Evaluation of Bridge Scour Research: Pier Scour Processes and Predictions (2011)

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44 CHAPTER 4 PARAMETER FRAMEWORK 4.1 Introduction Quantitative explanation of how pier flow field, erodibility of foundation material, and erosion processes influence pier scour depth requires a framework of parameters linking variables influencing pier scour. The framework’s central parameters can be determined by considering the variables associated with scour at a cylindrical pier in a single stratum of non-cohesive sediment. Additional parameters quickly arise when considering practical aspects of pier design and actual site conditions at bridge waterways. This chapter focuses on scour at cylindrical piers in non-cohesive sediment. Chapter 5 subsequently introduces pier site complications to be considered. The set of variables involved with the comparatively simple situation of a uniform cylindrical pier in a single stratum of non-cohesive sediment entails a surprising number of complexities, some of which have only become recognized during the past two decades. A few considerations must be kept in mind: 1. The pier flow field and scour depth vary substantially with pier form and dimensions, and approach flow variables; 2. Identifying the necessary full set of parameters at play, even for a cylindrical pier, is not as straightforward as prior publications on pier scour suggest; and, 3. As the number of variables considered increases, the parameter framework soon becomes intricate. Cross connections exist between parameters including common variables (e.g., nominal pier width), which appear in several parameters. The variation of one parameter along an axis of influence often incurs the variation of another parameter. Consequently, rational-type equations relating scour depth to parameter influences become more cumbersome and arguably less accurate (if indeed appropriate). Though most variables of importance had been identified before 1990, developments in instrumentation and computer-simulation techniques since 1990 (especially during the recent decade) have revealed additional parameters, or parameter influences, of importance to pier scour: 1. New fundamental parameters relating to the pier flow field (especially the turbulence structures in it) or boundary erosion characteristics (especially clay and rock); and, 2. Parameters describing the additional processes complicating scour (notably, soil cohesiveness, debris or ice accumulation, abutment proximity, and bridge-deck submergence).

45 The parameter framework must be consistent and comprise relevant parameters. As several variables exert multiple influences often interlinked, parameter sets can be formed of less relevant parameters. For example, bed material “size” affects flow field geometry and dynamics, relative roughness of bed, time evolution of scour, and the hydraulic deformation of the bed. The multiple influences of some variables cause them to be combined as alternative parameters, usually traditional hydraulic engineering parameters (Froude number or Reynolds number) some of which do not accurately express scour physics. Additionally, when the framework of parameters becomes overly intricate, because too many parameter influences cannot be treated independently, the framework loses its utility for formulating a general relationship for scour depth. Then, scour at a pier must be viewed in terms of a system of interconnecting influences, best treated by means of simulation (hydraulic modeling and numerical modeling). Simple relationships between scour depth and dominant parameters or variables (e.g., effective pier width) lead to practical, envelop curves useful for design estimation of the potential maximum scour depth, and for providing a tangible sense of scour depth magnitude. Few publications identify and discuss the full, interconnected framework of parameter influences. Melville and Coleman (2000) give a particularly comprehensive coverage of parameter influences known up to about the late 1990s. More recently, Sheppard and Millar (2006) provide a useful broad review. 4.2. Variables at a Cylindrical Pier in a Single Foundation Stratum The processes contributing to pier scour at a cylindrical pier in a single stratum of non- cohesive foundation material involve the basic variables shown in Figure 4-1. A contextual framework of non-dimensional parameters relating the magnitudes of length, time, and force associated with the processes, is discussed next. This framework brings together the findings reported in the extensive literature on pier scour, and reveals gaps where influences are inadequately explained or quantified. The functional relation between the depth of local scour, ys, and the pertinent variables can be stated as ݕ௦ ൌ ݂ݑ݊ܿݐ݅݋݊ ቈ݂݈݋ݓ ሺߩ, ߤ, ܸ, ݕ, ݃ሻ, ܾ݁݀ ݉ܽݐ݁ݎ݈݅ܽ ൫ܦ, ߪ௚, ߩ௦, ௖ܸ, ൯, ݌݅݁ݎ ሺܽ, ܾ, Ω, ߠሻ,ݐ݅݉݁ ሺݐሻ, ቉ (4.1) In Eq. (4.1),  and  = fluid density and molecular viscosity, respectively; V = depth-averaged velocity of approach flow; y = approach flow depth; and, g = gravity acceleration

46 Vc = critical shear velocity for bed sediment entrainment; D and σg = median size and geometric standard deviation of the foundation material particle size distribution; ρs = sediment density; and c = a parameter describing cohesiveness of the material. The ensuing discussion focuses on coheshionless foundation material, leaving consideration of material cohesiveness to Chapter 5, which addresses complications at pier sites. a = pier width; b = pier length; Ω = parameter describing the shape of the pier face (upstream side); θ = angle of the flow relative to pier alignment t = time The following set of non-dimensional parameters can be developed using a, V, and ρ as the normalizing (or independent) variables: ) a tV ,,,aV, ga V V V a D , a y, b a( function a y s g * 2 c s ρ ρ σ µ ρ θ ,,,,Ω= (4.2) At times it is useful to use shear velocity instead of depth-averaged velocity, V. Shear velocity V* fVV 8*= relates to the V, as , where )( y D functionf = is the Darcy- Weisbach resistance coefficient for fully turbulent flow. This relationship also expresses the critical shear velocity for bed sediment entrainment V*c in terms of the critical mean approach flow velocity for entrainment of bed sediment, Vc. Thus, V, and Vc could be replaced with V* and V*c in Eq. (4.2). However, use of V instead of V* affords greater clarity in describing the relationship between pier flow field and scour depth. Figure 4-1 Variables influencing pier scour at a cylindrical pier

