Why is particle movement considered random

Visually based characterization of the incipient particle movement in regular substrates: from laminar to turbulent conditions

Summary

Two different methods to characterize the incipient particle movement of a single bead as a function of the sediment bed geometry from laminar to turbulent flow are presented.

Abstract

Two different experimental methods for determining the threshold value of particle movement as a function of the geometric properties of the bed of laminar turbulent flow conditions are presented. For this purpose, the incipient movement of a single bead is examined on regular substrates, which consist of a monolayer of solid spheres of uniform size, which are regularly arranged in triangular and square symmetries. The threshold is marked by the critical number of shields. The criterion for the beginning of the movement is defined as the shift from the original equilibrium position with the neighboring one. The displacement and the mode of movement are identified with an imaging system. The laminar flow is induced using a parallel disk configuration using a rotary rheometer. The shear Reynolds number remains below 1. The turbulent flow is induced in the low-speed wind tunnel with an open jet measuring section. The air speed is regulated with a frequency converter on the blower fan. The speed profile is connected to a hot wire sensor and a hot film anemometer is measured. Reynolds' shear number varies between 40 and 150. The logarithmic velocity and the modified wall law presented by Rotta are used to derive the shear velocity from the experimental data. The latter is of particular interest when the mobile bead is partially exposed to the turbulent flow in the so-called hydraulic transition flow regime. The shear stress is estimated at the beginning of the movement. Some illustrative results show the strong influence of the angle of rest and exposure of the bead, flow to scissors are represented in both regimens.

Introduction

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Beginning particle movement has occurred in a wide variety of industrial and natural processes. Ecological examples are the first process of sediment transport in rivers and oceans, bed erosion or dune formation, among others. 1,2,3. Pneumatic conveyance4, Elimination of pollutants or cleaning of surfaces5,6 are typical industrial applications with the start of particle movement.

Because of the wide range of applications, particle motion has been in existence for more than a century, especially in turbulent conditions7,8,9,10,11studied extensively, 12,13,14,15. Many experimental approaches have been used to determine the threshold for starting movement. The studies include parameters such as the particle Reynolds number13,16,17,18,19,20who are relative submerge 21,22,23,24 or geometric factors as the angle of rest16,18,25, Exposure to the flow26,27,28,29, relative grain lead29 or streamwise bed slope30.

The current data for the threshold in turbulent conditions including12,31 are widely dispersed and the results often seem inconsistent24. This is mainly due to the complexity of controlling or determining flow parameters under turbulent conditions13,14. In addition, the threshold for sediment movement depends heavily on the type of movement, i.e. Sliding, rolling or lifting17 and the criterion of beginning movement31to characterize. The latter can be ambiguous in a bed of erosion-prone sediment.

Over the past decade, experimental researchers have studied incipient particle movement in laminar flows32,33,34,35,36,37,38,39,40,41,42,43,44where is the wide range of length scales avoiding interaction with the bed45. In many practical scenarios, sedimentation implies the particles are quite small and the particle Reynolds number remains lower than about 546. On the other hand, laminar currents are able to generate geometric patterns as waves and dunes, like turbulent currents42,47. Parables in both therapies have been shown to reflect analogies in the underlying physics47 So that important knowledge for the particle transport can be used a better experimental system48controlled.

In laminar flow, Charru noticed Et al. that the local rearrangement of a granular bed of uniformly sized beads, so-called bed reinforcement, led to a gradual increase in the threshold for the occurrence of movement until saturated conditions were reached 32. Literature, however, shows different threshold values ​​for saturated conditions in irregularly arranged sediment beds depending on the experimental setup36,44. This dispersion may be due to the difficulty of controlling particle parameters such as orientation, bulging level, and compactness of the sediments.

The main aim of this manuscript is to describe in detail how to characterize the incipient movement of individual spheres as a function of the geometric properties of the bed of horizontal sediment. For this purpose we use regular geometries, consisting of monolayers of solid spheres that are regularly arranged according to triangular or square configurations. Similar to regular substrates that we use are applications such as B. for template assembly of particles in microfluidic assays49, Self-assembly of the molding process in closed structured geometries50 or intrinsic particle-induced, please refer to transport in microchannels51. More importantly, using regular substrates allows you to highlight the effects of local geometry and orientation and avoid the moment of shock about the role of the neighborhood.

In laminar flow we observed that the critical shields of 50% only depend on the distance between the substrate and thus on the exposure of the bead to the flow38gone up. We also found that the critical number of shields was up to a factor of two depending on the orientation of the substrate on the flow direction38changed. We have found that immobile neighbors only affect the beginning of the mobile bead when they are closer than about three particles in diameter41were. Triggered by the experiment results, we recently presented a rigorous computational model that limits the critical number of shields in the creeping river40predicts. The model covers the beginning of the movement of hidden pearls very exposed.

The first part of this manuscript deals with the description of the experimental procedures used in previous studies on shear Reynolds number Re *, lower than 1. The laminar flow is induced with a rotary rheometer with a parallel configuration. In this low Reynolds number limit the particle should not be subject to any speed fluctuations20 experience and the system adapts the so-called hydraulically smooth flow, where the particle is submerged within the viscose underlayer.

As soon as there is incipient movement in laminar flow, the role of turbulence can become clearer. Motivated by this idea, we introduce a novel experimental procedure in the second part of the protocol. Use lowspeed wind tunnel Göttingen with open jet measuring section, the critical shields number in a wide range of Re * including hydraulic transition flow and the turbulent regime can be determined. The experimental results provide important knowledge about forces and torques on a particle due to the turbulent flow behavior depending on the substrate geometry. In addition, these results can be used as a benchmark for more complex models on high Re * in a similar way that previous work in laminar flow, semi-probabilistic models52 to feed or recent numerical models53has been used to check. We present some representative examples of applications at Re * between 40 and 150.

