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以基于纳米示踪平面激光散射技术的密度场测量方法获得的Ma = 3.0平板湍流边界层密度场实验数据为基础, 采用小波方法对湍流边界层密度脉动进行了多尺度与动态特性分析. 研究表明, 近壁区密度脉动概率密度呈偏离高斯分布, 大尺度分量对湍流边界层密度脉动起主导作用, 小尺度分量使概率密度呈“M”型分布; 采用希尔伯特变换对幅度调制效应进行分析, 结果表明超声速湍流边界层近壁处外层大尺度密度偏移会导致内层小尺度密度脉动的局部增强或减弱; 采用基于小波变换的时变谱密度估计对边界层不同高度的密度脉动进行分析, 结果表明脉动主要分布在1 MHz以内, 主导频段的密度脉动间歇性明显; 随着时间的发展, 大部分脉动存在频率从高频过渡到低频, 幅值先增加后减少的规律; 随着高度的增加, 对数区脉动主要分布在105 Hz以下, 尾迹区则集中在105 Hz以上, 边界层与主流交界处的脉动主要分布在两者相互作用形成的大尺度结构附近, 脉动能量从近壁面到主流区呈现先升高后降低的变化规律.In order to obtain the time-varying information and dynamic characteristics of density fluctuation in compressible turbulence, the wavelet method is used to analyze the flow density field of zero-pressure-gradient flat plate turbulent boundary layer at Ma = 3.0, which is measured based on Nano-tracer plane laser scattering technique. Utilizing Taylor’s frozen hypothesis, the spatial signal of density field converts into the temporal signal. The one-dimensional orthogonal wavelet multi-resolution analysis is used to reveal multi-scale turbulent structures, and the results suggest that large-scale structures play a leading role in the density fluctuation of turbulent boundary layer while the small-scale structures make the probability density function (PDF) of density fluctuation manifested as an “M” distribution. The density fluctuation scalar PDF deviates from Gaussian distribution. The Hilbert transformation is used to analyze amplitude modulation effects between large- and small-scale structure, and the results suggest that positive (negative) large scale density excursion in the outer layer induces local enhancement (suppression) of the small scale density fluctuation in the inner layer near the wall. The time-varying spectral density estimation method based on the wavelet transform is used to analyze the density fluctuation at different heights of turbulent boundary layer after proving its viability in time and frequency domain. The results suggest a wide range of frequencies throughout the turbulent boundary layer, mainly distributed within 1 MHz. The density fluctuation in the dominant frequency band is intermittent, most of which transits from high frequency to low frequency while the spectral density first increases and then decreases. Near the wall, the time-frequency distributions of density fluctuation in the logarithmic layer are similar. In the middle part of the turbulent boundary layer, the frequency distribution and spectral density of the density fluctuation each reach a peak. Near the mainstream region, the spectral density decreases obviously, which is mainly distributed near the structure formed by the interaction between the boundary layer and mainstream. The wall constraint, viscous dissipation, and uniform mainstream make the fluctuation nearby the region relatively weak. As a result, the spectrum amplitude of density fluctuation first increases and then decreases from the wall to the mainstream.
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Keywords:
- supersonic /
- turbulent boundary layer /
- density fluctuation /
- wavelet transform
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图 2 时频域有效性验证 (a) 密度脉动累积能量曲线; (b) 密度脉动功率谱估计
Fig. 2. Verification in time and frequency domain: (a) Husid of density wave; (b) power spectral density (PSD) of density fluctuation.
图 3
$Ma$ = 3.0湍流边界层流向剖面 (a) 瞬态密度场; (b) 瞬态密度脉动Fig. 3. Flow profile of turbulent boundary layer at
$Ma$ = 3.0: (a) Instantaneous density field; (b) instantaneous density fluctuation.
图 4 密度脉动时间分布曲线(
$y/\delta $ = 0.1)Fig. 4. Time distribution curve of density fluctuation (
$y/\delta $ = 0.1).
图 5 密度脉动信号多分辨率分析 (a)−(f) Level 1−Level 6代表组成原始信号6个不同尺度的分量
Fig. 5. Multiresolution analysis of density fluctuation signal: (a)−(f) Level 1−Level 6 represent 6 different sacles’s components of original signal.
图 6 原始信号和Level 1 + Level 2的概率密度函数
Fig. 6. Probability density function (PDF) of the original signal of density fluctuation and Level 1 + Level 2.
