Numerical Modelling of the flow and settling in a Trailing Suction Hopper Dredge

Transcription

1 Numerical Modelling of the flow and settling in a Trailing Suction Hopper Dredge C. van Rhee Faculty of Mechanical Engineering and Marine Technology, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. ABSTRACT A Trailing Suction Hopper Dredge (TSHD) is loaded by pumping a sand-water mixture in the cargo hold, the so-called hopper. Sand settles in the hopper and the excess water flows overboard. During the loading stage a part of the volume of sand pumped in the hopper will not settle but is lost with the overflow. In this paper a two-dimensional numerical model to simulate the flow and settling in a hopper will be presented. The model is based on the Reynolds Averaged Navier Stokes equations with a k-epsilon turbulence model. The model includes the influence of the overflow level of the hopper (moving water surface) and a moving sand bed due to the filling of the hopper. The influence of the particle size distribution (PSD) is included in the sediment transport equations. The computational results will be compared with experiments and prototype measurements. 1 INTRODUCTION A Trailing Suction Hopper Dredge (TSHD) (figure 1) is a sea-going vessel that is equipped with one or two suction pipes, which are lowered to the bottom during dredging. From the bottom a sand-water mixture is sucked up and discharged in the cargo hold, the so-called hopper. Sand settles and when the hopper is filled the ship sails to the locating where the load is being dumped or pumped ashore. When relative fine sand is loaded a part of the incoming sand will not settle, but will be lost overboard with the overflow. For production, sand quality and environmental reasons it is important to develop a method to predict these so-called overflow losses. For this reason a research program was initiated and financed by the Dutch dredging industry to improve knowledge and to develop a model that included the influencing factors. The research program consisted of three parts: Laboratory tests. Numerical modelling using one-dimensional (1DV) and two-dimensional approaches. Prototype verification of the numerical models. In this paper the developed two-dimensional numerical model is described. The model results are compared with laboratory and prototype tests.

2 Figure 1. Trailing Suction Hopper Dredge HAM PROCESS DESCRIPTION The research programme started with model hopper sedimentation tests (8). During these large-scale model tests a flow pattern was observed that could be schematised as indicated in figure 2. The inflow is located at the left side and the overflow at the right side of the hopper. The hopper area can be divided into 5 different sections: Inflow section, settled sand, density current over settled bed, horizontal flow at surface towards the overflow and the suspension in the remaining area. Figure 2. Schematised flow field inside hopper In the inflow section the incoming mixture flows towards the bottom owing to the high density. An erosion crater is formed below the inflow area. Over the sandbed a density current flows from which sand will settle. Depending on the operational parameters a part of the incoming sediment will not settle and will move upward into suspension. At the water surface a relative strong horizontal flow of low concentrated mixture towards the overflow is present.

3 2.1 Existing hopper sedimentation models Based on Camp s theory In the past a number of models to describe the hopper sedimentation process are published. These models were all based on the ideal settlement basin theory of Camp (1946) and Dobbins (1944). In this theory the hopper is divided in three areas: the inflow, outflow and settling sec-tion. It is assumed that the mixture flows horizontally from the inflow to the overflow section. Both the flowfield as the diffusion coefficient are based on a velocity distribution (uniform or logarithmic) over the total depth. Vlasblom and Miedema (1995) extended this theory with the influence of hindered settling and the influence of the rising sand bed. Due to the rising sand bed, horizontal velocity increases during loading which leads to increased scour. The effect on the overflow losses is however limited since the assumed velocity distribution leads to a low bottom shear stress apart from the very last moment of the loading stage The one-dimensional vertical hopper sedimentation model The laboratory test revealed that in a large part of the hopper the flow was in fact in vertical direction instead of the horizontal (figure 2). This motivated the development of a onedimensional model in vertical direction (8,9). The 1DV model supplies sand from the bottom equally distributed over the bottom (fed by the density current) and the overflow is located at the top. The model includes the important effect of the vertical velocity component and the mutual interaction of the different grain sizes of the particle size distribution in a relative simple way. The latter effect is totally absent in the Camp model; every fraction is calculated independently. The 1DV model was capable to simulate the laboratory hopper sedimentation tests quite well. 3 TWO DIMENSIONAL HOPPER SEDIMENTATION MODEL 3.1 Introduction This chapter presents the 2D model that has been developed to simulate the sedimentation process in a hopper. One might question why a two-dimensional model should be developed when the results of the above mentioned one-dimensional model already showed a good agreement with the experiments. The motivation is that although the 1D model predicts the experiments quite well, it is unsure whether the comparison on prototype scale would be as favourable. This is caused by the absence of horizontal transport in the 1D model. The horizontal transport is accompanied with a horizontal velocity near the bottom which can reduce sedimentation (7). This phenomenon did not play a major role during the experiments due to scale effects. 3.2 Summary of the process conditions to be included in the model In chapter 2 a very brief phenomenological description of the sedimentation process was given. Hereafter the most important issues influencing the process and therefore to be included in the numerical model will be summarised. Non-stationary process Due to the relative high sedimentation velocity the filling time will be of the same order of magnitude as the adaptation time of the process. So in most cases the hopper is fully loaded before a stationary situation is established. This discerns the loading process from the sedimentation process in clarifiers used in sewage and water treatment where the settled

