Recognition of an obstacle in a flow using artificial neural networks

In this work a series of artificial neural networks (ANNs) has been developed with the capacity to estimate the size and location of an obstacle obstructing the flow in a pipe. The ANNs learn the size and location of the obstacle by reading the profiles of the dynamic pressure $q$ or the $x$ component of the velocity $v_x$ of the fluid at a certain distance from the obstacle. Data to train the ANN were generated using numerical simulations with a two-dimensional lattice Boltzmann code. We analyzed various cases varying both the diameter and the position of the obstacle on the $y$ axis, obtaining good estimations using the $R^2$ coefficient for the cases under study. Although the ANN showed problems with the classification of very small obstacles, the general results show a very good capacity for prediction.

Figure 1

Magnitude of $v_x$ for the flow around a cylindrical obstacle with an incoming Poisseuille flow with $v_c=1.5$ m/s. The obstacle is fixed at $x=0.6$ m and the fluid flows over the $x$-positive direction along the pipe. Note that the vortices form after the obstacle is driven along the direction of the flow, generating the characteristic Karman vortex street.

Figure 2

Schematic of the 2D pipe, the cylindrical obstacle, and different measuring sites across the pipe, located at $\mathrm{A}=1.75$ m, $\mathrm{B}=2.10$ m, and $\mathrm{C}=2.45$ m, shown by dashed lines. The region within the dotted rectangle is the area used as target for the ANN.

Figure 3

(a) Plot representing the solid and fluid elements in the LBM mesh simulation; (b) plot representing the transformation from the simulation to the target grid. The lower number of cells used in the target grid in comparison with the LBM mesh causes a loss of resolution in the grid. The color box represents the solid-liquid ratio index (SLRI), where a 0 value means that a cell is composed of only fluid elements and 1 represents a cell fully occupied by solid elements. The grid lines shown are for reference only and do not represent the size of a target cell.

Figure 4

Time series of the $x$ component of the flow velocity at $y=0.105$ m (solid curve), 0.210 m (dashed curve), and 0.315 m (dash-dotted curve) at measurement site B, for an obstacle of $\beta =0.488$ centered at $y=0.21$ m.

Figure 5

$R^2$ coefficient for predictions produced by the ANN for obstacles located at the center of the pipe with different diameters. The ANN is trained with data extracted with a sensor located over the line in B and tested with measurements at A, B, and C. (a) The red curve with crosses shows the results using $v_x$, and $q$ is represented by the black curve with $X$'s at site A. (b) Input data were extracted at measurement site B, and the blue curve with asterisks and the green curve with squares represent $R^2$ using the profiles $v_x$ and $q$, respectively. (c) The red curve with triangles and the magenta curve with circles correspond to the results for $v_x$ and $q$ at measurement site C, respectively. All $R^2$ coefficients are above 0.6, which can be interpreted as a good correlation between the prediction and the target. Note how the results are independent of the measurement location.

Figure 6

(a) Target and (b) ANN prediction considering the profile of the $v_x$ of the fluid generated by an obstacle at the center of the pipe with $\beta =0.4758$. The color box represents the SLRI. The difference between the target and the prediction obstacle is almost imperceptible to the naked eye; the $R^2$ coefficient has reached a value of 0.979.

Figure 7

ANN prediction analysis by $R^2$ for several obstacle sizes and centered on the pipe. Measuring the time series on a single sensor for $v_x$ (crosses) and $q$ ($X$'s) at $y=0.205$ m and three detectors, at $y=0.105$, 0.210, and 0.315 m, for $v_x$ (asterisks) and $q$ (squares) at measurement site B. Note that by considering the time series of the three detectors the values of $R^2$ are close to 1 for diameters greater than $\beta =0.20$, while considering a single detector this happens for $\beta >0.25$.

Figure 8

Prediction analysis for obstacles in different positions on the $y$ axis, by training and testing the ANN with the profiles of (a) $v_x$ and (b) $q$ for detectors at measurement site B. The red curve with crosses shows the results for the obstacle with $\beta =0.122$; the black curve with $X$'s, $\beta =0.244$; the blue curve with asterisks, $\beta =0.366$; and the green curve with squares, $\beta =0.488$. Values of $R^2$ are similar for both approaches; the smallest obstacle has the worst prediction at the center and close to the borders of the pipe.

Figure 9

(a) Target and (b) ANN prediction considering the profile of the $v_x$ of the fluid generated by an obstacle at $y=0.35$ m with $\beta =0.122$. Contrasting with Fig. 6, we observe how the ANN shows two obstacles instead of a single one, resulting in bad prediction with $R^2=0.09$.

Figure 10

Prediction analysis for obstacles in different positions on the $y$ axis. The ANNs are trained with the time series of (a) $v_x$ and (b) $q$ at the center of the pipe at measurement site B. The red curve with crosses shows the results for the obstacle with $\beta =0.122$; the black curve with $X$'s, $\beta =0.244$; the blue curve with asterisks, $\beta =0.366$; and the green curve with squares, $\beta =0.488$. It is evident that with the ANN it is difficult to give a proper prediction of the location of the obstacle with only one sensor; this is more evident for small obstacles close to the borders of the pipe.

Figure 11

$R^2$ results obtained for obstacles in different positions at the $y$ axis. The ANNs are trained with the time series of (a) $v_x$ and (b) $q$ for three sensors at $y=0.105$, 0.210, and 0.315 m at measurement site B. The red curve with crosses shows the results for the obstacle with $\beta =0.122$; the black curve with $X$'s, $\beta =0.244$; the blue curve with asterisks, $\beta =0.366$; and the green curve with squares, $\beta =0.488$. Again, the worst adjustment coefficient was obtained with the smallest obstacle, $R^2=-0.102$, using $q$ as input data. However, there is a remarkable improvement for the other three obstacle sizes compared to the case where a single sensor was used.

Figure 12

Magnitude of $v_x$, for a simulation of a flow around a square obstacle with an incoming flow of $v_c=1.5$ m/s. The obstacle is located at 0.6 m on the $x$ axis. As with the cylindrical obstacle, the vortices formed after the square obstacle are driven along the direction of the flow, generating also a Karman vortex street, but with a different frequency.

Figure 13

Prediction for a square obstacle at the center of the diameter of the pipe, considering as input for the ANN the time series of $q$ at three sensors at site B. As expected, the ANN considers this obstacle with a shape similar to those which it was trained for, however, it has a size and location comparable to those of the target.