
Download
XML
0.00MB

Download
PDF
0.00MB

Read Online
HTML
0.00MB
Abstract
Erosion of solid material by flowing fluids plays an important role in shaping landforms, and in this natural context is often dictated by processes of high complexity. Here, we examine the coupled evolution of solid shape and fluid flow within the idealized setting of a cylindrical body held against a fast, unidirectional flow, and eroding under the action of fluid shear stress. Experiments and simulations both show selfsimilar evolution of the body, with an emerging quasitriangular geometry that is an attractor of the shape dynamics. Our fluid erosion model, based on Prandtl boundary layer theory, yields a scaling law that accurately predicts the body's vanishing rate. Further, a class of exact solutions provides a partial prediction for the body's terminal form as one with a leading surface of uniform shear stress. Our simulations show this predicted geometry to emerge robustly from a range of different initial conditions, and allow us to explore its local stability. The sharp, faceted features of the terminal geometry defy the intuition of erosion as a globally smoothing process.
We would like to thank Alexander Petroff, Tycho Sleator, Eric Keaveny, Adam Stinchcombe, Jacob Bedrossian, and Lisa Fauci for fruitful discussions. This work was partially supported by the NFS (DMS1103876 and MRI0821520) and the DOE (DEFG0288ER25053).
I. INTRODUCTION
II. THE EXPERIMENTS
A. Experimental apparatus and methods
B. Interfacial dynamics and flow visualization
III. THE MODEL
A. The Prandtl decomposition
B. Outline of simulation method
IV. ANALYTICAL RESULTS
A. Scaling analysis for vanishing rate
B. Exact solutions for shear stress on wedges
V. SIMULATION RESULTS
A. Erosion of an initially circular body and comparison to experiments
B. Opening angle of the wedge
C. Drag on the eroding body
D. Other initial geometries
E. Perturbing the attracting geometry
VI. DISCUSSION
Key Topics
 Erosion
 61.0
 Fluid flows
 14.0
 Flow visualization
 9.0
 Navier Stokes equations
 5.0
 Reynolds stress modeling
 5.0
Figures
Experimental study of the erosion of a clay cylinder in flowing water. (a) A water tunnel provides a unidirectional flow, and the crosssection of the cylinder is photographed every minute. Flow speed is monitored using a laser Doppler velocimeter (LDV). To visualize the flow, particles are added to the water and illuminated with a laser sheet. (b) Shrinking and shape change during erosion (adapted from Ristroph et al. ^{12} ). Interfaces extracted from photographs show the crosssection of the body at intervals of 8 min. Interfacial data are extracted through 115 min (indicated by the arrow on the color bar), and the projected time at which the body would vanish entirely is t f = 140 ± 2 min.
Click to view
Experimental study of the erosion of a clay cylinder in flowing water. (a) A water tunnel provides a unidirectional flow, and the crosssection of the cylinder is photographed every minute. Flow speed is monitored using a laser Doppler velocimeter (LDV). To visualize the flow, particles are added to the water and illuminated with a laser sheet. (b) Shrinking and shape change during erosion (adapted from Ristroph et al. ^{12} ). Interfaces extracted from photographs show the crosssection of the body at intervals of 8 min. Interfacial data are extracted through 115 min (indicated by the arrow on the color bar), and the projected time at which the body would vanish entirely is t f = 140 ± 2 min.
Visualizing the flow around a cylindrical body at different times in the erosion process. Streaklines are captured by 10 ms exposure time photographs of tracer particles illuminated by a laser sheet, and the initial diameter of the body is 3.6 cm. (a) Early in the process, t = 5 min, the incoming flow stagnates at the nose and conforms to the body until separating just upstream of the widest portion. The wake behind the body consists of a relatively slow and unsteady flow. (b) At t = 55 min, the body has formed a quasitriangular shape, yet the flow structure is qualitatively similar. The flow stagnates at the nose and separates near the body's widest portion, in this case near the back corners of the triangular shape. (c) and (d) Flow schematics.
