The problem in a horizontal plane

The problem in a horizontal plane expresses flow parameter distributions across the water conduit width which is vegetated near one or both banks like the situation shown in Fig. 1.8. It was assumed there that the local mean velocity was averaged over all the flow depth 0 z Ox,y). In fact, the flow field in such a geometry, even in the simplest configurations, is completely three-dimensional and is varied in the flow direction Ox, across the conduit Oy, and in the vertical direction Oz, i.e., U = U(x,y, z). It is natural, however, to average the velocities over the whole flow depth [540]  

Therefore the velocity vector [U, V is related to every point (x,y) of the flow plan. 

Bennovitsky suggested and validated the experimental evidence that velocity profiles followed the logarithmic law just over the vegetation in the free stream [56], So the approximation (1.1) may be employed in this problem again. Three empirical parameters required were adopted from the theory of atmospheric canopy flows [155], For instance, the relation d = 0.65h was found [56]. 

No approaches were known however for the velocity distribution within the vegetated layer. Bennovitsky stated that formulas (1.3) may be approximately applied. So both kinds of hydraulic problems turned out to be similar to the forest canopy problem. 

Theoretical approaches to the horizontal-plane hydraulic problem are often based upon momentum equations derived from the Navier—Stokes equations being averaged over the flow depth. This approach was developed by Rodi, Emtsev, Sherenkov, Beffa, and other authors [48, 171, 540], In the case where the vegetation is present, the resulting two-dimensional shallow-water equations for a time-dependent flow read as follows [540]  

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