Consider a turbulent fluid such as oil pumped through a pipeline with a rough interior wall, a river flowing between its irregular banks, or a weather front crossing mountainous terrain. For economic reasons, for storm prediction, and for public safety it is important to understand how turbulent friction converts some of the flow energy into heat. This project is aimed at understanding this phenomenon. Turbulence is also a fertile ground of understanding chaotic behavior. The laws of chaos govern fluctuations in fluid flow, stock market gyrations, and irregularities of the heart and the brain. In one set of proposed experiments, chaos and disorder are investigated in turbulent flows. The project has a strong educational element. A sizeable group of undergraduates receive lectures in fluid dynamics as they work in the laboratory during the summer. It has also proved possible to introduce ideas of chaos into the Pittsburgh school system.
Technical Fluid flowing through smooth or roughened pipes exert stress on the on the wall. The resulting frictional loss is a non-monotonic function of the Reynolds number (Re) and the pipe roughness. Drawing on ideas from critical phenomena and scaling, Nigel Goldenfeld has shown that all known measurements can be mapped into a single curve. Goldenfeld's approach will be put to an experimental test for a two-dimensional system, namely the flow of a soap film between two parallel, roughened vertical wires. Measurements are also made of the motion of particles that float on a strongly turbulent tank of water. These floaters sample the underlying flow at the surface. One can relate the velocity divergence of the floaters to the entropy S, or rather its rate dS/dt. This rate is a random variable and not identically zero as in the bulk. The results will be compared with computer simulations and with the "fluctuation theorem", which is an extension of the fluctuation-dissipation theorem, applicable only to systems near thermal equilibrium. The experiments provide a good training ground for both undergraduates and graduate students.
