Direct measurement of the inertial drag and lift forces on entrained coarse particles at various protrusion heights

Laboratory experiments were performed to study the impact of relative particle protrusion P/D (P is the protrusion height and D is the diameter of the spherical target particle) on the entrainment of sediment particles from a rough bed of spherical grains. The target particle to be entrained was instrumented with electronic sensors to measure the triaxial linear acceleration, and consequently the particle inertial forces, during the entrainment process. The velocity field around the particle was obtained using two‐dimensional particle tracking velocimetry (PTV) technique and the velocity data were synchronised with the force data relative to the moment of entrainment. Experimental results show that the inertial drag and lift forces systematically decreased as relative particle protrusion increased. The ratio of inertial lift force to drag force revealed that drag force slightly dominated the entrainment process at P/D > 0.7, while lift force slightly dominated at P/D < 0.62. Inertial drag coefficient was found to be independent of particle protrusion for low P/D, but rapidly declined for P/D > 0.62–0.7. Differently, inertial lift coefficient remained relatively constant within the tested protrusion ranges. Additionally, the critical Shields number was linearly related to the normalised inertial forces over the present particle protrusion range. In summary, the particle protrusion demonstrates a significant impact on the inertial forces and, further, the entrainment process.


| INTRODUCTION
Through the erosion and deposition process, sediment transport contributes significantly to the natural evolution of river bed morphology and also affects the aquatic habitat (Riebe et al., 2014). Sediment transport is a widely considered issue in water resources engineering such as the sediment deposition in a reservoir (Rahmani et al., 2018), erosion of the foundation of a hydraulic structure (Wang et al., 2018;Yang et al., 2019), and channel stability (Turowski et al., 2009). To address these issues, the critical condition for the initial movement of sediment particles is fundamentally important.
Generally, the critical condition of initial sediment movement can be determined using either deterministic (Ali & Dey, 2016;Zanke, 2003) or stochastic (Bose & Dey, 2013;Wu & Chou, 2003) approaches. Among the numerous studies on this topic, the classic work comprising the Shields parameter (Shields, 1936) has been widely used because of its simplicity of application. The critical Shields parameter, θ c , as a deterministic method for parametrising entrainment condition of a single grain, is calculated from the following equation: where u Ãc is the critical shear velocity, ρ s is the sediment density, ρ is the fluid density, g is the gravitational acceleration, and D is diameter of the median grain size. θ c is plotted against the particle Reynolds number Re * (Re * = u * D/v, where v is the kinematic viscosity) in the original Shields diagram. The Shields diagram is widely used as a standard method to obtain the threshold condition for incipient motion of sediments and predict bedload transport. Much considerable effort has been made to explain the data scatter in the diagram (Buffington & Montgomery, 1997;Paphitis, 2001). However, a critical Shields number for a given Re * is not suitable for predicting the wide range of threshold conditions for particle motion, even for an idealised particle. Many complex variables (bed irregularity, turbulence) need to be taken into account (Dey & Ali, 2017;Lamb et al., 2008;Wiberg & Smith, 1987). It is difficult to develop a universal criterion of entrainment for all circumstances. In this study, the impact of particle protrusion on initial particle movement is studied.
Bed irregularity may result in different protrusion heights of the sediment particles and affect the mobility of equivalent particles. A sediment particle with high protrusion is largely exposed to turbulent flow and more easily entrained than a lower protruding particle. Thus, the critical Shields number for a high protruding particle might be smaller than a lower protruding particle. Fenton and Abbott (1977) performed laboratory experiments to investigate the critical Shields number at different particle protrusion heights. They concluded that the critical Shields number significantly decreases with the increasing particle protrusion. Fenton and Abbott (1977) suggested that the Shields parameter has a minimum value of 0.01 at relative protrusion P/D = 0.82. This value was verified by Chin (1985), but Chin (1985) suggested that a smaller value may be obtained at a higher protrusion height. With similar experimental procedures, Coleman et al. (2003) investigated the influence of particle protrusion on initial particle movement for crushed rock particles. They found that the critical Shields number of cuboid particles was smaller than that of spherical particles at the same protrusion, and also had a decreasing trend as the protrusion height increased. Particle protrusion also affects the total bed sediment flux (Masteller & Finnegan, 2017;McKie et al., 2021). Protruding grains obstruct the flow and shelter downstream by preventing the shear region from approaching the bed surface (Raus et al., 2019). With flume experiments, Masteller and Finnegan (2017) highlighted a significant correlation between the fraction of highly protruding particles (determined from the variation of sediment elevation) and the total sediment transport rate. Overall, the influence of particle protrusion on sediment transport is significant from the grain scale to the channel reach scale.
