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37 CHAPTER 5 Local Scour Evolution Predictive Methods Introduction The initial shear stress is assumed to be 4 times the shear stress on the approach flow bed. Sediment nonuniformity Several methods have been proposed for predicting local and stratification are shown to have a significant effect on the scour evolution rates at bridge piers. Some of these methods scour depth. The effects of these elements, as well as that of and, where possible, the associated equations are shown in unsteadiness of flow, are taken into account in the proposed Table 12. The methods/equations vary in complexity. Some method. The method applies to scour development at circu- require computer programs to evaluate and therefore cannot lar piers and is based on experiments with pier widths rang- be displayed in the table. In all cases considered, the com- ing from 0.21 to 0.56 ft, uniform sediments with D50 equal to puter programs were obtained from the developer and used 0.4 to 4 mm, and sediment mixtures with D50 equal to 0.5 and in the evaluation process. A brief description of each of the 0.71 mm and standard deviations up to 7.8. 10 methods is presented in the following paragraphs. Melville and Chiew (1999) Shen et al. (1966) Melville and Chiew (1999) proposed Equation 32 for the Shen et al. (1966) fitted the Chabert and Engeldinger (1956) temporal development of local scour at cylindrical piers in data to an exponential function, Equation 29, to obtain an uniform sand beds. The method was based on the data of expression for scour depth as a function of time. The equation Ettema (1980), together with additional data collected at the is based on a narrow range of flow and sediment conditions University of Auckland and Nanyang Technological University. and was developed for circular piers. They defined an equilibrium time scale for the scour process and showed that the equilibrium time scale, t*, is subject to Sumer et al. (1992) the same influences of flow and sediment parameters as the equilibrium scour depth, ys. That is, t* is dependent on y1/a, Sumer et al. (1992) used measurements from 18 experiments V1/Vc, and a/D50. According to their method, the maximum of local live-bed scour at cylindrical piers to determine the time (in days) for the development of equilibrium scour depth exponent of an exponential function, Equation 30, for pre- is equal to 28.96 a/V1. dicting scour evolution rates. The equilibrium scour depth must be known a priori. If this value is not known, the authors Miller and Sheppard (2002) recommend a value of 1 to 1.5 times pier width based on the results of Breusers et al. (1977). The method comprises a semi-empirical mathematical model for the time rate of local scour at a circular pile located in cohesionless sediment and subjected to steady or unsteady Kothyari et al. (1992) water flow. The model is intended for use for both clear-water Kothyari et al. (1992) based their method on the scouring and live-bed scour conditions. A knowledge of the structure potential of the horseshoe vortex, which decreases as the scour dimensions, flow conditions, sediment properties, and the hole enlarges. They assumed that the shear stress under the equilibrium scour depth for the instantaneous flow conditions vortex at time t is a function of the initial shear stress, the is required as input to the model. The scour hole is assumed initial area of the vortex, and the area of the vortex at time t. to have the idealized geometry of an inverted frustum of a

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38 Table 12. Equations/methods for estimating local scour depth evolution with time. Reference Equations Notes No. 2.932 y1 m 0.026e , y1 in ft Shen et al. ys t 0 .4 mE 2 0 .3 3 29 (1966) 2.5Fr 1- e a V1t a E Fr 0.33 ln y1 y1 t T ys t ys 1 e yst = time-dependent scour Sumer et al. (1992) s 30 2.2 a2 2 Ss T 0.0005 y1 u* 3 g Ss 1 D5 0 a g Ss 1 D50 Kothyari et Computer program [see the discussion in the section "Kothyari et al. Computer Program 31 al. (1992) (2007)"] y s t (t) K t ys 1.6 yst = time-dependent scour Vc t Kt exp C1 ln ys = Melville's equilibrium Melville V1 te equation and Chiew 32 (1999) a V1 y1 V1 C1 = 0.03 t e ( days ) C 2 - 0.4 6 ,1.0 0. 4 V1 Vc a Vc C2 = 48.26 days/sec 0.25 a V1 y1 y1 V1 C3 = 30.89 days/sec t e ( days ) C3 - 0.4 6 ,1.0 0. 4 V1 Vc a a Vc Miller and Computer program [see the discussion in the section "Miller and Computer Program 33 Sheppard Sheppard (2002)"] (2002) Clear water: V1 Fd ys t 0.5 1.5 t Ss 1 gD50 Oliveto and 1/ 3 0.068 K s g Fd log y1a 2 tR Hagar (2002, Live bed: y1a 2 1/ 3 34 2005) ys t 0.5 1/ 4 t t tR 1/ 3 0. 4 4 g Fd Fdi log 300 1/ 3 Ss 1 gD50 Oliveto et y1a 2 tR tR g al. (2007) Fdi = densimetric particle ys t 1/ 4 t t g 0.5 Fd Fdi 0.80 0.12 log 300 105 Froude number for 2 1/ 3 tR tR y1a inception of scour Mia and Nago Computer program [see the discussion in the section "Mia and Nago Computer Program 35 (2003) (2003)"] Chang et Computer program [see the discussion in the section "Chang et al. 36 Computer Program al. (2004) (2004)"] yst = time-dependent scour dS S 0.37 2 S cot 1 = angle of repose of bed dTs Ts0.95 S2 cot S sediment ys t Ss = specific gravity of bed S sediment a t D50 Ss 1 gD50 0.5 g = geometric standard Yanmaz Ts deviation of particle 37 2 (2006) a size distribution 0.95 0.63 u* a u* = shear velocity 0.231 tan TD* 0.24 1.9 g D50 Ss 1 g D50 T = transport-stage 1/ 3 parameter Ss 1 g D* D50 2 Ts = dimensionless time = kinematic viscosity

