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In the near-field of a subsonic jet, complex energy transport and transformation processes occur between kinetic energy, thermal energy, and acoustic energy, which play a crucial role in jet instability and noise radiation. Accurately characterizing the transport features of each energy component is essential for developing effective noise suppression technologies. According to Myers’ exact energy equation for total disturbances in arbitrary steady flow [
1991 J. Fluid Mech. 226 383 ], the present study develops a modified energy equation based on hydro-acoustic mode decomposition to separate the contributions of vortical, entropic, and acoustic modes to the total disturbance energy. The method begins with the decomposition formulas for velocity, pressure, and density, following the hydro-acoustic mode decomposition method proposed by Han et al. [2023 Phys. Fluids 35 076107 ]. In Myers’ energy equation framework, the disturbances of primitive variables (velocity, pressure, and density) are expressed as linear combinations of their vortical, entropic, and acoustic components. With this formula, the vortical (entropic, acoustic) energy is defined as being contributed only by the disturbance of the corresponding mode, while the nonlinear energy is attributed to interaction between vortical, entropic, and acoustic components. This approach yields a modified energy equation capable of distinguishing the individual contributions of vortical, entropic, and acoustic modes to both total disturbance energy and energy flux, thus making it particularly suitable for analyzing energy transport characteristics in the near flow field. The developed equation is used to analyze a subsonic jet with a Mach number of 0.9, revealing different spatial distributions and transport mechanisms of hydrodynamic energy and acoustic energy. The results indicate that vortical energy and entropic energy are mainly concentrated in the near-field, convecting downstream at a velocity about 0.8 times the jet velocity. In contrast, acoustic energy exhibits dual propagation characteristics: it radiates outward to the far field through acoustic waves outside the potential core, while propagating upstream through trapped waves inside the potential core. The energy related to multi-mode nonlinear interactions is mainly concentrated in the jet wake and propagates without obvious directionality. The total disturbance energy is mainly contributed by vortical energy, while the acoustic energy only accounts for a small part of the total disturbance energy, approximately 10–3 of the total. This refined analysis provides deeper insights into the complex energy dynamics in subsonic jets and valuable information for predicting and controlling jet noise strategies. The modified energy equation provides a robust framework for understanding and quantifying the intricate energy transport processes in jet flows.
[1] Viswanathan K 2024 Int. J. Aeroacoust. 23 184
Google Scholar
[2] Jordan P, Gervais Y 2008 Exp. Fluids 44 1
[3] 李晓东, 徐希海, 高军辉, 何敬玉 2018 空气动力学学报 36 398
Li X D, Xu X H, Gao J H, He J Y 2018 Acta Aerodyn. Sin. 36 398
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Google Scholar
[5] Tam C K W, Viswanathan K, Ahuja K K, Panda J 2008 J. Fluid Mech. 615 253
Google Scholar
[6] Tam C K W 2019 Philos. Trans. R. Soc. London, Ser. A 377 20190078
[7] Obrist D 2011 Phys. Fluids 23 024901
[8] Cavalieri A, Jordan P, Lesshafft L 2019 Appl. Mech. Rev. 71 020802
Google Scholar
