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The decay of stably stratified grid turbulence in a viscosity-affected stratified flow regime

Published online by Cambridge University Press:  08 August 2022

Tomoaki Watanabe Open the ORCID record for Tomoaki Watanabe [Opens in a new window] ,
Yulin Zheng Open the ORCID record for Yulin Zheng [Opens in a new window]  and
Koji Nagata Open the ORCID record for Koji Nagata [Opens in a new window]
Tomoaki Watanabe*
Affiliation:
Education and Research Center for Flight Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
Yulin Zheng
Affiliation:
Department of Aerospace Engineering, Nagoya University, Nagoya 464-8603, Japan
Koji Nagata
Affiliation:
Department of Aerospace Engineering, Nagoya University, Nagoya 464-8603, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The decay of stably stratified turbulence generated by a towed rake of vertical plates is investigated by direct numerical simulations (DNS) of temporally evolving grid turbulence in a linearly stratified fluid. The Reynolds number $Re_M=U_0M/\nu$ is 5000 or 10 000 while the Froude number $Fr_M=U_0/MN$ is between 0.1 and 6 ($U_0$: towing speed; $M$: mesh size; $\nu$: kinematic viscosity; $N$: Brunt–Väisälä frequency). The DNS results are compared with the theory of stably stratified axisymmetric Saffman turbulence. Here, the theory is extended to a viscosity-affected stratified flow regime with low buoyancy Reynolds number $Re_b$, and power laws are derived for the temporal variations of the horizontal velocity scale ($U_H$) and the horizontal and vertical integral length scales ($L_H$ and $L_V$). Temporal grid turbulence initialized with the mean velocity deficit of wakes exhibits a $k^{2}$ energy spectrum at a low-wavenumber range and invariance of $U_H^2L_H^2L_V$, which are the signatures of axisymmetric Saffman turbulence. The decay of various quantities follows the power laws predicted for low-$Re_b$ Saffman turbulence when $Fr_M$ is sufficiently small. However, the decay of $U_H^2$ at $Fr_M=6$ is no longer expressed by a power law with a constant exponent. This behaviour is related to the scaling of kinetic energy dissipation rate $\varepsilon$, for which $\alpha =\varepsilon /(U_H^3/L_H)$ is constant during the decay for $Fr_M\leq 1$ while it varies with time for $Fr_M=6$. We also examine the experimental data of towed-grid experiments by Praud et al. (J. Fluid Mech., vol. 522, 2005, pp. 1–33), which is shown to agree with the theory of low-$Re_b$ Saffman turbulence.

JFM classification

Turbulent Flows: Stratified turbulence Wakes/Jets: Wakes Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A29
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© The Author(s), 2022. Published by Cambridge University Press

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