Journal of Hydraulic Engineering

Bed‐load particles tend to move downslope in response to gravity and downstream in response to fluid drag. The effect of gravity becomes particularly important when the downslope gravitational force has a component normal to the streamwise direction. Such phenomena as bank erosion, bed topography in river bends, sediment sorting, and stream braiding are strongly influenced by transverse bed‐load transport under the influence of gravity. Here, the problem is analyzed from a consideration of the dynamics of saltating grains. Simultaneous solution of the streamwise and transverse equations of motion for a grain saltating on a transversely tilting bed of moderate slope allows a determination of mean particle trajectory. This mean trajectory, while predominately directed in the streamwise direction, includes a transverse downslope component as well. The results are used to determine a relation for the ratio of transverse to streamwise bed‐load transport. This ratio increases with increasing transverse slope and decreases with increasing streamwise shear stress. The relation is compared with observations as well as existing relations obtained from simpler considerations.

A numerical model of saltation akin to those of previous researchers is presented to investigate the detailed nature of bed load motion in terms of the mechanics of saltation. The model is deterministic in the computation of particle trajectory, but probabilistic in terms of bed collision. This probabilistic element arises from the random structure of the bed. In the present analysis, a stochastic treatment of collision allows for a fairly complete model of saltation in water. Characteristic quantities, including the sediment transport rate itself, are derived and compared with experimental data. The model yields good agreement with data with a minimum of assumptions. The assumed three‐dimensional random structure of the bed is a unique feature of the present model. It allows for the prediction of several phenomena that cannot be obtained via the assumption of a regular two‐dimensional bed structure.

Experiments are conducted on the evolution by a bubble plume of the density distribution in a closed container. Two initial distributions are studied, two uniform layers and a continuous linear stratification. The structure of the induced liquid plume and layering of the environment are observed from shadowgraphs. The density profile in the environment is measured. In a step stratification, the liquid plume impinges on the interface and splits into two flows. A small flux is carried in the wake of the bubbles into the upper layer, but the larger flux entraines some upper‐layer fluid and then flows into the top of the lower layer. If the distribution is initially linear then three layers are formed. The upper and lower are similar to the two layers of a step distribution. In the middle layer the stratification is linear, and the liquid plume is of constant diameter. This linear layer reduces in thickness with time, and after it is gone, the evolution is that of the step distribution.

The paper deals with the characterization and the control of transient uplift generated by turbulent pressure fluctuations in spillway stilling basins. Theoretical and experimental analyses are presented, defining the maximum instantaneous uplift acting on the linings, which lead to a design criterion for suitable floor slabs. The thickness of concrete linings (not accounting for the possible contribution of anchors) required for stability is defined in this paper as a function of (1) The incoming kinetic head and/or Froude number; (2) a dimensionless function of the longitudinal and transversal dimensions of the lining slabs, the areal extent of extreme turbulent pressure fluctuations, and the anisotropic integral scales of the instantaneous spatial correlation of pressure fluctuations; (3) the ratio of specific weights of water and immersed concrete; and (4) a dimensionless pressure coefficient qualifying the maximum difference of positive and negative pressure fluctuations about the (stationary) mean value. It is concluded that safe design of protection works is significantly affected by the structure of instantaneous spatial distribution of pressures at the bottom of hydraulic jumps. Consideration of this factor leads to recommendations for the design of slabs in stilling basins.

The shape and dimensions of the cross section of a straight threshold channel are obtained by numerically solving the momentum balance equation for the fluid and the force balance equation for a sediment particle at the condition of impending motion. The first equation accounts for lateral momentum diffusion from the center of the channel toward its banks, that is caused by Reynolds stresses. The resulting bank profile is accurately described by a fifth‐degree polynomial that is quite different from the cosine, parabolic, or exponential profiles that have been traditionally assumed to represent the shape of a threshold bank. In fact, it is demonstrated here that the cosine and parabolic bank shapes are unstable, while the exponential is overly stable. Equations for the design of threshold channel cross sections are presented. Channel dimensions predicted by these equations are in good agreement with values obtained from laboratory experiments.

Laboratory data for local scour depth at bridge abutments are presented. These include sufficient data to demonstrate the effects on scour depth of abutment length, flow depth, and abutment shape and alignment. In addition, some data for the effects of sediment characteristics, flow intensity, and approach channel geometry are given. The data are rationalized in terms of the ratio of abutment length to flow depth, L/y. Two limiting cases are identified. For large and small values of L/y, the abutments are termed long and short, respectively, and it is shown that the maximum scour depths measured in the laboratory for these cases are 2L and 10 y. Most actual cases of abutment scour lie between the limiting cases, and it is demonstrated that, in such situations, scour depth is proportional to the square root of the product L/y. These relationships are formulated in a simple design method, which is used on some illustrative examples. Limitations to the design method are discussed.