Click to view
Visualizing the flow around a cylindrical body at different times in the erosion process. Streaklines are captured by 10 ms exposure time photographs of tracer particles illuminated by a laser sheet, and the initial diameter of the body is 3.6 cm. (a) Early in the process, t = 5 min, the incoming flow stagnates at the nose and conforms to the body until separating just upstream of the widest portion. The wake behind the body consists of a relatively slow and unsteady flow. (b) At t = 55 min, the body has formed a quasitriangular shape, yet the flow structure is qualitatively similar. The flow stagnates at the nose and separates near the body's widest portion, in this case near the back corners of the triangular shape. (c) and (d) Flow schematics.
HighRe flow past a bluff body in two dimensions. (a) The fluid flow is comprised of an outer and boundary layer flow, with the dashed curve indicating the thickness of the boundary layer. At the separation point, the boundary layer detaches and a wake is formed. The dotted curve represents the separating streamline. (b) Zoom into the boundary layer. The velocity profile inside the boundary layer approaches the outer tangential velocity U(s) at a characteristic distance δ(s). Fluid shear stress is proportional to the slope of the velocity profile at the surface (darkened).
Click to view
HighRe flow past a bluff body in two dimensions. (a) The fluid flow is comprised of an outer and boundary layer flow, with the dashed curve indicating the thickness of the boundary layer. At the separation point, the boundary layer detaches and a wake is formed. The dotted curve represents the separating streamline. (b) Zoom into the boundary layer. The velocity profile inside the boundary layer approaches the outer tangential velocity U(s) at a characteristic distance δ(s). Fluid shear stress is proportional to the slope of the velocity profile at the surface (darkened).
Body area versus time. (a) Area measurements of the experimental interfaces from Fig. 1(b) (black) show excellent agreement with the scaling law from Eq. (23) (dashed, red) (adapted from Ristroph et al. ^{12} ). This formula allows us to estimate the vanishing time, t f , from the experiments. (b) Loglog plot of the area measurements from the experiment (black) and simulation (gray). Both match the 4/3power prediction.
Click to view
Body area versus time. (a) Area measurements of the experimental interfaces from Fig. 1(b) (black) show excellent agreement with the scaling law from Eq. (23) (dashed, red) (adapted from Ristroph et al. ^{12} ). This formula allows us to estimate the vanishing time, t f , from the experiments. (b) Loglog plot of the area measurements from the experiment (black) and simulation (gray). Both match the 4/3power prediction.
FalknerSkan similarity solutions for flow past wedges. (a) Illustration of the outer and boundarylayer flow past a wedge with acute opening angle. The shear stress is highest near the nose as indicated by the surface slopes of the velocity profiles. This causes the wedge to broaden as it erodes, which we indicate by the white dotted wedge. (b) For an obtuse opening angle, the shear stress increases downstream, and the wedge tends to become more narrow at later times. (c) A rightangled wedge produces uniform shear stress, which allows the shape to be maintained during erosion.
Click to view
FalknerSkan similarity solutions for flow past wedges. (a) Illustration of the outer and boundarylayer flow past a wedge with acute opening angle. The shear stress is highest near the nose as indicated by the surface slopes of the velocity profiles. This causes the wedge to broaden as it erodes, which we indicate by the white dotted wedge. (b) For an obtuse opening angle, the shear stress increases downstream, and the wedge tends to become more narrow at later times. (c) A rightangled wedge produces uniform shear stress, which allows the shape to be maintained during erosion.
Erosion of an initially circular body. (a) Interfaces from the simulation at evenly spaced time intervals of 0.06 t f , with time indicated by the scale bar at right (color). As it shrinks, the body forms a quasitriangular shape with a wedgelike front that points into the flow. (b) Shifting the interfaces to have the same leading point and rescaling to have equal area more clearly reveals the shape change. (c) The same rescaling procedure applied to the experimental interfaces shows similar evolution and terminal shape.
Click to view
Erosion of an initially circular body. (a) Interfaces from the simulation at evenly spaced time intervals of 0.06 t f , with time indicated by the scale bar at right (color). As it shrinks, the body forms a quasitriangular shape with a wedgelike front that points into the flow. (b) Shifting the interfaces to have the same leading point and rescaling to have equal area more clearly reveals the shape change. (c) The same rescaling procedure applied to the experimental interfaces shows similar evolution and terminal shape.