图 7 幅度调制效应分析 (a) 大尺度分量
${d_{\rm{L}}}$ 与小尺度分量${d_{\rm{S}}}$ ; (b) 小尺度分量${d_{\rm{S}}}$ 与低通滤波包络${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ ; (c) 大尺度分量${d_{\rm{L}}}$ 与低通滤波包络${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ (为方便比较, 将${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ 平均值调整到0线并放大2倍)Fig. 7. Analysis of amplitude modulation effects: (a) Large scale component
${d_{\rm{L}}}$ and small scale component${d_{\rm{S}}}$ ; (b) small scale component${d_{\rm{S}}}$ and low-pass filtered envelope${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ ; (c) comparison between large scale component${d_{\rm{L}}}$ and low-pass filtered envelope${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ (for comparison, adjust the average of${E_{\rm{L}}}\left( {{d_{\rm{S}}}} \right)$ to the zero-line and magnify the magnitude by a factor of 2).
图 8 密度脉动的谱分析(
$y/\delta $ = 0.1) (a) 基于傅里叶变换的功率谱密度估计; (b) 基于小波变换的时变谱密度估计Fig. 8. Spectrum analysis of density fluctuation (
$y/\delta $ = 0.1): (a) Power spectrum based on Fourier transform; (b) time-varying power spectrum based on wavelet transform.
图 9 相近频率的时变谱密度 (a) (60 ± 3) kHz; (b) (113 ± 5) kHz
Fig. 9. Time-varying power spectrum of approximate frequency: (a) (60 ± 3) kHz; (b) (113 ± 5) kHz.
图 10 不同高度的密度脉动时变谱 (a)
$y/\delta $ = 0.4; (b)$y/\delta $ = 0.7; (c)$y/\delta $ = 1.0Fig. 10. Time-varying power spectrum of density fluctuation at different heights: (a)
$y/\delta $ = 0.4; (b)$y/\delta $ = 0.7; (c)$y/\delta $ = 1.0.表 1 风洞流场参数
Table 1. Flow conditions
$M{a_\infty }$ ${P_0}/{\rm MPa}$ ${T_0}/{\rm K}$ ${P_\infty }/{\rm Pa}$ ${T_\infty }/{\rm K}$ ${\rho _\infty }/{\rm kg \cdot {m^{ - 3} }}$ ${U_\infty }/{\rm m \cdot {s^{ - 1} }}$ $\mu /{\rm Pa \cdot s}$ ${R_e}/{\rm m^{ - 1} }$ 3.0 0.1 300 2750 107 0.089 622.5 7.43 × 10–6 7.49 × 106 
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[1] Lee C B, Jiang X Y 2019 Phys. Fluids 31 111301
Google Scholar
[2] Pan L B, Padoan P, Nordlund A 2018 Astrophys. J. Lett. 866 L17
Google Scholar
[3] Pan L B, Padoan P, Nordlund A 2019 Astrophys. J. 881 155
Google Scholar
[4] 王正魁, 靳旭红, 朱志斌, 程晓丽 2018 航空学报 39 122244
Google Scholar
Wang Z K, Jin X H, Zhu Z B, Cheng X L 2018 Acta Aeronaut. Astronaut. Sin. 39 122244
Google Scholar
[5] Tian Y F, Jaberi F A, Livescu D 2019 J. Fluid Mech. 880 935
Google Scholar
[6] Parziale N, Shepherd J, Hornung H 2015 J. Fluid Mech. 781 87
Google Scholar
[7] He L, Yi S H, Zhao Y X, Tian L F, Chen Z 2011 Chin. Sci. Bull. 56 489
Google Scholar
[8] 何霖, 易仕和, 陆小革 2017 物理学报 66 024701
Google Scholar
He L, Yi S H, Lu X G 2017 Acta Phys. Sin. 66 024701
Google Scholar
[9] Tian L F, Yi S H, Zhao Y X, He L, Cheng Z Y 2009 Sci. China, Ser. G 52 1357
Google Scholar
[10] Morkovin M V 1962 Mécanique de la Turbulence 367 380
[11] Berkooz G, Holmes P, Lumley J L 1993 Annu. Rev. Fluid Mech. 25 539
Google Scholar
[12] Smith T R, Moehlis J, Holmes P 2005 Nonlinear Dyn. 41 275
