Eindhoven University of Technology BACHELOR Angle of repose of a sediment cone differences between underwater and dry conditions Cooijmans, Werner

(1)

Eindhoven University of Technology

BACHELOR

Angle of repose of a sediment cone

differences between underwater and dry conditions

Cooijmans, Werner

Award date:

2019

Link to publication

Disclaimer

This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

(2)

Department of Applied Physics Turbulence and Vortex Dynamics

Angle of repose of a sediment cone:

differences between underwater and dry conditions

W. Cooijmans

R-1964-B

Supervisors:

dr. M. Duran-Matute A.S. Gonzalez MSc.

Eindhoven, February, 2019

(3)

The angle of repose underwater versus dry conditioned abstract

1 Abstract

The angle of repose is the critical angle at which granular particles in an inclined slope are set into motion due to gravity. The angle of repose is a material property and is useful, for example, whenever modeling granular materials. In this report, the difference of the angle of repose in different media is studied. A force balance is used to describe the angle of repose. The angle of a particle pile was measured through the reconstruction of the surface of the pile by using a photogrammetric technique. The angle of the pile was measured both in air and in water. The angle in air and water was measured to be respectively θ = 39.9 ± 1.9 degrees and θ = 42.3 ± 1.5 degrees. This difference in the range of values of the angle between air and water can be explained by the difference in the sediment velocity while being deposited, caused by a difference in buoyancy acting on the particles.

(4)

The angle of repose underwater versus dry conditioned CONTENTS

2 Introduction

Avalanches are extreme natural events that are capable of destroying everything they cross. These avalanches can occur in mountains, deserts and even underwater, as seen in figure 2.1.

(a) (b) (c)

Figure 2.1: (a) Example of an avalanche on a mountain [4]. (b) Example of an avalanche in the desert [11]. (c) Example of an avalanche underwater [5].

For avalanches to occur, particles need to accumulate on a surface. Through the accumulation, the slope of the particle surface will become steeper and when the slope is steep enough, gravity will lead to a net force and will therefore accelerate the particles into motion. The critical point of the slope for acceleration of the particles is expressed in the angle between the horizontal plane and the slope.

For granular materials, this critical angle is called the angle of repose [8]. The angle of repose is a material property and is, therefore, essential for modelling different phenomena occurring in granular material. For example, it can be important to describe this phenomenon for constructions to take into account while designing them, as this could lead to dangerous situations due to instability or wastage.

The angle of repose is present in all kinds of materials, for example, in daily usages such as storage piles of sand and, to some extent, in piles of waste. The angle could in these cases be used to optimize usage of the space, by calculating the heights of the piles which could be made and the area being covered. The angle is present as well in sediment transport on the seabed due to, for example, the tides, waves or vortices, whereas it will determine if the formed shape of the sediment will collapse or not.

A considerable amount of research has been done on the angle of repose. This includes a review on its theory and methods to measure the angle [6], its relation with gravity [7] and its relation with some material properties, such as shape and surface characteristics [2]. In [2], it is stated that the the values for subaqueous (in water) and subaerial (in air) values for the angle are similar, but not showing facts to prove this. Therefore, research is necessary to see if this similarity is really present and to what extent.

If this similarity is present, this could for example be used in future experiments to simplify them by excluding water, with the angle of repose being unchanged.

In this report, the change in angle of repose of the same material in dry and underwater circumstances is investigated. A photogrammetric technique is used to reconstruct the surface of a sediment cone. With this reconstruction the angle of the cone is determined. In section 3 of this report, a force balance is made to explain the theory behind the angle of repose. In section 4, a description of the experimental set-up is given, including the description of the apparatus and methodology used to produce a particle cone. In the same section, an explanation of the photogrammetric technique is given. With that, the calculations to find the angle of the cone with the measured data are explained. In section 5, the results are given and discussed. Lastly, in section 6 the results are compared to theoretical values and a conclusion and some recommendations for future experiments are given.

(6)

The angle of repose underwater versus dry conditioned theory

3 Theory

For the description of the angle of repose a force balance is made. The forces involved in this balance are shown in figure 3.1 and are described below.

Figure 3.1: A two dimensional schematic overview of the force balance of a particle on an inclined plane.

It consists of an inclined plane with an angle θ with respect to the horizontal plane. The circle presents a spherical particle, which has the following forces acting on it: the normal force FN, the gravitational force parallel and perpendicular to the inclined plane, respectively Fg,k and Fg,⊥, the friction force Fs and drag Fd.

