Journal of Hydraulic Engineering

A mathematical model is presented to simulate water temperatures of one‐dimensional unsteady flows in a river‐lake system having time‐varying lateral inflows and several cross‐sectional changes along its length. In the hydraulic part initial steady state conditions are computed by numerically integrating the gradually‐varied flow equation using the fourth‐order Runge‐Kutta method and the unsteady flow conditions by integrating the one‐dimensional Saint Venant equations using an implicit finite difference method. An upwind implicit finite difference scheme is used to solve the partial differential equation governing the water temperature of unsteady flows. Comparison of the computed water temperatures with those measured on the Columbia and Kootenay Rivers show good agreement.

The use of interpolations in time, rather than the more widely used spatial interpolations, demonstrates several benefits in the application of the method of characteristics to wave problems in hydraulics. Two such time‐line schemes are presented and analyzed in the solution of a linearized water hammer problem. Reachback time‐line interpolations, where the characteristic lines are projected back before the current time step, demonstrate less damping than the corresponding spatial interpolation scheme at the same discretization. Implicit time‐line interpolations, where the characteristic line is projected into the current time step, permit relaxation of the restrictive time step required by the Courant condition. An error analysis and numerical experiments demonstrate the degree of damping and dispersion introduced by both reachback and implicit time‐line interpolation methods. The error analyses and their verification may encourage further application of these methods in appropriate problem domains.

The hydraulic jump has been extensively studied using physical models; however, very little data are available for the internal flow in prototype hydraulic jump stilling basins. A mathematical' model has been developed to help in the prediction of prototype performance from physical models. The technique used here is an extension of the strip integral method of R. Narayanan. The strip integral method uses velocity shape functions to permit the partial integration of the equations of motion. A Gaussian velocity distribution is used in the mixing zone and the power law is used in the inner layer. The mathematical model includes the bed shear, turbulent shear, the potential core, entrained air, centrifugal force and turbulence pressure. The model gives a fairly good prediction of the jump length, roller length, velocity distribution, water surface and pressure at the bed. The prediction of the growth of the boundary layer was not very good. The model indicated that the main effect of entrained air is the bulking of flow. The effects of centrifugal forces and turbulence pressures increase as $F_1^2$ and start to become significant for $F_1<5\text{.}$

Mathematical models based on the Saint‐Venant equations for open channel flow often encounter serious difficulties when applied to natural channels. On the other hand, the approximate flood routing models used in practice may yield results which are in gross error. In this work five mathematical models are constructed based on equations ranging from the complete dynamic system to a simple normal‐depth kinematic wave equation. The results of the models are compared between themselves and with experimental data, in the form of free‐surface profiles and stage hydrographs. The models are then converted to dimensionless form, which reduces the number of independent parameters controlling their solution. The results of these calculations are presented in the form of dimensionless plots of maximum flood depth and time vs. distance along the channel, for various levels of truncation of the open‐channel flow equations. Estimates for the permissible range of application of the simplified routing models are given, and recommendations are made for their judicious application.

The flow characteristics of a turbulent underflow in a horizontal channel of limited depth are explored analytically and experimentally under a variety of upstream and downstream control conditions. Freshwater and salt water are used in the physical model to simulate a stable density stratification, and a contraction is adopted as a downstream control device. Three typical flow situations, respectively, distinguished as drowned internal jump, free internal jump, and flow instability, are observed depending on the nature of the interaction between the upstream and downstream controls. The hydraulic properties of each one of the flow situations are reviewed, and the experimental observations are compared with the results obtained from a hydraulic analysis. Maximum turbulent mixing is demonstrated to occur at a particular value of the discharge densimetric Froude number. Further increase of discharge densimetric Froude number is expected to result in a decrease in turbulent mixing and eventually to provoke a flow instability.

The effects of boundary permeability and undulation in turbulent pipe flow are observed in an air flow conduit lined with a granular permeable material. Boundary pressure, mean velocity distribution, and longitudinal turbulence intensity were measured at several sections along the sinusoidal boundary. The observed boundary pressure shows a phase shift relative to the boundary wave form and a magnitude similar to that found from a simple potential flow analysis. The mean velocity distributions differ drastically from those observed in impervious wavy pipes, implying an important effect of the aspiration from the porous boundary in the lee of the wave form. The observations of turbulence intensity show that boundary permeability dramatically alters the turbulence structure as compared to an impervious wavy pipe. In the permeable pipe, a zone of high intensity turbulence develops in the lee of the boundary wave form and expands to the central portion of the pipe. For impervious wavy pipes, the high turbulence intensity occurs only near the boundary as in the case of a straight pipe.

A general velocity distribution consisting of a logarithmic profile plus a wake function has been used to investigate the effect of suspended sediment on flow resistance. From the experimental evidence that the wake constant increases when sediment is in suspension, it is shown that flow resistance must be reduced. It is shown that a flow with suspended sediments should have lower velocities near the boundary and higher velocities further away from the boundary than a clear water flow, given that the discharge and slope are identical.