# Flow and Sediment Transport in a Laboratory Model of a stretch of the Elbe River

## Overview of CFD Simulations

A brief overview of what calculations are available for the Elbe river case has been given already in the introduction section. Here, attention is restricted to the 3D calculations of a laboratory model situation as documented in {1} using a numerical model described in greater detail in {2}. The flow and sediment transport in a laboratory model of a stretch of the river Elbe between km 506.0 and 513.0 has been simulated and the top view of the calculation domain is shown in Fig. 1. The bed was initially flat. The computer program FAST 3D developed at the Institute for Hydromechanics has been used which is a finite-volume method which allows the employment of body-fitted non-orthogonal (but structured) grids. The code can be run in steady or unsteady modes. In the calculation considered, the bed deforms and the grid is adjusted accordingly during the calculation. Turbulence effects are accounted for with the standard k-ε model. The sediment transport is calculated with a bed-load model (there is no suspended load in this case) and the bed-level change from the mass-balance equation for sediment. The bed-load model is based on a non-equilibrium bed-load equation involving an empirical relation due to van Rijn for equilibrium transport rate. In the original model of Wu et al 2, the non-equilibrium adaptation length Ls appearing in the bed-load equation was also determined from an empirical relation due to van Rijn, but in the present case this yielded unrealistically small values considerably smaller than the grid size. It was found earlier that Ls must be of the order of the grid size in order produce realistic results and hence Ls was taken as ${\displaystyle {\sqrt {\Delta x\Delta y}}}$ where ${\displaystyle \Delta x}$ and ${\displaystyle \Delta y}$ are the mesh sizes in the 2 horizontal directions. For determining the direction of bed load, in an extension of the original model of Wu et al {2} the effect of gravity on sloping beds is accounted for {1}.

The hydrodynamic flow model was tested first by carrying out calculations for a fixed bed at medium flow discharge corresponding to Q = 795 m3/s in the real river (i.e. these calculations were NOT for the laboratory-model situation). A comparison of water level variation along the river stretch and the distributions of surface velocities in two cross-sections with field measurements and laboratory experiments yielded good agreement [1]. The model was then applied to the laboratory situation with movable bed, starting with a flat bed as in the experiments. The flow discharge and the water level at the outflow boundary were set to the experimental values. The flow discharge is Q = 12.54 l/s and the waterlevel ${\displaystyle z_{s}}$ = 0.265 m (above datum) at the beginning of the experiment, increasing linearly to ${\displaystyle z_{s}}$ = 0.275 m at the end.

## Simulation Case CFD-1

Solution strategy

The main information on solution strategy has been given already in the CFD simulations section. The finite-volume method uses a non-staggered variable arrangement and a central/upwind hybrid differencing scheme for the convective fluxes. The pressure-velocity coupling is achieved by using the SIMPLE pressure-correction algorithm and the momentum interpolation procedure of Rhie and Chow. The discretized algebraic equations are solved with the strongly implicit procedure (SIP) of Stone. Further details are given in {2, 5, 6}.

Computational Domain

The geometry of the computational domain is given in Fig. 1. This was discretized by a numerical grid which has 62 points in the lateral, 701 points in the streamwise and 25 points in the vertical direction. This grid conforms to the geometry of the more than 50 groynes on each bank of the Elbe river. The grid points are given in the file grid_bott0.dat. The time step of the flow calculations was 30 s and the time step of the sediment transport calculations was 300 s.

Boundary Conditions

The experimental values of flow discharge and water elevation at the downstream end of the calculation domain were specified (see 3.1). The distributions of velocity and turbulence quantities k and ε were obtained from empirical formulae for the variation of these quantities over the depth in open-channel flow (see {7}). The rate of sediment inflow in the experiments was prescribed. At the outflow boundary, zero gradient conditions were applied except for the surface elevation given the experimental value there. At the bed, the wall-function approach is used (see {2} for details), relating the values of the horizontal velocity components, k and ε, at the first grid point (placed above the roughness elements) to the local bed-shear stress. For the velocity, a log law is used containing a roughness parameter. This is related to the Manning roughness coefficient which is an adjustable parameter. A value of n = 0.0285 s/m1/3 is adopted which was taken over from Minh Duc {4} who tuned it by performing 2D depth-average calculations of the flow in the real Elbe river at various discharges and adjusting it to give good agreement between calculated and measured water level distributions along the river stretch. A corresponding wall-function approach was also used at the wide walls, with the experimental wall roughness height entering in the log law. At the free surface, basically the vertical gradients of the horizontal velocity components and k are specified as zero and ε is related to the surface value of k and to the water depth (see {2}). The water level is calculated from a 2D Poisson equation for the surface height zs which is obtained from the depth-averaged 2D momentum equations {2}.

Application of Physical Models

Details on the application of the turbulence model and the sediment-transport model can be found in [2].

Numerical Accuracy

Calculations were obtained with only one grid so that numerical accuracy could not be investigated.

CFD Results

Results include distributions of surface velocity, bottom elevation distributions at the cross-sections given in Fig. 1 and the bed topography after equilibrium has been reached. These results are given in the graphical files: Fig1.jpg, Fig2.jpg, Fig3.jpg and Fig4.jpg.

## References

[1] Fang, H.W., Three-dimensional calculations of flow and bed-load transport in the Elbe river, Rept. Institute for Hydromechanics No. 763, University of Karlsruhe, Germany, 2000.

[2] Wu, W.M., Rodi, W., Wenka, Th., 3D calculations of flow and sediment transport in rivers, J. of Hydraulic Engineering, Vol. 126, pp. 4-15, 2000.

[3] Minh Duc, B., Wenka, Th., Rodi, W., Depth-average numerical modelling of flow and sediment transport in the Elbe river, Proc. 3rd Int. Conf. on Hydroscience and Engineering, Cottbus/Berlin, 1998.

[4] Minh Duc, B., Berechnung der Strömung und des Sedimenttransports in Flüssen mit einem tiefengemittelten numerischen Verfahren, Ph.D. Thesis, University of Karlsruhe, 1998.

[5] Majumdar, S., Rodi, W., Zhu, J., 3D finite-volume method for incompressible flows with complex boundaries, J. Fluids Engineering, Vol. 114, pp. 496-503 (1992).

[6] Zhu, J., An introduction and guide to the computer program FAST3D, Report Institute for Hydromechanics, University of Karlsruhe, Germany (1992).

[7] Fang, H.W., Rodi, W., Three-dimensional calculations of flow and suspended sediment transport in the neighborhood of the dam for the Three Gorges Project (TGP) reservoir in the Yangtze River, Rept. Institute for Hydromechanics No. 762, University of Karlsruhe, Germany, 2000.