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Kolmogorov’s hypotheses and global energy spectrum of turbulence
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3. It is also recognized gradually that K41 is flawed in that it is not suitable for higher order statistics. Even through a refinement in 1962 (K62) to depict the intermittency effect, K62 is still flawed. However, K41 works quite well for the second order statistics such as the energy spectrum that is of prime importance in theory and applications. We would be satisfied only with discussion on the energy spectrum and thus not be concerned with the more subtle issue of intermittency.
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30. Since a perfect plateau does not exist in a plotted compensated spectrum, to discern a definite inertial range and thus to determine the constant CK with no dispute are often difficult. The inertial range in our case is taken as 0.038 ≤ λη ≤ 0.062, obviously smaller than that in Ref. 29. The reason is the lowest wavenumber in the sphere is about 4.49 times high than that in the periodic box. If the same fitting range was taken for the case in Ref. 29, CK would be 1.81 rather than 1.64.
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36. Recalling that ψk vanishes at the poles of the sphere, there is no singularities here. It is important to note that, for most polar angles, varies only slowly.
39. Evidently, the energy spectrum can be altered due to the presence of strong rotation, magnetic fields, buoyancy, etc. See, e.g., P. D. Mininni and A. Pouquet, Phys. Fluids 22, 035105 (2010). Here, we intend to exclude these body forces with the emphasis on the underlying roles of geometry of the domain and the boundary conditions.
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We relate the justification of Kolmogorov’s hypotheses on the local isotropy and small-scale universality in real turbulent flows to an observed universality of basis independence for the global energy spectrum and energy flux of small-scale turbulence. To readily examine the small-scale universality, an approach is suggested that investigates the global energy spectrum in a general spectral space for which the nonlinear interscale interaction may not be Fourier-triadic. Specific verifications are performed based on direct numerical simulations of turbulence in a spherical geometry and reexaminations of several existing results for turbulent channel flows.
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