3.1 Gravity

The gravitational force F_g, also called weight, is a force pulling a particle towards the earth. If experienced by a particle on an inclined plane, which has an angle θ with the horizontal plane, it will act on the particle downwards parallel to the plane and its magnitude can then be described with:

F_g,k= ρpV g sin(θ), (3.1)

where F_g,k is the gravitational force parallel to the plane, ρp is the density of the particle, V is the volume of the particle and g is the gravitational acceleration. The force of gravity is directed parallel to the plane, towards the horizontal plane.

If the particle is submerged, an additional force should be added to the force balance, known as buoyancy.

Buoyancy is the force that is lifting a particle underwater, due to the pressure difference between the top and bottom of the particle, and it is equal to the weight that the water would have if it had the same volume as the particle. The magnitude of the buoyancy force is given by:

F_buo= ρ_waterV g sin(θ), (3.2)

with ρ_water describing the density of water. The direction of the buoyancy is opposed to the weight.

Therefore, the net gravitational force F_g,w on a particle under water is:

Fg,w = ρpV g sin(θ) − ρwaterV g sin(θ), (3.3) which can be rewritten with:

R = ρp

ρwater

− 1, (3.4)

where R can be called the fraction of density, yielding

F_g,w = ρ_waterRV g sin(θ). (3.5)

(7)

3.2 Friction

Whenever a particle makes contact with a surface, friction will slow the particle down or reduces the tendency for motion. If a particle is at rest on top of a surface, the static friction force Fsacting on the particle is:

Fs= µsFN, (3.6)

where µ_sis the coefficient of static friction and F_N is the normal force on the particle. The normal force is equal in size as the gravitational force, but in the opposite direction and can therefore be described as:

F_N = RV g cos(θ), (3.7)

where θ is the angle between the surface and the horizontal plane. If the particle is set into motion and then slides along the surface, which is called translation, the friction force acting on the particle will change and is described by:

Fk= µkFN, (3.8)

where Fk is the kinetic friction and µk the coefficient of kinetic friction. In general, this kinetic friction coefficient is smaller than the coefficient of static friction, thus µs> µk [9].

Besides translating, the particle can rotate as well. This leads to another force acting on the particle named rolling resistance. Rolling resistance is determined by the rate of deformation, roughness and shape of both the particle and surface, as well as the consistency of the particle rolling. Inconsistencies in rolling lead to slipping, which causes friction. For the case of a solid, flat and smooth surface and a solid, smooth, spherical particle which does not slip, rolling resistance is zero.

3.3 Drag

Drag F_D is the force acting on a particle in a fluid or gas, that acts in opposition to relative motion with respect to the surrounding fluid. Its magnitude is typically described as:

FD= 1

2ρwaterCDAu², (3.9)

where u is the velocity of the particle, A is the surface of the particle in the plane perpendicular to motion and CD is the drag coefficient. This drag coefficient is dependent on other properties of the particle and the medium it is moving in, such as the roughness of the surface and the viscosity of the medium.

(8)

3.4 Angle of repose

Making a force balance in the plane perpendicular to the surface for a particle at rest, as shown in figure 3.1, leads to:

F_s= F_g. (3.10)

Substituting equations (3.5) and (3.6) in (3.10) yields

µs= tan(θcr), (3.11)

where θcr is the critical angle, which is the minimum angle an inclined plane can have to set particles in motion. Since the coefficient of (static) friction is a material property and constant, this means that each material has a specific angle for which material will not be moving down the slope.

Whenever considering granular materials in a pile, the interface between each particle is different from the flat surface described before, as shown in figure 3.2 for 2 particles.

Figure 3.2: A schematic overview of granular material on a pile. The dark blue dots represent the particles. The green dashed line represents the average slope of the pile. The red dashed lines represent the slope of the interface of two particles. The normal force acting on these two particles is displayed with the arrows denoted by FN, 1 and FN, 2.

This leads to a change in the net force acting on each particle, dependent on the interface, due to the normal force acting on the particle changing in direction. For some particles this means that the net force stimulates motion. As for others, it might reduce its tendency to motion. If considering a large amount of particles, the surface of the pile can then be approximated with a flat surface, as the sum of all changes in net force with respect to the flat surface is small.

For granular materials, the critical angle then changes due to the variations in the interfaces and is then called the angle of respose θ_r. As these variations lead to a small difference in net force, the following relation can be made:

µs≈ tan(θcr) ≈ tan(θr). (3.12)

(9)

The angle of repose underwater versus dry conditioned experimental set-up

4 Experimental set-up

The following section describes the set-up used, the apparatus constructed for deposition of the sediment cone, the photogrammetric technique applied and the calculations needed to get the angle of repose.

4.1 Materials

A schematic overview of the set-up used is shown in figure 4.1.

