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Chapter 3 - Theoretical considerations
ENGELUND determined the deviation angle δ based on a simple analytical model of the flow in a
bend, which partially takes into account the roughness of the bed.
tanδ
7 h s
= ⋅ ----
r
(3.66)
From equations 3.65 and 3.66, ENGELUND finally obtained:
tanβ
7 h s
= ⋅ ---- ⋅ tanφ
r
(3.67)
Putting
tanβ
=
∂y ⁄ ∂r ≈ ∂h s
⁄ ∂r
and after integration, he got:
h s
= a⋅
r 7 ⋅ tanφ
(3.68)
describing the flow depth (and consequently the bed topography) in radial direction. The factor a
is an integration constant. By comparing his equation to model tests performed by HOOK (1974),
Engelund found values for the dynamic shear angle φ ranging between 27° and 33°.
ENGELUND’S formula, combined with the proposed flow model, does not only allow the determination
of the radial bed profile, but also a general computation of the bed topography in a bend.
ENGELUND extended his approach to bends with variable radius of curvature. It has to be mentioned
that his flow model is based on a rectangular cross-section with a fixed horizontal bed. Furthermore,
the flow model is restricted to channels with a high width to depth ratio and the radius
has to be significantly larger that the flow depth.
Under these mentioned conditions, tanδ depends only insignificantly on the roughness of the
bed and can be assumed to be almost constant. Consequently, equation 3.67 shows that the radial
bed slope depends only on the dynamic shear angle φ , which is constant for a given bed material.
Consequently, the radial bed topography depends only on the ratio h ⁄ r.
Results of laboratory tests and field observations (HOOK, 1974) show that tanβ
⁄ ( h ⁄ r)
is not
constant (ODGAARD, 1981). ODGAARD compared ENGELUND’s formula with lab and field data.
The correlation is relatively poor. According to Odgaard, this is due to the fact that ENGELUND’s
formula was established for weakly meandering channels, whereas his data sets were based on a
fully developed flow in bends for which the assumptions of ENGELUND are no longer valid.
4) Bridge (1976)
BRIDGE’S (1976) formula is similar to the one of ENGELUND, but using tanδ
= 11 ⋅ h s
⁄ r based
on the work of ROZOVSKII (1957) which is resumed in the present paragraph. The end of this
paragraph gives an overview of the study of BAGNOLD (1966) concerning the friction factor,
which ranges between tanφ
= 0.375 ÷ 0.75 = fct( θ,
Fr d
).
page 44 / November 9, 2002
Wall roughness effects on flow and scouring