Large-scale motions from a direct numerical simulation of a turbulent boundary layer

A study of large-scale motions from a direct numerical simulation database of the turbulent boundary layer up to ${\text{Re}}_{\theta }\sim 2500$ is presented. The statistics of large-scale streamwise structures have been investigated using two-dimensional and three-dimensional (3D) extraction procedures. The large-scale structures are abstracted using a robust skeletonization method usually applied to other research domains to simplify complex 3D objects. Different structure parameters such as the length, shape, or angle are investigated. The features of the detected structures are compared to their mean counterparts extracted from two-point correlations. Structures as large as ten boundary layer thicknesses are observed. The streamwise length of these structures follows a $-2$ power law distribution, similar to the experimental findings at higher Reynolds numbers.

Figure 1

Comparison of the friction coefficient $c_f$ with the DNS of Schlatter et al. [24].

Figure 2

Mean velocity and Reynolds stress (continuous) profiles of the current study in comparison with profiles from Jimenez et al. [32] (at ${\text{Re}}_{\theta }=1100$, 1551, 1968; dashed lines) and Schlatter et al. [24] at ${\text{Re}}_{\theta }=1000$, 1410, 2000, and 2540 (dash-dotted lines).

Figure 3

Streamwise energy spectra using spatial (solid line) and temporal (dashed line) data at different wall distances. The time spectra are computed at the streamwise position such that ${\text{Re}}_{\theta }=2068$ using the Taylor hypothesis and the spatial ones are computed in a domain such as $1764<{\text{Re}}_{\theta }<2348$ which corresponds to approximately 20 local boundary layer thicknesses at ${\text{Re}}_{\theta }=2068$.

Figure 4

Two-point spatial correlation function of the streamwise velocity fluctuations $\langle u(x-x_o,y-y_o)u(x,y)\rangle$ at wall distances (a) $y_o={50}^{+}$, (b) $y_o={100}^{+}$, and (c) $y_o={150}^{+}$, where $x_o$ is the streamwise position such that ${\text{Re}}_{\theta }=2068$.

Figure 5

Energy and volume distribution of the streamwise velocity fluctuations. Vertical dashed lines represent the threshold coefficient $C_{\mathrm{thr}}=1$, meaning that outer left and outer right regions are kept for low- and high-momentum regions, respectively.

Figure 6

Isovolume of the 3D streamwise velocity fluctuation structures with their skeletons extracted using thresholding [Eq. (1)] with $C_{\mathrm{thr}}=1.0$ on a $20\delta$-long subdomain centered at the Reynolds number ${\text{Re}}_{\theta }=2068$. Different colors are associated to separated structures.

Figure 7

Skeleton of a single structure. The black curve represents the longest curve (referred to as the main curve), and colorful curves are the branches. (a) Isometric, (b) top, and (c) side views of the geometry.

Figure 8

Profiles of (a) streamwise fluctuation and (b) Reynolds shear stress computed from the detected structures with $C_{\mathrm{thr}}=1.0$. Note that, in this figure only, structures not completely included in the $20\delta$-long investigated domain are kept. Statistics are given for $y^{+}>20$, which corresponds to the lower bound of the domain to detect structures.

Figure 9

PDF of the streamwise lengths of the detected structures from positive velocity fluctuations (continuous lines) and negative velocity fluctuations (dashed lines) for three different values of the detection threshold. Statistics are results of (a) 2D detection in the $XY$ planes and (b) 3D detection.

Figure 10

Joint PDFs of streamwise wall-normal sizes $P({\lambda }_x/\delta ,{\lambda }_y/\delta )$ and streamwise spanwise size $P({\lambda }_x/\delta ,{\lambda }_z/\delta )$ of the detected structures. Areas inside the contour lines correspond to 99%, 90%, 75%, 50%, and 25% of the detected structures. An indicative ratio between the two sizes of the joint PDFs is given by dashed lines. (a) and (b) are based on the binary volume ${\mathbb{B}}^{\ominus }$ while (c) and (d) are based on ${\mathbb{B}}^{\oplus }$.

Figure 11

(a) Histogram of the number of branches having a length of at least $10\%$ of its main curve for the structures longer than $4\delta$ and (b) PDF of the structure lengths computed from the main curve of the skeleton, $P(\lambda /\delta )$. The binary structures are extracted by Eq. (1) with $C_{\mathrm{thr}}=1.0$. 8500 main curves are used for the PDF.

Figure 12

Two-point spatial correlation function of the streamwise velocity fluctuations $\langle u(x-x_o,y-y_o)u(x,y)\rangle$ conditioned by $|u(x_o,y_o)|>C_{\mathrm{thr}}{\sigma }_u^{{100}^{+}}$ at wall distances (a) $y_o={50}^{+}$, (b) $y_o={100}^{+}$, and (c) $y_o={150}^{+}$, where $x_o$ is the streamwise position such that ${\text{Re}}_{\theta }=2068$ and $C_{\mathrm{thr}}=1$. The condition $|u(x_o,y_o)|>C_{\mathrm{thr}}{\sigma }_u^{{100}^{+}}$ represents only $32\%$ of volume but $80\%$ of the streamwise kinetic energy.

Figure 13

(a) PDFs of the pitch ($\alpha$) and yaw ($\beta$) angles from the main curve of the skeletons. (b) Mean pitch angle ($\alpha$) as a function of wall distance extracted from skeleton in comparison with the mean angles detected from the two-point spatial correlation functions (Figs. 4 and 12). The mean angle from the two-point correlation function is considered as the slope of the longest line between the fixed point and the $80\%$ isocontour.

Figure 14

PDF of the structure lengths at particular wall distances using 3D detection from negative velocity fluctuations (dashed lines) and positive velocity fluctuations (continuous lines).

Figure 15

Average mean squared streamwise velocity, $\overline{a^2}$, as function of structure length in (a), (d), (g) 3D structures and (b), (e), (h) 2D structures. (c), (f), (i) Streamwise energy spectra at three wall distances: $y^{+}=50,100,150$ (from top to bottom). All statistics are computed for ${\text{Re}}_{\theta }=2068$.

Figure 16

Graphical presentation of the workflow to extract the skeleton's curves for a single 3D spatially resolved data set. The red steps correspond to the structure detection method used prior to the bounding box statistics.