Coherent structures associated with interscale energy transfer in turbulent channel flows

We analyze the energy spectra, production spectra, and interscale energy transfer spectra for direct numerical simulation data of turbulent channel flows at ${\mathrm{Re}}_{\tau }=180$ and 550. The transfer spectra are characterized by a negative region and a positive region. We use the negative region as a spectral mask to separate the flow fields into large-scale and small-scale fields. The former are characterized by streaks and quasi-streamwise vortices, and the latter are characterized by hairpin-like vortical structures that are similar to the original flow fields. Proper orthogonal decomposition and linear stochastic estimation are utilized to further extract the flow structures of the decomposed fields. The outer large-scale low-momentum regions are flanked by large quasi-streamwise vortices that are attached to the wall. This coherent structure presents a high degree of similarity to the structure of the near-wall cycle. This result indicates that the outer large scales may present a self-sustaining mechanism similar to that of the near-wall region. We also extract the coherent structures associated with the real-space energy transfer. The formation of small-scale hairpin-like vortices is related to the large-scale shear layer. Within this process, the energy is transferred from large scales to small scales.

Figure 1

Statistical profiles of DNS datasets: (a) mean streamwise velocity $U^{+}$; (b) streamwise turbulence intensity ${\langle u^{\prime 2}\rangle }^{+}$; (c) wall-normal turbulence intensity ${\langle v^{\prime 2}\rangle }^{+}$; (d) spanwise turbulence intensity ${\langle w^{\prime 2}\rangle }^{+}$. The blue and red lines correspond to ${\mathrm{Re}}_{\tau }=180$ and 550, respectively. The black dashed line represents the DNS results downloaded from Ref. [53]; for details on the simulation, please refer to Álamo and Jiménez [54] and Hoyas and Jiménez [55].

Figure 2

Premultiplied two-dimensional spectra of $u^{\prime }$ (a), (d), $v^{\prime }$ (b), (e), and $w^{\prime }$ (c), (f) at $y^{+}=12$ for Reynolds numbers ${\mathrm{Re}}_{\tau }=180$ (a)–(c) and 550 (d)–(f). The parameters $k_x$ and $k_z$ are the streamwise and spanwise wave numbers, respectively. The solid gray lines correspond to contour levels from 0.1 to 0.9 with an interval of 0.2. For comparison, the spectra computed at ${\mathrm{Re}}_{\tau }$=550 with a larger domain of $L_x=8\pi \delta$ and $L_z=3\pi \delta$ are also presented as the red dashed lines. The spectra are normalized by the maximum magnitude and filtered by adopting a bandwidth moving filter of 20% [56].

Figure 3

Premultiplied two-dimensional spatial transfer spectra at different $y^{+}$ for ${\mathrm{Re}}_{\tau }=550$. The solid gray curves represent the corresponding TKE spectra, and the contour values increment by 0.2, starting with 0.1 and ending with 0.9. The black dashed lines represent ${\lambda }_x^{+}=2{\lambda }_z^{+}$. The spectra are normalized by the maximum magnitude and filtered by adopting a bandwidth moving filter of 20% [56].

Figure 4

Premultiplied two-dimensional production spectra at $y^{+}=5$ (a), (b), 15 (c), (d), and 30 (e), (f) for ${\mathrm{Re}}_{\tau }=180$ (a), (c), (e) and 550 (b), (d), (f). The solid gray curves represent the corresponding TKE spectra, and the contour values increment by 0.2, starting with 0.1 and ending with 0.9. The black dashed lines represent ${\lambda }_x^{+}=2{\lambda }_z^{+}$. The spectra are normalized by the maximum magnitude and filtered by adopting a bandwidth moving filter of 20% [56].

Figure 5

Premultiplied two-dimensional interscale transfer spectra at $y^{+}=5$ (a), (b), 15 (c), (d), and 30 (e), (f) for ${\mathrm{Re}}_{\tau }=180$ (a), (c), (e) and 550 (b), (d), (f). The solid gray curves represent the corresponding TKE spectra, and the contour values increment by 0.2, starting with 0.1 and ending with 0.9. The black dashed lines represent ${\lambda }_x^{+}=2{\lambda }_z^{+}$. The spectra are normalized by the maximum magnitude and filtered by adopting a bandwidth moving filter of 20% [56].

Figure 6

Premultiplied interscale transfer spectra (a), (b) and production spectra (c), (d) in the outer layer of $y^{+}=70$ (a), (c) and 250 (b), (d) at ${\mathrm{Re}}_{\tau }=550$. The solid gray curves represent the corresponding TKE spectra, and the contour values increment by 0.2, starting with 0.1 and ending with 0.9. The black dashed lines represent ${\lambda }_x^{+}=2{\lambda }_z^{+}$. The spectra are normalized by the maximum magnitude and filtered by adopting a bandwidth moving filter of 20% [56].

