Two different methods for characterizing the incipient particle motion of a single bead as a function of the sediment bed geometry from laminar to turbulent flow are presented.
Two different experimental methods for determining the threshold of particle motion as a function of geometrical properties of the bed from laminar to turbulent flow conditions are presented. For that purpose, the incipient motion of a single bead is studied on regular substrates that consist of a monolayer of fixed spheres of uniform size that are regularly arranged in triangular and quadratic symmetries. The threshold is characterized by the critical Shields number. The criterion for the onset of motion is defined as the displacement from the original equilibrium position to the neighboring one. The displacement and the mode of motion are identified with an imaging system. The laminar flow is induced using a rotational rheometer with a parallel disk configuration. The shear Reynolds number remains below 1. The turbulent flow is induced in a low-speed wind tunnel with open jet test section. The air velocity is regulated with a frequency converter on the blower fan. The velocity profile is measured with a hot wire probe connected to a hot film anemometer. The shear Reynolds number ranges between 40 and 150. The logarithmic velocity law and the modified wall law presented by Rotta are used to infer the shear velocity from the experimental data. The latter is of special interest when the mobile bead is partially exposed to the turbulent flow in the so-called hydraulically transitional flow regime. The shear stress is estimated at onset of motion. Some illustrative results showing the strong impact of the angle of repose, and the exposure of the bead to shear flow are represented in both regimes.
Incipient particle motion is encountered in a wide range of industrial and natural processes. Environmental examples include the initial process of sediment transport in river and oceans, bed erosion or dune formation among others 1,2,3. Pneumatic conveying4, removal of pollutants or cleaning of surfaces5,6 are typical industrial applications involving the onset of particle motion.
Due to the broad range of applications, the onset of particle motion has been extensively studied over a century, mostly under turbulent conditions7,8,9,10,11,12,13,14,15. Many experimental approaches have been applied to determine the threshold for the onset of motion. The studies include parameters such as the particle Reynolds number13,16,17,18,19,20, the relative flow submergence21,22,23,24 or geometrical factors as the angle of repose16,18,25, exposure to the flow26,27,28,29, relative grain protrusion29 or streamwise bed slope30.
The current data for the threshold including turbulent conditions are broadly scattered12,31 and the results often seem inconsistent24. This is mostly due to the inherent complexity of controlling or determining flow parameters under turbulent conditions13,14. Besides, the threshold for sediment motion strongly depends on the mode of motion, i.e. sliding, rolling or lifting17 and the criterion to characterize the incipient motion31. The latter may be ambiguous in an erodible sediment bed.
During the last decade, experimental researchers have studied incipient particle motion in laminar flows32,33,34,35,36,37,38,39,40,41,42,43,44, where the wide spectrum of length scales interacting with the bed is avoided45. In many practical scenarios implying sedimentation, the particles are quite small and the particle Reynolds number remains lower than about 546. On the other hand, laminar flows are able to generate geometric patterns as ripples and dunes as turbulent flows do42,47. Similitudes in both regimens have been shown to reflect analogies in the underlying physics47 so important insight for the particle transport can be obtained from a better controlled experimental system48.
In laminar flow, Charru et al. noticed that the local rearrangement of a granular bed of uniformly sized beads, so-called bed armouring, resulted in a progressive increase of the threshold for the onset of motion until saturated conditions were achieved32. Literature, however, reveals different thresholds for saturated conditions in irregularly arranged sediment beds depending on the experimental set-up36,44. This scattering may be due to the difficulty of controlling particle parameters such as orientation, protrusion level and compactness of the sediments.
The main goal of this manuscript is to describe in detail how to characterize the incipient motion of single spheres as a function of geometrical properties of the horizontal sediment bed. For that purpose, we use regular geometries, consisting of monolayers of fixed beads regularly arranged according to triangular or quadratic configurations. Regular substrates similar to that we use are found in applications such as for the template-assembly of particles in microfluidic assays49, self-assembly of microdevices in confined structured geometries50 or intrinsic particle-induced transport in microchannels51. More importantly, using regular substrates allows us to highlight the impact of local geometry and orientation and to avoid any dubiety about the role of the neighborhood.
