Analysis of the suspended load focuses on establishing the concentration profile above the bed in order to be able to directly compute the integral
Suspended load is operationally defined as those particles affected by a balance between turbulent diffusion and gravitational settling of particles.
Recall that the turbulent stress can be expressed in the form
By the law of the wall
We will assume that this eddy diffusivity is closely related to the turbulent diffusivity relevant to suspension of sediment:
where is a coefficient close to one. The coefficient would be exactly one if the tracer were a passive one. That particles settle on their own in spite of turbulence and thus will not exactly trace the patterns of turbulence suggest that the coefficient should be less than one. Conversely, the centrifugal forces acting on the particles due to turbulent stress will give them inertia allowing better mixing suggesting the coefficient may be greater than one. Empirical observation show and so it is reasonable to assume that the value is one.
The balance we seek is between the settling of particles downward and their diffusive flux upward (in the direction of lower concentration) due to turbulence:
In order to integrate this expression we need a boundary condition, but there are no likely candidates. The theories for bedload transport are not expressed in terms of sediment concentration but rather integrated transport in a layer of uncertain dimension. Generally then we consider only relative concentration, that is a comparison of concentration with that at some reference level. Integrating:
This is called the Rouse equation. The ratio in the exponent is called the Rouse parameter and is a measure of the relative size of the settling velocity and turbulent stress. In practice, the reference level is established by measuring sediment concentration and size distribution at some height off the bed in order to estimate the profile at all levels off the bed.
Because w depends on particle size, the Rouse parameter does as well. The vertical distribution of sediment reflects this, with fine material distributed more uniformly that coarse material. Increased shear also results in a more uniform distribution.
Figure 38-1. In the left hand panel, the grains are very fine sand; in the right hand panel the shear velocity is 5 cm/sec.
We can superimpose the kind of information in this diagram onto a Hjulstrom-type diagram:
The contours depict the ratio of concentration at 100 cm to that at 2 cm. Two features bear noting:
Toward the right side of the diagram, the load is concentrated near the bed. Thus, the Rouse parameter is often used as an operation definition for distinguishing bed load from suspended load. Various specific values have been suggested for this boundary, for example 0.8 and 2.0.
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