Analysis of the disequilibrium of the forces applied to a particle is the basis of determining the critical condition, especially for the establishment of theoretical models (Ali & Dey, 2016;Vollmer & Kleinhans, 2007). Therefore, some researchers have attempted to measure the hydrodynamic forces acting on a sediment particle under various circumstances (Bin Riaz et al., 2021;Cameron et al., 2019;Fischer et al., 2002) and tried to link them to the turbulent flow Lamb et al., 2017;Schmeeckle et al., 2007). However, there remain limitations regarding the force measurement methods. Load cells or pressure sensors are commonly employed to investigate hydrodynamic forces. The target particle was attached to a mechanical gauge that restricts the particle's movement during data collection. The target particle may vibrate slightly using this method, but it will not be completely dislodged from its setting location.
As a result, the load cell fails to acquire force data for a totally entrained particle, limiting a precise knowledge of particle entrainment process.
In recent years, advanced electronic techniques have brought new possibilities for sediment transport research. Some researchers placed accelerometers inside artificial or natural sediment particles (Abeywardana et al., 2009;Kularatna et al., 2005) to investigate particle entrainment. During particle movement, the embedded sensors record its acceleration data, which are transformed to the exerted forces by multiplying the particle mass. This new approach has been applied to measurement of impact force of test debris on infrastructures (Shafiei et al., 2016;Spreitzer et al., 2019), monitoring river bed scour and sediment transport (Pretzlav et al., 2020(Pretzlav et al., , 2021 Stephenson & Abazovi c, 2016), and particle entrainment mechanics (AlObaidi & Valyrakis, 2021;Frank et al., 2015;Maniatis et al., 2017Maniatis et al., , 2020. Use of instrumented accelerometers to measure the inertial forces exerted on target particle is distinctly different from the previously invasive methods. Since the accelerometers are instrumented inside the target particle, the flow in the vicinity of the target particle is not disturbed, and the actual entrainment condition is provided.
Without requiring the restriction of movement, the recorded acceleration can directly reflect the particle dynamics during the entrainment process. Compared with previous research, this method provides valuable information for the particle entrainment mechanism and direct measurement of the resultant force during the movement process.
The main objective of this study was to investigate the impact of particle protrusion on the entrainment process using instrumented sensors. The sensors can directly measure the inertial forces from the destabilised particle and monitor the instantaneous dynamic features after entrainment. The present study not only helps to strengthen the understanding of entrainment mechanism but also provides a new insight into data acquisition (direct access to particle dynamics during movement process) and analysis.
2 | EXPERIMENTAL 2.1 | Flume and particle protrusion setup Experiments were conducted in a 12 m long by 0.44 m wide recirculating tilting flume. A grid-shaped guide vane and straightener were installed to reduce water surface undulation at the inlet of the flume. The flume had glass sidewalls that made observation of sediment particle movement clear and convenient (Figure 1a). The slope of the flume can be adjusted through an in-built jacking system with a maximum slope of 1%. The adjustment of the flume slope ensures a uniform flow condition during the experiments, with a range of approximately 0.04-0.08%. A pavement layer of plastic spheres was used to create a rough bed in a close hexagonal-packed pattern. The interstice formed by three, adjacent, closely packed particles provided the pocket for the placement of target particle (as shown in Figure 1b). The term 'target particle' is used to refer to the spherical The velocity field was measured using two-dimensional particle tracking velocimetry (PTV). Tracer particles were fed into the Also, the light-box was covered with a black plastic cloth, which avoided extra ambient light from the outside environment. The camera operated at a frequency of 100 Hz, as was the sampling frequency of the acceleration measurement (described in the next section). The captured images were analysed for generating the flow field information by Streams, a software developed by Dr Nokes from the University of Canterbury, New Zealand. More information can be found in Nokes (2019).
As mentioned above, the spherical target particle was placed in the pocket formed by three adjacent spherical particles (Figure 1b).
From the known diameter sizes, the protrusion height P of the target particle above the top of surrounding roughness spheres can be computed. Two sizes of target particles, with diameters of 30 and 40 mm, and two sizes of bed spheres, with diameters of 40 and 27 mm, were used, resulting in four protrusion heights: P = 21.3 mm (30 mm target particle over 40 mm bed spheres); 25.4 mm (30 mm target particle over 27 mm bed spheres); 32.6 mm (40 mm target particle over 40 mm bed spheres); and 36.2 mm (40 mm target particle over 27 mm bed spheres). Additionally, some specially designed three-dimensional models were employed to increase the target particle protrusion range. A schematic design for the three-dimensional model supporting the 30 mm target particle at P = 16 mm is shown in Figure 2. The top part of the model has a spherical shape, and the bottom part is a cylinder. With this design, the downstream face of the target particle is supported by two adjacent bed spheres. The upstream face of the target particle is supported by the top sphere connected to the cylinder.