In laminar flow, we observed that the critical Shields number increased by 50% only depending on the spacing between the substrate spheres and thus on the exposure of the bead to the flow38. Similarly, we found that the critical Shields number changed by up to a factor of two depending on the orientation of the substrate to the flow direction38. We noticed that immobile neighbors only affect the onset of the mobile bead if they were closer than about three particle diameters41. Triggered by the experiment findings, we have recently presented a rigorous analytical model that predicts the critical Shields number in the creeping flow limit40. The model covers the onset of motion from highly exposed to hidden beads.
The first part of this manuscript deals with the description of the experimental procedure used in previous studies at shear Reynolds number, Re*, lower than 1. The laminar flow is induced with a rotational rheometer with a parallel configuration. In this low Reynolds number limit, the particle is not supposed to experience any velocity fluctuation20 and the system matches the so-called hydraulically smooth flow where the particle is submerged within the viscous sublayer.
Once incipient motion at laminar flow is established, the role of turbulence can become clearer. Motivated by this idea, we introduce a novel experimental procedure in the second part of the protocol. Using a Göttingen low-speed wind tunnel with open jet test section, the critical Shields number can be determined in a wide range of Re* including the hydraulically transitional flow and the turbulent regime. The experimental results can provide important insight about how forces and torques act on a particle due to the turbulent flow depending on the substrate geometry. Besides, these results can be used as a benchmark for more sophisticated models at high Re* in a similar way that past work in laminar flow has been used to feed semi probabilistic models52 or to validate recent numerical models53. We present some representative examples of applications at Re* ranging from 40 to 150.
The incipient criterion is established as the motion of the single particle from its initial equilibrium position to the next one. Image processing is used to determine the mode of onset of motion, i.e. rolling, sliding, lifting39,41. For that purpose, the angle of rotation of mobile spheres that were manually marked is detected. The algorithm tracks the position of the marks and compares it with the center of the sphere. A preliminary set of experiments was conducted in both experimental set-ups to clarify that the critical Shields number remains independent of finite size effects of the set-up and relative flow submergence. The experimental methods are thus designed to exclude any other parameter dependent on the critical Shields number beyond Re* and geometrical properties of the sediment bed. The Re* is varied using different fluid-particle combinations. The critical Shields number is characterized as a function of the burial degree, , defined by Martino et al.37 as where is the angle of repose, i.e. the critical angle at which motion occurs54, and is the exposure degree, defined as the ratio between the cross-sectional area effectively exposed to the flow to the total cross-sectional area of the mobile bead.
1. Incipient Particle Motion in the Creeping Flow Limit.
NOTE: The measurements are conducted in a rotational rheometer that has been modified for this specific application.
2. Incipient Particle Motion at the Hydraulically Transitional and Rough Turbulent Regime.
NOTE: The measurements are conducted in a customized low-speed Wind-tunnel with open jet test section, Göttingen type.
Figure 1(a) represents a sketch of the experimental set-up used to characterize the critical Shields number in the creeping flow limit, Section 1 of the protocol. The measurements are conducted in a rotational rheometer that was modified for this specific application. A transparent Plexiglas plate of 70 mm in diameter was carefully fixed to a parallel plate of 25 mm in diameter. The inertia of the measuring system was therefore readjusted before the measurement. A customized circular container of 176 mm in diameter with transparent walls was concentrically coupled to the rheometer. A vertical cut was performed in the front section. A microscope slide was carefully fixed to the front section to improve the imaging. The gap setting profile was readjusted to take into account the presence of the container. The plate velocity was minimized close to the fluid interface to avoid the bead movement before starting the measurement. In that system, the single bead can be optically tracked from the top through the transparent plate, see Figure 1(b), or from the side through the transparent sidewalls, see Figure 1(c). A Couette flow profile is induced between the rotating plate and the substrate. Therefore, the critical shear rate is given by . Accordingly, the critical Shields number and the shear Reynolds number can be defined as in Eq. 1 and Eq. 2, respectively. The set-up used in Section 2 of the protocol is illustrated in Figure 1(d). The measurements are conducted in a customized low-speed Wind-tunnel with open jet test section, Göttingen type. The regular substrates of 19 x 25 cm2 are located in the middle of the test section. The fan speed and thus the fluid velocity is regulated with a frequency converter connected to the blower fan. A turbulent boundary layer is induced above the regular substrate. The velocity profile is measured with a hot wire miniature probe specialized designed for measuring the boundary layer (see Figure 1(e)) coupled to a constant temperature anemometer (CTA). The wall-normal position, y, is controlled with a vertical stage that can be repositioned within approx. 0.01 mm. The position is measured with a digital level indicator with a resolution of 0.01 mm. In the fully rough turbulent regime (typically Re*>70), the shear velocity can be inferred from a fit of the experimental data to the logarithmic wall law, Eq. 559. In the hydraulically transition regime, the shear velocity is inferred from a fit to the modified wall law, Eq. 758. The critical Shields number and the shear Reynolds number can be obtained from the shear velocity as expressed in Eq. 8 and Eq. 9, respectively.
