Journal of Hydraulic Engineering

A nonuniform, fixed‐boundary, channel‐bend flow is formulated as two coupled sets of equations, each comprising two simultaneous equations, derived from the cross‐section‐integrated equations (in circular cylindrical coordinates) of conservation of flux of moment of momentum, and the depth‐integrated momentum and continuity equations. Boundary shear stresses are related to the primary‐flow shear stress and the primary and secondary velocities. The equations are solved numerically and the results found to be in satisfactory agreement with experimental data. The moment formulation elucidates the interplay among the secondary and primary flows, notably the observed flattening of primary velocity profiles, over‐shoot of the secondary flow, radial redistribution of the depth‐averaged primary velocity, and the secondary translational velocity. The erodible‐bed case is treated in the companion paper.

The hybrid integral‐differential moment and momentum model of fixed‐boundary channel‐bend flows developed in the companion paper is extended to erodible‐bed flows. The $\theta$ unit sediment discharge is formulated as a power of local depth‐averaged $\theta$ velocity; and the $r$ unit sediment discharge as proportional to the ratio of the excess of the $r$ bed shear stress, over the radial component of the submerged weight of the sediment bed layer, to the $\theta$ bed shear stress. Bed topography is computed from two‐dimensional (2D) sediment continuity. The resulting two coupled sets of simultaneous equations are solved numerically. The model yields accurate predictions of the velocities (primary and secondary), and 2D bed topography. The average radial bed slope, primary velocity‐profile exponent, and secondary rotational and translational velocities exhibit much larger variations along the channel than in fixed‐boundary bends. The increased variation and oscillatory behavior of these quantities are elucidated from moment‐of‐momentum conservation and interaction of bed topography and flow.

Two approaches are generally taken in deriving the stability limit for the Froude number $\left(F_s\right)$ for laminar sheet flow: The first approach uses the Orr‐Sommerfeld equation, while the second uses the cross‐section‐averaged equations of continuity and motion. Because both approaches are based on shallow‐water theory, the values of $F_s$ obtained from both approaches should be identical, yet in the literature they are not. This suggests that a defect exists in at least one of the two approaches. After examining the governing equations used in both approaches, one finds that the existing cross‐section‐averaged equation of motion is dependent on the frame of reference. To correct this defect, one can formulate a frame‐independent equation of motion relative to a coordinate system moving with constant velocity, then derive a new expression for $F_s$ that is generally applicable to both laminar and turbulent flows in prismatic channels of arbitrary cross‐sectional geometry. For laminar sheet flow, the new expression for $F_s$ obtained from the second approach yields $F_s\simeq 0.527$, which agrees with that obtained from the first approach.

As a preliminary phase in a predictive mass‐transport modeling study, ground‐water samples were collected from an unconfined, landfill‐contaminated aquifer over a five‐month time period. The dilution effect of recharge from a single precipitation event caused short‐term decreases of up to 56% in chloride concentrations measured at individual monitoring wells. A mass‐transport model was calibrated to contaminant plumes inferred from different sets of measured chloride concentrations. Sets of calibration parameters thereby attained were used to make separate, 50‐year contaminant distribution predictions. The resulting distributions contrasted significantly, particularly at the downgradient margin of the landfill where predicted chloride concentrations ranged from approximately 150–230 ppm. The results of this study suggest that: (1) Contaminant concentrations in shallow unconfined aquifers can fluctuate significantly in response to short‐term changes in hydrologic conditions; and (2) the selection of a set of “observed” concentrations, which is representative of an overall stage in the evolution of a plume, rather than indicative of short‐term extremes in contaminant concentrations, is an important step in calibrating a mass‐transport model.

A simple method is presented for determining aquifer transmissibility, $T$, and storativity, $S$, for a single well pumping in the vicinity of an impermeable barrier in a confined aquifer. The method makes use of the long‐term log‐linear relationship between time and drawdown of the piezometric surface at an observation well to determine $T$. The early time drawdown is then used for direct solution of $S$. The distance from the image well to the observation well, $r_2$, is determined using these estimates of $T$ and $S$ as well as the long‐term log‐linear drawdown. For the synthetic example used, $T$ and $S$ were determined within an error of 1% of their true values and $r_2$ was estimated within an error of 4.5% of the actual distance in the absence of errors in measured drawdown or time.

A stochastic model of particulate transport in a porous medium is presented. The model includes the influences of the random movements of the particulates and the fluid in its development, which in turn are based on measurable physical and hydraulic characteristics of the situation under consideration. Model performance was evaluated using 40 sets of data collected under laboratory conditions. Results were found to be quite good for the turbulent flow conditions in the tests. Comparison with an earlier model presented by Sakthivadivel for Darcy flow conditions showed the stochastic model much better for the turbulent flow conditions tested.

The Agricultural Research Service (ARS)‐type low‐drop structure was developed and utilized to stabilize incising channels in northern Mississippi. The structure consists of a sheet pile stabilizer weir and stilling basin protected by riprap. Field inspections and laboratory tests have indicated that the current sizing criteria for riprap protection in the impingement area of the stilling basin may be inadequate during flood events. A 1:12 Froude scale model study was conducted with prototype median riprap sizes ranging from 0.61 m (23.9 in.) to 0.JHEND8 119 m (36.5 in.), prototype discharges ranging from JHEND8 119.7 m³/s (3,308 cfs) to 221.1 m³/s7,801 cfs), and submergences ranging from 0.41 to 1033. The ratio of the approach unit discharge and median rock size was related to the submergence for calculating the stable riprap size. The unit discharge at the weir and the submergence appear to be indicators of riprap stability in the stilling basin..