Abstract
The phenomenon of lowering the riverbed level due to removal of sediment is known as scour. In general, scour is classified as general scour, contraction scour, and local scour. This chapter provides a comprehensive discussion on scour within channel contractions, downstream of structures, below horizontal pipelines, at bridge piers, and abutments. Further, scour countermeasures are of paramount importance to river engineers. This issue is also discussed. Numerical examples on prediction of scour depths are worked out.
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Keywords
10.1 General
Scour is a natural phenomenon of lowering the riverbed level due to removal of sediment by the erosive action of flowing stream. The magnitude of reduction in the riverbed level below an assumed natural level (or initial level) is termed scour depth. Scour is broadly classified as general scour, contraction scour, and local scour.
General scour in the river occurs as a result of the change in characteristics of the river. Based on the duration of scour development, general scour can be categorized as short-term scour and long-term scour. Short-term scour occurs during a single flood or several floods of shorter durations to appear in quick succession, while long-term scour takes a considerably long time, usually of the order of a number of years, and results in a progressive bed degradation and bank erosion. Short-term scour may also occur due to flow convergence, a shift in the meandering stream thalweg or braids within the stream and bedform migration. On the other hand, the long-term scour may be caused by the natural changes in the catchments, for example, channel straightening, volcanic activities, climate change, or by the human activities, for example, channel alterations, streambed mining, dam/reservoir construction, and land-use changes.
Contraction scour is the scour of streambed arising from accelerated flow through contraction of waterways, where flows over flood plains are converged by bridge causeways and channeled through the bridge waterways.
In contrast, local scour (also termed localized scour) is developed near the structures due to modification of the flow field as a result of obstruction to the flow by the structures. Scour within the contracted portion of rivers, scour downstream of structures, scour at bed sills, scour below horizontal pipelines, scour at bridge piers and abutments, and scour at other river training works are the examples of local scour.
Local scour is classified as clear-water scour and live-bed scour. Clear-water scour occurs when the sediment is removed from the scour hole but not supplied by the approaching flow. The equilibrium of scour is reached when the flow induced force can no longer dislodge the sediment particles from the scour hole. On the other hand, live-bed scour occurs when the scour hole is continuously fed with the sediment by the approaching flow. The equilibrium of scour is attained over a period of time, when the rate of removal of sediment out of the scour hole equals the rate of supply of sediment into the scour hole. Usually, the magnitude of scour depth in live bed is to some extent less than that in clear-water, if the flow condition is such that the approaching flow velocity U equals or is slightly less the threshold flow velocity U cr for the bed sediment motion.
This chapter summarizes the contraction scour and local scour at different structures including various aspects, such as mechanism of scour and design formulas for the prediction of scour depth. It is however pertinent to mention that despite large number of investigations, hydraulics of local scour is as yet not well established, because most of the studies came from the laboratory and only a few from the fields. As such, the scour prediction formulas can only provide a general guideline for the designers or engineers.
10.2 Scour Within Channel Contractions
A reduction in width of a watercourse by constructing parallel sidewalls is termed channel contraction. Contractions of river width to construct bridges, barrages, weirs, and cross-drainage works are common examples of channel contractions. The flow velocity in the contracted zone of the channel increases due to the reduction in flow area, and hence, the bed shear stress induced by the flow increases considerably. Consequently, the sediment bed within the channel contraction is scoured. Such localized scour in the contracted zone of the channel is called contraction scour.
Depending on the ratio of the length of the contraction L to the approaching channel width B 1, channel contractions are designated as long or short. According to Komura (1966) and Dey and Raikar (2005), a contraction becomes long when L/B 1 > 1, whereas Webby (1984) considered it as L/B 1 > 2. Figures 10.1 and 10.2 show schematic of scour in a channel contraction and a photograph of the scoured bed, respectively. Smith (1967) proposed the angles of upstream and downstream transitions as 12.5° for a smooth transition to the contracted zone.
Schematic of a rectangular channel contraction at equilibrium scour condition: a plan view and b elevation view
Photograph showing an equilibrium scoured bed within a channel contraction
Local scour in a channel contraction is usually studied considering a configuration of long rectangular contraction, as shown schematically in Fig. 10.1. Because of the simple geometrical configuration of the problem, various analytical investigations to predict the equilibrium scour depth in long contractions were attempted. Straub (1934) was the pioneer to present a simplified one-dimensional theory of the equilibrium scour in long contractions. His work was later extended and modified by Laursen (1963), Komura (1966), Gill (1981), Lim (1993), and Lim and Cheng (1998). Further, Dey and Raikar (2005, 2006) studied the scour in long contractions in gravel-beds and proposed analytical models for the estimation of scour depth under both clear-water and live-bed scour conditions.
10.2.1 Laursen’s Model
Laursen (1963) considered a channel contraction as shown in Fig. 10.1. The discharge in the channel is obtained from the continuity equation as
where U 1 is the approaching flow velocity, h 1 is the approaching flow depth, U 2 is the flow velocity in contracted zone, h 2 is the flow depth in contracted zone, and B 2 is the contracted width of the channel.
Using the energy equation between sections 1 and 2, the scour depth d s is obtained as
where h f is the head loss due to flow in transition, Fr 1 is the approaching flow Froude number [= U 1/(gh 1)0.5], and K L is the head loss coefficient.
When the scour in the contracted zone reaches an equilibrium, the bed shear stress becomes equal to its threshold value, that is, τ 0c = 0.628d 50 (in Pa), where d 50 is the median sediment size (in mm). The bed shear stress τ 01 in the uncontracted zone (section 1) can be estimated using the Manning equation and the Strickler’s relationship for Manning roughness coefficient n as
Taking the ratio of bed shear stress in the uncontracted zone to that in the contracted zone yields
Similar expression can also be written for the bed shear stress in the contracted zone. Hence, the flow depth ratio h 2/h 1 can be obtained from
where τ 02 is the bed shear stress in the contracted zone (section 2).
Using Eqs. (10.1) and (10.5), one can write
Substituting Eq. (10.6) into Eq. (10.2b) results
Neglecting the difference in the velocity heads and the loss through the transition, Eq. (10.7) reduces to
10.2.2 Dey and Raikar’s Model
Dey and Raikar (2005, 2006) developed analytical models for clear-water and live-bed scour cases.
10.2.2.1 Clear-Water Scour Model
Dey and Raikar (2005) analytically computed the equilibrium clear-water scour depth in two ways: Considering sidewall correction and without considering sidewall correction.
Determination of scour depth considering sidewall correction: In clear-water scour, the equilibrium scour depth d s reaches in a long contraction, when the flow velocity U 2 in the contracted zone becomes equal to threshold velocity U cr for the sediment motion. The flow velocity \( \left. {U_{2} } \right|_{{U_{2} = U_{\text{cr}} }} \) in the contracted zone can be obtained from the well-known equation of bed shear stress as a function of dynamic pressure (Eq. 3.54). It is
where ρ is the mass density of water, u *c is the threshold shear velocity for sediment, and λ D is the Darcy–Weisbach friction factor, which can be determined from Colebrook–White equation (Eq. 3.55).
In the contracted zone, the bed is rough consisting of sediment particles and the sidewalls are smooth. Hence, the friction factor λ D|w associated with the wall is considerably different from the friction factor λ D|b associated with the bed. Therefore, Vanoni’s (1975) method of sidewall correction can be applied for the contracted zone of the channel, as given in Sect. 3.9, where the solution for λ D|b was obtained from the solution of Eqs. (3.63) and (3.64), which are here expressed as
where k s is equivalent roughness height (=2d 50), Re|b is the flow Reynolds number associated with the bed, that is, \( 4\left. {U_{2} } \right|_{{U_{2} = U_{\text{cr}} }} A|_{\text{b}} /(\upsilon P|_{\text{b}} )\hbox{,}\;A|_{\text{b}} \) is the flow area associated with the bed, P|b is the wetted perimeter associated with the bed (=B 2), A is the total flow area of contracted zone (=h 2 B 2) P|w is the wetted perimeter associated with the wall (=2h 2), and υ is the kinematic viscosity of water.
In clear-water scour, at equilibrium scour condition, Eq. (10.1) becomes
For the given U 1, h 1, B 1, B 2, and d 50, the unknowns \( \left. {U_{2} } \right|_{{U_{2} = U_{\text{cr}} }} \), h 2, Re|b, and λ D|b can be determined numerically solving Eqs. (10.9), (10.10a, b) and (10.11). Then, neglecting the head loss in transition, energy equation [see the energy equation, Eq. (10.2a)] is used to determine equilibrium scour depth d s as
Determination of scour depth without considering sidewall correction: In this simplified approach, the depth-averaged flow velocity \( \left. {U_{2} } \right|_{{U_{2} = U_{\text{cr}} }} \) in the contracted zone for equilibrium scour is determined assuming the logarithmic equation of the depth-averaged velocity as
For the given U 1, h 1, B 1, B 2, and d 50, the unknowns \( \left. {U_{2} } \right|_{{U_{2} = U_{\text{cr}} }} \) and h 2 can be obtained numerically solving Eqs. (10.11) and (10.13). Then, equilibrium scour depth d s can be determined from Eq. (10.12).
10.2.2.2 Live-Bed Scour Model
Dey and Raikar (2006) proposed a live-bed scour model for the estimation of scour depth within channel contractions. In live-bed scour, the equilibrium scour depth is reached, when the sediment supplied by the approaching flow into the contracted zone is balanced by the sediment transported out of the contracted zone. Thus, at the equilibrium, the sediment continuity equation between sections 1 and 2 of Fig. 10.1 is
where q b is the bed-load transport rate of sediment. The bed-load transport rate q b can be estimated by the formula of Fredsøe and Deigaard (1992) as
Assuming the logarithmic equation of average velocity for approaching flow, the shear velocity u *1 at section 1 is obtained as
In the contracted zone, incorporating the logarithmic equation of average velocity in Eq. (10.1) yields
where u *2 is the shear velocity in the contracted zone.
For the given U 1, h 1, B 1, B 2, and d 50, the unknowns U 2 and h 2 can be determined numerically solving Eqs. (10.14), (10.15) and (10.17). Then, the energy equation [see the energy equation, Eq. (10.2a)] is used to determine equilibrium scour depth d s as
10.2.3 Maximum Scour Depth Prediction
The parameters that influence the scour within channel contractions are as follows (Dey and Raikar 2005):
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1.
Parameters relating to the channel contraction: Channel opening ratio and channel shape.
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2.
Parameters relating to the bed sediment: Median particle size, particle size distribution, angle of repose, and cohesiveness.
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3.
Parameters relating to the approaching flow condition: Approaching flow velocity, approaching flow depth, shear velocity, and roughness.
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4.
Parameters relating to the fluid: Mass density, viscosity, gravitational acceleration, and temperature. Note that the temperature may not be important in scour problems, unless free surface is frozen.
The functional relationship showing the influence of above parameters on the equilibrium scour depth d s in a long rectangular contraction can be given as
where ρ s is the mass density of sediment, g is the gravitational acceleration, and σ g is the geometric standard deviation of the particle size distribution.
Dey and Raikar (2005) argued that the scour in a long contraction starts when the excess approaching flow velocity \( U_{{ 1 {\text{e}}}} ( = U_{ 1} -\left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} ) \) is greater than zero. For no-scour condition, U 1e is less than or equal to zero. Here, \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \) refers to the approaching flow velocity U 1 that initiates scour in a contraction. Therefore, \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \)corresponds to U 1 for which U 2 becomes U cr for the undisturbed bed condition (d s = 0) in the contracted zone. The \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \) can be determined as follows:
Considering negligible head loss (h f = 0) and applying the energy equation between sections 1 and 2 for the bed sediments within contracted zone under threshold condition, that is U 2 = U cr, before initiation of scour (d s = 0), the following equation is obtained (Fig. 10.1) (Dey and Raikar 2005):
The continuity equation between sections 1 and 2 is
The threshold flow velocity at section 2 can be determined using the logarithmic equation of average velocity as
Therefore, for the given h 1, B 1, B 2, d 50, and u *c (determined from the Shields diagram), the approaching flow velocity \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \) required to initiate the sediment motion within the contracted zone can be estimated solving Eqs. (10.20)–(10.22) numerically.
