Frequency–wavenumber velocity spectra, Taylor's hypothesis, and length scales in a natural gravel bed river

Abstract

Macroscale turbulent coherent flow structures in a natural fast‐flowing river were examined with a combination of a novel 2 MHz Acoustic Doppler Beam (ADB) and a Maximum Likelihood Estimator (MLE) to characterize the streamwise horizontal length scales and persistence of coherent flow structures by measuring the frequency (f)‐streamwise‐wavenumber (ks) energy density velocity spectrum, E(f, ks), for the first time in natural rivers. The ADB was deployed under a range of Froude numbers (0.1–0.6) at high Reynolds numbers (∼106) based on depth and velocity conditions within a gravel bed reach of the Kootenai River, Idaho. The MLE employed on the ADB data increased our ability to describe river motions with relatively long (>10 m) length scales in ∼1 m water depths. The E(f, ks) spectra fell along a ridge described by V = f/ks, where Vis the mean velocity over depth, consistent with Taylor’s hypothesis. New, consistent length scale measures are defined based on averaged wavelengths of the low‐frequencyE(f, ks) and coherence spectra. Energetic (∼50% of the total spectral energy), low‐frequency (f < 0.05 Hz) streamwise motions were found. Mean length scales,Lm, compared with the depth, h, are significantly larger than previously suggested for macroturbulence with Lm/h∼ 28–118. Although the energy appears as low‐pass white noise, it is streamwise coherent along the length of the array. In fast flows with velocities > 1 m/s,Lmwere found to be significantly longer than their corresponding coherence lengths, suggesting that the turbulent structures evolve rapidly under these conditions. This is attributed to the stretching and concomitant deformation of preexisting macroturbulent motions by the ubiquitous bathymetry‐induced spatial flow accelerations present in a natural gravel bed river.