##### The non-Kolmogorov -5/3 spectra and the related scale-by-scale energy transfer in turbulent flows behind two side-by-side square cylinders

Wake flows behind two side-by-side square cylinders with the gap ratio, Ld/T0 = 6 (Ld is the separation distance between two cylinders and T0 is the cylinder thickness) are investigated by using direct numerical simulations. Two downstream locations, i.e. X/T0 = 6 and 26, at which the turbulent flows are highly non-Gaussian distributed and approximately Gaussian distributed, respectively, are analyzed in detail. A well-defined −5/3 energy spectrum can be found in the near-field region (i.e. X/T0 = 6), where the turbulent flow is still developing and highly intermittent and the Kolmogorov’s universal equilibrium hypothesis does not hold. We confirm that the approximate −5/3 power-law in the high-frequency range is closely related to the occurrences of the extreme events. As the downstream distance increases, the velocity fluctuations gradually adopt a Gaussian distribution, corresponding to a decrease in the strength of the extreme events. Consequently, the range of the −5/3 power-law narrows. In the upstream region (i.e. X/T0 = 6), the second-order structure function exhibits a power-law exponent close to 1, whereas in the far downstream region (i.e. X/T0 = 26) the expected 2/3 power-law exponent appears. The larger exponent at X/T0 = 6 is related to the fact that fluid motions in the intermediate range can directly ‘feel’ the large-scale vortex shedding. To shed light on the scale-by-scale energy transfer in physical space, we resort to the Karman-Howarth-Monin-Hill equation, which is directly derived from Navier-Stokes equations without any assumption and can be used to study the energy cascade process in any kind of turbulent flows. It can be seen that close to the inlet (i.e. X/T0 = 6) over significant intermediate length-scales up to the size of vortex shedding, the expected balance between the non-linear term and the dissipation term cannot be detected. Instead, the contributions from the non-local pressure, advection, non-linear transport and turbulent transport terms are significant. Moreover, the magnitudes of the non-local pressure, advection, non-linear transport and turbulence transport terms are significantly larger than that of the dissipation term. This observation indicates that these terms play significant roles in the scale-by-scale budgets and Kolmogorov’s equilibrium hypothesis does not hold. In contrast, at the far downstream location X/T0 = 26, the quasi-Richardson-Kolmogorov’s equilibrium energy transfer can be found for a short intermediate range.