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3.2 Flow equations and flow resistance
Flow equations and flow resistance
The flow resistance is influenced by the structure of the surface layer and the presence of sediment
load and suspended load transport. The roughness of the surface layer is due to the characteristics
of the surface layer and the presence of bedforms. Many works and textbooks give an overview
over the different flow equations and the flow resistance, like e.g. DUBOIS (1998), SINNIGER &
HAGER (1989), HUNZIKER (1995), ZARN (1997).
3.2.1 Flow resistance and friction laws
In his pioneer work, CHEZY (1768) related the average flow velocity to the bed slope and the
water depth.
V = C⋅
h ⋅ S
(3.10)
For the friction coefficient C , CHEZY made important simplifications. About 100 years later,
GANGUILLET & KUTTER published a formula, which allowed the computation of C as a function
of the bed roughness. Authors like CHÉZY, GAUKLER, FORCHHEIMER, MANNING, CHRISTEN,
(see SINNIGER & HAGER, 1989, §2.2) and many others published velocity power laws.
In Switzerland, the formula of STRICKLER (1923) found a wide application:
V = K s
⋅ R2 3
h
⋅
⁄ S 1⁄
2
(3.11)
The relation between STRICKLER’S roughness coefficient K s
and the size of the characteristic
roughness ε (usually the mean diameter of the surface layer d m
) is given by
K s
=
Cst
-------
(3.12)
6
ε
STRICKLER put the value of the constant Cst to 21.1 and used as characteristic roughness ε the
mean diameter. MEYER-PETER & MÜLLER (1948) used a value of Cst = 26 combined with
ε = d 90 (of the substrate). The higher constant Cst is partially compensated by the bigger grain
diameter. MEYER-PETER & MÜLLER justified the increased constant with a better fit to the results
of NIKURADSE. JÄGGI (1984) and HUNZIKER (1995) analyzed the constant Cst for the computation
of the bed friction. They concluded that the initially proposed value of STRICKLER fits better
to the observed values than the one of MEYER-PETER & MÜLLER. SCHÖBERL (1981) performed a
series of tests and proposed an intermediate value of 23.5. In the present study a value of
8.2 g = 25.7 is used in combination with d 90 .
DARCY-WEISSBACH (VANONI, 1975) proposed the following friction law:
V =
8 ⋅g ⋅R h
⋅ S
----------------------------
f
(3.13)
EPFL Ph.D thesis 2632 - Daniel S. Hersberger November 9, 2002 / page 27