2 The impulse threshold

Figure 1.

Sketch of the problem. (a) Three-dimensional rendered image of the setting consisting of four spheres of equal radii  $r$ , where the mobile particle rests on top of three fixed and densely packed spheres, which in turn lie on a horizontal plane; initial and final (dislodged) positions of the particle’s centroid are denoted by points  $A$ and  $B$ , coloured blue and red (online version), respectively. (b) Side view illustrating the trajectory $C$ followed by the particle during dislodgement, and the centroid’s angular displacement $\unicode[STIX]{x1D6FC}$ with respect to the pivot axis  $P$ . (c) Top view.

We are concerned with the conditions leading to full particle dislodgement, defined as the event when a sediment particle originally at rest on the bed surface is transported to a different location on the bed (hereinafter we avoid the alternative terms ‘incipient motion’ or ‘initiation of motion’, which may evoke local particle movement not necessarily leading to a different resting position). We focus on the bed configuration employed in previous works (Diplas et al. Reference Diplas, Dancey, Celik, Valyrakis, Greer and Akar2008; Celik et al. Reference Celik, Diplas, Dancey and Valyrakis2010; Valyrakis et al. Reference Valyrakis, Diplas, Dancey, Greer and Celik2010, Reference Valyrakis, Diplas and Dancey2013; Celik, Diplas & Dancey Reference Celik, Diplas and Dancey2013), where a mobile spherical particle rests on top of three, equal-sized, fixed, well-packed spheres, as depicted in figure 1 (the same diameter of both top and base spheres is assumed unless otherwise stated). A Cartesian frame of reference is adopted, where $x,y$ and $z$ denote streamwise, transverse and vertical coordinates, respectively. Fundamentally, to achieve full dislodgement, the work done on the particle by the net external forces must be sufficient to overcome the elevation threshold resulting from the local micro-topography. This happens when the particle originally at rest in stable equilibrium (point  $A$ ) moves to a higher, unstable position, where an infinitesimally small streamwise force acting at its centre of mass will lead to a new resting position (point  $B$ – a separatrix in the context of oscillators). Therefore, the minimum work required for full dislodgement can be defined for the condition of the sphere reaching point  $B$ with null kinetic energy. It is assumed that the particle is dislodged by rolling from $A$ to  $B$ , which is in agreement with experimental evidence that highlights this entrainment mode as the most common for near-threshold conditions (see e.g. Fenton & Abbott Reference Fenton and Abbott1977; Celik et al. Reference Celik, Diplas, Dancey and Valyrakis2010; Kudrolli et al. Reference Kudrolli, Scheff and Allen2016). Moreover, it may be argued that this trajectory and type of motion (as opposed to sliding) minimises the energy required to get to $B$ from  $A$ , and is thus in line with our objective of finding a fundamental energy threshold for dislodgement. Since the sphere rolls without sliding, static friction forces act at the points of contact between the mobile sphere and the base spheres, which in turn create a resultant torque about the rolling particle’s centre of mass. However, since the mobile sphere arrives at point  $B$ with no angular velocity (we assume null kinetic energy at point  $B$ ), no net work is done by this torque on the sphere, and so we exclude static friction forces from our derivation below. For partly exposed particles like the one under consideration, the net hydrodynamic force may not act exactly at the particle’s centre of mass. However, without much knowledge of the actual line of action of said force (Kudrolli et al. Reference Kudrolli, Scheff and Allen2016), we assume that it does for simplicity, as typically done in similar studies (e.g. Celik et al. Reference Celik, Diplas, Dancey and Valyrakis2010; Valyrakis et al. Reference Valyrakis, Diplas, Dancey, Greer and Celik2010; Kudrolli et al. Reference Kudrolli, Scheff and Allen2016).

Consider the net force acting on the mobile sphere’s centre of mass at any time $t$ during a dislodgement event of duration $T=t_{1}-t_{0}$ : $\boldsymbol{F}(t)=\boldsymbol{F}_{H}(t)+\boldsymbol{w}_{s}+\sum \boldsymbol{N}_{i}(t)$ ; where $\boldsymbol{F}_{H}(t)$ is the net hydrodynamic force (see discussions below);

