# About 3D Incompressible Flow Reconstruction from 2D Flow Field Measurements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Procedure

- -
- The mean value of the signal is time-invariant.
- -
- The signal variance is time-invariant.
- -
- The signal auto-correlation function depends only on the time-lag used for its computation.

## 3. Numerical Procedure

## 4. Experimental Results

## 5. Uncertainty Evaluation

## 6. Machine Learning-Based Metrics

_{1}, U

_{2}, and U

_{3}are the velocities along the three axes as reported in Table 2.

- where U represents the function of the indicator:
- $X=1$, if the $X\left(d\left({x}_{n}\right)\ne {j}_{n}\right)$ condition is verified;
- $X=0$, if the $X\left(d\left({x}_{n}\right)\ne {j}_{n}\right)$ is not verified;
- and $d\left(x\right)$ represents the classification model, as depicted in Figure 6.

## 7. Expanded Metrics for Comparison: Geostatistics and Sobol-Based Sensitivity

- -
- it calculates the variance necessary for accuracy;
- -
- it is denoted as an exact estimator because it delivers, in the points where we have the space information, its true value;
- -
- it is based on a model of probabilities by considering error and deviation.

^{2}

_{R}, of a set of k estimates can be written as:

_{1}with five measurements at different conditions. Figure 11 indicates all velocities reported on the left side according to the Sobol’s global description, and on the right, we can see the example for one case. The global interpolation encompasses all five curves related to the considered velocity. The sensitivity here demonstrated is close to the Chauvenet’s criterion. Hence, all measurement values are globally coherent, and they are from the same instrument.

## 8. Discussion and Conclusions

_{r}. For this reason, the dynamical system is fully predicted in time over all the vertical scales. The spatial distance between the measurement locations is significantly larger than the smallest vortex dimension developed into the flow. Thus, only large scales are resolved in spaces, whereas the smaller ones are filtered by the sampling procedure. This point can be useful for the industrial purposes in design since only the spatial average velocity is taken into account and the velocity fluctuations are neglected. The machine learning-based metrics have added a further contribution in terms of data clustering and accuracy. It could be another opportunity for a new paper on the use of machine learning for this kind of measurement. Additional metrics such as the Kriging estimator and Sobol sensitivity have been used to strengthen the work results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) NACA 0012 airfoil profile: acquisition probe Vectrino ADV 3D placed in the wake (

**left**), and its cross-section (

**right**). (

**b**) Longitudinal view (

**left**), and cross-view of the channel (

**right**): channel chassis/skeleton keeping the glass side (1), channel inner part allowing the flow (2), 3D instrument rail (3), head bringing the 3D instrument directed to the channel for measurement (4), vertical rail allowing an up-and-down displacement of the 3D instrument (5).

**Figure 3.**Subdivision of the fluid domain in 3 measuring grids of 25 points each with equal spacing steps Δx = Δy = Δz = 5 cm. The colored blue points, different from the red ones, are considered boundary points for the numerical integration of the continuity equation.

**Figure 5.**As an example in the central plane, the red cell means that the u component is computed from the continuity equation in function of the v and w components measured in the necessary grid points of the plane, according to the numerical scheme adopted.

**Figure 7.**Processing data of Table 2 as measured velocities/flow. The plots display the clustering process related to electrical and flow quantities. Sheet 1 (

**left**) depicts a major clustering versus sheet 2 (

**right**). Major clustering indicates a less turbulent flow. The conditions reported on the right indicate major turbulence.

**Figure 10.**Distribution of predicted velocities on the three sections of the channel: at the beginning (sez1), the middle (sez2), and the end (sez3).

**Figure 11.**Distribution speed variations (

**left**), and sensitivity analysis for U

_{1}acquisitions of Table 2 (

**right**).

**Table 1.**Vectrino ADV technical specifications [14] used in the experiment.

Water Velocity Measurements | |

Maximum profiling range | 0.05 m, 0.01 m (field probe) |

Distance from probe | 6 mm |

Sampling volume diameter | 3 ÷ 15 mm |

Velocity range | ±0.03, 0.1, 0.3, 1, 2.5, 4 m/s (software selectable) |

Accuracy | ±0.5% of measured value ±1 mm/s |

Sampling rate | 1 ÷ 200 Hz |

Echo Intensity | |

Acoustic frequency | 10 MHz |

Resolution | Linear scale |

Dynamic range | 25 dB |

U1 (cm/s) | V1 (cm/s) | W1 (cm/s) | ||||||||||||

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 |

16.9 | 16.9 | 17.3 | 17.7 | 16.3 | 5.10 × 10^{−1} | 5.80 × 10^{−1} | 1.70 × 10^{−1} | 2.80 × 10^{−1} | 4.60 × 10^{−1} | −1.82 × 10^{−1} | −3.20 × 10^{−1} | 1.90 × 10^{−1} | −5.00 × 10^{−1} | 9.10 × 10^{−1} |

