Orthogonal wavelet multiresolution analysis of the turbulent boundary layer measured with two-dimensional time-resolved particle image velocimetry

The turbulent boundary layer flow measured by two-dimensional time-resolved particle image velocimetry is analyzed using the discrete orthogonal wavelet method. The Reynolds number of the turbulent boundary layer based on the friction velocity is ${\mathrm{Re}}_{\tau }=235$. The flow field is decomposed into a number of wavelet levels which have different characteristic scales. The velocity statistics and coherent structures at different wavelet levels are investigated. It is found that the fluctuation intensities and their peak locations differ for varying scales. The proper orthogonal decomposition (POD) of different wavelet components reveals a cascade of scales of coherent structures, especially the small-scale ones that are usually difficult to be identified in POD modes of the undecomposed flow field. The interactions among the scales are investigated in terms of large-scale amplitude modulations of the small-scale structures. In previous studies the velocity fluctuations are separated into two parts, the large scale and the small scale, divided usually by the boundary layer thickness. In the present study, however, the scales smaller than the boundary layer thickness are further separated. Therefore, the modulation analysis is a refined investigation that differentiates the modulation effects on separated small scales. The results reveal that the modulation effects vary among the small scales.

Figure 1

Turbulent boundary layer profiles at ${\mathrm{Re}}_{\tau }=235$. Left axis: mean velocity $U^{+}$; Right axis: streamwise fluctuation intensity ${\mathrm{u}}_{\mathrm{rms}}^{+}$ (both quantities have been scaled by the friction velocity). The black circle represents the measured mean velocity, and the blue squares the measured fluctuation intensity. The red lines are corresponding profiles from the direct numerical simulation of a channel flow at ${\mathrm{Re}}_{\tau }=180$ [27]. The dashed lines represent the law of the wall $\left(U^{+}=y^{+}\right)$ and log law $\left(U^{+}=2.5ln{y}^{+}+5.5\right)$ of the turbulent boundary layer.

Figure 2

An example of a streamwise velocity fluctuation data and its wavelet decomposition at a single point. The chosen point was at $y^{+}=15.5$, the location of maximum streamwise fluctuation intensity $u_{\mathrm{rms}.\max }$. The first row is the measured fluctuation data, and the following rows are its components at different wavelet levels. Wavelet level increases from top to bottom.

Figure 3

Spectrum of the wavelet components in Fig. 2.

Figure 4

Wall-normal profiles of $u_{\text{rms}}$ at different wavelet levels.

Figure 5

Vectors of instantaneous flow fluctuations at different wavelet levels. (a) WL-9, (b) WL-7, (c) WL-5, (d) WL-3.

Figure 6

POD modes of the wavelet components. Left column, the mode for streamwise velocity $u$; Right column, the mode for vertical velocity $v$. (a), (h) WL-10; (b), (i) WL-9; (c), (j) WL-8; (d), (k) WL-7; (e), (l) WL-6; (f), (m) WL-5; (g), (n) WL-4.

Figure 7

POD mode energy for the measured and wavelet decomposed data. The mode energy is normalized by the total energy of all the modes in each POD realization.

Figure 8

(a) Spatial correlation of the velocity fluctuations for the measured and wavelet decomposed data in the streamwise direction. (b) Streamwise wavelength for flow components at different wavelet levels.

Figure 9

(a) Fluctuation of streamwise velocity $u$ at $y^{+}=5.5$. The fluctuation velocity has been normalized by the free-stream velocity $U_{\infty }$. (b) Large-scale component $u_L$ (dashed line) and small-scale component $u_S$ (full line). (c) Envelope $E_L\left(u_S\right)$ (red thick line) of $u_S$ (black thin line). (d) $E_L\left(u_S\right)$ (red full line) and $u_L$ (black dashed line). For comparison, the mean of $E_L\left(u_S\right)$ was adjust to zero and its magnitude was magnified by a factor of 5.

Figure 10

Large-scale modulation of small-scale structures at different wavelet levels.