4 sediment is scraped to a central location resulting in a constant water depth. This enables a stationary situation (as long as the inflow conditions remain constant obviously). Non-Hydrostatic pressure distribution The length and depth scales are of the same magnitude in a hopper. This implies that a correct representation of the flow in corners is important. A hydrostatic pressure distribution (commonly used for estuarine models) is therefore not permitted since this will lead to a very unrealistic flowfield in the corners. The hydrodynamic pressure must be taken into account. Moving Bed Since it is the intention to fully load the hopper with sediment, the interface between the settled sand and the mixture above must be movable, hence a morphological model is needed. The variation in flow depth is large, from maximum depth upto almost zero. Moving water surface The location of the water surface must be variable due to variation of the overflow level or filling of the hopper during the first phase of the loading when the water level is situated below the overflow level. The water level influences the sedimentation process in several ways. The most important influence is the entrainment in the inflow section. When the water level increases the jet in the inflow section must travel over a longer distance before the bottom is reached. This will lead to lower concentration near the bottom. Influence of sediment characteristics The results of the 1DV model have shown the importance of the shape of the particle size distribution (9). For the 2D model the same influence is expected and must hence be included in the model. Erosion / sedimentation boundary condition The sedimentation velocity depends on concentration, grain size and the local flow velocity expressed as turbulent energy or bottom shear stress. A correct description of this boundary condition is vital for the proper functioning of the model. Influence of discharge system A big advantage of the 2D model compared with the one-dimensional model is the possibility to include the influence of the inflow system in the model. The following variations are possible: Number of loading points, location of the inflow point(s), inflow velocity and turbulence intensity. Influence of overflow system The 2D model enables as well the influence of the overflow system on the process. The location, height and the number of overflows can be varied. 3.3 Motivation of the mixture model The equations used for the 2D model will be presented below. An Eulerian approach is followed. Since the model deals with a mixture of sand and water a multiphase flow approach looks obvious. In such a simulation the continuity and momentum equation are solved for every phase. In this case every fraction must be considered as a single phase. To represent a Particle Size Distribution accurately this would lead to order of ten different phases. Problem is the difficult (and often unknown) interaction between the different phases, which needs to be modelled. Despite of the increased computing power, in industrial applications multiphase

