Theoretical development on the effects of changing flow hydraulics
on incipient bed load motion
Received 11 January 2008; revised 5 December 2008; accepted 7 January 2009; published 1 April 2009.
Several decades of flume and field measurements have indicated that in rough
turbulent flows the critical Shields stress increases with increasing slope and associated
decreasing relative depth. This result contradicts the usual consideration of a decreased
critical Shields value on very steep slopes because of increased gravitational effects.
However, recent studies have demonstrated that these experimental results could be
reproduced with a force balance model if the classical logarithmic velocity profile was
replaced with a velocity profile that was more compatible with available velocity
measurements over gravel beds. These measurements indicate the existence of a roughness
layer that is a zone of almost constant velocity close to the bed, whose properties (mean
velocity and turbulence) depended on the flow’s relative depth. Unfortunately, velocity
profile measurements for low relative depth associated with steep slopes are scarce, and
it is still difficult to include such flow properties in a force balance model. Flow
resistance data (on the basis of depth average velocity measurements) are very common
and cover a wide range of slopes and relative depths. In this paper these data are used to fit
a velocity profile including a roughness layer. When used in a force balance model for
incipient motion, it adequately reproduced a data set composed of 270 critical Shields
values measured in a flume with near-uniform sediments. The relevance of this research to
field problems is discussed using a data set composed of 92 critical Shields stresses
obtained from field measurements. Finally, a model is proposed for field applications
taking into account the slope effect.
Citation: Recking, A. (2009), Theoretical development on the effects of changing flow hydraulics on incipient bed load motion,
Water Resour. Res., 45, W04401, doi:10.1029/2008WR006826.
] Bed load prediction is of primary importance for river
engineering, fluvial geomorphology, eco-hydrology, envi-
ronmental surveys and management, and hazard prediction.
Using similarity principles, Shields  established a
framework for bed load prediction that is still in use today.
He considered bed load a threshold phenomenon and
established a diagram relating the dimensionless critical
shear stress q
À r)gD] (where t
= rgHS is the
critical shear stress, r
is the sediment density, r is the water
density, g is the acceleration of gravity, D is the grain
diameter, H is the water depth and S is the energy slope) to
the roughness Reynolds number Re*=u*D/n (where u*=
is the shear velocity). Although this curve is not
easy to use (because both q
and Re* depend on the shear
velocity u*, which implies an iterative approach), one
interesting practical issue is that q
was hypothesized by
Shields to be constant when Re* > 1000, which is the case
for most natural flow conditions (rough and turbulent
flows). Thus, knowing the value of this constant, the
calculation of threshold flow conditions (characterized by
a flow depth H) for a given sediment (characterized by its
grain-size distribution curve) and a given energy slope S
should be straightforward.
] However, whereas Shields  proposed an as-
ymptotic value of 0.06 for q
, the appropriate value for this
constant has been widely and continuously discussed since
that time. For instance, the well-known bed load transport
equation proposed by Meyer-Peter and Muller 
= 0.047. Values as low as 0.01 were also
proposed [Fenton and Abbott, 1977; Carling, 1983; Mueller
et al., 2005] as well as values higher than 0.1 [Mizuyama,
1977; Church, 1978; Reid et al., 1985; Mueller et al., 2005].
More generally, values were proposed in the range 0.03
[Parker et al., 2003] to 0.07 (an exhaustive review was
provided by Buffington and Montgomery ), with a
mean value at approximately 0.045 [Gessler, 1971; Miller et
al., 1977; Yalin and Karahan, 1979; Saad,1989].The
importance attached to this question can easily be under-
stood when considering that in most natural gravel bed
rivers the Shields number q barely exceeds 20% of the
critical value q
[Parker, 1978; Andrews, 1983; Mueller et
al., 2005; Ryan et al., 2002; Parker et al., 2007] and that for
these flow conditions, transport rates increase by several
orders of magnitude for very small changes in shear stress,
which can lead to very large errors in bed load prediction if
is not correct.
Cemagref, UR Erosion Torrentielle Neige Avalanches, Saint Martin
Copyright 2009 by the American Geophysical Union.
WATER RESOURCES RESEARCH, VOL. 45, W04401, doi:10.1029/2008WR006826, 2009