2.1. General aspects

When an open channel flow impinges upon a cylindrical rigid structure, turbulence is generated in the form of a horseshoe vortex, a wake vortex and a surface roller (figure 1). The horseshoe vortex is the main factor for sediment entrainment since it causes a significant increase in the shear stress around the base of the structure. The wake vortex contributes to lifting the entrained sediment and displacing it outside the scour hole. The surface roller (i.e. a recirculating mass of turbulent water) develops near the free surface due to the formation of a bow wave. The influence of the surface roller on scouring is significant only at shallow flow conditions, namely when the flow depth is smaller than or comparable to the pier width (Melville & Coleman Reference Melville and Coleman2000; Ettema et al. Reference Ettema, Constantinescu and Melville2011).

Figure 1. Sketch of eddies and scour geometry induced by an open channel shear flow impinging on a cylindrical element inserted within an erodible bed.

Local scouring can occur in so-called clear-water or live-bed conditions depending on whether sediment transport occurs upstream of the cylinder. In both cases local scour is triggered by the horseshoe vortex at the base of the cylinder, provided that local shear stresses exceed the critical shear stress of the sediment (Ettema et al. Reference Ettema, Constantinescu and Melville2011). As the scour hole deepens, the erosive strength of the horseshoe vortex decreases until an equilibrium condition is reached. In the clear-water case such an equilibrium is reached when the shear stress at the base of the scour-hole approaches the critical shear stress associated with the sediment lying on the river bed. In live-bed conditions, instead, equilibrium conditions are dictated by a balance between ingoing and outgoing sediment fluxes (Melville Reference Melville1984). In both cases the vertical distance between the undisturbed bed level and the deepest point within the scour hole is commonly defined as the equilibrium scour depth (i.e.  $y_{s}$ ).

Figure 2 illustrates the typical evolution in time of the scour depth observed in live-bed and clear-water laboratory experiments. In live-bed conditions, equilibrium is reached very rapidly and $y_{s}$ oscillates due to the passage of bed-forms. In clear-water conditions the concept of equilibrium is not clear and is still a matter of controversy (Lança et al. Reference Lança, Fael, Maia, Pego and Cardoso2013). Some authors support the concept that equilibrium is reached in an arbitrarily defined finite time (Melville & Chiew Reference Melville and Chiew1999; Kothyari, Hager & Oliveto Reference Kothyari, Hager and Oliveto2007), whereas others argue that equilibrium can be reached only asymptotically and suggest that $y_{s}$ should be estimated by extrapolation of scour curves (like those reported in figure 2) at a time equal to infinity (Sheppard, Odeh & Glasser Reference Sheppard, Odeh and Glasser2004; Lança et al. Reference Lança, Fael, Maia, Pego and Cardoso2013). Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013) report that for identical experimental conditions $y_{s}$ can vary by 10–20 % depending on how it is defined, so care must be taken when comparing results from different experiments. We come back to this issue in § 3.

Figure 2. Conceptual description of the scour evolution in time for clear-water (dashed line) and live-bed (solid line) conditions.

2.2. Clear-water conditions

In clear-water conditions, provided that the ratio between the depth-averaged velocity in the undisturbed channel (i.e. $V_{1}$ ) and the sediment critical velocity is high enough (say $0.5\leqslant V_{1}/V_{c}\leqslant 1$ , where $V_{c}$ is the sediment critical velocity, which depends on both sediment diameter and flow depth), the horseshoe vortex erodes the sediment at the base of the structure until the shear stress generated within the scour hole approaches the critical shear stress value (Ettema et al. Reference Ettema, Constantinescu and Melville2011) and equilibrium conditions are reached. The point of maximum scour depth is normally located at the base of the scour hole in close proximity to the cylinder, either at its upstream face or at its flanks (Ettema et al. Reference Ettema, Constantinescu and Melville2011). After careful examination of the results from experiments and numerical simulations presented in the literature, we argue that the local slope of the sediment bed near the point of maximum depth is consistently zero (Unger & Hager Reference Unger and Hager2007; Kirkil, Constantinescu & Ettema Reference Kirkil, Constantinescu and Ettema2008; Ettema et al. Reference Ettema, Constantinescu and Melville2011). Therefore, at this location, the critical shear stress of the sediment (i.e. ${\it\tau}_{c}$ ) is presumably independent of local-slope (i.e. gravitational) effects and, assuming that the flow within the scour hole is in the fully rough regime (i.e. turbulence around the pier is fully developed and momentum transfer at the sediment–water interface is weakly influenced by viscosity), equilibrium conditions can be mathematically expressed as (Shields Reference Shields1936)