This design maintains a consistent geometry for the placement of the target particle (comparing Figure 1b and Figure 2a) and reduces the gap between the target particle and bed roughness caused by the particle protrusion. By this method, three particle protrusions (P = 16, 18, and 20 mm) were achieved for a 30 mm target particle on the bed of 40 mm spheres. Another two particle protrusions (P = 25, 30 mm) were achieved for the 40 mm target particle on the bed of 40 mm spheres. Overall, nine protrusions were used, resulting in nine relative particle protrusions P/D (where D is the diameter of the target particle).
During an entrainment experiment, the flow rate is slowly increased until initial particle movement is triggered. Here, the initial particle movement is defined as target particle dislodging from its setting pocket. After each incremental increase in flow, the flow rate remained constant for 10 s to check whether entrainment occurred.
Preliminary experiments were conducted prior to the formal entrainment experiments. The preliminary experiments were used to decide the appropriate flow range for the particle entrainment at a specific particle protrusion. Then, the formal entrainment experiments were started at a close-to-threshold condition flow rate, with each increment of flow rate small enough not to cause additional flow disturbance. For each case, five repeated experiments were carried out to obtain the data average and variance. The flow depth was measured from the roughness length scale of the bed, Z 0 , where the velocity approaches zero. In this study, the roughness length scale is defined as 0.2d (where d is the diameter of the spherical roughness elements). The value 0.2d follows previous research (Cameron, 2006;Dwivedi, 2014), which used a similar experimental setup to the present study, and is in the range of 0.15-0.3 k s , as suggested by Nezu and Nakagawa (1993). A flow depth of 203 mm was used for all experiment series, measured from Z 0 . An overflow pipe setting at the flume tank ensured the relatively steady flow depth. The coordinate system of measurement adopted the right-hand rule, with the positive X-axis along the mainstream direction and the positive Z-axis pointing to the flow surface. Since the target particle was always placed at the flume centre line, the origin of the coordinate system was at the flume centre line (Y = 0), 0.2d below the roughness top (Z = 0) and the target particle centre line (X = 0). The experimental setup and the threshold flow conditions for particle entrainment are summarised in Table 1. 2.2 | Measurement of inertial forces on the entrained particle The target particle was instrumented with an electronic board, including a tri-axis accelerometer, tri-axis gyroscope, and tri-axis magnetometer. With the instrumented sensors, the linear acceleration of the target particle was known during the dislodgement process.
The plan and side view of the electronic units are shown in Figure 3.
A cross mark was painted on the enclosure of target particle to ensure the direction of X-axis accelerometer aligned with flow direction, and  Note: Q denotes flow rate; H is flow depth measured from a datum of 0.2d below the grain tops of the surrounding bed surface; D is the diameter of the target particle; d is the diameter of the surrounding bed grains; P is the protrusion height of the target particle above the grain tops of the surrounding bed; P/D is defined as the relative protrusion height; U avg is the cross-sectional velocity calculated from the flow rate; Re is Reynolds number, and Fr is Froude number, calculated from the U avg .
F I G U R E 3 The embedded electronic board in the target particle: (a) plan view; (b) side view together at the centre plane. The bottom hemisphere of the enclosure was designed to keep the electronic board stable and on the centre plane. The seam of the two hemispheres was filled with a thin layer of silicon glue to prevent water leakage. The materials of the two enclosures were carbon fibre for the 30 mm target particle and aluminium for the 40 mm target particle. For the 30 mm target particle enclosure, three-dimensional printing technology was employed for the manufacture. However, the carbon fibre material was too light for the 40 mm particle enclosure, and it was difficult to fill the enclosure uniformly and maintain constant density using another type of material.
Therefore, aluminium was chosen to manufacture the 40 mm particle enclosure. Both particle enclosures had a smooth surface and were painted black to reduce light reflection. The particle enclosures were specifically designed to accommodate the electronic board at the centre plane. Also, the cavity between the enclosure and electronic board was filled with plastic modules to obtain a uniform weight distribution.