The beginning criterion is established as the movement of the particle from its initial equilibrium position to the next. Image processing is used to set the mode of the beginning of movement, i.e. determine rolling, pushing, lifting39,41. For this purpose, the angle of rotation of the mobile spheres, which were marked manually, is recognized. The algorithm determines the position of the marks and compares it with the center of the sphere. A first series of experiments resulted in both experimental arrangements to clarify that the critical number of shields remains independent of finite size effects of the set-up and relative submersion. The experimental methods are supposed to exclude thus another parameter depends on the critical shields number beyond Re * and geometric properties of the bed sediment. Re * is varied with various liquid-particle combinations. The critical number of shields is characterized as a function of the burial degree, , defined by Martino Et al.37 as Where is the angle of rest, d. H. the critical angle for which movement54 occurs, and is the exposure, defined as the ratio between the cross-sectional area effectively exposed to the flow around the total cross-sectional area of ​​the mobile bead.

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Protocol

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(1) Beginning of particle movement in the creeping flow limit.

Note: The measurements are carried out in a rotary rheometer that has been modified for this particular application.

  1. Preparation of the rheometer.
    1. Shut off the air supply to the rheometer to avoid damaging the air bearings. Open the valve next to the air filter until a pressure of approx. 5 bar is reached in the system.
    2. Manufacture the measuring plate fluid thermostat. Make sure that the tubes of the Peltier element are connected to the rheometer. Turn on the liquid circulator and set the desired temperature (20 ° C).
    3. Assemble the custom made container to the regular substrate on the rheometer.
      1. Take the regular substrate out of the container and carefully clean the surface with distilled water. Dry the surface with a lens cleaning tissue and remove any remaining dust with a blower.
        Note: The regular substrates are monolayers of 15 x 15 mm2 Constructed from spherical soda-lime glass beads (405.9 ± 8.7) µm.
      2. Using 0.4mm thick double-sided tape, secure the regular substrate in the container to ensure that the substrate center is at a distance of 21mm from the axis of rotation.
      3. Place custom adapters on the rheometer plate.
      4. Mount bespoke round containers into the panel to ensure that the flat face of the Image System stands for page recording.
        Note: Make sure that the container is completely horizontal with the water level (0.6 mm / m). To do this, place the water level on the container parallel to the rear of the device and level with the rheometer leveling feet. Repeat the process of rotating the water level 90 degrees.
    4. Turn on the rheometer. Wait until the startup process is complete and the status "Ok" appears on the device screen.
    5. Start the computer and the rheometer software. Initialize the rheometer and set the temperature controller from the software's control panel to the desired value (20 ° C).
    6. Assemble the tailor-made measuring system. Building the zero gap from the software.
      Note: Before setting the zero gap, ensure that there are no mobile beads on the substrate and that substrate boundaries are not bent. An error in establishing the zero distance leads to a systematic error in the calculation of the shear rate and therefore in the subsequent assessment of the critical shields number. An absolute uncertainty of 0.05 mm is assumed in the gap width when calculating the critical shields.
    7. Lift the measuring plate up to 30 mm and remove it.
    8. Fill the container with approx. 70 mL of 100 mPas silicone oil. Make sure that the level of the liquid in the container stays above 2mm. The silicone oil should not cover the top of the transparent plate. Wait about 15-20 minutes for thermal equilibrium. During this time, set up the imaging systems (see step 2 from the protocol).
      Note: The temperature, which is fixed (295.15 ± 0.5) K here, takes place with a Peltier element connected to the rheometer and measured with an external thermometer. Variations of less than 0.5 K are observed during the experiments.
  2. Adjust the imaging system.
    1. Turn on the 300 W xenon arc lamp. Adjust the flexible light guide to illuminate the grain from the side through the transparent walls of the container.
    2. Adjust the LED light intensity to avoid strong light reflection on the substrate.
    3. Adjust the imaging system designed to record the particle movement from above through the transparent measuring plate.
      1. Start up the imaging software from the computer and select monochrome profile from the start dialog.
      2. Open the 768 x 576 CMOS camera from the imaging system installed on top of the container. Start the live video.
      3. Adjust the horizontal stage positioning until the reference position that was previously marked in the center of the substrate is displayed in the center of the image.
      4. Adjust the vertical positioning stage to focus on the substrate.
