zbMATH — the first resource for mathematics

High wavenumber spectrum of a passive scalar in isotropic turbulence. (English) Zbl 0596.76067
Summary: The time-dependent solution is found for R. H. Kraichnan’s random- straining model [J. Fluid Mech. 64, 737-762 (1974; Zbl 0291.76022)] of the viscous-diffusive subrange. Temporal decay increases very rapidly with increasing spatial dimension. The Batchelor spectrum is recovered as the conditional variance of spectral amplitudes in excess of a threshold value. A by-product of the analysis is a trivial derivation of the inversion theorem for the Kontorovich-Lebedev transform.
76F05 Isotropic turbulence; homogeneous turbulence
Full Text: DOI
[1] H. Tennekes and J. L. Lumley,A First Course in Turbulence(MIT, Cambridge, MA, 1972). · Zbl 0285.76018
[2] Batchelor, J. Fluid Mech. 5 pp 113– (1959)
[3] Kraichnan, J. Fluid Mech. 64 pp 737– (1974)
[4] N. G. van Kampen,Stochastic Processes in Physics and Chemistry(North-Holland, Amsterdam, 1981). · Zbl 0511.60038
[5] In the Stratonovitch interpretation, the right-hand side of (1) is interpreted as the limit of a centered difference. The Stratonovitch interpretation is equivalent4 to taking the limit of a vanishingly small decorrelation time for the straining field. In the I to interpretation of (1), a forward difference is assumed. Ito theory is obtained if {\(\alpha\)} is replaced with {\(\alpha\)}-1 in (4).
[6] Gargett, J. Fluid Mech. 159 pp 379– (1985)
[7] G. N. Watson,Theory of Bessel Functions(Cambridge U.P., Cambridge, 1966). · Zbl 0174.36202
[8] F. Oberhettinger and L. Badii,Tables of Laplace Transforms(Springer, Berlin, 1973). · Zbl 0285.65079
[9] Measurements may also be affected by the sensitivity of the tail of the spectrum to large-scale spatial fluctuations in the effective time-integrated strain history. The author is indebted to Dr. R. H. Kraichnan for this remark and other helpful comments.
[10] N. N. Lebedev, I. P. Skalskaya, and Y. S. Ufland,Worked Problems in Applied Mathematics(Dover, New York, 1955).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.