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Transport equations for the normalized nth-order moments of velocity derivatives in grid turbulence

Published online by Cambridge University Press:  16 November 2021

S.L. Tang Open the ORCID record for S.L. Tang [Opens in a new window] ,
R.A. Antonia  and
L. Djenidi Open the ORCID record for L. Djenidi [Opens in a new window]
S.L. Tang*
Affiliation:
Center for Turbulence Control, Harbin Institute of Technology, Shenzhen518055, PR China
R.A. Antonia
Affiliation:
School of Engineering, University of Newcastle, NSW2308, Australia
L. Djenidi
Affiliation:
School of Engineering, University of Newcastle, NSW2308, Australia
*
†Email address for correspondence: [email protected]
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Abstract

Transport equations for the normalized moments of the longitudinal velocity derivative ${F_{n + 1}}$ (here, $n$ is $1, 2, 3\ldots$) are derived from the Navier–Stokes (N–S) equations for shearless grid turbulence. The effect of the (large-scale) streamwise advection of ${F_{n + 1}}$ by the mean velocity on the normalized moments of the velocity derivatives can be expressed as $C_1 {F_{n + 1}}/Re_\lambda$, where $C_1$ is a constant and $Re_\lambda$ is the Taylor microscale Reynolds number. Transport equations for the normalized odd moments of the transverse velocity derivatives ${F_{y,n + 1}}$ (here, $n$ is 2, 4, 6), which should be zero if local isotropy is satisfied, are also derived and discussed in sheared and shearless grid turbulence. The effect of the (large-scale) streamwise advection term on the normalized moments of the velocity derivatives can also be expressed in the form $C_2 {F_{y,n + 1}}/Re_\lambda$, where $C_2$ is a constant. Finally, the contribution of the mean shear in the transport equation for ${F_{n + 1}}$ can be modelled as $15 B/Re_\lambda$, where $B$ ($=S^*{S_{s,n + 1}}$) is the product of the non-dimensional shear parameter $S^*$ and the normalized mixed longitudinal-transverse velocity derivatives ${{S_{s,n + 1}}}$; if local isotropy is satisfied, $S_{s,n + 1}$ should be zero. These results indicate that if ${F_{n + 1}}$, ${F_{y,n + 1}}$ and $B$ do not increase as rapidly as $Re_\lambda$, then the effect of the large-scale structures on small-scale turbulence will disappear when $Re_\lambda$ becomes sufficiently large.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 930 , 10 January 2022 , A31
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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