Figure 1: Sketch of the experimental set-up used at laminar conditions (a). A mobile bead of (405.9 ± 8.7) µm diameter resting on the quadratic substrate made of spheres of same size with a spacing of 14 µm between them viewed from the top (b) and from the side (c), respectively. Sketch of the experimental set-up used at turbulent conditions (d). Two mobile beads of (3.00 ± 0.15) mm and (5.00 ± 0.25) mm resting on a quadratic substrate with no spacing between spheres of (2.00 ± 0.10) mm close to the miniature hot-wire probe (e). The probe is placed at a distance of approximately 0.05 mm from the top of the substrate sphere. Figure 1(d) is reproduced from Agudo et al. 2017a39, with the permission of AIP Publishing. Please click here to view a larger version of this figure.
An image process routine that analyzes marked beads was developed in previous studies39 to calculate the angle of rotation of the bead at onset of motion. Figure 2 and Figure 3 depict examples of applicability at laminar, Re* = 0.06, and turbulent conditions, Re* = 87.5, respectively. Using marked spheres, we obtained the same critical Shields number as for beads without marks within measurement uncertainty. Based on Canny edge detection and Hough transform, the routine is able to recognize the bead with relative uncertainties ranging between 1.2 and 4%39. The rotation angle is determined by tracking marks based on a gray-scale thresholding. The uncertainty, in this case, increases up to absolute values ranging from 7° to 17°, depending on the imaging system39. Snapshots in Figure 2(a) – (f) illustrate representative examples for the single glass bead of (405.9 ± 8.7) µm displacing from its initial equilibrium position to the next one on a quadratic substrate made of beads of same size with a gap of 14 µm between spheres. The video has been recorded from the top through the transparent measuring system as described in Section 1 (see step 1.2.3). Figure 2(g) shows the angle of rotation during the displacement as a function of the curved trajectory along the substrate (see inset of Figure 2(g)). The trajectory is normalized to the distance traveled by the bead along the curved path between two equilibrium positions, . The dotted line in Figure 2(g) represents the angle for pure rolling. The single bead experiences a total rotation of (140 ± 8.5)° which coincides with the angle for pure rolling motion, which also has a value of approximately 140°. Rolling is thus the mode of incipient motion and Eq. 1 can be used to characterize the incipient particle motion.
Figure 2: Snapshots during the incipient motion of a marked bead of (405.9 ± 8.7) µm diameter on the quadratic substrate with a spacing of 14 µm at Re* of approximately 0.06 (a)-(f). The red cross and the green line represent the center of the sphere, and the bead contour obtained from the algorithm, respectively. The blue circles represent the trajectory of the geometric center of the mark. Flow from left to right. The snapshots are reproduced from Agudo et al (2017)a39, with the permission of AIP Publishing. Angle of rotation as a function of the curved trajectory along two equilibrium positions (g). The time instances of snapshots are indicated in the diagram. The dotted line indicates the angle of rotation for a pure rolling motion. Figure 2(g) is reproduced from Agudo et al (2014)41, with the permission of AIP Publishing. Please click here to view a larger version of this figure.