These modules are three-dimensionally printed based on the shape of the electronic board, thus reducing the non-uniform distribution of mass. After assembly, the instrumented particles had the same density of 1140 kg/m 3 . The particle density was smaller than that of common sediment material (i.e., 2650 kg/m 3 ). The reduction of particle density was to ensure a wide range of laboratory experiments, because the critical flow condition was relatively low for a lightweight particle. Such a reduction of particle density is feasible under constrained laboratory flow conditions, which use an analogy from biological science: 'in vitro'. On the other hand, the threshold particle entrainment condition needs to be considered with care. Non-dimensionless parameters (e.g., Shields number) would be appropriate for the calculation.
Also, utilising a similarity law, the current results might be used and compared to a real-world case for a broader application (Cameron, 2006; Dwivedi, Melville, Shamseldin 2010).
The particle entrainment time was determined by inspecting the elevation change of the target particle among all the images. The entrainment time can also be determined from the time history of the acceleration data. The target particle was assumed to be entrained when its acceleration magnitude continuously surpassed the threshold noise level of the accelerometers. The threshold noise level was considered as the standard deviation of the acceleration when the accelerometer was kept stationary. Electronic sensors have noise even with the most accurate manufacturing techniques (Kok et al., 2017).
In spite of the inevitable noise, accelerometers still have good performance and potential for capturing particle dynamics during the movement process (Maniatis et al., 2017(Maniatis et al., , 2020. After determining the exact entrainment time, the velocity and acceleration measurements were synchronised. The tri-axis acceleration was then transformed to force data by multiplying by the target particle mass. The transformed force directly reflects the particle dynamics and is herein termed the inertial force.

| RESULTS
3.1 | Tri-acceleration time histories for various particle protrusions The tri-axis linear acceleration during a very short time (0.15 s) was investigated. The period of 0.15 s was chosen for two reasons. First, the accuracy of linear accelerations could be maintained for only a short time, and secondly, the acceleration during this time directly reflects the dynamic response of the particle at the instant of the entrainment event. Considering that the main objective of this paper is focused on the particle response at the instant of entrainment, analysing the tri-axis linear acceleration during that short period is appropriate. Figure 5 shows the time series of tri-axis linear acceleration at different particle protrusions, from 0.1 s before particle entrainment, to 0.15 s after particle entrainment. In Figure 5, a x , a y and a z represent Before t = 0, the target particle remains stationary. The fluctuations in the acceleration data before t = 0 are at the level of the signal noise (rather than being particle vibration), which was verified by inspecting the captured images. After t = 0, the variation of the acceleration magnitude exceeds the noise threshold, indicating that the target particle starts moving. In Figure 5, it is apparent that the Y-axis acceleration magnitude is small for all protrusions relative to the X-axis and Z-axis accelerations. The Y-axis acceleration magnitude tended to remain at the level of background noise (AE0.02 m/s 2 for all the axes). The relatively small value of the Y-axis acceleration indicates that the target particle has no or minimal lateral movement during the entrainment process. Generally, the X-axis and Z-axis the 40 mm target particle. This difference implies that the particle size itself also affects the magnitude of the accelerations, since a stronger force is required to 'move' a heavier object. However, this effect of particle size becomes less prominent for higher particle protrusions (P/D = 0.82, 0.85, and 0.9). The magnitude of tri-axis accelerations at these high protrusions are relatively similar. At high particle protrusion, the particle acceleration is small at entrainment compared to that at low particle protrusion. As presented previously, sensor noise can affect the output acceleration value. Thus, averaged acceleration in a time duration might be appropriate for a further quantified analysis, which will be presented in the next section.

| Averaged inertial force during entrainment
In this section, the mean inertial force data, obtained by averaging the acceleration data over the time period of 0.15 s following particle entrainment, are discussed. The mean inertial force acting on the particle during the 0.15 s time period is determined as where m is the mass of the target particle, with units of kg; a x , a y , a z , is the averaged acceleration during time period of 0.15 s in X, Y and Z direction respectively; the time interval of measurements is 0.01 s.
The computed mean forces in the X, Y and Z directions are plotted against the relative particle protrusions in Figure 6a-c. Also, the magnitude of resultant inertial forces F iR , calculated on the basis of the triaxis accelerations (shown in equation 2), are plotted in Figure 6d: The data in the figures are grouped by the target particle size. Following the findings from Figure 5, the role of the Y-axis inertial force F iy appears to be negligible compared with that of the other force components. For the current setup, the principal direction of movement of the entrained target particle is along the X-and Z-axis directions. This result shows that the entrainment path of the target particle is similar for different protrusion heights, as expected because of the method of setting the protrusion heights. As shown in Figures 1   (b) and 2, the direction of particle entrainment is towards the valley of two downstream bed particles and aligns with the main flow direction.