Figure 7

The zero-crossing boundary in the interscale transfer spectra at ${\mathrm{Re}}_{\tau }=550$. The wall-normal position $y^{+}$ increases from 1 to 250 as the line color changes from yellow to red.

Figure 8

Visualization of vortices and low-speed streaks for the original DNS data (a), the large-scale field (b), and the small-scale field (c) at a Reynolds number of ${\mathrm{Re}}_{\tau }=550$. The isosurfaces colored based on the wall-normal height $y^{+}$ indicate low-speed streaks of $u^{\prime }=-1.5$, and the isosurfaces colored by the streamwise fluctuation velocity $u^{\prime }$ represent vortical structures identified by the $Q$ criterion with $Q=0.006$.

Figure 9

Statistical profiles of the large scales (the solid symbols) and the small scales (the hollow symbols) at ${\mathrm{Re}}_{\tau }=180$ (blue) and 550 (red). The solid lines denote the profiles of the original DNS data.

Figure 10

Reynolds shear stress weighted joint PDF of $u^{+}$ and $v^{+}$ at $y^{+}=30$ (a) and 100 (b) for ${\mathrm{Re}}_{\tau }=550$. The gray colors represent the PDF of raw DNS data, and the red and blue contours represent the PDF of large scales and small scales, respectively. The contour levels are from 0.01 to 0.1 in a step of 0.03. The velocity is scaled by the root mean square of the original DNS velocity at this location.

Figure 11

The first two POD modes (the upper panel is for the first POD mode, and the bottom panel is for the second POD mode) of the large-scale flow fields in the regions of $0\le y^{+}\le 5$, $5\le y^{+}\le 30$, and $30\le y^{+}\le 180$ using the cross stream view. Here, the Reynolds number ${\mathrm{Re}}_{\tau }$ is 180. The contour represents the low- and high-speed (blue and red) $u^{\prime }$ component of the streaks, and the arrows denote the $v^{\prime }-w^{\prime }$ components. Only part of the spanwise range is shown in this figure for clarity.

Figure 12

The first two POD modes (the upper panel is for the first POD mode, and the bottom panel is for the second POD mode) of the large-scale flow fields in the regions of $0\le y^{+}\le 5$, $5\le y^{+}\le 30$, $30\le y^{+}\le 120$, and $120\le y^{+}\le 550$ using the cross stream view. Here, the Reynolds number is ${\mathrm{Re}}_{\tau }=550$. The contours represent the low- and high-speed (blue and red) $u^{\prime }$ components of the streaks, and the arrows denote the $v^{\prime }-w^{\prime }$ components. Only part of the spanwise range is shown in this figure for clarity.

Figure 13

Isosurfaces of conditionally averaged large-scale fields (a), (c), (e) and small-scale fields (b), (d), (f) given a low-velocity event in the original DNS data at reference location $y^{+}=30$ (a), (b), 60 (c), (d), and 100 (e), (f). The Reynolds number is 550. Condition is for $u^{\prime }\le -u_{\mathrm{rms}}^{\prime }$ at each reference location. The vortex structures are visualized using the $Q$ vortex detection criterion with $Q=0.2Q_{\max }$ and colored based on the wall-normal height. The blue isosurface indicates the low-speed streamwise velocity region at $u^{\prime }=-0.5u_{\max }^{\prime }$. The contour map of the streamwise velocity is plotted in the $x\text{−}z$ plane at the reference location.

Figure 14

Wall-normal distribution of interscale energy transfer $\mathrm{Tr}$ (the solid lines), $\langle T_L\rangle$ (the solid symbols), and $\langle T_S\rangle$ (the hollow symbols) at ${\mathrm{Re}}_{\tau }=180$ (blue) and 550 (red).

Figure 15

Conditionally averaged coherent structures corresponding to the events $T_L\ge T_{L,\mathrm{rms}}$ (a), (c), (e) and $T_S\le -T_{S,\mathrm{rms}}$ (b), (d), (f) at reference location $y^{+}=30$ (a), (b), 60 (c), (d), and 100 (e), (f) for ${\mathrm{Re}}_{\tau }=550$. The vortex structures are visualized using the $Q$ vortex detection criterion with $Q=0.2Q_{\max }$ (a), (c), (e) and $Q=0.04Q_{\max }$ (b), (d), (f), and colored based on the wall-normal height. The contour map of the streamwise velocity at $z^{+}=0$ is also plotted. The solid black lines in panels (a), (c), and (e) indicate the zero-cross boundaries of the streamwise velocity, and the solid and dashed black lines in panels (b), (d), and (f) represent $u^{\prime }=0.1$ and $-0.15$, respectively.