Environmental Engineering Reference
In-Depth Information
illustration of the concept. Einstein and Chien assumed the additional condition that :
UU U
(5.35)
w
b
Then one may deduce :
nPnp np (5.36)
In this case, the resistances on different boundary elements are distributed according to the 1.5 power of
the Manning coefficient.
In the study of flow resistance, the skin friction (i.e. grain friction) and form resistance (i.e. bed form
resistance) often exist simultaneously. The principle of summation of resistance is tenable if the faces on
which the resistance components act are fully separated, and if they partly overlap but have different
heights, so that the resistance components do not affect each other.
Two different approaches are used in calculations based on the principle of summation of resistance.
One is that under a certain velocity and flow depth without bed forms on the stream bed, the energy slope
can be smaller; however, if bed forms also exist, the flow requires a larger energy slope. Thus, one can
divide the energy slope into two parts corresponding to the grain friction and bed form, respectively
(Meyer-Peter and Muller, 1948). Because the grain friction and bed form resistance occur on the same
boundary, it is reasonable that an unit weight of water transmits energy simultaneously to the two
resistance components. Also in dealing with the grain friction and bed form resistance, Einstein divided
the hydraulic radius in the same way as for the bed resistance and bank resistance. Consequently, these
two approaches are currently used for dividing the total resistance of alluvial streams into corresponding
components in the following ways:
3/2
3/2
3/2
ww
b
b
A similar approach can be used to combine bed form resistance and the resistance induced by artificial
structures and other localized resistances. The floodplain resistance is discussed in the preceding sections.
In the Three Gorges region of the Yangtze River, narrow and wide gorges alternate so that the cross-
sectional geometry changes significantly in the longitudinal direction; the form resistance induced by
these changes accounts for some 5 to 60% of the total resistance. In dealing with this issue, Hui and Chen
(1981), using the approach of Einstein and Chien (1958), calculated the bed resistance and the bank
resistance, and then divided form resistance induced by the sudden enlargement or contraction of the
cross sections from the total resistance.
5.2 Sediment Transportation and Hyperconcentrated Flow
5.2.1 Fall Velocity
5.2.1.1 Fall Velocity
Fluvial process is a result of sediment deposition and erosion. To study sediment deposition and erosion
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