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On invariants in grid turbulence at moderate Reynolds numbers

  • T. Kitamura (a1), K. Nagata (a1), Y. Sakai (a1), A. Sasoh (a2), O. Terashima (a1), H. Saito (a1) and T. Harasaki (a2)...

Abstract

The decay characteristics and invariants of grid turbulence were investigated by means of laboratory experiments conducted in a wind tunnel. A turbulence-generating grid was installed at the entrance of the test section for generating nearly isotropic turbulence. Five grids (square bars of mesh sizes $M= 15$ , 25 and 50 mm and cylindrical bars of mesh sizes $M= 10$ and 25 mm) were used. The solidity of all grids is $\sigma = 0. 36$ . The instantaneous streamwise and vertical (cross-stream) velocities were measured by hot-wire anemometry. The mesh Reynolds numbers were adjusted to $R{e}_{M} = 6700$ , 9600, 16 000 and 33 000. The Reynolds numbers based on the Taylor microscale $R{e}_{\lambda } $ in the decay region ranged from 27 to 112. In each case, the result shows that the decay exponent of turbulence intensity is close to the theoretical value of ${- }6/ 5$ (for the $M= 10~\mathrm{mm} $ grid, ${- }6(1+ p)/ 5\sim - 1. 32$ ) for Saffman turbulence. Here, $p$ is the power of the dimensionless energy dissipation coefficient, $A(t)\sim {t}^{p} $ . Furthermore, each case shows that streamwise variations in the integral length scales, ${L}_{uu} $ and ${L}_{vv} $ , and the Taylor microscale $\lambda $ grow according to ${L}_{uu} \sim 2{L}_{vv} \propto {(x/ M- {x}_{0} / M)}^{2/ 5} $ (for the $M= 10~\mathrm{mm} $ grid, ${L}_{uu} \propto {(x/ M- {x}_{0} / M)}^{2(1+ p)/ 5} \sim {(x/ M- {x}_{0} / M)}^{0. 44} $ ) and $\lambda \propto {(x/ M- {x}_{0} / M)}^{1/ 2} $ , respectively, at $x/ M\gt 40{\unicode{x2013}} 60$ (depending on the experimental conditions, including grid geometry), where $x$ is the streamwise distance from the grid and ${x}_{0} $ is the virtual origin. We demonstrated that in the decay region of grid turbulence, ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{3} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{3} $ , which correspond to Saffman’s integral, are constant for all grids and examined $R{e}_{M} $ values. However, ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{5} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{5} $ , which correspond to Loitsianskii’s integral, and ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{2} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{2} $ , which correspond to the complete self-similarity of energy spectrum and $\langle {\boldsymbol{u}}^{2} \rangle \sim {t}^{- 1} $ , are not constant. Consequently, we conclude that grid turbulence is a type of Saffman turbulence for the examined $R{e}_{M} $ range of 6700–33 000 ( $R{e}_{\lambda } = 27{\unicode{x2013}} 112$ ) regardless of grid geometry.

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Email address for correspondence: nagata@nagoya-u.jp

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