Figure 4.1: A schematic overview of the setup used for the measurements. Which includes the table with a tank (3) of 1500 × 1000 × 300 mm on it. The tank is filled with water (5) to a height of z ≈ 150 mm.

A sediment cone (4) is located within the vision range of the camera (1), which is placed at height Hcam

and the projected area from the projector (2), which is placed at height Hproj. The sight of the projector had an angle α with respect to the sight of the camera.

The set-up consisted of a 1500 × 1000 × 300 mm transparent tank which was on top of a leveled table.

Two types of measurements were done: one where the tank was empty and one where the tank was filled with tap water to an minimal height of z = 150 mm to have the complete particle pile underwater.

The density of the water is close to 10³ kg/m³ [1].

The sediment which was used to make a cone had various shapes and sizes, being most similar to cylinders, as shown in figure 4.2. The average diameter and the density of the sediment were determined in the study by Van der Linden to be 2.1 mm and 1055 kg/m³, respectively [10]. This type of sediment was also used in the studies performed by Wilting [12] and De Zwart [3].

(10)

Figure 4.2: A photo of the used sediment, with a ruler for comparison.

To measure the particle cone, a camera was used to capture images of the cone. A JAI Inc. GigE Vision camera, with a resolution of 1600x1200 pixels was used. The camera was placed at height H_cam ≈ 184 cm and aimed at the surface of the tank. This height was chosen to leave the deformation of the image, produced by the camera lens, outside of the measurement area. The measurement area was approximately half the tank, so 750x1000 mm, as shown in figure 4.1.

To project images inside the tank, a BENQ digital projector was used, which has a resolution of 1920x1080 pixels. The projector was aimed at the surface of the tank with an angle α between the camera and the projector and placed at the height Hproj ≈ 160 cm. The projection was set to be within the measurement area.

4.2 Deposition of the cone

To produce a cone of sediment the hollow cylinder method was used [6]. This method consists of filling a hollow cylinder with sediment and then lifting it to release the sediment, causing the sediment to be deposited in a conical shape. A schematic overview of the apparatus used for this method is shown in figure 4.3.

(11)

Figure 4.3: Schematic overview of apparatus for creating a sediment cone, which is connected to the tank. It contains the support (6), with a pulley (5) attached to it. A rope (3) is connected through the pulley to the cylinder (2). The cylinder is pulled upwards (4) by the rope when forming a cone of sediment (1). The cone was formed on a bendable plate (7), which was on top of a larger plate (8).

The cylinder that was used had a height of 196 mm and a diameter of 75 mm. The wall of the cylinder was 4 mm thick. The cylinder center was set below the pulley as precise as possible in a way that pulling the rope would lead to a vertical motion of the cylinder. After this, the cylinder was filled from the top with the sediment. The volume of the sediment that was used for one cone was approximately 0,3 L.

This volume was chosen to be large enough to measure the cone. Furthermore, the volume was chosen to be small enough to prevent horizontal expansion of the cone, due to the weight in the middle of the cone. The cylinder was then lifted slowly, in order to set the least motion to the medium and sediment as possible. If raising the cylinder fast, many particles move at the same time and at a higher speed, which leads to the cone being less steep. When all the sediment was released, a cone was formed. When forming the cone in water, the motion of the cylinder and the particles caused motion of the water.

Therefore, whenever motion in water occurred, it was needed to wait for the water to be at rest, as it would otherwise cause the cylinder to move while depositing the sediment and lead to a deformed cone.

A large plate of 100×100×9 mm, from now on referred to as the bottom plate, was placed inside the tank.

This plate was more accurate to measure as this plate was not transparent in contradictory to the bottom of the tank. A smaller bendable plate of 50 × 30 × 1 mm was placed on top of the bottom plate. All of the cones were formed on the bendable plate, to use this plate for removing the cone after a measurement.

For comparison of the velocity of the particles between the different media, a video was made of the deposition of the sediment cone. One was made for a cone in water and one for a cone in air. This was done by a mobile phone and was used to make an estimation of the velocity of the particles. This was done by choosing a particle which was seen clearly enough to track. Two frames were then taken with 1 second between them. For each frame the particle was marked and its distance was used to determine its velocity.

