Journal of Fluid Mechanics

Papers

Steady free surface flows induced by a submerged ring source or sink

T. E. Stokesa1, G. C. Hockinga2 c1 and L. K. Forbesa3

a1 Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand

a2 Mathematics and Statistics, Murdoch University, Perth, WA 6150, Australia

a3 School of Mathematics and Physics, University of Tasmania, Hobart, TAS 7001, Australia

Abstract

The steady axisymmetric flow induced by a ring sink (or source) submerged in an unbounded inviscid fluid is computed and the resulting deformation of the free surface is obtained. Solutions are obtained analytically in the limit of small Froude number (and hence small surface deformation) and numerically for the full nonlinear problem. The small Froude number solutions are found to have the property that if the non-dimensional radius of the ring sink is less than $\rho = \sqrt{2} $, there is a central stagnation point on the surface surrounded by a dip which rises to the stagnation level in the far distance. However, as the radius of the ring sink increases beyond $\rho = \sqrt{2} $, a surface stagnation ring forms and moves outward as the ring sink radius increases. It is also shown that as the radius of the sink increases, the solutions in the vicinity of the ring sink/source change continuously from those due to a point sink/source ($\rho = 0$) to those due to a line sink/source ($\rho \ensuremath{\rightarrow} \infty $). These properties are confirmed by the numerical solutions to the full nonlinear equations for finite Froude numbers. At small values of the Froude number and sink or source radius, the nonlinear solutions look like the approximate solutions, but as the flow rate increases a limiting maximum Froude number solution with a secondary stagnation ring is obtained. At large values of sink or source radius, however, this ring does not form and there is no obvious physical reason for the limit on solutions. The maximum Froude numbers at which steady solutions exist for each radius are computed.

(Received January 07 2011)

(Reviewed July 29 2011)

(Accepted December 14 2011)

(Online publication January 24 2012)

Key Words:

  • critical layers;
  • surface gravity waves;
  • wave breaking

Correspondence:

c1 Email address for correspondence: G.Hocking@murdoch.edu.au

Metrics