Snapshots in Figure 3(b) – (e) depict an example for an alumina bead of (5 ± 0.25) mm displacing four positions over a quadratic substrate made of spheres of (2.00 ± 0.10) mm with no-gap between them. The video has been recorded from the side as in Section 2 (see steps 2.2.1-2.2.4). The measured angle agrees with the theoretical one during a path covering approximately the first two equilibrium position (see Figure 3(g)). Hence, rolling is assumed to be the mode of incipient motion and Eq. 8 can be used to calculate the critical Shields number. After the second equilibrium position, however, the measured rotation angle seems to deviate from pure rolling motion. The red line in Figure 3(f) represents the bead trajectory during a longer path of approximately 17 positions over the substrate. From the trajectory, it can be discerned how the particle experiences small flights during its motion along the substrate.
Figure 3: Snapshots during the incipient motion of a marked bead of (5.00 ± 0.25) mm diameter on the quadratic substrate with no spacing between spheres at Re* of approximately 87.5 (a) – (e). The red cross and the green line represent the center of the sphere, and the bead contour obtained from the algorithm, respectively. The blue circles represent the trajectory of the geometric center of the mark. The red crosses in (f) represent the trajectory of the bead center along approximately 17 positions along the substrate. Flow from left to right. Angle of rotation as a function of the curved trajectory along four equilibrium positions (g). The time instances of snapshots are indicated in the diagram. The dotted line indicates the angle of rotation for a pure rolling motion. Please click here to view a larger version of this figure.
Figure 4(a) illustrates the square-wave test to estimate the frequency response of the CTA at critical free-stream velocity for the alumina bead of (5 ± 0.25) mm (see step 2.3.5). The time required for the voltage to drop by 97%, , is about 0.1 ms. Accordingly, the frequency response, given by 60, results in approximately 7.7 kHz. From Figure 4(a), it can be discerned that the undershoot remains well below the 15% of the peak response. This indicates that the hot-wire CTA parameters, including the overheating ratio, are properly tunned61. The calibration curves for the illustrative example are shown in Figure 4(b) before (red squares), and after the measurements of the velocity profile (black circles). Both curves overlap each other indicating that no changes occurred during the experiment. For the alumina bead of (5 ± 0.25) mm, the time-averaged velocity and the root square velocity are plotted as a function of the normalized wall-normal component in Figure 4(c) and 4(d), respectively. They are obtained as described in steps from 2.5.1 to 2.6.1 of the protocol. Both velocities are normalized with the critical free-stream velocity. From the maximum value in , it can be shown that the measured viscous sublayer thickness is approximately 0.25 mm. The solid line in Figure 4(c) represents a fit to the experimental data according to the logarithmic velocity law, Eq. 5, while the blue line represents a fit of the data according to the modified velocity law proposed by Rotta20,58, Eq. 7. In this case, both fit are in good agreement since the viscous sublayer represents just a 5% of the mobile bead diameter. Accordingly, the shear velocity obtained from both fits differs by less than 8%. Figure 4(e) illustrates the action of fluctuating forces on the incipient motion from the energy criterion perspective as stated by Valyrakis et al. 201362. The solid line shows a portion of the temporal history of the cube of instantaneous streamwise velocity, , measured at a distance of half the mobile alumina bead diameter from the substrate. The velocity was stored at a sampling rate of 25 kSa for this specific measurement. The blue line represents the cube of the averaged velocity, . The red dotted line represents the cube of the critical velocity calculated as in Valyrakis et al. 201163
(10)
where is the hydrodynamic mass coefficient, approximately equal to 1 in our experiments, and is the drag coefficient assumed to be 0.9 as considered in Valyrakis et al. 201163. and are calculated as shown in Eq. 11 and 12, respectively. The instantaneous flow power is a linear function of the cube of velocity62. Therefore, peaks on above the critical value may be considered as a potential trigger for incipient particle motion if the duration of those flow events last enough62. The self-developed algorithm estimates the duration of energetic flow events by evaluating the intersection of with the horizontal line along the entire experiment. In the illustrative experiment depicted in Figure 4, the duration of energetic flow events is of the order of 1-2 ms with a maximum of 2.1 ms.
