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 1Aspects of Hydrodynamics and Turbulence with CryogensJ. NiemelaICTP, Trieste“Perhaps the fundamental equation that describes the swirling nebulae and the condensing, revolving, and exploding stars is just a simple equation for the hydrodynamic behavior of nearly pure hydrogen gas”.-- Richard Feynman, Nobel LaureateThe first scientific investigations of fluid turbulence are generally attributed to Leonardo Da Vinci.Turbulence is widespread, indeed almost the rule, in the flow offluids. It is a complex phenomenon, for which the development ofa satisfactory theoretical framework has been one of the greatest unsolved challenges of classical physics.Turbulence exists in a wide range of contexts such as the motion of submarines, ships and aircraft, pollutant dispersion in the earth's atmosphere and oceans, heat and mass transport in engineering applications as well as geophysics and astrophysics [See, e.g., D.J. Tritton, Physical Fluid Dynamics. Clarendon Press, Oxford (1988)]. The problem is also a paradigm for strongly nonlinear systems, distinguished by strong fluctuations and strong coupling among a large number of degrees of freedom. [G. Falkovich, K.R. Sreenivasan, Phy. Today 59, 43 (2006)]. The complexity of the underlying equations, however, has precluded much analytical progress, and the demands of computing power are such that routine simulations of large turbulent flows has not yet been possible. Thus, the progress in the field has depended more on experimental input. This experimental input in turn points in part to a search for optimal test fluids, and the development and utilization of novel instrumentation. These aspects will be discussed in this lecture.Turbulence is particularly useful because the equations of motion are known exactly and can be simulated with precision. And so, even distant areas--perhaps even market fluctuations [B.B. Mandelbrot, Scientific American 280, 50 (1999)]---may benefit from a better understanding of it.While it is not possible to predict turbulent motion in all details as a function of both time and position, turbulence can be well-characterized statistically, with reproducible average values of certain quantities. Let’s take a quick glance at Newton’s second law for hydrodynamics, including turbulence (Navier-Stokes)Flow past a circular cylinder. (a) Re = 26. (b) Re = 2000. rr'=Lvv'=UppU2'=ρttUL'=Re = UL/ν()v1vvv2∇+∇−=∇⋅+∂∂νρpt()vRe1vvv2′∇′+′∇′−=′∇′⋅′+∂′∂pt 2The Reynolds number is a manifestation of dynamical similarity. That is, if we consider two simple flows that are geometrically similar, then they are dynamically identical if the corresponding Re is the same for both, regardless of the specific velocities, lengths and fluid viscosities involved. Matching such parameters between laboratory testing of a model and the actual full-scale object (the prototype) is the principle upon which aerodynamic model-testing is based. Above left: wake behind a flat plate inclined 45 degrees to the direction of the flow (left to right). Above right: A foundered ship in the sea inclined 45 degrees to the direction of the current.Q. Two flows are dynamically similar if---a) the fluids have the same kinematicviscosity b) the flows have the same Rec) the flows have large Re>>1flowgridInitial turbulent forcing at scale of the mesh size M.Here, flow at speed U through a grid of crossed tines with mesh size M, generates aturbulent wake, injecting kinetic energy initially on that scale.Making Turbulence in the laboratoryTraditionally, large arrays of wake-producing “cylinders”-- such as we saw above--are placed in the flow in order to generate something approaching homogeneous and isotropic turbulence.Eddies produced by flow (e.g., through a grid at length scale M= mesh size)Cascade of energy in inertial range: local transfer of energy(due to nonlinear inertial term in NS eqn.).Viscous dissipation at small scales for which local Re~1.Where does the injected energy go? Richardson Energy CascadeIf there are any universal statistical properties of turbulence, it is reasonable to look for them as Re→∞, since the separation between the energy-injection scales and the dissipative scales increases with Re.For “inertial range” between large energy injection scale L and the smallest dissipation scale η the energy spectrum (in k-space) is E(k) = C ε2/3k-5/3where ε is the rate of energy transfer per unit mass. (E(k) is the energy contained in the wavenumber shell between k and k+dk.Smallest scale given by η∼ LRe-3/4.Q. As the viscosity gets smallera)The dissipation length scale growsb)The dissipation length remains the samec) The dissipation length becomes smaller. Leiden, 1908: Kamerlingh Onnes succeeds in liquifying helium, an element first identified spectroscopically in India in 1868 during a total eclipse of the sun. This leads to the discovery of superconductivity a few years later, superfluidity a few decades later, and many diverse applications in science and engineering, including the experimental study of fluid turbulence:100 years ago…ReULν=Arbitrarily increasing U can introduce Mach number (=U/c). Increasing L has its obvious limits.Helium has the lowest kinematicviscosity of any fluid. 3Direct numerical simulations (DNS) can be performed in which the appropriate equations are solved on a computer without making any approximation. The range of scales needing to be well resolved,however, grows as Re3/4 , and thus Re9/4 in 3 dimensions, severely hampering DNS efforts. The state of the art in DNS is about Re~104, or about 3-4 orders of magnitude lower than the Re corresponding to a typically commercial jet aircraft, and the same amount for most atmospheric and oceanic flows.Under certain circumstances, large eddy simulations (LES)--which compute only the large scales but model the small scales-- do better in terms of providing useful information, but they are not satisfactory as universal recipes. The state of the art in computer hardware is years away from allowing us to address the most important problems in natural and engineering fluid turbulence. It is reasonable to ask why nature’s “laboratories”, such as the atmosphere and oceans, cannot be instrumented and studied for turbulence dynamics. They can, but this is not a substitute for controlled laboratory studies when questions become sharp and a deeper understanding is required.Alternative


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UF PHY 4550 - Lecture notes

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