3.4 Gradually Varied, Unsteady Flow (One-Dimensional)

The assumptions of steady flow are not sufficiently accurate in all situations. The design engineer should consider an unsteady flow model when any of the following conditions are present:

Rapid changes in discharge or river elevation. These changes usually occur in conjunction with the operation of man-made structures, such as rapid opening or closing of a sluice gate, sudden start or stop of pumping, or a dam break flood. Rapid changes cause large increases in depth or discharge in very short times. For a dam break, this could be a change from a discharge of near zero to one of thousands of cubic feet per second in a few minutes. Figure 2.16b ((see page 34)) illustrates a similar situation.
A complex stream system where discharge leaves the main channel at various locations and then returns at downstream locations. Complex stream networks are often found in low-sloping, swampy, or wetland areas. This situation does not necessarily include split flow, islands, or diversions, which can be handled by steady flow assumptions for most channel slopes.

A looped rating relationship. Rivers with slopes less than about 0.0004 (2 ft/mi, 0.4 m/km) seldom have a unique relationship between stage and discharge, but rather exhibit different stages for the same flowrate, resulting in a looped rating curve (USACE, 1993b). A looped rating curve occurs when the discharge on the falling leg of the hydrograph passes at a higher elevation (and therefore a lower velocity) than the same discharge on the rising leg. Figure 3.1 illustrates a looped rating curve. The higher velocity on the rising limb means the stream can pass more discharge for a given water surface elevation (steeper slope of water surface profile at t1 shown on Figure 3.1), while exactly the reverse situation exists on the falling limb of the hydrograph (milder sloping water surface profile at time t2 shown on Figure 3.1). The peak stage may occur considerably later than the peak discharge.

Figure 3.1 Looped rating curve example.

Flood forecasting for major rivers. Providing stage forecasts for major river systems at numerous locations for different times is best performed with unsteady flow hydraulic modeling, particularly when the rivers have slopes less than about 2 ft/mi (0.4 m/km). Many major river systems are characterized by slopes in this range, including the Mississippi River (0.5 ft/mi or 0.1 m/km, or less) and the Missouri River (about 1 ft/mi, 0.2 m/km) in the United States. Forecasts for smaller watersheds and for larger streams with slopes exceeding 5 ft/mi (1 m/km) can be successfully made with hydrologic modeling, using a program such as HEC-HMS to compute the discharge and then converting the flowrate to a river stage within HMS, based on a rating curve developed from HEC-RAS. Rivers having a slope of 2-5 ft/mi (0.4-1 m/km) are in a transition range for which unsteady flow modeling would be a first choice, but steady flow modeling could also be satisfactory.
Backwater analysis at major river junctions. On low-sloping tributaries of major rivers, the maximum tributary river elevation is a function of both the discharge flowing downstream on the tributary and the stage in the main river at the mouth of the tributary. Continuous variation in the main river water surface elevations and in the tributary discharge requires unsteady flow modeling to completely analyze the situation. An example is an 80-mi (129-km) reach of the Illinois River upstream of its confluence with the Mississippi River. The tributary Illinois River has less than one-third the slope of the Mississippi River and the Illinois River's peak discharge is often separated from the peak stage by several days along this reach because of the flat slope and the great influence of Mississippi River backwater. While steady flow procedures are often applied to profile analysis at stream junctions, major backwater effects may be poorly modeled using steady flow assumptions. During the Great Flood of 1993 in the Midwestern United States, the peak discharge at Lock and Dam 25 on the Mississippi River, approximately 50 miles (80 km) upstream of the mouth of the Missouri River, occurred three days before the peak stage due to backwater effects from the Missouri River. A somewhat similar situation is a flow reversal (negative discharge) on a mild slope, another situation that cannot be adequately addressed by steady flow. Modeling rivers emptying into a bay or estuary subject to tidal influences normally requires unsteady flow modeling.

