Evaluation of Bridge-Scour Research: Abutment and Contraction Scour Processes and Prediction (2011)

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60 CHAPTER 6. CONTRACTION SCOUR FORMULAS 6.1 DEFINITION OF CONTRACTION SCOUR Contraction scour is caused by flow acceleration due to narrowing of the channel cross section whether by natural reduction in the width of the main channel for a bankline abutment, or by redistribution of floodplain flow in the contracted section as a result of flow blockage by the bridge embankment for a setback abutment. Although contraction scour will vary across the cross section in the field due to nonrectangular geometry and a nonuniform velocity distribution, it is often visualized and applied as a uniform decrease in bed elevation across the bridge opening. Floodplain contraction scour is usually treated separately from main channel contraction scour in compound channels. In this case, one of the difficulties in applying a contraction scour formula is the determination of the discharge distribution between the floodplain and the main channel in the bridge section. Both live-bed and clear-water contraction scour can occur in the field. The former commonly occurs in the main channel of a sand-bed river, while the latter is more likely to be found in a floodplain contraction or a relief bridge located on the floodplain. Contraction scour formulas have been developed analytically for an idealized long contraction as will be described subsequently. In the case of live-bed contraction scour, the limiting condition is continuity of sediment transport between the approach-flow section and the contracted section. For clear-water scour, the governing principle is that the depth of scour in the contracted section corresponds to the occurrence of critical velocity there as the scour approaches its equilibrium state. 6.2 DIMENSIONAL ANALYSIS Dimensional analysis provides a useful approach for evaluating contraction scour formulas. The approach is similar to that given in Section 5.1 for abutment scour. With reference to Figure 6-1, the dimensional analysis for contraction scour can be written as ( )       = ∗ ∗ g c c M B L B B Y B d BYV gY V u u Y Y σ µ ρϕ ,,,,,,,, 22 1 1 1111 1 2 11 2 1 2 (6.1) in which Y2 = maximum depth of flow after contraction scour; Y1 = upstream approach flow depth; B1 = width of approach flow channel; B2 = width of contracted section; Lc = length of contraction (streamwise); d = some measure of the sediment size; ρ and µ = density and viscosity of the fluid, respectively; V1 = approach flow velocity; u*1 and u*c = shear velocity of the approach flow and the critical value of shear velocity for initiation of sediment motion, respectively; g = acceleration of gravity; M =discharge ratio dependent on flow redistribution between main channel and floodplain; and σg = geometric standard deviation of sediment size distribution.

61 Figure 6-1. Definition sketch for idealized long contraction scour (Q1 = main channel flowrate for live-bed scour; Q2 = total flowrate in channel at contracted section; dsc = contraction scour depth. The first term on the right-hand side of Equation (6.1) can also be written as a ratio of shear stresses, τ1/ τc , since u* = (τ/ρ)1/2. This ratio is less than or equal to unity for clear-water contraction scour, while it plays no role in live-bed contraction scour which is governed by sediment transport continuity. The second term is the Froude number squared which reflects the influence of the drop in the water surface due to flow acceleration; it is important for larger values but is often neglected for smaller values. The third term is the flow Reynolds number which incorporates viscous effects, but it can be neglected for the large values typical of prototype turbulent flow. The fourth and fifth ratios define the relative sediment size and the aspect ratio of the approach flow, respectively, and can be neglected except in very small scale laboratory experiments. The ratio of channel widths, B1/B2, is a very important dimensionless ratio that determines the amount of geometric contraction of the flow and thus the degree of contraction scour. The discharge contraction ratio M is discussed below, and σg accounts for armoring in well-graded sediments. The parameter Lc /B2 expresses relative contraction length, which may affect the location of maximum contraction of flow and, thereby, scour development (this aspect of abutment scour has yet to be studied). All of the floodplain and main channel geometric and roughness characteristics from Equation (5.1) have been replaced in Equation (6.1) with a single discharge distribution factor, M, that differs based on whether the flow is main channel flow only or a compound channel flow, and whether the contraction scour can be classified as live-bed or clear-water. In the case of live-bed contraction scour, it will be shown below from the Laursen equation that for overbank flow with a contraction caused by a bankline abutment, M = Q1main channel / Q2total. If, on the other hand, live- B1 Q1 1 B2 Q2 2 Plan View Profile View Y1 Y2 dsc V1 Floodplain Floodplain Lc