      5. Carefully place a marked soda-lime glass ball (405.9 ± 8.7) µm.
      6. Make sure that at least one of the marks is placed at a distance of approximately 75% of the bead radius or greater from the axis of rotation. If this is not the case, manually shift the measurement of the plate in order to move the bead to reach the next equilibrium position (see Figure 2(a) for reference).
        Note: To ensure proper monitoring during movement, mobile beads are marked with several spots separated by about 45 ° (see Figure 3(a)). The code includes a simple cash flow statement control to minimize mark misassignment in order to calculate the angle of rotation. For further details we refer to Agudo Et al. 2017-39.
      7. Open the dialog box for setting the camera parameters and adjust the frame rate to 30 fps. Adjust the exposure time to make sure that the markings stand out correctly from the bead perimeter.
        Note: The soda-lime glass sphere immersed in a silicone oil of 100 mPas requires approximately 4 seconds to move the watershed into the adjacent equilibrium position from its original position. Therefore, a frame rate of 30 fps allows an uncertainty of less than 1%.
    4. Mount the measuring plate to the rheometer.
    5. Set the measuring distance to 2 mm
      Note: The focus of the top camera must be readjusted slightly due to the presence of the plexiglass plate.
    6. Adjust the imaging system designed to capture particle movement from the side through the transparent slide.
      1. Open the 4912 x 3684 CMOS camera from the imaging system installed on the front of the container and start up the live video.
      2. Adjust the vertical and horizontal positioning stage placed parallel to the rheometer until the highlighted bead appears in the center of the image.
      3. The field of view of the upper surface of the substrate, the grain and the lower part of the measuring disc contains the modular zoom lens adjusted.
      4. Adjust the horizontal positioning stage placed perpendicular to the rheometer, focus on the bead.
      5. Open the dialog box for setting the camera parameters and adjust the frame rate to 30 fps.
  3. Determine the critical rotating speed for the start of the movement.
    1. Increase the speed n, from 0.02 to 0.05 revolutions per second in small steps of 0.00025 revolutions in the second with the rheometer software linear.
      1. In the Measurement window, double-click the control type cell and edit the speed range from 0.02 to 0.05 revolutions per second.
      2. Double click on the time setting and enter the number of the measurement points, 60 and the duration of each measurement 5 s.
      3. Create a table that shows the speed as a function of time.
    2. Open the live video from the top and side cameras. Start a video sequence from both cameras using the imaging software.
    3. Start the measurement with the rheometer software.
      Note: A preliminary test with a larger step is recommended before step 1.3.1.1 to roughly estimate the speed range at which the beginning movement will occur. At a distance of 21 mm from the axis of rotation and with silicone oil of 100 mPa · s, for example, the glass bead moves at speeds of approx. 0.035 revolutions per second. Hence, a range of 0.02 to 0.05 revolutions per second appears to be suitable for the experiment.
    4. Pay close attention to the live video from above or from the side camera and stop the measurement when the bead is displaced from its equilibrium position.Note the speed at which the pearl crosses the separatrix in the neighboring equilibrium position. Known rotation speed represents the critical rotation speed, nC.. Stop the video sequences.
      Note: Make sure that the step size is very small, that the increase in speed during the time interval that the grain requires transition from its starting position in the neighboring does not exceed 1% of the critical value.
    5. Put the bead back in its original position. This can manually move the rotating plate until the pearl displaces a position again. Repeat the experiment five times with reference to the mean critical speed and the standard deviation.
    6. Repeat steps 1.3.1, 1.3.5 with another marked grain in 2 adjacent positions in the middle of the substrate.
  4. Analysis of the data.
    1. Determine the mode of movement: analyze the order of the images previously recorded from above or from the side using the algorithm such as Agudo Et al. 201739described.
    2. Determine the critical number of shields and the shear Reynolds number.
      1. Get the critical shields number from the following equation40
        (1)
        Where come from step 1.3.4, is the kinematic viscosity, and are particles and liquid density, is the acceleration of gravity and is the mobile bead diameter, known to all of you. is the gap width, defined as the distance from the upper edge of the substrate balls on the measuring plate, i.e. 2 mm and r is the radial distance of the particle from the axis of rotation, i.e. 21 mm.
      2. Obtain the shear Reynolds number, Re * based on the shear rate from the following equations:
        (2)
    3. Repeat the process from 1.1.3 to 1.4.2 with different regular substrate.
    4. Use different bead densities and different liquid viscosities to cover a wide range from the re * from creeping flow conditions to 1.

