1. An approximation of accuracy can be given for the erosion process.
2. Concerning free shear mixing, a qualitative agreement is found and the specific demand for more relevant experiments is described.
3. Deposition is revealed to be a non stationary process.

KEYWORDS: Estuary, Cohesive Sediments, Transport Simulations, Turbulence Model Verification, Accuracy Estimates.

INTRODUCTION

When comparing field measurements with numerical simulations the different models contribute in different ways.

The wish to quantify the uncertainty that is introduced into simulations of estuarine flows by the turbulence model is fed from three sources. When performing research on the neighbouring empirical models it is helpful to know how much of the deviation between simulation and reality is attributed to the turbulence model. In practical planning a quantified uncertainty is better manageable. When investigating yet unexplained phenomena of siltation with the help of numerical models, the accuracy of the model needs to be quantified to be sure about the identified reasons.

CONCEPT

Field measurements from real estuaries are limited. The mere size and the rough conditions prevent a sufficient spatial resolution and a good accuracy of the measurements. The need to calibrate four empirical models consumes so many data that an insight into one of the empirical models is mostly impossible. Therefor laboratory measurements are used for comparison in this study.

The comparison between schematic laboratory flows and their numerical simulation does not directly lead to an accuracy estimate for complex flows in real estuaries. Therefore it has to be assumed, that the schematic situations span a space and the real flows are somewhere inside this space. Here open channel flows in estuaries are considered to lie in between purely wall bounded and completely free shear flows.

Density stratification in estuaries result from differences in the concentration of suspended sediments and salt. The turbulence damping caused by stable density stratification needs to be accounted for by the turbulence model.

The two-phase effects between the solid sediment flocs and the liquid water can be estimated on the basis of the Stokes-number. The smaller the Stokes-number, the smaller the slip between solid and fluid phase. Here the Stokes-number is calculated with the Kolmogorov-timescale that occurs in typical estuarine situations and sediment flocs that are observed in real estuaries. Experiments [7] show, that higher Stokes-number are necessary to obtain a small influence. Therefor two-phase effects are neglected here.

Not all quantities calculated by numerical models have the same accuracy. For reasons of technical relevance the sediment transport rate is chosen as a measure here. Boundary conditions are chosen in a way to be typical for real estuaries like the Weser estuary.

IMPLEMENTATION

In neutrally stratified fluids free shear in a mixing layer results in a constant growth rate of the mixing zone. In stably stratified fluids the growth of the vertical thickness of the mixing layer reaches an end. The growth seizes gradually and the thickness becomes constant further downstream. In the state where the stably stratified mixing layer is not growing any more, the shear can no longer withdraw enough energy from the mean flow to lift the lower heavier fluid. This collapsed state can be characterised by a gradient Richardson number. Chu and Baddour [3] found values between Ri_g = 0.302 ... 0.469 in their experiments. Fig. 2 shows calculations with the standard k- turbulence model for a neutrally stratified and a stably stratified mixing layer. The collapse of the mixing layer growth is obvious in the stably stratified case. The gradient Richardson number of Ri_g = 0.454 lies within the experimental values. The experiments of Slessor [8] reveal that mixing layers can be significantly influenced by the inflow conditions. Therefore a practically relevant accuracy assessment would need a comparison with experiment representing estuarine conditions, i.e. turbulent inflow.

No experiments were found for the case where horizontal free shear meets vertical stratification, which is interesting for the shear layer developing between harbour and river.

TRANSPORT RATES

In the first test, that type of flow was simulated, which causes erosion of cohesive sediments (0.9 m/s depth averaged velocity, 10 m water depth).These are typical values for the navigation channel of the Weser estuary when the tide is running at maximum velocity. To simulate erosion, a constant flux of sediment concentration enters the flow domain across the bottom boundary. The density stratification introduced by the sediment causes a reduction of eddy viscosity and a decrease in the bottom friction. Here again two calculations are done, one with the Galperin stability function and the other without it. The bottom friction at the end of the channel is 15% higher when calculated without the stability function. The result obtained with the stability function is not necessarily exact, but it is better fitted to this type of flow and therefore can be used to find an estimate of the accuracy achievable with the standard k- turbulence model. When these values of bottom friction are taken as an input for the Krone erosion model, the mass flux eroded from the bottom is 42% higher for the calculation without the stability function.

The flow deposits sediments when slowing down around slack water. In order to investigate deposition, the situation in which Winterwerp [10] found a turbulence collapse is recalculated. The flow has a depth averaged velocity of 0.2 m/s in a 16 m deep channel. The inflow is loaded with 0.024 g/l sediment that settles at a speed of 0.5 mm/s. Fig. 3a shows the concentration calculated by Winterwerp with a 1DV model. Each vertical line of Fig. 3a represents one timestep. The 1D discretisation neglects horizontal gradients. In Fig. 3b four instances from the calculations done with "casu" are displayed. The turbulent kinetic energy k is plotted for the horizontal mid section. A transient formation of turbulence bubbles is calculated. The frequency of this bubbling is lower than the tidal frequency. In the tidal cycle slow velocities around slack water persist only a few minutes. They are not driven by a constant surface gradient but undergo acceleration or deceleration. The timescale of turbulence is above 30 sec. for places higher than 1.5 m above the bottom, according to the values Nezu and Nakagawa [6] give for turbulence in neutrally stratified 2D open channel flow.

CONCLUSION

The effect of turbulence damping caused by stable density stratification on vertical free shear mixing can be realistically simulated with the standard k- turbulence model. The differences between experiments and the flow in real estuaries prevent giving a reliable number of accuracy. To judge the interaction of horizontal shear and vertical density stratification, experimental evidence is lacking.

Flows depositing sediment in deep channels can not be compared with stationary channel flows. It is no longer possible to subtract the turbulent movement from the mean flow by Reynolds-averaging. Tide to tide variations in deposition may occur, not only depending on the tidal forcing. The calculations show a re-ignition of turbulence starting at the rough bottom. The general trend of the turbulence in such slow flows is to fade out. Small disturbances like bottom inhomogeneities might have large effect on the deposition. This would explain the large differences in deposition rates observed in the Weser estuary.

REFERENCES