The meandering channel is one of the classical research topics of hydraulics and sediment transport. The topic was chosen for the 3D CFD model with dynamic grid. Since the channel moves laterally during its formation, the algorithms to expand the grid and to provide smooth river banks would be tested. Also, the 3D flow field with secondary currents essential for the meandering process would be modelled.
The modelled case is formation of a meandering channel in a flume with sand. Initially, a straight channel is excavated in the sand, with one bend upstream. Then water is run through the channel, and a sequence of meanders are formed. A number of physical model studies of this case have been done over the years. A classical case is the study by Friedkin from 1945, at the US Army Engineer Waterways Experiment Station. The two photos below are from this case, showing the flume with the channel.
The two photos show the initial channel, before the experiment starts and after 3 hours of water flow. Regular meanders have formed. The view is in the downstream direction.
Reference: Friedkin, J. F. (1945) "A laboratory study of the meandering of alluvial rivers", US Army Engineer Waterways Experiment Station, Vicksburg, Mississippi.
The chosen test case for the computations was a physical model study done at Colorado State University, in a series of flume studies organized by Prof. Schumm. The present case was carried out by G. L. Zimpfer in 1975, for his PhD.
Reference:
Schumm, S. A., Mosley, M. P. and Weaver, W. E. (1987) "Experimental
Fluvial Geomorphology", John Wiley Publishers.
Initial grid. The dotted lines are limits for where the grid can expand horizontally.
The initial grid described a straight channel, with a water
discharge of 5.66 l/s, slope 0.008, and sediments with average
diameter of 0.5 mm. An initial upstream disturbance had to be
used to start the meandering in some of the experiments in the
physical model. In the CFD model, this was done by a 20 degrees
skewed inflow cross-section, as seen in the figures.
Velocity near the water surface
Velocity near the bed, shown in the grid
The meander wavelength computed by the CFD model is about 7 meters (the grid cells are 0.1 meters long). The wave length observed in the lab experiments was 7.5 meters.
The currently used algorithms are unstable close to the grid boundary, as seen by the boundary irregularity. Work is going on to improve the algorithms.
Velocity vectors near the bed (green) and close to the surface (black) after 772 iterations. The initial cross-section is skewed to the right instead of to the left, to test the program.
The velocity vectors above show the secondary currents, as the bed vectors points more to the inside of the curve than the surface vectors. Also note the vertical velocity profile at the inlet is uniform. Further downstream, the velocity at the surface becomes larger than at the bed, corresponding with the classical logarithmic profile.
Velocity vectors near the bed (green) and close to the surface (black) in a bend after 8000 iterations. The lower figure is an enlargement of a part of the upper figure.
The velocity vectors above show the secondary currents, as the bed vectors points more to the inside of the curve than the surface vectors. Also note only one vector is shown in the shallow areas, as there is only one cell in the vertical direction. The deeper parts has up to six cells in the vertical direction. The depth is also given in the figure below:
Contour map of water depth in a bend after 10 000 iterations. Red is large depths, and blue is shallow depths.
Contour map of depth-averaged water velocity in a bend after 10 000 iterations. Red is large velocities, and blue is low velocities.
Contour map of water depth in the whole flume after 10 000 iterations. Red is large depths, and blue is shallow depths.
In the physical model test, the meandering pattern started with shallows and pools alternating on each side of the initially straight channel. This can also be seen in the CFD results below, after 2200 iterations. The three pictures below show the whole geometry and enlargements of the downstream part of the channel.
Contour map of water depth in the whole flume after 2 200 iterations. Red is large depths, and blue is shallow depths.
Contour map of water depth in the downstream part of the flume after 2 200 iterations. Red is large depths, and blue is shallow depths.
Contour map of water depth in the downstream part of the flume after 2 200 iterations. Red is large depths, and blue is shallow depths.
Contour map of vertical velocity in the downstream part of the flume after 3 000 iterations. Red is large upwards velocities, and blue is downwards velocities.
In the physical model, shortcuts were formed in the meanders, creating islands. This is also given in the numerical model, as shown in the figures below: a meander bend after 4 000 iterations.
Velocity vectors close to the water surface in a bend after 4 000 iterations.
Contour map of water depth in a bend after 4 000 iterations. Red is large depths, and blue is shallow depths.
Contour map of water depth in the whole flume after 5 400 iterations. Red is large depths, and blue is shallow depths.
The images above have been obtained with an upstream disturbance given by a skewed inlet angle of the flow. The angle was set to 20 degrees initially. After correspondence with Prof. Schumm, he sent more detailed information about the inlet, showing that the angle actually was 40 degrees. The images below are from a new run with this parameter.
Velocity vector map from one of the bends from a case with 40 degree inlet angle. Green vectors are close to the bed and black vectors are close to the water surface. After 4800 iterations.
Grid from one of the bends from a case with 40 degree inlet angle, corresponding to the image above. After 4800 iterations.
Depth-averaged velocity contours, where the area inside the green lines are velocities above 8.5 cm/s. The image is after 5400 iterations.
Meander cutoffs are created, starting the process of making the meandering channel into a braided system. This transformation was also observed in the laboratory experiment.
The advice and help of Prof. Stanley Schumm at Colorado State University is greatly acknowledged. I also want to thank the US Army Engineer Waterways Experiment Station for allowing me to use the photos on this page.
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