4.3 Photogrammetric technique

Photogrammetry is the science of performing measurements by making photos, especially used for determination of positions of surface points. For this experiment, a photogrammetric technique was used to determine the angle of repose for a specific type of sediment for both in air and in water. By projecting a dot on the sediment and recording an image of it, the light trajectories of the projector to the dot and from the dot to the camera can be determined. These trajectories intersect at the surface of the sediment on the position of the dot and they can be described in spatial coordinates (x, y, z) by

(12)

calibration of the camera and the projector. The determined trajectories can then be used to determine the local heights of the surface by calculating their intersection points. For the sediment in water, these trajectories change when they intersect with the water surface due to refraction and are then redirected by applying Snell’s law. By doing this, measurements can also be done for surfaces underwater. This method is used as well and described in more specific details by Wilting [12]. A short description of the used technique is given below.

Camera calibration

For this technique to work, the camera needed to be calibrated first, as the results were used in the projector calibration. By calibrating the camera, its pixel trajectories for this set-up were determined.

For the sake of simplicity, the calibration is described in a two-dimensional space. Whenever a surface with a circle on it at x = x0 is placed inside the tank at z = z0, the camera will see this circle with its center being on pixel coordinate xpixel,0 = x0, as shown in figure 4.4. Whenever this surface is lifted to spatial height z = z₁, the circle will then be on the same spatial position x = x₀, but it will then be seen by the camera on a different pixel coordinate x_pixel,1. The only exception for this is for dots located directly in the center of view of the camera. If the spatial heights and positions of the circles are well defined, a two-dimensional spatial position can be connected to a pixel coordinate for both heights.

Figure 4.4: Schematic overview of the two dimensional principle of the pixel lines of the camera. With the gray bars representing the tank (6), the dots (4) representing circles on the surface and the crosses represent the circles seen by the camera and their respective trajectory (3), placed on the lower height for readability, since the pixel coordinates do not have a height and thus are all in the same plane. The camera (1) is presented with its outer boundaries of sight (2).

This can be done for n circles and m heights, which will lead to n times m spatial positions all being connected to a pixel coordinate. For each height, a third-order polynomial surface fit can be made to define the spatial coordinate for each pixel in that surface. After defining the m fits for the m heights, the same pixel coordinates can be substituted in the all of the fits to find m spatial coordinates for that pixel. With these spatial positions, a least-square fit can be used to determine the spatial trajectory of that pixel.

For the three-dimensional case, this would mean that there will not only be a shift in the xpixel-position, but also in the ypixel-position for a change in height. The spatial coordinates will then consist of the position in height, as well as the x, y-plane. Subsequently, pixel coordinates will be expressed as (xpixel, ypixel) and spatial coordinates in (x, y, z). This leads to three-dimensional trajectories for the pixels after fitting.

(13)

The calibration was done using a calibration plate. This plate is 12 mm thick. The plate has 23 columns of 23 rows of white dots on it, with their centers having a distance of 25mm in between. An example can be seen in figure 4.5. The plate was set on four different heights, defining the height by setting the bottom of the tank to z = 0, and making sure to have the plate in the exact same positions in the x, y-plane and orientation. After that, an image was recorded for every height.

Figure 4.5: A photo made by the camera of the calibration plate placed on the bottom of the tank z = 0.

MATLAB scripts were used to find the pixel coordinates of the center of the dots and the spatial coordinates of the dots were given, through the height and the dimensions of the plate. After defining the fits of the surface for each plane, an arbitrary grid was made to define the amount and the positions of the pixel trajectories to be determined. This grid was chosen to be equal in size to the dots found on the calibration plate. The result can be seen in figure 4.6.

(14)

Figure 4.6: The lines in space constructed by the camera calibration. Each line is connected to one pixel of the camera, whereas the line represents the spatial line which is seen by that pixel. The intersection point of all the lines is the middle of the lens of the camera. The black dots are the intersection points of the lines with the horizontal planes on the heights of the calibration. One line is plotted for each dot on the calibration plate.

Projector calibration

After the camera calibration, the projector calibration was done to determine its pixel trajectories as well. Both calibrations are needed to reconstruct a surface with this method. Instead of a circle being present at the surface, a predetermined projection is now visualized by a projector aiming at and being above the surface. For sake of simplicity, the projection is chosen to be completely black with a white dot in it, with the size of one pixel, on a random position. This dot is projected along its corresponding pixel trajectory. The pixel coordinate of this trajectory (xpixel,proj, ypixel,proj) is equal to the pixel coordinate of the dot and can be determined by analyzing the projection. The spatial height z of the surface at the position of the dot is known, as this is one of the chosen heights for the surface used for the calibration.

The camera pixel coordinate of the dot (xpixel, ypixel), and thus its corresponding trajectory, can be determined by capturing an image of the surface. With the pixel trajectory of the camera and the spatial height, the spatial coordinate of the dot (x, y, z) can be calculated by calculation the intersection point of the two. If projecting the same dot at different, well-known heights of a surface, the spatial position of the dot can be determined for each height. With these spatial positions a least-square fit is made to determine the spatial trajectory of the dot for that pixel of the projector. This is displayed in figure 4.7.