where ${\it\tau}$ is the shear stress at the point where the maximum scour depth occurs, ${\it\rho}_{s}$ is the density of the sediment material, ${\it\rho}$ is the density of the fluid, $g$ is the gravity acceleration, $d$ is the characteristic sediment diameter and the ${\sim}$ symbol means ‘scales as’.

We now aim to derive a simple analytical formula that links the scour depth of equilibrium with easily measurable properties of the impinging flow, the sediment bed and the geometry of the cylindrical structure. The following derivation is inspired by the work of Gioia & Bombardelli (Reference Gioia and Bombardelli2005) and Bombardelli & Gioia (Reference Bombardelli and Gioia2006), who have used an approach based on the phenomenology of fully developed turbulence (see e.g. Frisch Reference Frisch1995) to investigate local scouring induced by turbulent jets.

Following Frisch (Reference Frisch1995), the phenomenology of fully developed turbulence can be considered as a shorthand system that can be used to recover Kolmogorov’s scaling laws (Kolmogorov Reference Kolmogorov1991), which were originally derived in a much more systematic and (perhaps) rigorous way. In the present paper we make use of Kolmogorov’s theory of turbulence to derive an expression for ${\it\tau}$ that results from the interaction between large- and small-scale eddies impinging the scour-hole surface.

We start by recalling two important paradigms of turbulence phenomenology. (i) For fully developed turbulent flows, the turbulent kinetic energy (TKE) per unit mass is injected into the flow at scales commensurate with the largest eddies and is independent of viscosity. (ii) TKE, introduced at a rate ${\it\epsilon}$ , cascades from large to small scales at the same rate until eddies of sufficiently small scale dissipate it into internal energy still at the same rate ${\it\epsilon}$ . Following Kolmogorov’s theory (Kolmogorov Reference Kolmogorov1991), the length scale at which the energy cascade begins to be influenced by viscosity is ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ , where ${\it\eta}$ is the Kolmogorov length scale. Since TKE production occurs at large scales and is independent of viscosity, dimensional arguments suggest that ${\it\epsilon}\sim V^{3}/S$ , where $V$ and $S$ are the characteristic velocity and length scale of large eddies. At scales $l$ that are much smaller than $S$ but also much larger than ${\it\eta}$ (i.e. scales contained within the so-called inertial range), the energy cascade occurs inviscidly and ${\it\epsilon}\sim V^{3}/S\sim u_{l}^{3}/l$ , where $u_{l}$ is the characteristic velocity of eddies of size $l$ . This implies that

which is a well-known result of Kolmogorov’s theory of turbulence (Frisch Reference Frisch1995).

We can now go back to the turbulent flow generated within the scour hole forming at the base of a cylindrical structure. Under fully developed turbulence conditions and neglecting viscous components, the shear stress ${\it\tau}$ acting on the scour surface formed by sediment grains of diameter $d$ is the Reynolds stress ${\it\tau}={\it\rho}\overline{u^{\prime }w^{\prime }}$ , where $u^{\prime }$ and $w^{\prime }$ are defined as the velocity fluctuations parallel and normal to the mean flow direction, respectively, and the over-bar identifies turbulence-averaging (figure 3).

Figure 3. Schematic representation of the interaction between large-scale eddies and eddies scaling with the sediment diameter $d$ . $V$ is the characteristic velocity of large-scale eddies (i.e. eddies scaling with $S$ ) and $u_{d}$ is the characteristic velocity of near-bed eddies (i.e. eddies scaling with the sediment diameter $d$ );  $u^{\prime }$ and $w^{\prime }$ are velocity fluctuations along and normal to the main flow direction respectively; ${\it\eta}$ is the Kolmogorov length scale that quantifies the thickness of the viscous sublayer (Gioia & Chakraborty Reference Gioia and Chakraborty2006), as discussed in § 3.

