Expansion Reach Lengths

In some types of studies, a high level of sophistication in the evaluation of the transition reach lengths is not justified. For such studies, and for a starting point in more detailed studies, Table B-2 offers ranges of expansion ratios, which can be used for different degrees of constriction, different slopes, and different ratios of overbank roughness to main channel roughness. Once an expansion ratio is selected, the distance to the downstream end of the expansion reach (the distance Le on the figure below) is found by multiplying the expansion ratio by the average obstruction length (the average of the distances A to B and C to D from the figure below). The average obstruction length is half of the total reduction in floodplain width caused by the two bridge approach embankments. In Table B-2, b/B is the ratio of the bridge opening width to the total floodplain width, nob is the Manning n value for the overbank, nc is the n value for the main channel, and S is the longitudinal slope. The values in the interior of the table are the ranges of the expansion ratio. For each range, the higher value is typically associated with a higher discharge.

Table B-2 Ranges of Expansion Ratios

The ranges in Table B-2, as well as the ranges of other parameters to be presented later in this appendix, capture the ranges of the idealized model data from this study. Another way of establishing reasonable ranges would be to compute statistical confidence limits (such as 95% confidence limits) for the regression equations. Confidence limits in multiple linear regression equations have a different value for every combination of values of the independent variables (Haan, 1977). The computation of these limits entails much more work and has a more restricted range of applicability than the corresponding limits for a regression, which is based on only one independent variable. The confidence limits were, therefore, not computed in this study.

Extrapolation of expansion ratios for constriction ratios, slopes or roughness ratios outside of the ranges used in this table should be done with care. The expansion ratio should not exceed 4:1, nor should it be less than 0.5:1 unless there is site-specific field information to substantiate such values. The ratio of overbank roughness to main-channel roughness provides information about the relative conveyances of the overbank and main channel. The user should note that in the data used to develop these recommendations, all cases had a main-channel n value of 0.04. For significantly higher or lower main-channel n values, the n value ratios will have a different meaning with respect to overbank roughness. It is impossible to determine from the data of this study whether this would introduce significant error in the use of these recommendations.

When modeling situations which are similar to those used in the regression analysis (floodplain widths near 1000 feet; bridge openings between 100 and 500 feet wide; flows ranging from 5000 to 30000 cfs; and slopes between one and ten feet per mile), the regression equation for the expansion reach length can be used with confidence. The equation developed for the expansion reach length is as follows:

When the width of the floodplain and the discharge are smaller than those of the regression data (1000 ft wide floodplain and 5000 cfs discharge), the expansion ratio can be estimated by (2). The computed value should be checked against ranges in Table B-1. (2) is:

When the scale of the floodplain is significantly larger than that of the data, particularly when the discharge is much higher than 30,000 cfs, (1) and (2) will overestimate the expansion reach length. (3) should be used in such cases, but again the resulting value should be checked against the ranges given in Table B-1:

The depth at Section 2 is dependent upon the expansion reach length, and the Froude number at the same section is a function of the depth. This means that an iterative process is required to use the three equations above, as well as the equations presented later in this chapter for contraction reach lengths and expansion coefficients. It is recommended that the user start with an expansion ratio from Table B-1, locate Section 1 according to that expansion ratio, set the main channel and overbank reach lengths as appropriate, and limit the effective flow area at Section 2 to the approximate bridge opening width. The program should then be run and the main channel Froude numbers at Sections 2 and 1 read from the model output. Use these Froude number values to determine a new expansion length from the appropriate equation, move Section 1 as appropriate and recompute. Unless the geometry is changing rapidly in the vicinity of Section 1, no more than two iterations after the initial run should be required.

When the expansion ratio is large, say greater than 3:1, the resulting reach length may be so long as to require intermediate cross sections, which reflect the changing width of the effective flow area. These intermediate sections are necessary to reduce the reach lengths when they would otherwise be too long for the linear approximation of energy loss that is incorporated in the standard step method. These interpolated sections are easy to create in the HEC-RAS program, because it has a graphical cross section interpolation feature. The importance of interpolated sections in a given reach can be tested by first inserting one interpolated section and seeing the effect on the results. If the effect is significant, the subreaches should be subdivided into smaller units until the effect of further subdivision is inconsequential.