Standard step method

1.0 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 (Sf), channel slope (S0), 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).

2.0 Open Channel Flow Fundamentals

The energy equation used for in open channel hydraulics 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+ 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 (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 super critical and have Froude Numbers greater than . (For more information, see Dimensionless Specific Energy Diagrams for Open Channel Flow.)

F_N= V/√(g(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. 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’s Equation. 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.

2.2 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.)

These curves can be developed using the governing equation of Gradually Varied Flow (Equation 3). Essentially the derivative of the energy equation, this equation is based on the following assumptions (Chaudhry 2008): • The slope is relatively small • Channel is prismatic (dA/dy = 0) • There is a hydrostatic pressure distribution

3.0 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_vel+h_ele (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 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.)

3.1 Newton Raphson Numerical Method Adapted from (Chaudhry, 2008) Step 1: Calculate Total Head at Boundary Condition (H1). Step 2: Estimate depth (y*) at location 2. Step 3: Calculate total head as a function of y* at location 2 (H2*)

Step 4: Calculate the friction slope for location 1 (SF,1) and location 2 (SF,2*).Note, the friction slope at location 2 is based on the estimated height in Step 2.

Step 5: Calculate friction head (hf) by multiplying the distance between location 1 and 2 by the average of SF,1 and SF,2*

Step 6: Calculate total head (H2*) at location 2 by adding hf2 to H1.

Step 7: Repeat steps 2-6 until the difference in H2* in steps 2 and 6 is within an acceptable range. When this occurs, then y* = y.

3.2 Conceptual Surface Water Profiles (Sluice Gate)

Figure 4 illustrates the different surface water profiles associated with a sluice gate in a mild slope (top) and a steep slope (bottom). Note, the sluice gate induces a choke in the system, causing a “backwater” profile just upstream of the gate. In the mild slope 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 Jump in a Rectangular Channel page for more information. Below, an example problem will use conceptual models to build a surface water profile using the STM.