47 Some of the parameters in Eq. (4.2) can be combined to aid explanation of processes influencing scour; e.g., combination of y/a and D/a gives the length ratio y/D, of use when talking about relative roughness of the approach flow, and bedform presence in the approach flow. Because the variables can form alternative parameters, it is possible to assemble less meaningful parameters than those in Eq. (4.2). For example, flow Froude number (V/(gy)0.5 ) could arise, but it is not of immediate use other than distinguishing whether the approach flow is sub-critical or super-critical. 4.3. Primary and Secondary Parameters The parameter framework of the parameters in Eq. (4-2) comprises sets of primary and secondary parameters. The primary parameters define the structure and geometric scale of the pier flow field, and therefore determine the potential maximum scour depth. The secondary parameters characterize scour-depth sensitivities within the geometric scale limit, and normally lead to scour depths less than the potential maximum scour depth. The values of the secondary parameters are subject to considerable uncertainty at pier sites. The primary parameters relate directly to the pier flow field: y/a indicates the geometric scale of the pier flow field (in a vertical cross-sectional plain transverse to the pier, and streamwise to the pier) a/D relates the length scales of pier width and median diameter of bed particle Ω, a/b, and θ , define pier face shape, aspect ratio of pier cross-section, and approach flow alignment to pier, respectively. These parameters may be merged with pier width, a, to form the compound variable a* = effective pier width. It can be useful to express the two length-scale parameters as y/a* and a*/D The secondary parameters have magnitudes prescribed by the primary parameters, with regard to potential maximum scour depth: V/Vc expresses the extent or stage of sediment transport on the approach flow bed, often termed the flow intensity. When sediment diameter, D, is set, Vc is prescribed. This parameter distinguishes whether clear-water or live-bed scour (bedload movement in the approach flow) conditions prevail in the approach flow to the pier V2 /ga is an Euler number relating vorticity induced inertial forces in the pier flow field relative to gravity acceleration ρVa/µ is the pier Reynolds number. Though viscous effects are unlikely to have an effect on the scour depth at the pier because fully turbulent flow occurs around bridge piers, the inclusion of a Reynolds number accounts for the dependence of wake-vortex shedding on pier Reynolds number

48 σs is geometric standard deviation of bed particles, and characterizes sediment uniformity tV/a characterizes (in conjunction with other parameters) the temporal development of scour associated with pier flow field and nature of foundation material Before proceeding to discuss parameter influences, it is useful to make several comments: 1. Some equations for estimating scour depth (Appendix A) identify a group of primary parameters defining a potential maximum geometric scale of scour (y/a, a/D, Ω, a/b, and θ). Many others do not; 2. The coupled role of parameters V2 /ga and ρVa/µ has been identified quite recently. They express similitude in the flow power associated with large-scale turbulence structures generated in the pier flow field; 3. Only the methods developed by Melville 1997 (see also Melville and Coleman 2000) and Sheppard and Miller (2006) include the primary parameters; 4. Some variables appear in several parameters, and complicate the parameter framework. In particular, pier width, a, appears in several of the parameters influencing the pier flow field: i. Relative “shallowness” a/y of the flow field; ii. Relative “coarseness” a/D of the scour hole base, and the possibility of bedforms developing in a sufficiently large scour hole; iii. Vorticity (or intensity of circulation) and frequency of coherent turbulence structures in the pier flow field ( µ ρVa and ga V 2 ); iv. Variation of effective pier shape (with Ω and θ), and thereby all the above influences; and, v. The time rate of scour development, tV/a. 5. As noted later in Section 4.5, certain regions of the parameter framework lack data to confirm parameter influences, notably regions where parameter values are difficult to attain, such as live-bed scour at piers in the wide pier category. 6. An approach using semi-empirical equations based on selected key parameters is useful for approximate estimation of scour depth. The accuracy of such equations increases up to a point, as more parameters are considered. However, the utility and accuracy of such an approach then may diminish when attempting to account for parameter influences involving variables exerting several influences. Also, the large number of parameters to be considered may render such equations unwieldy. This situation quickly arises for piers formed of multiple components (e.g., column on a pile-cap with piles), and in pier sites complicated by additional

49 considerations. Moreover, acquiring the data to establish the quantitative relationships of parameter influences quickly entails very extensive programs of laboratory experiments and field observations. 4.4. Parameter Influences This section evaluates the known influences that individual parameters in Eq. (4.2) exert on scour, and points out knowledge gaps. Despite numerous studies aimed at describing the parameter influences of parameters, there remain significant gaps in the overall understanding of scour. The gaps are attributable to difficulties in conducting laboratory flume experiments at large geometric scales, with isolating some of the parameter influences, and the large number of parameter influences to be examined. Also, there is a paucity of reliable field data in certain parameter ranges. The evaluation extensively uses the substantial documentation provided by Melville and Coleman (2000), presently the most comprehensive publication explaining parameter influences on pier scour. Because certain parameter influences have come to light since 2000, the present evaluation extends beyond Melville and Coleman (2000). However, the evaluation is not intended to be highly detailed, but rather indicate parameter influences. An important consideration insufficiently recognized heretofore is that the parameter influences are not fully independent from each other. Further, the more parameters influencing scour at a pier, the more cross-connected become the parameter influences. In the following descriptions of parameter influences, when the cross-section of the pier is not specified, the cross-section is taken to be circular. 4.4.1 Flow-field Scale, y/a The parameter y/a defines the geometric scale of the pier flow field and, therefore, potential maximum scour depth. It is central to discussion of pier flow field and its variations. Melville and Coleman (2000) categorize pier scour flow field and processes in terms of three classes of y/a, and suggest the three categories in Table 4-1. The inequalities defining each were derived from plots of laboratory data as evident in Figure 4-2. They used the same laboratory data to define different functional relationships, ys = ƒ(y/a), describing the influence of flow shallowness on local scour depth (Table A-1 and Appendix A). For flow depths large compared to pier width (i.e., for narrow piers), scour depth increases proportionately with pier width, and is independent of y. Figure 3-4 illustrates the flow field commensurate with this category. Conversely, for the wide pier category, scour depth increases proportionately with y and is independent of a; Figure 3-6 illustrates the flow field. For intermediate depth flows, scour depth depends on both y and a (Figure 3-5). The solid line in Figure 4-2 envelops the data and applies, from left to right respectively, to wide-, intermediate-, and narrow-width piers at threshold conditions. The dashed lines in Figure 4-2 indicate scour depths for different values of V/Vc, and show that, for clear-