62 bed contraction scour occurs for flow in the main channel only, the value of M is unity but B1/B2 becomes the primary independent variable gauging the degree of flow contraction as determined by main-channel geometry alone. Because clear-water contraction scour tends to occur only on the floodplain, the effective B1/B2 can be replaced by the ratio of discharges per unit width, qf2/qf1, for a streamtube that passes through the contracted floodplain. The difficulty comes in estimating the value of qf2/qf1 . If it is assumed that there is no interaction between the floodplain and main channel flows, then it follows that all of the approach floodplain flow passes through the contracted floodplain so that qf1 = Qf1/Bf1 and qf2 = Qf2/Bf2. Then with the assumption that Qf1 = Qf2, the value of qf2/qf1 reduces back to Bf1/Bf2, the geometric floodplain contraction ratio. Between the two extremes of live-bed scour in the main channel for the contraction caused by a bankline abutment, and floodplain clear-water scour for a very short abutment that terminates on the floodplain at a large setback distance from the main channel, is the case of Qf1 ≥Qf2 . In this instance, the main channel flow entrains a portion of the floodplain flow as it travels from the approach-flow section to the contracted section. For this case, Sturm and Janjua (1994) and Sturm (2006) showed that qf2/qf1 can be estimated as 1/M where M = Q1unobstructed /Q2total. Another alternative is to estimate the values of Qf1 and Qf2, and thus qf1 and qf2, from the ratio of conveyances as in HEC-RAS, but Sturm and Chrisohoides (1998b) have shown that the latter estimate is not a good one because the flow is not one-dimensional at the contracted section. A better approach is to use a two-dimensional flow model. 6.3 IDEALIZED LONG CONTRACTION SCOUR Contraction scour has been estimated theoretically by assuming an idealized long contraction with uniform flow occurring in the approach section and in the contracted section. The theoretical development of ideal contraction scour occurred as early as the work of Straub (1934) who established the equilibrium condition for live-bed contraction scour as the scour depth that results in sediment continuity through the contracted flow section as shown in Figure 6-1. He applied the Duboys sediment transport formula (Vanoni 1975), which is generally considered a bed-load transport formula in which bed shear stress is the independent variable, for estimation of the sediment transport rate in the approach-flow and contracted sections. The work of Straub inspired several subsequent studies of contraction scour based on the idealized long contraction. More recently, additional experimental studies of long contractions have been reported in the literature. Several of the more prominent contraction formulas are given in Appendix B in Table B-1; they are discussed in the same order as given in the table. Laursen (1960) utilized a similar approach to that of Straub in which he applied his own sediment transport formula to the live-bed case with the result shown in Table B-1.as the Laursen live-bed contraction scour formula. In compound channels he assumed that all of the sediment transport occurs in the main channel. Laursen’s sediment transport formula considers both bed- load and suspended-load transport; the coefficient p varies according to the relative contribution of bed load and suspended load to the total sediment transport rate. For an overbank flow

63 contraction with a bankline abutment, it can be seen that dimensionless scour depth depends only on (Qt/Qc) or (1/M) as mentioned earlier, while for main channel flow alone it depends on B1/B2. Gill (1981) generalized the Straub formula for live-bed scour by assuming that sediment transport rate is proportional to excess shear stress, (τ − τc)β where β is a numerical exponent equal to 3 for the Einstein-Brown formula and 1.5 for the Meyer-Peter and Mueller formula, for example. The resulting live-bed contraction scour formula is given in Table B-1. Laursen (1963) also applied the assumption of a long contraction to the case of clear-water scour by assuming that the shear stress in the contracted section has reached its critical value τc at the end of the scouring process. Then using Manning’s equation for the approach flow and contracted flow, he obtained a ratio of τ1/τc that when combined with the continuity equation yielded the clear-water contraction scour formula given in Table B-1. 6.4 CONTRACTION SCOUR FORMULAS FROM LABORATORY DATA Komura (1966) emphasized the influence of armoring on live-bed scour depth by arguing that the ratio of the sediment sizes in the approach flow section and contracted section influence the contraction scour depth for large values of B1/B2 and σg1. He applied dimensional analysis to a series of laboratory experiments on live-bed and clear-water contraction scour in a long contraction (Lc/B1 ≥ 1.0) and proposed a formula based on his experimental results in which dimensionless scour depth depends on F1, B1/B2, and σg1 as shown in Table B-1. Lim and Cheng (1998a) derived a long contraction scour formula for live-bed scour along the same lines as that of Gill (1981) using a bedload formula in which β= 4, but then showed that the only solution of the equation was one in which the dimensionless live-bed contraction scour depth depends on B1/B2 alone as shown in Table B-1. They compared their formula with several sets of laboratory data for long contractions and concluded that it gave reasonable agreement not only with live-bed scour data but also with several sets of clear-water scour laboratory data. Briaud et al. (2005) conducted flume experiments on clear-water scour of a cohesive sediment (porcelain clay) in a long contraction. From their experimental results, they proposed a formula for maximum dimensionless contraction scour depth (dsc/Y1) that depends on F1, B1/B2, and the critical value of approach flow Froude number, F1c , as shown in Table B-1. They concluded that contraction length has no influence on the scour depth as long as Lc/B2 ≥ 0.25. In addition, their results showed no influence of the transition angle on scour depth. Dey and Raikar (2005) conducted a set of flume experiments on a long contraction using both sand and gravel beds and varied the geometric standard deviation of the sediments. They maintained the flow conditions such that 0.9 < V1/Vc <1.0, i.e. their formula in Table B-1 applies to maximum clear-water contraction scour. Their results showed a significant effect of sediment gradation for 1.4 < σg < 3 with a minimum value of scour depth due to armoring given as 25% of the value for uniform sediment. The value of the exponent on (B1/B2) in their formula is 1.26 which is somewhat different than the theoretical value and previous experimental values.

