No abstract

We explore the dynamics of turbulent bubble fountains produced when a descending stream of fresh water and air bubbles issues from a nozzle submerged in a tank of water. The bubbles have diameters of 2 to 5 mm and the fountains have source Froude numbers ranging from 10 to 240. The Reynolds numbers of the bubbly fountains range from 4000 to 24 000. The bubbles, carried into the tank by the downward jet of water, lead to a buoyancy force which reduces the downward momentum of the jet, thus producing a fountain. We find that $H_{F}$, the downward penetration distance of the bubbles into the water reservoir, may be characterised by two parameters: $\unicode[STIX]{x1D6EC}$, the ratio of the bubble rise speed to the characteristic fountain velocity, $u_{F}=f_{0}^{1/2}/m_{0}^{1/4}$, and $Fr_{0}$, the source Froude number, given by $m_{0}^{5/4}/(q_{W_{0}}f_{0}^{1/2})$, where $q_{W_{0}}$, $m_{0}$ and $f_{0}$ are the source volume, momentum and buoyancy fluxes. As $\unicode[STIX]{x1D6EC}$ increases, $H_{F}$ decreases, a result which is directly analogous to the height of rise of particles in a particle-laden fountain (Mingotti & Woods, J. Fluid Mech., vol. 793, 2016, R1). Also, we find that $H_{F}$ increases as $Fr_{0}$ increases, a result directly analogous to single-phase fountains (Turner, J. Fluid Mech., vol. 26, 1966, pp. 779–792). We present a model for the conservation of volume, momentum and buoyancy fluxes and use this to predict the penetration distance of the bubbles corresponding to that point at which the fountain liquid velocity equals the bubble rise speed. Using the best-fit value for the entrainment coefficient, $\unicode[STIX]{x1D6FC}=0.04\pm 0.004$, we find that our experimental measurements of the bubble penetration distance are in good accord with the model predictions for $10<Fr_{0}<240$ and $2<\unicode[STIX]{x1D6EC}<15$. In our experiments the bubble rise speed, $u_{slip}$, is large compared to the entrainment velocity of the descending fountain. Thus, only a small fraction of the rising bubbles are re-entrained, and so the buoyancy flux of the fountain is approximately independent of depth. Flow-visualisation experiments also show that the liquid momentum flux is not exhausted at the point of bubble separation and so the liquid in the fountain continues to travel downward, separated from the bubbles. We use the new theoretical model to estimate the flux of air entrained into plunging water jets.

We present new experiments and theoretical models of the motion of relatively dense particles carried upwards by a liquid jet into a laterally confined space filled with the same liquid. The incoming jet is negatively buoyant and rises to a finite height, at which the dense mixture of liquid and particles, diluted by the entrainment of ambient liquid, falls back to the floor. The mixture further dilutes during the collapse and then spreads out across the floor and supplies an up-flow outside the fountain equal to the source volume flux plus the total entrained volume flux. The fate of the particles depends on the particle fall speed, $u_{fall}$ , compared to (i) the characteristic fountain velocity in the fountain core, $u_{F}$ , (ii) the maximum upward velocity in the ambient fluid outside the fountain, $u_{u}(0)$ , which occurs at the base of the fountain, and (iii) the upward velocity in the ambient fluid above the top of the fountain associated with the original volume flux in the liquid jet, $u_{BG}$ . From this comparison we identify four regimes. (I) If $u_{fall}>u_{F}$ , then the particles separate from the fountain and settle on the floor. (II) If $u_{F}>u_{fall}>u_{u}(0)$ , the particles are carried to the top of the fountain but then settle as the collapsing flow around the fountain spreads out across the floor; we do not observe particle suspension in the background flow. (III) For $u_{u}(0)>u_{fall}>u_{BG}$ we observe a particle-laden layer outside the fountain which extends from the floor of the tank to a point below the top of the fountain. The density of this lower particle-laden layer equals the density of the collapsing fountain fluid as it passes downwards through this interface. The collapsing fluid then spreads out horizontally through the depth of this particle-laden layer, instead of continuing downwards around the rising fountain. In the lower layer, the negatively buoyant source fluid in fact rises as a negatively buoyant jet, but this transitions into a fountain above the upper interface of the particle-laden layer. The presence of the particles in the lower layer reduces the density difference between fountain and environment, leading to an increase in the fountain height. (IV) If $u_{fall}<u_{BG}$ then an ascending front of particles rises above the fountain and eventually fills the entire tank up to the level where fluid is removed from the tank. We compare the results of a series of new laboratory experiments with simple theoretical investigations for each case, and discuss the relevance of our results.

The radial size convergence of the E×B staircase pattern is addressed in local gradient-driven flux tube simulations of ion temperature gradient (ITG)-driven turbulence. It is shown that a mesoscale pattern size of ∼57–76 ρ is inherent to ITG-driven turbulence with Cyclone Base Case parameters in the local limit.