Energy Balance
For gradually varied flow, the solution is a matter of balancing the energy between the two ends of a pipe segment. The energy equation as it relates to each end of a segment is as follows (note that the pressures for both ends are zero, since it is free surface flow):
- Z1 = Hydraulic grade at upstream end of the segment (m)
- V1 = Velocity at the upstream end (m/s)
- Z2 = Hydraulic grade at the downstream end of the segment (m)
- V2 = Velocity at the downstream end (m/s)
- H1 = Loss due to friction - other losses assumed to be zero (m)
- g = Gravitational acceleration constant (m/s2)
The friction loss is computed based on the average rate of friction loss along the segment and the length of the segment. This relationship is as follows:
The conditions at one end of the segment are known through asinverttion or from a previous calculation step. Since the friction slope is a function of velocity, which is a function of depth, the depth at the other end of the segment can be found through iteration. There are two primary methods for this iterative solution, the Standard Step method and the Direct Step method.
Standard Step Method
The standard step method of gradually varied flow energy balance involves dividing the channel into segments of known length and solving for the unknown depth at one end of the segment, starting with a known or assumed depth at the other end. The standard step method is the most popular method of determining the flow profile because it can be applied to any channel, not just prismatic channels.
Direct Step Method
The direct step method is based on the same basic energy principles as the standard step method, but takes a slightly different approach towards the solution. Instead of assuming a segment length and solving for the depth at the end of the segment, the direct step method assumes a depth and then solves for the segment length.