5 simulations are still often abandoned because of the long computing times, especially in the case of several dispersed phases. Complete multiphase simulations are also often numerically unstable and simplified forms of the inter-phase forces are often needed (14). Various approximations have been developed in order to avoid full multiphase simulations. In Civil Engineering circumstances dispersed concentration is often low and emphasis is given to modelling of the water flow. When the flow field is solved the transportation of solids is computed using a transport equation for the sediment. Often this involves a one-way coupling: the density variations are not used in the momentum equations. The sediment is in fact regarded as neutrally buoyant solvent. In a hopper sand concentrations are too large to ignore density variations. On the other hand, a complex numerical model for hopper sedimentation only has added value when relative fine sand is loaded. In case of coarse sands everything settles and even the simplest model will predict that outcome. For fine sands the Stokes number is low, which means that the sand grains are following the mean flow. In this case the mixture model can be applied. The mixture model (or algebraic slip model) preserves the essential multiphase character of the flow, although it is an approximation of the full Eulerian model. It is based on the assumption of a local equilibrium between the continuous and dispersed phases, i.e. the particles are assumed to move always with the terminal slip velocity relative to the liquid phase (14) The mathematical model The horizontal momentum equation in conservative form reads: t Ω ρu dω + The vertical momentum equation: S r r ρu v n ds = S r τ i xj j r nds S r p i x r nds ( 1 ) t Ω r r r r r r ρw dω+ ρw v n ds = τ i nds p i nds ρg dω zj j z S S S Ω ( 2 ) In the left-hand side of the equation the total derivative is split in the unsteady and advection term. In the right hand side the shear stress term is still generic, the relation with the flow field is not yet defined. Beside the momentum equation the continuity equation is needed to solve the system. For an incompressible fluid in conservative notation this equation reads: v r ¼ n r ds = 0 ( 3 ) S To solve this system of equations a relation between the shear stresses and velocities must be known. Let us investigate the shear stress on the top side (north side) of a control volume. For a Newtonian fluid a linear relation exists between the shear stress and the velocity gradient. t u xz = rn z ( 4 ) In this relation is ν the kinematic viscosity of the fluid. The product ρν is called the dynamic viscosity. The shear stresses on the other surfaces of the control volume can be written

6 likewise. All stresses can now be expressed in term of velocity gradients leading to a system of three equations. The horizontal and vertical momentum equations and continuity equation with three unknowns (in case the density is constant): velocity components u, w and pressure p. With the appropriate boundary conditions a solution can be found in principle. For laminar flow (low Reynolds number) this is indeed the case. For higher Reynolds numbers however the flowfield becomes unstable (turbulent). Eddies are developing and the flow can only be described correctly if the smallest eddies are simulated in the calculation. The only way to solve the equations is numerically so this will lead to a very small grid size and hence huge computer memory demands and computational effort. This is the reason that this so-called Direct Numerical Simulation (DNS) approach will not be feasible for practical problems with large dimensions in the near future (5). The time scale at which the turbulent fluctuations take place are however very often much smaller that the time scale of the boundary conditions. Velocities can therefor be averaged over a period relative small to the changes in flow conditions (11). The relevant quantities can in that case be written as the sum of a time averaged value and a turbulent fluctuation. The substitution in the Navier-Stokes (NS) equations and time averaging and application of the eddy-viscosity concept leads to the following expressions fore the shear stress in the fluid. (Here only written for one component) τ ρν u u xz uw e z ρ ρ = = z ν + ν ( ) ( 5 ) The correlated velocity fluctuations are replaced with the product of the eddy viscosity n e and a velocity gradient. The NS equation after time averaging is called the Reynolds Averaged Navier-Stokes equation (RANS). 3.5 Turbulent closure The RANS-equations can be solved when a relation can be found between the flow field and the eddy-viscosity. The k ε model has been used for very different flow problems in the past with success. The eddy viscosity is computed using ( c m is a model constant): 2 k ν e = c µ ( 6 ) ε In order to compute the eddy-viscosity the problem is shifted towards determination of the values of the k (turbulent energy) and e (dissipation rate of turbulent energy). Both quantities are obtained from transport equations. The continuity and momentum equations together with these two transport equations form a strongly coupled system of equations. To solve the momentum equations the eddy viscosity must be known as function of time and space. To compute the eddy-viscosity, k and ε must be known and these values are dependent on the velocity distribution. 3.6 Sediment Transport In the Navier-Stokes equations the density of the mixture is present. The other important reason to know this density is that the sedimentation and hence filling of the hopper is directly related to the bottom concentration.