Computed flows at different stages of erosion. (a) The streamlines of the outer flow as determined by the FST method for the initial, circular geometry. The flow accelerates as it deflects around the body, as can be seen by the compression of streamlines. The small circle on the body indicates the separation point and the dotted curve shows the free streamline. The blank region behind the body represents the stagnant wake. (b) Boundary layer flow for the above body. The dashed curve shows the computed boundary layer thickness against normalized arc length, s/L, along the front of the body. This dashed curve ends at the point of flow separation. We also show the computed velocity profile at two points on the body (indicated by arrows in (a)), demonstrating that the shear stress varies along the surface of this body. (c) By time t/t f = 0.64, the body has developed a wedgelike front and the separation point has migrated backwards. (d) For this body, the velocity profiles indicate little variation in the shear stress.
Click to view
Computed flows at different stages of erosion. (a) The streamlines of the outer flow as determined by the FST method for the initial, circular geometry. The flow accelerates as it deflects around the body, as can be seen by the compression of streamlines. The small circle on the body indicates the separation point and the dotted curve shows the free streamline. The blank region behind the body represents the stagnant wake. (b) Boundary layer flow for the above body. The dashed curve shows the computed boundary layer thickness against normalized arc length, s/L, along the front of the body. This dashed curve ends at the point of flow separation. We also show the computed velocity profile at two points on the body (indicated by arrows in (a)), demonstrating that the shear stress varies along the surface of this body. (c) By time t/t f = 0.64, the body has developed a wedgelike front and the separation point has migrated backwards. (d) For this body, the velocity profiles indicate little variation in the shear stress.
Tendency towards uniform erosion rate. (a) The simulation interface velocity as it varies along the body's arc length at time t/t f = 0 (red) and 0.64 (blue). Also shown is the shear stress at the later time (dashed curve, blue, axis at right). Both interface velocity and shear stress become more uniform at the later time. (b) Measurements of the interface velocity against arc length from the experiments at time t = 4 min (red) and 90 min (blue), corresponding to t/t f = 0.03 and 0.64. The experimental measurement also shows the local erosion rate to become more uniform at later time (adapted from Ristroph et al. ^{12} ).
Click to view
Tendency towards uniform erosion rate. (a) The simulation interface velocity as it varies along the body's arc length at time t/t f = 0 (red) and 0.64 (blue). Also shown is the shear stress at the later time (dashed curve, blue, axis at right). Both interface velocity and shear stress become more uniform at the later time. (b) Measurements of the interface velocity against arc length from the experiments at time t = 4 min (red) and 90 min (blue), corresponding to t/t f = 0.03 and 0.64. The experimental measurement also shows the local erosion rate to become more uniform at later time (adapted from Ristroph et al. ^{12} ).
Opening angle measurements. (a) The opening angle is measured by fitting the upper and lower faces of the front with circle arcs, and finding the angle of intersection between the arcs. (b) The angle measurement, Ψ, versus dimensionless time, t/t*, for the simulation (gray) and experiment (black).
Click to view
Opening angle measurements. (a) The opening angle is measured by fitting the upper and lower faces of the front with circle arcs, and finding the angle of intersection between the arcs. (b) The angle measurement, Ψ, versus dimensionless time, t/t*, for the simulation (gray) and experiment (black).
Drag on an eroding body. (a) Drag versus time for the simulated body of Fig. 6(a) , normalized by the fixed value . After a transient increase, drag decreases with decreasing body size. (b) Drag coefficient for the same body, based on the largest transverse half width, a(t). The drag coefficient changes by less than 15% over the course of shape evolution, indicating that erosion is not a streamlining process.
Click to view
Drag on an eroding body. (a) Drag versus time for the simulated body of Fig. 6(a) , normalized by the fixed value . After a transient increase, drag decreases with decreasing body size. (b) Drag coefficient for the same body, based on the largest transverse half width, a(t). The drag coefficient changes by less than 15% over the course of shape evolution, indicating that erosion is not a streamlining process.
Simulated erosion of initially elliptical bodies. (a) Evolution of a broad ellipse (major axis perpendicular to the flow) with aspect ratio 3:2. Interfaces are shown at equally spaced time intervals of 0.06t f , with the same color coding as Fig. 6 . (b) Evolution of a narrow ellipse initialization (major axis parallel to the flow, same aspect ratio). (c) Overlaying the interfaces from three different initializations at time t/t f = 0.9 shows that all tend to a similar shape. Initializations are broad ellipse (dashed), narrow ellipse (dotted), and circle (solid). Running the simulations past t/t f = 0.9 leads to further collapse of the shapes.