Figure 4: Representative results obtained with the hot wire CTA in the test section of the low-speed wind tunnel at onset of motion of the alumina bead of (5 ± 0.25) mm resting on a quadratic substrate with no spacing between spheres (a) Frequency response of the CTA after a square-wave test (b) Calibration curves before (red squares), and after the measurements of the velocity profile (black circles). The solid line indicates a third polynomial trend fit to the data. The fit coefficients are depicted in the inset of the figure (c) Time-averaged streamwise velocity profile. The solid line and blue symbols indicate a fit according to the logarithmic, and modified wall law, respectively (d) Root mean square streamwise velocity profile within a small height range. The measured viscous sublayer is about 0.25 mm (e) A portion of the temporal history of the cube of instantaneous streamwise velocity measured at a distance of half the mobile alumina bead diameter from the substrate. The blue line indicates the cube of the time-averaged streamwise velocity. The red dotted line indicates the cube of the critical velocity calculated as in Valyrakis et al. 201164. Please click here to view a larger version of this figure.
Figure 5(a) represents the critical Shields number dependency as a function of the burial degree defined as Martino et al. 2009 by 37. The symbols marked in red are the threshold obtained from the illustrative examples in the protocol. The angle of repose and the exposure degree are geometrically coupled in our regular structures. The angle of repose can be analytically computed as follows:
(11)
where the superscript refers to the triangular geometry and refers to the quadratic geometry with spacing between spheres. Similarly, the exposure degree defined as the cross-sectional area exposed to the flow yields:
(12)
where is the angle the angle between the bead surface at effective zero level and the vertical axis (see inset of Figure 5). For the triangular and the quadratic substrate with spacing between spheres, it can be shown that:
(13)
where is an effective zero level below the top of the substrate (see inset of Figure 5). In the creeping flow limit, numerical simulations show that the effective zero level increases linearly with the spacing : . At larger Re*, the effective zero level is assumed to be constant as experimentally shown by Dey et al. 201264. For Re* ranging between 40 and 150, the shear stress was inferred using the modified wall law that includes the hydraulically transitional flow regime. The solid and dotted line are power trends fitted to the experimental data. As shown in Figure 5, the critical Shields number increases as a function of the burial degree showing the strong influence of partially shielding the particle to the shear flow. This includes comparing triangular to quadratic substrate configurations and different mobile beads diameters. The influence of the sediment bed geometry seems to be more pronounced at higher Re*. For the same degree of protrusion, the critical Shields number at Re* below 1 remains well above the value at Re* ranging between 40 and 150.
Figure 5: Dependence of the critical Shields number on the burial degree from laminar to turbulent flow conditions. At Re*<1, triangles, squares, circles and rhomboids indicate results obtained with triangular and quadratic substrates with a spacing of 14, 94 and 109 µm, respectively. Open and solid symbols represent experiments performed with less viscous and higher viscous oils, respectively. At 40<Re*<150, triangles and squares indicate experiments performed with triangular and quadratic substrates with no spacing, respectively. Black, blue, red, green and purple indicate experiments performed with glass, steel, alumina, polystyrene sulfonate, and Plexiglas, respectively. The data at Re*<1 are reproduced from Agudo et al (2012)38, with the permission of AIP Publishing. Please click here to view a larger version of this figure.
