Standard step method

The standard step method (STM) is a computational technique utilized to estimate one-dimensional surface water profiles in open channels with gradually varied flow under steady state conditions.

It uses a combination of the energy, momentum, and continuity equations to determine water depth with a given a friction slope

This is illustrated below in the plot of energy vs. flow depth, widely known as an E-y diagram.

This depth is analogous to the terminal velocity of an object in free fall, where gravity and frictional forces are in balance (Moglen, 2013).

Examples of this include the backwater behind an in-stream structure (e.g. dam, sluice gate, weir, etc.

In this case, hydrostatics relationships are not appropriate for analytical solutions, and continuity of momentum must be employed.

Examples of this include large changes in slope like a spillway, abrupt constriction/expansion of flow, or a hydraulic jump.

Mild reaches occur where normal depth is subcritical (yn > yc) while steep reaches occur where normal depth is supercritical (yn

This figure illustrates the different classes of surface water profiles experienced in steep and mild reaches during gradually varied flow conditions.

Below, an example problem will use conceptual models to build a surface water profile using the STM.

Step 4: Use the Newton Raphson Method to solve the M1 and M3 surface water profiles.

Table 1: Spreadsheet of Newton Raphson Method of downstream water surface elevation calculations

The first two figures below are the upstream and downstream water surface profiles modeled by HEC-RAS.

While the two different methods modeled similar water surface shapes, the standard step method predicted that the flow would take a greater distance to reach normal depth upstream and downstream of the gate.

This stretching is caused by the errors associated with assuming average gradients between two stations of interest during our calculations.

Smaller dx values would reduce this error and produce more accurate surface profiles.

HEC-RAS modeled the hydraulic jump to occur 18 meters downstream of the sluice gate.

Figure 1. Conceptual figure used to define terms in the energy equation. [ 2 ]
Figure 2. A diagram showing the relationship for flow depth (y) and total Energy (E) for a given flow (Q). Note the location of critical flow, subcritical flow, and supercritical flow.
Figure 4. Illustration of surface water profiles associated with a sluice gate in a mild reach (top) and a steep reach (bottom).