7 The grain size distribution is approximated with a finite number N of different grain diameters. The concentration of a certain fraction with grain diameter d j is c j. The mixture density follows from the total concentration:. ( 1 ) ρ cρ c ρ c c s w j j= 1 N = + = ( 7 ) For every fraction the distribution of the concentration is calculated with the transport equation for that fraction. This equation is used in conservative or integral form, which forms the basis of the Finite Volume Method. t W ( ) r v v c dw+ c v ¼ n ds = Ggrad c ¼n ds j j z, j j S S ( 8 ) The first term gives the rate of change of the concentration of a fraction, the second term represents the transport of that fraction due to advection. In the right-hand side the diffusive transport (caused by turbulence) is given. The diffusion coefficient Γ is related to the eddy viscosity with the Schmidt-Prandtl number: n G= e ( 9 ) s Generally this number is regarded as constant throughout the computational domain (11). Uittenbogaard (15) showed that in free turbulence, even in highly stratified conditions, σ = 0.6 can be used Coupling of the sediment fractions In horizontal direction it is assumed that the grain velocity is equal to the mixture velocity. In vertical direction a slip velocity between the fluid velocity and grain velocity is assumed. In (9) is shown that the relation between the vertical particle velocity, mixture velocity and slip velocity for a multi-sized mixture can be written as: N v = w+ c v v ( 10 ) z, j k s, k s, j k = 1 The mixture velocity w follows from the solution of the NS equations. Provided the concentration is known the slip velocity for every fraction can be calculated and consequently the grain velocities, which are used in the sediment transport equations can be computed. For the slip velocity the approach of Mirza & Richardson (4) is followed: s, j 0, j 1 ( ) n j 1 v = w c ( 11 ) In which w 0 is the fall velocity of a single grain and n j the hindered settling exponent for that grain size.

8 3.7 Overview of the model Now the three different parts of the model have been described the mutual relations can be shown in figure 3. The different modules contain transport equations for momentum, sediment and turbulent quantities. Figure 3. Model overview In the 2D RANS module the flowfield is solved using the spatial distribution of the eddyviscosity and the density. The sediment transport and turbulence module supply these quantities respectively. The Sediment transport and turbulence model both need the flowfield solved by the 2D RANS module. The turbulence and the sediment transport module exchange the eddy-viscosity and the density. The eddy-viscosity supplies the diffusion coefficient needed to compute the diffusive transport of sediment (equation ( 9 )). The density gradient is used in the turbulence model to take the extra dissipation (or production in case of unstable stratification) of turbulent energy into account. 3.8 Boundary conditions The hopper is schematised as a rectangular domain. Boundary condition must be applied at the bottom (interface between settled sand and mixture above), water surface, vertical walls, inflow section and outflow section Boundary conditions at walls, bottom and water surface The normal component of the mixture velocity at the vertical walls and bottom is equal to zero. For the bottom and vertical walls a wall-function approach is used for the velocity components parallel to the wall (11,12). At the water surface a rigid-lid assumption is used since surface wave phenomena are not important for the subject situation. A rigid-lid can be regarded as a smooth horizontal plate covering the water surface in the hopper. Depending on the total volume balance inside the hopper this plate will be moved up and down. The vertical velocity of the surface w follows from: H w H q q L in out = ( 12 ) In this simple equation is in q and out q the specific in- and outflowing discharge and L the length of the water surface. At the water surface the values of k and the normal gradient of ε