In the context of scour, it is appropriate that U 1, in Eq. (10.19), is to be replaced by U 1e. In sediment–water interaction, the parameters g, ρ, and ρ s are combined into a parameter Δg, where Δ = s − 1 and s is the relative density of sediment (= ρ s/ρ). Also, it is reasonable to use the channel opening ratio B 2/B 1 to account for the combined effect of B 1 and B 2. In addition, the influence of kinematic viscosity υ of water is insignificant for a turbulent flow over rough sediment beds (Yalin 1977). Therefore, applying these considerations, the Buckingham Π theorem (see Sect. 11.2.3) is used with U 1e and h 1 as repeating variables to obtain the following nondimensional equation:
where F 1e is the excess approaching flow Froude number \( [ = U_{{ 1 {\text{e}}}} /({\Delta} gh_{ 1} )^{0. 5} ] \). The condition \( U_{ 1} \to U_{\text{cr}} \) (that is the limiting condition for a clear-water scour, as \( U_{ 1} > U_{\text{cr}} \) corresponds to a live-bed scour) is recognized to be the most idealized condition for maximum equilibrium scour depth \( [d_{\text{s}} ]_{ \hbox{max} } ( = \left. {d_{\text{s}} } \right|_{{U_{1} = U_{\text{cr}} }} ) \) in long contractions under a clear-water scour condition (Gill 1981). Therefore, to determine the equation of maximum equilibrium scour depth [d s]max in long contractions, Eq. (10.23) is written for \( U_{ 1} \to U_{\text{cr}} \). Using the experimental data for clear-water scour, Dey and Raikar (2005) obtained the empirical equation of maximum equilibrium scour depth as follows:
where \( F_{{ 1 {\text{ec}}}} = U_{{ 1 {\text{ec}}}} /({\Delta} gh_{ 1} )^{0. 5} \) and \( U_{{ 1 {\text{ec}}}} = U_{\text{cr}} -\left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \). Equation (10.24) is written for uniform sediments, as it does not include \( \sigma_{\text{g}} \).
The equilibrium scour depth d s(\( \sigma_{\text{g}} \)) in nonuniform sediments can be estimated in terms of geometric standard deviation σ g of sediments using the following relationship:
where K σ is the coefficient due to sediment gradation. The coefficient K σ is defined as the ratio of equilibrium scour depth in nonuniform sediment (σ g > 1.4) to that in uniform sediment. The variation of K σ with σ g is shown in Fig. 10.3 (Dey and Raikar 2005).
Variation of K σ as a function of σ g (Dey and Raikar 2005)
10.2.4 Other Scour Depth Predictors
Phenomena involving scour in long contractions have been studied extensively in laboratories, from which a number of semianalytical and empirical equations have been developed to estimate the equilibrium scour depth under both clear-water and live-bed scour conditions. In general, they are based on a limited range of data. Table 10.1 furnishes the empirical equations of equilibrium scour depth proposed by different investigators.
10.3 Scour Downstream of Structures
10.3.1 Scour Below Drop Structures
Drops are provided in rivers for lowering the bed level when the slope of the river is smaller than the natural ground slope. These structures therefore artificially increase the slope of the rivers. The stream flow running over the drops is called an overfall. In addition, the scour is developed downstream of the bed protection provided to control the slope or elevation of the riverbed to create a drop. The water released from the drop structures impinges on the free surface of the tailwater as a jet, which is called plunging jet. This freely falling jet may have considerable potential to scour the bed downstream of the structures, and such scour is known as jet scour. Scour due to jets occurs very rapidly, which causes danger to the stability of the channel bed, in addition to the devastating effects on the hydraulic structures. Considerable portion of the energy of the flowing stream is dissipated through turbulent mixing in the pool due to plunging jet. Figures 10.4a, b show schematic of scour below weir type and free overfall type drop structures.
Schematic of scour below drop structures: a weir type and b free overfall type
The pioneering study on scour below a drop structure was due to Schoklitsch (1932). He proposed the following empirical relationship for the equilibrium scour depth for the flow over structures:
where q is discharge per unit width, H is the height between upstream and downstream water levels, d 90 is the 90 % finer sediment size, and h t is the tailwater depth.
Based on the dimensional analysis and using the experimental data, Kotoulas (1967) developed a relationship for the equilibrium scour depth downstream of a structure. It is
where d 95 is the 95 % finer sediment size.
For free overfall type drop structures, Dey and Raikar (2007b) proposed a procedure to calculate the jet velocity U 0 and the jet thickness l 0 at the entry of jet into the tailwater.Footnote 1
Fahlbusch (1994) proposed an empirical equation of equilibrium scour depth for weir type drop structures. He expressed scour depth d s as a function of q and jet velocity U 0 entering the tailwater depth h t at an angle θ j with the horizontal at the water level as
The coefficient K p is dependent on sediment size. For gravel, 3 < K p < 5; for sand, 5 < K p < 20; and for silt, K p ≈ 20.
Later, a more generalized relationship of equilibrium scour depth for weir type drop structures was recommended (Graf 1998). It is
Also, D’Agostino and Ferro (2004) suggested a simplified equation of equilibrium scour depth for weir type drop structures as
where Z is the crest height of the weir and h is the flow depth over a weir (Fig. 10.4a). Note that Eq. (10.30) does not take into account the effects of sediment size and tailwater depth.
Stein et al. (1993) developed an analytical equation to predict the equilibrium scour depth downstream of a headcut type drop structure for the condition of shallow tailwater depth (d s ≫ h t) (Fig. 10.5). Neglecting the effects of tailwater depth, they proposed
where C d is the jet diffusion coefficient (= 2.6).
Schematic of scour below a headcut
10.3.2 Scour Downstream of Grade-Control Structures
Grade-control structures are employed to prevent excessive riverbed degradation (Fig. 10.6). Bormann and Julien (1991) investigated the scour downstream of grade-control structures based on two-dimensional jet diffusion and particle stability. They put forward the following expression for the equilibrium scour depth:
where U 1 is the approaching velocity, Z p is the drop height of grade-control structure, θ j is the jet angle near the original bed level, and ϕ is the angle of repose of bed sediment.
Schematic of scour downstream of a grade-control structure
10.3.3 Scour Downstream of Bed Sills
Mountain streams are frequently subjected to channel incision. One of the methods to stabilize them is to employ a series of transverse structures called bed sills. Bed sills are generally preferred when the height of the riverbed is to be somewhat raised in order to reduce instability of the valley slope. The overfall plunging jet issued from a sill crest diffuses its energy in mixing process through rollers inside the downstream pool below. Further downstream, a uniform flow can be established if the riverbed has an equilibrium slope for a significant length. This condition is satisfied when the intermediate distance between two bed sills is adequately long. Flow over immediate upstream of a sill crest is characterized by a critical flow condition. At the edge of the sill, the flow becomes supercritical, being accelerated by that gravity as an overfall jet that has significant power to remove sediment from the bed downstream of a bed sill. Practically, scour downstream of a bed sill endangers its stability leading to the failure if the maximum scour depth is deep enough to expose the foundation (Fig. 10.7).
According to Gaudio et al. (2000), the maximum clear-water scour depth d s at a bed sill can be given as a functional form. It is
where h 1 is the uniform flow depth and a 1 is the morphological drop which is defined by
where S 0 is the initial streamwise bed slope, S eq is the equilibrium bed slope, and L is the distance between two neighboring sills. The equilibrium slope in clear-water condition can be expressed by the threshold Shields parameter Θc for the initiation of bed particle motion under fully developed turbulence flow condition. It is
From the Manning equation, one can write
where n is the Manning roughness coefficient. Using Eqs. (10.35) and (10.36), the uniform flow depth and the equilibrium bed slope can be obtained as
Equation (10.37a) provides the dependency of the uniform flow depth on other flow and sediment parameters. This allows h 1 to be dropped from Eq. (10.33).
The specific energy E s on the sill is given by
Applying Buckingham Π theorem (see Sect. 11.2.3) to Eq. (10.33) devoid of h 1, one can write
For fully developed turbulent flow, kinematic viscosity υ can be neglected. Further, Δ is assumed to be a constant. Equation (10.39) thus reduces to
Experiments by Gaudio and Marion (2003) revealed that only the first nondimensional parameter of the right-hand side of Eq. (10.40) influences the scour depth. The empirical equation of scour depth given by Gaudio and Marion is
The above equation is applicable for 1.3 ≤ a 1/(Δd 50) ≤ 9.1. The length of the scour hole is as follows (Gaudio et al. 2000):
Later, Lenzi et al. (2002) recognized that both the nondimensional parameters of the right-hand side of Eq. (10.40) affect the scour depth. They put forward expressions for the scour depth and the length of the scour hole as
10.3.4 Scour Due to Horizontal Jets Issuing from a Gate Opening
The scour phenomenon downstream of a sluice gate opening is complex in nature owing to the abrupt change of the flow characteristics on the sediment bed (Dey and Sarkar 2006a) (Fig. 10.8). Scour initiates at the downstream edge of the apron when the bed shear stress exerted by the submerged jet exceeds the threshold bed shear stress for the bed sediment. In the initial stage, the evolution of the vertical dimension of scour hole is faster than the longitudinal one, and the suspended load is the only mode of sediment transport. However, with the development of the vertical dimension of the scour hole, the mode of sediment transport changes to a combination of suspended load and bed load. As the flow separation takes place at the edge of the apron having a reattachment of flow at the deepest point of the scour hole, the movement of the sediment particles is divided into two parts. Some amount of sediments move along the downstream slope of the scour hole and ultimately go out of the scour hole. The other part moves back toward the upstream along the upstream slope of the scour hole by the reversed flow. The upstream portion of the scour hole achieves a steep slope.
Schematic of scour due to horizontal jet issuing from a sluice gate opening
The theory of submerged plane wall jet was developed by Dey et al. (2010). They derived the velocity and Reynolds stress distributions in submerged wall jets (see Sect. 10.8.1). The theory would help the future researchers to model scour downstream of an apron due to submerged wall jets. However, Dey and Sarkar (2006b) gave a semiempirical model of the same. The application of the theory of submerged wall jet to compute scour is discussed in Sect. 10.8.2.
Qayoum (1960) studied the scour downstream of a vertical gate without apron (L 0 = 0). Using the dimensional analysis, he proposed an empirical equation
Altinbilek and Basmaci (1973) proposed an equation of scour depth due to submerged jets issuing from a sluice opening as
where b 0 is the thickness of sluice opening and U 0 is the issuing velocity of jet.
Breusers and Raudkivi (1991) gave
Hoffmans (1998) [also Hoffmans and Verheij (1997)] derived equilibrium scour depth due to submerged jets (with L 0 = 0) by applying the momentum principle between the vertical section at the sluice opening and the section passing through the dune crest as
where λ s is the scour factor dependent on d 90, and U crest is the average velocity over the downstream dune crest. The values of λ s are λ s(d 90 = 0.1 mm) = 1.4, λ s(d 90 = 0.3 mm) = 2, λ s(d 90 = 0.5 mm) = 2.3, λ s(d 90 = 1 mm) = 2.95, λ s(d 90 = 3 mm) = 4.3, λ s(d 90 = 5 mm) = 5.1, λ s(d 90 = 10 mm) = 6.3, and λ s(d 90 ≥ 12 mm) = 6.8.
