Studies of Turbulence in Shallow Sediment Laden Flow With Superimposed Rainfall
B. J. Barfield
The structure of turbulence has been shown to affect the sediment carrying capability of streams. Due to the random nature of turbulence, sediment movement was analyzed as a stochastic process. Starting with the Langevin equation modified for a turbulent medium, a partial differential equation was developed as a mathematical model which describes the change in sediment concentration with time and space for two dimensional open channel flow with isotropic turbulence. The input parameters to the partial differential equation were the particle fall velocity and the turbulent diffusion coefficient. The diffusion coefficient used was the product of the mean square velocity and the Eulerian time scale of turbulence.
A 4O ft. recirculating research flume was used for the experimental investigations. The RMS velocity and Eulerian time scale profiles were determined by use of a hot-film anemometer and a random signal correlator. The effect of rainfall on the RMS velocities and time scale profiles was observed. Sediment concentration profiles were measured by withdrawing samples from the flow and were compared with values predicted by the derived mathematical model.
The history of mankind is interwoven with the search for suitable water supplies. From the early Egyptian empire along the Nile to the modern legal battles over water in the western United States and even to the war over land near the Jordan river, the availability of usable water has exerted a strong influence on the activities of mankind.
In man's quest for usable supplies of water, reservoirs have been built to store surplus runoff. One of the major problems encountered in reservoir storage of water is the loss of storage space due to siltation by eroded sediment. Of the 33.6 million acre-feet of initial storage available in reservoirs of the Upper Colorado River Basin, 17.8 million acre-feet have been included for sediment storage pools (6). Through control of sedimentation, this costly addition could be reduced.
One of the problems encountered in the removal of sediment from water is the inability to predict sediment profiles as a function of flow conditions. This difficulty generally limits sediment removal studies to trial and error procedures.
The movement of sediment from the initial point of detachment to its place of final deposition is effected through several processes. After its initial detachment, a sediment particle moves along in a shaLlow channel either as suspended load or as bed load. Bed load can be defined as that material which is moved by rolling or sliding at the stream bed. Suspended material is that material which is intermittently or continously detached from the bed, placed in suspension by the fluid turbulence, and redeposited in the channel.
After a particle is eroded and begins to be transported the flow is quite often subjected to intense superimposed rainfall containing a large amount of energy which must be dissipated. This energy dissipation can alter the sediment carrying capacity of the flow.
As these shallow channels merge to form deeper channels, the sediment is still carried as suspended material or bed load. It is conceivable that superimposed rainfall would have little effect on the sediment carrying capability in the deeper flows. The sediment is further carried along in the stream until it reaches the reservoir or large body of water where the absence of turbulence allows it to be finally deposited.
Most of the past research on detachment and movement of sediment has been oriented toward a look at average conditions and gross mechanics. The research reported herein had as its purpose a stochastic description of the movement of particles in a turbulent medium. Through this theoretical description, the movement of an ensemble of particles is investigated and finally an equation is formulated to describe the concentration of sediment as a function of time and space in a turbulent medium. Experimental investigations of the vaildity of the mathematical model were conducted in a 4O ft. research flume with rainfall simulation capability.
The research had as a further objective the study of the effect of superimposed rainfall on the sediment diffusive capability of shallow open channel flow.
Summary & Conclusions
The structure of turbulence in streams affects the suspended sediment carrying capability. Due to the random nature of turbulence, sediment movement can be analyzed as a stochastic process. Starting with the Langevin equation modified for a turbulent medium, a partial differential equation was developed which describes the change in sediment concentration with time and space in two dimensional open channel flow with isotropic turbulence. The input parameters to the equation were the fall velocity of the sediment particles and the turbulent diffusion coefficient. The diffusion coefficient used was the product of the mean square velocity and the Eulerian time scale of turbulence.
A 4O ft. recirculating research flume was used for the experimental investigations. The RMS velocity and Eulerian time scale profiles were determined by the use of a hot-film anemometer and a random signal correlator. The effect of rainfall on the RMS velocities and time scales was observed. Sediment concentration profiles were measured by withdrawing samples from the flow and were compared with the values predicted by the partial differential equation.
The following conclusions were made from this study:
- The Eulerian time scale is relatively constant with depth. In all of the observed profiles, no trend with depth could be detected.
- Rainfall on a free water surface tends to decrease the time scale of turbulence. This is most evident in shallow flows where the energy input of the rainfall has a large effect on the total energy of the system. The decrease in time scale will result in an increase in viscous dissipation of energy.
- The diffusion coefficient is a decreasing function of depth. An exponential function appears to best describe the relationship between diffusion and depth with the exception of points near the channel bed where a large amount of variation was observed.
- Turbulent diffusion is highly sensitive to RMS fluid velocities. Since the diffusion coefficient is proportional to the square of the RMS velocity, a variation in the RMS velocity is magnified in the equation for diffusion.
As a result of this investigation, the following areas are recommended for future study:
- The optimum sampling time for measuring representative correlograms should be investigated. Since the use of the correlograms assumes that the time average is the statistical average, a study of the times required for an accurate estimate of the correlogram of turbulent velocity fluctuations is needed. A report on the variability of correlograms with differing sample times would also be valuable in planning future research.
- Knowledge of the variation in RMS velocities between a smooth bed and a rough moveable bed would be useful in an analysis of the effects of a rough bed on turbulent diffusion and on the energy balance of a system. Past studies have been conducted on a fixed bed or an artificially roughened stationary bed. An alluvial bed does not fit either category.
- The structure of the diffusion coefficient at low relative depths warrants further study since it is closely tied to entrainment of sediment from the bed.
- An investigation of the possibility of describing the diffusion coefficient as a function of flow conditions should be undertaken. This information would eliminate the necessity of measuring the turbulence characteristics of a channel before attempting to predict the sediment carrying capability.
- The structure of the viscous drag coefficient for turbulent flow conditions should be investigated. Included in this study should be an investigation of the effects of turbulence on the fall velocity of a particle. In the alternate derivation of the theoretical model given in Chapter III, a Gaussian distribution of particle displacements with a non-zero mean equal to the product of the mean fall velocity and time was assumed. The value of the mean fall velocity was determined using Stokes law for a still medium; however, it is probable that the mean fall velocity is dependent on turbulent intensity.