$\unicode[STIX]{x1D70C}_{s}$

$\unicode[STIX]{x1D70C}$

$\boldsymbol{g}$

$\sum \boldsymbol{N}_{i}(t)$

$i=1,2,\ldots ,p$

$p$

$p$

$p=2$

$t_{0}<t\leqslant t_{1}$

$C=\boldsymbol{s}(t)$

$A$

$B$

$\int _{C}\boldsymbol{F}(t)\boldsymbol{\cdot }\text{d}\boldsymbol{s}=\int _{C}\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\text{d}\boldsymbol{s}+\int _{C}\boldsymbol{w}_{s}\boldsymbol{\cdot }\text{d}\boldsymbol{s}+\sum \int _{C}\boldsymbol{N}_{i}(t)\boldsymbol{\cdot }\text{d}\boldsymbol{s}=0$

$\boldsymbol{N}_{i}\boldsymbol{\cdot }\text{d}\boldsymbol{s}=0$

(2.1) $$\begin{eqnarray}\int _{C}\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\text{d}\boldsymbol{s}=\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{s}}{\text{d}t}\,\text{d}t>w_{s}\unicode[STIX]{x0394}z,\end{eqnarray}$$

where $w_{s}\equiv \Vert \boldsymbol{w}_{s}\Vert$ and $\unicode[STIX]{x0394}z$ is the elevation gained by the particle’s centre of mass (i.e. the vertical distance between points  $A$ and  $B$ ). Since the particle is at rest at  $t_{0}$ , its velocity $\boldsymbol{v}(t)\equiv \text{d}\boldsymbol{s}(t)/\text{d}t$ relates, through Newton’s second law, to its own mass and acceleration, $m$ and $\boldsymbol{a}(t)$ , respectively, and $\boldsymbol{F}(t)$ as follows:

(2.2) $$\begin{eqnarray}\boldsymbol{v}(t)\equiv \int _{t_{0}}^{t}\boldsymbol{a}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}=\frac{1}{m}\int _{t_{0}}^{t}\boldsymbol{F}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

Substitution of (2.2) in (2.1) yields the following dislodgement criterion in terms of time integrals of all relevant forces (i.e. impulses):

(2.3) $$\begin{eqnarray}\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\left[\int _{t_{0}}^{t}\boldsymbol{F}_{H}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}+\boldsymbol{w}_{s}(t-t_{0})+\mathop{\sum }_{i}\int _{t_{0}}^{t}\boldsymbol{N}_{i}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right]\text{d}t>mw_{s}\unicode[STIX]{x0394}z.\end{eqnarray}$$

Equation (2.3) represents an exact condition to be verified if dislodgement is to take place. However, in this form it is of no practical use. The above relation can be simplified under the assumption that the mobile sphere is highly exposed to the flow, such that (i) the angular displacement of the particle’s centroid during dislodgement with respect to the pivot axis $P$ is small (see $\unicode[STIX]{x1D6FC}$ in figure 1 b) and (ii)  $\boldsymbol{F}_{H}$ is dominated by the drag component acting predominantly in the streamwise ( $x$ ) direction, which is supported by experimental evidence for similar bed configurations (Fenton & Abbott Reference Fenton and Abbott1977; Schmeeckle, Nelson & Shreve Reference Schmeeckle, Nelson and Shreve2007; Celik et al. Reference Celik, Diplas, Dancey and Valyrakis2010). Under these assumptions, approximately $\boldsymbol{F}_{H}\bot \boldsymbol{w}_{s}$ and $\boldsymbol{F}_{H}\bot \boldsymbol{N}_{i}$ for all $t$ during the dislodgement event, thus leading to vanishing of the second and third dot products in the left-hand side of (2.3). (See appendix A for a more rigorous treatment of (2.3) and discussion of these assumptions.) Further noting that $\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\boldsymbol{F}_{H}(\unicode[STIX]{x1D70F})$ represents a symmetric function $f(t,\unicode[STIX]{x1D70F})$ , such that $f(t,\unicode[STIX]{x1D70F})=f(\unicode[STIX]{x1D70F},t)$ , we obtain

(2.4) $$\begin{eqnarray}\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\boldsymbol{\cdot }\int _{t_{0}}^{t}\boldsymbol{F}_{H}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\,\text{d}t=\frac{1}{2}\int _{t_{0}}^{t_{1}}\int _{t_{0}}^{t_{1}}f\,\text{d}\unicode[STIX]{x1D70F}\,\text{d}t=\frac{1}{2}\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\,\text{d}t\boldsymbol{\cdot }\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

Combination of (2.4) and the assumption of approximate orthogonality between $\boldsymbol{F}_{H}$ and both $\boldsymbol{w}_{s}$ and $\boldsymbol{N}_{i}$ discussed above permits significant simplification of (2.3) (see appendix A). This simplification represents an approximate estimate of the magnitude of the critical impulse imparted to the particle during  $T$ , $J_{c}$ , that a destabilising hydrodynamic force (capable of doing work on the particle) must overcome in order to achieve full dislodgement; namely

