Algebraic estimation of the three-dimensional radial energy spectrum in quasi-homogeneous and wall-bounded turbulence

Approximating the three-dimensional radial energy spectrum based on one-dimensional directional data is challenging, as experiments and reduced datasets provide limited spatial information. Additionally, spectral analysis is often restricted directionally by rapid spatial variations in flow properties. Classical relations, such as Batchelor's, formally connect one-dimensional and three-dimensional spectra, but their practical application requires numerical differentiation, which heavily amplifies noise in discrete spectral data. In addition, Batchelor's formula relies on the assumption of isotropy. To avoid these limitations, this study assesses a simple algebraic mapping for radial spectrum estimation. The parametrization and validation of the proposed function are based on direct numerical simulation data of Kolmogorov flow. The approximate radial spectrum is compared against the exact spectrum computed from the full three-dimensional velocity field. In addition, the resulting turbulent kinetic energy and dissipation values are also examined. The results show that the approximate spectrum reproduces the energy-containing and intermediate scales well, preserving the total turbulent kinetic energy. Larger discrepancies appear at high wavenumbers, indicating reduced accuracy near the smallest resolved scales. The conversion is extended to wall-bounded flows by introducing a local mapping function dependent on the nondimensional wall distance. Furthermore, the practical utility of the approximation is demonstrated by fitting the data to Pope's model spectrum to extract the integral length scale across the channel height. Overall, the proposed approach provides a simple, robust, and differentiation-free method for radial spectrum approximation.