Large- and very large-scale motions in open-channel flow: Spectra and structure functions*

Large-scale motions (LSMs) and very-large-scale motions (VLSMs) in open-channel flows are coherent structures that, in premultiplied streamwise velocity spectra, are represented by two “hills” at wavelengths of λ_x ≈ 2-4 and λ_x ≈ 20-50 flow depths, respectively. The origin of VLSMs remains unclear. Two main hypotheses suggest that VLSMs originate either from alignment of LSMs or as relatively independent structures.

Based on experimental data, it is found that the spectral correlation coefficient ρ_uw = C_uw / (S_uuS_ww)^0.5 (C is cospectrum, S is autospectrum, and u and w are streamwise and vertical velocity components) increases monotonically with scale, from LSMs to VLSMs, indicating greater coherence and efficiency in momentum delivery to the wall by the VLSMs compared to the LSMs. Furthermore, the negative sign of C_uw at the VLSM wavelength indicates energy extraction from the mean flow.

LSMs and VLSMs are also assessed via the streamwise second-order structure function (D_uu). In particular, we used the derivative of D_uu multiplied by the streamwise spatial separation r_x (i.e., r_x ∂D_uu / ∂r_x), which, similar to premultiplied spectra, represents energy distribution as a function of r_x. Analysis of the third-order structure function (D_uuu) indicates an energy flux from VLSMs towards LSMs and smaller scales.

The results are consistent with the hypothesis of VLSMs as independent structures, whereas the “LSM alignment” seems to be incompatible with the distributions of ρ_uw and D_uuu.