Click to view
Simulated erosion of initially elliptical bodies. (a) Evolution of a broad ellipse (major axis perpendicular to the flow) with aspect ratio 3:2. Interfaces are shown at equally spaced time intervals of 0.06t f , with the same color coding as Fig. 6 . (b) Evolution of a narrow ellipse initialization (major axis parallel to the flow, same aspect ratio). (c) Overlaying the interfaces from three different initializations at time t/t f = 0.9 shows that all tend to a similar shape. Initializations are broad ellipse (dashed), narrow ellipse (dotted), and circle (solid). Running the simulations past t/t f = 0.9 leads to further collapse of the shapes.
Measurements of the opening angle, Ψ, against time for initial geometries of a narrowside ellipse (dotted curve), broadside ellipse (dashed curve), and circle (solid curve) in the simulation. The opening angle tends to roughly 90° in each case.
Click to view
Measurements of the opening angle, Ψ, against time for initial geometries of a narrowside ellipse (dotted curve), broadside ellipse (dashed curve), and circle (solid curve) in the simulation. The opening angle tends to roughly 90° in each case.
The evolution of surface perturbations in experiments and simulations. (a) In experiments, a manually added protrusion (top arrow) and indentation (bottom arrow) are seen to even out as the body erodes. Interfaces are shown at intervals of 2 min, over a total of 20 min. (b) Adding a smallamplitude sinusoidal perturbation to the attracting geometry in the simulation shows similar perturbation decay. (c) For the simulated body, plotting the shear stress against arc length on top of the initial perturbation shows the stress to be highest on the front face of each crest. (d) Plotting the evolving perturbation relative to the background interface reveals that the perturbations propagate downstream in addition to the decay.
Click to view
The evolution of surface perturbations in experiments and simulations. (a) In experiments, a manually added protrusion (top arrow) and indentation (bottom arrow) are seen to even out as the body erodes. Interfaces are shown at intervals of 2 min, over a total of 20 min. (b) Adding a smallamplitude sinusoidal perturbation to the attracting geometry in the simulation shows similar perturbation decay. (c) For the simulated body, plotting the shear stress against arc length on top of the initial perturbation shows the stress to be highest on the front face of each crest. (d) Plotting the evolving perturbation relative to the background interface reveals that the perturbations propagate downstream in addition to the decay.
The conformal mapping method of LeviCivita. The outer flow domain, indicated by the gray region in the zplane, is conformal mapped to the upperhalfunit disk of the ζplane. The portion of the boundary not in the wake is mapped to the perimeter of the disk, ζ = e ^{ iξ}, with the stagnation point, A, mapped to ζ = i. The free streamlines are mapped to the diameter of the disk, −1 ⩽ ζ ⩽ 1, with the separation points, B ^{±}, mapped to ζ = ±1.
Click to view
The conformal mapping method of LeviCivita. The outer flow domain, indicated by the gray region in the zplane, is conformal mapped to the upperhalfunit disk of the ζplane. The portion of the boundary not in the wake is mapped to the perimeter of the disk, ζ = e ^{ iξ}, with the stagnation point, A, mapped to ζ = i. The free streamlines are mapped to the diameter of the disk, −1 ⩽ ζ ⩽ 1, with the separation points, B ^{±}, mapped to ζ = ±1.
Article metrics loading...
Abstract
Erosion of solid material by flowing fluids plays an important role in shaping landforms, and in this natural context is often dictated by processes of high complexity. Here, we examine the coupled evolution of solid shape and fluid flow within the idealized setting of a cylindrical body held against a fast, unidirectional flow, and eroding under the action of fluid shear stress. Experiments and simulations both show selfsimilar evolution of the body, with an emerging quasitriangular geometry that is an attractor of the shape dynamics. Our fluid erosion model, based on Prandtl boundary layer theory, yields a scaling law that accurately predicts the body's vanishing rate. Further, a class of exact solutions provides a partial prediction for the body's terminal form as one with a leading surface of uniform shear stress. Our simulations show this predicted geometry to emerge robustly from a range of different initial conditions, and allow us to explore its local stability. The sharp, faceted features of the terminal geometry defy the intuition of erosion as a globally smoothing process.
Full text loading...
Commenting has been disabled for this content