Oscillation in Height of a Negatively Buoyant Jet

Abstract

Jets and fountains with reversing buoyancy occur in numerous natural and man-made situations (1) including, for example, oxygen jets directed into molten metal furnaces (2), the overshoot stage of a smokestack discharging into a stratified atmosphere (3), and underwater volcanic eruptions (4). For simplicity, we assume an upward directed jet or fountain with downward buoyancy as shown in Fig. 1, although the results apply equally to downward directed jets and fountains with upward buoyancy. The flow structure consists of a rising central core, surrounded by an annular downward flow (5-6), and under appropriate conditions includes mixing (7-8) and phase mingling (9-10). The primary parameter governing this flow structure is the Richardson number (11), which is the ratio of negative buoyancy to inertial forcesRi=DgΔρρjU2where D is the vent diameter, ρj is the jet fluid density, and Δρ is the difference in density between the jet and surrounding fluids. For turbulent flows, the characteristic jet velocity is the volumetric flow rate divided by the cross-sectional area (U=UAve). For laminar flows, which have a higher total momentum for a similar average velocity, the characteristic velocity is the root-mean-square velocity (U=2UAve)(6). Apart from predicting the onset of turbulence, the Reynolds number has no effect. Regardless of the fluid pair selected, the maximum height (h—see Fig. 1) collapses to a single curve [h∕D=f(Ri)](6), and the flow transitions through distinct regimes, which are a function of Ri (11). At Ri>1, negative buoyancy dominates and the jet forms a shallow and stable penetration into the fluid above [Fig. 2], characterized by a nearly constant height with only slight fluctuations that result from waves on the interface. At Ri<1, the flow oscillates in height in a cycle that consists of the formation of a tall, steep, walled fountain [Fig. 2] followed by an asymmetric collapse [Figs. 2, 2]. Through dimensional analysis, the frequency of this oscillation is predicted to follow a functional relationship for Strouhal numberfDU=f(DgΔρρU2,ρUDμ)orSt=f(Ri,Re)As shown below, the effect of the Reynolds number is again limited to establishing the onset of turbulence and determining the appropriate characteristic velocity.