On invariants in grid turbulence at moderate Reynolds numbers

Abstract

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 ( <jats:inline-graphic xmlns:xlink="http: