Entrainment and suspension of sand and gravel

The entrainment and suspension of sand and gravel are important for the evolution of rivers, deltas, coastal areas, and submarine fans. The prediction of a vertical profile of suspended sediment concentration typically consists of assessing (1) the concentration near the bed using an entrainment relation and (2) the upward vertical distribution of sediment in the water column. Considerable uncertainty exists in regard to both of these steps, especially the near-bed concentration. Most entrainment relations have been tested against limited grain-size-specific data, and no relations have been evaluated for gravel suspension, which can be important in bedrock and mountain rivers. To address these issues, we compiled a database with suspended sediment data from natural rivers and flume experiments, taking advantage of the increasing availability of high-resolution grain size measurements. We evaluated 12 dimensionless parameters that may determine entrainment and suspension relations and applied multivariate regression analysis. A best-fit two-parameter equation (<span classCombining double low line"inline-formula">r2Combining double low line0.79</span>) shows that near-bed entrainment, evaluated at 10&thinsp;% of the flow depth, decreases with the ratio of settling velocity to skin-friction shear velocity (<span classCombining double low line"inline-formula"><math xmlnsCombining double low line"http://www.w3.org/1998/Math/MathML" idCombining double low line"M2" displayCombining double low line"inline" overflowCombining double low line"scroll" dspmathCombining double low line"mathml"><mrow><msub><mi>w</mi><mrow><mi mathvariantCombining double low line"normal">s</mi><mi>i</mi></mrow></msub><mo>/</mo><msub><mi>u</mi><mrow><mo>ĝ&circ;-</mo><mi mathvariantCombining double low line"normal">skin</mi></mrow></msub></mrow></math><span><svg:svg xmlns:svgCombining double low line"http://www.w3.org/2000/svg" widthCombining double low line"48pt" heightCombining double low line"14pt" classCombining double low line"svg-formula" dspmathCombining double low line"mathimg" md5hashCombining double low line"28edb121be821f54341da672db4b28c5"><svg:image xmlns:xlinkCombining double low line"http://www.w3.org/1999/xlink" xlink:hrefCombining double low line"esurf-8-485-2020-ie00001.svg" widthCombining double low line"48pt" heightCombining double low line"14pt" srcCombining double low line"esurf-8-485-2020-ie00001.png"/></svg:svg></span></span>), as in previous relations, and increases with Froude number (Fr), possibly due to its role in determining bedload-layer concentrations. We used the Rouse equation to predict concentration upward from the reference level and evaluated the coefficient <span classCombining double low line"inline-formula">βi</span>, which accounts for differences in the turbulent diffusivity of sediment from the parabolic eddy viscosity model used in the Rouse derivation. The best-fit relation for <span classCombining double low line"inline-formula">βi</span> (<span classCombining double low line"inline-formula">r2Combining double low line0.40</span>) indicates greater relative sediment diffusivities for rivers with greater flow resistance, possibly due to bedform-induced turbulence, and larger <span classCombining double low line"inline-formula"><math xmlnsCombining double low line"http://www.w3.org/1998/Math/MathML" idCombining double low line"M6" displayCombining double low line"inline" overflowCombining double low line"scroll" dspmathCombining double low line"mathml"><mrow><msub><mi>w</mi><mrow><mi mathvariantCombining double low line"normal">s</mi><mi>i</mi></mrow></msub><mo>/</mo><msub><mi>u</mi><mrow><mo>ĝ&circ;-</mo><mi mathvariantCombining double low line"normal">skin</mi></mrow></msub></mrow></math><span><svg:svg xmlns:svgCombining double low line"http://www.w3.org/2000/svg" widthCombining double low line"48pt" heightCombining double low line"14pt" classCombining double low line"svg-formula" dspmathCombining double low line"mathimg" md5hashCombining double low line"1ca89cd39931e861d7bdd8017205cca1"><svg:image xmlns:xlinkCombining double low line"http://www.w3.org/1999/xlink" xlink:hrefCombining double low line"esurf-8-485-2020-ie00002.svg" widthCombining double low line"48pt" heightCombining double low line"14pt" srcCombining double low line"esurf-8-485-2020-ie00002.png"/></svg:svg></span></span>; the latter dependence is nonlinear and therefore different from standard Rouse theory. In addition, we used empirical relations for gravel saltation to show that our relation for near-bed concentration also provides good predictions for coarse-grained sediment. The new relations extend the calibrated parameter space over a wider range in sediment sizes and flow conditions compared to previous work and result in 95&thinsp;% of concentration data throughout the water column predicted within a factor of 9.