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Asymptotic scaling laws for the irrotational motions bordering a turbulent region

Published online by Cambridge University Press:  05 May 2021

Ricardo P. Xavier Open the ORCID record for Ricardo P. Xavier [Opens in a new window] ,
Miguel A. C. Teixeira Open the ORCID record for Miguel A. C. Teixeira [Opens in a new window]  and
Carlos B. da Silva Open the ORCID record for Carlos B. da Silva [Opens in a new window]
Ricardo P. Xavier
Affiliation:
LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal
Miguel A. C. Teixeira
Affiliation:
Department of Meteorology, University of Reading, Meteorology Building, Whiteknights Road, Earley Gate, ReadingRG6 6ET, UK
Carlos B. da Silva*
Affiliation:
LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal
*
†Email address for correspondence: [email protected]
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Abstract

Turbulent flows are often bounded by regions of irrotational or non-turbulent flow, where the magnitude of the potential velocity fluctuations can be surprisingly high. This includes virtually all turbulent free-shear flows and also turbulent boundary layers, and is particularly true near the so-called turbulent/non-turbulent interface (TNTI) layer, which separates the regions of turbulent and non-turbulent fluid motion. In the present work, we show that in the non-turbulent region and for distances $x_2$ sufficiently far from the TNTI layer, the asymptotic variation laws for the variance of the velocity fluctuations $\langle u_i^{2} \rangle$ ($i=1,2,3$), Taylor micro-scale $\lambda$ and viscous dissipation rate $\varepsilon$ depend on the shape of the kinetic energy spectrum in the infrared region $E(k) \sim k^{n}$. Specifically, by using rapid distortion theory (RDT), we show that for Saffman turbulence ($E(k) \sim k^{2}$), we obtain the asymptotic laws $\langle u_i^{2} \rangle \sim x_2^{-3}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-5}$. Additionally, we confirm the classical results obtained by Phillips (Proc. Camb. Phil. Soc., vol. 51, 1955, p. 220) for Batchelor turbulence ($E(k) \sim k^{4}$), with $\langle u_i^{2} \rangle \sim x_2^{-4}$ ($i=1,2,3$), $\lambda \sim x_2$ and $\varepsilon \sim x_2^{-6}$. The new theoretical results are confirmed by direct numerical simulations (DNS) of shear-free turbulence and are shown to be independent of the Reynolds number. Therefore, these results are expected to be valid in other flow configurations, such as in turbulent planar jets or wakes, provided the kinetic energy spectra in the turbulence region can be described by a Batchelor or a Saffman spectrum.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Shear layer turbulence Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A3
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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