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Chapter 3 - Theoretical considerations
b) Formulae based on equilibrium considerations
The analytical formulae of VAN BENDEGOM, ENGELUND AND BRIDGE are valid for small bed
slopes, high width to depth ( B ⁄ h) ratios and for small curvatures of the bend ( R c » h ). VAN
BENDEGOM and ENGELUND established their equations for partially developed secondary flow
conditions. The equation of van Bendegom is based on equilibrium conditions of spherical grains
which are completely exposed to the flow. This leads to an underestimation of the bed slope in
radial direction. Furthermore the static lift was neglected leading to an additional underestimation
of the lateral bed slope. The formula of Engelund considered a rectangular cross-section with a
fixed bed and slightly meandering bends. Bridge adjusted the formula of Engelund, integrating the
results of the research works of Rozovskii; he assumed the ratio between tangential and radial
velocity components to be equal to 11.
KIKKAWA ET AL. introduced a flow model, which allows to determine the drag and lift force as a
function of characteristic flow and bed parameters. REINDL observed that the maximum flow also
depends on the sediment saturation Gs for river reaches influenced by a backwater curve. He
modified KIKKAWA’S formula, in order to take into account the quantity of sediment available for
transport.
c) Formulae based on control volume considerations
ZIMMERMANN, FALCON & KENNEDY and ODGAARD considered control volumes of different
size, but always with the thickness of the armoring layer. ZIMMERMANN used the whole cross-section
at the maximum scour location, while FALCON & KENNEDY and ODGAARD used a local control
volume. Since ZIMMERMANN’s equation is a function of h / R c , the bed geometry in the
cross-section is a straight line, which does not fit with the observed geometry of the cross-section.
All three formulae were established for sand bed rivers.
BAZILEVICH’S formula is very sensitive to the chosen parameters (see PETER, 1985). Since the
flow and geometric characteristics of a river often contain (considerable) uncertainties, this equation
does not seem very appropriate for engineering practice.
d) Empirical formula
PETER gives an empirical formula based on a dimensional analysis. In general, his formula gives
good results for alpine rivers (see PETER, REINDL). But unfortunately, it does not take into
account the sediment size. Therefore the formula should be used with care within the authorized
domain of radius to with ratios ⁄ B = 2 ÷ 6.
R c
3.5.4 Comparison with experimental data
In this paragraph, the mentioned scour formulae are compared to laboratory experiments for
smooth walls performed in the frame set of this study (also published in HERSBERGER &
SCHLEISS, 2002). The laboratory tests of PETER (1986), performed at the VAW (ETHZ) were
added to this data set. The so obtained extension of the data set (39 tests) to 71 tests covers a wide
page 58 / November 9, 2002
Wall roughness effects on flow and scouring