A model for measuring the velocity vector of bubbles and the pierced position vector in breaking waves using four-tip optical fiber probe, Part I: Computational method

A dual-tip optical fiber probe has been attempted to make measurement of bubble size in breaking waves, especially for the violent and aerated bubble plumes presented immediately after breaking in laboratory and field. However, the performance of field measurement based on such probe is not satisfactory due to the measurement error produced by the large numbers of escaped and missing bubbles. The objective of the present study and its companion are to evaluate the errors of bubble size measurement in breaking waves using a dual-tip optical fiber probe, and the errors were grouped as oncoming bubble errors, receding bubble errors and missing bubble errors in terms of the bubble motion relative to the probe. The study on errors shows that the incomplete velocity vector of bubble and missing pierced position vector where the tips contact the bubble are the major reasons for errors. In order to make error correction, this paper proposes a four-tip optical fiber probe to obtain these two vectors and other required parameters. The mathematical model of the recommended probe is derived, which enables the bubble velocity components in the x, y and z directions and the pierced position vector is the unique functions of several time intervals and probe's spatial distribution relationship. The optimized algorithm is suggested to calculate the velocity vector, the pierced position vector and other required parameters from time intervals taken from the output signals of the proposed probe. The model and algorithm claim that the velocity vector of bubbles and pierced position vector can be obtained as long as the time intervals can be captured by four-tip optical fiber probe. Verification and validation of the model is presented in the accompanying paper.