Provided that $d$ belongs to the inertial range of scales (i.e. ${\it\eta}\ll d\ll S$ ), Gioia & Bombardelli (Reference Gioia and Bombardelli2005) argue that eddies of size much larger than $d$ can hardly contribute to $w^{\prime }$ because they are too large to exchange momentum in the fluid space between two successive roughness elements. In contrast, eddies of size smaller than $d$ do fit within this space but are associated with lower characteristic velocities (i.e. recall Kolmogorov’s scaling $u_{l}\sim V(l/S)^{1/3}$ ). This implies that $w^{\prime }$ is dominated by eddies of size $d$ . Conversely, $u^{\prime }$ is influenced by the whole spectrum of turbulence length scales and therefore $u^{\prime }$ is dominated by $V$ . Therefore, the shear stress scales as ${\it\tau}\sim {\it\rho}u_{d}V$ , where $u_{d}\sim V(d/S)^{1/3}$ , and hence

Strictly speaking, Kolmogorov’s scaling (as applied in deriving the expression above) is valid if small-scale turbulence is homogeneous and isotropic. Turbulent flows within scour holes and near the sediment–water interface may not display these properties because of the significant strain rates of the mean flow imposed by the presence of the scour hole and the cylinder themselves. Nonetheless, the literature suggests that Kolmogorov’s predictions still hold for non-homogeneous and anisotropic flows (Knight & Sirovich Reference Knight and Sirovich1990; Moser Reference Moser1993). Moreover, Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994) and Saddoughi (Reference Saddoughi1997) show that for wall-bounded flows the energy spectra display Kolmogorov scaling across a range of wavenumbers at which local isotropy is not strictly valid, and this was observed in both equilibrium (i.e. canonical turbulent boundary layers) and non-equilibrium flows (i.e. flows characterized by complex mean strain rates as in the case of flows around a cylinder). Finally, Gioia and co-workers show that applying Kolmogorov’s scaling to the description of turbulent flows in close proximity to rough and smooth boundaries (these include near-wall flows in pipes and channels where turbulence is neither homogeneous nor isotropic) allows the recovery of many important empirical relations pertaining to classical hydraulics (Gioia & Bombardelli Reference Gioia and Bombardelli2002; Gioia & Chakraborty Reference Gioia and Chakraborty2006). It is therefore suggested that the validity of the phenomenological theory of turbulence (in the sense of Kolmogorov) to describe small-scale turbulence within a scour hole is, at least, a plausible hypothesis.

It is now assumed that the characteristic length scale of the energetic eddies forming within the scour hole (i.e. presumably the characteristic length scale of the horseshoe vortex) approximates the depth of the scour hole itself. This means that at equilibrium conditions it is $S\sim y_{s}$ , where $y_{s}$ is the scour depth of equilibrium. This is a reasonable assumption because at equilibrium conditions, the horseshoe vortex is notoriously fully buried within the scour hole, as reported by Unger & Hager (Reference Unger and Hager2007) and Kirkil et al. (Reference Kirkil, Constantinescu and Ettema2008).

Computing ${\it\tau}$ requires us to find a scaling formula for $V$ , which is derived following energetic principles. We recall that ${\it\epsilon}$ scales as ${\it\epsilon}\sim V^{3}/S$ . However, ${\it\epsilon}$ can also be estimated as the power associated with large-scale eddies (i.e. $P$ ) divided by the mass of the fluid contained within their characteristic volume, i.e. ${\it\epsilon}=P/M$ . $P$ can be estimated as the work of a drag force $F$ acting on the cylinder against the mean flow and, hence, as $P=FV_{1}$ , where $V_{1}$ can be taken as the depth-averaged velocity of the approaching flow. The drag force can thus be computed as $0.5{\it\rho}C_{d}aSV_{1}^{2}$ , where $C_{d}$ is a drag coefficient, $a$ is the cylinder diameter, and $aS$ is the frontal area of the cylinder exposed to scouring. The power of the localized turbulent eddies is estimated from the drag force acting on the exposed portion of the cylinder because, presumably, wake eddies forming above it do not contribute to the scour process.