(2) Particle movement starting the hydraulically transition and coarse Turbulent regime.

Note: The measurements are made in an adapted low-speed wind tunnel with an open jet measuring section, Göttingen type.

  1. Preparing the imaging system.
    1. Attach the square substrate in the middle of the measuring section.
    2. Place a 5 mm aluminum oxide bead previously marked on the desired starting position (110 mm from the front edge and 95 mm from the side).
    3. Connect the high-speed camera coupled to the macro lens to the computer and switch it on. The target bead clearly in the picture is adjusted with the macro lens.
    4. Initiate the imaging software on the computer. Activate "Live Camera" and "Samplerate" to 1000 fps.
    5. Turn on the LED light source and adjust the intensity as well as the focus of the camera in order to achieve a clear picture of the particle and its marks.
      Note: Make sure that at least one of the marks placed is at a distance of approx. 75% of the bead radius or larger from the axis of rotation (see Figure 3(a) for reference).
  2. Determination of the critical fan speed for the start of the movement.
    1. Set the speed of the fan well below the critical value (approx. 1400 rpm for the 5 mm aluminum oxide bead).
    2. Start the recording by pressing the trigger on the imaging software.
    3. Increase the speed in steps of approx. 4 to 6 rpm every 10 s until movement begins.
    4. Note: the critical bending speed value occurs at the beginning of the movement and stop the video sequence.
    5. Place a new marked bead on the same starting position and repeat the process from 2.2.1 to 2.2.4 ten times. Note the critical speed with every measurement.
    6. Repeat the process from 2.2.1 to 2.2.5 at the same distance from the front edge but at 65 and 125 mm from the side edge respectively. Note the critical speed with every measurement.
  3. Prepare the constant temperature hot wire anemometer (CTA).
    1. Stand by the CTA control function and set the resistance of the decade to 00.00. Turn on the main switch and wait about 15-20 minutes to warm up.
    2. Plug in the short-circuiting probe and change the CTA control function to resistance measurement. Adjust the zero ohms until the needle is in the red marking and switch back the control function to standby.
    3. Replace the shorting probe from the miniature hot-wire probe. Change the CTA control function to resistance measurement. Adjust the resistance switch until the needle is set in the red marker.
      Note: The measured resistance corresponds to the cold resistance of the miniature probe. The measured value should be in agreement with the value of the manufacturer (3.32 Ω).
    4. Turn on the CTA function and adjust the decade resistance to 5.5 Ω to achieve an overheating ratio of approx. 65%.
    5. Measure the frequency response of the CTA to the mean critical bending speed (step 2.2.4).
      1. Turn on the fan and set the speed of the fan to the critical value, approx. 1400 rpm. Turn on the oscilloscope.
      2. Turn on the square wave generator of the CTA.
      3. Initiate the oscilloscope software on the computer and open the CSV module to enable data acquisition. Select the channel (CH1) and save the recording of data i.e. time and voltage under the name of the desired file. Wait for the measurements to end (approx. 3 min).
        Note: The cut-off frequency is calculated from the response time at which the voltage dropped to a level of - 3db (see Figure 4(a)).
      4. Turn off the square wave generator and set the CTA function to standby.
  4. Calibration of the CTA.
    1. Turn on the CTA function to use. Make sure that the probe is set a sufficient height from the plate so that it is in the free online stream zone.
    2. The fan speed is set to 200 rpm. Measure the streamwise speed in the free online stream zone with the vane anemometer and read the voltage on the oscilloscope.
    3. Repeat step 2.4.2 for different speeds with a fixed step size from 50 rpm to approx. 1450 rpm (read a total of 26 times).
    4. A correlation between the RPM and the measured gratis stream streamwise speed, . Get the critical speed, , corresponding to the critical speed for the individual measurements from steps 2.2.5 to 2.2.6. Calculate the mean speed for critical free stream, and the standard deviation of the measurements.
    5. To establish a correlation between the speed and the voltage after a third degree polynomial fit:
      (3)
      Here, if you measure the streamwise speed in m / s, the voltage is measured in volts (V) and are the fit coefficients. The calibration curves are in Figure 4(b) the speed profile is displayed before and after the measurements.
  5. The streamwise speed to measure with the wall normal position under critical conditions.
    1. Remove the marked bead from the substrate.
    2. The hot wire temperature probe is placed on the desired starting position (110 mm from the front edge and 95 mm from the side), the handwheel is adjusted to the horizontal positioning stage.
    3. Carefully adjust the handwheel to the vertical positioning stage until the probe is placed as close as possible to the substrate surface. Look through the camera coupled to the macro lens to make sure the wire is not touching the substrate surface. Set the zero value in the digital display at this position.
      Attention: The hot wire is very delicate and if it touches the surface it will break. For safety reasons, we place the probe at a distance of 0.05 mm above the upper edge of the substrate sphere (see illustration 1(e) for reference). This corresponds to a normalized wall normal component Where is the measured value, is the shear rate and is the kinematic viscosity of air at operating temperature. Notice that the starting value under where the viscosity is dominant55.
    4. The speed of the fan to the mean speed at which the beginning movement occurs, see point 2.2.4. So the free stream speed is the same .
    5. Adjust the sampling rate to 1 kSa and the number of samples to 6000 on the oscilloscope (total sampling time of 6 s). Select the channel (CH1) and start the measurement. Save the recording data under the desired file name. Wait for the measurements to end (approx. 3 min).
    6. Increase the wall-normal position of the probe by an increment of 0.01 mm up to 0.4 mm, and an increment of 0.1 mm up to a height of 10 mm. This corresponds to a total of 137 points for the velocity profile curve. Save the recorded data for each altitude.
  6. Analysis of the data.
    1. Calculate the mean streamwise velocity and intensity of the turbulent for each wall-normal position.
      1. Run the self-developed algorithm to evaluate statistical quantities. Open the script and select the folder with the calibration curve and the saved data for each of the measured altitude.
        Note: The script first calculates the fit coefficients from the calibration curve as in GL. 3 shown. For each altitude, it calculates the current streamwise speed by using GL. 3 and the integral timescale calculated using the autocorrelation method56. Then he calculates on a time average, and the square root speed, , for samples that are separated by two times the integral necessary for the time averaged analysis.
      2. Plot the dimensionless perpendicular versus the dimensionless streamwise time average speed , Where is the diameter of the substrate balls. plot versus the dimensionless root square speed . Figure 4(CD) shows the results for the case of the 5 mm bead of alumina.
    2. Calculate the shear rate from the experimental data.
      1. Adjust the dimensionless time average speed with the logarithmic speed distribution57
        (5)
        Where is the shear rate is that of Kármán constant and is a constant that of Reynolds' Scissors number26depends. The solid line in Figure 4(c) is a logarithmic fit at present average speed.
        Note: From fitting to the experimental data, it can be shown that the shear forces Velocity, is given by:
        (6)
        Where is the logarithmic fit coefficient and 20.
        The viscose underlayer remains above the top of the substrate spheres in our experiments. In the strictest scenario, the law should change the speed, presented by Rotta20,58EQ. 5 to be replaced.
        (7)
        Where and . is the thickness of the viscose underlayer, which can be calculated approximately through 55.
        The algorithm directly calculates the shear rate from the accumulation of experimental data EQ. 5 and EQ. 7. The blue symbols in Figure 4(c) represent the fit to the experimental data according to GL. 7th
        For Re * over 70 represents up to 5% of the mobile bead diameter and with a bout of EQ. 5 and EQ. 7 includes a variation on within the assumed uncertainty. Compare solid line and blue symbols in Figure 4(c) at a Re * of about 87.5.
    3. Determine the mode of movement: analyze the order of the images previously taken from the page with the algorithm, as in Agudo Et al. 201739described.
    4. Determine the critical number of shields and the shear Reynolds number.
      1. Get the critical shields number from the following equation22
        (8)
        Where come from step 10.2, and are particles and liquid dense, is the acceleration of gravity and is the mobile bead diameter, known to all of them.
      2. Obtain the particle Reynolds number, Re *, from the following equations:
        (9)
      3. Repeat the procedure for measuring the velocity profile as a function of the wall-normal coordinate, step 2.5, at the same distance from the front edge but at 65 and 125 mm in the width direction, respectively.
      4. Repeat the process from 2.1 to 2.6.4.3 with different bead sizes and regular substrates.