We present two different experimental methods for characterizing the incipient particle motion as a function of the sediment bed geometry. For that purpose, we use a monolayer of spheres regularly arranged according to a triangular or quadratic symmetry in such a way that the geometrical parameter simplifies to a single geometry. In the creeping flow limit, we describe the experimental method using a rotational rotameter to induce the laminar shear flow as in previous studies39,40,41. Preliminary experiments showed that the incipient motion remained independent of finite-size effects of the substrate such as the radial location, or the distance from the upstream border of the substrate38. Similarly, the critical Shields number was found to be independent of the relative flow submergence within an interval ranging between 2 and 12 and independent of inertia up to a of about 338. Above this value, an increase in the critical Shields number was observed as a consequence of interferences due to a secondary flow induced by the rotating plate. This factor limited the maximum for the experimental procedure described in the first part of the manuscript. The second experimental method is designed to address the hydraulically transitional and the rough turbulent flow regime. The shear stress is induced by a low-speed wind tunnel. In order to establish a range of parameters independent of any size or boundary effect of the substrate, we conducted measurements of the turbulent boundary layer at distances of 50, 80, 110, 140, 170 and 200 mm from the leading edge. At 50, 80, 110 and 200 mm, the boundary layer was measured at 4 different positions in the width direction, 55, 65, 95 and 125 mm from one of the substrate borders. At 140 and 170 mm, the boundary layer was measured at two different positions in the width direction, 65 and 95 mm from one of the substrate border. All measurements were performed at critical free-stream velocity conditions, for a (5.00 ± 0.25) mm glass bead resting on a triangular substrate made of (2.00 ± 0.10) mm beads.Within the interval ranging between 80 and 200 mm, the shape factor ranged between 1.3 and 1.5 as expected for turbulent boundary layers57. The velocity profiles at the same distance from the leading edge were in good agreement with each other, revealing logarithmic coefficients that vary from 5% up to 10% independent of the width direction. The selected range of parameters in the description of the protocol is carefully chosen to ensure that the critical Shields number remained independent of any boundary effect of the experimental set-up. This holds true for both experimental methods.
The threshold for incipient motion depends on the mode of motion that in turn is a function of geometrical properties of the bed such as the exposure of the particle. At high Reynolds numbers, incipient motion is likely to happen by rolling if the particle is highly exposed to the flow14,65. For individual particles that are almost completely shielded by neighbors, however, lifting may be a more appropriate mode14. At laminar conditions, the situation simplifies since lift forces are usually neglected16,17,40,44,45,66 and rolling or sliding is assumed to be the main mode for incipient motion. To properly characterize the critical Shields number as a function of the bed geometry, the mode of motion must first be thoroughly analyzed. For that purpose, we recorded the particle motion and we used an image process algorithms that calculates the angle of rotation of the bead39. If this value matches the theoretical angle for pure rolling as depicted in Figure 2(g) or in the initial range of Figure 3(g), the critical Shields number can be inferred by using Eq. 1 and Eq. 8 for the Sections 1 and 2 of the protocol, respectively. The algorithm identifies particle positions and marks to study rotatory and sliding motion with a minimum of man interventions. The tracking of the particle is based on a Canny edge detector and Hough transform. This combination has been proven to provide a robust and reliable tool in studying granular transport processes1,39,67,68. On the other hand, the mark detection is based on simple gray-scale thresholding. The main disadvantage of the algorithm is that the threshold must be readjusted depending on the imaging system. Although the algorithm takes into account geometrical penalties s the marks, the tracking is more susceptible to errors caused by different threshold levels and light intensity fluctuations, as can be seen, for instance, from the blue circle indicating the centroid of a mark close to the bead center in the snapshot of Figure 3(e) and 3(f). For further applications, we propose to use cross-correlation techniques to detect mark displacements between subsequent frames. This may allow us to achieve a sub-pixel resolution69 and may improve the detection of the angle when many marks are present.
Different definitions for the particle threshold are found in literature. At laminar conditions, as considered in Section 1, the critical Shields number as a dimensionless parameter for the onset of motion is usually defined as stated in Eq. 1, i.e. with the characteristic shear stress as 32,34,36,70. Other dimensionless parameters as the Galileo number are also found in laminar flow37. This choice, however, might seem adequate at higher particle Reynolds numbers where inertia is more relevant than friction. The definition given in Eq. 1 seems to be particularly adequate in the creeping-flow limit where it has been shown that a deterministic modeling approach is valid when the geometrical parameter is simplified to a regular structure40. This statement is in agreement with maximum standard deviations of the order of 5-7% as measured with the experimental system described in Section 1. The standard deviation as estimated in the step 1.4.2.3, characterizes the random error associated with the rheometer and with fluctuations due to local imperfections on the substrate or in the bead size. Note that fluctuations in the hydrodynamic forces are not expected at Re* below one. Using the quadratic substrate with a spacing between beads of 14 µm, we obtained a critical Shields number equal to 0.040 ± 0.00238. The standard deviation was determined taking into account all individual measurements of Figure 5, i.e., five different runs for each material combination in three different local positions. Values up to 7% for the standard deviation are found for other substrate configurations demonstrating the precision of the method. It is worth here to remark, that apart from deviations in the wire mesh size, the substrates sometimes present larger local imperfections such as cavities where the fixed bead has been detached or such as variations in height. A visual inspection of both top and side camera is therefore recommended before starting the measurement. High-resolution laser 3D printing may be used to build up the substrates in further applications where a sub-micron resolution is required.