50 water scour at reduced flow velocities, lesser scour depths are developed. The lines have been plotted assuming a linear relation between clear-water scour depth and flow velocity. If other scour depth influences (especially sediment gradation, abutment shape and channel geometry) were present, actual scour depths would be further reduced from the maxima defined by the lines in Figure 4-2. Table 4-1 Classification of local scour processes at bridge piers in terms of y/a (Melville and Coleman 2000); the limits are approximate values beyond which different trends occur Pier Class y/a Pier Scour Dependence Narrow y/a > 1.4 ys Intermediate width ∝ a 0.2 ≤ y/a ≤ 1.4 ys ∝ (ay) Wide 0.5 y/a < 0.2 ys ∝ y Figure 4-2 Influence of y/a on local scour depth expressed as ys /a (Melville and Coleman 2000) Figure 4-2 shows the trends only for clear-water scour. A similar figure has yet to be developed for live-bed scour, especially for piers in the wide-pier category. This deficiency is a gap in data, insight, and ultimately in scour-depth formulation for a common condition of pier scour. For a pier in a broad, sand-bed channel of given intensity of sediment mobility V/Vc, particle size, and flow depth, increasing pier width modifies the parameters discussed immediately above. In most situations, pier widening may not lead to increased scour depth relative to effective width of pier. As a pier widens in a given flow, the flow field becomes shallower (relative to pier width), and the scour alters in depth and eventually in

51 geometry. Figure 3-6, in Chapter 3, illustrates a typical scour formation at a wide pier. The following factors come into play: 1. The proportion of the approach flow to be diverted into the evolving scour hole diminishes; 2. Some features of the pier flow field, notably the counter-clockwise surface roller, interfere with the down-flow into the scour hole and the clockwise horseshoe vortex; 3. The energy of the macro-turbulence structures (e.g., wake eddies) in the pier flow field decreases, because the diameter of the structures increases, but the rotational velocity remains approximately constant; 4. For wide, model-scale piers in sand beds, bedforms begin to develop at the base and exit slope of a scour hole. These bedforms increase the scour resistance of the scour boundary; 5. During live-bed conditions, bedform size diminishes relative to scour hole size, with as yet unclear net consequences for the time-averaged, and the extreme, values of scour depth; and, 6. More time is needed to develop the scour hole at wider piers. Under clear-water scour conditions, the asymptotic temporal approach to equilibrium scour depth can be a matter of hours (small cylinders in laboratory flumes) and of the order of weeks for large piers (even in laboratory flumes). Under live-bed scour, a median equilibrium scour condition is attained quite quickly, within a day or so for large piers, although the passage of bedforms may cause major fluctuations in scour depth. The duration of flow conditions is an increasingly important factor for wider piers, especially under clear-water scour conditions, as briefly discussed subsequently for the time-development of scour; and, 7. When y/a is less than about 1, the formation of the sediment deposition bar behind the pier affects scour hole development. The height of the bar height can extend over a large portion of the flow depth. The bar therefore alters the flow field at the pier’s rear by causing flow to be diverted to the sides of the bar and, thereby, reducing the erosive power of flow over the bar. The net effect is a widening of the bar, while the exit slope from the scour hole remains relatively steep. Over time, turbulence generated by flow around the pier gradually erodes the bar, and the scour hole may deepen somewhat further. A recognized weakness of the existing methods (Appendix A) for scour-depth estimation is their inadequate inclusion or articulation of the foregoing influences, even for the methods proposed since 1990; notably, Hoffmans and Verheij (1997), Johnson (1999), Melville and Coleman (2000), Richardson et al. (2001), Kohli and Hager (2001), Sheppard and Miller (2006, also Sheppard and Renna 2005). To varying extents, these

52 methods yield scour depth estimates that exceed observed depths at wide piers. Piers tens of feet wide, though, are reported as creating scour holes considerably shallower (relative to pier width) than the maximum depths found from laboratory flume tests with small circular cylinders. The methods proposed by Melville and Coleman (2000), Richardson et al. (2001), Sheppard and Miller (2006) indicate an increasing influence of flow depth on local scour depth for shallower flows. Melville and Coleman (2000) go furthest to differentiate the influence of y/a in terms of categories of scour depth sensitivity. Some methods show that the effect of flow depth on scour depth vanishes asymptotically with increasing flow depth (e.g., Breusers and Raudkivi 1991, Melville and Coleman 2000). 4.4.2 Relative Coarseness, a/D In accordance with the relative length scales it embodies, and considerations of similitude in hydraulic modeling of sediment transport and flow resistance, the parameter a/D expresses a relative coarseness of the flow boundary, but also links to other influences. In hydraulic modeling, exact replication of flow-depth, pier-width, and particle-diameter length scales is unusual. Consequently, scale effects in dynamic similitude are associated with a/D, because most hydraulic modeling (or laboratory flume experiments) maintain the length ratio y/a more-or-less the same as in field situations, but not so for the length ratio a/D. Data from small-scale laboratory experiments show that, for uniform sediment, local scour depths are affected by sediment coarseness when the sediment is either relatively large or relatively small. Several studies explain that, for smaller values of the sediment coarseness ratio, individual particles are large relative to the groove excavated by the down-flow and erosion is impeded because the rough and porous bed dissipates some of the energy of the down-flow. When a/D is less than about 8, individual particles are so large relative to the pier that scour is mainly due to erosion at the sides of the pier and scour is further reduced. Figure 4-3 shows the trend for clear-water scour depth at piers subject to varying a/D, indicating too that ripple formation for medium and fine sands (D ≤ 0.7mm) limits scour depths under clear-water conditions; for such sands, ripples form at sub-threshold conditions on the approach bed, and cause sediment transport into the scour hole. Figure 4-4 shows a comparable trend for live-bed scour depth; no separate cluster of data occurs for ripple-forming sands. The trends in Figures 4-3 and 4-4, obtained for piers of circular cross-section, show that scour depth is influenced by sediment size for a/D less than about 50. For a/D exceeding 50, the influence of the sediment size on scour depth is negligible. These trends used by several scour prediction methods to account for the effect of sediment coarseness are shown in the sketch in Figure 4-5.