The decreasing trends indicate that both F ix and F iz are inversely related to the relative particle protrusion height.
As pointed out in Figure 5, the particle size also impacts the magnitude of inertial forces. The magnitudes of inertial forces acting on the 40 mm target particle would be larger than those on the 30 mm target particle at the same P/D value, since the weight increases ( Figure 6a,c,d). Furthermore, the entrainment force decreases faster with increasing protrusion for the 40 mm particle, compared to that for the 30 mm particle. This observation could be linked to the rising rate of the particle's exposed area. As the particle protrusion increases, the 40 mm target particle is more likely to be exposed to the approaching flow than the 30 mm target particle. Thus, it is plausible to assume that the entrainment force for the 40 mm target particle drops noticeably as particle protrusion increases. We also noticed that the variability of inertial force increases for the larger target particle, particularly at P/D = 0.62 and 0.82. One possible explanation is that the 40 mm particle requires a stronger flow to be entrained while at the same protrusion with the 30 mm particle.
Accordingly, the strengthened turbulence causes more variability for the particle entrainment condition. The contribution of each axial component of force in the resultant force is expressed as a percentage and is plotted against relative protrusion for each case in Figure 7.
The percentage of the total entrainment force comprised by F ix and F iy varies slightly between cases, but is largely independent of particle protrusion. The F ix component fluctuates in a range of 40-50%, while F iy varies between 0 and 20% of the total entrainment force. However, the F iz component shows a slight decrease (from 50% to 40%) as the protrusion height increases.

| Inertial drag and lift forces
As discussed previously, F iy is relatively small compared to F ix and F iz -that is, the force components exerted on the X-Z plane. In the present study, the inertial drag force F iD is defined as the composition of inertial force in the X-and Y-axis direction, with magnitude as follows: Here, we will focus on discussing the magnitude of inertial forces only, and the direction of inertial force is beyond the scope of the present study. The inertial lift force F iL is represented by F iz . The inertial drag and lift forces for both sizes of the target particles, normalised using the diameter of each target particle (ρ s À ρ)gD 3 , are plotted as a function of P/D in Figure 8a,b, respectively. Such normalisation includes the effect of the different particle sizes in the plot. Here, F iD  Table 1, because the synchronised velocity data were not measured. For these two cases, the 30 mm target particle was used with the protrusion model shown in Figure 2. Therefore, the force data at P/D = 0.4 and 0.47 are only plotted in Figures 8 and 9 as additional information, to better demonstrate the data trend. The dashed lines (power functions), which demonstrate an inverse relationship between force and relative protrusion, are good fits to the data for the normalised inertial drag force (R 2 = 0.86) and lift force (R 2 = 0.84) within the range P/D = 0.4-0.9.In general, the normalised inertial drag force and lift force have similar magnitudes, but Figure 8a,b does not allow their individual effects on particle entrainment to be defined. For further analysis, the ratio of F iL to F iD is computed at each P/D value. The variation of F iL =F iD with P/D is presented in Figure 9, in which the horizontal line represents the value of F iL =F iD = 1-that is, the point at which the drag and lift components are equal. From the trend line in Figure 9, it is apparent that the value of F iL =F iD decreases as P/D increases. More specifically, F iL =F iD > 1 (with a range of [0.8, 1.5]) occurs for P/D < 0.62 and F iL =F iD < 1 (with a range of [0.7, 1]) for P/D > 0.67. These results imply that the inertial drag force commonly dominates the entrainment process at higher relative protrusions, while the inertial lift force becomes important at lower protrusions. When we take a closer insight into the data, F iL =F iD remains around 0.9 for 0.67 < P/D < 0.9 but climbs to 1.4 at P/D = 0.4. This finding indicates that inertial drag and lift force have comparable impacts on the particle entrainment in the high protrusion range, whereas the inertial lift force becomes a key factor in particle entrainment at low particle protrusion. However, inertial force measurements for P/D < 0.4 were not available in this study, indicating that further research into particle entrainment at a such a low protrusion height is required.