[9] Cavalieri A, Rodriguez D, Jordan P, Colonius T, Gervais Y 2013 J. Fluid Mech. 730 559
Google Scholar
[10] Papamoschou D 2018 Int. J. Aeroacoust. 17 52
Google Scholar
[11] Jordan P, Daviller G, Comte P 2013 J. Sound Vib. 332 3924
Google Scholar
[12] Doak P 1989 J. Sound Vib. 131 67
Google Scholar
[13] Liu Qilin, Lai Huanxin 2022 Phys. Fluids 34 045125
Google Scholar
[14] 刘琪麟, 赖焕新 2022 工程热物理学报 43 3266
Liu Q, Lai H. 2022 J. Eng. Thermophys., 43 3266
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Google Scholar
[16] Myers M 1986 J. Sound Vib. 109 277
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[17] Myers M K 1991 J. Fluid Mech. 226 383
Google Scholar
[18] Han S, Li H, Luo Y, Wang Y, Ma R, Zhang S 2023 Phys. Fluids 35 076107
Google Scholar
[19] Towne A, Dawson S, Brès G A, Lozano-Duran A, Saxton-Fox T, Parthasarathy A, Jones A R, Biler H, Yeh C, Patel H D, Taira K 2023 AIAA J. 61 2867
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[21] Brès G A, Jordan P, Jaunet V, Rallic M L, Cavalieri A V G, Towne A, Lele S K, Colonius T, Schmidt O T 2018 J. Fluid Mech. 851 83
[22] Brès G A, Jordan P, Colonius T, Rallic M L, Jaunet V, Lele S K 2014 Proceedings of the Summer Program. Stanford, CA: Center for Turbulence Research, Stanford University 2014 p221
[23] Freund J B 1997 AIAA J. 35 740
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Google Scholar
Han S, Wang Y, Wu C, Luo Y, Li H 2024 Chin. J. Theor. Appl. Mech. 56 3142
Google Scholar
[26] Mao F, Shi Y, Wu J 2010 Acta Mech. Sin. 26 355
Google Scholar
[27] Wu J, Ma H, Zhou M 2006 Vorticity and Vortex Dynamics (Berlin: Springer) p341
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Google Scholar
[31] Campos L 2007 Appl. Mech. Rev. 60 149
Google Scholar
[32] Zaman K B M Q, Fagan A F, Upadhyay P 2022 J. Fluid Mech. 931 A30
Google Scholar
[33] Schmidt O T, Towne A, Colonius T, Cavalieri A V G, Jordan P, Bres G A 2017 J. Fluid Mech. 825 1153
Google Scholar
[34] Suzuki T, Colonius T 2006 J. Fluid Mech. 565 197
Google Scholar
[35] Towne A, Cavalieri A V, Jordan P, Colonius T, Schmidt O, Jaunet V, Bres G A 2017 J. Fluid Mech. 825 1113
Google Scholar
[36] Colonius T, Lele S K 2004 Prog. Aerosp. Sci. 40 345
Google Scholar
[37] 赵鲲, 章荣平, 杨玫, 王勋年, 余荣科 2024 空气动力学学报 42 15
Google Scholar
Zhao K, Zhang P R, Yang M, Wang X N, Yu R K 2024 Acta Aerodyn. Sin. 42 15
Google Scholar
[38] 王益民, 马瑞轩, 武从海, 罗勇, 张树海 2021 物理学报 70 194302
Google Scholar
Wang Y, Ma R, Wu C, Luo Y, Zhang S 2021 Acta Phys. Sin. 70 194302
Google Scholar
[39] 马瑞轩, 王益民, 张树海, 武从海, 王勋年 2021 物理学报 70 104301
Google Scholar
Ma R, Wang Y, Zhang S, Wu C, Wang X 2021 Acta Phys. Sin. 70 104301
Google Scholar
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图 6 密度脉动及各模态频率-波数谱 (a)原始密度脉动; (b)声模态; (c)涡模态; (d)熵模态
Figure 6. Frequency-wavenumber diagram for the density perturbation and the decomposed components: (a) Raw density perturbation; (b) acoustic component; (c) vortical component; (d) entropic component.
图 2 三维流场$\theta = 0$位置处及轴对称模态压力脉动 (a)三维流场$\theta = 0$位置处压力脉动[19]; (b)轴对称模态压力脉动[19]
Figure 2. The distribution of pressure perturbation for 3D flow field at $\theta = 0$ and for asymmetric mode: (a) The distribution of pressure perturbation for 3D flow field at $\theta = 0$[19]; (b) the distribution of pressure perturbation for asymmetric mode[19].
图 3 轴对称模态密度脉动及其声涡熵模态空间分布 (a)原始密度脉动; (b)声模态; (c)涡模态; (d)熵模态
Figure 3. The distribution of density perturbation and the acoustic, vortical and entropic mode: (a) Raw density perturbation; (b) acoustic component; (c) vortical component; (d) entropic component.