Google Scholar
[13] Schmid P J 2010 J. Fluid Mech. 656 5
Google Scholar
[14] Schmid P J, Li L, Juniper M P, Pust O 2011 Theor. Comput. Fluid Dyn. 25 249
Google Scholar
[15] Ruppert-Felsot J, Farge M, Petitjeans P 2009 J. Fluid Mech. 636 427
Google Scholar
[16] 赵玉新, 易仕和, 田立丰, 何霖, 程忠宇 2010 中国科学: 计算科学 53 584
Google Scholar
Zhao Y X, Yi S H, Tian L F, He L, Cheng Z Y 2010 Sci. China Ser. E: Technol. Sci. 53 584
Google Scholar
[17] Rinoshika A, Omori H 2011 Exp. Therm. Fluid Sci. 35 1231
Google Scholar
[18] Zheng X B, Jiang N 2015 Chin. Phys. B 24 064702
Google Scholar
[19] Freund A, Ferrante A 2019 J. Fluid Mech. 875 914
Google Scholar
[20] 赵玉新, 易仕和, 何霖, 田立丰 2010 科学通报 55 2004
Google Scholar
Zhao Y X, Yi S H, Tian L F, He L, Cheng Z Y 2010 Chin. Sci. Bull. 55 2004
Google Scholar
[21] Chen Z, Yi S H, He L, Zhu Y Z, Ge Y, Wu Y 2014 J. Visualization 17 345
Google Scholar
[22] Liu X L, Yi S h, Xu X W, Shi Y, Ouyang T C, Xiong H X 2019 Phys. Fluids 31 074108
Google Scholar
[23] He L, Yi S H, Zhao Y X, Tian L F, Chen Z 2011 Sci. China, Ser. G 54 1702
Google Scholar
[24] Taylor G I 1938 Proc. R. Soc. London, Ser. A 164 476
Google Scholar
[25] Rinoshika A, Zhou Y 2005 J. Fluid Mech. 524 229
Google Scholar
[26] Van Doorne C, Westerweel J 2007 Exp. Fluids 42 259
Google Scholar
[27] Ganapathisubramani B, Lakshminarasimhan K, Clemens N T 2007 Exp. Fluids 42 923
Google Scholar
[28] Vétel J, Garon A, Pelletier D 2010 Exp. Fluids 48 441
Google Scholar
[29] 曹晖, 赖明, 白绍良 2004 工程力学 21 109
Google Scholar
Cao H, Lai M, Bai S L 2004 Eng. Mech. 21 109
Google Scholar
[30] Spanos P D, Failla G 2004 J. Eng. Mech. 130 952
Google Scholar
[31] Mouri H, Kubotani H, Fujitani T, Niino H, Takaoka M 1999 J. Fluid Mech. 389 229
Google Scholar
[32] Mallat S G 1989 IEEE Trans. Pattern Anal. Mach. Intell. 11 674
Google Scholar
[33] Rinoshika A, Watanabe S 2010 Exp. Therm. Fluid Sci. 34 1389
Google Scholar
[34] Hutchins N, Marusic I 2007 Philos. Trans. R. Soc. London, Ser. A 365 647
Google Scholar
[35] Mathis R, Hutchins N, Marusic I 2009 J. Fluid Mech. 628 311
Google Scholar
[36] Mathis R, Monty J P, Hutchins N, Marusic I 2009 Phys. Fluids 21 111703
Google Scholar
[37] He G S, Pan C, Feng L H, Gao Q, Wang J J 2016 J. Fluid Mech. 792 274
Google Scholar
[38] He G S, Wang J J, Rinoshika A 2019 Phys. Rev. E 99 053105
Google Scholar
[39] Gurley K, Kareem A 1999 Eng. Struct. 21 149
Google Scholar
[40] 白泉, 边晶梅, 康玉梅 2018 小波理论在工程结构振动分析中的应用 (北京: 清华大学出版社) 第32−49页
Bai Q, Bian J M, Kang Y M 2018 Application of Wavelet Theory in Vibration Analysis of Engineering structures (Beijing: Tsinghua Univesity Press) pp32−49 (in Chinese)
[41] 何霖 2011 博士学位论文 (长沙: 国防科学技术大学)
He L 2011 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[42] Eswaran V, Pope S 1988 Phys. Fluids 31 506
Google Scholar
[43] Antonia R A, Mi J 1993 J. Fluid Mech. 250 531
Google Scholar
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