9 are put equal to zero. For the vertical walls and bottom the value of these quantities must be consistent with the wall function approach (11). u u k = = ( 13 ) 2 3 * * ε c µ κy With Y as the normal distance from the first grid point to the wall and u * as the friction velocity. For the sediment transport equations the fluxes through vertical walls and water surface is equal to zero since no sediment enters or leaves the domain at these boundaries. At the bottom for every fraction the sedimentation flux S i is prescribed. The influence of the bottom shear stress on the sedimentation is modelled using a reduction factor R. S R = Rcw i i z, i θ 1 θ < θ0 θ = 0 > 0 θ θ 0 ( 14 ) This simple relation between the reduction factor and Shields parameter θ is based on flume tests (7,15). The critical value for the Shields parameter proved to be independent of the grain size for the sands tested (d 50 < 300 µ m). It will be clear that this approach can only be used when over all sedimentation (like in a hopper of a TSHD) will take place. When the Shields value exceeds the critical value erosion will occur in reality and this effect is not (yet) modelled Boundary condition at inflow and outflow section Usually the inflow is in vertical direction through the water surface in the hopper. The vertical velocity will therefore be prescribed over the width of the entering jet at the surface. The turbulent quantities have be prescribed at this location as well. For the turbulent energy it is common to prescribe (2,11,17) ( v in is the inflow velocity). k = αv = 1.5I v ( 15 ) in T in The value of the turbulence intensity I T can vary strongly on the situation. For the dissipation rate the following boundary condition is normally used (2): 3/4 3/2 cµ k ε = ( 16 ) l The mixing length l m is related to the width of the inflow section or inflowing jet. The following relation between the mixing length and inflow geometry is used (2): m l = c 0.5d ( 17 ) m For the sediment transport equations the influx of every fraction is prescribed at the inflow. µ in

10 The outflow boundary is only active when overflow is present, so when the mixture level in the hopper exceeds the overflow level. In that case the outflow velocity is prescribed, and follows simply from the ratio of the overflow discharge and the difference between the hopper and overflow level. For the other quantities the normal gradients are equal to zero (Neumann condition). 3.9 Numerical Approach The momentum and sediment transport equations are solved using the Finite Volume Method to ensure conservation. The transport equations for the turbulent quantities k and ε are solved using the Finite Difference method. A Finite Difference Method is allays implemented on a rectangular (cartesian) grid. Although a Finite Volume Method can be applied on any grid it is advantageous to use a cartesian approach for this method as well especially when a staggered arrangement of variables is used. In general the flow domain is however not rectangular. The water surface can be considered horizontal on the length scale considered, but a sloping bottom will not coincide with the gridlines. Different approaches are possible. The first method is to use a cartesian grid and to adjust the bottom cells (cut-cell method). Another method is to fit the grid at the bottom. In that case a boundary fitted non-orthogonal grid can be used. A third method is using grid transformation. By choosing an appropriate transformation the equations are solved on a cartesian domain in transformed co-ordinates. Although this transformation allows for a good representation of a curved topography the method has the disadvantage that due to truncation errors in the horizontal momentum equation artificial flows will develop when a steep bottom encounters density gradients. These unrealistic flows can be partly suppressed when the diffusion terms are locally discretized in a cartesian grid. (12,13). Since however in a hopper both large density gradients as steep bottom geometry can be present it was decided to develop the model in cartesian co-ordinates. Figure 4. Staggered grid arrangement near bottom In figure 4 a possible situation at the sand bed is shown. The location of the bed is indicated with the dotted line. The heights of the cells that intersect with the bed are adjusted. In the figure the staggered arrangement of variables is shown as well (star: horizontal velocity, rectangle: vertical velocity; dot: concentration, pressure). An overview of the model was shown in figure 3. The computational procedure can only be outlined here very roughly. The flow is not stationary hence the system is evaluated in time. The following steps are repeated during time:

11 Update the velocity field to time t n+1 by solving the NS-equations together with the continuity equation using a pressure correction method (SIMPLE-method (6)) using the density and eddy viscosity of the old time step t n. Update the turbulent quantities k and ε to time t n+1 using the velocity field of t n+1. Compute the eddy-viscosity for the new time. Use the flowfield of t n+1 to compute the grain velocities for the next time and update the concentrations for all fractions and hence the mixture density to time t n+1. Compute the new location for the bed level and mixture surface in the hopper. 4 APPLICATION OF THE MODEL 4.1 Introduction Because the code for the model was not based on available commercial CFD software the first step was to validate the results with known flow problems like the backward facing step, plane buoyant and non-buoyant jets, density currents on a slope and sediment transport in a canal. The code proved to be capable to simulate these types of flow. 4.2 Simulation of Laboratory Experiments In the framework of the research programme model laboratory tests were carried out at WL Delft hydraulics. A description of the experimental set up and test programme was given in (8). The model will be compared with the result of test 6. The operation parameters are summarised in the table below (average during the test): Discharge m3/s Bed level at start 0.18 m Density 1420 kg/m3 Length of hopper 12.0 m Overflow level 2.25 m Width of hopper 3.08 m Water level at start 1.25 m The measured inflow concentration and discharge as function of time was used as input for the model. Some variation of the particle size diameter in the inflow was observed, but during the calculations the PSD was kept constant. Figure 5. Computed concentration profiles during time

