Free - vs - forced bars
Alternate bars can be classified on their origin. ‘Free’ bars spontaneously develop because of an instability of the flow-bed system (Seminara and Tubino, 1989). Bars are called ‘forced’ when they are caused by a physical disturbance or constraint that may be introduced by obstacles (for example bridge piers, groynes) or a distributed forcing effect (gradual change in channel curvature). Because the origin of forcing is static, the forced bars will also become static. Below an example of a Delft3D simulation of both bar types is given.
In the past decades, several scientists have tried to explain the processes that are responsible for the formation of non-migrating alternate bars. In the 1960s, it was suspected that the occurrence of alternate bars can be explained by a stability analysis of the mathematical equations for flow and sediment transport. In 1985, two important linear theories were published, which are the base for the present theories: Struiksma et al (1985) and Blondeaux and Seminara (1985). The linear theories result in a free bar diagram, which depends on the shields parameter (a parameter which represents the sediment mobility), the diameter of the sediment and the chosen bed-load transport formula. Below one can see the free bar diagram for a shields parameter of 0.1 and a relative roughness (sediment diameter divided by the water depth) (Vanzo et al., 2011).
The vertical axis of the figure represents the half-width-to-depth ratio and the horizontal axis the dimensionless wavenumber (π*B/L), in which B is the channel width and L the length of the alternate bar. With this diagram one can investigate the behaviour of an alternate bar for a specific length and a specific water depth. The diagram contains the following information:
- The blue dotted line represents the length of forced alternate bars
- The red line represents free bars that are not migrating
- Bars below the red line (grey area) have a negative growth rate and will therefore not occur according to linear theory
- The grey dotted line represents the most unstable (with the largest growth rate) free bars. According to linear theory, the most unstable bar is dominant.
- The black dotted line represents bars that are not migrating. To the left of this line, bars will migrate in upstream direction, whereas bars on the right of this curve will migrate in downstream direction
- If β > βres (13.28 in this case) the river is super-resonant
- If β < βres: the river is sub-resonant
BLONDEAUX, P. & SEMINARA, G. 1985. A unified bar--bend theory of river meanders. Journal of Fluid Mechanics, 157, 449-470.
SEMINARA, G. & TUBINO, M. 1989. Alternate bars and meandering: Free, forced and mixed interactions. River meandering, Water Resourc.Monogr. Ser. , 12, 267-320.
STRUIKSMA, N., OLESEN, K. W., FLOKSTRA, C. & DE VRIEND, H. J. 1985. Bed deformation in curved alluvial channels. Journal ofHydraulic Research, 23, 57-79.
VANZO, D., SIVIGLIA, A., ZOLEZZI, G., STECCA, G. & TUBINO, M. 2011. Interaction between steady and migrating bars in straight channels.RCEM2011, Tsinghua University Press, Beijing, China.