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e . -< . . . e . ~ . . . e Chapter 5 Unsteady Flow 5-1. Introduction This chapter is presented in two sectinns. Section I presents guidance on the practical use of unsteady flow modeling and Section II presents some theoretical consid- erations regarding various routing techniques. Guidance regarding the application of unsteady flow models is presented first because the theoretical information, although important, is of a more general nature. Section I Application of Unsteady Flow Models 5-2. Steady versus Unsteady Flow Models The traditional approach 10 river modeling has been the use of hydrologic routing to determine discharge and steady flow analysis 10 compute water surface proftIes. This method is a simplification of true river hydraulics, which is more correctly represented by unsteady flow. Nevertheless, the traditional analysis provides adequate answers in many cases. This section identifies when to use unsteady flow analysis. a. Steady flow. Steady flow analysis is defmed as a combination of a hydrologic technique to identify the maximum flows at locations of interest in a study reach (termed a "flow profile") and a steady flow analysis 10 compute the (assumed) associated maximum water sur- face profIle. Steady flow analysis assumes that, although the flow is steady, it can vary in space. In contrast, unsteady flow analysis assumes that flow can change with both time and. space. The basics of steady flow analysis were given in Chapter 2; details may be found in Chapter 6. (I) The typical steady flow analysis determines the maximum water surface profile for a specified flood event. The primary assumptions of this type of analysis are peak stage nearly coincides with peak flow, peak flow can accurately be estimated at all points in the riverine network, and peak stages occur simultaneously over a short reach of channel. (2) The first assumption allows the flow for a steady state model to be obtained from the peak discharge com- puted by a hydrologic or probabilistic model. For small bed slopes (say less than 5 feet per mile), or for highly EM 1110-2.1416 15 Oct 93 transient flows (such as that from a dam break), peak stage does not coincide with peak flow. This phenome- non, the looped rating curve effect, results from changes in the energy slope. The change in slope can be caused by backwater from a stream junctinn, as shown in Figure 5-1, or by the dynamics of the flood wave, as depicted in Figure 5-2. Since coincidence of peak stage and flow does not exist in either of these cases, the proper flow 10 use in a steady flow model is not obvious. (3) The second assumption concerns the estimation of peak flow in river systems. For a simple dendritic system the flow downstream from a junction is not nec- essari1y equal to the sum of the upstream flow and the tributary flow. Backwater from the concentration of flow at the junction can cause water to be stored in upstream areas, reducing the flow contributions. Figure 5-2 shows the discharge hydrographs on the Sangmon River at the Oakford gage and at the mouth of the Sangmon River 21 miles downstream. The outflow hydrograph is attenu- ated and delayed by backwater from the Illinois River. Steady state analysis often assumes a simple summation of peak discharges. For steep slopes, once again, the assumption may be appropriate but its merit deteriorates as the gradient decreases. (4) A more difficult problem is that of flow bifurca- tion. Figure 5-3 shows a simple stream network that drains a portion of Terrebonne Parish in Louisiana. How can the flow in reach 3 be estimated? Figure 5-1 shows the hydrograph at mile 0.73 in reach 3; note the flow reversal. Hydrologic models and steady state hydraulics cannot predict that division of flow or the flow reversals. (5) The third assumption allows a steady flow model to be applied to an unsteady state problem. It is assumed that the crest stage at an upstream cross section can be computed by steady flow analysis from the crest stage at the next downstream cross section: hence, it is therefore assumed that the crest stage occurs simultaneously at the two cross sections. Because all flow is unsteady and flood waves advance downstream, this assumption is imprecise. As the stream gradient decreases and/or the rate of change of flow increases, the looped rating curve becomes more pronounced, and the merit of this assump- tion deteriorates. (6) The three assumptions are usually justified for simple dendritic systems on slopes greater than about 5 feet per mile. For bifurcated systems and for systems with a small slope, the assumptions are violated and the 5-1