| Relationship between inertial forces and velocity
In this section, we further discuss the relationship between the inertial forces and the velocities at different particle protrusion heights. The results can establish a link between the SSP measurement and flow field, which benefits the interpretation of SSP data. A protruding target particle leads to heterogeneous spatial distribution of velocity in the near bed region (Raus et al., 2019). Therefore, the doubleaveraged (both in time and space) velocity is considered appropriate to analyse its link with the mean inertial forces. In the doubleaveraged method, the time-averaged velocity can be decomposed into F I G U R E 7 Percentage of tri-axial inertial forces during entrainment for (a) 30 mm target particle and (b) 40 mm target particle at different P/D values. F I G U R E 9 Ratio of inertial lift force to drag force as a function of relative protrusion (P/D) during particle entrainment [Color figure can be viewed at wileyonlinelibrary.com] double-averaged and spatial averaged components (Nikora et al., 2001), as follows: where U and V refer to the streamwise and vertical velocity, respectively, the overbar and brackets denote the time average (over 10 s) and spatial average (from 2D upstream to 2D downstream of the target particle), respectively, and the circumflex represents spatial fluctuation of the time-averaged velocity over the double-averaged value.
The double-averaged velocity is extracted from the measurement field at 0.15D above the target particle centroid, following the research of The inertial drag force F iD and lift force F iL exerted on the sediment particle are generally expressed in the following forms: in which C iD is the inertial drag coefficient, C iL is the inertial lift coefficient, < U 0:15D > is the double-averaged streamwise velocity, A D is the cross-sectional area of the target particle projected on the transverse plane of the flow above the bed roughness (related to the particle protrusion P), and A L is the cross-sectional area of the target particle projected on the horizontal plane, which is not affected by the particle protrusion P (Hodge et al., 2020;Kirchner et al., 1990;Voepel et al., 2019). It is crucial to emphasise here that the inertial force coefficients are not the classical hydraulic coefficients defined in the literature of incipient grain motion (Papanicolaou et al., 2002;Valyrakis et al., 2010). The inertial coefficients C iD and C iL , parametrise partly the hydraulic forces but they are also affected by other forces exerted on the particle (e.g., support forces and gravity). In this context, the calibration presented here is not relevant to the relationship between the measured flow velocity and the hydraulic forces. Here, we parametrise the relationship between the mean velocity and the resultant interaction of the applied forces as captured by F iD and F iL . The entrainment point is defined at F iD or F iL exceeding 0, and hence the point where the hydraulic forces exceed the combined effect of gravity and particle support forces, with the latter being largely affected by different protrusion values (Coleman et al., 2003). The calculation of cross-sectional area above the top of the surrounding bed roughness is simplified here, following the research of Cameron (2006). The schematic graph of exposed area A D at the absolute particle protrusion P is shown in Figure 11, together with the computed results of A D versus P/D. Equations 7a and 7b show the calculation of the exposed area and the results are summarised in Table 2. A L is calculated with the diameter of the target particle, with a value of 7 Â 10 À4 m 2 for the 30 mm particle and 12.6 Â 10 À4 m 2 for the 40 mm particle.
where ω is the radian angle of the obstructed area of the target particle (shown in Figure 11), P is the particle protrusion over the bed sediments, D is the diameter of the target particle, A D is the exposed area of the protruding particle for calculating the drag coefficient, and π is a mathematical constant, defined as the ratio of the circumference of a circle to its diameter.
From Figure 11, it is seen that the exposed area of the target particle gradually increases with increasing P/D. The measured forces and the calculated values of 0:5ρ <U 0:15D > 2 A are plotted in Figure 12a,b.
Also, the values of 0:5ρ <V 0:15D > 2 A are calculated and plotted against the measured forces in Figure 12c,d, to demonstrate their correlation. From Figure 12, it is apparent that both the inertial drag and lift forces are positively related to < U 0:15D > . Also, the inertial drag and lift forces are positively related to < V 0:15D > .
However, the very small flow vertical velocity results in a steep fitting slope with inertial forces (Figure 12c,d). Consequently, the difference of fitting slope between drag and lift components is significant.
F I G U R E 1 0 Double-averaged velocity at the height of 0.15 D above the target particle: (a) longitudinal velocity; (b) vertical velocity [Color figure can be viewed at wileyonlinelibrary.com] In Figure 12, the dotted lines are plotted using regression analysis and indicate the linear relationship between the forces and 0:5ρ <U 0:15D > 2 A (or 0:5ρ <V 0:15D > 2 A) values. The scatter in the data, especially at higher inertial force values, is due to a number of factors, including the simplified method adopted to define the exposed areas, the variation of the inertial drag and lift coefficients, the calculation of the velocity at a specific location, and the velocity fluctuations. These simplifications facilitate the variation of the inertial drag and lift coefficients with the relative particle protrusion P/D being the focus of the analysis. Using Equations 6(a) and 6(b), the inertial drag coefficient C iD and lift coefficient C iD are calculated and plotted in Figure 13a,b as functions of P/D. The calculated results are summarised in Table 2.
