Dynamics of establishment of selective withdrawal of a stratified fluid from a line sink. Part 1. Theory

Abstract

An investigation is made of selective withdrawal of a linear stratified fluid from a line sink in a channel of depth d. The flow is characterized by a densimetric Froude number F = Q/Nd2, where Q is the discharge per unit width and N is the Väisälä frequency. The dynamics of establishment of flow are investigated theoretically. Analytic results are obtained from a linearized theory based on a systematic perturbation scheme for small values of F. These results lead to a proper identification of the successive arrival of ‘columnar disturbance modes’ as the mechanism responsible for the development of flow concentration in the withdrawal region. Viscous and diffusive effects are also examined. For larger times and higher discharges (higher values of F), nonlinear effects become important, and the full Navier-Stokes equations are now solved numerically by a finite-difference procedure in a ‘stretched’ co-ordinate system. The solutions indicate that the establishment of the steady flow field is due to the successive arrival and interaction of ‘columnar disturbance modes’. Steady-state solutions are also presented, and a similarity profile is obtained. Comparison of the theoretical findings with experimental results are presented in part 2. Agreement with experimental measurements is found to be excellent.