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K41 Kolmogorov scaling relation and structure functions (turbulence)

  1. Jan 16, 2013 #1
    Hi there.

    I am having trouble interpreting the Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence:

    [itex]S_{p}(l)[/itex] = <[itex]\delta u(l)^{p}[/itex]> = <[itex]|u(r+l) - u(r)|^{p}[/itex]> [itex]\propto[/itex] ([itex]\epsilon l[/itex])[itex]^{p/3}[/itex]

    where [itex]l[/itex] is the length of displacement between two points under consideration in the flow, [itex]\epsilon[/itex] is the mean energy dissipation, and [itex]u(r)[/itex] is the velocity at point [itex]r[/itex] in the flow.

    My question is:
    Since we assume the flow to be homogeneous and isotropic in a statistically averaged sense (fully developed turbulence), how come <[itex]\delta u(l)^{p}[/itex]> is not zero? If the turbulent flow is homogeneous and isotropic on average, then the (averaged) flow in the two point should be similar? I cannot see the [itex]l[/itex]-dependence in my mental picture of this situation?

    Regards Nicholas.
  2. jcsd
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