The inertial drag coefficient C iD ranged from 0.04 to 0.07, while the lift coefficient C iL tends to be a relatively constant value of 0.044.
The inertial lift coefficient has a smaller value than the inertial drag coefficient for all protrusion cases (Table 2). This result can be expected since different exposed areas of the target particle are used for inertial drag and lift force coefficients. As previously presented, the inertial drag force has comparable magnitude with the inertial lift force within the tested protrusion range (Figure 8). Due to a relationship of A L > A D existing for each protrusion case, the inertial lift force coefficients may have smaller values than the inertial drag force coefficients (Equations 6a and 6b). More specifically, the drag coefficient fluctuates around 0.07 at lower P/D (0.5-0.7) and shows an obvious F I G U R E 1 1 Schematic graph of the exposed area calculation and computed results of a versus P/D [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 2 Summarized exposed area and the inertial drag and lift coefficients Note: A D denotes cross-sectional area of target particle projected to the downstream; A L is cross-sectional area of target particle projected to the horizontal plane; C iD is inertial drag force coefficient; C iL is inertial lift force coefficient.
F I G U R E 1 2 Relationship between (a) F iD and 0:5ρ <U 0:15D > 2 A D ; (b) F iL and 0:5ρ <U 0:15D > 2 A L ; (c) F iD and 0:5ρ <V 0:15D > 2 A D ; (d) F iL and 0:5ρ <V 0:15D > 2 A L [Color figure can be viewed at wileyonlinelibrary.com] drop when P/D ≥ 0.7. This discrepancy could be related to the calculation of exposed area. According to Equation 4a, the drag force coefficient was determined by A D , < U 0:15D > ,F iD , and F iD . For P/D ≥ 0.7, the exposed frontal area of the target particle increased (Figure 11b) but the inertial drag forces and < U 0:15D > values remained relatively constant (Figures 8 and 10a). When the values of < U 0:15D > and F iD remained constant, the increased value of A D resulted in a decrease of the inertial drag force coefficient. In other words, the turning points of the inertial drag coefficients with P/D are correlated with those of the inertial drag forces with P/D. Here we noticed that the threshold P/D is 0.7 for drag coefficient variation ( Figure 13a). As discussed above, this difference indicates that the inertial drag force becomes stable when P/D > 0.7 (Figure 8).

| Inertial forces and critical Shields number
In this section, the critical Shields numbers θ c (defined in Equation 1) for different particle protrusions are investigated to clarify the particle protrusion effect. θ c is calculated on the basis of shear velocity u Ã .
There are many methods for the shear velocity calculation (Biron et al., 2004). In the research of Nikora et al. (2001), the ratio of H/k s (where H is the flow depth and k s is the roughness height) is used to define the flow type. In the present study, H/k s is 5.1 for flow over the 40 mm roughness bed and 6.7 for 30 mm roughness bed. The flow over the 40 mm bed is classified as Type II flow. For Type II flow, the Reynolds shear stress profile seems to be the most appropriate method to calculate the shear velocity, since the formation of the logarithmic region is uncertain in this flow type. Specifically, the doubleaveraged Reynolds shear stress profile is extrapolated to the bed for obtaining the shear stress and computing the shear velocity. The calculated u Ã and θ c value are listed in Table 3 and plotted in Figure 14a.
As expected, the effect of relative particle protrusion on the critical Shields number is significant. When P/D increased from 0.53 to 0.9, θ c decreased from 0.03 to 0.005. It is also noticed that θ c has a slower rate of decline starting from P/D = 0.65. The decreasing trend of θ c with P/D is consistent with the variation of inertial forces research results (Cameron, 2006;Chin, 1985;Dwivedi et al., 2012;Fenton & Abbott, 1977) in Figure 14b, showing good agreement. We noticed a scatter distribution of data points at each particle protrusion. This is expected since different target particle, bed roughness, and flow conditions were used among the various research studies.
However, a general trend of the critical Shields number with the particle protrusion can be observed. Coleman et al. (2003) proposed a best-fit expression to describe the relationship between θ c and the relative protrusion of spherical particles (P/D), as follows: For this equation, the constant 0.002 is adopted as the lower bound of the critical Shields number. As shown in Figure 14a, the lower bound of the Shields parameter, based on the present data, is about 0.005, larger than that suggested by Coleman et al. (2003).
