= Turbulence
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= Turbulent flow
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<Turbulence> is <fluid flow> with irregular, interacting <velocity> fluctuations over a range of lengths and times. An unsteady <laminar flow> also has a time-dependent <velocity field>; the important distinction is the organization and interaction of the fluctuations. In many flows a large <Reynolds number> allows nonlinear <advection> to distort the motion before <kinematic viscosity> smooths it, but transition depends on geometry and disturbances as well as the chosen <Reynolds number>.
The <Reynolds pipe-and-dye experiment> makes the distinction visible. A narrow band of colored water can remain coherent in a glass pipe, then break into irregular structures and spread across the pipe as the discharge increases. The inlet and pre-existing disturbances matter, so the observation does not establish a universal transition value.https://personalpages.manchester.ac.uk/staff/jdjackson/Osborne%20Reynolds/oreynb.htm[University of Manchester historical account, 1883 experiments and Figures 9–10]{ref}
In a three-dimensional forward <energy cascade>, nonlinear interactions transfer <kinetic energy> towards smaller scales until <viscous dissipation> converts it into heat. This transfer helps explain why a <turbulent flow> mixes a <passive scalar> efficiently, although final molecular homogenization still requires <molecular diffusion>. After <Reynolds averaging>, correlations of fluctuating <velocities> produce the <Reynolds stress>, an additional <momentum> flux. Predicting that flux from mean quantities is the <turbulence closure> problem; a model for it is an extra physical assumption.
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