(15)

Figure 4.7: Schematic overview of the two dimensional principle of the pixel lines of the projector.

With the gray bars representing the tank (7), the green and red dots (5) representing dots projected by the projector and the crosses (6) represent the dots seen by the camera and their respective trajectory (4), placed on the lower height for readability, since the pixel coordinates do not have a height and thus are all in the same plane. The camera (1) is presented with its outer boundaries of sight (8). The projector (2) is presented with trajectory lines (3) of two dots.

For the calibration, the bottom plate was set at three different heights. For each height, a specific projection was used. This projection consisted of multiple pictures with a black background and white ellipses in it. More details on this projection are given later. The pictures were projected in the same order for every height. For each projected picture an image was recorded. Two examples of these images and the shift in the positions of the ellipses can be seen in figure 4.8.

(a) (b)

Figure 4.8: Two image made of the first picture projected. In figure (a) the first picture is projected at an height z = 8 mm. In figure (b) the first picture is projected at an height z = 188 mm.

By using the images, all the spatial positions of all the ellipses were determined. With these positions the trajectories of the projector pixels were determined as described before. The result can be seen in figure 4.9

(16)

Figure 4.9: The spatial lines of the projector calibration. Each line is connected to one pixel of the projector, whereas the line represents the spatial line which is projected by the projector. The intersection point of all the lines is the middle of the lens of the projector. The black crosses on the lines are the intersection points of the lines with the horizontal planes on the heights of the calibration. One line is plotted for each dot on the calibration plate.

Patterns

As mentioned, a specific projection was used for this experiment. This projection consisted of white ellipses with a black background. The reason to project ellipses is that the projector was aimed at an angle, and its projection, therefore, was stretched. The projection was thus programmed to project ellipses, leading to circles being recorded by the camera. This stretch was small due to the projector settings being optimized as much as possible for the set-up. Therefore, the ellipses were seen by the bare eye as circles. In the remainder of the report, one projection is called a pattern. An example of a pattern is shown in figure 4.10. The patterns had a resolution of 1280x780 pixels.

(17)

Figure 4.10: An example of one pattern.

Multiple patterns were projected during one measurement. These patterns were made by a MATLAB script. This script is based on the script made by De Zwart [3]. This script determines semi-random positions for the white ellipses, from now on referred to as dots, and places them in a picture. For this experiment, all the dots together had 20000 different positions, spread over 156 patterns. An example of the distribution of these positions can be seen in figure 4.11.

Figure 4.11: An example of all the projected dots placed in one figure to show the distribution of dots.

With each red dot representing one projected dot.

The density of positions per pattern, or resolution, affects the amount of images needed to reach enough positions for a reconstruction. As each image takes a substantial amount of time to record, the amount of images and thus the amount of patterns must be minimalized. The resolution is dependent on the dot size and the distance between each dot. The dots had a diameter of 20 pixels, as this was the minimal

(18)

size for the dots to be detected in the photos by MATLAB. The size of one pixel on the bottom plate was measured by projecting a pattern and measuring the size of the pattern, divided by its amount of pixels. The size of a pixel on a height of 9 mm from the bottom of the tank was determined to be approximately 0.6 × 0.6 mm. This led to a dot having a diameter of approximately 12 mm on the bottom plate.

The maximum measurable height gradient is determined by the distance between the dots. As the dots have a different position in the picture for a different height, the position might intersect with the trajectory of another dot if the height difference is too large. For a displacement of 100 mm in height, a dot will shift approximately 50 pixels in the photo, so the minimal distance between dots was set to be 50 pixels, which is approximately between 30 mm (horizontally or vertically) and 42,5 mm (diagonally).

Reconstruction

Once the calibrations are performed, the height distribution of a surface can be measured by using the trajectories determined in the calibrations. This is done by projecting the patterns again. This does not need to be the same pattern as used in the calibration, but the pixel coordinates of the projected dots must be known. This is due to the fact that the trajectories determined by the calibrations are trajectories for the pixels from the projector, they are not connected to the pattern used for the calibration.

After projecting patterns and recording the corresponding images, the pixels of the dot centers are determined for the patterns and for the images. These pixels from respectively the projector and the camera correspond both to a trajectory determined by the calibration. The intersection point between these trajectories then relates to a spatial coordinate and represents a position on the surface. For every projected dot a position on the surface can then be calculated. With these positions the surface could then be visualized and used for further calculations.

