Take a circular cylindrical container with a small, short bell-mouthed orifice in the base center. Now fill it and remove the stopper. As John Falstaff would have it, “Broach that bloody keg!” What happens?
The water comes out the bottom – if I can be forgiven some complex technical verbiage! It does, indeed, and runs out the orifice in an approximately uniform flow, at a speed related to the height of the free surface above the outlet, the square root, to be precise. This was predicted by Galileo and has been supported by hundreds of 1,000s of experiments, including my own, conducted in steamy, malarial Kwa Zulu Natal, in the 40s one of the Last Outposts of the Empire. Our hydro lab instructor was a cultured, gentle Professore Italiano, washed ashore on the southern tip of this rough continent by the tides of the recent war. He told us, an unruly bunch of ranchers’ sons, that we should heed his hydrodynamics, since his ancestors had designed the aqueducts of the Roman Empire . We were duly impressed. This was class, of which we had little. Our ancestors had mainly built railroads, dug gold, imbibed cheap liquor and shot the local fauna (and, sadly, sometimes the inhabitants, too). I enjoyed doing hydro experiments with water that, only a few weeks ago, in its younger life, had harbored crocodiles and hippopotami. I don’t suppose too many of my fellow engineers thought like that. And I owe Dottore Luigi much for introducing me to the fountain designers of the Renaissance, maestros like Nicola Salvi who did the Fontana di Trevi. Hydraulics engineers calculate this outflow using a “constant” called the orifice coefficient, a function of the nozzle shapes involved. It’s tabulated in handbooks, and seldom less than 90%, so is hardly observable by jes lookin’. The orifice always seems to run full. One can also calculate the entire flow in the container, assuming there is not much dissipation, as is normally the case. Enter Signore Bernoulli, as the conservation of potential, kinetic and pressure energy. There is a lovely expression, called the Stokes Stream Function, that defines stream tubes within the container. These stream tubes have the shape similar to the outer container for the large radius flow, and, near the axis, smooth the kinks out to be roughly cylindrical. Particles cannot move from one stream tube to another, so that the velocity can be calculated. There is no problem in conceiving of the rules for this, but a computer is needed to take into account the details of the flow boundaries. Analytical functions particularly hate right angles, so tiresome computer calculations with teeny elements are required to nickel and dime the flow near the bottom corners. It the fluid is set into rotation by paddles (that, to minimize turbulence, should not be perforated) then a swirling flow is produced. There’s one in your car, called a fluid drive. Most of this rotary flow is irrotational, so the Laws of Bernoulli hold, roughly. But the energy of the flow is increased; you stirred it up, after all! The free surface, at constant pressure, has a dimpled depression near the axis. Its radius, r, and depth, h, satisfy the condition r 2 h equals constant, defining a kind of curved conical cavity like, and beautiful as, a Calla lily. Near the center, the depression gradient flattens out due to viscous effects that prohibit high central velocities, as would occur for the simple 1/r vortex flow model. The scale of this inner radius depends on the swirl rate and the kinematic viscosity of the fluid. It’s different for oil or water. Small, sticky flows, like filling your crankcase with oil, behave quite differently from vast, slick flows, as in the 500 km Florida Gyre of the Gulf Stream, on which I published in the grandiose, but doomed, Coriolis Project. If Bernoulli holds, then the speed of the outlet flow still obeys that Law. This implies that the exit speed is always higher than that of the non-swirling flow. But the angular momentum is conserved, so the direction of the flow is no longer vertical. The rate of flux out of the container is reduced by this angle. The question is whether the increased speed is offset by the increased inclination. You just gotta crunch the numbers to see what wins, and construct a flow model – inviscid is not too bad, laminar can be treated numerically with Navier-Stokes, but turbulence; “Oh, My!”, as Dick Feynman was wont to say. I think that the results are that introducing swirl has a bimodal effect. This world is neither monotonous nor monotonic! Sometimes swirl increases the outflow, sometimes not. I tried to express this in the Karman parable about the Prof who could explain anything, once he knew the result. Evidently this is a fertile field for inspired interpretation. Voodoo artistes love happenings that are sometimes one way and sometimes t’other. It’s like the Acts of God, sometimes they’re Good, sometimes the same things is Bad. That’s the fascinating puzzlement of religion. But the priests know the answer, ‘cause God tole ‘em. And the merit of philosophy is that it tries to resolve things to rational, universal truths. A worthy goal. Peter Lissaman, Da Vinci Ventures Expertise is not knowing everything, but knowing what to look for. 1454 Miracerros Loop South, Santa Fe, New Mexico 87505,USA tel:(505)983-7728
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