From dimensional considerations, the mass of the characteristic large-scale eddy can be computed as $M\sim {\it\rho}S^{3}$ . This implies that

and hence

Combining (2.3) and (2.5) leads to

When the scour process reaches equilibrium the sediment stops moving, and the shear stress approaches the value of the critical shear stress (i.e. incipient motion conditions, ${\it\tau}\approx {\it\tau}_{c}$ ), and hence, after some algebra,

or, alternatively,

Equation (2.8) shows that the scour depth of equilibrium, normalized with the kinetic head of the undisturbed approach flow, depends on the specific gravity of the sediment (i.e.  ${\it\rho}/({\it\rho}_{s}-{\it\rho})$ ), a drag coefficient (i.e.  $C_{d}$ ), and the so-called relative coarseness (i.e.  $a/d$ ). According to the literature (Ranga Raju et al. Reference Ranga Raju, Rana, Asawa and Pillai1983; Qi, Eames & Johnson Reference Qi, Eames and Johnson2014), the drag coefficient of cylinders impinged by open-channel flows depends on the cylinder shape, the blockage ratio (i.e. $a/B$ , where $B$ is the channel width), the ratio between flow depth and cylinder diameter (i.e. $y_{1}/a$ ), the Froude number of the impinging flow (i.e. $Fr=V_{1}/\sqrt{gy_{1}}$ ), and the cylinder Reynolds number (i.e. $Re=V_{1}a/{\it\nu}$ , where ${\it\nu}$ is the kinematic fluid viscosity). The dependence of $C_{d}$ on $Re$ is probably weak because of the turbulent nature of most open-channel flows, both in the laboratory and in the field.

2.3. Live-bed conditions

For the clear-water case, the equilibrium condition (i.e. the incipient motion condition) ${\it\tau}\approx {\it\tau}_{c}$ was used to derive a predictive formula for the maximum scour depth. For the live-bed case the equilibrium condition is different, as it involves a balance between the time-averaged flux of sediment transported within the scour hole, i.e. $Q_{in}$ , and the time-averaged flux of sediment removed, i.e. $Q_{out}$ (Melville Reference Melville1984) (averages must be taken over time scales much larger than those associated with the passage of bed forms). Therefore, the equilibrium condition for live-bed scour is $Q_{in}=Q_{out}$ . Unfortunately, this condition cannot be further developed to derive a formula for $y_{s}$ because of the difficulties in theoretically predicting sediment fluxes that occur within the scour hole (i.e. $Q_{out}$ ), which is characterized by a complex geometry and flow. However, the following arguments can be used to find a solution to the problem.

We start by pointing out that most of $Q_{in}$ must be in the form of bed-load because most of the sediment fluxes entering within the scour hole must occur next to the bed. Furthermore, most of (if not all) the laboratory experiments on live-bed scour that are available from the literature were carried out with bed-load only and hence we restrict our analysis to this transport regime. From the theory of sediment transport, the dimensionless bed-load sediment discharge per unit channel width (i.e. $q_{s}^{\ast }$ ) is commonly estimated through power laws of the type

where $q_{s}^{\ast }=q_{s}/\{d\sqrt{dg[({\it\rho}_{s}-{\it\rho})/{\it\rho}]}\}$ , $q_{s}$ is the dimensional sediment volumetric discharge per unit channel width, ${\it\tau}^{\ast }$ is the so-called Shields parameter defined as the ratio between the shear stress in the undisturbed bed and the critical shear stress of sediment ${\it\tau}_{c}$ ; ${\it\tau}_{c}^{\ast }$ , ${\it\alpha}$ and $n$ are constants (see e.g. Yang Reference Yang1996). Since shear stresses and depth-averaged velocities in the undisturbed channel can be related through a friction factor (i.e. ${\it\tau}=({\it\rho}V_{1}^{2}f)/8$ , where $f$ is the Darcy–Weisbach friction factor), $q_{s}^{\ast }$ can also be estimated as a function of $(V_{1}/V_{c})^{2}$ , where $V_{c}$ is the critical velocity for the sediment and $V_{1}/V_{c}$ is commonly referred to as the flow intensity parameter (Yang Reference Yang1996).