For every measurement, a reference object was placed in the tank within the range of the projected patterns to validate the measurement for having the proper dimensions. For this experiment, a cone was used as the reference object. The height of the cone was 56 mm and the diameter 103 mm. The cone was slightly tilted on one side. The tilt on this side was 3 mm. The side which was not tilted was marked with a black minus sign. A photo of the cone and a enlarged visualization of the cone acquired by a measurement can be seen in figure 4.13 .

Figure 4.12: The reference cone which was used every measurement to validate the measurement.

(19)

Figure 4.13: One example of the reconstruction of the reference cone. The reconstruction matches the actual dimensions of the reference cone.

The error in this photogrammetric method is related to the detection method of MATLAB for finding the centers of the dots, as well as the precision during the calibration in the place of the calibration plate and in the height of both the calibration and bottom plate. The error in height was determined by using the images made for the projector calibration as a measurement. Because all the dots are placed at the same height. The result of the reconstruction should have been a flat surface, but minor changes in height are present. The estimated error of this method is then determined by taking the average of the measurement error for the three heights used in the calibration. The measurement error is calculated by dividing the difference of the maximum and minimum value by 2. The measurement error in the height of the surface is ≈ 0.9 mm for this set-up. The average corrected sample standard deviation in the height for this set-up is σ = 0.3 mm.

4.4 Calculating the angle

After making a reconstruction of a sediment cone by doing a measurement as described before, the angle of the surface of the cone with the bottom plate was calculated with the positions obtained from the measurement. This was done through multiple steps. First, data around the sediment cone was separated. Only data within a square around the sediment cone was included, where each side of the cone was approximately 200 mm from the cone to include the complete cone and a part of the flat surface. Secondly, lines were defined in the x, y-plane, which intersected with the center under the top of the sediment cone (x_top, y_top) and had a different angle β with the line y = y_top, as shown in figure 4.14.

(20)

Figure 4.14: A schematic top view of a line used to separate data. The yellow dots represent the sediment pile. The black line represents the line used for the separation of data. The red dashed line represents the line y = ytop. β represents the angle between the red and black line.

For 17 angles, β = −80 degrees to β = 80 degrees with steps of 10 degrees, the lines were defined. The data which had a distance in the x, y-plane, thus independent of their height, less then 2 mm from the line was separated. The data connected to a line is then called a data band. Then the data of one band was plotted by defining new axes for the data by rotating the x, y-axes with the same angle as used for separating the data, which leads to all data being approximately on the new x-axis and could then be plotted in a two dimensional plot. This was done to be able to make a better fit. After that, a fit of the data could be made by using the smoothing spline fit. Lastly, the derivative of this fit could be taken to determine the slope everywhere on the fit. The maximum slope of the fit could then be used to determine the angle at the location of the maximum slope, whereas the angle is the arctangent of the slope. A line with the used slope was then plotted for validation, through the point where the maximum slope was located. This is shown in figure 4.15 for clarification. For every band, the angle was determined upwards and downwards, so on the left and the right side in the figure.

Figure 4.15: A schematic overview of the analysis of a band of data to determine the angle on one side of the sediment cone. The blue dots represent the data points from the reconstruction. The orange line is the fit. The green line is the line which represents the angle found. For clarification, the slope is being shown in the figure by ∆x and ∆z. Whereas _∆x^∆z is the same as the derivative for very small ∆x.

(21)

The angle of repose underwater versus dry conditioned results

5 Results

5.1 Sediment deposition velocity

The videos were analyzed as described in section 4.2 and an example is shown in figure 5.1.

(a) (b)

Figure 5.1: A example of tracking one particle. Two frames of the video of the deposition of the sediment cone in dry circumstances are shown. Frame b shows the situation one second later than frame a. The red dot is a particle marked to display its position more clearly.

For each video, two particles were tracked on two different time intervals. The speed of the particles in dry circumstances uair was estimated to be in the range of 5 · 10⁻³ ms⁻¹ to 15 · 10⁻³ ms⁻¹. The speed of the particles in water u_water was estimated to be in the range of 1 · 10⁻³ ms⁻¹ to 5 · 10⁻³ ms⁻¹.

5.2 Reconstruction

An example of a reconstruction is shown in figure 5.2.

Figure 5.2: An example of a reconstructed surface of a measurement in water. Every blue dot represents a data point calculated for the reconstruction. The left and right cone are the sediment and reference cone, respectively. The edge of the bendable plate is seen at y ≈ 300.

This shows the two cones: the left one being the sediment cone and the right one being the reference cone. On the right side in the figure, at y ≈ 300, the edge of the bendable plate is also visible. The flat surface is not at zero, due to it being on the bottom plate, which is 9 mm thick, and the bendable plate, which is 1 mm thick.