(2.5) $$\begin{eqnarray}\left\Vert \int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\,\text{d}t\right\Vert \equiv J_{c}\approx \sqrt{2mw_{s}\unicode[STIX]{x0394}z}.\end{eqnarray}$$

The elevation threshold $\unicode[STIX]{x0394}z$ is left as a free parameter in (2.5) for reasons that become clear in § 3.2. Note, however, that for certain ideal configurations, $\unicode[STIX]{x0394}z$ has analytical solutions. For instance, data employed for validation in § 3.1 are derived from the experimental setting depicted in figure 1, for which it can be shown that (see appendix B)

(2.6) $$\begin{eqnarray}\unicode[STIX]{x0394}z=\left(\frac{3-2\sqrt{2}}{\sqrt{3}}\right)r,\end{eqnarray}$$

where $r$ is the radius of the spheres. The mass of the mobile sphere, $m$ , appearing in (2.5) merits some discussion. If $\boldsymbol{F}_{H}(t)$ is taken as the total hydrodynamic force exerted on the mobile particle by the surrounding fluid (that is, excluding the buoyancy force, which is already accounted for in $\boldsymbol{w}_{s}$ ) – i.e. the integral of pressure and shear stresses over the surface of the sphere (minus the buoyancy force) – then $m$ represents the actual mass of the sphere, i.e.

$\boldsymbol{F}_{H}(t)$

$M$

$M=0.5$

$M\approx 0.5$

$M$

$\boldsymbol{F}_{H}$

(2.7) $$\begin{eqnarray}J_{c}\approx {\textstyle \frac{4}{3}}\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}r^{3}\sqrt{2(s+M)(s-1)g\unicode[STIX]{x0394}z},\end{eqnarray}$$

with $s\equiv \unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x1D70C}$ being the particle’s relative density and $g\equiv \Vert \boldsymbol{g}\Vert$ . The proposed criterion for critical impulse, derived for spherical particles, is solely dependent on the particle’s size and density (that is, if $M=0.5$ is adopted), and is applicable to a hydrodynamic time-varying force of arbitrary duration. This contrasts with expressions previously derived for the same bed configuration, as discussed in § 1. The (relative) lack of empirical coefficients in our derivation is in good measure achieved by the very definition of the criterion, which expresses the dislodgement condition as a function of the hydrodynamic force exerted on the particle, rather than the flow variables inducing said force. Naturally, coefficients such as drag will be necessary in practice when relating the fluid force exerted on the particle to local hydrodynamic parameters (see § 3.2). However, this is beyond the scope of the present paper, the main aim of which is to provide a theoretical analysis of existing experiments devoted to exploring the role of hydrodynamic impulse as a criterion for dislodgement.

Even though derivation of (2.5) implicitly assumes a horizontal channel (the simplifying assumptions invoked depend on this condition), the effect of the local slope is accounted for via $\unicode[STIX]{x0394}z$ , which will be affected by variations in the local micro-topography. However, use of (2.5) and (2.7) should be restricted to applications involving horizontal or nearly horizontal channels, given that terms neglected in (2.3) are anticipated to grow in importance for steep slopes. In deriving (2.5), we have also assumed exclusively forces capable of doing work on the particle (but note that this is not a restriction of the full condition, equation (2.3) – i.e. hydrodynamic forces too small to move the particle will simply not comply with the inequality). For the setting considered (figure 1), the maximum force opposing motion will be found at the initial position (see end of § 3.3).

3 Comparison against experiments

In order to test the validity of the theoretical impulse threshold for particle dislodgement proposed here, we compare predictions of (2.5) and (2.7) against experiments by Diplas et al. (Reference Diplas, Dancey, Celik, Valyrakis, Greer and Akar2008), Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010), Valyrakis et al. (Reference Valyrakis, Diplas, Dancey, Greer and Celik2010) and Celik et al. (Reference Celik, Diplas and Dancey2013). All these experiments deal with a setting similar to that illustrated in figure 1, where a mobile spherical particle rests on top of three, fixed, well-packed spheres. Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010, Reference Celik, Diplas and Dancey2013) employed identical-size top and base spheres, with $s=2.3$ and $r=6.35$  mm, subject to water flow. On the other hand, Diplas et al. (Reference Diplas, Dancey, Celik, Valyrakis, Greer and Akar2008) and Valyrakis et al. (Reference Valyrakis, Diplas, Dancey, Greer and Celik2010) investigated different combinations of top and base metallic spheres’ sizes subject to electromagnetic flux. For each of the above experiments, the parameters measured and methodologies vary, allowing us to test the derived theoretical prediction of critical impulse under different conditions, as detailed next.