53 Figure 4-3 Influence of sediment coarseness on local scour depth at piers for clear-water scour conditions (Melville and Coleman 2000) Figure 4-4 Influence of sediment coarseness on local scour depth at piers at different flow intensities for live-bed scour conditions (Melville and Coleman 2000)

54 Figure 4-5 Local scour depth variation with sediment coarseness (Melville and Coleman 2000) However, for much larger values of a/D, representative of prototype sized piers founded in sandy materials, recent data by Sheppard et al. (2004) and Lee and Sturm (2008) demonstrate significant scour depth reductions for increasing a/D. The reductions for a/D > 50 are shown schematically in Figure 4-6, which contains a larger data set and presents a more comprehensive trend than does Figure 4-4. Sheppard et al. (2004) used three different diameter circular piers (0.114, 0.305 and 0.914m), three different uniform non-cohesive sediment diameters (0.22, 0.80 and 2.90mm) and a range of water depths and flow velocities. The tests extended the range of ratios of a/D to 4,155. Lee and Sturm (2008) used the Sheppard et al. (2004) laboratory data, together with field measurements from three field sites monitored by the United States Geological Survey (Sturm et al., 2004) and field measurements of Landers and Mueller (1996) and Mueller and Wagner (2005) to extend the range of a/D to about 10,000 (see Figure 4-6).

55 Figure 4-6 Influence of sediment size a/D50 on local scour depth ys /a (Lee and Sturm 2008) 4.4.3 Pier Face Shape, Ω As scour depth is consequent to the flow field a pier develops, the shape of a pier’s face affects scour depth. Piers are constructed in a variety of basic face shapes, as Figure 4-7 illustrates. Numerous data show that blunter shapes induce slightly deeper scour. This effect occurs because blunter shapes increase flow contraction at the pier, and they increase flow-field vorticity. The more complicated overall shapes (pier column on piles or footing) are considered further in Chapter 5. Figure 4-7 Basic pier shapes Shape effects, for pier face, usually are given as a multiplying factor Ks that accounts for the difference in local scour between a particular pier shape and a simple circular cylindrical pier subject to the narrow pier category indicated in Table 4-1. Such factors have been proposed by several early studies (e.g., Tison 1940, Laursen and Toch 1956, Chabert and Engeldinger 1956), and are well accepted (Melville and Coleman 2000, Richardson et al. 2001, Sheppard 2006).

56 The recommended shape factors for cylindrical piers given in Table 4-2 are normally used (e.g., Melville and Coleman 2000, Richardson et al. 2001, Sheppard et al. 2006). The factors show that shape of the pier face is relatively insignificant for simple cylindrical piers. The form factors should only be used where the pier is aligned with the flow. A small change in pier alignment will eliminate any benefit from a streamlined shape. Also, the factors may become less accurate for transition piers, and may not apply to wide piers (as defined in Table 4-1). Table 4-2 Shape factors for uniform piers (Richardson et al. 2001) Pier Shape Factor, K Circular s 1.0 Round Nosed 1.0 Square Nosed 1.1 Sharp Nosed 0.9 4.4.4 Pier Aspect Ratio, a/b The factor Ks , for shape of pier face does not account for the influence of pier aspect ratio a/b (width to length ratio). The influence of this ratio for these shapes is illustrated by laboratory data in Table 4-3 for overall pier shapes and aspect ratios indicated in Figure 4-8. For the same projected width of pier (140mm), scour depth varies substantially. The variations are due to differences in the pier flow field generated by each cylindrical form. The flow field (Figure 3-4) adjusts in response to bluffness of pier face, sharpness of corners, and the overall structure and spacing of turbulence structures generated. Further research is needed to document such flow-field changes and relate them to scour depth. Figure 4-8 Cylinders differing in cross-sectional shape, but having the same projected width to the flow (Mostafa 1994)