64 6.5 FIELD DATA ON CONTRACTION SCOUR As with abutment scour, there is a paucity of reliable field data for comparison with the contraction scour formulas in Table B-1. Two major problems with such comparisons is that: (1) the formulas are based on a much simpler set of flow conditions in the laboratory than found in the field; and (2) existing field data are primarily based either on measurements of contraction scour long after the flood event for which the hydraulic parameters may not be known, or on “flood chasing” techniques in which the time of scour measurement may not coincide with the occurrence of maximum temporal scour depth. Furthermore, distinguishing contraction scour from other types of scour is not a straightforward process. Local pier scour is often separated from contraction scour using a concurrent ambient bed surface for the cross section which is essentially a graphical estimate of the cross section that would exist without pier scour at the time of the cross section measurement (Landers and Mueller 1996). After elimination of pier scour, field contraction scour is determined as the difference between the average bed elevation of the contracted bridge section and an assumed average bed elevation that would have existed without the bridge (uncontracted section). The uncontracted bed elevation can only be estimated from plots of the concurrent bed profile both upstream and downstream of the bridge. (Landers and Mueller 1996). Mueller and Wagner (2005) conducted a comprehensive analysis of the available field data for contraction scour even though it is limited. They compared field data with contraction scour estimates from the formulas of Straub, Laursen, and Komura which were discussed previously. In general, the results were mixed with overprediction in most cases, but instances of underprediction also occurred. More detailed real-time measurements of flow velocities and bed elevations were available for a flood in 1997 on the Pomme de Terre River in Minnesota. The velocity data were not reproduced well by HEC-RAS because the flow through the bridge opening was clearly not one-dimensional. The contraction scour for the bridge was significantly underestimated using the equations recommended in HEC-18; however, this comparison may have been biased by an attempt to separate abutment scour and contraction scour. Mueller and Wagner (2005) concluded that future efforts for computing contraction scour (and abutment scour) require a better balance between the complexity of field conditions and the simplicity of idealized laboratory conditions. Benedict (2003) measured clear-water contraction scour in the South Carolina Piedmont as the depth of remnant scour holes in the floodplain. Flow data was not available for many of the sites so the 100-year peak discharge was taken as representative for these sites while the historic peak discharge was used where it had been measured or could be estimated from surrounding gauges. The Laursen equation was shown to greatly over-predict the contraction scour under these assumptions. An envelope curve for contraction scour was recommended instead as a function of the geometric contraction ratio defined as (1 – B2/B1). The contraction scour depths were shown to vary from nearly zero to the limit of the envelope for all values of the geometric contraction ratio without any apparent trend. Benedict (2003) concludes that “because the envelope was developed from a limited sample of bridges in the (South Carolina) Piedmont, scour depths could exceed the envelope”.

65 In a follow-up study of live-bed contraction scour in the South Carolina Piedmont and the Coastal Plain, Benedict and Caldwell (2009) estimated the elevation of buried scour surfaces using ground-penetrating radar. They proposed eliminating Q2/Q1 from the Laursen live-bed scour equation by assuming that all flow remains in the main channel in order to justify an envelope curve for the contraction scour depth, which depends only on the geometric contraction ratio. By comparing the maximum depth of scour with soil boring data, they concluded that the Piedmont data for scour depth were limited by a scour-resistant subsurface layer that consisted primarily of bedrock, but in a smaller number of cases it was composed of gravel or clay. The Coastal Plain data exhibited a similar scour-resistant layer although some cutting into this layer of no more than 5 ft was evident. Hong and Sturm (2006) showed that field contraction scour can be modeled in the laboratory using Froude number similarity and equality of V1/Vc in model and prototype by judicious choice of the model geometric scale and the model sediment size. A 1:45 scale model of a bridge on the Ocmulgee River in Macon, Georgia was constructed in the hydraulics laboratory at Georgia Tech, and bathymetry of a 750 m reach of the river was reproduced. Good agreement was obtained between model and prototype velocity distributions for the 1998 historical flood of 1840 m3/s (50-year flood peak = 2,240 m3/s). The maximum clear-water contraction scour in the laboratory (V1/Vc = 1) agreed with the measured field live-bed contraction scour depth within 5%. 6.6 VERTICAL CONTRACTION SCOUR (PRESSURE SCOUR) As evidence continues to mount for a higher degree of variability in future climatic conditions, engineers must struggle with more frequent occurrences of submergence, and even overtopping of older bridges, and the need to develop new assessments of design risk for bridges to be built in the future. As a bridge first experiences inlet submergence prior to overtopping, there may be a critical design condition for maximum scour before overtopping relief begins. The work by Arneson and Abt (1998), Umbrell et al. (1998), Lyn (2008), and Guo et al. (2010) has advanced the state of knowledge on vertical contraction scour, but much remains to be done to integrate this information into a comprehensive abutment/contraction scour methodology.