When the bead is partially or fully exposed to the turbulent flow, as considered in Section 2, the role of peak turbulent-velocity values and its duration must be considered when we try to identify the incipient particle motion. The impulse14,71 or the energy criterion62 appear as a valuable alternative to the classical Shields criterion. They propose that apart from the hydrodynamic force, the characteristic time scale of flow structures must be properly parametrized71. For that purpose, the same algorithm that obtains time-averaged and root mean square velocities, estimates the duration of energetic flow events based on the condition . For the illustrative experiment of Figure 4(e), the duration of energetic flow events remains of the order of 1-2 ms. If we used a drag coefficient given by in Eq. 10 as suggested by Vollmer and Kleinhans 200713 or Ali and Dey 201620 based on Coleman's experiments72, the modified remains above the previous value, and the measured maximum duration decreases to about 1.6 ms. In any case, the duration remains well below the order of 10 ms as observed in previous experiments of Valyrakis, Diplas et al. 2013 conducted in a water channel62. In addition, the algorithm determines the integral length scale as shown by El-Gabry, Thurman et al. 201473 based on Roach's method74. At a distance of half the bead diameter from the substrate, the estimated macro-scale length scale is about 1.5 mm. It has been shown that most of the energetic events able to trigger the incipient motion should have a characteristic length of about two to four particle diameters62. This statement may thus indicate that the energetic events induced in our low-speed wind tunnel are not able to trigger the incipient motion. This is in agreement with an averaged velocity slightly above the critical value as shown in Figure 4(e), and with standard deviations below 8% in for 5 mm beads independent of the material as noticed in the experiments. The standard deviation in as calculated in steps 2.2.5-2.2.6 provides an estimation of the random fluctuation associated with the flow parameters but also to local imperfections on the regular substrate. For the alumina bead of 5 mm diameter, we obtained an equal to 12.30 ± 0.23 m/s. This standard deviation was determined taking into account 10 individual runs in three different positions at the same distance from the leading edge. For beads of 2 mm, the standard deviation increases up to approximately 14%. In view of this results, we decided to use the Shields criterion with a critical Shields number as defined in Eq. 8 to characterize the incipient motion. In addition, instead of presenting a probability of entrainment, we opt to provide a specific value of the critical Shields number with a representative degree of uncertainty. There are two main sources of uncertainty in Eq. 6 in order to evaluate the shear velocity: and . The relative uncertainty on is inferred from the standard deviation of the measurements. The relative uncertainty in is related to the measurement of the turbulent boundary layer. At the same distance from the leading edge, typical deviations on the fit coefficient range between 5 and 10% depending on the fan speed that in turns depends on the substrate geometry and the bead density. The relative uncertainty in was assumed to be 10% in the most conservative analysis. Accordingly, the uncertainty of ranges between 7 and 18% depending on the experiment. Error bars in Figure 5 display the uncertainty of the Shields number after applying the aforementioned analysis including relative uncertainties on the particle diameter, and air and particle densities.
