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 $(S_{f})$, channel slope $(S_{0})$, channel geometry, and also a given flow rate. In practice, this technique is widely used through the computer program HEC-RAS, developed by the US Army Corps of Engineers Hydrologic Engineering Center (HEC).

Open Channel Flow Fundamentals

Open Channel Flow Energy Lines

E-y Diagram

The energy equation used for open channel flow computations is a simplification of the Bernoulli Equation (See Bernoulli Principle), which takes into account pressure head, elevation head, and velocity head. (Note, energy and head are synonymous in Fluid Dynamics. See Pressure head for more details.) In open channels, it is assumed that changes in atmospheric pressure are negligible, therefore the “pressure head” term used in Bernoulli’s Equation is eliminated. The resulting energy equation is shown below:

$H=z+y+{\frac {v^{2}}{2g}}$            Equation 1

For a given flow rate and channel geometry, there is a relationship between flow depth and total energy. This is illustrated below in the plot of energy vs. flow depth, widely known as an E-y diagram. In this plot, the depth where the minimum energy occurs is known as the critical depth. Consequently, this depth corresponds to a Froude Number $(F_{n})$Fn of 1. Depths greater than critical depth are considered “subcritical” and have a Froude Number less than 1, while depths less than critical depth are considered supercritical and have Froude Numbers greater than . (For more information, see Dimensionless Specific Energy Diagrams for Open Channel Flow.)

$F_{n}={\frac {v}{{\sqrt {(}}g{\frac {A}{B}}}}$           Equation 2

Under steady state flow conditions (e.g. no flood wave), open channel flow can be subdivided into three types of flow: uniform flow, gradually varying flow, and rapidly varying flow. Uniform flow describes a situation where flow depth does not change with distance along the channel. This can only occur in a smooth channel that does not experience any changes in flow, channel geometry, or channel slope. During uniform flow, the flow depth is known as normal depth (yn). This depth is analogous to the terminal velocity of an object in free fall, where gravity and frictional forces are in balance (Moglen, 2013). Typically, this depth is calculated using the Manning Formula. Gradually varied flow occurs when the change in flow depth per change in flow distance is very small. In this case, hydrostatic relationships developed for uniform flow still apply. Examples of this include the backwater behind an in-stream structure (e.g. dam, sluice gate, weir, etc.), when there is a constriction in the channel, and when there is a minor change in channel slope. Rapidly varied flow occurs when the change in flow depth per change in flow distance is significant. . 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.

Water Surface Profiles (Gradually Varied Flow)

Typically, the STM is used to develop “surface water profiles,” or longitudinal representations of channel depth, for channels experiencing gradually varied flow. These transitions can be classified based on reach condition (mild or steep), and also the type of transition being made. Mild reaches occur where normal depth is subcritical (yn > yc) while steep reaches occur where normal depth is supercritical (yn<yc). The transitions are classified by zone. (See figure 3.)

Figure 3. This figure illustrates the different classes of surface water profiles experienced in steep and mild reaches during gradually varied flow conditions. Adapted from Chow 1959.

The above surface water profiles are based on the governing equation for gradually varied flow (seen below)

${\frac {dy}{dx}}={\frac {S_{0}-S_{f}}{1-F_{n}^{2}}}$           Equation 3

This equation (and associated surface water profiles) is based on the following assumptions:

• The slope is relatively small
• The Channel is prismatic $({\frac {da}{dy}}=0)$
• There is a hydrostatic pressure distribution
  

Standard Step Method Calculation

The STM numerically solves equation 3 through an iterative process. This can be done using the bisection or Newton-Raphson Method, and is essentially solving for total head at a specified location using equations 4 and 5 by varying depth at the specified location. (Chaudhry 2008).

$H_{2}=H_{1}-h_{f}$           Equation 4

$H_{2}=h_{v}el+h_{e}le$           Equation 5

In order to use this technique, it is important to note you must have some understanding of the system you are modeling. For each gradually varied flow transition, you must know both boundary conditions and you must also calculate length of that transition. (e.g. For an M1 Profile, you must find the rise at the downstream boundary condition, the normal depth at the upstream boundary condition, and also the length of the transition.) To find the length of the gradually varied flow transitions, iterate the “step length”, instead of height, at the boundary condition height until equations 4 and 5 agree. (e.g. For an M1 Profile, position 1 would be the downstream condition and you would solve for position two where the height is equal to normal depth.)

Newton Raphson Numerical Method

Conceptual Surface Water Profiles (Sluice Gate)

Figure 4. Illustration of surface water profiles associated with a sluice gate in a mild reach (top) and a steep reach (bottom).

Figure 4 illustrates the different surface water profiles associated with a sluice gate on a mild reach (top) and a steep reach (bottom). Note, the sluice gate induces a choke in the system, causing a “backwater” profile just upstream of the gate. In the mild reach, the hydraulic jump occurs downstream of the gate, but in the steep reach, the hydraulic jump occurs upstream of the gate. It is important to note that the gradually varied flow equations and associated numerical methods (including the standard step method) cannot accurately model the dynamics of a hydraulic jump (Chaudhry 2008). See the Hydraulic jumps in rectangular channels page for more information. Below, an example problem will use conceptual models to build a surface water profile using the STM.

Example Problem

Solution

Step 1: Determine if the reach is steep or mild

Step 2: Determine the effect of the Sluice Gate on flow:

Step 3: Develop a sketch of the surface water profile:

Step 4: Use the Newton Raphsom Method to solve the M1, M2, and M3 surface water profiles. The solution presented explains how to solve the problem in a spreadsheet, explaining how to calculate values column by column. Note, only calculations for the M1 profile will be shown below.

Step 5: Combine the results from the different profiles and display.

Step 6: Model the Reach in the HEC-RAS modeling environment:

It is beyond the scope of this Wikipedia Page to explain the intricacies of operating HEC-RAS. For those interested in learning more, the HEC-RAS user’s manual is an excellent learning tool and the program is free to the public. It is also important to note, the example problem is a simplified situation, where minor energy losses due to expansion/contraction and the sluice gate are not taken into account. To adequately model this within the HEC-RAS modeling environment, a sluice gate constant (Cs¬) was determined and minor loss coefficients were set to zero.

Failed to parse (syntax error): {\displaystyle Q = C_s*b*y_g*(2*g*H_{T,1})^{1/2}<\frac>}            Sluice Gate Equation

There are three figures below illustrating the “steady flow” HEC-RAS results. The first figure is a graphical output directly from HEC-RAS, showing the surface water profile of the modeled reach. The next two figures are comparing the surface water profiles developed by HEC-RAS and the STM calculation method. There is good agreement between HEC-RAS and the STM method, except for the M3 profile just downstream of the sluice gate. While HEC-RAS can model both subcritical and supercritical flows within the same reach, the did not produce an M3 Profile in this instance.