17.7 | 17 | 17.9 | 18.0 | 16.4 | 3.00 × 10^{−1} | 3.90 × 10^{−1} | 3.00 × 10^{−1} | 5.00 × 10^{−1} | 3.00 × 10^{−1} | 1.30 × 10 | 2.40 × 10^{−1} | 1.50 × 10^{−1} | −1.90 × 10^{−1} | 1.40 × 10^{−1} |

15.6 | 16 | 16.6 | 17.1 | 16.0 | 6.80 × 10^{−1} | 2.10 × 10^{−1} | 3.80 × 10^{−1} | 9.10 × 10^{−1} | 2.70 × 10^{−1} | 1.10 × 10 | 4.70 × 10^{−1} | −4.00 × 10^{−1} | 3.70 × 10^{−1} | 1.70 × 10^{−1} |

15.5 | 15.4 | 15.4 | 17.0 | 16.3 | 2.00 × 10^{−1} | 1.20 × 10^{−1} | 4.20 × 10^{−1} | 3.10 × 10^{−1} | 0.120 | −5.30 × 10^{−1} | −3.60 × 10^{−1} | 4.00 × 10^{−1} | 4.30 × 10^{−1} | 3.80 × 10^{−1} |

16.5 | 18 | 18 | 17.1 | 14.1 | 1.00 × 10^{−1} | 4.30 × 10^{−1} | 4.30 × 10^{−1} | 6.80 × 10^{−1} | 7.00 × 10^{−2} | 3.50 × 10^{−1} | 6.20 × 10^{−1} | 1.40 × 10^{−1} | 3.70 × 10^{−1} | 1.00 × 10^{−1} |

U2 (cm/s) | V2 (cm/s) | W2 (cm/s) | ||||||||||||

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 |

16.8 | 16.6 | 17.4 | 15.9 | 16.5 | −1.80 × 10^{−2} | −3.10 × 10^{−2} | 2.00 × 10^{−2} | −4.90 × 10^{−2} | 9.00 × 10^{−2} | 2.70 × 10^{−1} | −1.30 × 10^{−1} | 1.30 × 10^{−1} | 5.50 × 10^{−1} | 3.20 × 10^{−2} |

16.4 | 16.3 | 17.0 | 17.6 | 17.7 | 1.00 × 10^{−1} | 2.40 × 10^{−2} | 1.50 × 10^{−2} | −1.90 × 10^{−2} | 1.40 × 10^{−2} | 8.00 × 10^{−3} | 1.90 × 10^{−1} | −2.40 × 10^{−1} | 1.80 × 10^{−1} | −8.70 × 10^{−2} |

16 | 16.6 | 16.5 | 15.5 | 14.9 | 1.20 × 10^{−1} | 4.70 × 10^{−2} | −4.80 × 10^{−2} | 3.30 × 10^{−2} | 1.70 × 10^{−2} | 6.20 × 10^{−2} | −2.70 × 10^{−1} | 2.50 × 10^{−1} | 3.30 × 10^{−1} | 2.30 × 10^{−1} |

13.3 | 13.3 | 13.6 | 13.9 | 14.1 | −5.40 × 10^{−2} | −3.40 × 10^{−2} | 3.80 × 10^{−2} | 4.30 × 10^{−2} | 3.80 × 10^{−2} | 9.60 × 10^{−2} | −8.80 × 10^{−1} | −2.10 × 10^{−1} | −2.60 × 10^{−1} | 1.20 × 10 |

14.4 | 13.6 | 14.7 | 13.5 | 14.7 | 3.50 × 10^{−2} | 6.30 × 10^{−2} | 1.40 × 10^{−2} | 3.70 × 10^{−2} | 1.00 × 10^{−1} | 5.50 × 10^{−1} | 6.60 × 10^{−1} | −2.10 × 10^{−1} | 2.00 × 10 | 9.20 × 10^{−3} |

U3 (cm/s) | V3 (cm/s) | W3 (cm/s) | ||||||||||||

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 |

14.1 | 14.9 | 16.1 | 16.6 | 15.9 | −2.90 × 10^{−3} | 7.80 × 10^{−3} | 2.50 × 10^{−3} | 2.80 × 10^{−3} | −4.40 × 10^{−3} | 3.00 × 10^{−2} | −1.80 × 10^{−2} | 2.50 × 10^{−2} | 1.80 × 10^{−2} | 3.40 × 10^{−2} |

16.2 | 16.5 | 16.3 | 16.1 | 16.6 | 2.50 × 10^{−3} | 2.50 × 10^{−3} | 3.30 × 10^{−3} | 1.60 × 10^{−3} | 6.70 × 10^{−3} | 2.50 × 10^{−2} | 2.50 × 10^{−2} | 3.30 × 10^{−2} | 1.60 × 10^{−2} | 6.70 × 10^{−2} |