12 During the calculations the PSD is represented with 8 fractions, hence for the cumulative distribution 9 points were needed. Figure 5 shows the computed concentration profiles at four different positions for 9 different times. The inflow is located at the upper left side, the outflow at the right side. The overflow level was kept constant during the test at 2.25 m. The location of the sand bed is shown with the hatched lines. The erosion crater at the left side is clearly visible. The first profile in time was taken before the overflow level was reached. The other profiles were taken at the overflow phase. A strong similarity is present between the computed and measured concentration profiles as is shown in figure 6 where the measured and computed concentration profiles are compared for one location (x = 6 m). A sharp interface between the suspension and sand bed is present. The profiles are almost vertical over a large distance After some time (about 1000 s) the value of the concentration in suspension remains at a constant value. The measured value of the concentration is however larger than calculated, which can be partly explained by the larger computed vertical diffusion. In figure 7 the calculated velocity profiles at different times are plotted at 4 locations. It is clear that due to the density effect the flow is concentrated close to the bottom. Above this density current a return flow, caused by entrainment in the inflow section, is present. During time the magnitude of the horizontal velocity decreases, as can be seen from the maximum values of the profiles. (the same tendency was observed during the laboratory tests) Figure 6. Computed and measured concentration at x=6 m A very important quantity during the calculations is the overflow loss, in fact one of the major goals of the research program. The computed concentration in the overflow is compared with the measured value in figure 7. The agreement between theory and experiments is satisfactory. The maximum outflow concentration is somewhat smaller and the losses during the first part of the overflow phase are higher. The latter being caused by the larger diffusion during that time as already discussed above.

13 . Figure 7. Computed velocity profile in the hopper Figure 8. Measured and computed overflow concentration 4.3 Simulation of Prototype Measurements To test the model in a prototype situation, measurements were carried out onboard the Trailing Suction Hopper Dredge Cornelia owned by Royal Boskalis Westminster. The dredging area is near IJmuiden (Dutch Coast). The dimensions of the hopper are: Length = 52 m, Width = 11.5 m, Hopper Volume= 5000 m 3. The hopper has the typical silo shape. Since the model is based on a rectangular shape, the hopper area is simplified to a rectangular basin with a depth op 5000/(52*11.5) = 8.36 m. Typical operation parameters during loading (with two suction pipes) are. Discharge Q = 6 m3/s, Mixture Density = 1260 kg/m3, Water density = 1021 kg/m3 and Porosity (settled sand) n = Samples were taken from the seabed to determine the particle size distribution. d 50 = 235 micron. For the calculation 10 fractions were used. In figure 9 the cumulative overflow loss defined as the ratio between the total volume of sand lost overboard and the total volume of sand pumped into the hopper is plotted versus time. The measured overflow loss was based on the measured tonnage of the ship (from the

