Victor M. Ponce Professor of Civil Engineering San Diego State University [110807] ABSTRACT
INTRODUCTION This article revisits the Lane relation of fluvial hydraulics (Lane, 1955):
The new relation is derived from theory, and expressed as a dimensionless equation, with the particle size (d_{s}) replaced by the relative roughness function (d_{s}/R)^{1/3}. The derivation follows. THE FRICTION FUNCTION The quadratic friction law is (Ponce and Simons, 1977):
in which f is a friction factor equal to 1/8 of the DarcyWeisbach friction factor. The bottom shear stress in terms of hydraulic variables is (Chow, 1959):
Combining Eqs 2 and 3:
The Froude number is (Chow, 1959):
Combining Eqs. 4 and 5:
For a hydraulically wide channel: D ≅ R. Therefore:
THE SEDIMENT TRANSPORT FUNCTION A general sediment transport function is (Ponce, 1988):
According to Colby (1964), the exponent m varies in the range 3 ≤ m ≤ 7, with the lower values corresponding to high discharges, and the higher values to low discharges. Assume m = 3 as a first approximation (high water and sediment discharge). In this case, the sediment transport function is:
where k_{1} is a dimensionless constant. The unitwidth discharge is:
The sediment concentration is:
Combining Eqs. 9 and 11:
For a hydraulically wide channel, d ≅ D. Combining Eqs. 5 and 12, the sediment concentration is:
Combining Eqs. 7 and 13:
The relationship between f and Manning's n is, in SI units (Chow, 1959):
In U.S. Customary units:
Thus, in general:
In SI units:
In U.S. Customary units:
THE STRICKLER RELATION The Strickler relation between Manning n and mean particle size d_{50} is (Chow, 1959):
In SI units:
with d_{50} in meters. In U.S. Customary units:
with d_{50} in feet. Assume d_{s} = d_{50}:
Combining Eqs. 17 and 24:
THE SEDIMENT CONCENTRATION The sediment concentration is:
Substituting Eq. 25 on Eq. 14:
Thus:
Therefore:
and:
THE MODIFIED LANE RELATION The Lane relation (Lane, 1955) is:
Following Eq. 29, the Modified Lane relation is:
The sediment transport relation is:
In SI units:
In U.S. Customary units:
The sediment transport function is dimensionless; therefore, independent of the system of units. The sediment transport parameter k_{1} is the only one to be determined by calibration. Experience shows that this parameter varies typically in the range 0.001 ≤ k_{1} ≤ 0.01. APPLICATIONS Assume pre and postdevelopment cases, with subscripts 1 and 2, respectively. Further define:
From the modified Lane relation (Eq. 30):
Thus, the slope change is:
Example 1 A river reach downstream of a channel diversion intake, with d = 0.9, and a = 0.99, b = 1., and c = 0.95, will result in e = 1.12 (aggradation). Example 2 A river reach downstream of a sediment retention basin, with a = 0.3, and b = 1., c = 0.95, and d = 0.9, will result in e = 0.34 (degradation). In practice, the latter may be limited by geologic controls (armoring or bedrock) (Fig. 1). REFERENCES Chow, V. T., 1959. Openchannel hydraulics. McGrawHill, New York. Colby, B. R., 1964. Discharge of sands and mean velocity relations in sandbed streams. U.S. Geological Survey Professional Paper No. 462A, Washington, D.C. Lane, E. W., 1955. The importance of fluvial morphology in hydraulic engineering. Proceedings, American Society of Civil Engineers, No. 745, July. Ponce, V. M., and D. B. Simons, 1977. Shallow wave propagation in open channel flow. American Society of Civil Engineers Journal of the Hydraulics Division, Vol. 103, No. HY12, December. Ponce, V. M., 1988. Ultimate sediment concentration. Proceedings, National Conference on Hydraulic Engineering, Colorado Springs, Colorado, August 812, 1988, 311315.
NOTATION a, b, c, d, e = ratios of post and predevelopment hydraulic variables; C = Chezy coefficient; C_{s} = sediment concentration; d = flow depth; D = hydraulic depth; d_{s} = particle size; d_{50} = mean particle size; f = friction factor equal to 1/8 of DarcyWeisbach friction factor; F = Froude number; g = gravitational acceleration; k_{1} = dimensionless sediment transport parameter; k_{2} = friction parameter;
k_{3} = coefficient in the Strickler relation; n = Manning's friction coefficient; q = unitwidth water discharge; q_{s} = unitwidth sediment discharge; Q_{w} = water discharge; Q_{s} = sediment discharge; R = hydraulic radius; S_{o} = bottom slope;
v = mean velocity; γ = unit weight of water; ρ = density of water; and τ_{o} = bottom shear stress.