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Representative Results

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illustration 1(a) provides a sketch of the experimental setup used to characterize the critical number of shields in the creeping river boundary, Section 1 of the protocol. The measurements are made in a rotary rheometer that has been modified for this particular application. A transparent Plexiglas plate with a diameter of 70 mm was carefully attached to a parallel plate with a diameter of 25 mm. The inertia of the measuring system was therefore readjusted before the measurement. A tailor-made round container 176 mm in diameter with transparent walls was concentrically coupled to the rheometer. A vertical cut was made in the front area. The anterior area to improve imaging was carefully attached to a microscope slide. Gap setting profile was readjusted to take into account the presence of the container. The speed of the plate was minimized in the vicinity of the liquid interface in order to avoid the bead movement before starting the measurement. In this system the individual bead can be optically traced from above through the transparent plate, see illustration 1(b)or from the side through the transparent side walls, see Image 1(c). A Couette airfoil is induced between the rotating plate and the substrate. Hence the critical shear rate is given by . Accordingly, the critical number of shields and the shear Reynolds number can be as in GL. 1 and GL. 2, or can be defined. The setup used in section 2 of the protocol is in illustration 1(d)shown. The measurements are carried out in an adapted low-speed wind tunnel with an open jet measuring section, Göttingen type. The regular substrates of 19 x 25 cm2 are in the middle of the measuring section. The fan speed and thus the fluid speed is subject to a frequency converter connected to the fan. A turbulent boundary layer is induced over regular substrates. The velocity profile is measured with a hot wire miniature probe designed for measuring the boundary layer specialized (see illustration 1(e)) coupled with a constant temperature anemometer (CTA). The wall normal position, y, is controlled with a vertical stage that can be positioned within approximately 0.01mm. The position is measured with a digital display with a resolution of 0.01 mm. In the fully crude turbulent regime (usually Re *> 70), the shear rate can be derived from a bout of the experimental data of the logarithmic wall laws, EQ-559can be derived. In the hydraulically transition regime, the shear rate is changed from a fit to the wall law, EQ. 7th58derived. The critical number of shields and the shear Reynolds number can be taken from the shear rate expressed in GL. 8 and GL. 9, respectively.


Figure 1: Sketch of the experimental setup used in laminar conditions (a). a mobile bead (405.9 ± 8.7) µm in diameter rests on the square substrate made of spheres of the same size with a distance of 14 µm between them seen from above (b) and from the side (c), respectively. Sketch of the experimental setup used under turbulent conditions (d). Two mobile beads (3.00 ± 0.15) mm and (5.00 ± 0.25) mm rest on a square substrate with no spacing between the balls (2.00 ± 0.10) mm near the miniature hot -wire probe (s). The probe is located at a distance of approx. 0.05 mm from the upper edge of the substrate sphere. illustration 1(d) is from Agudo Et al. 2017a39, reproduced with permission from AIP Publishing. Please click here for a larger version of this figure.

An image process routine that analyzed labeled beads was used in previous studies39 , calculate the angle of rotation of the bead developed at the beginning of the movement. Figure 2 and Figure 3 show examples of applicability under laminar, Re * = 0.06 and turbulent conditions, Re * = 87.5, respectively. With marked areas, we have the same critical shield number as pearls with no markings within measurement uncertainty. Based on Canny edge detection and Hough transformation, the routine is able to determine the bead with relative uncertainties between 1.2 and 4%39to recognize. The angle of rotation is determined by tracking marks based on a gray level threshold value. The uncertainty in this case increases up to absolute values ​​of 7 ° to 17 °, depending on the imaging system39. Snapshots in Figure 2(a) (f) Illustrate representative examples of the single glass bead (405.9 ± 8.7) µm displacing from its initial equilibrium position to the next on a square substrate made of beads of the same size with a spacing of 14 µm between spheres. The video has been recorded from above through the transparent measuring system, as described in section 1 (see point 1.2.3). Figure 2(G) shows the angle of rotation during displacement as a function of the curved trajectory along the substrate (see box from Figure 2(G)). The trajectory is normalized to the distance traveled through the pearl along the curved path between two equilibrium positions, . The dashed line in Figure 2(G) the angle stands for pure rolling. The single pearl experiences a complete rotation of (140 ± 8.5) ° which coincides with the angle for pure rolling movement, has a value of approx. 140 °. Rolling is therefore the type of movement beginning and GL. 1 can be used to characterize the incipient particle movement.


Figure 2: Snapshots during the beginning of the movement of a marked bead (405.9 ± 8.7) µm diameter on the square substrate with a distance of 14 µm at Re * of approx. 0.06 (a) - (f). The red cross and the green line represent the center of the sphere, and the pearl contour obtained from the algorithm respectively. The blue circles represent the trajectory from the geometric center of the mark. Flow from left to right. The snapshots are taken from Agudo Et al (2017) a39, with permission from AIP Publishing. Angle of rotation as a function of the curved path along two equilibrium positions (g). The time instances of snapshots are shown in the diagram. The dotted line shows the angle of rotation for a pure roll motion. Figure 2(G) reproduced after Agudo Et al (2014)41, with permission from AIP Publishing. Please click here for a larger version of this figure.

