Propagating Holmboe waves at the interface between two immiscible fluids

Abstract

The instability of the surface of discontinuity separating two streams of immiscible constant-density fluids is studied experimentally and theoretically near onset when surface tension effects are significant. Following Thorpe's original idea, a tube filled with two immiscible fluids is tilted at an angle and returned to its horizontal position to produce a nearly constant velocity difference between both streams that can be varied continuously across threshold. In order to control the wavenumber near onset, the flow is spatially forced by periodically distributing small obstacles on the upper side of the tank. When the kinematic viscosities of each fluid are nearly equal, ones observes two counter-propagating waves of equal amplitude, which cannot be explained from a vortex sheet model. A linear stability analysis of a density discontinuity embedded within a piecewise-linear velocity profile demonstrates that such waves are Holmboe modes associated with the diffusive layers above and below the interface. Good agreement is obtained between the measured and predicted values of the critical velocity difference, propagation velocity and growth factors of the waves. The instability analysis of the asymmetric velocity profile reveals that the breaking of reflectional symmetry gives rise to a single propagating wave near onset. When the kinematic viscosities of each fluid differ, the first destabilized wave is observed to propagate in the same direction as the less-viscous fluid, in agreement with the theoretical results, and the dominant direction of propagation can be manipulated by adjusting the viscosities accordingly.