Shalash (1959) and Dey and Sarkar (2006a) conducted experiments and proposed empirical equations of equilibrium scour depth downstream of an apron due to submerged jets issuing from a sluice opening. According to Shalash (1959),
and according to Dey and Sarkar (2006a),
Eggenberger and Müller (1944) investigated scour downstream of hydraulic structures due to a combined overfall and submerged jet (Fig. 10.9). They gave a general relationship for equilibrium scour depth as
where \( \hat{q} = q_{ 1} /q_{0} \hbox{,}\; q_{ 1} \) is the overfall discharge per unit width, and q 0 is the submerged jet discharge per unit width through a sluice opening. By definition of continuity, total discharge per unit width is q = q 1 + q 0. For overfall only, c 0 (\( \hat{q} \gg 1 \)) = 22.88 s0.6 m−0.3; and for submerged jets only, c 0 (\( \hat{q} \ll 1 \)) = 10.35 s0.6 m−0.3.
Schematic of scour due to a combined action of overfall and submerged jet allowed by a sluice gate
10.4 Scour Below Horizontal Pipelines
Local scour below pipelines, laid on and across the riverbeds to convey water, oil, gas, or any fluid, commonly occurs by the erosive action of flowing stream. Scour may leave a pipeline unsupported over a considerable distance resulting in fatigue failure due to flow-induced oscillation by wake vortex shedding. Therefore, one of the imperative aspects of pipeline design is to predict the magnitude of scour below pipelines.
When a pipeline is laid on an erodible bed with a little embedment e (Fig. 10.10), it is subjected to a hydrodynamic force, and a pressure gradient is set up between the two sides of the pipeline. At the same time, the pressure gradient between upstream and downstream contact points (A and B) is also set up in the sediment bed immediately below the pipeline. Due to this pressure gradient, the subsurface seepage flow is driven below the pipeline. As the flow velocity increases, the pressure gradient is simultaneously enhanced, because the pressure intensity is proportional to the quadratic of the flow velocity. With an increase in pressure gradient dp/dx (where p is the pressure intensity and x is the distance along the pipe perimeter), a condition is reached when the sediment below the pipeline starts to dislodge and is called scour threshold. The mechanism of scour below a pipeline under a steady flow is described as follows (Sumer and Fredsøe 2002):
Schematic of seepage flow induced below a pipeline
Once a threshold point is reached, the seepage flux increases rapidly than the order of the pressure gradient by which the seepage is driven. At the same time, the surface of the sediment bed in the immediate downstream of the pipeline bulges. Eventually, the sediment–water mixture breaks through the space underneath the pipeline, which is called piping (Fig. 10.11). Sumer and Fredsøe (2002) derived the threshold condition of piping through a simple calculation by balancing the upward seepage pressure force and the submerged weight of sediment at the exit point B (see Fig. 10.10). It is
where ρ 0 is the porosity of the sediment. Further, Sumer et al. (2001) determined the threshold condition empirically in terms of velocity considering a small embedment e of the pipeline having an external diameter D. Using the above relationship and the laboratory experimental data, they proposed
where U gcr is the threshold velocity of scour below a pipeline. Note that the scour does not initiate all along the length of the pipeline simultaneously, but occurs locally.
Piping
After the piping process, a small gap is developed between the pipeline and the bed. A considerable magnitude of flow is diverted to the gap leading to a flow concentration in the gap. It enhances the shear stress acting on the bed immediately below the pipeline. The enhanced bed shear stress is of the order of magnitude of three times the bed shear stress of the approaching flow (Jensen et al. 1990). As a result, a large amount of sediment is scoured underneath the pipeline. The sediment–water mixture is spouted from the enlarged gap. Such scour process is known as tunnel erosion (Fig. 10.12). With an increase in gap size, the gap velocity decreases and the tunnel erosion gradually seizes. This stage is followed by lee-wake erosion.
Tunnel erosion
As a result of tunnel erosion, a dune is formed on the downstream end of the pipeline and it gradually migrates further downstream. Finally, the dune disappears as the scour progresses. At this stage, the scour is governed by the lee-wake erosion, which occurs due to the vortex shedding on the downstream end of the pipeline (Fig. 10.13). At the end of the tunnel erosion, when the gap between the pipeline and the bed enlarges to a certain magnitude, the vortex shedding begins. The vortices that shed from the pipe wall sweep the sediment as they get convected downstream. During this period, the bed shear stress increases by about four times having a greater scour potential at the lee end of the pipe. However, the bed shear stress gradually reduces with the enlargement of the scour hole size. The equilibrium is reached when the bed shear stress underneath the pipeline reaches the value being equal to the threshold bed shear stress for sediment motion in clear-water case or equal to the approaching bed shear stress in live-bed case. Figure 10.14 shows the photograph of an equilibrium scour below a pipe in a flume.
Lee-wake erosion
Photograph of an equilibrium scour below a pipe in an experimental flume. Flow took place from the right to left
10.4.1 Estimation of Gap Discharge
In studying scour below a pipeline, estimation of gap discharge is an important aspect. It can be determined analytically as described here. The schematic of a scour hole below a pipeline and the coordinate system are shown in Fig. 10.15, where the origin of the coordinate system lies at the contact point of the circular pipe with the initial bed level. According to Chao and Hennessy (1972), the image method of potential flow theory is applicable assuming a steady flow around the pipeline and neglecting the curvature effect of the scour hole. The potential function ϕ and the stream function ψ governing the flow are
where U 0 is the depth-averaged velocity up to the elevation of the horizontal diameter of the pipeline, that is, z = 0.5D. It can be estimated assuming a logarithmic law of approaching velocity as U 0 = 5.75u * log (2.765D/d 50), where u * is the approaching shear velocity. Here, it is intuitive that the flow through the gap is contributed by the approaching flow velocity up to the elevation of the horizontal diameter of the pipeline from the bed level. Hence, the horizontal velocity component \( \bar{u} \) within the scour hole is given by
Schematic of a scour hole below a pipeline and the coordinate system
Integrating \( \bar{u} \) within limits z = –d s and z = 0, the gap discharge q g is obtained as
Alternatively, one can also estimate the gap discharge q g from the graphical solution given by Chiew (1991) as h/D is a function of q g/q, where h is the flow depth and q is the free stream discharge per unit width in the channel. However, it would be convenient to use his graphical solution if it is expressed as follows (Dey and Singh 2007):
10.4.2 Scour Depth Estimation
Phenomena involved in local scour below underwater pipelines have been studied most extensively in the laboratory experiments, from which a number of empirical equations and the methodologies have been developed to estimate the equilibrium scour depth below pipelines. The important predictors of scour depth are summarized below:
Kjeldsen et al. (1973) were the pioneer to give an empirical relationship for equilibrium scour depth below pipelines. It is
The Dutch research group (Bijker and Leeuwestein 1984) put forward the following empirical equation of scour depth below an underwater pipeline:
The empirical equations proposed by Ibrahim and Nalluri (1986) for the estimation of scour depth below pipelines in clear-water and live-bed conditions are
where Fr is the Froude number [= U/(gh)0.5] and U cr is the threshold velocity for sediment motion.
Chiew (1991) proposed the following iterative method to predict equilibrium scour depth below pipelines:
-
1.
For a given value of h/D, determine the gap discharge q g.
-
2.
Assume a scour depth d s and estimate the average gap velocity U g below the pipeline by U g = q g/d s.
-
3.
Compute the shear stress τ 0 on the scoured bed using \( \tau_{0} = \lambda_{\text{D}} \rho U_{\text{g}}^{2} /8 \), where the friction factor λ D can be estimated from the Moody diagram for a relative roughness (= d 50/d s) and a Reynolds number (= U g d s/υ).
-
4.
Compare τ 0 with the threshold bed shear stress τ 0c obtained from the Shields diagram. Continue the iteration for different values of d s until τ 0c = τ 0.
Moncada-M and Aguirre-Pe (1999) suggested the following empirical equation of equilibrium scour depth below an underwater pipeline:
With consideration of an initial gap e between the original bed level and the pipe bottom above the bed level, Moncada-M and Aguirre-Pe (1999) proposed
Based on wall correction method (Sect. 3.9), Dey and Singh (2007) put forward a simplified iterative method for the computation of equilibrium scour depth. Dey and Singh (2008) conducted an extensive experimental study to explore the influence of various parameters on equilibrium scour depth in clear-water condition (U/U cr ≈ 0.9). The equilibrium scour depth d s increases with an increase in approach flow depth h for shallow flow depths, becoming independent of higher flow depths when h/D > 5. The curve of scour depth versus pipe Froude number F D [= U/(ΔgD)0.5] has a maximum value of d s/D = 1.65 at F D = 0.58 (Fig. 10.16).
Influence of pipe Froude number F D on scour depth d s (Dey and Singh 2008)
Dey and Singh (2008) also studied the influence of sediment gradation on scour depth. The influence of sediment gradation was found to be prominent for nonuniform sediments, which reduce scour depth to a large extent due to the formation of armor layer within the scour hole. The variation of K σ with σ g is shown in Fig. 10.17. Further, the influence of different shaped cross sections of pipes on the scour depth was investigated, where the shape factors Footnote 2 for circular, 45° (diagonal facing), and 90° (side facing) square pipes were obtained as 1, 1.29, and 1.91, respectively.
Variation of K σ as a function of σ g (Dey and Singh 2008)
10.5 Scour at Bridge Piers
At bridge sites, localized scour in the vicinity of piers, over which the bridge superstructure rests, poses a challenging problem to the hydraulic engineers. Figure 10.18 shows the photograph of bridge piers, where the scour takes place even in the summer season at a lean flow discharge condition. Failure of bridges due to scour at pier foundations is a common occurrence, if the magnitude of scour is too large to uncover the pier foundations. The obstruction to the approaching flowing stream by a bridge pier causes a three-dimensional separation of flow forming a vortex flow and a periodical vortex shedding downstream of the pier. The complexity in flow structure increases with the development of the scour hole. To be more explicit, the flow separates at the upstream face of the pier as it travels by the side of the pier, creating a vortex trail, termed horseshoe vortex. The vortex moves downstream and as a result of which local scour takes place around the pier due to the removal of bed sediment. The scour at bridge piers has been studied extensively by various researchers. Reviews of the important experiments and field studies were given by Breusers et al. (1977), Dey (1997a, b), Melville and Coleman (2000), and Richardson and Davis (2001). Figure 10.19 shows the photograph of an equilibrium scour hole at a circular pier in an experimental flume. It gives an idea about the shape of the scour hole that occurs at a pier.
Scour taking place at bridge piers in a field condition. Photograph by the author
Photograph of an equilibrium scour at a circular pier in an experimental flume
According to Raudkivi (1986), the approaching flow, which is stationary at the upstream face of the pier, is maximum at the free surface and decreases to zero at the bed in the free flow (unobstructed by the pier) reach. The stagnation pressure at the upstream face of the pier decreases in the downward direction causing the flow to be driven down along the face of the pier, producing a downflow. The downflow along the vertical plane of symmetry of the pier has a velocity profile with zero at the pier wall and again at some distance upstream of it. The maximum magnitude of downflow, at any elevation in the upstream, measured by Ettema (1980) occurs at 0.02–0.05b from the pier face being closer to the pier lower down. Here, b is the pier width across the flow (or pier diameter in case of a circular pier). The horseshoe vortex developed due to the flow separation at the upstream edge of the scour hole rolls to form a helical flow, which is similar to the ground roller downstream (leeside) of a dune crest (see Sect. 8.2.2). It migrates downstream by the side of the pier for a few pier diameters before losing its existence becoming a part of the general turbulence. The horseshoe vortex is a consequence of scour, not the cause of scouring. It also pushes the downflow inside the scour hole closer to the pier. The bow wave, formed at the free surface adjacent to the pier face rotating in a direction opposite to that of the horseshoe vortex, becomes pertinent in relatively shallow flows where it can interfere with the approaching flow causing a reduction in downflow velocity. The stagnation pressure also accelerates the flow by the side of the pier, resulting in a flow separation at the side and creating a wake with the cast-off vortices at the interfaces to the main flow. The cast-off vortices travel downstream with the flow and interact with the horseshoe vortex at the bed causing it to oscillate. Figures 10.20a, b show the typical geometry of a scour hole and the components of the flow field at a pier.
