Scalings and decay of fractal-generated turbulence

Abstract

A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence. The resulting turbulent flows have been studied using hot wire anemometry. Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax∕tmin. With relatively small blockage ratios, as low as σ=25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage 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’s center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v′∕U and w′∕U): (i) for fractal cross grids (Df=2), (u′∕U)2=tr2CΔPfct(x∕Meff); (ii) for fractal I grids, (u′∕U)2=tr(T∕Lmax)2CΔPfct(x∕Meff), where T is the tunnel width and Lmax is the maximum bar length on the grid; (iii) for space-filling (Df=2) fractal square grids, the turbulence intensity builds up as the turbulence is convected downstream until a distance xpeak from the grid is reached where the turbulence intensity peaks and then decays exponentially, u′2=upeak′2exp[−(x−xpeak)∕lturb], where upeak′2 increases linearly with tr, xpeak∝tminT∕Lmin (Lmin being the minimum bar length on the grid), and lturb∝λ2U∕ν (ν being the kinematic viscosity of the air and λ being the Taylor microscale); λ remains approximately constant during decay at x≫xpeak. The longitudinal and lateral integral length scales also remain approximately constant during decay at x≫xpeak.