Seeing through water: diffuse image-based depth measurements in three-dimensional dam-break flows

TL;DR

This work addresses the challenge of measuring water depth in fully three‑dimensional dam‑break flows by implementing a large, diffusely illuminated facility and an absorption‑based image‑depth method. Depth is inferred from transmitted light through dyed water using a bi‑exponential relation $g_n = C_1 e^{-c_1 l_p} + C_2 e^{-c_2 l_p}$, with in‑situ calibration yielding $C_1=0.38$, $c_1=0.12$, $C_2=0.51$, $c_2=0.016$ and an optical path $h_o = l_p/2$ after refraction correction. The breach discharge is modeled as $Q(H) = \frac{2}{3} b C_c \sqrt{2g}\, H^{3/2}$, enabling explicit time dependencies for $H(t)$, $Q(t)$, and $V(t)$, and the imaging volumes are validated against ultrasonic sensor data and a simple emptying model. The approach yields consistent depth fields $h(x,y,t)$ and total imaged volumes $V^c$ in good agreement with theory, establishing a robust framework for high‑fidelity validation of three‑dimensional dam‑break simulations and offering a pathway to broader applications in rapidly varying shallow flows.

Abstract

In this work we present a dedicated experimental facility and an image-based method for measuring water depth in a radially spreading dam-break wave propagating over a horizontal plane. The facility consists of a prismatic reservoir containing a known volume of water dyed with a soluble colorant and equipped with a removable vertical breach whose geometry can be varied, and a 6.4 m x 3.4 m plane that can be inclined from 0° to 30°. The plane is enclosed within a light box providing highly uniform illumination through an array of 60 LED floodlights. Wave propagation is captured by two scientific CMOS cameras mounted on the ceiling of the light box, which record the spatial and temporal evolution of the dye-induced color intensity associated with the advancing water layer. Preliminary dry calibration tests were conducted to assess the spectral compatibility between the broadband white-LED emission, the CMOS sensor sensitivity, and the absorption properties of several dyes at different concentrations. This analysis identified the dye providing the highest attenuation within the effective spectral band of the imaging system, ensuring sensitivity to very small optical path lengths. Based on this characterization, a bi-exponential model is introduced to relate the normalized gray level to the optical path length. A series of dam-break experiments with five initial reservoir levels was performed to assess statistical repeatability. The high consistency observed across the repeated tests confirms the robustness of the measurement procedure. The validity of the reconstructed depth fields is further supported by independent estimates of the water volume released from the reservoir, obtained from an array of ultrasonic level sensors and from a calibrated analytical emptying model. Together, these comparisons confirm the reliability and accuracy of the proposed methodology.

Figures (12)

• Figure 1: The experimental setup for three-dimensional dam-break problem. Panel a) picture of the back side with the reservoir, b) picture from the inside on the light box from the downstream end of the plane, c) picture of the whole facility from the top-upstream corner and d) longitudinal (on the left) and transverse (on the right) sections of the whole experimental facility with the most meaningful dimensions and cameras location.
• Figure 2: A schematic of the instrument setup. a) Side view with the dam released, b) captured dam-break wave with $H_o=0.30 \ m$ at $t=1.6s$, c) top view with the positions of the measuring instrumentation and reference system and d) illustration of the tilting kinematic.
• Figure 3: Results of spectral investigation on light attenuation for three food grade dyes as function of light wavelength $\lambda$. On column from left to right: yellow, blue and green dyes. On rows from top to bottom: spectral light intensity $i$ as $\mathrm{W/m^2/nm}$ (left axis) for various attenuation path lengths $l_p$ (color lines) and CMOS camera sensors' sensitivity $QE$ (dashed lines, values on right axis), best fit attenuation coefficient $a(\lambda)$ from Eq. \ref{['eq:beer-lambert_law']}, coefficient of determination $R^2(\lambda)$ of the fit.
• Figure 4: On panel a) imaged normalized gray level $g_{n,m}$ along with attenuation path length $l_p$ for four food grade dye (green, blue, red, yellow). On panel b) $g_{n,m}$ along with the same quantity $g_{n,t}$ modeled with Beer Lambert law Eq. \ref{['eq:beer-lambert_law']}. On Panel c) measured $g_{n,m}$ and modeled $g_{n,t}$ with here proposed bi-exponential model \ref{['eq:h_gray_tfun']}. On panels b) and c) solid black lines represent perfect agreement with models.
• Figure 5: Examples of imaged gray level field $G$. On panel a) background gray level, on panels b) and c) two water depths $h_v= 13.7, \, 44.6 \, \mathrm{mm}$ respectively. The reference patch is visible in images' top corner.