Jens Wyrwa
ABSTRACT Transport of fine and cohesive sediments in estuaries interacts with man made structures. To manage siltation, numerical simulation is used as a tool for planning. Between numerical simulations and measurements there are still differences, some of which are not fully understood yet. The contribution of the turbulence model to these differences is quantified in this study. The transport rate of sediments is used as a measure for the differences. The turbulence model utilized in this study is the k- -model. Turbulence damping caused by stable density stratification is an important feature that needs to be covered by the turbulence model. Schematic situations of boundary and free shear flow in stably stratified fluids are compared with laboratory measurements. The results are:- An approximation of accuracy can be given for the erosion process.
- Concerning free shear mixing, a qualitative agreement is found and the specific demand for more relevant experiments is described.
- Deposition is revealed to be a non stationary process.
KEYWORDS: Estuary, Cohesive Sediments, Transport Simulations, Turbulence Model Verification, Accuracy Estimates.
INTRODUCTION Suspended fine sediment in estuaries consist of cohesive material (clay). These sediments have the tendency to gather in turbidity zones where they are frequently deposited and resuspended by the tidal movement. Man made structures like navigation channels and harbours in these zones are threatened by siltation. To minimise dredging costs and to safeguard navigation, a forecast of the behaviour of suspended sediments by numerical simulation is desirable.At least four empirical models need to be utilised to simulate suspended sediment transport: - A model that describes the settling velocity of cohesive sediments (flocculation model).
- A model that describes the erosional and depositional mass fluxes over the bottom boundary of the fluid volume.
- A model that describes the consolidation which takes place inside the resting sediment.
- A turbulence model describing turbulent mixing of momentum and mass.
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 This study focuses onto the aspect of accuracy evaluation and develops methods for testing. The k- -turbulence-model as the most popular 2-equation-model is just the first model for this investigation.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 The 3D-hydrostatic model coded in this study is based on the ideas of Casulli et al. [2]. A standard k- turbulence model [10] including the buoyancy term is added. The code is named "casu" and can be downloaded under http://www.wyrwa.de/casu. The input data for the test cases discussed here can be downloaded there as well.
At the ERCOFTAC workshop Oct. 2001 in Darmstadt different numerical results were presented for the same turbulence model (
STABLE STRATIFIED FLOWS Stably stratified wall boundary layers can be produced by cooling the bottom of a wind tunnel. Experiments of this kind have been performed for meteorological purposes [9]. The Monin-Obukhov similarity hypothesis [5] applies to the lower part of the wall boundary layer that is logarithmic in neutrally stratified flows. This hypothesis claims, that velocity profiles from flows with different strength of stratification fall on one curve, if the wall distance is scaled with the stability length L_{m} and the velocity is scaled with the friction velocity.
Fig. 1
shows two such curves that were given by Wier and Römer [9] and Businger [11] as a fit to their measurements.
Fig. 1
displays two calculations made with "casu". One is using the
k-
turbulence model in its standard form, the other additionally employs the stability function of Galperin [4]. Stability functions are empirical functions that introduce additional damping into the turbulence model. Burchard and Petersen [1] successfully tested the Galperin stability function for vertical mixing in different boundary layers. Here the results of the calculations with the stability function approximate the measurements better, too.
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 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 order to quantify sediment transport rates, two test calculations are made. Both use a model of a 2000 m long straight channel with a rectangular cross section and a horizontal bottom, which has no side wall friction.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.
CONCLUSION For the erosion process an estimate of accuracy is found by comparing with a model fitted better for this situation. In combination with the erosion model the error of 15% in the bottom friction results in a 42% error in the erosional mass flux.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 |

[1] |
Burchard, H.; Petersen, O. : Models of turbulence in the marine environment - a comparative study of two-equation models, (1999) Journ. Marine Syst. vol. 21, pp. 29-53 |

[2] |
Casulli, V. ; Cheng, R.T. : Semi-implicit finite difference methods for three-dimensional shallow water flow, (1992) Int. J. Num. Meth. Fluids, Vol. 15, pp 629-648 |

[3] |
Chu, V.H. Baddour, R.E. : Turbulent gravity-stratified shear flow (1984) J. Fluid Mech. vol. 138 pp 353-378 |

[4] |
Galperin, B.; Kantha, L.H.; Hassid, S.; Rosati, A. : A quasi-equilibrium turbulent energy model for geophysical flows, (1988) J. Atmos. Sci. 45, pp. 55-62 |

[5] |
Monin, A.S.; Obukhov, A.M. : Fundamentale Gesetzmäßigkeiten der turbulenten Vermischung in der bodennahen Schicht der Atmosphäre, (1958) in: Goering, H. (Hrsg.) Sammelband zur statistischen Turbulenz, S.199-226, Akademie-Verlag Berlin |

[6] |
Nezu, I. ; Nakagawa, H. : Turbulence in open-channel flow, (1993) IAHR Monograph, Balkema Rotterdam |

[7] |
Ruck, B. : Einfluß der Tracerteilchengröße auf die Signalinformation in der Laser-Doppler-Anemometrie, (1990) Technisches Messen, Heft 7/8, pp. 284-295 |

[8] |
Slessor, M.D.; Bond, C.L.; Dimotakis, P.E. : Turbulent shear-layer mixing at high Reynolds numbers: effects of inflow conditions, Journ. Fluid Mech., (1998), Vol. 376, pp115-138 |

[9] |
Wier, M. ; Römer, L. : Experimentelle Untersuchungen von stabil und instabil geschichteten turbulenten Plattengrenzschichten mit Bodenrauheit, (1987) Z. Flugwiss. Weltraumforsch. 11, pp78-86 |

[10] |
Winterwerp, J.C. : On the dynamics of high-concentrated mud suspensions, Thesis(1999) Technische Universiteit Delft |

[11] |
Businger, J.A.; Wyngaard, J.C.; Izumi,Y.; Bradley,E.F. : Flux profile relationships in the atmospheric surface layer, (1971) J. Athmos. Sci., 28, pp 181-189 |