Snapshots in Figure 3(b) (e) show an example of an alumina bead (5 ± 0.25) mm displacing four positions over a square substrate made of spheres (2.00 ± 0.10) mm with no-gap in between. The video was recorded from the side as in Section 2 (see steps 2.2.1-2.2.4). The measured angle agrees with the theoretical one while a path covers roughly the first two equilibrium positions (see Figure 3(G)). Therefore roles are believed to be the mode of incipient movement and GL. 8 can be used to calculate the critical shield number. After the second equilibrium position, however, the measured angle of rotation 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 above the substrate. From the path it can be seen how the particle experiences small flights during its movement along the substrate.


Figure 3: Snapshots during the beginning movement of a marked caterpillar (5.00 ± 0.25) mm diameter on the square substrate with no distance between the balls at Re * of about 87.5 (a) - (e). The red cross and the green line represent the center of the sphere, and the pearl contour obtained from the algorithm respectively. The blue circles represent the trajectory from the geometric center of the mark. The red crosses (f) represent the trajectory of the center bead along approximately 17 positions along the substrate. Flow from left to right. Angle of rotation as a function of the curved path along four equilibrium positions (g). The time instances of snapshots are shown in the diagram. The dotted line shows the angle of rotation for a pure roll motion. Please click here for a larger version of this figure.

Figure 4(a) illustrates the square wave test to estimate the frequency response of the CTA with critical gratis stream speed for the alumina bead (5 ± 0.25) mm (see step 2.3.5). The time required for the tension of 97%, will fall , it's about 0.1 Ms. corresponding to the frequency response given by 60, results in approx. 7.7 kHz. Out Figure 4(a)it can be seen that the shortfall remains well below 15% of the top answer. This shows that the hot wire CTA parameters, including the superheat ratio, are properly tunned61. The calibration curves for illustrative example are in Figure 4(b) Shown before (red squares), and after the measurements of the speed (black circles) profile. Both curves overlap indicating that there were no changes during the test. For the alumina bead (5 ± 0.25) mm, the time average velocity and the square root velocity as a function of the normalized wall-normal component in Figure 4(c) and 4 letter d, or shown. You will get as described in steps from 2.5.1 to 2.6.1 of the protocol. Both speeds are normalized with the critical speed gratis stream. From the maximum value in the It can be shown that the measured viscose sub-layer thickness is approximately 0.25mm. The solid line in Figure 4(c) puts a fit on the experimental data according to the logarithmic rate law, EQ. 5, while the blue line a bout of data corresponding to the changed speed of Rotta20 proposed ,58, EQ. 7. In this case both fits are in good agreement as the viscose underlayer represents only 5% of the mobile bead diameter. Accordingly, the shear rate obtained by both fits differs by less than 8%. Figure 4(e) shows the effect of swaying forces on incipient motion from the energy criterion perspective, as suggested by Valyrakis Et al. 201362specified. The solid line shows part of the cube's temporal history of instantaneous streamwise speed, , measured at a distance of half the mobile alumina bead diameter from the substrate. The speed was stored at a sampling rate of 25 kSa for this specific measurement. The blue line represents the cube of the average speed, . The red dotted line represents the cube's critical speed calculated as in Valyrakis Et al. 201163

(10)

Where is the hydrodynamic mass coefficient corresponds approximately to 1 in our experiments and the CW value is considered to be 0.9 than in Valyrakis Et al. 201163accepted. and are calculated as in GL. 11 and 12, respectively, shown. The instantaneous flow power is a linear function of the cube of the speed62. Hence tips on Above the critical value, the duration of this flow events are enough as possible triggers for the beginning of particle movement62last. The self-developed algorithm estimates the duration of the energetic flow events by evaluating the point of intersection of the with the horizontal line along the entire experiment. In the illustrative experiment in Figure 4shown is the duration of the energetic flow events of the order 1-2 ms with a maximum of 2.1 ms.


Figure 4: Representative results obtained with the hot wire CTA in the measuring section of the low-speed wind tunnel at the beginning of the movement of the aluminum oxide bead (5 ± 0.25) resting on a square substrate with no distance between the spheres mm (a) Frequency response of the CTA after a square wave test (b) calibration curves before (red squares), and the measurements the speed profile (black circles). The solid line shows a third trend polynomial to match the data. The fit coefficients are shown in the inset of Figure (c) time averaged streamwise velocity profile. The solid line and blue symbols show a seat within the meaning of the law logarithmic and modified wall or (d) Root-Mean-Square streamwise velocity profile within a small height area. The measured viscous sublayer is measured at a distance of half a mobile alumina bead diameter from the substrate over 0.25 mm (e) A serving of the temporal history of the cube from instantaneous streamwise velocity. The blue line shows the cube of time averaged streamwise speed. The red dotted line indicates that the cube of the critical speed as in Valyrakis Et al. 201164calculated. Please click here for a larger version of this figure.

Figure 5(a) represents the critical shields number depending on the burial degree defined as Martino Et al. 2009 through 37. The symbols that are highlighted in red are the threshold taken from the illustrative examples in the protocol. The angle of rest and the degree of exposure are geometrically coupled in our regular structures. The angle of repose can be calculated analytically as follows:

(11)
where the superscript refers to the triangular geometry and refers to the square geometry with spacing between the spheres. Likewise, the exposure degree defined as the cross-sectional area exposed to the flow provides:

(12)
Where is the angle between the surface of the pearl with zero effect height and the vertical axis (see box of the Figure 5). For the triangular and square substrate with spacing between the spheres, it can be shown that:

(13)
Where is an effective zero level below the upper edge of the substrate (see box of Figure 5). To limit the creeping flow, numerical simulations show that the effective zero level increases linearly with distance : . With larger Re *, the effective zero level is assumed to be constant Dey Et al. 201264as shown experimentally. For Re * between 40 and 150, the shear stress was derived using the modified wall law, which contains hydraulic transition flow regimes. The solid and dotted line power trends are fitted to the experimental data. As in Figure 5shown flow the critical number increases shields depending on the burial degree shows the strong influence of the partial shielding of the particle, the scissors. These include triangular square substrate configurations and different mobile bead diameters to compare. The influence of the sediment bed geometry seems to be more pronounced at higher Re *. For the same degree of lead, the critical number of shields at Re * below 1 remains well above the value at Re * between 40 and 150.


