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.
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 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.
(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 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) , (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.
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 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].
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.
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.
Conditions of computational cases are presented.
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.
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