3.2 Indirect comparison (pseudo-impulse)

Next, we consider the experiments by Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010), who measured the streamwise component of the fluid velocity, $u$ , one particle diameter upstream of a test (mobile) sphere. Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) studied particle dislodgement as a function of pseudo-impulse, defined as the product $\langle u^{2}\rangle T$ , where the angle brackets denote time-averaging over the interval  $T$ . The referred researchers employ this surrogate for impulse based on the proviso that the prevailing hydrodynamic force is drag, $F_{D}$ , which is proportional to  $u^{2}$ . Considering the time-average of the net hydrodynamic force acting on the particle over the interval  $T$ ,

(3.1) $$\begin{eqnarray}\langle \boldsymbol{F}_{H}\rangle \equiv \frac{1}{T}\int _{t_{0}}^{t_{1}}\boldsymbol{F}_{H}(t)\,\text{d}t,\end{eqnarray}$$

the impulse imparted to the particle by the flow over $T$ can also be expressed as $\boldsymbol{J}=\langle \boldsymbol{F}_{H}\rangle T$ . We can then find an approximate equivalence between the real impulse, $\boldsymbol{J}$ , and pseudo-impulse employed by Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010), $\boldsymbol{J}_{ps}$ , as follows:

(3.2) $$\begin{eqnarray}\Vert \boldsymbol{J}_{ps}\Vert \equiv \langle u^{2}\rangle T=\frac{\langle F_{D}\rangle T}{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}A_{p}C_{D}}\approx \frac{\Vert \boldsymbol{J}\Vert }{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}A_{p}C_{D}},\end{eqnarray}$$

where the conventional parametrisation of the drag force has been employed; i.e. $F_{D}=0.5\unicode[STIX]{x1D70C}A_{p}C_{D}u^{2}$ , with $C_{D}$ representing the drag coefficient and $A_{p}$ being the projected area of the spherical particle. The latter may be approximated, due to the assumption of the particle being highly exposed to the flow (also employed by Celik et al. Reference Celik, Diplas, Dancey and Valyrakis2010), as $A_{p}=\unicode[STIX]{x03C0}r^{2}$ . The assumption of drag being the predominant hydrodynamic force acting on the particle underpins (3.2).

In their experiments, Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) observe a range of values of pseudo-impulse of 0.0034 to $0.0095~\text{m}^{2}~\text{s}^{-1}$ , for which both dislodgement and no-dislodgement events are observed for all values of $\langle u^{2}\rangle$ considered. In other words, below (above) this range of pseudo-impulse, the test particle was never (always) dislodged by the flow. Therefore, the lower limit of this range ( $0.0034~\text{m}^{2}~\text{s}^{-1}$ ) can be interpreted as the fundamental pseudo-impulse threshold, below which no particle dislodgement is observed.

To test our prediction of critical impulse (2.7), we first convert it to pseudo-impulse via (3.2), and use $\unicode[STIX]{x0394}z=0.4$  mm. The reason for fixing $\unicode[STIX]{x0394}z$ to the value reported by Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) is that their experimental setting included the presence of a retention pin downstream of the mobile particle that ensured that once ‘fully dislodged’, the particle could return to its original position, thus automating the experiment. To transform impulse to pseudo-impulse, a value of $C_{D}$ must be assumed (see (3.2)). To this end, we employ the value of $C_{D}$ from run U8 in Celik et al. (Reference Celik, Diplas and Dancey2013) (namely, $C_{D}=0.818$ ), who carried out very similar experiments to those of Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010), reporting values of $C_{D}$ for a resting sphere under diverse flow conditions. As in § 3.1, our focus is on run U8 because this run represents the experimental conditions which are closest to a fundamental dislodgement threshold. Use of (2.7) in (3.2), with $C_{D}=0.818$ , $M=0.5$ and $\unicode[STIX]{x0394}z=0.4$  mm, yields a prediction of critical pseudo-impulse of $0.0035~\text{m}^{2}~\text{s}^{-1}$ , which shows a very good agreement with the lower limit experimentally found by Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) (i.e. $0.0034~\text{m}^{2}~\text{s}^{-1}$ ), thus supporting the argument that (2.7) describes accurately the fundamental impulse threshold for particle dislodgement.

Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) also proposed an algorithm to predict the critical pseudo-impulse, which yielded an estimate of $0.0033~\text{m}^{2}~\text{s}^{-1}$ for this experimental setting. This estimate is very close to our prediction of $0.0035~\text{m}^{2}~\text{s}^{-1}$ , but it is important to highlight that the method proposed by Celik et al. (Reference Celik, Diplas, Dancey and Valyrakis2010) (i) represents a methodology, rather than an expression, to estimate the critical pseudo-impulse; (ii) lacks rigour by requiring a hypothetical initial velocity of the resting particle; and (iii) needs input of certain geometric variables (such as lever arms) not required in (2.7).

3.3 Recovery of trend $F\propto T^{-1}$ obtained empirically

Diplas et al. (Reference Diplas, Dancey, Celik, Valyrakis, Greer and Akar2008) were the first to demonstrate experimentally the importance of force duration by plotting normalised drag force, $\hat{F}_{D}$ , versus its normalised duration, $\hat{T}_{D}$ . A metallic particle subject to electromagnetic flux was employed to achieve a highly controllable flow. Measurements by Diplas et al. (Reference Diplas, Dancey, Celik, Valyrakis, Greer and Akar2008) (328 in total) collapsed remarkably well into a curve of the form $\hat{F}_{D}=K{\hat{T}_{D}}^{n}$ , where $K$ and $n$ are constants obtained from best-fit curves, the latter of which takes a value of $n\approx -1$ ( $n=-0.99$ , to be precise). Later, Valyrakis et al. (Reference Valyrakis, Diplas, Dancey, Greer and Celik2010) extended this study and provided different values of coefficients $K$ and $n$ arising from best-fit curves for different combinations of top and base spheres’ diameters (a total of 1709 data points was obtained); the value of $n\approx -1$ was confirmed: their reported values of $n$ range from $-0.89$ to $-1.07$ with a mean of $-0.99$ (Valyrakis et al. (Reference Valyrakis, Diplas, Dancey, Greer and Celik2010) employ squared voltage across the electromagnet as proxy for force). We conclude our experimental comparisons by noting that the value of $n=-1$ is to be expected since, by rewriting (2.5) as $J_{c}=\Vert \langle \boldsymbol{F}_{H}\rangle \Vert T\approx \sqrt{2mw_{s}\unicode[STIX]{x0394}z}$ , we obtain

(3.3) $$\begin{eqnarray}\Vert \langle \boldsymbol{F}_{H}\rangle \Vert \approx \left(\!\sqrt{2mw_{s}\unicode[STIX]{x0394}z}\right)\frac{1}{T},\end{eqnarray}$$

where $\sqrt{2mw_{s}\unicode[STIX]{x0394}z}$ is indeed a constant for a local bed configuration (i.e.  $K$ ) and $n$ is precisely equal to  $-1$ . Combinations of $\Vert \langle \boldsymbol{F}_{H}\rangle \Vert$ and $T$ falling above the curve given by (3.3) will result in particle dislodgement, so long as the net hydrodynamic force is capable of doing work on the particle during  $T$ . For the problem considered, where $\boldsymbol{F}_{H}$ equals a virtually time-independent constant $\widetilde{F_{H}}$ over $T$ (a pulse), the condition to verify is $\widetilde{F_{H}}\cos \unicode[STIX]{x1D6FC}_{0}>w_{s}\sin \unicode[STIX]{x1D6FC}_{0}$ , where $\unicode[STIX]{x1D6FC}_{0}=\unicode[STIX]{x1D6FC}(t=t_{0})$ ; or $\widetilde{F_{H}}\gtrapprox w_{s}/3$ for the case of equal-sized top and base spheres (in figure 1, the maximum force opposing motion is found at  $t_{0}$ , where $\sin \unicode[STIX]{x1D6FC}_{0}=1/3$ ). It is worth remarking that (3.3) is only approximate. In appendix A, an exact expression is derived for this experimental setting (A 5), which is completely determined if the time history of $\boldsymbol{a}(t)$ (or the particle’s position) is known. However, as discussed in appendix A, this exact equation reduces to the approximation proposed here (2.5) for small angular displacement of the particle during dislodgement, as is assumed to be the case with the highly exposed sphere under consideration. A quantitative comparison between the approximate constant predicted (i.e. $\sqrt{2mw_{s}\unicode[STIX]{x0394}z}$ ) and $K$ empirically found by Valyrakis et al. (Reference Valyrakis, Diplas, Dancey, Greer and Celik2010) cannot be carried out without more detailed information on the experiment, especially pertaining to the electromagnet (e.g. resistance, number of turns in the coil, etc.), which unfortunately is unavailable.