Typical geometry of a scour hole and the components of flow field at a circular pier: a elevation view and b plan view
Conducting an experimental study, Dey (1991) described the clear-water scouring process at a circular pier. According to him, the horseshoe vortex is formed due to the diving mode of approaching flow into the scour hole. In the upstream of a pier, the sediment particles are mainly dislodged by the action of the downflow and subsequently pulled up along the upstream slope of the scour hole by the upward velocity of the horseshoe vortex. The scour hole is also fed by the sediment due to the collapse of the slant bed of scour hole, and finally, the sediment particles are drifted downstream by the arms of the horseshoe vortex along the circumference of the pier. The process of digging by the downflow along with the slant bed erosion continues until a quasi-equilibrium state is reached (Dey 1995). In this state, the scour hole is continually fed by a small amount of sediment particles due to slant bed erosion without a noticeable change in scour depth. The equilibrium, when the erosion ceases, is reached after a long period of time. In quasi-equilibrium state, the average upstream slope of the scour hole, termed dynamic angle of repose ϕ d, is about 10–20 % greater than the angle of repose ϕ of sediment in still water. A flat semicircular rim is formed around the upstream pier base (Fig. 10.20a). On downstream, the scour initiates at the pier base due to wake vortices (which act like a tornado) in the form of a spontaneous lifting of sediment particles by the action of suction from the each core of wake vortices. In the further downstream, a dune is progressively formed by the substantial deposition of sediment and side scouring. The dune, thus formed, slowly migrates downstream with the development of the scour hole. The erosion continues on both sides of the dune to form a shallow channel on either side having an adverse longitudinal slope that is flatter than the upstream slope.
Melville (1975), Dey et al. (1995), Dey (1995), Graf and Istiarto (2002) and Dey and Raikar (2007a) and Raikar and Dey (2008) measured the flow field in a scour hole at a pier. The flow measurement was done by hot-wire anemometry (Melville 1975), five-hole pitot sphere (Dey et al. 1995), and acoustic Doppler velocity profiler (Graf and Istiarto 2002). On the other hand, Dey and Raikar (2007a) and Raikar and Dey (2008) measured the flow by an acoustic Doppler velocimeter (ADV) and studied the characteristics of turbulent horseshoe vortex flow within the developing scour hole at cylindrical piers. Figure 10.21 shows the time-averaged velocity vectors at the upstream axis of symmetry of a pier in an equilibrium scour hole. The horseshoe vortex flow is well evident within the scour hole.
Velocity vectors at the upstream axis of symmetry of a pier in an equilibrium scour hole (Dey and Raikar 2007a)
10.5.1 Kinematic Model of Horseshoe Vortex
Dey et al. (1995) developed a kinematic model for the horseshoe vortex flow in a scour hole at a pier.
In the upstream, the azimuthal section of a quasi-equilibrium scour hole, as shown in Fig. 10.20a, is divided into zone 1, that is the zone vertically above the sloping bed, and zone 2, that is the zone vertically above the flat bad (semicircular rim) of the scour hole adjacent to the pier. The upstream edge of the scour hole can be approximately represented by a circular arc up to the azimuthal angle θ = ±75°. The width δ s of zone 2 is expressed as ε w(R − 0.5b), where ε w is a factor (≈0.1), R is the radius of the scour hole, that is, [d scotϕ d/(1 − ε w)] + 0.5b, and d s is the scour depth. In cylindrical polar coordinates, the velocity components (u r , u θ , u z ) are in r, θ, and z-direction, respectively.
The tangential velocity u θ is represented by a power law preserving the no-slip condition at the bed
where
where k 1 = k 1(U, h, b, d s), U is the depth-averaged approaching flow velocity, h is the approaching flow depth, and m and n are the exponents. The values of m and n were obtained as 2.1 and 3.9, respectively.
The radial velocity u r changes direction in the scour hole. The u r -distribution along z varies linearly in the scour hole (z ≤ 0) and follows a power law above it (z > 0). On the other hand, the u r -distribution along r is parabolic. Thus, the expression for u r is
where
where k 2 and k 3 are the coefficients similar to k 1, α = κU/u *, κ is the von Kármán constant, and u * is the shear velocity. Note that Eqs. (10.61) and (10.62) violate the no-slip condition at the pier wall and the bed, respectively, and cannot be applicable to the immediate vicinity of them.
The expression for vertical velocity u z , obtained by integrating the continuity equation (Eq. 2.164), is
where
Equation (10.63) produces a strong downflow along the upstream face of the pier and a flow reversal in the scour hole. It however violates the no-slip condition at the pier wall and the bed. Dey et al. gave the expressions for the coefficients as
Dey and Bose (1994) used the expressions for u r , u θ , and u z to compute the bed shear stress in the scour hole by applying the turbulent boundary-layer approach.
10.5.2 Scour Depth Prediction
Scour at piers is influenced by various parameters (Breusers et al. 1977), which are grouped as follows:
-
1.
Parameters related to the pier: Size, shape, spacing, number, and orientation with respect to the approaching flow direction.
-
2.
Parameters related to the bed sediment: Median particle size, particle size distribution, angle of repose, and cohesiveness.
-
3.
Parameters related to the approaching flow condition: Approaching flow velocity, approaching flow depth, shear velocity, and roughness.
-
4.
Parameters related to the fluid: Mass density, viscosity, gravitational acceleration, and temperature. Note that the temperature may not be important in scour problems, unless free surface is frozen.
-
5.
Parameters related to the time: Time of scour for an evolving scour case.
A large number of empirical equations were proposed by various investigators to estimate the maximum scour depth at piers based on the data from the laboratory experiments and the field measurements. In general, these equations were derived from a limited range of data and are applicable to the conditions similar to those for which they are valid. Though the number of proposed equations for the estimation of maximum scour depth is overwhelming (Dey 1997a; Melville and Coleman 2000), it is however difficult to confirm their adequacy for the design purposes due to limited field measurements. Nevertheless, the design equations proposed by Melville and Coleman (2000), HEC18 (Richardson and Davis 2001), and Sheppard et al. (2014) seem to provide satisfactory estimations.
Melville and Coleman (2000) recommended a design equation for the estimation of maximum scour depth at piers based on empirical factors, called K-factors, which account for the effects of pier, flow, and sediment characteristics. The K-factors were determined by fitting the curves that overlap the data plots. Thus, the proposed K-factors potentially remain adequate from the viewpoint of a safe pier foundation design. The maximum scour depth d s at a bridge pier formulated as a product of various K-factors is given as
where K h is the flow depth–pier size factor, K I is the flow intensity factor, K d is the sediment size factor, K s is the pier shape factor, K α is the pier alignment factor, and K t is the time factor. The relationships for the K-factors are as follows:
The flow depth–pier size factor K h is the scour depth d s at a pier of width b for a given value of flow depth h. It is given by
The flow intensity factor K I is the ratio of the scour depth d s for a given flow velocity U to that for the threshold flow velocity U cr for the bed sediment motion. Thus, the flow intensity factor K I represents the effects of flow intensity on scour depth. It also accounts for the nonuniformity of sediment in terms of average approaching velocity U a at armor peak. It is given by
where U a = 0.8U cra and U cra is the maximum average velocity for the bed to armor. Note that for uniform sediment, U a(σ g < 1.3) = U cr. Under varied stream flow velocities over a bed of nonuniform sediment, a process of armoring prevails resulting in an exposure of coarser particles due to washing out of the finer fraction. The armoring effect is to reduce the scour depth. The U cr and U cra can be obtained from
where u *ca is the threshold shear velocity for median size d 50a of armor particles, d 50a = d max/1.8 and d max is the maximum bed sediment size. Melville and Coleman suggested that the u *c can be empirically calculated as \( u_{{*{\text{c}}}} (0. 1\;{\text{mm}} \le d_{ 50} < 1\;{\text{mm}}) = 0.0 1 1 5+ 0.0 1 2 5d_{50}^{1.4} \) and \( u_{{*{\text{c}}}} ( 1\,{\text{mm}} \le d_{ 50} < 100\;{\text{mm}}) = 0.0 30 5d_{50}^{0.5} -\,6. 5\times 10^{{- 3}} d_{50}^{ - 1} \).
Here, u *c is in m s−1 and d 50 in mm. For u *ca, same expressions can be used replacing u *c by u *ca and d 50 by d 50a.
The sediment size factor K d is the ratio of the scour depth d s for a given value of b/d 50 to that for b/d 50 for which d s becomes maximum and beyond which, there is no effect of b/d 50 on d s. For nonuniform sediment, d 50 is to be replaced by d 50a. It is
However, for the piers embedded in gravel-beds, Raikar and Dey (2005b) proposed an additional set of sediment size factor. It is
The shape factor K s is defined as the ratio of the scour depth d s for a particular pier shape to that for the circular pier having a diameter same as the pier width. The shape factors K s for different piers are furnished in Table 10.2.
The alignment factor K α is the ratio of the scour depth d s at an oblique pier to that at an aligned pier. In case of noncircular piers, the scour depth increases with an increase in the effective projected width of the piers. The K α is given by
where b p is the projected width of a rectangular pier normal to the approaching flow (= Lsinα + bcosα), α is the pier orientation with respect to the approaching flow or skewness, and L is the pier length (Fig. 10.22). For circular piers, K α = 1.
Oblique alignment of a rectangular pier with respect to approaching flow direction
The time factor K t is the ratio of the scour depth d s for a given time t to the equilibrium scour depth. It depends on the scour condition, such as clear-water scour and live-bed scour. For live-bed scour, K t can be approximated as unity, as equilibrium is attained rather rapidly, while for clear-water scour, K t is given by
where t e is the time to reach equilibrium scour depth. It can be calculated from
At threshold condition (U = U cr), t e being maximum, when h > 6b, is given as
where b is in m and U in m s−1.
According to HEC18 (Richardson and Davis 2001), the scour depth at a pier in both clear-water and live-bed scour conditions is given by
where K bed is the factor for bed condition (Table 10.3) and K a is the factor for armoring of bed sediment. Further, for a maximum value of d s at a round-nosed pier with aligned flow, if Fr ≤ 0.8, then d s/b ≤ 2.4 and if Fr > 0.8, then d s/b ≤ 3 (Table 10.4).
The K a that takes into account of armoring of bed sediment can be given as
where \( \left. {U_{\text{crs}} } \right|_{{d_{i} }} \) is the approaching flow velocity required for threshold of scour at a pier with sediment size d i , \( \left. {U_{\text{cr}} } \right|_{{d_{i} }} \) is the threshold flow velocity for bed sediment of size d i , and d i is the i-percent finer sediment size. Note that in Eq. (10.75b), U r should be positive (U r > 0). The \( \left. {U_{\text{crs}} } \right|_{{d_{i} }} \) is estimated as
Further, Sheppard et al. (2014) proposed a method of scour depth prediction as
where b e is the effective pier diameter and U peak is the live-bed peak flow velocity,
In the above, α is in radians and K s = 1 for circular piers.
They suggested the empirical relationship for the computation of U cr as
where
Besides the aforementioned scour predictors that can produce potentially safe results for the scour depth, Table 10.5 furnishes some empirical equations of equilibrium scour depth proposed by different investigators.
Regarding the maximum limit of scour depth at circular piers, Melville and Coleman (2000) [also Melville and Sutherland (1988)] reported d s ≤ 2.4b for flow Froude number Fr < 1. On the other hand, d s ≤ 3b was recommended by HEC18 (Richardson and Davis 2001) and Jain and Fischer (1979). In the experiments conducted by Jain and Fischer (1979), the Fr was as high as 1.5 for the bed conditions of antidunes. However, for noncircular piers, these maximum limits increase and are to be corrected for pier shape and skewness, if any.
