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Publication# Saturation of curvature-induced secondary flow, energy losses, and turbulence in sharp open-channel bends: Laboratory experiments, analysis, and modeling

Abstract

The paper investigates the influence of relative bend curvature on secondary flow, energy losses, and turbulence in sharp open-channel bends. These processes are important in natural streams with respect to sediment transport, the bathymetry and planimetry, mixing and spreading of pollutants, heat, oxygen, nutrients and biological species, and the conveyance capacity. Laboratory experiments were carried out in a configuration with rectangular cross section, consisting of a 193 degrees bend of constant radius of curvature, preceded and followed by straight reaches. This somewhat unnatural configuration allows investigating the adaptation of mean flow and turbulence to curvature changes in open-channel bends, without contamination by other effects such as a mobile bed topography. Experiments were carried out for three different values of the curvature ratio, defined as the ratio of centerline radius of curvature over flow depth, which is the principal curvature parameter for hydrodynamic processes. Commonly used so-called linear models predict secondary flow to increase linearly with the curvature ratio. The reported experiments show that the secondary flow hardly increases in the investigated very sharp bends when the curvature ratio is further increased. This phenomenon is called saturation. Similar saturation is observed for the energy losses and the turbulence. This paper focuses on the analysis and modeling of the saturation of energy losses and turbulence. Secondary flow is found to be the dominant contribution to the curvature-induced increase in turbulence production, which leads to increased energy losses. The curvature-induced turbulence is explained by the fact that the turbulence dissipation lags behind the turbulence production, in agreement with the concept of the turbulence energy cascade. A 1-D model is proposed for the curvature-induced energy losses and turbulence. It could extend 1-D or depth-averaged 2-D models that are commonly used in long-term (scale of a flood event to geological scales) or large-scale (scale of a river basin) investigations on flood propagation, hazard mapping, water quality modeling, and planimetric river evolution.

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Curvature

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point.

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space.

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

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