The hypotheses and the arguments underpinning the derivation of the shear stress formula for the clear-water case (see (2.6)) are also applicable to live-bed conditions. It is now easy to show that, at equilibrium, the scour depth function $S_{e}$ , defined as

represents the ratio between the critical shear stress (i.e. ${\it\tau}_{c}\sim ({\it\rho}_{s}-{\it\rho})gd$ ) and the shear stress acting at the point of maximum scour as obtained from (2.6) using $S=y_{s}$ . Equation (2.10) is essentially the inverse of a Shields parameter. In live-bed conditions, $S_{e}$ must depend on the sediment discharge in the undisturbed flow. Since $S_{e}$ is effectively the inverse of a local Shields parameter, its value at equilibrium should depend on $q_{s}$ rather than on $Q_{in}$ , which is an integral quantity that includes contributions of sediment fluxes from flow regions away from the point of maximum scour that do not contribute to the local sediment mass balance. Since $q_{s}$ is essentially dictated by $V_{1}/V_{c}$ (or ${\it\tau}^{\ast }$ : see (2.9)), we assume that in live-bed conditions the dimensionless scour depth is related to $V_{1}/V_{c}$ by the equation

where the functional relation ${\it\phi}$ must be found experimentally. We have chosen to use $V_{1}/V_{c}$ instead of ${\it\tau}^{\ast }$ because, for validation purposes, $V_{1}/V_{c}$ is readily available from the literature reporting live-bed scour experiments, unlike ${\it\tau}^{\ast }$ .

3.1. Clear-water conditions

The validation of (2.8) is carried out by using experimental data for the case of cylindrical structures with circular cross-section, uniform sediment beds and steady, turbulent shear flows. Data of this kind are largely available from the literature.

The linear dependence of $(y_{s}g)/V_{1}^{2}$ on ${\it\rho}/({\it\rho}_{s}-{\it\rho})$ cannot be tested because this parameter is practically constant in most of the available experiments. Similar difficulties apply to testing the scaling derived for $C_{d}$ because, in general, $C_{d}$ values are contained within a range that is too small to test the occurrence of a power law with confidence. Instead, the proposed scaling for the relative roughness $a/d$ can be extensively validated from experimental data. To this end it is important to further clarify under which conditions (2.8) is applicable. Equation (2.8) was derived under the assumption that the sediment diameter is within the range of length scales pertaining to the inertial range, i.e. ${\it\eta}\ll d\ll S$ . At equilibrium, this condition becomes ${\it\eta}\ll d\ll y_{s}$ . In order to find predictive conditions at which (2.8) can be applied, it is necessary to replace $y_{s}$ , which is not known a priori, with a known parameter that is of the same order of magnitude as $y_{s}$ . It is well known from the literature that $y_{s}$ scales well with $a$ , more precisely $a<y_{s}<3a$ (see e.g. Lee & Sturm Reference Lee and Sturm2009). Assuming $y_{s}\approx a$ , the TKE production can be estimated from (2.4) as ${\it\epsilon}\sim (C_{d}V_{1}^{3})/a$ , and hence the order of magnitude of the bulk Kolmogorov length scale can be estimated as

and therefore, (2.8) is valid if

or, analogously, if

Owing to the small exponent of the drag coefficient, it is fair to assume that $C_{d}^{1/4}\approx 1$ , and therefore the range of validity of the proposed theory can be expressed as

Since it was assumed that $S\sim y_{s}\approx a$ , $a/d$ can now be physically interpreted as the ratio between characteristic scales associated with energy containing eddies (i.e. $a$ ) and roughness elements (i.e. $d$ ) within the scour hole.

The validity of (2.8) is now tested against the laboratory data provided by Ettema (Reference Ettema1980), Sheppard et al. (Reference Sheppard, Odeh and Glasser2004), Ettema, Kirkil & Muste (Reference Ettema, Kirkil and Muste2006) and Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013). This data set is also utilized to further constrain the limits of validity of the proposed theory as expressed by (3.4).

Table 1 provides a summary of the relevant experimental conditions associated with each referenced source. The definition of equilibrium scour depth is, in general, arbitrary and not consistent over these four studies: in the experiments by Ettema (Reference Ettema1980) and Ettema et al. (Reference Ettema, Kirkil and Muste2006) equilibrium conditions were considered to be reached when no appreciable change of the maximum depth was observed over a minimum period of four hours. Instead Sheppard et al. (Reference Sheppard, Odeh and Glasser2004) and Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013) applied the concept of equilibrium as an asymptotic condition, as discussed in § 2.1. In order to avoid fictitious scatter of data, the validity of the proposed scaling for $a/d$ is tested by plotting $y_{s}g/V_{1}^{2}$ versus $a/d$ for each data set individually (figure 4).