14.1 | 16.0 | 17.6 | 15.7 | 15.2 | 7.00 × 10^{−3} | 8.60 × 10^{−3} | −4.70 × 10^{−3} | 6.50 × 10^{−3} | −3.00 × 10^{−4} | 5.00 × 10^{−2} | 8.60 × 10^{−2} | −4.30 × 10^{−2} | −2.50 × 10^{−2} | −1.10 × 10^{−3} |

13.6 | 13.2 | 14.4 | 13.9 | 13.0 | 2.70 × 10^{−3} | 4.40 × 10^{−3} | 6.40 × 10^{−3} | 1.80 × 10^{−3} | 9.90 × 10^{−3} | 3.70 × 10^{−2} | 6.30 × 10^{−2} | 5.40 × 10^{−2} | 1.80 × 10^{−2} | 1.00 × 10^{−3} |

14.8 | 13.1 | 15.3 | 15.4 | 15.5 | 7.80 × 10^{−3} | 8.10 × 10^{−3} | 1.10 × 10^{−2} | 6.30 × 10^{−3} | 2.20 × 10^{−3} | 6.60 × 10^{−2} | 7.00 × 10^{−2} | 1.30 × 10^{−1} | 4.30 × 10^{−2} | 1.20 × 10^{−2} |

u_{2−calculated} (cm/s) | |||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

1 | 16.6 | 16.7 | 17.3 | 17.5 | 16.5 |

2 | 17.4 | 17.6 | 18.1 | 18.5 | 16.7 |

3 | 15.9 | 16.6 | 16.0 | 17.1 | 16.8 |

4 | 14.9 | 15.4 | 15.7 | 15.9 | 16.0 |

5 | 16.2 | 17.1 | 18.9 | 16.9 | 13.3 |

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1 | −0.01 | 0.01 | −0.01 | 0.10 | 0 |

2 | 0.06 | 0.08 | 0.06 | 0.05 | −0.06 |

3 | −0.01 | 0 | −0.03 | 0.10 | 0.13 |

4 | 0.12 | 0.16 | 0.15 | 0.14 | 0.13 |

5 | 0.13 | 0.26 | 0.29 | 0.25 | −0.10 |

${\mathit{S}}_{{\mathit{u}}_{\mathbf{1}}}$ (cm/s) | |||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

1 | 0.020 | 0.021 | 0.021 | 0.021 | 0.021 |

2 | 0.022 | 0.020 | 0.021 | 0.019 | 0.019 |

3 | 0.030 | 0.022 | 0.022 | 0.022 | 0.024 |

4 | 0.029 | 0.034 | 0.037 | 0.032 | 0.040 |

5 | 0.024 | 0.017 | 0.019 | 0.023 | 0.030 |

**Table 6.**Computed ${u}_{j,k}$ component in the middle plane and its conservative uncertainty (rounded to the 2nd decimal digit).

1 | 2 | 3 | 4 | 5 | ± | 1 | 2 | 3 | 4 | 5 | cm/s | ||

1 | 16.6 | 16.7 | 17.3 | 17.5 | 16.5 | 1 | 0.19 | 0.19 | 0.19 | 0.19 | 0.19 | ||

2 | 17.4 | 17.6 | 18.1 | 18.5 | 16.7 | 2 | 0.19 | 0.19 | 0.19 | 0.20 | 0.19 | ||

3 | 15.9 | 16.6 | 16.0 | 17.1 | 16.8 | 3 | 0.19 | 0.19 | 0.19 | 0.19 | 0.19 | ||

4 | 14.9 | 15.4 | 15.7 | 15.9 | 16.0 | 4 | 0.18 | 0.19 | 0.19 | 0.19 | 0.20 | ||

5 | 16.2 | 17.1 | 18.9 | 16.9 | 13.3 | 5 | 0.19 | 0.19 | 0.20 | 0.19 | 0.18 |

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**MDPI and ACS Style**

Fabbiano, L.; Oresta, P.; Lay-Ekuakille, A.; Vacca, G.
About 3D Incompressible Flow Reconstruction from 2D Flow Field Measurements. *Sensors* **2022**, *22*, 958.
https://doi.org/10.3390/s22030958

**AMA Style**

Fabbiano L, Oresta P, Lay-Ekuakille A, Vacca G.
About 3D Incompressible Flow Reconstruction from 2D Flow Field Measurements. *Sensors*. 2022; 22(3):958.
https://doi.org/10.3390/s22030958

**Chicago/Turabian Style**

Fabbiano, Laura, Paolo Oresta, Aimé Lay-Ekuakille, and Gaetano Vacca.
2022. "About 3D Incompressible Flow Reconstruction from 2D Flow Field Measurements" *Sensors* 22, no. 3: 958.
https://doi.org/10.3390/s22030958