# Estimating void fraction in a hydraulic jump by measurements of pixel intensity

## Abstract

A hydraulic jump is a sudden transition from supercritical to subcritical flow. It is characterized by a highly turbulent roller region with a bubbly two-phase flow structure. The present study aims to estimate the void fraction in a hydraulic jump using a flow visualization technique. The assumption that the void fraction in a hydraulic jump could be estimated based on images’ pixel intensity was first proposed by Mossa and Tolve (J Fluids Eng 120:160–165, 1998). While Mossa and Tolve (J Fluids Eng 120:160–165, 1998) obtained vertically averaged air concentration values along the hydraulic jump, herein we propose a new visualization technique that provides air concentration values in a vertical 2-D matrix covering the whole area of the jump roller. The results obtained are found to be consistent with new measurements using a dual-tip conductivity probe and show that the image processing procedure (IPP) can be a powerful tool to complement intrusive probe measurements. Advantages of the new IPP include the ability to determine instantaneous and average void fractions simultaneously at different locations along the hydraulic jump without perturbing the flow, although it is acknowledged that the results are likely to be more representative in the vicinity of sidewall than at the center of the flume.

## Keywords

Void Fraction Pixel Intensity Hydraulic Jump Conductivity Probe Image Editing## Abbreviations

*a*,*b*Parameters of the Fuzzy logic S function

- C
Void fraction defined as the volume of air per unit volume of air and water; it is also called air concentration

*AvPI*Averaged pixel intensity matrix (pi)

*AvPIt*Time average matrix (pi)

*Fr*Froude number

- g
Acceleration due to gravity: g = 9.81 m

^{2}/s*h*_{1}Upstream water depth (m)

*i*,*j*Matrix indexes

*I*_{T1},*I*_{T2},*I*_{T3}Thresholding functions

*limS*Water surface upper limit (i)

*limSt*Water surface lower limit (i)

*lmf*Fuzzy logic linear function

*n*,*m*Matrix dimensions

*p*,*q*Factors of

*m*and*n*- pi
Pixel intensity defined as a single point in a gray scale image; pi = 0 for a black pixel and pi = 255 for a white pixel

- \( PI_{i,j} \), \( PI_{i,j}^{1 \ldots 4} \)
Matrices of pixel intensity (pi)

- \( PI_{i,j}^{f} \)
Transformed matrix (pi)

*Ptr*,*Ptr*2Threshold values (pi)

*Q*Flow rate (l/s)

*Re*Reynolds number

*RPI*Resized pixel intensity matrix (pi)

- \( RPI(:,:)_{i,j} \)
(i, j)th submatrix of RPI (pi)

*Smf*Fuzzy logic S function

*U*_{1}Upstream mean velocity (m/s)

*x*_{1}Horizontal distance between the gate and the jump toe (m)

*x*−*x*_{1}Horizontal distance between the jump toe and the conductivity probe (m)

*x, y*Horizontal and vertical coordinates (m)

*y*_{1}*, y*_{2}Parameters of the Fuzzy logic linear function

*Y*_{90}Characteristic depth (m) where air concentration is 90%

*We*Weber number

## Greek symbols

- Δ
*x*, Δ*y* Horizontal and vertical grid resolution (pi)

*δ*Boundary layer thickness (m)

- Ø
Diameter (m)

## Notes

### Acknowledgments

The first and second authors acknowledge the support of the Foundation for Science and Technology, the *Operacional Temático Factores de Competitividade* (COMPETE) program, and the *Fundo Europeu de Desenvolvimento Regional* (FEDER) through project PTDC/AAC-AMB/101197/2008. The authors wish to acknowledge the anonymous reviewers for their helpful comments.

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