The problem in a vertical plane

The problem in a vertical plane is formulated for a wide channel, ideally for an infinitely wide one, of a constant depth assuming that the flow pattern is approximately the same in each vertical plane along the flow direction, as it has been depicted in Fig. 1.5. A usual vertical velocity distribution that follows formula (1.1) takes place only above the vegetation submerged in the water flow. This distribution becomes complex and uncommon within the vegetation layer where a significant motion of the water still takes place. A number of experimental data given in [69, 172, 347, 370, 617] confirms 

Kouwen seemed to be the first among hydraulicians who doubted yet in 1969 the ability of traditional resistance formulas to correctly represent the role of vegetation, its possible features (solidity or flexibility) and characteristics like its height and spacing (density) [86], In his experimental work, he introduced an empirical slip velocity taken proportionally to the velocity gradient outside the penetrable block . Later, he carried out the experiment with a flexible artificial grass in a laboratory flume [350], 

The significant contribution to the field under study was also done by Rouve, Pasche, Nuding [470, 553], Tsujimoto and Kitamura [581, 617, 618, 619] (both experiment and theory), by Garcia [377] and Ginsberg [231] (mainly experiment). 

As for the terrestrial vegetation, the difficulty appeared here concerns to the problem of the estimate of the vegetation stiffness that may be also called the density of the obstruction layer (the canopy). In the field measurements of Bennovitsky [53, 54], the natural vegetation (the reed) provided the blockage 7,40, and 120 plants on m2. In the measurements of Nepf [463], the blockage of plants was 330 plants/m2. Another measure for the canopy density, the frontal area per unit volume s, 1 /m, has been more universal. Its typical vertical distribution is shown in Fig. 1.7, [463], 

This brief overview of the vertical-plane hydraulics problem may be accomplished with references to non-stationary problems. It is important for oceanography problems that the flow evolves in time because of the seagrass-current periodicity. In article [172], the one-dimensional problem (1.10) accounted also for the time- and space-periodicity of the sea water surface, g(x, t) = A sin(u xx - oj,t). The final motion equation was written as 

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