2; −· · − n = 2; ▲ n = −3; and ■ n = 3.

The pressure profiles of varied pairs of azimuthal wavenumber and frequency on the cylindrical surface of r = 10r 0. (a) SJ0, St = 0.66; (b) SJ1, St = 0.66; (c) SJ2, St = 0.64; (d) SJ0, St = 0.33; (e) SJ1, St = 0.33; (f) SJ2, and St = 0.32. —– n = 0; −− − n = −1; −· − n = 1; · · · n = −2; −· · − n = 2; ▲ n = −3; and ■ n = 3.

Root mean square of the filtered sound source by multiplying radius. (a) SJ0, 10 contours [0.05,0.55]; (b) SJ1, 10 contours [0.05,0.6]; and (c) SJ0, 10 contours [0.05,0.6].

Root mean square of the filtered sound source by multiplying radius. (a) SJ0, 10 contours [0.05,0.55]; (b) SJ1, 10 contours [0.05,0.6]; and (c) SJ0, 10 contours [0.05,0.6].

Space-time correlation of the sound source at the point of z = 20r 0 on the nozzle lip line (r = r 0) in SJ0 (a) and (b), SJ1 (c) and (d), and SJ2 (e) and (f). (a), (c), and (e) full sound source: 10 contours [0,0.9]; (b), (d), and (f) filtered sound source: 11 contours, [−0.1,0.9], solid lines present positive values, whereas negative values are represented by dashed lines.

Space-time correlation of the sound source at the point of z = 20r 0 on the nozzle lip line (r = r 0) in SJ0 (a) and (b), SJ1 (c) and (d), and SJ2 (e) and (f). (a), (c), and (e) full sound source: 10 contours [0,0.9]; (b), (d), and (f) filtered sound source: 11 contours, [−0.1,0.9], solid lines present positive values, whereas negative values are represented by dashed lines.

Intermittency factor γ along (a) the centerline and (b) the nozzle lip line. “—–” SJ0, “■” SJ1; and “▲” SJ2.

Intermittency factor γ along (a) the centerline and (b) the nozzle lip line. “—–” SJ0, “■” SJ1; and “▲” SJ2.

The profiles of the noise source with a frequency of St = 0.33 are displayed, while the amplitude A is scaled by each own maximum. (a) SJ0; (b) SJ1; and (c) SJ2.

The profiles of the noise source with a frequency of St = 0.33 are displayed, while the amplitude A is scaled by each own maximum. (a) SJ0; (b) SJ1; and (c) SJ2.

## Tables

Conditions of computational cases are presented.

Conditions of computational cases are presented.

Some parameters of mean flow quantities, z c is the end of the potential core.

Some parameters of mean flow quantities, z c is the end of the potential core.

Variations of the location and peak RMS values of fluctuating velocities and the peak magnitudes of Reynolds shear stress.

Variations of the location and peak RMS values of fluctuating velocities and the peak magnitudes of Reynolds shear stress.

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