10.6 Scour at Bridge Abutments
Abutments, located at either end of a bridge span, are the substructures over which the bridge superstructure rests (Fig. 10.23). Akin to bridge piers, they also help to transmit the weight of the bridge including traffic to the foundation bed. Piers are located within the bridge span (Sect. 10.5). Scour at bridge abutments is also equally or even more responsible for failure of bridges as compared to scour at piers. Similar to bridge piers, the flow separates at the upstream of the abutment as it travels by the side of the abutment, creating a vortex trail to move downstream. The result is that the sediment bed in the vicinity of the abutment is scoured, exposing the abutment foundation that may lead to the failure of the bridge.
Photograph of a wing-wall abutment. Photograph by the author
A study of the US Federal Highway Administration in 1973 concluded that of 383 bridge failures, 25 % involved pier damage and 72 % involved abutment damage (Richardson et al. 1993). In a report submitted to the National Roads Board of New Zealand, Sutherland (1986) pointed out that of the 108 bridge failure records, 29 were attributed to abutment scour during 1960–1984. According to Kandasamy and Melville (1998), 6 out of 10 bridge failures that occurred in New Zealand during Cyclone Bola were related to the abutment scour. In another report of the Department of Scientific and Industrial Research of New Zealand, Macky (1990) mentioned that about 50 % of total expenditure was made toward the bridge damage repairing and maintenance, out of which 70 % was spent toward the abutment scour. Therefore, abutment scour, due to its practical importance, was studied extensively by various researchers [see the review by Barbhuiya and Dey (2004)]. Figure 10.24 shows the photograph of an equilibrium scour hole at a wing-wall abutment in an experimental flume. It gives an idea about the shape of the scour hole that occurs at an abutment.
Photograph of an equilibrium scour hole at a wing-wall abutment in a flume
The flow field at an abutment is complex in detail, and the complexity increases with the development of scour hole involving flow separation to develop three-dimensional vortex flow. Kwan and Melville (1994) [also in Kwan (1988)] used a hydrogen bubble technique to measure the three-dimensional flow field in a scour hole at a wing-wall abutment. They identified a primary vortex, which is quite similar to the horseshoe vortex at a pier, along with the downflow being the principal mechanism of scour at an abutment. On the upstream face of an abutment, a vertical pressure gradient is developed due to the stagnation of approaching flow by the abutment. The pressure gradient drives the fluid downward to roll up. Thus, primary vortex is developed and it enlarges its size with the development of the scour hole. They also reported that the primary vortex and the downflow are confined mainly in the scour hole beneath the line of the original bed level. The primary vortex is elliptical in shape with an inner core region as that of a forced vortex and an outer core region as that of a free vortex. The maximum velocity and downflow component in the vicinity of the abutment were measured as 1.35 and 0.75 times the approaching flow velocity, respectively. They also identified a secondary vortex with counter rotational direction to that of the primary vortex occurring next to the primary vortex. The secondary vortex is believed to have the effect of restricting the scouring capacity of the primary vortex. In the downstream of abutment, wake vortices are created due to the separation of flow at the upstream and downstream of the abutment corners. The unstable shear layers created due to the flow separation roll up to form eddy structures, termed wake vortices. The wake vortices drifted downstream by the mean flow act like small tornadoes lifting up sediment particles from the bed. The wake vortices are rather weak as compared to the primary vortex. Bow wave having a rotational motion opposite to that of primary vortex appears on the upstream face of the abutment near the free surface. The major flow components at a wing-wall abutment identified by Kwan (1988) are shown schematically in Fig. 10.25.
Schematic of flow field at an abutment (Kwan 1988)
Dey and Barbhuiya (2005a, b, 2006) investigated the three-dimensional turbulent flow fields at semicircular, vertical-wall and 45o wing-wall abutments, embedded in a stabilized equilibrium scoured bed by using an ADV [also in Barbhuiya (2003)]. A cylindrical polar coordinate system, as shown in Fig. 10.26, was used by Dey and Barbhuiya (2006) to represent the normalized velocity vectors at azimuthal sections of a 45o wing-wall abutment with a scour hole. Figure 10.27 shows velocity vectors at θ = 10o and 30o (Dey and Barbhuiya 2006). Here, z is the vertical distance, r is the radial distance, and l is the abutment length transverse to the flow. The characteristics of vortex flow inside the scour hole (that is, in the flow zone z ≤ 0) together with the strong downflow along the upstream face of the abutment are evident. Note that as the length scales of the axes (ordinate and abscissa) are different in Fig. 10.27, the vortices are apparently stretched horizontally, but they are actually not so. The circulation is strong at the upstream of the abutment and decreases with an increase in θ. Above the scour hole (that is, in the flow zone z > 0), the flow is horizontal for r > 3l, and then, it gradually curves down toward the abutment.
Coordinate system for representation of flow and schematic of a scour hole at a 45o wing-wall abutment
Normalized velocity vectors at azimuthal sections (θ = 10o and 30o) of a 45o wing-wall abutment after Dey and Barbhuiya (2006)
10.6.1 Scour Depth Prediction
Parameters involved in scour phenomenon at abutments can be grouped in the similar way as those of pier scour (Sect. 10.5.2), except the following two parameters:
-
1.
Parameters related to the abutment: Size, shape, and orientation with respect to the approaching flow direction.
-
2.
Parameters related to the geometry of channel: Width, cross-sectional shape, and slope.
Most of the scour predictors were preliminarily obtained, as functional relationships, using the dimensional analysis based on the aforementioned parameters. Then, experimental and field data were used to give them a final shape as empirical equations. Proposed empirical equations for the estimation of maximum scour depth at abutments are overwhelming (Barbhuiya and Dey 2004), but their application is limited to the conditions similar to those for which they were validated. The design approaches proposed by Melville and Coleman (2000) and Froehlich (1989), which was recommended by HEC18 (Richardson and Davis 2001), for the estimation of maximum scour depth at abutments based on K-factors seem to be adequate from the viewpoint of the safety of abutment foundation.
According to Melville and Coleman (2000), the maximum equilibrium scour depth d s at an abutment for both clear-water and live-bed conditions is given by
where K G is the channel geometry factor and other K-factors are designated similar to that in Eq. (10.64). The relationships for K-factors, in case of abutment scour, are given as follows:
The flow depth–pier size factor K h for abutment scour is
The above K h-factor is based on Melville (1992) who distinguished short and long abutments. For short abutments (l/h ≤ 1), the scour depth d s is independent of flow depth h and dependent on abutment length l. For long abutments (l/h ≥ 25), the d s is dependent on h and independent of l. For 1 < l/h < 25, the d s is dependent on both l and h. Note that l is projected length for skewed abutments (Fig. 10.28).
Oblique alignment of an abutment (plan view)
The flow intensity factor K I given by Eq. (10.66) for the case of pier scour is also applicable for the abutment scour.
The sediment size factor K d for abutment scour is
However, for the abutments embedded in gravel-beds, Raikar and Dey (2005a) proposed new sediment size factors as
The shape factor K s is defined as the ratio of the scour depth d s for a given abutment shape to that for the vertical-wall abutment having a same length. The shape factors K s for different abutments are given in Table 10.6.
For abutments, the values of alignment factor K α for abutment scour are
The values of \( K_{\alpha }^{*} \) in the above equation are obtained from Table 10.7.
The channel geometry factor K G is defined as the ratio of the scour depth d s at an abutment to that at the same abutment in the equivalent rectangular channel. In case of rectangular channels, K G = 1. However, for abutments in compound channels, K G depends on the abutment position in the compound channel. The K G is
where l* is the abutment length spanning the flood channel, h* is the flow depth in the flood channel, and n and n* are the Manning roughness coefficients in the main and flood channels, respectively. In case of an inclined abutment, l and l* are the projected abutment length and that spanning the flood channel, respectively.
For live-bed scour, the time factor K t is unity; while for clear-water scour, K t for abutment scour is given by
The time t e to reach equilibrium scour depth is given by
HEC18 (Richardson and Davis 2001) recommended Froehlich’s (1989) equation of live-bed scour at abutments. The estimation of maximum scour depth is
where Fr = U 0/(gh)0.5, U 0 = Q 0/A 0, Q 0 is the flow rate obstructed by the abutment and approach embankment, and A 0 is the flow area of the approach cross section obstructed by the embankment. The values of shape factors K s are furnished in Table 10.8.
The alignment factor K α is given by
where α is the downstream angle of inclination of abutment with bank (Fig. 10.28), such that α < 90° is for the abutment pointing downstream and α > 90° for pointing upstream.
Table 10.9 furnishes some additional empirical scour depth predictors at abutments proposed by various investigators.
10.7 Scour Countermeasures
Engineering devices for countermeasure of scour at bridge piers are generally classified into two categories: Flow-altering and bed-armoring countermeasures. The working principle of flow-altering countermeasures is to diminish the strength of the downflow and the horseshoe vortex, which are the primary cause of pier scour. Of various types of flow-altering countermeasures, slot in a pier (Grimaldi et al. 2009), spirally wrapped cables on a pier (Dey et al. 2006), collars and horizontal plates attached to a pier (Kim et al. 2005; Parker et al. 1998), arrangement of sacrificial piles (Melville and Hadfield 1999; Chiew and Lim 2003; Haque et al. 2007), and flow deflection by upstream vanes or plates (Odgaard and Wang 1991; Lauchlan 1999) are pertinent. Tafarojnoruz et al. (2010) compiled a review of literature on flow-altering countermeasures at piers. On the other hand, bed-armoring countermeasures provide a physical barrier against scour. In practice, these barriers often consist of large and heavy units, which cannot be easily removed by the flow at piers (Melville and Coleman 2000; Lagasse et al. 2007; Melville et al. 2008). The most commonly employed protection of bridge piers (and also abutments) is the use of a riprap layer around the piers. In this section, riprap protection at piers is mainly discussed.
Figure 10.29 shows a schematic of the placement of a riprap layer at a pier of width b, considering the flow to be the normal to the plane of the drawing. Parameters involved in riprap protection design at piers are as follows:
Typical placement of a riprap layer at a pier
-
1.
Thickness t r of riprap layer.
-
2.
Coverage b r of riprap layer at sides, upstream, and downstream of a pier.
-
3.
Placement z r of riprap layer with respect to the original bed level.
-
4.
Median size d 50r of riprap stones and their gradation.
The thickness of the riprap layer is recommended typically in the range t r = 2–3d 50r. Thicker layer can resist higher flow intensity (Chiew 1995). The general recommendation for riprap coverage is to place riprap around a pier extending up to b r = 3b from the pier wall in all directions (Parola 1995; Croad 1997; Parker et al. 1998; Lauchlan 1999). Parker et al. (1998) suggested b r for rectangular piers is b r = 1.5b/cosα. Regarding the placement of a riprap layer z r, the surface of the riprap layer to be placed at the original streambed level is the common recommendation (Richardson and Davis 2001). Another recommendation is to place the riprap layer below the possible general scour depth level (Neill 1973; Breusers et al. 1977). Further, to improve the performance of a riprap protection, the use of a filter layer beneath the riprap layer is generally proposed. Filters that can be granular filters or synthetic filters prevent the passage of finer bed sediment through the highly porous riprap layers, but also have sufficient permeability to prevent building up water pressure within the underlying bed sediment. Regarding riprap stones, it is important that they should be well graded, such that the maximum stone size should not exceed twice the median size of riprap stones, that is, d max ≤ 2d 50r (Richardson and Davis 2001), and the median size should not exceed twice the 15 % finer stone size, that is, d 50r ≤ 2d 15r (Croad 1997). To determine the riprap stone size d 50r, HEC-18 (Richardson and Davis 2001) and HEC-23 (Lagasse et al. 2001) recommended using the reorganized form of Isbash (1936) equation. It is
where K is the pier shape coefficient. The values of K are that for round-nosed piers, K = 1.5 and for rectangular piers, K = 1.7.