One-dimensional, unsteady flow programs use the momentum and continuity equations (St. Venant equations) to perform a simulation through time and space. Changes in velocity and depth with time and distance result in accelerations and forces that cannot be adequately modeled using the energy equation. One-dimensional unsteady flow models can be used for long stream reaches and long time periods and are most appropriate for streams for which velocity vectors can be assumed to be approximately parallel to the direction of flow. The U.S. National Weather Service (NWS) and the USACE have used unsteady flow models to provide river forecasts and operate navigation or flood storage dams, respectively. The following sections provide information on some of the most common models in this category.

HEC-UNET

Robert Barkau, formerly with the USACE, developed the Unsteady Network Program (UNET) in the late 1980s (Barkau, 1992). UNET was later adopted by the USACE as its preferred model for one-dimensional, unsteady flow. Future program upgrades, modifications, maintenance, and documentation were made the responsibility of the USACE's Hydrologic Engineering Center. HEC-UNET (USACE, 1997) is the current version of the program and it is used to perform subcritical, gradually varied unsteady flow analysis.

The program was originally designed to use HEC-2 cross-section data as input to both UNET and HEC-UNET, and they both used a preprocessor (CSECT) to convert the cross-section data to tables of hydraulic properties, such as elevation-area and elevation-conveyance. These tables were then interpolated during the unsteady flow computations for the appropriate values at each section. The HEC-UNET program incorporated dam and spillway analysis, levee overtopping and breaching, and off-channel storage (ponding). The program has since been superceded by the addition of unsteady flow capability in HEC-RAS.

HEC-RAS, Unsteady Flow

HEC-RAS can now incorporate both steady and unsteady, one-dimensional flow computations using the same set of geometry data for either analysis. Unsteady flow computations use the full equations of motion (St. Venant equations), presented in detail in Chapter 14. The unsteady flow equation solver is taken directly from the HEC-UNET program, but all other unsteady flow procedures in HEC-RAS are different from those in HEC-UNET.

The unsteady portion of the program accepts geometric input in the form of standard HEC-2 or HEC-RAS cross sections and then converts each section to tables of hydraulic properties, using a preprocessor program (HTAB) to facilitate the computations in the unsteady flow engine (UNET). The engineer must specify all cross sections, inflow hydrographs (not just peak discharge) for all tributaries, upstream and downstream boundary conditions (flow or stage hydrographs or discharge rating curves), and various coefficients. A postprocessor program is available to facilitate the output review. The postprocessor provides all the tables and plots for unsteady flow that are available for steady flow computations. Without the postprocessor, only graphical output consisting of stage and/or discharge hydrographs at all cross sections is available.

The unsteady flow portion of the program can perform subcritical, supercritical, or mixed-flow computations. Dam breaching and levee break algorithms are also included, as is the ability to model pumping stations. A maximum of ten pump groups, with each group consisting of up to 20 identical pumps, can be modeled in the unsteady flow portion of HEC-RAS. The modeling of flap-gated culverts (allowing only one-way flow) is also an option. As is the case when modeling steady flow in HEC-RAS, the unsteady analysis is limited to 500 profiles. A maximum of 6000 cross sections may be used in the model, with up to 500 elevation-station points for each cross section.

Both steady and unsteady flow analysis in HEC-RAS begin with the same set of geometric data, but the water surface profiles for the same actual or hypothetical event are normally somewhat different. Differences are primarily due to three key features:

For eddy or other losses, steady flow computations use the absolute difference in velocity heads at adjacent cross sections multiplied by an expansion or contraction coefficient. In unsteady flow computations, these eddy losses are computed within the momentum equation.
Steady flow computations find the average friction slope between cross sections based on the average conveyance method (HEC-RAS default method). Unsteady flow computations use the average friction slope between cross sections directly from a simple average of the computed friction slopes. Tests using both UNET and HEC-UNET (Brunner, 2002) have shown that the unsteady flow program is more stable when using average friction slope, rather than the average conveyance method that is applied in steady flow computations.
For a given discharge, steady flow computations compute losses through bridges, culverts, and other obstructions directly from the obstruction geometry and the type of flow conditions through the bridge (low flow, pressure, weir, momentum, or combination, as discussed in more detail in Chapter 6). In unsteady flow, a family of curves is developed for defining the headwater-tailwater-discharge relationships through each obstruction for a full range of flow. Unsteady flow analysis in HEC-RAS interpolates the headwater elevation for computed discharge and tailwater data for each time period. Depending on the number of discharges used to set up the family of curves, there could be differences in computed losses at obstructions between steady and unsteady state computations.

Given the computational differences between steady and unsteady state analysis, there is generally a small difference between results for a selected flood discharge for a steady flow solution compared to the unsteady flow solution. The steady flow solution is generally 0.1-1 ft (0.03-0.3 m) higher than the unsteady flow solution, but the difference can be outside this range, depending on how the engineer developed his input data and the degree of variation in flow expansion and contraction through a reach. The difference does not necessarily mean that one method is more accurate than another; it simply means that a difference may exist because the computation procedures are different between steady and unsteady flow analysis.

The unsteady flow analysis routing in HEC-RAS has been successfully applied for a variety of rivers and streams, ranging from more than 2000 mi (3200 km) of the Mississippi and Missouri River system for 100 years of daily discharge data, to analyzing a hypothetical design flood for small, swampy streams experiencing flow reversals.

FLDWAV

In the late 1980s, the U.S. National Weather Service (NWS) developed the FLDWAV (pronounced floodwave) computer program (NOAA, 2000) for unsteady flow analysis using the full equations of motion. The program performs hydraulic simulations for real-time forecasting of natural floods or dam-break events and supplies information for the design of waterway improvements and for flood inundation mapping for dambreak flood planning. The flow can be subcritical, supercritical, or mixed throughout the downstream reach. The flood being modeled can be interconnected through a river system (main stem and tributaries). Levee overtopping and breaching are handled, along with split flow (island) situations and the modeling of mud-debris flow. Bridges can be modeled with the program, but not culverts. Planned improvements include the addition of culvert modeling, the operation of movable gates, sediment transport, additional routing methods, and the modeling of landslide-generated waves in reservoirs.

The FLDWAV program is a combination of two popular NWS programs: the Dynamic Wave Operation Network Model (DWOPER) and the Dam-Break Forecasting Model (DAMBRK). DWOPER was developed by Danny Fread (Fread, 1982) of the NWS for use in the river forecasting program. It is a general model with many added features to allow simulation of river structures and levees. The NWS used the program to routinely provide daily stage and discharge predictions for the Lower Mississippi River prior to FLDWAV. The NWS also developed DAMBRK (Fread, 1984) specifically for simulating the failure of a dam and the resulting flood wave through the downstream valley. The model has been used in numerous dam break simulations to determine the maximum crest height to be expected and the warning time available for downstream inhabitants.

FEQ

The Full Equations (FEQ) computer program (Franz and Melching, 1997) simulates flow in a stream system by solving the full equations of motion for one-dimensional, subcritical unsteady flow. The effect and/or operation of structures including bridges, culverts, dams, spillways, weirs, and pumps may be simulated with the program. A companion program (FEQUTL) operates as a preprocessor to convert cross-section data into hydraulic tables for use during the unsteady computations. FEQ uses the continuity and momentum equations to determine the flow and depth throughout the stream system following the specification of initial flow and boundary conditions.

The program was initially developed in 1976 for simulation of flow through the Sanitary and Ship Canal in Chicago, Illinois. The program has been improved and modified in many versions over the years. Documentation was published by the USGS in 1997, as referenced in the previous paragraph. The program has been used for a wide variety of rivers and streams, ranging from 600 mi (966 km) of the Mississippi River to small creeks in DuPage County, Illinois. The program is distributed by the USGS and additional information may be obtained at the USGS web site: http://water.usgs.gov/.