Figure 4. Dimensionless scour depths versus relative coarseness: (a) data from Ettema et al. (Reference Ettema, Kirkil and Muste2006); (b) data from Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013); (c) data from Ettema (Reference Ettema1980); (d) data from Sheppard et al. (Reference Sheppard, Odeh and Glasser2004). For (a–d) white squares and black circles refer to values of $a/d$ that are within and outside the limits imposed by (3.5), respectively. In these panels the solid lines represent a $2/3$ power law that best fits the white squares, whereas dashed lines represent the associated $\pm 36$  % error lines. (e) All the experimental points contained within the limits imposed by (3.5) are plotted together to provide a general overview: vertical triangles, Ettema (Reference Ettema1980); left-pointing triangles, Sheppard et al. (Reference Sheppard, Odeh and Glasser2004); circles, Ettema et al. (Reference Ettema, Kirkil and Muste2006); stars, Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013). In (e) the solid line represents a power law with a $2/3$ exponent.

Before commenting on figure 4 we further discuss the uncertainties associated with the assumptions underpinning the proposed theory in relation to the experimental data reported in table 1. Equation (2.8) was derived under the assumption of fully rough conditions and hence it was possible to assume that, at equilibrium, the critical shear stress scaled as ${\it\tau}_{c}\sim ({\it\rho}_{s}-{\it\rho})gd$ and the shear stress ${\it\tau}$ had only a turbulent component (see (2.3)). Fully rough conditions are typical of flows over gravel beds or coarse sands (Shields Reference Shields1936; Buffington & Montgomery Reference Buffington and Montgomery1997). However, sediment diameters reported in table 1 mostly pertain to sand beds, for which the transitionally rough regime is more likely to occur at equilibrium conditions. In principle, for such a regime, the proposed scaling for both ${\it\tau}_{c}$ and ${\it\tau}$ does not hold, because both shear stresses should also depend upon viscosity. Ignoring viscosity effects, therefore, introduces some uncertainty, which is discussed in the following two points. (i) According to the pioneering work of Shields on sediment entrainment, viscosity effects in transitionally rough flows can account for variations of ${\it\tau}_{c}/({\it\rho}_{s}-{\it\rho})gd$ (i.e. the Shields parameter) contained within a range of $\pm 33$  % (Shields Reference Shields1936; Buffington & Montgomery Reference Buffington and Montgomery1997) around an intermediate value. (ii) Similarly, according to the seminal work of Nikuradse on turbulent flows over granular walls, within the transitionally rough regime and for a given relative roughness, viscosity effects can account for variations of friction factors (and hence of bed shear stress ${\it\tau}$ ) contained within a range of $\pm 10$  % (Yang & Joseph Reference Yang and Joseph2009) around an intermediate value.

Some uncertainty is also introduced by the drag coefficient $C_{d}$ . In order to isolate the effects of $a/d$ on $y_{s}g/V_{1}^{2}$ , we discarded all the experimental data associated with flow conditions that could include significant variations in $C_{d}$ . In particular, all the experiments characterized by $y_{1}/a<1.4$ were discarded, because for these cases the surface roller interacts with the near-wall horseshoe vortex and therefore it is likely to alter significantly the drag coefficient of the cylinder and consequently the equilibrium scour depth (Melville & Coleman Reference Melville and Coleman2000). All the remaining experiments were characterized by flow conditions with blockage ratio and Froude numbers which, according to Ranga Raju et al. (Reference Ranga Raju, Rana, Asawa and Pillai1983), induce variations of bulk drag coefficients $C_{d}$ of $\pm 15$  %. This means that assuming a constant $C_{d}$ in (2.8) introduces a relative error of $\pm 10$  % on $C_{d}^{2/3}$ .

Combining the relative errors of $C_{d}$ , ${\it\tau}_{c}$ and ${\it\tau}$ in (2.8) implies that $y_{s}g/V_{1}^{2}$ can be estimated with a maximum relative error of about $\pm 36$  %, with ${\it\tau}_{c}$ providing the largest contribution.

