Revisiting Townsend’s spatial energy-density function

Fourier analysis is the standard tool of choice for quantifying the distribution of kinetic energy amongst the eddies in a turbulent flow. The resulting spectral energy-density function is the well-known energy spectrum. And yet, because eddies are distinct from waves, alternative approaches to finding energy-density functions have long been sought. Townsend (1976) outlined a promising approach to finding a spatial energy-density function, $V\!(r)$ , where $r$ is the eddy size. Notably, this approach led to two distinct and mutually inconsistent formulations of $V\!(r)$ in homogeneous, isotropic turbulence. We revisit Townsend’s proposal and derive the corresponding three-dimensional $V\!(r)$ as well as introduce its one-dimensional variants (which, to our knowledge, have not been explicitly discussed before). By training our focus on the associated dimensionality of the function, we resolve the discrepancies between the previous formulations. Additionally, we generalise our analysis to include anisotropic flows. Finally, by means of concrete examples, we illustrate how one-dimensional spatial energy-density functions are useful for analysing empirical data. Some notable findings include new insights into the $k_1^{-1}$ scaling (where $k_1$ is the streamwise wavenumber) and a possible resolution of the enigmatic sizes of organised motions at large scales.