Linear Free Surface of Steady and Unsteady Flow Theoretical Study

Abstract

This study investigates the relationship between the crucial cone point in limiting steady and unsteady simulation flows inside a simple two-dimensional, vertical sand column subjected to water withdrawal. The analysis employs a versatile technique applicable to both linear steady and linear unsteady problems. By utilizing Fourier series techniques, the method finds the solution for a line sink in a vertical duct and extends the steady technique to achieve unsteady solutions. The evolution of the beneath-the-water table surface towards steady states is monitored when they exist. The key findings highlight that as the impermeable base moves further away, the distance between the interface and the sink increases. With increasing flow rates, the critical depth elevates; however, a minor decrease in the critical flow rate is observed when the sink approaches the base. The prior behaviour of the flow impacts the critical flow rate value for each interface height. The study provides steady and unsteady solutions, employing linearized formulations, which demonstrate good agreement with observed phenomena. The length scale of the spatial separation between the interface and the sink emerges as a critical factor in determining cone. These findings contribute to a better understanding of cone phenomena and emphasize the significance of considering specific dimensions in the analysis of flow dynamics within porous media.