Figure 4 illustrates $y_{s}g/V_{1}^{2}$ as a function of $a/d$ in log–log plots for all data sets. The figure shows that, for each data set, $a/d$ varies over at least one order of magnitude and hence the proposed power-law scaling for $a/d$ (dashed line in figure 4) can be validated with confidence. Overall, figure 4 shows that the majority of the experimental data pertaining to intermediate values of $a/d$ agree well with the proposed theory.

The experimental data are now used to better constrain the lower and higher bounds of the intermediate range of $a/d$ values identified by (3.4) which, in turn, identifies the limits of validity of the proposed theory. The lower bound can be found from the data by Ettema (Reference Ettema1980) (figure 4 c), which shows that the $2/3$ scaling holds for $a/d>20$ . Below this threshold the theory over-predicts the normalized scour depth. In fact, when $a/d<20$ , there is poor scale separation between roughness elements and large eddies of the flow. Therefore, $d$ becomes comparable to the energy-containing eddies and, with respect to the case of $d$ belonging to the inertial range, the flow resistance offered by the roughness elements is enhanced. In other words, the effective roughness of the scour hole increases and therefore $y_{s}$ decreases. The phenomenon of enhanced effective roughness of rough-walled flows with poor scale separation is well known in hydraulics (see e.g. Chow Reference Chow1988). For example, when the scale separation between flow depth and roughness (i.e. $y_{1}/d$ ) is not large enough, Manning’s coefficients of rough beds underlying turbulent open-channel flows increase with decreasing $y_{1}/d$ (see e.g. Ferguson Reference Ferguson2010). The flow depth $y_{1}$ and the cylinder diameter $a$ quantify the scale of energy-containing eddies in open-channel flows and flows around cylinders, respectively. Therefore, for both flow types the ratios $y_{1}/d$ and $a/d$ have the same physical meaning. Interestingly, and consistent with the results reported herein, Ferguson (Reference Ferguson2010) shows that Manning’s coefficient begins to be influenced by the relative submergence for $y_{1}/d<20$ .

The data from Sheppard et al. (Reference Sheppard, Odeh and Glasser2004) and Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013) help to identify the upper bound in (3.4), which is Reynolds-number-dependent and therefore cannot be visually found from figure 4. The data points not respecting the proposed $2/3$ scaling are associated with experimental conditions for which the sediment diameter was less than five times the Kolmogorov length scale (i.e. for $d/{\it\eta}<5$ ), as estimated with (3.1). In wall turbulence, ${\it\eta}$ is closely related to the viscous length scale of the flow (see figure 3 and Gioia & Chakraborty Reference Gioia and Chakraborty2006) and therefore, if $d/{\it\eta}<5$ , sediment grains are likely to be of a size comparable to the viscous sublayer thickness. This, in turn, means that the shear stress at the water–sediment interface becomes predominantly viscous, so the eddies of size $d$ no longer dominate the turbulent momentum transfer (figure 3) and hence the proposed theory no longer holds. Furthermore, figure 4 shows that all the data points for which $d/{\it\eta}<5$ are associated with values of $y_{s}g/V_{1}^{2}$ that are smaller than those predicted by the $2/3$ power law. This is to be expected because the viscous sublayer shelters the sediment grains from the turbulent fluctuations of the flow above and therefore reduces their erosive power.

The upper bound of (3.4) can therefore be identified from the condition $d/{\it\eta}>5$ , which, in terms of bulk Reynolds number and relative coarseness, corresponds to $a/d<0.2Re^{3/4}$ . It is concluded that (2.8) is valid under the following approximate conditions:

The data of Ettema et al. (Reference Ettema, Kirkil and Muste2006), which are all contained within this range, agree very well with our proposed theory.

Excluding the data points outside the limits imposed by (3.5) leads to a striking agreement between theory and experiments (see figure 4). Furthermore, the $\pm 36$  % error bounds, predicted by the uncertainty analysis presented previously, correspond to the level of scatter appearing in figure 4(a–d).

Figure 5. Comparison between scaling laws pertaining to $y_{s}$ versus $a$ (a) and $y_{s}g/V_{1}^{2}$ versus $a/d$ (b). The dashed lines represent a power law with a $2/3$ exponent. Data from Ettema (Reference Ettema1980).

