Physical Modelling in Fluvial Geomorphology

232 SCIENTIFIC NATURE OF GEOMORPHOLOGY
The supercritical to subcritical flow transition may also produce hydraulic jumps which
have been observed to cause headward erosion in alluvial fan models (Parker 1996) and
produce abrupt gravel to sand size sorting in downstream fining models (Paola et al.
1992). Significant headward erosion of anabranches has also been observed within braided
river FSM studies (e.g. Ashmore 1982, p. 218), but it is unknown how common hydraulic
jumps are in the field, particularly in braided channels with high width: depth ratios.
Particle Settling
The particle fall velocity in a stationary fluid can be considered using two different
equations. Stokes' law considers only viscous resistance forces and is of the form
Uf ∝ D 2 (15)
where Uf is the fall velocity of a particle and D is the grain size. For particles smaller than
0.1 mm in water this relationship holds very well, but Stokes' law does not account for
boundary layer separation behind a falling particle and a consequent increase in the fluid
drag. For large particles, Newton derived an expression, known as the impact law, that
incorporates the effects of boundary layer separation
Uf ∝ D 0.5 (16)
The impact law is not a particularly good approximation of experimental results, even for
particles larger than 1 mm (see Figure 9.5), and has many inadequacies when the particles
are non-spherical. The combined experimental curve does, however, break into two
distinct linear segments, one characterised by Stokes' law and the other broadly delineated
by the impact law. When both model and prototype grain sizes fall exclusively within
either of these two areas, a linear scaling ratio can be applied. However, if the prototype
grain size is in the 'impact region' and the model is in the field of Stokes' law, then the
function is nonlinear and consequently cannot be perfectly scaled. The main result of this
nonlinearity is that the fall velocities of the particles relative to the downstream velocity
are much slower in the model than the prototype, although the particle time constant (the
ratio of the particle response time to the characteristic eddy turnover time, see Elghobashi,
1994) and relationship between particle size and turbulence in the model and prototype
must also be taken into account. Saltating particles in the model may consequently have
larger hop lengths and heights than the geometric scale ratio would suggest. This
limitation could be overcome by altering the grain-size distribution as suggested by Jaeggi
(1986) for initiation of sediment movement, but this solution would also affect the mode
of sediment transport.
Bedload Transport and Deposition
A scale ratio for sediment transport rate in models with fully turbulent flow, (λt)s can be
derived by dimensional analysis (Yalin 1963, 1971),
(λt)s = (λL) 1.5 (17)
However, Yalin (1971) has demonstrated that if Re* is below the critical threshold for a
fully turbulent boundary and the fluid and temperature are the same in the prototype and