On fully self-preserving solutions in homogeneous turbulence

Abstract

Mathematical attributes of full self-preserving solutions (‘full’ meaning self-similar at all scales with non-zero viscosity) for isotropic decay as well as for homogeneous shear flow turbulence are examined from a fundamental theoretical standpoint. Fully self-preserving solutions are those wherein the two-point double and triple velocity correlations are self-similar at all scales. It is shown that the full self-preserving solutions for isotropic decay corresponds to a t ⁻¹ asymptotic power law decay, consistent with earlier studies. Fully self-preserving solutions for homogeneous shear flow correspond to a production-equals-dissipation equilibrium, with bounded turbulent kinetic energy and dissipation. It is then shown that the fully self-preserving solutions of isotropic decay and of homogeneous shear flow both require severe constraints on the behavior of the low-wavenumber energy spectra. These constraints render the full self-preserving solutions as mathematical consistent but having no physical relevance. The implications of the dependence of the full self-preserving solutions on the low-wavenumber spectra for one-point (engineering) turbulence models is discussed.

Invited speaker, Santa Fe CNLS-LANL Workshop on Turbulence and Cascade.

Acknowledgements

CZ and TTC were supported by the United States Department of Energy LDRD Program, no. LDRD-94135. This work by the second author was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract no. W-7405-Eng-48.

Notes

Invited speaker, Santa Fe CNLS-LANL Workshop on Turbulence and Cascade.

^†We are indebted to Professor Speziale for this analysis.

^‡Equation (3) and Equation (71) are obtained from different approaches. They lead to the same α when n = 1, but provided different values of α when n ≠ 1.

^†The authors are indebted to J. Herring, of the National Center for Atmospheric Research for this interpretation.

^‡Several papers have reported a tendency to a t ⁻¹ power-law decay, see [22, 23, 24, 25]. For related turbulence modeling issues, please see [28, 29, 30, 31, 32, 33, 34].