Scaling Of One-Dimensional Non-Equilibrium Sediment Transport While Maintaining Material Properties

The governing physical process of sediment transport can be represented in a scale model without the simplifying assumptions required in numerical modeling. However, the current methods utilized in scaling sediment transport in unsteady open-channel flow result in a number of model and scale effects, which decrease the accuracy and application of scale models. Commonly, researchers apply Froude similitude, vertical distortion by scaling channel width and depth differently, and scaled sediment density to sediment transport models. These practices result in: bank inclination angles which are not conserved, errors in secondary current representation in the model, velocity distortion, inaccurate representation in the location and extent of erosion and deposition, inaccurate bed-porosity, inaccurate description of incipient motion, and dissimilar suspended sediment concentration. A number of criteria for sediment transport scaling are available that aim to decrease the error that results from scaling flow and sediment transport processes. These criteria often solve one issue of similitude in scale models, while simultaneously decreasing similitude of another flow or sediment transport parameter. For example, the scaling of sediment density is a common solution when scaling of the sediment diameter alone would result in cohesive sediments with fundamentally different behavior from non-cohesive sediment. Scaling of the sediment density may increase the similitude of the particle Reynolds number. However, the lighter sediments cause an over-estimation of suspended sediment concentration, and an accelerated time scale for bed deformation. The implications of these errors are not trivial in light of the goal of physical scale modeling to represent prototype conditions accurately enough to use scale models as tools for design and performance prediction, and environmental impact assessment. I propose the use of the one-parameter Lie group of point scaling transformations as a method of scaling sediment transport without the errors encountered in traditional scaling methods. The conditions under which the governing equation for non-equilibrium sediment transport in unsteady flows is self-similar are examined by applying the one-parameter Lie group of point scaling transformations. The conditions are further examined to determine the case in which the sediment material properties need not be scaled. It is proposed that channel depth and width be scaled equally, and the scaling ratio of channel length should equal the square of the scaling ratio of channel depth. The proposed method is compared to the traditional method of vertically distorted Froude similitude and sediment transport scaling. It is shown that under Lie group scaling, the unsteady suspended sediment transport process as an initial- boundary value problem in the prototype domain, can be self-similar with that of a variety of different scaled domains. The scaled values of flow and sediment variables at specified temporal and spatial locations can then be upscaled to the corresponding values in the prototype domain with little to no scale effect. When the geometric scaling is set as I propose, not only are sediment diameter and density unscaled, but so too are the critical and total shear, kinematic viscosity and particle Reynolds number. The similarity of sediment transport is increased through more accurate representation of incipient motion, the time-rate of change of bed morphology, bed porosity, suspended sediment concentration and carrying capacity of flow. The proposed method, detailed herein, meets the needs of modelers by maintaining the benefits found from distortion such as reduced cost, space and model run-time, while simultaneously avoiding the scale effects and resulting errors of traditional flow and sediment transport scaling.