Numerical investigation of the reservoir filling with a Newtonian fluid using the VoF-method

In this paper, the filling of a plane reservoir, having a central rod, with a fluid under gravity is studied. The flow is described by the Navier-Stokes and continuity equations. On the solid walls, the no-slip boundary conditions are satisfied. The fluid with a fixed flow rate enters the reservoir through the channel on the top. The problem is solved numerically using the control volume method and SIMPLE procedure. The free surface location is determined using the PLIC-VOF method. Mathematical formulation of the problem includes two non-dimensional parameters, namely, the Reynolds number (Re) and the ratio of the gravity forces to the viscous forces in a fluid (W).
As a result, four regimes of the reservoir filling have been discovered. When viscous effects dominate over gravitational effects, the regime characterized by draining all-over the lateral grooves is observed, wherein the free surface completely overlaps the lateral grooves. An increase in the gravitational effects leads to the regime characterized by draining down the central rod. In this case, the fluid flows along the central rod walls, then reaches the bottom, and fills in the rest part of the reservoir. Further increase in the inertial effects gives rise to the regime characterized by draining as a jet, and, subsequently, to the regime of draining down the lateral grooves. The kinematic characteristics of the flows are presented. It is shown that in the case of draining all-over the lateral grooves, one-dimensional flow occurs in the lateral grooves and two-dimensional flow occurs in the vicinity of the central rod. In other regimes, dead zones are formed in the vicinity of the bottom. The behavior of the fluid mass distribution over the reservoir is studied. In the regime of draining fluid all-over the lateral grooves, a fountain flow is observed, and in other regimes, the portions of the fluid are distributed in parallel layers. The deformation of the fluid portions during the filling is analyzed. The diagrams illustrating the reservoir filling regimes are presented at various values of Reynolds number and parameter W.