图 4 x/D = 4.0位置处密度脉动及其流声模态的均方根随径向位置r的变化
Figure 4. The root mean square of density perturbation and the hydrodynamic and acoustic component along r at x/D = 4.0.
图 5 总脉动能量以及各模态能量的瞬态和时间平均值的空间分布 (a) $E$; (b) $\bar E$; (c) ${E_{\text{r}}}$; (d) ${\bar E_{\text{r}}}$; (e) ${E_{\text{a}}}$; (f) ${\bar E_{\text{a}}}$; (g) ${E_{\text{s}}}$; (h) ${\bar E_{\text{s}}}$; (i) ${E_{\text{n}}}$; (j) ${\bar E_{\text{n}}}$
Figure 5. Spacial distribution of instantaneous and time-averaged total energy perturbation and the decomposed components: (a) $E$; (b) $\bar E$; (c) ${E_{\text{r}}}$; (d) ${\bar E_{\text{r}}}$; (e) ${E_{\text{a}}}$; (f) ${\bar E_{\text{a}}}$; (g) ${E_{\text{s}}}$; (h) ${\bar E_{\text{s}}}$; (i) ${E_{\text{n}}}$; (j) ${\bar E_{\text{n}}}$.
图 7 各模态时均能流线(灰色背景为各模态时均能量) (a)涡模态; (b)熵模态; (c)非线性模态; (d)声模态
Figure 7. Streamline of time-averaged intensity for the decomposed components: (a) Vortical component; (b) entropic component; (c) nonlinear component; (d) acoustic component.
图 8 时均声模态能流线脉动及对流部分 (a)脉动部分Iap; (b)对流部分Iac
Figure 8. Streamline of time-averaged acoustic intensity: (a) Perturbation component; (b) convective component.
表 1 流声模态能量的空间积分及占比
Table 1. Spatial integral of total energy and the decomposed components.
$E$ ${E_{\text{r}}}$ ${E_{\text{a}}}$ ${E_{\text{s}}}$ ${E_{\text{n}}}$ 总能量 $1.27 \times {10^{ - 3}}$ $1.12 \times {10^{ - 3}}$ $1.74 \times {10^{ - 6}}$ $5.28 \times {10^{ - 5}}$ $9.20 \times {10^{ - 5}}$ 百分比/% — 88.4 0.14 4.16 7.26 
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[1] Viswanathan K 2024 Int. J. Aeroacoust. 23 184
Google Scholar
[2] Jordan P, Gervais Y 2008 Exp. Fluids 44 1
[3] 李晓东, 徐希海, 高军辉, 何敬玉 2018 空气动力学学报 36 398
Li X D, Xu X H, Gao J H, He J Y 2018 Acta Aerodyn. Sin. 36 398
[4] Chu B T 1965 Acta Mech. 1 215
Google Scholar
[5] Tam C K W, Viswanathan K, Ahuja K K, Panda J 2008 J. Fluid Mech. 615 253
Google Scholar
[6] Tam C K W 2019 Philos. Trans. R. Soc. London, Ser. A 377 20190078
[7] Obrist D 2011 Phys. Fluids 23 024901
[8] Cavalieri A, Jordan P, Lesshafft L 2019 Appl. Mech. Rev. 71 020802
Google Scholar
[9] Cavalieri A, Rodriguez D, Jordan P, Colonius T, Gervais Y 2013 J. Fluid Mech. 730 559
Google Scholar
[10] Papamoschou D 2018 Int. J. Aeroacoust. 17 52
Google Scholar
[11] Jordan P, Daviller G, Comte P 2013 J. Sound Vib. 332 3924
Google Scholar
[12] Doak P 1989 J. Sound Vib. 131 67
Google Scholar
[13] Liu Qilin, Lai Huanxin 2022 Phys. Fluids 34 045125
Google Scholar
[14] 刘琪麟, 赖焕新 2022 工程热物理学报 43 3266
Liu Q, Lai H. 2022 J. Eng. Thermophys., 43 3266
[15] Unnikrishnan S, Gaitonde D V 2016 J. Fluid Mech. 800 387
Google Scholar
[16] Myers M 1986 J. Sound Vib. 109 277
Google Scholar
[17] Myers M K 1991 J. Fluid Mech. 226 383
Google Scholar
[18] Han S, Li H, Luo Y, Wang Y, Ma R, Zhang S 2023 Phys. Fluids 35 076107
Google Scholar
[19] Towne A, Dawson S, Brès G A, Lozano-Duran A, Saxton-Fox T, Parthasarathy A, Jones A R, Biler H, Yeh C, Patel H D, Taira K 2023 AIAA J. 61 2867
Google Scholar
[20] Vreman A 2004 Phys. Fluids 16 3570.