One might argue that the good agreement between theory and experimental data, as reported above, could be the result of a fortuitous correlation between $y_{s}$ and $a$ . This is because, as already discussed and well reported in the literature, these two parameters are strongly correlated in local scour experiments. To remove this suspicion and further substantiate the validity of the proposed approach, we show that $y_{s}$ and $a$ display a scaling relation but the associated exponent is different from $2/3$ . To this end we first note that the experiments by Ettema et al. (Reference Ettema, Kirkil and Muste2006) and Lança et al. (Reference Lança, Fael, Maia, Pego and Cardoso2013) were carried out with a weak variation in $d$ and $V_{1}$ while the cylinder diameter $a$ was varied extensively (see table 1). This means that plotting $y_{s}g/V_{1}^{2}$ versus $a/d$ taken from these data sets is essentially the same as plotting $y_{s}$ versus $a$ , and therefore they cannot be used to validate our approach. Instead, the data sets by Ettema (Reference Ettema1980) and Sheppard et al. (Reference Sheppard, Odeh and Glasser2004) were obtained by extensively varying $V_{1}$ , $d$ and $a$ . However, only Ettema (Reference Ettema1980) provides enough points within the limits imposed by (3.5) to perform a robust statistical analysis (see figure 4 c,d).

Figure 5 shows the data from Ettema (Reference Ettema1980) plotted as $y_{s}$ versus $a$ (a) and $y_{s}g/V_{1}^{2}$ versus $a/d$ (b) together with a line corresponding to a power law with exponent equal to $2/3$ . For both panels a best-fit analysis was carried out by minimizing least-squares errors over both the $x$ - and $y$ -coordinate. The best fit of $y_{s}$ versus $a$ resulted in exponents equal to 0.94 (minimizing errors over the $y$ -coordinate) and 1.1 (minimizing errors over the $x$ -coordinate), which suggests a linear rather than power-law relation between the two variables. Instead, the best fit of $y_{s}g/V_{1}^{2}$ versus $a/d$ resulted in exponents equal to 0.67 (minimizing errors over the $y$ -coordinate) and 0.81 (minimizing errors over the $x$ -coordinate), which are reasonably close to the theoretically predicted value of $2/3\approx 0.67$ . We therefore conclude that the proposed scaling is not the result of a fortuitous correlation between $y_{s}$ and $a$ .

3.2. Live-bed conditions

In live-bed conditions the proposed theory essentially suggests that if scour depth function $S_{e}$ is plotted against the flow intensity parameter $V_{1}/V_{c}$ , experimental data should collapse around a curve identified by a functional relation ${\it\phi}$ (see (2.11)). In order to substantiate these hypotheses, experimental data were extracted from Chiew (Reference Chiew1984) and Sheppard & Miller (Reference Sheppard and Miller2006) for a total of 167 data points. As in the clear-water case, these experiments were carried out with circular cylinders and uniform quartz sand. Only very few data points with $y_{1}/a<1.4$ were filtered out. Table 2 provides a summary of the relevant experimental data, which include a wide range of hydraulic conditions.

Since in live-bed conditions the maximum scour depths oscillate in time due to the passage of bed-forms, here $y_{s}$ is taken as the time-averaged value of maximum scour depths as reported by the authors of the referenced papers. Contrary to the clear-water case, in live-bed conditions there is no ambiguity about the definition of $y_{s}$ , so the proposed theory can be tested against all data sets at once and not for each data set individually.

Figure 6 shows that the agreement between theory and experiments is striking. The experimental data, with exception of a few points, collapse nicely onto a power-law function of the type

with ${\it\beta}=0.47$ and ${\it\theta}=-1.89$ . Interestingly, the points that do not collapse on (3.6) are associated with $a/d<20$ . This suggests that, although (3.5) was obtained from the analysis of experimental data pertaining to clear-water flows, it seems to be applicable to live-bed flows as well. Furthermore, consistent with the clear-water case, for these experimental points the proposed theory over-predicts scour depths. As discussed earlier, this is an effect associated with an increase in flow resistance due to the poor scale separation between sediment diameter and energy-containing eddies.

Figure 6. $S_{e}$ versus the flow intensity parameter $V_{1}/V_{c}$ . Experimental data are taken from Chiew (Reference Chiew1984) and Sheppard & Miller (Reference Sheppard and Miller2006). White squares and black circles refer to values of $a/d$ that are within and outside the limits imposed by (3.5), respectively. The solid line is the best fit to the white square data (see (3.6)).