Figure 5: Dependence of the critical shields number of burial degrees on laminar turbulent flow conditions. At Re * < 1,="" dreiecke,="" quadrate,="" kreise="" und="" rhomben="" zeigen="" ergebnisse="" mit="" dreieckigen="" und="" quadratische="" substrate="" mit="" einem="" abstand="" von="" 14,="" 94="" und="" 109="" µm="" bzw..="" offene="" und="" feste="" symbole="" repräsentieren="" experimente="" durchgeführt="" mit="" weniger="" zähflüssig="" und="" höher="" viskosen="" ölen,="" beziehungsweise.="" bei="" 40="">< re="" *="">< 150,="" dreiecke="" und="" quadrate="" zeigen="" experimente="" durchgeführt="" bzw.="" mit="" dreieckigen="" und="" quadratische="" substrate="" mit="" kein="" abstand.="" schwarz,="" blau,="" rot,="" grün="" und="" lila="" geben="" sie="" experimente="" mit="" glas,="" stahl,="" aluminium,="" polystyrol="" sulfonate="" und="" plexiglas,="" bzw.="" durchgeführt.="" die="" daten="" im="" re="" *="">< 1="" von="" agudo="">Et al. (2012)38, reproduced with permission from AIP Publishing. Please click here for a larger version of this figure.

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Discussion

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We present two different experimental methods to characterize incipient particle movement as a function of sediment bed geometry. For this purpose we use a monolayer of spheres regularly ordered according to a triangular or square symmetry in such a way that the geometric parameters of a single geometry are simplified. In the creeping flow limit we describe the experimental method using a rotatory rotameter to induce the laminar thrust flow as in previous studies39,40,41. First experiments showed that the incipient movement has effects of the substrate such as the radial position or the distance from the upstream boundary of the substrate, regardless of finite size.38stayed. Similarly the critical number of shields was found to be independent of the relative submersion within an interval ranging between 2 and 12 years and regardless of indolence up to one from approx. 338. Above this value an increase in the critical shields was observed as a result of the disturbances by a secondary current induced by the rotating plate. This factor limits the maximum for carrying out the experiment described in the first part of the manuscript. The second experimental method is intended to address the hydraulic transition and the rough turbulent flow regime. The shear stress is induced by a slow wind tunnel. In order to establish a number of parameters independent of any size or limit effect of the substrate, we took measurements of the turbulent boundary layer at a distance 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 the substrate edge. At 140 to 170 mm, the boundary layer was measured at two different positions in the width direction, 65 and 95 mm of one of the substrate boundary. All measurements were made under critical free stream speed conditions, for a (5.00 ± 0.25) mm glass bead rests on a triangular substrate made (2.00 ± 0.10) beads (mm). Within the interval between 80 and 200 mm, the form 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 between 5% to 10% regardless of the width direction. The choice of parameters in the description of the protocol is carefully chosen to ensure that the critical shields remained independent of any limit effect of the experimental number. This also applies to both experimental methods.

The threshold for the beginning movement depends on the type of movement, which in turn is a function of the geometric properties of the bed, such as the exposure of the particle. At high Reynolds numbers, incipient motion is likely to have happened by rolling as the particle flowed14,65is exposed to high levels. For the individual particles, which are almost completely shielded from their neighbors, an appropriate mode is allowed to lift14however. In laminar conditions, the situation is simplified, since the elevator usually neglects forces16,17,40,44,45,66 are and rolling or sliding supposedly the main mode for starting motion. In order to correctly characterize the critical number of shields as a function of the bed geometry, the type of movement must first be thoroughly analyzed. To do this we recorded the particle movement and we used an image process algorithms that set the rotation angle of the bead39calculated. If this value corresponds to the theoretical angle for pure rolling as shown in Figure 2 (g) or in the first row of Figure (3.g), the critical shield number can be derived using GL. 1 and GL. 8 for Sections 1 and 2 of the Protocol, respectively. The algorithm identifies particle positions and markings for rotational and sliding motion with a minimum of man-made interventions. The tracking of the particle is based on a Canny edge detector and Hough transformation. This combination has been proven to be a robust and reliable tool in the study of granular transportation processes1,39,67,68to disposal. On the other hand the marker recognition are based on simple gray level threshold values. The biggest disadvantage of the algorithm is that the threshold has to be readjusted depending on the imaging system. Although the algorithm takes account of geometric penalties s marks, the tracking is more prone to errors due to different thresholds and fluctuations in light intensity, as seen for example from the blue circle indicating the center of gravity of the mark near bead in the snapshot Figure 3 letter e and 3 letter f. For further applications, we suggest using cross-correlation techniques to detect mark shifts between subsequent images. This can allow us to have a sub-pixel resolution69 and can improve the detection of the angle when there are many markings.