[21] Brès G A, Jordan P, Jaunet V, Rallic M L, Cavalieri A V G, Towne A, Lele S K, Colonius T, Schmidt O T 2018 J. Fluid Mech. 851 83
[22] Brès G A, Jordan P, Colonius T, Rallic M L, Jaunet V, Lele S K 2014 Proceedings of the Summer Program. Stanford, CA: Center for Turbulence Research, Stanford University 2014 p221
[23] Freund J B 1997 AIAA J. 35 740
Google Scholar
[24] Mani A 2012 J. Comput. Phys. 231 704
Google Scholar
[25] 韩帅斌, 王益民, 武从海, 罗勇, 李虎 2024 力学学报 56 3142
Google Scholar
Han S, Wang Y, Wu C, Luo Y, Li H 2024 Chin. J. Theor. Appl. Mech. 56 3142
Google Scholar
[26] Mao F, Shi Y, Wu J 2010 Acta Mech. Sin. 26 355
Google Scholar
[27] Wu J, Ma H, Zhou M 2006 Vorticity and Vortex Dynamics (Berlin: Springer) p341
[28] Sagaut P, Cambon C 2008 Homogeneous Turbulence Dynamics (Cambridge: Cambridge University Press) p80
[29] Kovasznay L S G 1953 J. Aeronaut. Sci. 20 657
Google Scholar
[30] Chu B T, Kovásznay L S G 1958 J. Fluid Mech. 3 494
Google Scholar
[31] Campos L 2007 Appl. Mech. Rev. 60 149
Google Scholar
[32] Zaman K B M Q, Fagan A F, Upadhyay P 2022 J. Fluid Mech. 931 A30
Google Scholar
[33] Schmidt O T, Towne A, Colonius T, Cavalieri A V G, Jordan P, Bres G A 2017 J. Fluid Mech. 825 1153
Google Scholar
[34] Suzuki T, Colonius T 2006 J. Fluid Mech. 565 197
Google Scholar
[35] Towne A, Cavalieri A V, Jordan P, Colonius T, Schmidt O, Jaunet V, Bres G A 2017 J. Fluid Mech. 825 1113
Google Scholar
[36] Colonius T, Lele S K 2004 Prog. Aerosp. Sci. 40 345
Google Scholar
[37] 赵鲲, 章荣平, 杨玫, 王勋年, 余荣科 2024 空气动力学学报 42 15
Google Scholar
Zhao K, Zhang P R, Yang M, Wang X N, Yu R K 2024 Acta Aerodyn. Sin. 42 15
Google Scholar
[38] 王益民, 马瑞轩, 武从海, 罗勇, 张树海 2021 物理学报 70 194302
Google Scholar
Wang Y, Ma R, Wu C, Luo Y, Zhang S 2021 Acta Phys. Sin. 70 194302
Google Scholar
[39] 马瑞轩, 王益民, 张树海, 武从海, 王勋年 2021 物理学报 70 104301
Google Scholar
Ma R, Wang Y, Zhang S, Wu C, Wang X 2021 Acta Phys. Sin. 70 104301
Google Scholar
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