57 Table 4-3 Comparison of local scour depths for the pier shapes shown in Figure 4-8 (Mostafa 1994) Shape (Figure 4-17) width/length ratio, a/b Projected width of pier (mm) )( )( circulars rnoncirculas y y A 4 1.50 B 4 1.33 C 1 1.29 D 200 140 1.28 E 1 1.28 F 1 1.07 G 1 1.00 4.4.5 Pier Alignment, θ The depth of local scour for all shapes of pier, except circular, is strongly dependent on pier alignment θ to the approach flow. As θ increases, scour depth increases because the effective frontal width of the pier is increased. A figure or chart of multiplying factors, Kθ, is recommended for use to account for the influence of flow alignment on scour depth at non-circular piers. Figure 4-9 shows the importance of alignment. For example, the local scour depth at a rectangular pier b/a = 8 is nearly tripled at an angle of attack of 30o . The angle of attack at bridge crossings may change significantly during floods for braided channels, and it may change progressively over a period of time for meandering channels. The use of circular piers, a row of piles or other shapes of low (length-to-width) aspect ratios is beneficial, where such changes in flow alignment are possible. The Kθ values in Figure 4-9 were obtained (Laursen and Toch 1956) for rectangular cylindrical piers by normalising the measured scour depths with the value at θ = 0°. Richardson et al. (2001) provides a table of values drawn from the curves in Figure 4-9. Ettema et al. (1998) show that the curves in Figure 4-9 are reasonably consistent with new laboratory data (Mostafa, 1994), but note that the maximum scour depth at skewed piers of low aspect ratio (small b/a) occurs at skew angles slightly less than 90°. This latter phenomenon arises because the projected width ap (ap = b sinθ + a cosθ for rectangular piers) of such piers is larger than for θ = 90°. For example, the maximum projected width (dap/dθ = 0) for a rectangular pier, with b/a = 6, occurs at θ = tan-1 (b/a) = 80.5°. The combined influences of pier alignment and shape are sometimes combined with pier width, so as to be expressed as an effective pier width, a*. As shown in Chapter 6, the leading methods for predicting scour depth commonly use a* .

58 The scour depth related to effective pier width, though, may not increase, because other parameters may exert influences; e.g., y/a and a/D, as discussed in subsequent sub- sections. Research since 1990 indicates that the use of the curves is not without complication in this regard (e.g., Ettema et al. 1998). The influences of y/a and a/D on scour depth may vary with skew angle, which affects the upstream-projected width of the pier obstructing the flow. Although the influences of alignment, water depth and sediment size are connected, existing methods of estimating scour depth treat them as mutually independent. Figure 4-9 Local scour depth variation with pier alignment (Laursen and Toch, 1956) 4.4.6 Flow Intensity, V/V Local scour at piers can occur under live-bed or clear-water conditions. Clear-water scour occurs for velocities up to the threshold velocity for general bed movement, for which there is no supply of sediment to the scour hole from upstream. Clear-water conditions are typically encountered on the flood channel of a compound river channel. Live-bed scour occurs when sediment is continuously supplied to the scour hole and the equilibrium depth is attained when there is a balance between the sediment supply and that transported out of the hole. The differences between clear-water and live-bed scour are highlighted in this section. c The variation of local scour depth at piers with flow intensity (and approach flow velocity), as evident from laboratory data, is shown in Figure 4-10.

59 Figure 4-10 Local scour depth variation with flow intensity, V/Vc (Melville and Coleman 2000) Under clear-water conditions, the local scour depth in uniform sediment increases almost linearly with velocity to a maximum at the threshold velocity. The maximum scour depth is called the threshold peak. As the velocity exceeds the threshold velocity, the local scour depth in uniform sediment first decreases and then increases again to a second peak, these changes being relatively small, but the threshold peak is not exceeded providing the sediment is uniform. The second peak occurs at about the transition flat bed stage of sediment transport on the channel bed and is termed the live-bed peak. These trends have been observed by others, including studies conducted over fifty years ago: Chabert and Engeldinger (1956), Garde et al. (1961), Shen et al. (1966), Maza Alvarez (1968), Gill (1972), Raudkivi and Ettema (1983), Chiew (1984), Baker (1986) and Dongol (1994). Consequently, in the laboratory, the maximum local scour depth in uniform sediments occurs at the threshold condition and the live-bed scour depth is largely independent of flow velocity. This fact is acknowledged in early studies, including Laursen and Toch (1956), Shen et al. (1969), Shen (1971), Hancu (1971) and Breusers et al. (1977). The weak dependency of scour depth on flow velocity is an important consideration for establishing a method for design estimation of scour depth. The scour depth variations under live-bed conditions are a consequence of the size and steepness of the bed features occurring at particular flow velocities (Chee, 1982; Chiew,

60 1984; Melville 1984; Raudkivi, 1986; Melville and Sutherland, 1988; and Dongol, 1994). The steeper and higher the bedforms, the lesser the observed scour depth because the sediment supplied with the passage of a given bedform is not fully removed from the scour hole prior to the arrival of the next bedform. The live-bed peak occurs at about the transition flat bed condition when the bedforms are very long and of negligible height. Anti-dunes dissipate some energy at higher velocities and the local scour depth appears to decrease again. The magnitude of the scour depth fluctuations due to bedform migration is approximately equal to the half-amplitude of the bedforms (see Melville and Coleman, 2000, Section 4.7), indicating that the scour depth due to bedforms is about one-half the bedform height (Shen et al., 1966; Chee, 1982; Chiew, 1984; and Dongol, 1994). For non-uniform sediments, the scour depth maxima are termed the armour peak and the live-bed peak. Armouring occurs for V<Va and the scour depth is limited accordingly. Beyond the threshold velocity for the transition from clear-water to live-bed conditions for non-uniform sediments, Va , the armouring diminishes and live-bed conditions pertain. The live-bed peak, which typically exceeds the armour peak, occurs at the transition flat bed condition when all particle sizes in the non-uniform sediment are in motion. At the live-bed peak, the scour depth is about the same for uniform and non-uniform sediments of the same median size. Equilibrium is reached much more rapidly under live-bed conditions than under clear- water conditions (Figure 4-10). Thus, the live-bed peak may be the critical condition for design because clear-water conditions may not last long enough for the scour depth, associated with the threshold peak, to be attained. Figure 4-11 (uniform sediments) and Figure 4-12 (non-uniform sediments) are plots of laboratory data from many sources for local scour at piers in terms of the flow intensity parameter KI. The flow intensity parameter is defined, for each set of data, as the scour depth at a particular flow intensity, ys, divided by the maximum scour depth for the data set, ysmax (see also Figure 4-10), where V is systematically varied for each data set and all other dependent parameters are held constant. The scour maxima used occur at the threshold peak for uniform sediments and the live-bed peak for non-uniform sediments. The non-uniform sediment data are plotted in terms of a transformed velocity parameter, [V-(Va-Vc)]/Vc. The transformed velocity parameter aligns the armour peaks (that is V = Va) for non-uniform sediments with varying σg with the threshold peak for uniform sediments. For uniform sediments, Va ≡ Vc, and [V-(Va-Vc)]/Vc ≡ V/Vc. The transformed velocity parameter incorporating Va largely accounts for the effects of sediment non- uniformity as well as those of flow velocity, although the smaller values of scour depth at [V-(Va-Vc)]/Vc≈1, as σg increases, remain. Therefore, the effects of sediment nonuniformity are mostly accounted for in the flow intensity factor.

