Characteristics of sources and sinks of momentum in a turbulent boundary layer

In turbulent boundary layers, the wall-normal gradient of the Reynolds shear stress identifies momentum sources and sinks ($T=\partial [-uv]/\partial y$). These motions can be physically interpreted in two ways: (1) as contributors to the turbulence term balancing the mean momentum equation, and (2) as regions of strong local interaction between velocity and vorticity fluctuations. In this paper, the space-time evolution of momentum sources and sinks is investigated in a turbulent boundary layer at the Reynolds number (${\text{Re}}_{\tau }$) = 2700, with time-resolved planar particle image velocimetry in a plane along the streamwise and wall-normal directions. Wave number-frequency power spectra of $T$ fluctuations reveal that the wave velocities of momentum sources and sinks tend to match the local streamwise velocity in proximity to the wall. However, as the distance from the wall increases, the wave velocities of the $T$ events are slightly lower than the local streamwise velocities of the flow, which is also confirmed from the tracking in time of the intense momentum sources and sinks. This evidences that momentum sources and sinks are preferentially located in low-momentum regions of the flow. The spectral content of the $T$ fluctuations is maximum at the wall, but it decreases monotonically as the distance from the wall grows. The relative spectral contributions of the different wavelengths remains unaltered at varying wall-normal locations. From autocorrelation coefficient maps, the characteristic streamwise and wall-normal extents of the $T$ motions are respectively 60 and 40 wall units, independent of the wall distance. Both statistics and instantaneous visualizations show that momentum sources and sinks have a preferential tendency to be organized in positive-negative pairs in the wall-normal direction.

Figure 1

Profile of ${\overline{T}}^{+}$ (black asterisks) and ${\overline{v{\omega }_{\mathrm{z}}}}^{+}$ (red circles) at different wall-normal locations within the flow.

Figure 2

Wave number-frequency power spectral density of the $T$ fluctuations, at different wall-normal distances. Lines represent the local mean streamwise velocity.

Figure 3

Power spectra of T, at different wall-normal distances, $*$: $y^{+}=60$, $+$: $y^{+}=140$, $•$: $y^{+}=240$, $\diamond$: $y^{+}=440$, $\square$: $y^{+}=740$, $\circ$: $y^{+}=1040$.

Figure 4

Characteristic wave velocities of sources and sinks of momentum (blue squares) at different wall-normal locations, calculated from the wave number-frequency power spectra. Black filled spot indicates the mean streamwise velocity.

Figure 5

Autocorrelation coefficient map of $T$ events, at different wall-normal distances.

Figure 6

Snapshot of sources and sinks of momentum. Black dashed lines mark intense momentum sinks, while black continuous lines mark intense momentum sources, both identified with the criterion shown in Eq. (5). Spots of different sizes and shades mark the time evolution of the weighted centroids of the intense events.

Figure 7

Joint probability density function between the wall-normal locations of the positive (a) and negative (b) $T$ events and their intensities averaged over the respective areas.

Figure 8

Probability density functions of the wall-normal component of the local velocities ($V_{\mathrm{c}}$) of the intense momentum sources (a) and sinks (b) (continuous line) and a two-term Gaussian fitting curve (dashed line). The vertical lines in the insets represent the mean values, which are respectively of 0.20 for the momentum sources and of 0.25 for the momentum sinks.

Figure 9

Joint probability density functions between the streamwise component of the local velocities ($U_{\mathrm{c}}$) of the positive (a) and negative (b) $T$ events and their wall-normal locations ($Y_{\mathrm{c}}$). The black thicker line is a contour of the $50\%$ of the peak value at each wall-normal location. The blue dashed line represents the streamwise component of the mean velocity profile of the flow ($\overline{U}/U_{\infty }$). Asterisks ($*$) mark the mean of the local streamwise velocities of the intense $T$ events.