The Structure Functions of the Velocity and Temperature Fields from the Perspective of Dimensional Scaling |
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Authors: | Gerhard Kramm Fritz Herbert |
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Institution: | (1) Geophysical Institute, University of Alaska Fairbanks, 903 Koyukuk Drive, P.O. Box 757320, Fairbanks, AK 99775-7320, USA;(2) Theoretical Meteorology, J.W. Goethe-University, Robert-Mayer-Strasse 1, D-60325 Frankfurt am Main, Germany |
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Abstract: | In this paper, dimensional scaling is used to describe the turbulence structure of the velocity and temperature fields in
the inertial range and the far dissipation range as well as the intermediate transition range under locally isotropic conditions
at sufficiently large Reynolds numbers. This kind of scaling is expressed in a strictly mathematical manner employing dimensional
π -invariants analysis. It is shown that in the case of the asymptotic solutions for either the inertial range or the far
dissipation range only one π number occurs that has to be considered as a non-dimensional universal constant. This π number
may be determined theoretically or/and empirically. In the case of the transition range two π numbers occur. Consequently,
a universal function is established that has to be derived theoretically or/and empirically, too. Here, Batchelor's 7] classical
interpolation formula for the turbulence structure of the velocity field and the empirical one of Stolovitzky et al. 59],
both may serve as universal functions, are compared with the results provided by numerical solutions of Kolmogorov′s 32]
structure equation for the velocity field. It is shown that these interpolation formulae match not only the asymptotic solutions
of the inertial range and the far dissipation range, respectively, but also these numerical results in an excellent manner.
The former may be considered as necessary condition and the latter as sufficient condition. In the case of the temperature
field results of the corresponding universal function are predicted using Yaglom's 63] structure equation. These results
also match the corresponding asymptotic solutions of both the inertial range and the far dissipation range. However, in contrast
to the case of the velocity field, the predicted universal function for the temperature field may notably overshoot its asymptotic
solution for the inertial range. This overshooting occurs in the transition range and may be considered as an analogue to
the so-called Hill ‘bump’ that usually occurs in the high-wave number portion of the temperature spectrum. |
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Keywords: | dimensional scaling isotropic turbulence large Reynolds numbers structure function of the velocity field structure function of the temperature field Kolmogorov's structure equation Yaglom's structure equation dimensional π -invariants analysis Hill ‘ bump’ |
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