61 Figure 4-11 Influence of flow intensity on local scour depth in uniform sediment (Melville and Coleman 2000) Figure 4-12 Influence of flow intensity on local scour depth in non-uniform sediment (Melville and Coleman 2000) 4.4.7 Sediment Non-uniformity, σ Research since 1990 has not significantly advanced the insights obtained from early laboratory studies (e.g., Ettema 1976, 1980, Chiew 1984 and Baker 1986) regarding the effects of sediment-diameter non-uniformity on local scour depth under clear-water conditions at piers. Some of the data from these studies are given in Figure 4-13 for clear-water scour. g

62 Figure 4-13 Influence of sediment non-uniformity on local scour depth at piers subject to clear water scour (Melville and Coleman 2000) The broader context of the data is given in Figure 4-14, which schematically summarises the trends associated with varying intensity of bed material motions, V/Vc. Around the threshold condition, V/Vc ≈ 1, armouring occurs on the approach flow bed and at the base of the scour hole. The armoured bed in the erosion zone at the base of the scour hole significantly reduces the local scour depth. Conversely, at high values of V/Vc, when the flow is capable of entraining most grain sizes within the non-uniform sediment, sediment non-uniformity has only a minor effect on the scour depth. At intermediate values of V/Vc, the effect of σg reduces progressively with increasing flow velocity between these two limits, as more and more of the grains are transported by the flow.

63 Figure 4-14. Local scour depth variation with sediment non-uniformity (Melville and Coleman 2000) 4.4.8 Power of Turbulence Structures, Eu and Re The energy associated with turbulence structures in the pier flow field can be characterized in terms of a pier Euler number, Eu = V2 /ga, and the frequency of vortex formation and break-up/shedding in terms of pier Reynolds number, Re = ρVa/µ.; Ettema et al. (1998) and Ettema et al. (2006). A substantial scale effect occurs in loose-bed models used to investigate local scour at various hydraulic structures. It arises because of inconsistencies in flow field similitude incurred when intensity of bed sediment movement is used as the primary criterion for similitude, as elaborated briefly below. This effect can be seen in the study reported by

64 Ettema et al. (2006) for piers, and by Sturm and Chrisochoides (1997), who investigated scale effects in two hydraulic models of bridge abutments. It arises because of the difficulty in simultaneously satisfying three length scales (y, a, and D). The parameters V2/ga and ρVa/µ. can be interpreted as, expressing similitude in the energy and frequency (hence power) of eddies shed from a pier. This is because, for flow as an eddy, V2 /ga is a normalized expression of eddy energy. Given the range of length scales commonly used in scour experiments, Reynolds number in terms of viscous effect is unlikely to have direct bearing on pier-scour depth. However, Reynolds number also influences the frequency of shedding, n, which can be estimated using the relationship between Strouhal number (St) and Reynolds number (Re) for flow around circular cylinders;       = µ ρVa function V na (4.3) For the typical values of cylinder widths and flow velocities used in flume experiments, as well as for cylindrical piers in rivers, Re lies between 103 to 105 , for which St ≈ 0.2. Thus for cylinders in the same approach flow (V = constant), the frequency of vortex shedding, n, is inversely proportional to cylinder width (diameter). In other words, smaller cylinders in the same flow generate eddies at a greater rate. The capacity of vortices to erode sediment has been shown by several studies. For instance, Dargahi (1989, 1990) began by studying the vortex systems formed by a cylinder on a flat bed. He looked at the interaction of the horseshoe vortex and wake vortices, observing that the main oscillatory frequency of the horseshoe vortices is close to the shedding frequency associated with the cylinder’s Strouhal number, and concluded that the two mechanisms are connected. Also, he observed that wake vortices cause bursts to occur downstream of the cylinder. Dargahi (1990) studied vortex shedding for a cylinder placed in a deformable bed, and found essentially the same vortex shedding behaviour as for the fixed flat bed. His observations, together with those reported from scour development around circular cylinders (e.g., Hjorth 1975, Melville 1975, Ettema 1980) and bed-sediment transport generally (e.g., Müller and Gyr 1986, Yalin 1992), indicate that the passage of vortices increases sediment entrainment and movement. Ettema et al. (1998) present data suggesting that scour depth at piers does not scale linearly with pier width unless there is more-or-less complete geometric similitude of pier, flow and bed sediment particles. The non-linearity can result in laboratory flume studies of local scour (at scale-reduced model piers) leading to deeper scour holes relative to pier width than any likely to occur in the field. Many laboratory experiments have been undertaken using sands to model sand bed rivers. Consequently, the model bed material relative to the pier size is larger than its scaled counterpart in the field. To ensure similitude of the state of bed mobility requires that the value of V/Vc be maintained the same in the laboratory and the field, implying that the flow velocity used in the laboratory may need to be larger than that derived from Froude

