A mathematical model of two-layer stratified flow is presented. One-dimensional and radial flows are considered. The numerical solution is based on the method of characteristics. The reliability of the mathematical model is investigated with the help of one-dimensional laboratory experiments and a hydraulic model of a real bay. Agreement between calculations and experiments is achieved but since the numerical method of solution does not allow Courant numbers exceeding one the calculations are time consuming.

Based on the analysis of laboratory and field data according to similarity methods, it is found that the following variables are important in determining the bed forms of sand bed streams: flow depth d and mean velocity, V, density, ρ, and viscosity, ν, of the water; median size of sediment d₅₀ and the acceleration of gravity, g. The data thus analyzed are presented in six graphs of d/d₅₀ against Froude number F for each of six ranges of R_g = d₅₀ (gd₅₀)¹/²/ν which accounts for the effects of sediment size and water viscosity or temperature. Flume data indicate that the size distribution of the bed sediment also affects bed form, but the extent of this effect could not be determined. The effect of grain shape and channel width could not be determined with the available data. A relation is developed between R_g and u⋆( gd₅₀)¹/² which predicts quite well the occurrence of ripple and dune beds in which ^⋆ = shear velocity.

A two-space dimensional mathematical model of computing steady flow distributions and water levels near contractions in natural waterways (e.g., highway bridges) is developed. The two-dimensional model provides a theoretical basis for such one-dimensional concepts as expansion losses by accounting for all losses by friction along properly located two-dimensional streamlines. The finite element solution algorithm locates these streamlines and integrates energy loss due to friction along them. Stagnation zones downstream from contractions can be predicted and located by the algorithm. A necessary condition for separation is that the velocity head at the point of separation exceeds head loss due to friction along some possible path for a free streamline. The absence of separation which is associated with scouring currents along the downstream side of bridge approach embankments can be predicted where the necessary condition is not satisfied.

The pressures and flows in a water distribution system are solved by a modified Newton-Raphson method. This method removes the problems of pressure initialization and greatly increases convergence particularly in systems containing pumps or pressure reducing valves, or both. The program will automatically amend pipe resistances so that calculated pressures agree with measured pressures. Both the system equations and those for the automatic balancing routine are derived. Examples are shown of the improved rate of convergence and the automatic balancing.

The transient controlling gate and valve-stroking concept recently developed and verified by Streeter and Wylie at the University of Michigan is extended to the operation of the Sacramento weir gates during periods of high flows in the river. The weir is located several miles upstream from Sacramento, Calif., and is only operated at high stages to maintain a 29.0-ft (8.8-m) maximum stage at the Sacramento I Street gage. High inflow from the regulated American River between the weir and the gage often causes flow to reverse in a portion of the Sacramento River but this complication is easily handled by the method of characteristics. This paper is divided into three sections in which the following are analyzed: (1) the basic theory of unsteady flow and the development and application of the characteristic equations; (2) the geometry of the lower Sacramento River; and (3) transient control of the lower Sacramento River.

Typical solutions for conduits of various shapes are presented. For turbulent flow the solution is based on the validity of the use of an eddy viscosity concept to account for the Reynolds stresses. The solution procedure allows the quantitative isolation of the effects of secondary flows on the velocity and boundary shear distributions. However, the effect of secondary flows has been considered herein only for a square duct because of the present dearth of knowledge about their distribution in other conduit shapes. For the square duct the solution shows that the secondary flow is responsible for a 6% increase in the friction factor and an 11% decrease in the maximum boundary shear. The inclusion of the secondary flows in the analysis allows satisfactory agreement to be achieved between the calculated and experimental velocity and boundary shear distributions and illustrates the limitations that must accompany any solution that ignores the effects of secondary flows.

Reduction of computation in reservoir operation optimization problems can be made through a modification of the optimization technique instead of limiting development of the system models. Considerations are presented herein which lead to the development of a modified application of deterministic optimization techniques. The modification enables reduction of computation to take place while achieving results that approximate the optimum. The modified application of dynamic programming is made for a single reservoir system illustrating the technique and the achievement of near optimum performance.

The paper illustrates a systematic method to evaluate alternative water resources systems with particular reference to the development of water resources in the Lower Mekong Basin. This method has been inspired by Kazanowski's cost-effectiveness approach. In the evaluation process, the development goals that may be of an economic or social nature are explicitly stated and transformed into specifications. The approach requires that distinct alternatives be sought and compared using both quantitative and qualitative criteria. A decision to select an alternative can be reached by use of a fixed-cost or fixed-effectiveness approach. Sensitivity analysis in the approach takes into account uncertainty in hydrologic and economic data.