(22)

The figure shows inconsistencies in the flat surface. This is due to the actual surface not being flat, but slightly bent due to floating of the plate or held upwards by a sediment particle below it. This is a disadvantage of the bendable plate.

5.3 Calculating the angle

When the data is separated for a band, the data is being analyzed as described in section 4.4. An example of such an analysis is shown in figure 5.3.

Figure 5.3: An example of an analysis of a band of data from a measurement with water. The blue dots represent data points calculated. The blue line on the bottom represents the lowest value found for the height and is then plotted for the whole line for clarification. The black line is the fit of the data points.

The light blue and red line are the lines for displaying the calculated angle, plotted through the point where the maximum value for the fit derivative is located.

When looking at the analyses from all the bands, it was noticed that the data on the left side of the cone, with respect to its top, was less dense than the data on the right side of the cone, as seen in figure 5.3. The side of the cone which is closer to the projector will be referred to as the front, which is the right side in the figure, and the other side as the back. The less dense data led to the fit being less consistent on the back. It can be seen that the angles located on the backside vary much more than the angles located on the front side. For this reason, only the angles calculated on the front side will be used in the further calculations.

After analyzing all the bands, the result was plotted. Figure 5.4 shows an example of a final result, where all the calculated angles are displayed by the red and black lines, including the angles located on the back of the cone.

(23)

Figure 5.4: An example of the final results from one measurement. The blue dots represent the calculated data points. The black lines represent the angles downwards. The red lines represent the angles upwards.

In total eleven measurement were done, five measurements underwater and six dry measurement. Two of the dry measurements were excluded in the calculations of the average angle of the cone. This is due to the sediment cone not being properly formed. In the reconstruction, it can be seen that the cone is skewed, probably due to the cylinder not being straight while depositing the sediment. After that, every band was checked for consistency and having enough data for the fit to be correct. The bands which did not meet that requirement were excluded from the average angle of the cone. This lead to less then 17 angle values per measurement.

The average is taken from all of the angles measured for each medium, after applying the mentioned exclusions. The average angle calculated for the measurements without water was based on 59 measured angles and is calculated to be θ = 39.9 ± 1.9 degrees, with the error being presented by the corrected sample standard deviation for this average. The average angle calculated for the measurements underwater was based on 60 angles measured and is calculated to be θ = 42.3 ± 1.5 degrees, with the error being presented by the corrected sample standard deviation for this average.

(24)

The angle of repose underwater versus dry conditioned conclusion

6 Conclusion

The angle of repose for the measurement in air was found of θ = 39.9 ± 1.9 degrees and the angle of repose for the measurements in water was found of θ = 42.3 ± 1.5 degrees. As the range of these values intersect, the results should be interpreted with caution. According to the theory described in section 3, there should not be a difference in the angle of repose for different media, as the angle of repose is not dependent on the medium that it is located in.

If looking at the method of deposition, the sediment is set into motion when being released from the cylinder. The motion is generated by particles settling due to gravity as shown in figure 6.1.

Figure 6.1: Schematic overview of a particle being released from the cylinder in water. The net force of gravity is being displayed, as described in (3.5). The grey bars represent the cylinder. The yellow dots represent the sediment. The blue background is the water. The light grey bar represents the tank.

As the particles are set into motion due to gravity, there is a difference between underwater and dry conditions. The net gravitational force acting on the particles in water is substantially smaller than the net gravitational force acting on the particles in air. As the density of the particles is close to the density of water, buoyancy is close in size to the gravitational force. Buoyancy is acting in opposed direction to gravity and therefore the net force acting on the particles is small. The difference in the net force between underwater and dry conditions leads to a difference in the velocity of the particles. As the particles are in motion, drag is acting on the particles in opposed direction to their motion, similar to buoyancy. This drag is substantially large for the particles in water, as the density of the particles is close to the density of water. This leads to the maximum velocity of the particles in water being reduced and therefore lowering the average velocity of the particles in water. The difference in their velocity was relatively large as the velocity of the particles in air could be up to 15 times as fast as in water 5.1. With a lower velocity, the particles come at rest more easily through friction, which leads to the particles settling at an steeper slope of the surface. Therefore, the particles in water settle at a larger angle in comparison to particles in air.

Another phenomenon to take into consideration is that the particles come at rest through kinetic friction as they are in motion. Kinetic friction is smaller than static friction due to the lower coefficient of friction. This means that the particles could be at an even steeper incline when held at rest through the static friction. Therefore, this could be one of the causes that the measured angles are lower than the angle of repose.