65 scaling of the flow velocity in the field. Hence, the Froude number used in laboratory experiments may be larger than that for the corresponding field conditions. Ettema et al.’s (1998) data show that scour depth, relative to pier width, may increase with pier Euler number. The parameter V2/ga is useful for describing energy gradients for flow around a pier. It can be considered to express the ratio of stagnation head V2/2g to pier width. Flow-field similitude requires preservation of flow patterns such that pressure heads along flow paths scale directly with the geometric scale relating a model pier in the laboratory to a pier in the field. For the same stagnation head V2/2g, steeper gradients occur at narrower piers. A narrower pier will induce a larger value of V2/ga , and thereby larger ys /a, than will a wider pier in the same flow field. Further data, in addition to those presented by Ettema et al. (1998), are needed to quantify the influence of the Euler number on local scour depth at piers. Nevertheless, the data do indicate that using a value for the maximum scour depth at circular piers of 2.5a is conservative for all pier sizes larger than about 0.1m. A recent study by Ettema et al. (2006) suggests the need for the addition of a factor for Eu effects in scour prediction methods (see variation of ys /a with the Euler number in Figure 4-15). Based on their laboratory data, Ettema et al. (2006) defined a correction factor that could be used to account for Euler number effects in laboratory-scale tests. Figure 4-15 Variation of ys/a with Euler number (Ettema et al., 2006)

66 4.4.9 Time Rate of Scour, tV/a Though the objectives set for the present evaluation do not expressly include evaluation of the time-rate of scour development, it is useful to mention that differences in pier flow fields and in foundation material affect the rate of scour development. The parameter tV/a relates scour duration to pier width a, and approach velocity V. Many studies focus on the time-development of scour at piers of simple cylindrical form. However, relationships expressing the time rate of scour development for the narrow-, transition-, and wide-pier categories (Table 4-1) likely differ, as do relationships expressing scour in foundation material comprising non-cohesive, cohesive, or weak rock. It is well known that local scour under clear-water or live-bed conditions develops at quite different rates, and are subject to different equilibrium balances (erosion of sediment from scour hole, and sediment inflow and outflow rates, respectively for the two conditions). Figure 4-16, a simplification of Figure 4-10, conceptually illustrates the time-development trends. Under clear-water scour conditions, the scour depth develops asymptotically towards an equilibrium depth of scour. Under live-bed conditions, the equilibrium depth is reached more quickly and thereafter the scour depth oscillates due to the passage of bed features past the pier or abutment. Therefore, in most cases when scour occurs under live-bed conditions, the time–development of scour usually is not of major significance. Still, there are bridge sites where limits in the duration of scouring flows constrain scour depths. This is true especially for scour developing under clear- water conditions in which no significant bedload occurs, but sometimes even for scour under live-bed conditions. For instance, time can be a constraint for scour associated with storm-surge flows at tidal or estuary bridge sites, and for sites a short distance downstream from large dams. There exist much laboratory data on the time-development of scour at piers, most being for clearwater scour (e.g., Chabert and Engeldinger 1956, Shen et al 1969, Hjorth 1975, Nakagawa and Suzuki 1975, Ettema 1980, Yanmaz and Altinbilek 1991, Bertoldi and Jones 1998, Melville and Chiew 1999, Miller 2003, Dey and Raikur 2005, Yanmaz 2006, Oliveto and Hager, 2005, Kothyari et al., 2007). Also, quite a few formulations have been proposed for scour development at piers (e.g., Hjorth 1977, Ettema 1980, Johnson and McCuen 1991, Miller and Sheppard 2002 (also Miller 2003), Faruque and Nago 2003, Dey and Raikur 2005, Yanmaz 2006, Oliveto and Hager, 2005, Kothyari et al., 2007, Lai et al. 2009), or for indicating a temporal scale for scour development (e.g., Melville and Chiew 1999). However, the vast majority of the data, and most of the formulations, are for scour at circular cylindrical piers in the narrow-pier category. The methods proposed by Oliveto and Hager (2005) and Kothyari et al. (2007) can be applied for piers of various shape and predict an unbounded logarithmic increase in the local scour depth with time. Predicting the time-development of scour at piers in situations of limited flow duration remains a challenge. Such situations include piers on floodplains that inundate only during relatively rare flood events; piers in tidal environments subject to storm-surge effects; and piers subject to tsunami effects. During these conditions, the scour condition arguably is predominantly a clearwater-scour condition, though site variations may occur.

67 Several papers outline the difficulties of scour estimation for these conditions; e.g., Hughes (1999), Normets et al. (2003), Tonkin et al. (2003), and Kothyari et al. (2007). It is useful to point out there is little information on the time development of clear-water local scour at piers considered wide or of complex geometry. The data and the formulations upon which the time-development relationships are based do not take into account θΩ , influences, nor do they adequately account for values of a/y and a/D in ranges found for wide piers, and therefore are of questionable validity when scaled, or applied, to wide piers and long skewed piers. The expressions are largely semi-empirical, or essentially empirical, in nature. Some of them begin with rational expressions of sediment transport rate out of a conical scour hole. These expressions eventually rely on empirical calibrations connecting the formulation to laboratory data. Other expressions simply normalize scour depths, as scour progresses, with an equilibrium scour depth. However, the expressions demonstrate the difficulty of arriving at a general predictive formulation for scour development that will be valid over several orders of magnitude of cylindrical pier diameter, not to mention a method sufficiently general to handle piers of complex or unusual geometry. They also suggest that, when estimating time-dependent scour at piers of complex geometry, the viable approach is to develop relationships for common (or comparatively similar) pier geometries. Figure 4-16 Time-development of scour at a cylindrical pier subject to clear-water or live-bed conditions Though field data on the temporal development of pier scour are scarce, measurements during routine bridge surveys (such as conducted by state departments of transportation) indicate that multiple flood events may be required before scour reaches a maximum depth at a pier site, especially for piers founded in material than does not erode in a particle-by-particle manner as does non-cohesive sediment; notably, foundations comprising cobble-armoured beds, or beds comprising cohesive soils and/or weak rock.