Different definitions for the threshold in particles are found in the literature. In laminar conditions, as in Section 1 as the critical number of shields as the dimensionless parameter for the start of the movement is usually defined as in GL. 1, d. H. with the characteristic shear stress as 32,34 ,36,70. Laminar flow37there are also other dimensionless parameters such as the number of Galileo. However, this choice might seem appropriate at higher particle Reynolds numbers where inertia is more relevant than friction. The definition in GL. 1 seems particularly appropriate in the creeping-flow limit where it has been shown that a deterministic modeling approach is valid when the geometric parameters are of a regular structure40is simplified. This statement is in agreement with maximum standard deviations of the order of 5-7%, measured with the experimental system described in Section 1. The standard deviation, as estimated in step 1.4.2.3, characterizes the random errors associated with the rheometer and fluctuations due to local bumps on the subsurface or in the bead size. Note that fluctuations in hydrodynamic forces are not expected to be Re * below one. With the square substrate with a spacing between beads 14 µm, we obtained a critical shield number 0.040 ± 0.00238corresponds to. The standard deviation was determined taking into account all individual measurements of Figure 5, ie., five different runs for each material combination in three different local positions. Values ​​up to 7% for the standard deviation for other substrate configurations show the precision of the method to be found. It is worth noting here that, apart from variations in the wire size mesh, the substrates sometimes have larger local imperfections, such as voids where the solid bead was separated, or variations in height. A visual inspection of the top and side cameras is therefore recommended before starting the measurement.High resolution 3D laser printing can be used to build up the substrates in other applications where a sub-micron solution is required.

If the bead is wholly or partially exposed to the turbulent flow as in Section 2, the role of the turbulent velocity peak and its duration must be taken into account when trying to identify incipient particle movement. The impulse14,-71 or energy criterion62 appear as a valuable alternative to the classic shields criterion. They suggest that in addition to the hydrodynamic the characteristic time scale of the flow structures are correctly parameterized71have to be. To this end, the same algorithm that gets time averaged and root mean square velocities, estimates the duration of energetic flow events based on the condition . For the illustrative experiment of the Figure 4If we used a cw value of given, the duration of the energetic flow events remains of the order of 1-2 ms in GL. 10 as proposed by Vollmer and Kleinhans 200713 or Ali and Dey 201620 based on Coleman's experiments72who have favourited modified remains above the previous value and the measured maximum duration drops to about 1.6 Ms. in any case, the duration remains well below the order of 10 ms, as observed in previous years by Valyrakis, Experiments Diplas Et al. 2013 in a water canal62. In addition, the algorithm determines the integral length scale, such as El-Gabry, Thurman Et al. 201473 based on Roach's method74shown. At a distance of half a pearl diameter from the substrate, the estimated macro-level scale length is approximately 1.5 mm. It has been shown that most energetic events able to trigger the incipient movement have a characteristic length of about two to four particles in diameter62should have. This statement can thus show that the energetic events induced in our wind tunnel low-speed are not able to trigger the beginning movement. This is done in agreement with an averaged speed slightly above the critical value as in Figure 4(e)and standard deviations below 8% in for 5 mm beads regardless of the material as noted in the experiments. The standard deviation in as in the calculated steps, 2.2.5-2.2.6 provides an estimate of the random risk associated with the flow parameters but also local imperfections on the regular substrate. For the 5mm diameter aluminum oxide bead, we get one 12.30 corresponds to ± 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 pearls 2 mm, the standard deviation increases up to approx. 14%. In view of these results, we have decided on the shield criterion with few critical shields in the sense of GL. Use 8 to characterize the incipient movement. Instead of a probability of entrainment, we also decide to offer a certain value the critical number of shields with a representative uncertainty. There are two sources of uncertainty in GL for assessing the shear rate. 6: and . The relative uncertainty on is derived from the standard deviation of the measurements. The relative uncertainty in relates to the measurement of the turbulent boundary layer. At the same distance from the leading edge, typical deviations on the fit coefficient between 5 and 10% depending on the speed of the fan, which in turn depends on the substrate geometry and the pearl density. The relative uncertainty in was assumed to be 10% in the most conservative analysis. Accordingly, the uncertainty of the is between 7 and 18% depending on the experiment. Error indicators in Figure 5 show the uncertainty of the shields number after applying the aforementioned analysis including the relative uncertainties about the particle diameter and air and particle density.

The experimental protocol enables the characterization of the incipient particle movement depending on the burial degree in different flow forms. 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 movement is satisfied when the bead moves from its original position of the closest equilibrium one. The use of image processing algorithm clarifies the mode of beginning movement. The experimental method described in Section 1 of the protocol was used in previous studies to indicate the strong influence of the local bed arrangement on incipient movement under laminar conditions38,39,40,41. the system, however, was limited to Re * below 3. At the higher Re * we propose a new experimental method that allows to tackle the hydraulic transition and the rough turbulent flow regime. Interestingly, the turbulence properties of the system in conjunction with a simplified geometric parameter allows us to characterize the incipient movement with few critical shields with uncertainties ranging between 14 and 25%. We present just a few representative examples of application at Re * between 40 and 150. As a future area of ​​study, a wider range of Re * needs to be covered with particular emphasis on the hydraulically transition flow regimes where there is less data in the literature. Likewise, experiments should be carried out at larger grave mound degrees. These results can be used as a benchmark for more complex models. For example, the realistic model recently proposed by Ali and Dey 2016 is based on an obstacle coefficient from experimental results only for the case of densely packed sediment beads20is derived. Experimental results for particles that are less exposed to the flow because treated in the creeping flow boundary can trigger an extrapolation of the model at larger burial mound degrees. In addition, the proposed experimental method allow us to emphasize the role of turbulent coherent structures on the incipient particle movement with a great simplification of the geometrical factor. This is still poorly understood in the literature.

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Disclosures

The authors have not revealed anything

Acknowledgments

The authors are grateful to unknown referees for valuable advice and Sukyung Choi, Byeongwoo Ko, and Baekkyoung Shin for collaboration in setting up the experiments. This work was supported by the Brain Busan 21 project in 2017.

Materials

SurnameCompanyCatalog NumberComments
MCR 302 Rotational RheometerAnton pairInduction of shear laminar flow
Measuring plate PP25Anton pairInduction of shear laminar flow