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亚声速射流的流场中存在着动能、热能、声能等多种形式能量的输运与转化, 影响射流的稳定性和噪声辐射等特性, 准确认识射流中各模态能量的输运特性是发展高效降噪措施的重要基础. 基于拓展型亥姆霍兹分解流声分离方法, 发展了基于流声分离的脉动能量方程, 可有效地分离脉动能量及能流矢量中涡、熵、声及非线性模态的贡献, 为揭示流动近场的能量输运特性提供了分析工具. 将该方程应用于射流马赫数0.9的亚声速射流, 获得并分析了流声模态能量的空间分布特征和输运特性. 研究发现: 亚声速射流脉动能中的涡模态能量和熵模态能量分布于射流近场并向下游输运; 声模态能量在势核外向远场辐射, 在势核内则由束缚波携带传播至上游; 多模态非线性相互作用相关的能量集中于射流尾迹区内, 输运无显著方向性.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’ precise energy equation for total disturbances in any 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.
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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
Fig. 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]
Fig. 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
Fig. 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
Fig. 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) ${\bar E_{\text{n}}}$
Fig. 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
Fig. 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
Fig. 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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