The theoretical/simulational research focuses on liquid and granular jets that impact at a free surface. For liquid flow, the research will examine the jet formed upon collapse of a cavity. The work will focus on the generic situation where the dynamics is not axisymmetric but instead fully three-dimensional. For granular flow, the research will focus on the ejecta sheet formed when a densely packed granular jet collides with a solid target.
Intellectual Merit: In both problems, numerous small-scale deviations are present in the initial state giving rise to jetting. In jetting from a collapsing cavity, recent studies revealed that the shape of the interface which first forms into a jet is generically fully three-dimensional, characterized by strong azimuthal distortions. In granular impact, each non-cohesive grain in the jet collides with the target and introduces a small disturbance to the mean motion. Intuitively one would expect that these deviations would prevent jetting and create a disordered response. The strongly distorted cavity should break up into a shower of air bubbles instead of forming a strong central jet. The granular jet, upon colliding with the target, should break up into a diffuse spray instead of forming a thin ejecta sheet with a well-defined ejection angle. The counter-intuitive outcomes in both cases connect them to a central puzzle in fluid dynamics. Under some circumstances, strong forcing appears to act in such a way as to organize the dynamics, producing a coherent structure (a central jet or a thin ejecta sheet) out of a disordered background. Insights from research into these two particular examples, which are novel, conceptually simple, and experimentally accessible can potentially transform our understanding of wide class of phenomena.
Broader Impacts: Collapse-induced jets are believed to cause erosion of turbines in submarines, failure in steam turbines as well as wave-damage on shore breaks. Spray and atomization technologies rely on the formation of thin, fast moving jets that break-up into many droplets. Microfluidic devices that rely on collapse-induced jets to cleave bubbles with precision are being developed. Snow avalanches, or rock falls, propagating at high speeds can end up ejecting a great deal of materials upwards when they encounter an obstacle. On a slower scale, the craters formed by raindrop impact in a heavy downpour can be an important mechanism for soil erosion. Finally, many drugs and food products are processed as streams of particles. The proposed research also provides an excellent education for graduate students. It combines close collaboration between theory and experiment. Undergraduates will be recruited to work on both the experimental and simulation aspects of the project as part of the University of Chicago?s Research Experience for Undergraduates (REU) program. Results from the research will also be incorporated into the curriculum of two elective courses (one undergraduate / one graduate) that the PI has developed.
Key outcomes from research sponsored by CBET divide into two sections. First, our study of underwater cavity collapse has shed new light on the nonlinear dynamics near pinch-off. Scientists' understanding of how nonlinearity organizes continuous fields in motion are informed by two processes: nonlinear resonance and singularity formation. Nonlinear resonance generates a wealth of complex motion from a few, initially simple ingredients. In contrast, singularity formation often causes an initially complicated state to evolve into a simple form. As stresses in the neighborhood of the singularity diverge, the dynamics becomes dominated by the presence of the singularity. In the most extreme cases, the singularity dynamics approaches a universal form, one independent of initial or boundary conditions. At first sight, these two processes seem confined to mutually exclusive regimes. The full effect of nonlinear resonance requires many iterations and a spatially-extended system. Finite-time singularity formation occurs when the characteristic length- and time-scales go to 0. However, there are many examples where nonlinear resonance gives rise to the formation of a singularity. In wave breaking, the generation and amplification of higher harmonics by nonlinear resonance among the different modes are so efficient that a finite-time singularity results. The onset of a period-doubling cascade as a nonlinear system's parameter is tuned towards a critical value, such as occurs in Rayleigh-Benard convection or in the dripping of water drops from a faucet, are other examples. In contrast, few phenomena corresponding to the obverse scenario, singularity formation being perturbed by, or even cut-off, by nonlinear resonance, have been identified. This study presents and analyzes evidence from experiments by Keim and Nagel at the University of Chicago in collaboration with Fezzaa at the X-Ray Science Division of the Argonne Laboratory, theory and simulation by Lai and Zhang (PI CBET support) that the familiar phenomenon of a bubble breaking into several bubbles while underwater provides such an example. Specifically the researchers use high-speed X-ray imaging to fully visualize the three-dimensional evolution that characterizes the final stage of underwater bubble break-up and its aftermath. Their results demonstrate that the nonlinear distortions from shape vibrations indeed become important in the final stage of break-up. High-speed visible-light photography studies are not able to provide information about the surface evolution in this regime because the air-water surface becomes re-entrant, thus obscuring key features. They also used weakly nonlinear analysis and boundary integral simulations to assess how the final dynamics varies as a function of the initial perturbation. The nonlinear resonance between an initial perturbation and the first higher harmonic it generates gives rise to two different types of break-up modes: a coalescene-like mode dominated by the initial perturbation and a strongly nonlinear evolution resulting in nearly singular cusps at the air-water interface. A numerical/theoretical analysis has appeared in Physics of Fluids (24, 102106, 2012) while a full study comparing numerics, theory with experiments is currently under review at the Journal of Fluid Mechanics. The second key outcome analyzes granular jet impact. The impact of two colliding objects is the rudimentary process that underlies splashing and coalescence at the human-size scale, as well as cratering and even planet formation on the celestial scale. Impact leads to catastrophic deformation as theincoming objects distort and change shape. Similar impact geometry has also been used to create the quark-gluon plasma, the primordial constituents of the universe, in high energy collisions in accelerators. The seemingly complicated physics of impact however can sometimes lead to elegant results that, while counter-intuitive, can be simply understood as we demonstrate in the present study. When a high-density jet of granular particles strikes a fixed target, the beam is ejected into a highly collimated cone with a well-defined opening angle reminiscent of the water bells in fountains created when a water jet hits a target. This conical structure is robust even though, in the case of the granular jet, the collisions are highly dissipative while, in the case of water, the liquid behaves like a nearly perfect fluid. Thus, the scattering from these dense jets imparts little information about the internal dynamics in the collision region. However, the similarity between these two cases provides a decisive clue about what physics controls this impact phenomenon. Both processes are controlled by the limiting scenario of perfect fluid jet impact. In this limit, the ejecta pattern is robust and changes little in response to internal structure. A short report describing these results has appeared in Physical Review Letters (111, 168001, 2013). Disclaimer: This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
