Identifying Turbulent Energy Distributions in Real, Rather than Fourier, Space

It has been suggested that the equilibrium-range properties of high-Reynolds number turbulence are more readily observed in spectral space, using E(k) or T(k), than in real space, using second- or third-order structure functions. For example, the -5/3 law is usually easier to see in experimental data than the equivalent 2/3 law. We argue that this is not an implicit feature of a real-space description of turbulence. Rather, it is because the second-order structure function mixes small and large-scale information. To remedy this problem we adopt a real-space function, the signature function, which plays the role of an energy density, somewhat analogous to E(k). In this Letter we determine the form of the signature function in a variety of turbulent flows. We find that dissipation-range phenomena, such as the so-called bottleneck effect, are evident in the signature function, while absent in the structure function.