Estimation of Reynolds number for flows around cylinders with lattice Boltzmann methods and artificial neural networks

The present work investigates the application of artificial neural networks (ANNs) to estimate the Reynolds (Re) number for flows around a cylinder. The data required to train the ANN was generated with our own implementation of a lattice Boltzmann method (LBM) code performing simulations of a two-dimensional flow around a cylinder. As results of the simulations, we obtain the velocity field $\left(\vec{v}\right)$ and the vorticity $\left(\vec{\nabla }\times \vec{v}\right)$ of the fluid for 120 different values of Re measured at different distances from the obstacle and use them to teach the ANN to predict the Re. The results predicted by the networks show good accuracy with errors of less than $4\%$ in all the studied cases. One of the possible applications of this method is the development of an efficient tool to characterize a blocked flowing pipe.

Figure 1

A Karman vortex street is a sequence of alternated rotating vortices caused by the unsteady separation of the boundary layer of a flow passing over submerged bodies. This plot was obtained using our own LBM code for a $\text{Re}=100$.

Figure 2

Magnitude of $v_x$ for the flow around a cylinder with $\text{Re}=30$ (a) and $\text{Re}=100$ (b). It is noted that the vortices formed at the cylinder at low Reynolds disappear while for high Reynolds generate the characteristic Karman vortex street.

Figure 3

A schematic representation of the flow around the cylinder and locations of the detectors where 1, 4, 6, 11, 21, 41, and 82 values of $v_x$ and the $z$ component of the vorticity were extracted. The fluid moves from left to right and the measurement is performed when the neutral stability is reached. The times vary from $t=20\mathrm{s}$ to $t=50\mathrm{s}$ depending on the value of Re.

Figure 4

Example of the measurements obtained for different locations of the detectors for $\text{Re}=100$. We plot the $x$ component of the velocity field (a) and the $z$ component of the vorticity (b) versus the position in $y$, for detectors at $0.3\mathrm{m},1.1\mathrm{m}$, and $1.9\mathrm{m}$ over the $x$ axis.

Figure 5

Same as described in the caption of Fig. 4, but in this case we fix the detector at $x=0.3\mathrm{m}$ and change the value of the Reynolds number. In (a) is presented the $x$ component of the velocity field and in (b) the $z$ component of the vorticity.

Figure 6

RMSE for the average of the prediction set, measuring at $0.3\mathrm{m},0.5\mathrm{m},0.7\mathrm{m},1.1\mathrm{m}$, and $1.9\mathrm{m}$ using $4,6,11,21,41$, and 82 sampling points using $v_x$ in (a) and the $z$ component of the vorticity in (b). In the plots the RMSE using a single point as input in the ANNs is not considered since that case is out of scale; see Tables 1 and 2.

Figure 7

Relative errors on predictions using 82 values of $v_x$ and the $z$ component of the vorticity of the fluid as inputs. In (a) the predictions using $v_x$ are better for fluids with a moderate Re independently of the measurement location. However, using the vorticity as input, the accuracy is better for low Re and distances of the measurement closer to the obstacle instead of high Re and farther distances (b). In both situations, the error increases in the extrapolation regime (Re 115–120).

Figure 8

Relative errors on predictions using 82 values of $v_x$ (a) and the $z$ component of the vorticity (b) of the fluid as inputs for $\text{Re}=30$ along the simulation time. We observe the fluctuations on the percentage error, as expected since the flow patterns change on time.

Figure 9

Relative errors on predictions using 82 values of $v_x$ (a) and the $z$ component of the vorticity (b) of the fluid as inputs for $\text{Re}=99$ along the simulation time. We observe some fluctuations on the percent error, as expected since the flow patterns change on time.