Statistical sand wave dynamics in one-directional water flows

Abstract

Moving sand waves and the overlying tubulent flow were measured on the Wilga River in Poland, and the Tirnava Mica and Buzau Rivers in Romania. Bottom elevations and flow velocities were measured at six points simultaneously by multi-channel measuring systems. From these data, the linear and two-dimensional sections of the three-dimensional correlation and structure functions and various projections of sand wave three-dimensional spectra were investigated.It was found that the longitudinal wavenumber spectra of the sand waves in the region of large wavenumbers followed Hino's −3 law (S(Kx) ∝K−3x) quite satisfactorily, confirming the theoretical predictions of Hino (1968) and Jain & Kennedy (1974). However, in contrast to Hino (1968), the sand wave frequency spectrum in the high-frequency region was approximated by a power function with the exponent −2, while in the lower-frequency region this exponent is close to −3.A dispersion relation for sand waves has been investigated from analysis of structure functions, frequency spectra and the cross-correlation functions method. For wavelengths less than 0.15–0.25 of the flow depth, their propagation velocity C is inversely proportional to the wavelength λ. When the wavelengths of spectral components are as large as 3–4 times the flow depth, no dispersion occurs. These results proved to be in good qualitative agreement with the theoretical dispersion relation derived from the potential-flow-based analytical models (Kennedy 1969; Jain & Kennedy 1974). We also present another, physically-based, explanation of this phenomenon, introducing two types of sand movement in the form of sand waves. The first type (I) is for the region of large wavenumbers (small wavelengths) and the second one (II) is for the region of small wavenumbers (large wavelengths). The small sand waves move due to the motion of individual sand particles (type I, C∝λ−1) while larger sand waves propagate as a result of the motion of smaller waves on their upstream slopes (type II, C∝λ0). Like the sand particles in the first type, these smaller waves redistribute sand from upstream slopes to downstream ones. Both types result in sand wave movement downstream but with a different propagation velocity.The main characteristics of turbulence, as well as the quantitative values characterizing the modulation of turbulence by sand waves, are also presented.