To conclude, the difference found between the range of angles measured of the cone underwater and in air is 2.4 degrees and might be due to the deposition method causing motion to the particles. As the particles are really light and have a density comparable to water, the buoyancy acting on the particles in water has a large influence on their behaviour. The buoyancy makes the net gravitational force

(25)

The angle of repose underwater versus dry conditioned conclusion

acting on the particles smaller. As the particles are in motion, kinetic friction is the force leading to the particles come at rest. Kinetic friction is smaller than static friction and therefore the particles come at rest at a smaller angle. The angles that were calculated do not represent the actual angle of repose of the sediment, due to their motion. The calculated angles are lower, but close to the actual value.

As the particles in water have a smaller velocity, the value of the angle measured in water is the most representative for the angle of repose.

6.1 Recommendations for future research

Two other ways to measure the angle of repose are, for example, with a mould or with an plate with an adjustable incline [6]. The mould should be able to be filled with sediment and then be lifted to release the sediment. As the angle of repose is not known in many cases, multiple moulds are needed or one which is adjustable. The mould should be shaped to a form where the sediment will form an inclined surface, for example the shape of an funnel upside down. This inclined surface should start as steep as possible and be flattened every time the sediment collapses after releasing. In this method the mould is lifted and thus causes motion in the medium where it is located in, which means two options should be considered: either the sediment should be heavy enough to not be influenced by the motion or the mould should be lifted slowly to make sure the motion is not large enough to influence the sediment.

The method with the plane with an adjustable incline has the sediment on top of the plate, placed there as flat as possible. The incline should be increased until the sediment starts rolling down the surface.

In this method motion is present as well. Therefore the same options should be considered, but in this case the plate should be moving slow enough.

Another improvement could be implemented through the patterns. The patterns can be adjusted to have more space in between the dots, this means that higher changes in the height gradient of the surface could be measured. This means more patterns are needed to find the same amount of data points and will take more time to do one measurement. This could be further extended to making even more patterns, to make even more data points. This will lead to more precise measurements, especially for the calculations of the angle. As in this experiment only 20000 data points were made, while in fact 1 million could be used, since there are 1280 × 780 pixels, a lot more data points could be made.

(26)

The angle of repose underwater versus dry conditioned REFERENCES

References

[1] P. Ball. “Water - an enduring mystery”. In: (Mar. 2008). url: http://dx.doi.org/10.1038/

452291a.

[2] M. A. Carrigy. “Experiments on the angle of repose of granular materials.” In: (Jan. 1970). url:

http://dx.doi.org/10.1111/j.1365-3091.1970.tb00189.x.

[3] R.F.P.G. De Zwart. “Sediment transport by tidal dipolar vortices at the inlet of a semi-enclosed basin”. In: (June 2017).

[4] GNS IT Desk. “Avalanche warning”. In: (Feb. 2014). url: https://gnskashmir.com/2018/02/

12/avalanche-warning/.

[5] diogenes.ukr.net. “Onderwater lawine op een rotsachtige helling”. In: (). url: https : / / nl . depositphotos.com/108997834/stockvideo-onderwater-lawine-op-een-rotsachtige.html.

[6] H. Al-Hashemi and O. Al-Amoudi. “A review on the angle of repose of granular materials”. In:

(May 2018). doi: 10.1016/j.powtec.2018.02.003.

[7] M. G. Kleinhans et al. “Static and dynamic angles of repose in loose granular materials under reduced gravity”. In: (Nov. 2011). url: http://dx.doi.org/10.1029/2011JE003865.

[8] A. Mehta and G. C. Barker. “The dynamics of sand”. In: (Apr. 1994). doi: 10 . 1088 / 0034 - 4885/57/4/002. url: https://iopscience.iop.org/article/10.1088/0034-4885/57/4/002/.

[9] S.D. Sheppard, B.H. Tongue, and T. Anagnos. “Statics: Analysis and Design of Systems in Equilibrium”. In: (May 2006).

[10] S.J.A. Van der Linden. “Particle dynamics in boundary layers below a swirl flow”. In: (Oct. 2014).

[11] M. A. Wilson. “Grain flows (sand avalanches) on the slip faces of sand dunes”. In: (Mar. 2010). url:

https://en.wikipedia.org/wiki/Grain_flow#/media/File:KelsoDunesAvalancheDeposits.

JPG.

[12] T.J.S. Wilting. “Implementation of pattern matching and photogrammetric measurement techniques for bed evolution through a moving water surface”. In: (Jan. 2019).

Eindhoven University of Technology BACHELOR Angle of repose of a sediment cone differences between underwater and dry conditions Cooijmans, Werner