14 draught), the measured hopper volume and the total measured influx of sand. This method has the disadvantage that at the beginning of the loading process the outcome is very sensitive on the draught and hopper volume and therefore unreliable. The results improve however during time. This is phenomenon is visible in the results of the procedure. Before t=1500 s the (indirectly) measured overflow loss varies strongly Figure 9. Discharge and cumulative overflow loss The upper figure shows the measured discharge into the hopper and the calculated overflow discharge (dotted line). The agreement between the measurements and model is encouraging, especially taking into account that no calibration of the numerical model (like k - e equations) was undertaken. 5 CONCLUSIONS The two-dimensional model based on the Reynolds Averaged Navier-Stokes for mixture flow simulates the sedimentation process quite well. The model includes all important phenomena of the hopper sedimentation process. The numerical model has been compared with laboratory experiments and a limited number of prototype measurements. Differences between measured and computed overflow losses are small. The concentration and velocity profile in the model hopper differ from the simulated values. These differences are all caused by the behaviour of the model in the inflow zone where the velocity of the mixture is over predicted. This phenomenon is most probably caused by a deficiency of the turbulence model in strong buoyant situations. The higher flow velocity in the jet over predicts entrainment and hence dilution of the inflowing mixture which results in a lower suspension concentration. Owing to the larger entrainment the circulation in the hopper is slightly exaggerated which enlarges the mixing in the hopper. ACKNOWLEDGEMENT The financial support of the VBKO (Vereniging van Waterbouwers in Bagger-, Kust en Oeverwerken) is gratefully acknowledged. I want to thank Dr. F. van Kats for the preparation,

15 execution and processing of the prototype measurements and Royal Boskalis Westminster for making available the results. REFERENCES 1. Camp, T.R.,1946, Sedimentation and the Design of Settling Tanks, Trans. ASCE, 111(2285), pp Celik, I., Rodi, W., Stamou, A. I, Prediction of hydrodynamic characteristics of rectangular settling tanks. Turbulence measurements and flow modelling. ASCE, New York, N.Y 3. Dobbins, W.E., 1944, Effect Of Turbulence On Sedimentation. ASCE Trans., Vol 109, no. 2218, pp Mirza, S., and Richardson, J. F., 1979, Sedimentation of suspensions of particles of two or more sizes, Chem. Eng. Sci., 34, Nieuwstadt, F.T.M, 1992, Turbulentie; inleiding in de theorie en toepassingen van turbulente stromingen., Utrecht Epsilon. 6. Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York. 7. Rhee, C. van, Talmon, A.M., 2000, Entrainment of sediment (or reduction of sedimentation) at high concentration, 10th Int. Symposium on Transport and Sedimentation of Solid Particles, Wroclaw, Poland. 8. Rhee, C. van, 2001, Numerical Simulation of the Sedimentation Process in a Trailing Suction Hopper Dredge, 16th World Dredging Congress and Exhibition (WODCON), Kuala Lumpur. 9. Rhee, C. van, 2001c, Modelling the sedimentation process in a trailing suction hopper dredge, 4th Int. Conference on Multiphase Flows, New Orleans. 10. Rhee, C. van, 2002, The influence of the bottom shear stress on the sedimentation of sand, 11 th Int. Symposium on Transport and Sedimentation of Solid Particles, Gent, (to be published) 11. Rodi, W., 1993, Turbulence Models and their Application in Hydraulics, A state of the art review, IAHR, Third Edition. 12. Stansby, P.K., 1997, Semi-Implicit Finite Volume Shallow-Water Flow and Solute Transport Solver with k-epsilon Turbulence Model, Int. Journal for Numerical Methods in Fluids, Vol. 25, pp Stelling, G. S., Kester, J. A. Th. M. van, 1994, On the approximation of Horizontal Gradients in Sigma Co-ordinates for Bathymetry with steep bottom slopes,, Int. Journal for Numerical Methods in Fluids, Vol. 18, pp Taivassalo, V., Manninen, M., 2001, Formulation and Validation of the Mixture Model for Multiphase Flows, 4th Int. Conference on Multiphase Flows, New Orleans. 15. Uittenboogaard, R.E., 1995, The importance of Internal Waves for Mixing in a Stratified Estuarine Tidal Flow, PhD-Thesis, Delft University of Technology 16. Vlasblom, W.J.,Miedema, S.A., 1995, A theory for determining sedimentation and overflow losses in hoppers, Proc. of the 14th World Dredging Congress, Amsterdam. 17. Zijlema, M., 1994, Finite volume computation of 2D incompressible turbulent flows in general co-ordinates on staggered grids, Technical University Delft