Further, Lauchlan (1999) suggested an equation of riprap stone size taking into account the placement depth z r below the original bed level. It is
where f SF is a safety factor that has a minimum recommended value of 1.1.
In case of riprap protection at abutments, the coverage b r of riprap layer around an abutment, called launching apron, is extended up to b r = 1.5d s from the abutment wall in all directions. It should have a minimum thickness of t r = 2d 50r. The spill-through abutments are additionally protected by stone-pitching on the slant face of the abutment. The median stone size d 50r of the riprap layer can be obtained from the equation given by Austroads (1994) as
According to Atayee et al. (1993) and Richardson and Davis (2001), the median stone size can be obtained as
where h 2 is the flow depth in the contracted section of the bridge, Fr 2 is the flow Froude number in the contracted section [= U 2/(gh 2)0.5], and U 2 is the average flow velocity in the contracted section. The values of the shape factor K s are 0.89 (Fr ≤ 0.8) and 0.61 (Fr > 0.8) for spill-through abutments and 1.02 (Fr ≤ 0.8) and 0.69 (Fr > 0.8) for vertical-wall abutments.
10.8 Appendix
10.8.1 Submerged Wall Jets (Dey et al. 2010)
Submerged plane wall jet is described as a fluid jet that impinges tangentially (or at an angle) on a solid boundary surrounded by the same fluid (still or moving) progressing along the solid boundary (Fig. 10.30a). By virtue of the initially abounding momentum, the streamwise velocity \( \bar{u} \) in the shear flow of jet exceeds that in external stream over a downstream reach (Launder and Rodi 1981). In a submerged wall jet, the flow zone of principal importance is the fully developed zone existing after the developing zone of jet (Fig. 10.30b). The jet is confined to a solid boundary on one side, and the other side is fluid bounded. The jet layer is overlain by a circulatory flow having an enormous mixing of fluid with a flow reversal. Since the boundary conditions for a submerged wall jet are such that the velocities at the solid boundary and on the separation line are zero (\( \bar{u} = 0 \)), the velocity distribution has a peak within the jet layer. Below the peak velocity level (in the inner layer), the flow is characterized by a boundary-layer flow, and the upper flow zone is structurally similar to a free jet. Therefore, a submerged wall jet, characterized by a shear flow influenced by the solid boundary and an overlying circulatory flow layer, is of self-similar type of flow (Dey et al. 2010).
a Typical sketch of ū-distribution superimposed on the flow field in the fully developed flow zone and b flow zones in a submerged wall jet (Dey et al. 2010)
A typical \( \bar{u} \)-distribution in the fully developed zone describing various layers of flow is shown in Fig. 10.30a. The inner layer and outer layer of jet refer to the zones below and above the point of occurrence of peak velocity u 0, called the jet velocity. Precisely, the jet layer (0 ≤ z ≤ δ) that comprises of inner layer and outer layer extends up to the inflection point (that is, the point of change of sign of slope, \( {\text{d}}\bar{u}^{2} /{\text{d}}z^{ 2} = 0 \)) of a \( \bar{u} \)-distribution. Above the jet layer, there exists a circulatory flow layer that is divided by the separation line \( \bar{u} \) = 0 into inner layer and outer layer of the circulatory flow. The flow in the outer layer of the circulatory flow is directed toward upstream. Momentum exchange takes place through the separation line within the circulatory flow layer of the jet. The jet layer thickness δ is important from the viewpoint of scaling the vertical distance z (Dey et al. 2010).
A two-dimensional submerged plane wall jet issuing from a sluice opening is considered as the jet emerges in the form of a bunch of diverging streamlines. Another bunch of streamlines constitutes a circulatory flow above the jet in the circulatory flow layer. The limiting streamline on the solid boundary has a velocity \( \bar{u} = 0 \) due to no-slip. Let the equation of the jet layer be z = δ(x 1), where x 1 = x + x 0, extending up to the inflection point of \( \bar{u} \)-distribution. The jet layer is assumed as a boundary layer. Due to finite size of issuing jet, the point of emergence (that is the origin) of the jet is located upstream of the sluice opening at a certain distance x 0. Applying the boundary-layer approximation to the two-dimensional Reynolds-averaged Navier–Stokes (RANS) equations of steady flow and eliminating pressure term, the following equation is obtained (Rajaratnam 1976):
The continuity equation is
The flow in submerged plane wall jets is characterized by the self-similar class. To obtain the similarity solutions of Eqs. (10.93) and (10.94) by the transformation η = z/δ(x 1), where the horizontal length scale x 1 is dimensional for the theory, the solutions are of the form
where u 0 = u 0(x 1). It is pertinent to mention that a wall jet boundary layer is not amenable to similarity analysis, unless different scaling laws are assumed for the inner layer and the outer layer of the jet (Barenblatt et al. 2005). Inserting the above expressions into Eq. (10.93) and using Eq. (10.94), one obtains
where R δ = u 0 δ/υ. The right-hand side of Eq. (10.96) vanishes, as the terms containing the difference of streamwise and vertical Reynolds normal stresses represented by σ are negligible and R δ is too large. For a similarity solution, Eq. (10.96) must be independent of x or x 1 (Schwarz and Cosart 1961), that is
where β and α are constants. Hence, by integration, one can write
where β 0 is a constant. Noticeably, the δ increases linearly with x 1, and u 0 varies as \( x_{1}^{ - \alpha } \). For a free jet, α is 0.5 (Schlichting 1979).
The velocity distribution obtained using Eqs. (10.96) and (10.98) is given by
Setting φ(η) = f ′(η), Eq. (10.99) becomes
By definition of the turbulence diffusivity ε t, one can write
Using Eq. (10.95), Eq. (10.101) yields
In Eq. (10.102), the left-hand side being independent of x 1 implies that ε t is proportional to \( x_{1}^{1 - \alpha } \), and hence, \( \varepsilon_{\text{t}} = \bar{\varepsilon }_{\text{t}} x_{1}^{1 - \alpha } \). In the narrow turbulent jet layer, \( \bar{\varepsilon }_{\text{t}} \) may be assumed to be an averaged value of ε t over η. Thus, one obtains
The velocity distribution contains an arbitrary constant β 0. Replacing β 0 by \( 4\bar{\varepsilon }_{\text{t}} /\beta^{2} \) in Eq. (10.103), one gets the following equation for f:
The boundary conditions applicable for the solution of Eq. (10.104) are that at the peak velocity of the jet f′(η = η 0) = φ(η = η 0) = 1, f(η = η 0) = 0 (that is \( \bar{w} = 0 \)), and f′(η → ∞) = 0 (that is \( \bar{u} = 0 \)). For a free jet, η 0 = 0 and α = 0.5 (Schlichting 1979), and the solution of Eq. (10.104) is f = tanh η. It is anticipated that due to the submergence, α is modified as
where α 1 is an additional term mainly due to submergence. The solution of Eq. (10.104) can be given by
Substituting Eq. (10.106) into Eq. (10.104) and equating the coefficients of α 1, the differential equation for G is given by
with boundary conditions \( G(\eta = \eta_{0} ) = 0{\hbox{,}}\;G^{\prime}(\eta = \eta_{0} ) = 0 \), and \( G^{\prime}(\eta \to \infty ) = 0 \). Equation (10.107) that has highly nonlinear coefficients is a linear differential equation. Galerkin’s method is applied to obtain an approximate analytical solution. For this purpose, it is recognized that a function of the pattern of the leading term of Eq. (10.106) that satisfies the boundary condition is
Substituting Eq. (10.108) into Eq. (10.107) and taking the weighted average with the weight appearing in the equation yield
Numerically evaluating the integrals in Eq. (10.109), one obtains a 0 ≈ 32/15. Differentiating Eq. (10.106) with the value of a 0 in Eq. (10.108), one gets
Giving the velocity distribution of a submerged wall jet by Eqs. (10.95) and (10.98), the profile holds for η ≥ η 0, because below the point of η 0 (that is, within the inner layer of the jet), boundary effects come into account.
In the near-boundary zone (that is, within the inner layer of the jet) 0 ≤ η ≤ η 0, the 1/m-th power law for φ(η) can be assumed as in case of a flow over a solid plate. Noting that as φ(η = η 0) = 1, φ′(η = η 0) = 0, such a law is
The Reynolds shear stress τ xz is given by using Eqs. (10.101) and (10.102). It is
where \( \xi = \beta_{0} \bar{\varepsilon }_{\text{t}} /(\beta U_{0}^{2} b_{0}^{2\alpha } ) \), \( \hat{x} = x/b_{0} \), and \( \hat{x}_{0} = x_{0} /b_{0} \).
From Eqs. (10.110) and (10.111), the following expressions for φ′ are obtained:
The Reynolds shear stress vanishes at the solid boundary. Equation (10.113b) cannot be applicable to the very thin viscous sublayer in the vicinity of the boundary, where viscous shear stress prevails. Using the experimental data, Dey et al. (2010) estimated the values of coefficients and exponents as \( \hat{x}_{0} = 1 1. 3 4 \), β = 0.078, β 1 = 3.17, α = 0.455, α 1 = –0.045, η 0 = 0.3, m = 6, and ξ = 79.87. Figures 10.31a, b show φ(η) and φ′(η) curves for U = 0.639 m s−1 and b 0 = 40 mm and the comparisons with the experimental data.
a φ as a function of η and b φ′ as a function of η for U = 0.639 m s−1 and b 0 = 40 mm in submerged plane wall jet (Dey et al. 2010)
10.8.2 Computation of Scour Due to Submerged Wall Jets
The bed shear stress can be determined from Eq. (10.113b) by applying it to the near-bed level. In case of an erodible sediment bed, the bed is initially horizontal (before scour), and the fluid jet flows in the direction parallel to the horizontal bed surface of sediment. The jet erodes the bed forming a scour hole as shown in Fig. 10.32. The scour profile can be calculated by considering the bed shear stress distribution along the surface of the bed. To determine the bed shear stress τ 0 in the scour hole, Eq. (10.113b) along with Eq. (10.112b) is thus applied to the particle level with an introduction to a shape function due to scour. With modification, the equation of bed shear stress τ 0 can then be given by
where η b = k s/δ and \( G(\eta {\hbox{,}}\,\hat{x}) \) is a shape function to account for the bed shear stress variation when the bed is no longer horizontal. Here, k s can be assumed as d 50. Note that the value of α = 0.5 + α 1 depends on submergence ratio S [= (h t − h j)/h j], where h j is the conjugate tailwater depth of free jump \( \{ = 0. 5b_{0} [( 1+ 8F_{0}^{2} )^{0. 5} - 1]\} \) and F 0 is the jet Froude number [= U 0/(gb 0)0.5].
Definition sketch of scour due to a submerged wall jet
Initially, the bed is horizontal (η = 0) and so \( G(0{\hbox{,}}\; \hat{x}) = 1 \). A Gaussian-like stress distribution with vertical distance can be assumed (Hogg el al. 1997):
where C 0 is the coefficient to be determined from the experimental data.
For equilibrium scour, Eq. (4.115) is reorganized as
where \( \hat{\delta } = \delta /b_{0} \). Using Eqs. (10.114) and (10.115), Eq. (10.116) which is a differential equation can be solved by Runge–Kutta method to determine the variation of scour depth \( \hat{z}( = \eta \hat{\delta }) \) with \( \hat{x} \), that is the nondimensional profile of equilibrium scour hole.
