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A note on third-order structure functions in turbulence. (English) Zbl 0952.76026
Summary: Starting from the Navier-Stokes equation, we rigorously prove that a modified third-order structure function, \(S_3(r)\), asymptotically equals \(-{4\over 3}\varepsilon r\) (\(\varepsilon\) is the dissipation rate) in an inertial regime. From this result, we rigorously confirm the Kolmogorov four-fifths law, without the Kolmogorov’ assumption of isotropy. Our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector \(y\), besides space-time-averaging. Direct numerical simulation for a highly symmetric flow for a Taylor-Reynolds number of up to 155 shows that the flow remains significantly anisotropic and that, without solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of \(r=|y|\) for some orientation of \(y\), but not for another.

76F02 Fundamentals of turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
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