Gradually varied flow water-surface profiles are expressed in terms of the critical slope dy/dx is shown to be strictly limited to values outside
the range encompassed by S and _{c}S, in which _{o}S is the bed slope.
This new approach improves and completes the definition of flow-depth-gradient ranges in the analysis
of water surface profiles. Online calculators are provided to round up the experience.
_{o}
Computations of gradually varied flow (GVF) are part of the routine practice of hydraulic
engineering. The GVF equation describes steady gradually varied flow in open channels
(Chow 1959; Henderson 1966). The conventional GVF equation is expressed in terms of bed
slope S, and Froude number _{f}F.
Herein, the GVF equation is alternatively expressed in terms of bed slope S,
critical slope _{o}S, and Froude number _{c}F. Analysis of this equation reveals
that the flow-depth gradient dy/dx is strictly limited to values outside the range
encompassed by S and _{o}S. This improves and completes the
definition of flow-depth-gradient ranges in the analysis of water surface profiles.
_{c}
The GVF equation is (Chow 1959, p. 220; Henderson 1966, p. 130):
in which C is (Chow 1959):
in which The Froude number in terms of discharge is (Chow 1959):
Combining Equations 2 and 3 leads to:
At normal critical flow,
Combining Equations 1, 4, and 5:
which is strictly valid for the following condition: T . This latter
condition is generally satisfied in a hydraulically wide channel,
for which _{c}T is asymptotically equal to P.
For ease of expression, the flow-depth gradient is renamed dy/dx. Solving for Froude number from Equation 6:
Since
which effectively limits the flow-depth gradient to values outside the range encompassed by S. Furthermore, Equation 6 can be alternately expressed as follows:
_{c}
Equation 10 is the GVF equation in terms of bed slope S , and Froude number _{c}F. The bed slope could be positive
(steep, critical, or mild), zero (horizontal), or negative (adverse). The critical
slope (Equation 5) and Froude number squared (Equation 3) are always positive.
Equation 10 is used to develop a classification of water-surface profiles based solely
on the three dimensionless parameters: S ,_{c}S /_{o}S ,_{c}F. For the sake of completeness,
subcritical flow is defined as that for which the flow depth is greater than the critical
depth F ^{2} < 1)subnormal flow is defined as that for which the flow depth is greater than
the normal depth F ^{2} < S /_{o}S ]._{c}Supernormal flow is defined as that for which the flow depth is smaller than
the normal depth F ^{2} > S /_{o}S ]_{c}
A summary of the twelve possible water-surfaces profiles is shown in Table 2.
The classification follows directly from the governing equation (Equation 10).
It is seen that the general type of profile (Type 1, 2, or 3)
determines the sign of S_{c}S /_{o}S_{c}S /_{y}S_{c}S /_{o}S_{c}
Unlike the description available in standard references (Chow 1959; Henderson 1966),
the flow-depth gradient ranges (Table 2, Columns 7 and 8) are now complete for all
twelve water-surfaces profiles. Significantly, the flow-depth gradient S and _{c}S .
_{o}
Figure 1 shows a graphical representation of flow-depth gradient ranges in the water-surface
profiles. The arrow shows the direction of computation. For instance, the depth gradient for
the C_{1} (subcritical/subnormal) and C_{3}
(supercritical/supernormal) profiles is constant and equal to S = _{o}S .
Online water-surface profile calculators are enabled in Table 2.
_{c}
Figure 1. Graphical representation of flow-depth gradient ranges in water-surface profiles.
The gradually varied flow equation is expressed in terms of the critical
slope dy/dxS and _{o}S. This completes the definition of
depth-gradient ranges for all water-surface profiles. For instance,
the flow-depth gradient for the _{c}S_{3} profile decreases
from S (a finite positive value) to 0 (asymptotic to
normal depth). Likewise, the flow depth gradient for the _{c}C_{1}
and C_{3} profiles is constant and equal to S = _{o}S_{c}
Chow, V. T. (1959).
Henderson, F. M. (1966).
USDA Soil Conservation Service. (1971). Classification system for varied flow in prismatic channels. |

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