10.9 Examples
Example 10.1
Estimate the maximum equilibrium scour depth within a long contraction using the empirical equation given by Dey and Raikar for the following data:
Approaching flow depth, h 1 = 5 m
Approaching channel width, B 1 = 70 m
Channel width at contracted zone, B 2 = 40 m
Median size of sediment, d 50 = 2.6 mm
Geometric standard deviation of sediment, σ g = 2.2
Consider coefficient of kinematic viscosity of water υ = 10−6 m2 s−1 and relative density of sediment s = 2.65
Solution
Use van Rijn’s empirical formula for the determination of threshold shear velocity (see Table 4.1):
Particle parameter, \( D_{*} = d_{ 50} ({\Delta} g/\upsilon^{ 2} )^{ 1/ 3} = 2. 6\,\times\,10^{{- 3}} \left[ 1. 6 5\,\times\,9. 8 1/\left( { 10^{{- 6}} } \right)^{ 2}\right]^{ 1/ 3} = 6 5. 7 7 \)
Threshold Shields parameter, \( {\Theta}_{\text{c}} ( 20 < D_{*} \le 1 50) = 0.0 1 3D_{*}^{0.29} = 0.0 1 3\,\times\,6 5. 7 7^{0. 2 9} = 0.0 4 4 \)
Threshold bed shear stress, \( \tau_{{0{\text{c}}}} = {\Theta}_{\text{c}} {\Delta} \rho gd_{ 50} = 0.0 4 4\,\times\,1. 6 5\,\times\,10^{ 3}\,\times\,9. 8 1\,\times\,2. 6\,\times\,10^{{- 3}} = 1. 8 5 2\;{\text{Pa}} \)
Threshold shear velocity, \( u_{{*{\text{c}}}} = (\tau_{{0{\text{c}}}} /\rho )^{0. 5} = \left( { 1. 8 5 2/ 10^{ 3} } \right)^{0. 5} = \;0.0 4 3\;{\text{m}}\;{\text{s}}^{{- 1}} \)
The approaching flow velocity \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} \) that corresponds to the threshold of sediment motion within contraction is estimated from the solution of Eqs. (10.20)–(10.22) as follows:
Numerically solving above three equations, \( \left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} = 0. 5 6 5\;{\text{m}}\;{\text{s}}^{{- 1}} \)
The threshold velocity of approaching flow is determined as follows:
Then, for \( U_{{ 1 {\text{ec}}}} = U_{\text{cr}} -\left. {U_{1} } \right|_{{U_{2} = U_{\text{cr}} }}^{{d_{\text{s}} = 0}} = 1 - 0. 5 6 5= 0. 4 3 5\;{\text{m}}\;{\text{s}}^{{- 1}} \),
Using B 2/B 1 = 40/70 = 0.571 and d 50/h 1 = 2.6 × 10−3/5 = 5.2 × 10−4 in Dey and Raikar’s equation (Eq. 10.24), the [d s]max is estimated as
The maximum scour depth in uniform sediment is [d s]max = 0.59h 1 = 0.59 × 5 = 2.95 m
For nonuniform sediment with σ g = 2.2 for which K σ = 0.38 (Fig. 10.3), the maximum equilibrium scour depth in nonuniform sediment is [d s]max = 0.38 × 2.95 = 1.121 m.
Example 10.2
Calculate the equilibrium scour depth below a drop structure using the equations of various investigators for the following data:
Height between upstream and downstream water levels, H = 1.5 m
Tailwater depth, h t = 0.9 m
Discharge per unit width, q = 1.4 m2 s−1
Angle of jet entering the tailwater, θ j = 60°
Thickness of jet at the tailwater level, l 0 = 0.1 m
Sediment size, d 90 = 32 mm and d 95 = 38 mm
Relative density of sediment, s = 2.65
Solution
Schoklitsch’s equation
Kotoulas’ equation
Fahlbusch’s equation
Velocity of jet entering the tailwater, U 0 = q/l 0 = 1.4/0.1 = 14 m s−1
Assume K p = 3.5 (for gravels)
Graf’s equation
Eggenberger and Müller’s equation
Example 10.3
Calculate the equilibrium scour depth downstream of a grade-control structure for the following data:
Approaching flow depth, h = 1.2 m
Drop height, Z p = 0.5 m
Discharge per unit width, q = 1.55 m2 s−1
Angle of jet near the original bed level, θ j = 55°
90 % finer size of sediment, d 90 = 4 mm
Angle of repose of sediment, ϕ = 40°
Relative density of sediment, s = 2.65
Solution
Bormann and Julien’s equation is used to calculate equilibrium scour depth downstream of a grade-control structure.
Approaching flow velocity, U 1 = q/h = 1.55/1.2 = 1.292 m s−1
Example 10.4
Calculate the equilibrium scour depth due to a horizontal jet issuing from a sluice gate opening using the equations of various investigators for the following data:
Height between upstream and downstream water levels, H = 0.5 m
Tailwater depth, h t = 2 m
Discharge per unit width, q = 1.6 m2 s−1
Sluice gate opening, b 0 = 0.5 m
Sediment size, d 50 = 10 mm and d 90 = 18 mm
Angle of repose of sediment, ϕ = 42°
Relative density of sediment, s = 2.65
Dune height, η d = 0.1h t
Solution
-
Use van Rijn’s empirical formula for the determination of threshold shear velocity (see Table 4.1):
-
Particle parameter, D * = d 50(Δg/υ 2)1/3 = 10 × 10 -3[1.65 × 9.81/(10−6)2]1/3 = 252.95
-
Threshold Shields parameter, Θc(D * > 150) = 0.055
-
Threshold bed shear stress, τ 0c = ΘcΔρgd 50 = 0.055 × 1.65 × 103 × 9.81 × 10 × 10-3 = 8.903 Pa
-
Threshold shear velocity, u *c = (τ 0c/ρ)0.5 = (8.903/103)0.5 = 0.094 m s−1
Qayoum’s equation
Altinbilek and Basmaci’s equation
Jet velocity, U 0 = q/b 0 = 1.6/0.5 = 3.2 m s−1
Breusers and Raudkivi’s equation
Haffmans’ equation
Averaged velocity over dune, U crest = q/(h t − 0.1h t) = q/(0.9h t) = 1.6/(0.9 × 2) = 0.889 m s−1
Scour factor, λ s = 6.8 for d 90 = 18 mm
Eggenberger and Müller’s equation
Example 10.5
Determine the equilibrium scour depth downstream of an apron of length L 0 = 3 m due to a horizontal jet issuing from a sluice gate opening for the data given in Example 10.4.
Solution
Shalash’s equation
Dey and Sarkar’s equation
Example 10.6
Calculate the equilibrium scour depth downstream of a hydraulic structure due to a combined overfall and submerged jet for the following data:
-
Height between upstream and downstream water levels, H = 1.2 m
-
Tailwater depth, h t = 1.5 m
-
Total discharge per unit width, q = 1.6 m2 s−1
-
Discharge through sluice opening, q 0 = 0.6q
-
Sediment size, d 90 = 12 mm
Solution
Eggenberger and Müller’s equation
Submerged jet discharge through sluice opening, q 0 = 0.6q = 0.6 × 1.6 = 0.96 m2 s−1
Overfall discharge, q 1 = q − q 0 = 1.6 − 0.96 = 0.64 m2 s−1
Discharge ratio, \( \hat{q} = q_{ 1} /q_{0} = 0. 6 4/0. 9 6= 0. 6 7 \)
Example 10.7
Given pipe diameter, D = 1.2 m; embedment, e = 0.1 m; porosity of sediment, ρ 0 = 0.4; and relative density of sediment, s = 2.65, what is the threshold velocity of scour underneath the submarine pipeline?
Solution
From Eq. (10.51)
Therefore, U gcr = 1.979 m s−1
Example 10.8
Compute the equilibrium scour depth below a 0.1 m diameter underwater pipeline, laid on a sediment bed of d 50 = 0.6 mm in a laboratory flume, subjected to a steady flow velocity of 0.35 m s−1 having a flow depth of 0.4 m. Take coefficient of kinematic viscosity of water υ = 10−6 m2 s−1 and mass density of water ρ = 103 kg m−3.
Solution
Given data are as follows:
Pipe diameter, D = 0.1 m; flow velocity, U = 0.35 m s−1; flow depth, h = 0.4 m; sediment size, d 50 = 0.6 mm; kinematic viscosity of water, υ = 10−6 m2 s−1; and mass density of water, ρ = 103 kg m−3
Use van Rijn’s empirical formula for the determination of threshold bed shear stress (see Table 4.1):
Particle parameter, D * = d 50(Δg/υ 2)1/3 = 0.6 × 10−3[1.65 × 9.81/(10−6)2]1/3 = 15.18
Threshold Shields parameter, \( {\Theta}_{\text{c}} \left( { 10 \, < D_{*} \le { 2}0} \right) = 0.0 4D_{*}^{ - 0.1} = 0.0 4\times 1 5. 1 8^{{-0. 1}} = 0.0 3 \)
Threshold bed shear stress, τ 0c = ΘcΔρgd 50 = 0.03 × 1.65 × 103 × 9.81 × 0.6 × 10−3 = 0.291 Pa
Estimation of scour depth by Chiew’s method
Discharge per unit width, q = Uh = 0.35 × 0.4 = 0.14 m2 s−1. Then, q g is
Chiew proposed to determine friction factor λ D from the Moody diagram. It can however also be determined from the Colebrook–White equation (Eq. 3.55). Remembering that the Colebrook–White equation is an implicit equation, it is therefore preferred here to use Haaland’s (1983) explicit equation that gives an approximate solution for the Colebrook–White equation and can be used as a substitute. The original Haaland’s equation is given for a pipe flow case having an average flow velocity U with an internal pipe diameter D i as
In this case, for the pressurized flow beneath the pipeline, the scour depth d s and the average gap velocity U g are analogous to D i and U, respectively. The roughness height k s can be assumed as d 50 (= 0.6 mm)
For the first trial, assume d s = 0.12 m and then calculate the average gap velocity U g = q g/d s = 0.0367/0.12 = 0.306 m s−1 and the Reynolds number Re = U g d s/υ = 0.306 × 0.12/10−6 = 36,720. The friction factor is determined from Haaland’s equation as
Thus, the bed shear stress in the scour hole beneath the pipeline is
Following a trial-and-error method, the value of d s that satisfies the condition τ 0 = τ 0c (= 0.291 Pa) is 0.136 m. Therefore, the equilibrium scour depth d s is 0.136 m.
Example 10.9
Calculate the maximum scour depth at a rectangular pier for the following data:
-
Pier width, b = 2 m
-
Pier length, L = 8 m
-
Approaching flow depth, h = 8 m
-
Discharge per unit width, q = 12 m2 s−1
-
Flow skewness, α = 15°
-
Median size of sediment, d 50 = 0.9 mm (uniform sediment)
Use (1) Melville and Coleman’s method (2) HEC18 method and (3) Sheppard et al.’s method.
Also, determine the size of riprap stone for the scour countermeasure at the pier. Assume the riprap to be placed at the original bed level.
Solution
Approaching flow velocity, U = q/h = 12/8 = 1.5 m s−1
(1) Calculation of scour depth by Melville and Coleman’s method
The threshold shear velocity and threshold approaching flow velocity are as follows:
For uniform sediment, U a = U cr
Computation of K-factors is as follows:
-
1.
For b/h = 2/8 = 0.25 < 0.7, K h = 2.4b = 2.4 × 2 = 4.8 m ⇐ Eq. (10.65)
-
2.
For \( \frac{{U - (U_{\text{a}} - U_{\text{cr}} )}}{{U_{\text{cr}} }} = \frac{1.5}{0.593} = 2. 5 3> 1 {\hbox{,}}\;K_{\text{I}} = 1\)⇐ Eq. (10.66)
-
3.
For b/d 50 = 2/(0.9 × 10−3) = 2222.2 > 25, K d = 1 ⇐ Eq. (10.68)
-
4.
For a rectangular pier (square nosed), K s = 1.1 (Table 10.2)
-
5.
