Transient impact of a liquid column on a miscible liquid surface

B. Kersten, C. D. Ohl, A. Prosperetti
2003 Physics of Fluids  
The flow induced by a liquid column falling on an undisturbed liquid surface is studied with the aid of a high-speed camera. The falling liquid spreads over the receiving liquid forming a cavity which eventually pinches off due to the action of gravity. It is only at this point that the normal flow pattern in which the impacting liquid penetrates below the free surface is established. The same process-at a scale smaller by four orders of magnitude-is encountered in the jetting behavior of
more » ... sing cavitation bubbles. It is also observed that the cavity dynamics is strikingly similar to that found when a disturbance is induced on a steady jet falling on a liquid. This observation supports a generic mechanism for air entrainment hypothesized in an earlier paper. At sufficiently large Reynolds number, the streamlines of a jet impinging steadily on the surface of a pool of a miscible liquid separate from the free surface and ͑aside from the effect of well-known mixing and instability mechanisms͒ penetrate into the pool in the same general direction as the jet. This flow pattern is quite different from that predicted by the inviscid theory, according to which the jet fluid would spread over the pool fluid without penetrating it. The latter behavior is in fact observed in the case of a drop: it has been known at least since the classic work of Worthington, 1 that the drop liquid initially spreads along the surface, and only later penetrates below it and mixes with the receiving liquid. In this Brief Communication we show that, for a sufficiently high-speed jet, the reconciliation of the irrotational flow prediction and the observed steady flow is intimately related to the mechanism by which the jet entrains air when it first strikes the free surface. Furthermore, the present results confirm a generic mechanism for air entrainment in free-surface flow recently postulated in connection with a somewhat different experiment. 2 We carried out a simple experiment in which a 4-mmdiam column of water falls vertically on a quiescent water surface in a small tank (heightϫwidthϫbreadthϭ0.19 ϫ0.19ϫ0.29 m 3 ). A glass tube ͑0.2 m in length with a 4 mm inner diameter͒ is filled with water and the upper opening of the tube closed with the thumb. The diameter is small enough that surface tension stabilizes the liquid surface at the tube exit and prevents the water from falling. The tube is positioned vertically above the pool surface, the thumb lifted, and the liquid column falls onto the pool. The ensuing process is observed with a high-speed camera ͑Imager CR 1000, Roper Scientific͒ at a frame rate of 1000 fps under diffusive backillumination. By comparing sequences taken under nominally identical conditions it was found that, in spite of the simplicity of the procedure, the flow generated was quite reproducible. In some of the experiments, the water was colored with blue ink. Figure 1 shows a typical sequence of events. The liquid column enters the pool and generates a cavity which then collapses from the sides entrapping a toroidal bubble. Here the Froude number, defined by FrϭU 2 /gd ͑U impact velocity of the water column, d tube diameter, g acceleration of gravity͒, is 36.7; the Weber WeϭdU 2 / ͑ liquid density, surface tension coefficient͒ and Reynolds number Re ϭdU/ ͑ viscosity coefficient͒ are 79 and 4800, respectively. A sequence taken in very similar conditions (Fr ϭ39.2, Weϭ84, Reϭ4960), but with the addition of ink to the water, is shown in Fig. 2. Here the cavity appears thicker. It is revealing to look at the images obtained by subtracting pixel by pixel the digitized gray levels of the two sets of images ͑Fig. 3͒. These processed images clearly show that the dyed fluid remains attached to the surface of the cavity, as predicted by irrotational flow theory. The results of a similar subtraction in which the dyed fluid images were obtained with Frϭ34.9, Weϭ75, Re ϭ4680 are shown in Fig. 4 . It is seen here that, after the toroidal bubble at the bottom of the cavity pinches off and the whole cavity collapses against the jet, the separated flow pattern described at the beginning and associated with a steady jet flow is established. We have observed the same process every time the impact velocity of the liquid column was large enough to generate a cavity and entrain air. In our experiments the threshold for this occurrence corresponded to a Froude number of about 10. At lower Froude numbers the entrapped bubble was very small or nonexistent and the stagnation pressure insufficient to deflect the incoming liquid stream. A sequence with Frϭ3.2, Weϭ6.8, and Reϭ1412 is shown in Fig. 5 . Reference 3 presented the results of a similar experiment, which differed in that the mass of falling water was much greater although the Froude number was in a comparable range. In that study it was argued that, for sufficiently high Froude numbers Fr * ͑based on the free-fall velocity U * a͒ Permanent address:
doi:10.1063/1.1542614 fatcat:3elayvsdbrcebhffdlpibeytde