Analytical solution for supercritical flow in rectangular channel contraction

Abstract

Analysis of the supercritical flow pattern in nonprismatic open channels is an important issue in engineering practice and often a challenging question principally due to the oblique standing wave structure. In this paper, an original analytical solution for supercritical free-surface flow in rectangular channel contraction is proposed. The two-dimensional steady depth-averaged shallow water equations are simplified to an asymptotic one-dimensional (1D) unsteady flow problem in the transverse direction. A first-order approximation is then obtained using small perturbation theory, and the 1D asymptotic model is solved analytically using the Laplace integral transformation. The two-dimensional flow field solution is obtained following a compilation of sequence planes in the time domain. Free-surface profiles along the side wall and the channel axis were compared against experimental data. For this, laboratory tests were performed in addition to a comparison with classical experimental results available in the literature. Within the limits underlying the model, good agreement in water surface profiles is obtained by comparing experimental and theoretical predictions. Principal flow features, such as wave amplitude and positions, are well captured, which makes the solution model of interest for the design and analysis of contracted open channels. HIGHLIGHTS An asymptotical quasi-2D mathematical model and analytical solution is proposed.; 2D steady shallow water model is converted to 1D unsteady transverse flow problem.; Mathematical model is solved analytically using Laplace transformation.; Predicted water surface profiles are in satisfactory agreement with experimental data.;

Keywords

• analytical solution
• channel contraction
• chute spillway
• laplace transform
• standing waves
• supercritical flow