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Similarity of energy structure functions in decaying homogeneous isotropic turbulence

Published online by Cambridge University Press:  25 June 2003

R. A. ANTONIA
Affiliation:
Discipline of Mechanical Engineering, University of Newcastle, NSW 2308, Australia
R. J. SMALLEY
Affiliation:
Energy and Resources Research Institute, University of Leeds, Leeds, LS2 9JT, UK
T. ZHOU
Affiliation:
Department of Mechanical & Production Engineering, Nanyang Technological University, Singapore 639798
F. ANSELMET
Affiliation:
IRPHE, Université d'Aix-Marseille I & II, 13384 Marseille, France
L. DANAILA
Affiliation:
CORIA, Avenue de l'Université, BP12, 76801, Saint Etienne du Rouvray, France

Abstract

An equilibrium similarity analysis is applied to the transport equation for $\langle(\delta q)^{2}\rangle$ (${\equiv}\,\langle(\delta u)^{2}\rangle + \langle(\delta v)^{2}\rangle + \langle(\delta w)^{2}\rangle$), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy $\langle q^{2}\rangle$ decays with a power-law behaviour ($\langle q^{2}\rangle\,{\sim}\,x^{m}$), and the characteristic length scale, which is readily identifiable with the Taylor microscale, varies as $x^{1/2}$. This solution is identical to that obtained by George (1992) from the spectral energy equation. The solution does not depend on the actual magnitude of the Taylor-microscale Reynolds number $R_{\lambda}$ (${\sim}\,{\langle q^{2}\rangle}^{1/2} \lambda/\nu$); $R_{\lambda}$ should decay as $x^{(m+1)/2}$ when $m < -1$. The solution is tested at relatively low $R_{\lambda}$ against grid turbulence data for which $m \simeq -1.25$ and $R_{\lambda}$ decays as $x^{-0.125}$. Although homogeneity and isotropy are poorly approximated in this flow, the measurements of $\langle(\delta q)^{2}\rangle$ and, to a lesser extent, $\langle(\delta u)(\delta q)^{2}\rangle$, satisfy similarity reasonably over a significant range of $r/\lambda$, where $r$ is the streamwise separation across which velocity increments are estimated. For this range, a similarity-based calculation of the third-order structure function $\langle(\delta u)(\delta q)^{2}\rangle$ is in reasonable agreement with measurements. Kolmogorov-normalized distributions of $\langle(\delta q)^{2}\rangle$ and $\langle(\delta u)(\delta q)^{2}\rangle$ collapse only at small $r$. Assuming homogeneity, isotropy and a Batchelor-type parameterization for $\langle(\delta q)^{2}\rangle$, it is found that $R_{\lambda}$ may need to be as large as $10^{6}$ before a two-decade inertial range is observed.

Type
Papers
Copyright
© 2003 Cambridge University Press

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