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39 Table 12. (Continued). Reference Equations Notes No. = a/B B = rectangular channel ys t 0.5 2/ 3 t width 0.272 g Fd Fd log 2 1/ 3 tR R = hydraulic radius y1a Kothyari et 1/ 6 te = time to "end scour" 0.25 R 1/ 3 38 al. (2007) Fd Fdi 1.26 g tR = reference time D50 V1 te Fd log 4.8 Fd0.2 Ss 1 gD50 tR Fdi = Fd for inception of scour right circular cone that maintains a constant shape through- Mia and Nago (2003) out the scour process. The slope of the sides of the scour hole is uniform and equal to the submerged angle of repose for the This method comprises a mathematical model for the sediment. Removal of sediment from the scour hole is limited development of local scour at cylindrical piers in cohesionless to a narrow band adjacent to the cylinder where the effective sediment subjected to steady flows in the clear-water scour shear stress is greatest. The sediment transport function used in regime. Input parameters to the model are structure dimen- the model is based on commonly used functions for transport sions, flow conditions, and sediment properties. The scour on a flat bed. hole is assumed to be the frustum of an inverted cone with The effective shear stress in the scour hole, used in the the angle of the frustum being the angle of repose of the bed sediment transport equation, is a function of the normalized sediment. The change in shear stress at the nose of the pier scour depth (scour depth/equilibrium scour depth) and the with increasing scour depth is estimated using a modified form structure, flow, and sediment parameters. The function for of an equation by Kothyari et al. (1992) for the temporal vari- the shape of the effective shear stress versus normalized scour ation of bed-shear velocity. The bed-load sediment transport depth and its dependency on structure, flow, and sediment function used is attributed to Yalin (1977). This method does parameters was determined empirically using data from not require knowledge of the equilibrium scour depth but a number of clear-water and live-bed scour experiments rather, according to the authors, can compute the equilibrium conducted at the Universities of Florida and Auckland and scour depth and the time required to reach this depth. the USGS Laboratory in Turners Falls, Massachusetts. These This model was able to predict the data obtained by the experiments cover a wide range of structure, flow, and sediment model developers but did not accurately predict the data from conditions. This method must be programmed to obtain the other researchers. time history of the scour depth. Chang et al. (2004) Oliveto and Hagar (2002, 2005) This method comprises a mathematical model for the devel- and Oliveto et al. (2007) opment of local scour at circular piers in cohesionless non- Oliveto and Hager (2002, 2005) proposed a method for the uniform diameter sediments subjected to steady or unsteady evolution of clear-water scour, while Oliveto et al. (2007) gave flows in the clear-water scour regime. Input parameters for this a method for the development of scour under live-bed con- model are structure dimensions, sediment properties (size, size ditions. The methodology, which is given as Equation 34, is distribution, mass density), and flow parameters. A sediment based on an extensive set of experiments conducted at ETH mixing layer thickness is computed along with an equivalent Zurich, Switzerland. In formulating the equations, Oliveto et al. sediment size in the mixed layer. The dimensionless scour assumed that scour depth varies logarithmically with time. depth is expressed in terms of time normalized by Melville and The clear-water equation is based on data for uniform sediments Chiew's expression for time to equilibrium. The scour rates ranging from D50 equal to 0.55 to 5.3 mm and cylindrical pier are divided into three normalized time intervals as the scour widths ranging from 0.066 to 1.64 ft. Their live-bed equation depth progresses toward an equilibrium time. This method was is based on Chabert and Engeldinger (1956) and Sheppard and developed for steady flows, but according to the authors can be Miller (2006) data. applied in a finite step-wise manner to unsteady flows. As with