Application of a self-organizing map to identify the turbulent-boundary-layer interface in a transitional flow

Existing methods to identify the interfaces separating different regions in turbulent flows, such as turbulent/nonturbulent interfaces, typically rely on subjectively chosen thresholds, often including visual verification that the resulting surface meaningfully separates the different regions. Since machine learning tools are known to help automate such classification tasks, we here propose to use an unsupervised self-organizing map (SOM) machine learning algorithm as an automatic classifier. We use it to separate a boundary layer undergoing bypass transition into two distinct spatial regions, the turbulent boundary layer (TBL) and non-TBL regions, the latter including the laminar portion prior to transition and the outer flow which possibly contains weak free-stream turbulence. Both regions are separated by the turbulent boundary layer interface (TBLI). The data used in this study are from a direct numerical simulation and are available on an open database system. In our analysis of one snapshot in time, every spatial point is characterized by a 16-dimensional vector containing the magnitudes of the components of total and fluctuating velocity, magnitudes of the velocity gradient tensor elements, and the streamwise and wall-normal coordinates, all normalized by their global standard deviation. In an unsupervised fashion, the SOM classifier separates the points into TBL and non-TBL regions, thus identifying the TBLI without the need for user-specified thresholds. Remarkably, it avoids including vortical streaky structures that exist in the laminar portion prior to transition as well as the weak free-stream turbulence in the turbulent boundary layer region. The approach is compared quantitatively with existing methods to determine the TBLI (vorticity magnitude, cross-stream velocity fluctuation). Also, the SOM classifier is cast as a linear hyperplane that separates the two clusters of data points, and the method is tested by finding the TBLI of other snapshots in the transitional boundary layer data set, as well as in a fully turbulent boundary layer with similar levels of free-stream turbulence. Variants in which the approach failed are also summarized.

Figure 1

(a) Flow configuration of the current transitional boundary layers with free stream turbulence. Flow is in the $x$ direction and from left to right. The region covered by data provided in JHTDB is shown as a black box. Instantaneous coherent structures are identified by isosurface of $Q$ criterion, colored by their wall-normal heights. The boundary layer thickness ${\delta }_{99}$ is shown as a blue line, and the value at the inlet of the stored region, ${\delta }_{{99}_0}$, is used as reference length scale. (b) Skin-friction coefficient $C_f$ and (c) boundary layer thickness ${\delta }_{99}$ as a function of streamwise position $x$. The lengths are normalized by ${\delta }_{{99}_0}$.

Figure 2

Contours of streamwise velocity on a single plane at height $y=0.50$. In the laminar region, there are streaks with streamwise vorticity that should not be counted as a turbulent region.

Figure 3

Sketch of original self-organizing map (SOM) as applied to a problem with two classes. Data (gray points) are described by two coordinates $\mathbf{X}=(X_1,X_2)$ (a 2D state vector). The green and red circles represent the two nodes. The open circle represents the first randomly chosen datapoint with position ${\mathbf{D}}_1$ toward which the closest of the initial nodes is drawn most strongly. After a few (N) iterations over the training data and nodes, the two nodes move to locations representing the clusters.

Figure 4

(a) Wall contour of ${|\partial u/\partial y|}_s$ and identified TBLI using SOM. (b) Scatter plot of ${|\partial u/\partial y|}_s,{|\partial u/\partial y|}_s$ and $x_s$: blue circles, non-TBL data points; green circles, TBL data points; $+$, weight (position) vectors of two $\sout{\mathrm{neurons}}$ nodes. The TBLI is a bisecting plane of the two weight vectors in the space constructed by the input features ${|\partial u/\partial y|}_s,{|\partial u/\partial y|}_s$, and $x_s$. For convenience, the original $x$ is also shown at the top axis.