Using the present data, combined with the results from the relevant literature, an alternative expression is derived (Equation 9). The Rsquare value for this relationship is about 0.88, indicating a good match with the measured data: Fenton and Abbott (1977) suggested that the critical Shields number has a minimum value of 0.01 at P/D = 0.82. P/D = 0.82 is the maximum relative protrusion for a grain placed on a uniform bed of densely packed spheres that are the same size as the test grain. In the present study, θ c has a value of 0.009 for P/D = 0.82 (representing the 40 mm target particle on the 40 mm bed), matching the results from Fenton and Abbott (1977). Despite methodological differences between studies, the effect of particle protrusion on the critical entrainment condition is apparent in this study and consistent with previous results.
The normalised inertial drag force and lift force are plotted against the critical Shields number in Figure  for Figure 15a; R 2 = 0.71 for Figure 15b). The values of normalised inertial drag force are slightly higher than those for normalised inertial lift force at lower θ c , and vice versa at higher θ c . According to Figure 14, lower values of θ c are relative to higher P/D, and higher values of θ c are relative to lower P/D. Thus, the inertial lift force is higher than the inertial drag force at low particle protrusion, which is consistent with the previous result (Figure 8). At high θ c values, the linear relationship between the critical Shields number and normalised inertial forces performs poorly. One possible reason for such data scatter might be the deviation of critical Shields number.
At lower particle protrusion, the flow strength required for destabilising target particle is high, which is associated with strong flow fluctuation. Consequently, the critical Shields number shows much deviation (Figure 14b).

| SUMMARY AND CONCLUSIONS
Laboratory experiments to study the impact of relative particle protrusion on entrainment mechanisms are reported. A tri-axial accelerometer was used to measure the inertial forces acting on the target particle during the entrainment process. The measured inertial forces were averaged over the time period of 0.15 s after particle entrainment. The mean Y-axial inertial force F iy shows similar magnitude for different P/D. Both mean X-axial inertial force F ix and Z-axial inertial force F iz are inversely related to P/D. The F ix and F iy forces are combined to represent the inertial drag force F iD , while F iz is regarded as the inertial lift force F iL . Both F iD and F iL decrease as the protrusion height increases. The data are also interpreted in terms of F iL =F iD , which indicates that the inertial lift force plays a more significant role in the entrainment process for low particle protrusion (F iL =F iD > 1) and the effect of drag force becomes predominant for high particle protrusion (F iL =F iD < 1). The double-averaged velocity at 0.15D above the test grain, in the stream-wise direction < U 0:15D > and vertical direction <V 0:15D > , is found to be inversely related to the protrusion height, and F iD and F iL are strongly correlated with both < U 0:15D > and <V 0:15D > . The inertial drag coefficient C iD and lift coefficient C iL are computed using <U 0:15D > . The values of C iD fluctuate around 0.07 for P/D < 0.7 but decline sharply for higher values of P/D.
Values of C iL are relatively constant at about 0.044 for all protrusion ranges. Critical Shields number θ c is computed and compared with previous research results, confirming the significance of particle protrusion in the entrainment process. The present results show good agreement with previous studies and extend the data for medium to high relative protrusion. Also, the relationship between the normalised inertial forces and θ c is investigated. Generally, both normalised inertial drag force and lift force are linearly correlated with θ c , with lower confidence in this relationship for lower protrusions.
Overall, the experimental results reveal that protrusion height profoundly impacts the acting inertial forces during the entrainment process, including the magnitude, allocation of force components, and correlation with local velocity and critical flow conditions. The instrumented sensor allows data collection during the particle movement, which is intrinsically distinct from the traditional force measurement method of connecting a fixed particle with a force gauge. The uncertainty of determining exact forces associated with particle entrainment may be eliminated with such direct measurement of inertial forces. Also, the link between flow parameters and particle inertial dynamics can be established through the simultaneous measurement of inertial forces and flow velocities. Thus, the dataset presented in Yifan Yang: writingreviewing and editing.

DATA AVAILABILITY STATEMENT
The link to download the experimental data is as follows: https:// drive.google.com/drive/u/1/folders/1H_hiijxC7ZVkonpbYlSYR3k1de Od378G.

NOTATION
A D cross-sectional area of target particle projected to the downstream, m 2 A L cross-sectional area of target particle projected to the horizontal plane, m 2 a x , a y , a z acceleration for X, Y, and Z direction, respectively, m 2 /s a x , a y , a z averaged acceleration during time period of 0,15 s for X, Y, and Z direction, respectively, m 2 /s C iD coefficient of inertial drag force C iL coefficient of inertial lift force D diameter of the target particle, mm