Scalings and Decays of Fractal-generated turbulence

21 planar fractal grids from 3 fractal families have been used in 2 wind tunnels to generate turbulence. This turbulence and its homogeneity, isotropy and decay properties are strongly dependent on the grid's fractal dimension $D_{f}\le 2$, the effective mesh size $M_{eff}$ (which we introduce and define) and the ratio $t_r$ of largest to smallest bar thicknesses, $t_{r}=t_{max}/t_{min}$. With blockage ratios as low as $\sigma = 25$\%, these grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than higher blockadge ratio classical grids in similar wind tunnels and wind speeds $U$. The scalings and decay of the turbulence intensity $u'/U$ in the $x$-direction along the tunnel centre line are (in terms of the normalised pressure drop $C_{\Delta P}$ and with similar results for $v'/U$ and $w'/U$): (i) for fractal cross grids ($D_{f}=2$), $(u'/U)^{2}= t_{r}^{2}C_{\Delta P} fct (x/M_{eff})$; (ii) for fractal I grids, $(u'/U)^{2}= t_{r} (T/L_{max})^{2} C_{\Delta P} fct (x/M_{eff})$ where $T$ is the tunnel width and $L_{max}$ is the maximum bar length on the grid; (iii) for $D_{f}=2$ fractal square grids, the turbulence builds up till a distance $x_{peak}$ from the grid where the turbulence intensity peaks and then decays exponentially, $u'^{2}=u'^{2}_{peak} exp[-(x-x_{peak})/l_{turb}]$ where $u'^{2}_{peak}$ increases linearly with $t_r$, $x_{peak}\propto t_{min}T/L_{min}$ ($L_{min}$ being the minimum bar length on the grid) and $l_{turb}\propto \lambda^{2}U/\nu$ ($\nu$ being the air's kinematic viscosity and $\lambda$ being the Taylor microscale); $\lambda$ and the longitudinal/lateral length-scales remain approximately constant during decay at $x\gg x_{peak}$.