68 For example, Hopkins and Beckham’s (1999) comprehensive study of rock scour at bridge piers and abutments in Kentucky shows that for scour of weak rock may occur over several years before becoming noticeably severe. They also found no clear relationship between rock scour and bridge age. The equilibrium scour depth in live-bed conditions fluctuates due to the effects of bed-form migration. The dashed lines in Figure 4-16 represent the temporal average scour depth under live-bed conditions. As Figure 4- 10 infers, variations in the duration of approach velocity magnitude, or variation in the resistance and nature of foundation material erosion, may complicate the time development of scour at a pier site. Most equations for depth of local scour give the equilibrium depth and are, therefore, conservative regarding temporal effects. For the live-bed conditions that typically occur in floods, equilibrium scour depths are appropriate. However, where clear-water scour conditions exist, the equilibrium depth of scour may be overly conservative. The actual scour may be only a small fraction of the equilibrium scour depth, which could take weeks to fully develop. To achieve equilibrium conditions in small-scale laboratory experiments of clear-water scour depth development at bridge foundations, it is necessary to run the experiments for several days. Data obtained after lesser times, say 10 to 12 hours, can exhibit scour depths less than 50% of the equilibrium depth. The shape of the flood flow hydrograph is important as well as the flood duration. Typically, the flood duration determines if the equilibrium live-bed scour will develop. After the flood peak passes, the flow recedes. The duration of the recession period is also important. With flow recession, clear-water conditions may prevail, which could induce additional scour, especially if near-threshold conditions are maintained over a considerable period. Chiew and Melville (1996), also Melville and Chiew (1997), presents extensive laboratory data that describe the temporal development of local scour at circular bridge piers (of diameter a) under clear-water conditions. Their data show that local clear-water scour depth increases asymptotically towards the equilibrium local scour depth, ys, Figure 4-17 shows that local scour depths at the same stage of development (t/te, where te is the time to develop the equilibrium depth of scour) are reduced at lower values of V/Vc . Melville and Chiew (1997) show that both te and ys are subject to similar influences of flow and sediment properties, as might be expected because they are inherently interdependent. The main trends from their plots, which show the dependence of a (dimensionless) equilibrium time scale t* (=Vte/a) on flow shallowness (y/a), flow intensity (V/Vc ) and sediment coarseness (a/D), are illustrated in the sketches in Figure 4- 18. Flow shallowness effects are depicted in the upper diagram of Figure 4-18. The equilibrium time scale increases with y/a for shallow flows and becomes independent of y/a for deep flows. The apparent limit to the influence of flow shallowness on t* occurs at y/a ∼ 6. The maximum value of t* is about 2.5x106. The middle diagram shows that the equilibrium time scale increases rapidly with flow intensity for clear-water scour conditions, attaining a maximum value at the threshold condition. At higher live-

69 bed flows, t* is expected to rapidly decrease again, as shown by the dashed line (refer also to Figure 4-10). The lower diagram indicates that t* increases asymptotically with a/D. The limit to sediment coarseness influence on t* occurs at a/D ≈ 100. Figure 4-17 Temporal development of local scour at piers, clear-water conditions (Melville and Chiew 1997)

70 Figure 4-18 Equilibrium time-scale variation with flow shallowness, flow intensity and sediment coarseness

71 4.5 Data Quality and Gaps The parameter trends are based largely on data obtained from laboratory flume experiments, and to a far lesser extent on field measurements at actual pier sites. The flume experiments normally use similitude considerations to interpret the data obtained. At times the quality of data is uncertain (or not documented), and data gaps exist for uncommon or hard-to-model pier situations. Moreover, laboratory experiments have limitations, which arise for several reasons: 1. The vast majority of laboratory studies have used small-scale, simple cylindrical forms to replicate piers. Consequently, the difficulty of satisfying the similitude requirements associated with three length scales (flow depth, pier width, and boundary material) typically have not been satisfied, and thereby requires the interpretation of laboratory data so as to account for scale effects; 2. A large number of variables influence scour depth, even for cylindrical piers. Therefore, a great number of tests are needed to define the influences. Most studies have involved series of tests examining the influence of a variable cast in terms of one independent parameter at a time. Though useful for some variables (e.g. pier form), this approach may yield incomplete insights when a variable exerts multiple influences, or when different variables in a parameter are adjusted. For instance, varying flow depth y in the parameter a/y may produce somewhat different results than varying pier width a, because other parameters containing these variables are hard to keep constant; 3. Laboratory techniques for flow velocity measurement and visualization have not been able to reveal the full temporal variations of the complex flow field at piers. A new approach, numerical modeling is needed for this purpose; 4. The manner whereby the pier flow field entrains and transports material from the scour hole still is not adequately understood and reflected in the semi-empirical equations for scour-depth estimation; 5. Limitations on flume size and flow capacity have meant that there are few data from tests done with large diameter piers (say a > 0.75m) in live-bed scour conditions. Remarkably few flumes have the capacity to accommodate such tests; and, 6. The difficulties of replicating cohesive or rocky foundation material, along with the considerable period needed to run scour tests with these materials, has resulted in only a modest amount of data for scour in these foundation materials. For design estimation of scour depth, however, the potential maximum scour depths for the narrow- and transition-pier categories can be determined reliably from laboratory data. The same cannot yet be said for wide-pier scour.