Figure 5

(a) Top view of TBLI identified by the SOM algorithm. The surface is colored by its local wall-normal height. In panels (b), (c), and (d), $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ contours are shown together with the TBLI (black line) and the boundary layer thickness ${\delta }_{99}\left(x\right)$ (white dashed line). Three different cuts are shown: streamwise at $z=122.6$ between $x=200$ and 500 (b), between $x=500$ and 800 (c), and a spanwise plane at $x=613.1$ (d). Panel (e) shows the TBLI obtained by applying the SOM trained from panel (a) to another temporal snapshot, separated by a time $1175{\delta }_{{99}_0}/U_{\infty }$. The color map in panel (e) is the same as in panel (a).

Figure 6

Average height of TBLI $\langle y_I\rangle /{\delta }_{99}$.

Figure 7

(a) Contour map of $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ at $y=0.50$. (b) The PDF of $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$. The blue filled profile shows the overall PDF on the entire plane and a plateau is seen in the range of values indicated by the orange band. The black dashed and solid lines are the PDFs in the SOM-determined TBL/non-TBL regions. The threshold picked by Otsu's method is shown as the orange line. Panels (c)–(d) show the TBL/non-TBL regions (blue, non-TBL region, yellow, TBL region) identified using (c) the SOM algorithm, (d) the threshold on $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ chosen within the PDF plateau, and (e) the threshold identified by Otsu's method.

Figure 8

Results when applying various methods on a streamwise vertical plane at $z=122.6$ in the transitional boundary layer data set. Panels (a) and (b) show TBL/non-TBL regions identified by the SOM algorithm and Otsu's method applied to $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$, respectively. The background shows $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ contours, the black line is the TBLI, and the white dashed line is the ${\delta }_{99}\left(x\right)$. Panel (c) shows the PDF of $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ on the entire plane. See Fig. 7 for legend.

Figure 9

PDFs of (a) ${\left|\omega \right|}_s$ and (b) $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ in the entire 3D domain. See Fig. 7 for legend.

Figure 10

Joint PDFs of (a) $|v^{\prime }{|}_s$ vs $|w^{\prime }{|}_s$ and (b) ${|\partial u/\partial z|}_s$ vs ${\left|u\right|}_s$ in the TBL (color contour) and the non-TBL (black line) regions in the entire 3D domain. The lines from top right to bottom left in panel (a) are isolines with PDFs equal to 0.05, 0.07, 0.5, 1, 1.5, and 2 respectively. The lines from top to bottom in panel (b) are isolines with PDFs between 0.01 and 0.1 with a constant step of 0.01.

Figure 11

Fully turbulent boundary layer without free stream turbulence. (a) PDF of ${log}_{10}\left({\omega }^{*}\right)$ at different $y/{\theta }_{in}$. (b) PDF of ${log}_{10}\left({\omega }^{*}\right)$ in the whole 3D domain. [(c), (d)] ${log}_{10}\left({\omega }^{*}\right)$ contour, TBLIs identified using SOM (black solid line), and ${\delta }_{99}$ (blue dashed line) at $z/{\theta }_{in}=1.875$ and $x/{\theta }_{in}=900$, respectively

Figure 12

Three cross-flow planes at different streamwise locations in a fully turbulent boundary layer with free stream turbulence: (a) $x/{\theta }_{in}=125$, (b) $x/{\theta }_{in}=500$, and (c) $x/{\theta }_{in}=875$, where ${\theta }_{in}$ is the momentum thickness at $x/{\theta }_{in}=0$. The left column shows PDF of vorticity magnitude ${\omega }_s$ at different wall normal heights, middle column shows the vorticity magnitude ${\omega }_s$ contours, and right column shows $|v^{\prime }{|}_s+{\left|w^{\prime }\right|}_s$ contours. The TBLI identified using SOM are shown as black solid lines.

Figure 13

TBLI detection results using SOM with noise. The input data is at $y=0.50$ (same as Fig. 7), but 10% white Gaussian noise has been added to all input variables in the whole domain. The yellow and blue colors are the TBL and non-TBL regions with the original input, while the black line is the TBLI with the noisy data.