The experimental protocol allows the characterization of the incipient particle motion as a function of the burial degree in different flow regimes. The use of regular geometries simplifies the geometrical factor to a single geometry and avoids any doubt about the role of the neighborhood. The criterion for incipient motion is satisfied when the bead moves from its initial position to the next equilibrium one. The use of an image processing algorithm clarifies the mode of incipient motion. The experimental method described in Section 1 of the protocol has been used in previous studies to point out the strong impact of the local bed arrangement on the incipient motion under laminar conditions38,39,40,41. The system, however, was limited to Re* below 3. At higher Re*, we propose a new experimental method that permits us to address the hydraulically transitional and the rough turbulent flow regime. Interestingly, the turbulence characteristics of the system in conjunction with a simplified geometrical parameter allows us to characterize the incipient motion with a critical Shields number with uncertainties that ranges between 14 and 25%. We present just some representative examples of the application at Re* ranging between 40 and 150. As a future scope of the research study, a broader range of the Re* must be covered with special emphasis on the hydraulically transitional flow regime where fewer data are available in literature. Similarly, experiments at larger burial degrees should be conducted. These results may be used as a benchmark for more complex models. For example, the realistic model recently proposed by Ali and Dey 2016 is based on a hindrance coefficient that is inferred from experimental results only for the case of closely packed sediment beads20. Experimental results for particles that are less exposed to the flow as addressed in the creeping flow limit may trigger an extrapolation of the model at larger burial degrees. In addition, the proposed experimental method may permit us to emphasize on the role of turbulent coherent structures on the incipient particle motion with a strong simplification of the geometrical factor. This is still poorly understood in literature.
The authors have nothing to disclose.
The authors are thankful to unknown referees for valuable advice and to Sukyung Choi, Byeongwoo Ko and Baekkyoung Shin for the collaboration in setting up the experiments. This work was supported by the Brain Busan 21 Project in 2017.
MCR 302 Rotational Rheometer | Antoon Par | Induction of shear laminar flow |
Measuring Plate PP25 | Antoon Par | Induction of shear laminar flow |
Peltier System P-PTD 200 | Antoon Par | Keep temperature of silicon oils constant in the system at laminar flow |
Silicone oils with viscosities of approx. 10 and 100 mPas | Basildon Chemicals | Fluid used to induced the shear in the particles |
Soda-lime glass beads of (405.9 ± 8.7) μm | The Technical Glass Company | Construction of the regular substrates for laminar flow conditions |
Opto Zoom 70 Module 0.3x-2.2x | WEISS IMAGING AND SOLUTIONS GmbH | Imaging system for recording the bead motion in the rheometer |
2 x TV-Tube 1.0x, D=35 mm, L=146.5 mm | WEISS IMAGING AND SOLUTIONS GmbH | Imaging system for recording the bead motion in the rheometer |
UI-1220SE CMOS Camera | IDS Imaging Development Systems GmbH | Imaging system for recording the bead motion in the rheometer |
UI-3590CP CMOS Camera | IDS Imaging Development Systems GmbH | Imaging system for recording the bead motion in the rheometer |
Volpi IntraLED 3 – LED light source | Volpi USA | Imaging system for recording the bead motion in the rheometer |
Active light guide diameter 5mm | Volpi USA | Imaging system for recording the bead motion in the rheometer |
300 Watt Xenon Arc Lamp | Newport Corporation | Imaging system for recording the bead motion in the rheometer |
Wind-tunnel with open jet test section, Göttingen type | Tintschl BioEnergie und Strömungstechnik AG | Induction of turbulent flow |
Glass spheres of (2.00 ± 0.10) mm | Gloches South Korea | Construction of the regular substrates for turbulent flow conditions |
Alumina spheres of (5.00 ± 0.25) mm | Gloches South Korea | Targeted bead for experiments |
CTA Anemometer DISA 55M01 | Disa Elektronik A/S | Measurement of flow velocity in the wind tunnel |
Miniaure Wire Probe Type 55P15 | Dantec Dynamics | Measurement of flow velocity in the wind tunnel |
HMO2022 Digital Oscilloscope, 2 Analogue. Ch., 200MHz | Rohde & Schwarz | Measurement of flow velocity in the wind tunnel |
Phantom Miro eX1 High-speed Camera | Vision Research IncVis | Imaging system for recording the bead motion in the wind-tunnel |
Canon ef 180mm f/3.5 l usm macro lens | Canon | Imaging system for recording the bead motion in the wind-tunnel |
Table LED Lamp | Gloches South Korea | Imaging system for recording the bead motion in the wind-tunnel |