For b p = Lsinα + bcosα = 8 × sin15° + 2 × cos15° = 4 m, \( K_{\alpha } = \left( {{{b_{\text{p}} }}/{b}} \right)^{0.65} = \left( {4}/{2} \right)^{0.65} = 1. 5 6 9\Leftarrow {\text{Eq}}.\;\left( { 10. 70} \right) \)
-
6.
For an equilibrium scour (t = t e), K t = 1 ⇐ Eq. (10.71)
Then, the scour depth is
-
(2)
Calculation of scour depth by HEC18 method
Computation of K-factors is as follows:
-
1.
For a rectangular pier, K s = 1.1 (Table 10.2)
-
2.
For L/b = 8/2 = 4 and α = 15°, K α = 1.5 (Table 10.4)
-
3.
For Fr = 1.5/(9.81 × 8)0.5 = 0.169 and U (= 1.5 m s−1) > U cr (= 0.593 m s−1), the possible bedforms are small dunes (assumed). Thus, K bed = 1.1 (Table 10.3)
-
4.
For d 50 = 0.9 mm < 2 mm, K a = 1 ⇐ Eq. (10.75a)
Then, the scour depth is
-
(3)
Calculation of scour depth by Sheppard et al.’s method
The threshold shear velocity and threshold approaching flow velocity are calculated as follows:
The effective pier diameter is calculated as follows:
The functions are calculated as follows:
Then, the scour depth is calculated as follows:
For 5U cr < 0.6(gh)0.5, U peak = 0.6(gh)0.5 = 0.6(9.81 × 8)0.5 = 5.315 m s−1. ∴ U cr ≤ U ≤ U peak
Calculation of riprap stone size
By HEC-23 formula:
Note: For a rectangular pier, K = 1.7
By Lauchlan’s equation:
Placement depth, z r = 0
Note: f SF = 1.1 is considered in the above calculation.
Example 10.10
Determine scour depth in the end of the second day for q = 4 m2 s−1 and other data same as in Example 10.9.
Solution
Approaching flow velocity, U = q/h = 4/8 = 0.5 m s−1 < U cr (= 0.519 m s−1)
For U/U cr = 0.5/0.519 = 0.963 < 1 (clear-water scour), K I = 0.963 ⇐ Eq. (10.66)
For h/b = 8/2 = 4 < 6 and U/U cr = 0.963 > 0.4, the time t e to reach equilibrium is
For t = 2 days, K t is
Then, the scour depth is
Example 10.11
Calculate the maximum scour depth at a circular pier for the following data:
Pier diameter, b = 2.5 m
Approaching flow depth, h = 3.4 m
Discharge per unit width, q = 11.9 m2 s−1
Sediment size, d 50 = 20 mm, d 95 = 85 mm and d max = 99 mm
Use (1) Melville and Coleman’s method and (2) HEC18 method.
Also, determine the size of riprap stone for the scour countermeasure at the pier. Assume the riprap to be placed 0.5 m below the original bed level.
Solution
Approaching flow velocity, U = q/h = 11.9/3.4 = 3.5 m s−1
(1) Calculation of scour depth by Melville and Coleman’s method
The threshold shear velocity u *c and threshold approaching flow velocities, U cr, U cra, and U a are calculated as follows:
Computation of K-factors is as follows:
-
1.
For b/h = 2.5/3.4 = 0.735 (0.7 ≤ b/h ≤ 5), K h = 2(hb)0.5 = 2(3.4 × 2.5)0.5 = 5.831 m ⇐ Eq. (10.65)
-
2.
For \( \frac{{U - (U_{\text{a}} - U_{\text{cr}} )}}{{U_{\text{cr}} }} = \frac{3.5 - (2.634 - 2.325)}{2.325} = 1. 3 7 2> 1{\rm ,}\;K_{\text{I}} = 1\Leftarrow {\text{Eq}}.\;\left( { 10. 6 6} \right) \)
-
3.
For b/d 50a = 2/(55 × 10−3) = 36.36 > 25, K d = 1 ⇐ Eq. (10.68)
-
4.
For a circular pier, K s = 1 (Table 10.2)
-
5.
For a circular pier, K α = 1 ⇐ Eq. (10.70)
-
6.
For an equilibrium scour (t = t e), K t = 1 ⇐ Eq. (10.71)
Then, the scour depth is
-
(2)
Calculation of scour depth by HEC18 method
Computation of K-factors is as follows:
-
1.
For a circular pier, K s = 1 (Table 10.2)
-
2.
For a circular pier, K α = 1
-
3.
For Fr = 3.5/(9.81 × 3.4)0.5 = 0.6 and U (= 3.5 m s−1) > U cr (= 2.06 m s−1), the possible bedforms are large dunes (assumed). Thus, K bed = 1.3 (Table 10.3)
-
4.
For d 50 = 20 mm > 2 mm and d 95 = 85 mm > 20 mm, \( K_{\text{a}} = 0. 4U_{\text r}^{0.15} = 0. 4\times 10. 2 8 9^{0. 1 5} = 0. 5 6 7\Leftarrow {\text{Eq}}.\;\left( { 10. 7 5 {\text{b}}} \right) \)
Then, the scour depth is
Calculation of riprap stone size
By HEC-23 formula:
Note: For a circular pier, K = 1.5
By Lauchlan’s equation:
Placement depth, z r = 0.5 m
Note: f SF = 1.5 is considered in the above calculation.
Example 10.12
Calculate the maximum scour depth at a spill-through abutment for the following data:
Abutment length = 10 m
Abutment length spanning to flood channel = 95 % of abutment length
Abutment slope, S a = 0.5 (horizontal) : 1 (vertical)
Abutment alignment, α = 80°
Flow depth in main channel, h m = 8 m
Flow depth in flood channel, h* = 2 m
Discharge per unit width in main channel, q = 20 m2 s−1
Discharge per unit width in flood channel, q* = 5 m2 s−1
Manning coefficient in main channel, n = 0.022 SI units
Manning coefficient in flood channel, n* = 0.03 SI units
Sediment size, d 50 = 20 mm, d 95 = 85 mm, and d max = 99 mm
Assume the flow depth reduction to be 2 % in the contracted portion.
Use (1) Melville and Coleman’s method and (2) HEC18 method.
Also, determine the size of riprap stone for scour countermeasure at the abutment.
Solution
Calculation is based on the flow in the flood channel, where 95 % of abutment length exists.
Approaching flow velocity in flood channel, U = q*/h* = 5/2 = 2.5 m s−1
Projected abutment length, l = 10 sinα = 10 sin80° = 9.848 m
Projected abutment length in flood channel, l* = 10 × 0.95 sinα = 10 × 0.95 sin80° = 9.356 m
The threshold approaching flow velocities, U cr, U cra, and U a, are as follows:
u *c = 0.136 m s−1; u *ca = 0.226 m s−1; d 50a = 55 mm (see Example 10.11)
Calculation of scour depth by Melville and Coleman’s method
Computation of K-factors is as follows:
-
1.
For l/h* = 9.848/2 = 4.924 (1 ≤ l/h* ≤ 25), K h = 2(h*l)0.5 = 2(2 × 9.848)0.5 = 8.876 m ⇐ Eq. (10.80)
Note: As the flow depth in flood channel is applicable, h is replaced by h* in Eq. (10.80)
-
2.
For \( \frac{{U - (U_{\text{a}} - U_{\text{cr}} )}}{{U_{\text{cr}} }} = \frac{2.5 - (2.394 - 2.145)}{2.145} = 1.0 4 9> 1{\rm ,}\;K_{\text{I}} = 1\Leftarrow {\text{Eq}}.\left( { 10. 6 6} \right) \)
-
3.
For l/d 50a = 9.848/(55 × 10−3) = 179.05 > 25, K d = 1 ⇐ Eq. (10.81)
-
4.
For a spill-through abutment with slope S a = 0.5:1, K s = 0.6 (Table 10.6)
-
5.
For α = 80°, \( K_{\alpha }^{*} = 0. 9 9 3 \) is obtained from Table 10.7. Then, for l/h* = 4.924 > 3, K α = 0.993 ⇐ Eq. (10.83)
-
6.
K G is calculated, considering h = h m, that is, the flow depth in main channel, as
-
7.
For an equilibrium scour (t = t e), K t = 1 ⇐ Eq. (10.85)
Then, the scour depth is
Calculation of riprap stone size
By Austroads formula:
Fr = U/(gh*)0.5 = 2.5/(9.81 × 2)0.5 = 0.564
By Lauchlan’s equation:
Shape factor, K s = 0.89
Flow depth in contracted portion, h 2 = 8 − 0.02 × 8 = 7.84 m
Flow velocity in contracted portion, U 2 = q/h 2 = 20/7.84 = 2.551 m s−1
Froude number in contracted portion, Fr 2 = U 2/(gh 2)0.5 = 2.551/(9.81 × 7.84)0.5 = 0.291 < 0.8
Example 10.13
Determine scour depth in the end of the second day for the discharge per unit width in flood channel q = 4 m2 s−1 and other data same as in Example 10.12.
Solution
Approaching flow velocity in flood channel, U = q*/h* = 4/2 = 2 m s−1
For \( \frac{{U - (U_{\text{a}} - U_{\text{cr}} )}}{{U_{\text{cr}} }} = \frac{2 - (2.394 - 2.145)}{2.145} = 0. 8 1 6< 1{\rm ,}\;K_{\text{I}} = \;0. 8 1 6\Leftarrow {\text{Eq}}.\;\left( { 10. 6 6} \right) \)
For l/h* = 9.848/2 = 4.924 > 1.2, the time t e to reach equilibrium is
For t = 2 days, K t is
Then, the scour depth is
Example 10.14
Calculate the maximum scour depth at a vertical-wall abutment for the following data:
Abutment length, l = 12 m
Abutment alignment, α = 90°
Flow depth, h = 8 m
Discharge per unit width, q = 24 m2 s−1
Median size of sediment, d 50 = 0.9 mm (uniform sediment)
Solution
Approaching flow velocity, U = q/h = 24/8 = 3 m s−1
Threshold approaching flow velocity, U cr = 0.593 m s−1 (see Example 10.9)
It is a live-bed flow condition (U > U cr). Hence, Froehlich’s (HEC 18) method is applicable
Calculation of scour depth by HEC18 method:
Computation of K-factors is as follows:
Flow Froude number, Fr = U/(gh)0.5 = 3/(9.81 × 8)0.5 = 0.339
Then, the scour depth is
Notes
- 1.
Dey and Raikar (2007b) considered section 0 at the upstream of the drop where the critical depth h c occurs and section 0 at the entry of jet into the tailwater (Fig. 10.4b). The continuity equation applied between sections 1 and 0 is
$$ U_{\text{c}} h_{\text{c}} = U_{0} l_{0} $$where U c is the critical velocity of the flow upstream of the drop. According to Bakhmeteff (1932), the jet velocity U 0 is given by
$$ U_{0} = C_{0} [2g(h_{0} + 1.5h_{\text{c}} )]^{0.5} $$where C 0 is the velocity coefficient and h 0 is the height of drop above the tailwater level.
Using the value of end-depth-ratio (= h e/h c, where h e is the end depth) for rectangular channels equaling 0.715 as given by Rouse (1936), the above equation becomes
$$ U_{0} = C_{0} [2g(h_{0} + 2.1h_{\text{e}} )]^{0.5} $$Inserting into the continuity equation, the expression for jet thickness l 0 can be written as
$$ l_{0} = \frac{{1.17h_{\text{e}}^{1.5} }}{{C_{0} (h_{0} + 2.1h_{\text{e}} )^{0.5} }} $$Using the experimental data, the value of C 0 was estimated as 0.672.
- 2.
Shape factor is the ratio of the equilibrium scour depth for a given non-circular pipe to that for a circular shaped pipe having a same diameter to the vertical cross-sectional length of the non-circular pipe.
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Dey, S. (2014). Scour. In: Fluvial Hydrodynamics. GeoPlanet: Earth and Planetary Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19062-9_10
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